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  F![](media/image1.png){width="0.4875in"   [American Institutes for Research^®^]{.smallcaps}
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> **[A Review of the Literature in Adult Numeracy: Research and
> Conceptual Issues]{.smallcaps}**

**[Prepared By]{.smallcaps}:**

[The American Institutes for Research^®^]{.smallcaps}

[1000 Thomas Jefferson Street, N.W.\
Washington, D.C. 20007]{.smallcaps}

**[Prepared For]{.smallcaps}:**

Office of Vocational and Adult Education\
Division of Adult Education and Literacy\
U.S. Department of Education\
Washington, D.C.

**[Project Staff]{.smallcaps}:**

+------------------------------+---------------------------------------+
| **Principal Investigator:**  | **Larry Condelli, AIR**               |
+------------------------------+---------------------------------------+
| **Co-Project Director:**     | **Kathy Safford-Ramus,**              |
|                              |                                       |
|                              | **Saint Peter's College,**            |
|                              |                                       |
|                              | **Jersey City, New Jersey**           |
+------------------------------+---------------------------------------+
| **Co-Project Director:**     | **Renee Sherman, AIR**                |
|                              |                                       |
|                              | **Diana Coben, King's College,**      |
|                              |                                       |
|                              | **London**                            |
|                              |                                       |
|                              | **Iddo Gal, University of Haifa,**    |
|                              |                                       |
|                              | **Haifa, Israel**                     |
|                              |                                       |
|                              | **Anestine Hector-Mason, AIR**        |
+------------------------------+---------------------------------------+

**March 27, 2006**

> [American Institutes for Research]{.smallcaps}
>
> [1000 Thomas Jefferson Street, NW \| Washington, DC
> 20007-3835]{.smallcaps}

Table of Contents

[1. Introduction 1](#__RefHeading___Toc130973115)

[Organization of the Review 2](#__RefHeading___Toc130973116)

[2. Issues in Conceptualizing Adult Numeracy
5](#__RefHeading___Toc130973117)

[Definitions of Numeracy 5](#__RefHeading___Toc130973118)

[Formative Phase 6](#__RefHeading___Toc130973119)

[Mathematical Phase 7](#__RefHeading___Toc130973120)

[Integrative Phase 7](#__RefHeading___Toc130973121)

[Theories on Learning and Knowing Mathematics
11](#__RefHeading___Toc130973122)

[Behaviorism 11](#__RefHeading___Toc130973123)

[Constructivist Theories of Learning 11](#__RefHeading___Toc130973124)

[Socio-Cultural Perspectives on Mathematics
13](#__RefHeading___Toc130973125)

[Numeracy and Cognition: Experience and Situations
14](#__RefHeading___Toc130973126)

[Affective Factors, Anxiety, and Learning Styles
16](#__RefHeading___Toc130973127)

[Learning Styles 18](#__RefHeading___Toc130973128)

[Brain Research: "A New Science of Learning"
19](#__RefHeading___Toc130973129)

[Summary 20](#__RefHeading___Toc130973130)

[3. Adult Numeracy and Mathematics Instructional Approaches and
Interventions 21](#__RefHeading___Toc130973131)

[Professional Society Instructional Standards
21](#__RefHeading___Toc130973132)

[National Council of Teachers of Mathematics Standards
21](#__RefHeading___Toc130973133)

[Crossroads 22](#__RefHeading___Toc130973134)

[Other Frameworks 22](#__RefHeading___Toc130973135)

[Research Evaluating Instructional Approaches
23](#__RefHeading___Toc130973136)

[Methodology for Identifying Research 23](#__RefHeading___Toc130973137)

[Quality of Research 25](#__RefHeading___Toc130973138)

[Effective Instruction for Adult Learners: Research Findings
25](#__RefHeading___Toc130973139)

[Use of Technology in Instruction 26](#__RefHeading___Toc130973140)

[Computer Use in ABE Instruction 27](#__RefHeading___Toc130973141)

[Computer Use in Correctional Education
28](#__RefHeading___Toc130973142)

[Computer Use in GED Instruction 29](#__RefHeading___Toc130973143)

[Computer Use in Developmental Mathematics Instruction
30](#__RefHeading___Toc130973144)

[Research Evaluating Instruction Based on Constructivist Theories
31](#__RefHeading___Toc130973145)

[Cooperative Learning 31](#__RefHeading___Toc130973146)

[Discovery Learning 32](#__RefHeading___Toc130973147)

[ABE Instructional Interventions 33](#__RefHeading___Toc130973148)

[Summary 34](#__RefHeading___Toc130973149)

[4. Assessment Issues in Adult Numeracy
35](#__RefHeading___Toc130973150)

[The Purposes of Assessments in Adult Numeracy Education and Related
Research 35](#__RefHeading___Toc130973151)

[Purposes of Assessment 35](#__RefHeading___Toc130973152)

[Related Research 36](#__RefHeading___Toc130973153)

[Overview of Widely Used Instruments for ABE Numeracy Assessment
38](#__RefHeading___Toc130973154)

[TABE 38](#__RefHeading___Toc130973155)

[CASAS 39](#__RefHeading___Toc130973156)

[A Critical Look 40](#__RefHeading___Toc130973157)

[Defining the Meaning of Assessment and How It Can Be Improved
42](#__RefHeading___Toc130973158)

[Considerations for Developing Improved Assessments for Adult Numeracy
44](#__RefHeading___Toc130973159)

[What Should Be Assessed in Adult Numeracy Instruction?
45](#__RefHeading___Toc130973160)

[Conclusions 47](#__RefHeading___Toc130973161)

[5. Professional Development in Adult Numeracy
49](#__RefHeading___Toc130973162)

[Issues in Professional Development 49](#__RefHeading___Toc130973163)

[Need For Professional Development in Numeracy
50](#__RefHeading___Toc130973164)

[Overlapping Relationship Between Literacy and Numeracy
52](#__RefHeading___Toc130973165)

[Research in Professional Development and Adult Numeracy
53](#__RefHeading___Toc130973166)

[International Efforts 56](#__RefHeading___Toc130973167)

[Summary 57](#__RefHeading___Toc130973168)

[6. Summary and Implications for Future Research
59](#__RefHeading___Toc130973169)

[Definitions and Theories of Adult Numeracy
59](#__RefHeading___Toc130973170)

[Instructional Approaches and Interventions for Adult Numeracy
59](#__RefHeading___Toc130973171)

[Implications for Further Research 61](#__RefHeading___Toc130973172)

[Further Research on Instruction 61](#__RefHeading___Toc130973173)

[Research on Mathematics Instruction for Children
62](#__RefHeading___Toc130973174)

[Development of Improved Assessment 63](#__RefHeading___Toc130973175)

[Research to Improve Professional Development
65](#__RefHeading___Toc130973176)

[Conclusion: Moving Forward 66](#__RefHeading___Toc130973177)

[References Cited 67](#__RefHeading___Toc130973178)

[References Consulted 84](#__RefHeading___Toc130973179)

[Appendix A -- Definitions of Numeracy
A-1](#__RefHeading___Toc130973180)

[Appendix B -- Professional Society Standards
B-1](#__RefHeading___Toc130973181)

[Appendix C -- List of Web Sites and Journals Searched
C-1](#__RefHeading___Toc130973182)

[Appendix D -- Summary of Mathematics and Numeracy Intervention Studies
Reviewed D-1](#__RefHeading___Toc130973183)

[]{#__RefHeading___Toc130973115 .anchor}1. Introduction

The increasing need for numeracy skills in all aspects of adult
life---family, employment, community---has made numeracy a requisite
skill for success in today's society. In the context of Adult Basic
Education (ABE), more emphasis is needed on providing quality numeracy
instruction to adults to help them achieve the mathematical knowledge
and skills that will enable them to adjust to this growing societal
demand. The Office of Vocational and Adult Education (OVAE) recognized
the need for learners to improve their numeracy skills when it
identified mathematics as a core academic area for the development of
rigorous content standards. As adult learners are encouraged to move
beyond the General Educational Development (GED) and into postsecondary
education, the development of numeracy skills will become more critical.
However, as educational assessments have shown, 35 percent of all U.S.
students are scoring "below basic" on the National Assessment of
Educational Progress (NAEP) (NCES, 2002, Math Assessment), with even
higher proportions of Hispanic, African American, and low-income
students scoring "below basic."

This problem is of significant concern to adult educators because an
increasing number of 18- to 25-year-olds are enrolling in adult
education programs: the very same students who lack numeracy skills. The
concern about the numerous skill deficiencies in today's adult learners
is exacerbated by the fact that adult education programs are not
adequately prepared to provide numeracy education to a diverse student
population that bring different needs, interests, skills, and behavior
and attitudes toward numeracy.

Although numeracy instruction plays a significant role in adult
education in many countries---notably Australia, the Netherlands, and,
more recently, the United Kingdom---the United States has experienced
limited attention to numeracy instruction and little research on how
local adult education programs teach mathematics or numeracy. There are
many reasons for this lack of focus: little agreement on what
constitutes numeracy; poor professional development in numeracy; limited
understanding of how adults with diverse characteristics, needs, and
backgrounds obtain numeracy skills; and the lack of alignment among
content standards, curricula and instruction, and assessments. Schmitt
(2002) points out that GED preparation has been the driving force in
mathematics instruction in most adult education programs. Workbooks
focusing on standard computational rules, fractions, whole numbers,
decimals, percentages, and prealgebra drive instruction. Exercises tend
to emphasize repetitive problems, word problems, and problems with
real-life applications.

The Adult Numeracy Initiative is the first major effort of the U.S.
Department of Education to improve the research and practice of adult
numeracy. This project has several goals:

-   Develop a thorough understanding of the current state of the field
    of adult numeracy.

-   Identify the gaps in knowledge about common strategies for teaching
    adult numeracy and how these strategies differ across different
    types of learners.

-   Identify the type of professional development and teacher
    certification that should be required for teachers of adult
    mathematics.

-   Identify the type of assessment instruments that might be
    appropriate for measuring adult quantitative skill acquisition.

The project will achieve these goals through two phases of activity. The
first phase, consisting of a literature review, an environmental scan, a
technical working group convened to identify critical issues, and
commissioned papers to address some of these issues, will distill the
limited body of knowledge about current research and practice on adult
numeracy instruction, assessment, and professional development. The
information from this first phase will inform the second part of the
project, which entails the design of curriculum and professional
development materials and activities and the design of a research-based
intervention or demonstration program that can be tested with a rigorous
methodology.

These activities of the Adult Numeracy Initiative seek to answer the
following research questions posed by OVAE for the project:

1.  How does adult numeracy develop and how does it differ from the
    development of quantitative literacy in children?

2.  What are the social variables that affect quantitative skill
    acquisition in adults? How should programs address these social
    variables to enhance skill acquisition?

3.  What instructional practices exist in mathematics education for
    adult learners that are worthy of replication?

4.  What outcomes are most important to address in the evaluation of
    adult education programs in mathematics? What are the best tools or
    assessments for evaluating these outcomes?

5.  What practices exist in professional development and certification
    requirements for teachers of adult mathematics education that are
    worthy of replication?

6.  What types of programs have been implemented at the state and local
    levels through federal funding that incorporate or focus on adult
    mathematics instruction?

7.  What types of programs have been implemented at the state and local
    levels through federal funding that focus on adult mathematics
    instruction related to adult English language acquisition learners?

This review addresses the first five research questions by summarizing
key findings from the literature. The report that will incorporate the
findings from the environmental scan will address the final two research
questions concerning mathematics instructional programs.

[]{#__RefHeading___Toc130973116 .anchor}Organization of the Review

This literature review is the first report of the Adult Numeracy
Initiative and lays the foundation for the other substantive activities
in the project's first phase. It strives to answer the first five
research questions posed by the Statement of Work. The review summarizes
the definitions, theories, and research around adult numeracy to
organize what is known and to point the way for future research and
development. We have organized the literature review into five parts:

-   **Issues in Conceptualizing Adult Numeracy** addresses the first and
    second research questions and presents an overview of the competing
    approaches to defining adult numeracy. The section then reviews the
    main theoretical approaches toward teaching and learning mathematics
    for adults that reflect these definitions. The concepts and theories
    in this section provide a background and context for the research in
    instruction and assessment presented in subsequent sections.

-   **Adult Numeracy and Mathematics Instructional Approaches and
    Interventions** reviews the small number of studies done on ABE
    students, supplemented with an also small number of studies on
    adults in community college developmental education programs, on the
    effects of different types of instructional approaches on
    mathematics learning among adults. In response to the third research
    question, this section presents the instructional approaches,
    findings, and methodologies of these studies.

-   **Assessment Issues in Adult Numeracy** endeavors to respond to the
    fourth research question as it summarizes the existing knowledge
    base regarding assessment in adult numeracy, reviews the uses of
    assessment, analyzes the nature of assessment and how it can be
    improved, and summarizes the principles for designing effective
    assessments for adult numeracy.

-   **Professional Development in Adult Numeracy** deals with the fifth
    research question and discusses the state of professional
    development in ABE around numeracy. The section notes the low-level
    background of ABE teachers to teach mathematics, presents
    professional development approaches, and briefly discusses research
    on the characteristics of effective professional development.

-   **Summary and Implications for Future Research** summarizes the
    findings of the review and suggests future research and how the
    field of adult numeracy practice and research might progress. It
    also briefly suggests how research on children's learning of
    mathematics may inform research on adults, thus also addressing the
    first research question.

Writing a literature review is one of the researcher's more difficult
tasks. Unless it is to be a life's work, lines must be drawn around the
topic to identify what is important, what is to be included, and how it
is to be analyzed. This task is even more difficult in the field of
adult numeracy, which Diana Coben has aptly called a "moorland," where
the lines, where they exist at all, are often indistinct and vaguely
drawn. The Adult Numeracy Initiative's research questions identified the
general topics we were to cover. Within these topics, two goals guided
our approach: (1) to identify the areas of greatest interest to
furthering research and practice of adult numeracy instruction and
professional development within the U.S. adult education system and (2)
to set the stage for the other activities of the project.

The review suggests areas to pursue further in the environmental scan
and commissioned papers, including ongoing work on the development of
instruction approaches and curricula and professional development. The
technical working group and the commissioned papers will also address
many of the conceptual and theoretic issues identified in the review.

[]{#__RefHeading___Toc130973117 .anchor}2. Issues in Conceptualizing
Adult Numeracy

The construct "numeracy" does not have a universally accepted
definition, nor agreement about how it differs from "mathematics" (Gal,
van Groenestijn, Manly, Schmitt, & Tout, 2005).

This statement, by the authors of a recent international report on adult
numeracy, indicates a fundamental problem for anyone reviewing the
research literature in this area: there is as yet no consensus about the
nature of adult numeracy. Numeracy is a deeply contested concept, beset
by terminological confusion, especially when referring to adults. A
plethora of similar and loosely related terms compete for attention:
*mathematical literacy, techno-mathematical literacy, quantitative
literacy, functional mathematics, mathemacy,* and so on. The resultant
complexities are discussed in depth in *Adult Numeracy: Review of
Research and Related Literature* (Coben, 2003).

The issues of the definition of *numeracy* may seem to be an academic
exercise, with little practical value. However, how numeracy is defined
has profound implications for all issues of concern to the Adult
Numeracy Initiative. Definitions of numeracy have implications for what
adults need to know, what should be taught, how students should be
assessed, and what professional development teachers need, as a recent
international comparative study of adult numeracy frameworks makes clear
(Hagedorn et al., 2003). In this section, we summarize the
conceptualizations of numeracy and learning theory related to how adults
learn mathematics and numeracy.

[]{#__RefHeading___Toc130973118 .anchor}Definitions of Numeracy

The term *numeracy* originated in the United Kingdom in the *Crowther
Report* on the education of children ages 15--18. As "the mirror image
of literacy," numeracy was a way of bridging scientific and literary
cultures (Ministry of Education, 1959, ¶. 389). The definition entailed
"not only the ability to reason quantitatively but also some
understanding of scientific method and some acquaintance with the
achievement of science." Literacy and numeracy, at a basic rather than
an advanced level, have been yoked ever since, with numeracy often
subsumed within literacy.

Definitions of numeracy have proliferated. One view equates numeracy
with mathematics and computational skills, in much the same way that
literacy is viewed as mastery of basic reading and writing. A much
broader view focuses on people's capacity and propensity to interact
effectively and critically with the quantitative aspects of the adult
world (Gal, 2002a). Similarly, in relation to literacy, some argue that
numeracy is subsumed in literacy, whereas others argue that debates
about numeracy within the context of literacy limit the full
operationalization of both concepts. Gal and Schmitt (1994) reported
that "some people prefer to use the term 'mathematical literacy,'
believing that 'numeracy' is too...limiting in scope. Others feel just
the opposite, taking 'numeracy' to be the mirror image of
literacy...while viewing 'mathematical literacy' as a sub-area of
mathematics" (p. ii). Appendix A presents a sampling of definitions of
numeracy.

Maguire and O'Donoghue's (2002) organizing framework (Exhibit 1),
developed through discussions with researchers and practitioners in
Adults Learning Mathematics -- A Research Forum (ALM), offers a way of
bringing some order into the conceptual confusion surrounding adult
numeracy. In the framework, concepts of numeracy are arranged along a
continuum of increasing levels of sophistication. In the *formative*
phase, numeracy is considered to be basic arithmetic skills; in the
*mathematical* phase, numeracy is "in context," with explicit
recognition of the importance of mathematics in everyday life. The third
phase, the *integrative* phase, views numeracy as a multifaceted,
sophisticated construct incorporating the mathematics, communication,
cultural, social, emotional, and personal aspects of each individual in
context.

Exhibit 1.\
Adult Numeracy Concept Continuum of Development

+----------------------+-----------------------+----------------------+
| [Phase               | > [Phase              | [Phase               |
| 1]{.smallcaps}       | > 2]{.smallcaps}      | 3]{.smallcaps}       |
+----------------------+-----------------------+----------------------+
| Increasing levels of |                       |                      |
| sophistication       |                       |                      |
+----------------------+-----------------------+----------------------+
| FORMATIVE            | MATHEMATICAL          | INTEGRATIVE          |
|                      |                       |                      |
| **(basic arithmetic  | **(mathematics in     | **(mathematics       |
| skills)**            | context of everyday   | integrated with the  |
|                      | life)**               | cultural, social,    |
|                      |                       | personal, and        |
|                      |                       | emotional)**         |
+----------------------+-----------------------+----------------------+
| A continuum of       |                       |                      |
| development of the   |                       |                      |
| concept of numeracy  |                       |                      |
| showing increased    |                       |                      |
| level of             |                       |                      |
| sophistication from  |                       |                      |
| left to right (from  |                       |                      |
| Maguire &            |                       |                      |
| O'Donoghue, 2002)    |                       |                      |
+----------------------+-----------------------+----------------------+

[]{#__RefHeading___Toc130973119 .anchor}Formative Phase

Conceptions of numeracy following the *Crowther Report* lost the
sophistication of the original definition. For example, "numeracy" first
appeared in the UNESCO International Standard Classification of
Education in 1997 as "Literacy and numeracy*:* Simple and functional
literacy, numeracy." The designations "simple," with respect to content
and skills, and "functional," with respect to purpose and application,
are telling: numeracy in these definitions refers to basic mathematical,
or sometimes specifically numerical or quantitative, skills, which
adults are deemed to need to function effectively in society. In this
view, numeracy is a basic skill normally acquired in childhood; in some
versions of numeracy, what adults are deemed to need is simple
arithmetic. Evans (2000) calls this the *limited proficiency* model of
numeracy, a hangover from the Victorian period when the "3Rs" of
reading, (w)riting, and (a)rithmetic held sway in elementary education.
As a corollary, because the content is seen as simple, numeracy may also
be thought to be easy to learn, a view roundly rejected by Ma (1999).
This view of numeracy is located in Maguire and O'Donoghue's formative
phase.

[]{#__RefHeading___Toc130973120 .anchor}Mathematical Phase

A broader view of numeracy may also be traced back to the United
Kingdom, in the 1982 *Cockcroft Report.* This view epitomizes Maguire
and O'Donoghue's *mathematical* phase, with its emphasis on the use of
mathematics in daily life:

We would wish "numerate" to imply the possession of two attributes. The
first of these is "at-homeness" with numbers and an ability to make use
of mathematical skills, which enable an individual to cope with the
practical mathematical demands of his everyday life. The second is
ability to have some appreciation and understanding of information which
is presented in mathematical terms, for instance in graphs, charts or
tables or by reference to percentage increase or decrease. (Department
of Education and Science/Welsh Office, 1982,¶. 39)

In this phase, numeracy often includes number, money, and percentages;
aspects of algebraic, geometric, and statistical thinking; and problem
solving based on the mathematical demands of adult life. This view of
numeracy has been influential in the Anglophone world, including the
United Kingdom's Adult Numeracy Core Curriculum (Basic Skills Agency,
2001). In the United States, this approach appears as part of
"functional literacy" approaches, exemplified in the CASAS framework
(2004), in several states' mathematics instructional content standards,
and in the National Adult Literacy Survey (NALS) (NCES, 1992) and it
successor, the National Assessment of Adult Literacy (NAAL) (NCES,
2005). These national surveys measure "quantitative literacy," a concept
that clearly falls within this applied mathematics phase.

However, the issue of functionality is not straightforward. Numeracy
could be functional with respect to a wide range of contexts and
purposes, and the practical mathematical demands of everyday life may
require more than basic numeracy. This complexity is acknowledged in
Maguire and O'Donoghue's *integrative* phase.

[]{#__RefHeading___Toc130973121 .anchor}Integrative Phase

All of the most recent, influential approaches to defining adult
numeracy fall into Maguire and O'Donoghue's *integrative* phase. In this
phase, numeracy is viewed as a complex, multifaceted, and sophisticated
construct, incorporating the mathematics, communication, cultural,
social, emotional, and personal aspects of each individual in context.
FitzSimons and Coben (in press) argue that numeracy in this sense may
empower individuals as "knowledge producers" as well as "knowledge
consumers"that is, to become technologically, socially, personally,
and/or democratically numerate.

Steen (1990) exemplifies this phase when he outlines five dimensions of
numeracy, distinguished in terms of their purposes and associations:

-   *Practical*, concerning mathematical and statistical skills that can
    be put to immediate use in the routine tasks of daily life

-   *Civic*, where the focus is on benefits to society

-   *Professional*, because many jobs require mathematical skills

-   *Recreational*, for the appreciation of games, puzzles, sports,
    lotteries, and other leisure activities

-   *Cultural*, concerned with mathematics as a universal part of human
    culture

Maguire and O'Donoghue's integrative phase also encompasses critical
concepts of numeracy that eschew any automatic association with
low-level mathematics. For example, Johnston (1995) proposes that

To be numerate is more than being able to manipulate numbers, or even
being able to succeed in school or university mathematics. Numeracy is a
critical awareness, which builds bridges between mathematics and the
real world, with all its diversity. \[\...\] in this sense \... there is
no particular level of mathematics associated with it: it is as
important for an engineer to be numerate as it is for a primary school
child, a parent, a car driver or gardener. The different contexts will
require different mathematics to be activated and engaged in. (p. 34)

Coben (2000a) also emphasizes the individual's judgments about the use
(or not) of mathematics in a given situation:

To be numerate means to be competent, confident, and comfortable with
one's judgments on *whether* to use mathematics in a particular
situation and if so, *what* mathematics to use, *how* to do it, what
*degree of accuracy* is appropriate, and what the answer means in
relation to the context. (p. 10)

Although integrative conceptions of adult numeracy dominate almost all
current theorizing and thinking in adult numeracy, this view has only
just begun to move beyond a limited core of numeracy researchers and
practitioners. Most mainstream practice continues to reflect formative
and mathematical approaches to numeracy. However, more integrative
approaches to numeracy have become influential over the last few years,
as illustrated by projects to define numeracy instructional content
standards, the Program for International Student Assessment (PISA), and
the Adult Literacy and Lifeskills (ALL) Survey. The numeracy definitions
in these projects specify the intended cognitive outcomes of adult
numeracy education and/or emphasize the need for the individual to
adjust to the increasing technological demands of the knowledge economy.

The introduction of the National Council of Teachers of Mathematics
*Curriculum and Evaluation Standards for School Mathematics* (NCTM,
1989) helped fuel the instructional standards reform movement in
mathematics and numeracy (discussed further in the next section). These
standards emphasized conceptual understanding and the development of
problem-solving and decision-making skills, rather than rule-based
learning. This view of mathematics and numeracy has had an impact not
only in the teaching of mathematics to children but also in adult
education and has helped prompt a movement toward the development of
content standards for teaching adult mathematics. The state of
Massachusetts (Leonelli & Schwenderman, 1994) and the Adult Numeracy
Network's (ANN) mathematics standards framework were among the first
attempts at developing an integrative numeracy framework for adult
instruction. The following seven themes were proposed to serve as the
foundation for the development of numeracy standards (Curry, Schmitt, &
Waldron, 1996):

-   Relevance/Connections

-   Problem-Solving/Reasoning/Decision-Making

-   Communication

-   Number and Number Sense

-   Data

-   Geometry: Spatial Sense and Measurement

-   Algebra: Patterns and Functions

Several states have moved toward developing mathematics content
standards, using basic computational, functional, or integrative
approaches, and eight states have already developed mathematics
standards. OVAE's Adult Education Content Standards Warehouse project
(http://www.adultedcontentstandards.org) has supported states' efforts
to develop standards, as has the National Institute for Literacy's
(NIFL) Equipped for the Future (EFF) project. EFF's Math Content
Standard states that adults should be able to "Use Math to Solve
Problems and Communicate" (see Exhibit 2) after participation in adult
basic education.

Exhibit 2.\
EFF Standard: Use Math to Solve Problems and Communicate

+-----------------------------------------------------------------------+
| -   Understand, interpret, and work with pictures, numbers, and       |
|     symbolic information.                                             |
|                                                                       |
| -   Apply knowledge of mathematical concepts and procedures to figure |
|     out how to answer a question, solve a problem, make a prediction, |
|     or carry out a task that has a mathematical dimension.            |
|                                                                       |
| -   Define and select data to be used in solving the problem.         |
|                                                                       |
| -   Determine the degree of precision required by the situation.      |
|                                                                       |
| -   Solve problems using appropriate quantitative procedures and      |
|     verify that the results are reasonable.                           |
|                                                                       |
| -   Communicate results using a variety of mathematical               |
|     representations, including graphs, charts, tables, and algebraic  |
|     models.                                                           |
+-----------------------------------------------------------------------+

From National Institute for Literacy (2000)

The Organization for Economic Co-operation and Development's (OECD) PISA
also focuses on using mathematics. It is designed to assess the
readiness of 15-year-olds for life beyond school, focusing on the extent
to which students are able to use their knowledge and skills to meet
real-life challenges. This reflects a change in curricular goals and
objectives in many countries, which are increasingly concerned with what
students can do with what they learn at school (OECD, 2003).

Mathematical literacy is defined in PISA as

an individual's capacity to identify and understand the role that
mathematics plays in the world, to make well founded judgments and to
use and engage with mathematics in ways that meet the needs of that
individual's life as a constructive, concerned and reflective citizen.
(p. 24)

It encompasses four domains or subscales:

1.  Space and shape, which includes recognizing shapes and patterns

2.  Change and relationships, which includes data analysis needed to
    specify relationships or translate between representations

3.  Quantity, which focuses on quantitative reasoning and understanding
    of numerical patterns, counts and measures

4.  Uncertainty, which includes statistics and probability

The second PISA survey, which included the United States, covered
reading, mathematical and scientific literacy, and problem solving, with
a primary focus on mathematical literacy. Conducted in 2003 in 41
countries, it is usually referred to as PISA 2003. The United States
ranked 28th out of 40 countries in terms of the percentage of students
at each level of proficiency on the mathematics scale in PISA 2003.

In the most recent international survey of adult numeracy (including the
United States), the Adult Literacy and Lifeskills (ALL) Survey, also
conducted under the auspices of the OECD, the emphasis is on using
mathematics in real contexts, including, but not limited to, everyday
life. Numeracy is considered in the ALL Survey as **"**the knowledge and
skills required to manage and respond effectively to the mathematical
demands of diverse situations"; in addition,

Numerate behavior is observed when people manage a situation or solve a
problem in a real context; it involves responding to information about
mathematical ideas that may be represented in a range of ways; it
requires the activation of a range of enabling knowledge, factors, and
processes. (Gal et al., 2005, p. 142)

The issue of level of difficulty experienced by the individual adult
when engaging in numerate behavior is tackled in the ALL Survey through
an analysis of assessment items in terms of their textual and
mathematical complexity. The inclusion of a textual dimension allows for
the fact that numeracy is often mediated through text and difficulties
in reading may impede numeracy performance. Text may be presented in
digital form or on paper, and in the integrative phase, numeracy is
often associated with information and communication technologies (ICT).
For example, Steen (2001) describes numeracy as a natural tool for the
computer age, encompassing the capacity to communicate using digital
data.

[]{#__RefHeading___Toc130973122 .anchor}Theories on Learning and Knowing
Mathematics

Each of the main definitions of numeracy has ties to adult learning
theory. In turn, the learning theories have profound implications for
the content of mathematics instruction, its pedagogy, and how learning
should be assessed (Forman & Steen, 1999). Definition, theory, and
instruction are thus tied together: one's view of what numeracy is leads
to a theory of learning, and this theory affects preferred approaches to
instruction. There remains controversy around the implication of theory
to practice because there is little empirical research demonstrating the
effects of instructional approaches implied by the theories on how
adults know and learn about mathematics.

[]{#__RefHeading___Toc130973123 .anchor}Behaviorism

Up until the mid-1990s, behaviorist approaches dominated adult
mathematics instruction. In the behaviorist approach, learning is
defined as a change in behavior observed when a stimulus results in a
response. In behaviorist mathematics instruction, the teacher conveys
knowledge, such as a number fact embedded in a word problem (the
stimulus), to the students who absorb it and produce a solution (the
response). Learning is considered to have occurred when the correct
solution is given consistently. Learning mathematics in this mode
entails immediate recall, retention, and transfer, and understanding is
equated with computation and operations, as measured by achievement
tests or performance tasks. This approach is associated with an
absolutist view of mathematics, that is, the belief in the certainty and
truth of mathematics. In the absolutist view, mathematics is a set of
absolute truths determined by authority; doing mathematics means
following the rules correctly (Coben, 2000a). Behaviorist methods
dominated U.S. educational practice until the late 1950s.

[]{#__RefHeading___Toc130973124 .anchor}Constructivist Theories of
Learning

The last 10 years have brought a major shift in ideas about learning
mathematics, from a behaviorist perspective to a constructivist
perspective (Kieran, 1994) so that constructivism now has a markedly
greater influence on contemporary mathematics education. The shift to
constructivist theories of learning corresponds to the adoption of
integrative definitions of adult numeracy. For example, constructivism
underpins the NCTM mathematics standards discussed above. The keystone
of constructivism is the notion that all knowledge is constructed by
individuals acting upon external stimuli and assimilating new
experiences by building a knowledge base or altering existing schemas.
Learners actively construct knowledge by integrating new information and
experiences into what they have previously come to understand, revising
and reinterpreting old knowledge in order to reconcile it with the new
(Billett, 1996). Two main strands in constructivism have emerged,
following, on the one hand, Piaget (focusing on ways in which individual
learners make sense of mathematics) or, on the other hand, Vygotsky
(seeing learning as an activity in which shared mathematical meanings
are constructed socially). Jaworski (1994) notes that debates between
radical and social constructivists parallel debates between these two
positions.

Piagetian theories in adult numeracy focus on the importance of an
individual's cognitive developmental stage in learning. Piaget proposed
four major developmental stages through which a child progresses
intellectually from birth to adolescence. Much of the research at the
K--8 level has proposed materials and methods that promote student
learning at the penultimate stage, the *concrete level,* and progress
toward the *formal operations* level, the final stage. The *concrete
operational* and *formal operations* levels have been the subjects of a
few studies specific to adult populations. These studies include Mayta
(1990), who correlated achievement in mathematics to the concrete stage
among a group of incarcerated males, and Brockbader (1992), Wolfe
(1999), and Martelly (1998), who found the same relationship among
community college students enrolled in developmental mathematics
courses. Although these studies do not suggest either materials or
instructional approaches, they do refute, to varying degrees, Piaget's
conviction that the evolution to formal operations is complete by age
15; they also validate the use of concrete materials and manipulatives
for adult students.

