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A Travel Excursion of the Mind

(from Homeschooling and the Voyage of Self-Discovery)

A group of homeschooling mothers gathered together in a circle to discuss unschooling approaches to their children's education.

"I can't get mine to do any math," moaned one, and heads began to nod.

"Mine neither," whined another. "She never wants to."

The heads rolled and shook more vigorously, and soon I found myself sitting -- metaphorically, of course, and with no offense intended -- amidst a Greek chorus of heartrending laments, sighs, and whimpers, perhaps something like a modern homeschooling rendition of Euripedes' The Trojan Women.

"I've tried to convince her that math is a skill she'll really use later in life, but she isn't buying it."

I've pondered this for some time now. Perhaps the kids have a sixth sense about them. They somehow know it is a lie. Most of the math I learned in school I have never used. Not once. Nary a differential equation, nor a logarithm, nor the area of a scalene triangle has wriggled or waddled across my path in more than 30 years, and I use a significant amount of quantitative analysis in my day job. My carpenter friend Bill, who flunked geometry and dropped out of high school, makes use of angles and sides all the time; I am yet to encounter a colleague who still uses a sliderule.

Consider the dukes and duchesses, counts and countesses, marquis and marquises, earls and earlesses of earlier times. They didn't learn math so they could balance their checkbooks (there were no checkbooks!), or so they could become accountants - they hired people to do that for them. They didn't use math in shopping; they had stewards for such mundane activities, who paid the grocer's and haberdasher's bills. And unless they were real misers (or getting ready to flee), they didn't spend a lot of time counting money. They didn't study their Euclid so they could become architects. They did so because it added meaning and beauty to their existence, rather like the required "continental tour", only this one a travel excursion of the mind.

Preaching future utility is futility (for math, or for any other subject) -- it is a wrong-headed approach. It's not only based on a lie, one of many my teachers told me (they may have believed them, too, for all I know), but an ineffective one to boot. The young child comes into the world as a princess. The whole world is there, and is hers, waiting to be revealed, fully investigated, and, ultimately, inhabited. She is a "stout Cortez when with eagle eye/He star'd at the Pacific--/Silent, upon a peak in Darien." What use worrying about some wholly inscrutable future time, when this glittering oyster of a world lay opening before you!

Don't attempt to brainwash your kids into contemplating something that is ultimately unknowable. All that can be known with certainty about the future is that it will be unlike today (and checkbooks will probably have gone the way of sliderules.) Teach them (yes, unschoolers, I'm using the forbidden "T" word) that mathematics is one of the most beautiful creations of the human spirit. String necklaces of colored beads in varying mathematical patterns, and wear them with pride. Provide allowances in wampum (convertible to hard currency, of course). Get out the old magnifying glass and count centipede legs (are there really 100?) Give your child a set of pattern blocks (as soon as you are sure she won't swallow them) - chances are that if you provide them at 3, she'll still be playing with them when she's 12. Count the sections of oranges and tangelos, plot them on a graph, and see if the distribution falls in any particular pattern. Read books about Archimedes and see how a lever, properly placed, can move the world (don't let the kids try this without adult supervision.)

When they are ready, show them the Fibonacci numbers, and where they can be found throughout the natural order: in the spirals of shells, branching plants and leaf arrangements, flower petals and seed heads, pineapples and pine cones. (Check out the book Fascinating Fibonaccis: Mystery and Magic in Numbers by Trudi Hammel Garland, and her wonderful posters - www.iguanagraphics.com/fibonacci/ ). Make beaded bracelets in the pattern of the Fibonacci-related Golden String (1011010110101101.... you can have fun for hours on the best Fibonnaci website: www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci ) If you know the Fibonnaci series, you may be able to sit in a field of daisies and figure out whether "she loves you, or loves you not" without picking a single petal!

Go to the library and get a copy of the extraordinary Arthur C. Clarke video "Fractals: The Colors of Infinity" on the Mandelbrot sets, those extraordinary patterns of fractal geometry to be found in nature that may remind you of the wall projections during an '60s Grateful Dead concert (my age is showing, but the soundtrack really is by Pink Floyd!) (This was the public television show I referred to in “The Code Cracker and the Information Hound.”)

Find a set of Zometools (www.zometool.com), sophisticated tinkertoys updated for use by architects, research biochemists, and hobbyists, and which are just plain fun! (Your daughter or son may end up making "truncated icosahedrons", also known as "Buckyballs" after Buckminster Fuller, or "clustering Kepler solids" - whatever they are!) Be forewarned, however: Zometools are outrageously addictive, and will quickly supplant all other forms of youthful human activity.

Okay - sold on beauty but still want to ensure that the usefulness of mathematics seems like a plausible hypothesis? Well, you probably learned that one in Unschooling 101. But to review: bake cakes. Go to the supermarket and figure out the per ounce costs of all the breakfast cereals; convert the ounces to grams, too. Compare distances to various friends' houses using the odometer, and compute how much the gas costs to get there and back. Does your son want to purchase something with his savings? Construct bar charts with the goal, and calculate and plot the percentages of how much has been socked away thus far. Have your daughter balance the checkbook as one of her chores (she might learn to do a better job than you would anyway, and she'll have learned a vanishing art.) Figure out how many jars (by volume) it's going to take to can all the peaches from the tree in the backyard. Concerned about your weight? Have your son manage the calorie counter - he'll keep you honest!

