Archived Information

# 

**STRENGTHENING MATHEMATICS SKILLS AT THE**

**POSTSECONDARY LEVEL:**

**LITERATURE REVIEW AND ANALYSIS**

U.S. Department of Education

Office of Vocational and Adult Education

Division of Adult Education and Literacy

2005

**STRENGTHENING MATHEMATICS SKILLS AT THE**

**POSTSECONDARY LEVEL:**

**LITERATURE REVIEW AND ANALYSIS**

Prepared for:

U.S. Department of Education

Office of Vocational and Adult Education

Division of Adult Education and Literacy

Prepared by:

The CNA Corporation

Contributors:

Peggy Golfin, The CNA Corporation, Alexandria, VA

Will Jordan, The CNA Corporation, Alexandria, VA

Darrell Hull, Center for Occupational Research and Development, Waco, TX

Monya Ruffin, American Institutes for Research, Washington, D.C.

This report was produced under U.S. Department of Education Contract No.
ED-01-CO-0037 with The CNA Corporation. Carolyn Lee served as the
contracting officer's representative. The views expressed herein do not
necessarily represent the positions or policies of the Department of
Education. This publication includes information about and references to
products, commodities, services or enterprises from other organizations,
both public and private. Inclusion of these does not constitute an
endorsement of them by the U.S. Department of Education.

**U.S. Department of Education**

Margaret Spellings

*Secretary*

September 2005

This report is in the public domain. Authorization to reproduce it in
whole or in part is granted. While permission to reprint this
publication is not necessary, the citation should be: U.S. Department of
Education, Office of Vocational and Adult Education, *Strengthening
Mathematics Skills at the Postsecondary Level: Literature Review and
Analysis,* Washington, D.C., 2005.

On request, this publication is available in alternate formats, such as
Braille, large print, or computer diskette. For more information, please
contact the Department's Alternate Format Center at (202) 260-0852 or
(202) 260-0818.

CONTENTS

[**Tables v**](#tables)

[Abbreviations vi](#abbreviations)

[Executive Summary 1](#executive-summary)

[Background 1](#background)

[Major Findings 1](#major-findings)

### [Institutions providing developmental math instruction 1](#institutions-providing-developmental-math-instruction) {#institutions-providing-developmental-math-instruction-1 .TOC-3}

### [What constitutes adequate math preparation? 2](#what-constitutes-adequate-math-preparation) {#what-constitutes-adequate-math-preparation-2 .TOC-3}

### [Best instructional practices 3](#best-instructional-practices) {#best-instructional-practices-3 .TOC-3}

[Introduction 5](#introduction)

[What Skills and Knowledge Do Students Need to Pursue College-level
Mathematics?
11](#what-skills-and-knowledge-do-students-need-to-pursue-college-level-mathematics)

[Crossroads 11](#crossroads)

[Standards for Success 14](#standards-for-success)

[The Vision Report 15](#the-vision-report)

[American Diploma Project 16](#american-diploma-project)

[Summary 17](#summary)

[Assessment and Placement Policies
19](#assessment-and-placement-policies)

[Tests Commonly Used 20](#tests-commonly-used)

### [ASSET and COMPASS 21](#asset-and-compass) {#asset-and-compass-21 .TOC-3}

### [ACCUPLACER 21](#accuplacer) {#accuplacer-21 .TOC-3}

### [TABE 22](#tabe) {#tabe-22 .TOC-3}

[State-specific Policies 22](#state-specific-policies)

[What Instructional Methods Work Best for Adult Learners?
27](#what-instructional-methods-work-best-for-adult-learners)

[The Role of Technology 27](#the-role-of-technology)

### [Computer-Assisted Instruction 28](#computer-assisted-instruction) {#computer-assisted-instruction-28 .TOC-3}

### [Computer Algebra System 32](#computer-algebra-system) {#computer-algebra-system-32 .TOC-3}

[Pedagogical Issues 33](#pedagogical-issues)

### [How people learn 33](#how-people-learn) {#how-people-learn-33 .TOC-3}

### [Learner-centered environment 36](#learner-centered-environment) {#learner-centered-environment-36 .TOC-3}

### [Small-group instruction 38](#small-group-instruction) {#small-group-instruction-38 .TOC-3}

### [Contextual learning 39](#contextual-learning) {#contextual-learning-39 .TOC-3}

### ["Systems Thinking"---DevMap Project 40](#systems-thinkingdevmap-project) {#systems-thinkingdevmap-project-40 .TOC-3}

### [Accelerated courses 41](#accelerated-courses) {#accelerated-courses-41 .TOC-3}

[Metrics of Program Effectiveness 43](#metrics-of-program-effectiveness)

[A Survey of Community Colleges' Practices
47](#a-survey-of-community-colleges-practices)

[The U.S. Military 51](#the-u.s.-military)

[General Eligibility Requirements 51](#general-eligibility-requirements)

[Armed Services Vocational Aptitude Battery
51](#armed-services-vocational-aptitude-battery)

### [Basic skills 53](#basic-skills) {#basic-skills-53 .TOC-3}

### [Postsecondary VolEd 54](#postsecondary-voled) {#postsecondary-voled-54 .TOC-3}

[Businesses and Organized Labor 57](#businesses-and-organized-labor)

[Corporate Math Skill Trainers 58](#corporate-math-skill-trainers)

[Adult Education and Workforce Development
61](#adult-education-and-workforce-development)

[Summary and Conclusions 63](#summary-and-conclusions)

[Appendix A: Mathematics Knowledge and Skills for Success From Conley
and Bodone (2002)
65](#appendix-a-mathematics-knowledge-and-skills-for-success-from-conley-and-bodone-2002)

[Appendix B: Summary of Studies Related to Developmental Mathematics
Reviewed in This Report
71](#appendix-b-summary-of-studies-related-to-developmental-mathematics-reviewed-in-this-report)

[Appendix C: Comparable ASSET, ACT, and COMPASS Cutoff Scores For
Student Placement into Mathematical Courses
75](#appendix-c-comparable-asset-act-and-compass-cutoff-scores-for-student-placement-into-mathematical-courses)

[Appendix D: Level of Proficiency Associated With ACCUPLACER Cutoff
Scores
77](#appendix-d-level-of-proficiency-associated-with-accuplacer-cutoff-scores)

# Tables

[Table 1: Summary of Required Minimum Mathematics Assessment Test Scores
for\
Selected States 25](#__RefHeading___Toc114303160)

Table B-1: Summary of Studies Related to Developmental Mathematics
Reviewed in

This
Report..............................................................................71

[Table C-1: Asset, Act, and Compass Cutoff Scores for Student Placement
75](#__RefHeading___Toc114303161)

[Table D-1: Level of Proficiency Associated With ACCUPLACER Cutoff
Scores 77](#__RefHeading___Toc114303162)

# Abbreviations

AACC American Association of Community Colleges

AASCU American Association of State Colleges and Universities

AAU Association of American Universities

ABE Adult Basic Education

ACCUPLACER College Placement Exam

ACES Adult Classroom Environment Scale

ACT American College Testing Program

AECF Advanced Electronics/Computer Field

AFQT Armed Forces Qualifications Test

ALEKS Assessment and LEarning in Knowledge Spaces

AMA American Management Association

AMATYC American Mathematical Association of Two-Year Colleges

AP Advanced Placement

AR Arithmetic Reasoning

ARG Association Review Group

AS Auto Shop Information

ASI Adaptive Style Inventory

ASSET Assessment of Skills for Successful Entry and Transfer

ASTD American Society for Training and Development

ASVAB Armed Services Vocational Aptitude Battery

BLS Bureau of Labor Statistics

BSEP Basic Skills Education Program

CAI Computer-Assisted Instruction

CAS Computer Algebra Systems

CBE Computer-Based Education

CBI Computer-Based Instruction

CEI Computer-Enriched Instruction

CMI Computer-Managed Instruction

CNA Center for Naval Analyses

CNAC CNA Corporation

COMAP Consortium for Mathematics and its Applications

COMPASS Computerized Adaptive Placement Assessment and

Support System

COTS Commercial off the Shelf

CPT Computerized Placement Test

CRAFTY Curriculum Renewal Across the First Two Years

DOD Department of Defense

DTIC Defense Technical Institute Center

EI Electronics Information

EIT Employee-in-training

ESL English as a Second Language

ETS Educational Testing Service

FY Fiscal Year

GED General Equivalency Diploma

GS General Science

GT General Technical

HSDG High School Degree Graduate

IEP Individualized Education Plans

JDCC Jefferson Davis Community College

KET Kentucky Educational Television

KSAs Knowledge, Skills and Abilities

MAA Mathematical Association of America

MASP Military Academic Skills Program

MC Mechanical Comprehension

MK Math Knowledge

NAEP National Assessment of Education Progress

NALS National Adult Literacy Survey

NCES National Center for Education Statistics

NCLC Navy College Learning Centers

NCLP Navy College Learning Program

NCPACE Navy College Program for Afloat College Education

NCRVE National Center for Research in Vocational Education

NCTM National Council of Teachers of Mathematics

NHSDG Non-High School Diploma Graduate

OJT On-the-job training

OVAE Office of Vocational and Adult Education

PALS Principles of Adult Learning Scale

PC Paragraph Comprehension

RCT Randomized Controlled Trial

ROI Return on Investment

SAT Scholastic Assessment Test

SOC Servicemembers Opportunity Colleges

TA Tuition Assistance

TABE Test of Adult Basic Education

TiPS Tutorials in Problem Solving

UAW United Auto Workers

UVSC Utah Valley State College

VE Verbal Expression

VolEd Voluntary Education

VTC Video Teleconferencing

WASL Washington State Assessment of Student Learning

WIA Workforce Investment Act

WIB Workforce Investment Board

WK World Knowledge

#  {#section-1 .unnumbered}

# Executive Summary

## Background

The nature of America's workforce has changed dramatically in the past
several decades, due in large part to the infusion of rapidly changing
technology. This trend has resulted in an increased need for workers
with greater mathematical skills and higher education.

The U.S. Department of Education Office of Vocational and Adult
Education (OVAE) contracted with The CNA Corporation (CNAC) and its
partners to identify promising strategies within community colleges,
businesses, organized labor, and the military that enable adult learners
to strengthen their math skills and abilities and to transition into
higher-level math courses or work assignments requiring higher-level
mathematics.

This literature review is the first step in this process. In order to
establish a baseline understanding of postsecondary developmental
mathematics programs we examine the following three issues:

> 1\. What is the definition, or skill threshold, of adequate student
> preparation in mathematics at the postsecondary level?
>
> 2\. What institutions provide developmental math education, and how
> does the education provided differ across these institutions?
>
> 3\. What approaches and strategies appear to hold promise for enabling
> adult learners to strengthen their mathematical skills and to progress
> into college-level math courses or work assignments requiring
> higher-level mathematical abilities?

Strategies identified in this review will provide the basis for the
second phase of this project, the purpose of which is to identify math
programs in community colleges, business, labor organizations, and the
military that have supporting evidence that such strategies are, indeed,
successful.

## Major Findings

### Institutions providing developmental math instruction

Community colleges are the largest source of developmental math
instruction, and virtually all public two-year colleges offer at least
one developmental course. However, these colleges vary in the number of
developmental courses they offer, how many of these courses each student
may take, and the type of credit awarded. In addition, 56 percent offer
developmental education to local businesses, and basic math courses are
offered by 93 percent of the colleges that extend courses to businesses
(NCES 2003).

The military services offer basic skills instruction to members who
qualify on the basis of low entrance test scores or because they do not
possess a General Equivalency Diploma (GED). Their remediation efforts
vary across the services in both the length and method of instruction,
yet more than 37,000 service members receive basic skills instruction
each year (U.S. Department of Defense 2004).

Adult education and workforce development programs also provide basic
skills remediation. There is growing interest in developing
opportunities for people who lack skills and resources around career
pathways that integrate education, training, and skill development in
targeted high-wage, high-demand employment areas. Career pathways
provide developmental, adult, or English as a Second Language (ESL)
classes in the context of students' lives and the work-specific skills
they need for employment in particular industries or sectors.

Central to federal government efforts to strengthen the skills of
displaced or dislocated workers is Title I of the *Workforce Investment
Act of 1998*. This program supports a network of One-Stop Career Centers
that provide access to a full range of services pertaining to
employment, training and education, employer assistance, and guidance
for other types of assistance.

Businesses also are involved in the remediation of basic skills.
Companies spend an average of 1.8 percent of payroll on training. Of
this amount, 5 to 7 percent is in basic skills, including literacy,
reading, comprehension, writing, math, ESL, and learning how to learn.
By far, the largest category of training is in technical processes and
procedures, which totals approximately 13 percent of all training
expenditures (Van Buren 2001). The most often cited sources of external
education and training used by business are community colleges,
technical and vocational schools, business and industry associations,
consultants, and universities.

### What constitutes adequate math preparation?

We found that the knowledge necessary for successfully pursuing
college-level math programs depends on the education and career goals of
the individual. For instance, adult learners in community colleges would
require somewhat different knowledge if the first college-level course
were calculus rather than business math. Regardless of whether this is a
contributing factor, we have found that no consistent definition of math
standards for college-level preparation exists. However, a number of
studies indicate the need to have a good foundation in arithmetic,
geometry, trigonometry, and algebra I and II. Emerging work also
indicates the increasing need for basic statistics and the ability to
analyze data.

There is less uncertainty or ambiguity in the skills necessary to pursue
college-level math and to succeed in the highest-paid and
highest-skilled jobs. In particular, there seems to be agreement on the
need to think critically, to solve problems, and to communicate
mathematically. Both businesses and postsecondary institutions that were
surveyed as part of large curriculum reform efforts indicate that they
want people who can identify a problem, determine whether it can be
solved, know which operations and procedures are required to solve it,
use multiple representations (such as graphs and words) to describe
problems and solutions, and understand and apply mathematical modeling.
However, these are the skills that are the most difficult to teach and
to assess (American Diploma Project 2004).

Whether community colleges are universally adopting these
recommendations---in terms of the specific knowledge, skills, and
abilities---remains to be seen. It is also uncertain whether community
colleges adequately assess the knowledge and skills necessary to pursue
postsecondary level math or succeed in the workplace. Regardless, the
majority of two-year colleges require incoming students to take and pass
an assessment test before they are allowed to enroll in college-level
math courses. Given their prevalence, this may be the most relevant
benchmark for whether a student can successfully transition to
college-level mathematics. While minimum scores vary, we note a range of
score thresholds for the most common of these tests.

### Best instructional practices

Our extensive search of the developmental education literature yielded
only a limited number of studies pertaining to adult developmental
mathematics instruction, the majority of which has been conducted in
two-year colleges. Of these, we reviewed 15 studies of postsecondary
institutions, with a majority based on programs in community colleges.
Unfortunately, none were based on randomized controlled trial
experiments, which have been elevated to a position of the "gold
standard" for research because this experimental design is relatively
unbiased in evaluating the effect of programmatic interventions in the
field of education. To augment research on developmental mathematics
programs for adult learners, particularly aspects of developmental math
courses, we relied on a broader base of research to provide guidance as
to what may hold promise for developmental mathematics specifically.
However, we were not able to locate published research on developmental
mathematics programs outside of academic institutions. In addition to
scholarly sources, we searched Web sites of businesses and labor
organizations. For example, the Web site of the AFL-CIO, with a
membership of over 13 million, has a section concerning education issues
and legislation, but it contains no information about specific education
programs in general, or developmental mathematics in particular.

Although we did not identify existing studies containing scientifically
based evidence of promising practices, salient themes concerning
pedagogy emerged, suggesting promising but unproven instructional
practices that are frequently implemented. These may warrant further
study. Among the recommendations in the literature are: greater use of
technology; integration of classroom and laboratory instruction; giving
students the option to select from among different instructional
methods; use of multiple approaches to problem solving; project-based
instruction; low student to faculty ratios; assessment and placement of
students into the appropriate mathematics courses; and integration of
counseling, staff training, and professional development.

Underscoring these recommendations, our review found that a number of
studies sought to evaluate the impact of various teaching delivery
methods on student success, including traditional lecture,
computer-assisted courses, self-paced instruction, Internet-based
courses, and accelerated programs. No clear consensus of the
effectiveness of technology-based delivery methods emerged. Using
various metrics, some studies found no effects, some found higher levels
of success, and some found lower levels of success for students using
technology-based or technology-enhanced instruction versus traditional
lecture. However, a number of research projects, such as those from the
American Mathematical Association of Two-Year Colleges (AMATYC) (1995
and 2002) and the American Diploma Project (2004), conclude that all
students should be familiar with technology, including graphing
calculators, and spreadsheets, and should be able to understand the
benefits and limitations of each. Further, there is general consensus
that technology should be a supplement to, as opposed to a replacement
of, more traditional delivery methods. However, given the inconsistency
in study findings, we believe that this is one area that warrants
further investigation.

Finally, we have found research that indicates that the types of
problems used in teaching the material is important. In particular, it
is important to use activities that engage students in the learning
process, particularly in small collaborative group processes, most of
which reflect the real-world problem solving done in businesses. These
activities should require the student to actively plan, design,
research, model, and report findings for projects or case studies. Some
argue that students require contextual learning and real-world problems
to help make coursework and training relevant and meaningful.

We summarize key components of best practice approaches to postsecondary
developmental mathematics programs below:

-   **Instructional and pedagogical**: adjuncts to traditional
    instruction; multiple delivery options from which students may
    choose; computer-assisted instruction; Internet-based; self-paced;
    distance learning; calculators; computer algebra systems;
    spreadsheets; laboratories; small group instruction; learning
    communities; contextual learning; linkages to and examples from the
    workplace; and career pathways.

```{=html}
<!-- -->
```
-   **Curriculum content**: nonstandard topics covered in developmental
    math courses or topics that vary by career path; length of
    instruction; and types of activities used to reinforce the material.

-   **Professional development**: faculty training and development;
    full-time versus part-time instructors; and proportion of faculty
    that are adjuncts.

-   **Supporting strategies**: counseling, assessment, placement, and
    exit strategies.

-   **Learner and institutional characteristics:** full-time versus
    half-time community college student; socioeconomic attributes of
    learner; workplace versus academic learner; and having private or
    military employment versus preparing for a new career.

# Introduction

The nature of America's workforce has changed dramatically in the past
several decades, due in large part to the infusion of rapidly changing
technology. This trend has resulted in an increased need for workers
with greater skills and higher education. For instance, the Bureau of
Labor Statistics (BLS) (BLS 2001) predicts that jobs requiring at least
a bachelor's degree will grow 21.6 percent between 2000 and 2010, and
those requiring an associate degree or vocational certificate will
increase 24.1 percent. In contrast, jobs requiring only work-related
experience will increase just 12.4 percent during the same time period.

In spite of these trends, employers are finding that their workforce is
simply not prepared to meet even the most basic skill requirements,
including reading, writing, and mathematics. The National Association of
Manufacturers found in a 2001 survey sent to its members that 80 percent
of manufacturers experience a moderate to serious shortage of qualified
job candidates, and that 26 percent of employers listed inadequate math
skills among the most serious skill deficiencies (National Association
of Manufacturers Center for Work Success 2001). Further, 20 percent said
they rejected applicants for hourly production positions due to
inadequate math skills. A survey conducted in 2001 by the American
Management Association (AMA), based on responses from 1,627 human
resource managers in AMA member and client companies, found that 41
percent test job applicants in basic literacy and or math skills; of
those tested, 34 percent lacked sufficient skills for the positions they
sought (AMA 2001). Less than 9 percent of the respondents said that they
hired those found to be deficient. If interested in hiring, respondents
either assigned applicants to obligatory developmental training or
offered voluntary developmental training, and this was true across
business sectors. Manufacturing offered remediation to the most, 8.6
percent, while wholesale and retail offered remediation to the least,
just 2.8 percent. Instead, the overwhelming majority of companies simply
refuse to hire those who do not pass the basic skills requirements---the
fate of over 80 percent of these applicants in all business sectors.

This growth in deficient skills is in large part a function of the
rapidly increasing skill requirements and the changes in these
requirements over the past few years. In other words, it is not just a
matter of employers requiring more workers to know math; the type of
math required is also changing.

The most recent statistics on adult literacy[^1] confirm the deficit in
skilled workers. The 1992 National Adult Literacy Survey (NALS),
conducted by the National Center for Education Statistics (NCES),
assessed the prose, document, and quantitative literacy proficiency of
adults (Kaestle et al. 2001). The study defined the following five
hierarchical levels of quantitative proficiency (Kaestle et al. 2001, p.
11):

-   Level 1: Tasks require performing single, relatively simple
    arithmetic operations, such as addition. The numbers to be used are
    provided and the arithmetic operation to be used is specified.

-   Level 2: Tasks typically require performing a single operation using
    numbers that are either stated in the task or easily located in the
    material. The operation to be performed may be stated or easily
    determined from the format of the material (for example, an order
    form).

-   Level 3: Two or more numbers are typically needed to solve the
    problem, and these must be found in the material. The operation(s)
    needed can be determined from the arithmetic relation terms used in
    the question or directive.

-   Level 4: The fourth level requires performing two or more sequential
    operations or a single operation in which the quantities are found
    in different types of displays, or the operations must be inferred
    from semantic information given or drawn from prior knowledge.

-   Level 5: Level 5 requires performing multiple operations
    sequentially. The features of the problem must be disembedded from
    text or based on background knowledge to determine the quantities or
    operations needed.

Their results indicated that 22 percent of adults demonstrated skills in
the lowest level of quantitative literacy proficiency, and an additional
25 percent demonstrated skills at the second lowest level. In other
words, almost half of all adults could not perform tasks at the level
necessary as defined by the third level of quantitative proficiency,
such as using a calculator to calculate the difference between the
regular and sale price of an item in an advertisement (Kaestle et al.
2001, p. 208).

Kaestle and his colleagues (2001) concluded that there is a strong
relationship between the level of literacy and education attainment. For
instance, the quantitative proficiency of 79 percent of adults who were
high school dropouts and completed nine to 12 years of school was below
the third level, compared with 51 percent of those who completed a high
school degree. They also found that those high school dropouts who went
on to earn a GED fared as well as high school diploma graduates, with 54
percent scoring below the third level (Kaestle et al. 2001, p. 17).

Low levels of literacy have fairly serious negative economic
consequences. The NALS study found employed workers who scored in the
lowest two levels of literacy tended to be employed in the lowest wage
occupations, such as food service, childcare, and maintenance (Kaestle
et al. 2001, p. xxxviii). Other studies have also found that high school
dropouts experience higher rates of unemployment and are more likely to
receive public assistance than high school diploma graduates who did not
go on to college (National Center for Family Literacy 2003).

The fact that half of all high school graduates do not possess
quantitative literacy skills at least at the third level, as defined by
the NALS study, is a clear indication that a high school diploma is not
enough to meet the increasing need for highly skilled workers. In part
as a response to the needs of employers and the higher wages that
high-tech jobs offer, the rate of college enrollment of graduating high
school seniors has increased significantly since the last half of the
20th century, from 45 percent in 1960 to 62 percent in 2001 (NCES 2002,
table 184). Even so, studies find that a large proportion of those who
enroll in college are not prepared to pursue college-level courses. A
recent study concluded that more than one million students entering
postsecondary education each year require participation in developmental
courses, representing 42 percent of the student population (McCabe
2000). This same research concluded that successfully remediated
students do perform well in standard college-level courses, noting that
82 percent of a nonrandom sample of remediated students included in the
study passed college-level mathematics classes. This is a striking
finding considering that many developmental courses are described by
students as dull and poorly taught, and emphasize low-level drill and
practice (Grubb 1999).

Recently, states have established higher standards for high school
graduation, have increased admission requirements at colleges and
universities, have structured open admissions programs at community
colleges, and have used testing and evaluation to assess education
outcomes (Bandy 1985; Fonte 1997; Merisotis and Phipps 2000; Thacker
2000). According to the U.S. Department of Education, however, only four
states[^2] required students to have four Carnegie Units (each unit is
roughly equivalent to one academic year of study) in mathematics for
high school graduation in 2001, seventeen of the remaining states only
required two units, and the rest required three (NCES 2002, table 152).

Clearly, some students can take more math than what is mandated by state
law for high school graduation, and some states recommend more math for
college-bound students. According to Barth (2002), the percentage of
students completing algebra II in high school, the minimum content
typically required to enroll in college-level mathematics, has grown
from 40 percent to 62 percent between 1982 and 1998.

