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.
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!"
• Lucy, in the UK, wrote about her son who has just turned 5: "He counted to one hundred once just for the fun of it whilst getting dressed. It was the first time I realized he could do that! He loves to line up number magnets and get me to tell him what the number is, particularly when the number goes into the millions! He can work out what a number is into the thousands from playing with fridge magnets. He has learned about odd and even numbers from walking around locally and noticing the numbers on houses. He can recognize them in other contexts now. He also learned to count in twos by predicting the next house number. We have never done any formal arithmetic or written anything down."
• Kathy wrote: "Our oldest son, who is 6, has always been fascinated by numbers. He could count to 199 before the age of 4. He loved to count, and to have me count, and to do rhythmic things with his body. He would jump while I counted, or bounce on the couch. He started on math when he wanted to know how many things he would have if he doubled them. We went through a doubling phase!"
In their continued math play, young children often discover the basic concepts of adding, subtracting, multiplying, dividing, and more. Once they have the concepts, the actual ways of performing these operations come easily. Here are a few quotations from the many stories that reinforced this idea:
• Janet wrote, of her young daughter: "She developed counting skills as most toddlers do, using fingers, food and toys, and game pieces and spaces on game boards and computer games. ... That naturally led to adding and subtracting with fingers and objects, and then doing that in her head. ... Often, seemingly out of the blue, she would ask questions like, ‘Does four plus ten equal fourteen?' Me,‘Yes.' She, ‘Then does five plus ten equal fifteen and seven plus ten equal seventeen?' She quickly found patterns in the adding and subtracting of numbers and would apply these rules, which she discovered on her own, and increase the values used. This genuine interest in the patterns numbers create was most noticeable in her seventh year. ... I myself was quite terrified of math as a school child and teenager. But I have to say that [my experience watching and talking with my daughter] has given me a new appreciation for math and a sharpness of mind, with regard to calculations, that I had never previously felt. I also see real beauty in the unfolding of her relationship with numbers."
• Unschooling mom Lori wrote: "One thing just happened two minutes ago. My younger son [age 5] was building with Legos while I was in another room, and he called out to me with a smile on his face, while jumping on the couch, ‘Mom! What is 4 plus 4 plus 4 plus 4?' I said, ‘16.' He smiled and said, ‘What is 8 plus 8?' I said, ‘16.' He smiled more and said, ‘What is 2 plus 2 plus 2...' and he got exactly the right number of 2's to go to 16. It was clear that he knew the answers to these questions before he asked. These were not memorized from having been taught, but concepts that he figured out from working with Legos and playing around with the numbers in his head and on his fingers. And he was thrilled to manipulate the numbers, all on his own. To him, it was a game."
• A-L wrote of her young son: "When he was 3 or 4, one day he went into our living room where we have a large window and noticed that there were four rows of seven panes. ‘So,' he said, ‘if I count to seven four times then it's 28.' I don't think we'd ever talked about multiplication at that point, but he'd essentially figured out how it worked and how to do it on his own from looking at the arrangement of squares. He began experimenting with it on his own, [putting] buttons in rows arrayed like the panes of glass. He still had to count up most of his answers because he hadn't committed them to memory, but he understood how it worked and what it meant."
• And Barbara wrote this about her unschooled young daughter: "She had just been telling me what games she and her friend had been playing, and then we were both quiet for several minutes. All of a sudden she exclaimed, quite excitedly, ‘Oh, I get it!!!' I asked her what she meant, and she replied, ‘I understand division.' ... She then proceeded to explain that when you have a whole of something and you want to break it up into some number of equal parts, that's division. Then she asked me to quiz her, and she indeed knew how to do simple division. Before this moment we had never played around with division. I had never given her any problems to solve, nor had I even tried to explain what it was. ... My story doesn't explain how she has learned these math concepts. But I do know that our lifestyle gives her the time to integrate, ponder, and wonder about the things she sees and hears in the world around her. In her own way, she gets to make the connections, puzzle things out, and test her theories. And I am certain that when she ‘gets' something she will remember it and use it because it is truly her discovery."
• Aurore wrote of her son: "One evening, at age 7, he had brought home a pack of Skittles. Like many kids, he likes to put them on a plate, sort them by color and play with them. On this day he had nine left and arranged them into three rows of three. He said, ‘you know, the number nine is a square.' I told him that's what it's called, a square number, and that he could also make a square with four rows of four. He ended up making bigger and bigger squares ... When it became impractical to keep making squares with skittles (too big), or perhaps because he was just getting bored with doing that, he used a calculator to find more square numbers and wrote them down."
