Difference between revisions of "Lesh & Zawojewski (2007)"
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This chapter, from [[NCTM]]'s[[Second Handbook of Research on Mathematics Teaching]] summarizes the existing research on problem solving and points to promising directions for future research. | |||
==Quotes== | ==Quotes== | ||
<blockquote>"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 | <blockquote>"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."</blockquote> | ||
<blockquote>"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."</blockquote> | <blockquote>"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."</blockquote> | ||
<blockquote> | <blockquote>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 <i>draw a picture</i> assumes that a student would know what pictures to draw when, under what circumstances, and for which type of problems. Therefore, mastering <i>draw a picture</i> (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.</blockquote> | ||
<blockquote>"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 <i>do</i> when "stuck" during a problem attempt."</blockquote> | <blockquote>"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 <i>do</i> when "stuck" during a problem attempt."</blockquote> | ||
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<blockquote>"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."</blockquote> | <blockquote>"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."</blockquote> | ||
Revision as of 09:57, 27 May 2015
This chapter, from NCTM'sSecond Handbook of Research on Mathematics Teaching summarizes the existing research on problem solving and points to promising directions for future research.
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."