Problem Solving Is More Than Solving Problems
THE CURRICULUM AND EVALUATION Standards for School Mathematics (NCTM 1989) states that one of its five general goals is for all students to become mathematical problem solvers. It recommends that "to develop such abilities, students need to work on problems that may take hours, days, and even weeks to solve" (p. 6). Clearly the authors have not taught my students! When my students first encountered a mathematical problem, they believed that it could be solved simply because it was given to them in our mathematics class. They also "knew" that the technique or process for finding the solution to many problems was to apply a skill or procedure that had been recently taught in class. The goal for most of my students was simply to get an answer. If they ended up with the correct answer, great; if not, they knew that it was "my job" to show them the "proper" way to go about solving the problem.
I began to notice an unwritten contract between my students and me when they experienced difficulties in solving a mathematics problem-I would come to their rescue. At first, I encouraged them to think about things that we had just learned or advised them to reread the problem. If these suggestions failed to help an individual or a group of students, I gave them specific hints that could guide them to find the answer. I might say, "Use a table to organize your data, and look for patterns." I would ask them specific questions that I knew would lead directly to the solution of the problem. When none of these teaching strategies was effective in helping my students get the answer, I worked the problem on the chalkboard with the whole class.
I cannot remember ever taking more than about fifteen minutes on any problem that I gave my students to solve. Imagine my discomfort and confusion when I first read the passage from the Curriculum and Evaluation Standards regarding the need for students to spend hours, days, and weeks on a single problem. I tried to think of mathematical problems that I could give my students to keep them thinking and interested for hours, but I was uncomfortably unsuccessful in this quest.
I began to read articles in mathematics education journals (the Arithmetic Teacher, Teaching Children Mathematics, Mathematics Teaching in the Middle School, and the Mathematics Teacher) and attended local professional-development meetings in search of a new view of problem solving. At one presentation on problem solving, the speaker said that she would share a great mathematics problem that could be used for students of all ages. She put this "problem" on the overhead projector for us to read:
The Elmwood School caught fire today. After reading this "problem," I became disappointed and confused. I looked around the room and noticed that others were chatting about this problem. After discussing my reaction with those around me, I realized that many other teachers thought the same way I did. But as we discussed it, I began to get a different view of problem solving. One person in our group declared, "It was a science experiment that exploded, causing the fire." Another person stated, "The sun caused the fire. As it shined through the window of a classroom, it was caught by a magnifying glass, which was left on a stack of old newspapers." During this presentation, the speaker helped us see how our own creativity, reasoning, and prior mathematical knowledge could be used in the problem-solving process to facilitate our creating meaningful mathematics for ourselves. I came away with some good ideas on how to proceed with problem solving in my classes. I also collected good problems from these sources and resolved to practice patience with my students' problem solving.
As I planned during the summer, I vowed to integrate problem solving into all areas of the mathematics curriculum. Armed with the problems I had gathered and a firm resolve based on my summer reflection, I began the new year with problem solving as my focus. What occurred during the first month of school was far from enjoyable.
During the first few weeks of the school year, my students wanted me to show them how to solve all problems. Furthermore, because I did not give them hints or strategies on how to solve the problems, they believed that I was not doing my job--as did some of their parents! I learned that many of my students had no real problem-solving experiences. As my students and I have struggled with problem solving over the years, I have developed some techniques and strategies to help them become more powerful problem solvers.
GETTING STARTED MY FIRST SET OF GOALS FOCUSES ON HELPING my students learn what tools are available for the mathematical problems they encounter. I introduce a wide variety of problems that facilitate the use of multiple strategies, manipulatives, and calculators. As my students become familiar with these tools, I emphasize creativity and novelty in solving mathematics problems. During the beginning of our problem-solving experiences, I notice that some students experience anxiety and frustration, which I believe occurs because I do not give in to their requests to show them "how to do it." I try to be compassionate but strive to wean them from this behavior. I find that it is difficult to create the desired atmosphere for problem solving in a classroom without first changing the students' attitudes and beliefs about mathematics and problem solving. This attitude shift begins to occur in my classroom when I demonstrate to my students that I value problem solving and that patience and persistence are necessary to be a successful problem solver. Good problems are also essential in the process of changing my students' beliefs about mathematical problem solving.