Yet, according to the U.S. Department of Education, the average scores
on the National Assessment of Education Progress (NAEP) of 17-year-olds
whose highest level of mathematics is algebra II are at a level that
enables them to perform reasoning and problem solving involving
fractions, decimals, percents, elementary geometry, and simple algebra
(NCES 2002, table 125). But the problem does not begin in high school.
The most recent news of the math competency of the nation's
schoolchildren shows that scores on the NAEP are up in mathematics, but
a fairly large number still do not meet the proficiency standards set by
the National Assessment Governing Board (Plisko 2003). Among
fourth-graders, 77 percent are at or above a basic level of proficiency,
up from 50 percent in 1990. For eighth-graders, 68 percent are at a
basic level of proficiency or higher, up from 52 percent in 1990. But as
impressive as these gains are, they still show that almost one-third of
eighth-graders are not at a basic level of math proficiency.

Adult learners who are not recent high school graduates who seek to
improve their basic skills literacy, earn a GED, or pursue postsecondary
education face more difficulties in obtaining higher level math skills
than recent graduates do. In particular, they often face more financial
(often as sole household earner) and logistical (such as daycare and
time off from work) challenges. And in many cases, they have a history
of education failure and of long-term functioning at low levels of
quantitative literacy.

The *Adult Education and Family Literacy Act of 1998*, Title II of the
*Workforce Investment Act of 1998*, authorizes a program of national
leadership activities to enhance the quality of adult education and
literacy programs nationwide, including collecting data and
disseminating best practices information. The U.S. Department of
Education Office of Vocational and Adult Education (OVAE) is sponsoring
a number of studies that address the growing need for adult education in
general and basic literacy in particular.

OVAE has sponsored this project to provide better information on current
and new strategies under development in the field, and to use that
information as guidance for future research efforts into promising
practices in developmental mathematics for adult learners. One specific
goal is to inform and enhance adult basic education programs to ensure
that participants have the math knowledge and skills necessary to pursue
college-level mathematics when they transition from Adult Basic
Education (ABE) to postsecondary education or to workforce programs that
require higher-level math.

The goals of this literature review---the first phase of this
research---are to: (a) define the mathematical knowledge and skills
necessary to pursue college-level mathematics and (b) identify the
elements of developmental mathematics programs within community
colleges, the military, businesses, and organized labor that enable
adult learners to transition from developmental to college-level
mathematics. To that end, we address the following three issues:

> 1\. What is the definition and skill threshold of adequate student
> preparation in mathematics at the postsecondary level?
>
> 2\. What institutions provide developmental math education, and how
> does the education provided differ across these institutions?
>
> 3\. What approaches and strategies appear to hold promise in enabling
> adult learners to strengthen their mathematical skills and to progress
> into college-level math courses or work assignments requiring
> higher-level mathematical abilities?

We begin our literature review with a discussion of the knowledge and
skills necessary to pursue college-level mathematics, and of common
tests and their cutoff scores used to assess the ability of students to
pursue college-level mathematics. Next we discuss a review of the
literature concerning components of successful developmental mathematics
programs in postsecondary settings. Because of the paucity of research
on many aspects of developmental mathematics instruction for adult
learners, we also review literature of more common themes within
developmental education.

We then review some common practices in colleges in terms of these
components, to determine the progress of postsecondary institutions in
addressing the recommendations from the literature. Finally, we examine
what other organizations are doing in developing the mathematics skills
of their workforce. In particular, we look at the military, businesses,
labor organizations, and other adult education efforts.

#   {#section-2 .unnumbered}

# What Skills and Knowledge Do Students Need to Pursue College-level Mathematics?

The primary goal in this literature review is to define the skill
threshold necessary to pursue college-level mathematics and to discover
promising strategies for helping students to progress to this threshold.
We have discovered two salient themes in the literature concerning what
this means precisely. The first is the knowledge, or content, required.
This includes a detailed description of the specific math facts or
subjects to be covered, such as ratios, decimals, or, more broadly,
arithmetic. The second theme concerns the skills and abilities necessary
to pursue college-level math. By skills we refer to observable
competencies to perform a function. For instance, critical thinking,
generating ideas, and determining which tool is necessary to do a job
are considered skills. Abilities are attributes that affect the ability
to perform a task, such as manual dexterity and inductive and deductive
reasoning.

Based on or review, we conclude that there is less uncertainty or
ambiguity in these necessary skills and abilities than there is in the
required knowledge, or content. In fact, community colleges and
businesses are in general agreement concerning these skills and
abilities. However, skills and abilities are often more difficult than
knowledge to teach and assess. In particular, there seems to be
widespread consensus as to the need to think critically, to solve
problems, and to communicate mathematically. Several studies provide
more precise definitions of these skills, which we summarize below.

## Crossroads

*Crossroads in Mathematics: Standards for Introductory College
Mathematics Before Calculus* (hereafter referred to simply as
Crossroads), published in 1995 by the American Mathematical Association
of Two Year Colleges, established goals and standards for preparation
for college-level mathematics that are the most oft-cited of any study
of developmental mathematics at the postsecondary level (American
Mathematical Association of Two Year Colleges 1995). AMATYC developed on
six guiding principles upon which it based its standards:

1.  All students should grow in their knowledge of mathematics while
    attending college.

2.  Students should study mathematics that is meaningful and relevant.

3.  Mathematics must be taught as a laboratory discipline.

4.  The use of technology is an essential part of an up-to-date
    curriculum.

5.  Acquiring mathematics knowledge requires balancing content and
    instructional strategies recommended in the AMATYC standards along
    with the viable components of traditional instruction.

6.  Increased participation in mathematics and in careers using
    mathematics is a critical goal in our heterogeneous society.

The standards are divided into three categories: intellectual
development, content, and pedagogy. Because Crossroads is the seminal
work in this area, we summarize its standards, which provide goals for
introductory college mathematics and guidelines for selecting content
and instructional strategies for accomplishing the principles.

**Intellectual development standards**

Students will:

-   Engage in substantial mathematical problem solving;

-   Learn mathematics through modeling real-world situations;

-   Expand their mathematical reasoning skills as they develop
    convincing mathematical arguments;

-   Develop the view that mathematics is a growing discipline,
    interrelated with human culture, and understand its connections to
    other disciplines;

-   Acquire the ability to read, write, listen to, and speak on
    mathematics subjects;

-   Use appropriate technology to enhance their mathematical thinking
    and understanding and to solve mathematical problems and judge the
    reasonableness of their results; and

-   Engage in rich experiences that encourage independent, nontrivial
    exploration in mathematics, develop and reinforce tenacity and
    confidence in their abilities to use mathematics, and inspire them
    to pursue the study of mathematics and related disciplines.

**Content standards**

Students will:

-   Perform arithmetic operations and will reason and draw conclusions
    from numerical information;

-   Translate problem situations into their symbolic representations and
    use those representations to solve problems;

-   Develop a spatial and measurement sense;

-   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;

-   Use discrete mathematical algorithms and develop combinatorial
    abilities in order to solve problems of finite character and
    enumerate sets without direct counting;

-   Analyze data and use probability and statistical models to make
    inferences about real-world situations; and

-   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.

**Pedagogy standards**

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;

-   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;

-   Actively involve students in meaningful mathematics problems that
    build on their experiences, focus on broad mathematical themes, and
    build connections within branches of mathematics and other
    disciplines so that students will view mathematics as a connected
    whole relevant to their lives;

-   Model the use of multiple approaches---numerical, graphical,
    symbolic, and verbal---to help students learn a variety of
    techniques for solving problems; and

-   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.

The AMATYC is revising its Crossroads curriculum standards to "create a
product that communicates a renewed vision and guidelines" (American
Mathematical Association of Two Year Colleges 2002a). To that end, the
AMATYC conducted two activities to assess the impact of the original
Crossroads standards. It sent a survey to 150 AMATYC members and 250
potential members; respondents numbered 42 and 13, respectively. In
addition, an Association Review Group (ARG) was established consisting
of 63 AMATYC affiliates, academic committee chairs, and members. While
the survey response rate is low, we note the findings because they may
be indicative of a larger trend. In particular, the survey responses
suggest that respondents made the following changes to mathematics
curricula in response to the original Crossroads recommendations:

-   Greater use of technology;

-   More emphasis on contextual experiences, problem solving, or
    modeling;

-   More collaborative work in the classroom; and

-   Increased awareness of different teaching and learning styles.

Many respondents attributed these curricula changes to Crossroads. Those
who did not said that either Crossroads reaffirmed such principles or
the National Council of Teachers of Mathematics (NCTM) Standards had a
greater effect on causing these changes. NCTM provides comprehensive
guidelines covering curricula, professional teaching standards, and
assessment standards targeted toward K--12 mathematics curricula
(National Council of Teachers of Mathematics 1989, 1991, 1995, 2000).

Further, survey respondents noted that the most significant barriers to
implementing the Crossroads recommendations were time, overcoming
faculty resistance to change, money, scarcity of texts and materials,
and lack of convenient and affordable professional development
opportunities. Finally, the top six issues that faculty members believe
should be addressed in their current reform efforts are instructional
delivery, technology, pedagogy, content, adjunct faculty, and training
new and retaining current faculty.

## Standards for Success

Conley and Bodone (2002) provide one of the more comprehensive studies
of mathematical content necessary for success for entry-level college
students. This study reports findings from a collaborative group of 400
representatives from numerous universities that generated the so-called
Standards for Success, in which they formulated key knowledge and skills
necessary for university success in entry-level courses compared with
just high school preparation. The Pew Charitable Trusts and the
Association of American Universities (AAU) sponsored the work of Conley
and Bodone, and the authors note that it is the first and only
comprehensive statement of university entrance-level skills that is
presented in terms of standards rather than simply in terms of course
names or broad content statements.

The authors point out several important findings emanating from this
work. For instance, academic content standards in the K--12 system are
not set in consultation with higher education personnel, and no state's
standards correlate with college admission criteria. These criteria are
expressed in terms of class rank, GPA, and required courses---but not in
competencies. The authors suggest that it is important to align the
K--12 standards with academic expectations so that there are not two
distinct educational systems (K--12 versus postsecondary) with vastly
different knowledge and skills expectations and outcomes. However, they
also note that, while the basic content knowledge standards proposed by
this group do align well with individual states' standards for high
school, a real divide exists in the types of intellectual development
that should accompany the mastery of content knowledge.

Because the Standards for Success included the knowledge and skills
necessary, which goes beyond the Crossroads standards, it is important
to include the full set of standards in this report. However, because
they are so detailed and lengthy, we provide the full list in Appendix
A. In summary, the standards established by this group indicate that
before pursuing college-level mathematics, students should have the
following knowledge: basic arithmetic, including fractions and
exponents; basic algebra, including manipulation of polynomials and
solutions for systems of linear equations and inequalities; basic
trigonometric principles; basic pictorial and coordinate geometry,
including the relationship between geometry and algebra; and statistics
and data analysis.

Further, the standards stipulate abilities similar to those in
Crossroads, under mathematical reasoning. For instance, they state that
students should have the ability to: (a) use inductive reasoning and a
variety of strategies to solve problems; (b) use a framework or
mathematical logic to solve problems that combine several steps; and (c)
determine mathematical concept from the context of a real-world problem,
solve the problem, and interpret the solution in the context of a
real-world problem.

Related to this literature, we note some emerging research below on math
content and skill requirements of students in two-year colleges. This
work has important implications. It may mean a shift in the knowledge
and skills that are necessary to pursue math at two-year colleges. It
also reinforces some common themes in the two studies cited, as well as
other literature that we reviewed and that we discuss later.

## The Vision Report

First, we note the work that is being conducted by AMATYC under the
National Science Foundation grant, "Technical Mathematics for Tomorrow:
Recommendations and Exemplary Programs." At a recent national
conference, over 80 educators, technical personnel from business and
industry, and technical faculty from two-year colleges identified what
they defined as exemplary practices in mathematics programs that serve
highly technical curricula, such as biotechnology, computerized
manufacturing, electronics, information technology, semiconductors, and
telecommunications. Their work, summarized in *A Vision: Final Report
from the National Conference on Technical Mathematics for Tomorrow*
(AMATYC 2002b) and hereafter referred to as the Vision Report, built on
that conducted by those of the Mathematical Association of America's
(MAA) subcommittee on Curriculum Renewal Across the First Two Years
(CRAFTY), who also participated in this conference.

The recommendations that emerged from this conference cover several
topics. Underlying the discussion of content is the necessity for
students to possess certain abilities that the Crossroads and Standards
for Success research highlighted, as well as those emphasized by other
studies we discuss later. The recommendations are uniform, regardless of
the learner's age, level of mathematics, or organization noting the
requirement, and are equally important as, or perhaps even more
important than, the knowledge requirements themselves. These are
critical thinking skills, the ability to communicate mathematics, and
the ability to select an appropriate method to solve a problem---from
fairly basic word problems to those that are much more complex and may
require research, development of a new process, data collection, or use
of technology to organize data.

The Vision Report's discussion of content includes arithmetic, algebra,
geometry, trigonometry, calculus, and statistics. While these topics and
their subcategories refer to college-level mathematics, a level beyond
what concerns us here, they indicate the types of courses in which
students should have some background before pursuing college-level math.
It is worth highlighting that both the Vision Report's recommendations
and those contained in the Standards for Success (Conley and Bodone
(2002)) include some knowledge and skills in statistics, particularly
the knowledge of analytic tools and the ability to analyze, interpret,
and display real data. They also note the need to be able to use
spreadsheets, graphing calculators, and Computer Algebra Systems. These
studies suggest that, in preparing adult learners for college-level
mathematics, it would be useful to introduce them simultaneously to the
technology and the fundamental concepts of statistics.

## American Diploma Project

The last study on which we report was conducted by the American Diploma
Project (2004), a joint venture of Achieve, Inc., the Education Trust,
and the Thomas Fordham Foundation, and supported in part by a grant from
the William and Flora Hewlett Foundation. The National Alliance of
Business also was an original partner of the project. We refer to this
publication by project name rather than by title.[^3]

This is a particularly relevant study for our work for a number of
reasons. First, this research confirms the findings of the studies we
have just cited concerning the knowledge, skills, and abilities that are
necessary for high school graduates, and therefore adult learners, who
want to pursue postsecondary education. It goes further by providing
precise examples of the types of problems that students should be able
to solve. In addition, these recommendations are not just based on
feedback from postsecondary institutions; they have been developed in
partnership with frontline managers in occupations with the highest
projected pay and skill requirements for the next decade.

The goal of this work was to realign high school diploma requirements
with the expectations of employers and postsecondary institutions to, in
their terms, reestablish the value of a high school diploma. The
starting point was to describe the English and math skills that high
school graduates need to succeed in postsecondary education or in
high-performance, high-growth jobs. The work of the American Diploma
Project was based on a close collaboration with K--12, postsecondary,
and business leaders in such occupations as healthcare, information
technology, telecommunications, high-tech manufacturing, semiconductor
technology, law, energy, retail, and financial services.

The math content necessary reflects what is typically taught in algebra
I and II, geometry, and data analysis, the latter echoing the
recommendations for statistics and data analysis skills noted
previously. American Diploma Project partners also include analytic and
reasoning skills that they suggest have been associated traditionally
with honors or Advanced Placement (AP) courses, but that they now assert
are considered to be essential skills by colleges and employers.

The American Diploma Project found a surprising amount of consistency in
the skills and content standards established by business and
postsecondary institutions involved in this work, both within and across
states. It concludes that this work confirms the notion that
postsecondary and workplace expectations are converging.

Once again, the researchers note that it is not just specific knowledge
that is important but the ability to think critically. They emphasize
skills to develop and analyze an argument, to define and research a
problem, to present a well-reasoned solution to the problem, and to
apply basic knowledge and skills in new and unfamiliar contexts.

However, unlike many of the standards we have reviewed, the American
Diploma Project differentiates the knowledge and skills necessary by
whether the person intends to major in math or in math-dependent fields.
This is an important distinction in terms of what defines adequate
preparation to pursue college-level mathematics. Certainly, the answer
to this question would depend on whether the person plans to pursue a
career in, for example, electronics engineering versus law.

The project's research includes specific benchmarks and actual workplace
tasks and postsecondary assignments that illustrate each of these
benchmarks. Examples of these tasks can be found on its Web site at:
www.achieve.org.

## Summary

The four studies that we reviewed---Crossroads, Standards for Success,
the Vision Report, and the American Diploma Project---together provide
consistent and comprehensive guidance on specific topics and the skill
threshold necessary to pursue college-level mathematics. The consensus
is that students should have a basic foundation in geometry,
trigonometry, algebra I and II, and some basic statistics. All four
studies emphasize the importance of mathematical skills, particularly
critical thinking skills. This research also indicates the necessity of
tailoring the preparation to the types of college math and career path
that the person intends to pursue.

While the conclusions and recommendations of these studies are based on
careful research and collaboration with postsecondary institutions or
businesses or both, they have yet to be adopted universally by two-year
colleges. Even where they are in use, assessment of incoming students'
skills may not reflect this new approach to content or abilities. As we
noted previously, it is much easier to assess in a paper-and-pencil test
whether a person has command over certain basic skills that require
memorization than it is to determine whether a person has the critical
thinking skills to perform higher-level math that postsecondary
education and the workplace require.

An examination of the adequacy of assessment tests in determining
whether incoming students possess adequate skills and abilities required
for these emerging trends is beyond the scope of this study. However, we
turn to a discussion of assessment tests because, more than anything
else, they are currently the most common requirement of students who
wish to pursue college-level mathematics at community colleges.

#   {#section-3 .unnumbered}

# Assessment and Placement Policies

The use and misuse of placement tests is central to this review for two
reasons. First, according to a recent survey by the American Association
of Community Colleges, 58 percent of its 400 respondents required adult
learners transitioning from an ABE program to pass an assessment of
basic skills in order to enroll in a college-level math course (Schults
2001). Studies that we reviewed suggest that mandatory student
assessment and placement tests have a positive impact on student
performance. Young (2002) argues that requiring mandatory placement
tests is a good policy because numerous studies have shown that students
who take mandatory placement and assessment tests and subsequently
enroll in developmental courses perform better in college-level courses
than similar students who do not take developmental courses. And
according to Boylan and Saxon (2002), fewer than 10 percent of students
who require remediation will be successful in college without getting
it. Their work, based on a synthesis of over 200 studies on
developmental education, finds that only the most motivated students
will enroll when assessment and placement into developmental courses is
voluntary. They conclude that placement and assessment should be made
mandatory. However, we note that, if unmotivated students are not
seeking remediation, making remediation mandatory will not necessarily
increase their motivation level or their course performance.

The other reason for focusing on placement tests is that an
understanding of test content and cutoff scores to bypass developmental
mathematics may help to create curriculum guidelines for enhancing ABE
programs. In particular, they may provide useful information not only
about what students should know but what level of comprehension is
required. Using other tests, such as standardized tests or high school
exit exams, may not be sufficient in many cases. Several studies we
reviewed argued that there is a disconnect between scores on
standardized tests used by postsecondary institutions and scores on exit
exams required for high school graduation, or on other measures of
mathematics knowledge acquired in high school.

For instance, a study conducted in the mid-1990s looked at how high
school preparation affected placement rates in developmental courses at
Utah Valley State College (UVSC) (Hoyt and Sorensen 2001). In that
study, the researchers surveyed high school transcripts from five high
schools in two districts for 1995 through 1997 to determine the
relationship between high school preparation and college placement test
scores. Of those students who took algebra II and geometry, the nominal
prerequisites for college algebra, the average score on the American
College Test (ACT)[^4] math component was 20 in one district and 19 in
another. During the time frame under study, UVSC required students to
score 24 or higher on the ACT math component to be eligible to enroll in
college algebra. In fact, over half of all students in these districts
who had completed that level of high school math were subsequently
placed into developmental math at UVSC. Thus, taking the presumed
prerequisites for college algebra is no guarantee that the student will
acquire the level of competency to take college algebra.

Hoyt and Sorensen (2001) found that their results were consistent with
those obtained from ACT for nationwide trends. The publisher of ACT
(ACT, Inc.) reported that, for those in the class of 1998 who took the
ACT, totaling nearly 1 million students, the average math ACT score was
18 for those who had completed algebra II---still below the standard
used by most colleges for college-level mathematics.[^5]

Similarly, in a study conducted by the State of Washington, scores on
the 10th- grade Washington State Assessment of Student Learning (WASL)
test were compared with scores on college placement tests for students
taking both tests in the same spring term (Pavelchek, Stern, and Olsen
2002). The authors found that, while there was significant correlation
in the content of the tests, the college placement tests tended to
include some higher-level material than that in the WASL. Even so, WASL
scores and college placement scores were moderately correlated. Further,
while a score between 400 and 424 on the WASL is sufficient to "meet
standards," they found that only 33 percent of students who scored 400
on the WASL had placement scores high enough to place them in
college-level math. Further, students who scored 442 on the WASL,
considered to be exceeding standards, had only a 75-percent chance of
being placed in a college-level math course based on their standardized
placement test scores.

We turn now to a discussion of the tests most commonly used by community
colleges for the purpose of placing students in mathematics courses.

## Tests Commonly Used[^6]

The most well-known tests are the two standard college entrance exams
typically taken by students seeking admission to a four-year college:
the ACT and the Scholastic Assessment Test (SAT). Each test assesses
students' verbal and mathematical reasoning skills, and virtually all
four-year colleges require results from at least one of these tests for
admission.

Community colleges do not require the ACT or the SAT for entrance to the
institution. Instead, admission typically is based on high school GPA
or, even less stringent, admission is open to all those with either a
high school degree or GED. In the case of those who are not seeking a
degree but want to take vocational courses or simply want to take a
limited number of courses, admission is typically open to everyone.[^7]
While two-year colleges do not require the ACT or the SAT, some will
accept scores from either one of these tests in lieu of institutionally
required tests to determine whether the student has the necessary basic
skills in mathematics. In some cases, however, the student may still be
required to take the basic skills assessment test because the tests are
required to place the student in the correct college-level mathematics
course.

Three tests that are more widely used and have been developed
specifically to assess basic skills are the Assessment of Skills for
Successful Entry and Transfer (ASSET) and the Computerized Adaptive
Placement Assessment and Support System (COMPASS), both of which are
published by ACT, and ACCUPLACER, published by the College Board. A less
commonly used test is the Test of Adult Basic Education (TABE), produced
by CTB McGraw-Hill. We describe these tests next. We gathered the
information reported in this section from telephone conversations with
corporate representatives, from e-mail exchanges with the same, and from
Web sites. The contact list is included in the bibliography.

### ASSET and COMPASS

The ASSET and COMPASS tests cover much the same material, except that
the COMPASS test is a computerized adaptive test, while ASSET is a paper
and pencil test. The math component of the ASSET test covers numerical
skills, elementary algebra, intermediate algebra, and college algebra,
and COMPASS covers pre-algebra, algebra, college algebra, and
trigonometry.

According to ACT, about 1,000 institutions use COMPASS, 85 percent of
which are two-year colleges and the remaining 15 percent are four-year
colleges. ASSET is used by about 500 institutions, some of which use
both ASSET and COMPASS. And to their knowledge, no two-year college uses
ACT exclusively to assess students' basic skills (Roth 2003).

ACT has compiled data that reflect the most typical cut scores that
institutions report they are using to determine adequate preparation for
various levels of mathematics, which we provide in Appendix C (Roth
2003). The table includes comparable scores for all three tests: ACT,
ASSET and COMPASS. The courses at the top of the list are those that are
usually considered to be developmental mathematics. ACT notes that
colleges generally require a score of at least 23 on the mathematics
component of the ACT to be allowed to enroll in a college algebra class.
For reference, 66 percent of all people who took the ACT in 2003 scored
22 or below on the mathematics component (ACT 2003). However, as we note
later, it appears that several states have established an ACT math score
of 19 or 20 as the minimum score necessary for students to enroll in
college-level mathematics courses. For reference, 44 percent of all who
took the ACT in 2003 scored below 20 on the math component.

### ACCUPLACER

According to the College Board, the publisher of ACCUPLACER, about 800
postsecondary institutions use this test, roughly 40 percent of which
are four-year colleges; the rest are two-year colleges. In addition,
some high schools use this test in conjunction with the local community
college (Murphy 2003).

The College Board provided us with information pertaining to the level
of proficiency in each of the three categories (arithmetic, elementary
algebra, and college-level mathematics) that scores represent. They are
reproduced in Appendix D.

### TABE

The TABE battery encompasses five graduated levels of difficulty across
five different content areas that include reading, language, and applied
mathematics. In addition to its regular forms, the TABE is available in
a special version suitable for use in a work environment and in a
Spanish edition designed to measure the basic skills of Spanish-speaking
adults in their primary language.

The TABE was developed through a review of adult curricula and follow-up
meetings with content specialists to determine common educational goals,
knowledge, and skills emphasized in the curricula. The TABE is available
in both paper-and-pencil and software formats. The software is PC-based
to permit electronic administration and scoring.

