Trying to cram all the subjects needed to cobble together an undergraduate introduction to a scientific discipline into forty-odd courses arranged to flow, a pack at a time, into one another in some sort of meaningful way is no easy task. As a result, certain little gems of scheduling arise to befuddle students, and one such idiosyncrasy in my physics education was the haphazard insertion of my first linear algebra course into the first semester of my second year.
When you first come to it, linear algebra is a totally mysterious exercise. In it, boxes of numbers ('matrices') are introduced, and a few well-defined but seemingly totally arbitrary rules for how to combine them are presented; the professor then moves on to spend three months proving a series of theorems of ever more inscrutable purpose, before a final exam is administered, barely passed, and never thought of again by the students, as if waking from a terrible and bizarre dream.
I would have none of it. Up until that class, college math was all operations on numbers --- albeit increasingly dexterous operations, but with nonetheless fairly transparent interpretations and meanings. Linear algebra, on the other hand, was witchcraft. Why were we learning this weird number box game, and who made it up in the first place? The powers that be seemed to think there was a good reason --- the course was mandatory for all physics students, and a strict prerequisite for many later classes, though it was dreaded by many undergrads for its seemingly useless nature. So, I stomped into the office hours of a particularly embattled-seeming TA, and demanded to know what the deal, in fact, was.
"We do it because... we just do alright?," was the enlightening tautology offered --- and that moment, with its utter abandonment of reason and the realization that I was completely alone in my desire to see more deeply into what I was being spoon fed, stuck with me forever as the germinating moment of a lifelong motivation to understand not just the mechanics of the mathematics I was to learn, but its deeper meaning as well.
I spent all my free time over the next several months dauntlessly trying to sleuth out the secrets of linear algebra on my own, having been failed by my institutional apparatus. And, little by little, I came to understand what a rich, expressive, elegant device those number boxes really were for describing geometry and transformations in any space. A deeper understanding of linear algebra helped rip the blinders off of what can be meant by spaces not just Euclidean, but with all kinds of intricate contours and features and behaviors --- it was beautiful, and empowering, and liberating. I had crossed a mighty gap; it was a leap propelled by finding the right motivation, which as per usual in my case, came from sheer belligerent refusal to accept the superficial.
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In grad school, I had the opportunity to TA the senior lab course in my department for several years. This lab was the last experimental course offered to undergrads before graduating, making it our last opportunity to assess and reinforce their foundational skills of hardware debugging, data analysis, paper composition, and conference-style presentation. As such, I came to the position determined to make sure my students had a solid grounding in all these skills.
For the most part, it was an easy gig --- the class wasn't mandatory, so the students that took it were more motivated than most, and arrived fluent in their skills; mostly, they just needed someone to help them train with real professional standards inside spitting distance, so they could confidently enter graduate programs with as smooth a transition as possible. My students were the physics department's best upcoming experimentalists --- all of them, except for S.
The term consisted of two experiments, with a paper at the end of each, and S' first was an unmitigated disaster. They were unable to get their experiment to work and thus had no results to speak of, failed to document their work in any way that would allow them to discuss what went wrong with their methodology, and showed no grasp of the underlying theory. At that rate, not only would S have failed the class, but there were serious questions how a student with such weak skills wasn't stopped for remedial action much earlier. In other words, it would have been pretty much impossible to be in worse shape in a class than S was, with half the semester already gone.
The instructors laboured long over strategies to try and patch S' skills up to a place where we could at least pass them, but with no solid plan and shortly after the second experiment began, patience began to crack. I caught one of the other instructors taking S to task over the quality of their work, berating them in a tirade that quickly metastasized into verbal abuse. I stepped in, sent the other instructor away, forbade them to talk to my students like that. S looked like they had had about all they could take; that was a dressing down that would be almost impossible to come back from. But as it turned out, that day would be S' lowest point.
After that particularly repellent incident, the instructors met, and I told them to back off, to leave it to me. I would handle all of S' needs in the lab from then on, to make sure a repeat did not occur. Every day in the lab I would come in, and ask S how they were doing --- most days, the answer was 'fine, fine, no problems!'; once in a while, they would ask a question about some minor detail of the equipment or the theory, which I would answer for them, and leave them to keep puttering along. They worked industriously and made consistent progress, so I mostly just left them be.
When the semester came to an end and the instructors were meeting to discuss and grade the second round of papers, the whole table was totally flabberghasted with S' paper. It was pretty much perfect. How could such a clear methodology, astute grasp of theory and dexterous handling of data come from someone they had written off as a lost cause just six weeks earlier? The obvious allegations of cheating were leveled and immediately refuted --- S' lab book contained the development of all their ideas, they were able to competently answer all probing questions in their final oral exam, and I had seen several drafts of their paper as it developed; the work was doubtlessly theirs. But how?
It certainly wasn't me. While I did take over responsibility for their instruction, I barely spoke to them; all I really gave S was space to do their own thing. And no student could possibly cover that much ground in just six lousy weeks; no herculean Rocky-esque training montage could produce results that big, that fast.
The only hypothesis that remained, was that S had it in them from the start, despite what my colleagues thought. As the semester progressed, I heard a lot of mumbling from staff regarding S' reputation in other classes as an underachiever; but all I heard, was that S was constantly beset by negativity, low expectations, and prejudice. To go through school like that must have been enormously oppressive, and I believe it undermined S to the point that even when faced with a task they were capable of, they failed miserably at it for no other reason than the want of an environment that didn't treat them with contempt.
As it turned out, S was by no means unique, particularly in the sciences. I have encountered student after student who believed with fiery conviction that they couldn't do math, that they were incapable, incompetent, destined to fail at anything with a number in it. But when they found the courage and personal conviction to put those fears aside, and they were given nothing but a bit of space to try, they all succeeded brilliantly; no matter how intensely they doubted their own ability and no matter how shocked they were to find it, they all had it in them --- every one.