[Numeracy 159] Concrete physical reality vs. negative integers
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Fri Feb 12 19:29:10 EST 2010
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Greetings Brooke and everyone,
Possibly the overarching reason negative values so often present a pedagogic challenge is because they do not exist in physical (concrete, material) space. Units of measurement have no innate existence. On the other hand, concrete objects most certainly do. When we say one pound of apples, for example, we do not see the "pound," but rather a number of apples. We can lift those apples and get a sense of their weight. The attribution of a measurement of weight, however, is an abstraction. Similarly, when we say $5.00, we can see that amount of money in any combination of bills and coins, even if its meaning or significance is subjective.
On the other hand, we can count 5 apples, share them among five people, after which none (zero) will be left. There are no negative apples, except in the abstract mathematical sense that if two more people asked us for apples, we would be 2 apples short. Those 2 apples are positive apples.
When we are in debt, our understanding of its amount is an absolute value (one that knows no positives or negatives). Yes, the balance sheet will reflect a negative balance, but the amount we owe is not negative (if we owed, for example, negative $10, we would be $10 in the black (per the double negative rule which serves well in English, vs., say, Romance languages, to establish a mathematics-language link).
Using bank accounts to teach negative values (as the abstraction that they are) works well, because most of us have bank accounts, or at least budgets that mirror or can mirror our income and expenses. We must understand those budgets in positive or absolute terms in order to grasp when it is that we may be spending more than we have and why. That's why I find coordinate graphs so helpful - and also a great tool to build cognitive skills.
I'd be delighted to be led to view matters differently and in new ways - that's wherein learning occurs. We can always choose to agree to disagree. However, before we do so, I believe we need to agree on our understanding of premises and terms first for the sake of exploring current and perhaps new teaching and learning practices. Agreeing to disagree at this juncture doesn't appear to be useful for the purposes of a discussion list.
Michael A. Gyori
Maui International Language School
From: "Denney, Brooke" <denneyb at cowley.edu>
To: numeracy at nifl.gov
Sent: Fri, February 12, 2010 1:42:47 AM
Subject: [Numeracy 155] The double negative language - math link
You and I will have to agree to disagree on this issue.
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