Do your kids surf the Internet? Help them explore the Boolean logic operators behind their searches (AND, NOT, OR), and use them to solve some of the marvelous puzzles by Charles Dodgson, otherwise known as the author of Alice in Wonderland. (For a primer, try www.albany.edu/library/internet/boolean.html) Choose a neighborhood tree and try to find three ways to figure out its height without climbing or employing the aid of a helicopter. Sort potatoes - see if you can come up with a volume rule for Mr., Mrs., and Baby PotatoHead. Measure absolutely everything - from the size of the living room rug that needs replacing to the relative girth of olives, from small to super colossal (that's the kind with St. Louis stuffed inside it.) Use this one as an introduction to "fuzzy set theory" (ever see a fuzzy olive?)

Oh, I know. You still want them to understand that the math they learn today might be of use later in life. Mrs. Blum, the 9th grade algebra teacher with the voice of one of the Harpies, has infected your bloodstream and there's no known cure. Well, don't preach - visit! If your child seems interested, meet with an architect, an air traffic controller, a computer software designer, an epidemiologist, an astronomer, a baseball statistician, a physicist, my mother's stockbroker Larry, anyone who uses math as part of her daily work - anyone, that is, but Mrs. Blum! Don't know any? That's okay, that's what phone books are for. Work with your child to develop a list of questions she might actually like to ask. If you're still stuck, go visit my friend Bill the carpenter.

Whenever we are stuck in our homeschooling routines, whether it be around math or anything else, I am learning not to be frustrated with my children, but to step back and ask myself three questions: Have I provided what is necessary so that my kids can discover the beauty in what they learning? Have I given them opportunities in the present to use it? Do they have models in front of them to which they can aspire if they put in the necessary learning effort? And, my experience has shown me that when I can answer these questions affirmatively, there's not an awful lot left to worry about. My kids, bless them, can take care of the rest.

Except maybe for the centipede legs....

All right – I understand – the first part of this essay worked for some of you, but others of you read it and started to sweat. Admit it, your blood pressure went up, your heart began to race, and you began to worry. First of all, what went through your heads was that if it doesn’t seem like real work, how can you be sure the kids are learning anything? After all, that was the way you had to do it, right? And then you flashed just briefly on how in your own school career, joy was systematically leeched from mathematics, and fear instilled in its place. From your first memory of “This little piggy went to market” forward, it was all a downward spiral, from which you’ve never entirely recovered. They kept on trying to find out what was wrong with you, probing and testing for all your mathematical weaknesses, and finally – with SATs – presenting you with a test where they expected you to get a whole passel of wrong answers, and that no matter how well you prepared, you were going to feel inadequate.

So you’ve decided to use a curriculum. Nothing wrong with that, if such is your proclivity. We’ve used them ourselves – tried various book versions (Singapore Math – www.singaporemath.com --being by far the least offensive), and ended up having the kids do their high school math through the Federal Way Internet Academy (www.iacademy.org). We liked the Internet Academy primarily because it gave the kids a pretest before trying to do any instruction – that way if the kids already knew the material, they didn’t have to repeat it. No busywork! And following a “test”, the computer would isolate only those areas where problems were still occurring, and only require a review of these. Slick and efficient, and it left time for us, as parents, to focus on the all-important context in which math education occurs, which is what the first part of this essay is all about.

But how one uses a curriculum, we’ve discovered, is just as important, maybe more important, than the curriculum itself. I can remember those hours – long hours! – of mindless homework, when I already knew what I needed to know, and really – for educational purposes – I should have been out playing stickball. Or, worse, on those rare occasions that I didn’t absorb a concept quickly enough, there would be pages and pages of rote, boring, plaguing problems that made me remember how much I would have preferred a trip to the dentist.

The wonderful folks at the Sudbury Valley School – a democratically managed, child-directed learning environment in Massachusetts that has now been operating for more than 30 years (www.sudval.org) – have demonstrated rather conclusively that all the mathematics taught in public schools from kindergarten through twelfth grade can be learned by average, normal, healthy kids in about eight weeks, when the child has expressed a real interest in doing so. (No kidding – check out some of the books on the School.) They use curricula for this purpose, but the real issue is not whether or what curriculum to use, but one of interest and motivation and timing. So now I’ve got your palms wet.

