Problem Solving Heuristics , Affect , and Discrete Mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students’ possible affective pathways and structures. Kurzreferat: Vielfach wird vorgeschlagen, dass Inhalte der Diskreten Mathematik geeignet sind, Lehrern und Schülern ein neues Bild der Mathematik zu vermitteln. Mathematische Entdeckungen an Problemen, die nicht zur Unterrichtsroutine gehören, sind hier leichter möglich als in vielen anderen Gebieten der Mathematik. Das gilt selbst für Schülerinnen und Schüler, die eher als weniger leistungsstark angesehen werden können. Damit Lehrerinnen und Lehrer allerdings die möglichen Vorteile optimal nutzen können, sollten sie wissen, welche Art von Denken und mathematischer Argumentation bei solchen Aufgaben gefordert ist. Der Artikel diskutiert das Thema und geht dabei insbesondere auf das Modellieren des allgemeinen Falls auf der Basis eines speziellen Falls ein. Einige Bemerkungen befassen sich mit der affektiven Komponente des Problemlösens. ZDM-Classifikation: C20, C30, D50, N70 1. Possibilities for Discrete Mathematics In their accompanying article, DeBellis and Rosenstein (2004) describe a vision for discrete mathematics in the schools of the United States, one toward which they have both contributed substantially over a decade and a half. They see the domain of discrete mathematics—a looselydefined term that includes combinatorics, vertex-edge graphs, iteration and recursion, and many other topics— as providing teachers with “a new way to think about traditional mathematical topics and a new strategy for engaging their students in the study of mathematics.” Through experiences in discrete mathematics, they suggest that teachers may better be able to help children “think critically, solve problems, and make decisions using mathematical reasoning and strategies.” And they cite Gardiner (1991) in cautioning, “If instead discrete mathematics is introduced in the schools as a set of facts to be memorized and strategies to be applied routinely ... [its qualities] as an arena for problem solving, reasoning, and experimentation are of course destroyed.” It is certainly an appealing idea that students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find something quite new here. Evidently the novel possibilities have to do less with particular formulas and techniques of combinatorics, search or sorting algorithms, theorems about graphs, and so on, than with the opportunities for interesting, nonroutine problem solving and for mathematical discovery that discrete mathematics may provide (cf. Kenney & Hirsch, 1991; cf. Rosenstein, Franzblau, & Roberts, 1997). Here I would like to emphasize the importance of characterizing these opportunities more specifically, both mathematically and cognitively. The following questions deserve consideration: (1) What especially desirable ways of thinking, powerful problem-solving processes, or other important mathematical competencies do discrete mathematical situations naturally evoke? (2) Why are these particular processes or capabilities evoked in students? Under what problem conditions do we expect them to occur? Why might we anticipate the emergence of previously hidden mathematical capabilities in some students, and which capabilities are these? (3) How can we consciously structure students’ activities so as to best encourage the further development of the mathematical capabilities we have identified? Can we do this through discrete mathematics more readily or naturally than we could through a comparable commitment to conventional arithmetic, algebra, or geometry? (4) How can or should we assess the degree to which student performance is enhanced—in discrete mathematics particularly, and in the mathematical field generally? Of course, in a short article one can discuss only small parts of these questions. Here I shall focus on just two aspects: (a) heuristic processes for mathematical problem solving, especially the way of thinking we may call “modeling the general on the particular”, and (b) students’ affective pathways and structures. 2. A Problem for Discussion Let us consider a specific and rather well-known “nonroutine” problem-solving activity, in order to lend concreteness to the discussion. You are standing on the bank of a river with two pails. One pail holds exactly 3 liters of water, and the other holds exactly 5 liters. The pails are not marked for measurement in any other way. How can you carry exactly 4 liters of water away from the river? This is a problem I have given many times, to children and to adults. While it does not fall clearly into one of the above-mentioned domains of discrete mathematics, it has features in common with problem activities drawn from many of those domains. It is “discrete” in the sense of involving discrete steps that are permitted at any point by the problem conditions. It is sufficiently “nonroutine” for ZDM 2004 Vol. 36 (2) Analyses 57 even the mathematical interpretation of the problem conditions to be challenging to many. Like “counting problems” in combinatorics, it requires the problem solver to devise some means of keeping track of what has already been done. Like “coloring problems” and “shortest route” problems, it invites successive trials that may fail to satisfy the problem conditions. Like many problems in discrete mathematics, this problem is also suggestive of a possible hidden structure that would, if recognized, make the solution easy to see. Schematically, one may represent this problem by means of a state-space diagram as in Fig. 1, where each node (or problem state) corresponds to a configuration with a fixed, known volume of water in each of the pails, and each arc corresponds to the step of filling a pail at the river, emptying a pail, or pouring water from one pail into the other until the latter is full (Goldin,1984). In Fig. 1 the ordered pair of numbers (0,2), for example, refers to the configuration where the 3-liter pail is empty and there are 2 liters of water in the 5-liter pail. Fig. 1. Schematic representation of paths through the state-space for the problem of the two pails