Ever since the 1970s, mathematics courses and textbooks have been inexorably moving away from a reasoning-based pure mathematics and towards a more algorithmic sort of applied mathematics. As a result, students, and sometimes even teachers, are often unable to explain why a method or theorem works, and therefore are forced to memorize several other methods for similar operations that would otherwise be evident. For example, if students understood at an intuitive level why polynomials behave like their highest degree term as they approach infinity or negative infinity, then they wouldn’t have to memorize three separate rules for horizontal asymptotes of rational functions. Why don't they understand this? It is being taught as three separate rules, and any attempt to simplify them is brushed off with the excuse that "this is easier to understand". A student once asked me how to solve a system of one linear and one quadratic equation. I asked him why he wasn't able to perform linear combination just like he would in a system of linear equations. He replied that he had never learned how to do it with a quadratic system. After I went through the problem with him, he remarked that he had "never thought of it that way before". What he had learned was an arithmetic procedure to solve a simple system of two linear equations by combination, not a general approach to solving all types of systems. Due to the textbook, the teacher, and the student having never made the connection, the class would have to spend an entire day studying how to solve a system of quadratic equations, again in a mindless, stepwise manner. The end result of this mindset is that students are unable to recognize quadratic equations unless they are in the form ax2+bx+c and taught a false dichotomy between rational expressions based on their degrees.
Without deduction and inference, mathematics is nothing more than rote memorization. With it, it becomes a tool for revealing the only universal truths this universe holds. Students learn mathematics for two reasons: to supply basic tools like arithmetic, algebra, geometry, and calculus that can be applied in the real world, and to learn the process and intuition for deducing truth from the chaos that is the universe we live in. The majority of college graduates have already forgotten most of the former, and they never even had the chance to learn the latter.
Note: some of this was written in February for an assignment of Mr. Stueben's. He's the only math teacher I've had in my decade in school who understands the importance of both of these reasons. If you want to start learning mathematics, one of the best things you can do is to tell your math teacher to read his 1998 book detailing how he is able to perform this feat while keeping to the curriculum and keeping his job (entitled Twenty Years Before the Blackboard, available here).
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2 years ago
Interesting Opinion.
ReplyDeleteI tend to disagree, seeing as mathematics, as well as any other type of education is based on the teaching methods of the teacher. Although most POS programs are based only on a narrow range of concepts that are convenient in a formative assessment, it is the teachers(as well as the students) job to expand upon the knowledge.
Most teachers follow to the letter the POS, and therefore the quality of education goes down. However, Mathematics teachers that do expand on the POS often create a basis on which the concept can be applied, followed by connection to previously learned methods.
The main problem with implementing "quality" mathematics education is that it is very hard to show connections such as these, or a basic understanding of the underlying concepts on a formative assessment.
DFB