I heavily skimmed the middle parts of this book when I bought it in May 2012. [Thanks to Christopher Danielson for recommending it.]

I’m re-reading some parts now, and the sub-section *Teaching Number Sense* [pages 124 – 128] resonates with me, not just in elementary school mathematics, but in K-12 mathematics.

If my hypothesis is correct, innumeracy is with us for a long time, because it reflects one of the fundamental properties of our brain: its modularity, the compartmentalization of mathematical knowledge within multiple partially autonomous circuits… The numerical illiterate performs calculations by reflex, haphazardly and without any deep understanding.

… A good teacher is an alchemist who gives a fundamentally modular human brain the semblance of an interactive network. Unfortunately our schools often do not quite meet this challenge. All too often, far from smoothing out the difficulties raised by mental calculation, our educational system increases them… But our schools are often content with inculcating meaningless and mechanical arithmetical recipes into children.

This state of affairs is all the more regrettable because… most children enter preschool with a well-developed understanding of approximation and counting. In most math courses, this informal baggage is treated as a handicap rather than as an asset. Finger counting is considered a childish activity that a good education will quickly do away with. How many children try to hide when they count on their fingers because “the teacher said not to”?

Despising children’s precocious abilities can have a disastrous effect on their subsequent opinion of mathematics.

… It seems more likely that many of these “mathematically disabled” children are normally abled pupils who got off to a false start in mathematics. Their initial experience unfortunately convinces them that arithmetic is a purely scholastic affair, with no practical goal and no obvious meaning. They rapidly decide that they will never be able to understand a word about it. The already considerable difficulties posed by arithmetic to any normally constituted brain are thus compounded by an emotional component, a growing anxiety or phobia about mathematics.

… We need to help children realize that mathematical operations have an intuitive meaning, which they can represent using their innate sense of numerical quantities. In brief, we must help them build a rich repertoire of “mental models” of arithmetic… The day the teacher introduces negative numbers and asks pupils to compute 3 – 9, a child who only masters the set scheme judges this operation impossible. Taking 9 apples from 3 apples? That’s absurd! Another child who relies exclusively on the distance scheme concludes that 3 – 9 = 6, because indeed the distance from 3 to 9 is 6. If the teacher merely maintains that 3 – 9 equals “minus six,” the two children run the risk of failing to understand the statement. The temperature scheme, however, can provide them with an intuitive picture of negative numbers. Minus six degrees is a concept that even first-graders can grasp.

But let us leave this chapter with a note of optimism… In the United States, the national council of teachers of mathematics is now de-emphasizing the rote learning of facts and procedures and is focusing instead on teaching an intuitive familiarity with numbers… Number sense — indeed, common sense — is making a comeback.

In fact, most children are only too pleased to learn mathematics if only one shows them the playful aspects before the abstract symbolism. Playing snakes and ladders may be all children need to get a head start in arithmetic.

By the way, I don’t read this and put the blame entirely on myself and my colleagues. As long as we place more emphasis on test scores than we do on learning, we are at best hypocrites.

## One Comment

Thank you for this. Your last line nicely encapsulates my thinking about teaching. I shamelessly steal from you in my work, so keep it coming.