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Topic: What are the "basic" facts?
Replies: 20   Last Post: Jul 6, 1995 8:34 PM

 Messages: [ Previous | Next ]
 Norm Krumpe Posts: 53 Registered: 12/6/04
Re: What are the "basic" facts?
Posted: Jul 1, 1995 2:11 AM

>How can students understand the above concepts if they don't know their times
>tables?
>By pushing buttons on a calculator so that the magic black box can give them
>some numbers?
>
>marge
>

Imagine asking a child if, in the multiplication of two numbers, it matters
what order the two factors are written, and imagine that the child says,
"No, it doesn't matter what order the numbers are in."

Now ask the child, "How do you know this?"

Suppose the child responds, "Because in my times tables, 7 x 8 = 56, and 8 x
7 = 56. And because 7 x 4 = 28 and 4 x 7 = 28. etc." Is this substantial
evidence that multiplication is commutative? Certainly not. Even if the
child continues responding with, "and besides, it even works with big
numbers, like 17 x 23 = 391 and 23 x 17 = 391." This still does not
constitute substantial evidence, and I really don't care at this point if
the student demonstrated this multiplication on paper or with a "magic black
box." (BTW, many students don't approach calculators with the mysticism you

Suppose, on the other hand, the student says, "Here, let me show you. See,
to show a multiplication like 7 x 8, we can draw a rectangle on graph paper
that measures 7 centimeters across and 8 centimeters tall. It has an area
of 56 square centimeters. Then, if I wanted to show 8 x 7, I could just
turn my drawing sideways, which doesn't really change how many square
centimeters are in it. So, changing the numbers in a multiplication is just
like turning a rectangle sideways. It doesn't change the area, so it
doesn't change the answer in the multiplication either."

Granted, this is hypothetical, and neither proof is presented with the rigor
that we would expect in an abstract algebra course, but I maintain that in
the second scenario, the student demonstrates a truer understanding of the
commutativity of multiplication than the student in the first scenario. I
also maintain that the student in the second scenario can demonstrate this
understanding *without knowing the times tables*.

You asked the question, "How can students understand the above concepts if
they don't know their times tables?"

I pose to you this question, "How can students really understand (not just
memorize) the times tables if they don't understand some of the 'basic'
properties of multiplication?"

Why make students memorize the entire 12 by 12 multiplication table when, if
they understand the commutativity of multiplication, they can eliminate the
need to memorize such facts as 8 x 7 = 56, because they already know that 7
x 8 = 56.

Once again, I want to stress that IMHO, yes, knowing the "times tables" can
be helpful, but there are concepts more basic than these which should be
understood before learning the times tables.

Norm Krumpe
Indiana University of Pennsylvania
Indiana University at Bloomington

Date Subject Author
6/21/95 Norm Krumpe
6/22/95 Norm Krumpe
6/23/95 A. Karassowitsch
6/23/95 Norm Krumpe
6/29/95 Janet V Smith
7/1/95 Arthur Howard
6/23/95 Norm Krumpe
6/23/95 Norm Krumpe
6/23/95 Norm Krumpe
6/30/95 Norm Krumpe
7/6/95 Chi-Tien Hsu
7/1/95 MCotton@aol.com
7/1/95 Kevin J.Maguire
7/1/95 Norm Krumpe
7/1/95 Norm Krumpe
7/2/95 MCotton@aol.com
7/2/95 Norm Krumpe
7/2/95 Steve Means
7/4/95 MCotton@aol.com
7/4/95 Norm Krumpe
7/6/95 Eileen Abrahamson