HOTtopics EFF

VOL. 4/NO. 1

ON THE INSIDE

3 What Is the Use

Math to Solve

Problems and

Communicate

Performance 

Continuum?

5 The Use Math to

Solve Problems

and Communicate

Performance 

Continuum: 

It’s Not Just for 

Summative 

Evaluations

9 Conducting a

Math Class the

EFF Way

12 Using Math in the

ESOL Classroom

15 Questions for 

Dr. Hot Topics

16 Online Resources

What does research say about teaching

math to adults? How can EFF help me 

help my students with math? Where do 

I start?

T

his issue of Hot Topics focuses on the

EFF standard Use Math to Solve Problems and Communicate, addresses these

questions and provides examples of

activities that will help teachers incorporate the use of this Standard and its five performance levels into their teaching and assessment

activities.

If you are like many teachers, the EFF math standard may challenge your notions of how math is

‘supposed’ to be taught. Studies tell us that adult

basic education teachers are usually not well-prepared to teach mathematics to adults who are

preparing for the GED (Ward, 2000; Mullinex,

1994), and fewer than 5% are certified in mathematics (Gal and Shuh, 1994). Many HOT Topics

readers may fall into these categories, and look to

Equipped for the Future for resources and ideas to

teach mathematics in ways that are both meaningful to the student and understandable to the

teacher. The EFF math standard calls for the introduction of number sense, algebra, geometry and

data and statistics at all levels of instruction rather

than for the teaching only of arithmetic at the beginning levels. In this way, the standard is in line

with current theories of learning and with current

approaches to math instruction (NCTM, 2000). In

other ways, though, the EFF math standard will be

familiar, because it, like all EFF standards, shifts the

emphasis from rote learning of procedures to learning for understanding and for application in contexts important for adults.

The articles in this issue provide background in

mathematics learning and describe how to use the

EFF Use Math to Solve Problems and Communicate

standard and performance continuum as tools to

help design, implement, and assess mathematics instruction. Lynda Ginsberg gives an overview of the

standard and continuum and describes how teachers can use the continuum’s five performance levels.

Ellen McDevitt’s article constructs an EFF math lesson. Suzanne Elston describes her experiences

teaching math in an ESOL class, and Donna Curry

and Beth Bingman tell one GED teacher’s story of

using EFF to strengthen her math instruction. ●

References

Gal, I. & Schuh, A. (1994). Who counts in adult literacy

programs? A national survey of numeracy education.

Philadelphia: National Center on Adult Literacy.

Ward, J.E. (2000). Arkansas GED mathematics instruction: History repeating itself. In M.J. Schmitt & K.

Stafford-Ramus (Eds.), A conversation between researchers and practitioners: Proceeding of ALM-7. Cambridge, MA: National Center for the Study of Adult

Learning and Literacy.

WINTER 2005

Use Math to Solve Problems 

and Communicate

STRATEGIES 

AND TOOLS 

FOR TEACHING

WITH EFF

See Bringing Math Research into the Classroom, pg. 2

Using EFF to Bring Math

Research into the Adult

Ed Classroom

by Lynda Ginsburg

Arecent study in the United Kingdom found

that large numbers of adults return to study

math, not for practical purposes, but to

prove to themselves that they can indeed learn the

mathematics that they were unsuccessful at during

their schooling. Others, particularly those at the

lowest levels, are interested in improving their math

skills to help them better manage daily life tasks, as

preparation for entry into a training or credentialing program, and/or so they can best support their

children’s learning (The Guardian, June 8, 2004).

The motivations of adults in the US are certainly

very similar.
2 HOTtopics Winter 2005

To help adults learn the mathematics

that was formerly so difficult for them,

EFF looked at existing research on mathematics learning and applied it to the particular needs of adult ABE and ASE

populations. While there is much research

on how children of all ages learn math,

unfortunately, there is currently only a

small body of research available focused

specifically on how adults learn this same

math. An accepted summary document

of math research equivalent to that of the

Reading Research Working Group

(Kruidenier, 2002) currently does not

exist. Thus development of the EFF math

standard was informed by research on

how people of all ages learn (Bransford,

Brown, & Cocking, 1999), how they learn

math in particular (Kilpatrick, Martin, &

Schifter, 2003), by utilizing the rich standards document developed by the National Council of Teachers of

Mathematics (2000), and through extensive observations of adult learners.

The Standard, Use Math to Solve Problems and Communicate, includes the

words “use” and “solve problems” to inform learners and teachers that math is

more than numbers and computational

procedures. The Standard reflects the fact

that adults can actually use the mathematics they learn, and it also reflects the

research-based theories of learning indicating that effective learning builds off of

prior knowledge and experience that is

understood and meaningful. For adults,

much of that prior knowledge and experience is grounded in their everyday lives

and tasks.

This Standard and the level descriptors

of the performance continuum differ in

important ways from many traditional

curriculum materials and textbooks.

Often math is thought of as a linear progression, in which students move step by

step through a series of topics, starting as

a blank slate and ending at a mastery

level, ready to proceed on to the next

topic. Each topic is assumed to “build” on

the previous one. But, unlike children,

adults don’t come to education as a

“blank slate!” They come to study math

with some previous knowledge of fractions, decimals or percents and have experiences estimating, measuring and

making decisions with numerical information.

For example, we know that virtually all

adults (including those in ABE) know

that 50% is equal to 1/2 and they know

that 50% of $10.00 is $5.00. Many adults

can compute 25% of $80 with different

people using different strategies. Some

people mentally divide $80 in half getting

$40 and then divide it in half again, getting $20. (They recognize that these familiar fractions and percents are

equivalent, know that half of a half is a

fourth, and can compute mentally.) Others will change 25% to a decimal and

multiply, using the traditional school procedure. Still others will estimate the answer, arriving at a reasonably close

amount (Ginsburg, Gal, & Schuh, 1995).

Rather than thinking of math instruction as a ladder, a better metaphor might

be a math web—expanding out from the

center of adults’ previous knowledge,

with many connecting threads. Equipped

for the Future provides tools and support

for spinning this web.

■ The EFF level descriptors take into account the kinds of knowledge adults

bring with them, giving them “credit” for

what they likely know, but also as a way of

Bringing Math Research

into the Classroom,

continued from page 1

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STANDARD

Use Math to 

Solve Problems

and Communicate

• Understand, interpret, and work

with pictures, numbers, and 

symbolic information.

• Apply knowledge of mathematical

concepts and procedures to figure

out how to answer a question, 

solve a problem, make a prediction,

or carry out a task that has a 

mathematical dimension.

• Define and select data to be used in 

solving the problem.

• Determine the degree of precision

required by the situation.

• Solve problem using appropriate

quantitative procedures and verify

that the results are reasonable.

• Communicate results using a variety

of mathematical representations, 

including graphs, chart, tables, and

algebraic models.

Family

Citizen

Worker

In order to fulfill responsibilities as parents/family members, 

citizens/community members, and workers, adults must be able to:
Winter 2005 HOTtopics 3

encouraging teachers to find out what

their learners know and can do; and

teachers should then build upon this

knowledge when planning instruction.

We all know that we are much more likely

to understand and remember new information if it relates to that which we already know.

