Ideals

I find it difficult to view a mathematics education objectively at times. If I try to ponder what should be taught or what the weaknesses of the current curriculum are, I encounter a simple perception-based problem--I enjoyed math at every level. If I'd had my druthers while I was in school, then we'd have seen more sophisticated mathematics courses than what we did have. It is one of my great hopes in becoming a teacher to learn how to transfer some of my success and ardor in the field to my students, and it would be excellent if the way I look at mathematics knowledge worked to do so. I doubt it, but it would be nice. For starters, let me describe this view.

As one advances in mathematics, one continues to be exposed more and more to the theoretical foundations of the field and less and less to the numerical calculations of its use. In some ways, it seems to me that advanced math is more like a language than a set of tools. It's something with which a student can become fluent. It is certainly possible to speak math! When a student learns to do some calculation or to find some important number or point in a math problem, in the language of math it's as though they are speaking rote phrases--the same sorts of phrases one would learn to "get by" in another country. Please, where is the bathroom? How much does this cost? May I have a glass of water? The student can describe some intent but they do not know what the words mean in their statements!

The most successful math teachers I've ever had reserved a significant portion of their exams and quizzes for vocabulary. Starting with Algebra, this is how I was exposed to many years of math. An equation is a relation between two statements. A solution to an equation is the value of the variable that makes the equation true (since clearly in beginning algebra one is not exposed to multiple independent variables). I was armed with the actual language of math, rudimentary as it was at that point, but I knew how to string together sentences and make sense. When a student can bring topics to a point where they make sense, they can understand why they work. Yes, you still have to teach an algebra student to factor equations. The methods of approaching a problem are like the grammar for the language! But, when 'equation', 'root', and 'solution' have proper definitions in the student's mind, I think there will be much less time spent re-teaching the material later because the students will have internalized the concepts more than they do now.

Though I don't teach formally now, I've done my share of tutoring. It is almost always this problem of not knowing what one is actually working with that is the underlying difficulty that students have with applying what they know. Students always seem to be applying methods blindly, without considering whether those methods work with the objects they're talking about. It's worth knowing whether something is essentially a set, or a function, or a logical statement. You cannot grab elements of a function, there's nothing in an equation. The methods and properties that apply to one kind of object will not apply to another. I've seen many people describe the operation of addition or multiplication as being one-to-one, or describe a function as being commutative. It's great that they have enough of a grasp to know that commutativity means things can swap positions, but there's a fundamental misunderstanding if a student is not shown that it's a property that describes some things but not others.

So, I think that one excellent way of approaching mathematics education is to teach it through the language of the field. The language doesn't have to be as sophisticated as it ever gets. No language is learned in such a way. At least you won't be in a position where you expect the student to distill the underlying concept behind a section, which is frequently the definition for a term, without ever actually giving them the chance to understand that definition as it is. They'll start to develop an intuition for how something works. It's wonderful if it gets to the point where they are just almost grasping it, but they can't quite put it to words. Give them the words. They'll keep that grasp that they developed, but might also see further.

This is definitely a more traditional view in the reformed-traditional scale. I'm not saying that one shouldn't pursue getting students to develop their own framework for knowledge, but refining their concepts and providing them with the tools they'll need to actually communicate about the field seems obvious to me. Obvious... that's another term that just gets thrown around. Some take it to mean "I don't know, either!"