Thinking About Mathematical Thinking

Psychologists should be paying more attention to mathematics, not only because of that discipline's importance for our statistical analyses, but also to better understand how it can be taught. In his review of Raymond Nickerson's book Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs, Gordon Pitz says, "Perhaps no academic subject elicits stronger or more varied reactions than mathematics." Those reactions, positive and negative, presumably are based on our varied experiences doing mathematics. My guess is that most doctoral-level psychologists never got beyond number crunching statistics into the real mathematics that is the subject of Nickerson's book.

I present my own case as an example. In high school and college I was pretty good in algebra and geometry. Statistics at that level (central tendency and correlation) was easy and fun, and I did it by hand with paper and pencil. In grad school, analysis of variance was done on a machine (crunch). But that was arithmetic. Then I had to take a course in mathematical statistics (which included stuff like the Neyman-Pearson lemma), and most of that was nearly incomprehensible to me. Gordon Pitz also was a student in that class, and I recall that he actually enjoyed it and earned an A grade. In his review, my friend says it is meaningless to try to explain our difference in terms of innate and experiential factors. What else is there?

Given the importance of learning real math, is there much help for current students whose brains have the same deficiency that mine had? Nickerson and Pitz don't seem to offer much hope at that level.

Read the Review

Thinking About Mathematical Thinking
By Gordon Pitz
      PsycCRITIQUES, 2010 Vol 55(23)

Comments

[Gordon Pitz]

Indeed, what else is there besides innate and experiential factors? I fear, though, that Jim Korn misread an important point in my review. When explaining the behavior of a single individual, it is meaningless to discuss the relative importance of innate and experiential factors. It makes no sense to ask if Korn’s achievements in mathematics, or my own, were due more to our ancestors or to our upbringing. We can ask, though, to what degree such factors account for any differences in our achievements (if there were any—one should not trust the memory of a pair of septuagenarians for events that happened 50 years ago). It’s a subtle difference, but a crucial one.

Unfortunately, although we may ask the question, we are not in a position to answer it in a very useful way. Heritability estimates vary widely, and we have only a few hints as to which environmental variables might influence performance. Psychologists and educators can offer a few ideas for improving mathematics education. The Newcombe et al (2009) report that I mentioned in my review is as good a summary as any. Yet psychologists need to devote more energy to understanding the fundamental genetic, cognitive, and other mechanisms by which mathematical skills are acquired.

[James Korn]

My comment was not about the "relative importance of innate and experiential factors," but rather how we might better understand how our experiences with mathematics affect our ability to learn math at higher levels. Surely Gordon and I could compare what we still can remember about how we were encouraged or not, what our different pre-grad school courses and curricula were, and how we approached studying for a demanding math stats course. Gordon essentially says that in his last sentence.