Another aspect of Piaget's theory of intellectual development has
received less attention in the adult numeracy field, his notion of
intellectual growth as involving three fundamental processes:
assimilation, accommodation, and equilibration. Assimilation is the
process through which new events are incorporated into pre-existing
cognitive structures. Accommodation involves changes in these structures
to accommodate new information. Through this dual process of
assimilation-accommodation, the learner forms *schemata*. Equilibration
refers to the balance the learner strikes between his or her schemata
and the environment and between assimilation and accommodation. A new
experience causes disequilibrium until the learner is able to assimilate
and accommodate the new information, thereby attaining equilibrium.
Llorente's (1996) study of the problem-solving behavior of adults in
Argentina with little formal education in work situations uses Piaget's
theory of equilibration to highlight the interactive and constructive
nature of everyday knowledge and the social constraints that influence
problem solving.

Vygotsky (1978) emphasized the social aspect of learning and the
interplay of speech and action in children's learning activities. To
Vygotsky, affect, motivation, and will are central to learning. Two of
his major contributions to constructivist theory were the ideas of a
"zone of proximate development" (ZPD) and "scaffolding," although the
latter name was not actually used by Vygotsky (Wilson, Teslow, & Taylor,
1993). Vygotsky defined the ZPD as "the distance between the actual
developmental level as determined by independent problem solving and the
level of potential development as determined through problem solving
under adult guidance or in collaboration with more capable peers."
Scaffolding is the process in which the student masters a skill under
the guidance of an expert. There are four basic steps in the process
(Wilson, Teslow, & Taylor, 1993):

-   The student observes the teacher modeling an activity.

-   The student tries the activity under the guidance of the teacher.

-   The teacher prompts with cues only when needed.

-   The student is free to practice the skill independently.

Vygotsky's work has many applications to the teaching of mathematics and
has been referenced by advocates of cooperative learning and
problem-solving activities. Like Piaget, Vygotsky studied children, but
his theories of ZPD and scaffolding both translate smoothly to the
design of instruction for adult mathematics students.

Constructivists maintain that learning is an active process, that
collaboration enhances learning, that learning should be situated in
realistic settings, and that testing should be integral to the
experience, not a separate activity (Merrill, 1991; Wilson, Teslow, &
Taylor, 1993). At its most radical sense, constructivist theory holds
that each person discovers truth and constructs his or her own unique
knowledge base (von Glasersfeld, 1990). Constructivism is associated
with fallibilist views of mathematics, in which mathematics is seen as a
social construct and therefore value laden, culturally determined, and
open to revision. Benn (1977) argues that fallibilist approaches lead to
more inclusive, adult-friendly teaching and learning; by contrast, the
absolutist view is associated with the product view of mathematics, in
which mathematical skills and concepts are seen as external to the
learner. In constructivist epistemologies of mathematics education,
mathematics education is viewed as a process whereby knowledge of
mathematics is gained by doing mathematics.

[]{#__RefHeading___Toc130973125 .anchor}Socio-Cultural Perspectives on
Mathematics

The belief in a socially constructed base of knowledge does not negate
the idea of the universal truth of mathematical ideas. But although the
concept that 2 + 2 = 4 is true wherever and whoever you are, Bishop
(1991) illustrates concepts such as the idea of negative numbers and the
angles of a triangle adding to 180 degrees (as opposed to 100 or 150) as
evidence of the cultural basis for mathematics. Bishop's work is an
example of ethnomathematics. Ethnomathematicians address the question of
whose knowledge counts, challenging the hegemony of the Western model of
mathematics from a variety of perspectives (Powell & Frankenstein,
1997). Bishop has identified six pan-cultural mathematical activities:
counting, locating, measuring, designing, explaining, and playing; the
Western model of mathematics represents one approach to these, but there
are others. Against this background, research and practice in
ethnomathematics focus on the mathematics of cultural groups and the
development of pedagogies that take these different forms of mathematics
into account, establishing comparisons between academic mathematics
knowledge and local knowledge and analyzing the power relations involved
in the use of both kinds of knowledge (Knijnik, 1996).

An interesting example relative to ABE was conducted by Masingila (1992)
in a work-place literacy setting. She contrasted the mathematics
practices that carpet layers use when estimating and installing their
product with the school-based knowledge of general mathematics students.
Masingila found that the carpet layers engaged in conceptually deep
mathematical thinking as they solved problems encountered during
installations. These constraint-filled situations differed substantially
from the textbook area problems that the students were required to
solve. The straightforward school problems did not prepare the novice
carpet layers for the realities of area, ratio and proportion, and
measurement experienced on the job.

Feminist studies of mathematics represent another approach from the
ethnomathematical, social constructivist view. This work has been
influential in the mathematics education of girls and women and in
research in gender studies. Becker (1995) sets out two types of knowing
developed in feminist research: separate knowing (associated with men
and concerned with such things as logic, rigor, abstraction,
rationality, and certainty) and connected knowing (associated with women
and concerned with intuition, creativity, hypothesizing, conjecture, and
experience). She contends that in mathematics and science, separate
knowledge is valued over connected knowledge and that this disadvantages
girls. She advocates a connected approach to teaching and learning
mathematics, through which, she contends, more girls would enjoy
mathematics, succeed at it, and choose to study it to advanced levels.
Feminists have also taken issue with assumptions about women's allegedly
"unmathematical" minds and sought to explain women's and girls'
underachievement, where it exists or seems to exist, in cultural rather
than biological terms, while seeking to understand and improve the
teaching of mathematics to women and girls and their take up of and
performance in mathematics (L. Burton, 1995).

A representative study in this vein is Crittenden (2000), who followed
eight women, volunteers from an intact class, as they progressed through
a preprofessional mathematics review course designed to prepare them for
placement in nontraditional jobs in the building trades. Crittenden
reports the following:

-   The women had difficulty learning mathematics in a classroom
    setting.

-   Although work-related mathematics tasks are often more complex than
    classroom-based mathematics problems, the women in the study had an
    easier time learning and using mathematics on the job.

-   The women in the study did not perceive any difficulties using
    mathematics in common budgeting and shopping chores because of the
    repetitive nature of the tasks.

-   The women in the study were unable to use relevant mathematics
    skills for nonroutine personal finance tasks, such as the evaluation
    of investment options for retirement planning, because they had
    either incomplete or unhelpful schemas for the financial services
    industry.

Ethnomathematics and feminist approaches demonstrate that school
mathematics and "street" mathematics differ substantially. Further, the
affective environment of the traditional classroom setting can impede
student learning despite the best-intentioned efforts of the instructor.
Both points should be taken seriously when interventions are designed
for the adult numeracy student.

[]{#__RefHeading___Toc130973126 .anchor}Numeracy and Cognition:
Experience and Situations

Despite the importance of understanding cognition---what and how people
know what they know---for instruction, such studies in adult numeracy or
mathematics education are rare; most studies of cognition and
numeracy/mathematics in the education and psychology fields have been
developed through research with children. However, there is clear
research evidence that mathematical knowledge develops both in and out
of school, for adults and children, and is profoundly influenced by
experience and cultural practice, as socio-cognitive theorists have
shown (Lave, 1988; Saxe, 1991; Schliemann & Acioly, 1989). Such studies
emphasize the ability of people to control and regulate their own
behavior in relation to their experience in their environment, rather
than react automatically to stimuli, as behaviorist psychologists
predict.

Adults bring this prior knowledge and life experience to the classroom
and apply it to their use of mathematics in a wider range of situations.
Effective instruction must be responsive to these experiences, but there
is little theory designed toward understanding prior experience and how
these experiences affect learning. Duffin and Simpson (1993, 1995) have
developed a theory of learning that attempts to classify learners'
experiences into three categories: natural, conflicting, and alien
(Exhibit 3).

Exhibit 3.\
Adult Experiences and Learning Responses

+---------------+--------------------------+---------------------------+
|               | **Experiences**          | **Responses to            |
|               |                          | Experience**              |
+---------------+--------------------------+---------------------------+
| **Natural**   | Fits the learner's       | Strengthens the current   |
|               | mental structures        | way of thinking           |
|               |                          |                           |
|               | Is expected and          | Extends the scope of the  |
|               | unsurprising             | internal mental structure |
+---------------+--------------------------+---------------------------+
| **            | Is inconsistent with the | Destroys an internal      |
| Conflicting** | learner's internal       | mental structure          |
|               | mental structures        |                           |
|               |                          | *or*                      |
|               | Jars with expectations   |                           |
|               |                          | Limits a way of thinking  |
|               | Highlights limitations   |                           |
|               | or contradictions        | *or*                      |
|               |                          |                           |
|               |                          | Merges two mental         |
|               |                          | structures                |
+---------------+--------------------------+---------------------------+
| **Alien**     | Has no connection with   | Ignores the experience    |
|               | the learner's internal   |                           |
|               | mental structures        | *or*                      |
|               |                          |                           |
|               | Is meaningless for the   | Avoids the experience     |
|               | learner                  |                           |
|               |                          | *or*                      |
|               | Cannot be coped with     |                           |
|               |                          | Absorbs the experience as |
|               |                          | a new mental structure    |
+---------------+--------------------------+---------------------------+

From Simpson & Duffin (1995)

Duffin and Simpson suggest that the natural-based learner aims to build
on existing internal mental structures, developing connections between
new and previous experiences. Such learners are likely to develop highly
flexible, widely applicable learning because of their highly
interconnected mental structures, but they are likely to build up their
understanding of complex methods slowly. Meanwhile, for an alien-based
learner, new experiences are accepted separately and either are left
unconnected or are connected only later in response to any conflicts
that may arise. Such learners should be able to master techniques
quickly, without the need to build on existing structures, but such
skills would be bound by the context in which they were learned and
would be difficult to reconstruct if they went unused and became
forgotten.

Gal (2000) takes a different approach, beginning from the learner's
perspective. He notes that real-life numeracy *situations* are always
embedded in the life stream with personal meaning to the individual
involved. Adults need numerate skills to enable them to manage diverse
types of quantitative situations, and he identifies three types of
numeracy situations that adults must manage. *Generative situations*
require people to count, quantify, compute, or otherwise manipulate
numbers and generate a response. Examples are dealing with simple
operations, such as calculating a total price of products while shopping
or measuring a shelf; dealing with multistep operations embedded in
text, such as completing a tax form; and making reasonable decisions,
such as choosing the best home mortgage loan. Resulting responses have
clear right or wrong answers.

*Interpretive situations* require people to make sense of verbal or
text-based messages based on quantitative data but do not require them
to manipulate numbers. Examples include interpreting a chart in a
newspaper article reporting crime statistics or reading a report of a
survey with poll results. The response expected in such situations is an
opinion or the activation of critical questions that have no clear right
or wrong answers.

*Decisions situations* require people to find and consider multiple
pieces of information to determine a course of action. Such situations
include identifying ways to use limited resources, such as money or
time, and choosing among alternatives (renting the right apartment,
purchasing the right car, or deciding on the best insurance, for
example). Not only is there no clear correct answer in such situations,
but the person may sacrifice precision or accuracy to save time or
mental load when deciding on a response and may reach the response in an
inefficient or nonstandard way (Gal et al., 2005).

The three numeracy situations are not necessarily distinct categories,
and Gal (2000) notes that other types of numeracy situations and hybrid
situations are possible. Numerate behavior is also enabled by
dispositional elements (prior belief, attitudes, and habits) that
motivate and support effective behavior in any situation.

[]{#__RefHeading___Toc130973127 .anchor}Affective Factors, Anxiety, and
Learning Styles

Dispositional elements are central to constructivist theories, which not
only posit the importance of experience and situations in learning but
also include characteristics of the learner as integral to learning.
Research on the role of learners' affect, attitudes, and beliefs in
learning mathematics, although widespread for children, has been limited
on adults, focusing on attitudes and anxiety about mathematics. Singh
(1993) surveyed adults' attitudes toward mathematics and found that
abstraction and perceived lack of relevance are common reasons students
cited for their dislike for mathematics and that fear of failure induced
by instruction and testing in mathematics is a main cause of anxiety.

Sheila Tobias has conducted extensive research on math anxiety in
adults. She authored *Overcoming Math Anxiety*, in which she explores
the economic impact, both personal and societal, of poor mathematics
skills. Using her work with math-phobic individuals, particularly women,
Tobias shares her insights into the causes of math anxiety and the myths
of math ability (or not), and she suggests ways to help students
overcome their anxiety and open the door to disciplines and occupations
that require strong mathematical skills. She recommends the following:

-   Students need to take charge of their mathematics learning, refusing
    to be intimidated by their history and the culture of previous
    mathematics classrooms.

-   Teachers must create environments where math anxiety can be
    discussed openly, help students recognize their mathematics
    strengths, and provide opportunities for success and difficulty
    within their zone of proximal development.

-   Mathematics books are not an easy read, and part of the
    desensitization process compels teachers to help students develop
    appropriate reading skills.

-   Teaching styles must be adjusted to include methods that recognize
    differences invoked by gender and culture.

-   Talking and writing about feelings and strategies must permeate the
    course. (Tobias, 1993)

Several other studies have investigated the existence and effect of
anxiety on learning in adult students. Altieri (1987) surveyed
developmental studies students (*n* = 89) at a community college using
Kolb's Learning Styles Inventory followed by interviews with 17 of the
students and four faculty members. Although the study was not specific
to mathematics and anxiety was not the focal question, analysis revealed
that anxiety and remembering were the central learning problems of the
students. The researcher proposes that the first may be a root cause of
the second. Both the students and the faculty named anxiety as the
dominant impediment to learning, an impediment that fostered problems
with remembering, course pacing, and testing.

Using the Brief Math Anxiety Rating Scale (BMARS) and the Learning Style
Inventory-Adapted (LSI-A), Cook (1997) found a connection between
anxiety level and perceptual learning style in more than 500 community
college students. Cook found that students with audio and
tactile/kinesthetic learning styles were more likely to have math
anxiety. Female students had a higher math anxiety level, but age was
not significantly correlated.

Jost (1997) studied the way that anxiety about mathematics and
computer-assisted instruction affected 40 students in an adult education
class. He used the Computer Attitude Scale and Mathematics Anxiety
Rating Scale along with a demographic questionnaire and conducted
analyses to determine the interaction of computer anxiety and
demographic variables. He found significant gender differences for
computer experience, more negative attitudes, and higher computer
anxiety. There was no gender difference for achievement as measured by
the final exam in the course.

There are other, correlational studies of math anxiety on adults in
developmental mathematics courses. Peskoff (2001) evaluated the
relationship between students' level of math anxiety and the strategies
they employ to cope with it, using 279 developmental mathematics
students. A multivariate statistical analysis related the effects of
math anxiety, gender, and course enrollment on 10 coping strategies
rated for frequency of use and helpfulness. Peskoff found that students
with low math anxiety use and value a wider variety of coping strategies
than peers with high math anxiety. High-anxiety students used tutoring
services and met with their counselors significantly more than
low-anxiety students, and males used the avoidance strategy of
exercising or engaging in physical activity significantly more than
females. However, students and the faculty considered this strategy one
of the least useful strategies. Two strategies considered more helpful
by all students and the faculty, completing homework assignments on time
and letting the instructor know they do not understand the material,
were used significantly more often by females. Students identified two
additional strategies that they felt were more helpful: asking questions
in class and allowing extra study time before exams.

Parker (1997) interviewed 12 developmental mathematics students who
previously had math anxiety to describe the steps they took to overcome
their anxiety. Analysis of the data identified six stages in the
transition from math anxious to confident. First, each spoke of a
personal realization that he or she had to become comfortable with
mathematics because some personal goal depended on it. Often that goal
was related to job advancement, although that was not the only reason
cited. Next, students made a personal commitment to achieving the goal.
For some of the subjects, past success mastering difficult tasks spurred
belief that they could also be successful at mathematics. They spoke of
determination and positive thinking as keys to victory. They then mapped
out a strategy to master the subject and took action. Many described the
recognition of a significant turning point in both their aptitude and
their attitude, which led to positive feelings about themselves and
about the value of mathematics, eventually becoming a part of the math
support system to help other struggling with mathematics.

Duffin and Simpson (2000) have also explored the tensions between the
cognitive and the affective aspects of learning, as has Evans (2000),
who goes beyond the focus in some earlier studies on math anxiety to
focus on the interrelationship between adults' mathematical thinking and
both positive and negative emotions. He argues that thinking and emotion
are inseparable, so mathematical activity is always emotional and
teachers should encourage adult students to seek to understand what
might be emotional blocks to actively seeking out possible applications
of their learning.

[]{#__RefHeading___Toc130973128 .anchor}Learning Styles

Along with affect, recent research has examined the role of individual
learning styles in adult education. Two influential examples are David
Kolb's (1984) learning style "inventory" that he termed experiential
learning and Howard Gardner's (1993) theory of multiple intelligences
(MI). Kolb describes learning as an ongoing, circular process based in
concrete experiences that the learner reflects upon, abstracts concepts
from, and then actively experiments with to enrich the learning base.
Learners are classified according to the point in the process where they
seem most comfortable. "Divergers" ponder concrete experiences, imagine
possibilities, and ask "What if...?" questions, whereas "convergers"
start with concepts and seek out a solution. "Assimilators" focus on
concepts, reflecting on their abstract qualities with little concern for
practical applications. Finally, "accommodators" take concrete
experiences and experiment with them to create new experiences and build
knowledge.

Gardner's theory combines psychology with neuroscience and identifies
eight intelligences: musical, bodily-kinesthetic, logical-mathematical,
linguistic, spatial, interpersonal, intrapersonal, and naturalist.
These, he contends, are located in different regions of the brain.
Someone who is "left brained" is thought to be logical and structured,
whereas someone who is "right brained" is more creative and spontaneous.
The idea is that by teaching to these "intelligences," teachers can make
their lessons more effective.

Learning styles approaches are popular among some adult numeracy
educators, and a group of adult literacy teachers in New England has
investigated the utility of MI theory to their work. Two of the teachers
focused on mathematics instruction, reporting benefits for both teacher
and student from the project (Costanzo, 2001).

However, recent research by Frank Coffield and colleagues warns against
stereotyping people on the basis of their learning styles (Coffield,
Moseley, Hall, & Ecclestone, 2004). Coffield's team surveyed more than
70 instruments designed to identify people's learning styles. They then
undertook a rigorous scrutiny of 13 of these instruments, together with
a literature review of the main theories on learning styles. They
concluded that the idea that various types of "intelligence" are located
in diverse parts of the brain is not confirmed by neuroscience. Although
certain parts of the brain do seem to control particular activities, the
brain is far more flexible and more robust than some theorists had
assumed. They also found that some of the most widely used instruments
have low reliability, poor validity, and a negligible impact on teaching
and learning. Similarly, John White describes learning styles approaches
as deterministic and potentially leading people to restrict their own
possibilities (White, 1998).

[]{#__RefHeading___Toc130973129 .anchor}Brain Research: "A New Science
of Learning"

The study of cognitive factors and individual differences that affect
learning has recently taken a radically different turn, toward brain
research. The question of how our understanding of adult numeracy and
mathematics teaching and learning might relate to this new research in
cognitive neuroscience has been explored since 1999 in the Numeracy
Network, which is part of OECD's Brain and Learning project. This
project has several aims:

-   Develop a "new science of learning" through creative dialogue
    between cognitive neuroscience, psychology, education, health and
    policy.

-   Discover what insights cognitive neuroscience might offer to
    education and educational policy and vice versa.

-   Identify questions and issues in the understanding of human learning
    where education needs help from other disciplines. (Organization for
    Economic Co-operation and Development \[OECD\] 2004, p. 9)

In the present, second phase (2002--06), the project is focusing on
three main issues: literacy, numeracy, and lifelong learning; the
findings and policy recommendations will be published in 2006.

The Numeracy Network focuses primarily on brain mechanisms related to
fundamental educational skills that enable comprehension of mathematical
thought. This includes basic work on numeracy skills and symbolic
thinking, with emphasis on the cognitive psychology and neuropsychology
of mathematics operations.

An example of work discussed in the project that may have far-reaching
implications for understanding mathematics learning at all ages is
Dehaene's triple code theory. This theory describes a modular system of
brain areas that are active when a child is learning or performing
arithmetical operations: addition, subtraction, multiplication, and
division. The basic idea is that when manipulating a number, a child
does one of three actions, each involving a different region of the
brain:

-   Performs some visual manipulation (seeing the number as a visual
    digit, such as "3"),

-   Performs some linguistic manipulation (hearing or reading the number
    as a word, such as "three"), and

-   Represents it as a quantity (such as "3 is bigger than 1") (OECD
    2004, p. 64).

On the basis of this theory, Dehaene (1992) contends that:

Adult human numerical cognition can therefore be viewed as a layered
modular architecture, the preverbal representation of approximate
numerical magnitudes supporting the progressive emergence of
language-dependent abilities such as verbal counting, number
transcoding, and symbolic calculation. (p. 35)

If Dehaene is right, his theory may explain why some adults have
difficulty in one or more of these areas: recognizing, manipulating, or
representing numbers. It also tells us that we should not assume that an
adult who can do one of these actions will be able to do the others and
shows that language is deeply implicated in some, but not all,
arithmetical operations. This is just one example; the work of the OECD
Numeracy Network shows that there is much debate among cognitive
neuroscientists, psychologists, and educationalists. For example,
Karmiloff-Smith (2004) argues that modules might pre-exist in the
neonate, but that this cannot be assumed from studies of children or
adults because domain specificity can emerge over developmental time.
Also, it is a long way from producing a theoretical model to proving its
explanatory power and then working out the implications for teaching and
learning. It is important to recognize the present limitations, as well
as the power and potential of brain research with respect to education,
and to avoid a determinism that sees brain research as the final arbiter
in matters of education. Nevertheless, the development of a "new
learning science" is promising, especially if it takes the form of a
creative multidisciplinary collaboration.

[]{#__RefHeading___Toc130973130 .anchor}Summary

This brief review of the competing conceptualizations of adult numeracy
and learning has identified a rich and active body of theoretical work.
There is substantial debate within the field on how to define and
characterize adult numeracy, and we used Maguire and O'Donohue's
conceptual framework to organize the competing definitions. Integrative
definitions of numeracy have the most influence on recent instructional
frameworks and standards developed for adult numeracy. These frameworks
also reflect constructivist views of learning that posit that learners
actively construct knowledge by integrating new information and
experiences into what they have previously come to understand. There is
a substantial body of research inspired by constructivist theories, and
we reviewed relevant research on the role of learners' prior experience,
numeracy situations, math anxiety, and learning styles on adult
mathematics learning, along with new, promising brain research.

[]{#__RefHeading___Toc130973131 .anchor}3. Adult Numeracy and
Mathematics Instructional Approaches and Interventions

As just illustrated, there is a rich and lively debate on definitions of
adult numeracy and on how adults learn mathematics. Integrative
definitions of numeracy and constructivist theories have been
particularly influential, and current approaches to numeracy instruction
for adults reflect this thinking. In this section we briefly review the
predominant instructional frameworks about teaching mathematics to
adults. We then present a review of the empirical research evaluating
instructional approaches toward teaching mathematics to students in
adult education and literacy classes.

[]{#__RefHeading___Toc130973132 .anchor}Professional Society
Instructional Standards

Several professional societies concerned with mathematics instruction
have developed standards that influence recent practice in ABE
instruction. These standards define instructional content, including the
specific facts or subjects to be covered; skills needed, such as problem
solving and critical thinking; and process or pedagogy. There tends to
be agreement among the frameworks and standards on the need for specific
skills such as problem solving, but there is less agreement on specific
content and teaching methods (Safford-Ramus, 2006, in press). In
particular, there is agreement on the need for critical-thinking and
problem-solving skills within mathematics instruction.

[]{#__RefHeading___Toc130973133 .anchor}National Council of Teachers of
Mathematics Standards

Although developed for teaching primary and secondary school
mathematics, the framework developed by the National Council of Teachers
of Mathematics (NCTM) has been among the most influential in adult
mathematics teaching. Three documents present the NCTM approach:
*Curriculum and Evaluation Standards for School Mathematics* (1989)*,
Professional Standards for Teaching Mathematics* (1991), and *Assessment
Standards for School Mathematics* (1995)*.* The intent of the
*Standards* series was the identification of the best way to teach
mathematics the first time. A revised edition of the book, *Principles
and Standards for School Mathematics,* was published in 2000.

The NCTM principles provide a valuable model for instructors, balancing
content and methodology in light of the needs of the workforce in the
twenty-first century and the technology available to workers. Six
principles are stipulated for school mathematics:

-   ***Equity.*** Excellence in mathematics education requires
    equity---high expectations and strong support for all students.

-   ***Curriculum.*** A curriculum is more than a collection of
    activities: it must be coherent, focused on important mathematics,
    and well articulated across the grades.

-   ***Teaching.*** Effective mathematics teaching requires
    understanding what students know and need to learn and then
    challenging and supporting them to learn it well.

-   ***Learning.*** Students must learn mathematics with understanding,
    actively building new knowledge from experience and prior knowledge.

-   ***Assessment.*** Assessment should support the learning of
    important mathematics and furnish useful information to both
    teachers and students.

-   ***Technology.*** Technology is essential in teaching and learning
    mathematics; it influences the mathematics that is taught and
    enhances student's learning (National Council of Teachers of
    Mathematics, 2000).

The six principles undergird the ten standards that are split between
content and process. The *content* standards*,* which focus on what
students should learn, include number and operation, algebra, geometry,
measurement, and data analysis and probability. The *process* standards
define the ways of acquiring and using that content knowledge and
include problem solving, reasoning and proof, communications,
connections, and representation. Appendix B presents the NCTM standards.

[]{#__RefHeading___Toc130973134 .anchor}Crossroads

In September 1995, the American Mathematical Association of Two-Year
Colleges (AMATYC) released *Crossroads in Mathematics: Standards for
Introductory College Mathematics before Calculus*. This document defines
the mathematics that students need to be able to pursue collegiate
mathematics courses. Such core mathematics may have to be retaught to
students who did not meet or master it during their youth. *Crossroads*
also suggests methods for implementing instructional improvements.
Because virtually all students in U.S. postsecondary studies are adults,
*Crossroads* can be looked at as an andragogical document. Also,
two-year colleges in the United States are a primary vehicle for adults
to reenter further education, so a substantial percentage of that
population consists of adults under the most restrictive definition of
that term. A revision of *Crossroads,* titled *Crossroads Revisited,* is
scheduled for release as this literature review is being written.

AMATYC based its standards development in *Crossroads* on six
principles, including that mathematics should be meaningful and relevant
and that the use of technology in instruction is essential. *Crossroads*
offers three categories of standards: intellectual development, content,
and pedagogy. The recommendations for intellectual development emphasize
modeling and problem solving, reasoning and communicating
mathematically, and the judicious use of technology to accomplish those
tasks. The standards for content include number and operation sense,
pattern, symbolism and algebra, geometry, functions, discrete
mathematics, probability and statistics, and deductive proof. The third
focus, pedagogy, examines the place of technology, interactive and
collaborative learning, connection to other disciplines and real-world
applications, and multiple approaches that embrace all learning styles.
Appendix B includes a summary of the recommendations from the 1995
AMATYC document. The revision includes and expands on the earlier lists
with substantial attention to math anxiety, learning and teaching
styles, assessment, and professionalism.

[]{#__RefHeading___Toc130973135 .anchor}Other Frameworks

In ABE, state- and practitioner-led projects have developed standards.
The Adult Numeracy Network's (ANN) mathematics standards framework
(Curry et al., 1996) and the National Institute for Literacy's (2000)
Equipped for the Future (EFF) Math Content Standard have been most
directly influential. We have already discussed these in Section 2.
Eight states have developed, or are in the process of developing,
content standards for mathematics for their ABE programs. The conceptual
bases for these efforts vary widely, with some states following an NCTM
or EFF model, some employing competency-based approaches, and others
defining only basic mathematics content knowledge.

The Mathematical Association of America (MAA) has for many years
supported inquiry into the content and teaching methods used in
undergraduate mathematics courses. Its Committee on the Undergraduate
Program in Mathematics (CUPM) offered recommendations on the mathematics
curriculum for all undergraduates in a report published in 2004, *CUPM
Curriculum Guide 2004*. The report recommends standards for content
selection as well as the conduct of university mathematics courses from
the introductory level through the mathematics major. These standards
have had influence in adult education, though to a lesser extent that
the NCTM and *Crossroads* standards.

[]{#__RefHeading___Toc130973136 .anchor}Research Evaluating
Instructional Approaches

Although there has been significant work toward developing instructional
frameworks and content standards, there has not been a similarly intense
focus on evaluating the effectiveness of these frameworks or the
theories underlying them for adult mathematics instruction.
Unfortunately, very few research studies have used ABE students to study
the effects of adult numeracy instruction, and the research that does
exist is neither theory-driven nor guided by any systematic approach.
There has been no agenda or systematic model guiding research in adult
numeracy (Coben, 2000a). Two previous reviews of this research (Tout &
Schmitt, 2002; Coben, 2003) identified fewer than 20 studies on
instructional impact on adults; only 9 of the studies identified in
these reviews were conducted in the United States. Reviews focusing on
identifying methodologically rigorous studies of instructional effects
on ABE and ESL students concur. For example, Torgerson, Brooks, and
colleagues (2003) identified only one well-conducted randomized control
study that used mathematics as an outcome measure, a finding verified by
Condelli and Wrigley (2004), who also found that only 9 studies using
ABE and ESL students demonstrated a statistically significant impact of
instruction.

A central goal of the Adult Numeracy Initiative is to identify
instructional practices worthy of replication (Research Question 3).
Although the body of field-based, research-practitioner studies on
numeracy instruction within ABE classes has been growing, this research
uses primarily introspective and qualitative methods. Although the
knowledge of experienced practitioners is an important resource in any
effort to improve the effectiveness of adult numeracy teaching and
professional development, this knowledge must be verified through
more-rigorous methodologies. Consequently, our approach is to identify
whether rigorous research has found instructional approaches or
interventions that have been effective in enhancing the mathematics and
numeracy skills of adult learners.

[]{#__RefHeading___Toc130973137 .anchor}Methodology for Identifying
Research

Our review examined all studies conducted between 1985 and 2005 that
tested an instructional approach or intervention, used an outcome
measure that assessed skills in mathematics or numeracy, and employed
some type of comparison group. This review includes only intervention
studies with adult students that addressed mathematics skill levels that
would be considered basic elementary level through the secondary level.
These skills are the basic mathematics skills typically taught to
students enrolled in ABE programs and adult students in developmental
mathematics programs (typically taught in community college settings).

While we recognize the differences between ABE programs other
instructional settings, ABE classes are often taught within community
colleges. In addition, developmental mathematics students are in many
ways similar to ABE students, and the content of instruction to these
students is also similar to what is taught in ABE. We believe that
including these studies increases the potential knowledge base and thus
can inform ABE instruction and address the goals of the Adult Numeracy
Initiative. However, we excluded studies of developmental mathematics
interventions that examined instruction of higher level (postsecondary)
mathematics skills to adults, such as college algebra, because these
skills are beyond what is taught in ABE.[^1]

To identify studies, we searched research databases (Proquest, ERIC,
EBSCO, MATHS4Life, Dissertation Abstracts International, NALD, and
Reference Manager) and generic Web sites (Google.com, Yahoo.com,
MSN.com, Askjeeves.com, Webcrawler.com, Altavista.com, Excite.com, and
AOL.com), using different permutations of the keywords *adult numeracy
basic education* and *adult mathematics basic education.* We also
visited numerous national and international mathematics- and
numeracy-related Web sites to try to identify additional adult numeracy
and mathematics--related sources. Appendix C lists all Web sites and
publications included in our search. Reference Manager connected us to
catalogs from more than 500 universities, and several features from the
software enabled us to categorize all articles by title, author, link to
PDF, URL, abstract, link to full text, and keywords, among several other
categorization parameters.

This search identified 223 studies, of which only 91 related
specifically to adult numeracy. We next deleted studies that

-   were not empirical research on instructional interventions,

-   did not include adults in ABE classes,

-   were conducted prior to 1985,

-   did not have outcomes related to learning mathematics,

-   did not have a comparison group, and

-   included fewer than five students per group.

At the end of this process, our search criteria left only 15 studies for
the review, all of which included ABE students.

To identify additional studies, we included doctoral dissertations,
identified through the mathematics dissertation database developed by
Safford (2000) and updated to 2002.[^2] Safford developed this database
by searching *Dissertation Abstracts International (DAI)* using a
Boolean search for *Adult AND Mathematics AND Education* in the
descriptor field. After we applied our selection criteria, this search
resulted in an additional 9 studies, which gave us a total of 24 studies
for our review. Appendix D includes a matrix that summarizes all the
intervention research studies we have included in this review.