At the beginning of the year, I start with a problem that is easy to understand and that can be approached in various ways. We call one such problem that has generated productive discussions in my classes the "shoe store problem":
A man goes into a shoe store and buys a pair of shoes for $5 and pays with a counterfeit $20 bill. The shoe store owner does not realize it. Not having change for the $20 bill, he runs to the grocer next door. The grocer gives the shoe store owner four $5 bills for the bogus $20. The shoe store owner returns to his store and gives the man the shoes and $15 change. Later the grocer comes back to the shoe store owner with the FBI and informs him that the $20 bill was bogus. So the shoe store owner gives the grocer $20, and the FBI keeps the bogus bill. How much did the shoe store owner lose? (adapted from Sobel and Matelesky 1988).
I recommend that my students read the problem in a number of different ways. First we read the problem aloud as a whole group. Then we read it again and have certain groups read alternate sentences to help students interpret the problem in a variety of ways. This step leads into a brainstorming activity in which students offer what they believe is important information in the problem. We do not discredit any ideas at this point; we only list possible relevant and irrelevant information. For example, my students usually list all participants involved, the order of the events, the amounts of money, and any other information that they believe is important in the shoe store problem. The purpose of this part of the activity is to generate direction for the small-group work that follows.
After a few minutes of brainstorming, students work in small groups of three or four to generate strategies that can be used to solve the problem. They carry out their methods until the group members are satisfied with their solution or all agree that no solution is possible. As they are working, I rotate among the groups and ask questions to stimulate their thinking and communication. As they attempt to solve this problem in their small groups, it is more important to me that students exchange points of view about their own thinking than that they conform to someone else's thinking. However, each group must at some point agree on the most convincing approach and be prepared to defend its conclusion to the whole class.
After students have had sufficient time to agree on an interpretation, a method, and a solution for the problem, the groups reconvene into one large group to share their results. Here the insightful discussions usually take place. As they are sharing, I keep track of the types of strategies used by groups and their solutions and ask questions that require students to reconsider their solutions or their methods for solving the problem. For the shoe store problem, students typically try to present a "logical sequential" argument or one that contains visual information--acting it out, charts, diagrams, and so on--in an attempt to convince others in the class. A typical student argument follows: "The shoe store owner gives the man $15 change and the shoes, then must give the $20 back to the grocer and the bogus bill to the FBI. He is therefore out $15 + $20, or $35, plus the shoes." As these arguments are shared, students frequently use pictures or actions to illustrate the "flow" of money in their solutions.
One reason that I like the problem is the significant disagreement that it creates among my students. As students convince their peers of the logic of their solutions, I restate these arguments as closely as possible, often with some confusion in my voice. When I use this approach, I notice that many students in the class nod in agreement with my expressed confusion. The most difficult teacher "action" for me during this period of our development together is not answering students' questions. My students frequently look to me to verify their answers or settle their disagreements. Answering their questions was how I used to think that I was helping my students. I have found that this help is only superficial and temporary.
Rather, by asking questions that keep my students focused and thinking about central mathematical ideas, I give them the boost they need to develop confidence in their own abilities to think mathematically. I ask questions like these: "Which solution makes the most sense to you after considering these arguments? Are these arguments convincing to you? How would you defend your thinking to those who may not agree with you?" As my students become more independent and creative, I ask questions that encourage them to investigate their own problems and ideas in mathematics, for example, "How could you use what you have found in this problem to solve a related problem, if, for example, a part of the original problem was changed in some way? How does changing parts of this problem affect your problem-solving process and solution?" I can tell when my students are progressing because they begin asking me these types of questions before I ask them.
