“There is a substantial degree of homogeneity in ways that expert problem solvers approach new problems” (Schoenfeld, 1985, p. 71). Here “problem” is meant as a task that is not routine for the solver (see Problems and problem solving). The methods expert problem solvers use are known as heuristics or heuristic strategies. These include:
Drawing a figure includes specific strategies such as representing a word problem by a bar model (see Line and bar models). For example, one can represent the following problem in various ways using the bar model: “There are 36 students in a class. There are eight more boys than girls. How many boys and how many girls are in the class?”
Using related problems includes two more specific strategies: reducing to a simpler problem (see HK3 [0:33] and J1 [4:32–10:10]) and reducing to a previously solved problem (see HK3). Introducing auxiliary elements includes introducing an auxiliary line (see J1, HK4, US3).
Working backwards, “taking the problem as solved,” includes different specific strategies. In algebra, letting a variable (often called x) be the unknown and working with it as if it were known in order to solve an equation has its origins in the idea of working backwards. Descartes’s predecessor Vieta introduced a radically new symbolization for algebra (Mahoney, 1994, p. 35). In his algebraic work, Vieta was conscious of using techniques of the ancient Greeks (Grabiner, 1995; Mahoney, 1994), in particular “analysis.”
A[n] . . . antecedent of Descartes’ work was the invention of symbolic algebra as a problem-solving tool, a tool that was explicitly recognized as a kind of “analysis” in the Greek sense by its discoverer, Vieta in 1591. . . . To say “let x =” the unknown, and then calculate with x—square it, add it to itself, etc., as if it were known—is a powerful technique when applied to word problems, both in and outside of geometry. Vieta recognized that naming the unknown and then treating it as if it were known was an example of what the Greeks called “analysis.” (Grabiner, 1995, p. 86)
What Vieta called “analysis” is literally “solution backward” (Grabiner, 1995, p. 85) and is now known as the heuristic “working backward.”
Ma provides examples of decomposing and recombining in elementary mathematics (1999). For example (p. 12), to calculate 53 – 26:
N2 shows an incomplete proof of the Pythagorean theorem based on decomposing and recombining squares.
Heuristic reasoning is usually thought of as incomplete reasoning that suggests why something is true. Sometimes one can make this reasoning into a proof, but in other cases it only suggests what might be true. “Heuristic reasoning is good in itself. What is bad is to mix up heuristic reasoning with rigorous proof. What is worse is to sell heuristic reasoning for rigorous proof” (Polya, 1973, p. 113).
For more examples and discussion of heuristics, see Polya (1973, 1990) and Schoenfeld (1985, 1992). For elaborated examples of drawing a figure, exploiting a related problem, working backwards, and working forwards in a classroom setting, see Arcavi et al. (1998).
Arcavi, A., Kessel, C., Meira, L., & Smith, J. (1998). Teaching mathematical problem solving: A microanalysis of an emergent classroom community. In Alan Schoenfeld, Ed Dubinsky, & James Kaput (Eds.), Research in Collegiate Mathematics Education III (pp. 1–70). Providence, RI: American Mathematical Society.
Grabiner, J. (1995). Descartes and problem-solving. Mathematics Magazine, 68(2), 83–97.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Mahoney, M. (1994). The mathematical career of Pierre de Fermat, 1601–1665 (2nd ed.). Princeton, NJ: Princeton University Press.
Polya, G. (1973). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press (2nd ed.).
Polya, G. (1990). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
| School Improvement |
Research and Evaluation |
International Studies |
RBS Publications |
Education Resources |
About RBS |
| Contact Us | Home |
