What Counts?

Out of a Pickle: Setting the Stage for Math

Martha Merson
Adult Literacy Resource Institute

In every literacy setting I’ve worked in, I’ve wanted to change the rules.
The new rules would be: this is a class where evaluation doesn’t hurt self-esteem, where learners’ knowledge about the world gets counted and woven into the learning at hand, where the work that is done in the class has a purpose or use.

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 worksheets.

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 haven’t heard of dialogue journals or cooperative learning puzzles, I shouldn’t be surprised that students aren’t asking for them. We have to be clear about our own views and the reasons. But sometimes such a forthright approach doesn’t work. I need to make my point concretely without getting on a soapbox, lecturing, or preaching.

Meeting Math Goals
This paper lays out a strategy for familarizing students with a broad view of mathematics, for opening the dialogue about which topics should get covered during math class, and for using assessment as an opportunity to build expectations for a new or continuing class. In this article I am recommending a way to meet these goals. Listen, it’s CHEAP – not a lot of materials, not a lot of time. Teachers can add this strategy to a repertoire of ways to create situations in which students become familiar with a wider range of instructional options and are therefore more informed when they exercise their right to choose.

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.

Topics on Cards
I gave students in pairs or groups of three, a set of twelve cards. I used this Topic on Cards approach three different times. In one case I worked with the whole class. In the other two cases, I asked the teachers to choose two students they thought would be interested.

Each of the twelve cards had a math topic on it. (See Figures 1-3.) For example:

Estimation
Guessing about how much something will be

Whole numbers
(Computation)
+ - x :

FIGURE 1
ASSESSMENT CARDS USED IN RABEM STUDY

Estimation Guessing about how much something will be	Whole Numbers Computation + - x ÷
Fractions	Decimals

I simply said, “I’m 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 didn’t 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 “Didn’t 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 don’t 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.

FIGURE 2
Assessment Cards Used in RABEM Study

Patterns and Relationships	Measurement
Problem Solving	Reasoning Does it make sense?

 

What Emerged
This activity opened three new doors. A rush of new ideas flowed in to mingle with the typical expectations of math in adult ed. They are:
• 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. I’ve 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 don’t know where that decimal is, you can’t 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? Isn’t that jumping the gun?

FIGURE 3
Assessment Cards Used in RABEM Study

Communication Symbolic, graphic, text How do you read and say different math symbols?	Geometry and Spatial Sense Understanding shapes and spaces
Algebra	Statistics and Probability Collecting and organizing information, reading graphs, etc.

No, it’s 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, they’ll pace themselves more slowly. Like people anticipating an ordeal, they’ll 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.

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I noticed learners initially reach for the Whole Numbers card. One student asked “Where’s addition. You’re going to need that even if you aren’t 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 wouldn’t know whether to add two numbers or multiply them. Without communication, you couldn’t explain your answer, couldn’t explain that you’d been given you the wrong change. Without problem solving, “you can’t solve the problem; you’re 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 wouldn’t 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 ask:

Which topics would you like to spend most time on in class? Put the cards in order from most time to least time.

or

Which topics do you think are most important for us to cover during the next cycle?

and

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, we’ll be doing measurement with whole numbers. As we get into it, you’ll 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 you’ll 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.

Conclusion
Now when I hear about students’ negative response to an innovative math class (like “This isn’t 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 don’t 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 I’ve learned that to have a real conversation about what learners want, I need to provide some structure or some scaffolding. Otherwise, I’ll hear the internalized messages from a lifetime of encounters with a traditional approach to schooling. It’s 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. Let’s build a repertoire.

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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.