■ The EFF performance continuum emphasizes elementary ideas, regardless of

mathematical topic, before more complex ones (for example, calculations with

benchmark numbers — 50%, 25%, 1/2 —

before calculations with 12.5% or 5/6). It is

important for learners to feel confident

about the interconnections and equivalence among concepts such as 1/2, 50%, .5

and .50 because we frequently make decisions about which form of a number is

easiest to use. It is easiest to develop this

flexibility when the numbers are familiar

and meaningful. The traditional focus on

adding, subtracting, multiplying and dividing fractions followed by a similar pattern with decimals, and then percents,

does little to help learners see relationships

between them or gain flexibility moving

between alternative representations of the

same quantity.

The identification of a sequence that

progresses from elementary ideas and

(continued on page 4)

The EFF Use Math to Solve Problems

and Communicate Performance Continuum is:

A developmental description of 

performance on the standard...

This continuum is based on recent research on how people develop and use

mathematics in settings outside the

classroom. Thus its descriptions of

adult performance on the standard look

different from the kinds of linear sequencing of mathematical learning traditionally described and taught in

school. The continuum describes performance in math from Level 1

(roughly corresponding to the exit

point for the beginning ABE levels on

the National Reporting System) to Level

5 (corresponding to the exit point for

adult secondary education); in doing

so, it also describes simultaneous connections within and across mathematical content and procedures, and

maintains a focus on the use of multiple strategies to meet practical realworld goals using mathematics.

A multidimensional description of 

performance on the standard... 

Performance levels on Use Math to Solve

Problems and Communicate are defined

by key features called dimensions of

performance. They include:

• the structure and depth of the knowledge base;

• fluency of performance;

• independence of performance;

• range of performance.

These dimensions reflect an understanding that the development of expertise in mathematics requires the

interrelated development of:

• conceptual understanding,

• the procedural knowledge of how to

apply that conceptual understanding,

and 

• the strategic competence to choose

the most appropriate strategies for accomplishing a given task.

A set of benchmarks for the 

development of proficiency on 

the standard...

Performance level descriptors define

points along the continuum that serve

as benchmarks for key stages in development on the standard and for the increasing ability of adults to accomplish

important activities that require the

ability to use math to solve problems

and communicate.

A set of measures based on applied skills and processes...

The performance continuum reflects

the kinds of applied knowledge and

skills adults need to meet real world

purposes and goals in their lives as

workers, citizens, and family members.

See page 6 for the Level 2 Performance

Continuum for Use Math to Solve Problems and Communicate.

What is the Use Math to Solve Problems and

Communicate Performance Continuum?

S

everal of the articles in this issue of Hot Topics refer to the EFF Use Math to

Solve Problems and Communicate Performance Continuum. The EFF Continua of Performance are multidimensional, developmental descriptions of

performance on the EFF Content Standards. They allow for descriptions of

performance ranging from the novice level to the expert level by identifying key

features of performance at various points along the continuum.

This article describes the continuum and is excerpted from the EFF Assessment Resource Collection found on-line at

http://eff.cls.utk.edu/assessment/index.htm 

For more about the Use Math to Solve Problems and Communicate Performance Continuum and to see the entire 

continuum go to http://eff.cls.utk.edu/assessment/math1.htm or

http://eff.cls.utk.edu/eff_docs/EFFMathPC.doc
4 HOTtopics Winter 2005

skills through more complex ideas and

skills points to the important responsibility of the teacher to assess a learner’s

knowledge and skills. Because of time

away from formal schooling, spotty attendance, or other factors, many adults exhibit patchy knowledge or gaps in their

understanding of various concepts. Such

gaps are not identified with a standardized test score and may not be readily apparent but will have an impact on

successful performance. For example, a

learner who does not have an understanding of the meaning of area (and

multiplication) will always confuse formulas for area and perimeter. Similarly, a

learner who cannot skip count with 10s

and 5s will have difficulty counting

money.

■ Two different types of “strands” flow

across all the levels of the performance

continuum: problem-solving and content strands. Learners’ problem-solving

abilities include:

• deciding on the necessary degree of

precision;

• making decisions about the appropriate

mathematical tools, data or information, reasonableness, and computational procedures;

• communicating the solution(s) of the

problem in understandable and useful

ways.

The gradually higher expectations of

problem-solving skills reflect not only

the increasing complexity of the mathematical information and procedures, but

also the increasing levels of reasoning or

“mathematical thinking.” As people become more independent problem

solvers, they need to be able to juggle

multiple representations of the underlying mathematics.

For example, if a learner wants to

know if the costs of phone service have

decreased over the last 10 years, the

learner needs to gather the appropriate

data, decide how to characterize that data

(as the average total bill, as a percentage

of household income, as compared to cell

phone costs, etc.), and decide how to

communicate findings (visually with a

bar graph, circle graphs, or a line graph;

verbally in a paragraph, etc.). This is a

much more complex problem-solving activity than one in which the phone costs

of all the members of a class are gathered

and the mean, median and mode are

found to decide which is the best “average” to describe current phone costs.

■ The content strands of 1) number and

number sense; 2) data, data analysis and

statistics; 3) measurement and spatial

sense; and 4) patterns, functions and algebraic thinking are found in all levels of

the continuum. This approach is in line

with the adult education standards documents of Australia, the U.K., and the numeracy component of the international

ALL survey, as well as the NCTM K-12

standards (2000). Within each of these

content strands, elementary ideas are developed before more complex ones, and

instructional activities can easily span

more than one strand, as they do in the

real world.

■ The EFF math standard and performance continuum promote learning and

the application of mathematics within

real life, meaningful contexts. For most

adults, numbers rarely appear without

some kind of context. But, while adults

often bring years of experience using

numbers for many different life purposes,

their understanding of the underlying

mathematics is often quite limited. Their

knowledge may be closely tied to the contexts in which they learned or developed

their numerical strategies. Ultimately,

adults need to be able to apply their

knowledge to all kinds of contexts, both

familiar and new. A good example of this

is many adults’ comfort and ability calculating and estimating with decimals when

dealing with money but not with equivalent calculations in less familiar contexts

such as population statistics or weights

expressed in decimals.

Traditionally it was thought that by

working with numbers outside of any

context (such as workbook computational exercises with stand-alone numbers), students would be able to apply

their knowledge in “real-life” contexts,

but this does not seem to be true for

many people. After years of schooling,

many are left with disconnected ideas and

procedures that have little meaning. Although lots of people know how to divide

fractions (though often they can’t remember which number to invert, or

whether to invert when you multiply or

divide), they have difficulty when asked to

estimate the answer or describe a situation in which you would have to do such

a computation.

It makes more sense to be certain that

adults can actually apply their knowledge

in actual useful and meaningful situations. As teachers provide a rich variety of

problem contexts and spend time disEFF Math Standard and 

Level Descriptors,

continued from page 3

While the Math Standard and Level Descriptors compliment the

Read With Understanding Standard, it differs in important ways. 

• There is always new, different content to learn in every math content

strand.

• Learning the “basic skills” is not enough.

• Mastery does not come from acquiring a fixed set of skills and

processes that are then applied to new texts, but rather acquiring 

a deep conceptual understanding that allows for flexible problem

solving and prepares one for seeing meaning and relationships in

new topics.
Winter 2005 HOTtopics 5

cussing how the activities are similar and

different and discussing the underlying

mathematics involved, learners gain the

ability to generalize their mathematics experience. Individuals gain power by being

able to take the knowledge and skills they

have and use them whenever they are appropriate.