## State-specific Policies[^8]

Not all states mandate whether, or which, assessment tests must be
administered to incoming students. We searched the Internet and
contacted several officials in states that had policies governing the
placement of incoming freshman to learn more about what tests and
cutoffs they use. In correspondence with the Florida Department of
Education, we learned that Florida requires all entering freshmen in a
degree program at any public community college or state university to
take a common placement test in math, reading, and writing (Sherry
2003). Florida uses a computerized, adaptive test, called the
Computerized Placement Tests (CPT) that was developed by the College
Board and is part of the ACCUPLACER System. The score required on the
elementary algebra, a minimum of 72, is all that is required for a
student to be enrolled in college-level mathematics. Among the various
exemptions stipulated by the state are completion of an associate degree
or higher, or a score of 440 or better on the math portion of the SAT or
19 or better on the Enhanced ACT.

The Oklahoma State Regents for Higher Education have a policy that
governs entry-level assessments at all public institutions within
Oklahoma, with only certain specifics and other requirements determined
by each institution. Their policy concerning mathematics is that all
students at all institutions are required to score a minimum of 19 on
the math portion of the ACT prior to placement in a college-level math
course. Students who do not meet that requirement are given the
opportunity to take an additional test that is selected by the
institution. In this system, 11 institutions use ACCUPLACER, 14 use
COMPASS, one uses ASSET, and one uses the ETS for the secondary
mathematics assessment (Moss 2003). Some of these institutions use
combinations of tests, including locally developed instruments. The
cut-scores are determined by the institution and vary greatly among
institutions. The Oklahoma state regents\' policy has no requirement for
either the method of assessment or the cut-scores.

The Tennessee Board of Regents requires an assessment of all students
admitted with less than 19 on the ACT in math, or 460 on the math
portion of the SAT. Those students who have not taken the ACT or SAT, or
whose scores are below the cutoffs and want to challenge the results,
must take the COMPASS test for appropriate placement (Bradley 2003).

In the University System of Georgia, students must take a placement test
if they have not completed the college preparatory curriculum (four
years of high school math) or have SAT math scores below 400. To be
exempted from or exit required remediation, a student must have a
COMPASS algebra score of at least 37. Institutions may set higher scores
(Burk 2003).

All first-time entering freshmen at all state-supported colleges and
universities in Arkansas who are admitted to enroll in all associate or
bachelor\'s degree programs must be tested by the admitting institution
for placement purposes. Students must score 19 or above on the math
section of the Enhanced ACT, 460 or above on the quantitative portion of
the recentered SAT, 39 or above on the ASSET intermediate algebra test,
or 41 percent or above on the COMPASS algebra test to enroll in
college-level mathematics courses. Students not meeting the standard
must successfully complete a developmental math program demonstrating
achievement at a level at least as sophisticated as intermediate
algebra. These scores are considered to be the minimum math requirement,
and individual institutions may elect to follow a higher standard (Bird
2004).

The South Dakota Board of Regents developed a standardized placement
process that requires all entering freshmen seeking a bachelor's degree,
or enrolling students in English and mathematics, to either score 20 or
higher on the ACT or take the COMPASS exam. For COMPASS, they must score
45 or greater on the algebra test in order to enroll in a college-level
math course.

According to the West Virginia Higher Education Policy Commission Web
site, in order to enroll in college-level math, students must score 460
or above in the SAT math test, 19 or higher on the ACT math test, 40 or
higher on the numerical and 38 on the elementary algebra ASSET tests, 59
on the pre-algebra and 36 on the algebra COMPASS tests, or 85 on the
arithmetic and 84 on the elementary algebra ACCUPLACER tests (West
Virginia Higher Education Policy Commission 2003).

The cutoff score required to enroll in college-level math course in
Colorado is 440 or higher on the math SAT test or 19 or higher on the
ACT math test (Colorado Community College Online Placement Testing
2003).

For comparison, in table 1 we summarize the minimum scores necessary for
the ACT, SAT, ACCUPLACER, COMPASS, and ASSET tests provided as
stipulated by state policies or a test publisher. We note that, in all
cases, the minimum scores provided by the test publisher are higher than
those mandated by the states included in our survey.

[]{#__RefHeading___Toc114303160 .anchor}Table 1: Summary of Required
Minimum Mathematics Assessment Test Scores of Selected States

+--------+--------+---------+-------------+------------+------------+
| **S    | *      | **SAT** | **A         | *          | **ASSET**  |
| tate** | *ACT** |         | CCUPLACER** | *COMPASS** |            |
|        |        |         |             |            |            |
|        |        |         |             | *          |            |
|        |        |         |             | *Algebra** |            |
+--------+--------+---------+-------------+------------+------------+
| AR     | 19     | 460     |             | 41         | Inter.     |
|        |        |         |             |            | Alg: 39    |
+--------+--------+---------+-------------+------------+------------+
| CO     | 19     | 440     |             |            |            |
+--------+--------+---------+-------------+------------+------------+
| FL     | 19     | 440     | Elem. Alg:  |            |            |
|        |        |         | 72          |            |            |
+--------+--------+---------+-------------+------------+------------+
| GA     |        | 400     |             | 37         |            |
+--------+--------+---------+-------------+------------+------------+
| OK     | 19     |         |             |            |            |
+--------+--------+---------+-------------+------------+------------+
| SD     | 20     |         |             | 45         |            |
+--------+--------+---------+-------------+------------+------------+
| TN     | 19     | 460     |             |            |            |
+--------+--------+---------+-------------+------------+------------+
| WV     | 19     | 460     | Arith: 85   | Algebra 36 | Num.       |
|        |        |         |             |            | Skills: 40 |
|        |        |         | Elem. Alg:  | Pre-Alg 59 |            |
|        |        |         | 84          |            | College    |
|        |        |         |             |            | Alg: 23    |
+--------+--------+---------+-------------+------------+------------+
| Pub    | 23     |         | Arith: 93   | 66         |            |
| lisher |        |         |             |            |            |
|        |        |         | Elem. Alg:  |            |            |
|        |        |         | 82          |            |            |
|        |        |         |             |            |            |
|        |        |         | College     |            |            |
|        |        |         | Math: 63    |            |            |
+--------+--------+---------+-------------+------------+------------+

While the value of placement tests is known, some research has shown
that not all assessment tests accurately place students in developmental
mathematics courses. For instance, faculty at the United States Air
Force Academy wanted to determine whether providing appointees to the
academy with a practice placement exam before the actual placement exam
would reduce the number of students required to take developmental
mathematics, and whether those who took the practice exam and were
subsequently placed into college-level calculus did worse in that course
(Revak, Frickenstein, and Cribb 1997). Revak and his colleagues found
that the placement scores for those who took the practice exam were
significantly higher than for those who did not. Further, those who took
the practice exam and were placed into calculus I were as successful in
that course as those who did not take the practice exam. Their
experiment also resulted in fewer students being placed into remediation
math (precalculus at the Air Force Academy). Thus, the practice
placement exam allowed more students to place into, and successfully
pass, calculus I their first semester.

We turn now to a review of the literature pertaining to successful
instructional strategies.

#   {#section-4 .unnumbered}

# What Instructional Methods Work Best for Adult Learners?

Our review of the literature includes 15 studies of developmental
mathematics programs in postsecondary institutions.[^9] We chose these
studies because they addressed developmental mathematics in particular,
they covered a number of different strategies, and they were
representative of the body of literature in general. However, we were
not able to identify any studies of developmental mathematics that were
based on randomized controlled trial (RCT) experiments. RCT has become
the gold standard for research (National Research Council 2002) and is
the method that has been found to be the most unbiased in evaluating the
effect of programmatic interventions in the field of education. Instead,
the studies we found are based on nonscientific methods, with students
self-selecting[^10] into the course, and no attempt is made to control
for factors that are correlated with such choices. Even so, we cover
these studies in some detail because we believe that their findings help
to inform our initial understanding of programmatic structures and
practices that may hold promise. We also include findings from studies
of developmental education in general that serve to reinforce the
findings from the developmental mathematics literature, or to suggest
promising practices in areas that have not been addressed for
developmental mathematics for adult learners specifically.

## The Role of Technology

The literature that we reviewed pertaining to instructional methods in
developmental education in general, and in developmental mathematics in
particular, revealed that a significant body of research has been
devoted to the question of the relative value of technology versus the
traditional instructor-led modality, and the extent to which technology
should be used as an instructional tool. We concentrate on this debate
in this section, and review other classroom strategies in the section on
pedagogy.

Research on education technology has increased tremendously in this
arena in the past several years, with studies on the effectiveness
lagging behind. For instance, while calculators have been around for
decades, controversy still exists over the appropriate use of this far
less complicated and far more pervasive technology.

Perhaps the greatest topic of debate and uncertainty in the
effectiveness of various strategies in developmental education for adult
learners is in the appropriate use of technology. The debate includes
questions of how extensively it should be used, as well as the
appropriate choice of technology. For instance, MacDonald et al. (2002)
summarize the debate as follows: "The debate is over whether or not to
utilize technology that is capable of conducting the very skill that the
developmental mathematics student is trying to obtain" (p. 36). They
note that one aspect of the debate concerns the type of calculator that
students should use---scientific versus graphing. Unlike graphing
calculators, scientific calculators do not allow the learner to see the
connection between input parameters and output results. However, the
debate cannot be conducted apart from the content of the course, as the
authors point out. A number of developmental mathematics instructors are
finding greater success in using calculators when they change their
emphasis from basic skills to problem solving, using real-world problems
or emphasizing development of critical thinking skills.

There is even a debate about the precise definition of certain terms,
including computer-based education (CBE), computer-based instruction
(CBI), computer-assisted instruction (CAI), computer-managed instruction
(CMI), and computer-enriched instruction (CEI). For the purpose of this
review, the term CAI will be used to refer to instruction that is
typically a supplement to traditional instructor-led instruction and
most commonly includes drill-and-practice, tutorial, or simulation
activities.

A number of studies evaluate the impact of various teaching delivery
methods on student success, such as traditional lecture,
computer-assisted courses, self-paced instruction, Internet-based
courses, and accelerated programs. Several studies reported inconsistent
conclusions as to whether technology-assisted or technology-based
instruction is superior to instructor-led approaches. Conclusions are
based on different definitions of success in each study, such as
receiving a passing grade in the course, persistence to higher-level
mathematics, or scores on final exams. However, there is general
agreement that, while students may not necessarily be more competent
with one particular type of instructional mode, their persistence in
developmental math and beyond may be enhanced by the option of
instructional choice. Several researchers contend that allowing students
to choose the instructional method that they feel best suits their
particular learning style makes them more likely to complete the course
and perhaps take higher-level mathematics.

### Computer-Assisted Instruction

Our first studies on this topic include computer-assisted instruction as
an option to the traditional instructor-led modality. Cartnal (1999)
examined success, retention, and persistence in several math courses
between those offering traditional instructor-led methods and
computer-assisted courses. The study found that students who took
computer-assisted courses in elementary algebra and intermediate algebra
had higher retention rates, but students in the traditionally taught
courses had a higher success rate, defined as receiving a passing grade
(C or higher). However, of those who successfully completed the
computer-assisted algebra courses, a greater percentage went on to take
higher math, including trigonometry and precalculus. Because the study
did not control for self-selection, and results are not robust, Cartnal
suggests the need to do further research in this area.

McClendon and McArdle (2002) conducted a study of the effectiveness of
three delivery modalities of developmental instruction in mathematics in
the Mathematics Department of the Winter Park Campus of Florida's
Valencia Community College: traditional lecture, Academic Systems (an
Internet-accessed software curriculum that combines lecture, practice
programs, and self-administered assessment tests), and Assessment and
LEarning in Knowledge Spaces (ALEKS) (a nonlinear, nontraditional
Internet-based course). Students were aware of which modality was used
in each of the classes, and their advisors were told by the researchers
which modality was optimal for various learning styles. However,
students self-selected into each course, and not all students used
advisors when selecting courses. In addition, ALEKS was the only
modality available for students who registered late.

Using raw percentages of students who completed the course (defined as a
grade of C or higher), the study found that students who attended the
traditional lectures had the highest completion rate, while those who
attended ALEKS had the lowest. Because of the significantly higher
withdrawal rate in the ALEKS courses versus the traditional lecture
method, which they believe was due to the inability of a large number of
students to self-select out of ALEKS, the researchers recalculated the
completion rate only of those who did not withdraw from the course.
Netting out withdrawals, the authors found no significant difference in
outcomes by modality. Nevertheless, the study concluded that
institutions should consider making various learning modalities
available.

Similar to the study by McClendon and McArdle, Kinney (2001) analyzed
the difference in effectiveness of various approaches to teaching
elementary algebra and intermediate algebra at the University of
Minnesota--General College. Two methods were used, with students
choosing the one that they believed would meet their learning
preferences best: direct instruction classes or a computer-mediated
instruction using what is known as a full implementation model. In this
latter model, students met at the same time and followed the same
schedule, but the software delivered the instruction while the
instructor provided individual or small group assistance on request. The
advantage of this type of instruction for developmental education is
that it provides students with an alternative to direct instruction, and
gives them more control over their learning. However, this is not a
self-paced model.

The study found no significant difference in scores on common final
exams between the two methods of instruction. However, students in both
groups reported an increase in confidence to succeed in math, and their
attitudes toward math had improved. And, similar to McClendon and
McArdle, Kinney also concluded that it is important to provide students
with alternative instructional formats and to give them guidance in
choosing the format that best suits their particular learning style.

In a similar vein, a study conducted by Creery (2001) compared the
outcomes of developmental math students taught basic math using lecture,
self-paced, and online methods. Not all students were aware of the
different modalities when they signed up for the course, although
descriptions were available in the bulletin. They also conclude that
many of the students in the nontraditional modalities were in those
classes because the traditional ones were already closed, again implying
that these types of students were enrolling relatively late. Creery
noted that many of the students in the online courses had relatively few
computer skills.

Creery uses grades at the end of the semester for those who did not drop
out by the end of the first week of class to evaluate the outcomes of
the three methods. No statistically significant difference in the
outcomes for the three different delivery methods was found. Following
students into the next level math course, elementary algebra, the
results were the same---no statistically significant difference for the
three delivery methods. Even so, Creery argues that it is important to
offer various methods of instruction.

A seven-year study conducted by Waycaster (2001) in five Virginia
colleges examined 10 instructors and 15 developmental math classes whose
primary instruction was either lecture with lab or individualized
computer-aided instruction. The goal was to determine the most effective
ideas and teaching methods being used in developmental mathematics,
looking specifically at such factors as course credit hours, class size,
attendance, student and teacher gender, class participation rates,
method of instruction, success rates in developmental and subsequent
college-level mathematics courses, and retention and graduation rates.

Results indicated that the success rate in the developmental classes,
defined as a passing grade, was independent of the manner of instruction
used, although no test for statistically significant difference in
proportion passing was conducted. It also was found that students who
took developmental mathematics had higher retention rates (although
retention is not defined, it often means persistence at the college from
one semester to another), again by simply looking at the retention rates
across programs for those who took developmental mathematics versus
those who did not.

Several studies included in our review addressed the effectiveness of
specific software in developmental mathematics programs. Two of the
studies were based on a popular computer-based instructional tool, the
PLATO Adult Learning Technologies. Each study found the use of software
to be effective, either as a self-paced program, or as a
computer-assisted component of an instructor-led course.

Quinn (2003) reported on the success of adult learners using this system
at Miami-Dade Community College. As part of the admission process, all
students at Miami-Dade Community College must take a CPT, developed by
the College Board as part of its ACCUPLACER system, to assess their
competency in reading comprehension, sentence skills, elementary
algebra, and arithmetic. Students must achieve a certain score to be
able to take college-level courses. If students do not pass the CPT,
they are required to meet with counselors and receive guidance as to
what they must do to pass the CPT, which includes assignments to
complete certain PLATO Adult Learning Technologies courseware modules.

Following 82 students who used the PLATO software for arithmetic skills,
and 79 who used it for elementary algebra, Quinn found that student
scores on the CPT showed a statistically significant increase for both
subjects, with an average gain of about one standard deviation.[^11]
Quinn also was able to correlate the time spent on the software to
increases in CPT scores and found that, for every hour students spent on
the courseware instruction, they gained 0.61 percent to 1.86 percent on
the CPT retake test.

Lancaster (2001) studied the effectiveness of using PLATO
computer-assisted instruction in developmental courses at Jefferson
Davis Community College (JDCC). All students are placed in the
appropriate class based on scores on the COMPASS test, which they take
at enrollment. Students whose scores place them into two or more
developmental courses are required to take a study skills class.

JDCC's developmental courses use computer-assisted instruction in
combination with classroom instruction, and some instructors choose to
use additional Web-based programs to enhance other skills or for career
awareness purposes. For instance, the career awareness software allows
students to explore career opportunities, conduct job skills
assessments, and develop job preparation skills.

In the beginning of each term, students take an initial assessment on
the PLATO software program, which then forms the basis for their
Individualized Education Plans (IEPs). These form the basis for
specialized modules that are established to remediate those
deficiencies. To increase mastery of the material, instructors also
often conduct traditional classroom instructions and group activities.

The CAI approach was initiated during the summer 2000 term, and the
study compared the performance of those in developmental courses the
previous summer term, using the traditional classroom format, with those
using the CAI approach during the summer 2000 term. This is a
quasi-experimental design since students could not necessarily
self-select into the academic term based on instructional modality
differences.

The study found that using CAI resulted in a 7-percent decrease in the
number of withdrawals in elementary algebra, a 12-percent increase in
the number of satisfactory grades, and an 11-percent decrease in the
number of unsatisfactory grades in elementary algebra.

The U.S. Navy recently evaluated the effectiveness of Tutorials in
Problem Solving (TiPS), an intervention for training arithmetic and
problem-solving skills in adult populations, using adult learners
requiring developmental mathematics instruction at Mississippi State
University (Atkinson 2003). The material in TiPS is provided in the
context of word problems, providing students with a set of
diagrammatical tools to analyze the problem. It has an interactive
interface tool, a help system, and diagnostic and feedback capabilities.

This study used a rigorous approach to evaluate the effectiveness of the
software, using pretest and posttest measures, and program-comparison
group strategies in which students were chosen for the program based
solely on a pretest cutoff score. The author notes that this approach is
useful when courses are given on the basis of need or merit, and it is
considered to be as robust in inferences as those drawn from randomized
designs.

Their metric was test performance change across time for students using
TiPS, compared to the test performance change of a control group of
students in developmental mathematics who had similar scores on the
pretest, but who were not chosen to use TiPS. They also sought to
determine whether the length of instruction varied systematically
according to a student's mathematical ability before using TiPS. They
found that the average posttest score for TiPS participants was
significantly higher than those of their peers in the control group.
TiPS was originally designed for use in training of enlisted sailors,
but the results indicate that it has much wider applicability for middle
schools, adult literacy programs, and workplace training programs.

### Computer Algebra System

Another type of technology that is the subject of some debate is the
Computer Algebra System (CAS), which in general refers to a system or
software that is used in manipulating mathematical formulae in both
symbolic and numeric form, unlike traditional calculators that only
allow manipulation of numeric equations. In addition, CAS automates some
of the more tedious or difficult algebraic manipulations, with the
intent to reduce the amount of time spent on drill exercises, allowing
more time to spend on greater comprehension of the subject matter.

Livingston (2001) investigated the impact of a computer algebra system
on six intermediate algebra classes at Orange Coast College in
California. Using a quasi-experimental nonrandomized control group
pretest-posttest design, he examined whether classes taught using a
graphing calculator with a computer algebra system (the CAS TI-89,
produced by Texas Instruments) performed as well as classes taught using
traditional methods with scientific calculators. Livingston does not
mention how the control and experimental groups were chosen, though he
notes that it was a nonrandomized design. Also, he does not control for
student characteristics, thereby further reducing the value of his
results. The findings indicated that there was no statistically
significant difference in the pretest and posttest scores of the two
groups, nor was there a difference in the ability to perform mathematics
by hand. However, the group taught using the computer-based system did
perform better at solving higher-order reasoning skills by hand.

In general, CAS has not met with overwhelming success. Leinbach,
Pountney, and Etchells (2002) argue for the value of such systems. Their
argument is that for technology such as CAS to be successful, it must be
a tool used in otherwise good pedagogy that allows students to become
active participants in their learning experiences and to plan and carry
out problem-solving strategies, echoing conclusions drawn by MacDonald,
Vasquez, and Caverly (2002).

The Standards for Success study cited earlier (Conley and Bodone 2002)
did not provide much detail in its recommendations concerning
technology, but the authors did state that students should understand
both the use and the limitations of calculators, including graphing
calculators. The Crossroads standards (American Mathematical Association
of Two-Year Colleges 1995), described earlier, stated that the use of
technology is an essential part of an up-to-date curriculum; students
should use appropriate technology to enhance their mathematical thinking
and understanding and to solve mathematical problems.

The most important point concerning the role of technology is that it
appears to be most useful as a supplement to, rather than a replacement
for, regular classroom instruction. This point is reiterated by Boylan
and Saxon (2002), as well as by Leinbach, Pountney, and Etchells et al.
(2002) and the AMATYC Vision Report (American Mathematical Association
of Two Year Colleges 2002b). The latter report also concludes that, for
technology to be effective, instructors must have adequate professional
development in the appropriate use of technology. This conclusion
applies to adjunct as well as full-time faculty. The authors suggest
that all types of technology, including graphing calculators,
spreadsheets, and CAS, should be used to give students the chance to
become familiar with the technology and to understand its benefits and
limitations.

Even so, the studies we reviewed that specifically addressed the
effectiveness of technology have found that, relative to the traditional
instructor-led format, CAI and CAS resulted in higher, lower, or no
difference in pass rate, no difference or higher rates of persistence to
higher-level math, and no difference in final grades. Clearly, this is
an area ripe for further study.

## Pedagogical Issues

We turn now to addressing research on other pedagogical techniques. We
begin with a discussion of the research concerning how people learn.
This field of study provides a context for much of the literature that
we reviewed on pedagogy.

### How people learn

The AMATYC (American Mathematical Association of Two-Year Colleges 1995,
2002b), Standards for Success (Conley and Bodone 2002), and American
Diploma Project (2004) research that we discussed previously defined the
knowledge necessary to pursue college-level mathematics. Their
recommendations also included several skills and abilities, as well as
pedagogical strategies. These strategies, as well as the skills and
abilities themselves, are best understood in the context of how people
learn. The National Research Council compiled an important body of
research on this topic. According to *How People Learn: Bridging
Research and Practice* (Donovan et al. 2000), because students have
certain preconceived notions, they will fail to grasp new concepts if
their initial understanding is not engaged. Further, students need a
deep foundation of factual knowledge and a strong conceptual framework
if they are to develop competence in a particular area, and they need to
monitor their own understanding and progression in problem solving. If
learners are able to make analogies to what is known when confronted
with new material, they can better advance their understanding of new
material. While the study was initially intended for primary and
secondary education, the authors indicate that the larger design
framework for children's learning environments applies to adult learning
as well.

Other research on learning confirms that there are new ways to introduce
adult students to traditional subjects, such as mathematics. In concert
with strategies outlined by the National Research Council (Donovan et
al. 2000), these new approaches make it possible for the majority of
individuals to develop a deep understanding of important subject matter
(Bransford et al. 2000).

The National Research Council has developed a notion that there are four
perspectives on learning environments. These seemingly separate
perspectives, however, should be interconnected to mutually support one
another (Brown and Campione 1996). Specifically, the perspectives are as
follows:

1.  **Learner-centered environment**. Accounting for the perspective of
    the adult learner requires paying careful attention to the learner's
    knowledge, skills, attitudes, and beliefs. Adult learners need to be
    treated as adults who are responsible for their own lives and who
    are capable of self-direction (Knowles 1989). Also, adults acquire
    knowledge about things they need to know---that is, to cope
    effectively with their real-life situations. Postsecondary programs
    for adults should be designed with the understanding that adults
    undertake most learning efforts in response to life transitions
    (Aslanian and Brickell 1980).

2.  **Knowledge-centered environment**. Adult learners require a
    well-organized structure of concepts, such as those defined by
    AMATYC (1995), that organize the presentation of subject matter and
    help students: (1) develop substantial mathematical problem-solving
    abilities; (2) learn to develop models involving real-world
    situations; (3) expand their mathematical reasoning skills; and (4)
    use technology. Students need help to become metacognitive by
    expecting new information to make sense and asking for clarification
    when it doesn't (Palinscar and Brown 1984; Schoenfeld 1983, 1985,
    1991). Students should learn to compute, but they should also learn
    other things about mathematics, especially the fact that it is
    possible for them to make sense of mathematics and to think
    mathematically (Cobb, Yackel, and Wood 1992).

