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Topic: neg * neg = pos; why?
Replies: 26   Last Post: Jul 31, 1997 3:22 PM

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 Boulet Posts: 84 Registered: 12/3/04
Re: neg * neg = pos; why?
Posted: Jul 25, 1997 6:51 AM

You are right about different learning strategies needed by different
people. However, I also believe that the strategies must lead to
understanding. And, I'm not sure that the strategy you described does that.
Let me explain:

The problem starts with the initial labeling of one token "plus" and the
other "minus": why? How is this different from simply writting +1 and -1 on
a piece of paper? In other words, it does not provide a conceptual basis
for the negative, nor the positive numbers for that matter.

Then, a rule is introduced. a "plus" token and and "minus" token make a
zero. And this in spite of the fact that there are two tokens on the table.

>What is artificial about using a number and its inverse to
>produce the additive identity? Conceptually aren't we trying to show that
>multiplication is another way of adding?

The additive identity on this token model is defined (i.e., it is because
you say so). It is artificial in the sense that no argument is provided for
it, it is not deduced as a consequence from a conceptual base. For example,
comparing it to: adding 1 object and removing 1 object leaves us with no
object. A child can understand that. But, here we have two tokens and we
have to "pretend" this represents a zero.

In fact, the whole token approach does not explain anything. It simply
replaces the marks on the paper by tokens:

>>>Let a "+" represent +1 and a "-" a -1: Why? (I hear my old high school
teacher saying:"Because I say so"!)

>>>A zero is represented by the combination, "+ -": Why?

Also, another problem with this model is the connection between this model
and that of the number line. And, that is a very big problem. Take for
example, the measurement of temperature. How to explain -30 degrees celcius
with tokens? In other words, the idea of "relative number", a crucial
historical step in the right direction, is not expressed by the token model.

Consequently, I have no doubt that the token model has helped teachers
remember the rules and definitions of operating with negatives on paper, but
do they understand the underlying concepts? I do not believe so.

Genevieve Boulet
Professeure
Departement d'enseignement au prescolaire et au primaire
Faculte d'education
Universite de Sherbrooke
Sherbrooke, Quebec
J1K 2R1

tel: (819)821-8000 ext. 1207
fax: (819)821-8048
email: gboulet@courrier.usherb.ca

Date Subject Author
7/22/97 Randolph Philipp
7/22/97 Bob Quinn
7/22/97 Marilyn Simon
7/23/97 MARJ@mth.pdx.edu
7/23/97 Boulet
7/23/97 Mark Klespis
7/23/97 Susan E Enyart
7/23/97 Loren Johnson
7/23/97 Boulet
7/23/97 Boulet
7/24/97 MATHSTUFF@aol.com
7/24/97 Loren Johnson
7/24/97 mark snyder
7/25/97 Boulet
7/25/97 Stuart Moskowitz
7/25/97 Frances Rosamond
7/25/97 Boulet
7/26/97 Boulet
7/28/97 Raymond E. Lee
7/28/97 Boulet
7/29/97 mark snyder