During this past year I worked with 16 ABE Math Team teachers,
all intent on changing the rules to make our classes look more like
a hands-on math lab of the 21st century. (To read more about our
vision and the results of the shifts in math instruction, see The
Massacusetts ABE Math Standards Project Vol. 1 and Vol. 2.)
Much of the time, adult learners will politely go along with whatever
the teacher has in mind. Often the learners who come and stay, however,
are empowered enough to ask that the class meet some felt need.
How frustrating it is, though, when learners request something reminiscent
of the traditional schooling I was working against. And it happens
in every setting. The class wants to read orally round robin instead
of practicing silent reading. The students in one-to-one want a
pure phonics curriculum, the math class wants me to correct their
Bonnie Mullinix and the research we did together are responsible
for the ideas that have at least temporarily gotten me out of this
pickle jar. On an abstract level, the answer is continuing education.
We have to educate students to the alternatives. If they havent
heard of dialogue journals or cooperative learning puzzles, I shouldnt
be surprised that students arent asking for them. We have
to be clear about our own views and the reasons. But sometimes such
a forthright approach doesnt work. I need to make my point
concretely without getting on a soapbox, lecturing, or preaching.
The strategy is simple and could be adapted to serve as an initial
classroom assessment, as an interim evaluation to get student feedback
on next steps, or as an ongoing or final evaluation to compare students
ideas and comfort with mathematics with their ideas at the beginning
of the program.
Each of the twelve cards had a math topic on it. (See Figures 1-3.)
Guessing about how much something will be
+ - x :
ASSESSMENT CARDS USED IN RABEM STUDY
Guessing about how much
something will be
+ - x ÷
I simply said, Im going to give you a bunch of cards.
Each of these cards has a math topic on it. I want you to look
through them and put them in order. Put the thing you think is
most important on the top. Put what you think is the least important
on the bottom.
With basic level readers, I read the topics with them. Usually
the topics didnt need elaboration, although I found myself
offering some examples. The students needed little else to work
on the task.
Some learners did need help working well together. Although there
were no stated rules about coming to consensus, I wanted both
students in a pair to participate actively. I checked in with
the quieter student, saying, Do you agree with that?
or Didnt you want to keep estimation at the top?
When one student seemed to have lost track of the purpose, I reminded
her This is about your opinion. What do you think is most
important for an adult to know? There is no right answer.
Once students had finished and I had recorded the outcome, I
asked them to use the same cards to show me which topics
you spend the most time on in class. Put the ones you do the most
at the top and the ones you dont do much or barely touch
at the bottom. In both tasks, I said that ties between two
of the topics were allowed. In other words, two cards could occupy
the same place in the order by being positioned side by side.
Assessment Cards Used in RABEM Study
Does it make sense?
This activity opened three new doors. A rush of new ideas flowed
in to mingle with the typical expectations of math in adult ed.
Relevance of math to life rather than to the test
New awareness of the range and breadth of math
Placement of computation in perspective.
These results are true for all 49 learners Bonnie and I interviewed.
Ive chosen to focus here on the nine learners I interviewed.
Students cited relevance to life as one criterion for a high
ranking. Two of the three groups rated decimals as more important
than fractions. One student explained that if you dont
know where that decimal is, you cant tell about how much
to expect. In a bank or a store, they could give you forty dollars
instead of four hundred. You have to know where that decimal is.
Without a secure knowledge of decimals, adults are vulnerable
in the world of money. The third group gave decimals and fractions
equal importance, but none of the groups invoked the GED as their
criteria for importance.
In another instance, measurement was consistently above geometry
for all the groups. Though geometry figures on the GED, measurement
is a skill called for in daily life. In one class, a learner asked
me to explain what geometry is. On reflection, I realized this
lack of information is indicative of a much larger gap. Many adults
lack an overall sense of mathematics as a discipline, as an area
of study. I can hear impatient voices asking, why should someone
who needs to learn addition and subtraction be burdened with explanations
of geometry or calculus? Isnt that jumping the gun?
Assessment Cards Used in RABEM Study
Symbolic, graphic, text
How do you read and say
different math symbols?
Collecting and organizing
information, reading graphs, etc.
No, its not. Our students are consumers of education. It
is critical for adults as students, parents, and citizens, to
have some sense of the topics that lie ahead of them, the kinds
of math their children will study, the disciplines that comprise
scientific endeavor. Furthermore, an attitude toward math informed
by limited experience with computation may well have an adverse
effect on their drive to learn. If they feel that math will only
get harder and be even more tedious than the times tables, or
more complicated than adding fractions with different denominators,
theyll pace themselves more slowly. Like people anticipating
an ordeal, theyll make the tasteless thing they are working
on drag on as long as possible. Subconsiously they may hope that
time will run out before they have to go on to the next thing,
and in the meantime, at least their struggle will be familiar.
Top of page
I noticed learners initially reach for the Whole Numbers card.
One student asked Wheres addition. Youre going
to need that even if you arent going to know anything else.
