Wiggins questions #2

Question 2

“Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

This question is a strange one. It really isn’t how I would define problem solving, and I certainly wouldn’t include equality as a major component underlying problem solving.

Nonetheless…

I suppose he is getting at the idea that expressing equations in equivalent forms sometimes reveals different details of a problem.

For instance, I have created a new measure for cylinders: the circumradial measure. You add the radius and height. Then multiply this sum by the circumference.

In exploring this measure, one might end up restating this formula in equivalent terms, as:

This is more recognizable as a formula for surface area of a cylinder. The form of the equation affects how we think about the relationship it expresses.

What does the equal sign mean?

This is an important question. There is lots of research about it (CGI folks have worked on it, for instance). Three quick points:

1. The equal sign means that the two things on either side have the same value as each other.
2. We often teach in ways that lead students to think that the equal sign means and now write the answer.
3. You can’t really understand much about algebra with the conception that (2) fosters. You need (1).

Finally, there are deep ideas underlying the equal sign. Equivalence is the mathematical way of talking about sameness. Stating the meaning of sameness precisely in mathematics turns out to be tricky and interesting work, and is a foundation of modern algebra.

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Wiggins questions #1

Math folks online have been all atwitter (heh) about a recent post by Grant Wiggins on conceptual understanding in math. Within that post (which I have not read in its entirety for reasons to be explained later), he proposed a series of questions that we should offer students as a way of opening our minds to what conceptual understanding means in mathematics.

Max Ray expressed a wish for some math ed bloggers to answer these questions in writing. I am obliging. One question at a time. One per week. I have not read the post so as not to bias myself.

I reserve the the right to critique the questions along the way.

Question 1

“You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)

Fact families.

Division is defined in relation to multiplication. For every one multiplication fact, there are two division facts.

3•2=6 is matched with 6÷2=3 and 6÷3=2.

Zero is a special case. 0•2=0, 0•5=0, 0•a=0 for all possible values of a.

This is no problem for multiplication. But it is a problem for division.

0÷0=2 would be a fact from the fact families. 0÷0=5 is another one. 0÷0=a for all possible values of a.

That is, 0÷0 can equal anything. And if it equals anything, it actually equals nothing. So 0÷0 is undefined.

More generally, though, 2÷0, 5÷0 and a÷0 for all possible values of a are problematic. Let’s say we decide that a÷0=12 (and let’s say that a isn’t 0, since we took care of that case already). Then the fact family tells us that 0•12=a. But 0 multiplied by anything is 0. So a÷0 can’t be 12. But it can’t be anything else either. So a÷0 is undefined.

Conclusion: We cannot divide by zero for two reasons.

1. Division is defined in relation to multiplication, and
2. Zero has a special role in multiplication: 0•a=0 for all values of a.

We can use intuitive notions to establish that division by zero is a strange beast, but we can’t really firm up why without these more formal mathematical ideas.