Turns out, though, that learning algebra is actually pretty hard and teaching it is very tricky indeed. It’s full of little “threshold concepts” — those hard-to-teach, hard-to-learn, see-the-world-differently-after-you-know-them ideas that frustrate learners and teachers alike. And oddly enough, it is also a topic where the math the student already knows can turn out to be full of deep-seated misconceptions, lying in wait to trip her up and make new learning painful.

Take the humble, ubiquitous equals sign for example.

*C’mon man!* Every elementary student knows what the equals sign means. It shows you where to put the answer, of course.

2 + 5 = __

What does the equals sign mean above? It just tells you to write down a 7 on the right, right? Well, sure. In elementary school.

But when you hit algebra, in middle school, the equals sign matures, as your understanding of it is supposed to mature, into a “balance symbol.” For example, what does the equals sign mean here?

2 + 5 = __ + 4

It means you are supposed to balance the equation and put a 3 to the right of the equals, because 3 + 4 is the same as 2 + 5.

But that’s NOT obvious to every student. And it’s up to the instructor to figure out, if Susie completes the problem above with an answer like:

2 + 5 = 7 + 4

that Susie did NOT make a mistake in arithmetic. Susie does not think that 7 + 4 is 7 or 2 + 5 is 11. Susie has an immature relationship with the equals sign. Or, more correctly, Susie has a misconception about the equals sign — that, frankly, has served her well till now — and it’s Teacher’s job to identify that problem and root it out.

(Later in life, if Susie is lucky enough to take a programming course, she’ll stand at another small threshold and see the equals sign in its guise as the assignment operator:

x = x + 1

Which is likely to be the first of many difficult thunks she has to get past to be a coder. This time, it will be the job of a programming instructor to coach her across the understanding boundary and help her accept that an old friend sometimes needs to be known and understood in a new light. With luck, she and her old pal equals will work through that transition, too.)

There’s actually quite a lot of lot of scholarly research on how preconceptions and misconceptions affect students learning in the sciences and in math. It turns out that the old Tolstoy rule

* All happy families are alike; each unhappy family is unhappy in its own way.*

does not apply. In these unhappy cases, the misconceptions are often very much alike. It is, of course, theoretically possible for a student to come up with a completely unique, idiosyncratic misconception to derail his learning of algebra or chemistry or physics. But, in practice, most students get confused in one of a fairly narrow range of ways.

The hard part, though, with algebra students, is that the misconceptions can go back such a long way, and be buried so deeply, that it can be wicked hard for an instructor without specialized training to understand what’s actually going wrong under the surface.

Attempts were made, in the late 1970’s and early 1980s, to auto-diagnose students’ “procedural bugs” using artificial intelligence techniques. Even with a fairly sophisticated approach to the problem, the resulting software product, charmingly named BUGGY, was quite limited in its ability to actually tease out the effects of multiple, overlapping problems. But, oddly, it did turn out to be a relatively effective tool for helping *student teachers* understand that the mistakes their future pupils were going to make conformed to certain patterns and could be traced back to relatively atomic misconceptions.