Dreaming of math class
Commenter “Socratease” calls my calculus class “the stuff of nightmares.” That is quite literally true, because after experiencing the horror of that class, and the ensuing and resounding “F” I received (my first and only in a long student career), I proceeded to have the classic student anxiety dream on a regular basis for about three decades.
And it almost always involved a math exam.
You know the drill: you’re sitting in a classroom, bluebook in front of you, and don’t know a single answer (which is what actually had happened to me in that class). Sometimes the dream begins earlier in time, when you know an exam is coming and you know you’re unprepared, and you know there’s not nearly enough time to learn the material. Sometimes (usually, in my case) you’ve even forgotten you were taking this course and haven’t even attended the lectures or read the book.
There are variations on the theme, but I had that dream every few months for most of my life, despite having been a stellar student. Or maybe it’s because I was a stellar student.
Then there’s this comment from Cato Renasci in that same thread:
I had difficulty with calculus as an undergraduate – passing with a then “gentleman’s C”, but not really confident that I understood it and not willing to pursue further mathematics courses. It was only several years later – when I was a graduate student – that a friend doing a doctorate in mathematics suggested that I should approach learning mathematics as I would a foreign language. That insight made all the difference for me. YMMV, of course.
Yep, my mileage most definitely differs—because it was actually two subjects that were the foci of my recurrent student anxiety dream: math (only after the calculus situation) and foreign languages, which I detested learning and did more poorly in than any other subject I took. I managed by some near-miracle to place out of foreign language study in college, so I’m only talking about high school. And I did well in terms of grades, even then. But I hated my Spanish courses, never felt confident about the subject matter, and couldn’t wait for them to end.
I don’t think I’ve had a student anxiety dream in many years. Maybe there are some perks to getting this old. I do recall that, a decade or two ago, I had the student anxiety dream as usual—about math, like most of the others. And then, as I was sitting in my chair in that classroom, the following thought dawned on me: I have a law degree. I must have graduated from high school, and from college, and from law school. So I don’t have to have this dream anymore.
In your face, Clyde Rutherford!
Wow! It was certainly not my intention to trigger with a reference to foreign language learning! Like mathematics, learning foreign (including dead) languages seems to be something that some excel at and others are frustrated by. I suppose music is the same way — both performance and theory, though perhaps for different reasons — some people hear pitch accurately and find singing or playing intuitive (not that they don’t have to work hard to do well….) and others, as the saying goes “can’t carry a tune in a bucket.”
It’s often said that musical ability and mathematical ability are closely related. I’m not sure that’s entirely true, but I think both of them represent ways of perceiving/thinking. In that context one can think of another language as another way of perceiving/thinking.
Perhaps the reason my friend’s insight resonated with me was that I was already (close to) fluent in a modern language and was a journeyman musician.
Another related factor among the three ways of perceiving/thinking is that each of them requires a substantial amount of rote work early on. Each has a grammar and vocabulary that must be mastered before the learner can move to the level of the really fun stuff.
For mathematics, I think that means having the fundamental axioms down cold at an early age and doing the tedious (for me) work of becoming facile with computation and algebra. My greatest failing in math was always computational, but computational errors often took me down dark and fruitless paths….
Similarly, music has its grammar and vocabulary of rhythm, pitch, notes and scales and keys, etc., and its various physical and technical coordination skills.
And, of course, unless one learns very naturally as a child, languages require learning their grammars and vocabularies…
Perhaps this puts my comment in a bit larger (and less triggering) context….
My most common college grade was A, second most common was F, I really didn’t care and did what interested me. As a senior I got a B+ in a difficult class for which I hadn’t the prerequisites — I changed my major to physics in my senior year — and was proud of that. Some might say I was beginning to grow up.
The education dream I always have involves needing to take one more final to graduate, and not being able to remember which class or where the exam room is located.
