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Topic: Standards
Replies: 1   Last Post: Jul 10, 1996 11:10 PM

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 Kreg A. Sherbine Posts: 26 Registered: 12/6/04
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 vice-versa. 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
28-people-and-four-in-a-car-so-how-many-cars 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

Date Subject Author
7/10/96 Kevin Maguire
7/10/96 Kreg A. Sherbine