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Alan H. Schoenfeld

It is widely accepted that cultures are reflected by, and are reflections of, their technologies, and also that changes in technology may both reflect and catalyze significant changes in the societies that employ them. Perhaps the generic case in point is agriculture, where advances in crop cultivation led to fundamental changes in the organization of society. Crop cultivation “encouraged settlement of stable farm communities, some of which grew to be towns and city states. … Early agricultural implements—the digging stick, the scythe, the hoe, and the plow—developed slowly over the centuries, each innovation … causing profound changes in human life.” (Bridgwater & Kurtz, 1963, p. 29.) Similar effects can be traced to the current era: The existence of techniques for mass agricultural production is a necessary condition for the existence of a highly urbanized society, as well as a contributor to the decrease in the percentage of the farming population, and the kind of life associated with it.

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Alan H. Schoenfeld

Radical Consfructivism in Mathematics Education contains a dozen attempts to deal with one of the most difficult philosophical and pedagogical issues of our time, the reconciliation of a radical constructivist perspective with the demands of individual and classroom instruction. To see why these problems are so difficult, we begin by identifying what distinguishes the radicals represented in this book from their mainstream counterparts.

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Alan H. Schoenfeld

Ohlsson. Ernst, and Rees (this issue) have produced a wonderfully lucid description of their paradigmatic approach to issues of cognition and instruction. They illustrate their approach by presenting the details of a well worked out computational model. Then, on the basis of simulation runs on the model, they derive some implications for prac tice. The authors have also laid down some rather stringent constraints for commentary. Do not critique our paradigms, they say, unless you can offer a replacement that does better. Do not critique the choice of knowledge representation (production systems) or modeling assumptions (e.g., limitations on working memory) unless you have compelling data to offer in service of your argument and in contradiction of our assumptions. Argument about details is useful, they say, but that won't change the conclusions we draw. So what's a reviewer to do?

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Alan H. Schoenfeld

A review of the 2013 book titled The International Handbook of Collaborative Learning, edited by Cindy E. Hmelo-Silver, Clark A. Chinn, Carol K. K. Chan, and Angela O'Donnell.

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Alan H. Schoenfeld

A questionnaire with 70 closed and 11 open questions was administered to 230 mathematics students enrolled in Grades 10 through 12, the majority of whom were enrolled in the traditional year-long 10th-grade course in plane geometry. Sections of the questionnaire dealt with the students' attributions of success or failure; their comparative perceptions of mathematics, English, and social studies; their view of mathematics as a discipline; and their attitude toward mathematics. The data, which are closely tied to a series of classroom and protocol studies, suggest the resolution of contradictory patterns of data in other attitude surveys, where students simultaneously claim that “mathematics is mostly memorizing” but that mathematics is a creative and useful discipline in which they learn to think.

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Alan H. Schoenfeld

Life was simpler when I was a mathematician. As Gertrude Stein might have said, “a proof is a proof is a proof”: Once validated by the community, a mathematical result provides a rock-solid foundation on which one can build. By way of contrast, Henry Pollak once said that “there are no theorems in mathematics education.” In the social sciences, we seek to substantiate rather than to prove.

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Alan H. Schoenfeld

This experiment examined the impact that explicit instruction in heuristic strategies, above and beyond problem-solving experience, has on students' problem-solving performance. Two groups of students received training in problem solving, spent the same amount of time working on the same problems, and saw identical problem solutions. But half the students were given a list of five problem-solving strategies and were shown explicitly how the strategies were used. The heuristics group significantly outperformed the other students on posttest problems that were similar to, but not isomorphic to, those used in the problem sets. This lends credence to the idea that explicit instruction in heuristics makes a difference--an idea further supported by the transcripts of students solving the problems out loud.

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Alan H. Schoenfeld

There has been much discussion of standards for conducting and reporting research in mathematics education (see references). This note deals with an issue that, although certainly implicit in these discussions, has not received enough explicit attention: How can a research report be made truly useful to its readers? That is, how can one write a report in such a way that (a) researchers can replicate or build on the work, or (b) teachers who read it can take something directly into the classroom with them?

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Alan H. Schoenfeld

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Alan H. Schoenfeld

This article presents and discusses three paper-and-pencil tests of students' performance on nonroutine mathematics problems at the college level. These tests assess the following: 1. the frequency with which students apply various problem-solving heuristics, and how far and with what success they pursue them; 2. students' subjective assessments of their own problem-solving behavior; and 3. transfer of heuristic behavior on problems related in various degrees to ones that students have studied. These measures were used to assess students' performance before and after a month-long intensive problem-solving course. They provide evidence that such a course can produce strong changes in students' problem-solving behavior.