Experimenting with classroom formats to encourage problem solving
ABSTRACT: In this article, I describe how modifying familiar classroom formats in a college geometry class helped to encourage student problem solving. These modified formats, which resemble the problem-solving settings of mathematicians, are described in the context of problems my students and I explored. It is hoped that this paper’s success stories encourage the reader to consider experimenting with classroom formats as a resource for encouraging student problemsolving.
KEYWORDS: Problem solving, geometry, cooperative learning.
In the spring of 2000, I was faced (once again) with the challenge of teaching geometry to prospective teachers from a problem-solving perspective.1 As I looked over the syllabus, I struggled (once again) with the “coverage versus reasoning” conflict: how could I cover so many topics in 14 weeks while providing opportunities for genuine reasoning and problem solving amongst my students? In this article, I describe how I resolved this conflict by experimenting with two teaching formats or classroom arrangements. To illustrate the formats’ usefulness, I describe them in the context of problems my students and I explored. I begin by briefly reviewing what I intend by a problem-solving approach to teaching mathematics. I then describe the formats I used and their relation to the goals of problem solving.
A PROBLEM-SOLVING APPROACH TO TEACHING
In the past twenty years, problem solving has taken on a renewed significance in the teaching of mathematics [4, 5, 6, 7, 8]. Among other things, this interest emphasizes the (sometimes myriad) paths traversed in obtaining mathematical results. Over the years, mathematicians and educators have offered advice on how to encourage and investigate such twists and turns in classrooms. In his classic work How to Solve It, George Polya  describes strategies that mathematicians use during problem solving (like considering a simpler problem or searching for patterns) and suggests that teachers also, can ask questions that encourage the use of these strategies in classrooms. Alan Schoenfeld [10, p. 26] encourages solving problems “‘fresh’ at the blackboard” so that students witness first hand the starts and stops, obstacles and fallibility that are a natural part of problem-solving.
Interestingly, Schoenfeld also touches on the subject of the present article: varying classroom formats (like groupwork and lecture). Despite Schoenfeld and Polya’s sound advice, I nevertheless encountered difficulties, some of which, I conjectured, were due to differences between genuine and classroom problem-solving situations. For example, where mathematicians take on the role of “teacher” and “student” in problem solving, classrooms generally have one or a few “teachers” (the instructor and “smart” students) and several students. The classroom expectation that the teacher or select students are there (solely) to dispense knowledge, while the rest of the class is there (solely) to absorb it can hinder classroom problem solving (in group or whole-class settings). Where mathematicians can arrange flexible time periods to discuss topics of their own selection, classroom teachers and students are constrained by time periods and syllabi restrictions. Again, such considerations can interrupt the problem-solving process with its many starts, stops, insights, reversals and modifications.
During my second attempt at teaching geometry, I accidentally stumbled upon certain teaching formats that successfully encouraged genuine problem solving. These formats, which were modifications of familiar formats, served to minimize some of the obstacles cited above and to promote a more genuine problem-solving atmosphere between my students and myself.
THE “WHOLE-CLASS GROUPWORK” FORMAT
The “whole-class groupwork” format is a union of formats represented in its title. In it, the teacher serves as liaison for students working in groups to communicate ideas across groups in a whole-class setting. I discovered this format and its benefits the first day of class while my students and I worked on the following problem:
“Someone has drawn 1000 points in a plane. No three of them lie on a straight line. You are to connect all the points with straight line segments. How many of these segments will you have to draw?” [3, p. 9]
Discussion of the “Whole-Class Groupwork” Format
From my experience as a student and a teacher, I had been conditioned to believe that whole-class and groupwork formats were mutually exclusive. That is, while I led discussion, students sat individually at their desks and while students worked in groups, I helped individual groups. The “Wholeclass, groupwork” format minimizes traits that can hinder problem solving in individualized settings (like the teacher as sole knowledge dispenser or too many student ideas being suggested at once) and enhances traits cited earlier as characteristic of problem solving among mathematicians. Not only is my role as knowledge dispenser minimized, but students appear to derive comfort and direction from belonging to a smaller group which is part of a larger one working toward a common goal. In my experience, students also seem more likely to determine the issues to be resolved and how to resolve them in this format. I negotiate turn-taking and provide direction when necessary. A word of caution: the format can be intimidating at first. It has a life of its own as students go to the board, and join or leave groups in accordance with their interests. With practice however, I’ve learned to manage these changes and discovered a different classroom format for encouraging problem solving among my students.
THE OFFICE-HOURS GROUPWORK FORMAT
Some of the problems in problem-solving courses are too involved to discuss in class. Nevertheless, they are so rich with ideas that it is important to provide students an opportunity to delve into them. In my course, I learned to make new use of office hours to provide such opportunities. As students visited me individually, I made note of their thoughts on a given problem. I then started bringing selected students together in my office when I noted similarities or distinctions in approaches that could enhance one another.
