Breakin'

It's Spring Break.

Which doesn't, sadly, mean I have nothing to do. It only means that what I've got to do (and there is a great deal of it) needn't be done on a rigid schedule.

I've got several job-related tasks to take care of in the next week, ranging from the quotidian (prepping for class once school resumes a week from tomorrow) to the leviathan (going through a stack of roughly 75 applications for this coming summer's REU). I've got a couple of meetings tomorrow, one with a student (my independent study in order theory), another with a colleague (Writing Intensive stuff). After that, I'm looking at a nearly completely unstructured week.

I'm in need of some unstructured time, after the busyness of this past week. Dr. Robert P. Moses, noted civil rights leader and founder of the Algebra Project, came for his visit this past Wednesday, and between the public lecture on Wednesday evening (a talk about the degree to which the Constitution ensures a quality education, at which I was delighted to see several of my students!) and the ensuing Math Literacy Summit held on Thursday, there was no shortage of excitement and things to do in our department.

The session I chaired at the summit (a talk on numeracy as it relates to health issues, given by a psychologist at the Duke University Medical Center) led me to the book I'm now reading, Stanislas Dehaene's The number sense: how the mind creates mathematics (Oxford University Press, 1997). This is proving a truly fascinating read!

Dehaene is a psychologist specializing in the neurobiology of mathematical acquisition, his book is a record of many of the facts that have been discovered concerning the way in which people learn mathematics, they way they organize its ideas in our minds, the way math is retrieved from memory. At its most basic level, our sense of mathematics is very little advanced beyond that of many animals, who share with us a precise sense only of the numbers 1, 2, and 3; beyond this is a roughly-reckoned haze of numeric quantities. Dehaene compares our mental conception of number as an "accumulator" with approximate graduations allowing us to give rough estimates of large quantities, but which fails to give precise values for these same quantities.

A few snippets:

• Even as soon as a few days after birth, babies are able to discern between the numbers 2 and 3. (See p. 50.)
  
• We (adults included!) are susceptible to "the magnitude effect": it's harder for us to discern the difference between 90 objects and 100 than it is the difference between 10 objects and 20. Various factors (symmetry, density, etc.) militate and mitigate this effect. (See pp. 71 ff.)
  
• Studies show that when asked to compare numbers, such as 5 and 7, and state which is the larger, instead of behaving reflexively and answering based upon our knowledge that the symbol "7" represents a larger quantity than the symbol "5," we instead convert each of these abstract digits into collections of the requisite number of objects before performing the comparison on these collections. (See pp. 75 ff.)
• We have a tendency to "compress" numbers as they grow, storing them in our minds as though on a logarithmic scale. One corollary of this behavior is that when asked to provide a random sample of numbers in a certain range, people will tend to elect an overrepresentation of smaller values, as though these were more widely spaced than their larger compatriots. (See pp. 77 ff.)
• Since adults compute sums and products (for example) by retrieving the resultant quantity from a memorized table, those whose native languages have exceedingly short names for the ten numerals (like Chinese and Japanese) are able to more efficiently memorize the desired sums and products, and so perform much more quickly and with fewer errors than their counterparts speaking other tongues. (See pp. 130 ff.)
  

These are just a few of the fascinating facts I'm learning about the development and refinement of mathematical thought processes in and by the human mind. Ultimately, one of Dehaene's primary points is summed up nicely on pp. 118-119: "Although our knowledge of this issue is still far from complete, one thing is certain: Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources."

To be continued, I'm sure.

For now, I'm off to enjoy some more of this wonderfully unstructured time, probably by knocking off a few more pages of Dehaene's book. Highly recommended!