1 A Retrospective Account of the Past 25 Years of Research on Teaching Mathematical Problem Solving Kilpatrick, J. (1985). A retrospective account of the past 25 years of research on teaching mathematical problem solving. In E. A. Silver (ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp.1-15). Hillsadle, N. J: Lawrence Erlbaum. Reporter: Lee Chun-Yi Advisor: Chen Ming-Puu

2 What is a problem? Perspectives on Mathematical Problems The Roles of Mathematical Problems in Teaching Task Variables in Mathematical Problem Solving

3 Perspectives on Mathematical Problems Psychological perspective: a problem is defined generally as a situation in which a goal is to be attained and a direct route to the goal is blocked (problem as an activity of a motivated subject). Social-anthropological perspective: the mathematics classroom is a social situation jointly constructed by participants, in which teacher and students interpret each other’s actions and intentions in the light of the their own agendas (problem as a task). Mathematical perspective: problem as construction. Pedagogical perspective: problem as vehicle.

4 The Roles of Mathematical Problems in Teaching Polya’s classification of problems from a pedagogical perspective  One rule under your nose  Application with some choice  Choice of a combination  Approaching research level

5 The Roles of Mathematical Problems in Teaching a.7/25=12/x, what is x? b.See the right figure? c.If a 7-oz. cup of cola costs 25c, what is the cost of a 12-oz. cup?

6 The Roles of Mathematical Problems in Teaching d. A group of three sixth-graders has been given the problem of planning a picnic. They have been told that 7- oz. cups of cola cost 25c each, and they are trying to find out what 12-oz. cups should cost. e. Your neighborhood association is having a picnic and is hoping to make some money selling cups of cola. One of the officers has set a price of 25c for the 7-oz. cups and has asked you what price would be fair for the 12-oz. cups. f. If a 7-oz. cup of cola costs 25c, the proportional cost of a 12-oz. cup is not a whole number. What is the smallest whole number one could add to the cost of the 7-oz. cup to make the proportional cost of the 12-oz. cup a whole number?

7 Task Variables in Mathematical Problem Solving Why certain problems gave students so much difficulty?  Readability  Solvability  Syntax variables There is not a theoretical framework to guide the selection of variables.  Information processing perspective  Interaction between task characteristics and the characteristics of the problem solver

8 How are Problems Solved? Polya’s view of problem solving: the study of the processes students use in solving mathematical problems. A scheme for classifying problem-solving processes and some appreciation of the complexity of problem-solving behavior.  Assess whether their instructional treatments are influencing students’ approaches to problems  Turn our attention to cognitive behavior such as monitoring one’s progress and reflecting on one’s performance.

9 How are Problems Solved? Studies of expert problem solvers and computer simulation models have shown that the solution of a complex problem requires  A rich store of organized knowledge about the content domain  A set of procedures for representing and transforming the problem  A control system to guide the selection of knowledge and procedures. It is easy to underestimate  Deep knowledge of mathematics and extensive experiences in solving problems  The sophistication of the control processes used by experts to monitor and direct their problem-solving activity.

10 How is Problem Solving Learned? Osmosis Memorization Imitation Cooperation Reflection We do not have a comprehensive scheme for classifying various features of different methods.

11 Osmosis Practice: master thinking strategy  It may underestimate the content-specific techniques  No instructional program can be successful that does not deal with the effects of students’ negative attitudes and beliefs about themselves as problem solvers.

12 Memorization Task analyses that decompose the solution of a problem into atomic procedures, each of which is then taught. Treat heuristic suggestions as procedures to be followed and attempt to organize them into an algorithm. Such approaches appear to be difficult for students to manage and may even be counterproductive.

13 Imitation Modeling the master problem solver  Analyze the difference between their solutions for remedial instruction.  Work with an adult on various mysteries and puzzles  The effectiveness of a teacher who playacts ignorance, uncertainty, and then a growing assurance in trying various problem-solving procedures in a dialogue with the class has not been given systematic study.

14 Cooperation Small-group discussion may help students clarify concepts and rehearse procedures in ways that are difficult to do alone. Polya has used discussion groups to give teachers practice in guiding problem- solving instruction; such groups can also be instruments for developing one’s problem-solving skills.

15 Reflection Deway: learning by doing Paper: children learn by doing and by thinking about what they do Schoenfeld and Silver: metacognition (cognition of one’s own cognition)

16 What Have We Learned about Problem Solving? Be clear about your goals Understand that problem solving highly complex Be prepared to find problem-solving performance difficult to improve Get students to shift to an active stance Provide a congenial environment for problem solving

17 Be clear about your goals A teacher should be clear about what sorts of problems he or she wants students bo be able to solve. Research over the last tow and a half decades has not provided much explicit guidance to the teacher in matching teaching techniques to goals.

18 Understand that problem solving highly complex Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected. Successful problem solving in a given domain depends upon the possession of a large store of organized knowledge about that domain, techniques for representing and transforming the problem, and metacognitive processes to monitor and guide performance.

19 Be prepared to find problem-solving performance difficult to improve Successful treatments take a long time. In addition to providing some sort of regular, guided practice in problem solving, successful programs appear to have two other features in common:  They get the students to adopt an active stance toward problem solving.  They provide a congenial setting in which problems can occur.

20 Get students to shift to an active stance Active learning (Polya, 1981): let the students discover by themselves as much as feasible under the given circumstances.  Have them formulate and solve their own problems,  Have them rewrite problems  Format changes (adding a diagram, removing extraneous information and reordering information).

21 Provide a congenial environment for problem solving The children see the problem as a school task rather than as an intellectual challenge that is worth accepting, they grab at answers so as to escape from the task as fast as possible. Successful problem-solving instruction often needs to transform the terms of the school situation that previous instruction has negotiated and reinforced.

22 Coda We do not have a final vision of what problem solving is and how to teach it, but we are much more keenly aware of the complexity of teachers and researchers. The emerging view of the mathematics teacher as a full partner in the research process, rather than an imperfect system for delivering instruction, can only yield research that will better mediate between theory and practice.