When I look up reviews of classes and professors online, or discuss them with other students, one complaint always seems to surface about mathematics professors in particular: professors don’t explain topics well enough, but at the same time, they explain some things too much. From what I’ve gathered from online reviews, it seems that students are not very fond of professors going through proofs of mathematical theorems – many students view proofs as a waste of time.
This is understandable, as not all students in math classes are interested in math for its own sake, and thus not willing to stomach the arcane and onerous language that is used to describe mathematics in full generality. Many students, especially the ones who are not majoring in math, but math-related fields like engineering, just want to know the end product of a theorem and apply it. Even if students will not be required to prove theorems on exams, or their future careers will not depend on proving theorems, complaints about professors spending considerable class time on demonstrating proofs is unwarranted.
In terms of problem solving, students benefit directly from being taught proofs. Understanding the reasoning behind a theorem or formula is particularly advantageous when confronted with unconventional questions that do not seem solvable by applying a memorized formula. These kinds of questions trip up students because they involve a knotty nuance that doesn’t fit a predetermined formula. It is by having an intimate familiarity with a theorem, and the reasoning behind it, that students can grasp the nuance of the problem.
However, what if students are not confronted with such questions? What good is understanding proofs if students can reasonably expect that exam questions will only require straightforward application of theorems? Even in such cases, it is still vital that professors teach proofs in math classes. Showing proofs of theorems is an exercise of an indispensable intellectual tradition and a convention of inquiry. It reminds us that no one in any position of authority can pass off something as knowledge by hand waving and imperious assertion. It is a reminder that in the enterprise of inquiry, there is no authority figure, only participants. Therefore, an insistence on teaching proofs protects inquiry from enshrining thoughts into mere dogmas.
To show mathematical proofs is to adhere to the standard that what is being passed off as knowledge is agreed to be internally consistent with the audience. Students come to a class with an accumulation of past knowledge. Professors show that theorems do not float in air, but instead sit atop of a structure of knowledge assembled from what students already know. From this view, showing the poof of a theorem is an expression of respect of the student as a living, independent, thinking entity. Understanding proofs is a bulwark against blind acceptance of information.
It is disappointing not that many students are fond of this tradition and convention when it is practiced in math classes, but seem to defend it vehemently in discourse with direct societal significance. What if a sociology professor says that immigration wrecks social cohesion among local people? If that strikes you as contrary to what you like to believe you would certainly demand the evidence that led the professor to say that. It seems that people are more insistent on justification when the discourse has an emotional dimension.
One may defend this difference by arguing that social situations are fluid and unique, while students can afford to accept theorems by their conclusions only because it has always worked, thus it must be reasonably safe to continue doing so. However, is that not merely accepting the status quo?
Whether you will or will not have to write them yourself, simply learning the proofs of mathematical theorems is indispensable. It is an exercise of an intellectual tradition that enabled Western civilization to become what it is today.
Heikal Badrulhisham ([email protected]) is a freshman majoring in economics.
