Changing Approaches to Math

As the GED changes, so must instruction. Cynthia Zengler describes how her approach to math has evolved.

by Cynthia J. Zengler
When I began teaching mathematics in the mid-1970s, I taught students new mathematical concepts and followed up with a set of word problems using these concepts. Of course, the problems fit the new concepts and only required the students to identify which new concept was used. I emphasized key words such as of' for multiplication and grouped' for division. To improve as a teacher, I joined the National Council of Teachers of Mathematics (NCTM) and began to read their literature, but did not really apply it to my practice.

As my career evolved, I had the opportunity to attend meetings and workshops on mathematics. In addition to teaching mathematics part time for a community college, I became an editor, involved in developing mathematics textbooks. I worked with authors who were actively involved in research on teaching and learning mathematics. I had to think about trends in math and what math should look like in the future.

In 1987, the National Council of Teachers of Mathematics published standards for the mathematics curriculum that included more than just computation. The Council suggested including skills such as developing conjectures, reasoning through phenomena, building abstractions, validating assertions, and solving problems (NCTM, 1987). My worlds began to clash. How could I develop textbooks that suggested one method of developing problem-solving skills and use an antiquated method in my own practice? The NCTM recommendations made me rethink my teaching, which has not been easy. It is a continuing process that involves reading, reflection, and experimentation.

The Change

I taught business mathematics at a community college and math for the tests of General Educational Development (GED) at a community-based program to students who had been less than successful with mathematics. My students were typically older than 25 and trying to improve themselves so they could qualify for a promotion at work. I heard the normal anxieties about mathematics. Students would say, "I could never do mathematics," "I hope you have patience. You will need it," "I really need help," and "I hate math." The business course emphasized applying mathematics to various business topics such as mark up and mark down, trade discounts, and interest. When I emphasized strategies in problem solving, I received blank stares and panic.

The NCTM literature I had been reading suggested a four-step model, which I began using. The steps, which all need to be completed, are read the problem, decide what to do, solve the problem, and answer the question. They seem easy, but disciplining myself to follow them was hard.

I noticed that, as students encountered numbers in word problems, they wrote them down, regardless of whether they had completely read the problem or not. Well, I did that myself some times. I decided to model the method to ensure that the students and I would apply the steps. Now, as the first step, I have someone read the problem aloud to the class and we all listen.

The second step decide what to do is the most difficult to teach. To help students focus their thinking, and to wean them from dependence upon key words, I started using three questions: What do you want to find? What do you know? How are they related? Once the students answer these questions, they tend to understand the problem better. For example, a baseball team won 17 more games than they lost. If they played 52 games and tied three, how many games did they lose? The answer to: "What do you want to find?" is how many games did the team lose? You know that the team played 52 games, tied three games, and won 17 more games than they lost. This information is related in this way: games won plus games lost plus games tied equals total games, or (X + 17) + X + 3 = 52.

The third step is often the easiest because it is usually a calculation. The fourth step, answering the question, is not redundant. Often the initial solution only responds to part of the question asked. Getting my class to use the model often caused stress for my students and me. Since I was continuing to refocus myself to include more strategies for problem solving, I had to reinforce my own thinking. I began using the model whenever we did a problem in class and found this helpful in reinforcing the importance of the process. Sometimes I forgot to apply the method. My class would tell me to stop and ask myself the three questions. This helped give my students confidence in their own abilities. The atmosphere of the classroom changed from "the dreaded math class" to "a fun class." Including myself in the learning process created an atmosphere in which students felt freer to share their ideas.

I read all the NCTM journals, finding Mathematics: Teaching in the Middle School the most useful. The more I read about problem solving, the more I encouraged my students to think and trust themselves. They could use various strategies to solve problems if they could explain how they arrived at their solution. One strategy the students were amazed that I would allow was guess-and-check. They soon realized, however, that guessing takes longer than a more mathematical solution. As we went over their work, I had the students share their approaches. This provided them with an opportunity to develop trust in themselves and to feel less anxious about finding the' way to do a problem.

Example

Here is an example of a set of problems I have used as part of the initial lesson on problem solving.

1. If each letter of the alphabet has a price attached to it (A=0.01, B=0.02, C=0.03, and so on), find the price of your name. For example, the name FRED is worth 33 cents because F is worth 6 cents, R is worth 18 cents, E is worth 5 cents, and D is worth 4 cents. 2. Given that the alphabet has the above values, find five words that are worth exactly $0.50. [One solution is joy = 10 (j) + 15 (o) + 25 (y) = 50 cents].

Once the students have determined the value of their names, I form groups to work on the second question. After allowing a reasonable time for the groups to complete the activity, we discuss the various approaches they used, emphasizing the fact that many approaches exist. One strategy used by the groups is to find one word and see if rearranging the letters can form other words. For example, since joy is worth 50 cents, do "jyo", "yoj", " yjo", "ojy", or "oyj" also make words? This activity can be extended by changing the values of the letters or by finding words that equal other values.

Using a model for solving problems has given my students a tool for solving any problem, not just those with key words they can identify. The students in my classes have begun to solve more difficult problems consistently and with more confidence.

Assessment

My approach to assessment has also evolved. Once I started using various approaches to teach problem solving in my practice, I realized I had to refocus my assessment to include more open-ended questions that challenge the thought process of my students. As I have developed assessment over the last several years, I have kept in mind the need to assess areas such as thinking critically, analyzing how to do a problem, making conclusions that are valid, and evaluating the advantages and disadvantages of a method.

As I began my teaching career, my favorite assessment method was the typical question and answer method that requires students to produce a number. Now, I include essay questions. Sometimes the students are amazed when I want them to describe their approaches to a problem rather than produce an answer. "I thought this was math class, not English class," said one student.

I also use group quizzes in which group members are responsible for ensuring that each member understands the concept. Each group gives me one paper with all the answers. I choose a member of the group to present one of their solutions to the class. Initially students thought that they would have nothing to offer the group, but all students, regardless of their mathematical level, contribute to the discussion. The group setting often provides students a safe' way to discuss their approach to a problem. These quizzes have given some students a new confidence in their mathematical ability. John had barely passed any math class he had taken and truly believed he could offer nothing to the group discussion. After one such quiz, he told me that he was surprised that he could actually teach the others something about the problem. He was so proud. So was I.

The New GED

The current GED mathematics test emphasizes problem solving that requires the application of computational skills to life situations. The GED 2000 Series Test, being developed, is shifting the content of the mathematics tests to more involvement with data analysis, statistics, and probability (Woodward, 1998). The GED Testing Service mathematics specifications committee has even recommended the use of calculators for the major part of the mathematics test (Manly, 1998). As a GED instructor, I found that the students had the same anxieties and panic as my business mathematics students. I see the need for those preparing for the GED to learn how to approach a problem. The new GED will expect the students to analyze a problem and to make conclusions based on their findings. The more confident students become in their mathematical abilities, the better prepared they will be for not only the GED but for the changing work place.

References

Manly, M. (1998). "Calculators in the ABE/GED classroom: Gift or curse?" Adult Learning 9(2), 16 - 17.

NCTM (1987). Curriculum and evaluation/standards for school mathematics. Washington, D.C.: NCTM.

Woodward, K. S. (1998). "Next generation of GED tests: Blue prints leaving drawing board." GED Items 15(1). 1,3,6.

About the Author
Cynthia J. Zengler has a masters degree in Mathematics Education and is working toward a doctorate in adult education. She is the project manager for a multi-year evaluation design project for the Ohio Department of Education, Adult and Basic Literacy Education programs. She has taught mathematics at many levels, including GED preparation classes.