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I teach algebra to 7th and 8th graders. As someone with a degree in computer science and a keen amateur's interest in cognitive science and evolutionary psychology, I am very interested in what is going on in my students' brains when they get an answer wrong (or right, in fact). I've recently started to work my way through a stack of books on mathematical cognition, and I may try to spend some of my summer at the local university library reading relevant papers. But I'd also like to share some of my observations with others who think about mathematics, and see what insights you can offer.
Let's say that prior experience has shown that a student can add and subtract fractions correctly with probability p and can solve one step addition and subtraction equations with integers correctly with probability q. Then it seems reasonable to assume that this student will solve an equation like x+2/3=4/5 correctly with probability pq. In fact, for many students, the probability of success with such an equation is significantly less than pq.*
Why is this? I have three non-mutually exclusive theories:
- The interface between the fraction addition module and the addition equation module in the student's brain is imperfect.
- Running the fraction addition module as a subroutine of the addition equation module degrades the performance of one or both, like a computer with low memory attempting to run multiple applications.
- The assumption that there is a general addition equation module that works for any addable quantities the student is familiar with is wrong; there may be separate modules for integer addition equations and fraction addition equations, maybe even different modules depending on where the signs are. Thus q simply is not a relevant probability.
The computer scientist in me would start debugging by looking for error #1, but based on my reading, I suspect errors #2 and (to a lesser extent) #3 are closer to the truth. Of course, there may be other qualitatively different possibilities I haven't considered, and that's one of the reasons I'm looking for input. I am also interested in any links or references people can provide to research on cognition in higher math; most of the materials I have seen focus on more fundamental concepts like numerosity and basic arithmetic.
*Two points need to be made in the interest of intellectual honesty: First, attentive students with a few days practice with one-step equations actually don't really make errors with these types of problems any more often than they err in straightforward fraction arithmetic; I am using this as an easy to describe example of the general phenomenon of students being less successful at multi-step procedures than would be predicted by their competence at the individual steps. Second, the probability language here is metaphorical. That is, my question is based on anecdotal observations; I have never tried to quantify the phenomenon I'm describing or to examine it under controlled conditions. |