Tribute to Invariance

The Evolution of the Brain

In the preface to Conversations on Mind, Matter, and Mathematics [Cha] one finds the following exciting paragraph:

But the brain also owes its special characteristics to the fact that it is an evolving system. While everyone is familiar with Darwin's theories about the evolution of living species, it is not generally recognized that the development of the brain itself, both during the embryonic stage and then after birth, constitutes an evolution in the course of which the connections among nerve cells are themselves subject to a process of selection. The fact that this process continues through further stages of evolution, at higher levels of organization, may account for the unfolding of thought and mathematical reasoning, perhaps even for imagination.

Given the widely spread innumeracy of the population, it is natural to suspect the environment in which the development of most people's brains takes place.

Experiments (1978, [Dhe]) by Mark Ashcraft and colleagues at Cleveland State University indicate that young adults hardly ever solve addition and multiplication problems by counting. Instead, they generally retrieve the result from a memorized table. Retrieval time from the memorized table increases as the operands get larger. Several reasons are available to explain dependency of the retrieval time on the problem size. A very plausible explanation stems from the drilling practices used in schools. Researchers have tallied up how often each addition or multiplication problem appears in children's textbooks. The outcome is surprisingly inane: children are drilled far more extensively with multiplications by 2 and 3 than by 7,8, or 9, although the latter are more difficult.

Dehaene [Dhe] continues:

The hypothesis that memory plays a central role in adult mental arithmetic is now universally accepted. ... It does mean, however, that a major upheaval in the mental arithmetic system occurs during preschool years. Children suddenly shift from an intuitive understanding of numerical quantities, supported by simple counting strategies, to a rote learning of arithmetic. It is hardly surprising if this major turn coincides with the first serious difficulties that children encounter in mathematics. All of a sudden, progressing in mathematics means storing a wealth of numerical knowledge in memory. Most children get through as best as they can. But as we will see, they often lose their intuitions about arithmetic in the process.

Most children enter preschool with a well-developed (perhaps even innate) understanding of basic counting operations, even if it is finger based. Their intuitive math concepts beg to be developed. Instead, the practice is to subject them to senseless drill. At that age the capacity for abstract reasoning is still lacking. Drill does away with whatever vestiges of abstract math concepts children might have on an intuitive level.