# Tribute to Invariance

## Brain's Evolution

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

Given a widely spread *innumeracy* of the population, it is natural to suspect the environment in which brain's evolution 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 memory 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. 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:

Most of the 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 drilling. At that age the abstract reasoning is still lacking. Drilling does away with whatever vestiges of abstract math concepts children might have on an intuitive level.

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