In the course of reviewing the other cards, however, learners
began to place whole numbers in perspective. All three groups
pulled communication out and put it on the top. Without an understanding
of symbols, you wouldnt know whether to add two numbers
or multiply them. Without communication, you couldnt explain
your answer, couldnt explain that youd been given
you the wrong change. Without problem solving, you cant
solve the problem; youre lost. The learners were clearly
still attached to computation, but they were seeing it in the
context of other mathematical skills. Since members of the Math
Team and other teachers are experimenting with the math curriculum,
I was interested to see that students wouldnt necessarily
be wedded to tradition. The responses to this task indicate that
learners expectations of math class can be quite flexible.
Using this Strategy
To adapt this activity for initial or ongoing assessment, I would
Which topics would you like to spend most time on in class? Put
the cards in order from most time to least time.
Which topics do you think are most important for us to cover
during the next cycle?
Which topics do you feel most comfortable with? Put those at
the top, the least comfortable at the bottom.
I would phrase follow-up questions to get at the thinking behind
the order. To maximize the open lines of communication about instruction,
I would connect what I had planned to do to what the student was
asking for. For example, I see you want to learn fractions.
When you start class, well be doing measurement with whole
numbers. As we get into it, youll be able to begin fractions
and see how they work using rulers. Or You said that
problem solving is important. We will do that in two ways during
math class. We do problems together as a group that you have to
talk about (point to communication) and youll have problems
to take home as well.
The Context: Where This Activity Comes From
This activity came about as part of a larger study from February
to May, 1993. I worked with Bonnie Mullinix of World Education
on the Research into Adult Basic Education Mathematics (RABEM)
project sponsored by the Federal Department of Education. The
primary purpose of RABEM was to get a really good picture of math
instruction in Massachusetts ABE programs. To do that, program
coordinators and teachers filled out surveys. Bonnie and I interviewed
teachers, observed classes and interviewed students.
One aspect of the larger goal was to figure out where Massachusetts
adult ed classes are in relation to the Curriculum and Evaluation
Standards for School Mathematics published by the National Council
of Teachers of Mathematics. The cards we gave students came in
part from the standards NCTM put forth. A few minor changes were
made based on teacher interviews which preceded the learner interviews.
For example, Number Sense was too vague for teachers, so we left
it out of the student interview. A few other changes came from
learners themselves. When students from the Jackson-Mann ABE class
helped design the learner interview, they identified terms which
needed rewording or clarification.
The card part of the learner interview came after we had asked
students questions about their instructors, present and past (how
do you think your teacher feels about math? how do you know?);
to define math (what do you think math is anyway?); and how they
liked math (what they liked most and least). While we made no
effort to convert anyone to a holistic view of math, the types
of questions we asked certainly geared students to think about
math from many different angles.
Now when I hear about students negative response to an innovative
math class (like This isnt math. When are we going
to get back to long division?), I think this kind of exercise
would really help. By doing it, students are reminded that math
is bigger than the whole number computation they are used to.
By the act of choosing problem solving or reasoning in their top
five important topics, they commit themselves to learning concepts
that will help them in those areas.
The excitement I felt at discovering a successful new strategy
to work my way out of a pickle was undoubtedly intensified by
the use of a manipulative, in this case, the cards. I suppose
some readers may already be planning to turn this idea into a
checklist. In this case, a checklist would turn a hands-on activity
into a two-dimensional task. It will shut off creative thinking.
It will limit ownership because if the teacher holds it, she will
retain control and if she gives the page to the student, that
act will turn the activity into a written task, like a test, a
medium which literacy students generally find nerve-wracking.
Cards are more open-ended. The task gives students a chance to
be creative in their lay-out and to exhibit their organizational
strategies in a way reading and writing assessments dont
usually. The cards allow for pyramids, diamonds as well as stairs
or a linear arrangement. Because one can talk while moving the
cards, they encourage an external thinking process. Physically
moving the cards around made the impact of each decision on the
order as a whole a visible fact. In one case, learners kept adding
topics to the top of the order and we watched as Algebra, originally
placed toward the top sank down further and further. Manipulatives
and tactile learning belong in assessment.
As more and more teachers implement the NCTM standards or the
Massachusetts ABE Math Standards by emphasizing communication
and problem solving with calculators and estimation, some will
encounter resistance from their students. Countering student expectations
in an understanding and empowering way will be key to a smooth
transition to new and fun activities in the classroom. We need
to react in ways that will further the adoption of a new way of
doing math as well as to reassure students and give them a feeling
of control over what is happening in math class. Over the years
Ive learned that to have a real conversation about what
learners want, I need to provide some structure or some scaffolding.
Otherwise, Ill hear the internalized messages from a lifetime
of encounters with a traditional approach to schooling. Its
not fair to ourselves as teachers or to our students as consumers
of education to present choices without providing learners with
the information they need to make reasoned and informed choices.
Topics on cards are one simple strategy. Lets build a repertoire.
Top of page
This article was published in Adventures
in Assessment, Volume 6 (Spring 1994), SABES/World Education,
Boston, MA, Copyright 1994.
Funding support for the publication of this document
on the Web provided in part by the Ohio State Literacy Resource
Center as part of the LINCS
Assessment Special Collection.