The upcoming February 2012 issue of the Notices of the American Mathematical Society contains an interesting essay titled "A Modest Proposal" by Alan H. Schoenfeld, a Professor of Education and Mathematics at UC Berkeley. He writes of some "horror stories" of how kids being taught mathematics respond to questions that don't match the usual pattern:
Reusser  asked ninety-seven ﬁrst- and second-grade students the following question:
There are 26 sheep and 10 goats on a ship. How old is the captain?
More than 3/4 of the students “solved” the problem, obtaining their answers by combining the integers 26 and 10.( Note: The cited paper is K. Reusser, Problem solving beyond the logic of things, Instructional Science 17 (1988), 309–338. )
Schonfeld concludes that kids are not being taught how to think correctly in classrooms and advocates teaching in such a way that they come to view mathematics as providing "a set of sensible answers to a set of reasonable questions".
This is certainly commendable, and I like the example he gives of how to make the arc length formula simple and intuitive. But, as I've argued repeatedly on this blog, the problem runs deeper.
Over-schooling, by its very nature, so constrains young children's thinking that they are unable to see outside the patterns that they have been drilled on and made to repeat. They need to learn in a more carefree environment, ideally one where they have plenty of time to think of many "reasonable" questions and possible solutions on their own through their natural habits of play and curiosity-driven exploration, free of the overwhelming pressures of adult expectations.
Here is a relevant piece from John Holt's foreword to his classic "How Children Fail"; the italicized words in brackets are mine, connecting the quote to the above setting:
...there is a more important sense in which almost all children fail: Except for a handful, who may or may not be good students, they fail to develop more than a tiny part of the tremendous capacity for learning, understanding, and creating with which they were born and of which they made full use during the first two or three years of their lives.
Why do they fail?
They fail because they are afraid, bored, and confused.
They are afraid, above all else, of failing, of disappointing or displeasing the many anxious adults around them, whose limitless hopes and expectations for them hang over their heads like a cloud. [Every question posed to them at school has a clear and definite answer; and failure to answer disappoints the teacher. So they know once they are asked a question, they *must* set out to solve it.]
They are bored because the things they are given and told to do in school are so trivial, so dull, and make such limited and narrow demands on the wide spectrum of their intelligence, capabilities, and talents. [Face it. At the end of the day, solving word problems about addition are simply not as interesting as any of the dozen other things related to playing and exploring and being with friends that little kids would rather be doing.]
They are confused because most of the torrent of words that pours over them in school makes little or no sense. It often flatly contradicts other things they have been told, and hardly ever has any relation to what they really know — to the rough model of reality that they carry around in their minds. [If much of what is presented to them in their workbooks or by the teacher doesn't make sense anyway, most of the time, why should this puzzle be any different? What all this confusion sadly does is to breed a special form of intellectual laziness --- it's easier just to play along and mindlessly apply some known formula, rather than say over and over again, "I don't understand".]I asked my eight-year old the same question. The amazing school he goes to leaves him to his resources almost entirely. I do teach him a bit myself from time to time, mostly because of my own interest in mathematics, but I try hard not to over-do it. And when we do talk about math, our discussions are very much in the spirit of Schonfeld's "reasonable questions" approach. So I was quite sure how he would respond.
Nevertheless, it was with some relief that I heard him say, "Wait, that doesn't make sense! What does that have to do with anything?"
This post seems as good place a place as any to put in a pointer to one of my favorite songs: