## 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:

- The equal sign means that the two things on either side have the same value as each other.
- We often teach in ways that lead students to think that the equal sign means
*and now write the answer*. - 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.