Bad advice

July 30, 2010 at 8:12 pm (Blogs, School)

I was told recently by my (substitute) math teacher that I wouldn’t learn anything from watching him solve problems.

That statement was false. I have only really been studying math for a year now, but one thing that I have learned is that anytime I go to a teacher with a question, I will inevitably spend more time trying to convince them to answer my question rather than the answering question that they assume that I am asking.

I am a good student. I don’t cut corners, ever. I do all the work that is required of me, and if that’s not enough to give me a strong grasp of the subject, I study even more. This work ethic is especially useful in math, because math, as I see it, is a combination of two things. I’ll call them mechanical skills and conceptual skills. By mechanical skills I mean the ability to perform operations on a problem. Examples of mechanical skills are addition, subtraction, multiplication, distribution and so on. Mechanical skills were once conceptual skills. Meaning at one point, someone has to show the student how and why distribution works. At a certain point however, the student has used the distributive property so many times that it is no longer an action that requires any conceptual thought. Building math skills seems to follow that pattern more or less to a tee: demonstrate a concept, use it until it no longer has need for the concept to be recalled and used, and then move on to the next concept. Mechanical skills become the means to solve a problem. Without a solid base of mechanical skills, the student has no hope of ever solving a problem, except by accident.

If mechanical skills are the “hows” of problem solving, then the conceptual skills are the “whats and whys.” In order to be able to apply mechanical knowledge, a student has to recognize what they’re looking at and why the problem is relevant to what they already know. This is easy when a problem asks to add two and four, but as math becomes complex, the whats and whys of the problem tend to be well hidden.

This is where the communication between my teachers and I always breaks down. When I come to them with a problem, I literally never have the question, “how do I solve this problem?” I am always asking, “why can’t I solve this problem.” In other words, “what pieces of the puzzle am I missing that are keeping me from making the necessary connections here?”

Here is how the conversation usually goes down: I say, “Hi professor, I’ve been having problems with these types of problems. Like this one. Can you give me some insight?” They say, “sure, let me see your work.” and I show it to them. Then, they look at my work and become stumped, “you got the right answer.” I say, “well yeah, I worked on it for a very long time and I got the math to work out, but if you changed the problem just a little bit I’d be at square one.” This is usually followed by either a blank stare, or an attempt to show me all of the individual operations that took place, which of course, I already knew how to do.

I’ve started to think that this problem may not be the teacher’s individual fault, but perhaps the collective student’s fault. So few students requisite work in a math class that I think teachers may have forgotten how to teach student to actually solve problems, rather than just perform operations. As for myself, the closest thing that I’ve seen to what I think I need is watching a skilled mathematician think their way through a problem. It seems to me to be the most important and useful aspect of going to a math class and now, according to this particular teacher, it is no longer useful. I say he’s wrong.