Philosophy of Mathematics Education
I believe that learning mathematics has three components: one is that of learning a new language as one might learn French or Spanish. Another component is learning mathematical ideas, which are most frequently expressed in mathematical language or symbolism. Finally, there is the formal operations executed by logical reasoning. I believe that mathematics teaching and learning should then closely resemble instruction in foreign language as well as instruction in concepts and logical reasoning.
For the first, mathematics requires memorization. Just as one might remember that cosa means thing in Spanish or that fahren is the German verb for to drive, we also learn that "=" means equals or that x most often means some variable expression or term. By stringing together mathematical words, we create mathematical sentences. These statements are either true (correct) or false (incorrect). Learning the vocabulary of mathematics and the rules of making mathematical statements is just like learning vocabulary and grammatical rules in German or any other foreign language. Definitions and rules require memorizing, which is achieved through repetition. Although knowing the vocabulary of mathematics can help us analyze the truth of mathematical statements, just knowing the vocabulary won't help us solve problems. Only by using the rules of the language - formal operations that are grounded in logical reasoning - can we solve problems by creating new statements from other ones.
One the other hand, logically deducing or inducing concepts and ideas is best learned not by memorizing, but rather by incorporating those concepts and ideas into one's already existent understanding of the world, specifically mathematical understanding. The pathway from one concept to another is logical reasoning. Contrary to learning vocabulary and rules, logical reasoning is something that develops with practice and with time spent thinking. Staring at a problem without picking up your pencil can mean that you're doing the extremely difficult task of thinking! More specifically, one is trying to make sense of the problem. This making sense is essentially creating a logical chain between the beginning of the problem and the end of the problem. To get better at this, one must spend a lot of time thinking about and doing practice problems. Here, too, one must be cautious not to solely focus on logical operations; knowing only the formal operations of mathematics leads to only a partial understanding of the problem. One must understand the statements in order to understand how the ideas and concepts are relating to each other. The concepts, the vocabulary, and the formal operations (logical thinking) are all interdependent; knowing only one facet will mean one only has a partial understanding of the material.
Cramming concepts into short-term memory might help one pass one exam, but soon the material will be forgotten and one will not be able to use it for all further material, condemning one's self to future unnecessary difficulty. Therefore, to learn the concepts and reasoning of mathematics, one must extrapolate from one's known understanding of mathematics, i.e. one must build on an already existent foundation of mathematical ideas, paying particular attention to logical thinking, using the language and symbolism of mathematics, and exploiting the logical abilities innate to human beings.
Translating this into the classroom, it means that the instuctor must make a concerted effort to express new ideas and concepts within the context of previous learned concepts, to use the new vocabulary explicitly when demonstrating its usage within the context of problem solving and concept instruction, and to explain the logical reasoning behind the mathematical operations used in problem solving. The student's responsibility is to spend time memorizing the vocabulary and to spend time solving problems and thinking about problems until their execution makes sense and the dynamic processes and static concepts integrate into the student's framework of understanding.
It is imperative that the student spends at least two hours outside of class studying these things for each hour of in-class instruction. Although the instructor can convey material and demonstrate solutions in a manner which makes sense to students, students must then spend time practicing generating the solutions to problems themselves; although what is demonstrated in class might make sense to the student, the generation of solutions by her requires extra time and practice.
Finally, it follows that the assessment of student understanding must include assessment of knowledge of vocabulary, understanding of concepts, the ability to execute formal operations to solve problems, and the integration of these three components.Copyright 2005 Seamus Mulryan, All Rights Reserved.