3.  **Assessment-centered environment**. Postsecondary programs should
    have clear learning goals and assessment methods, procedures, and
    items that are congruent with those goals. The primary element of
    importance is that assessment be used for feedback and revision of
    the program, including teaching and learning. In many classrooms,
    opportunities for feedback appear to occur infrequently, resulting
    in grades on tests, papers, worksheets, homework, and final reports
    that represent summative assessment only, since students typically
    move on to a new topic without opportunity to revise their
    thinking---in particular, higher-order thinking. Assessment of
    higher-order thinking in mathematics includes among factors: methods
    of examining nonalgorithmic (not fully specified in advance) problem
    solving; opportunities for multiple solutions (each with costs and
    benefits); nuanced judgment and interpretation; application of
    multiple criteria that sometimes conflict with one another;
    self-regulated thinking processes; and imposing meaning and
    structure in situations with apparent disorder (Romberg, Zarinnia,
    and Collis 1990). Critics argue that in contrast to assessments
    built around higher-order thinking, many of the typical assessments
    in postsecondary mathematics developed by teachers emphasize memory
    for procedures and facts (Porter et al. 1993).

4.  **Community-centered environment**. The extent to which students,
    teachers, or even administrators feel connected in several aspects
    of community is reflected in classrooms as communities, institutions
    as communities, and even larger communities such as those within the
    military and businesses. The importance of communities in learning
    cannot be emphasized strongly enough with adult learners. In the
    development of higher mental functions, such as planning and
    numerical reasoning, "internalization" of self-regulatory activities
    first takes place in the social interaction between adults and more
    knowledgeable others (Vygotsky 1978). Studies of mathematical
    problem solving, for example, by Noddings (1985), Pettito (1984a,
    1984b), and Schoenfeld (1985), indicate how useful dialogues among
    mathematics problem solvers can be in learning to think
    mathematically. Small group dialogues prompt disbelief, challenge,
    and the need for explicit mathematical argumentation; the group can
    bring more previous experience to bear on the problem than can any
    individual, and it highlights the need for an orderly
    problem-solving process. In addition, computers can serve to enhance
    communities of learning by functioning as mediational tools that
    promote dialogue and collaboration on mathematical problem solving.

The Vision Report (American Mathematical Association of Two Year
Colleges 2002b) also provides an extensive discussion of what
participants considered to be the best teaching and learning methods for
the first two years of college, and it specifically addresses the needs
of adult learners. It recommends that lectures be supplemented by a
number of student-centered methods, such as computer simulations and
collaborative learning activities, including working in teams. The need
to be able to work collaboratively, particularly in teams, is consistent
with requirements of businesses as stated in the American Diploma
Project (2004*)*. The Vision Report also contends that it is not just
the pedagogy but the curriculum content that can be effective in
teaching mathematics to adults. In particular, the report states that it
is important to use activities that engage students in the learning
process, such as the use of case studies and projects that require
designing, modeling, researching, and presenting findings.

Several studies note similar factors as being effective in adult
education programs. Alamprese (1998), Alamprese, Labaree, and Voight
(1998), and Alamprese (2001) detail program-level factors in adult
education and their relation to student outcomes in their review of the
literature. According to their review, exemplary adult education
programs feature the following five characteristics:

-   Effective program management and instructional leadership;

-   A commitment to staff development;

-   Conscious attention to appropriate instructional strategies;

-   A focus on learner assessment; and

-   Extensive supports for learning, especially for students with low
    levels of literacy proficiency.

The themes discussed above are present in many of the studies that we
reviewed. We summarize a number of these studies that specifically
address the role that learner-centered instruction and metacognition
strategies, small group instruction, and collaborative learning play in
developmental math education. In addition, we summarize a few studies
that suggest additional strategies or highlight important
considerations.

### Learner-centered environment

Lending support to the potential benefits of learner-centered
instruction, Miglietti, Strange, and Carney (2002) investigated the
relationship between learning and teaching styles in developmental
English and mathematics courses in a two-year branch of a four-year
Midwestern college. Instructors rated their own teaching styles using a
specifically designed instrument for this purpose, the Principles of
Adult Learning Scale (PALS) tool. Students assessed their learning
styles with two tools: the Adult Classroom Environment Scale (ACES),
which the authors state is the only scale designed to measure adult
students' perception of the classroom environment in general, and the
Adaptive Style Inventory (ASI), which measures students' emphasis on
styles of learning. A total of 185 students chose to participate, but
only 159 completed the courses. Of these, 59 percent were enrolled in a
developmental math course.

The study found no age or gender effects on classroom environment and
learner style preferences. In terms of age and teaching style on
classroom outcomes, the authors note that they could conduct an analysis
of the effects of teaching style on developmental English classes only
because, within the mathematics sections, none of the five mathematics
instructors reported a learner-centered teaching style. The authors'
findings in terms of developmental English led them to conclude that
adult underprepared students in learner-centered classrooms achieved
higher grades than similar students in teacher-centered classrooms.

Higbee and Thomas (1999) reviewed the literature concerning important
factors pertaining to a learner-centered environment, and how they
relate to achievement in mathematics. The authors note that the
following affective variables are important: student's academic
self-concepts, attitudes toward success in mathematics, confidence in
their ability to learn mathematics, math anxiety, text anxiety,
perceptions of the usefulness of math, motivation, self-esteem, and
locus of control. Further, these researchers also have examined the
relationship between performance in mathematics and cognitive factors,
such as preferred learning styles, visual and spatial ability, the use
of specific cognitive strategies, and critical thinking skills. Based on
this body of research, educators have begun to research various
techniques to reduce or eliminate some of the barriers so far
identified, including the use of collaborative learning and
verbalization during the problem-solving process. Finally, Higbee and
Thomas note that there is an increasing shift from a focus on learner
characteristics to a more integrated and holistic approach,
incorporating the role of the teacher and course content, including
different types of tests, grading systems, the use of mathematics
applications, and collaborative learning.

Higbee and Thomas also explore the relationship between noncognitive
variables and success in a two-quarter developmental algebra sequence
designed for high-risk students at the University of Georgia. The
two-quarter sequence covered the same material as a one-quarter course,
except at a slower pace. One day per week a counselor taught with the
math instructor and introduced special learning-promotion topics, such
as relaxation exercises and metacognition strategies, as well as
strategies for solving word problems in collaborative groups. Students
also were required to attend the mathematics laboratory weekly to take
computer tests that paralleled those administered in the core algebra
course.

The results indicated a significantly lower test anxiety and an increase
in students' confidence to succeed in learning math as measured at the
beginning and end of the two-quarter class sequence. In terms of course
outcomes and affective variables, they found negative correlations
between pretest scores on general test anxiety and math test anxiety.
They note that there was no relationship between posttest scores on
tests of anxiety and any of the test, homework, or final GPAs. In other
words, on average, students experienced a reduction in math and test
anxiety over the two-quarter sequence, but the reduction in anxiety was
not correlated with greater math competency, as measured by a variety of
course outcomes.

Consistent with these findings and with a learner-centered classroom
approach in general, Boylan and Saxon (2002) conclude that remediation
programs require counseling as an integral part of the program. They
find that remediation programs with counseling that is integrated into
the entire remediation program have better results. They report that
counseling should be based on stated goals and objectives of the program
and undertaken early in the program. The counseling should use sound
principles of student developmental theory, and should be carried out by
counselors who are trained to work with developmental students.

Different sets of student perceptions were the subject of a recent study
by Wheland, Konet, and Butler (2003). They looked at five perceived
inhibitors to student success in intermediate algebra at a public
university with an undergraduate enrollment of 24,000 students.
Student-perceived factors inhibiting success were as follows:

(a) Nonnative English-speaking instructors had a detrimental impact on
    their success;

(b) Teaching assistants resulted in lower success than adjunct
    professors;

(c) Student performance in intermediate algebra was not reflective of
    overall performance in nonmathematics courses;

(d) Student success in intermediate algebra did not affect performance
    in subsequent math courses; and

(e) Attendance had no significant impact on performance.

Faculty, however, perceived that factors (c), (d) and (e) all had a
potential negative impact on performance.

Using midsemester tests, final exam scores, and GPA to investigate the
effect of each of these factors, the study found, contrary to students'
perceived notions, that nonnative instructors and teaching assistants
did not have a negative impact on their success in intermediate algebra,
but their performance in intermediate algebra correlated quite highly
with overall GPA that semester, their attendance also was highly
correlated with success in the course, and their grade in intermediate
algebra did have a fairly high predictive value on their performance in
subsequent math courses. However, these conclusions are based on
examination of the effect size (the difference in means of two groups
divided by the standard deviation) for the various metrics under study;
they do not attempt to control for self-selection or other potentially
confounding effects. For instance, students who do not perceive
nonnative instructors as having a negative impact on their learning may
disproportionately select into courses that have these types of
instructors.

The authors conclude by noting that a misconception of many students
(i.e., that factors contributing to their failure in intermediate
algebra are in large measure perceived to be out of their control) only
serves to make the course material more difficult to master.

### Small-group instruction

DePree (1998) examined differences in outcomes for students at a large
urban community college who were instructed in preparatory algebra
classes delivered either by instructor or by small-group instruction.
Small-group instruction is one strategy in community-centered learning
environments that is specifically recommended in the Crossroads and
Vision documents (American Mathematical Association of Two Year Colleges
1995, 2002b). Students were not aware of which type of instruction would
be used at the time they enrolled, enabling a quasi-experimental design.

The results indicated that those taking the course via small-group
instruction had statistically higher confidence in their mathematical
ability, as measured by the Fennema-Sherman Mathematics Attitude Scales.
Improvements were greatest for students who have been traditionally
underrepresented in mathematics: Hispanic, Native American, and female
students. Further, students who received the small-group instruction
were more likely to complete the course than students in the
instructor-led course. However, Depree did not find any difference in
achievement between the two teaching methods. Even so, the fact that
students increased their confidence and were more likely to complete the
course when administered by small-group instruction led the author to
conclude that a larger number would ultimately be successful in this
type of class.

### Contextual learning

Consistent with other literature that we have cited concerning the need
for adults to have contextual learning experiences, Mazzeo, Rab, and
Alssid (2003) describe the efforts of five community colleges that have
created bridges between basic skills development and entry-level work or
training in high-wage, high-demand career sectors. Mazzeo and his
colleagues argue that contextualized basic skills instruction is often
more successful than traditional models of adult education for engaging
disadvantaged individuals and linking them to work. Each program they
describe uses contextualized teaching and learning experiences, which
means that courses incorporate material from specific fields into course
content, and employ projects, laboratories, simulations, and other
experiences that enable students to learn by doing. Further, they
contend that workforce and education systems should be reorganized
around "career pathways" that integrate education, training, and work,
and are targeted to high-wage, high-demand employment to address the
growing needs for skilled workers and workers' needs for economic
self-sufficiency. However, they state that further research is needed to
determine whether contextualized basic skills instruction is more
effective than more traditional instructional approaches.

We have found that other authors also suggest that a strong connection
between education and employment increases earnings and placement rates
(Grubb 1996; Jenkins and Fitzgerald 1998). Both Grubb (1999) and Murphy
and Johnson (1998) argue that one essential characteristic of effective
programs is a focus on employment-related goals through instruction that
integrates basic and occupational skills training with work-based
learning. Rogoff (1990), Lave and Wenger (1990), Lave (1991), and Wenger
(1998) emphasize the important role of context in shaping student
learning. Greeno et al. (1999) describe one of the most important
contexts for adult learning as the world of work itself and the specific
tools, practices, and social relations embedded in the work setting.

The five programs that Mazzeo and his colleagues reviewed also exhibited
these additional characteristics:

-   Integration of developmental and academic content.

-   Development of new curriculum materials and provision of
    professional development to learn to teach in a new way.

-   Maintenance of active links with employers and industry
    associations.

-   Identification of resources to fund the programs, at least in the
    short run.

-   Production of promising program outcomes, especially in terms of job
    placement and earnings.

We highlight their finding that these programs emphasize professional
development for faculty. Boylan and Saxon (2002), a study by the
Education Commission of the States (Spann 2000), and the reviews of
adult programs conducted by Alamprese (1998), Alamprese, Labaree, and
Voight (1998), and Alamprese (2001) cited earlier all conclude that
staff training and ongoing professional development are very important
components of successful adult developmental efforts.

### "Systems Thinking"---DevMap Project

We turn now to a project that addresses the National Research Council's
notion that students must be able to make analogies to what is known
when confronted with new material. In a 1997 National Center for
Research in Vocational Education (NCRVE) workshop, "Beyond Eighth
Grade," industry representatives emphasized the need for "systems
thinking" that allows workers to recognize complexities in various
situations subject to multiple inputs, and for those in certain fields
to formulate a problem and design experiments to determine the influence
of various factors (Consortium for Mathematics and Its Applications
2003). The DevMap Project at the Consortium for Mathematics and Its
Applications (COMAP), which is funded by the Adult Technology Education
Program of the National Science Foundation, is intended to address the
disconnect between traditional college developmental programs in math,
that have been just a replication of high school math, and the needs of
industry as stated in the 1997 workshop. To that end, COMAP is
developing a one-year sequence, Developing Mathematics Through
Applications. This program, based in large part on the Crossroads
recommendations, includes the following unique features:

-   While all the major components of algebra, geometry, and
    trigonometry will be included, the program will not be divided into
    these distinct and separate topics.

-   The program will be based on applications, which should be
    particularly relevant to adult learners who can use applications
    that draw on areas in which they may be working or in which they
    aspire to work.

-   Problem solving will require integrating technology in a natural
    way, as opposed to the "drill-and-practice" use of technology found
    in many developmental mathematics programs.

### Accelerated courses

Lastly, we note an interesting approach that is not based on any of the
other strategies or theories that we have covered so far but may be a
new strategy that holds promise. The University of Maryland, College
Park (Adams 2003) began a program in the fall of 2001, in which students
who required math remediation (representing 20 to 25 percent of entering
freshman, or about 1,000 students) and scored in the top 60 percent of
the placement tests were placed into a combination course that met five
days a week and covered the material for both the developmental
mathematics and the introductory college-level course. At the end of the
intensive first five weeks of the course, in which all of the
developmental material was completed, students retook the placement
test. If they passed the test, they continued in the course, which had
as many contact hours during the remainder of the semester as those
students who enrolled in the regular college-level course. Students
successful in the latter part of the class were then able to complete
both their developmental and first college-level math requirement in
just one semester. Those who did not pass, only 11 percent, were placed
into the regular developmental course, which is a more traditional six
hours-per-week self-paced course, using a computer platform.

Both the accelerated developmental course and the regular college-level
course used the same final exam. Adams found that the final test scores
were about the same for the two groups; in fact, they were often higher
for the students in the accelerated developmental course. Adams also
found that the two classes had about equivalent A, B, and C grade rates.
Precise statistics are not cited, however, nor is it noted which
statistical tests were conducted.

The study also followed those students who successfully completed the
accelerated program into higher math courses. The pass rate (grade of C
or higher) of those students in elementary calculus was about 7
percentage points higher than regular students. The results were not as
good for those students who took the engineering calculus, however;
their pass rate was sometimes far worse than that of regular students.

In summary, we note that the research we reviewed consistently indicates
that certain skills and abilities are as important as the specific
knowledge to successfully pursue college-level math and to succeed in
the workplace. These include the ability to:

-   Understand the connection of math to other disciplines;

-   Perform inductive reasoning;

-   Communicate math orally and in writing;

-   Model real-world problems; and

-   Work collaboratively.

Some pedagogical strategies to achieve these skills and abilities
include:

-   Addressing students' perceptions;

-   Incorporating counseling into the program;

-   Using small-group instruction;

-   Using collaborative learning;

-   Using contextual learning and real-world examples; and

-   Requiring students to conduct research and modeling exercises.

Finally, professional development of developmental math educators is
necessary to ensure that they keep current with technology and
pedagogical research. Absent gold-standard research on the value of
these strategies for adult mathematical literacy education, we believe
that they warrant further study.

# Metrics of Program Effectiveness

In this section, we discuss metrics of program effectiveness noted in
the literature. We do so because, in our search of promising practices,
it may be difficult to locate programs that track the metric of greatest
interest to this study---that is, the ability of adult learners to
successfully progress through and out of a developmental mathematics
program and into their first college-level mathematics course. It is
important to understand whether this is in fact a well-established goal
or stated objective of programs in order to understand whether the
strategy is aimed at improving this metric or at some other equally
valid outcome.

Our review has found that numerous metrics are used in evaluating the
effectiveness of college developmental mathematics courses or programs.
However, no clear consensus emerged concerning optimal metrics for
impact evaluations. This lack of consensus may be due to a more general
lack of consensus about the ultimate role of developmental courses. In
other words, should the goal be to ensure that those who complete the
course achieve some heightened mathematics competency, or is it better
to ensure that a larger number complete the course, but at a slightly
lower yet acceptable level of competency?

We have already discussed that some researchers, for instance, when
looking at whether a particular type of instruction or material is more
effective, examine the pass rate (typically a C or higher), or average
exam scores, or final class GPA of the various approaches. These studies
are intended to determine whether students who ultimately complete the
course have learned more of the material or are more competent in the
subject. This may be at the expense of a higher withdrawal rate,
however.

Others are concerned with the completion rate of the course for which
the Grade Point Average (GPA) does not count since only data for those
students who completed the course can be included in the GPA. These
studies are concerned less with whether those who pass the course are
more knowledgeable in the subject than with whether more students are
able to pass the course. This is an important consideration because,
even if the particular approach does not improve the overall
understanding of those who pass, a larger number of underprepared
students may be able to succeed in the basic skills instruction.

Still other research concerns the level of anxiety or satisfaction with
the developmental course because that directly relates to the
willingness of students to either persist in the developmental program
or pursue higher-level mathematics. Again, this is important not so much
because each "successful" student is more knowledgeable but because more
developmental students are able to pursue even higher mathematics in
order to ultimately achieve their career goals. Even in these cases, as
we have noted, researchers rarely follow students beyond the immediate
semester or, at best, they follow them one additional semester.

We also note that, because of the subjective nature of grading, the use
of grades and pass rates are not necessarily valid or reliable measures
of the knowledge and skills of students. This means that a comparison of
pass rates or the average GPA of students in institutions using
different strategies, both across institutions and even within
institutions that do not use common exams, often is not useful. It is
more meaningful when common placement exams and cutoff scores are used,
say, as a requirement to transition out of developmental, or into
college-level, mathematics courses or when performance within the same
college-level math course at a particular institution is compared for
students subjected to various strategies in lower-level developmental
math courses.

For instance, two studies we reviewed looked at differences in the GPA
of students in college-level mathematics between those who took
developmental courses and those who did not. One study investigated
whether students who successfully complete an exit-level developmental
course and enroll immediately in a college-level course in the same
subject do better than those who do not enroll immediately afterward
(Sinclair Community College 2003). It found that, in the case of
developmental math, average GPA is significantly lower when students
delay three terms, but the course pass rate is not affected. While this
study uses two metrics---GPA and passing the course---the analysis fails
to take into consideration factors that influence a student's choice in
delaying additional math courses. In other words, students who are not
as confident in math, and may not have completed their last
developmental math course with a high grade, may be more likely to delay
taking a subsequent math course. Their lack of confidence, as well as
low GPA in the developmental math course, may have a direct impact on
their performance in subsequent math courses.

The University of Wisconsin--Madison Mathematics Department has been
publishing results of its developmental math program for the past
several years (University of Wisconsin 2003). In 1999, five criteria
department staff developed to evaluate the overall program effectiveness
in quantifiable terms:

-   Success in developmental math, defined as percentage of students who
    remained after the initial add/drop period and earned a grade of C
    or better in the course.

-   Progress from developmental math to degree credit math courses,
    defined as the proportion of students who completed elementary
    algebra who enrolled in the first college-level math course,
    intermediate algebra, within one year.

-   Success equal to other students overall, as defined by three
    metrics: average number of course completion attempts, grade of C-
    or better, and average grade of students enrolled in intermediate
    algebra for those who took developmental math courses versus those
    who did not.

-   Better preparation for college credit courses than other students,
    defined as the cumulative GPA of students currently enrolled in
    intermediate algebra of those who took developmental math courses
    versus those who did not.

-   Achievement of designated course proficiencies, defined as
    percentage meeting expectations in seven different proficiency goals
    for elementary algebra.

These metrics are examples of what may be commonly tracked by community
colleges. Some are good proxies for the metric of greatest interest to
us, while others are not. For instance, in the University of Wisconsin
example, the fourth metric is not necessarily a good proxy for
persistence in developmental education, whereas the second metric is
precisely that in which we are most interested. Yet we have found few
studies that specifically use that metric as a measure of program
effectiveness.

#   {#section-5 .unnumbered}

# A Survey of Community Colleges' Practices

In our survey of community colleges in the second phase of this project,
we need to be aware of certain policies or relevant components of
programs that may have a direct impact on the success of their
particular strategies. Such factors as professional development for
developmental mathematics instructors or the organizational location of
developmental math (e.g., whether it is part of the mathematics
department, in a separate developmental education department, and so on)
may be highly correlated with the value that the institution places on
developmental education in general. This section summarizes key
components of community college practices that are relevant to
developmental education in general and developmental mathematics
specifically.

A recent study by the American Association of Community Colleges (AACC)
involved sending surveys to over 1,100 community colleges, with a nearly
40-percent return rate (Schults 2001). The study covered a number of the
most important features of developmental education, including the
student body involved, and the colleges' approaches and policies. Many
of the questions concerned approaches that have been made as
recommendations for optimal developmental training that we have outlined
in previous sections. In some of these cases, it is clear that community
colleges have not universally adopted the recommendations. Below are
some highlights of their findings:

-   Every college responding to the survey offered at least one
    developmental course. Math, reading, and writing were offered by 94
    to 96 percent, and adult basic education was offered by less than
    half.

-   Of the institutions that responded, 33 percent of their faculty at
    public community colleges who teach developmental education classes
    were full-time, roughly equivalent to the overall proportion of
    faculty that are full-time.

-   The majority, or 58 percent of responding institutions, required
    assessment of basic skills for all students. Many allow exemptions
    from these tests; 76 percent of those that allow exemptions use
    college entrance exam scores instead. Other criteria for waiver of
    the tests include high school GPA, statewide high school exam
    scores, advanced placement scores, and transfer from another
    postsecondary institution.

-   The most commonly used tools for assessing skills were a
    computerized test (63 percent) and a paper-and-pencil test (60
    percent). Other measures included college entrance tests (36
    percent), institutionally developed measures (24 percent), and
    state-developed measures (16 percent).

-   A large percentage of the institutions---77 percent---set their own
    cutoff scores on the assessment tests, while the state sets the
    standards in the remaining 23 percent.

-   Of the 58 percent of institutions that mandate assessment, 75
    percent require placement in courses based on the testing. Of these,
    almost two-thirds set this policy, and the remaining one-third
    reports that the standards are set by the state.

-   Developmental courses are predominantly offered within relevant
    departments (61 percent), while 25 percent report that developmental
    courses are housed in a separate developmental department. The
    remaining 13 percent report that courses were offered through just
    one academic department.

-   The majority of institutions responding indicated that ESL and ABE
    courses were typically offered by departments that are separate from
    those offering developmental education, typically through a
    noncredit department.

-   The median number of levels of developmental math offered by
    colleges is three. More levels of remediation were offered in
    institutions located in large cities, and enrollment in
    developmental education also was typically higher in these
    institutions.

-   Three-fourths of the institutions offered only institutional credit
    (not toward graduation but counting toward full-time status for the
    purpose of financial aid) for all developmental courses, 5 percent
    offered degree credit only, and 5 percent offered no credit. The
    remaining institutions offered multiple forms of credit.

-   Developmental courses in math had the highest median class
    size---25---of any other type of developmental class surveyed. For
    comparison, the median class size for developmental reading and
    developmental writing were both 20. However, almost two-thirds of
    the institutions report having a policy concerning limits to class
    sizes with 95 percent of these reporting that the state did not
    mandate such limits.

-   Partly because of limits on federal student financial aid that a
    person may receive for developmental education, almost one-quarter
    of institutions use various methods to limit the number of
    developmental courses a student may take. Of those, 20 percent
    increase tuition after students attempt multiple times to take
    developmental courses, 32 percent simply restrict students from
    taking additional developmental courses, 30 percent cease nonfederal
    funding, and 19 percent use other methods. Of those that set limits,
    45 percent do so by state mandate.

