We fear it and loathe it; we admire but are also suspicious of those who are good at it; we place it in such high esteem that we make children study (or pretend to study) it almost every day of every year that they are in school; and we use it as a major criterion for college entry. We put math on a pedestal and then we avert our eyes, or else we spit at it--as happens with most things that we put on pedestals.
Math is that school subject that we can't BS our way through. That's one thing that makes it so scary to so many. There are right and wrong answers to every question, no partial credit. It also seems to many people that math performance reflects basic intelligence. To do badly is to come across as logically inept, so fear of failure is even greater in math than in other school subjects, and fear of failure always inhibits learning. I suppose the reason math counts so much on the SAT and ACT college admissions tests is that people think it is an index of general reasoning ability. But they are wrong.
The first step in coming to grips with math is to knock it off its pedestal. The real-life problems that are important to us are problems like these: Whom should I marry? Should I marry? Should gays be allowed to marry? What career should I go into and how should I prepare for it? If I invent gizmo X, will people buy it? Should corporations have the same constitutional rights as individuals? What's the best way to unplug the toilet? Math plays little if any role in solving such problems, nor do such problems have clear-cut right or wrong answers, demonstrable by some formula. Human intelligence and reasoning reside in wisdom, not math. Wisdom is the ability to bring one's values, likes and dislikes, knowledge about other people and their likes and dislikes, and general knowledge of the world together in a manner that leads to workable solutions to the problems that confront us--solutions that promote our own and others' happiness and decrease our own and others' miseries. Math has its purposes, indeed it has some valuable purposes in our modern world, but it is far from the core of intelligence. Humans were intelligent long before math was invented. Some of the smartest people I know--even some of the best scientists I know--are not particularly good at math.
The second step in coming to grips with math is to realize that math is not particularly difficult. There is nothing magical about it. You do not need some natural gift beyond that of a normal human brain to do it. Nor does it require the thousands of hours of study that we try to force upon school children. In fact, those thousands of hours of forced work at math, done for a grade and not for fun or for any practical use, are what make math seem so difficult and intimidating.
The best evidence I know that math is not hard comes from the experiences of people involved in the unschooling movement and the Sudbury "nonschool" school movement. I have written about these movements in previous posts. Unschoolers are homeschooling families that do not provide a curriculum for their kids or evaluate their learning in any formal way. Sudbury schools are those that are modeled after the Sudbury Valley School, where kids of all ages are free all day to interact with whomever they choose and pursue their own interests. Unschoolers and Sudbury schoolers defy our cultural beliefs about what kids must do to succeed in our society. All available evidence shows that the kids in these settings grow up to become happy, productive, ethical members of the larger society, who continue to take charge of their own lives and learning throughout adulthood (for references to research on Sudbury Valley graduates, see my post of Aug. 13, 2008).
Several weeks ago I invited readers of this blog to send me stories about the self-directed learning of math. A total of 61 readers kindly responded, some with beautifully written pieces that could be stand-alone essays. I am extraordinarily grateful. Most of the stories came from unschooling parents who described math learning that they observed in their kids. It has taken me several days to organize and analyze qualitatively these stories to extract the common themes, but now I have completed that task in at least a preliminary way and am ready to relay those themes to you.
I have found it convenient to organize the stories into four categories based on the primary motive that seemed to underlie the math learning that was described. I have labeled the four categories as: playful math (which might also be called "pure math"), instrumental math (math learned as a tool to solve problems encountered in daily life), didactic math (math studied according to some curriculum or plan set out by someone other than the learner), and college admissions math (math learned for the explicit purpose of performing well or adequately on the SAT, or ACT, or some other test used for college admissions). As I relay the stories about each of these categories of math learning my convention will be to use only the first names of the storytellers and not to use children's names at all, as some requested anonymity. In what follows, I have put my own comments in Italics and the words of contributors in Roman after bullets, so as to help you guide your reading if you choose to skim.
Playful Math
I've chosen to start, most joyfully, with playful math. Playful math is what some call "pure math." It is what real mathematicians do, and it is also what 4-year-olds do. Playful math is to numbers what poetry is to words, or what music is to sounds, or what art is to visual perception. I will write later about math that is used as a tool in play, but now I am writing about math that is play--math done for no other purpose than the sheer fun and beauty of it. Playful math involves the discovery or production of patterns in numbers, just as poetry involves the discovery or production of patterns in words, and music involves the discovery or production of patterns in sounds, and art involves the discovery or production of patterns in visual space.
Four-year-olds have a knack for bringing the whole world around them into the realm of play. They play with words, so they are poets. They play with sounds, so they are musicians. They play with crayons, paints, and clay, so they are artists. And they play with numbers, so they are pure mathematicians. I've noticed that students at Sudbury Valley, who are free of any imposed curriculum, don't stop such play as they grow older. They continue to play with words, sounds, paints, and numbers and often become really good at such play. The same seems to be true for kids growing up in unschooling homes.
The earliest math play, by little kids, commonly involves the discoveries that numbers come in a fixed sequence, that the sequence repeats itself in a regular (base-ten) way, and that once you understand the pattern there is no end to how high you can count. Here are three quotations from unschoolers' stories that nicely illustrate this point:
• Evelyn wrote, of her 4 ¾-year-old son (who "insists that the ¾ be included"): "When he found out about connect-the-dot drawings, it started to click for him how numbers proceed in order. He started counting aloud all the time, when walking, when lying in bed, etc. ... The other day he was playing with one of his in-school friends, and her mother expressed shock that he did so well with the ‘teen' numbers. . .. He counted to thirty for her in Spanish and then told her he could count to one million in English. So, since then, he has been counting morning, noon and night. This, as you can imagine, can sometimes be hard on others, and we have to remind ourselves it's a good thing! ... He is now at 5068 .... And when I tell people he is counting to one million, he says, ‘No, ten million.' I hope I can survive it!"