Some readers are no doubt thinking, "Well, a good teacher can use these sorts of demonstrations to teach math and thereby help children learn more quickly and efficiently than they could through self-discovery." But the problem with such reasoning is that every child is different and no teacher, no matter how brilliant, can get into every child's mind and come up with just the trick that will engage that mind at that exact time. That's why self-learning--learning in which the child is in charge--is almost always, in the long run, more efficient and enduring than anything that can be taught by even the most brilliant teacher.
Math is not just play. It is also a useful instrument (tool) in our daily lives, and to that extent we naturally learn it in our daily lives. Most of the math stories sent to me included at least some account of learning math as a tool in daily living. Here are a few choice quotations from those stories:
• Amy, a homeschooling mother of seven, wrote: "They all know how to divide and multiply, calculate percentages, add and subtract, just by handling money and cooking. I'm sure it helps that they have to share limited amounts of yummy snacks not only among the 7 of them but with various friends who are always around. Food and money teach kids a LOT of math, and it highly motivates them."
• Anne wrote: "All five kids learned to read recipes, measurements, how to divide and how to double or triple a recipe's ingredients. They read maps and figured out the mileage. They all played various card games and board games that use numbers and/or reasoning skills -- Uno, Skip-bo, Pinochle, etc. As they became involved in local sports, they learned how to keep the scorebook and figure out averages. One son learned how to make a spreadsheet to keep track of his team's batting averages. They all kept their own ledgers in their bank savings accounts."
• Vincente, a staff member at a Sudbury school, sent me this cute story: "Somehow we always end up with a lot of loose change, which needs to be rolled to be deposited. One of our very young students chose to do this [with my help]. We make stacks of five and count to fifty, stack and roll. This is just the beginning; it gets better. ... A week later I'm dodging vampires. Another of our mega-young invites me to play in one of the first role-playing adventures he's running. ... The penny-counter and others are watching us, learning. The dungeon master rolls 4 fives in a dexterity check, his vampire executing a superb jump and spin landing on one hand on a pencil thin branch. Out of secondary hearing the penny-counter's words clink into the space reserved for Peter's blog: ‘four times five is twenty, five four times is twenty.' Commutative property of multiplication--check."
• And this, from Jennifer: "Three years ago, my son [at age 8] was diagnosed with Type 1 Diabetes. Now, every meal is math. We calculate total carbohydrates from nutritional labels, total carbs for a meal, carb to insulin ratios by time of day, correction factors, percentages, etc. Now he NEEDS to know math to stay alive. He still hates memorizing times tables. ... If I asked him, "What is 3x6?" I just got a blank stare. Then one day at lunch he wanted cookies, so I said, ‘OK, if each cookie has 6 grams of carb and you are going to eat 3, how much carb is that all together?' Without even blinking, he replied, ‘18'.
But it's not just food and money. Here's another example:
• Beatrice wrote: "Playing the piano, my daughter told me she was doing math. She was encountering fractions--half notes, quarter notes, eighth notes, sixteenth notes, all in musical notation as well as in patterns and rhythm."
Many of the stories sent to me about instrumental math had to do with games. Most of the games that kids play today involve numbers, at least to keep score; and many of them involve really complicated math, which the players pick up eagerly in order to play the game. Here are a few representative quotations:
• H wrote: "I have 3 kids attending a democratic free school with no imposed curriculum. My kids have spent a lot of time playing online games. Real games, not those stupid educational ones. My 11-year-old son plays MapleStory and has figured out complex mathematical structures to play the game. ‘If I want to buy this helmet for this amount, how many hours do I have to play making this amount per hour in order to buy the helmet? If I sell this item in the market and the fee to sell is a certain percentage, how much will I have left after the fee? If I have this percentage of experience and I make a certain percentage per hour of experience, how many hours will it take to level up?' ... Plus in the game you work with three different currencies and have to be able to translate back and forth among them regularly. Put these problems isolated from the game context to a bunch of 5th graders in ‘real' school and ask them to show their work and see what you get."
• Rebecca wrote: "Before my oldest son was ‘school age' he learned to solve basic math problems so that he could save the world from enemy invaders."
• Gillian wrote: "My 10 year old and 5 year old are unschooled and there is no way to avoid them being exposed to math if they live a stimulating life. In particular, the computer and PS3 games that my son plays -- World of Warcraft, Second Life, Uncharted, City of Heroes -- have math concepts built into them in a completely natural way. I do not particularly like games that are deliberately 'educational' and my children have never liked them. Any time I have tried to direct them to those games they lose interest very quickly, perhaps because those games are often condescending in tone and less complex than a well-designed game. But give them intelligent games to play and almost inevitably they learn lots of things that schools try to cover in the school syllabus, and they learn them in a much more natural and effortless way."