Clearly, when using mathematics to

solve problems, computation is only one

of many necessary skills. Effective problem solving requires many decisions—

among them are 

• Identify the problem,

• Select or find the data or information

needed for the solution,

• Determine the degree of precision

needed for the situation,

• Decide which problem solving strategies to use (computation, estimation,

graph, diagram, etc.),

• Decide how to present, represent, or

communicate the solution,

• Decide if the solution should be verified.

Computation is only one component

of numeracy and, in fact, there are many

numerical problems that do not require

computation at all, such as interpreting

graphical or diagrammatic information.

Indeed, the GED test also emphasizes

problem solving, estimation, multiple

representations—not just computation.

In this age of calculators and computers,

it is much more important to understand

the meaning of operations and calculations than to just be adept at doing them.

As teachers can use the EFF Standard

Use Math to Solve Problems and Communicate to guide effective instruction, they

may have to change some of their instructional practices. Traditional workbooks

will not be enough—real materials will be

used frequently, and teachers may well

have to search beyond existing textbooks

and/or develop their own activities to

challenge learners to solve problems and

think mathematically. This type of teaching requires that the teacher facilitate class

discussions and explore the meaning of

the mathematics rather than just computational rules. A teacher’s effectiveness

will depend on his/her own understanding of the mathematics, and thus, some

teachers may have to augment their own

math content knowledge.

Students find instruction based on the

EFF Standard to be more engaging, more

useful, and more accessible (building

from what they know and can do). However, to learners, it may not resemble

what they expect math class “should”

look like. They will not be sitting passively listening and writing but will be

expected to participate actively. They will

not be answering the teacher’s questions

but will be asking questions and then trying to find the answers. They will be

learning to be an active participant in

their own learning. ●

References

Bransford, J.D., Brown, A.L., & Cocking, R.R. (Eds.) (1999). How people learn: Brain, mind,

experience, and school. Washington, D.C.: National Academy Press.

Ginsburg, L., Gal, I. & Schuh, A. (1995). What does “100% Juice” mean? Exploring adult

learners’ informal knowledge of percent. Technical Report TR95-06. Philadelphia, PA: University of Pennsylvania, National Center on Adult Literacy.

Kilpatrick, J., Martin, W.G., & Schifter, D. (2003). A research companion to principles and

standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Kruidenier, J. (2002). Research-Based Principles for Adult Basic Education Reading Instruction. Washington, DC.: The Partnership for Reading, National Institute for Literacy.

National Council of Teachers of Mathematics (2000). Principles and standards for school

mathematics. Reston, VA.

The Use Math to Solve Problems and Communicate

Performance Continuum:

It’s Not Just for Summative Evaluations

by Donna Curry and Beth Bingman

Allison teaches in a small rural ABE/GED program in Virginia.

Twelve students are enrolled in her class, and nine or ten usually attend. The class meets three evenings a week for threehour sessions. All the students have named passing the GED as

a goal and most also share concerns about getting a job (or a

better job) and giving their children more opportunities. The students’ tested

abilities in reading and math vary widely with most students falling within

NRS levels 1 – 3.

While Allison had always tried to do

some group work with her class and often

brought in articles to discuss and write

about, the students spent most of their

class time working as individuals or small

groups in workbooks, especially for math.

After she attended an EFF workshop, Allison tried using the EFF Teaching/Learning cycle to structure her classes, but lack

of planning time and her students’ irregular schedules made it difficult.

When she learned about the EFF Use

Math to Solve Problems and Communicate

performance continuum, Allison decided

to use it to help her figure out some common content that her students would be

able to work on together. She found that

(continued on page 6)
6 HOTtopics Winter 2005

at Performance Level 2 adults should be

able to use “simple ways to interpret and

represent data (tables, bar graphs, line

graphs and pie graphs).” So for several

weeks she worked with her students on

basic line and bar graphs. They completed an activity in which they had to decide whether to use a bar graph or line

graph to best communicate the information from data. Allison looked back at the

Use Math to Solve Problems and Communicate performance continuum and realized that students needed practice not

just in reading and interpreting charts

and graphs but also in organizing the data

that was to then be displayed.

She decided to continue to focus on

“simple ways to interpret and represent

data” and also to work to help them “define, select, and organize simple data,” another key skill for students at Level 2. She

realized that her students had been doing

some basic organization but she hadn’t

really given students the opportunity to

explore how different organizational

strategies affect the interpretation of the

data.

From the Use Math to Solve Problems

and Communicate performance continuum, Allison knew that her students

should be able to “independently accomplish simple, well-defined, and structured

math tasks in a range of comfortable and

familiar settings.” She knew that she

needed to work with her students to develop an activity that began with what is

familiar to students and involved numbers that are fairly easy to manipulate

(“friendly” numbers rather than “messy”

ones, if possible).

To determine what her students already knew about how organization of

data impacts interpretation, Allison

began a discussion about why data are organized as they are. She first asked

whether students thought their lives are at

all influenced by data. They seemed

stumped at first but then they began to

give examples. One student mentioned

that her doctor suggested that she get a

mammogram every year because of her

age bracket. Another student then shared

that her doctor recommended specific dietary restrictions because of her age and

risk of developing a particular disease.

The doctors were influenced by researchers’ interpretation of data.

Allison then asked whether it makes a

difference how data are organized. Students, with a little prompting, were able

to build on their health examples. They

explained that their insurance rates are

based on data because they know that

they pay different rates based on their

age and sex. The younger students in the

class picked up the discussion by moving

from health insurance to car insurance

rates. They were clearly not happy that

their rates are so much higher than the

“older” students. Allison reeled them in

a bit. She wanted them to understand

that health and life insurance policies as

well as car insurance rates are all based

on data and that these data are grouped

by ages.* 

*NOTE: What’s important for the students to understand is that key factors

in their lives are based on samplings

of data that is categorized and interpreted and that decisions are then

made based on those interpretations.

For example, life insurance policies

It’s Not Just for Summative 

Evaluations, continued from page 5

Use Math to Solve Problems and Communicate

PERFORMANCE LEVEL 2

Adults performing Level 2 are able to:

Use Key Knowledge, Skills, and 

Strategies 

• Read, write, and interpret some 

common types of mathematical 

information such as whole numbers,

monetary values and prices, benchmark fractions (3/4, 1/10), decimals

(.25, .50, .75, .10) and percents (25%,

75%, 10%, 100%); simple patterns,

probability and proportions (1:4, 4:1);

commonly used standard units of

measurement, common geometric

shapes, and the concept of “area”;

and simple ways to interpret and 

represent data (tables, bar graphs

with and without gridlines, line 

graphs and pie graphs).

• Recall and use mathematic 

procedures such as addition and 

subtraction (with or without use of 

calculator), grouping, comparing 

2 numbers, and basic estimating; 

and measure length, weight, and

areas of standard and non-standard

shapes using tools calibrated with

whole numbers and benchmark 

equivalents (rulers, manipulatives).

• Evaluate the degree of precision

needed for the solution.

• Define, select and organize simple 

data, and measure with appropriate

tools, describe patterns, and/or use

computational procedures effectively, 

to solve a problem and to verify that

the solution is reasonable.

• Communicate the solution to the 

problem orally, in pictures, or in 

writing.

Show Fluency, Independence, and 

Ability to Perform in a Range of 

Settings

Select and apply the knowledge, 

skills and strategies at this level to 

independently accomplish simple,

well-defined, and structured math

tasks in a range of comfortable and 

familiar, or highly structured, settings.

