>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 implied in your response.)
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