• And Erica wrote: "My sons (ages 11 and 7) made up a game together called ‘Draw Fight'. It's a strategy game that uses addition and subtraction. Each of them draws their own character and ... each character gets 50 points to spend at the beginning of the game towards his fighting skills, weapons, health, and armor. Choosing where to spend your points is very important because some of the things are worth more than others. After each player has had a turn to attack, you must add up your damage points done to the enemy character and subtract points that were taken from your character. The player with the most points remaining at the end of the game wins."
Beyond the world of food, games, and handling your own money, math is also an essential tool in some careers--such in physics, engineering, and accounting. People who freely choose such careers eagerly learn the math they need as part of their self-training, regardless of any deficiency in their previous math education. Here are quotations from three stories about math for careers:
• Terry, a homeschooling (but not unschooling) mom, wrote: "My oldest always balked at math.... He fought me about doing any math workbooks and I started asking for less and less in the way of math. ... We stopped after 5th grade. He had always enjoyed pretty much unlimited computer time and enjoyed writing games and programs in a self-taught way. He was offered [at age 17] an internship doing programming at a company that auctions municipal bonds. He did so well that they hired him and he is still working there at age 20. He really has a knack for programming and finds the bond and tax stuff fascinating. He is often on the phone with big-time bank executives who have no idea that they are talking to someone so young. He still can't tell you what 6 times 7 is without having to add it in his head. He took placement tests to get into community college and did badly on the math part and was supposed to take a remedial math class. This bothered him because you have to pay to take the remedial math, but you don't get credit for it. So ... he did two days of math study [my emphasis] and re-took the test. This time he placed out of both the remedial math and the basic math courses. If he sees a reason to learn something, he will do it. Otherwise, forget it!"
• Dan, a Ph.D. candidate in anthropology, wrote to me explaining that the out-of-context math courses he took in college left him poorly trained for the statistics he needed in his graduate work. He added: "Through a lot of self-teaching and a little mentorship, I am [now] better at statistics than most of the professors I encounter."
• A colleague of mine, a highly esteemed biologist whose work includes the development of mathematical models, wrote in an autobiographical sketch that he performed poorly in math in high school and college and learned little. He wrote: "I took one year of math in college, freshman calculus, and it almost killed me. In graduate school I had strong reason to learn math so I did. I purchased Calculus for Dummies, practiced hard, and pestered more knowledgeable graduate students when I got stuck. It wasn't exactly fun, but every time I figured something out I had a feeling of triumph that motivated me to take the next step. I published my first theoretical paper while still a graduate student and now I'm a well-known theoretical biologist."
If this were a typical article about math education, it would be entirely about didactic math--math as it is taught by "expert" educators to naïve students. Our society is so convinced that this is how math must be learned that even parents who become unschoolers are often reluctant, at first, to give up the formal or semi-formal teaching of math. They tend, for awhile, to succumb to the cultural beliefs that (a) math must be learned to be successful in our society and (b) math is no fun, so most people will not learn it on their own. But over time, watching their kids, they change their minds and stop the instruction. Here are two quotations that nicely express these points:
• Rebecca wrote: "With my son's apparent agreement, we succumbed to using a packaged program, with a video component.... And then it happened. Both my son and I lost our enthusiasm. He was bored. I didn't like the way things were going in the material ... repetition, repetition, and more repetition. So, after internal writhing, I pried my white-knuckled hands from the crutch of the packaged, predictable, lock-step curriculum and told my son that I was done with my part in making that happen. ... Letting go of the math curriculum (and expectations) has been a huge weight off of my mind. For so many years I had a split home-learning personality--‘we unschool, except in math.' I was all tied up in knots about math and felt I had to strongly encourage (coerce?) my son to adopt a traditional approach to learning it." Rebecca went on to explain that her initial concern for teaching math had to do with expectations about college. For years she couldn't let go of the idea that her son must attend college to have a good life and he must learn math in order to get into college (even though he was not yet 9 years old!).
• Carin2Learn wrote: "I confess that it was a moment of almost-panic that motivated me to show my son the math worksheet website. ... He also claimed he wanted a math workbook. I bought him one, and it remains unused. Thankfully there is more to math than sitting down and writing."
A number of other respondents pointed out that math lessons and programs are easy for kids who choose to do them and are allowed to do them in their own ways, on their own schedules. Here are several quotations to that effect:
• Carlotta wrote, about her son who did no formal math lessons until age 12: "He then shot through Key Stage 3 Maths in about 3 weeks of doing just a little bit here and there. He found it almost ridiculously easy, doing things like memorizing his tables (with some interest in the various patterns that he spotted) in less than an hour. Trigonometry easy peasy, equations no problem. ... OK, so he had spent a considerable amount of his younger years playing the markets on Runescape and solving other mathematical problems in various (fun) games...but that had been it. SOO much less sweat."