Apply Math Knowledge to 

Accomplish a Variety of Purposes

Select, record and work with simple

mathematical information to accomplish a variety of purposes, such as:

• Stay within a set budget while 

shopping for work clothes.

• Determine the area of a room by 

grouping the square units within 

the shape and performing repeated

addition.
Winter 2005 HOTtopics 7

vary for individuals based on their

ages, whether or not the individual

smokes and the health of the individual, just to name a few of the factors

involved. Risk factors are used to determine whether an individual is likely

to live long or not.

Allison and her students discussed the

underlying assumptions about how insurance rates change for different ages.

While students were in some agreement

about why the cost of life insurance is

more expensive for someone in their 50s

than in their 20s, the younger students

disagreed that younger drivers have more

accidents than older drivers and, therefore, should not have to pay higher insurance premiums.

Allison saw this disagreement as an

opportunity for an application of data

collection, organization, analysis, and decision-making. She suggested that they

collect data from all the classes in the program to see whether their assumptions

about accident rates of young drivers—or

those of the insurance companies—were

more accurate.

In order to test their assumptions, Allison and the students had to figure out a

plan (See EFF Standard Plan above). They

created the following list of steps:

1. Decide what question(s) we want to

ask.

2. Decide who we’re going to ask.

3. Figure out how we’re going to ask

everybody.

Once the data are collected, then

4. Decide how to group the data into different age groups.

5. Count the number in each group

6. Make a graph to show how many in

each group.

7. Decide what our information tells us

in comparison with the data from insurance companies.

Students asked other learners in the

program whether they had ever had an

accident. They used a simple check sheet

to record their data, placing a check for

Yes (I have had an accident) or No (I have

not had an accident.) They also recorded

the ages at which people had had accidents. Students collected information

from many students but decided to focus

on those individuals who had had accidents. They wanted to see the ages at

which individuals seemed to have accidents. They wanted to challenge the belief

that younger drivers have more auto accidents than older drivers, hence the more

expensive insurance rates.

Students found that there were 15 individuals who had had prior auto accidents. The ages at which they had an

accident were as follows: 27, 54, 35, 47, 32,

36, 17, 21, 22, 23, 45, 53, 54, 16, 19. Allison

asked them to work in pairs to organize

and represent their data using whatever

reasonable organizational strategy that

they wanted. The students created the following graphs (on page 8).

Once the graphs were done, they

shared their results and were amazed at

the different representations of the same

data. Allison asked how this could be. The

students realized that how the data are organized does influence how the data are

represented and interpreted. They are

able to see that in the second chart, it appears that there are more older drivers

having accidents than in the first chart,

where it appears that more young drivers

have accidents.

Allison wants her students to be critical thinkers so she pushed them to look

beyond their graphs. She asks them to

consider whether they believe their data is

a representative sample. They ponder

whether their data would be similar to

(continued on page 8)

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Plan

• Set and prioritize goals.

• Develop an organized approach

of activities and objectives.

• Actively carry out the plan.

• Monitor the plan’s progress while

considering any need to adjust the

plan.

• Evaluate its effectiveness in

Family achieving the goals.

Citizen

Worker
other samples collected in other parts of

the country or in places other than adult

education programs. The students realize

that the sample size and representation

are important factors in working with

charts and graphs. They decide that they

need to question information they view

in graphs and not necessarily believe

everything that they see represented in

graphs.

As they discuss ways to group and represent their data, one student suggested

they could figure the percentage that had

accidents at various ages. She suggests

that maybe they should look at their data

and use percents, too. Several students

agreed, and then added that they wanted

to understand percents anyway because

they see it all the time; they just can’t remember “how to do them.” This concerned Allison because she knew that

according to the Use Math to Solve Problems and Communicate performance continuum her students should be able to

handle percents. She thinks to herself that

this is not possible because not all of her

students know their multiplication facts.

According to the workbook she used to

follow, her students had to know multiplication and division of whole number,

plus everything about fractions before

they were ready to handle percents.

She returned to the Use Math to Solve

Problems and Communicate performance

continuum for an answer. While the performance continuum is not a curriculum

framework detailing what she is to teach,

it does serve to give her some ideas on the

sort of strategies and skills students

should have at each level. Students should

be comfortable with benchmark percents.

This means that she doesn’t have to teach

them everything about percents. Instead,

she can focus on ensuring that her students are grounded in what percents are

and focus only on percents that are already a part of students’ lives.

As Allison was considering how to introduce percents and particularly

whether to continue to work with the

data about insurance rates or start on a

different project, she looked at a copy of

the EFF Teaching/Learning Cycle. She

thought she might find ideas about next

steps. When she read over the cycle she

realized that in their work on graphs and

insurance data, the class had gone

through the cycle. She had taken most of

the cycle steps without focusing on them:

She had determined individual’s goals

and the group had identified a purpose.

She had designed learning activities that

addressed real-life concerns and then

carried out her plans, capturing evidence

through observations. They had reflected

on what they had learned. What she had

not done was involve her students systematically in the process. She decided to

engage in Step 8, determining next steps

with her students. She shared her assessment of their understanding of percents,

how those skills connected to their goal

of passing the GED as well as their dayto-day needs, and together the group decided to continue using the data that they

were familiar with to learn more about

percents. ●

8 HOTtopics Winter 2005

It’s Not Just for Summative Evaluations, continued from page 7

0

1

2

3

4

5

6

7

16-25 26-35

Age of Drivers

Number of People

36-45 46-54

0

1

2

3

4

5

6

7

8

16-21 22-34

Age of Drivers

Number of People

35-55

0

0.51

1

2

3

4

16-19 20-24 25-29 30-34

Age of Drivers

Number of People

35-39 50-54 40-44 45-49

1. Accidents by Age

2. Accidents by Age

3. Accidents by Age

Once the graphs were done, they shared their results and were amazed

at the different representations of the same data. Allison asked how this

could be. The students realized that how the data are organized does 

influence how the data are represented and interpreted.
Winter 2005 HOTtopics 9

Conducting a Math Class the EFF Way by Ellen McDevitt

P R E P A R A T

I

O

N

P L A N N I

N

G

C

A

R

R Y

I

N G

O U T

T H E P L A N

R E F L E C T I O N

The EFF

Teaching/Learning

Cycle

STEP 1. Determine 

individual learner’s 

goals and purposes 

and identify the 

Standards that will help 

him/her achieve them.

Identify student’s prior 

knowledge about these 

goals and Standards.

STEP 2. In a group identify 

a shared interest, purpose, 

or goal and determine the

group’s prior knowledge of 

this topic. Identify the 

Standard that will help the 

group address this shared 

goal. Make clear the 

connection between the 

class focus and individual 

needs.

STEP 3. Design a learning 

activity to address the real-life 

concerns of the learners.

STEP 4. Develop a plan 

to capture evidence and 

report learning.

STEP 5. Carry out 

the learning activity.

STEP 6. Observe and 

document evidence of 

performance of the 

Standard.

STEP 7. With students, evaluate 

and reflecton how what was 

learned is transferable to 

real-life situations.

STEP 8. With students, 

determine next steps to help 

learners meet their goals.

 

(Return to Step 1 and/or 2.)

Teaching/Learning Cycle

Remember that you’re teaching

learners, not lessons. This sentence is the one that describes

the major difference between

the traditional math lesson and an EFF

math lesson. Adult educators care deeply

about their students, want them to succeed, and believe that they can succeed.

