Abstract
This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectives—mathematics, mathematics education, cognitive psychology, and evolutionary psychology—each offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research).
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Notes
Gigerenzer (2005), p. 1.
See also Nisan and Schocken (2005) for such a multilevel view of computer science.
This is meant to describe a typical view in the community. Some individuals or subcommunities may, of course, hold different views.
We refer here mainly to the reasoning and decision-making subcommunities.
The quote is from Cosmides and Tooby (1996), p. 2.
This means that 5% of the people who test positive do not have the disease.
Here is Cosmides and Tooby’s (1996, p. 4) explanation:
In science, what everyone really wants to know is the probability of a hypothesis given data—p(H|D). That is, given these observations, how likely is this theory to be true? This is known as an inverse probability or a posterior probability. The strong appeal of Bayes' theorem arises from the fact that it allows one to calculate this probability:
\({\text{p}}\left( {{\text{H}}\left| {\text{D}} \right.} \right) = {\text{p}}\left( {\text{H}} \right){\text{p}}{{\left( {{\text{D}}\left| {\text{H}} \right.} \right)} \mathord{\left/ {\vphantom {{\left( {{\text{D}}\left| {\text{H}} \right.} \right)} {{\text{p}}\left( {\text{D}} \right)}}} \right. \kern-\nulldelimiterspace} {{\text{p}}\left( {\text{D}} \right)}}\), where \({\text{p}}\left( {\text{D}} \right) = {\text{p}}\left( {\text{H}} \right){\text{p}}\left( {{\text{D}}\left| {\text{H}} \right.} \right) + {\text{p}}\left( {{\text{\~H}}} \right){\text{p}}\left( {{\text{D}}\left| {{\text{\~H}}} \right.} \right)\).
Bayes’ theorem also has another advantage: it lets one calculate the probability of a single event, for example, the probability that a particular person, Mrs. X, has breast cancer, given that she has tested positive for it.
Adapted from Hazzan and Leron (1996).
In the present context, these conditions are equivalent to the standard definition of a group.
The elements of the groups Z n are often taken to be equivalence classes, not numbers as in our definition, which would lead to a different mathematical analysis of the task. The present analysis, however, is the one relevant for the version that our students have learned.
Hazzan and Leron (1996) discuss data on two more tasks, which show that this misuse of Lagrange’s theorem is deeper and more prevalent than might appear merely from the data presented here.
See Stanovich (2008) for a recent attempt to formulate a tri-process theory.
In the following quotation, Evans uses heuristic processes instead of Stanovich & Kahneman's System 1 and analytic processes instead of System 2.
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Acknowledgements
The first author gratefully acknowledges a Bellagio Residency grant from the Rockefeller Foundation, where some of the ideas presented in this paper were developed.
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Leron, U., Hazzan, O. Intuitive vs analytical thinking: four perspectives. Educ Stud Math 71, 263–278 (2009). https://doi.org/10.1007/s10649-008-9175-8
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DOI: https://doi.org/10.1007/s10649-008-9175-8