Algebra and Arithmetic Education

March 16th, 2008 | View Comments

Jonah Lehrer writes this about algebra education:

Why is algebra so hard to learn? Because it’s so abstract. No other high school subject is as disconnected from the real world. When students open their algebra textbook, they enter into a world of pure ideas, with page after page of elusive equations and intangible theories. In fact, supporters of mandatory algebra classes tout this as one of the subject’s benefits: it is often a student’s only introduction to abstract thinking.

Well, sort of. Yes, algebra is abstract*, but I think that’s a relatively small part of the problem, and I question the premise that algebra is significantly more abstract than say, analyzing the symbolism of Dante’s Inferno.

I’m going to argue that algebra is so hard to learn because students never really understand arithmetic to begin with. Math is inherently abstract**. If you only learn to do math in concrete terms, in the way that most elementary math is taught, you are not really learning how to do math. You are learning to be a calculator (and there’s no reason to learn to be a calculator, because we have perfectly good ones already). But if you really understand the arithmetic, then algebra should be pretty easy.

In fact, students are doing simple algebra way before junior high.

5 + 7 = ?

This is something we typically see in elementary school math.

5 + 7 = x. Solve for x.

This is something we typically see in early algebra. It is the exact same problem.

When you do long multiplication, you are really applying the distributive property.

When you work with decimals, percentages, fractions, and ratios, you are really working with linear functions.

And so on and so forth.

I think that putting the blame on algebra’s abstractness is really selling students short. I really think the underlying problem is the quality of mathematics teaching in the early grades.

An elementary math teacher is responsible for developing students’ number and operation sense, which sets the stage for all future mathematics achievement. An elementary math teacher must have what Liping Ma calls a “profound understanding of [elementary] mathematics” and be able to see the connections between basic arithmetic and higher math. However, elementary math teachers in the US do not need to demonstrate this level of knowledge to teach math, and generally do not have it. Subject-specific education and certification requirements for math teachers don’t appear until middle school, and by then it’s probably too late.

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*I also question Jonah’s use of the word “abstract.” Abstract doesn’t really mean something is a “pure idea” and that it’s divorced from the concrete world. Abstract means that it transcends concreteness—it can be made concrete, but it doesn’t have to be. Take linear functions, for example. An apple can only be an apple, but a linear function could be used to represent:

Distance (d = rt)
Mass (m = dv)
Force (f = ma)

etc., all of which are common real-world phenomena. Or you can just write out the standard form of a linear function, y = mx + b, and use that in whatever context.

This does, however, get at the silliness of treating math and science (well, mostly physics) as two separate subjects.

**Numbers themselves are abstract concepts. You can have three of something, but as in the linear function example above, the number three is just a symbol and isn’t attached to any particular concrete referent. Three apples, three oranges, three blue-haired aliens—all have the quality of “three-ness”, but none are actually the number three. If you want to get technical, they’re subsets of three—the mathematical definition of the number three is the set of all sets that contain three elements.