Yet most continue to follow the lesson

templates they acquired in college classes

many years ago—the templates that put

teachers as the source of knowledge with

students as receptacles of that knowledge.

It’s a deficit model: teacher has all the

knowledge, student falls short in knowledge, teacher shares knowledge, student

absorbs knowledge and makes up the

deficit. The focus is on completing a lesson and taking a test in which the

learner’s success is measured according to

how well s/he is able to remember what

the teacher has said is important.

It’s a model that doesn’t take into account the fact that learners have goals beyond learning facts. It doesn’t make room

for the fact that learners already have

(continued on page 10)
knowledge and skills that they can apply

to learning new material. It doesn’t acknowledge that teachers can learn from

students. It relies heavily on the lecture

format for instruction assuming a onelecture-fits-all standard. And it wraps up

with a workbook or worksheet practice of

many problems that are thought to incorporate the newly-taught information. A

traditional math class tends to be quiet

and static.

By contrast, the EFF math lesson centers on the learner and his/her needs. Beginning by identifying learner goals, an

EFF lesson proceeds by helping learners

identify and apply their prior knowledge

and experiences to new learning, building

an opportunity for the learner “to reflect

on and monitor their own developing

knowledge, skills, and learning strategies.”

Throughout the lesson, the teacher adjusts the learning activities to reflect

changes in learner goals and emerging

learning needs. An EFF math class tends

to be noisy and dynamic. The table at the

right describes the differences between

the two kinds of lessons.

What might a lesson plan look like for

an EFF math class? Here is a suggested

outline of a lesson for an EFF algebra

class. For purposes of illustration, assume

that the class is a typical ABE/GED class,

with learners of different levels of ability

in the same session. They attend class two

nights a week for three hours, and you alternate reading and math instruction.

Most of the learners have a goal of getting

the GED, with most needing that credential so they can get a job.

In the traditional lesson, the objective

is for learners to master algebra word

problems, and the only measure of success is a test score. Lesson content might

begin with substituting letters such as x

and y in very simple equations, with the

difficulty of the equations increasing

with each lesson. Learners would practice these substitutions on workbook

pages of rows of problems. The vocabulary of algebra would be used and explained somewhat, but the focus would

be on solving equations where letters are

used to represent “things we don’t

know.” Teacher would use phrases such

as “Let x equal the number of...” and

would offer simple steps to solving algebra word problems. Students would copy

what the teacher wrote, practice solving

problems in their workbooks or on

teacher-generated sheets, and take tests

to see how well they were doing.

The following EFF lesson was

planned using the eight-step EFF Teaching/Learning Cycle, and it reflects the EFF

Math Standard. The steps in the cycle

help you devise lessons that have importance to learners, beginning with setting

goals based on what the learners need to

learn or be able to do. The performance

components of the Standard provide descriptors of what success looks like. Look

at the steps in the cycle (below) and the

performance descriptors of the Standard

and see if you can find them in the lesson

plan that follows.

10 HOTtopics Winter 2005

Conducting a Math Class the

EFF Way, continued from page 9 Traditional Math Class EFF Math Class

What do I need to teach? What do they need to learn?

Linear—discrete topics Spiral—integrated topics

Goal—teacher generated Goal—student generated

Based on Standards

Lessons—teacher centered Lessons—learner centered

Curriculum driven Driven by student purposes

Deficit model—fill the gap Identify and apply prior knowledge

Follow the book Use contextual, authentic materials

Teach to the test Teach to student goals

Individualized work Collaborative and cooperative learning

Worksheets Project-based learning

Seat work Student teams

Little or no interaction with each other Sharing, comparing, explaining

Competitive Cooperative, mutually affirming

Teacher as expert, student as Students and teachers co-construct 

receptacle knowledge

Materials = workbooks Materials = authentic, relevant to learner

Proof of learning = test score Proof of learning = test scores as well as 

demonstration of the skill in a meaningful 

context—student or group project

Review the The EFF Teaching/ Learning

Cycle (see page 8)

Step 1: Determine individual learner’s

goals and purposes and his/her prior

knowledge about them. Identify the 

Standards that will help learner achieve

these goals. Identify learner knowledge 

of the Standards.

Step 2: In a group, identify a shared 

interest, purpose, or goal to determine 

the group’s prior knowledge of this topic.

Identify the Standard that will help the

group address the shared goal. Make

clear the connection between the class

focus and individual’s needs.

Step 3: Design a learning activity to 

address the real-life concerns of the 

learners.

Step 4: Develop a plan to capture 

evidence and report learning.

Step 5: Carry out the learning activity.

Step 6: Observe and document evidence

of performance of the Standard.

Step 7: Evaluate and reflect on how what

was learned is transferable to real-life 

situations.

Step 8: Determine next steps to help

learners meet their goals. (Return to 

Step 1 and/or 2)
Winter 2005 HOTtopics 11

Performance objective: Students will

demonstrate algebraic thinking by creating a table of values, a graph, and an

equation to describe a situation that they

encounter in daily life.

Rationale: Establish a context for learning—when/where do students use algebra? The ability to see patterns and

describe those patterns in generalized

terms allows learners to predict future

patterns. This algebraic thinking is useful

for making decisions in the workplace

and in everyday life. Review of student

goals provides more specific contexts for

the use of algebraic thinking in life: at

work (e.g., using tables of values to see

patterns in production), as a parent (e.g.,

understanding patterns describing student standing after high-stakes testing), as

a member of the community (e.g., examining the differences in two political candidates’ positions on an issue of relevance

to the learners to see if what they say

matches what the numbers say).

Lesson content:

• Discussion to elicit specific needs for

using algebra—identify a situation in

which choices are made based on data.

• Information gathering activity—learners provide facts based on situations at

work, at home or in the community.

• Data collection activity—converting

information in the situation to data by

interpreting information provided by

learners.

• Mini-lesson on ways in which algebra

presents—as words, as a table of values,

as a graph, as an equation.

• Teacher-led translation of one student-generated situation to a Table of

Values (TOV), a graph and an equation.

Class discussion.

• Small group work—students use another student-generated situation and

practice creating a TOV, a graph and an

equation.

• Coming back to words—reflection on

student-created work to determine

what stories they tell.

• Connect to the way algebra presents

on the GED.

Instructional procedure:

• Focus event to activate prior knowledge, e.g., “When can you remember

having to use algebra?” “Why did you

have to use it?” “What did it look like?”

• Define algebraic thinking, and then ask,

“When do you have to make decisions?”

“How do you compare information?”

Guide learners in identifying a situation

at work, at home, or in the community

in which they make choices based on

data. Relate that need to using algebraic

thinking.

• Mini-lesson on ways in which algebra

presents—as words in a situation, as Tables of Values (TOV), as graphs, as

equations. Use the Styrofoam cups activity to demonstrate conversion of

words to each of the other three examples: I have to buy a coffee urn for our

church kitchen. I want to buy one that’s

big enough so we don’t have to worry

about refilling it all the time, but not so

big that we have coffee left over. We use

6-ounce Styrofoam cups. How many

ounces of coffee will we use for 10 cups,

20 cups, 30 cups, and 50 cups of coffee?

How big an urn should I buy? Or should

I change the size of the cups? How

many ounces of coffee would I use for

the same amounts with cups that hold 4

ounces of liquid? 8 ounces of liquid? 

• After the Styrofoam cups activity, use

one of the learners’ own situations in

the same way. Instruct group that you

will be using wait time between asking

the question and getting the answer—

15 seconds for people to think before

someone gives the answer.