One of my more successful experiences with this format occurred with the problem of finding the 100th pyramidal number P(100). The 100th pyramidal number is related to the 100th triangular number T(100) which can be found by counting the total number of pennies in a 100-sided triangle or a triangle with 100 pennies on each side (see Figure 3). P(100) can be pictured using a 100-layer pyramid. The pyramid is constructed by replacing each penny in the 100-sided triangle by a sphere of the same radius, thereby generating a 100-sided triangle of spheres (see Figure 4). A 100-layer pyramid is formed by stacking a 1-sided triangle of spheres onto a 2-sided triangle of spheres, the 2-sided triangle of spheres onto a 3-sided triangle of spheres, and so on, until the 99-sided triangle of spheres is stacked onto a 100-sided triangle of spheres (see Figure 5). The number of spheres in the resulting pyramid is P(100).
Discussion of Office-Hours Groupwork Format
In my experience as a student, office hours always meant an individual visit to a professor in which this professor would answer my questions and move me along so the next students’ concerns could be addressed. Bringing a small number of students together during office hours introduces a new interactional dimension to this setting that is more manageable than whole-class interaction. In our four meetings, we succeeded in discovering a geometric interpretation for a student who questioned the value of her thinking and we shed some insight into why another student’s method worked. We were all impressed by Carlos’ solution, noting that it could make this problem accessible to algebra students and Carlos, Jessica, and Lynn conveyed their belief that my solution reflected most closely “what was happening in the pyramid.”
In contrast to the whole-class, groupwork format, office-hours groupwork provides select students opportunities for more focused and individualized thinking. It can incorporate different ideas without the constraint of classroom time schedules and it doesn’t take up any class time. It also provides the teacher a smaller setting in which to practice asking questions. The obvious drawback to this format is the exclusion of certain students. However, it is possible to provide all students at least one such discussion during a semester. Since adopting this format into my teaching, I have further modified it to help me when the line outside my office door reaches three students or more. That is, I now group students together in the study space outside my office according to their strengths and questions so that they can help one another. This modification has served to cut significantly the time I spend helping students and conveys the strong message that students can be resources for one another.
As a teacher, I have had moments where my students and I became genuine problem solvers, that is, where my students and I became intent on communicating unexpected mathematical ideas to one another and we worked together to clarify, justify and record the ideas. Even without the research to support it [1, 2], I recognize that these are the moments that most empower students mathematically. In my spring geometry class, my goal became to increase the number of such moments in my teaching. I believe that experimenting with familiar classroom formats so that they more closely resemble the kinds of settings mathematicians work in was a positive step in achieving this goal. I hope this paper’s success stories provide the reader the impetus to consider such experimentation in encouraging genuine problem solving with his or her students.
1 This paper was written by Professor Fernandez (hence the first person “I” throughout the paper). The three co-authors were students in her geometry class who participated in writing their solutions for the second half of the paper and also helped to edit several versions of the final draft.
2Throughout this paper, I focus on the use of problem-solving strategies and how they manifest themselves within a given teaching format during my discussions with students. The students investigated more formal proofs of their conjectures in homework exercises.
1. Battista, Michael T. and Carol N. Larson. 1994. The role of JRME in advancing the learning and teaching of elementary school mathematics. Teaching Children Mathematics. 1: 1778-181.
2. Borasi, Raffaella. 1994. Capitalizing on errors as “springboards for inquiry”: a teaching experiment. Journal for Research in Mathematics Education. 25(2): 166-208.
3. Gay, David. 1998. Geometry by Discovery. New York: John Wiley and Sons, Inc.
4. National Council of Teachers of Mathematics. 1980. An Agenda for Action: Recommendations for School Mathematics of the 1980’s. Reston VA: NCTM.
5. National Council of Teachers of Mathematics. 1989. Curriculum and Evaluation Standards. Reston VA: The Council.
6. National Council of Teachers of Mathematics. 1991. Professional Standards for Teaching Mathematics. Reston VA: The Council.
7. National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston VA: The National Council of Teachers of Mathematics, Inc.
8. National Research Council. 1989. Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington DC: National Academy Press.
9. Polya, G. 1973. How to Solve It. Princeton: Princeton University Press. (Original work published in 1945).
10. Schoenfeld, Alan H. 1983. Problem solving in the mathematics curriculum: A Report, Recommendations and an Annotated Bibliography. MAA Notes, No. 1. Washington DC: The Mathematical Association of America.
11. Schoenfeld, Alan H. 1992. Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In Douglas A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. (pp. 334-369). New York NY: Macmillan Publishing Company.
ADDRESS: Department of Mathematical Sciences, Montclair State University, Upper Montclair NJ 07043 USA.
Eileen Fernandez is an assistant professor in the Department of Mathematical Sciences at Montclair State University. She received her doctorate in mathematics education in 1998 from the University of Chicago and adopted her first baby boy Matthew from Guatemala in April 2001. In addition to experimenting with classroom formats, she is currently experimenting with furniture arrangements that will facilitate Matthew’s graduation from crawling to walking without taking too many years off his mother’s life. Suggestions are welcome.
Jessica Kazimir graduated from Montclair State in January 2001 with a major in mathematics (concentration in mathematics education) and a minor in computer science. She is currently a business intelligence analyst at PSEG.
Lynn Vandemeulebroeke graduated from Montclair State in May 2001 with a major in mathematics. She is currently a pricing specialist at BlisTech.
Carlos Burgos is a senior at Montclair State. He is majoring in mathematics with a concentration in mathematics education and plans to be a high school teacher upon graduation.
Copyright PRIMUS Sep 2002
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