Of course, some of us insist, against the entire tide of our own personal experience, that the way a child should be made to learn a particular mathematical operation with which she has struggled is by assigning several dozen additional problems where use of the same skill is required. Well, maybe, or maybe not. I remember once being told an anecdote related by the great anthropologist and systems thinker Gregory Bateson. Bateson had met an experimental psychologist who had substituted a ferret for laboratory rats in his learning experiments, as ferrets in their natural state, unlike rats, do actually hunt for their prey in the maze of rabbit warrens. The psychologist placed the ferret in the maze and, after turning down every blind alley, the ferret found the haunch of a dead rabbit in the reward chamber and promptly chowed down. The next day, the psychologist placed the ferret in the same maze. This time, the ferret turned down every blind alley, but the one place he did not go was the same location where now a new rabbit haunch had been placed. This, concluded the psychologist, proved that the ferret had not learned anything. On the contrary, said Bateson, no self-respecting ferret is going to expect to find dead rabbits in the exact same spot twice on consecutive days. Do I detect a nervous facial tick or a little bit of tremor in your lower limbs?

My wife and I happened upon a strategy that we’ve now used with both children. It might on the surface seem counterintuitive, but it works! Meera would master multiple-digit multiplication, but then all of a sudden when faced with a problem involving multiplying decimals, it would be as if the final decimal point would fall from the sky like a meteor, and wherever it landed would be where it ended up. Ellen and I would look at each other and instantly (based on our experience with our older daughter Aliyah) knew what to do. Failing fifth grade math? Give her seventh grade math! Sure enough, Meera would move on to the new, more interesting concepts and, usually sooner rather than later, the difficulties in accomplishing particular mathematical operations would clear themselves up of their own accord, with little help from us whatsoever.

You’ll never see this attempted in a public school environment. Imagine the parent-teacher conference: “I’m sorry, Mrs. Johnson, but Susie is failing fifth grade math. However, instead of making her do extra homework, or signing her up to work with a tutor while the other kids are enjoying themselves in the schoolyard, or suggesting you take her to the nearest Sylvan or Kumon learning center while her friends are out riding their bikes, or leaving her back or putting her in the “slow learners group”, we’ve decided to give her seventh grade math instead. Is that okay with you?”

This begs the whole question of what exactly is “fifth grade material” and what “seventh”? I’m sure the “scope and sequence” people are convinced there is a logic to this business – it is, after all, an entire industry! -- but what difference does it make if the children, who are, after all, the “end users”, lose interest along the way? I find it is the exception rather than the rule to find children who learn math in a linear fashion, which is one of the reasons so many of us ended up hating math, isn’t it?

Using the school model for our homeschooling endeavors generally speaking is extremely limiting. Gregory Bateson’s daughter Mary Catherine Bateson once wrote that, “Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos.” The reason the “skipping” method worked for both of my children was not that they moved “ahead” in material, but rather that they left concepts they had already mastered for a new, more interesting mathematical universe, one where the rote operations they formerly had been struggling with now had a larger purpose, embedded as they were in an area which fed their expanding mathematical view of the world.

Now I’ve got some of you feverishly mopping your brows. “This is just too challenging. I can’t deal with it, much as I couldn’t deal with math when I was in school back in the dark ages.” Okay. I’ll keep it simple: the single most important thing you can do for your kids around math is to help them avoid “math anxiety”. And one best avoids “math anxiety” by preventing “math trauma”. Be a physician, and apply the first principle, “Do no harm.” Without trauma, anything remains possible. With trauma, your kids may end up with certain skills, but they will also end up with wounds that may take a long time to heal.

The June 2001 issue of the Journal of Experimental Psychology--General includes an article titled “The Relationships Among Working Memory, Math Anxiety, and Performance” by Drs. Mark Ashcraft and Elizabeth Kirk. In their study, the authors found that, “Fear of math can cause a temporary brain glitch that may explain why an otherwise glib person stumbles and stammers over the simple matter of adding two numbers.” In experiments with university students, the researchers found that those with “math anxiety” suffered a fleeting lapse in working memory when asked to do some mental arithmetic. These memory problems failed to crop up in tests that did not involve numbers, meaning that the phenomenon is “very specific to math… It’s a learned, almost phobic reaction to math,” explained Dr. Ashcraft. He noted that research indicates that people need not be anxious types in general to harbor a fear of math. The mere specter of doing sums has been shown to send a person’s blood pressure and heart rate skyward.

Math-phobic students were often stumped when it came to remembering basic math rules like “carrying over” a number when adding, or “borrowing” from a number when subtracting. An explanation for the memory problem, Ashcraft proposed, is that when math anxiety takes hold, a rush of thoughts goes through a person’s head. This leaves little room for the task at hand. And this makes for a “vicious cycle” for students. Once they develop math anxiety, the fear gets in the way of learning, which leads to waning self-confidence in their ability to ever conquer arithmetic. Part of the problem, according to Ashcraft, may rest in how math is taught – at least in the U.S. Students may be taught math rules, but they rarely know why a certain approach to a math problem works. Giving students a “deeper understanding” of math may help fight phobias, he suggested.

Getting kids to develop deeper, problem-solving skills in school may be important, argues Ashcraft, but that may be easier said than done. In one study of math anxiety among college students, he noted, fear of math was most rampant among elementary education majors.

Hmm.

Glad we’re homeschooling!