Dear Caroline — and anyone else who has struggled with linear algebra at Tech,
I read your recent article in the fifth issue of the Technique entitled “Eigenvalue issues,” in which you said that you “never learned how to calculate an eigenvalue.” As a math major, a former student of MATH 1564 and MATH 3406, and a current teaching assistant (TA) for MATH 1553, I would like to contextualize and provide a response. When it comes down to it, there are three eigenvector issues.
Many students feel that, to them, linear algebra is not and never will be useful, and that taking a semester-long
class in it is pointless.
There are, in fact, many uses of linear algebra (see CT scans and Leontief models, to name a few), which we often neglect to discuss in entry-level classes in favor of foundational information, like vectors and n-dimensional spaces, simply because there isn’t time to cover such things.
This may, in some cases, be detrimental to student learning, because when students feel unmotivated to study a topic, they typically won’t do as well in it. Adding a relation of the topics to a student’s major might improve success in the course. Yet, there simply isn’t time.
Even with these applications, however, there is often still dissonance between a student’s experience and their future career; while linear algebra may be applicable to nearly every STEM subject, that doesn’t mean every STEM professional will use it. For those who don’t, or who know they won’t, the question is as follows: why should a student learn to integrate, to diagonalize or, yes, even learn to calculate an eigenvalue when there are hundreds of websites online which can just as easily do it for them?
Beyond the issue of being able to verify that the formulas and eigenvalues provided are correct, most people outside of the academic offices or math department don’t realize the following: the point of requiring linear algebra isn’t entirely to learn linear algebra.
It’s to teach you how to learn and how to study a subject which you are entirely unfamiliar with, and, should you fail to do so, how to accept that failure and recover from it.
In the same way that our ENGL classes are focused on multimodal skills which will prepare us for future conversations and careers, rather than focusing on english composition, general required math classes at Tech (for those who aren’t studying subjects in which said math is necessary) are here to teach you how to succeed at Tech. Congratulations, Caroline, you passed!
All of that being said, the linear algebra struggle is real! For many students at Tech, it may be a subject that they have never even heard of before, and vectors are equally as unknown. This brings me to my second point.
Mathematics education in the United States is heavily calculus-based. There was, in the 1960s, a push for mathematics education to include topics in set theory and algebra, deemed New Math. It was established to counter the advanced mathematics comprehension of students in the Soviet Union. Largely, it was a failure, due to criticism that math should be taught and used primarily as applicable, rather than conceptual. Calculus, of course, was entirely applicable to the scientific and engineering pursuits of the time and a mathematics pipeline was established, with calculus as the end goal.
Calculus is, in some ways, easier to learn than linear algebra, because you have spent your whole academic life preparing to learn calculus. It seems — and is — a herculean task to absorb the mindset for thinking about linear algebra in a single semester when you have spent 12 years learning to think about math differently.
So how can a student begin to reformulate (ha, math joke!) their mathematical thinking? For one, try talking to mathematicians. The instructors which teach your classes and the TAs who lead your studios are intimately familiar with all of these concepts and may be able to break them down in a way that makes more sense to you.
However, I do advise you to be cautious: there will be some TAs who will not want to work with you or who cannot conceptualize the problems from your perspective.
Just because they are good at math does not mean they are good at teaching math. Be patient and remember that TAs are students too — they struggle with certain concepts just as much as you — and try to find one who is able to explain concepts in a way that fits into your frame of reference.
The same goes for instructors. I am nearly certain that every student currently at Tech — and perhaps every student who has ever attended the Institute — has had at least one instructor who was much better at being a researcher than they were at being a teacher.
Tech is an R1 research university, so it is inevitable that there will be some people hired to do research who, while being experts in their disciplines, have spent so long studying these disciplines that they have forgotten what it was like to learn them in the first place.
There is nothing wrong with that, but it isn’t all that helpful for students. Beyond the academic professionals which work here at Tech, most instructors never receive any formal training in instruction. It does make one wonder, though, why, if it is quite literally in someone’s job description to teach, they don’t know how to do so effectively. Why aren’t instructors required to learn how to instruct?
Not only is it very difficult to implement such a requirement (why should a researcher learn how to teach when they can just as easily find another university which doesn’t require them to do so), it is just “not how college is” (with many a stigma about learning independently, not babying students through college courses and blaming high schools for not “adequately” preparing their students).
It is also simply because, well, Tech doesn’t consider education a science.
The best you can do for studying education at Tech is speak to Pre-Graduate and Pre-Professional Advising or reach out to one of the aforementioned academic professionals for advice. Said academic professionals, not only at Tech, but nationally, are often paid far less than their research counterparts because research generates far more money and clout for institutions than actually teaching one’s students does.
The teaching profession itself has been in decline for years because the skills required to grasp a concept on a fundamental level, understand another person’s point of view, see exactly which parts they are struggling with and explain it to them in a patient, kind manner are devalued.So no, it isn’t entirely the fault of some instructors that they aren’t very good at instructing, as they too are victims of a more systemic national problem.
But to think that being able to instruct is less valuable than researching begs the question: beyond the allure of capitalism, what is the point of discovering new information if you are unable to effectively and efficiently share it with others?
There is another problem too, that has arisen with the decline of the teaching profession. If a student has never had a math teacher who was able to explain concepts in a way that made sense to them, then they are guaranteed to believe that they are bad at math.
And if a student believes they are bad at math and encounters an entirely new math subject — linear algebra — under the instruction of someone who doesn’t conceptualize math in the same way that they do, then they are guaranteed to struggle.
This is the context from which we must view the eigenvalue issue: Tech, and our nation itself, has devalued teaching, leading to a gap between student comprehension and instructor explanation which is only worsened by the current subpar standards for mathematics education in the United States.
Paired with the seeming futility of studying linear algebra in the first place, this gap has led to entry level linear algebra courses at Tech being hard at best and, at worst, leave students feeling “totally lost.” That, dear Caroline, is why, through very little fault of your own, you never learned how to calculate an eigenvalue.
All the best,