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Re: Standards
Posted:
Jul 10, 1996 11:10 PM


Kevin alluded to a phase between intuitive development of concepts and formal expressions of algorithms. I wouldn't place this phase, whatever it's called, *between* these two "extremes" so much as I would describe it as (a) overlapping both significantly, and (b) serving to bring them closer together.
In other words, it might be a reasonable goal (and one that I think is consistent with the philosophy of the Standards) to try to make formal expressions of algorithms a significant part of intuitions about concepts, and viceversa. After all, this is how many professional mathematicians behave: ax^2 + bx + c has meaning not only as an indication of action, where appropriate, but also as an indication of relationships among quantities, including the recognition that at least one of these quantities is unknown.
It looks like I'm returning to a favorite theme: the relationship between *structural* ways of seeing and *procedural* ways of seeing. Conjecture: intuition is intricately related to structure, and algorithms are intricately related to procedure. (I'm pretty sure the second part is true.) That's why kids can solve problems like the 28peopleandfourinacarsohowmanycars one via all the different ways mentioned by Ron and others: they understand the structures involved in having people in cars. By extension, the *processes* which we could help the kids see with algorithms could serve to represent the *action* of *putting* people in cars, which is, of course, obviously (even for kids) intricately related to the *result* of *having* people in cars.
These are somewhat delicate distinctions, but I think they really get to the heart of a lot of understanding of how kids think and how adults think.
To summarize, Kevin's notion of abandoning intuition too early, and indeed at all, is as he says dangerous and discouraging. But the value in having kids be able to generate and use algorithms requires that we teach them to organize their intuitions and to check them efficiently and, when necessary, to abandon them and, eventually, to create new types of intuitions. Hence we need to explicitly, carefully, and fully develop the relationship between intuition and algorithm, recognizing that each can provide a rich supplement to the other in students' understandings.
Kreg A. Sherbine  To doubt everything or to believe Apollo Middle School  everything are two equally convenient Nashville, Tennessee  solutions; both dispense with the sherbine@math.vanderbilt.edu  necessity of reflection. H. Poincare



