# Lesh & Zawojewski (2007)

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Problem Solving and Modeling

## Summary

This chapter of the NCTM research handbook summarizes the existing research on problem solving and points to promising directions for future research. In general, it argues for the insufficiency of past attempts to find (and instruct students in) globally applicable problem solving strategies or heuristics. The piece advocates for a "Models and Modeling" approach to research on learning and teaching, a perspective that Lesh elaborates on in this piece and his other publications. (See, for example, Lesh & Lehrer (2003).)

## Outline of Headings

• An Overview of Past Research on Mathematical Problem Solving
• Early Problem-Solving Research
• Task Variables and Problem Difficulty
• Expert/Novice Problem-Solver Studies
• Instruction in Problem-Solving Strategies
• Research on Higher-Order Thinking
• Metacognition
• Habits of Mind
• Beliefs and Dispositions
• Summary and Reflections on Research in Higher-Order Thinking
• New Directions and Perspectives for Research in Mathematical Problem Solving
• A Fresh Perspective on Problem Solving
• Beyond School Mathematics
• A Definition for Problem Solving
• Learning Problem Solving and Learning Mathematics
• Model-Eliciting Activities
• An Evolving View of Problem Solving
• Promising Lines of Research on Mathematics Learning
• Research on Situated Cognition
• Research on Communities of Practice
• Research on Representational Fluency
• Directions for Future Research
• The Need for New Research Tools and Methodologies
• An Illustration: The Evolving Models-and-Modeling Perspective

## Quotes

"Begle (1979) reviewed early stages of this research. He concluded that "no clear-cut directions for mathematics education are provided by the findings of these studies...Similarly, Schoenfeld in his 1992 review of the literature concluded that attempts to teach students to use general problem-solving strategies (e.g. draw a picture, identify the givens and goals, consider a similar problem) generally had not been successful...In fact, even a decade later when Lester and Kehle (2003) compared a current list of issues to those described by Lester in 1994, they concluded that, still, little progress had been made in problem-solving research and that the literature on problem solving had little to offer to school practice."
"Krutetskii found that gifted students tend to perceive the underlying mathematical structure of problem situations very rapidly, whereas others tend to notice and remember relatively superficial problem characteristics. Notice that the characteristics of expert problem solvers identified by Krutetskii cannot be directly taught to the nonexperts. One cannot directly teach novice problem solvers to "generalize broadly," "identify the underlying structure," or "look ahead to skip steps," -- especially assuming that novices often view mathematical problem solving as keeping track of and carefully processing pieces of information."
"Why do after-the-fact descriptions of past activities not necessarily provide guidelines for next steps during future problem-solving activities? Perhaps one reason is because using a strategy, such as draw a picture assumes that a student would know what pictures to draw when, under what circumstances, and for which type of problems. Therefore, mastering draw a picture (in general) depends on the interpretation abilities--not just execution abilities. Thus, developing systems for interpreting problem situations is as important, if not more important, than developing processes for doing particular strategies."
"Our interpretation of Polya's heuristics is that the strategies are intended to help problem solvers think about, reflect on, and interpret problem situations, more than they are intended to help them decide what to do when "stuck" during a problem attempt."
"Recent research on problem solving in complex problematic situations suggests that the abilities involved in "seeing" are as important as abilities involved in "doing."
"We propose the following definition: a task...becomes a problem (or problematic) when the "problem solver" (which may be a collaborating group of specialists) needs to develop a more productive way of thinking about the given situation."

## Corrolary

APA
Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte, NC: Information Age.
BibTeX
@incollection{Lesh2007,
address = {Charlotte, NC},
author = {Lesh, Richard and Zawojewski, Judith},
booktitle = {Second handbook of research on mathematics teaching and learning},
chapter = {17},
editor = {Lester, Frank K.},
pages = {763--804},
publisher = {Information Age},
title = {{Problem solving and modeling}},
year = {2007}
}