[note: this is a math-centric post but most of the concepts can, on some level, be generalized to other subjects]

There's a curiously inverted quality to the education debate. We spend a great deal of time discussing revolutionary changes to the educational system and almost no time talking about what we should be teaching, as if the proper combination of reforms and incentives can somehow overcome the rule of garbage in, garbage out.

I spent a lot of my time as a teacher thinking about which parts of the mathematics curriculum were good and which parts were garbage and I came up with a list of reasons why a topic might be worth the student's time. The list isn't in order (I'm not sure it's even orderable) but it is meant to be comprehensive -- everything that belongs in the curriculum should qualify under one or (generally) more of these criteria.

1. Students are likely to need frequent and immediate access to this for jobs and daily life.

and

2. Students are likely to need to know how to find this (Samuel Johnson level knowledge).

(These are the only two mutually exclusive reasons on the list.)

3. This illustrates an important mathematical concept

4. This helps develop transferable skills in reasoning, pattern-recognition and problem solving skills

5. Students need to know this in order to understand an upcoming lesson

6. A culturally literate person needs to know this

Most topics can be justified under multiple reasons. Some, like the Pythagorean Theorem can be justified under any of the six (though not, of course, under one and two simultaneously).

Where a topic appears on this list affects the way it should be taught and tested. Memorizing algorithms is an entirely appropriate approach to problems that fall primarily under number one. Take long division. We would like it if all our students understood the underlying concepts behind each step but we'll settle for all of them being able to get the right answer.

If, however, a problem falls primarily under four, this same approach is disastrous. One of my favorite examples of this comes from a high school GT text that was supposed to develop logic skills. The lesson was built around those puzzles where you have to reason out which traits go with which person (the man in the red house owns a dog, drives a Lincoln and smokes Camels -- back when people in puzzles smoked). These puzzles require some surprisingly advanced problem solving techniques but they really can be enjoyable, as demonstrated by the millions of people who have done them just for fun. (as an added bonus, problems very similar to this frequently appear on the SAT.)

The trick to doing these puzzles is figuring out an effective way of diagramming the conditions and, of course, this ability (graphically depicting information) is absolutely vital for most high level problem solving. Even though the problem itself was trivial, the skill required to find the right approach to solve it was readily transferable to any number of high value areas. The key to teaching this type of lesson is to provide as little guidance as possible while still keeping the frustration level manageable (one way to do this is to let the students work in groups or do the problem as a class, limiting the teacher's participation to leading questions and vague hints).

What you don't want to do is spell everything out and that was, unfortunately, the exact approach the book took. It presented the students with a step-by-step guide to solving this specific kind of logic problem, even providing out the ready-to-fill-in chart. It was like taking the students to the gym then lifting the weights for them.

Long division and logic puzzles are, of course, extreme cases, but the same issues show up across the curriculum. Take factoring trinomials. A friend and former boss of mine wrote a successful college algebra text book that omitted the topic entirely. I had mixed feelings about the decision but I understood his reasoning: this is one of those things you will almost certainly never have to do outside of a math class (what fraction of trinomials are even factorable?).

You can justify teaching the factoring of trinomials because it illustrates important mathematical concepts and because it gives students practice manipulating algebraic expressions, but the way you teach this concept has got to reflect the reasons for teaching it. Having students memorize a step-by-step algorithm would be the easiest way to teach the students to answer these questions (and improve their standardized test scores) but it completely miss the point of the lesson.

The point about standardized test scores is significant and needs to be revisited a post of its own. By evaluating teachers and schools on standardized test scores, we put pressure on teachers to treat all subjects as if they fell solely under reason one. This is not a good outcome.

Even more important than how we should teach something is the question of what we should be teaching. Current curricula tend to be broad and shallow with a tragic evenhandedness that often grants the same amount of time to trivial techniques as it does to fundamental concepts. This is bad enough when a class on grade level and everything is going well but it's disastrous when a large part of the class is struggling. There is tremendous pressure under those circumstances to leave the stragglers behind (a pressure that actually increases under many proposed reforms).

In addition to being overstuffed, the current curriculum omits subjects that are arguably more important than most of what we cover. The obvious example here is statistics, a topic that everyone actually does need on a daily basis (as informed citizens and consumers if nothing else). Perhaps even more relevant is what we might call spreadsheet math (customized worksheets, recursive functions, graphs, macro programming). You could also make a case for discrete mathematics, particularly graph theory (I might even put this one up there with statistics and spreadsheets but that's a subject for another post).

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