I also had this dream for a few years after graduating, and it was equally traumatic (although for engineering/math classes in general). They finally stopped after I began realizing quickly during/after the dream that I ALREADY had my degree, both of them in fact, and they were in a little tube in my closet. After one or two realizations of that, I think I was finally able to let that ghost go (flash to Maverick throwing Goose’s dog tags into the ocean).
Neo,
I empathize. The only “C” I ever received in college was in a freshman year Trig course which I had just taken in my senior year in high school. Prior to Trig I was a B student in algebra and geometry. Cato Renasci’s comment about math as language is, IMO, absolutely spot-on. I had an epiphany experience when I was studying for an exam at age forty-three; I looked at a formula we were supposed to learn and I understood it; I realized what it was doing and, more importantly, what it was communicating. I didn’t need to memorize it because I could now recreate it as needed. The humbling part of this was to suddenly realize that there were fellow students in my high school who could do this when were all age 18.
I blame math teachers (speaking generally, not absolutely). Most of them in my experience teach computation. They, themselves, may be able to compute but very rare is the secondary math teacher who comprehends the concept behind the discipline, and even rarer is the math teacher (secondary or university level) who can successfully communicate and inspire understanding of that concept in others. I certainly never had such a math teacher and it took me to age forty-three to intuit that concept myself.
Let me, perhaps dangerously, jump into a math example of my point. Einstein’s well-known E=MC squared. The extraordinary insight in this equation is in “E=M”, energy and mass as two ends of the same spectrum; “C squared”, although hardly a trivial number, is nothing but the conversion ratio between the two. It’s impact is even more astounding in its simplicity, yet in my very limited knowledge, I know of no math teacher or professor who was ever willing to actually point that out. IMO <i?that is how one gets students excit3ed about the math behind the physics.
Two final notes, sorry for being so long-winded here. I am not a mathematician or physicist, so if anyone feels the need to correct or tweak my humble example above–feel free. Also, for those interested in Einstein and his Theory of Relativity, I highly recommend a wonderful book The Universe and Dr. Einstein by Lincoln Barnett which, IMO, is the best explanation of Einstein in layman’s terms that I have ever read.
CatoRenasci:
Don’t worry one little bit. I don’t get “triggered” that easily, and your comment actually sparked a thought process that led to this post.
I thought it was kind of amusing and interesting that some people would find an analogy to foreign language learning a help, and some a hindrance.
I think the commonality of foreign language and math learning is that, once you get behind, it’s nearly impossible to catch up. Before I took integral calculus it was never an issue for me with math at all; I aced it. But it was always an issue with foreign languages for me, for some reason.
“… I proceeded to have the classic student anxiety dream on a regular basis for about three decades.
And it almost always involved a math exam.”
Mine was that it was time at least for mid-terms, if not finals, and I suddenly remembered that I’d forgotten to attend an important class all term, and I couldn’t even find out where the class was held.
But I hated my Spanish courses, never felt confident about the subject matter, and couldn’t wait for them to end.
I took no foreign languages in college, as my U didn’t require it for my major. I did, however, have a good command of Spanish, courtesy of a Spanish summer school in Mexico after 3 years of high school Spanish. I later spent over 4 years in Latin America, initially as a backpacking tourist and later working in drilling engineering services in the oil field.
I had a short second career as a math teacher. Lack of classroom management skills did me in. After gaining initial certification in a subject, I could gain additional certifications by passing an exam in the subject. By exam, I became a certified Spanish teacher without having taken a single college course in Spanish.
BTW, while I was taught in Spanish class to use “tú ” for the second person familiar pronoun, I worked in areas that used “vos”. As a result of speaking “vos” for years, I prefer using “vos” instead of “tú ” .
These“vos” speaking areas tended to have been backwaters of the Spanish Empire, and thus less likely to be affected by post-1700 changes in Spanish as spoken in Spain.You can see “vos” in Don Quixote, published circa 1600.Bing Search: tú versus vos in Spanish.
