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Re: Ratsense
Posted:
Aug 17, 2013 1:21 PM



On Sat, 17 Aug 2013 01:53:54 0600, Robert Hansen <bob@rsccore.com> wrote:
> This type of reasoning shows up often in education science. Arguments > that young children understand advanced concepts. I think it took me a > year to unravel the >basic principle behind this scam and why it is > effective. But first, why did I question these arguments to begin with? > Simple. My basic problem with all of these >arguments is  If young > children understand these things so well then why don't they understand > them when they get to algebra? Some educationalists simply ignore >that > question, like a comedian ignores a heckler, while some educationalists > offer a conspiracy theory. They propose that there is some dark force > that stifles that >innate understanding. Then of course I ask "Why would > anyone do that?" and then they resort to the heckler approach.
You are scamming yourself here. "An intuitive understanding" (the words Dehaene used) isn't the same thing as a "formal understanding". It is the step from intuitive to formal that kids have trouble making.
Most kids can tell you that it doesn't matter which order you add a pair of numbers in. There's the intuitive understanding. And you haven't dealt at all with Dehaene's observation that kids, by the age of five, already intuitively select the larger of two numbers to begin counting from when they add by counting. If that doesn't demonstrate a grasp of commutativity, I don't know what would.
But it's a big step from that simple understanding of the commutativity of addition to an understanding of the statement "For every a and for every b, a + b = b + a" \it{and its connection with the simple, intuitive statement.} One of the conceptual difficulties here is the formal notion of a fixed, but *unspecified,* number. That notion appears twice in the "For every" statementpossibly even three times, if we think about the sum itself. That notiona fixed but unspecified numberis one of the biggest hurdles kids have to pass in going from arithmetic to algebra.
Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver
<http://rowdy.msudenver.edu/~talmanl>