-   Virtually all colleges surveyed report that students could take
    college-level courses not related to a degree or certificate program
    while in developmental courses.

-   Slightly over half (56 percent) of institutions report using more
    than one measure to assess whether a student can transition out of
    developmental work. The largest percentage (91 percent) use
    successful completion of the developmental course for assessment.

-   Almost half (45 percent) of institutions offered self-paced
    developmental courses to students, and 26 percent offered distance
    education for developmental courses.

-   Approximately 80 percent of institutions responding indicated that
    they sometimes or frequently use computers in instruction.

-   Forty-five percent of institutions provide contract developmental
    training to business and industry, with 65 percent of these
    reporting that they do not award college credit for those classes.

The U.S. Department of Education recently released an updated summary of
remedial education in postsecondary institutions for the year 2000 (NCES
2003). Many of its findings are similar to those of the AACC report, but
the Department surveyed all postsecondary institutions, not just
two-year colleges, and it also measured other phenomena. The major
findings include:

-   Seventy-one percent of all institutions surveyed and 97 percent of
    public two-year institutions that enrolled freshmen, offered
    developmental mathematics.

-   Of those that offered developmental mathematics, 60 percent offered
    between two and four courses, with an average of 2.5 courses. The
    average for public two-year colleges was 3.4 courses.

-   Of the institutions that did not offer any developmental courses, 34
    percent said that they did not because either institutional or state
    policy or law prohibits them from offering such courses. This is an
    increase from 27 percent in 1995.

-   Fifty-six percent of all public two-year colleges provided remedial
    education services to local businesses and industry, an increase
    from 50 percent in 1995. Twenty-one percent of all institutions did
    so.

-   Of the public two-year institutions that offered remedial education
    to employers, 93 percent offered math skills; 85 percent offered
    instruction on site, 80 percent offered instruction at the business
    or industry, and 16 percent offered instruction via distance
    learning.

-   Sixty-one percent of all institutions required all entering freshmen
    to be given placement tests in mathematics, while 64 percent of all
    public two-year colleges did so.

-   Twenty-six percent of all institutions limit the amount of time a
    student may spend in developmental courses. Of those, 71 percent
    state that the policy is set by the institution, while 24 percent
    say that it is set by state policy. For public two-year colleges, 20
    percent have limits, and their reasons are roughly divided between
    state law or policy (46 percent) and institutional policy (43
    percent).

-   Forty percent of all two-year colleges stated that computers were
    used frequently, and 44 percent said students used them occasionally
    as a hands-on instructional tool for on-campus developmental
    mathematics courses. This compares to 31 percent of all institutions
    reporting that they were used frequently, and 40 percent stating
    they were used occasionally.

Community colleges and four-year institutions are not the only source of
developmental mathematics or even postsecondary education. All of the
military services provide training and education, as do many businesses
and business organizations. In addition, numerous basic skills services
offered in adult education and in employment training efforts are
outside the traditional postsecondary arena. We turn to these next.

# The U.S. Military

## General Eligibility Requirements

All of the services have basic enlistment eligibility criteria that
include age, weight and height, moral background (e.g., arrest histories
and prior drug use), medical conditions, education credentials, and
mental ability. Among the eligibility criteria is a cap on recruiting
non-high-school-diploma graduates (NHSDGs). No service is allowed to
have more than 10 percent of its total accessions[^12] per year in this
category. The reason for the cap is that there is a significant body of
evidence showing that NHSDGs are much less likely to complete their
enlistment contract than are high school diploma graduates. Also for the
purpose of this cap, recruits with GEDs are considered in the NHSDG
category because GED graduates perform more like dropouts than like
graduates in terms of their attrition. Within the cap, each service is
permitted to manage its own mix of recruits. While the Army and the Navy
strive to cap their NHSDG accessions at 5 percent, during the very
difficult recruiting period of the late 1990s, both services had to
increase their cap to 10 percent. In recent years, they have decreased
their caps steadily. The Marine Corps has consistently accessed 5
percent or fewer NHSDG accessions, and the Air Force typically accesses
less than 1 percent NHSDGs, and almost all of these possess GEDs.

## Armed Services Vocational Aptitude Battery

The services assess the mental ability of potential recruits with a test
known as the Armed Services Vocational Aptitude Battery (ASVAB).
According to the ASVAB Web site (2003):

> The ASVAB is the most widely used multiple aptitude test battery in
> the world. It was originally designed to predict future academic and
> occupational success in military occupations. Since its introduction
> in 1968, the ASVAB has been the subject of extensive research.
> Numerous validation studies indicate the ASVAB assesses academic
> ability and predicts success in a wide variety of military and
> civilian occupations.

The test consists of eight components: General Science (GS), Arithmetic
Reasoning (AR), Word Knowledge (WK), Paragraph Comprehension (PC), Math
Knowledge (MK), Electronics Information (EI), Auto and Shop Information
(AS), and Mechanical Comprehension (MC). The ASVAB scores are
standardized to a nationally representative sample of American youths
(18- to 23-year-olds) who took the ASVAB in 1980,[^13] with each test
normalized[^14] to have a mean of 50 and a standard deviation of 10.

The scores on various subtests are used to screen recruits for specific
occupations, and four of the tests are used for basic enlistment
eligibility. Specifically, the Armed Forces Qualifications Test (AFQT)
is composed of the standardized scores on the Arithmetic Reasoning test
plus the Math Knowledge test, plus two times the Verbal Expression (VE)
measure, with VE being the standardized score for the sum of the
Paragraph Comprehension and Word Knowledge components. The AFQT is then
expressed as a percentile.

Congress has stipulated that the services cannot recruit people who
score in the lowest 10 percentile of AFQT scores, and only 20 percent
may be in the 10th through 30th percentile. Further, NHSDGs must score
above the 30th percentile (Title 10 United States Code, Section 520).
However, most of the services impose even higher standards. For
instance, the Navy currently does not access anyone who scores below the
31st percentile on the AFQT.

Thus, all of the services do assess the basic skills competency of their
recruits in the areas of reading, writing, and mathematics, and they
establish certain criteria for admission. Beyond the screens for
admission, however, the services also screen recruits for their military
occupation, based on ASVAB scores. For instance, to enlist in one of the
Navy's most technical programs, the Advanced Electronics/Computer Field
(AECF), in addition to stricter screening criteria that pertain to
citizenship, color perception and moral background, they impose
guidelines in terms of mental ability. The sum of a recruit's scores on
the MK, EI, and GS components of the ASVAB must meet a minimum
threshold, and scores must be above a prescribed minimum for the AR and
MK components (U.S. Navy 2002). These thresholds have been established
to ensure that the recruit will have a satisfactory chance of
successfully completing the training, which takes as long as 18 months.

Because of these ASVAB requirements, the services typically do not
recruit people who are significantly deficient in their math skills who
also will require higher-level math to perform their job. Instead, those
with inadequate math skills either are not recruited, or are chosen for
occupations that do not require those skills.

Servicemembers who need minimal remediation in math may receive some as
part of their training, but this would typically involve only very short
courses of a few days in length. Instead, servicemembers with deficient
math skills would pursue developmental mathematics education only to
enhance their own personal goals, such as to improve their ASVAB scores
in order to requalify for another occupation, or to earn a GED, or to
pursue a college education. All of these pursuits would be under what
the military terms Voluntary Education (VolEd). VolEd in the military
consists of numerous programs, each leading to various education
outcomes and each with a variety of financial support to the
servicemember. In fiscal year (FY) 2003, Department of Defense (DoD)
spent \$477.5 million on all VolEd programs (U.S. Department of Defense
2004).

### Basic skills

The lowest level of VolEd consists of basic skills training---that which
helps servicemembers (primarily enlisted personnel) master reading,
writing, and mathematics skills necessary to either do their job or
accomplish their personal education goals. In FY03, 37,346
servicemembers, or 3.1 percent of the total, enrolled in noncredit basic
skills courses:

-   18,025 in the Army,

-   11,768 in the Navy,

-   2,392 in the Marine Corps, and

-   5,161 in the Air Force.

By service these numbers range from 4.4 percent for the Army to1.5
percent for the Marine Corps (U.S. Department of Defense 2004).

All of the services pay 100 percent of the cost of studies and testing
that lead to a GED. In FY03, 84 soldiers, 64 sailors, 22 Marines, and 36
airmen received high school diplomas or GEDs while on active duty (U.S.
Department of Defense 2004).

Each of the services provides basic skills education independently. The
Army's program is called the Basic Skills Education Program (BSEP),
designed to help soldiers master the functional reading, writing, and
mathematical skills required of their jobs. In 1999, the undersecretary
of the Army directed that BSEP should be automated and fully deployed
(Bilodeau 2003). The Army found in an evaluation of three commercial
off-the-shelf (COTS) products,[^15] students performed as well on TABE
posttests as soldiers who took traditional BSEP courses.

BSEP is open only to soldiers who score below the 10th-grade level on
TABE, or whose General Technical (GT) scores are below 100 (U.S. Army
2005). GT scores are computed as the sum of VE plus AR standard ASVAB
scores (each has a mean of 50 and a standard deviation of 10 in the
youth population). All services except the Navy then standardize this
sum to have a mean of 100 and a standard deviation of 20.

The Marine Corps basic skills program is called Military Academic Skills
Program (MASP), and is offered in a variety of delivery methods: via
videoconferencing classroom, traditional classroom, or online. This is a
four-week program targeted toward Marines who score 99 or below on the
GT and 10.2 or below on TABE; there is an additional MASP preparatory
course for those who score 8.5 or below on the TABE. In addition,
Marines who have been referred by their commander also may enroll (U.S.
Marine Corps 1998).

Thus, both the Army and the Marine Corps restrict enrollment in their
basic skills education courses to servicemembers who score below a
threshold of the TABE that is addressed by our research---that is, below
the 10th-grade level.

The Navy's VolEd program is called Navy College Program. Basic skills
are available either through online instruction in the Navy College
Learning Centers (NCLC) located on numerous bases or in teacher-led
instruction under the Navy College Learning Program (NCLP). NCLC
instruction is provided using PLATO software, and sailors are
administered a pretest to determine proper placement. The content of the
PLATO instruction covers math beginning with second grade material,
through the twelfth grade, including algebra (U.S. Navy 2003).

We consulted with members of the Navy and the Army research staff, and
conducted academic literature searches and a search in the Defense
Technical Institute Center (DTIC) repository (DTIC 2003) and were not
able to identify research that has been conducted concerning the
effectiveness of the services' basic skills education in developmental
mathematics. The only study that emerges on this topic in general is one
that was conducted in the late 1990s by the Center for Naval Analyses
(CNA) (Garcia 1998). The Navy wanted to conduct an analysis of the
effectiveness of VolEd in general because so little was known about the
returns to this program, including basic skills and postsecondary
education. The study found that, relative to nonparticipants,
participants in the VolEd program got promoted faster and farther, had
fewer disciplinary problems, and had higher retention even after
controlling for relevant factors, such as military specialty and
demographic factors. In sharp contrast to other studies in this review,
this study uses an econometric technique that adjusts for the
self-selection bias inherent in a voluntary program. Furthermore, this
study calculates a return on investment (ROI) for various components of
the program by comparing the implied reduction in recruiting and
training costs from high retention to the cost of the program. The ROI
on the adult basic education component of the program was even higher
than the ROI on more advanced skill components.

### Postsecondary VolEd

For servicemembers who want to pursue postsecondary education, there are
a number of avenues, and they vary by service. Tuition Assistance (TA),
which is offered by all the services, provides 100 percent of the
mandatory tuition and fees at accredited institutions of higher
education, up to \$250 per semester hour. All of the services except the
Navy cap this assistance at \$4,500 per year. The Navy limits funding to
12 semester hours per year (Military.com 2003).

The Army, Navy, and Marine Corps each have a group of colleges that
together compose the Servicemembers Opportunity Colleges (SOC), a
consortium of approximately 1,700 colleges, which is cosponsored by the
American Association of State Colleges and Universities (AASCU) and the
AACC. Among other requirements, members of the consortium agree to
accept credit awarded from other members. This feature is an important
consideration for servicemembers who move frequently and so may not be
able to complete a degree at the institution that originally granted
credit. The Air Force does not have a comparable consortium because it
has the Community College of the Air Force that serves a similar
function.

Servicemembers may take courses at any accredited college, regardless of
membership in SOC, on their own time. For many, this includes distance
learning or Video Teleconferencing (VTC), but for those located near
large bases, a number of community colleges will actually provide
instruction on base.

The cost of these courses is subsidized for servicemembers through TA,
including developmental mathematics courses that confer credit
(including institutional) and are required by the college. Hence, any
developmental mathematics that would be required by the institution
would be similar to those for other adult learners enrolled at that
institution and are, therefore, included in our analysis of community
colleges.

# Businesses and Organized Labor

Many corporations and moderate-sized companies support continuing
education and training of their workforce. A study conducted by the
American Society for Training and Development (ASTD), in which they
surveyed 365 organizations in 1999, found that companies spend an
average of 1.8 percent of payroll on training (Van Buren 2001). Of the
amount spent on training, between 5 and 7 percent is in basic skills,
including literacy, reading comprehension, writing, math, English as a
second language, and learning how to learn. By far, the largest category
of training was in technical processes and procedures, totaling
approximately 13 percent of all training expenditures. The National
Association of Manufacturers survey cited earlier found that 7 percent
of employers offered advanced math education opportunities, while 15
percent offered basic math training. The study also found that just 4
percent of respondents in companies with at least one union said that
the union offered training to employees.

In concurrence with the NCES (2003) findings previously cited, the most
often cited sources of external education and training in the survey
were community colleges (45 percent), technical and vocational schools
(46 percent), business and industry associations (45.9 percent),
consultants (39 percent), and universities (17 percent).

Our search of the literature revealed no definitive research on business
or organized labor education programs in general. In addition, a search
of the Web sites of major unions and large corporations also did not
reveal more information about specific education programs in
developmental mathematics. For example, the Web site of the AFL-CIO,
with a membership of over 13 million, has a section concerning education
issues and legislation, but it contains no information about specific
education programs in general, or developmental mathematics in
particular (AFL-CIO 2003). We did find one Web site that mentions an
education program offered by a joint partnership between the United Auto
Workers (UAW) and General Motors Corporation (GM), although little
information about the specific program is provided (UAW-GM Center for
Human Resources 2004). They do note, however, that there are more than
97 UAW-GM Skill Centers across the country that operate within a local
education agency, and offer opportunities in: Adult Basic Education,
General Educational Development, educational enrichment services,
English as a second language, academic advising services, and high
school completion.

Wolfe (2001) described a business-based program in Kentucky that is a
collaboration with the local community college, Kentucky Educational
Television (KET), and Hospitality Television. This School-to-Work
program brings basic math and developmental classes to the workplace via
television, thereby bringing the education to the worker rather than
requiring them to come to the college. The program was launched in 1998
for workers in the hospitality sector, but by 2001 it had expanded to
include retail, manufacturing, and local government, especially school
districts. Although no evaluation of effectiveness of the program was
noted, Wolfe stated that one official envisioned this program as being a
feeder for community colleges and technical schools because of its
potential to get learners started on the right path.

Dougherty and Bakia (2000) discussed the role that business and
organized labor have in contract training to community colleges. They
describe contract training in the five areas they investigated: auto
manufacturing, construction, apparel marketing, banking, and auto
repair. They note that entry-level skills training across the five
industries they studied were focused primarily on machinists,
carpenters, and auto repair technicians. They found far less contract
training for entry-level semiskilled workers.

These entry-level skills training programs combine a large amount of
classroom and on-the-job (OJT) training. Among the most significant of
these programs are the apprenticeships in auto manufacturing and the
nonapprenticed employee-in-training (EIT) programs that are primarily
conducted at two-year postsecondary institutions, the control of both
being dominated by employers and the UAW. In fact, Dougherty and Bakia
note that many are joint union-management apprenticeship programs and
union members play a key role in decisions concerning curriculum, choice
of providers, and evaluation of the program.

Because of the absence of unions in the auto repair industry, they note
that training in this area is primarily sponsored by GM, Ford, Chrysler,
and Toyota, with the community college as a much more equal partner.

Also, construction contract training tends to be shorter in duration if
it is sponsored by industry than if it is sponsored by joint
union-management apprenticeship training committees.

Finally, they note that contract training has the potential to change
not only the content of courses to meet the particular needs of the
business or union but also the pedagogy, particularly when the corporate
customer uses newer instructional techniques or technology than the
college uses. In those cases, college instructors who teach
corporate-sponsored courses often pick up these technologies and then
import them back into their regular courses.

## Corporate Math Skill Trainers

Much of our review so far has focused on developmental college programs.
Businesses also report using business and industry associations staff,
and corporate math skills trainers to remediate the math skills of their
workforce. We gathered information via Web sites and exchanges with
personnel from a number of corporate math skill trainers, including four
of the largest---MathWorks Inc. (Maher 2004); Thomson NETg (Rollins
2004); SkillSoft Inc. (Jordan 2004); and Mathsoft (Schindler 2004)---and
two smaller organizations---Math Learning Institute Inc. (Charles 2004)
and the Alinea Group (Martin 2004) in order to obtain a more
comprehensive representation of corporate training.

The research revealed that a variety of businesses rely on the
contractual services of corporate math skill training companies to
provide math skill development tools, products, and training for their
employees. Although most companies cannot give detailed information
about their clients, they indicated that clients have included U.S.
government agencies, major auto manufacturers, aerospace companies,
major airlines, IT companies, and large discount chain stores. This
inquiry yielded several key observations:

-   A large number of companies rely primarily on the contractual
    services of local colleges and universities, particularly services
    provided by community colleges, for basic math skill (addition,
    subtraction, division, multiplication, fractions, etc.) training.

```{=html}
<!-- -->
```
-   Of the four largest math skill training companies interviewed and
    researched, all required basic math skill knowledge before taking
    classes, or using their products, or both. Although courses were
    offered to those with little to no advanced level math knowledge, in
    almost all cases, the training companies' clientele consisted
    primarily of individuals with a strong mathematics background, at
    least a baccalaureate degree, or both, often in the areas of
    mathematics, engineering, the sciences or technology. Many of their
    clients also have advanced degrees.

-   The two smaller companies did provide customized basic math skill
    training.

-   Several of the larger math skill training companies provide referral
    services to clients to local colleges and universities that provide
    basic math skill training to employees to ensure that their clients
    are adequately prepared to succeed in their training classes, or use
    their math software, or both.

-   Several corporations that provide some math skill training have
    partnered with educational institutions (e.g., University of
    Phoenix) to give employees the opportunity to receive college credit
    for courses they take through their training programs.

-   Most corporate consulting and training organizations provide
    coursework through e-learning (Web-based), CD ROMs, instructor lead
    training, onsite training**,** or training at their own facilities.

-   The math skill courses are designed to meet the needs of the client
    and therefore vary in length and format. Some classes are as short
    as one to two hours or may meet regularly over the course of several
    months.

-   Math skill training is generally only one component of the services
    provided by companies that provide math skill training through
    various course offerings (finance, budget, etc.). Most of the
    companies that provide training also have developed products and
    tools that they market, train others to use, and sell.

#  {#section-6 .unnumbered}

# Adult Education and Workforce Development

New economic realities have created a need for workforce and education
policies that better meet employer demands for skilled workers. Current
efforts to develop opportunities for people lacking skills and resources
are focusing on career pathways that integrate education, training, and
skill development in targeted high-wage, high-demand employment areas
(Mazzeo, Rab, and Alssid 2003). Career pathways provide developmental,
adult education, or ESL classes in the context of students' lives and
the work-specific skills they need for employment in particular
industries or sectors. With this type of approach, courses such as
mathematics are modified to incorporate materials from specific fields
into the actual course content. Programs that promote contextual
teaching and learning make heavy use of projects, laboratories,
simulations, and other experiences that enable students to learn by
doing (Jenkins 2002). By integrating instruction in basic skills with
instruction in technical content, contextualized teaching and learning
also enable academically unprepared students to obtain career training
at the same time that they enroll in basic education.

Central to federal efforts to remediate displaced or dislocated workers
is the *WIA*. Title I of *WIA* is the cornerstone legislation in the
federal arsenal as it provides workforce investment services and
activities through a network of One-Stop Career Centers and strategic
planning and oversight by business-led workforce investment boards
(WIBs). Available workforce development activities provided in local
communities can benefit job seekers, laid off workers, youths, incumbent
workers, new entrants to the workforce, veterans, persons with
disabilities, and employers. The purpose of these activities is to
promote an increase in the employment, job retention, earnings, and
occupational skills improvement by participants. This, in turn, is
intended to improve the quality of the workforce, reduce welfare
dependency, and improve the productivity and competitiveness of the
nation.

Adult and laid-off worker services are provided through locally based
One-Stop Career Centers. Comprehensive one-stop centers provide access
to a full range of services pertaining to employment, training and
education, employer assistance, and guidance for obtaining other
assistance. While *WIA* requires one-stop centers to provide specific
services, local areas may design programs and provide services that
reflect the unique needs of their area. *WIA* Title I funds may be used
to support adult education and other literacy activities only if this
instruction is provided in combination with occupational skills or
on-the-job training. Title II of *WIA* authorizes the *Adult Education
and Family Literacy Act*, which provides formula funding to states to
support adult education and literacy services (including workplace
literacy services), family literacy services, and English literacy
programs. Eligible individuals may access adult education programs
funded by *WIA* Title II through One-Stop Career Centers.

Community colleges are also key players in adult education and workforce
development. A recent study by the Education Commission of the States
(Jenkins and Boswell 2002) found that community colleges frequently
offer training to upgrade the skills of workers, which is often provided
under contract to employers and typically does not confer college
credit. They note that a study done by Columbia University's Teachers
College (Bailey et. al. 2004) estimated that, in 1999, 2.3 million
students were enrolled in noncredit, job-related training programs at
community colleges. In addition, community colleges play a key role in
helping unemployed and underemployed adults in basic skills training,
such as ESL, in programs linked to training for jobs. The authors also
offer extensive programs to help welfare recipients enter the workforce.

The study notes that community colleges are designated as the lead
agency to provide workforce training in at least 19 states: Alaska,
Alabama, Arkansas, Colorado, Delaware, Iowa, Kansas, Kentucky, Maine,
Missouri, Mississippi, Nebraska, New Hampshire, Nevada, North Carolina,
North Dakota, Virginia, Washington, and Wisconsin.

The Education Commission of the States (ECS) administered a survey in
2001 to the state agency responsible for community colleges in all 50
states. Five states did not respond: Hawaii, Idaho, Maryland, Montana,
and South Dakota. According to the ECS (2002) report, the majority of
states indicated that the lack of workforce development funding is a
challenge, particularly in terms of making investments in technology to
prepare a technically competent workforce. Further, several states
pointed out the inconsistency in the decreased willingness of
policymakers to support workforce development programs while they stress
the growing importance of a skilled labor force. Major highlights of the
findings were:

-   Eighteen states provide state funding (in addition to federal
    funding) to support occupational training of disadvantaged students
    by community colleges. In all but one of these states, welfare
    recipients are targeted specifically for such training. Other
    targeted groups include low-income adults, displaced workers,
    veterans, the disabled, and at-risk youth.

```{=html}
<!-- -->
```
-   Thirty-two states provide state funding to support customized
    training for employers, with most states imposing some restrictions
    on the use of these funds. The level of funding ranged from under
    \$1 million in several states to \$50 million in New Jersey. In
    general, community colleges compete with other training providers
    for these funds.

-   Twenty states fund noncredit occupational training (separate from
    funding for customized training) at community colleges.

# Summary and Conclusions

This literature review set out to examine research on promising
strategies for strengthening math skills at the postsecondary level. Our
work has indicated that research into instructional practices and
curriculum content methodologies that are specific to developmental
mathematics is largely flawed, lacking in the scientific rigor necessary
to make sound inferences. Most of the studies we reviewed are
methodologically limited by the absence of control or comparison groups,
which makes it virtually impossible to gauge the interventions' true
impact on learning.

In terms of the knowledge necessary for successfully pursuing
college-level math, we have found that no consistent definition of math
standards for college-level preparation exists. However, a number of
studies indicate the need to have a good foundation in arithmetic,
geometry, trigonometry, and algebra I and II. Emerging work also
indicates the increasing need for basic statistics and the ability to
analyze data.

We found that there is less uncertainty or ambiguity in the skills
necessary to pursue college-level math and to succeed in the
highest-paid and highest-skilled jobs. In particular, there seems to be
widespread agreement on the need to think critically, to solve problems,
and to communicate mathematically. Both businesses and postsecondary
institutions indicate that they want people who can identify a problem,
determine whether it can be solved, know which operations and procedures
are required to solve them, use multiple representations (such as graphs
and words) to describe the problems and solutions, and understand and
apply mathematical modeling. However, these are the skills that are the
most difficult to teach and to assess.