My no-longer-recurring nightmares mirrored yours, except they were never math class. I was great in math, finishing all 3 high school higher math classes by my Junior year. The once/week calculus class, taught by a college professor who happened to be suffering from dementia of some form, was my utter downfall. He would show up, often late, to a class of 10 high school seniors, go directly to the blackboard and begin writing a formula of some type or another, seemingly unrelated to what he wrote the week before. I was painfully shy and never said anything to anyone except my parents about how awful it was. I never finished a problem, never understood any concept, not surprising since he rarely explained anything, and still managed to get a “C”. A complete waste of time and angst. I did go on to become a more than decent computer programmer, in machine language, no less.
T
Courtesy of the Illinois Math/UICSM version of New Math, I worked proofs from the beginning of 9th grade. Geometry proofs , of course. Inductive proofs in 11th grade. For N and then for N+1. Etc. I was a bit shocked in college to find more emphasis on computation and less emphasis on proofs. Math was more fun in high school, courtesy of all those proofs.
When I was back in school in my 40s, and took a Linear Algebra course that had a heavy dose of proofs, I had no problem once again cranking out proofs, as it was a language I had learned in high school. Most of the class wasn’t as familiar with proofs as I was.
My *other* recurring student anxiety nightmare was that a decade or more after college, I decided to go back and get that one last advanced (*) math class that would have allowed me to graduate with a double major.
The nightmare was like a serial; a couple of years passed over the recurring course of the dream … for instance, I moved apartment a few times, including at least twice to the same place.
(*) Probability and/or statistics, I think; those being the two that I had had so much trouble with that I’d opted in real life *not* to go for the double major.
“I did go on to become a more than decent computer programmer, in machine language, no less.”
What a small world! In my first job, I coded almost entirely in BAL … including sometimes toggling switches on the front of the computer to change the contents of memory, both the date and the executing code.
I think proofs is a weak spot in modern education. Classic geometry, with an emphasis on deriving results starting with a few axioms was the traditional introduction to the subject, and my impression is that it has been replaced with computation. In high school we had to write the text ourselves, it was a workbook that stated the propositions and we had to prove them one by one by construction with ruler, compass, and reference to previous results.
Neo: your point on the difficulty (neigh unto impossibility….) of catching up if one falls behind in math or language (or music, actually) is well-taken. Except with those who fall behind from sloth, indifference or neglect of work, I think the otherwise good student in any of those disciplines who falls behind despite trying her or his best to do the work, often results from instruction which fails to realize the importance of all of the elements necessary for learning. For example, the instructor for whom computation was intuitive may not have patience for those who need to be more methodical in mastering computation.
Like others, I occasionally have the student anxiety nightmare, but I also have another nightmare which I think might be related.
When I was in high school I was quite involved in theater. Although I haven’t been in any sort of theater production since high school (I’m 67 years old), I sometimes have a terrifying dream that it’s opening night, I’m about to go on stage, and I haven’t memorized a single line! I wonder if anyone else who has been involved in theater has had similar nightmares?
“Two final notes, sorry for being so long-winded here. I am not a mathematician or physicist, so if anyone feels the need to correct or tweak my humble example above–feel free.”
T: I often get annoyed at the attitude that physics “is just a bunch of math”. You have pointed out exactly what we always try to do with physics students: point out to them in each equation what that equation is saying about nature, and to try and get them to do that themselves. Maybe that’s why so many struggle with the subject. One has to be both conversant with a number of high level mathematical topics, and then at the same time look into the math for the physical meaning it’s expressing.
Chris B:
The actor’s nightmare—which you describe—is almost as common as the student anxiety nightmare.
The other common one is going out without realizing you don’t have your pants on.
CatoRenasci:
I never fell behind in math until integral calculus, and that happened in that course because for whatever reason (perhaps the bad teacher) it just was unintelligible almost from the start.
