Wednesday, March 01, 2006

The Case For A Math Specialist

So, you're a teacher, and you have a class full of children who are at multiple ability levels in math. What do you do? Well, conventionally, you let the more advanced children rot in the muck of boredom while you give your attention to the students who need it most. Bzzzzzz! Wrong answer.

A better idea would be to have a multi-tiered lesson plan, one where you are prepared to give more advanced work to students who finish the required work early. By "more advanced" I don't mean "work that you will be doing in the future with the class," I mean "work that enriches the students' experience." This is not that difficult to implement, once you have such a lesson plan. Kids who have mastered addition can generate Pascal's Triangle and color in all the odd numbers. Those bored with multi-digit multiplication can be taught to multiply ancient Egyptian style (by doubling), use gelosia arrays for multiplication, or do multiplication in binary. There are also plenty of mathematical yet non-curriculum-related activities: the Tower of Hanoi puzzle comes to mind.

For example, I had an algebra class where the students were learning to factor quadratic trinomials. My lesson plan consisted solely of three pages of quadratic trinomials, sorted into types (two positive roots, two negative roots, one positive one negative, leading coefficient not 1, etc.) All the students were started off factoring trinomials with two positive roots at the board; each student worked by himself. When a student successfully completed one of that type and had had enough practice, he or she was given problems of the next type. By the end of the lesson, the "lowest" students were mastering the two positive root case. There were several students who mastered that in the first five minutes of class; by the end of the lesson they were factoring quadratic form trinomials with non-1 leading coefficients and fractional exponents. (They could handle fractional exponents because in an earlier lesson, while their classmates were learning the exponent laws with whole-number exponents, I'd given them fractional exponents to keep their minds expanding.) Thus I kept the entire class busy and learning, without letting anyone in the class get "ahead" to the point where they'd be bored later on in the semester.

The main reason why such lesson plans are not regularly implemented is not that they're rocket science; it's that they're mathematics. Few, if any, elementary level teachers (and few enough secondary teachers) have enough of a math background that they could even conceive of mathematical activities that lie off to the side of the curriculum's beaten path. How many teachers have done, just to take one example, multiplication in binary themselves? How many can generate Pascal's Triangle or know what happens when you color all the odd numbers? It would simply not be practical to expect elementary school teachers to take the math courses they'd need to learn how to do modular arithmetic and all the wonderful things they could be teaching their students.

That's why schools and/or districts really ought to have a math specialist in their employ. Schools already have reading specialists; why not math? The math specialist would be someone whom teachers could consult for ideas such as these; someone with the necessary training to conduct inservice for teachers, come up with multi-tiered lesson plans of this sort, and generally serve as a resource for teachers to expand their ability to teach math to the above-average students (or even, in the ideal case, to have the specialist teach these students himself).

There is also a lack of curricula in the area of real math enrichment at the elementary level. One of these days, when I run out of things to do (ha!), I'll have more time to work on the one I was designing, a supplemental curriculum for the upper grades that introduces some of the history of math and shows how math was done in ancient cultures.