It remains to be seen whether community colleges are adopting these
recommendations, in terms of the knowledge, skills, or abilities. It is
also uncertain whether community colleges adequately assess the
knowledge and skills necessary to pursue postsecondary-level math or
succeed in the workplace. Regardless, the majority of two-year colleges
require incoming students to take and pass an assessment test before
they are allowed to enroll in college-level math courses. Given their
prevalence, this may currently be the most relevant benchmark for
whether a person may successfully transition into college-level
mathematics.

While we did not identify existing studies based on gold-standard
research in developmental mathematics at the postsecondary level,
salient themes concerning pedagogy emerged, suggesting promising but
unproven instructional practices that are frequently implemented. These
may warrant further study. We summarize promising key components of
strategies or approaches to developmental mathematics programs at the
postsecondary level into the categories and topics below:

-   **Instructional and pedagogical:** traditional instruction; multiple
    delivery options for students to choose from; computer-assisted
    instruction; Internet-based; self-paced; distance learning;
    calculators; computer algebra systems; spreadsheets; labs;
    small-group instruction; learning communities; contextual learning;
    linkages to and examples from the workplace; and career pathways.

-   **Curriculum content:** nonstandard topics covered in developmental
    math courses or topics that vary by career path; length of
    instruction; and types of activities used to reinforce the material.

-   **Professional development:** faculty training and development; and
    full-time versus part-time instructors.

-   **Supporting strategies:** counseling; and assessment, placement,
    and exit strategies.

-   **Learner and institutional characteristics:** full-time versus
    half-time community college student; socioeconomic attributes of
    learner; workplace program; and servicemember.

**Implications for Further Research**

Additional research is necessary to understand what works in
developmental mathematics. In particular, we need to understand more
precisely why students drop out of developmental math courses. Is it
because of the material covered, the instructional methods used,
challenges outside of the classroom such as financial or family
constraints, or some combination of all of these factors? As a first
step in enhancing that understanding, researchers need to gather more
information concerning (a) a variety of outcomes, such as developmental
math course pass rates, persistence to and pass rates of developmental
mathematics students in higher-level math courses, transfer rates to
other institutions, and graduation rates, (b) student characteristics
(e.g., race and ethnicity, age, gender, highest education credential,
socioeconomic status), and (c) the relationship of each of these
characteristics to the various outcomes. This information could then
begin to address important questions, such as whether a particular
pedagogical approach benefits all students equally regardless of their
education credential, age, and other characteristics, and whether the
benefits persist to higher-level math courses and ultimately, to
graduation.

# Appendix A: Mathematics Knowledge and Skills for Success From Conley and Bodone (2002)

The key knowledge and skills considered to be necessary for success in
mathematics include the following:

-   Computation

    -   The student will know basic mathematics operations by being able
        to:

        -   Use arithmetic operations with fractions;

        -   Use exponents and scientific notation;

        -   Use whole numbers to perform all basic arithmetic
            operations, including long division with and without
            remainders;

        -   Use radicals correctly;

        -   Understand relative magnitude and absolute value;

        -   Know terminology for real numbers, such as irrational
            numbers, natural numbers, integers, and rational numbers;
            and

        -   Use the correct order of arithmetic operations;

    -   The student will know and carefully record symbolic
        manipulations.

    -   The student will know and demonstrate fluency with mathematical
        notation and computation by being able to:

        -   Perform addition, subtraction, multiplication and division;

        -   Perform appropriate basic operations on sets; and

        -   Recognize alternative symbols (e.g., Greek letters).

-   Algebra

    -   The student will know and apply basic algebraic concepts by
        being able to:

        -   Use the distributive property to multiply polynomials;

        -   Multiply and divide polynomials;

        -   Factor polynomials;

        -   Add, subtract, multiply, divide, and simplify rational
            expressions including finding common denominators;

        -   Understand properties and basic theorems of roots and
            exponents; and

        -   Understand properties and basic theorems of logarithms.

    -   The student will use various techniques to solve basic equations
        and inequalities by being able to:

        -   Solve linear equations and absolute value equations;

        -   Solve linear inequalities and absolute value inequalities;

        -   Solve systems of linear equations and inequalities using
            algebraic and graphic methods;

        -   Solve quadratic equations using various methods and
            recognize real solutions by being able to:

            -   Use factoring and zero products;

            -   Use completing the square; and

            -   Use the quadratic formula.

    -   The student will be able to recognize and use basic algebraic
        forms by being able to:

        -   Distinguish between expression, formula, equation, and
            function and recognize when simplifying, solving,
            substituting in, or evaluating is appropriate;

        -   Determine whether a relation is a function;

        -   Understand applications;

        -   Use a variety of models to represent functions, patterns,
            and relationships;

        -   Understand terminology and notation used to define
            functions; and

        -   Understand the general properties and characteristics of
            many types of functions (e.g., direct and inverse variation,
            general polynomial, radical, step, exponential, logarithmic,
            and sinusoidal).

    -   The student will understand the relationship between equations
        and graphs by being able to:

        -   Understand slope-intercept form of a equation of a line and
            graph the line;

        -   Graph a quadratic function and recognize the intercepts as
            solutions to a corresponding quadratic equation; and

        -   Know the basic shape of the graph of an exponential
            function.

    -   The student will know how to use algebra both procedurally and
        conceptually by being able to:

        -   Recognize which type of model best fits the context of a
            situation.

    -   The student will demonstrate ability to algebraically work with
        formulas and symbols by being able to:

        -   Understand formal notation and various applications of
            sequences and series.

-   Trigonometry

    -   The student will know and understand basic trigonometric
        principles by being able to:

        -   Know the definitions of the trigonometric ratios---sine,
            cosine, and tangent---using right triangle trigonometry and
            position on the unit circle;

        -   Understand the relationship between a trigonometric function
            in standard form and its corresponding graph;

        -   Know and use identities for sum and difference of angles;

        -   Recognize periodic graphs;

        -   Understand concepts of periodic and exponential functions
            and their relationships to trigonometric formula, exponents,
            and logarithms;

        -   Solve problems using exponential models; and

        -   Understand and use double and half angle formulas.

-   Geometry

    -   The student will know synthetic (i.e., pictorial) geometry by
        being able to:

        -   Use properties of parallel and perpendicular lines in
            working with angles;

        -   Know triangle properties;

        -   Understand the concept of mathematical proofs, their
            structure and use;

        -   Use geometric constructions to complete simple proofs, to
            model, and to solve mathematical and real-world problems;
            and

        -   Use similar triangles to find unknown angle measurements and
            lengths of sides.

    -   The student will know analytic (i.e., coordinate) geometry by
        being able to:

        -   Know geometric properties of lines;

        -   Know the equations for conic sections;

        -   Use the Pythagorean Theorem and its converse and properties
            of special right triangles to solve mathematical and
            real-world problems;

        -   Use transformations of figures to graph simple variations of
            equations for basic graphs;

        -   Set up appropriate coordinate system for applications; and

        -   Understand vectors in mathematical settings.

    -   The student will understand the relationships between geometry
        and algebra by being able to:

        -   Know how to manipulate conics;

        -   Understand that objects and relations in geometry correspond
            directly to objects and relations in algebra; and

        -   Solve real-world problems using three-dimensional objects.

    -   The student will demonstrate geometric reasoning by being able
        to:

        -   Prove congruency of triangles; and

        -   Use inductive and deductive reasoning to make observations
            about and to verify properties of and relationships among
            figures.

    -   The student will be able to combine algebra, geometry, and
        trigonometry by being able to:

        -   Understand and use the law of sines and the law of cosines;
            and

        -   Use properties of and relationships among figures to solve
            mathematical and real-world problems.

-   Mathematical Reasoning

    -   The student will demonstrate an ability to solve problems by
        being able to:

        -   Use inductive reasoning;

        -   Demonstrate ability to visualize;

        -   Use multiple representations to solve problems;

        -   Use a framework or mathematical logic to solve problems that
            combine several steps;

        -   Use a variety of strategies to understand new mathematical
            content and to develop more efficient solution methods or
            problem extensions; and

        -   Construct logical verifications or counter examples to test
            conjectures and to justify algorithms and solutions to
            problems.

    -   The student will understand various representations by being
        able to:

        -   Understand abstract mathematical ideas in word problems,
            pictorial representations, and applications.

    -   The student will demonstrate a thorough understanding of
        mathematics used in applications by being able to:

        -   Understand the concept of a function.

    -   The student will demonstrate strong memorization skills by being
        able to:

        -   Know a variety of formulas and short proofs.

    -   The student will know how to estimate by being able to:

        -   Understand the relationships among equivalent number
            representations;

        -   Know when an estimate or approximation is more appropriate
            than an exact solution for a variety of problem situations;
            and

        -   Recognize the validity of an estimated number.

    -   The student will understand the appropriate use of technology by
        being able to:

        -   Know the appropriate uses of calculators and their
            limitations;

        -   Perform difficult computations using a calculator;

        -   Know how to use graphing calculators;

    -   The student will be able to generalize (e.g., to go from general
        to abstract and back and to go from specifics to abstract and
        back) by being able to:

        -   Determine the mathematical concept from the context of a
            real-world problem, solve the problem, and interpret the
            solution in the context of the real-world problem.

    -   The student will be willing to experiment with mathematics by
        being able to:

        -   Understand that math problems can have multiple solutions
            and multiple methods to determine the solution(s).

    -   Students will emphasize process over mere outcome(s) by being
        able to:

        -   Understand the various steps to a solution.

    -   The student will show ability to modify patterns and
        computations for different situations by being able to:

        -   Compare a variety of patterns and sequences.

    -   The student will use trial and error to solve problems by being
        able to:

        -   Find the way(s) that did not work to solve a problem and
            finally find the one(s) that do work.

    -   The student will understand the role of mathematics by being
        able to:

        -   Know the relationship between the various disciplines of
            math; and

        -   Understand the connections between mathematics and other
            disciplines.

    -   The student will use mathematical models by being able to:

        -   Use mathematical models from other disciplines.

    -   The student will understand the need to be an active participant
        in the process of learning mathematics by being able to:

        -   Ask questions throughout multistep projects, recognizing
            natural questions arising from a mathematical solution;

        -   Use appropriate math terminology; and

        -   Understand that mathematical problem soling takes time.

    -   The student will understand that mathematics is a symbolic
        language and that fluency requires practice by being able to:

        -   Translate simple statements into equations; and

        -   Understand the role of written symbols in representing
            mathematical ideas and the precise use of special symbols of
            mathematics.

-   Statistics

    -   The student will understand and apply concepts of statistics and
        data analysis by being able to:

        -   Select and use the best method of representing and
            describing a set of data;

        -   Understand measures of central tendency and variability and
            their application to specific situations; and

        -   Understand different methods of curve-fitting and various
            applications.

# Appendix B: Summary of Studies Related to Developmental Mathematics Reviewed in This Report

+--------+----------+-----------+----------------+-------------------+
| **S    | **Ap     | -   **    | -              | -   **Research    |
| tudy** | proach** | Metrics** |    **Outcome** |     Issues**      |
+========+==========+===========+================+===================+
| Adams  | Combined | -   Top   | -   Final test | -   No control    |
| 2003   | two-     |     60    |     scores     |     for student   |
|        | semester |           |     similar    |                   |
|        | course   |   percent |                |   characteristics |
|        | in one   |     of    | -              |                   |
|        | semester |           |  Developmental | -   Precise       |
|        |          |   scorers |     students   |     measures of   |
|        |          |     in    |     had 7      |     differences   |
|        |          |     a     |     percentage |     not stated    |
|        |          | ssessment |     points     |                   |
|        |          |     test  |     higher     |                   |
|        |          |           |     pass rate  |                   |
|        |          | -   C     |     in         |                   |
|        |          | omparison |     elementary |                   |
|        |          |     of    |     calculus   |                   |
|        |          |     final |                |                   |
|        |          |     test  | -   Results    |                   |
|        |          |           |     not as     |                   |
|        |          |    scores |     good for   |                   |
|        |          |     for   |                |                   |
|        |          |     those |    engineering |                   |
|        |          |     in    |     calculus   |                   |
|        |          |     devel |                |                   |
|        |          | opmental/ |                |                   |
|        |          |           |                |                   |
|        |          |   regular |                |                   |
|        |          |           |                |                   |
|        |          |    course |                |                   |
|        |          |     vs.   |                |                   |
|        |          |     those |                |                   |
|        |          |     in    |                |                   |
|        |          |           |                |                   |
|        |          |   regular |                |                   |
|        |          |           |                |                   |
|        |          |    course |                |                   |
|        |          |     not   |                |                   |
|        |          |           |                |                   |
|        |          | requiring |                |                   |
|        |          |     re    |                |                   |
|        |          | mediation |                |                   |
|        |          |           |                |                   |
|        |          | -   Pass  |                |                   |
|        |          |     rates |                |                   |
|        |          |     in    |                |                   |
|        |          |           |                |                   |
|        |          |    higher |                |                   |
|        |          |     math  |                |                   |
|        |          |           |                |                   |
|        |          |   courses |                |                   |
|        |          |           |                |                   |
|        |          |  compared |                |                   |
|        |          |     for   |                |                   |
|        |          |     two   |                |                   |
|        |          |           |                |                   |
|        |          |    groups |                |                   |
+--------+----------+-----------+----------------+-------------------+
| At     | T        | -         | -   Average    | -   Use of        |
| kinson | utorials |   Pretest |     posttest   |     regress       |
| 2003   | in       |     and   |     score was  | ion-discontinuity |
|        | Problem  |           |                |     design        |
|        | Solving  |  posttest |  significantly |     ensures       |
|        | (TiPS)   |           |     higher for |     validity of   |
|        | for      |  measures |     students   |     results       |
|        | ar       |           |     using TiPS |                   |
|        | ithmetic | -   Rel   |     than       |                   |
|        | and      | ationship |     similarly  |                   |
|        | problem  |           |     assessed   |                   |
|        | -solving |   between |     students   |                   |
|        | skills   |           |     in an      |                   |
|        | in       |    length |     untreated  |                   |
|        | adults   |     of    |     control    |                   |
|        |          |     in    |     group      |                   |
|        |          | struction |                |                   |
|        |          |     and   | -   While      |                   |
|        |          |           |     originally |                   |
|        |          |  assessed |     intended   |                   |
|        |          |     mat   |     for Navy   |                   |
|        |          | hematical |     sailors,   |                   |
|        |          |           |     results    |                   |
|        |          |   ability |     indicated  |                   |
|        |          |     prior |     TiPS has   |                   |
|        |          |     to    |     much wider |                   |
|        |          |     using |                |                   |
|        |          |     TiPS  |  applicability |                   |
|        |          |           |     to adult   |                   |
|        |          |           |     literacy   |                   |
|        |          |           |     programs   |                   |
|        |          |           |     and        |                   |
|        |          |           |     workforce  |                   |
|        |          |           |     training   |                   |
|        |          |           |     programs   |                   |
+--------+----------+-----------+----------------+-------------------+
| C      | Tra      | -         | -   Retention  | -                 |
| artnal | ditional |   Success |     and        |    Self-selection |
| 1999   | vs       |           |                |     bias          |
|        | c        | -         |    persistence |                   |
|        | omputer- | Retention |     highest in | -   No control    |
|        | assisted |           |     com        |     for student   |
|        | ins      | -   Pe    | puter-assisted |                   |
|        | truction | rsistence |                |   characteristics |
|        | in       |           |    instruction |                   |
|        | el       |           |                | -   No            |
|        | ementary |           | -   Success    |     statistical   |
|        | and      |           |     highest in |     test of       |
|        | inte     |           |                |     significance  |
|        | rmediate |           |    traditional |     reported      |
|        | algebra  |           |                |                   |
|        |          |           | -   Recommends |                   |
|        |          |           |     further    |                   |
|        |          |           |     research   |                   |
+--------+----------+-----------+----------------+-------------------+
| Creery | Lecture  | -   Final | -   No         | -                 |
| 2001   | vs.      |           |     difference |    Self-selection |
|        | se       |    grades |     in final   |     bias          |
|        | lf-paced |           |     grades or  |                   |
|        | vs.      | -   Pe    |                | -   No control    |
|        | online   | rsistence |    persistence |     for student   |
|        | ins      |           |     among the  |                   |
|        | truction |           |     three      |   characteristics |
|        |          |           |     methods    |                   |
|        |          |           |                | -   No analysis   |
|        |          |           | -   Recommends |     of            |
|        |          |           |     offering   |     differentials |
|        |          |           |     multiple   |     in            |
|        |          |           |                |     withdrawals   |
|        |          |           |  instructional |                   |
|        |          |           |     modes      |                   |
+--------+----------+-----------+----------------+-------------------+
| DePree | In       | -   Math  | -              | -   Q             |
| 1998   | structor |     c     |    Small-group | uasi-experimental |
|        | vs.      | onfidence |                |     design with   |
|        | sma      |           |    instruction |     students not  |
|        | ll-group | -   Ac    |     had        |     aware of      |
|        | ins      | hievement |                |     modality used |
|        | truction |           |  statistically |     at time of    |
|        |          | -         |     higher     |     registration  |
|        |          |    Course |                |                   |
|        |          |     c     |    confidence, |                   |
|        |          | ompletion |     more       |                   |
|        |          |           |     likely to  |                   |
|        |          |           |     complete   |                   |
|        |          |           |     the course |                   |
|        |          |           |                |                   |
|        |          |           | -              |                   |
|        |          |           |   Improvements |                   |
|        |          |           |     were       |                   |
|        |          |           |     greatest   |                   |
|        |          |           |     for        |                   |
|        |          |           |                |                   |
|        |          |           |  traditionally |                   |
|        |          |           |     un         |                   |
|        |          |           | derrepresented |                   |
|        |          |           |     students   |                   |
|        |          |           |     in         |                   |
|        |          |           |                |                   |
|        |          |           |   mathematics: |                   |
|        |          |           |     Hispanics, |                   |
|        |          |           |     Native     |                   |
|        |          |           |     Americans, |                   |
|        |          |           |     and        |                   |
|        |          |           |     females    |                   |
|        |          |           |                |                   |
|        |          |           | -   No         |                   |
|        |          |           |     difference |                   |
|        |          |           |     in         |                   |
|        |          |           |                |                   |
|        |          |           |    achievement |                   |
+--------+----------+-----------+----------------+-------------------+
| Higbee | Rela     | -   Test  | -   Math and   | -   Authors       |
| and    | tionship |           |     test       |     acknowledge   |
| Thomas | between  |   anxiety |     anxiety    |     that design   |
| 1999   | nonc     |           |     decreased  |     does not      |
|        | ognitive | -   C     |     for        |     allow for     |
|        | v        | onfidence |     students   |     separating    |
|        | ariables |     to    |     in course  |     out effects   |
|        | and      |           |     with       |     of individual |
|        | success  |   succeed |     relaxation |     treatments,   |
|        | in       |     in    |     exercises  |     teaching      |
|        | two      |           |     and        |     ability, and  |
|        | -quarter |  learning |                |     personality   |
|        | algebra  |     math  |  metacognition |     traits of     |
|        | sequence |           |     strategies |     professor     |
|        |          |           |                |                   |
|        |          |           | -   Posttest   | -   No control    |
|        |          |           |     measures   |     group         |
|        |          |           |     of anxiety |                   |
|        |          |           |     were not   | -   Withdrawals   |
|        |          |           |     correlated |     are not       |
|        |          |           |     with       |     accounted for |
|        |          |           |     higher     |                   |
|        |          |           |     course     |                   |
|        |          |           |     grades     |                   |
+--------+----------+-----------+----------------+-------------------+
| Kinney | Tra      | -         | -   No         | -                 |
| 2001   | ditional |    Scores |     difference |    Self-selection |
|        | vs.      |     on    |     in final   |     bias          |
|        | c        |     final |     exam       |                   |
|        | omputer- |     exams |     scores     | -   No control    |
|        | mediated |           |                |     for student   |
|        | ins      | -   C     | -   Both       |                   |
|        | truction | onfidence |     groups     |   characteristics |
|        | in       |     in    |     reported   |                   |
|        | el       |     math  |     increased  | -   No            |
|        | ementary |           |     confidence |     information   |
|        | and      |           |                |     pertaining to |
|        | inte     |           | -   Recommends |     metrics and   |
|        | rmediate |           |     offering   |     tests         |
|        | algebra  |           |     multiple   |     conducted     |
|        |          |           |                |                   |
|        |          |           |  instructional |                   |
|        |          |           |     modes      |                   |
+--------+----------+-----------+----------------+-------------------+
| Lan    | PLATO    | -         | -   Computer   | -   Q             |
| caster | Software |    Course |     Assisted   | uasi-experimental |
| 2001   | vs.      |     wi    |                |     design with   |
|        | tra      | thdrawals |    Instruction |     control group |
|        | ditional |           |     (CAI)      |     consisting of |
|        | lecture  | -   Sat   |     resulted   |     students in   |
|        | in       | isfactory |     in         |     prior year    |
|        | el       |           |     7-percent  |                   |
|        | ementary |    grades |     decrease   | -   No control    |
|        | algebra  |           |     in number  |     for student   |
|        |          |           |     of         |                   |
|        |          |           |                |   characteristics |
|        |          |           |    withdrawals |                   |
|        |          |           |                |                   |
|        |          |           | -   12-percent |                   |
|        |          |           |     increase   |                   |
|        |          |           |     in number  |                   |
|        |          |           |     of         |                   |
|        |          |           |                |                   |
|        |          |           |   satisfactory |                   |
|        |          |           |     grades     |                   |
|        |          |           |                |                   |
|        |          |           | -   11-percent |                   |
|        |          |           |     decrease   |                   |
|        |          |           |     in         |                   |
|        |          |           |                |                   |
|        |          |           | unsatisfactory |                   |
|        |          |           |     grades     |                   |
+--------+----------+-----------+----------------+-------------------+
| Livi   | Computer | -         | -   No         | -   Q             |
| ngston | Algebra  |   Ability |                | uasi-experimental |
| 2001   | System   |     to    |  statistically |     nonrandom     |
|        | (CAS)    |           |                |     control group |
|        | vs.      |   perform |    significant |     design        |
|        | tra      |     ma    |     difference |                   |
|        | ditional | thematics |     in ability | -   No control    |
|        | method   |     by    |     to perform |     for student   |
|        | with     |     hand  |                |                   |
|        | cal      |           |    mathematics |   characteristics |
|        | culators | -         |     by hand    |                   |
|        | in       |   Ability |                | -                 |
|        | inte     |     to    | -   CAS group  |    Self-selection |
|        | rmediate |     solve |     performed  |     bias          |
|        | algebra  |     hig   |     better in  |                   |
|        |          | her-order |     ability to |                   |
|        |          |           |     solve      |                   |
|        |          | reasoning |                |                   |
|        |          |     by    |   higher-order |                   |
|        |          |     hand  |     reasoning  |                   |
|        |          |           |     by hand    |                   |
+--------+----------+-----------+----------------+-------------------+
| McC    | Tra      | -         | -   Without    | -                 |
| lendon | ditional |    Course |                |    Self-selection |
| and    | vs.      |     c     |    eliminating |     bias          |
| M      | le       | ompletion |                |                   |
| cArdle | cture/co |           |   withdrawals, | -   No control    |
| 2002   | mpu-ters |   (letter |                |     for student   |
|        | vs.      |     grade |    traditional |                   |
|        | As       |     of C  |     had        |   characteristics |
|        | sessment |     or    |     highest    |                   |
|        | LEarning |           |     completion | -   No analysis   |
|        | in       |   better) |                |     of why ALEKS  |
|        | K        |           | -   Netting    |     had higher    |
|        | nowledge |           |     out        |     withdrawal    |
|        | Spaces   |           |                |     rate          |
|        | (ALEKS)  |           |   withdrawals, |                   |
|        |          |           |     no         |                   |
|        |          |           |     difference |                   |
|        |          |           |                |                   |
|        |          |           | -   Recommends |                   |
|        |          |           |     offering   |                   |
|        |          |           |     multiple   |                   |
|        |          |           |                |                   |
|        |          |           |  instructional |                   |
|        |          |           |     modes      |                   |
+--------+----------+-----------+----------------+-------------------+
| Mig    | Rela     | -   P     | -   Adult      | -   Withdrawals   |
| lietti | tionship | rinciples |                |     are not       |
| et al. | between  |     of    |  underprepared |     accounted for |
| 2002   | age and  |     Adult |     students   |                   |
|        | gender   |           |     in         |                   |
|        | on       |  Learning |     le         |                   |
|        | learning |           | arner-centered |                   |
|        | styles,  | -   Adult |                |                   |
|        | and      |           |  developmental |                   |
|        | between  | Classroom |     English    |                   |
|        | teacher  |     En    |     classrooms |                   |
|        | style    | vironment |     achieved   |                   |
|        | and      |     Scale |     higher     |                   |
|        | c        |           |     grades     |                   |
|        | lassroom | -         |     than       |                   |
|        | outcomes |  Adaptive |     similar    |                   |
|        |          |     Style |     students   |                   |
|        |          |           |     in         |                   |
|        |          | Inventory |     te         |                   |
|        |          |           | acher-centered |                   |
|        |          |           |     classrooms |                   |
|        |          |           |                |                   |
|        |          |           | -   No age or  |                   |
|        |          |           |     gender     |                   |
|        |          |           |     effects on |                   |
|        |          |           |     classroom  |                   |
|        |          |           |                |                   |
|        |          |           |    environment |                   |
|        |          |           |     or         |                   |
|        |          |           |     learning   |                   |
|        |          |           |     style      |                   |
|        |          |           |     preference |                   |
|        |          |           |                |                   |
|        |          |           | -   Effect of  |                   |
|        |          |           |     age and    |                   |
|        |          |           |     teaching   |                   |
|        |          |           |     style on   |                   |
|        |          |           |                |                   |
|        |          |           |  developmental |                   |
|        |          |           |                |                   |
|        |          |           |    mathematics |                   |
|        |          |           |     was not    |                   |
|        |          |           |     possible   |                   |
+--------+----------+-----------+----------------+-------------------+
| Quinn  | PLATO    | -         | -              | -   No comparison |
| 2003   | Software |  Absolute |  Statistically |     group         |
|        | for      |           |                |                   |
|        | ar       |  increase |    significant | -   Does not      |
|        | ithmetic |     in    |     increase   |     control for   |
|        | and      |           |     for all    |     student       |
|        | el       |    scores |     students   |                   |
|        | ementary |     on    |     in pretest |   characteristics |
|        | algebra  |     com   |     and        |                   |
|        |          | puterized |     posttest   |                   |
|        |          |           |     CPT scores |                   |
|        |          | placement |                |                   |
|        |          |     test  | -   Each       |                   |
|        |          |     (CPT) |     additional |                   |
|        |          |           |     hour spent |                   |
|        |          | -   Rel   |     results in |                   |
|        |          | ationship |     0.61 to    |                   |
|        |          |           |     1.86       |                   |
|        |          |   between |     percent    |                   |
|        |          |     time  |                |                   |
|        |          |     spent |    improvement |                   |
|        |          |     on    |     in CPT     |                   |
|        |          |           |     posttest   |                   |
|        |          |  software |     score      |                   |
|        |          |     and   |                |                   |
|        |          |           |                |                   |
|        |          |  increase |                |                   |
|        |          |     in    |                |                   |
|        |          |     CPT   |                |                   |
|        |          |           |                |                   |
|        |          |    scores |                |                   |
+--------+----------+-----------+----------------+-------------------+
| Si     | Rela     | -         | -   Average    | -   Factors       |
| nclair | tionship |   Average |     GPA is     |     affecting     |
| Com    | between  |     GPA   |                |     choice to     |
| munity | time     |     in    |  significantly |     delay         |
| C      | since    |           |     lower when |     subsequent    |
| ollege | last     |    course |     students   |     math courses  |
| 2003   | devel    |           |     delay 3    |     are not       |
|        | opmental | -         |     terms      |     measured or   |
|        | mat      |   Success |                |     controlled    |
|        | hematics |     rate  | -   Success    |     for           |
|        | course   |           |     rate is    |                   |
|        | and      |           |     not        |                   |
|        | first    |           |     affected   |                   |
|        | colle    |           |                |                   |
|        | ge-level |           |                |                   |
|        | math     |           |                |                   |
|        | course   |           |                |                   |
+--------+----------+-----------+----------------+-------------------+
| Way    | Lecture  | -   Pass  | -   Success    | -                 |
| caster | with lab |     rate  |     was        |    Self-selection |
| 2001   | vs.      |           |                |     bias          |
|        | indivi   | -         |    independent |                   |
|        | dualized | Retention |     of         | -   No control    |
|        | comput   |           |                |     for student   |
|        | er-aided | -   G     |    instruction |                   |
|        | ins      | raduation |     method     |   characteristics |
|        | truction |     rate  |                |                   |
|        |          |           | -              | -   No            |
|        |          |           |  Developmental |     statistical   |
|        |          |           |                |     test for      |
|        |          |           |    mathematics |     differences   |
|        |          |           |     students   |     conducted     |
|        |          |           |     had higher |                   |
|        |          |           |     retention  | -   Metrics not   |
|        |          |           |     than       |     well-defined  |
|        |          |           |     regular    |                   |
|        |          |           |     students   |                   |
+--------+----------+-----------+----------------+-------------------+
| W      | P        | -   Mi    | -   Nonnative  | -                 |
| heland | erceived | dsemester |                |    Self-selection |
| et al. | in       |     tests |    instructors |     bias          |
| 2003   | hibitors |           |     do not     |                   |
|        | to       | -   Final |     have       | -   Does not      |
|        | student  |     exam  |     negative   |     control for   |
|        | success  |           |     impact on  |     student       |
|        | in       |    scores |     success    |                   |
|        | inte     |           |                |   characteristics |
|        | rmediate | -   GPA   | -              |                   |
|        | algebra  |           |    Performance |                   |
|        |          |           |     in         |                   |
|        |          |           |                |                   |
|        |          |           |   intermediate |                   |
|        |          |           |     algebra    |                   |
|        |          |           |     correlated |                   |
|        |          |           |     with       |                   |
|        |          |           |     overall    |                   |
|        |          |           |     semester   |                   |
|        |          |           |     GPA        |                   |
|        |          |           |                |                   |
|        |          |           | -   Attendance |                   |
|        |          |           |     highly     |                   |
|        |          |           |     correlated |                   |
|        |          |           |     with       |                   |
|        |          |           |     success    |                   |
|        |          |           |                |                   |
|        |          |           | -   Grade in   |                   |
|        |          |           |                |                   |
|        |          |           |   intermediate |                   |
|        |          |           |     algebra    |                   |
|        |          |           |     predictive |                   |
|        |          |           |     of         |                   |
|        |          |           |                |                   |
|        |          |           |    performance |                   |
|        |          |           |     in higher  |                   |
|        |          |           |     math       |                   |
+--------+----------+-----------+----------------+-------------------+