For foreign language, I can’t recall exactly what was going on, but I had no interest in it to begin with, and it was mostly drill, drill, drill and memorization. So I probably didn’t drill as much as necessary to retain the information. I also think that some people find foreign languages easier than others do, and I don’t think I was one of those people.
I suspect that the problem lay not in you neo but in the teacher…
“If you can’t explain it simply, you don’t understand it well enough.” Albert Einstein
Most ‘teachers’ I’ve found assume that their ability to memorize and pass tests on the way to their ‘credentials’ equates to understanding the subject.
History is notorious for this as memorizing dates is useless without context. As example, the 300 Spartans and 7000 Thespians stopping the massive Persian invasion at Thermopylae is NOT of historical importance because it gains the time for the Athenian navy to defeat Xerxes’ navy at Salamis, which prevents Xerxes ability to resupply his army and saves the Greek city states.
It’s important because it saves the beginning of the process that leads to Western civilization and the founding of America’s democratic republic.
I’ve known a number of people who have related prior difficulties in mastering a subject until they met someone who explained it in terms that created the ‘light-bulb effect’ in their suddenly making a breakthrough in understanding a previously difficult to grasp subject.
Physicsguy,
If physics is just “a bunch of math”, then language is just a bunch of words, and no one would ever make such a preposterous statement about language. To further illustrate my point above, I’ve never heard a math teacher point out when studying quadratic equations and parabolae that when you throw a ball, or when you water your garden you are creating a segment of a parabola. The lack of such real-world examples renders both math and physics academic exercises that are divorced from everyday life. And as you, of course know, nothing could be further from the truth.
It’s my impression that the people who have the exam dream are people who were good and conscientious students. That would explain why I’ve never had it.
Couldn’t these bad dreams be called post traumatic student disorder? You could claim to be a victim of PTSD.
It is rather astonishing that you came to be a lawyer without having to know differential and infernal calculus.
On the other hand, as far as language goes, res ipsa loquitur, a posteriori (about that behind), casus belli (time for a diet), &c (et cetera……)……
Once upon a time, long, and even longer ago, I was studying law at Georgetown. At that time in many courses you were given only one exam. Uh oh. Gotterdammerung. Well, I showed up for one such exam and picked up the list of questions. When I read the questions, I was beyond horrified. The questions were impenetrable. I sat there in stupified panic. Wondering if I were the only one, I sneakily cast my eyes about the room. I discovered that most of my classmates were similarly horrified and also engaged in furtive surveillance– the rest simply gave every indication of being suicidal. We had been given the Wrong Exam.
The TA giving the exam was nonplussed. But he told us to close out eyes, take one for Olde England, and do the best we could. Not a course of action to inspire confidence in the result.
But we shouldn’t have worried. The whole fiasco was swept under the rug, and it seems a dartboard or a like device of allocation was used to sprinkle about passing grades or better.
So, no Post Traumatic Student Syndrome for us!
Boy, I wasn’t the same after scoring 33% on a calculus test last summer. I had never gotten a mark like that before. Nightmare! It shook me. I’m only now getting back into the saddle of taking tests without panic.
The good news was after the teacher curved my 33%, it became a D — leading me to wonder if maybe she hadn’t quite calibrated the test very well when she composed it.
At that point about a third of the class had already dropped out and 80% of those remaining had taken the class before. I was wondering why most everyone else was copping on faster than I was.
I don’t think I’ve had a student anxiety dream in many years. Maybe there are some perks to getting this old. I do recall that, a decade or two ago, I had the student anxiety dream as usual—about math, like most of the others. And then, as I was sitting in my chair in that classroom, the following thought dawned on me: I have a law degree. I must have graduated from high school, and from college, and from law school. So I don’t have to have this dream anymore.
This reminds me of what happens when Sages awaken from this dream that life is real. When a Sage awakens from the Illusion of Maya, what you all call “life” and the “world”, they have a similar reaction.
