"Why do some children find Math hard to learn? I suspect that this is often caused by starting with the practice and drill of a bunch of skills called Arithmetic—and instead of promoting inventiveness, we focus on preventing mistakes. I suspect that this negative emphasis leads many children not only to dislike Arithmetic, but also later to become averse to everything else that smells of technology. It might even lead to a long-term distaste for the use of symbolic representations."
Why should children learn only “fixed-point” arithmetic, when “floating point” thinking is usually better for problems of everyday life! More generally, we need to find out more about how each child regards each subject. How might it answer questions like “What am I doing here, and why? ”What can I expect to happen next?” “Where and when am I likely to use this?
Until the 20th century, mathematics was mainly composed of Arithmetic, Geometry, Algebra, and Calculus; then Logic and Topology started to rapidly grow. Then the 1950s saw a great explosion of new ideas about the nature of computation—ideas that are now so indispensable that our primary school curriculum is out of date by a century.
Today, our new computational concepts have become so useful and powerful that we should start teaching them in earlier years. We usually think of Arithmetic as a subject in itself. But we can also think of it, instead, as just a certain bunch of algorithms—which suggests that we could begin with simpler and more interesting ones!
...How can we encourage children to invent and carry out more elaborate processes in their heads? Teachers often insist that pupils “show their work”—which means to make them “write down every step.” This is convenient for making grades, as well as for diagnosing mistakes, but I suspect that this focus on ‘writing things down’ could lead to mental slowness and awkwardness, by discouraging pupils from trying to learn to perform those processes inside their heads—so that they can use mathematical thinking in ‘real time’. It isn’t merely a matter of speed, but of being able to keep in mind an adequate set of alternative goals and being able to quickly switch among different strategies and representations.
In the case of school-mathematics, the vocabulary is remarkably small. The children do learn the names of various objects and processes—such as addition, multiplication, fraction, quotient, divisor, rectangle, parallelogram, and cylinder, equation, variable, function, and graph. However, they learn only a few such terms per year—which means that, in mathematics, our children are mentally starved, by having to live in a “linguistic desert.” It really is hard to think about something until one learns enough terms to express the ideas in that subject. Specifically, it isn’t enough just to learn nouns; one also needs adequate adjectives! What's the word for when you should use addition? It’s when a phenomenon is linear. What's the word for when you should use multiplication? That’s when something is quadratic or bilinear. How does one describe processes that change suddenly or gradually: one needs terms like discrete and continuous. To talk about similarities, one needs terms like isomorphic and homotopic. Our children all need better ways to talk about, not only Arithmetic and Geometry, but also vocabularies for the ideas one needs to think about statistics, logic, and topology.