A persistent topic in mathematics education is whether to focus on conceptual or procedural knowledge. After reading Kris Boulton’s recent post that argues, “It depends,” I found myself thinking about the disconnect between arithmetic and algebra.

What is needed to understand algebra? The first leap that students need to be able to make is from the concrete to the abstract: The fact that 3 + 4 always equals 7 suggests that, for any two numbers x and y, their sum is always predictable. This is conceptual knowledge, and without it, algebra is very difficult to master. We can certainly do basic algebra problems procedurally, and we usually teach it this way.

Consider x + 5 = 7. The typical way to “solve” this equation in first year algebra is to subtract the same number (5) from both sides, turning an algebraic equation (x + 5 = 7) into an arithmetic one (7 – 5 = ?). But students who have mastered the basic concept of algebra don’t necessarily need to do this step: We’re saying, “There’s some number which, when added to five, yields seven. What is that number?” Students who can go straight to that concept, and realize that the unknown number is 2 without explicitly rewriting the equation, are often punished for failing to show their work, even though their thinking is more efficient.

It seems to me that much of Algebra I education is focused on such procedures, rather than focusing on the key concept of what the purpose of algebra is in the first place. Algebra allows us to generalize relationships between numbers and the effect of operations, with the ultimate goal of finding patterns in those relationships.

In his entry, Kris Boulton points out that we don’t need to understand the concept of what a logadeon is in order to parse and repeat the procedure. This is true. But at the same time, the sample exercise requires a conceptual understanding of how operations work, so we can determine how this specific function works. We don’t need the specific concept, but we need more general conceptual understanding of the premises of abstraction that is the foundation of algebra.