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Re: [xmca] a minus times a plus

Thanks for that, Linda! It would be interesting to know at how early an age a child can grasp these various non-math concepts that have been mentioned, and a couple I have added, which utilize negative versus positive:

A) Placement: 1) leave it alone (stay in town), 2) move it away (leave town) 3) put it back (come back into town); 4) keep away (stay out of town) B) Mirror images: 1) reflect it in a mirror 2) reflect a reflection in a mirror (that one sounds hard for young children to grasp - is it? C) Direction: 1) go toward/away, 2) go up/down 3) go in/out 4) go left/right etc. Left and right are also difficult for the young, yes?
D) Value:  1) yes/no  2) good/bad  3) want it/don't want it  etc.

I can see by just making up such a list that I am butting right into the problem of culturally-based so-called "intelligence" with such questions - mirrors, towns, etc. etc. The entire way that we are construing negative and positive at all in this discussion seems to be based on modern cultural concepts and artifacts. The number line is a cultural artifact, as are the clever and intriguing ways changing patterns of symbols in equations are being shown here regarding how they can demonstrate mathematical relations. Very effective to some (me, for example), but very heavy on the usage of text and abstract symbols - very culturally specific. Seeing the variety of ways, symbolic and otherwise that these relations can be learned is intriguing.

My question regarding the above is, in contrast to the above "non- maths" examples of positive/negative, when do children schooled in such things begin to grasp the principles of the number line (addition, subtraction, zero, positive numbers, negative numbers, etc.)? Better put, as a general rule, in modern cultures, is one of these forms of positive/negative thinking really absorbed before the others? Or do they tend to emerge together, so to speak?

My intuition tells me that in modern society there is an integral cultural connection between all these negative/positive concepts, where sooner or later, in various orders, most schooled children in modern society learn their locally normal and class-influenced cultural forms, certainly by adolescence. Mechanically "undoing" something, returning something to its original form, might be a common example of this principle. Is there evidence to back this hunch up - that there is an underlying, ubiquitous, common principle of "positive/ negative" continuums and operations in modern society that gets applied in many different ways?

In contrast, learning the arithmetical/mathematical forms of these positive/negative paradigmatic ideas, such as -8 * +8 = -64, (and we haven't even touched on something like multiplying negative fractions!), is not so ubiquitous - doing so takes a fair amount of specialized training to master, is a potent learning and teaching tool itself as schooling gets more complex, and is therefore fairly indicative of class upbringing and challenges.

My sense is that lots of people crash and burn in math when they get to negative numbers and fractions. Loads of people have math "horror" stories like that. I noticed at Boeing that the one of the clearest differentiators between the various class layers in the factory (engineers vs operators, blue versus white collar, etc.) was arithmetical/mathematical ability and confidence.

My point here is that a common essential method of teaching things like "why" -8 * -8 = +64 to children and adults - and please correct me if I am wrong about this - is to find examples in their everyday lives where they have conceptualized the positiveness/negativeness paradigm, and show them how to transfer that understanding to how to manipulate negative numbers in arithmetic, equations, etc. This suggestion has some implications.

If a child has a family/class background and future anticipations that combine in some way to motivate them to learn arithmetical and mathematical kinds of things - a motivation that is extending to more US females and minorities today than in the past, but is still far behind that of white males, reflecting class, racist and sexist employment, property and power patterns in general society - then that so-motivated child will probably make the effort to grasp these more complex, symbolically-based, arithmetical/mathematical forms of the negative/positive paradigm, and perhaps go from there to algebra, geometry, trig, etc.

If this takes place, I am surmising, then they are going to need to draw on their general knowledge of negative/positiveness, from whatever form that knowledge happens to have taken in their lives - mechanical, visual, symbolic, logical, verbal, etc. My question here is, am I anywhere close to what actually takes place? I've taught lots of working class adults, but not children ...

In a sense, I am arguing, perhaps incorrectly, that at certain points, everyday development leads academic learning, a counterpoint to Vygotsky's paradigm that learning always leads development, but certainly not in opposition to his central argument that learning academic concepts significantly leads everyday development.

I don't mean to sidetrack from this cool discussion of how to explain to kids how to multiply negative numbers! Just tossing out some of the questions this interesting discussion has me thinking about ...

- Steve

On Apr 27, 2009, at 9:13 PM, Linda Williams wrote:

Hello folks. You're inspiring this lurker to post. Here's how I, another literacy person, learned it.

Good guy = + (positive)
Bad guy = - (negative)
Coming to town = +
Leaving town = -

Good guy comes to town (+*+) = +
Good guy leaves town (+*-) = -
Bad guy comes to town (-*+) = -
Bad guy leaves town (- * -)= +

Add a background scenario of the stereotype of the old West in America, and you have the whole picture. I guess this is just a narrative around the "leave it alone/reflect it through a mirror" pattern Jerry Balzano spoke about. I found a narrative like this really helped my 12 year old former students get this pattern. . .

Linda Williams

Linda Williams, Ph.D.
Assistant Professor
College of Education
Department of Teacher Education
Reading Program Area
313C Porter Hall
Eastern Michigan University
Ypsilanti, MI 48197
734-487-7120 , Ext. 2635
Fax:  734-487-2101


On Apr 27, 2009, at 11:53 PM, Duvall, Emily wrote:

My non-math vision.
If I multiply a positive number by 0, I still have 0.
If I multiply a positive number by less than 0, a negative number, I
will still have less than 0... only it will be a more specific amount of
less than zero.
Ergo multiply a positive number by a negative number and it will have to
be a negative result.
P.S. I teaching reading.

-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca- bounces@weber.ucsd.edu]
On Behalf Of Mike Cole
Sent: Monday, April 27, 2009 3:48 PM
To: eXtended Mind, Culture, Activity
Subject: [xmca] a minus times a plus

Since we have some mathematically literate folks on xmca, could someone
please post an explanation of why

multiplying a negative number by a positive numbers yields a negative
number? What I would really love is an explanation
that is representable in a manner understandable to old college
and young high school students alike.

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