[bslorence]

> there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables.

Sure, but imagine learning multiplication tables without having any idea what it means to "multiply"; literally just memorizing sequences of symbols, without ever looking at piles of coins or whatever.

Multiplication is so basic that this is hard to imagine, but I'm sure that, say, difference-of-squares rules feel like this to most beginning algebra students -- and for most people that probably never changes.

I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.

[BeetleB]

> I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.

Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretation is merely a manifestation of it that just happens to work in some domain.

As an example, the formula is equally valid for complex numbers, but I doubt you'd get there from Euclid's. No doubt some geometric interpretation can be found for that as well, but then I'd pick some other algebraic field where it's true and you'd have to search yet again for a geometric interpretation.

[bslorence]

Fair enough. Maybe I should have just said that it's possible to give concrete illustrations of basic algebraic concepts, and that doing this would probably help some students learn algebra, and might help others retain it.

But for whatever reason this is generally skipped in middle/high-school algebra.

[BeetleB]

I'm torn. I think it's always good to show these - it certainly makes the subject more interesting!

At the same time, if one is to use algebra for future studies/work, one really needs to be able to manipulate those symbols in the abstract, without feeling a need for some deeper understanding. I can see teachers not wanting to deal with "But what does that really mean?" for every detail in algebra.

[bslorence]

This goes to the question of why anything other than basic arithmetic is compulsory. The famous 10th-grader's whine "what are we going to use this for", which infuriated my own 10th-grade Algebra 2 teacher, and even made me roll my eyes at the time, is actually a fair question when algebra is taught as abstractly as it typically is.

I think you've explained exactly why here -- because the emphasis on abstract manipulation presupposes that this is useful for something that we need to get on to. But that's just false for almost all students. And yet they're required to take the class to get a diploma.

My vote would be to treat any math beyond basic arithmetic as a liberal art, and do a lot less of it in compulsory curricula, but spend a lot more time on deep understanding. This would benefit everyone. The current approach pretends that everyone in the class is going to be a certain kind of engineer or scientist some day.