• Use guided questioning to help learners

convert the words in the situation to a

TOV, a graph, and an equation. “What

are we comparing in this situation?”

“How should we label the columns on

our table of values in order to compare

the elements we want to compare?”

“Does everyone agree?” “What pattern

do you see in the table we’ve created?”

“Does anyone see a different pattern?”

“Can you describe that pattern in

words?” “Now, can you describe it in

graph form?” “Which of the columns is

the horizontal axis? Which is the vertical axis?” “Does everyone agree?”

“What do we call the axes?” “Looking at

the TOV and the graph, can you write a

word sentence that describes the pattern?” “Can you now translate those

words to symbols?”

• Small group work—assign learners to

groups. (Combine according to skill?

According to interests? Decide according to who is present in class.) Each

group selects another student-generated situation and creates a TOV, a

graph, and an equation. Circulate and

offer assistance or guidance where

needed.

• Return to the words—groups exchange

their TOVs, graphs, and equations with

another group. New group describes in

Components of Performance

for the EFF Standard Use 

Math to Solve Problems and 

Communicate

• Understand, interpret, and work with

pictures, numbers, and symbolic 

information.

• Apply knowledge of mathematical

concepts and procedures to figure out

how to answer a question, solve a

problem, make a prediction, or carry

out a task that has a mathematical 

dimension.

• Define and select data to be used in

solving the problem.

• Determine the degree of precision 

required by the situation.

• Solve problem using appropriate

quantitative procedures and verify that

the results are reasonable.

• Communicate results using a variety

of mathematical representations, 

including graphs, charts, tables, and

algebraic models.

(continued on page 12)

EFF Lesson
12 HOTtopics Winter 2005

words what the three elements describe

in other forms.

• Guide reflection on the lesson. What are

they uncomfortable with? What do they

feel they need more work on? What

makes sense? 

• Relate to how algebra presents on tests

and in workbooks—go through some

workbooks with learners and focus on

some problems. In small groups ask

them to describe the situation presented in the problem in words, as a

TOV, as a graph, and, finally, as an

equation. Compare answers by sharing

with the larger group.

• More practice if necessary.

Formative check:

• Monitor student discussions to identify

understanding.

• Monitor small group discussions to

identify learners who are unsure.

• Evaluate content of student-generated

TOVs, graphs, and equations.

• Evaluate student explanations of TOVs,

graphs, and equations.

• Help students monitor their growing

understanding of the process.

Make connections: 

• Relate student-generated situations,

TOVs, graphs, and equations to textbook situations.

• Students explain similarities between

the two.

• Students explain how using algebra and

algebraic thinking is related to their

broader goals.

Evaluation procedures:

• Evaluate student ability to describe in

words, as a TOV, as a graph, and as an

equation a situation they encounter at

work, at home, or in the community.

• Use a traditional test to evaluate

whether learner can transfer understanding to the typical test format.

EFF Lesson, continued from page 11

Suzanne Poteet Elston has been a GED instructor for over 14 years and an ESOL instructor for 5 years. Her adult education

program in Tennessee began employing

Equipped for the Future principles and

strategies two years ago following an EFF

training workshop. Recognizing her ESOL

students’ need to understand and use math

in their daily lives, Suzanne began to focus

on the Use Math to Solve Problems and

Communicate Standard with her Intermediate/Advanced ESOL class. She used the

EFF Teaching/Learning Cycle as she and

her class explored what they did and didn’t

know about using math in an Englishspeaking environment. As Suzanne and her

students completed an eight-week unit of

study they had developed to focus on their

math and language needs, they discovered

that while we may do math differently from

one culture or language to the next, we still

need and use math for the same purposes.

In the following article, Suzanne examines

some of the challenges and rewards she and

her students discovered while Using Math

to Solve Problems and Communicate in

the ESOL classroom.

•••••••

Teaching and learning math in the

ESOL classroom offers some

unique challenges. One challenge

comes from the students themselves,

whose math education and experience

may vary widely. Another challenge often

lies with the teacher, who may be uncomfortable with addressing mathematics in

the ESOL classroom. In addition, the

teacher may be unprepared for the differUsing Math in the ESOL Classroom

by Suzanne Poteet Elston

It is even more interesting writing a

plan for an EFF lesson than for a traditional lesson! It isn’t hard to imagine

learners finding an EFF lesson more interesting to participate in; and when we have

learner interest—when learners find that

what they’re learning has impact in their

lives—they tend to stick around longer.

But what about the time factor? Many

adult educators worry that an EFF lesson

is more time-consuming than a traditional lesson. If you assume that each lesson should take only one class period to

complete—the old tyranny of the bell

from school days—then the EFF lesson

would take longer. The focus of an EFF

lesson is on understanding something in

depth, and on integrating multiple skills

in one lesson. So an EFF lesson might

take several sessions to complete, but

those sessions would be filled with discussion and sharing and learning from

one another and learning math concepts

so that when the formulas are long forgotten the learner can still remember

what the math was all about. Does planning the lesson take longer? Maybe the

first few times you do it. But if you allow

the learners to be responsible for their

own learning, and require them to supply

the materials that are authentic to their

needs, an EFF lesson should take no

longer to plan.

If we return to the first sentence—remember that you’re teaching learners, not

lessons—we see that the algebra class

we’ve described above teaches the learners

by making the lesson relate to what is important to them. Rather than impose information, it builds understanding of the

underlying principles of algebraic reasoning. Students in this class are less likely to

ask the standard question, “When am I

ever ‘gonna’ use this?” Such a lesson requires a willingness on the part of the

teacher to let go—to allow situations to

evolve in which s/he is not the all-knowing source of knowledge. It means that

every algebra lesson will be different, because the learners in each class will be

different. As a result, it’s a much more interesting class to teach—and a much

more interesting class for learners to participate in. ●
Winter 2005 HOTtopics 13

ences in the way math is often taught and

performed from one culture to the next.

Furthermore, ESOL students themselves

may not see the relevance of learning

math to learning English. To meet these

challenges, the ESOL teacher must approach teaching math with a learner’s attitude and an open mind, eager to

discover the students’ needs and to find

strategies to teach numeracy rather than

math as a set of discrete computational

skills. In other words, math must have

real meaning in the lives of English language learners.

Teaching math to ESOL students was

unprecedented in my experience, and I

found myself without a lot of models to

follow. Fortunately, the EFF Teaching/

Learning Cycle became a familiar and

reassuring guide as my students and I

explored new academic territory. As we

worked our way through the Cycle, we

discovered how they used math in their

daily lives and identified the math and

language skills they needed to perform

these math-related tasks. We devised our

own plans for how to teach and learn

these tasks, and how to evaluate our

learning (for it was indeed a learning

process for all of us.) The Cycle kept us

on-task in what might have been an

overwhelming project and kept us 

mindful of how the learning related 

to their lives.

Diversity of Needs and Experience in

ESOL Math Learners

Some of the problems we face in teaching

adult ESOL numeracy skills derive from

the same complexities we face in teaching

adult ESOL literacy. Students from widely

disparate educational backgrounds and

experience with math tend to be placed

together in an ESOL class based on their

oral English proficiency. It is not unusual

to have students with university degrees

placed alongside students with only a few

years of formal schooling. Plus, the multilevel ESOL classroom is still more the

norm than the exception, so the ESOL

teacher is often dealing with varying degrees of English proficiency as well. One

of the first challenges the teacher faces is

determining what the students already

know about a variety of math-related topics. This needs assessment is often not

easily obtained since students who actually understand and use math proficiently

in their own language may lack the English skills necessary to interpret and perform math-related tasks within an English

language scenario. Since typical standardized assessments often test more on the

students’ ability to read and understand

the problem rather than on their ability to

actually use math, an alternative approach

to assessment may be more useful.