Waiting for the students to take charge of verifying a solution is still a bit uncomfortable for me, but contributions from student observers reflecting on the solution presented occur spontaneously. When I gave them time to respond in one class, a student suggested from the corner of the room that "the $20 given to the grocer by the shoe store owner was not lost." The room then erupted with talk from all groups until a different student blurted out, "Let's act it out." After such reenactments and further discussion, I review all previous arguments by asking new questions for all my students to consider, such as, "How much was the shoe store owner out after he got back from the grocery store and gave the man his change and shoes?" By reviewing students' arguments and asking them questions, I affirm the viability of their techniques and encourage them to think about different ways of viewing the problem.
This method helps not only those who are confused by a problem or someone else's line of reasoning but the presenters as they reconsider their own thinking. In the process of looking for an alternative technique or perspective for the problem, my students either come away with confidence in their original thinking or shift toward a more convincing perspective.
It is important during these initial experiences with problem solving that my students view problems from many different perspectives and present and listen to convincing arguments. After initial arguments to the shoe store problem are presented and reviewed, I call for any new solutions or different arguments not yet presented. By asking for further discussion, I want my students to understand that the real, important issue in problem solving is considering a variety of possible strategies in solving problems.
Furthermore, students must learn that becoming a good problem solver requires that each person alone must decide whether his or her thinking about a given problem is correct. This requirement makes me reluctant to contribute my personal views on individual problems until I believe that they will not overly influence my students' thinking or confidence to do mathematics. The rule that I use for knowing when to respond with my viewpoint during our problem-solving activities involves listening to my students' comments. If they regularly listen and openly discuss one another's ideas in class, and offer arguments to support or refute these ideas, I will begin to offer my thinking about a particular problem. If after I offer my thinking on a problem, students respond to me in the same way that they respond to their classmates, I will continue to contribute my ideas. This type of student response indicates that my students' confidence in doing their own mathematics is strong. I have found that the length of time I spend doing these initial activities varies with my students' prior experiences and age and with my ability to communicate these new problem-solving goals consistently.
Eventually my students figure out that I am really interested in the process of problem solving. Unfortunately, they also begin to believe that anything they say or do is correct and that answers to mathematical problems are not important or necessary. It is common during this time for my students to say, "He isn't saying I'm wrong, I must be right! As long as I write down anything, he'll give me full credit." Piaget believed that all students will eventually arrive at the truth of mathematical ideas if they argue long enough, but therein lies the problem (Inhelder and Piaget 1958). If I tell my students that their answers are wrong, I become their mathematical power. If I do not tell them that some of their answers are wrong, they seem to be satisfied with their first attempts.
I find that my students pursue problems more rigorously when I require them to create arguments to communicate and to convince others that their thinking and solutions are correct. I have each group present its version of a problem to the class; afterward I ask such questions as"Does anyone have another way to think about this problem?" When my students ask me to tell them which answer or strategy is correct, I ask, "Which answer do you think it is? Why do you believe that that solution or strategy is correct?" This response does not satisfy some students in my classes at first, and they insist that I verify the correct answer. I tell them that if they have any doubt about the solution and method they used, their problem-solving process is not complete. Anyone who has doubt remaining should find yet another view of the problem and solution.
Doubt usually lingers with the shoe store problem. After much discussion and some encouragement to seek another way to view this problem, one group presented a simpler problem and a working-backward approach. They stated, "Instead of thinking about how much money the shoe store owner lost, consider who in the problem ended up with money and how much they have. From this we can say how much the shoe store owner lost. The grocer gave $20 and got that back, so he is not ahead.
The FBI only has the counterfeit bill. The only other person in the story (besides the owner) is the person who wanted to 'buy' the shoes. He left the store with the shoes and $15." These varied presentations, discussions, and reflections help us understand what problem solving is all about.