########  {#section-7 .unnumbered}

# Appendix C: Comparable ASSET,[^16] ACT,[^17] and COMPASS[^18] Cutoff Scores For Student Placement into Mathematical Courses

This table provides information concerning the most typical cutoff
scores used by colleges on the ASSET, ACT, and COMPASS tests for
placement into various levels of mathematics courses. For instance,
reading across the top row, colleges that use the ASSET test generally
place a student who scores between 23 and 40 on the Numerical Skills
component of that test into a course that provides a basic arithmetic
review. Institutions that use the ACT Mathematics subtest instead would
place students scoring below 18 into a similar course, and institutions
that use the COMPASS would place a student into a similar course if he
or she scored less than a 44 on the Pre-algebra component.

[]{#__RefHeading___Toc114303161 .anchor}Table C-1: Asset, Act, and
Compass Cutoff Scores for Student Placement

+-----------------+---------+---------------+------------------------+
| ASSET Scores    | ACT     | COMPASS       | Course Recommendations |
|                 | Math    | Scores        |                        |
+=================+=========+===============+========================+
| ####### N       |         | #######       |                        |
| umerical Skills |         |  Pre--algebra |                        |
+-----------------+---------+---------------+------------------------+
| 23--40          | 0--17   | 0--43         | Arithmetic review      |
+-----------------+---------+---------------+------------------------+
| 41--55          | 18--20  | 44--100       | Elementary algebra or  |
|                 |         |               | courses with           |
|                 |         |               | arithmetic             |
|                 |         |               | prerequisite           |
+-----------------+---------+---------------+------------------------+
| Elementary      |         | Algebra       |                        |
| Algebra         |         |               |                        |
+-----------------+---------+---------------+------------------------+
| 23--40          | 18--20  | 0--45         | Elementary algebra or  |
|                 |         |               | courses with           |
|                 |         |               | arithmetic             |
|                 |         |               | prerequisite           |
+-----------------+---------+---------------+------------------------+
| 41--55          | 21--22  | 46--65        | Intermediate algebra   |
|                 |         |               | or courses with        |
|                 |         |               | elementary algebra     |
|                 |         |               | prerequisite           |
+-----------------+---------+---------------+------------------------+
| Intermediate    |         |               |                        |
| Algebra         |         |               |                        |
+-----------------+---------+---------------+------------------------+
| 23--40          | 21--22  | 46--65        | Intermediate algebra   |
|                 |         |               | or courses with        |
|                 |         |               | elementary algebra     |
|                 |         |               | prerequisite           |
+-----------------+---------+---------------+------------------------+
| 41--55          | 23--25  | 66--100       | College algebra or     |
|                 |         |               | courses with           |
|                 |         |               | intermediate algebra   |
|                 |         |               | prerequisite           |
+-----------------+---------+---------------+------------------------+
| #######         |         | ####### Co    |                        |
| College Algebra |         | llege Algebra |                        |
+-----------------+---------+---------------+------------------------+
| 23--40          | 23--25  | 0--45         | College algebra or     |
|                 |         |               | courses with           |
|                 |         |               | intermediate algebra   |
|                 |         |               | prerequisite           |
+-----------------+---------+---------------+------------------------+
| 41--55          | 26--27  | 46--100       | Trigonometry or        |
|                 |         |               | business calculus or   |
|                 |         |               | courses with college   |
|                 |         |               | algebra prerequisite   |
+-----------------+---------+---------------+------------------------+
|                 |         | #######       |                        |
|                 |         |  Trigonometry |                        |
+-----------------+---------+---------------+------------------------+
|                 | 26--27  | 0--45         | Trigonometry or        |
|                 |         |               | business calculus or   |
|                 |         |               | courses with college   |
|                 |         |               | algebra prerequisite   |
+-----------------+---------+---------------+------------------------+
|                 | 28--36  | 46--100       | Calculus 1 or courses  |
|                 |         |               | with college algebra   |
|                 |         |               | and trigonometry       |
|                 |         |               | prerequisites          |
+-----------------+---------+---------------+------------------------+

Source: Roth 2003

# Appendix D: Level of Proficiency Associated With ACCUPLACER[^19] Cutoff Scores

[]{#__RefHeading___Toc114303162 .anchor}Level of Proficiency Associated
With ACCUPLACER Cutoff Scores

+-----------------------+-----------------------+-----------------------+
| Arithmetic\           | Elementary Algebra    | College-level         |
| Proficiency           | Proficiency           | Mathematics           |
|                       |                       | Proficiency           |
+=======================+=======================+=======================+
| **Score of            | **Score of            | **Score of 39 or      |
| 38**--**-64**         | 28**--**43**          | less**                |
+-----------------------+-----------------------+-----------------------+
| Students at this      | Students at this      | These students should |
| level have minimal    | level have minimal    | take the elementary   |
| arithmetic skills.    | pre-algebra skills.   | algebra test before   |
| These students can:   | These students        | any placement         |
|                       | demonstrate:          | decisions are         |
| -   perform simple    |                       | finalized.            |
|     operations with   | -   a sense of order  |                       |
|     whole numbers and |     relationships and |                       |
|     decimals          |     the relative size |                       |
|     (addition,        |     of signed numbers |                       |
|     subtraction, and  |                       |                       |
|     multiplication)   | -   the ability to    |                       |
|                       |     multiply a whole  |                       |
| -   calculate an      |     number by a       |                       |
|     average, given    |     binomial          |                       |
|     integer values    |                       |                       |
|                       |                       |                       |
| -   solve simple word |                       |                       |
|     problems          |                       |                       |
|                       |                       |                       |
| -   identify data     |                       |                       |
|     represented by    |                       |                       |
|     simple graphs     |                       |                       |
+-----------------------+-----------------------+-----------------------+
| **Score of 65--92**   | **Score of            | **Score of            |
|                       | 44**--**81**          | 40**--**62**          |
+-----------------------+-----------------------+-----------------------+
| Students at this      | Students scoring at   | Students scoring at   |
| level have basic      | this level have       | this level can:       |
| arithmetic skills.    | minimal elementary    |                       |
| These students can:   | algebra skills. These | -   identify common   |
|                       | students can:         |     factors           |
| -   perform the basic |                       |                       |
|     arithmetic        | -   perform           | -   factor binomials  |
|     operations of     |     operations with   |     and trinomials    |
|     addition,         |     signed numbers    |                       |
|     subtraction,      |                       | -   manipulate        |
|     multiplication,   | -   combine like      |     factors to        |
|     and division      |     terms             |     simplify complex  |
|     using whole       |                       |     fractions         |
|     numbers,          | -   multiply          |                       |
|     fractions,        |     binomials         | These students should |
|     decimals, and     |                       | be considered for     |
|     mixed numbers     | -   evaluate          | placement into        |
|                       |     algebraic         | intermediate algebra. |
| -   make conversions  |     expressions       | For further guidance  |
|     among fractions,  |                       | in placement, have    |
|     decimals, and     |                       | these students take   |
|     percents          |                       | the elementary        |
|                       |                       | algebra test.         |
+-----------------------+-----------------------+-----------------------+
| **Score of            | **Score of            | **Score of            |
| 93**--**109**         | 82**--**108**         | 63**--**85**          |
+-----------------------+-----------------------+-----------------------+
| Students at this      | Students at this      | Students scoring at   |
| level have adequate   | level have sufficient | this level can        |
| arithmetic skills.    | elementary algebra    | demonstrate the       |
| These students can:   | skills. By this       | following additional  |
|                       | level, the skills     | skills:               |
| -   estimate products | that were beginning   |                       |
|     and squares of    | to emerge at a "total | -   work with         |
|     decimals and      | right score" of 57    |     algebraic         |
|     square roots of   | (i.e. 57 correct)     |     expressions       |
|     whole numbers and | have been developed.  |     involving real    |
|     decimals          | Students at this      |     number exponents  |
|                       | level can:            |                       |
| -   solve simple      |                       | -   factor polynomial |
|     percent problems  | -   add radicals, add |     expressions       |
|     of the form p% of |     algebraic         |                       |
|     q = ? and ?% of q |     fractions, and    | -   simplify and      |
|     = r               |     evaluate          |     perform           |
|                       |     algebraic         |     arithmetic        |
| -   divide whole      |     expressions       |     operations with   |
|     numbers by        |                       |     rational          |
|     decimals and      | -   factor quadratic  |     expressions,      |
|     fractions         |     expressions in    |     including complex |
|                       |     the form ax^2^ +  |     fractions         |
| -   solve simple word |     bx + c, where a = |                       |
|     problems          |     1                 | -   solve and graph   |
|     involving         |                       |     linear equations  |
|     fractions, ratio, | -   factor the        |     and inequalities  |
|     percent increase  |     difference of     |                       |
|     and decrease, and |     squares           | -   solve absolute    |
|     area              |                       |     value equations   |
|                       | -   square binomials  |                       |
|                       |                       | -   solve quadratic   |
|                       | -   solve linear      |     equations by      |
|                       |     equations with    |     factoring         |
|                       |     integer           |                       |
|                       |     coefficients      | -   graph simple      |
|                       |                       |     parabolas         |
|                       |                       |                       |
|                       |                       | -   understand        |
|                       |                       |     function          |
|                       |                       |     notation, such as |
|                       |                       |     determining the   |
|                       |                       |     value of a        |
|                       |                       |     function for a    |
|                       |                       |     specific number   |
|                       |                       |     in the domain     |
|                       |                       |                       |
|                       |                       | -   a limited         |
|                       |                       |     understanding of  |
|                       |                       |     the concept of    |
|                       |                       |     function on a     |
|                       |                       |     more              |
|                       |                       |     sophisticated     |
|                       |                       |     level, such as    |
|                       |                       |     determining the   |
|                       |                       |     value of the      |
|                       |                       |     composition of    |
|                       |                       |     two functions     |
|                       |                       |                       |
|                       |                       | -   a rudimentary     |
|                       |                       |     understanding of  |
|                       |                       |     coordinate        |
|                       |                       |     geometry and      |
|                       |                       |     trigonometry      |
|                       |                       |                       |
|                       |                       | These students should |
|                       |                       | be considered for     |
|                       |                       | placement into        |
|                       |                       | college algebra or a  |
|                       |                       | credit-bearing course |
|                       |                       | immediately preceding |
|                       |                       | calculus.             |
+-----------------------+-----------------------+-----------------------+
| **Score of 100+**     | **Score of 109+**     | **Score of            |
|                       |                       | 86**--**102**         |
+-----------------------+-----------------------+-----------------------+
| Students at this      | Students at this      | Students scoring at   |
| level have            | level have            | this level can        |
| substantial           | substantial           | demonstrate the       |
| arithmetic skills.    | elementary algebra    | following additional  |
| These students can:   | skills. These         | skills:               |
|                       | students can:         |                       |
| -   find equivalent   |                       | -   understand        |
|     forms of          | -   simplify          |     polynomial        |
|     fractions         |     algebraic         |     functions         |
|                       |     expressions       |                       |
| -   estimate          |                       | -   evaluate and      |
|     computations      | -   factor quadratic  |     simplify          |
|     involving         |     expressions where |     expressions       |
|     fractions         |     a = 1             |     involving         |
|                       |                       |     functional        |
| -   solve simple      | -   solve quadratic   |     notation,         |
|     percent problems  |     equations         |     including         |
|     of the form p% of |                       |     composition of    |
|     ? = r             |                       |     functions         |
|                       |                       |                       |
| -   solve word        |                       | -   solve simple      |
|     problems          |                       |     equations         |
|     involving the     |                       |     involving:        |
|     manipulation of   |                       |                       |
|     units of          |                       | -   trigonometric     |
|     measurement       |                       |     functions         |
|                       |                       |                       |
| -   solve complex     |                       | -   logarithmic       |
|     word problems     |                       |     functions         |
|     involving         |                       |                       |
|     percent, average, |                       | -   exponential       |
|     and proportional  |                       |     functions         |
|     reasoning         |                       |                       |
|                       |                       | These students can be |
| -   find the square   |                       | considered for a      |
|     root of decimal   |                       | pre-calculus course   |
|     numbers           |                       | or a non-rigorous     |
|                       |                       | course in beginning   |
| -   solve simple      |                       | calculus.             |
|     number sentences  |                       |                       |
|     involving a       |                       |                       |
|     variable          |                       |                       |
+-----------------------+-----------------------+-----------------------+
|                       |                       | **Score of 103+**     |
+-----------------------+-----------------------+-----------------------+
|                       |                       | Students scoring at   |
|                       |                       | this level can        |
|                       |                       | demonstrate the       |
|                       |                       | following additional  |
|                       |                       | skills:               |
|                       |                       |                       |
|                       |                       | -   perform algebraic |
|                       |                       |     operations and    |
|                       |                       |     solve equations   |
|                       |                       |     with complex      |
|                       |                       |     numbers           |
|                       |                       |                       |
|                       |                       | -   understand the    |
|                       |                       |     relationship      |
|                       |                       |     between exponents |
|                       |                       |     and logarithms    |
|                       |                       |     and the rules     |
|                       |                       |     that govern the   |
|                       |                       |     manipulation of   |
|                       |                       |     logarithms and    |
|                       |                       |     exponents         |
|                       |                       |                       |
|                       |                       | -   understand        |
|                       |                       |     trigonometric     |
|                       |                       |     functions and     |
|                       |                       |     their inverses    |
|                       |                       |                       |
|                       |                       | -   solve             |
|                       |                       |     trigonometric     |
|                       |                       |     equations         |
|                       |                       |                       |
|                       |                       | -   manipulate        |
|                       |                       |     trigonometric     |
|                       |                       |     identities        |
|                       |                       |                       |
|                       |                       | -   solve             |
|                       |                       |     right-triangle    |
|                       |                       |     problems          |
|                       |                       |                       |
|                       |                       | -   recognize graphic |
|                       |                       |     properties of     |
|                       |                       |     functions such as |
|                       |                       |     absolute value,   |
|                       |                       |     quadratic, and    |
|                       |                       |     logarithmic       |
|                       |                       |                       |
|                       |                       | These students should |
|                       |                       | be considered for     |
|                       |                       | placement into        |
|                       |                       | calculus.             |
+-----------------------+-----------------------+-----------------------+

