A Foundation for Learning Math

by Jan Phillips
Every time I enter my multilevel mathematics classroom, I'm reminded of the truly diverse adult student population that love encountered during the past 30 years of teaching. In the northwest suburbs of Chicago, at William Rainey Harper College, where I teach, this diversity seems to be increasing exponentially and is infinitely more obvious now than ever before.

It's not my imagination that these present day students are more diverse. Not only do they have a wide range of math skills and very differing needs; but they have a diversity of character, whether it's age, ethnicity, race, sex, socialization, economics, maturity, mental ability, emotional stability, or any combination of these assorted factors. As a multilevel teacher I have to take all of these factors into account, and take one student from point A to point B to point C at the same time I'M taking another student from point C to point D, and another from point B to point C, and another and another 15 times over. Hey, what's the big deal? Here's how I do it.

Structure

I build a structure that the students can rely on. It may not seem very important, but this structure gives the students a foundation; it gives them a base of operation they can trust. They know what to expect from me and from each other. My classroom structure may not seem to be that innovative, but it is a structure that I use to foster communication, interaction, and verbalization in a mathematical context and to promote the importance of the correct application of math principles and skills in our lives.

My students attend math class one day a week for three hours. The time just flies. During the first 30 to 40 minutes, as they are coming in, learners work individually on computers, following a series of problem solving programs that I've recommended. I hop from student to student, giving each one individual attention, commenting on his or her progress, listening to family and work experiences, sharing my own personal experiences, emphasizing math skills, looking at homework, and generally getting to know as much as I can about how he or she is progressing in the class. Each student is aware that we are focusing on math, making connections that relate his or her life to what we are studying, and above all, developing a communication process in an mathematical context.

Recently, one student named Isidro received a promotion to head custodian at an exclusive private school. Although he was extremely happy to have this opportunity, he realized that this new responsibility required a new level of math skills, including those needed for purchasing cleaning products, keeping an inventory, and scheduling other employees. In class we spent a considerable amount of time reviewing the necessary math problem solving skills he would need to be successful at his job. His enthusiasm for learning job-related math skills was contagious. Other students could hear what we were talking about and asked that we continue the topics in our large group segment that followed. Pretty soon the whole class got involved sharing their experiences on the job or in the home and wanted to apply what we were studying to their own lives.

Engagement

Following the individual computer work, we spend a good part of class time building and strengthening student engagement in a large group setting. In this segment students of all levels move their chairs and gather closely as a large group around a board, to discuss a topic - for instance adding, subtracting, multiplying, and dividing decimals. Each student has paper and pencil and a calculator, and whatever notes he or she wishes. After I write a problem on the board, I ask one student to tell us all what the rule is that we should follow to solve the problem. Usually the answer is a combination of several student responses. They all write the rule on their papers. Then I ask another student to put the numbers in a real- life context. For example, the problem might be 12.99 x .08. "Can you imagine doing a problem like this in real life?" With a little coaxing we relate the problem to finding sales tax on a purchase at the mall. Now we calculate the answer, following the rule and using the context to help us put the decimal point in the right place. The students back up their calculations with their calculators and see that they need their understanding of the numbers in context to help them choose the right answer even when they use a calculator. (The answer is 1.0392. In our context, we would choose $1.04 as a reasonable response.) We usually go through all four operations in this manner, so that we can compare and contrast the methods of calculation. For some students this is a review and for others it's all new. I encourage all students to use their resources, whether that means using a times table, helping each other, or asking for teacher support.

After about 30 minutes, I pass out a sheet of what I call real-life problems and ask them to follow the rules, use their life experience, and solve the problems. Then they compare answers with each other and make changes and corrections as they see fit. Finally, they all have the same answers - or have agreed to disagree - and they check with my answer key to see if I agree. The students count on this time of interaction and communication. By working together, the students overcome their differences, find new associations, develop verbal skills, and understand how they can help each other. They're often surprised how they can enjoy solving math problems even though they sometimes struggle in the process. At this point it's time for a coffee break and it's not unusual to hear them continue their math conversations outside of class.

Skills

After this break the class reassembles and forms small groups to work on particular skills. One group might be working on operations with fractions using a ruler. They draw lines of certain lengths and then add, subtract, multiply, or divide the lines according to instructions. They discuss how to perform the operations visually and then mathematically. Some students say that they have avoided using rulers since they were in elementary school because they were not sure how to read the units or how to manipulate them. This particular activity increases the understanding of fractions and allow students to practice using a measuring instrument. Quite often, this leads to measuring all kinds of things in the room.

A second group could be using the rulers to measure objects and then use ratio and proportion with the measurements to solve problems. For instance, they measure pictures, triangles, distances on maps, and reduce and enlarge them proportionately. We then extend these concepts to a discussion of similarity and how we can use it in problem solving. I encourage all students to use a GED textbook as a reference and source of practice problems pertaining to the topic. In these small groups the students find the math skills to be the bond between them, and while they share feelings of anxiety and frustration, they also share strategies for learning and problem solving. They have the opportunity to verbalize their thoughts and processes as they help each other, ask questions, and explain the steps to solve the problems.

My adult math students have challenged and continue to challenge every brilliantly conceived and well-planned teaching strategy that I've devised. Sometimes, I ask myself, where is that "Teachable moment" and more specifically, how do I get there? How do I address their very differing needs and wide range of skills? How do I engage them in meaningful, thought-provoking activities without scaring them away? The National Council of Teachers of Mathematics Standards, The Adult Numeracy Practitioners Network Frameworks, and the Massachusetts ABE Math Standards all agree that communication, participatory problem solving, and connecting number sense to relevant life skills are the keys to success in the study of mathematics. Nowhere is this more true than in an adult education multilevel classroom. By developing a structure that addresses these concepts we are giving our students the foundation they need to build a future that includes confidence in their mathematical abilities. By employing a wide variety of teaching activities and strategies to engage the students, to create an interactive environment, and to encourage them to communicate mathematically, we can soften the effect of differing levels of mathematical abilities and enhance the effect of shared experience.

Last week, Ed, a student who failed the GED test on his first try, reminded me that he only needs one more point to pass the GED test. He thought if he just spent a little more time practicing adding and subtracting fractions he could get that point. I reminded him that the GED test was not a computation test but a problem solving test and that he and the other students were better served by practicing problem solving and sharing strategies with each other. So he said, "Now I get what we're doing - we're learning how to figure out the solutions and explain them to other people! Wow! I hope it work."

Well, so do I, Ed; so do I. I'M counting on it.

About the Author

Jan Phillips, Associate Professor of Adult Education at William Rainey Harper College, a community college near Chicago, teaches ABE/GED mathematics full-time. She has been working in the field of adult education for more than 25 years, and is the secretary and a co-founder of the Adult Numeracy Practitioners Network and a member of the National Council of Teachers of Mathematics.

For a list of computer programs, see the Blackboard.