Problem Solving with Discrete Mathematics
Friedler, Louis M., Teaching Children Mathematics
The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) calls for an emphasis on problem solving at all levels and recommends introducing discrete mathematics topics. In fact, the first standard for grades K-4 states that "the study of mathematics should emphasize problem solving . . ." (p. 23). The section on high school standards recommends that discrete mathematics topics be included because so much current "information processing requires the use of discrete (discontinuous) mathematics" (p. 176). Is this material appropriate for elementary school? The 1991 NCTM Yearbook (Kenney 1991) contains articles concerning this material at all levels. Although Graham (1991) makes several excellent suggestions for specific discrete topics for elementary students, most authors refer to the same two or three problems on Euler paths. (See Kaczmar .) So the questions remain. How can elementary school teachers implement these ideas? What are other sources for appropriate problems? What methods of presentation will work? How do we use a problem-solving approach with this material?
I come to these questions with a perspective different from that of the typical elementary school teacher. I am a research mathematician and college mathematics teacher and had never thought about elementary school mathematics until my daughter entered school. This background as a mathematician affects both the topics I choose and my method of presentation. I have recently been working with groups of children at Swarthmore-Rutledge (S-R) Elementary School for two and one-half years. During the 1992-93 school year, eighteen fifth graders came to school forty-five minutes early once every other week to pursue mathematics.
My objectives in working with the children have been to use discrete-mathematics topics to explore problem solving, to introduce the students to mathematics that they will find stimulating and that is related to current "real" mathematics, to give them challenging problems yet let them feel successful, to have them explain their reasoning, and to encourage group interaction. In Polya's (1945, 1966) paradigm for problem solving, students are given difficult problems and told to look first at simple or special cases. When I work with elementary school students, we start with simple examples. That is, we first consider challenging problems and then break them down into smaller parts. For example, in the class-1 problems that follow, most students attempting problem 7 would find it extremely difficult without first doing problems 1-6. Once students discover the patterns in problems 1-5 and have solved problem 6, they can successfully solve item 7. It is relatively unimportant whether students learn discrete mathematics in elementary school; it is important that they understand that mathematics is more than arithmetic and that they know how to, and wish to, attack new problems.
To begin, I ask students to work in groups of four to five. Although that number was originally determined by the configuration of the room, I have found that it works well, and this finding is supported by other authors (see, e.g., Slavin [1988, 12]). When asked a question, I am more likely to ask another question than to answer it. It is difficult to avoid helping, but students will turn to each other to work out answers if allowed. Our classroom is noisy; students argue with one another about solutions. When students explain their reasoning, they frequently suggest ideas different from what I had in mind. They are encouraged to believe that more than one correct way may be possible to solve a problem. The real-life context for each problem appeals to the students.
The students who regularly attend the mathematics group are motivated and bright. Yet I have successfully tested the class-2 problems and several similar problems with a more heterogeneous group. Students respond to the excitement of mathematics. The following describes two typical classes, which might vary in length from thirty to fifty minutes. …