[]{#__RefHeading___Toc130973138 .anchor}Quality of Research

The Institute for Education Sciences (IES) has established evidence
standards for research, embodied in the procedures of the What Works
Clearinghouse (WWC; www.whatworks.ed.gov). Applying the evidence
standards allows reviewers to rate studies as (1) meeting the standards,
(2) meeting the standards with reservations, or (3) failing to meet
standards. Studies that meet standards have the strongest evidence of
effects. These studies are primarily well-conducted randomized
controlled trials and regression discontinuity studies, or
quasi-experimental studies of especially strong design.

Although we were not required to follow the WWC procedures or use the
evidence standards in this review, we applied them to the adult numeracy
studies we identified to evaluate the overall quality.[^3]
Unfortunately, the methodologies were generally of such low quality that
very few could meet the evidence standards. Fewer than 5 studies used
random assignment; among these, many had severe differential attrition,
which made the resulting groups nonequivalent. Other studies used
nonequivalent comparison groups, used very small sample sizes (fewer
than 10), used nonstandardized or qualitative outcome measures, did not
adequately explain the intervention, or did not adequately explain
statistical analyses.[^4]

Rather than eliminate all studies that did not meet evidence standards
(which would have left us with very little to report), we included all
intervention studies that used some type of comparison group and had at
least five students per group. In the discussion of all studies, we
describe the methodology to allow an assessment of overall study
quality.

[]{#__RefHeading___Toc130973139 .anchor}Effective Instruction for Adult
Learners: Research Findings

Of greatest interest to adult educators is whether the research can
identify the characteristics of effective instruction. The instructional
frameworks, constructivist theories, and integrative conceptions of
numeracy are compelling and beg the question of whether they are
effective on adult learners. Unfortunately, our review revealed that
very little research examining these approaches met our criteria for
rigorous methodology.

The 15 studies of the effects of instruction on mathematics learning for
ABE students investigated a remarkably narrow range of topics: 13
compared the effects of technology or computer-assisted instruction
(CAI). Of the 2 remaining studies, 1 examined inference training (a
contextual learning method for literacy development) and 1 examined
participation in family literacy. Among the 9 studies on interventions
with developmental mathematics students, 5 studies evaluated cooperative
learning methods, which are based on constructivist views of learning,
and 4 studies evaluated the use of technology and learning.

[]{#__RefHeading___Toc130973140 .anchor}Use of Technology in Instruction

Most of the research (13 ABE studies and 4 development mathematics
studies) on the effect of instruction of adult mathematics in ABE has
examined the effects of the use of instructional technology. The content
standards of the three major mathematics education organizations endorse
using technology for mathematics instruction. In its *Principles and
Standards for School Mathematics*, the NCTM (2000) states:

Electronic technologies---calculators and computers---are essential
tools for teaching, learning, and doing mathematics. They furnish visual
images of mathematical idea, they facilitate organizing and analyzing
data, and they compute efficiently and accurately. They can support
investigations by students in every area of mathematics, including
geometry, statistics, algebra, measurement, and number. (pp. 24--25)

In *Beyond Crossroads,* the AMATYC authors (2005) enumerate the
technologies used in the community college environment: graphing
calculators, mathematics software, spreadsheets, multimedia, computers,
and the Internet. They also introduce distance learning, a concept with
powerful potential for adult educators. The MAA (2004), although
endorsing the use of technology, warns that such use must be judicious:

The use of technology can help students develop mathematical skills and
understanding. However, the use of technology must be focused on
students' needs rather than on the capabilities of the technology
itself. Instructors must first decide what mathematics is to be learned
and how students are to learn it. The answers to these questions will
determine whether and how students should use technology. (p. 22)

The research on the use of technology in instruction has not
demonstrated that it improves the learning of mathematics by adults over
instruction that does not use technology. Among the 13 studies of ABE
students and the 4 developmental mathematics studies, only B. S. Burton
(1995) and Wilson (1987) found statistically significant increases in
achievement for CAI. Lavery, Townsend, & Wilton (1998) reported an
impact of CAI, but used a sample of only six students per group. Nurss
(1989) and the Indiana Opportunities Industrialization Centers (O.I.C.)
of America State Council (1990) also showed a positive effect for CAI,
but these studies have severe methodological problems that cloud
interpretation. In addition, the instructional approaches represented
within the technology and the type of technology studied vary so widely
that it is difficult to draw conclusions about instruction from this
research.

Although there seems to be no clear advantage or disadvantage to the
incorporation of CAI into ABE, GED, or developmental mathematics courses
from this research, technology, particularly the Internet, has evolved
rapidly and is quite different from what it was when most of the studies
were conducted. The studies span 20 years, years that have seen a great
change in the quality and sophistication of educational software. The
degree of change in technology has been so profound that is questionable
whether the early studies have much applicability to the more
technologically knowledgeable adults of the 21st century. The adult
student cohort in 2005 is more knowledgeable about and dependent on
technology than those of 10 years ago. We summarize each study below,
beginning with the ABE studies, which include correctional education and
GED courses, and concluding with studies of students in developmental
mathematics courses.

[]{#__RefHeading___Toc130973141 .anchor}Computer Use in ABE Instruction

Among the earliest studies in ABE, Barnett (1985) and Reid (1986)
reported studies that used Program Logic for Automated Training
Operations (PLATO) computer-assisted instruction. The Barnett study had
a pretest-posttest design comparing students in two juvenile
correctional facilities. There was no indication of the number of
participants or the instruments used to measure the variables.
Achievement and attitude of students in the PLATO group were not
significantly different from those of students who received only
traditional instruction.

Reid (1986) compared three teaching methods: CAI using PLATO, tutoring
using Laubach materials, and traditional teaching. Subjects (*n* = 30)
were members of existing ABE/GED classes. The TABE M and D levels served
as pre- and posttest instruments. There was no significant difference in
mathematical achievement among the three groups, although the CAI group
gained 1.9 grade levels while the traditional group gained 1.1.
Robichaud (1985) also compared students in traditional settings with
those whose regular instruction was supplemented by CAI. No details
concerning instrument or evaluation were supplied, but she reported that
statistical analysis revealed no significant difference in skills
gained. However, there was a significantly positive change in attitudes
toward computers and the instructional use of CAI by the CAI users. In a
later study, B. S. Burton (1995) revisited the same question of
traditional versus CAI instruction with more definitive results. He
compared CAI and traditional instruction with 200 adults at a vocational
technical adult education center, using the TABE M and D as its
assessment measure with a nonequivalent control group design. Students
using CAI were found to do significantly better in mathematics than
those in the control group. Age and gender had no effect, but student
ethnicity and extent of prior formal education did affect results.

Nurss (1989) assessed the effectiveness of the Principle of Alphabet
Literacy Systems (PALS) CAI program on the literacy skills of adult
nonreaders, compared with traditional adult basic education. This trial
showed a significant, positive effect for the traditional adult basic
education classes (i.e., the control group). Attrition, however, was
extremely high in both groups. Of the 74 students assigned to the
control group, 15 percent (*n* = 11) remained at the posttest; 32
percent of the 135 students in the experimental group (*n* = 43)
completed the program. One could conjecture that the "cream" of the
control remained and therefore performed well on the test. In addition
there was differential attrition, with more of the control group staying
to completion, which clouded the results. However, it might be argued
that the greater retention of the experimental group indicated higher
student satisfaction with the instruction.

The Indiana O. I. C. of America State Council (1990) reported the
results of a study that examined the effectiveness of traditional
classroom instruction versus computer-assisted instruction in raising
the competency levels of adults one grade level for every 80 hours of
instruction. Of the 149 individuals who were pretested, only 50 attended
more than 30 hours and remained at the time of posttesting. Evaluation
was done using a randomized methodology with the ABLE test as the pre-
and posttest instrument. No information was supplied concerning the
statistical analysis used to evaluate the results. However, the report
indicates that the overall average grade change for CAI students was 2.6
grades compared with an average of 1.84 grades for non-CAI students. The
high attrition again makes these findings difficult to interpret.

The objective of a study by Nicol and Anderson (2000) was to evaluate an
experiment that compared CAI and teacher-implemented instruction in
numeracy. It is unclear whether the same two teachers taught the two
intervention groups. The researchers randomized the adult students into
three groups of eight students. The method of random allocation was not
described, but stratification by gender was implied. The researchers
reported no difference in improvement between the teacher-led
intervention and the CAI, but given the very small numbers in each
group, there is a high possibility of a Type II error in this study.

Lavery, Townsend and Wilton (1998) conducted a randomized control trial
of 12 students in New Zealand to compare the learning outcomes
associated with basic literacy education programs conducted through
traditional instruction with computer-assisted instruction. The students
and instructional approaches in these programs are similar to those in
U.S. adult education courses. The study measured the gains in reading
and numeracy skills in two "training opportunities" classes. Six
students received traditional teaching, and another six used Readers'
Workshop, Math Concepts and Skills, and Computer Curriculum
Corporation's Computer Assisted Learning (CAL) software packages.
Participants' reading and numeracy skills were measured by the Burt Word
Reading Test, the Neale Analysis of Reading Ability, and the KeyMath
Revised Test. The results show that significantly greater achievements
were made in reading (word recognition, word accuracy, and
comprehension) and numeracy (mathematical concepts, operations, and
applications) under CAI than under traditional instruction. The students
who used the CAL made a 3-year gain on the Burt, over 1 year on the
Neale, and 16 months on the mathematics test in less than 2 months of
instruction. During the same time, the students who received traditional
teaching made no gains in reading skills and showed a slight decline in
mathematics performance.

[]{#__RefHeading___Toc130973142 .anchor}Computer Use in Correctional
Education

In addition to Barnett (1985), three studies focused specifically on
incarcerated populations. Winters, Teslow and Taylor (1993) studied the
effect of a CAI-supplemented program on ABE/pre-GED and GED students in
an adult correctional facility. Five students were assigned to either
the experimental or the control group (*n* = 10), and the researchers
pre- and posttested with the TABE. The statistical methods used to
analyze the data were unclear, but the results favored the CAI
intervention: 86 percent of the students in the pilot study advanced in
level in mathematics in contrast to a 50 percent gain in the control
group. A comparison of students advancing one year or more showed 43
percent for the pilot study versus 14.5 percent for the control group.
Once again, the small sample size limits the utility of this study.

Batchelder and Rachel (2000) studied the effect of skill-and-drill
tutorial software to enhance mathematics and language skills. They
randomly assigned 71 male inmates in the prison's GED program to receive
either the regular classroom instruction offered or classroom
instruction supplemented by CAI using the tutorial software. The
classroom instruction consisted of four hours per day in English,
mathematics, history, and science. Students in the experimental group
received three hours of instruction per day but spent the fourth hour
using the CAI software for mathematics and reading. Inmates were
posttested with the CASAS reading and mathematics tests after receiving
80 hours (four weeks) of instruction. There was no significant
difference between the groups on these tests.

Burnham (1985) examined the effect of a televised curriculum on an
incarcerated ASE population. Subjects in a nonequivalent control group
research design (*n* = 40) were pretested using the General Educational
Performance Index (Form AA) and posttested using Form BB of the same
test. The experimental group used an instructional televisions series,
Adult Math, as a reinforcement resource, viewing the "telelessons" under
supervision and then completing workbook exercises tied to the program.
The control group completed self-paced workbooks and used other
instructional materials but did not view the television series. The
researcher found no difference in achievement between the groups,
although he cautions that Adult Math is more effective when the subjects
have grade-level scores of at least 5.8 in arithmetic and reading and
that the literacy levels of incarcerated populations are noticeably
lower than those of the general population.

[]{#__RefHeading___Toc130973143 .anchor}Computer Use in GED Instruction

Two other studies evaluating computer-based instruction (CBI) were GED
specific. Wilder (1994) compared the effects of a CBI simulation-test
treatment, a CBI drill-and-practice program, and a traditional workbook
drill-and-practice class on retention, completion time, and elevation of
test scores on the mathematics section of the GED. The research design
was a three-group, posttest-only design with unequal sample sizes, where
a total of 564 students self-selected into the classes. Wilder followed
the students for five years, with 308 students retained long enough to
get a GED diploma. In addition, 94 percent of the simulation group was
retained compared with 65 percent in the CBI drill group and 36 percent
in the workbook-only group. Completion time was also considerably less
in both CBI groups. Scores on the test were not significantly different.

Wardlaw (1997) studied the effect of CAI on achievement and attitude for
a group of pre-GED and GED adults. The study was conducted in
established classes with 60 students each in the treatment and control
groups. Pre- and posttesting was done using the TABE and the Semantic
Differential Attitudinal Questionnaire. Wardlaw found no significant
difference on either achievement or attitude. He does offer an important
caveat for developers planning to incorporate CAI into a program.
Wardlaw surveyed ABE facilities and found that although some were well
equipped, many others had few or outdated workstations. One facility had
banned student use of the equipment because the director believed that
the students were using it to arrange dates rather than study. Wardlaw
suggests that these environmental issues may have contributed to the
failure of CAI to effect positive attitudinal change.

[]{#__RefHeading___Toc130973144 .anchor}Computer Use in Developmental
Mathematics Instruction

Four studies examined the use of CAI with developmental mathematics
students in tertiary institutions. In an early study, Wilson (1987)
described a diagnostic and tutorial program that was conducted at a
vocational school in Kentucky. The researchers designed a diagnostic
test for pre- and posttesting, which they normed against the TABE at the
8.75 grade equivalent. The results of the experiment showed a
significant effect in favor of the experimental group. Toet (1991)
studied a randomly selected sample of students who had been placed into
remedial reading, English, or mathematics at a community college. Using
the TABE, she compared achievement between students who completed
assignments based on textbook use and students who worked in a CAI
laboratory. The group taking basic mathematics showed a statistically
significant cognitive gain. Those studying beginning algebra were
retained longer at a statistically significant level of .05. There was
no significant retention difference for the basic mathematics group.

Hsieh (1992) examined the effect of two specific features of CAI,
animation and manipulation, on 54 students participating in two
computer-based laboratory (CBL) sections of a developmental mathematics
course. The students were randomly assigned to receive instruction with
or without animation and with or without manipulation. The outcome
measures were overall achievement, retention of content, and motivation,
measured through a questionnaire. The researcher listed five findings:

-   Animation enhanced retention when the tasks required high-level
    cognitive processes such as analysis or synthesis.

-   Animation did not help learning or retention when the tasks were
    comprehension of mathematical concepts.

-   Animation increased continuing motivation.

-   Manipulation helped the transference of mathematical concepts
    learned through a computer to paper-and-pencil tests.

-   Manipulation did not promote intrinsic motivation.

In the most recent and most thoroughly defined study, Costner (2002)
examined the effectiveness of a computer algebra system (CAS) on
achievement and attitudes of students in a college remedial algebra
course. Students in the treatment group (*n* = 26) used the CAS to
discover algorithms, explore algebraic manipulation, and identify
misconceptions, while students in the control group (*n* = 25) did not
have access to the CAS. Several instruments were used in the study: a
researcher-designed pretest and periodic section tests; a departmental
final exam; the Fennema-Sherman Attitude Toward Success in Mathematics
Scale, the Confidence in Learning Mathematics Scale, and the Mathematics
Usefulness Scale; a researcher-designed questionnaire and semistructured
interview (*n* = 5); and periodic writing assignments. There was no
statistically significant effect on achievement or surveyed attitudes.
However, the qualitative data gathered through the questionnaire
revealed significant differences in attitudes and in classroom culture
issues. Students in the treatment group cited the helpfulness of group
work and classroom discussions more often than students in the control
group. With respect to the use of CAS, the treatment group welcomed the
ability to check their work and get immediate feedback. They felt that
the CAS helped them see mathematics differently, yet they attributed
little of their new mathematical understanding to technology. One
criticism was the unavailability of the computer in testing situations.
The researcher suggests that assessment needs to be altered if CAS is an
integral part of the course.

[]{#__RefHeading___Toc130973145 .anchor}Research Evaluating Instruction
Based on Constructivist Theories

As discussed in Section 2, constructivist theories of instruction and
learning---the hypothesis that all knowledge is constructed by
individuals acting upon external stimuli---has had a great influence on
recent work in adult mathematics and numeracy instruction. Although no
studies have directly examined the effect of constructivist approaches
of mathematics learning on ABE students, researchers have studied two
constructivist models, cooperative learning and discovery learning, on
students enrolled in developmental mathematics courses.

We identified three studies on cooperative learning and two studies on
discovery learning. Only one of the studies on cooperative learning
found a positive effect, but none indicated that cooperative learning
has a detrimental effect on achievement for adult students. Findings
also suggested that cooperative learning may contribute positively to
student attitude while decreasing math anxiety. The studies on discovery
learning were more positive, with all showing positive effects on either
affective measures or measures of mathematics achievement or
understanding.

[]{#__RefHeading___Toc130973146 .anchor}Cooperative Learning

Cooperative learning embraces a number of classroom organization styles,
all of which group students in learning teams for some or all of the
instructional time. Several models, according to Neil Davidson, share
the following characteristics (Davidson in Slavin, 1985):

-   The class is divided into small groups composed of two to six
    members.

-   Each group has its own working space, which may include a section of
    the blackboard.

-   The group is involved in discussing mathematical concepts and
    principles, practicing mathematical techniques, and solving
    problems.

-   The teacher moves from group to group, checks the students' work,
    and provides assistance in varying degrees.

-   The groups sometimes gather outside of class to work on projects.

-   Within in each group, certain leadership and management functions
    must be performed.

Students in Costner's study (2002) cited above found the use of group
work and classroom discussion helpful. Two other studies have reported
findings from research investigating the use of cooperative learning
with adult populations.

Peer tutoring is cooperative learning between two individuals where each
learns from and with the other. Berry (1996) studied the effect of peer
tutoring in dyads on adult students in a remedial algebra class. Two
studies were conducted: a 6-week program and a 12-week semester.
Students self-selected the classes but had no knowledge of the planned
intervention. Instructors were randomly assigned and trained in the
intervention after assignment. In each case, three peer-tutoring
sections were contrasted with three traditional lecture sections.
Pre-and posttests were given using the Suinn Mathematics Anxiety Rating
Scale, the Fennema-Sherman Mathematics Attitude Scales, a profile
questionnaire, and an abbreviated version of the institutional Freshman
Skills Assessment Program test. An open-ended survey was also used.
Sections had an average of 35 students (*n* = approximately 210). Of the
variables measured, only attitude increased significantly during the
6-week study. For students in the 12-week semester, the intervention
group showed significant improvement in mathematics achievement and
attitude as well as reduced anxiety.

In a study by Ellis (1992), each of seven instructors at a community
college taught one developmental algebra section that incorporated the
use of in-class study groups and one section that did not use groups.
She compared the achievement and completion rates and found no
significant difference between the experimental and control groups for
the group neither as a whole nor on the basis of age or gender.

[]{#__RefHeading___Toc130973147 .anchor}Discovery Learning

One method of using cooperative learning in the adult mathematics
classroom is termed small-group discovery (Safford, 1998; Davidson,
1985). The NCTM standards include the use of discovery learning for
mathematics. Davidson (1985) described the method in the following way:

The instructor introduced new material with brief lectures at the
beginning of class, during which he posed problems and questions for
investigation. For most of the class time, the students worked together
cooperatively at the blackboard in four-member groups. The students
discussed mathematical concepts, proved theorems, made conjectures,
constructed examples and counter-examples, and developed techniques for
problem solving. The instructor provided guidance and support for the
small groups.

Although discovery learning has not been studied in ABE research, three
studies using developmental mathematics students have examined
approaches that allowed students to use discovery learning to construct
their own knowledge.

Bartlett (1993) used a guided discovery approach to teaching mathematics
in one section of a developmental mathematics course at a university.
She defined the methodology in the following way: "Under the guidance of
the teacher, students find and use their own rules and generalizations
to solve other problems." The experimental group (*n* = 27) was a class
taught with this approach and was compared with the same class taught in
a previous quarter (*n* = 52) without the approach. Outcome measures
were mathematics performance measured by a researcher-developed test and
mathematics anxiety measured by the Math Anxiety Rating Scale (MARS).
Students in the experimental class performed better on the outcomes, and
Bartlett reported that the experimental method was effective in
improving the mathematics performance of adult students.

Ramus (1997) reported similar findings from a course with 13
developmental mathematics students that used discovery learning methods.
She supplemented quantitative measures (course tests) with qualitative
interviews of 8 students. Students reported a sense of ownership of the
rules of algebra because they had discovered them from classroom
exercises and also self-reported a positive change in attitude toward
mathematics and increased confidence that transferred to other
activities outside the classroom. Quantitative measures, derived from
the course examination, were less conclusive. Examination results were
scored using two rubrics, one to measure correctness and one to measure
the use of problem-solving strategies. An ANOVA showed that the
experimental section performed as well as the evening section but less
well than the daytime class. The author suggested that the different
demographic composition of the evening and experimental classes may have
affected the outcome as much as the intervention.

Pace (1989) explored the applicability of constructivist methods to the
teaching of geometry concepts in a remedial mathematics class at an
urban community college. Students (*n* = 67) were pretested using the
Applied Geometry Test, the Van Hiele Geometry Test, and the New Jersey
College Basic Skill Placement Test. They were randomly assigned to four
sections of the course, two experimental and two control, all taught by
the same instructor. The treatment class instruction consisted of five
80-minute sessions during which students explored concepts of area and
perimeter using activities embedded in applied problem-solving settings.
Students were posttested and delayed posttested. The data were assessed
using single and multivariate linear regression models. Those in the
treatment program performed significantly better than their
counterparts.

[]{#__RefHeading___Toc130973148 .anchor}ABE Instructional Interventions

Our review identified only two additional studies of instructional
interventions on the ABE student population that included a measure of
mathematics as an outcome measure. In neither of these studies was
numeracy or mathematics instruction the main focus of the intervention.
One study examined inference training, a meaning-making strategy used
for reading development. This study may be relevant to adult mathematics
instruction to the extent that numeracy may be considered an integrative
skill that includes literacy abilities. The second study examined the
general impact of family literacy instruction compared with general ABE
instruction and therefore provides an indication of the overall effect
of the programmatic model, although not mathematics instruction
specifically.

Farr (1987) investigated the effects of inference training in learning
vocabulary on verbal abilities and mathematics problem solving among 40
ABE students. Half the students had inference training, a predictive
reading strategy where they were taught vocabulary skills and reasoning
training, and the other students received traditional ABE instruction
without the training. Although the main focus of the study was literacy
development, mathematics problem solving was included as a dependent
variable to ascertain whether training in inferencing in language
acquisition would be reflected in other areas. The results showed a
correlation between mathematics performance and reading performance. The
results also showed that verbal ability correlated with the ability to
solve analogies and neologisms.

Irby et al. (1992) conducted a randomized control trial of approximately
25 predominantly black and Hispanic students in an ABE setting; 15
students were in a family literacy project (intervention group), and 10
were enrolled in GED classes only (control). The objective of the study
was to evaluate the effectiveness of a family literacy project on the
numeracy and literacy levels of adults. The intervention was conducted
in a family literacy project comprising several components, and ABE
classes were offered twice a week for 12 weeks. Instructors developed
individualized educational plans for each student to work at his or her
own pace. The results indicated that students in the family literacy
project showed a higher average gain in reading and mathematics compared
with the GED class.

[]{#__RefHeading___Toc130973149 .anchor}Summary

In a recent review of the research on the effects of instruction of ABE
and ESL students, Condelli and Wrigley (2004) concluded that the
research:

...reflect\[s\] a haphazard and unorganized approach toward studying
adult literacy and \[is\] not guided by any theory, approach or school
of thought about good pedagogy. They do not provide a comprehensive body
of knowledge on the impacts or literacy interventions in ABE. (p. 22)

The same can be said of the research studying the effects of instruction
in mathematics and numeracy on ABE students. With only 15 studies
examining mathematics interventions, almost all of them dealing with the
use of technology in instruction, we cannot consider this research a
meaningful guide toward directing future efforts in practice or
research. The additional 9 research studies on developmental mathematics
students suggest promising directions, particularly in studying
constructivist approaches toward teaching and learning. However, this
research is also limited and indicates a very early state of inquiry.

Besides the need for theory- and standards-driven research, our review
has identified a lack of research on instruction for adults that
addresses individual learning differences. There are descriptive and
theoretical studies on how adults learn, including cognitive influences,
learning disabilities, gender differences and motivation, but we found
no studies of how learning and instruction interact with these
differences to influence the development of numeracy. In addition, we
found no research of any type examining instruction to adult ESL
learners. There exists no research base at all on how numeracy is taught
in ESL classes, let alone studies that examine instructional approaches
and their impact on these learners.

[]{#__RefHeading___Toc130973150 .anchor}4. Assessment Issues in Adult
Numeracy

As we have just demonstrated, the research on the impact of instruction
on adult mathematics learning has not kept pace with the strides in
theory development or conceptualizations in numeracy. The same situation
exists in regard to the assessment of adults' mathematics and numeracy
skills. There exist very few assessments of mathematics skills for ABE
students, and those that do exist do not address critical-thinking and
problem-solving skills or other aspects of numeracy. Assessment in adult
literacy programs is driven more by the GED tests and the need to meet
reporting needs of the National Reporting System (NRS), the adult
education program's accountability system.

In this section, we not only review the scant research base and practice
on adult numeracy assessment but also reflect on key issues related to
the development of effective assessments in adult numeracy education. We
bring to the surface issues and dilemmas that can inform the goals of
the Adult Numeracy Initiative and a future research and development
agenda in this area. We begin with an overview of the purposes of
assessments in adult numeracy and the limited related research and take
a closer look at the most widely used instruments. We then analyze the
nature of an assessment and what can be improved about it and summarize
principles and ideas to inform the conceptualization of "good"
assessments in numeracy education.

[]{#__RefHeading___Toc130973151 .anchor}The Purposes of Assessments in
Adult Numeracy Education and Related Research

A discussion of the assessment of numeracy in adult education is a
complex and at times daunting matter. The very heterogeneous array of
adult education programs encompasses varying sizes and formats; the
programs operate in different contexts with different degrees of
independence; and many of their operations are undocumented. Numeracy
education is often subsumed as part of "basic education," "literacy
education," "workplace education," "prison education," and other titles,
unlike "mathematics education," which normally occupies a prominent and
separate space in K--12 school contexts. Hence, numeracy is often *not*
discussed as a stand-alone topic.

[]{#__RefHeading___Toc130973152 .anchor}Purposes of Assessment

Assessments related to the learning of numeracy (and literacy) are
undertaken for many purposes associated with needs that learners,
teachers, and program administrators face in various stages of activity
(Sticht, 1990; Ananda, 2000), as follows:

-   *Entrance stage.* Assessments are conducted as part of an initial
    diagnosis of incoming students' skills, capacities, and work habits.
    Assessments inform decisions about placement, help set learning
    goals, and influence the choice of curricula and teaching/learning
    methods. In the context of numeracy learning, entrance-stage
    assessments may also have to detect whether learners have informal
    mathematical knowledge, math anxiety, or other characteristics that
    may contribute to or affect further learning. Additional assessments
    may be needed for new students who show evidence of having special
    learning needs; the results of these assessments may have an impact
    on decisions about special accommodations or adaptations so that
    these learners can demonstrate their full range of capacities (Sacks
    & Cebula, 2000).

-   *Teaching / learning stage.* After a period of learning, assessments
    may be used for formative evaluation or feedback, that is, to
    monitor learners' progress, examine areas of strength and weakness,
    and help with designing a further course of study or deciding on
    needed interventions. When learning is undertaken with the goal of
    eventually passing a formal, external test (e.g., GED, workplace
    certification), selected exams or practice tests may be used
    intermittently to determine whether learners are ready to engage in
    the real test. At the entrance stage and at this stage, assessments
    may involve a mix of teacher-developed or locally designed methods,
    both written and oral, as well as formal instruments from external
    sources (e.g., commercial standardized assessments, GED practice
    tests). The assessments should enable *both* the teacher and the
    learners to understand the interpretation of the results of the
    assessment and the logic underlying decisions.

-   *End/exit stage.* When learners end their prescribed program of
    studies, or when programs need to measure learners' progress because
    of accountability reporting demands by sponsors (e.g., the NRS) or
    as part of accreditation schemes, assessments may be used to
    document overall learning gains and thus serve a summative function
    or help with program evaluation. At times, the end stage may be a
    stepping stone into a new cycle of learning numeracy or mathematics
    at a higher level; hence, summative assessments may also serve a
    diagnostic function.

-   *Other assessments.* Although partially overlapping with the uses of
    assessment above, sometimes assessments are conducted as part of
    research projects by academic researchers or by state and federal
    agencies. They may be part of program evaluation initiatives that
    emerge from the needs of stakeholders other than the learners,
    teachers, or administrators linked with a specific program. These
    external demands may affect the time allotted to assessments or the
    level of motivation of those being assessed.

[]{#__RefHeading___Toc130973153 .anchor}Related Research

Our review of the literature shows that it is difficult to locate
publications relating to adult numeracy assessment because this topic is
usually intertwined with assessments pertaining to literacy and other
skill areas. Hence, the knowledge base we uncovered is quite limited and
patchy. Below are examples illustrating the dearth of attention to
assessment related directly to numeracy learning.

-   Of a total of 88 presentations, workshops, and discussion groups in
    the proceedings of the last three conferences of ALM (Adult Learning
    Mathematics, an international forum of researchers and
    practitioners) held in 2002, 2003, and 2004, only 4 discussed
    assessment; only 2 of these presented empirical data (both on the
    same single study).

-   In the 2002, 2003, and 2004 issues of *Adventures in Assessment*, a
    widely read online magazine published once a year by SABES/World
    Education in Massachusetts for adult education practitioners
    nationwide, only 1 of a total of 20 articles examined mathematics
    assessment.

-   In the *Research Companion to Principles and Standards for School
    Mathematics,* published by NCTM in 2003 (Kilpatrick, Martin, &
    Schifter, 2003), out of 23 core chapters, only 1 directly examined
    assessment issues.

Few authors in the sources above directly addressed numeracy assessment,
yet quite a few touched on more general issues with *relevance* to
numeracy assessment, such as problem-solving processes, task
authenticity, the role of context demands on performance, the impact of
dispositions such as beliefs or attitudes, the nature of mathematical
practices in workplace versus school contexts, and many more. This
finding should not be too surprising because assessment serves many
useful functions in any learning and teaching context and affects what
learners, teachers, and programs do, yet it is a *support* activity in
the service of larger goals.

Indeed, in the literature search we conducted for this review, numerous
articles presented opinions and models for assessment or reported on
studies that used assessments of numeracy skills (e.g., to evaluate
learning gains). Few sources, however, focused on the characteristics of
adult numeracy assessments or on their improvement as an object of
scholarly inquiry and empirical research. Most of those that did were
*not* related to adult education per se, but to the development of
policy-driven research projects and large-scale assessments (Kirsch,
Jungeblut, Jenkins, & Kolstad, 1993; Gillespie, 2004; Gal et al., 2005;
Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005; Brooks, Heath, &
Pollard, 2005). On a broad systemic level, assessments of adults have
been examined in connection with the development of the NRS and the
search for performance measures that can satisfy accountability
requirements or in the context of planning for the National Assessment
of Adult Literacy (Stites, 2000). These efforts looked across a range of
skills, with literacy (i.e., language-related skills) at the forefront
and numeracy in the background, sometime in a restricted sense, such the
notion of "Quantitative literacy" as defined in the NALS (Kirsch et al.,
1993) and subsequently in the NAAL.

Over the last 10 to 15 years, several monographs have examined aspects
of assessments in adult education programs. Some looked at assessment
issues in general (Rose & Leahy, 1998) or reviewed a broad range of
assessment tools (Sticht, 1990), whereas others focused on methods that
can be used for assessment in specific contexts, such as teaching
workplace literacy (Taylor, 1997; Sticht, 1999) or English language
learners (Van Duzer & Berdan, 1999). Some projects focused less on
instruments and more on ways the assessments can be incorporated to
support teachers' ongoing work (Ananda, 2000) or on considerations
affecting the decisions of administrators or teachers to adopt one
assessment scheme over another (Nagel, 1999).

Only two reports were found on issues in numeracy assessment in the
context of adult education programs, yet they are of partial use in the
present context. Cumming, Gal, and Ginsburg (1998), as part of the
Numeracy Project at the National Center on Adult Literacy (NCAL) and
with the goal of informing practitioners and program personnel more than
policymakers, examined principles for good assessment and limitations of
methods in field use, especially of the TABE test. Although informative,
this review referred to TABE versions 5-6, which have since been
replaced by versions 7-8 and, starting in 2003, by TABE 9-10 whose
content has changed somewhat.

As part of a project of the National Literacy Secretariat in Canada,
Hagedorn and colleagues (2003) examined several numeracy frameworks and
provided a detailed description of the features of the "math" component
of three tests in widest use (GED, TABE, and CASAS). They also examined
the EFF mathematics standards, already introduced in Section 2, and the
numeracy aspects of the NRS. Both the EFF and the NRS descriptors are
not tests; rather, they outline the content areas or skills that adults
are expected to develop during instruction and thus have to be
eventually assessed. Although useful as a summary of existing frameworks
with implications for assessment, the Hagedorn report was meant to be
descriptive and does not provide a critical analysis of tests or
frameworks.

