Eighth Grade Algebra

American education has long been a fashion industry, usually with deplorable results. A currently surging notion is that of making algebra mandatory for all eighth grade students. A few good articles on the subject can be found here, here, here and here.

Algebra is a critical gateway to all of higher mathematics and therefore to science, technical skills, engineering, economics and business. The essence of algebra is abstraction, the manipulation of symbols in place of numbers (we are talking elementary algebra here). In the age of the calculator, this ability is far more useful and general than the merely calculational skills of elementary arithmetic.

Those arithmetic skills are not quite superfluous though. The algorithms learned there serve as models for their algebraic counterparts. Few can understand the abstract without being able to understand more concrete counterparts first.

The objection to eighth grade algebra is that a lot of students enter eighth grade without mastering the seventh, sixth, or even second grade skills of ordinary arithmetic. How can you follow the derivation of the roots of a quadratic equation if you can't compute with fractions and don't know what a square root is? It is pretty natural to doubt that students so unprepared can benefit from algebra.

The notion that all students can benefit from algebra is hard to doubt. But eighth grade algebra is one of those Procrustean ideas that, like no child left behind, will do as much or more harm than good. Some students are quite ready for algebra in sixth grade or earlier. Many others, unfortunately, never will be. Rather than throw sloganized non-solutions at students, we need to deal with the reasons they aren't learning the more elementary skills.

Some of those reasons are largely beyond the scope of the school. Some students are not bright. It's hard to focus on school when mom is in prison and dad is a crack whore. Other reasons are squarely the fault of the schools, and in particular I'm thinking of the last bad idea but one in math education, often called "math investigations." The key notion here is to forget about learning math facts like the multiplication tables and algorithms, and let students discover their own ways of soving problems.

American education has a long standing aversion to memorization in any form, and math is one of the places hardest hit by this particular error. The idea that "rote memorization" is bad is derived from the notion that memorization is not an acceptable substitute for understanding. Well, actually, sometimes it is. Every job requires a certain amount of memorization, and some jobs, like physician and lawyer, require a very great deal. Even more importantly, memory is an essential substrate for understanding. It can be pretty hard to make sense of history if you can't remember which of two interlinked events happened first. Memory is perhaps the most developed and fundamental of human intellectual qualities, and ignoring it is absurd. Behind every great folly there is a germ of truth, of course. Ignoring understanding and only relying on memorization is equally misguided.

From time to time I have worked with elementary school children on math or science. In many cases they are interested in the kinds of mathematical ideas I present (like how to calculate pi), but have trouble following because they don't know elementary addition or multiplication facts (like 7 + 9 or 4 x 5).

Math is the most sequential and hierarchical of disciplines. If you don't know addition, subtraction and multiplication are incomprehensible. If you don't know multiplication and division, you can't understand fractions, factors and factorization, primes, powers, or roots. Once you know those things, you are ready for algebra.