There is more than sufficient useful mathematics, and I don't mean merely
arithmetic, to educate our students beyond any level we could hope for. We
are in the mathematics game because we like solving abstract puzzles that
have no particular meaning. There is a whole world of people out there who
don't share our passion, nor should they. I don't require my students to
master my other passions, karate for instance, just because I have. They
want tools they can use and who are we to say they want the wrong thing and
what we want for them is better. As the educational free market opens up,
we'd better start listening to the customers if we want to stay in business.
Doug Cardell
----- Original Message -----
From: Bob Leibman <bleibman@io.com>
To: George Alexander <galexand@UWC.EDU>; <mathedcc@archives.math.utk.edu>
Sent: Monday, November 08, 1999 8:51 PM
Subject: RE: [MATHEDCC] Marilyn Vos Savant
> I began this reply earlier, but I think it is not entirely redundant with
> some of the most recent replies, so I will go ahead and add my thoughts.
>
>
> At 10:14 AM -0600 11/8/99, George Alexander wrote:
> >I like to address this question about multiplying two negative numbers by
> >examining what we mean by the following questions:
> >
> >"You buy 5 books priced at $10 each. How much did you spend?" Most
students
> >will readily agree that this asks for the calculation (5 books) x (-$10)
> >= -$50. The book price and the total cost are negative numbers because
they
> >represent expenses. The number of books is positive because the buyer is
> >receiving the books while spending the money.
> >
> >Now what about the opposite statement:
> >"You sell 5 books priced at $10 each. How much did you earn?" Note that
the
> >book price remains the same (negative $10!), so changing "buy" to its
> >opposite "sell" introduces a second negative sign. The number of books is
> >now negative since the seller is giving up the books. Thus we have the
> >calculation
> >(-5 books) x (-$10) = +$50. We know the result must be positive because
it
> >represents income. Most people will automatically change the double
> >negatives and only write 5 x 10 = 50. This verifies that the product of
two
> >negative numbers must be positive in a common sense argument that
students
> >find convincing.
> >
> >
> >George Alexander
> >University of Wisconsin Colleges
> >Developmental Math Coordinator
> >UW Rock County
> >2909 Kellogg Ave.
> >Janesville, WI 53546-5699
> >(608) 758-6627
>
>
> First, with respect to the example given above, I am unconvinced. There
are
> 5 books (+5) whether they are coming or going. Moreover, you are not
really
> dealing with the books, but with the money which was handed over in the
> other direction. Outgoing = expense = negative. Incoming = revenue =
> positive.
>
> Buying 5 books @$10 means there is an outward flow of $10 for each of 5
> books, hence 5(-$10) = -$50.
>
> Selling 5 books @$10 means there is an inward flow of $10 for each of 5
> books, hence 5(+$10) = +$50
>
> I see no double negative here.
>
>
> To me, a more reasonable example (although I'm sure there might be better
> ones) is to consider the activies involved in keeping a ledger of some
sort
> in which debits are negative and credits are positive. The activity of
> entering might be considered positive while that of removing is negative.
>
> This produces a sort of 2x2 matrix of possible activities in which the
> entries are the net effect on the account of multiplying the sign of
> (enter/remove) with the sign of (credit/debit):
>
> credit debit
>
> enter (+,+) = + (+,-) = -
>
>
> remove (-,+) = - (-,-) = +
>
> Thus entering a credit of $10 has the same net effect on the value of the
> account as removing a debit of $10.
>
> If we can equate taking the "opposite of" with multiplying by -1, then
> there are many real life examples of two-state systems in which taking the
> opposite twice returns you to the original state, -e.g., a light switch
> (particularly in those situations where you can control the light from
> either switch).
>
> I think it is harder to find simple real life examples for products like
> -3(-4).
> Clearly they are not immediately obvious or the question wouldn't arise.
>
>
> I would like to address the question somewhat differently. I am somewhat
> bothered by the need to justify everything in terms of its immediate
> application to "real life," whatever that is.
>
> First of all, would a business calculus problem involving limited growth
> modeled by an equation of the form P = L - ae^(-kt) qualify as "real
world"
> whether or not everyone is going to want to be able solve this? Looking at
> an example in some book of the form
> S = 5500 - 5000e^(-.04t), letting S = 2750, the solving leads to t = (ln
> 0.55)/(-0.04) = approx 15 days. Of course, real life is probably much more
> complicated.
>
>
>
> It seems to me that in teaching math we are also teaching something about
> the different number systems and how and why they arise. It seems to me
> that it is worth pointing out to students that math often takes basic
> simple ideas which are clearly real life and available to us at an
> intuitive level, abstracts them, and then "pushes" that abstraction as far
> as it can - way beyond the bounds of intuition.
>
> Thus, if multiplication arises as repeated multiplication, n*m meaning the
> sum of n numbers all of which are m (or more concretely, the total number
> of objects when you combine n piles of m objects each.) It is still
> reasonably intuitive to talk about 3(-400) once we have given some
concrete
> meaning to -400. (The total debit due to 3 identical debits of 400, or the
> total descent in feet achieved by descending in three stages of 400 feet
> each.
>
> It is less obvious what -3(400) or even -400(3) means. We tend to justify
> this on the basis of extending the commutative property, or by showing the
> pattern in a sequence of products. Finally, to justify (-3)(-2) most books
> now look at sequences of products. Of course, if we accept that -3
= -1(3),
> etc.
>
> Likewise, when we move into negative exponents in Elem. Algebra (or even 0
> as an exponent) we are moving from a basically intuitive notion (shorthand
> for repeated multiplication) in which the exponent represents the number
of
> identical factors to a purely formal definition which is intended to
expand
> the notion (and notation) in a way which is consistent with the
> already-established rules of exponents and also the results of working
with
> ratios of exponentials with the same base but higher exponent in the
> denominator.
>
> Just some thoughts.
>
> Bob Leibman
> Austin Community College
>
>
>
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