Abstract
A recurrent concern in mathematics education—both theory and practice—is a family of mathematical tasks which elicit from most people strong immediate (“intuitive”) responses, which on further reflection turn out to clash with the normative analytical solution. We call such tasks cognitive challenges because they challenge cognitive psychologists to postulate mechanisms of the mind which could account for these phenomena. For the educational community, these cognitive challenges raise a corresponding educational challenge: What can we as mathematics educators do in the face of such cognitive challenges? In our view, pointing out the clash is not enough; we’d like to help students build bridges between the intuitive and analytical ways of seeing the problem, thus hopefully creating a peaceful co-existence between these two modes of thought. In this article, we investigate this question in the context of probability, with special focus on one case study—the Medical Diagnosis Problem—which figures prominently in the cognitive psychology research literature and in the so-called rationality debate. Our case study involves a combination of theory, design and experiment: Using the extensive psychological research as a theoretical base, we design a new “bridging” task, which is, on the one hand, formally equivalent to the given “difficult” task, but, on the other hand, is much more accessible to students’ intuitions. Furthermore, this new task would serve as a “stepping stone”, enabling students to solve the original difficult task without any further explicit instruction. These design requirements are operationalized and put to empirical test.
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Notes
- 1.
This term comes from an ongoing work with Abraham Arcavi.
- 2.
Wikipedia: A proof of concept […] is realization of a certain method or idea(s) to demonstrate its feasibility, or a demonstration in principle, whose purpose is to verify that some concept or theory is probably capable of being useful. A proof-of-concept may or may not be complete, and is usually small and incomplete.
- 3.
This means that 5 % of the healthy people taking the test will test positive.
- 4.
In our experience, this solution can be easily understood even by junior high school students.
- 5.
A third kind of simplification—rounding off 999 to 1000 and 1/51 to 1/50—is more specific to the numbers given here, and is not similarly generalizable.
- 6.
These two modes of helping the students are termed, respectively, bridging down and bridging up in a forthcoming paper, where they are treated more thoroughly.
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Ejersbo, L.R., Leron, U. (2014). Revisiting the Medical Diagnosis Problem: Reconciling Intuitive and Analytical Thinking. In: Chernoff, E., Sriraman, B. (eds) Probabilistic Thinking. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7155-0_12
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