[]{#__RefHeading___Toc130973154 .anchor}Overview of Widely Used
Instruments for ABE Numeracy Assessment

With the implementation of the NRS, all states must now use for
reporting purposes a standardized test to measure educational gain in
ABE and ESL students. *NRS* *Implementation Guidelines* require that
tests must be reliable and valid; allowable tests include the Tests of
Adult Basic Education (TABE), the Comprehensive Adult Student Assessment
System (CASAS), the Basic English Skills test (BEST), and the Adult
Basic Learning Examination (ABLE )(OVAE, 2005). Although these
requirements do not preclude the use of other types of assessments
(e.g., nonstandardized) for other needs of teachers and programs as long
as they are reliable and valid, most programs have adopted the
mathematics sections of the TABE and CASAS (Gal & Schuh, 1994) to
measure the mathematics abilities of ABE learners.

We now take a look at these two assessment instruments and critically
evaluate them in light of the numeracy definitions and instructional
frameworks presented earlier. In addition, we also note the GED test,
which for many adult education programs serves as an important target or
milestone as they prepare ABE learners.

[]{#__RefHeading___Toc130973155 .anchor}TABE

The TABE is a family of widely used standardized tests published by
CTB/McGraw-Hill. The most recent revision of the tests, version 9-10,
was put into use in 2003 (TABE 2003a, 2003b, 2003c). In addition to
various subtests related to literacy, two of the TABE subtests assess
"mathematics computation" and "applied mathematics," each with 40 to 50
multiple-choice questions presented with four response options. The test
is norm-referenced (i.e., scores are computed in reference to a norming
group of adults), and results are usually reported as grade-level
equivalents divided into five levels, from preliteracy (level L -- equal
to grades 0--1.9) to Advanced (level A -- equal to grades 8.6--12.9).
The TABE system includes a Locator test and a short Survey form that
according to the publisher can be used to help place students at
different levels. At each level, two equivalent forms exist, presently 9
and 10, which according to the publisher can be used to measure growth
and educational gains as pre- and posttest measures. It is important to
note that the TABE is a skill-based test; that is, most items measure
specific skills (e.g., whole number operations, conversion of
fractions), often with relatively little contextual information. The
publisher claims that the revised forms 9-10 offer better coverage of
key areas of mathematics. The training materials for teachers, also from
the publisher, indicate that the higher skill levels use not only items
in arithmetic but also some that touch on measurement, geometry,
statistics, and algebra. That said, no systematic analysis has been
published of the actual content by which we could evaluate these claims,
and presently it appears that teachers and programs are expected to use
the test on the assumption that it is "better" but without having access
to much detailed information regarding its alignment with instructional
goals.

[]{#__RefHeading___Toc130973156 .anchor}CASAS

The CASAS is widely used for assessing adult basic reading, mathematics,
listening, writing, and speaking skills. The CASAS (2004) manual claims
to measure some 300 separate *competencies* by using items couched in
realistic *functional* contexts. CASAS was designed from the outset to
offer an integrated system for assessment, training, and evaluation. It
offers more than 100 separate tests, which measure groups of
competencies at different levels of ability by using items that examine
proficiency in performing specific tasks involving solving life-skill
problems or applying general reading and mathematics skills. Like the
TABE, the CASAS uses only multiple-choice items, although the emphasis
is on dealing with such real-world situations as workplace, community,
or family tasks, all of which may involve the activation of multiple
skills. The CASAS manual does not explain exactly how the tests are
scored, but it appears that test scores are based on the difficulty
estimates for each item, which were derived through IRT (Rasch) scaling.
Scores are usually reported on a numerical scale ranging between 150 and
250, commonly divided into five levels from A (beginning literacy) to E
(advanced adult secondary).

Both the TABE and the CASAS claim that using the tests enables programs
to establish measurable goals, conduct diagnostic assessments, document
learning outcomes, and report program gains to students, staff, and
external stakeholders. Practitioners who want to use the tests in
educational contexts are offered various suggestions for
*accommodations* in test administration for students with special
learning needs. The test publishers claim that these accommodations do
not change what the tests intend to measure. Further, numerous tables of
"correlations" are available from the publishers, which link specific
test items or scale scores to various chapters or units in a diverse
range of instructional materials, thus enabling teachers to assign
learning resources to students who fail specific test items. In
particular, the CASAS (2004) technical manual provides numerous tables
showing linkages or mapping between CASAS levels and levels or score
ranges for other systems or measures, such as NRS, EFF, GED, and NALS.

The developers of both assessment systems claim that their instruments
undergo rigorous test development and validation procedures and meet
common psychometric standards. The CASAS technical manual lists various
studies that have provided information about the tests' validity, such
as content validation through various procedures involving different
stakeholders, positive association between CASAS scores and learning
gains, or positive correlations in linking studies between CASAS scores
and GED mathematics scores. The TABE Web site presents more modest
information (undated report) on the same topics, such as positive gains
in programs that used CTB/McGraw-Hill instructional materials and
measured performance by TABE forms and positive correlations between
TABE scores and grade levels in school.

It is important to emphasize that the recent GED 2002 versions have been
developed on the basis of the NAEP Mathematics Framework, and item types
and their proportions in the GED are designed to reflect NAEP content
guidelines. Both the TABE and the CASAS are *not* expressly designed to
reflect NAEP, yet both show as part of their validation data that scores
on the TABE or the CASAS are highly predictive of scores on the GED.
Given that the CASAS is competency-based and the TABE is skill-based,
further analysis is warranted to understand what cognitive and other
processes are at work when learners work on all these different
instruments yet achieve highly correlated results.

[]{#__RefHeading___Toc130973157 .anchor}A Critical Look

The TABE and the CASAS are widely employed, yet they have differences in
logic, construction, and use. Nevertheless, they both appear to be
reliable and valid as far as standard techniques and published sources
allow us to evaluate. Various concerns and questions can be raised that
touch on validity, construct coverage, permissible interpretations of
the meaning of test scores, and alignment between assessments and
instruction. These concerns are presented below regarding the TABE and
the CASAS, but they are just as relevant when evaluating the quality of
other assessments:

-   Both tests claim to have good content validity, in part ascertained
    by interviewing diverse educators and learners, yet little
    information is published regarding their underlying construct. Both,
    and especially the TABE, purport to cover key mathematical
    subdomains such as number, measurement, algebra, and statistics (in
    the higher skill levels). These appear to be linked to well-known
    frameworks of mathematical knowledge, such as the one from NCTM.
    However, little published information exists (from test publishers
    or other sources) about the extent to which the tests cover broader
    facets of numeracy and acknowledge the nature of the everyday
    mathematical tasks and problems that adults have to solve. We are
    *not* claiming here that they do not do so---only that there is no
    published materials on the underlying construct and its coverage.
    For example, test manuals do not report the percentage of items
    representing the key subdomains or having high versus low literacy
    demands. The Massachusetts Department of Education (2005) compared
    the content of TABE, AMES, CASAS and ABLE to the\
    state curriculum frameworks. The analysis found that the TABE is
    best aligned to the state's ELA and Math standards, but the TABE
    items aligned with only 50% of the framework and the test has
    limited ability to provide information deemed essential for teachers
    and students.

-   How is it possible to reconcile the fact that one test (TABE) is
    skill based and one is competency based (CASAS)? On the one hand, a
    test that emphasizes assessing skills in a decontextualized context
    may not provide much information on learners' capacities and
    competencies in a functional context. This is a serious issue from
    the point of view of skill transfer because there is no valid
    evidence that skill transfer can happen. Indeed, research on
    everyday mathematics, noted above, suggests that the transfer of
    school or decontextualized learning is difficult. On the other hand,
    the demonstration of competencies in a functional context may be
    domain-specific (Strasser, 2003) and learners may not have more
    generalizable skills. A related research question asks to what
    extent are the practitioners who are using the tests aware of the
    tests' different logic and the implications for the interpretations
    about skills that can be drawn from each test?

-   There is a long-standing awareness of the limitations of
    forced-choice items, used by both the TABE and the CASAS, to reflect
    reasoning, problem-solving, or communication skills (Sticht, 1990).
    A further limitation of the TABE results from its extensive reliance
    on problems that involve little or no text, ignoring the inherent
    links between numeracy and literacy skills in everyday functional
    contexts. The CASAS involves more stimuli with text in them, but
    test-takers do not have to produce any text. Thus, the tests do not
    address the expectation that learners of mathematics are also able
    to "communicate mathematically" (NCTM, 2000; U.S. Department of
    Labor, SCANS, 1991).

-   Both tests expect accurate results that can be clearly classified as
    right or wrong. However, many real-life situations call for
    approximate answers, estimates, or opinions or judgments rather than
    for accurate results. Further, both tests score performance in terms
    of correct/incorrect responses. They ignore the possibility of a
    partially correct response. Yet, recent large-scale assessments of
    mathematical skills employ scoring systems that give credit to
    partial responses (such as TIMSS and PISA; Ginsburg et al., 2005) or
    accommodate various levels of accuracy in responses (such as ALL;
    Gal et al., 2005). The upshot is that because of their forced-choice
    format, both the TABE and the CASAS lose information about the
    skills of the adults being tested---it is technically feasible to
    accommodate more levels of performance in ways that better reflect
    the nature of adults' numeracy.

-   There are claims that a hierarchy of skill levels cannot be
    ascertained in all areas of mathematics (Kilpatrick, 2001; Gal et
    al., 2005). It is unclear how the developers of the two tests have
    coped with this issue in their suggestions for interpretation of
    test results. This is especially true of the TABE, where raw scores
    are converted into grade-level equivalents, given known limitations
    of grade-level equivalents (Spruck Wrigley, 1998; Sireci & Zenisky,
    2003).

Overall, the TABE and the CASAS assessments reflect definitions of adult
numeracy within the *formative* and *mathematical* phases in the Maguire
and O'Donohue framework discussed in Section 2. Neither assessment is
adequate for conceptions of numeracy in the *integrative* phase, such as
those embodied in many of the standards and frameworks influencing the
field. The NRS descriptors for numeracy also reflect the formative and
mathematics phases of numeracy, with most skills related to arithmetic
or "number." Additional numeracy-related skills are listed under
"functional and workplace skills" and mainly call for the ability to
read graphs and charts, deal with forms, and "solve multi-step
problems." This last statement implies an integration of skills, giving
the NRS descriptors a "split personality" when it comes to numeracy
issues.

Tests that assess skills such as problem solving and critical thinking
are difficult to develop. They are also hard to implement in the ABE
system because they are time-consuming to administer and difficult to
score. However, assessments reflecting the integrative approach to
numeracy are used in the Netherlands as part of its national Realistic
Mathematics Education (RME) curriculum for adult students and in
Australia for its Certificates in General Education for Adults (CGEA)
framework (Tout & Schmitt, 2002). The implication is that it is possible
to create assessments that align with overall instructional models.
Assessment should focus educators on broader instructional goals and be
designed in a way that will not encourage teachers to "teach to the
test." Many programs already use the GED test as a de facto definition
of what students need to know. This situation may become more ingrained
with the increased use of both the TABE and the CASAS, which are
sanctioned for use by many states and programs to meet NRS demands.

[]{#__RefHeading___Toc130973158 .anchor}Defining the Meaning of
Assessment and How It Can Be Improved

The absence of recent studies and critical analyses related to
assessment in adult numeracy programs justifies the current review of
assessment issues in the Adult Numeracy Initiative. As a starting point,
we briefly reflect in this subsection on the nature of assessments and
how they can be improved before discussing in the next subsection the
criteria for "good" assessments in the area of adult numeracy education.

Definitions of assessments may vary, yet in one way or another, they all
discuss a systematic process for collecting and evaluating information
about people (Anastasi, 1997). In the context of learning mathematical
topics, Wilson and Kenney (2003), using the NCTM *Assessment Standards*
(1995), explain that the information collected pertains to "a student's
knowledge of, ability to use, and disposition towards, mathematics." As
Wiggins (1992) has aptly warned, however, all too often there is a
tendency to equate assessment with testing and to assume that testing is
a simple, complication-free process that yields clear-cut results.
Instead, it should be acknowledged at the outset that assessment is far
from being a simple or unitary construct and that the complex nature of
an assessment has implications for how it can be improved.

An analysis of various claims and ideas raised in the testing and
assessment literature over several decades (e.g., Cronbach & Gleser,
1965; Messick, 2000), coupled with an analysis of the multiplicity of
purposes of assessments in adult numeracy education and of the diverse
ways in which they can be used by learners, teachers, and programs,
shows that an "assessment" as used in adult education has at least four
key components.

First, all assessments use an *information collection* component, a
process or sequences of steps and activities for gathering data in
diverse ways (Anastasi, 1997). The collection may involve a tool, a
person, or both (e.g., a written GED test, a teacher using a scoring
rubric to grade a learner's written solution to a word problem or the
contents of a cumulative portfolio of a learner's class work over a
period of time, an observation form for rating an employee's performance
on a work sample used for workplace licensing, a computer-based
questionnaire). The tools themselves may have different degrees of
formality or complexity. Artifacts or technical aids, such as a
hand-held calculator, a ruler, or computer software, may be involved and
used by the learner.

Second, an assessment usually activates a *judgment/evaluation
component* in which information from different sources (e.g., subtest
scores, performance on functional tasks, teacher observations) may first
be integrated or combined and then evaluated. In this component,
*meaning* is given to collected data, and *interpretations* are reached
on the basis of expectations, performance standards, norms, and the
like, whether implicit (e.g., a teacher's informal evaluation) or
explicit (e.g., number of correct responses compared with national
norms, IRT scaling of responses of groups of test-takers). When the data
collection involves a simple test (e.g., a single TABE booklet), the
evaluation of a person's performance may appear simple and free of human
judgment, as in the calculation of a total raw score on a test and its
conversion to a grade level equivalent. However, when we examine the
overall range of occasions when information about a specific learner is
collected and evaluated, we clearly see that many judgments and
interpretations are involved. As the psychological testing literature
has repeatedly emphasized (e.g., Goldberg, 1986), people's judgments and
interpretations in such situations may be affected by multiple factors
and may have different degrees of accuracy or validity. Consider the
challenge of evaluating whether a learner "knows" or "does not know" a
certain mathematical construct or procedure when the learner is not a
native speaker of English or is anxious. In these and similar
circumstances, the teacher/assessor has to decide what (if any)
allowance to make for language or learning style differences, and the
judgment process may be affected by a host of seemingly external but at
times very relevant factors.

Third, the assessment will often involve a *decision component* that may
be interwoven with the first two components (Cronbach & Gleser, 1965).
For example, a student may be routed to different test forms on the TABE
on the basis of the TABE Locator test or may receive different items as
part of a computerized adaptive test system that is based on preliminary
item parameters chosen by researchers. Another example occurs when a
teacher has to decide whether to introduce an accommodation (e.g., add
testing time, rephrase a question) because of personal characteristics
of a specific learner, and if so, how far to go in this regard.

Fourth, assessments in educational contexts always involve a *human
component* because people may be the object of the assessment, conduct
the assessment, or examine its results and reach some conclusions.
Teachers or administrators involved in the assessment pathway have
beliefs, attitudes, emotions, and both personal and organizational
motivations. The performance of learners undergoing assessment may be
influenced by dispositional factors discussed in earlier sections (e.g.,
math anxiety or test anxiety, beliefs about testing) or by motivational
factors (the perceived importance of the learning of mathematical
topics, or of the assessment event itself, in the context of the
learner's life course and goals). The learner-related factors and
teacher-related factors may operate separately or interact with each
other, and they may have an impact on the behavior, attention level, or
degree of effort invested by those being assessed or doing the
assessment.

Finally, *value judgments* affect all the other assessment components in
explicit or implicit ways. Test planners, administrators, teachers, and
learners may operate in different social or organizational ecologies and
may have different perspectives on issues such as what is *important* to
measure or what is a *fair* assessment. Adult education staff who
administer the assessments, often lack sufficient training on the
purpose, administration and scoring of tests and impose their own
judgment and values on assessment. This lack of training and
understanding may lead to improper administration, invalid scores and
incorrect interpretation of outcome measures. Experts interested in
psychological and educational testing have time and again argued and
demonstrated that values affect choice of data collection methods,
interpretations, judgments, and behaviors of stakeholders involved in an
assessment (Messick, 2000). The recognition of the centrality of values
and the influence of the human component on all aspects of the
assessment pathway (e.g., test planning, selection of a data collection
method, interpretations of performance, decisions) has led to a strong
emphasis on the need for thorough training for those involved in
assessing human capabilities (Alfonso & Pratt, 1997). This recognition
has also led various professional associations to establish professional
standards in this area.

It follows that assessment is a complex, dynamic system. It can combine
technical elements and supporting artifacts. It involves human
interaction as well as internal and external actions and choices of
individuals. It unfolds over time and is conducted in a social context
that presents diverse demands on those being assessed. Both the inner
workings (setup, artifacts used or not, evaluation of performance, time
limits imposed) *and* the outcomes of an assessment (scores, written
evaluation, pass/fail decisions) are influenced by criteria or values
imposed by the assessment designer, by the beliefs and training of the
humans conducting the assessment, by the motivation and reactions of
those being assessed, and by organizational stakeholders.

These considerations lead us to two important realizations:

-   **Judgments** regarding the "goodness" or "quality" of an assessment
    can refer to *any* of its components---not only to the collection
    methods (e.g., content and format of test items or tasks used) but
    also to the stages involving integration and interpretation, or
    decisions. Further, given that assessments always involve humans and
    are embedded in a certain context, the quality of an assessment may
    also be related to such associated factors as the demand
    characteristics of the testing context, time limits, artifacts used,
    training of the people involved, and the values or criteria applied
    when interpreting performance.

-   **Improvements** in an assessment can relate to any of its
    components or associated factors described above, either alone or in
    combination. Further, assessments in adult numeracy education
    usually occur in an organizational context (e.g., ABE program,
    community college, workplace); hence, broader institutional or
    systemic issues can also be examined and improved, such as how
    assessment processes are scheduled or budgeted, the criteria that
    program directors use for choosing one assessment method over
    another, or the professional qualifications and preparation of those
    administering assessments or interpreting their results.

[]{#__RefHeading___Toc130973159 .anchor}Considerations for Developing
Improved Assessments for Adult Numeracy

If we have concerns about current assessments, we should first consider
what criteria to use to make value judgments about them and then reflect
on whether these criteria are the right ones. Discussions of what counts
as a good assessment often begin with a description of a set of general
principles or criteria. The literature on psychological and educational
assessment invariably states that assessments should be examined in
terms of their *reliability* and *validity* and specifies multiple types
or forms of each of these two key psychometric properties (Anastasi,
1997). The testing literature has also referred, using somewhat
different terminology, to various other important aspects of
assessments, such as *degree of standardization* of an instrument (seen
as reflecting its degree of subjectivity or objectivity); *cost* of
testing; *ease of scoring and usage*; *fairness* or *discrimination*;
and *utility*, a concept related to the contribution of a test to an
organization making acceptance/rejection or placement decisions
(Cronbach & Gleser, 1965). These attributes of assessment are typically
required of all high-stakes educational assessments. Indeed, the NRS
*Guidelines* require all adult education programs to use standardized,
reliable, and valid assessments.

However, in discussing the assessment of *numeracy*, we must note
additional principles or ideas, especially by those involved in teaching
the relevant subject matter. The U.S. mathematics education community
has articulated several principles for assessment as part of the reform
movement, which started two decades ago. The Mathematical Sciences
Education Board (1994) enumerated three principles of good assessment in
mathematics: the *Content Principle* (assessment should reflect the
mathematics that is most important for students to learn), the *Learning
Principle* (assessment should enhance mathematics learning and support
good instructional practices), and the *Equity Principle* (assessment
should support every student's opportunity to learn important
mathematics). NCTM's (1995) *Assessment Standards* raised additional
points, later reiterated in its *Principles and Standards for School
Mathematics* (2000), regarding *openness*, *valid inferences*, and
*coherence* of an assessment with the curriculum and instruction.

The mathematics-oriented assessment principles echo several key ideas
found in the general testing literature, especially regarding validity,
fairness, and utility. Yet, these principles go above and beyond
technical demands for validity and reliability and bring new elements
into the discourse about good assessments of numeracy skills. They also
encompass the nature of mathematical knowledge, the intended goals of
the learning process, and the varied uses of an assessment or its
influence regarding the learner, the teacher, and the program. Messick
(1989) has argued that validity should be viewed broadly as the extent
to which empirical findings and theoretical rationales support the
*appropriateness of inferences* and *actions* based on test scores or
performance on the assessment instrument. However, the actions taken by
a teacher and a learner *after* they know an assessment's results are
seldom based on the assessment itself. Rather, such actions are
influenced by the broader ecology or context within which the assessment
takes place and within which the teacher and the learner operate. In
particular, an assessment becomes meaningful when there is a linkage
among the assessment, the curriculum, and the learner's own goals (which
are especially important for adult learners; Donovan, 2002), as well as
when both teachers and learners believe that an assessment measures
valued skills or capacities. It thus becomes paramount to examine what
content areas related to mathematics and numeracy are important for
adults to know.

[]{#__RefHeading___Toc130973160 .anchor}What Should Be Assessed in Adult
Numeracy Instruction?

The answer to this question will affect our judgment of the quality of
different assessments in numeracy education. However, the answer is
complex because it depends on the breadth of the analysis being
attempted and on the extent to which we go beyond traditional ideas that
equate mathematical knowledge with mastery of basic arithmetic concepts
and accurate execution of computational procedures.

Section 2 has already reviewed several perspectives on the concept of
numeracy, clustered into three groups or phases: formative,
mathematical, and integrative. Below we summarize these and also
consider additional key influences that help us sketch the nature of the
skills, knowledge, and dispositions that are part of the constructs that
could be the focus of teaching/learning, and hence also of assessments,
in the field of adult numeracy education:

-   Curriculum frameworks and statements regarding the goals of
    mathematics learning for adults and young adults (e.g., Australian
    Committee for Training Curriculum, 1994; Curry et al., 1996; NCTM,
    2000; Stein, 2000) present a broad set of areas for learning:
    knowledge and understanding of numbers and operations that involve
    computation, as well as number sense and estimation; algebra and
    modeling; geometry and shape; measurement; and data and chance. In
    addition, knowing mathematics involves problem solving, reasoning,
    and communicative capacities as well as an understanding of
    connections and representations of various mathematical ideas.

-   Conceptions of workplace skills (e.g., U.S. Department of Labor,
    SCANS, 1991; Mayer, 1992; Packer, 1997; Forman & Steen, 1999)
    further show that effective functioning on the job involves not only
    a diverse set of mathematical and arithmetical skills but also
    broader knowledge and skills related, for example, to the ability to
    allocate resources, handle scheduling, understand the role of
    quantitative information in the operation of systems, and use
    technological tools to quantify or display quantitative information.

-   Teachers and the professional literature have mostly addressed the
    acquisition, teaching, and learning of language skills and
    mathematical skills as two separate areas of inquiry and practice
    with little crossover (Gal, 2000). However, many real-world
    functional tasks require adults to integrate numeracy and literacy
    skills. Examples are interpretive tasks that do not require the
    manipulation of numbers but do require the expression of opinions
    (e.g., statistical literacy tasks, decision tasks involving notions
    of chance) and other tasks that involve numbers or quantitative
    statements embedded in text (e.g., forms, schedules, manuals,
    technical and financial documents, statistics in the media). Kirsch
    and Mosenthal (1993) integrate numeracy and literacy in their
    literacy model, which comprises prose, document, and quantitative
    components. The integration of these skill areas underlies the
    design of various large-scale surveys of numeracy and mathematical
    skills, such as the NALS, the International Adult Literacy Survey
    (IALS), NAAL, and the recent ALL survey (Gal et al., 2005), and the
    design of some quantitative literacy tasks in the Trends in
    International Mathematics and Science Study (TIMSS).

-   The research on ethnomathematics, briefly discussed in Section 2,
    has focused on the nature of everyday mathematics. One of the
    conclusions in this literature is that differential performance can
    occur when assessments are divorced from, as opposed to
    contextualized in, realistic settings (Lave, Murtagh, & de la Rocha,
    1984). Problem solving in contextualized real-life and work
    activities may be complex and different from solving school-like
    problems (Resnick, 1987; Greeno, 2003). Many researchers (e.g.,
    Strasser, 2003; Wedege, 2003) discuss the hidden aspects of
    workplace mathematics and point to the fact that workers often have
    trouble identifying the "math" in what they do or encounter. These
    and other observations suggest that good assessments need to enable
    learners, teachers, and programs to identify the ability to transfer
    and apply learned numeracy skills in real, functional contexts, but
    not only those where the mathematics is explicit and obvious. At the
    same time, the ability to handle functional numeracy tasks will not
    necessarily imply that a learner has acquired generalizable skills
    or the ability to handle abstract or more formal mathematical
    concepts and ideas. Thus, a mix of both functional and more abstract
    tasks may be needed.

-   Finally, many studies and curricular frameworks emphasize the
    critical role that positive *attitudes, beliefs, and habits of mind*
    (collectively called here *dispositions*) can play in supporting
    effective and confident numerate behavior and in enhancing adults'
    ability to manage the mathematical demands of different situations
    (MacLeod, 1996; Gal, 2002b). Indeed, items measuring selected
    dispositions associated with mathematical activities and feelings
    about mathematics are part of major large-scale assessments in the
    mathematical areas (e.g., NAEP, PISA), and more elaborate scales
    have been included in the ALL survey and shown (Gal et al., 2005) to
    correlate positively with performance.

Overall, the picture emerging from this review is that it is presently
unclear how well statements about target skills or curricular standards
(as reflected, for example, in the EFF and ANN standards, or in the NRS
skill specification) are linked to the structure and content of key
assessment schemes and how all these are aligned with instructional
practices. Too little is known about the types and uses of assessments
developed locally by teachers or programs, and no systematic analyses
have been conducted of the mathematics or numeracy components of the
most recent versions of the TABE, the CASAS, and other assessment
systems. Clearly there is room for states and programs to develop or
experiment with assessment systems that offer better or at least more
explicit alignment with curricular goals and teaching methods, and some
states (such as Massachusetts or Ohio) have recently begun to take steps
in this direction.

[]{#__RefHeading___Toc130973161 .anchor}Conclusions

Multiple factors converge to challenge anyone trying to summarize the
"state of the field" in numeracy assessment or to offer any quick fixes.
The scope of the objects to be assessed (e.g., skills, capacities,
dispositions) and the deficiencies of current assessment methods depend
to some extent on the eye of the beholder. Definitions of numeracy vary,
from relatively narrow notions that focus on computational or procedural
skills to broader, integrative views that go well beyond computations
and encompass a broad set of mathematical skills, communicative and
interpretive abilities, supporting dispositions, and skill transfer. The
teaching/learning contexts in which adults engage with numeracy-related
topics vary greatly given the diversity of backgrounds, ages, or
purposes for learning that adults bring. Conceptions regarding the goals
of adult numeracy education and of the competencies to be developed
evolve over time and are affected by multiple stakeholders (Curry et
al., 1996; Forman & Steen, 1999; Kilpatrick, 2001).

Policymakers' pressures for accountability have an impact on what skills
are deemed important and assessed (or not) by programs and teachers
(Stites, 2000). Funding and program accountability requirements, and
especially the NRS, place different emphases on preferred forms of
assessment, although they allow the concurrent use of tests that provide
different types of information of varying qualities. This approach may
provide information that learners have made some progress, but what
exactly that progress is and whether the progress is toward valued goals
cannot be fully ascertained from the reported information provided by
current instruments. The reliance on only standardized tests for
reporting, tests that presently rely on multiple-choice format, runs the
danger of pushing more teachers to "teach for the test" and stay away
from using assessment approaches that rely more on performance in
diverse forms, such as oral presentations, portfolios, or various
locally developed assessments (which are quite popular with adult
educators in countries such as United Kingdom and Australia).

Given NRS demands, there is likely to be a tendency to use only one or
two styles of information gathering for multiple purposes (Nagel, 1999;
Messemer & Valentine, 2004). The use of only one or two instruments may
in fact have some advantages; for example, it can enable learners,
teachers, and funders to use a consistent system of concepts and
interpretations when reporting and discussing assessment results. Yet,
the use of only one or two instruments may be problematic, given on the
one hand the different needs of learners, teachers, and programs and on
the other hand the breadth of the numeracy construct (at least according
to the integrative views presented in Section 2) that is at present only
partially addressed by existing large-scale assessments. The
multiplicity of purposes or contexts of the uses of assessment outlined
earlier implies a need for different forms and types of information
gathering and interpretation at different points during a learner's
progress in a program. The reports covered in our review do not contain
much information on the extent to which the assessment methods in use in
programs are sufficiently differentiated in terms of these multiple
purposes―they may be, but this issue requires further investigation,
given that in each assessment only a limited number of items are used.

As Cumming & Gal (2000) have argued, teachers have a key role in
changing assessment practices. Teachers need to reflect on their goals
and curricula in light of extant perspectives on the goals of
mathematics learning for adults and how well assessment practices are
aligned with such goals to better cover the full range of skills and
dispositions expected of numerate adults. Eventually, however, most
instruction is delivered at the individual teacher level. The teacher is
thus the main player and a strategic partner whose knowledge, values,
decisions, practices, and training need to be considered when thinking
of improving assessments in ways that can better address instructional,
validity, and accountability needs.

[]{#__RefHeading___Toc130973162 .anchor}5. Professional Development in
Adult Numeracy

One of the major thrusts of the Adult Numeracy Initiative and of the
states is to improve the quality of instruction in adult education.
Several states have moved in this direction by developing content
standards, many of which model the ANN and the NCTM standards, and by
implementing numeracy initiatives and policy changes to foster better
adult numeracy teaching. The vehicle by which states can implement these
changes into practice, however, is professional development that
improves an instructor's knowledge and skills to teach the mathematical
content and processes reflected in the standards (Gal & Schmitt, 1995).
Professional development is regarded as a key mechanism for
strengthening the instructor's content knowledge and teaching and
learning processes.

This section explores issues in numeracy professional development,
examines the gaps in professional development in adult numeracy, and
sheds light on how numeracy-based professional development is positioned
in literacy-driven programs. It also offers a brief review of research
on professional development in general with attention drawn more
specifically to research related to numeracy professional development,
including professional development quality.

Very little research in this area has been conducted on adult education
teachers. We supplement this sparse research with studies from the K--12
research literature, although we recognize the vast differences between
the characteristics of teachers, the teaching environment, and
professional development opportunities in K--12 and those in ABE.

[]{#__RefHeading___Toc130973163 .anchor}Issues in Professional
Development

Professional development is considered a principal vehicle for advancing
educational improvement on both a national and a local level. As long as
there is a need to restructure education, professional development will
remain an imperative device for steering educators to effective practice
and for improving their instructional capacity. Through professional
development, teachers gain opportunities to collaborate with others in
their field, initiate practitioner research, model instructional
techniques, and even apply theories and concepts to real classroom
activities.

Primary issues related to professional development in adult numeracy are
vast and varied, which has implications for defining quality
professional development practices for adult numeracy. The scant
research on professional development in the context of adult numeracy
and basic education reveals little about well-tested, professional
development--driven directions for improving ABE numeracy practice in
general and highlights gaps and needs in professional development in ABE
numeracy in relation to teacher capacity. The close relationship between
literacy and numeracy shrouds a clear picture of the quality and
quantity of numeracy professional development that is offered.
Furthermore, the lack of a standard definition of numeracy thwarts
opportunities for a proper conceptualization of ABE numeracy
professional development, which presents a challenge in examining
literature to identify quality, common practices, and the potential for
replicable interventions.

[]{#__RefHeading___Toc130973164 .anchor}Need For Professional
Development in Numeracy

Adult education teachers appear to be especially unprepared to teach
mathematics, which makes professional development in the area of
numeracy essential. Many adult numeracy teachers come from the K--12
arena where there is already a shortage of qualified teachers
(Darling-Hammond, 2000). For example, the California Commission on
Teacher Credentialing (CCTC, 2004) reported that in the 2002--2003
school year, more than 20,000 teachers in California were teaching with
emergency permits and credential waivers. To complicate matters, a huge
percentage of ABE instructors teach on a part-time basis, which gives
them very little opportunity to receive adequate professional
development and training. Gal's (2002a) review found little evidence of
mathematics training among adult education teachers, with fewer than 10
percent of teachers reporting that they were certified in mathematics.
Gal also observed that there are no university-based graduate programs
on teaching mathematics to adults, although in Australia, some efforts
have been made to provide adult numeracy teachers with opportunities for
professional development (Tout & Johnston, 1995; Marr & Helme, 1991;
DEET, 1993).

