In a historical review focusing on the role of problem solving in the - PDF document

In a historical review focusing on the role of problem solving in the mathematics curriculum, Stanic and Kilpatrick (1989, page 1) provide the following brief summary: Problems have occupied a central place in the school mathematics curriculum since antiquity, but problem solving has not. Only recently have mathematics educators accepted the idea that the development of problem solving ability deserves special attention. With this focus on problem solving has come confusion. The term problem solving has become a slogan encompassing different views of what education is, of what schooling is, of what mathematics is, SOLUTION: -- $3.75 is 3/8 of $10. find the product (A x B), where A is given as a two-digit decimal that corresponds to a price in dollars and cents. The decimal values have been chosen so that a simple ratio is implicit in the decimal form of A. That is, A = r x C, where r is a simple fraction and C is a power of 10. Hence (A x B) can be computed as r x (C x (We do, commonly, see prices such as "3 for $20.00.") The numbers used in problem 55, and others, were clearly selected so that students could successfully perform the algorithm taught in this lesson. On the one hand, choosing numbers in this way makes it easy for students practice the technique. On the other hand, the choice makes the problem itself implausible. Moreover, the problem settings (cords of wood, price of sheep, and so on) are soon seen to be window dressing designed to make the problems appear relevant, but which in fact have no real role in the problem. As such, the artificiality of the re a broad range of problems and problem situations, ranging from exercises to open-ended problems and expl introspectionist techniques were shown to be methodologically unreliable, and the concept of mentalism came under increasing attack. In Russia, Pavlov (1924) achieved stunning results with conditioned reflexes, his experimental work requiring no concept of mind at all. Finally, mind, consciousness, and all related phenomena were banished altogether by the behaviorists. John Watson (1930) was the main exponent of the behaviorist stance, B. F. Skinner (1974) a zealous adherent. The behaviorists were vehement in their attacks on mentalism, and provoked equally strong coun -to-late 1980's) "metacognition" became a major research topic. Here too, the literature is quite confused. In an early paper, Flavell characterized the term as follows: Metacognition refers to one's knowledge concerning one's own cognitive processes or anything related to them, e.g. the learning-relevant properties of information or data. For example, I am engaging in metacognition... if I notice that I am having more trouble learning A than B; if it strikes me that I should double-check C before accepting it as a fact; if it occurs to me that I should scrutinize each and every alternative in a multiple-choice task before deciding which is the best one.... Metacognition refers, among other things, to the active monitoring and consequent regulation and orchestration of those processes in relation to the cognitive objects or data on which they bear, usually in the service of some concrete [problem solving] goal or objective. (Flavell, 1976, p. 232) This kitchen-sink definition includes a number of categories which have since been separated into more functional categories for exploration: (a) individuals' declarative knowledge about their cog Suppose a person finds him or herself in a situation that calls for the use of mathematics, either for purposes of interpretation (mathematizing) or problem solving. In order to understand the individual's behavior -- e.g. which options are pursued, in which ways -- one needs to know what mathematical tools the individual has at his or her disposal. Simply put, the issues related to the individual's knowledge base are: What information relevant to the mathematical situation or problem at hand does he or she possess, and how is that inf tends to become less salient, and ultimately negligible, as the vertex angle is made larger.) Of course, Euclidean geometry is a formal game; these informal understandings must be exploited within the context of the rules for constructions. As noted above, the facts, definitions, and algorithmic procedures the individual brings to the problem situation may or may not be correct; they may be held with any degree of confidence from absolute (but possibly incorrect) certainty to great unsureness. Part of t n " k/(k+1)! i=1 Problem 2. Let P(x) and Q(x) be polynomials whose coefficients are the same but in "backwards order:" P(x) = a0 + a1x + a2x2 + ... anxn , and Q(x) = an + an-1 them in advance. By the end of t ve extraordinarily powerful (and often negati ¥ The teacher must create and maintain an open and informal classroom atmosphere to insure the students' freedom to ask questio knowledgeable participant -- a representative of the mathematical community who was not an all-knowing authority but rather one who could ask pointed questions to help students arrive at the correct mathematical judgments. Her pedagogical practice, in deflecting undue authority from the teacher, placed the burden of mathematical judgment (with constraints) on the shoulders of the students. Balacheff (1987) exploits social interactions in a different way, but with similar epistemological goals. He describes a series of lessons for seventh graders, concerned with the theorem that "the sum of the angles of a triangle is 180¡." The lessons begin with the class divided into small groups. Each group is given a work sheet with a copy of the same triangle, and asked to compute the sum of its angles. The groups then report their answers, which vary widely -- often from as little as 100¡ to as much as 300¡! Since the students tension that must be resolved; they work on it until all students agree on a value. Balacheff then hands out a different triangle to each group, and has the group conjecture the sum of the angles of its triangle before measuring it. Groups compare and contrast their results, and repeat the process with each other's triangles. The conflicts within and across groups, and the discussions that result in the resolutions of those conflicts, make the relevant mathematical issues salient and meaningful to the students, so that they are intellectually prepared for the theoretical discussions (of a similar dialectical nature) that follow. In a classic st Teaching and learning mathematical problem solving: Multiple research perspect Teaching and learning mathematical problem solving: Multiple research perspectives. Hillsdale, NJ: Erlbaum. Silver, E. A. (1987). Foundations of cognitive theory and research for mathematics problem solving instruction. In A. Schoenfeld (Ed.), Cognitive Science and Mathematics Education (pp. 33-60). Hillsdale, NJ: Erlbaum. Silver, E. A., Branca, N., & Adams, V. (1980). Metacognition: The missing link in problem solving? In R. Karplus (Ed.), Proceedings of the IV international Congress on Mathematical Educa