#### This article is a reprint.

##### The original article by Aaron Falbel can be found in Growing Without Schooling 63 published on June 1, 1988. This version was posted by Pat Farenga on the Growing Without Schooling blog and shared with Tipping Points as one of the most widely read articles from the publication.

*Note from Pat: This is a longer article than usual, but I think it is one that homeschoolers and unschoolers, or anyone interested in math, will enjoy. The author, Aaron Falbel, is a long-time contributor to GWS and a key volunteer in many projects over the years at Holt Associates. I look forward to sharing more of his work with you in the future.*

# The Mathematics of the Ordinary

One of the things that has become painfully obvious to me, over the past few years I have spent working with Seymour Papert (author of *Mindstorms*) at MIT, is the enormous difference between the type of mathematics foisted on children in schools and the sort of intellectual activity enjoyed by actual mathematicians. School math is basically a dead subject, containing ossified, decontextualized facts and symbol manipulation techniques that are, by themselves, neither much fun nor very useful. One has to wonder whether most schoolchildren grow up believing that mathematicians sit around all day doing word problems and long division exercises!

What is it that mathematicians do, if not this? They wonder about the world. They are particularly sensitive to patterns or regularities they observe, and they make conjectures based on them — or as I like to say, they “have hunches.” Then they try to find out whether or not their conjectures make sense by constructing arguments (called proofs) using already-proven ideas (called theorems) or agreed-upon definitions and assumptions (called axioms or postulates), piecing these elements together according to the rules of deductive logic to demonstrate that their conjecture “follows” from what is already known, or, alternatively, that one encounters a contradiction.

This is not quite as straightforward as it sounds. Mathematicians are constantly in danger of taking a misstep and falling into the trap of making unwanted assumptions, using faulty logic, or just “getting stuck” and not knowing quite where to turn. Much of what guides mathematicians in their work can only be called intuition or aesthetics, making mathematics a truly creative, human activity like any other. Other mathematicians are constantly looking out for flaws in each other’s reasoning, trying to avoid pitfalls, or, when they occur, picking each other up out of self-dug holes they have fallen into. In a way, constructing a mathematical argument is like building a house of cards. One misplaced card can cause the whole building to come tumbling down.

When asked to define “mathematics,” many mathematicians might say, “Mathematics is what is contained in all those books over there.” This sort of definition makes mathematics into a thing, nearly a commodity, or at any rate some type of *stuff*. I am taking a different tack in saying that mathematics is a *human endeavor* — it’s what mathematicians *do*. The stuff that ends up in the textbooks is the *result* of their work. Moreover, the ability to understand the formalisms of mathematics (the stuff in the books) does not make one a mathematician any more than the ability to read musical notation (the dots and lines) makes one a musician. Both mathematics and music are activities. One *does* mathematics. One *makes* music. As John Holt argued in *Instead of Education*, academic disciplines ought to be thought of not as nouns but as verbs.

“A mathematician,” wrote G.H. Hardy (himself one), “like a painter or poet is a master of pattern.” This statement, which so well captures what doing mathematics means, provides the key to seeing mathematics not as an esoteric activity, but as an ordinary one. We are everywhere surrounded by pattern; patterns of shape, of motion, of language, of behavior, of sound, of history. . .of practically anything that has a *form* of one sort or another. The act of perceiving, recognizing, classifying, describing, capturing, analyzing, mulling over, or playing with patterns involves a type of formal thinking that lies at the heart of the mathematics of the ordinary.

I have for a long time been meaning to reply to Nancy Wallace’s provocative “Why Study Math?” in GWS 54. I want to question Nancy’s and Bill Hoyt’s description of mathematics as a language. I would say that mathematics often involves a special language or symbolism, but I would not say that that’s what mathematics is. As the semanticist, A.H. Korzybski once said, “The map is not the territory.” Mathematicians use a special language to aid their thinking, to give a sort of robustness or rigor to their wondering, but it’s the wondering that’s the mathematics, not the language or symbolism used to represent it. To say that mathematics is a language is, for me, to turn it back into a noun. Language is a type of tool, a very expressive one at that, but tools themselves are not activities. They help us to do things better, as, indeed, the language or “grammar” (to use Bill Hoyt’s excellent term) of mathematical thought does.