• Fawn wrote: "My 11-year-old daughter was home schooled for grades 2-5. We did very little formal math instruction, maybe an hour a week total. She had a workbook she could do when she felt like it, and if she had a question I would briefly explain, but she was pretty much on her own. At the end of 4th grade she scored way above grade level on a standardized math test. She is now in 6th grade in a traditional school, at her request, and she has a 94 average in math."
• Leslie wrote: "We did some hands-on stuff, but honestly, I was handicapped by my own math education to the point that when I would try to explain to my kids how to do something, one of them would interrupt me and say, ‘you're confusing me--this is how I do it' and then explain some much more elegant way of coming to the right answer that showed me that they had a much better understanding of HOW math worked than I ever did. It always humbled me."
• An anonymous commenter on my last post wrote: "One friend of mine was an unschooler and the extent of her son's math education was reading Murderous Maths when he felt like it. At 14 he decided he'd like to take algebra at the community college. He picked up a textbook and learned all of arithmetic in a few weeks. Another friend put her son into school at 5th grade. After the testing the school said her son would never be at grade level by the end of the year. He caught up in a month."
• Chris, whose daughter went to a traditional school, wrote: "[She] was diagnosed with learning disabilities. In grade school she could intuitively give me the answers to complex homework math problems involving large fractions or long division, but she didn't consciously know how she got the answers. She would weep loudly when I tried to show her the steps to ‘write out' the problem for her homework, wailing, ‘That is not how the teacher told me to do it!' Then she would try to work out what to her were the meaningless magical steps of long division that she could not remember correctly and so never gave her the right answer. She refused to accept my version of the ‘steps,' even though it gave the right answer, because it was not the way her teacher told her to do it."
College Admission Math
And now, finally, we come to the math that middle-class parents most worry about. For some odd reason we have decided, as a society, that all young people who go to college--even those who want to become poets or linguists--must show their mettle on a test of ability to do a certain amount of algebra, geometry, and trigonometry that they will never again use as long as they live. And so, some companies make lots of money tutoring kids--kids who have already ‘taken' thousands of hours of math in school--to do those tests. And quite often the tutoring does the trick because the young people at this point want to learn what they must to get into the college of their choice. Then they can promptly forget, forever, the math that they had put into their temporary memory banks. Here are two pieces about how unschooled kids prepare for the math SAT or ACT.
• Leslie wrote this about her son who was entirely unschooled until he went to college: "The first real formal math he did was when he studied for the ACT test. When he was younger, we had math workbooks and even a couple textbooks around the house, but they barely got looked at. ...The ‘dirty little secret' about math is that it just doesn't take as long to learn it as we're culturally indoctrinated to believe it takes. My son learned enough math in just a few weeks to get a 33 on the ACT test just by studying some ACT test prep books." [Note: In the United States, the ACT is most commonly used in the middle states and the SAT is most commonly used on and near the two coasts.]
• To find out more about how kids with no formal math training deal with college admissions math, I interviewed Mikel Matisoo, the Sudbury Valley staff member who is most often sought out by students who want help in preparing for the math SAT. He told me that the kids who come to him are usually those who have relatively little genuine interest in math; they just want to do well enough on the SAT to get into the college of their choice. He said, "The way the SAT is structured it is relatively easy to prepare directly for it; there are certain tricks for doing well." Typically, Mikel meets with the students for about 1 to 1 ½ hours per week for about six to ten weeks and the students may do another 1 to 1 ½ hours per week on their own. That amounts to a range of about 12 to 30 hours, total, of math work for kids who may never before have done any formal math. The typical result, according to Mikel, is a math SAT score that is good enough for admission to at least a moderately competitive college. Mikel explained that the kids who are really into math, and who get the top SAT scores, generally don't seek him out because they can prepare on their own.
And so, dear parents, please stop worrying about your kids' learning of math. If they are free to play, they are likely to play with math and learn to enjoy its patterns. If they live real lives that involve calculations, they will learn, in their own unique ways, precisely the calculations that they need to live those lives. If they choose to go to college, they can learn quickly--from a test preparation book, program, or tutorial--the specific math tricks necessary to do well enough on college admissions math. If they choose some career that involves math, they will eagerly find ways to learn the specific kinds of math that they need for that career. Your worry is only a hindrance.
And so, dear educators, please step out of your boxes and take a look at these remarkable educational movements--the unchooling and Sudbury movements--and study them to see, from a different point of view, how education can work in such a painless and joyful manner when kids are free and in charge of their own learning. Nobody, at least no student, benefits from the thousands of hours of forced math "study" that we put kids through in our schools. The same amount can be learned in a small fraction of that time by kids who are free.
See new book, Free to Learn