## 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.

Wiggins’ question does indeed seem oddly put, but your answer to it seems solid. The issue about “=” meaning “Hey, kids, here comes the answer to the right of me” is VERY problematic. It impedes understanding of what the real meaning is, at least for many people. I know a lot of students who think that you can’t solve an equation and write the answer as 7 = x, because 7 is the answer, not the “question,” which is, for them, “What is x?” With that idea in the forefront of your mind, you want to write “x = 7”; so students bend over backwards to avoid operations on equations that would result in the variable on the right and “the answer” on the left. When dealing later with solving inequalities where one solution method would be multiplying/dividing both sides by a negative number – a procedure that frequently leads to errors because students don’t remember that this entails changing the direction of the inequality itself, I try to get students to turn, say, -x > 7 into -7 > x by strictly using addition/multiplication and then reading from right to left. This REALLY bothers a lot of them. But I think that’s in no small part a carryover from not wanting to read 7 = x as the same thing as

x = 7, where it should be even more obvious that the symmetric property allows that interpretation (demands it, even).

Forgive me for its being oddly put. I didn’t want to just say “define the equal sign” because that would be too easy and not really probe at the misunderstanding. The aim was to get at the practical value of equivalence. Otherwise kids might define the equal sign but still think that math is just math facts.

I meant “addition/subtraction,” not multiplication.

That’s odd. I thought he was talking about the properties of equality. I teach proof through algebra, not geometry, and so I teach the kids the various properties of equality and creating equivalent expressions through the properties. Thus we discuss the meaning of the “=” signs means that the two sides are bound by the properties.