Lack of preparation is compounded by low teacher retention (National
Commission on Teaching and America's Future \[NCTAF\], 2003), especially
in low-income urban areas likely to have high numbers of adult numeracy
students. Many teachers in these areas are responsible for teaching
multiple subjects for which they are not trained to teach. This seems to
be an international trend. The U.K.'s Advisory Committee on Mathematics
Education (ACME, 2002) points out that with a shortage of professional
development opportunities in the K--12 setting in general, finding
quality professional development could be a hard test for those who
teach adult numeracy. For K--12 teachers who also teach ABE part-time,
hardly any content-based professional development opportunities are
available. The same situation can be said to exist in the United States.

The lack of adequately trained teachers has consequences for student
achievement and for educational reform. Although no research has
measured the relationship between teacher preparation and the
achievement of adult numeracy students, research done with children
(Rice, 2003) shows that teacher preparation had the strongest
correlation to student achievement in reading and mathematics. Coben
(2003) notes that:

Adult numeracy teacher education is currently undergoing major
transformation. Some teachers' inadequate subject knowledge is a
continuing concern. Studies with children suggest that: initial and
ongoing teacher education increases subject knowledge, facilitates
career development and encourages future research and development;
effective teaching correlates with engagement in continuing professional
development. (p. 7)

Inadequately trained teachers will be unable to help students achieve in
numeracy if they do not know mathematical content, if they are not
trained in pedagogical principles specific to or appropriate for the
adult learners represented in most adult education programs, or if they
have been inadequately prepared to meet state standards (ACE, 1999).
Research shows that a substantial number of teachers learned to teach by
using a model of instruction that focuses on memorizing facts
(Darling-Hammond & McLaughlin, 1995; Porter & Brophy, 1988) without
integrating procedural and conceptual instruction (Desimone, Smit,
Baker, & Ueno, in press). They may not have been taught to integrate
procedural and conceptual instruction and thus focus on rote
memorization (Leonelli & Schwendeman, 1994) and decontextualized
situations (Lionelli, 2005), strategies deemed futile for adult numeracy
students (Coben, 2003; Skemp, 1986). Owing, in part, to the inadequacy
of adult numeracy training, teachers tend to pivot back on traditional
pedagogical techniques (Little, 1993; Loucks-Horsely, Love, Stiles,
Mundry, & Hewson, 1998; Sparks & Loucks-Horsely, 1989). Some teachers
are not trained adequately in the mathematics principles (Massachusetts
ABE Math Team, 1994) and concepts needed to teach numeracy; they often
teach numeracy the way it was taught to them in school and incorporate
methods they learned in college (Leonelli, 1999). Adults need "good
numeracy provision" (Newmarch, 2005) to be able to make tangible use of
abstract numeracy concepts. Because adults construct information
differently than children do (Rogers, 2003), teachers will need much
more than K--12 knowledge and experience to help adults achieve in
numeracy class.

The ACME made the case that professional development should help
teachers "unpack" mathematics by motivating teachers and helping them
increase their knowledge of innovative ways to promote mathematics to a
diverse body of students. Organizations in the United States make
comparable statements, focusing on practitioner development that will
"encourage the learning and teaching of mathematics in a manner which is
interesting and appropriate to adults" (Massachusetts ABE Math Team,
1994) so that "educators can refocus the adult numeracy curriculum in a
meaningful way" (Curry et al., 1996) in an effort *"*to change the way
math is actually taught and learned in an adult literacy community"
(Brover, Deagan, & Farina, 2005). Teachers also need to know adult
learning theories (Hansman & Wilson, 1998; Brookfield, 1986; Knowles,
Holton, & Swanson, 1998; Lawler, 1991; Merriam & Caffarella, 1999;
Mezirow, 1991). Moreover, they need to know about adult numeracy
learning. Teacher attitude also plays a key role in adult numeracy
practice. Instructors need to consider their own attitudes about
numeracy (Stoudt, 1994), especially because their outlook and viewpoints
about numeracy are likely to influence how they teach it to adults.

The National Research Council's Mathematics Learning Study Committee
(2001) identified the kinds of knowledge that K--8 mathematics teachers
need in order to teach mathematics proficiently. Teachers, quite
obviously, need to be aware of the individual needs of their students,
need to know how to manage their classrooms, and should have an
"elaborated, integrated knowledge of mathematics" so they can anticipate
and understand students' different interpretations of concepts (p. 381).
According to the committee's analysis of K--8 research, proficient
teaching of mathematics consists of the following components:

-   Teachers must possess a deep understanding of core mathematical
    concepts and of the ways their students' understanding matures.
    Teachers must be able to see and make connections between their
    knowledge and their classroom practices.

-   Teachers must possess a repertoire of instructional and classroom
    management routines that they can implement fluently.

-   Teachers must possess a strategic competence of mathematics in order
    to respond, on the fly, to students' questions or statements.
    Professional development programs can give teachers practice in
    analyzing and dealing with instructional problems.

-   Teachers must exercise adaptive reasoning. They must reflect
    frequently on topics such as the difficulties their students are
    encountering.

-   Teachers must bear a disposition that is productive: they must view
    their own knowledge, practice, and learning as valuable, and they
    must feel confident in their own ability to learn from their
    students' thinking.

These principles can easily be applied to adults. ABE mathematics
students represent diverse characteristics, and ABE numeracy teachers
need to be able to design lessons, manage ABE numeracy classrooms,
create an environment that fosters quality numeracy teaching, and teach
mathematics to adults who will need to integrate and transfer such
knowledge to multiple aspects of their lives. This could be a daunting
task, especially for inexperienced ABE mathematics teachers, so the need
for quality professional development grounded in principles and research
that correlate not only to mathematics achievement in general but to ABE
mathematics achievement in particular is urgent. The burgeoning ABE
population, as well as the increasing workforce demand for numerate
employees, underscores the urgency for quality professional development
in the area of ABE mathematics.

[]{#__RefHeading___Toc130973165 .anchor}Overlapping Relationship Between
Literacy and Numeracy

Professional development and teacher training in adult numeracy are
often included under the umbrella of "literacy," thereby resulting in
both the appearance of a nonessential need for effective teacher
training in numeracy and underdeveloped numeracy professional
development programs. A consequence of this indirect numeracy
professional development is literacy-centered numeracy classroom
practice through which teachers teach numeracy merely as a tangential
element of adult literacy.

Because of the intricate relationship between numeracy and literacy, it
is quite easy for practitioners to deemphasize the need for professional
development for teachers who teach numeracy to adult learners (Maguire &
O'Donoghue, 2002). Numeracy teachers who have knowledge of literacy
might be able to integrate numeracy with literacy instruction
(Kallenbach, 1994; Lucas, Dondertman, & Ciancone, 1991) and teach in a
way that is "relevant, contextualized, and essentially linked to overall
literacy" (Stout, 1994, p. 11), but the call for numeracy-specific
teacher professional development (Gal & Schmitt, 1995) may not be
answered.

Certainly, there is some value for teachers to have pedagogical
knowledge of literacy in an adult numeracy class. In their examination
of the pedagogical relationship between language and literacy,
researchers found numeracy to be embedded in, and intricately
overlapping with, literacy (Lee, Chapman, & Roe, 1994). Mathematical
knowledge is ingrained in language, and teaching numeracy requires the
use of literacy skills. The literacy activities that occur in numeracy
classes, such as reading word problems and decoding word meanings, could
be strengthened through professional development that shows teachers how
to use reading to help students maximize their ability to use
mathematics in their everyday life, as well as how to incorporate
numeracy and literacy activities that build on student experience.
Because numeracy and literacy are often taught within the same academic
context, it will be helpful for teachers to understand the correlations
between numeracy and literacy and design appropriate curricula and
assessment for adults (Buchanan & Helman, 1997). Of benefit also will be
their ability "to find different ways to assess literacy students\'
progress in mathematics" (Buchanan & Helman, 1997, p. 19). Numeracy
professional development in this regard needs to be deliberate and
direct with guided intent to help teachers complement their knowledge in
literacy with strong pedagogical techniques in numeracy.

Definitional issues of numeracy, as discussed in Section 2, and
numeracy's relationship to mathematics and literacy become issues here.
Without clarity about what numeracy means, it is difficult to formulate
consistent approaches to professional development in a way that fully
measures teacher instructional capacity and instructional change.

[]{#__RefHeading___Toc130973166 .anchor}Research in Professional
Development and Adult Numeracy

As the review in Section 3 shows, adult numeracy instruction is an
underresearched area of inquiry. This lack of research extends to
professional development in adult numeracy. Indeed, identifying
components of quality professional development and measuring the impact
on instructor knowledge and skills are areas that we have only begun to
explore in the last decade. Recent research has examined the impact of
participation in professional development on instructors' knowledge,
skill levels, and instructional practices. Much of this research has
occurred at the K--12 level and focuses on teacher learning and teacher
change (Corcoran, 1995; Hargreaves & Fullan, 1992; Lieberman, 1996;
Little, 1993; Loucks-Horsely, Love, Stiles, Mundry, & Hewson, 1998;
Sparks & Loucks-Horsely, 1989; Stiles, Loucks-Horsley, & Hewson, 1996),
with only limited research in the adult education arena. The research
literature is a mix of large- and small-scale studies (Dias, 1997; Falk
& Kilpatrick, 1998; Gray, 2003; Porter, Birman, Garet, Desimone, & Yoon,
2004), teacher surveys of preservice and in-service experiences (Joseph,
1997; Maguire & O'Donoghue, 2002), and evaluations of professional
development programs (Brown, 2002; Earl, Fullan, Leithwood, & Watson,
2000; Earl, Levin, Leithwood, Fullan, & Watson, 2001; Earl, Watson,
Levin, Leithwood, Fullan, & Torrance, 2003) designed to improve teaching
and learning.

Although different approaches have been tried in teaching numeracy to
adults (Tout & Johnston, 1995; Marr & Helme, 1991; DEET, 1993), the
state of numeracy professional development is such that priorities vary.
With little research available to identify quality professional
development practices, teachers are left with a very small repertoire of
proven pedagogical techniques to use and inspire changes in the adult
numeracy classroom.

Nonetheless, to provide guidance for numeracy professional development,
we can draw from two empirical studies conducted in the United States
that identify characteristics of quality professional development: an
evaluation of the Eisenhower Professional Development Program, actually
a coordinated set of studies designed to evaluate the effectiveness of
the federal government's professional development program for elementary
and secondary school teachers (Porter et al., 2004, p. 4), and How
Teachers Change: Study of Professional Development in Adult Education,
conducted by the National Center for the Study of Adult Learning and
Literacy (NCSALL).

The Eisenhower study is the first large-scale empirical comparison of
the effects of different characteristics of professional development on
teachers' learning. The evaluation of the program includes three
quantitative studies: (a) telephone interviews with a national
probability sample of Eisenhower coordinators in 363 school districts;
(b) a mail survey to collect a national probability sample of 1,027
teachers who participated in 657 Eisenhower-assisted activities; and (c)
a third study in which 287 mathematics and science teachers
participated. Researchers augmented these studies with a set of case
studies in five states with districts representing regional, ethnic, and
economic diversity. Their evaluation of available literature about best
professional development practices revealed five key features of quality
professional development:

1.  **Duration** --- is sustained over time (including the total number
    of contact hours and the span over which the activity takes place)

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5.  **Content Knowledge** --- focuses both on content in the subject
    area and on how students learn that content

6.  **Active Learning** --- promotes active learning, gives teachers
    opportunities for hands-on work, and includes opportunities for
    teachers to observe expert teachers, to link ideas learned in
    professional development to the teaching context, to examine and
    review student work, and to make presentations, lead, and write

7.  **Collective Participation** --- emphasizes collective participation
    of groups of teachers from the same school, department, or grade
    level

8.  **Coherence** --- forms part of a coherent program for teacher
    learning and development (e.g., consistent with teachers' goals and
    aligned with state and district standards and assessments) (Porter
    et al., 2004)

A major finding in the Eisenhower study is that to improve professional
development, it is more important to focus on duration, collective
participation, content, active learning, and coherence than on the type
of development (e.g., mentoring, teacher networks, individual research
project, traditional workshop or conference). The type of delivery has
an effect on teacher outcomes only insofar as the activities reflect the
key features (Garet, Porter, Desimone, Birman, & Yoon, 2001).

Professional development does not occur in a vacuum; contextual factors
play a role in the delivery of high-quality professional development and
have an impact on change (Desimone, Porter, Garet, Yoon, & Birman, 2002;
Porter, Garet, Desimone, & Birman, 2003). In general, the Eisenhower
studies showed that effective professional development involves the
proper alignment of various program components, including management,
funding sources, standards and assessments, and collective
participation. At the K--12 level, school district practices regarding
the alignment of professional development, standards and assessments,
and teacher involvement in planning affect the quality of the
professional development. In adult education, the most important system
factors affecting teacher change are access to preparation time, access
to benefits as part of their adult education jobs, and a voice in
decision making (Smith, Hofer, Gillespie, Solomon, & Rowe, 2003).

The How Teachers Change Study (Smith et al., 2003, p. 11) was designed
to help professional development decision makers understand the impact
of professional development (including system, program, and individual
factors) on teacher change. The study examined how adult education
teachers changed after participating in three models of professional
development: (a) a *multisession workshop* (experiential, active
learning activities), (b) a *mentor teacher group* (study circles, peer
coaching, and observation), and (c) a *practitioner research group*
(teachers investigate their classroom practice and collect and analyze
data). The study also investigated the most important individual,
professional development, program, and system factors that influenced
the type and amount of teacher change.

Of a sample of 106 adult education teachers from three New England
states (Maine, Massachusetts, and Connecticut), 100 teachers took part
in 18 hours of one of the three models of professional development and
provided researchers data through questionnaires and interviews before,
after, and one year after participating in the study. Six teachers
served as a comparison group. Also, six teachers were randomly selected
from each group to serve as a subsample. Participants who completed 12
out of 18 hours were considered completers; 16 participants dropped out
of the study. The research team designed all three professional
development models, using the best methods and accepted principles of
adult learning and effective professional development, and recruited and
trained experienced teachers or professional development leaders in each
state to facilitate the professional development.

An analysis of data showed implications for the *amount* of teacher
change, the *roles* in which teachers changed, the *ways* in which they
changed, and also the *factors* that influenced teacher change.
Regarding the amount of teacher change, the study suggested that most
teachers, including those who dropped out, changed at least minimally
through knowledge and action gains, with relatively few experiencing no
change. The majority of teachers changed in their role as classroom
teachers as opposed to other academic roles they may have played. In
looking at the *ways* the teachers changed, the findings showed that the
majority (72 percent) of the 83 completers demonstrated change, most of
which was nonintegrated (thinking or acting) change.

Other factors that demonstrated teacher change were motivation to attend
professional development, years of experience in the field of adult
education, venue of first teaching experience, and level of formal
education. Factors that led to teacher change were hours of professional
development attended, quality of professional development, and
collaborative participation. As in the Eisenhower study, the model of
professional development was not a factor. Contextual factors that
affected change were teacher access to benefits and prep time, the
program's history in addressing learner persistence, and teacher access
to decision making.

The K--12 literature has introduced several ways to develop teachers'
abilities to teach with mathematical proficiency. The National Research
Council's Mathematics Learning Study Committee (2001) identified four
models of professional development that allow teachers' "knowledge,
conceptions and practice \[...\] to grow and evolve" (p. 385). The
committee sorted professional development programs into those that
focused on mathematics, those that focused on student thinking, those
that focused on cases, and those that focused on "lesson study," a
practice that originated in Japan.

-   **Focus on mathematics**: Instructors seek to develop teachers'
    proficiency with core content from elementary school mathematics.
    Teachers are asked to unpack familiar content and make explicit the
    underlying procedures they use. The instructors, in the meantime,
    expose teachers to concepts that belie familiar procedures.

-   **Focus on student thinking**: Teachers read the written work and
    problem-solving strategies of real students, consider how students'
    thinking can inform their lessons, and discuss how they interact
    with their own students. This model is based on Cognitively Guided
    Instruction (CGI), a professional development program that helps
    teachers develop instructional materials after they watch and listen
    to their students solve problems.

-   **Focus on cases**: Teachers study a mathematics topic and the
    contexts surrounding it (e.g., the assignment, the students' work,
    the teacher's materials). Studying cases "serves as a basis for
    discussion and inquiry rather than as \[a\] \[model\] of instruction
    for the teachers to emulate" (p. 395).

-   **Focus on lesson study**: Teachers create a lesson plan together,
    and then one teaches it to the group. The group revises it together,
    and the next person teaches the lesson. The teachers conduct very
    detailed analyses of effective mathematics instruction and revise
    their own teaching practices in the process.

The foregoing models should inform adult educators as they think about
designing professional development programs for adult numeracy
instructors. Regardless of which models are delivered, however, the
content of professional development is likely to be driven by content
standards. The NCTM standards for mathematics reform and EFF, for
example, have led to a call for teachers to develop strong capacity in
mathematical content. This interest resulted in heavy emphasis not only
on the inclusion of numeracy in adult basic education (Tout & Schmitt,
2002) but also on how to provide quality professional development to
adult numeracy teachers (Gal & Schuh, 1994), including the use of adult
numeracy curricula content as a vehicle for professional development
(Massachusetts ABE Math Team, 1994).

Several organizations, such as the Arkansas Adult Numeracy Campaign
(ANC), the Math Education Group (MEG) in New York City, and the
Massachusetts Adult Basic Education Math Standards project, have
modified and extended the K--12 *Principles and Standards for School
Mathematics* to address content in adult numeracy professional
development. Although these efforts provide guidance, more long-term
research is needed to draw safe conclusions about which professional
development initiatives work specifically in the context of adult
numeracy and which do not.

[]{#__RefHeading___Toc130973167 .anchor}International Efforts

While issues about the nature of quality and content of adult numeracy
professional development are debated in the United States, other
countries seem to have taken a lead. In the United Kingdom, Australia,
and the Netherlands, numeracy has been well established as a separate
area of inquiry, and numeracy teachers are given more structured
opportunities for professional development than in the United States.
For example, urgent calls in Australia for reformed numeracy teacher
training (Marr & Helme, 1991) resulted in the development of Adult
Numeracy Teaching: Making Meaning in Mathematics (Tout & Johnston,
1995*),* also called the ANT, which is an 84-hour professional
development program designed to support teachers of adult numeracy in
the context of ABE. The program comprises a coherent system of
strategies and readings to enable teachers to learn how to use
manipulatives and design their teaching to meet the everyday needs of
students. However, the efficacy of the program is still not clear
because there has been no intervention to verify such information.

Also produced in Australia, *Breaking the Math Barrier: A Kit for
Building Staff Development Skills in Adult Numeracy* (Marr & Helme,
1991) is a book designed for a two- to three-day numeracy teacher
training course. The book is also useful for teacher self-study as well
as for mathematics teachers who are interested in learning more about
teaching adult numeracy, literacy teachers who are interested in
teaching numeracy, and numeracy teachers who want to develop specific
numeracy teaching skills and strategies. The book supports a
participatory model of staff development and provides teachers with
background theory, including activities and materials that they could
use with students. Like the ANT, it does not appear that any
interventions have tested the effectiveness of the strategies prescribed
in this book.

What is clear is that the field of adult numeracy is facing major
transformation, and so is the area of adult numeracy professional
development. The call for curriculum developers to produce more
professional development materials for numeracy teachers is being slowly
answered, although the call for researchers to respond with more
efficacy research to verify the effectiveness of the techniques and
strategies prescribed in the books and curriculum available remains
unanswered.

[]{#__RefHeading___Toc130973168 .anchor}Summary

The search for research on effective professional development in adult
numeracy produced scant examples of best practices grounded in rigorous
research and no examples addressing the effect of adult numeracy
profession development. Advances in this area first require agreed-on
principles to inform adult numeracy teaching and instruction and then
research-based standards of professional development directly related to
these principles. The move toward developing content standards for adult
numeracy instruction holds promise in filling this need. However, what
adult numeracy professional developers need to know to facilitate the
development of adult numeracy teachers has not been articulated clearly,
nor has it emerged in any research. Professional developers must also
conceptualize the anticipated end result of training and engage teachers
in activities that will enable them to help students meet their goals.

According to an evaluation of the available research, Web sites related
to adult numeracy, and practitioner issues discussed in articles, the
state of numeracy professional development is such that teachers who are
not adequately prepared to teach adult numeracy may have some difficult
identifying good practice because of the relatively limited repertoire
of information currently available. Adult numeracy teachers have
relatively little time, limited compensation, and few resources to learn
on their own. Many resources put limited emphasis on teacher training
and development, and of the information provided, much of it is grounded
in opinion and limited experience, rather than in research.

The future does hold promise. Several ABE-focused professional
development initiatives in adult numeracy are currently under way. These
projects include the Adult Numeracy Campaign (AANC) in Arkansas; the
Adult Literacy Resource Center (ALRC) and the Adult Volunteer Literacy
Tutoring Project in Illinois; the Center for Adult Learning and Literacy
(CALL) and Math Instructional Ideas for College Transitions Teachers
(MIICTT) project in Maine; the ABE Math Standards Project in
Massachusetts; the Math Education Group (MEG); the Ocean Sciences and
Math Collaborative Project in Oregon; Making Math Real Institute and
Clinic (MMR) in California; TERC in Massachusetts, and South Carolina's
Workplace Resource Center (WRC). The next activity of the Adult Numeracy
Initiative project will gather more information from these projects and
address other key issues identified in this review to further adult
numeracy instruction and research within the ABE program.

[]{#__RefHeading___Toc130973169 .anchor}6. Summary and Implications for
Future Research

This review of the literature on the research and conceptual issues of
adult numeracy has examined definitions and theories of numeracy,
reviewed the research on instruction and assessment, analyzed commonly
used assessments, provided a framework for evaluating and improving
assessment, and reviewed what we know about professional development for
teaching mathematics to ABE adults. The review addresses five of the
research questions presented in Section 1 that OVAE posed for the Adult
Numeracy Initiative and builds a foundation for the project by
describing the current state of the field. In this section, we summarize
the findings of the review and present implications for future research
to promote progress in research and practice.

[]{#__RefHeading___Toc130973170 .anchor}Definitions and Theories of
Adult Numeracy

The construct of numeracy has no single, universally accepted
definition, and there remains considerable debate on how best to define
it, especially when referring to adults. We briefly traced the
definitions from the origin of the concept in the United Kingdom in 1959
along a continuum of phases of development. These definitions include
views of numeracy as basic arithmetic and computations skills;
functional definitions, where numeracy is mathematics in "context" to
cope with the demands of everyday life; and views of numeracy as an
integrative skill, incorporating mathematics, communication, cultural,
social, emotional, and personal aspects of individuals in context. This
view of numeracy as a multidimensional, integrated skill is dominant in
all current theorizing and thinking on adult numeracy.

As in other areas of learning, theories on how adults learn mathematics
have gone through a shift from behaviorist theories, where the teacher
is the conveyor of objective knowledge to students who absorb it to
create a response, to constructivist theories, which now predominate.
The adoption of constructivism as the guiding theory to approaches to
mathematics instruction corresponds to the adoption of integrative
definitions of numeracy. Constructivist theories argue that all
knowledge is constructed by individuals acting on external stimuli and
assimilating new experiences by building a knowledge base or altering
existing schemas. Learners actively construct knowledge by integrating
new information and experiences into what they have previously come to
understand, revising and reinterpreting old knowledge in order to
reconcile it with the new. Research inspired by constructivist theories
has examined the role of social and cultural perspectives, personal
experience and situations, affective factors and individual learning
styles on how adults construct meaning about and learn mathematics.

[]{#__RefHeading___Toc130973171 .anchor}Instructional Approaches and
Interventions for Adult Numeracy

Since the 1990s, several professional societies and other organizations
developed frameworks and standards for teaching mathematics. All of
these frameworks reflect integrative definitions of numeracy and
principles of constructivist theories toward learning. The most
influential for instructing adults have been the frameworks developed by
the National Council of Teachers of Mathematics (NCTM), the American
Mathematical Association of Two-Year College's (AMATYC) *Crossroads in
Mathematics,* the Adult Numeracy Network's (ANN) mathematics standards
framework, and the National Institute for Literacy's Equipped for the
Future (EFF) Math Content Standard. Each of these frameworks or
standards projects defines instructional content, including specific
facts or subjects to be covered; skills needed, such as problem solving
and critical thinking; and instructional processes or pedagogy. There
tends to be agreement among the frameworks and standards on the need for
specific skills such as critical thinking and problem solving, but there
is less agreement on specific content and teaching methods.

The recent blooming of work advancing numeracy concepts, theory, and
instructional frameworks, however, has not guided empirical research
examining instructional practice. Very few studies have used ABE
students to conduct research on the effects of adult numeracy
instruction. In our review of all research from 1985 to 2005 on the
effects of instructional interventions on mathematics teaching for ABE
students, we found only 24 studies. Most of these studies evaluated the
impact of computer-assisted instruction and did not provide a definitive
answer on the effects of this technology on adults' mathematics
learning. The research is neither theory-driven nor guided by any
systematic approach. Only five studies examined approaches based on
constructivist theory, and findings were inconclusive, though suggestive
that cooperative and discovery learning might be effective. With so few
studies, there clearly has been no agenda or systematic model guiding
research in adult numeracy instruction.

**Assessment and Professional Development in Adult Numeracy**

Assessment used by ABE programs for measuring knowledge of mathematics
is also undeveloped and not researched. We found the research base
limited and patchy with only a few studies over the last five years even
addressing ABE mathematics assessment. Current assessments are not
aligned with the predominant instructional frameworks. Accountability
demands, such as NRS requirements, drive most assessment in ABE
programs. The NRS framework does not address problem-solving or
critical-thinking skills, and neither do two widely used assessments
that measure mathematics: the TABE and the CASAS. These assessments
measure basic mathematics content knowledge and functional skills but
are not adequate for measuring integrative conceptions of numeracy.
Assessments geared toward passing the GED tests are also inadequate for
measuring critical-thinking skills.

Professional development is the main mechanism for advancing effective
instruction, but we found almost no research on the characteristics of
effective professional development approaches for ABE teachers of
mathematics. Yet research does show that ABE teachers are especially in
need of training in mathematics instruction, having little prior
training or experience. This lack of preparation is compounded by the
relatively low retention of teachers within ABE.

Research in effective professional development for K--12 teachers
suggests that several factors must be aligned for professional
development to have significant impact: duration, collective
participation, content, active learning, and coherence. Changes in K--12
mathematics education materials and methods have begun to percolate into
the adult education system, but without professional development that
provides both explanations for and examples of the new content, there is
danger that teachers will continue to teach what and as they were
taught.

[]{#__RefHeading___Toc130973172 .anchor}Implications for Further
Research

It is perhaps not surprising that there is little rigorous research
studying the effects of numeracy instruction, assessment, or
professional development on adults. The field has not received attention
from policymakers, nor has there been funding for research or a research
agenda, especially in the United States. Further, there has been little
meaningful instruction in adult numeracy that researchers could study.
The ABE system is difficult to study, with mostly part-time teachers
with little background or training in mathematics and widely varying
program arrangements and approaches toward instruction among the states.
ABE students are also difficult to study, entering as they do with
varying goals and initial skills and attending for relatively short
times. These conditions make the implementation and evaluation of
instructional approaches and professional development difficult and also
inhibit advancements in assessment. Work on developing and aligning
assessment to instruction can make little progress in the absence of
generally accepted instructional models.

However, the recent proliferation of instructional frameworks and
content standards holds promise for improving this situation. These
standards and frameworks not only suggest directions for research but
also can provide a guide for developing meaningful instruction that will
help adult learners cope with the demands of modern life. Varied and
meaningful instruction creates a classroom environment that researchers
can evaluate. We present a brief discussion of the implications for
further research that our review suggests for theory and instruction,
assessment and professional development.

[]{#__RefHeading___Toc130973173 .anchor}Further Research on Instruction

Our review of theories on how adults learn mathematics found a dominant
influence of constructivist theories that could inform a research
agenda. These theories propose that students take ownership of concepts
that they construct under expert guidance. How adults perform this
construction depends not only on the content and skills taught but also
on their cultural background, past experience, the learning situation,
and their attitudes and affect. Educational experiences are affected by
students' mathematics histories, which can have a profound effect on
students' ability to learn mathematics as adults.

The little research to date has not addressed these issues in any
organized way. The limited research identified---15 studies of ABE
students and 9 studies in developmental mathematics---seems like
guerrilla warfare far more than an organized victory campaign toward
improving adult numeracy instruction. To move the field forward, we
suggest the following five areas for further research.

1.  **Evaluate instructional frameworks and theories of adult
    mathematics learning.** The NCTM and other influential instructional
    frameworks suggest content knowledge, skills, and strategies that
    learners need and suggest ways of delivering this content through
    instruction. They all to some extent include constructivist
    principles of learning. However, very limited research exists
    evaluating these instructional approaches with ABE students. A few
    of the studies of technology and discovery and cooperative learning
    that we identified touched on concepts embodied in the frameworks,
    but more systematic and organized research evaluating the
    instructional components of these frameworks is needed. Research is
    also needed evaluating other theoretical and instructional
    approaches to mathematics learning.

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9.  **Identify and evaluate specific instructional practices.** The
    instructional framework and standards are often prescriptive about
    how instruction should be provided, but the suggested approaches are
    often phrased generally. Teachers need more specific guidance on how
    to implement instructional practices, and these practices require
    evaluation. Without this guidance, teachers must continually invent
    practice from scratch, using only general advice. For example,
    directing that students "discuss the solutions to a problem"
    provides little specificity. Research needs to identify and validate
    what a "discussion" entails (National Research Council, 2001).

10. **Study how adults learn mathematics in class.** There is little
    research on classroom dynamics that would facilitate classroom
    learning. We assume implicitly that instruction acts on students and
    that opportunities to learn are actually moments of learning, but if
    and how this occurs has not been studied. Research that describes
    what students have to know and do and what teachers can do to help
    students learn is needed. Constructivist theories that suggest ways
    adults construct meaning can guide this research, but empirical
    research needs to verify these approaches in the ABE mathematics
    classrooms.

11. **Explore the role of learner attitudes, affect, and experience.**
    There is currently no research on ABE students' attitudes toward
    mathematics, affective factors such as math anxiety, or the role of
    students' prior knowledge and how these elements affect learning.
    However, there is evidence from research on how child learn
    mathematics and general learning research that understanding such
    personal and instructional interactions is essential for
    understanding how adults learn mathematics.

12. **Examine ESL learners and students with learning disabilities.**
    Two completely neglected areas of research in adult mathematics have
    been instruction for adult ESL learners and instruction for students
    with learning disabilities. We found no research on how to provide
    instruction to these learners, on how they learn, or on how to
    address the challenges these learners face in learning mathematics.
    Research needs to pay particular attention to instruction for adult
    ESL learners, who make up over 40 percent of students in the adult
    literacy programs.

[]{#__RefHeading___Toc130973174 .anchor}Research on Mathematics
Instruction for Children

Because there is virtually no systematic research in ABE identifying
effective mathematics instruction, the field could benefit from
examining some of the research on mathematics instruction for children.
The principles of instruction and learning in K--8 mathematics are
rooted in the same theories of constructivism as instruction and
learning in the field of adult mathematics.

The National Research Council's Mathematic Learning Study Committee
(2001) synthesized the research on teaching and learning mathematics in
the K--8 classroom. The committee's findings, presented in *Adding it
Up: Helping Children Learn Mathematics*, bolster and inform the small
body of research in adult mathematics education. The committee divided
the research into three areas: how teachers interact with mathematical
content, how teachers interact with their students, and how students
interact with mathematical content. Although there are clear and
profound differences between the K--12 instructional setting and ABE, as
well as differences between children and ABE learners, a brief review of
some of the key findings suggest practices that may be worthy of
research for adult mathematics.

-   Students learn best when they are given academically challenging
    work that focuses on making sense of problems, solving problems, and
    building skills. Teachers should support their students' cognitive
    activities without reducing the difficulty of the task.

-   Teachers will keep students interested if they engage students'
    background knowledge, scaffold appropriately, emphasize multiple
    solution strategies, and privilege explanation and meaning-making by
    asking students to explain their thought processes.

-   Teachers will keep students motivated if they set high expectations
    for their students, assign tasks that their students can succeed at
    with reasonable effort, and tap into students' intrinsic motivation
    by helping them view the content as fun, relevant, and worth their
    time.