I found that Steps One and Two of the

Teaching/Learning Cycle offered an 

excellent approach to discover what

ESOL students know about using Math

and what they want to learn. I presented

my students with a variety of everyday

contexts requiring an element of math;

this encouraged them to talk about how

they would address these situations.

These contexts may focus on any 

number of math-related tasks such as

reading a bus schedule (telling time),

estimating a restaurant food bill, taxes,

and tip (money, addition, percents,

making change), or following a recipe

(measurements). In my class, we looked

at workplace contexts, such as figuring

wages earned based on pay rate and

hours worked, plus estimating paycheck

deductions. We also talked about family

contexts, like following a budget and

planning for major purchases or 

projects. From these discussions, we

were able to identify several areas that

students wanted to work on.

Recognizing the Need for Math in

ESOL Learners’ Lives

The type of math education ESOL students need may vary from simple vocabulary acquisition to learning basic math

concepts and operations. This disparity of

math needs may lead the teacher to believe that math education is unnecessary

for some students and unapproachable

for others. Yet, as I talked with my students about their need to use and understand math in their lives, I realized that

their successful adaptation to life in the

U.S. depended, not just on being able to

perform math skills, but in being able to

communicate their findings in their new

language. Thus the argument for including math instruction in the ESOL curriculum became clear to me when I

considered how pervasive math is in our

language and everyday life. Surely no

ESOL teacher would question the necessity of teaching numbers, dates, telling

time, money values, and standard English

measurement systems to ESOL students.

These numeracy skills are survival skills

and are a part of any basic ESOL curriculum. Yet, the more complex math skills my

students wanted to master and the language skills they needed to talk about

them are no less important to their survival. They wanted help with everyday

numeracy tasks such as comparison-shopping, estimating earnings based on time

and pay rate, making choices about saving

and borrowing money, and working

within a budget. They wanted to understand more about workplace decisions,

such as pay rates for different employees

and how and why such business decisions

are made. They wanted to understand

more about the language of math and

business so they could communicate better with their employers about pay discrepancies, pay raises, and promotions.

Through our explorations and discussions, I began to recognize how complex

and pervasive their needs for math were.

As my students and I worked together

through Steps Three and Four of the
14 HOTtopics Winter 2005

Teaching/Learning Cycle, we began to

develop a larger contextual framework

that would offer them opportunities to

practice the skills and knowledge they

wanted to learn. We decided to create

our own imaginary business, ESOL 

Designs, and to go through the business

cycle of interviewing and hiring 

employees, setting pay rates, estimating

and graphing business expenses, creating

and submitting a decorating design 

proposal, accounting for all costs and

purchases associated with the proposal,

figuring payroll and deductions, and 

determining and graphing company

profits. Thus, they would be given the

chance to learn more about how 

American business works while they

used math and the language of math

and business.

Learning about ESOL Students’

Math Methodologies

Perhaps one of the most surprising challenges a teacher working with ESOL students will face is learning how to cope

with the variety of approaches and

methods ESOL students may use to do

math. The “universal language” of math

may actually vary widely from one language or culture or educational background to the next. Some ESOL students

use commas where we use decimals and

vice versa. Some students have an entirely different way of setting up and performing basic math operations, such as

division or multiplication. ESOL students may not recognize some of our

common operation symbols or must be

taught the words for these symbols in

order to be prepared to communicate

about subjects that have a mathematical

element. Furthermore, like most adults

with real-life survival experience, these

students may have developed their own

strategies for accomplishing mathematical tasks. Many of these students, though

limited in formal school math, may actually be gifted math strategists.

The teacher should be prepared to accept

the students’ different math procedures

and to realize that demonstrating how a

problem is worked according to our commonly taught algorithms may have little

meaning for ESOL students who were

taught to work the problem another way.

Add these factors to the different levels of

math ability and training likely to exist in

any ESOL classroom, and the teacher has

some enormous challenges to work with.

As my students and I worked through

Steps Five and Six of the Teaching/

Learning Cycle, we made some of these

discoveries along the way. Often we had

to stop in the middle of a business

process to review the mathematical 

concepts and skills needed to perform

the task. Through this process, we 

discovered some very talented (though

sometimes unorthodox) accountants

and money managers. We found that

the forms and worksheets we had 

created for our “company” kept us on

task as we collected data about materials

costs and employee payroll accounting.

In addition, they served as excellent

rubrics to document our success at 

performing the related math tasks.

Addressing Mathematical Content 

in the ESOL Classroom

The divergence of students’ needs once

again challenges the teacher to find a

meaningful way to address math content

with ESOL students. While English language learners may be well aware of their

needs for language education, they may

be less than enthusiastic about using their

“English time” to develop math skills.

Certainly, this concern would be justified

if the teacher’s approach to teaching math

is to “drill and kill” on math computation

and processes. In order to make math

meaningful as a necessary component of

language acquisition, the teacher needs to

approach math as an integrated part of

language and life. Math activities should

allow students to use their reasoning skills

and language skills to explore how they

would solve problems with a mathematical component. For example, teaching

multiplication with decimals as a discrete

skill will have little meaning for many

ESOL students; but putting this activity in

the context of a real life situation, such as

approaching the boss at work about a discrepancy in a paycheck, gives students a

meaningful task for using math. In this

context, they can practice figuring total

pay by multiplying pay rate times hours

worked (or by “adding on” or using a calculator or any other workable strategy the

student may have) and then communicating those results to another person.

As my students and I completed our

project and examined the success of our

"company," we used Steps Seven and

Eight of the Teaching/Learning Cycle to

guide our reflections about the unit of

study. We discussed what we had

learned and how we might use that

knowledge. All students thought they

were better prepared, both in language

and math skills, to deal with some of the

problems they face in their everyday

lives. I think I am better prepared to

understand and meet the numeracy

needs of my ESOL students.

While using math in the ESOL classroom may seem new and challenging at

first, it can be ultimately rewarding for

both teacher and students. Upon reflection, the ESOL teacher will recognize that

learning to use math within the context of

learning English is valid, necessary, and

already a part of what we do in the ESOL

classroom. The challenge for the ESOL

teacher is to expand this learning beyond

the basics of language into the building of

competence in life skills. In this way, we

prepare our students to better meet the

challenges they will face beyond our classrooms. ●
Winter 2005 HOTtopics 15

1. What if my students don’t know their

multiplication facts? 

Students who don’t know their times tables have other strategies to compute.

Adding on, doubling, using a calculator are

all examples of other strategies that work

for students. 

While many problems are easier to solve if

multiplication facts are known, it is not a

requirement to solve any problem. That is

not to say that you shouldn’t encourage

students to try to learn their multiplication

facts. 

Adult students have to solve problems involving math on an almost daily basis;

they should not be denied access to learning other relevant math content while they

are struggling to master their facts. In fact,

an effective way to practice multiplication

facts is in the context of other contents,

such as geometry and measurement.

2. But this [the traditional way] is how I

learned math!

