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.

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I agree with everything you say about why you are better than (almost all) high-school students but I think you’ve left out some important aspects about not only what you know and can do as well as HOW you know what you know. First, consider that you haven’t included anything about geometry or spatial/visual reasoning. Surely, there are a host of concepts and techniques you’ve acquired in your life as a doer of math that include geometry, visualization, etc. For example, anytime you confront a problem (or should I say a task to keep from raising the issue of what counts as a problem?) involving right triangles you immediately are able to draw on a wealth of knowledge about right triangles that most high schoolers don’t have. Second, there is the matter of how you know what you know. In his interesting book “Adventures of a Mathematician,” Stan Ulam makes a comment to the effect that truly good mathematicians have very well organized memories. He wrote his book more than 35 years ago and he was no cognitive scientist but I think he was onto something. Simply put, experts’ knowledge is organized differently from non-experts. So it isn’t merely a matter of you knowing more than HS kids, you also know know it in a qualitatively different way. You notice features of problems that they don’t notice, you ignore things they pay attention to, etc. Third, there is the matter of the “habits of mind” (I put quote marks here because so many folks are talking about HoM these days and in so may different ways). One of the HoM that you undoubtedly have developed is to draw pictures when you don’t know what to do. You’ve learned that by doing this you gain a better sense of what the situation you are confronted with is about. How did you pick up this HoM? Who knows? But you did pick it up and most HS kids haven’t yet. Another HoM is being skeptical of your answers. Another is . . . . you get the idea. What I’m getting at is that what makes you better isn’t just that you know more algebra and number concepts. It’s much more than this. A lot more.

Thanks for getting the discussion started.

Ouch. After reading your comments, I’m embarrassed at the shallowness of my thinking. I have a way of rushing in where angels fear to tread. Still, I don’t think I missed as much as you say!; let me try again.

It seems to me the main problem is the word “why” in my title. The _skills_ a competent math teacher has are one thing, but, as you say, there’s also the issue of how they acquired them, which is far more complex. I didn’t intend to address that question, but only to make a list of what the skills are, and I really should have said so. However, the fact that students’ knowledge is organized differently, so that a decent teacher sees things they don’t notice and vice-versa, is obviously critical. In his Pulitzer-Prize-winning book _Goedel, Escher, Bach_, Hofstadter discusses (p. 286) studies by psychologist Adriaan de Groot of “how chess novices and chess masters perceive a game situation”. In brief, from five-second glances at the board, the masters could reproduce an actual position from a game far more quickly than the novices. “Highly revealing was that masters’ mistakes involved placing whole _groups_ of pieces in the wrong place, which left the game strategically almost the same, but to a novice’s eyes, not at all the same.” Furthermore, in the same experiment but with pieces randomly assigned to squares on the board, the masters did no better than the novices. I’d bet a large sum (or product 🙂 ) that math experts would be much better than novices at reproducing complicated expressions or equations, but not at reproducing random strings of mathematical symbols.

Regarding Ulam’s claim about the organization of good mathematicians’ memories, I’m not a cognitive scientist, either, but I’ve been on the fringes of cognitive science for many years, and I have no doubt that he’s correct; that claim probably applies also to chess masters, or masters of almost any discipline.

And yes, I learned long ago to draw pictures to clarify things I don’t understand. Mathematical HoMs are important!

Thanks for pointing out the problems in my post. I’d love to see someone else’s list of skills — or comments on how they acquired those skills, though the latter seems like an overwhelming question.