-   Instruction should allow students to make connections and organize
    knowledge; should build on what students already know and take
    advantage of their informal, everyday knowledge of mathematics;
    should not involve "overly abstract instruction that proceeds too
    quickly"; and should not rely on rote memorization.

-   Teachers should show students the value of using multiple ways to
    solve problems and give students the freedom to explore different
    problem-solving methods.

-   Cooperative groups can be an effective way to facilitate learning,
    develop students' social skills, and keep more students
    intellectually engaged.

-   The use of manipulatives can be valuable if students are shown how
    to make connections between the object, the mathematical symbol, and
    the idea that each represents.

The practices discussed above---presenting cognitively demanding
content, setting high expectations, keeping students motivated, and
creating a safe and supportive learning community---suggest a wide
variety of practices that can be tried with adults and evaluated through
a rigorous research agenda.

[]{#__RefHeading___Toc130973175 .anchor}Development of Improved
Assessment

The adult numeracy field clearly needs to design better assessments for
large-scale use that not only examine procedural skills and focus
primarily on arithmetic but also are broader and call for more diverse
forms of learner reactions and performances. Currently available
assessments do not align with the standards or frameworks recently
developed and do not reflect integrative conceptions of numeracy. The
field needs a new approach toward numeracy assessment where assessments
examine connected knowledge, communication skills, and interpretative
abilities; they could examine learners' capacity to cope with nonroutine
problems, text-rich tasks, and realistic, ill-structured problems.
Having information about such capacities could provide much needed
information about learning gains that would be more directly linked to
adults' needs.

The essence of a good assessment is that it is appropriate for the
context, purpose, and interpretation made or needed. Assessment has to
be fully linked with the learning and teaching goals of a program and
must reflect, for teachers, students, and policymakers, what is valued
in a student's performance and learning in numeracy. Overall, the ideas
expressed in this review point to the existence of a set of high
expectations regarding both the desired breadth of the curriculum and
the richness of the teaching and learning process in adult numeracy
education. We list suggested characteristics to guide development of
assessments for adult numeracy that are better aligned with
instructional standards and frameworks.

1.  Offer a balanced coverage of key domains deemed part of a broad view
    of adult numeracy, including numbers and operations, algebra and
    modeling, geometry and shape, measurement (time, money,
    length/volume, etc.), and statistics and probability.

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13. Not only examine knowledge and skills in the key domains listed
    above but also more broadly provide information about learners'
    problem-solving, reasoning, and communicative capacities, as well as
    learners' understanding of connections between different
    mathematical and statistical ideas and their ability to explain and
    justify their reasoning.

14. Include types of items and stimuli that provide information about
    ability to

    a.  interpret, critically react to, explain, and communicate about
        quantitative and statistical information embedded in print or
        media messages with varying types of documents and displays;

    b.  cope with a mix of well-structured and ill-structured problems
        that require different forms of response and interaction (e.g.,
        by mixing multiple-choice and constructed-response items,
        written and oral reports, group activities) and use tasks for
        which the reactions and responses required are choices and
        decisions and do not necessarily have clear answers; and

    c.  use different tools or technologies relevant for learners' life
        circumstances or work goals.

15. Provide information about learners' ability to demonstrate confident
    numerate behavior in life and work contexts of value to learners and
    programs. This point relates to skill transfer and points to the
    need to examine the realism and authenticity of tasks.

16. Document beliefs, attitudes, or habits of mind that may affect both
    learning in class and numerate behavior outside the classroom.

17. Enable teachers and programs to identify the mathematical
    experiences and strategies that adults bring and that may contribute
    to, or otherwise hamper, their learning of more formal types of
    mathematics.

18. Ensure they are appropriate for adults with special learning needs,
    for low-literacy students, or for students from other cultures and
    languages.

We need to grapple with the development of broader and more
sophisticated assessments in the Adult Numeracy Initiative project and
in the field in the coming years. It is not at all clear that states,
programs, teachers, or even learners will be open to such ideas, owing
to beliefs, conceptions of mathematics or numeracy, or practicalities of
adult education programs and limited time and energies. Nonetheless, the
criteria and influences discussed here, along with the analysis of the
four components of assessment, can serve as a basis for examining the
quality of assessments beyond a focus on technical aspects of
reliability and validity.

[]{#__RefHeading___Toc130973176 .anchor}Research to Improve Professional
Development

Research on professional development in ABE has been neglected in
general and is almost nonexistent where adult numeracy is concerned. The
professional development approaches and principles we did identify as
effective come from research on K--12 teachers. Yet the limited research
does identify a clear need for better trained teachers in ABE and more
systematic, ongoing training on principles and approaches to mathematics
instruction. The dedication of the numeracy instructors cannot offset
the weak mathematical backgrounds that the majority of them possess.
Part-time employment, low wages, and uncompensated leave compound the
problem of attracting to professional development the very teachers who
most likely need content enrichment.

We offer three suggestions for further research that may advance
professional development for ABE educators.

1.  **Study the relationship between teachers' knowledge of mathematics
    and instructional effectiveness.** We reviewed research that showed
    that few ABE teachers have training in mathematic instructions.
    Given ABE program realities, this state of affair is unlikely to
    change in the next several years. Research that explores how much
    mathematics knowledge teachers really need to be effective
    instructors will help us understand training needs the field can
    address. This research needs to be coupled with research on how
    teachers use their knowledge of mathematics and knowledge of adult
    learners in planning instruction.

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19. **Study the effectiveness of professional development delivery
    systems**. Because ABE programs are likely to continue to have
    limited resources to provide professional development, research
    could help improve the field by evaluating the effectiveness of
    different approaches toward training and ongoing professional
    development. This research could focus both on *how* to provide
    effective training and on the effectiveness of *different
    mechanisms* for professional development. For example, given the
    limited programmatic resources and time available to ABE teachers,
    the use of technology and distance learning as a professional
    development mechanism may be a promising area to research.

20. **Study the impact of numeracy professional development on teacher
    knowledge, behaviors and learner outcomes**. Given the adult
    education delivery system and parttime nature of both teachers and
    learners, what is the best way to design a research study that can
    focus on providing scientifically based evidence that specific
    professional development strategies actually change behavior and
    outcomes. These types of studies would provide definitive evidence
    of what works in professional development.

Changes in materials and methods in K--12 mathematics education have
begun to percolate into the adult education system. But without
professional development that provides both explanations and examples of
the new content, there is the danger that teachers will continue to
teach what and as they were taught. As the agents of instruction, ABE
teachers cannot change or improve their numeracy instruction without
sound, systematic, and ongoing professional development.

[]{#__RefHeading___Toc130973177 .anchor}Conclusion: Moving Forward

This literature review provides a portrait of the current state of
theory, instruction, assessment, and professional development of
numeracy education for ABE students. The project team identified the
relevant research in adult mathematics education and grouped the studies
within the framework of the Adult Numeracy Initiative research
questions. The driving philosophy was for this report to serve as a
research baseline. To determine a direction for moving the field
forward, it was critical to develop an understanding of how far adult
mathematics education has come and where it is now.

A technical working group of experts in mathematics and numeracy
research and practice will build on the information and recommendations
of the review to identify the key critical issues on which to focus
further efforts and will commission papers to study some of the issues
identified here in greater depth. The review will also guide another
project activity, an environmental scan of current professional
development and instructional projects focusing on adult numeracy. We
offer the review as the first piece of the puzzle that will present a
clearer picture and direction of the field by the end of the project.

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[]{#__RefHeading___Toc130973180 .anchor}Appendix A -- Definitions of
Numeracy

  ------------- --------------------------------------------------------------
  Sources       Definitions

  AAMT, 1997    Numeracy "involves using some mathematics to achieve some
                purpose in a particular context."

  Brown, 2002   Numeracy is the "competence and inclination to use number
                concepts and skills to solve problems in everyday life and
                employment."

  Coben, 2000b, To be numerate means to be competent, confident, and
  p. 35         comfortable with one's judgments on whether to use mathematics
                in a particular situation and if so, what mathematics to use,
                how to do it, what degree of accuracy is appropriate, and what
                the answer means in relation to the context.

  Evans, 2000,  The "limited proficiency" vision of numeracy prevails. Against
  p. 236        this vision, he offers a "provisional working definition for a
                reconstituted idea of numeracy" as meaningful social practice:
                the ability to process, interpret, and communicate numerical,
                quantitative, spatial, statistical, even mathematical,
                information, in ways that are appropriate for a variety of
                contexts, and that will enable typical members of the culture
                to participate effectively in activities that they value.

  Gal, 2000     He characterizes numeracy as a semiautonomous area at the
                intersection between literacy and mathematics (p. 23). He
                describes three different types of "numeracy situations":
                "generative," "interpretive," and "decision." Generative
                situations require people to count, quantify, compute, and
                otherwise calculate. Interpretive situations demand that
                people make sense of verbal or text-based messages that may be
                based on quantitative data but require no manipulation of
                numbers. Decision situations "demand that people find and
                consider multiple pieces of information in order to determine
                a course of action, typically in the presence of conflicting
                goals, constraints or uncertainty" (p. 15).

  Johnston &    We believe that numeracy is about making meaning in
  Tout, 1995;   mathematics and being critical about maths. This view of
  Yasukawa,     numeracy is very different from numeracy just being about
  Johnston, &   numbers, and it is a big step from numeracy or everyday maths
  Yates, 1995   that meant doing some functional maths. It is about using
                mathematics in all its guises -- space and shape, measurement,
                data and statistics, algebra, and of course, number -- to make
                sense of the real world, and using maths critically and being
                critical of maths itself. It acknowledges that numeracy is a
                social activity.

  Johnston &    Numeracy is "the ability to situate, interpret, critique and
  Yasukawa,     perhaps even create mathematics in context, taking into
  2001          account all the mathematical as well as social and human
                complexities which come with that process."

  O'Donoghue,   Numeracy and mathematics are not interchangeable terms;
  2003, p. 8    numeracy is seen as encompassing some elements of mathematics,
                rather than vice versa: Mathematics and numeracy are not
                congruent. Nor is numeracy an accidental or automatic
                by-product of mathematics education at any level. When the
                goal is numeracy, some mathematics will be involved, but
                mathematical skills alone do not constitute numeracy.

  McDevitt,     Numeracy has been defined as the kinds of math skills needed
  2001          to function in everyday life ― not one fixed set of skills but
                rather a continuum of skills that an adult draws from to meet
                different needs. And it's numeracy that we want for our
                learners, not just math.
  ------------- --------------------------------------------------------------

[]{#__RefHeading___Toc130973181 .anchor}Appendix B -- Professional
Society Standards

+-----------------------------------------------------------------------+
| **NCTM's Content Standards**                                          |
+=======================================================================+
| **Number and Operations**                                             |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Understand numbers, ways of representing numbers,             |
|         relationships among numbers, and numbers systems;             |
|                                                                       |
|     -   Understand meanings of operations and how they relate to one  |
|         another;                                                      |
|                                                                       |
|     -   Compute fluently and make reasonable estimates.               |
+-----------------------------------------------------------------------+
| **Algebra**                                                           |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Understand patterns, relations, and functions;                |
|                                                                       |
|     -   Represent and analyze mathematical situations and structures  |
|         using algebraic symbols;                                      |
|                                                                       |
|     -   Use mathematical models to represent and understand           |
|         quantitative relationships;                                   |
|                                                                       |
|     -   Analyze change in various contexts.                           |
+-----------------------------------------------------------------------+
| **Geometry**                                                          |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Analyze characteristics and properties of two- and            |
|         three-dimensional geometric shapes and develop mathematical   |
|         arguments about geometric relationships;                      |
|                                                                       |
|     -   Specify locations and describe spatial relationships using    |
|         coordinate geometry and other representational systems;       |
|                                                                       |
|     -   Apply transformations and use symmetry to analyze             |
|         mathematical situations;                                      |
|                                                                       |
|     -   Use visualization, spatial reasoning, and geometric modeling  |
|         to solve problems.                                            |
+-----------------------------------------------------------------------+
| **Measurement**                                                       |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Understand measurable attributes of objects and the units,    |
|         systems, and processes of measurement;                        |
|                                                                       |
|     -   Apply appropriate techniques, tools, and formulas to          |
|         determine measurements.                                       |
+-----------------------------------------------------------------------+
| **Data Analysis and Probability**                                     |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Formulate questions that can be addressed with data and       |
|         collect, organize, and display relevant data to answer them;  |
|                                                                       |
|     -   Select and use appropriate statistical methods to analyze     |
|         data;                                                         |
|                                                                       |
|     -   Develop and evaluate inferences and predictions that are      |
|         based on data;                                                |
|                                                                       |
|     -   Understand and apply basic concepts of probability.           |
+-----------------------------------------------------------------------+
| **Problem Solving**                                                   |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Build new mathematical knowledge through problem solving;     |
|                                                                       |
|     -   Solve problems that arise in mathematics and in other         |
|         contexts;                                                     |
|                                                                       |
|     -   Apply and adapt a variety of appropriate strategies to solve  |
|         problems;                                                     |
|                                                                       |
|     -   Monitor and reflect on the process of mathematical problem    |
|         solving.                                                      |
+-----------------------------------------------------------------------+
| **Reasoning and Proof**                                               |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Recognize reasoning and proof as fundamental aspects of       |
|         mathematics;                                                  |
|                                                                       |
|     -   Make and investigate mathematical conjectures;                |
|                                                                       |
|     -   Develop and evaluate mathematical arguments and proofs;       |
|                                                                       |
|     -   Select and use various types of reasoning and methods of      |
|         proof.                                                        |
+-----------------------------------------------------------------------+
| **Communication**                                                     |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Organize and consolidate their mathematical thinking through  |
|         communication;                                                |
|                                                                       |
|     -   Communicate their mathematical thinking coherently and        |
|         clearly to peers, teachers, and others;                       |
|                                                                       |
|     -   Analyze and evaluate the mathematical thinking and strategies |
|         of others;                                                    |
|                                                                       |
|     -   Use the language of mathematics to express mathematical ideas |
|         precisely.                                                    |
+-----------------------------------------------------------------------+
| **Connections**                                                       |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Recognize and use connections among mathematical ideas;       |
|                                                                       |
|     -   Understand how mathematical ideas interconnect and build on   |
|         one another to produce a coherent whole;                      |
|                                                                       |
|     -   Recognize and apply mathematics in contexts outside of        |
|         mathematics.                                                  |
+-----------------------------------------------------------------------+
| **Representation**                                                    |
|                                                                       |
| -   Students should be able to:                                       |
|                                                                       |
|     -   Create and use representations to organize, record, and       |
|         communicate mathematical ideas;                               |
|                                                                       |
|     -   Select, apply, and translate among mathematical               |
|         representations to solve problems;                            |
|                                                                       |
|     -   Use representations to model and interpret physical, social,  |
|         and mathematical phenomena.                                   |
+-----------------------------------------------------------------------+

+-----------------------------------------------------------------------+
| **AMATYC *Crossroads in Mathematics***                                |
+-----------------------------------------------------------------------+
| **Standards for Content**                                             |
+-----------------------------------------------------------------------+
| 1.  Students will perform arithmetic operations, as well as reason    |
|     and draw conclusions from numerical information.                  |
|                                                                       |
| 2.  Students will translate problem situations into their symbolic    |
|     representations and use those representations to solve problems.  |
|                                                                       |
| 3.  Students will develop a spatial and measurement sense.            |
|                                                                       |
| 4.  Students will demonstrate understanding of the concept of         |
|     function by several means (verbally, numerically, graphically,    |
|     and symbolically) and incorporate it as a central theme into      |
|     their use of mathematics.                                         |
|                                                                       |
| 5.  Students will use discrete mathematical algorithms and develop    |
|     combinatorial abilities in order to solve problems of finite      |
|     character and enumerate sets without direct counting.             |
|                                                                       |
| 6.  Students will analyze data and use probability and statistical    |
|     models to make inferences about real-world situations.            |
|                                                                       |
| 7.  Students will appreciate the deductive nature of mathematics as   |
|     an identifying characteristic of the discipline, recognize the    |
|     roles of definitions, axioms, and theorems, and identify and      |
|     construct valid deductive arguments (Cohen, 1995).                |
+-----------------------------------------------------------------------+
| **Standards for Pedagogy**                                            |
+-----------------------------------------------------------------------+
| 1.  Mathematics faculty will model the use of appropriate technology  |
|     in the teaching of mathematics so that students can benefit from  |
|     the opportunities it presents as a medium of instruction.         |
|                                                                       |
| 2.  Mathematics faculty will foster interactive learning through      |
|     student writing, reading, speaking, and collaborative activities  |
|     so that students can learn to work effectively in groups and      |
|     communicate about mathematics both orally and in writing.         |
|                                                                       |
| 3.  Mathematics faculty will actively involve students in meaningful  |
|     mathematics problems that build upon their experiences, focus on  |
|     broad mathematical themes, and build connections within branches  |
|     of mathematics and between mathematics and other disciplines so   |
|     that students will view mathematics as a connected whole relevant |
|     to their lives.                                                   |
|                                                                       |
| 4.  Mathematics faculty will model the use of multiple                |
|     approaches\--numerical, graphical, symbolic, and verbal\--to help |
|     students learn a variety of techniques for solving problems.      |
|                                                                       |
| 5.  Mathematics faculty will provide learning activities, including   |
|     projects and apprenticeships that promote independent thinking    |
|     and require sustained effort and time so that students will have  |
|     the confidence to access and use needed mathematics and other     |
|     technical information independently, to form conjectures from an  |
|     array of specific examples, and to draw conclusions from general  |
|     principles (Cohen, 1995).                                         |
+-----------------------------------------------------------------------+

[]{#__RefHeading___Toc130973182 .anchor}Appendix C -- List of Web Sites
and Journals Searched

+-----------------------------------+-----------------------------------+
| **Web Sites**                     | **Journals (Online and Print)**   |
+===================================+===================================+
| 1.  [ABC                          | 1.  Academic Leadership           |
|     Ca                            |                                   |
| nada](http://www.abc-canada.org/) | 2.  [Academic press electronic    |
|                                   |     journal                       |
| 2.  Adult Community and Further   |     libr                          |
|     Education (ACFE)              | ary](http://www.idealibrary.com/) |
|                                   |                                   |
| 3.  Adult Learning Math (ALM)     | 3.  Adult Basic Education: An     |
|                                   |     Interdisciplinary Journal for |
| 4.  Adult Literacy and Numeracy   |     Adult Literacy Educators      |
|     Australian Research           |                                   |
|     Consortium (ALNARC), The      | 4.  Adult Education Quarterly     |
|                                   |                                   |
| 5.  Adult Numeracy Down Under     | 5.  [American Journal of          |
|                                   |     Mathematics](http://www.      |
| 6.  Adult Numeracy Network        | press.jhu.edu/journals/american_j |
|                                   | ournal_of_mathematics/index.html) |
| 7.  [American Association for the |                                   |
|     Advancement of Science        | 6.  [American Mathematical        |
|     (AAAS                         |     Monthly](gopher://gopher      |
| )](http://www.aaas.org/aaas.html) | .maa.org:70/11/pubs/Journals/AMM) |
|                                   |                                   |
| 8.  [American Mathematical        | 7.  [American Mathematical        |
|     Association of Two-year       |     Society                       |
|     Colleges                      |     Journa                        |
|                                   | ls](http://www.ams.org/journals/) |
| (AMATYC)](http://www.amatyc.org/) |                                   |
|                                   | 8.  Annals of Mathematics         |
| 9.  [American Mathematical        |                                   |
|     Society                       | 9.  Arizona State University -    |
|     (AMS)](http://www.ams.org/)   |     Bilingual Research Journal    |
|                                   |                                   |
| 10. [American Statistical         | 10. [Australian Mathematical      |
|     Association                   |     Society                       |
|                                   |     Gazette](http:                |
|    (AMS)](http://www.amstat.org/) | //solution.maths.unsw.edu.au/htdo |
|                                   | cs.ams/Publ/Gazette/gazette.html) |
| 11. [Association for Women in     |                                   |
|     Mathematics                   | 11. [Bulletin of the London       |
|                                   |     Mathematical                  |
|  (AWM)](http://www.awm-math.org/) |                                   |
|                                   |   Society](http://www.cup.cam.ac. |
| 12. Association of Mathematics    | uk/Journals/JNLSCAT/blm/blm.html) |
|     Teacher Educators (AMTE),     |                                   |
|     Canadian Mathematical Society | 12. Canadian Applied Mathematics  |
|     (CMS)                         |     Quarterly                     |
|                                   |                                   |
| 13. [Australian Council for Adult | 13. [Canadian Journal of          |
|     L                             |     Mathematics](                 |
| iteracy](http://www.acal.edu.au/) | gopher://camel.cecm.sfu.ca/11/Eng |
|     (ACAL)                        | lish_Switch/CMS/Abstract/journal) |
|                                   |                                   |
| 14. [Australian Council of TESOL  | 14. [Canadian Mathematical        |
|     Assoc                         |     Bulletin](g                   |
| iations](http://www.acta.edu.au/) | opher://camel.cecm.sfu.ca/11/Engl |
|     (ACTA)                        | ish_Switch/CMS/Abstract/bulletin) |
|                                   |                                   |
| 15. [Australian Literacy          | 15. [Canadian Mathematical        |
|     Educators\'                   |     Society                       |
|     Asso                          |     Note                          |
| ciation](http://www.alea.edu.au/) | s](gopher://camel.cecm.sfu.ca/11/ |
|     (ALEA)                        | English_Switch/CMS/Abstract/note) |
|                                   |                                   |
| 16. [Australian Literacy          | 16. Central European Journal of   |
|     Fe                            |     Mathematics                   |
| deration](http://www.alf.edu.au/) |                                   |
|     (ALF)                         | 17. Chreods                       |
|                                   |                                   |
| 17. [Basic Skills                 | 18. Comparative Education Review  |
|     Agency                        |                                   |
| ](http://www.basic-skills.co.uk/) | 19. Current Issues in Education   |
|     (BSA)                         |                                   |
|                                   | 20. Education Next                |
| 18. [Canadian Mathematical        |                                   |
|     Society](http:                | 21. Educational Studies in        |
| //camel.math.ca/CMS/Publication/) |     Mathematics                   |
|                                   |                                   |
| 19. Catalog of Mathematics        | 22. Effective Teaching            |
|     Resources on the WWW and the  |                                   |
|     Internet                      | 23. [Electronic Library of the    |
|                                   |     European Mathematical         |
| 20. [Center for Applied           |     Society](http:/               |
|     Linguistics                   | /www.emis.de/journals/index.html) |
|     (CAL)](http://www.cal.org/)   |                                   |
|                                   | 24. [Electronic Research          |
| 21. [Center for Educational       |     Announcements of the          |
|     Change in Mathematics and     |     AMS](                         |
|     Science]                      | http://www.ams.org/journals/era/) |
| (http://www.stolaf.edu/other/exte |                                   |
| nd/Resources/programs.html#CECMS) | 25. [Euromath                     |
|                                   |     Bulletin]                     |
| 22. [Center for Research in       | (http://sophie.helsinki.fi/~emb/) |
|     Mathematics and Science       |                                   |
|     Education                     | 26. Harvard Education Publishing  |
|     (CRMSE                        |     Group                         |
| )](http://public.sdsu.edu/CRMSE/) |                                   |
|                                   | 27. Harvard Educational Review    |
| 23. [Center for Research on       |                                   |
|     Education, Diversity &        | 28. [Indiana University           |
|     Excellence                    |     Mathematics                   |
|     (CR                           |     Journ                         |
| EDE)](http://www.crede.ucsc.edu/) | al](http://www.iumj.indiana.edu/) |
|                                   |                                   |
| 24. Center for Research on        | 29. International Education       |
|     Evaluation, Standards, and    |     Journal (IEJ)                 |
|     Student Testing (CRESST)      |                                   |
|                                   | 30. Journal for Research in       |
| 25. [Center for Research on the   |     Mathematics Education         |
|     Education of Students Placed  |                                   |
|     at Risk                       | 31. Journal of Industrial Teacher |
|     (CRESPAR)](                   |     Education                     |
| http://www.csos.jhu.edu/crespar/) |                                   |
|                                   | 32. Journal of Science Education  |
| 26. [Center for Science,          |                                   |
|     Mathematics and Engineering   | 33. [Journal of the London        |
|     Education                     |     Mathematical                  |
|     (CSM                          |                                   |
| EE)](http://www2.nas.edu/center/) |   Society](http://www.cup.cam.ac. |
|                                   | uk/Journals/JNLSCAT/jlm/jlm.html) |
| 27. [Center for the Enhancement   |                                   |
|     of Science and Mathematics    | 34. Journal of Vocational and     |
|     Education                     |     Technical Education           |
|     (CESAME)                      |                                   |
| ](http://www.dac.neu.edu/cesame/) | 35. [Mathematica                  |
|                                   |     Journal](gopher://gopher.wri  |
| 28. [Center for the Study of      | .com/11/MathSource/Publications/P |
|     Teaching and Policy           | eriodicals/TheMathematicaJournal) |
|     (CTP)](http                   |                                   |
| ://depts.washington.edu/ctpmail/) | 36. [MathUser                     |
|                                   | ](ftp://mathsource.wri.com/pub/Pu |
| 29. [Center on English Learning & | blications/Periodicals/MathUser/) |
|     Achievement                   |                                   |
|                                   | 37. National Forum Journals       |
|  (CELA)](http://cela.albany.edu/) |                                   |
|                                   | 38. [New York Journal of          |
| 30. Centre for the Study of       |     Mathemati                     |
|     Mathematics Education (CSME)  | cs](http://nyjm.albany.edu:8000/) |
|                                   |                                   |
| 31.                               | 39. Northeastern Mathematical     |
|  Clearinghouse on Adult, Career,\ |     Journal                       |
|                                   |                                   |
|  and Vocational Education  (ACVE) | 40. Phi Delta Kappan, The         |
|                                   |                                   |
| 32. Commission on Adult Basic     | 41. Reading Online                |
|     Education (COABE)             |                                   |
|                                   | 42. State University of West      |
| 33. Conference Board of the       |     Georgia ― Online Journal of   |
|     Mathematical Sciences (CBMS)  |     Distance Learning             |
|                                   |     Administration                |
| 34. [Consortium for Mathematics   |                                   |
|     and Its Applications          | 43. [Teacher Education            |
|                                   |     Quarterl                      |
|   (COMAP)](http://www.comap.com/) | y](http://rds.yahoo.com/S=87037:D |
|                                   | 1/CS=87037/SS=52438583/SIG=10u5nl |
| 35. [Consortium for Policy        | 7v9/*http%3A/www.teqjournal.org/) |
|     Research in Education         |                                   |
|     (CPRE)](http://www.cpre.org/) | 44. Teachers College Record       |
|                                   |                                   |
| 36. Department for Education and  |                                   |
|     Skills Standards Site (DfES   |                                   |
|                                   |                                   |
| 37. [European Mathematical        |                                   |
|     Society                       |                                   |
|     (EMS)](http://www.emis.de/)   |                                   |
|                                   |                                   |
| 38. Extend Resources for          |                                   |
|     Mathematics Education         |                                   |
|                                   |                                   |
| 39. [International Commission on  |                                   |
|     Mathematical Instruction      |                                   |
|     (ICMI)](http://eli            |                                   |
| b.zib-berlin.de/imu.icmi.bull.44) |                                   |
|                                   |                                   |
| 40. [International Literacy       |                                   |
|     Institute                     |                                   |
|     (ILI)](http://litse           |                                   |
| rver.literacy.upenn.edu/ili.html) |                                   |
|                                   |                                   |
| 41. [Literacy                     |                                   |
|     Online](http://il             |                                   |
| i2.literacy.upenn.edu/index.html) |                                   |
|                                   |                                   |
| 42. LiteracyNet                   |                                   |
|                                   |                                   |
| 43. Massachusetts ABE Standards   |                                   |
|                                   |                                   |
| 44. Maths4life                    |                                   |
|                                   |                                   |
| 45. Math Forum                    |                                   |
|                                   |                                   |
| 46. [Mathematical Association of  |                                   |
|     America                       |                                   |
|     (MAA)](http://www.maa.org/)   |                                   |
|                                   |                                   |
| 47. [Mathematical Sciences        |                                   |
|     Education Board               |                                   |
|     (MSEB)](h                     |                                   |
| ttp://www.nas.edu/mseb/mseb.html) |                                   |
|                                   |                                   |
| 48. [Mathematical Sciences        |                                   |
|     Research Institute            |                                   |
|     (MSRI)](http://www.msri.org/) |                                   |
|                                   |                                   |
| 49. Mathematics Hotlist           |                                   |
|                                   |                                   |
| 50. [National Adult Literacy      |                                   |
|                                   |                                   |
|    Database](http://www.nald.ca/) |                                   |
|     (NALD)                        |                                   |
|                                   |                                   |
| 51. [National Center for Early    |                                   |
|     Development and Learning      |                                   |
|     (NCEDL)                       |                                   |
| ](http://www.fpg.unc.edu/~ncedl/) |                                   |
|                                   |                                   |
| 52. [National Center for          |                                   |
|     Improving Student Learning    |                                   |
|     and Achievement in            |                                   |
|     Mathematics and Science       |                                   |
|     (NCISLA)](                    |                                   |
| http://www.wcer.wisc.edu/NCISLA/) |                                   |
|                                   |                                   |
| 53. National Center for the Study |                                   |
|     of Adult Learning and         |                                   |
|     Literacy , The (NCSALL)       |                                   |
|                                   |                                   |
| 54. National Centers, The         |                                   |
|                                   |                                   |
| 55. [National Council of Teachers |                                   |
|     of Mathematics                |                                   |
|     (NCTM)](http://www.nctm.org/) |                                   |
|                                   |                                   |
| 56. [National Institute for       |                                   |
|     Literacy                      |                                   |
|     (NIFL)](http://www.nifl.gov/) |                                   |
|                                   |                                   |
| 57. [National Literacy            |                                   |
|     Trust](http://www.liter       |                                   |
| acytrust.org.uk/About/index.html) |                                   |
|                                   |                                   |
| 58. National Numeracy Network     |                                   |
|                                   |                                   |
| 59. National Research and         |                                   |
|     Development Center (NRDC)     |                                   |
|                                   |                                   |
| 60. National Science Teachers     |                                   |
|     Association                   |                                   |
|                                   |                                   |
| 61. [NSW Board of Adult and       |                                   |
|     Community Education           |                                   |
|     (BA                           |                                   |
| CE)](http://www.bace.nsw.gov.au/) |                                   |
|                                   |                                   |
| 62. [Research and Practice in     |                                   |
|     Adult                         |                                   |
|     Literacy](http://www.lite     |                                   |
| racy.lancs.ac.uk/rapal/rapal.htm) |                                   |
|     (RaPAL)                       |                                   |
|                                   |                                   |
| 63. [Shell Centre for             |                                   |
|     Mathematical                  |                                   |
|     Education](http://acorn.      |                                   |
| educ.nottingham.ac.uk/ShellCent/) |                                   |
|     [Society for Industrial and   |                                   |
|     Applied Mathematics (SIAM     |                                   |
|     )](http://www.siam.org/)      |                                   |
|                                   |                                   |
| 64. Supporting Adult Applied      |                                   |
|     Learning and Teaching (SAALT) |                                   |
|                                   |                                   |
| 65. Teachers College Record       |                                   |
|                                   |                                   |
| 66. TERC                          |                                   |
|                                   |                                   |
| 67. Victorian Adult Literacy and  |                                   |
|     Basic Education Council       |                                   |
|     (VALBEC)                      |                                   |
+-----------------------------------+-----------------------------------+

[]{#__RefHeading___Toc130973183 .anchor}Appendix D -- Summary of
Mathematics and Numeracy Intervention\
Studies Reviewed

Studies Discussed in Section 3, Adult Numeracy Instructional Approaches
and Interventions

