# How It Came To Pass That Students In Linear Algebra III Did Not Know What A Basis Is

Once upon a time, an instructor (whom I shall call Professor “A”) went on sabbatical leave. As a result, another instructor (whom I shall call part-time instructor “b”) was called upon to teach Linear Algebra III, which was normally taught by Professor “A.”

All 16 of the students who attended Linear Algebra III had successfully completed Linear Algebra I and Linear Algebra II, which are prerequisites for Linear Algebra III.

Everything seemed to be going just swimmingly, until part-time instructor “b” graded the first quiz. Many students had a lot of trouble with a question involving a basis. “Well,” thought “b,” we shall have to discuss this in the next class.”

At the next class, as “b” was in the process of discussing the problematic question, he figured that he ought to back up a little, to toss them an easy one, so that they could get a solid hit, feel good about themselves, and get on a roll. So he asked the class to give him an example of a basis for $\mathbb{R}^3$. An awkward silence ensued. Perhaps the students had somehow misunderstood this very straightforward request? “Just any basis for $\mathbb{R}^3$ whatsoever,” said “b.” Silence. “Just give me the simplest basis for $\mathbb{R}^3$ that you can think of.” Silence.

Did I mention that all of these students had already passed TWO one-semester courses (the equivalent of a full-academic-year course) in linear algebra?

Part-time-instructor “b” was quite bewildered about this state of affairs, but naively soldiered on, trying to help students remember this essential concept, while exhorting them to spend significant time on review to make sure they were up to speed, because Linear Algebra III goes considerably beyond I and II. (Did I mention that “b” was naive?)

After this puzzling class, one of the students (whom I shall call “c”), who was wise to the situation, kindly explained all to naive Mr. “b,” who was appalled by what he heard. Professor “A” uses the same tests and exams every year for Linear Algebra I, II, and III, and even though he closely guards the questions, students over the years have learned how to play the game. The game being how to pass a course by writing correct answers on the final exam without understanding anything. (It turns out that some of the students, “c” being one of them, were indeed able to answer the question about the basis, but remained silent for some reason.)

Why would any students wish to play such a game? Well, it turns out that many students in the class were training to be high-school mathematics teachers, and needed a certain number of mathematics credits to receive their certification. Evidently they did not see any point in actually learning any mathematics beyond the high school level, because they would never actually need it (right?) in order to teach high-school mathematics. Rather, they sought out the easiest math credits that would satisfy the requirements of their degree, and this series of courses seemed like a sure thing.

After the class with the awkward silences, some students quickly realized that they would have to do some serious remedial work if they wanted to pass the course, because they correctly inferred that Mr. “b” would not be playing by the rules they were expecting. Either that or wait until “A” returned from sabbatical leave. Eight of the sixteen registrants immediately withdrew from the course.

All was not lost, though. Five of the eight remaining students passed the course, and some of them went on to have success in graduate mathematics programs. And “b” had his eyes opened by the very kind student “c.”

* * *

Before we pile onto Professor “A”, we should recognize the enormous pressure professors are under to produce publishable research results, and to attract external funding; these two goals are, of course, linked, because success in the first activity is essential for success in the second activity. Although some top researchers are also brilliant and inspiring teachers, and consider their teaching to be an intrinsically worthwhile activity, I suppose many professors find it challenging to balance the twin tasks of producing good research and teaching well. Anyone who has done either will realize how much time and effort are required to do either well; to do both well requires a truly impressive level of commitment and energy.

I suppose it would shock most parents to know that the primary purpose of the university (and the primary responsibility of professors) is not to educate their children. A number of prominent members of the media don’t understand this either, which explains why they think professors are under-worked (“They only work for SIX hours per week!!”). The reality is that most professors are workaholics, who are under enormous internal and external pressure to produce more, more, more, and who have no boundaries between work and the rest of their lives.

Until governments (for they ultimately pay the bills, even in private universities (via research grants)) decide that this situation is unsatisfactory, expect things to get worse for undergraduate students. Unless, of course, we do something about it.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)