“If you can’t explain it simply, you don’t understand it well enough.” Albert Einstein
Hah, says the guy who needed dark matter to patch up his wrong equations about Relativity, light speed constant, and galaxy spin rates due to gravity wells.
His explanations are so simple his peers failed to understand the math and just took Einstein as Gospel. Although some of his peers did question him. That didn’t make it out to this century however.
When I am tormented by irrational fears and my existence weighs heavily on me, I sometimes imagine that my fellow human beings are blessed with a strength and serenity I somehow lack. Look at them! On the street. In the train. In the office. They smile, chat, laugh, express opinions, go about their business, and in general appear to be playing the game of life with an easy grace that always manages to elude me. In moments such as these, I try to remind myself of a math exam I took when I was a graduate student at Stanford….The subject was matrix algebra and vectors, and although I was very familiar with the material, I was having a great deal of difficulty that day solving the problems at hand. From time to time in the course of the exam, I would look up and glance around the auditorium at the other students, hoping to detect in some of them the panic that was stirring within me. But I saw no signs of panic or even worry, just a roomful of people working away with apparent ease. It was only when the test was over and a number of us started to compare notes that it became clear to me that the problems had indeed been tough and that probably no one had done well. The unruffled exteriors of the other students had deceived me; beneath those placid surfaces there had almost certainly been agitation and fear. I speculated: Suppose everyone in that auditorium had felt as I had; suppose every other person had on several occasions looked up, glanced around, and returned resignedly to his task. Suppose, further, that each head had popped up and sunk down at different moments (each time undetected by the other students) so that each student, surveying a room filled with self-composed test takers, had then concluded that he was alone in his misery. How deliciously comic and ironic that would have been! And how instructive too – a math exam serving as a parable of human loneliness and isolation….Alas, the memory of this experience does little to sustain me in times of emotional distress. It titillates my mind but does not lift my spirits; it instructs but does not comfort. Head and heart each go their own ways, as is so often the case. As La Rochefoucauld put it: “Philosophy triumphs with ease over misfortunes past and to come; present misfortune triumphs over it.”
“The other common one is going out without realizing you don’t have your pants on.”
My twist on that one was *intentionally* going to work nude … for months! … and everyone else just rolled their eyes about it.
I can say I only had one academic nightmare, and it happened a few months AFTER I graduated. It took a while to calm myself down and realize I had actually graduated.
The suggestion that math is another language is not only useful, it is quite literally true.
Math is a mental model for handling certain kinds of questions, especially those like “related rates” – really hard in words, easy in functions. Surely you’ve seen legal language in financial contracts; these can be incredibly convoluted with lots of colons and semicolons.
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Math is to physics as grammar is to poetry.
The rest is math models.
Aristotle wasn’t wrong. In real life things DO slow down.
Newton set the astonishing base case of “no friction,” then went and explained WHY things slow down. He also switched perspective and said the earth falls UP to meet the apple falling down. And that math got us to the moon and back.
To explain certain anomalies, Einstein set light as the speed limit. Minkowski already had the equations to express that. https://en.wikipedia.org/wiki/Minkowski_space#Lorentz_transformations_and_symmetry
We faced a new problem with protons and electrons, viz. why don’t they just collide and end everything in a purple flash? Planck assumed a minimum unit of energy, so that last step was forbidden. Just like when the needle stops at the last groove of the record and never reaches the spindle.
Now some complain we need “dark matter” or other kludges. They miss the point that we never know reality. We just refine the math, to make new and more subtle errors.
College was decades ago, of course, but I cannot recall ever having that kind of nightmare… maybe I did but I don’t remember it.
Then a couple of years ago I did a pretty intensive course of study for Series 7 and 66 licensing exams (financial securities) and didn’t have the dream then either.
I guess some people just don’t have that dream, while others have it over and over. Definitely not complaining though!