+---+------------+-------------------------------+-------------+--------+
|   | **Study    | **Description of Intervention | **Outcome   | > **F  |
|   | Name**     | and Population Measured**     | Measure and | urther |
|   |            |                               | Effects     | >      |
|   |            |                               | Found**     | Conclu |
|   |            |                               |             | sions/ |
|   |            |                               |             | > Im   |
|   |            |                               |             | plicat |
|   |            |                               |             | ions** |
+===+============+===============================+=============+========+
| * |            |                               |             |        |
| * |            |                               |             |        |
| [ |            |                               |             |        |
| A |            |                               |             |        |
| d |            |                               |             |        |
| u |            |                               |             |        |
| l |            |                               |             |        |
| t |            |                               |             |        |
| B |            |                               |             |        |
| a |            |                               |             |        |
| s |            |                               |             |        |
| i |            |                               |             |        |
| c |            |                               |             |        |
| E |            |                               |             |        |
| d |            |                               |             |        |
| u |            |                               |             |        |
| c |            |                               |             |        |
| a |            |                               |             |        |
| t |            |                               |             |        |
| i |            |                               |             |        |
| o |            |                               |             |        |
| n |            |                               |             |        |
| ] |            |                               |             |        |
| { |            |                               |             |        |
| . |            |                               |             |        |
| s |            |                               |             |        |
| m |            |                               |             |        |
| a |            |                               |             |        |
| l |            |                               |             |        |
| l |            |                               |             |        |
| c |            |                               |             |        |
| a |            |                               |             |        |
| p |            |                               |             |        |
| s |            |                               |             |        |
| } |            |                               |             |        |
| * |            |                               |             |        |
| * |            |                               |             |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Barnett,   | The study used Program Logic  | Achievement |        |
| . | T. L.      | for Automated Training        | and         |        |
|   | (1985). A  | Operations (PLATO)            | attitude of |        |
|   | c          | computer-assisted             | students in |        |
|   | omparative | instruction. The study had a  | the         |        |
|   | analysis   | pretest-posttest design with  | e           |        |
|   | of the     | random assignment of subjects | xperimental |        |
|   | PLATO      | in two juvenile correctional  | group were  |        |
|   | compute    | facilities. There was no      | not         |        |
|   | r-assisted | indication of the number of   | si          |        |
|   | ins        | participants or of the        | gnificantly |        |
|   | tructional | instruments used to measure   | different   |        |
|   | delivery   | the variables.                | from those  |        |
|   | system and |                               | in the      |        |
|   | the        |                               | control     |        |
|   | t          |                               | group.      |        |
|   | raditional |                               |             |        |
|   | indi       |                               |             |        |
|   | vidualized |                               |             |        |
|   | ins        |                               |             |        |
|   | tructional |                               |             |        |
|   | program in |                               |             |        |
|   | two        |                               |             |        |
|   | juvenile   |                               |             |        |
|   | co         |                               |             |        |
|   | rrectional |                               |             |        |
|   | facilities |                               |             |        |
|   | owned by   |                               |             |        |
|   | the        |                               |             |        |
|   | Co         |                               |             |        |
|   | mmonwealth |                               |             |        |
|   | of         |                               |             |        |
|   | Pen        |                               |             |        |
|   | nsylvania. |                               |             |        |
|   | *Di        |                               |             |        |
|   | ssertation |                               |             |        |
|   | Abstracts  |                               |             |        |
|   | Inte       |                               |             |        |
|   | rnational, |                               |             |        |
|   | 46* (09),  |                               |             |        |
|   | 2668A.     |                               |             |        |
|   | (UMI No.   |                               |             |        |
|   | 8525658)   |                               |             |        |
+---+------------+-------------------------------+-------------+--------+
| 2 | B          | Batchelder and Rachal studied | Inmates     |        |
| . | atchelder, | the effect of skill and drill | were        |        |
|   | J. S., &   | tutorial software to enhance  | posttested  |        |
|   | Rachal, J. | mathematics and language      | with the    |        |
|   | R. (2000). | skills. They randomly         | CASAS       |        |
|   | Effects of | assigned 71 male inmates in   | reading and |        |
|   | a          | the prison's GED program to   | mathematics |        |
|   | computer-  | receive either the regular    | tests after |        |
|   | assisted-i | classroom instruction offered | receiving   |        |
|   | nstruction | or classroom instruction      | 80 hours    |        |
|   | program in | supplemented by CAI using the | (four       |        |
|   | a prison   | tutorial software. The        | weeks) of   |        |
|   | setting:   | classroom instruction         | i           |        |
|   | An         | consisted of four hours per   | nstruction. |        |
|   | ex         | day in English, mathematics,  | There was   |        |
|   | perimental | history, and science.         | no          |        |
|   | study.     | Students in the experimental  | significant |        |
|   | *Journal   | group received three hours of | difference  |        |
|   | of         | instruction per day but spent | between the |        |
|   | Co         | the fourth hour using the CAI | groups on   |        |
|   | rrectional | software for mathematics and  | these       |        |
|   | Education, | reading.                      | tests.      |        |
|   | 51,*       |                               |             |        |
|   | 324--332.  |                               |             |        |
+---+------------+-------------------------------+-------------+--------+
| 3 | Burnham,   | The study examined the effect | The         | Compu  |
| . | P. T.      | of a televised curriculum on  | researcher  | ter-As |
|   | (1985). A  | an incarcerated ASE           | found no    | sisted |
|   | field test | population. Subjects in a     | difference  | Instr  |
|   | of a       | nonequivalent control group   | of          | uction |
|   | structured | research design (*n* = 40)    | achievement |        |
|   | television | were pretested using the      | between the |        |
|   | curriculum | General Educational           | groups,     |        |
|   | on the     | Performance Index (Form AA)   | although he |        |
|   | m          | and posttested using Form BB  | cautions    |        |
|   | athematics | of the same test. The         | that Adult  |        |
|   | a          | experimental group used an    | Math is     |        |
|   | chievement | instructional televisions     | more        |        |
|   | of         | series, Adult Math, as a      | effective   |        |
|   | in         | reinforcement resource,       | when the    |        |
|   | carcerated | viewing the "telelessons"     | subjects    |        |
|   | high       | under supervision and then    | have at     |        |
|   | school     | completing workbook exercises | least 5.8   |        |
|   | e          | tied to the program. The      | entry       |        |
|   | quivalency | control group completed       | grade-level |        |
|   | learners.  | self-paced workbooks and used | scores in   |        |
|   | *Di        | other instructional materials | arithmetic  |        |
|   | ssertation | but did not view the          | and reading |        |
|   | Abstracts  | television series.            | and that    |        |
|   | Inte       |                               | the         |        |
|   | rnational, |                               | literacy    |        |
|   | 46* (05),  |                               | levels of   |        |
|   | 1175A.     |                               | i           |        |
|   | (UMI No.   |                               | ncarcerated |        |
|   | 8514722)   |                               | populations |        |
|   |            |                               | are         |        |
|   |            |                               | noticeably  |        |
|   |            |                               | lower than  |        |
|   |            |                               | those of    |        |
|   |            |                               | the general |        |
|   |            |                               | population. |        |
+---+------------+-------------------------------+-------------+--------+
| 4 | Burton, B. | This study revisited the same | The study   |        |
| . | S. (1995). | question of traditional       | used the    |        |
|   | The        | versus CAI instruction with   | TABE M and  |        |
|   | effects of | more definitive results. Two  | D as its    |        |
|   | a          | hundred adults at a           | instrument  |        |
|   | compute    | vocational technical adult    | and a       |        |
|   | r-assisted | education center participated | combination |        |
|   | i          | in the study.                 | of a        |        |
|   | nstruction |                               | no          |        |
|   | and other  |                               | nequivalent |        |
|   | selected   |                               | control     |        |
|   | variables  |                               | group       |        |
|   | on the     |                               | design and  |        |
|   | academic   |                               | a causal    |        |
|   | p          |                               | comparative |        |
|   | erformance |                               | design.     |        |
|   | of adult   |                               | Students    |        |
|   | students   |                               | using CAI   |        |
|   | in         |                               | were found  |        |
|   | m          |                               | to do       |        |
|   | athematics |                               | si          |        |
|   | and        |                               | gnificantly |        |
|   | reading.   |                               | better in   |        |
|   | *Di        |                               | mathematics |        |
|   | ssertation |                               | than those  |        |
|   | Abstracts  |                               | in the      |        |
|   | Inte       |                               | control     |        |
|   | rnational, |                               | group. Age  |        |
|   | 57* (07),  |                               | and gender  |        |
|   | 2798A.     |                               | had no      |        |
|   | (UMI No.   |                               | effect.     |        |
|   | 9639904)   |                               | Ethnicity   |        |
|   |            |                               | and extent  |        |
|   |            |                               | of formal   |        |
|   |            |                               | education   |        |
|   |            |                               | did affect  |        |
|   |            |                               | the         |        |
|   |            |                               | results.    |        |
+---+------------+-------------------------------+-------------+--------+
| 5 | Farr, C.   | Farr investigated the effects |             |        |
| . | W. (1987). | of inferencing training in    |             |        |
|   | Effects of | learning vocabulary on verbal |             |        |
|   | i          | abilities and mathematics     |             |        |
|   | nferencing | problem solving among 40 ABE  |             |        |
|   | training   | students. Half the students   |             |        |
|   | on verbal  | had inference training, a     |             |        |
|   | abilities  | method where they were taught |             |        |
|   | and        | vocabulary skills and         |             |        |
|   | m          | reasoning training. The other |             |        |
|   | athematics | students received traditional |             |        |
|   | probl      | ABE instruction without the   |             |        |
|   | em-solving | training. Although the main   |             |        |
|   | among      | focus of the study was        |             |        |
|   | adult      | literacy development,         |             |        |
|   | basic      | mathematics problem solving   |             |        |
|   | education  | was included as a dependent   |             |        |
|   | students*. | variable to ascertain whether |             |        |
|   | Di         | training in inferencing in    |             |        |
|   | ssertation | language acquisition would be |             |        |
|   | Abstracts  | reflected in other areas. The |             |        |
|   | Inter      | results showed that there was |             |        |
|   | national*, | a correlation between         |             |        |
|   | *48* (08), | mathematics performance and   |             |        |
|   | 2021A.     | reading performance. The      |             |        |
|   | (UMI No.   | results also showed that      |             |        |
|   | 8725351)   | verbal ability correlated     |             |        |
|   |            | with ability to solve         |             |        |
|   |            | analogies and neologisms.     |             |        |
+---+------------+-------------------------------+-------------+--------+
| 6 | Indiana    | Study examined the            | No          |        |
| . | O.I.C. of  | effectiveness of traditional  | information |        |
|   | America    | classroom instruction versus  | was         |        |
|   | State      | computer-assisted instruction | supplied    |        |
|   | Council.   | (CAI) in raising the          | concerning  |        |
|   | (1990). *A | competency levels of adults   | the         |        |
|   | c          | one grade level for each 80   | statistical |        |
|   | omparative | hours of instruction. Of the  | analysis    |        |
|   | study of   | 149 individuals who were      | used to     |        |
|   | adult      | pretested, only 50 attended   | evaluate    |        |
|   | education  | more than 30 hours and        | the         |        |
|   | In         | remained at the time of       | results.    |        |
|   | dianapolis | posttesting. Evaluation was   | However,    |        |
|   | /Richmond. | done using a randomized       | the report  |        |
|   | T          | methodology with the ABLE     | indicates   |        |
|   | hird-party | test as pre- and posttest     | that the    |        |
|   | evaluation | instrument.                   | overall     |        |
|   | final      |                               | average     |        |
|   | report.*   | Population: Economically      | grade       |        |
|   | Ind        | disadvantaged adults,         | change for  |        |
|   | ianapolis, | predominantly female, ages    | CAI         |        |
|   | IN:        | 17--67 (mean age = 32.3       | students    |        |
|   | Author.    | years) who tested below 12th  | was 2.6     |        |
|   |            | grade/GED competency levels   | grades      |        |
|   |            | on the ABLE. Un- or           | compared    |        |
|   |            | underemployed.                | with an     |        |
|   |            |                               | average of  |        |
|   |            |                               | 1.84 grades |        |
|   |            |                               | for non-CAI |        |
|   |            |                               | students.   |        |
|   |            |                               | The high    |        |
|   |            |                               | attrition   |        |
|   |            |                               | again makes |        |
|   |            |                               | these       |        |
|   |            |                               | findings    |        |
|   |            |                               | difficult   |        |
|   |            |                               | to          |        |
|   |            |                               | interpret.  |        |
+---+------------+-------------------------------+-------------+--------+
| 7 | Irby, T.   | Irby conducted a randomized   | The results |        |
| . | R., et al. | control trial of              | indicated   |        |
|   | (1992).    | approximately 25              | that        |        |
|   | *The       | predominantly black and       | students in |        |
|   | Joliet     | Hispanic students in a ABE    | the family  |        |
|   | Junior     | setting; 15 students were in  | literacy    |        |
|   | College    | a family literacy project     | project     |        |
|   | Center for | (intervention group), and 10  | showed a    |        |
|   | Adult      | were enrolled in GED classes  | higher      |        |
|   | Basic      | only (control). The objective | average     |        |
|   | Education  | of the study was to evaluate  | gain in     |        |
|   | and        | the effectiveness of a family | reading and |        |
|   | Literacy's | literacy project on the       | mathematics |        |
|   | ''Families | numeracy and literacy levels  | compared to |        |
|   | about      | of adults. The intervention   | the control |        |
|   | success'': | was conducted in a family     |             |        |
|   | Interge    | literacy project comprising   |             |        |
|   | nerational | several components and ABE    |             |        |
|   | p          | classes were offered two      |             |        |
|   | rogramming | times per week for 12 weeks.  |             |        |
|   | that       | Instructors developed         |             |        |
|   | works.*    | individualized lesson plans   |             |        |
|   | Joliet,    | for each student to work at   |             |        |
|   | IL: Joliet | his/her own pace.             |             |        |
|   | Junior     |                               |             |        |
|   | College.   |                               |             |        |
+---+------------+-------------------------------+-------------+--------+
| 8 | Lavery,    | Lavery and colleagues         | The results |        |
| . | L.,        | conducted a randomized        | show that   |        |
|   | Townsend,  | control trial of 12 students  | si          |        |
|   | M., & K.   | in New Zealand to compare the | gnificantly |        |
|   | Wilton     | learning outcomes associated  | greater     |        |
|   | (1998).    | with basis literacy education | a           |        |
|   | Compute    | programs conducted via        | chievements |        |
|   | r-assisted | traditional instruction with  | were made   |        |
|   | i          | computer-assisted instruction | in reading  |        |
|   | nstruction | (CAI). The study measured the | (word       |        |
|   | in         | gains in reading and numeracy | r           |        |
|   | teaching   | skills in two "training       | ecognition, |        |
|   | literacy   | opportunities" classes. Six   | word        |        |
|   | skills to  | students received traditional | accuracy,   |        |
|   | adults not | teaching and another six used | and         |        |
|   | in paid    | Readers' Workshop, Math       | com         |        |
|   | e          | Concepts and Skills, and      | prehension) |        |
|   | mployment. | Computer Curriculum           | and         |        |
|   | *New       | Corporation CAL software      | numeracy    |        |
|   | Zealand    | packages. Participants'       | (m          |        |
|   | Journal of | reading and numeracy skills   | athematical |        |
|   | E          | were measured by the Burt     | concepts,   |        |
|   | ducational | Word Reading Test, the Neale  | operations, |        |
|   | Studies,   | Analysis of Reading Ability,  | and         |        |
|   | 33*(2),    | and the KeyMath Revised Test. | ap          |        |
|   | 181--192.  |                               | plications) |        |
|   |            |                               | under CAI   |        |
|   |            |                               | than under  |        |
|   |            |                               | traditional |        |
|   |            |                               | i           |        |
|   |            |                               | nstruction. |        |
|   |            |                               | The         |        |
|   |            |                               | students    |        |
|   |            |                               | who used    |        |
|   |            |                               | the CAL     |        |
|   |            |                               | made three  |        |
|   |            |                               | years' gain |        |
|   |            |                               | on the      |        |
|   |            |                               | Burt, over  |        |
|   |            |                               | one year on |        |
|   |            |                               | the Neale,  |        |
|   |            |                               | and 16      |        |
|   |            |                               | months on   |        |
|   |            |                               | the         |        |
|   |            |                               | mathematics |        |
|   |            |                               | test in     |        |
|   |            |                               | less than   |        |
|   |            |                               | two months  |        |
|   |            |                               | of          |        |
|   |            |                               | i           |        |
|   |            |                               | nstruction. |        |
|   |            |                               | During the  |        |
|   |            |                               | same time,  |        |
|   |            |                               | the         |        |
|   |            |                               | students    |        |
|   |            |                               | who         |        |
|   |            |                               | received    |        |
|   |            |                               | traditional |        |
|   |            |                               | teaching    |        |
|   |            |                               | made no     |        |
|   |            |                               | gains in    |        |
|   |            |                               | reading     |        |
|   |            |                               | skills and  |        |
|   |            |                               | showed a    |        |
|   |            |                               | slight      |        |
|   |            |                               | decline in  |        |
|   |            |                               | mathematics |        |
|   |            |                               | p           |        |
|   |            |                               | erformance. |        |
+---+------------+-------------------------------+-------------+--------+
| 9 | Nicol, M.  | Objective of study was to     | The         |        |
| . | M., &      | evaluate an experiment that   | researchers |        |
|   | Anderson,  | compared computer-assisted    | reported no |        |
|   | A. (2000). | and teacher-implemented       | difference  |        |
|   | Compute    | instruction in numeracy. It   | in          |        |
|   | r-assisted | is unclear whether the same   | improvement |        |
|   | vs.        | two teachers taught the two   | between the |        |
|   | teache     | intervention groups. The      | teacher-led |        |
|   | r-directed | researchers randomized the    | i           |        |
|   | teaching   | adult students into three     | ntervention |        |
|   | of         | groups of eight. The method   | and the     |        |
|   | numeracy   | of random allocation was not  | CAI, but    |        |
|   | in adults. | described, but stratification | given the   |        |
|   | *Journal   | by gender is implied.         | very small  |        |
|   | of         |                               | numbers in  |        |
|   | Computer   |                               | each group, |        |
|   | Assisted   |                               | there is    |        |
|   | Learning,  |                               | high        |        |
|   | 16,*       |                               | possibility |        |
|   | 184--192.  |                               | of a Type   |        |
|   |            |                               | II error in |        |
|   |            |                               | this study. |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Nurss, J.  | This study assessed the       | This trial  |        |
| 0 | R. (1989). | effectiveness of the PALS CAI | showed a    |        |
| . | *PALS      | program on the literacy       | s           |        |
|   | evaluation | skills of adult nonreaders    | ignificant, |        |
|   | project.*  | compared with traditional     | positive    |        |
|   | Atlanta,   | adult basic education.        | effect for  |        |
|   | GA:        |                               | the         |        |
|   | Georgia    |                               | traditional |        |
|   | State      |                               | adult basic |        |
|   | U          |                               | education   |        |
|   | niversity, |                               | classes     |        |
|   | Center for |                               | (i.e., the  |        |
|   | the Study  |                               | control     |        |
|   | of Adult   |                               | group).     |        |
|   | Literacy.  |                               | Attrition,  |        |
|   | (ERIC      |                               | however,    |        |
|   | Document   |                               | was         |        |
|   | Re         |                               | extremely   |        |
|   | production |                               | high in     |        |
|   | Service    |                               | both        |        |
|   | No.        |                               | groups. Of  |        |
|   | ED313573)  |                               | the 74      |        |
|   |            |                               | students    |        |
|   |            |                               | assigned to |        |
|   |            |                               | the control |        |
|   |            |                               | group, 15%  |        |
|   |            |                               | (*n* = 11)  |        |
|   |            |                               | remained at |        |
|   |            |                               | the         |        |
|   |            |                               | posttest;   |        |
|   |            |                               | 32% of the  |        |
|   |            |                               | 135         |        |
|   |            |                               | students in |        |
|   |            |                               | the         |        |
|   |            |                               | e           |        |
|   |            |                               | xperimental |        |
|   |            |                               | group (*n*  |        |
|   |            |                               | = 43)       |        |
|   |            |                               | completed   |        |
|   |            |                               | the         |        |
|   |            |                               | program.    |        |
|   |            |                               | One could   |        |
|   |            |                               | conjecture  |        |
|   |            |                               | that the    |        |
|   |            |                               | "cream" of  |        |
|   |            |                               | the control |        |
|   |            |                               | remained    |        |
|   |            |                               | and         |        |
|   |            |                               | therefore   |        |
|   |            |                               | performed   |        |
|   |            |                               | well on the |        |
|   |            |                               | test. In    |        |
|   |            |                               | addition    |        |
|   |            |                               | there was   |        |
|   |            |                               | d           |        |
|   |            |                               | ifferential |        |
|   |            |                               | attrition,  |        |
|   |            |                               | with more   |        |
|   |            |                               | of the      |        |
|   |            |                               | control     |        |
|   |            |                               | group       |        |
|   |            |                               | staying to  |        |
|   |            |                               | completion, |        |
|   |            |                               | clouding    |        |
|   |            |                               | the         |        |
|   |            |                               | results.    |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Reid, M.   | Study compared three teaching | The TABE M  |        |
| 1 | N. B.      | methods: CAI using PLATO,     | and D       |        |
| . | (1986). A  | tutoring using Laubach        | levels      |        |
|   | c          | materials, and traditional    | served as   |        |
|   | omparative | teaching. Subjects (*n* = 30) | pre- and    |        |
|   | study of   | were members of existing      | posttest    |        |
|   | three      | ABE/GED classes.              | i           |        |
|   | teaching   |                               | nstruments. |        |
|   | methods    |                               | There was   |        |
|   | used in    |                               | no          |        |
|   | Adult      |                               | significant |        |
|   | Basic      |                               | difference  |        |
|   | Education  |                               | in          |        |
|   | and        |                               | m           |        |
|   | General    |                               | athematical |        |
|   | E          |                               | achievement |        |
|   | ducational |                               | among the   |        |
|   | D          |                               | three       |        |
|   | evelopment |                               | groups,     |        |
|   | m          |                               | although    |        |
|   | athematics |                               | the CAI     |        |
|   | programs.  |                               | group       |        |
|   | *Dis       |                               | gained 1.9  |        |
|   | sertation* |                               | grade       |        |
|   | *Abstracts |                               | levels      |        |
|   | Inte       |                               | while the   |        |
|   | rnational, |                               | traditional |        |
|   | 47* (08),  |                               | group       |        |
|   | 2853A.     |                               | gained 1.1. |        |
|   | (UMI No.   |                               |             |        |
|   | 8626463)   |                               |             |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Robichaud, | Study compared students in    | No details  |        |
| 2 | K. K.      | traditional settings with     | concerning  |        |
| . | (1986).    | those whose regular           | instrument  |        |
|   | The use of | instruction was supplemented  | or          |        |
|   | computer   | by CAI.                       | evaluation  |        |
|   | assisted   |                               | were        |        |
|   | i          | The variables chosen for      | supplied.   |        |
|   | nstruction | comparison of two such groups | Robichaud   |        |
|   | with Adult | were skills gains in          | reports     |        |
|   | Basic      | mathematics and reading and   | that        |        |
|   | Education  | change in attitudes toward    | statistical |        |
|   | students:  | computers. The attitudes of   | analysis    |        |
|   | A          | CAI users toward CAI also     | revealed no |        |
|   | c          | were assessed.                | significant |        |
|   | omparative |                               | difference  |        |
|   | study.     |                               | in skills   |        |
|   | *Di        |                               | gained.     |        |
|   | ssertation |                               | There was a |        |
|   | Abstracts  |                               | si          |        |
|   | Inte       |                               | gnificantly |        |
|   | rnational, |                               | positive    |        |
|   | 47* (06),  |                               | change in   |        |
|   | 1985A.     |                               | attitudes   |        |
|   | (UMI No.   |                               | toward      |        |
|   | 8621681)   |                               | computers   |        |
|   |            |                               | and the     |        |
|   |            |                               | in          |        |
|   |            |                               | structional |        |
|   |            |                               | use of CAI  |        |
|   |            |                               | by the CAI  |        |
|   |            |                               | users.      |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Wardlaw,   | Wardlaw studied the effect of | Wardlaw     | He     |
| 3 | R. (1997). | CAI on achievement and        | found no    | does   |
| . | Effects of | attitude for a group of       | significant | offer  |
|   | computer   | pre-GED and GED adults. The   | difference  | an     |
|   | assisted   | study was conducted in        | on either   | imp    |
|   | i          | established classes with 60   | achievement | ortant |
|   | nstruction | students each in the          | or          | caveat |
|   | on         | experimental and control      | attitude.   | for    |
|   | a          | groups. Pre- and posttesting  |             | pr     |
|   | chievement | was done using the TABE and   |             | ograms |
|   | outcomes   | Semantic Differential         |             | pl     |
|   | of adults  | Attitudinal Questionnaire.    |             | anning |
|   | in         |                               |             | to     |
|   | dev        |                               |             | incor  |
|   | elopmental |                               |             | porate |
|   | education  |                               |             | CAI    |
|   | programs:  |                               |             | into a |
|   | A          |                               |             | pr     |
|   | c          |                               |             | ogram. |
|   | omparative |                               |             | W      |
|   | study.     |                               |             | ardlaw |
|   | *Di        |                               |             | su     |
|   | ssertation |                               |             | rveyed |
|   | Abstracts  |                               |             | ABE    |
|   | Inte       |                               |             | faci   |
|   | rnational, |                               |             | lities |
|   | 58* (10),  |                               |             | and    |
|   | 3804A.     |                               |             | found  |
|   | (UMI No.   |                               |             | that   |
|   | 9811694)   |                               |             | al     |
|   |            |                               |             | though |
|   | []{.mark}  |                               |             | some   |
|   |            |                               |             | were   |
|   |            |                               |             | well   |
|   |            |                               |             | equ    |
|   |            |                               |             | ipped, |
|   |            |                               |             | many   |
|   |            |                               |             | others |
|   |            |                               |             | had    |
|   |            |                               |             | few or |
|   |            |                               |             | ou     |
|   |            |                               |             | tdated |
|   |            |                               |             | w      |
|   |            |                               |             | orksta |
|   |            |                               |             | tions. |
|   |            |                               |             | One    |
|   |            |                               |             | fa     |
|   |            |                               |             | cility |
|   |            |                               |             | had    |
|   |            |                               |             | banned |
|   |            |                               |             | s      |
|   |            |                               |             | tudent |
|   |            |                               |             | use of |
|   |            |                               |             | the    |
|   |            |                               |             | equ    |
|   |            |                               |             | ipment |
|   |            |                               |             | b      |
|   |            |                               |             | ecause |
|   |            |                               |             | the    |
|   |            |                               |             | di     |
|   |            |                               |             | rector |
|   |            |                               |             | be     |
|   |            |                               |             | lieved |
|   |            |                               |             | that   |
|   |            |                               |             | the    |
|   |            |                               |             | st     |
|   |            |                               |             | udents |
|   |            |                               |             | were   |
|   |            |                               |             | using  |
|   |            |                               |             | it to  |
|   |            |                               |             | a      |
|   |            |                               |             | rrange |
|   |            |                               |             | dates  |
|   |            |                               |             | rather |
|   |            |                               |             | than   |
|   |            |                               |             | study. |
|   |            |                               |             | He     |
|   |            |                               |             | su     |
|   |            |                               |             | ggests |
|   |            |                               |             | that   |
|   |            |                               |             | these  |
|   |            |                               |             | e      |
|   |            |                               |             | nviron |
|   |            |                               |             | mental |
|   |            |                               |             | issues |
|   |            |                               |             | may    |
|   |            |                               |             | have   |
|   |            |                               |             | contr  |
|   |            |                               |             | ibuted |
|   |            |                               |             | to the |
|   |            |                               |             | f      |
|   |            |                               |             | ailure |
|   |            |                               |             | of CAI |
|   |            |                               |             | to     |
|   |            |                               |             | affect |
|   |            |                               |             | po     |
|   |            |                               |             | sitive |
|   |            |                               |             | attit  |
|   |            |                               |             | udinal |
|   |            |                               |             | c      |
|   |            |                               |             | hange. |
+---+------------+-------------------------------+-------------+--------+
| 1 | Wilder, M. | Wilder compared the effects   | The         |        |
| 4 | (1994).    | of a computer-based           | research    |        |
| . | The        | instructional (CBI)           | design was  |        |
|   | effects of | simulation-test treatment, a  | a           |        |
|   | a          | CBI drill and practice        | t           |        |
|   | simulation | program, and a traditional    | hree-group, |        |
|   | test model | workbook drill and practice   | posttest    |        |
|   | of the     | class on retention,           | only design |        |
|   | General    | completion time, and          | with        |        |
|   | Education  | elevation of test scores on   | unequal     |        |
|   | D          | the mathematics section of    | sample      |        |
|   | evelopment | the GED.                      | sizes,      |        |
|   | (GED)      |                               | where a     |        |
|   | program as |                               | total of    |        |
|   | compared   |                               | 564         |        |
|   | to the     |                               | students    |        |
|   | effects of |                               | se          |        |
|   | a drill    |                               | lf-selected |        |
|   | and        |                               | into the    |        |
|   | practice,  |                               | classes.    |        |
|   | both       |                               | Wilder      |        |
|   | comp       |                               | followed    |        |
|   | uter-based |                               | the         |        |
|   | and        |                               | students    |        |
|   | work       |                               | for five    |        |
|   | book-based |                               | years, with |        |
|   | on GED     |                               | 308         |        |
|   | m          |                               | students    |        |
|   | athematics |                               | retained    |        |
|   | scores,    |                               | long enough |        |
|   | retention, |                               | to get a    |        |
|   | and time.  |                               | GED         |        |
|   | *Di        |                               | diploma;    |        |
|   | ssertation |                               | 94% of the  |        |
|   | Abstracts  |                               | simulation  |        |
|   | Inte       |                               | group was   |        |
|   | rnational, |                               | retained    |        |
|   | 57* (07),  |                               | compared    |        |
|   | 2808A.     |                               | with 65% in |        |
|   | (UMI No.   |                               | the CBI     |        |
|   | 9639896)   |                               | drill group |        |
|   |            |                               | and 36% in  |        |
|   |            |                               | the         |        |
|   |            |                               | workbook    |        |
|   |            |                               | only group. |        |
|   |            |                               | Completion  |        |
|   |            |                               | time was    |        |
|   |            |                               | also        |        |
|   |            |                               | c           |        |
|   |            |                               | onsiderably |        |
|   |            |                               | less in     |        |
|   |            |                               | both CBI    |        |
|   |            |                               | groups.     |        |
|   |            |                               | Scores on   |        |
|   |            |                               | the test    |        |
|   |            |                               | were not    |        |
|   |            |                               | si          |        |
|   |            |                               | gnificantly |        |
|   |            |                               | different.  |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Winters C. | Winters and colleagues        | Pre- and    |        |
| 5 | A.,        | studied the effect of         | posttesting |        |
| . | Matthew,   | CAI-supplemented program on   | was         |        |
|   | M.,        | ABE/pre-GED and GED students  | conducted   |        |
|   | Booker,    | in an adult correctional      | using the   |        |
|   | F., &      | facility. Five students were  | TABE. The   |        |
|   | Fleeger,   | assigned to either the        | statistical |        |
|   | F. (1993). | experimental or control group | methods     |        |
|   | The role   | (*n* = 10).                   | used to     |        |
|   | of a       |                               | analyze the |        |
|   | comput     |                               | data were   |        |
|   | er-managed |                               | unclear,    |        |
|   | ins        |                               | but the     |        |
|   | tructional |                               | results     |        |
|   | system's   |                               | favored the |        |
|   | pr         |                               | CAI         |        |
|   | escriptive |                               | in          |        |
|   | curriculum |                               | tervention: |        |
|   | in the     |                               | 86% of the  |        |
|   | basic      |                               | students in |        |
|   | skill      |                               | the pilot   |        |
|   | areas of   |                               | study       |        |
|   | math and   |                               | advanced in |        |
|   | reading    |                               | level in    |        |
|   | scores for |                               | mathematics |        |
|   | co         |                               | as opposed  |        |
|   | rrectional |                               | to a 50%    |        |
|   | pre-trial  |                               | gain in the |        |
|   | detainees. |                               | control     |        |
|   | *Journal   |                               | group. A    |        |
|   | of         |                               | comparison  |        |
|   | Co         |                               | of students |        |
|   | rrectional |                               | advancing   |        |
|   | Education, |                               | one year or |        |
|   | 44*(1),    |                               | more showed |        |
|   | 10--17.    |                               | 43% for the |        |
|   |            |                               | pilot study |        |
|   |            |                               | versus      |        |
|   |            |                               | 14.5% for   |        |
|   |            |                               | the control |        |
|   |            |                               | group. Once |        |
|   |            |                               | again, the  |        |
|   |            |                               | small       |        |
|   |            |                               | sample size |        |
|   |            |                               | limits the  |        |
|   |            |                               | utility of  |        |
|   |            |                               | this study. |        |
+---+------------+-------------------------------+-------------+--------+
| > |            |                               |             |        |
|   |            |                               |             |        |
| * |            |                               |             |        |
| * |            |                               |             |        |
| [ |            |                               |             |        |
| D |            |                               |             |        |
| e |            |                               |             |        |
| v |            |                               |             |        |
| e |            |                               |             |        |
| l |            |                               |             |        |
| o |            |                               |             |        |
| p |            |                               |             |        |
| m |            |                               |             |        |
| e |            |                               |             |        |
| n |            |                               |             |        |
| t |            |                               |             |        |
| a |            |                               |             |        |
| l |            |                               |             |        |
| > |            |                               |             |        |
|   |            |                               |             |        |
| M |            |                               |             |        |
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| t |            |                               |             |        |
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| t |            |                               |             |        |
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| } |            |                               |             |        |
| * |            |                               |             |        |
| * |            |                               |             |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | >          | Bartlett used a guided        | Students in |        |
| 6 |  Bartlett, | discovery approach to         | the         |        |
| . | > L. E.    | teaching mathematics in one   | e           |        |
|   | > (1993).  | section of a developmental    | xperimental |        |
|   | > The      | mathematics course at a       | class       |        |
|   | > e        | university.                   | performed   |        |
|   | valuation, |                               | better on   |        |
|   | > im       | The experimental group (*n* = | the         |        |
|   | provement, | 27) was a class taught with   | outcomes.   |        |
|   | > and      | this approach and was         | Bartlett    |        |
|   | > dis      | compared with the same class  | reports     |        |
|   | semination | taught in a previous quarter  | that the    |        |
|   | > of a     | (*n* = 52) without the        | e           |        |
|   | > guided   | approach.                     | xperimental |        |
|   | >          |                               | method was  |        |
|   |  discovery |                               | "very       |        |
|   | > method   |                               | effective   |        |
|   | > for      |                               | in          |        |
|   | > teaching |                               | improving   |        |
|   | > dev      |                               | the         |        |
|   | elopmental |                               | mathematics |        |
|   | > ma       |                               | performance |        |
|   | thematics. |                               | of adult    |        |
|   | > *Di      |                               | students."  |        |
|   | ssertation |                               | Outcome     |        |
|   | >          |                               | measures    |        |
|   |  Abstracts |                               | were        |        |
|   | > Inte     |                               | mathematics |        |
|   | rnational, |                               | performance |        |
|   | > 54*      |                               | measured by |        |
|   | > (12A).   |                               | a           |        |
|   | > (UMI No. |                               | researche   |        |
|   | > 9411836) |                               | r-developed |        |
|   |            |                               | test and    |        |
|   |            |                               | mathematics |        |
|   |            |                               | anxiety     |        |
|   |            |                               | measured by |        |
|   |            |                               | the Math    |        |
|   |            |                               | Anxiety     |        |
|   |            |                               | Rating      |        |
|   |            |                               | Scale       |        |
|   |            |                               | (MARS).     |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | Berry, A.  | Berry studied the effect of   | Pre-and     |        |
| 7 | J. (1996). | peer tutoring in dyads on     | posttests   |        |
| . | The        | adult students in a remedial  | were given  |        |
|   | effects of | algebra class. Two studies    | using the   |        |
|   | peer       | were conducted: a 6-week      | Suinn       |        |
|   | tutoring   | program and a 12-week         | Mathematics |        |
|   | on adult   | semester. Students            | Anxiety     |        |
|   | students   | self-selected the classes but | Rating      |        |
|   | in         | had no knowledge of the       | Scale, the  |        |
|   | remedial   | planned intervention.         | Fenn        |        |
|   | algebra at | Instructors were randomly     | ema-Sherman |        |
|   | an urban   | assigned and trained in the   | Mathematics |        |
|   | community  | intervention after            | Attitude    |        |
|   | college.   | assignment. In each case,     | Scales, a   |        |
|   | *Di        | three peer tutoring sections  | profile     |        |
|   | ssertation | were contrasted with three    | que         |        |
|   | Abstracts  | traditional lecture sections. | stionnaire, |        |
|   | Inter      |                               | and an      |        |
|   | national*, |                               | abbreviated |        |
|   | *57* (08), |                               | version of  |        |
|   | 3433A.     |                               | the         |        |
|   | (UMI No.   |                               | in          |        |
|   | 9701475)   |                               | stitutional |        |
|   |            |                               | Freshman    |        |
|   |            |                               | Skills      |        |
|   |            |                               | Assessment  |        |
|   |            |                               | Program     |        |
|   |            |                               | test. An    |        |
|   |            |                               | open-ended  |        |
|   |            |                               | survey was  |        |
|   |            |                               | also used.  |        |
|   |            |                               | Sections    |        |
|   |            |                               | had an      |        |
|   |            |                               | average of  |        |
|   |            |                               | 35 students |        |
|   |            |                               | (*n* =      |        |
|   |            |                               | ap          |        |
|   |            |                               | proximately |        |
|   |            |                               | 210). Of    |        |
|   |            |                               | the         |        |
|   |            |                               | variables   |        |
|   |            |                               | measured,   |        |
|   |            |                               | only        |        |
|   |            |                               | attitude    |        |
|   |            |                               | increased   |        |
|   |            |                               | si          |        |
|   |            |                               | gnificantly |        |
|   |            |                               | during the  |        |
|   |            |                               | 6-week      |        |
|   |            |                               | study. For  |        |
|   |            |                               | students in |        |
|   |            |                               | the 12-week |        |
|   |            |                               | semester,   |        |
|   |            |                               | the         |        |
|   |            |                               | i           |        |
|   |            |                               | ntervention |        |
|   |            |                               | group       |        |
|   |            |                               | showed      |        |
|   |            |                               | significant |        |
|   |            |                               | improvement |        |
|   |            |                               | in          |        |
|   |            |                               | mathematics |        |
|   |            |                               | achievement |        |
|   |            |                               | and         |        |
|   |            |                               | attitude as |        |
|   |            |                               | well as     |        |
|   |            |                               | reduced     |        |
|   |            |                               | anxiety.    |        |
+---+------------+-------------------------------+-------------+--------+
| 1 | > Costner, | Costner examined the          | Several     | One    |
| 8 | > B. G.    | effectiveness of a computer   | instruments | cri    |
| . | > (2002).  | algebra system (CAS) on       | were used   | ticism |
|   | > The      | achievement and attitudes of  | in the      | was    |
|   | > effects  | students in a college         | study:      | the    |
|   | > on       | remedial algebra course.      | research    | un     |
|   | > student  | Students in the treatment     | er-designed | availa |
|   | > a        | group (*n* = 26) used the CAS | pretest and | bility |
|   | chievement | to discover algorithms,       | periodic    | of the |
|   | > and      | explore algebraic             | section     | co     |
|   | >          | manipulation, and identify    | tests, a    | mputer |
|   |  attitudes | misconceptions, while         | d           | in     |
|   | > of       | students in the control group | epartmental | t      |
|   | > inc      | (*n* = 25) did not have       | final exam, | esting |
|   | orporating | access to the CAS.            | the         | situa  |
|   | > a        |                               | Fenn        | tions. |
|   | > computer |                               | ema-Sherman | The    |
|   | > algebra  |                               | Attitude    | rese   |
|   | > system   |                               | Toward      | archer |
|   | > into a   |                               | Success in  | sug    |
|   | > remedial |                               | Mathematics | gested |
|   | > college  |                               | Scale,      | that   |
|   | > m        |                               | Confidence  | asse   |
|   | athematics |                               | in Learning | ssment |
|   | > course.  |                               | Mathematics | needs  |
|   | > *Di      |                               | Scale, and  | to be  |
|   | ssertation |                               | Mathematics | a      |
|   | >          |                               | Usefulness  | ltered |
|   |  Abstracts |                               | Scale, a    | if CAS |
|   | > Inter    |                               | researcher  | is an  |
|   | national*, |                               | designed    | in     |
|   | > *63*     |                               | qu          | tegral |
|   | > (07),    |                               | estionnaire | part   |
|   | > 2483A.   |                               | and         | of the |
|   | > (UMI No. |                               | sem         | c      |
|   | > 3059227) |                               | istructured | ourse. |
|   |            |                               | interview   |        |
|   |            |                               | (*n* = 5),  |        |
|   |            |                               | and         |        |
|   |            |                               | periodic    |        |
|   |            |                               | writing     |        |
|   |            |                               | a           |        |
|   |            |                               | ssignments. |        |
|   |            |                               | There was   |        |
|   |            |                               | no          |        |
|   |            |                               | st          |        |
|   |            |                               | atistically |        |
|   |            |                               | significant |        |
|   |            |                               | effect on   |        |
|   |            |                               | achievement |        |
|   |            |                               | or surveyed |        |
|   |            |                               | attitudes.  |        |
|   |            |                               | However,    |        |
|   |            |                               | the         |        |
|   |            |                               | qualitative |        |
|   |            |                               | data        |        |
|   |            |                               | gathered    |        |
|   |            |                               | via         |        |
|   |            |                               | qu          |        |
|   |            |                               | estionnaire |        |
|   |            |                               | revealed    |        |
|   |            |                               | significant |        |
|   |            |                               | differences |        |
|   |            |                               | in          |        |
|   |            |                               | attitudes   |        |
|   |            |                               | and in      |        |
|   |            |                               | classroom   |        |
|   |            |                               | culture     |        |
|   |            |                               | issues.     |        |
|   |            |                               | Students in |        |
|   |            |                               | the         |        |
|   |            |                               | treatment   |        |
|   |            |                               | group cited |        |
|   |            |                               | the         |        |
|   |            |                               | helpfulness |        |
|   |            |                               | of group    |        |
|   |            |                               | work and    |        |
|   |            |                               | classroom   |        |
|   |            |                               | discussions |        |
|   |            |                               | more often  |        |
|   |            |                               | than        |        |
|   |            |                               | students in |        |
|   |            |                               | the control |        |
|   |            |                               | group. With |        |
|   |            |                               | respect to  |        |
|   |            |                               | the use of  |        |
|   |            |                               | CAS, the    |        |
|   |            |                               | treatment   |        |
|   |            |                               | group       |        |
|   |            |                               | welcomed    |        |
|   |            |                               | the ability |        |
|   |            |                               | to check    |        |
|   |            |                               | their work  |        |
|   |            |                               | and get     |        |
|   |            |                               | immediate   |        |
|   |            |                               | feedback,   |        |
|   |            |                               | they felt   |        |
|   |            |                               | that the    |        |
|   |            |                               | CAS helped  |        |
|   |            |                               | them to see |        |
|   |            |                               | mathematics |        |
|   |            |                               | differently |        |
|   |            |                               | yet         |        |
|   |            |                               | attributed  |        |
|   |            |                               | little of   |        |
|   |            |                               | their new   |        |
|   |            |                               | m           |        |
|   |            |                               | athematical |        |
|   |            |                               | un          |        |
|   |            |                               | derstanding |        |
|   |            |                               | to          |        |
|   |            |                               | technology. |        |
+---+------------+-------------------------------+-------------+--------+

