In talking with my students, I often half-seriously refer to my personal “Top 10” list of useful topics that students don’t remember, or don’t remember well, even though most of them are not difficult (by the students’ own evaluation). I never have actually compiled such a list, but in preparation for this coming year (starting Thursday!) I thought about it a little more. The ones that immediately come to mind are:

- Similar triangles. This comes up just often enough to be important to remember, but just rarely enough so that the students don’t leap to it as a tool. Once it’s mentioned, most students don’t have too much trouble remembering it or applying it.
- The triangle inequality. In my experience, inequalities in general are a black hole for student retention, and I wish we could do better at showing students the power and applicability of inequalities. The triangle inequality is a great example of a totally intuitive fact that kids have no trouble believing, but a lot of trouble using.
- Completing the square. This is a bit different from the previous two topics, as students actually do have trouble understanding and performing this. It’s still a big project for me to figure out exactly why students are so mystified by this—perhaps wiser heads can enlighten me. But like the previous topics, it is definitely something that is very useful, but that students never apply without prodding.
- Trig identities. It’s amazing to me how often students (in calculus, mainly) stare blankly at equations or expressions involving trig without even thinking of trying to apply an identity. It’s certainly not trivial to apply them, since there are many subtle permutations of how they can be used, but they often don’t think to even try.
- Common denominators. Students often get stuck in manipulating rational expressions at a point where they have a sum or difference of fractions. If the context is complicated at all (such as a compound fraction) they tend not to try common denominators.

OK, so it’s not 10 items yet, but maybe I’ll expand it later, or maybe I’ll get comments to flesh it out. But there are some themes:

- I find that students are reluctant to try something if they don’t see how it will completely solve the problem. That applies strongly to (4) and (5) above. Applying a trig identity, or using common denominators, may not get directly to the solution, but it is *something* to try, and may well lead in the right direction. I feel that this points out how important it is to constantly make them do multistep problems.
- All of these items point, of course, more towards failings in curriculum and pedagogy than to failings in the students. I want to know how to teach these topics to students in a way that they understand them as tools to be used in future problems, not as isolated bits of knowledge. I think the solution is to always drive more towards connecting the topics, using them in complex, multistep problems, and not letting them lie fallow for months before they come up again.

I wandered my way to your top ten list via a re-tweet. My comment is regarding completing the square. I have a theory about why students can’t remember how to CTS, and it has to do with the fact that it is nearly always taught in far too abstract a fashion. It is seldom approached even representationally with square Real numbers before they are confounded with square polynomials. This never gives them a hook in their previous schema about numbers or even squares. Euclid showed the process (or at least a variation on the process) very nicely, and geometrically in the Elements. Which brings to mind a question that plagues me in this discussion: Why do we teach 2000 year old processes still, when it is very difficult to justify what good they do a student? Euclid was attempting to summarize everything he thought was known regarding mathematics at the time, his format is what has made his book important, not so much its specific contents any longer.

Great point about doing it with numbers first…I might try that when I re-teach it (which I always have to do, to all but the very best students). No matter how many times I get whacked in the head with “be more concrete!” it doesn’t completely sink in. Thanks!

Ah, and don’t get me started on Euclidean geometry, especially the way it’s usually taught in high school. I’m a geometer by training so I have some pretty deep feelings about how the usual HS curriculum screws up a beautiful subject….