Source: Murphy, S. 2002

########   {#section-8 .unnumbered}

######## Bibliography

> ACT. 2003. *The 2003 national score report.*
> http://www.act.org/news/data/03/ index.html (accessed Oct. 20, 2003).
>
> Adams, W. 2003. Developmental mathematics: A new approach. *Focus* 23,
> No. 9*.* Mathematical Association of America. (Dec. pp. 4--5).
>
> AFL-CIO Web site. http://www.aflcio.org. (accessed Oct. 25, 2003).
>
> Alamprese, J. S. 1998. *Adult basic education: Strategies for
> supporting learning*. Bethesda, MD: Abt Associates, Inc.
>
> ---------. 2001. Teaching reading to first-level adults: Emerging
> trends in research and practice. *Focus on Basics*. 5(A).
>
> Alamprese, J., S. Labaree, and J. Voight. 1998. *Case studies of
> organizational and instructional strategies for first-level learners
> in adult basic education*. Bethesda, MD: Abt Associates, Inc.
>
> Alinea Group Web site. http://www.alineagroup.com (accessed Nov. 25,
> 2003).
>
> American Diploma Project. *Ready or Not: Creating a High School
> Diploma That Counts.* The American Diploma Project. Achieve, Inc. 2004
>
> American Management Association. *2001 AMA Survey on Workplace
> Testing: Basic Skills, Job Skills, Psychological Measurement.*
> amanet.org/research/ pdfs/bjp_2001.pdf (accessed Oct. 20, 2003).
>
> American Mathematical Association of Two Year Colleges. 1995.
> *Crossroads in mathematics: Standards for introductory college
> mathematics before calculus*. D. Cohen, ed. Memphis, TN: American
> Mathematical Association of Two Year Colleges.
>
> ---------. 2002a. *The Crossroads Revisited Project: The American
> Mathematical Association of Two-Year Colleges Executive Summary of
> Research on the Effect of Crossroads.*
> www.amatyc.org/Crossroads/execsummaryARG.html (accessed Nov. 20,
> 2003).
>
> ---------. 2002b. *A Vision: Final Report from the National Conference
> on Technical Mathematics for Tomorrow.*
> www.wake.tec.nc.us/\~rlkimbal/CRAFTY/Webvision.pdf (accessed Feb. 10,
> 2004).
>
> Armed Services Vocational Aptitude Battery Web site.
> http://www.asvabprogram.com/index.cfm?fuseaction=overview.test
> (accessed Oct. 22, 2003).
>
> Aslanian, C. B., and H. M. Brickell. 1980. *Americans in transition:
> Life changes as reasons for adult learning*. New York: College
> Entrance Examination Board.
>
> Atkinson, R. 2003. *An experimental evaluation of tutorials in problem
> solving (TiPS): A remedial mathematics tutor*. Office of Naval
> Research ONR N00014-02-1-0191.
>
> Bailey, T. et al 2004. The Characteristics of Occupational
> Sub-Baccalaureate Students. New York, NY: Community College Research
> Center, Teachers College, Columbia University.
>
> Bandy, I. G. 1985. Ready or not, high school graduates are going to
> college. *NASSP Bulletin* 69(479): 87--90.
>
> Barth, P. 2002. *Add it up: Mathematics education in the U.S. does not
> compute.* Education Trust. ERIC, ED 470539.
>
> Bilodeau, D. 2003. *Automating Army's Basic Skills Program* briefing,
> July 9, 2003. ww-003.com/Session%2046/46_bilodeau.pdf (accessed Oct.
> 23, 2003).
>
> Bird, LeAnne. YOU Program Coordinator, Arkansas Higher Education
> Coordinating Board. In correspondence with Peggy Golfin of the CNA
> Corporation, Oct. 21, 2003.
>
> Bloom, B. S. 1964. *Stability and change in human characteristics*.
> New York: Wiley.
>
> Boylan, H. R., and D. Saxon. 2002. *What works in remediation: Lessons
> learned from 30 years of research.*  Boone, NC:  Appalachian State
> University, National Center for Developmental Education.
>
> Bradley, Linda. Associate Executive Director for Academic Affairs,
> Tennessee Higher Education Commission. In correspondence with Peggy
> Golfin of the CNA Corporation, Oct.21, 2003
>
> Bransford, J. D., A. L. Brown, R. R. Cocking, M. S. Donovan, and J. W.
> Pellegrino. 2000. *How people learn: Brain, mind, experience, and
> school.* Washington, D.C.: National Research Council.
>
> Bransford, J. D., with Cognition and Technology Group at Vanderbilt.
> 1998. Designing environments to reveal, support, and expand our
> children's potentials. In S. A. Soraci and W. McIlvane (eds.)
> *Perspectives on fundamental processes in intellectual functioning*
> (Vol. 1). Greenwich, CT: Ablex.
>
> Brown, A. L., and J. C. Campione. 1996. Guided discovery in a
> community of learners. In K. McGilly (ed.) *Classroom lessons:
> Integrating cognitive theory and classroom practices* (pp. 229-270).
> Cambridge, MA: MIT Press.
>
> Bureau of Labor Statistics (BLS). 2001*. News release: Employment and
> total job openings by education or training category, 2000--2010*.
> Washington, DC: U.S. Department of Labor.
> www.bls.gov/news.release/ecopro.toc .htm (accessed Oct. 31, 2003).
>
> Burk, Kathleen. Director of Regents Testing, University System of
> Georgia Board of Regents. In correspondence with Peggy Golfin, The CNA
> Corporation. Oct. 21, 2003.
>
> Cartnal, R. 1999. *Preliminary success and retention rates in selected
> math courses.* Cuesta College Matriculation and Research Service. San
> Louis Obispo, CA. ERIC, ED 442480.
>
> Charles, Richard. President and CEO, Math Learning Institute. In
> correspondence with Monya Ruffin, American Institutes for Research,
> Education & Human Development, Jan.6, 2004.
>
> Cobb, P., E. Yackel, and T. Wood. 1992. A constructivist alternative
> to the representational view of mind in mathematics education.
> *Journal for Research in Mathematics Education* 19:99-114.
>
> Colorado Community College Online Placement Testing Information Web
> site. ccconline.org/studentinfo/placementassess.htm (accessed Nov. 28,
> 2003).
>
> Conley, D., and F. Bodone. 2002. *University expectations for student
> success: Implications for system alignment and state standard and
> assessment policies.* Center for Education Policy Research, University
> of Oregon. ERIC, ED464922.
>
> The Consortium for Mathematics and its Applications*. Developmental
> Mathematics and its Applications (DevMap) Project.*
> http://www.comap.com/undergraduate/ projects/devmap (accessed Oct. 21,
> 2003).
>
> Creery, K. 2001. *Comparison of lecture, self-paced, and on-line
> courses.* National Association for Developmental Education (NADE)
> Selected Conference Papers, Vol. 7. nade.net/h3_scp.htm (accessed Nov.
> 13, 2003).
>
> Defense Technical Information Center Web site. www.dtic.mil. (accessed
> Oct. 2003)
>
> DePree, J. 1998. Small-group instruction: Impact on basic algebra
> students. *Journal of Developmental Education* 22(1): 2--5.
>
> Donovan, M., J. Bransford, and J. Pellegrino (eds.). 2000. *How People
> Learn: Bridging Research and Practice.* National Research Council
> (U.S.) Committee on the Sciences, National Research Council (U.S.)
> Committee on Learning Research and Education.
>
> Dougherty, K., and M. Bakia. 2000. Community Colleges and Contract
> Training: Content, Origins and Impact. *Teachers College Record*
> 102(1): 197--243.
>
> Education Commission of the States (ECS). The Center for Community
> College Policy. 2002. State policies on community college remedial
> education: Findings from a national survey.
> http://www.ecs.org/ecsmain.asp?page=/html/issuesPS.asp (accessed Oct.
> 16, 2003).
>
> Evans, J. 1993. *Adults and numeracy*. Unpublished doctoral
> dissertation, University of London.
>
> ---------. 2000. *Adults' mathematical thinking and emotions: A study
> of numerate practices*. New York: RoutledgeFalmer.
>
> Fonte, R. 1997. Structured versus laissez-faire open access:
> Implementation of a proactive strategy. *New Directions for Community
> Colleges* 25(4): 43--52.
>
> Fort Hood Basic Skills Education Program information.
> http://esd.hood.army.mil/bsep.htm.
>
> Gal, I., and M.J. Schmitt. 1995. NCAL Brief: Proceedings. Conference
> on Adult Mathematical Literacy. Philadelphia, PA. National Center on
> Adult Literacy.
>
> Garcia, Federico. 1998. *Effectiveness of the Voluntary Education
> Program.* CNA Research Memorandum 98-40.
>
> Greeno, J., et al. 1999.  *Learning in and for participation in work
> and society.* Institute for Research on Learning.
>
> Grubb, N. W. 1996. *Learning to work: The case for reintegrating job
> training and education.* New York: Russell Sage.
>
> ---------. 1999. Innovative practices: The pedagogical and
> institutional challenges. In W. N. Grubb (ed.), *Honored but
> invisible: An inside look at teaching in community colleges* (pp.
> 245-279). New York: Routledge.
>
> Hasselbring, T. S., L. Goin, and J. D. Bransford. 1987. Effective
> mathematics instruction: Developing automaticity. *Teaching
> Exceptional Children* 19(3):30--33.
>
> Higbee, J., and P. Thomas. 1999. Affective and cognitive factors
> related to mathematics achievement. *Journal of Developmental
> Education* 23(1), Fall 99: 8--16.
>
> Hoyt, U., and C. Sorensen. 2001. High school preparation, placement
> testing, and college remediation. *Journal of Development Education*
> 25(2), winter 2001: 26--34.
>
> Jenkins, D. 2002. *The potential of community colleges as bridges to
> opportunity for the disadvantaged: Can it be achieved on a large
> scale?* Chicago, IL: Great Cities Institute, University of Illinois at
> Chicago.
>
> Jenkins, D., and K. Boswell. 2002. *State policies on community
> college workforce development: Findings from a national survey.*
> Education Commission of the States, Center for Community College
> Policy. http://www.ecs.org/clearinghouse/40/82/4082.pdf (accessed Oct.
> 15, 2003).
>
> Jenkins, J., and J. Fitzgerald. 1998. *Community colleges: Connecting
> the poor to good jobs.* Education Commission of the States. Community
> College Policy Center, Policy Paper.
>
> Jordan, Kathy. SkillSoft, Inc. In correspondence with Monya Ruffin,
> American Institutes for Research, Education & Human Development,
> Jan.6, 2004.
>
> Kaestle, C., et al. 2001. *NCES Adult Literacy and Education in
> America*. National Center for Education Statistics. NCES document
> 2001-534. http://nces.ed.gov/ pubs2001/2001534.pdf (accessed Dec. 3,
> 2003).
>
> Kinney, P. 2001. *Implementation models for interactive multimedia
> software in developmental mathematics.* National Association for
> Developmental Education (NADE) Selected Conference Papers, Vol.7.
> http:// www.nade.net/h3_scp.htm (accessed Nov. 13, 2003).
>
> Knowles, M. S. 1989. *The making of an adult educator: An
> autobiographical journey*. San Francisco: Jossey-Bass.
>
> Lancaster, M. 2001. *Jefferson Davis Community College and
> developmental education: A partnership for student success*. ERIC, ED
> 454892.
>
> Lave, J., and E. Wenger. 1991. *Situated learning: Legitimate
> peripheral participation.* Cambridge, United Kingdom: Cambridge
> University Press.
>
> Lave, J. 1991. Situated learning in communities of practice. In L.
> Resnick, S. Levine, and L. Teasley (eds.). *Perspectives on socially
> shared cognition.* Washington, D.C.: American Psychological
> Association.
>
> Leinbach, C., D. Pountney, and T. Etchells. 2002. Appropriate Use of a
> CAS in the Teaching and Learning of Mathematics. *International
> Journal of Mathematical Education in Science and Technology*
> 33(1):1-14.
>
> Livingston, T. 2001. *The cognitive impact of a computer algebra
> system on intermediate algebra students at the community college
> level.* Orange Coast College. ERIC, ED 458901.
>
> MacDonald, L., S. Vasquez, and D. Caverly. 2002. Techtalk: Effective
> Technology Use in Developmental Mathematics. *Journal of Developmental
> Education* 26(2): 36-37.
>
> Maher, Maureen. Director of Education Marketing, MathWorks. In
> correspondence with Monya Ruffin, American Institutes for Research,
> Education & Human Development, Jan.13, 2004.
>
> Martin, Dan. Alinea Group. In correspondence with Monya Ruffin,
> American Institutes for Research, Education & Human Development,
> Jan.6, 2004.
>
> Math Learning Institute Web site. http://www.mathlearningi.com
> (accessed Nov. 24, 2003).
>
> MathSoft Web site. http://www.mathsoft.com (accessed Nov. 24, 2003).
>
> Mathworks Web site. http://www.mathworks.com (accessed Nov. 24, 2003).
>
> Mazzeo, C., S. Rab, and J.Alssid. 2003. *Building bridges to college
> and careers: Contextualized basic skills programs at community
> colleges.* The Workforce Strategy Center: New York, San Francisco.
> http:// www.workforcestrategy.org (accessed Oct. 28, 2003).
>
> McCabe, R. 2000. *No one to waste: A report to public decision-makers
> and community college leaders.* Washington, D.C.: Community College
> Press.
>
> McClendon, M., and M. McArdle. 2002. *Comparing alternative algebraic
> modalities for remedial students*. Paper presented at the Chair
> Academy Leadership Conference, Kansas City, MO, Feb. 28--March 2,
> 2002. ERIC, ED 464658.
>
> McLeod, D. 1992. Research on affect in mathematics education: A
> reconceptualisation. In D. A. Grouws (ed.) *Handbook of research in
> mathematics education teaching and learning* (pp. 575-596). New York:
> Macmillan.
>
> Merisotis, J., and R. Phipps. 2000. Remedial education in colleges and
> universities: What is really going on? *The Review of Higher
> Education* 24 (1): 67-85.
>
> Miglietti, C. L., C. Strange, and C. Carney. 2002. Learning styles,
> classroom environment preferences, teaching styles, and remedial
> course outcomes for underprepared adults at a two-year college.
> *Community College Review 26*(1): 1--19.
>
> Military.com Web site.
> http://www.military.com/Resources/ResourcesContent/
> 0,13964,32879,00.html (accessed Nov. 10, 2003).
>
> Minnesota MATYC Web site.
> willmar.ridgewater.mnscu.edu/mnmatyc/Conference2005/technology_demonstrations.htm
> (accessed July 2005).
>
> Moss, Phillip, Interim Vice Chancellor for Academic Affairs, Oklahoma
> State Regents for Higher Education. In correspondence with Peggy
> Golfin, The CNA Corporation. Oct. 22, 2003.
>
> Murphy, G., and A. Johnson. 1998. *What works: Integrating basic
> skills training into welfare-to-work*. Washington, D.C.: National
> Institute for Literacy. ERIC, ED 424359.
> http://www.nifl.gov/whatworks.htm (accessed Oct. 20, 2003).
>
> Murphy, Suzanne. Associate Director, ACCUPLACER, The College Board. In
> correspondence with Peggy Golfin, The CNA Corporation. Oct. 21, 2003.
>
> National Association of Manufacturers Center for Work Success. *The
> skills gap 2001.*
> nam.org/docs/centerforworkforcesuccess/23975_namskillsgapforpress.pdf
> (accessed Nov. 2, 2003).
>
> National Center for Education Statistics (NCES). 2002. *Digest of
> Higher Education Statistics*. Washington DC: U.S. Department of
> Education. http://www.nces
> .ed.gov/pubsearch/pubsinfo.asp?pubid=2003060 (accessed Oct. 30, 2003).
>
> ---------. 2003. *Remedial education at degree-granting postsecondary
> institutions in fall 2000*. Washington, DC: U.S. Department of
> Education. http://nces.ed.gov/pubs2004/2004010.pdf (accessed Dec. 1,
> 2003).
>
> National Center for Family Literacy. 2003. Literacy facts and figures:
> Updated January 2003.
> http://www.famlit.org/loader.cfm?url=/commonspot/security/
> getfile.cfm&PageID=2996 (accessed Dec. 4, 2003).
>
> National Council of Teachers of Mathematics. 1989, *Curriculum and
> evaluation standards for school mathematics.* Reston, VA: National
> Council of Teachers of Mathematics.
>
> ---------. 1991. *Professional standards for teaching mathematics.*
> Reston, VA: National Council of Teachers of Mathematics.
>
> ---------. 1995. *Assessment standards for school mathematics.*
> Reston, VA: National Council of Teachers of Mathematics.
>
> ---------. 2000. *Principles and standards for school mathematics.*
> Reston, VA: National Council of Teachers of Mathematics.
>
> National Research Council. 2002. *Scientific research in education.*
> Committee on Scientific Principles for Education Research. R. J.
> Shavelson and L. Towne (eds.). Center for Education. Division of
> Behavioral Social Sciences and Education. Washington, DC: National
> Academy Press.
>
> NETg Web site. http://www.netg.com (accessed Nov. 24, 2003).
>
> Noddings, N. 1985. Small groups as a setting for research on
> mathematical problem solving. In E. A. Silver (ed.) *Teaching and
> learning mathematical problem solving: Multiple research perspectives*
> (pp. 345-359). Hillsdale, NJ: Lawrence Erlbaum.
>
> Palinscar, A. S., and A. L. Brown. 1984. Reciprocal teaching of
> comprehension monitoring activities. *Cognition and Instruction*
> (1):117--175.
>
> Pavelchek, D., P. Stern, and D. Olsen. 2002. *Relationship between the
> WASL and placement tests used by community and technical colleges*.
> Social and Economic Research Center, Washington State University.
> sbctc.ctc.edu/ewag/ewag0203/2002%20March%20WASL%20Placement%20Study%20v3.doc
> (accessed Nov. 3, 2003).
>
> Pea, R. D. 1987. Cognitive technologies for mathematics education. In
> A. H. Schoenfeld (ed.) *Cognitive science and mathematics education*.
> Hillsdale, NJ: Lawrence Erlbaum.
>
> Pettito, A. L. 1984a. Beyond drill and practice: Estimation and
> collaboration in a microcomputer environment. Paper presented at the
> American Educational Research Association, New Orleans, LA.
>
> ---------. 1984b. Discussant's comments on misconceptions and comments
> on important issues for research in science and mathematics education.
> Paper presented at the meeting of the American Educational Research
> Association, New Orleans, LA.
>
> Plisko, V., Associate Commissioner of the National Center for
> Education Statistics. 2003. *The release of the National Assessment of
> Educational Progress (NAEP): The nation\'s report.* Nov. 13, 2003.
> http://www.nces.ed.gov/ commissioner/remarks2003/11_13_2003.asp
> (accessed Nov. 13, 2003).
>
> Porter, A. C., M. W. Kirst, E. J. Osthoff, J. S. Smithson, and S. A.
> Schneider. 1993. *Reform up close: A classroom analysis*. Draft final
> report to the National Science Foundation on Grant No. SPA-8953446 to
> the Consortium for Policy Research in Education. Wisconsin Center for
> Education Research, University of Wisconsin-Madison.
>
> Quinn, D. 2003. Report on the PLATO adult learning technologies
> implemented for developmental education at Miami-Dade Community
> College, Miami, Florida. PLATO Learning, Inc.
>
> Revak, M., S. Frickenstein, and D. Cribb. 1997. *The effects of a
> practice placement exam on placement exam scores and success in
> calculus I*. U.S. Air Force Academy publication TR-97-4, August 1997.
>
> Rogoff, B. 1990. *Apprenticeship in thinking: cognitive development in
> social context*. New York: Oxford Press.
>
> Rollins, Shawn. Thomson NETg. In correspondence with Monya Ruffin,
> American Institutes for Research, Education & Human Development,
> Jan.8, 2004.
>
> Romberg, T. A., E. A. Zarinnia, and K. F. Collis. 1990. A new world
> view of assessment in mathematics. In G. Kulm (ed.) *Assessing higher
> order thinking in mathematics*. Washington, D.C.: American Association
> for the Advancement of Science.
>
> Roth, John. Principal Consultant, ACT Placement Programs. ACT. In
> correspondence with Peggy Golfin, The CNA Corporation, Oct. 21, 2003.
>
> Schindler, Steve. MathSoft. In correspondence with Monya Ruffin,
> American Institutes for Research, Education & Human Development,
> Jan.8, 2004.
>
> Schoenfeld, A. H. 1983. Problem solving in the mathematics curriculum:
> A report, recommendation and annotated bibliography. *Mathematical
> Association of America Notes*, No. 1.
>
> ---------. 1985. *Mathematical problem solving*. Orlando, FL: Academic
> Press.
>
> ---------. 1991. On mathematics as sense making: An informal attack on
> the unfortunate divorce of formal and informal mathematics. In J. F.
> Voss, D. N. Perkins, and J. W. Segal (eds.) *Informal reasoning and
> education*. Hillsdale, NJ: Erlbaum.
>
> Schults, C. 2001. *Remedial education: Practices and policies in
> community colleges*. American Association of Community Colleges.
> http://www.aacc.nche.edu/
> Content/NavigationMenu/AboutCommunityColleges/Trends_and_Statistics/
> EnrollmentInfo/Remedial_Education.htm (accessed Oct. 20, 2003).
>
> Sherry, Heather. Director of Articulation and Student Services,
> Division of Community Colleges, Florida Department of Education. In
> correspondence with Peggy Golfin, The CNA Corporation. Oct. 22, 2003.
>
> Sinclair Community College. 2003*. Final report---Impact on
> performance when DEV students defer taking college-level courses.*
> Office of Institutional Planning and Research. Sinclair Community
> College, Dayton, OH.
> sinclair.edu/departments/ipr/pub/survey_reports/DEV_students
> \_impact_on_performance.pdf. (accessed Nov. 6, 2003).
>
> SkillSoft Web site. http://www.skillsort.com (accessed Nov. 25, 2003).
>
> South Dakota Board of Regents Placement Process for Undergraduate
> Students into Initial English and Mathematics Courses Administration
> Guidelines. www.ris.sdbor.edu/Academics/PlacementGuidelinesRevised.doc
> (accessed Nov. 7, 2003).
>
> Spann, M. 2000. *Remediation: A must for the 21st century learning
> society*. Education Commission of the States.
> http://www.ecs.org/html/offsite.asp
> ?document=http%3A%2F%2Fwww%2Ecommunitycollegepolicy%2Eorg
> %2Fpdf%2F3347%5FSpann%5Fremediation%2Epdf (accessed Oct. 7, 2003).
>
> Thacker, J. 2000. Raising student test scores through comprehensive
> school improvement. *ERS Spectrum* 18(3): 25--29.
>
> UAW-GM Center for Human Resources Web site. http://uaw-gm.org/
> (accessed Feb. 2, 2004).
>
> United States Army Fort Hood Web site. http://esd.hood.army/bsep.htm
> (accessed Nov. 5, 2003).
>
> United States Code, Title 10, Section 520.
>
> United States Department of Defense, Defense Activity for
> Non-Traditional Education Support (DANTES) Web site.
> dantes.doded.mil/dantes_Web/library/docs/voledfacts/fy03.pdf (accessed
> Nov. 20, 2004).
>
> United States Marine Corps. 1998. MARADMIN number 057/98.
>
> United States Marine Corps Web site. www.usmc-mccs.org/perssvc/Life
> \_Learn/prgms_masp.asp (accessed Nov. 6, 2003).
>
> United States Navy College Web site. http://www.navycollege.navy.mil
> (accessed Nov. 6, 2003).
>
> United States Navy 2002. COMNAVCRUITCOMINST 1130.8F,\
> Chapter 3.
>
> University of Wisconsin, Developmental Mathematics Program.
> http://rock.uwc.edu/ galexand/devmath/devmath.htm (accessed Oct. 31,
> 2003).
>
> Van Buren, M. 2001. *State of the industry report 2001*. American
> Society for Training and Development. Alexandria, VA.
>
> Vygotsky, L. S. 1978. *Mind in society*. M. Cole, V. John-Steiner, S.
> Scribner, and E. Souberman (eds.), Cambridge, MA: Harvard University
> Press.
>
> Waycaster, P. 2001. Factors impacting success in community college
> developmental mathematics courses and subsequent courses. *Community
> College Journal of Research and Practice* 25(5/6).
>
> Wenger, E. 1998. *Communities of practice: Learning, meanings and
> identity*. Cambridge, England: Cambridge University Press.
>
> West Virginia Higher Education Policy Commission Web site. http://www
> .hepc.wvnet.edu/resources/133-21.pdf (accessed Nov. 7, 2003).
>
> Wheland, E., R. Konet, and K. Butler. 2003. [Perceived inhibitors to
> mathematics
> success](http://web23.epnet.com/citation.asp?tb=1&_ug=dbs+0+ln+en-us+sid+A7E7DA92-0BCF-42B2-807F-81A01D77748E@sessionmgr3-sessionmgr4+B360&_us=bs+TX++perceived++And+++inhibitors++And+++mathematics+db+0+ds+TX++perceived++And+++inhibitors++A ).
> *Journal of Developmental Education* 26(3):18--24.
>
> Whitehead, A. N. 1929. *The aims of education*. NY: Macmillan.
>
> Wolfe, C. 2001. Kentucky tuning in to education at work. *Community
> College Week.* 13(12):14--15.
>
> Young, K. 2002. *Retaining underprepared students enrolled in remedial
> courses at the community college*. Champaign, IL: Parkland College.
> ERIC, ED 467850.

[^1]: A number of definitions exist for numeracy and mathematical, or
    quantitative, literacy. We prefer the definition provided in the
    1994 Conference on Adult Mathematical Literacy. That source states
    that numeracy and mathematical literacy are interchangeable notions,
    and define them as "the aggregate of skills, knowledge, beliefs,
    patterns of thinking and related communicative and problem-solving
    processes individuals need to effectively interpret and handle
    real-world quantitative situations, problems, and tasks" (Gal and
    Schmitt 1995)

[^2]: Alabama, Georgia, North Carolina, and South Carolina.

[^3]: The title is *Ready or Not: Creating a High School Diploma That
    Counts.*

[^4]: The ACT is one of the two most popular tests used for four-year
    college admissions. We discuss the various placement tests later in
    this section.

[^5]: We discuss the ACT and commonly used cutoff scores later, but
    generally a score of 19 or higher on the math component of the ACT
    is required.

[^6]: For reference, the maximum mathematics component score is 800 on
    the SAT, 36 on the ACT, 100 on the COMPASS test, 55 on the ASSET
    test, and 120 on the ACCUPLACER test.

[^7]: For instance, in 1999-2000, 62 percent of public two-year
    institutions had a policy of open admissions (NCES 2002, table 308).

[^8]: For reference, the maximum mathematics component score is 800 on
    the SAT, 36 on the ACT, 100 on the COMPASS test, 55 on the ASSET
    test, and 120 on the ACCUPLACER test.

[^9]: See Appendix B for a summary of the studies we reviewed.

[^10]: The difficulty with self-selection is that it can cause a
    statistical bias resulting in a misattribution of cause and effect.
    For example, consider a system that allows each student to select
    his or her own method of instruction. Those who are more motivated
    and more confident would be more likely to select computer-based
    instruction because they are more familiar with computers. In a
    simple statistical test, it would not be surprising to find that
    students using computer-based instruction have a higher pass rate in
    the course than those using an alternative method of instruction. It
    would be incorrect to conclude that it is the mode of instruction
    that is causing the better results unless it is possible to adjust
    for the motivation and confidence of the entering students. It is
    rare in this context to be able to make such an adjustment. In an
    RCT experiment, however, in which each student is randomly assigned
    to either an experimental or traditional class (the latter
    constitutes the control group), , there is no reason to believe that
    students assigned to one mode of instruction are any different from
    students assigned to another mode. Thus, if there are differences in
    outcomes, they can be more confidently attributed to the mode of
    instruction.

[^11]: Approximately 68 percent of all student scores fall within plus
    or minus one standard deviation of the mean.

[^12]: The term "accession," as used by the military, is reserved for
    those who begin military active duty. This is a smaller number than
    those who are recruited, since the entire recruiting process has a
    number of phases. A certain proportion of recruits never make the
    transition to active duty. To access is to successfully make that
    transition to active duty.

[^13]: The services are in the process of renorming the ASVAB based on a
    sample from the late 1990s.

[^14]: Normalized means related to a normal or bell-shaped curve
    distribution.

[^15]: Lifetime Library, Passkey, and PLATO.

[^16]: Assessment of Skills for Successful Entry and Transfer

[^17]: American College Testing Program

[^18]: Computerized Adaptive Placement Assessment and Support System

[^19]: College Placement Exam