We teachers must begin to try to look at our ideas and our teaching through the eyes of someone who knows nothing, can accept nothing unproven, and cannot tolerate inconsistency and paradox. We must try to free our teaching from ambiguity, confusion, and self-contradiction.

Cuisenaire rods:

The beauty of Cuisenaire rods is not only that they enable the child to discover, by himself, how to carry out certain operations, but also that they enable him to satisfy himself that these operations really work, really describe what happens. One can *see* strong connections between the world of rods and the world of numbers. We need to use these materials to enable children to make for themselves, out of their own experience and discoveries, a solid and growing understanding of the ways in which numbers and the operations of arithmetic work. Our aim must be to build soundly, and if this means that we must build more slowly, so be it. The work of the children themselves will tell us.

Knowledge, learning, understanding, are not linear. They are not little bits of facts lined up in rows or piled up one on top of another. A field of knowledge, whether it be math, English, history, science, music, or whatever, is a territory, and knowing it is not just a matter of knowing all the items in the territory, but of knowing how they relate to, compare with, and fit in with each other.

With thought, practice, and luck we should be able to devise problems that children can do in ways which, being their own, will be of use to them. Such problems could make up a kind of self-adjusting learning-machine, in which the child himself makes the program harder as he becomes more skillful. But this approach to mathematical learning, and other kinds as well, will require teachers to stop thinking of *the *way or *the best *way to solve problems. We must recognize that children who are dealing with a problem in a very primitive, experimental, and inefficient level, are making discoveries that are just as good, just as exciting, just as worthy of interest and encouragement, as the more sophisticated discoveries made by more advanced students. When Dorothy discovers, after long painful effort, that every other number can be divided into 2 equal rows, that every third number can be divided into 3 equal rows, she had made just as great an intellectual leap as those children who, without being told, discovered for themselves some of the laws of exponents.

In other words, the invention of the wheel was as big a step forward as the invention of the airplane–bigger, in fact. We teachers will have to learn to recognize when our students are, mathematically speaking, inventing wheels and when they are inventing airplanes; and we will have to learn to be as genuinely excited and pleased by wheel inventors as by airplane inventors. Above all, we will have to avoid the difficult temptation of showing slow students the wheel so that they may more quickly get to work on the airplanes. In mathematics certainly, and very probably in all subjects, knowledge which is not genuinely discovered by children will very likely prove useless and will be soon forgotten.

What then should we do about making the world of numbers and math accessible, interesting, and understandable to children?

A few good principles to keep in mind: (1) Children do not need to be “taught” in order to learn; they will learn a great deal, and probably learn best, without being taught. (2) Children are enormously interested in our adult world and what we do there. (3) Children learn best when the things they learn are embedded in a context of real life called “the continuum of experience.” (4) Children learn best when their learning is connected with an immediate and serious purpose. What this means in math is simply this: the more we can make it possible for children to see how we use numbers,* and** to use them as we use them*, the better.

We should introduce children to numbers by giving them or making available to them as many measuring instruments as possible–rulers, measuring tapes (in both feet and meters), scales, watches and stopwatches, thermometers, metronomes, barometers, light meters, decibel meters, scales, and so on. Whatever we measure in our lives and work, we should try to measure so that children can see us doing it, and we should try to make it possible for them to measure the same things, and let them know how we are thinking about the things we have measured.

Children are interested in themselves, their own bodies, their growth, quickness, strength. Children might measure their own size, strength, and speed, and how these things change over time and vary with different conditions. Thus children might measure their own respiration and pulse rate, then exercise violently for a while, then measure their breathing and pulse rate again, then measure it at intervals to see how long it takes it to get down to normal. Or children might do various tests of speed and strength, running timed distances or lifting weights or doing other exercises, and see what happens when they try to do this a second time, and how their performance varies with the amount of rest they have, and how their speed or strength, and their recovery times, vary from week to week or from month to month.

Aside from involving numbers, all this is *true *science.