We now have a much better understanding of how people learn math. Our earlier

approaches to teaching math were based

on the math that was needed in the 1800’s

and not based on research. Just because

we were able to learn math the “old” way

does not mean that it works for everyone,

nor is it necessarily the most effective way

to teach math.

3. How can I be sure that I’m teaching

what the students need to know? How

do you decide what is considered prerequisite for students? 

For example, if students want to learn

about fractions as ratios, don't they have

to know how to add, subtract, multiply,

and divide with fractions first? The EFF

Use Math to Solve Problems and Communicate performance continuum is a great

first step in guiding you in what should be

taught. Using the continuum helps you

make decisions about what content to

teach at each level—and to what extent.

For example, at Level 2, students need to

be able to use benchmark fractions such

as 1/2, 1/4, and 3/4. That does not mean

that they have to be able to add 1/2 and

1/4 and 3/4, but rather that they know how

to figure out what 1/2 of something is.

4. My students don't have a lot of time.

They only want to know the shortcuts

and tricks to doing math so they can

pass the TABE or GED. 

You’re right. None of us has a lot of time.

That’s why it’s critical that what we do

teach is meaningful and stays with us long

after the class is over. 

The GED and TABE tests are very different

so it’s important that you (and your students) know what you’re “teaching” to.

The TABE test is decontextualized computation. You can help students transfer their

learning to decontextualized problems

once they have had opportunities to explore content that is meaningful. The GED

is much more real-world related; there isn’t

the decontextualization that there is on the

TABE. The numbers are actually quite

“friendly” because the test assesses understanding of concepts rather than computation.

If you use the EFF Use Math to Solve

Problems and Communicate Standard

and performance level descriptors, you’ll

be teaching students skills that can be

used on the TABE as well as the GED. In

fact, at the lower levels of the continuum,

you’ll see that you’re encouraged to

tackle content such as proportional reasoning, algebraic thinking, and geometry

and measurement—much sooner than

traditional workbooks suggest.

5. My students are math phobic. I don't

know if they can learn math.

I wonder if they’re afraid of the memories

that are conjured up every time they are

introduced to math using the same methods that were used when they were in

grade school and didn’t “get it.” Have you

ever noticed that the topics for all adult

education workbooks look very similar—

and they all follow the traditional scope

and sequence? Students keep getting the

same content, taught the same way. And

if, year after year, they didn’t understand

what was taught, no wonder they’re beginning to dread math!

If you think your students are math phobic,

then it’s critical that you introduce math in

friendly ways that allow them many opportunities for success. Students, whether

they want to admit it or not, often use

math. Build on what they know and understand. Don’t jump from benchmark numbers to messy ones right after they’ve

begun to get comfortable with the concept. Give them lots of time to explore

concepts using just benchmark numbers

(including fractions, decimals, and percents). The EFF Use Math performance

continuum highlights what benchmark

numbers to use first, including what strategies should be comfortable for students

beginning to grapple with math content.

6. I wasn’t trained as a math teacher. I

need the structure of a workbook. 

Structure is one thing; holding on to traditional ways of doing things when you

know they weren’t necessarily successful

is another. If you take the attitude of being

a learner yourself, you’re more likely to do

more for your students than hanging on to

the structure now used. Pose dilemmas

for students (and yourself), then figure out

new strategies to address the dilemmas.

It’s amazing how creative our students can

be when given a problem that can be

solved in many different ways—and you

might be surprised at how much you can

learn from your students. 

The EFF Standard Use Math to Solve

Problems and Communicate does provide

you with some structure. The EFF Use

Math performance level descriptors provide additional structure. And, the EFF

Teaching/Learning Cycle is a useful organizing tool to help you and your students

develop meaningful lessons.

7. Are there materials available that go

along with the EFF Use Math performance continuum? 

No, not yet. But there is the EFF Teaching/Learning Cycle that you can use as an

organizing tool to create your own activities—with your students’ input. Students’

lives should be the basis for the curriculum, not a workbook. 

The EFF Toolkit 

http://eff.cls.utk.edu/toolkit/index.htm 

and the Guides to Using the 

Performance Continua

http://eff.cls.utk.edu/assessment/guides.

htm 

have more examples and tools.

Questions for Dr. HOT Topics
16 HOTtopics Winter 2005

■ Guide to Using the Use Math to Solve Problems and Communicate

Performance Continuum The purpose of this guide is to introduce the performance

continuum for the EFF Standard Use Math to Solve Problems and Communicate and

to show you how to use it to plan instruction and classroom-based assessment.

Site address: http://eff.cls.utk.edu/assessment/math1.htm

■ The EFF Toolkit Examples for Use Math to Solve Problems and 

Communicate describe math lessons and provide links to tools and supports for

these lessons.

Site address: http://eff.cls.utk.edu/toolkit/examples.htm

■ A Framework for Adult Numeracy Standards: The Mathematical Skills and

Abilities Adults Need to Be Equipped for the Future. This research project,

funded by the National Institute for Literacy, informed the EFF Standard Use Math to

Solve Problems and Communicate.

Site address: http://shell04.theworld.com/std/anpn//framewkTOC.html 

■ Developing Adults’ Numerate Thinking: Getting Out from Under the 

Workbooks (Focus on Basics: V4B, Sept.2000). This article suggests that we seriously

reflect on what and how we presently teach “math” to adults.

Site address: http://gseweb.harvard.edu/~ncsall/fob/2000/schmitt.html

■ The Effects of Continuing Goal-Setting on Persistence in a Math 

Classroom (Focus on Basics: V4A, March 2000). An adult education math instructor

shares the results of her own research on the effects of goal-setting.

Site address: http://gseweb.harvard.edu/~ncsall/fob/2000/meader.html

■ The LINCS Science and Numeracy Collection, organized by the four major

strands: number sense, data and statistics, spatial sense and measurement, algebraic

thinking, as well as another strand: problem-solving.

Site address: http://literacynet.org/sciencelincs/home.html

■ The Math Forum homepage contains a wealth of resources on math education.

While there is only a limited number of resources specific to adult numeracy, it does

provide a variety of activities and articles on topics that are relevant to adults as well 

as all other age groups.

Site address: http://mathforum.org

■ EMPower Math: The Empower materials, which include a series of eight books,

have been developed for and field tested in adult education classrooms by TERC.

EMPower is published by Key College Press and is also available from

http://www.peppercornbooks.com/catalog/

■ The GED Math Problem Solver: Reasoning Skills to Pass the Test. Myrna

Manly. Contemporary Publishing (2002). A resource to support students working on

the math portion of the GED Test Battery.

■ Adult Numeracy Development: Theory, Research, Practice (ed. I. Gal)

Hampton Press. Inc. Cresskill, NJ. (2000). This book focuses on numeracy as a 

component of adult literacy and includes approaches to instruction, reflections 

on practice and learning in adult education classrooms, and assessment.

EFF HOT Topics (Highlights on 

Teaching Topics) is a publication 

of the EFF Center for Training and 

Technical Assistance. Each issue 

concentrates on a specific topic, 

highlighting examples of actual 

practice and offering strategies and

tools for using EFF in the classroom. 

Please direct all correspondence 

including editorial comments to:

Diane Gardner

EFF Center for Training and 

Technical Assistance

Center for Literacy Studies

600 Henley St, Suite 312

Knoxville, TN 37996

or dgardner@utk.edu

This issue was edited by

Beth Bingman, EFF Center for 

Training and Technical Assistance, 

assisted by the authors.

Online Resources 

National Insti

NIFL

tute for Literacy

Funded by