Schools have denatured math. – Seymour Papert

School math has been designed to fit into school’s way of doing things: it can be written on a blackboard, copied into a notebook, easily made into quizzes and tests, and graded. As Seymour Papert has argued, schools have *denatured* math — they have taken something that was once an activity, that was once do-able, and turned it into a commodity that needs to be administered. Getting out from under the shadow of conventional school should be sufficient to escape from the penumbra of school math. Yet, some of my colleagues at MIT have been quite critical of free schools and some homeschooling families who seem to be compromising their principles when it comes to mathematics, falling back to the old worksheets of arithmetic problems. Even A.S, Neill of *Summerhill* once remarked in an interview, “You’d have to keep mathematics (as it is). You have to learn mathematics from books.” The error here is in taking school math as a given instead of treating it as a creation of school. It’s hard to see how computing quadratic equations in and of itself can capture the imagination of any reasonable person. Someone, and it might as well be me, has to come out and say this. Using such equations to think about how objects fall under the influence of gravity (when one comes to wonder about that) is another matter altogether.

The important question for me, given my view of mathematics of the ordinary, is not how it can be taught, but how it can be lived. I don’t think we need to find special “math-speaking” tutors when it comes to the mathematics of the ordinary. Most young children are not interested in becoming specialists in mathematics. But it is in their nature, indeed, in practically everyone’s nature, to wonder about the patterns and regularities they observe in the world. What we can do is to make this wondering, this particular style of thinking, more visible and accessible to children — to bring it out in the open, to wonder out loud.

I want to emphasize again that the mathematics of the ordinary goes beyond just thinking about how we use numbers. Thinking about a knitting or weaving pattern not only involves mathematics, it is mathematics. Thinking about games like chess, checkers, Go, tic-tac-toe, Nim, and various card games is mathematics. Wondering about how the Rubik’s cube works, or various other puzzles, is mathematics. Thinking about and inventing magic tricks is mathematics. So is thinking about 20 paradoxes, making maps, building models, inventing and deciphering secret codes, learning how to juggle, experimenting in the kitchen, making an animated film, thinking about musical rhythms (to use Ishmael’s and Vita’s beautiful example in Nancy Wallace’s piece). Our main task, then, is to open our eyes and realize how much of the mathematics of the ordinary can be found in the activities we are *already* doing.

As children get older, some of them might want to do mathematics in a more focused and deliberate way. They might want to go beyond the mathematics of the ordinary and concentrate on one or another of the subspecialties of mathematics. Here it might make sense to seek out a ‘specialist’, as one would with any skill that requires a special (i.e. out of the ordinary) degree of competence.

This is, in fact, the only way that people in mathematics and the sciences ever really learn anything about their true work: by participation in an intellectual community, a community of practitioners. As more and more sociologists and philosophers of science are beginning to find out, these activities are inherently social and community-based. This is precisely the point David Deutsch makes in GWS 29 (“Becoming Experts”–PF).

At 20, I was lucky enough to obtain a summer job at a major high-tech research laboratory in Connecticut. I was hired to construct a computer animation system, and later on, a “paint” program, using the most advanced, state-of-the-art equipment available at the time. Though film animation had been a hobby of mine for some time, I had no experience whatsoever with computer graphics. However, between consulting a few reference books, looking at other people’s programs, and asking my coworkers for help and feedback when I needed it, I got the job done. I would guess that I acquired the equivalent of several graduate level courses, in terms of the knowledge and skill involved, in those few months. While university students “studying” computer graphics were slaving away^{1} at their problem sets and cramming for exams, I was doing real work (I was getting something done) and, needless to say, learning very much in the process. (Some of the most important things I learned were that I grew bored with that type of work after a few months, got disgusted with the way “corporate politics” was treating people, and made the important decision that computer graphics was not something I wanted to pursue as a career. These are things I would not likely have learned in school.)

I want to emphasize one very important aspect of my short apprenticeship at that research lab: No one “exposed” me to the things I experienced. To expose someone to something means to do something to them. That’s not what happened at that lab in Connecticut. What did happen was that they gave me access to their facilities and to themselves, that is to say, to their research community. That they did so in spite of my lack of “experience” was to their credit. They got their projects accomplished, and I learned some new skills and important things about myself.

I would advise homeschooling parents not to look for apprenticeships for their children because they think “mathematics will be good for them” or because they want to “expose their children to mathematics” but do not themselves feel competent to do the exposing. I maintain that exposing in this sense is wrong. What we can and ought to do is provide access to various types of work for young people, lf and when they want such a challenge, if and when they are ready to make the sort of commitment that such an arrangement entails.

Otherwise, with respect to mathematics, we can rely on the mathematics of the ordinary, the mathematics all around us, to give children plenty of food for formal thought.

###### 1. Note from Patrick Farenga: “Slaving away” was a commonly used phrase at the time the article was written in 1988, but today we understand that using the word “slave” as a metaphor disrespects the history of this violent social practice. If written today, I would replace this phrase with “laboring.”

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