My goal is accomplished when my students and I interact as equals in the problem-solving process. When this situation occurs, I am viewed as one among many members of our community of problem solvers. My suggestions and ideas are valuable, but no more so than those of any other member of our community. My role is to introduce problems and questions that stimulate mathematical thinking. I am successful in stimulating thinking if my students ask questions that I have not previously considered. For example, I gave my students a problem about finding the area of a set of isosceles triangles. During the process of solving this problem, one group conjectured that the product of a leg with one-half the base was always greater than the area of these triangles. I had not previously encountered, or could not remember, this conjecture. At first I felt the need to be able to verify this conjecture immediately, but instead I devised questions that helped the students and me carefully consider the impact that this conjecture might have on other things that we know about area and isosceles triangles: "How are isosceles triangle special? How does the height of an isosceles triangle compare with either of its legs? How is finding the area of an isosceles triangle related to the length of its sides?"
When my students are confident in their problem-solving abilities, they automatically create mathematical arguments to support or refute hypotheses and understand the need to share these arguments with the whole class. Needless to say, this phase does not occur overnight. It takes first, a teacher willing to participate in the problem-solving process with students, and then, time, persistence, and good problems.
Finding a few good problems to use in my classes helped me begin changing the way in which I taught and interacted with my students. I include a few of my favorite problems in the hope that they may be useful.
|1. If you fold a sheet of paper in half one time, the result is two sheets half the size. If you fold the half sheets again in half, you now have four sheets. If you can continue to fold this piece of paper in half twenty times, how many sheets thick would it be? If you sat it on the floor, how high would it reach?|
|2. Joe and Esther were making the same salaries when the boss came in and told Joe that he was getting a 10 percent cut in pay. While he was there, he told Esther that she was getting a 10 percent raise. After six months of complaining by Joe, the boss came back and gave Joe a 10 percent raise and Esther a 10 percent cut in pay. Compare their current salaries to their original salaries, after all these cuts and raises. Are they making more? Less? Explain your answer.|
|3. Two cars are traveling in the same direction on a road. The car that leads is traveling at 46 miles an hour. The other car is traveling at 70 miles an hour. How many miles apart will they be thirty minutes before the faster car catches the slower one?|
|4. Five cats can catch 5 mice in 5 minutes. At this rate, how many cats are needed to catch 100 mice in 100 minutes? (Hint: it is not 100.)|
|5. Of twelve coins in a bag, one is counterfeit and weighs less than the others. Use a balance scale to find the fake coin in the fewest number of weighings. Explain.|
6. Tennis balls are packed tightly, three to a can. Is the can taller than the distance around, or the reverse? Explain how you know that your answer is true.
In the end, my students abstract a definition of mathematics and problem solving through their interaction with the problems I select, the discussions we have, and the things I assess and value. In the process of changing my classroom, I have learned to be more concerned with the subtle qualitative changes in my students and classroom environment than with actual numbers of problems correctly completed. I have found that it is conceivable for my students to spend hours, days, and even weeks on a single problem on their way toward learning important mathematics.
Inhelder, Barbel, and Jean Piaget. The Growth of Logical Thinking from Childhood to Adolescence. New York: Basic Books, 1958.
National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.
Schoenfeld, Alan H. "Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics." In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws, 334-70. New York: MacMillan Publishing Co., 1992.
Sobel, Max A., and Evan M. Maletsky. Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies. 2d ed. Englewood Cliffs, N.J.: Prentice Hall, 1988.
AUTHOR:MICHAEL MIKUSA, firstname.lastname@example.org, teaches at Kent State University, Kent, OH 44242-0001. His research interests include investigating how students establish the truth of their ideas in mathematics and studying the dynamics of change in implementing reform in teaching mathematics.
SOURCE: Mathematics Teaching in the Middle School 4 no1 20-5 S '98. Reproduced with permission from Mathematics Teaching in the Middle School, copyright 1998 by the National Council of Teachers of Mathematics. All rights reserved.