+---+------------+-------------------------------+------------+--------+
|   | **Study    | **Description of Intervention | **Outcome  | > **F  |
|   | Name**     | and Population Measured**     | Measure    | urther |
|   |            |                               | and        | >      |
|   |            |                               | Effects    | Conclu |
|   |            |                               | Found**    | sions/ |
|   |            |                               |            | > Im   |
|   |            |                               |            | plicat |
|   |            |                               |            | ions** |
+===+============+===============================+============+========+
| 1 | Ellis, N.  | In the study, each of seven   | Ellis      | Older  |
| 9 | F. (1992). | instructors at a community    | compared   | adult  |
| . | The effect | college taught one            | the        | st     |
|   | of         | developmental algebra section | a          | udents |
|   | in-class   | incorporating the use of      | chievement | had a  |
|   | study      | in-class study groups and one | and        | s      |
|   | groups on  | section where groups were not | completion | ignifi |
|   | a          | used.                         | rates and  | cantly |
|   | chievement |                               | found no   | g      |
|   | and course |                               | s          | reater |
|   | completion |                               | ignificant | re     |
|   | rates in   |                               | difference | sidual |
|   | dev        |                               | between    | gain   |
|   | elopmental |                               | the        | than   |
|   | algebra    |                               | ex         | tradi  |
|   | classes.   |                               | perimental | tional |
|   | *Di        |                               | and        | st     |
|   | ssertation |                               | control    | udents |
|   | Abstracts  |                               | groups for | rega   |
|   | Inte       |                               | the group, | rdless |
|   | rnational, |                               | neither as | of the |
|   | 53* (07),  |                               | a whole    | method |
|   | 2283A.     |                               | nor on the | emp    |
|   | (UMI No.   |                               | basis of   | loyed. |
|   | 9237220)   |                               | age or     | Women  |
|   |            |                               | gender.    | had a  |
|   |            |                               |            | s      |
|   |            |                               |            | ignifi |
|   |            |                               |            | cantly |
|   |            |                               |            | g      |
|   |            |                               |            | reater |
|   |            |                               |            | re     |
|   |            |                               |            | sidual |
|   |            |                               |            | gain   |
|   |            |                               |            | than   |
|   |            |                               |            | men in |
|   |            |                               |            | the    |
|   |            |                               |            | entire |
|   |            |                               |            | study  |
|   |            |                               |            | and in |
|   |            |                               |            | the    |
|   |            |                               |            | c      |
|   |            |                               |            | ontrol |
|   |            |                               |            | group. |
|   |            |                               |            | In the |
|   |            |                               |            | experi |
|   |            |                               |            | mental |
|   |            |                               |            | group, |
|   |            |                               |            | the    |
|   |            |                               |            | diffe  |
|   |            |                               |            | rences |
|   |            |                               |            | b      |
|   |            |                               |            | etween |
|   |            |                               |            | men    |
|   |            |                               |            | and    |
|   |            |                               |            | women  |
|   |            |                               |            | were   |
|   |            |                               |            | not    |
|   |            |                               |            | s      |
|   |            |                               |            | tatist |
|   |            |                               |            | ically |
|   |            |                               |            | signif |
|   |            |                               |            | icant. |
|   |            |                               |            |        |
|   |            |                               |            | The    |
|   |            |                               |            | group  |
|   |            |                               |            | as a   |
|   |            |                               |            | whole, |
|   |            |                               |            | tradi  |
|   |            |                               |            | tional |
|   |            |                               |            | stu    |
|   |            |                               |            | dents, |
|   |            |                               |            | older  |
|   |            |                               |            | adult  |
|   |            |                               |            | stu    |
|   |            |                               |            | dents, |
|   |            |                               |            | and    |
|   |            |                               |            | men    |
|   |            |                               |            | had    |
|   |            |                               |            | sl     |
|   |            |                               |            | ightly |
|   |            |                               |            | higher |
|   |            |                               |            | comp   |
|   |            |                               |            | letion |
|   |            |                               |            | rates  |
|   |            |                               |            | in the |
|   |            |                               |            | experi |
|   |            |                               |            | mental |
|   |            |                               |            | cl     |
|   |            |                               |            | asses. |
|   |            |                               |            | In     |
|   |            |                               |            | none   |
|   |            |                               |            | of     |
|   |            |                               |            | these  |
|   |            |                               |            | cases, |
|   |            |                               |            | ho     |
|   |            |                               |            | wever, |
|   |            |                               |            | was    |
|   |            |                               |            | the    |
|   |            |                               |            | diff   |
|   |            |                               |            | erence |
|   |            |                               |            | in     |
|   |            |                               |            | comp   |
|   |            |                               |            | letion |
|   |            |                               |            | rates  |
|   |            |                               |            | s      |
|   |            |                               |            | tatist |
|   |            |                               |            | ically |
|   |            |                               |            | signif |
|   |            |                               |            | icant. |
|   |            |                               |            | Women  |
|   |            |                               |            | had    |
|   |            |                               |            | vir    |
|   |            |                               |            | tually |
|   |            |                               |            | ide    |
|   |            |                               |            | ntical |
|   |            |                               |            | comp   |
|   |            |                               |            | letion |
|   |            |                               |            | rates  |
|   |            |                               |            | in the |
|   |            |                               |            | two    |
|   |            |                               |            | types  |
|   |            |                               |            | of     |
|   |            |                               |            | c      |
|   |            |                               |            | lasses |
+---+------------+-------------------------------+------------+--------+
| 2 | Hsieh, F.  | Hsieh (1992) examined the     | The        |        |
| 0 | J. (1992). | affect of specific features   | outcome    |        |
| . | Effects of | of CAI, animation and         | measures   |        |
|   | animation  | manipulation, on 54 college   | were       |        |
|   | and        | students participating in two | overall    |        |
|   | ma         | computer based laboratory     | ac         |        |
|   | nipulation | (CBL) sections of a           | hievement, |        |
|   | on adult   | mathematics course. The       | retention  |        |
|   | learning   | students were randomly        | of content |        |
|   | of         | assigned to receive           | and        |        |
|   | ma         | instruction with/without      | motivated, |        |
|   | thematical | animation and with/without    | measured   |        |
|   | concepts.  | manipulation. It is not clear | through a  |        |
|   | *Di        | from the design whether this  | ques       |        |
|   | ssertation | was a four-group design or a  | tionnaire. |        |
|   | Abstracts  | two-group.                    | The        |        |
|   | Inte       |                               | researcher |        |
|   | rnational, |                               | list five  |        |
|   | 53* (09),  |                               | findings:  |        |
|   | 3094A.     |                               |            |        |
|   | (UMI No.   |                               | Animation  |        |
|   | 9301313)   |                               | enhanced   |        |
|   |            |                               | retention  |        |
|   |            |                               | when the   |        |
|   |            |                               | tasks      |        |
|   |            |                               | required   |        |
|   |            |                               | high level |        |
|   |            |                               | cognitive  |        |
|   |            |                               | processes  |        |
|   |            |                               | such as    |        |
|   |            |                               | analysis   |        |
|   |            |                               | or         |        |
|   |            |                               | synthesis, |        |
|   |            |                               |            |        |
|   |            |                               | Animation  |        |
|   |            |                               | did not    |        |
|   |            |                               | help       |        |
|   |            |                               | learning   |        |
|   |            |                               | or         |        |
|   |            |                               | retention  |        |
|   |            |                               | when the   |        |
|   |            |                               | tasks were |        |
|   |            |                               | com        |        |
|   |            |                               | prehension |        |
|   |            |                               | of         |        |
|   |            |                               | ma         |        |
|   |            |                               | thematical |        |
|   |            |                               | concepts,  |        |
|   |            |                               |            |        |
|   |            |                               | Animation  |        |
|   |            |                               | increased  |        |
|   |            |                               | continuing |        |
|   |            |                               | m          |        |
|   |            |                               | otivation, |        |
|   |            |                               |            |        |
|   |            |                               | Ma         |        |
|   |            |                               | nipulation |        |
|   |            |                               | helped the |        |
|   |            |                               | tr         |        |
|   |            |                               | ansference |        |
|   |            |                               | of         |        |
|   |            |                               | ma         |        |
|   |            |                               | thematical |        |
|   |            |                               | concepts   |        |
|   |            |                               | learned    |        |
|   |            |                               | through a  |        |
|   |            |                               | computer   |        |
|   |            |                               | to         |        |
|   |            |                               | paper-     |        |
|   |            |                               | and-pencil |        |
|   |            |                               | tests, and |        |
|   |            |                               |            |        |
|   |            |                               | Ma         |        |
|   |            |                               | nipulation |        |
|   |            |                               | did not    |        |
|   |            |                               | promote    |        |
|   |            |                               | intrinsic  |        |
|   |            |                               | m          |        |
|   |            |                               | otivation. |        |
+---+------------+-------------------------------+------------+--------+

+---+------------+-------------------------------+------------+---------+
|   | **Study    | **Description of Intervention | **Outcome  | > **    |
|   | Name**     | and Population Measured**     | Measure    | Further |
|   |            |                               | and        | > Concl |
|   |            |                               | Effects    | usions/ |
|   |            |                               | Found**    | >       |
|   |            |                               |            | Implica |
|   |            |                               |            | tions** |
+===+============+===============================+============+=========+
| 2 | > Pace, J. | This study explored the       | Students   | By the  |
| 1 | > P.       | applicability of              | were       | measure |
| . | > (1989).  | constructivist methods to the | posttested | of      |
|   | > A model  | teaching of geometry concepts | and        | g       |
|   | > for      | in a remedial mathematics     | delayed    | eometry |
|   | > teaching | class at an urban community   | p          | achi    |
|   | > area and | college. Students (*n* = 67)  | osttested. | evement |
|   | >          | were pretested using the      | The data   | used in |
|   |  perimeter | Applied Geometry Test, the    | were       | this    |
|   | > concepts | Van Hiele Geometry Test, and  | assessed   | re      |
|   | > from a   | the New Jersey College Basic  | using      | search, |
|   | > cons     | Skill Placement Test. They    | single and | the     |
|   | tructivist | were randomly assigned to     | mu         | exper   |
|   | > p        | four sections of the course,  | ltivariate | imental |
|   | erspective | two experimental and two      | linear     | program |
|   | > to adult | control, all taught by the    | regression | of      |
|   | >          | same instructor. The          | models.    | t       |
|   |  community | experimental treatment        | Those in   | eaching |
|   | > college  | consisted of five 80-minute   | the        | signif  |
|   | > students | sessions during which         | ex         | icantly |
|   | > through  | students explored concepts of | perimental | in      |
|   | > applied  | area and perimeter using      | program    | creased |
|   | > probl    | activities embedded in        | performed  | st      |
|   | em-solving | applied problem-solving       | sig        | udents' |
|   | > and      | settings.                     | nificantly | short-  |
|   | > acti     |                               | better     | and     |
|   | vity-based |                               | than their | lo      |
|   | > in       |                               | cou        | ng-term |
|   | struction. |                               | nterparts. | perfo   |
|   | > *Di      |                               |            | rmance. |
|   | ssertation |                               |            | Every   |
|   | >          |                               |            | linear  |
|   |  Abstracts |                               |            | reg     |
|   | > Inte     |                               |            | ression |
|   | rnational, |                               |            | model   |
|   | > 51*      |                               |            | su      |
|   | > (02),    |                               |            | pported |
|   | > 0442A.   |                               |            | the     |
|   | > (UMI No. |                               |            | con     |
|   | > 9008820) |                               |            | clusion |
|   |            |                               |            | of a    |
|   |            |                               |            | signi   |
|   |            |                               |            | ficant, |
|   |            |                               |            | uncon   |
|   |            |                               |            | founded |
|   |            |                               |            | i       |
|   |            |                               |            | ncrease |
|   |            |                               |            | in      |
|   |            |                               |            | student |
|   |            |                               |            | perfo   |
|   |            |                               |            | rmance. |
|   |            |                               |            | Vid     |
|   |            |                               |            | eotaped |
|   |            |                               |            | in      |
|   |            |                               |            | terview |
|   |            |                               |            | data    |
|   |            |                               |            | p       |
|   |            |                               |            | rovided |
|   |            |                               |            | add     |
|   |            |                               |            | itional |
|   |            |                               |            | insight |
|   |            |                               |            | into    |
|   |            |                               |            | what    |
|   |            |                               |            | a       |
|   |            |                               |            | pparent |
|   |            |                               |            | changes |
|   |            |                               |            | had     |
|   |            |                               |            | o       |
|   |            |                               |            | ccurred |
|   |            |                               |            | in      |
|   |            |                               |            | st      |
|   |            |                               |            | udents' |
|   |            |                               |            | geom    |
|   |            |                               |            | etrical |
|   |            |                               |            | conce   |
|   |            |                               |            | ptions. |
|   |            |                               |            |         |
|   |            |                               |            | This    |
|   |            |                               |            | study   |
|   |            |                               |            | p       |
|   |            |                               |            | rovides |
|   |            |                               |            | a       |
|   |            |                               |            | well-   |
|   |            |                               |            | defined |
|   |            |                               |            | model   |
|   |            |                               |            | for the |
|   |            |                               |            | t       |
|   |            |                               |            | eaching |
|   |            |                               |            | and     |
|   |            |                               |            | l       |
|   |            |                               |            | earning |
|   |            |                               |            | of area |
|   |            |                               |            | and     |
|   |            |                               |            | pe      |
|   |            |                               |            | rimeter |
|   |            |                               |            | through |
|   |            |                               |            | applied |
|   |            |                               |            | problem |
|   |            |                               |            | s       |
|   |            |                               |            | olving. |
|   |            |                               |            | In      |
|   |            |                               |            | ad      |
|   |            |                               |            | dition, |
|   |            |                               |            | it has  |
|   |            |                               |            | at its  |
|   |            |                               |            | core a  |
|   |            |                               |            | con     |
|   |            |                               |            | ceptual |
|   |            |                               |            | model   |
|   |            |                               |            | that is |
|   |            |                               |            | ad      |
|   |            |                               |            | aptable |
|   |            |                               |            | for use |
|   |            |                               |            | in the  |
|   |            |                               |            | t       |
|   |            |                               |            | eaching |
|   |            |                               |            | and     |
|   |            |                               |            | l       |
|   |            |                               |            | earning |
|   |            |                               |            | of      |
|   |            |                               |            | other   |
|   |            |                               |            | mathe   |
|   |            |                               |            | matical |
|   |            |                               |            | co      |
|   |            |                               |            | ncepts. |
+---+------------+-------------------------------+------------+---------+
| 2 | Ramus, K.  | In this hybrid study (*n* =   | Students   |         |
| 2 | S. (1997). | 13), students were            | reported a |         |
| . | How the    | interviewed by a third party  | sense of   |         |
|   | m          | using an open-ended interview | ownership  |         |
|   | athematics | protocol.                     | of the     |         |
|   | education  |                               | rules of   |         |
|   | reforms    |                               | algebra    |         |
|   | pertain to |                               | because    |         |
|   | und        |                               | they had   |         |
|   | ergraduate |                               | discovered |         |
|   | c          |                               | them from  |         |
|   | urriculum: |                               | classroom  |         |
|   | An         |                               | exercises  |         |
|   | in         |                               | and also   |         |
|   | troductory |                               | sel        |         |
|   | study of   |                               | f-reported |         |
|   | an         |                               | a positive |         |
|   | ex         |                               | change in  |         |
|   | perimental |                               | attitude   |         |
|   | dev        |                               | toward     |         |
|   | elopmental |                               | m          |         |
|   | algebra    |                               | athematics |         |
|   | course for |                               | and        |         |
|   | adults.    |                               | increased  |         |
|   | *Di        |                               | confidence |         |
|   | ssertation |                               | that       |         |
|   | Abstracts  |                               | t          |         |
|   | Inte       |                               | ransferred |         |
|   | rnational, |                               | to other   |         |
|   | 57* (12),  |                               | activities |         |
|   | 5090A.     |                               | outside    |         |
|   | (UMI No.   |                               | the        |         |
|   | 9717243)   |                               | classroom. |         |
|   |            |                               | Qu         |         |
|   |            |                               | antitative |         |
|   |            |                               | measures,  |         |
|   |            |                               | derived    |         |
|   |            |                               | from the   |         |
|   |            |                               | course     |         |
|   |            |                               | ex         |         |
|   |            |                               | amination, |         |
|   |            |                               | were less  |         |
|   |            |                               | c          |         |
|   |            |                               | onclusive. |         |
|   |            |                               | E          |         |
|   |            |                               | xamination |         |
|   |            |                               | results    |         |
|   |            |                               | were       |         |
|   |            |                               | scored     |         |
|   |            |                               | using two  |         |
|   |            |                               | rubrics,   |         |
|   |            |                               | one to     |         |
|   |            |                               | measure    |         |
|   |            |                               | c          |         |
|   |            |                               | orrectness |         |
|   |            |                               | and one to |         |
|   |            |                               | measure    |         |
|   |            |                               | the use of |         |
|   |            |                               | probl      |         |
|   |            |                               | em-solving |         |
|   |            |                               | s          |         |
|   |            |                               | trategies. |         |
|   |            |                               | An ANOVA   |         |
|   |            |                               | showed the |         |
|   |            |                               | ex         |         |
|   |            |                               | perimental |         |
|   |            |                               | section    |         |
|   |            |                               | performed  |         |
|   |            |                               | less well  |         |
|   |            |                               | than the   |         |
|   |            |                               | daytime    |         |
|   |            |                               | comparison |         |
|   |            |                               | class and  |         |
|   |            |                               | as well as |         |
|   |            |                               | the        |         |
|   |            |                               | evening    |         |
|   |            |                               | section.   |         |
+---+------------+-------------------------------+------------+---------+
| 2 | Toet, J.   | Toet studied a randomly       | Using the  |         |
| 3 | A. (1991). | selected sample of students   | TABE, she  |         |
| . | A          | who had been placed into      | compared   |         |
|   | c          | remedial reading, English, or | a          |         |
|   | omparative | mathematics at a community    | chievement |         |
|   | study of   | college.                      | between    |         |
|   | two        |                               | students   |         |
|   | ins        |                               | who        |         |
|   | tructional |                               | completed  |         |
|   | modalities |                               | a          |         |
|   | on the     |                               | ssignments |         |
|   | a          |                               | based on   |         |
|   | chievement |                               | textbook   |         |
|   | level of   |                               | use versus |         |
|   | und        |                               | students   |         |
|   | erprepared |                               | who work   |         |
|   | community  |                               | in a CAI   |         |
|   | college    |                               | l          |         |
|   | students.  |                               | aboratory. |         |
|   | *Di        |                               | The group  |         |
|   | ssertation |                               | taking     |         |
|   | Abstracts  |                               | basic      |         |
|   | Inte       |                               | m          |         |
|   | rnational, |                               | athematics |         |
|   | 52* (12),  |                               | showed a   |         |
|   | 4191A.     |                               | sta        |         |
|   | (UMI No.   |                               | tistically |         |
|   | 9215165)   |                               | s          |         |
|   |            |                               | ignificant |         |
|   |            |                               | cognitive  |         |
|   |            |                               | gain.      |         |
|   |            |                               | Those      |         |
|   |            |                               | studying   |         |
|   |            |                               | beginning  |         |
|   |            |                               | algebra    |         |
|   |            |                               | were       |         |
|   |            |                               | retained   |         |
|   |            |                               | longer at  |         |
|   |            |                               | a          |         |
|   |            |                               | sta        |         |
|   |            |                               | tistically |         |
|   |            |                               | s          |         |
|   |            |                               | ignificant |         |
|   |            |                               | level of   |         |
|   |            |                               | .05. There |         |
|   |            |                               | was no     |         |
|   |            |                               | s          |         |
|   |            |                               | ignificant |         |
|   |            |                               | retention  |         |
|   |            |                               | difference |         |
|   |            |                               | for the    |         |
|   |            |                               | basic      |         |
|   |            |                               | m          |         |
|   |            |                               | athematics |         |
|   |            |                               | group.     |         |
+---+------------+-------------------------------+------------+---------+
| 2 | Wilson, O. | Describes a diagnostic and    | The        |         |
| 4 | D. (1987). | tutorial program that was     | results of |         |
| . | An         | conducted at a vocational     | the        |         |
|   | automated  | school in Kentucky. The       | experiment |         |
|   | diagnostic | researchers designed a        | showed a   |         |
|   | test and   | diagnostic test for pre- and  | s          |         |
|   | tutorial   | posttesting, which they       | ignificant |         |
|   | package    | normed against the TABE at    | effect in  |         |
|   | for basic  | the 8.75 grade equivalent.    | favor of   |         |
|   | skills of  |                               | the        |         |
|   | m          |                               | ex         |         |
|   | athematics |                               | perimental |         |
|   | in post    |                               | group.     |         |
|   | secondary  |                               |            |         |
|   | vocational |                               |            |         |
|   | education  |                               |            |         |
|   | of         |                               |            |         |
|   | Kentucky:  |                               |            |         |
|   | Co         |                               |            |         |
|   | nstruction |                               |            |         |
|   | and        |                               |            |         |
|   | v          |                               |            |         |
|   | alidation. |                               |            |         |
|   | *Di        |                               |            |         |
|   | ssertation |                               |            |         |
|   | Abstracts  |                               |            |         |
|   | Inte       |                               |            |         |
|   | rnational, |                               |            |         |
|   | 49* (01),  |                               |            |         |
|   | 0055A.     |                               |            |         |
|   | (UMI No.   |                               |            |         |
|   | 8804685)   |                               |            |         |
+---+------------+-------------------------------+------------+---------+

[^1]: The U.S. Department of Education (2005) commissioned a separate
    review of these higher level postsecondary mathematics classes in
    community college settings, which examined 15 studies of
    instructional interventions.

[^2]: Safford's database contains abstracts of 129 dissertations that
    met her search criteria from 1980 to 2002. The online version of
    *Dissertation Abstracts International* has abstracts only for the
    previous year (2003--04), so the two databases together allowed us
    to retrieve information on all dissertations.

[^3]: One of the authors of this review (Condelli) received training on
    the use of the evidence standards as the senior content advisor on
    adult literacy for the WWC.

[^4]: Further evidence of the low quality of methodology comes from a
    review of 17 adult ESL instructional intervention studies (Condelli
    & Wrigley, 2004) that used WWC procedures. That review included
    several of the same ABE studies reported here. Only 4 studies met
    the evidence standards.