Math! Part 2
by David Albert
All right, I know 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 fully 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 being by far the least offensive), and ended up having the kids do their high school math through the Federal Way Internet Academy. 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 should have been out playing stickball. Or, worse, in those rare times 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 going for more than 30 years have demonstrated rather conclusively that all the mathematics taught in public schools from kindergarten through twelfth grade can be taught to 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 she has struggled with 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 the rabbit in the reward chanber 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, the psychologist concluded, 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 same place 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 actually it is not. 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 one 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 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 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 shows 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 said, 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, Ashcraft noted. 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 said.
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!
copyright 2002 David Albert
About the Author:
David H. Albert is a homeschooling father of two, and author of Homeschooling and the Voyage of Self-Discovery, And the Skylark Sings with Me: Adventures in Homeschooling and Community-Based Education, editor of two books on storytelling called The Healing Heart, and co-author of the new book about his Indian parents The Color of Freedom. He speaks all over the country, and if you would be interested in having him speak in your community, contact him through his website at www.skylarksings.com or e-mail him directly at firstname.lastname@example.org
Growing Together Family Learning Newsletter, Vol. 1, No. 1, page 15