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The Challenge of Discovering Mathematical Structures : Some research based pedagogical recommendations for the secondary classroom

This contribution addresses the use of number theoretic and combinatorial problems in facilitating the discovery of deeper mathematical structures. In particular, the paper reports on the outcome of teaching experiments in secondary classrooms with the use of problems involving Steiner triple systems, the Dirichlet principle and Diophantine n-tuples. Data collected from the teaching experiments are analyzed and used to construct research based pedagogical recommendations for the secondary classroom, particularly reflections on the types of mathematical challenges that are accessible to secondary students. Introduction The ICMI 16 Study on challenging mathematics in and outside the classroom comes at an opportune time, given the growing discrepancy between what constitutes “appropriate” K-12 mathematics among professional mathematicians and mathematics educators in many countries. A particularly troubling manifestation of this is evidenced in the “math wars” in the United States and Canada with each “side” prescribing mathematics content and pedagogy for schools. The continual clash of political and ideological forces in many countries on what is mathematics has resulted in schools being unable to implement any mathematics curriculum for a sufficiently long period to time, nor to train their teachers on effectively implementing the particular curriculum in use (traditional, reform or otherwise). The student is ultimately the one whose learning experiences are compromised. Given this broad preamble, this ICMI focus on the role of mathematical challenges with a broad ensemble of mathematicians and educators in participation offers an opportunity to tackle core issues which when answered offer teachers and students the possibility of experiencing afresh the joy of mathematics, its teaching and learning. As a former public school teacher, research mathematician and an active mathematics educator, I find the first four questions particularly appealing. More importantly, I have conducted numerous studies in high school classrooms, which offer a start at understanding how challenges contribute to the learning process, how challenges can be used in the classroom, and opportunities to challenge that enhance the teaching and learning experience in the regular classroom. In the remainder of this paper, I present same research based pedagogical recommendations for the secondary classroom, relevant for this particular ICMI study based on the results of various teaching experiments conducted in high school classrooms. Mathematical Challenges in the secondary classroom The author conducted several teaching experiments at a rural mid-western high school with ninth grade students enrolled in various sections of a beginning algebra course, consisting of a heterogeneous group of learners. The author was also the teacher of each of these algebra courses. The goal of these teaching experiments were to offer mathematical challenges not provided by the regular curriculum and to study how students abstract and generalize advanced mathematical concepts, with implications for new ways of differentiating the curriculum for all learners. Each teaching experiment typically lasted one semester, involved a different group of students and was designed as follows. The students in each class had to maintain a mathematics journal, in which they solved one non-routine problem every other week. Students were asked to restate the problem in their own words, devise a strategy to solve the problem, and then write a summary of their solution. They were also encouraged to include all their scratch work in their journals. Full credit was granted to all students who followed the cues and included all their work, irrespective of the correctness of their solutions. The problems for each of these experiments were carefully chosen to facilitate the discovery and formulation of mathematical generalizations. The problems were chosen from combinatorics and number theory, which were not covered by the 9 grade curriculum. The author chose these topics in particular because (a) counting and arrangement problems were easy to understand, and (b) previous mathematics education research indicated that students brought considerable intuition to such problem situations. Experiment 1 In the first experiment four problems that resulted in the construction of Steiner Triple Systems (STS) were assigned over the course of three months (Sriraman, 2004a). A STS is an arrangement of ‘n’ objects in triplets such that every pair of objects appears in a triplet exactly once. Formally speaking, a STS consists of a set X of n points, and a collection B of subsets of X called blocks or triples, such that each block contains exactly 3 points, and any two points lie together in exactly one block. One of the oldest results in Block Design Theory is Kirkman's existence theorem for STS, which states that a STS of order n exists if and only if n = 0 or n is congruent to 1 or 3 mod 6. The sequence of possible values for n are 3, 7, 9, 13, etc. There exists 1⁄2 n (n-1) such pairs and the number of required triplets is one-third the number of pairs It can be mathematically proven that there exist 1⁄2 n(n-1) such pairs and that the number of required triplets is one-third the number of pairs. It is possible to construct a Steiner Triple system only when each object is in 1⁄2 (n-1) triples, with the restriction that these numbers are integers. The goal was to present students problem solving situations which would facilitate student discovery of this beautiful mathematical structure and the formulation of the statement of this theorem. Experiment 2 In the second experiment, five problems whose solutions were characterized by the pigeonhole principle were assigned over the course of three months (Sriraman, 2003a, 2004b). The pigeonhole principle states that if ‘m’ pigeons are put in ‘n’ pigeon holes and if m>n, then at least one pigeonhole will have more than one pigeon. The beauty of this obvious principle is the wide range of mathematical situations to which it can be applied.