So why is the person whose mathematical ability I’m most familiar with (me) better at high-school-level math than nearly all high-school students I’ve encountered? Knowing the answer might help me help the other students! I think the rather short list below pretty much explains things. (Of course I’m better in a lot of cases because I already know exactly how to solve the problem, but of course there’s more to it.)
I’m certain I’m better at:
1. Understanding “word problems”, i.e., translating into mathematical form the kind of verbal descriptions of situations that commonly appear in textbooks and on standardized exams. However, it’s not clear how much this has to do with math skill as such, as opposed to reading skill; Jo Boaler has some interesting comments on this point in What’s Math Got to Do With It?.
Outside of translating words to math, I’m better at:
2. Factoring small integers, especially recognizing perfect squares.
3. Seeing simple algebraic relationships (e.g., in equations), simplifying expressions, etc. To a considerable extent, this is just a matter of being really comfortable with mathematical notation.
4. Intuition for what’s reasonable — intermediate values and results: among other things, estimating numerical values.
5. Avoiding specific common mistakes, e.g., “everything is commutative”, confusing what works for addition/subtraction vs. multiplication/division.
Is this a good list? How would you describe why you’re better than students?
Now, accepting my list as a starting point for discussion, what can be done to help students with these skills? Or are all of these skills truly important, and if not, what can be done to reduce their effect on students’ grades and mathematical confidence?
To keep this post short, I’ll just mention one item in my list that strikes me as both not very important and easily mitigated: #2. I discovered last year that many of the students in both of my Algebra II classes — both honors classes, in fact — were struggling on tests with what I thought were easy problems simply because they didn’t recognize small perfect squares as such. When I realized that, I started putting the above table of Powers of Small Integers on the whiteboard, and that seemed to do the trick.