by Professor Michael Butler
To the reader,
I am delighted to write a foreword to this lovely book on mathematics for younger children. The work described here reflects a more general Farm School approach, but thinking about how people come to like and be good at mathematics played an important part in developing that approach.
Years ago I was a young professor at UC Irvine, and although I had long been fascinated by the act of learning, this was my first teaching job. Among other things I taught mathematics. The experience was immensely rewarding but unsettling. I thought of math as beautiful, richly ordered, and fun. Most of my students in those required courses appeared to think of it, at least at first, as arbitrary, impenetrable, incoherent, and dull; some of them found it scary.
A few students did not, cheerfully pushing and pulling at a formula, for instance, and asking: What would happen if this part of the denominator were in the numerator? What would happen if I reversed this and that part? What would happen if I made this piece very large or very small? What would be a simpler form or a more general form of the expression? They engaged in this systematic play for the fun of it, but their reinventing or recasting of the material of mathematics also helped them see why something was the way it was; it helped them understand. In fact, the students I started listening to each seemed to carry with them a kind of 'understanding kit.' They had an expectation that math would make sense; they knew when a particular expression or idea did not yet make sense to them, and when it did; and they had developed skills and stamina for getting from the first state to the second, and the habit of doing so. The math they came to know in this way, they owned.
These happy few were regarded by the others (and by most of my colleagues) as having a peculiar knack. There was no shame in not having it; that was just the luck of the genetic draw. Or did the attitude of the rest of the class toward math have to do with the way they had been educated? Their reports of their pre-college math study matched what I found when I started visiting schools, especially elementary schools, and reading texts of that era: my students had been spending most of their time memorizing calculation recipes and learning to run them more or, often, less well.
But that wasn't at all what the kind of people who had discovered the math did. Mathematicians look for and find patterns in formal objects, extend them, seek counterexamples, figure out why the patterns work, and then, finally, publish an account of one way that they work. The last is the public part, but the rest is what they do. Almost none of my undergraduate students seemed to have had much experience with that. There was an odd disjunction between what practitioners did and what schools asked students to do, a disjunction that was deeper and odder the more you looked at it. It was as though we had plucked the fruit "mathematics" for use in schools, peeled it, and fed students the rind instead of the flesh.
Much the same thing seemed to be true in other areas. What working historians did, for example, or scientists was rarely much like what school children did, so it wasn't surprising that undergraduates found it hard to think that way when asked. Again, it was as though teachers had discovered what it was that delighted practitioners, that drew them to their discipline-and in fact kept it a discipline, a thing that people were willing to spend their lives in, over generations-and having found these sources of delight in practice, schools threw them away and taught the residue.
So we and the times and UCI being young, and there being some farmhouses available on the edge of the campus, some of us made a school to redress these wrongs. We wanted a school where children would learn to do what finders and makers do, not just master more or less badly and mechanically some scattered things they had worked out. The students would ideally acquire some of the skills and habits of mind of mathematicians and historians and writers and scientists and artists, and even learn to do what good thinkers do when they are thinking well, independent of a particular practice; and they would learn to find matter of interest in and around themselves, and to develop and sustain those interests, as creators of new art and knowledge must do. These were not the only aims of the school, called the Farm School, but they were central. In this sense we were elitist in our ambitions for children, but populist in our belief that most children could realize those ambitions.
We are all older now, but thanks to the dedicated work of people like Suki Glenn, Susan Carpenter, and Alysia Krafel over the years, the Farm School's approach has endured and evolved. If you also want your children to learn to do mathematics, this book will prove a subtle, wise, pleasure-giving and compassionate guide.
Professor Emeritus Michael Butler
Former Director, UCI Farm Elementary School