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What are some interesting mathematical things you have learned while grading student work (or marking, if you prefer)?

It is final exams time here, so if anyone can help cast a more positive light on the grading experience, it would be most welcome.

Answers can be things that students wrote, or inspired by something a student wrote, or just something we learned during the grading process in some way. For example, clever proofs that students came up with; nice counterexamples or insights; interesting new questions inspired while grading; even just something you looked up to find out if a student's work was valid. However, for an answer to be interesting, it should be something beyond just a different way to solve a problem.

Zach Teitler
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I once asked students to find the derivative of $x^x$ (with respect to $x$). One student figured that if the exponent were a constant then the answer would be $xx^{x-1}$ which is to say $x^x$, while if the base were constant the answer would be $x^x\log x$, so she added the two together to get $x^x+x^x\log x$. I was just about to mark the answer as wrong, when I realized that she had arrived at the correct answer – and, later, realized that it wasn't a coincidence, her unorthodox method actually works in a more general setting.

Gerry Myerson
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    So basically she is using df(x,x)/dx=partial_1 f+ partial_2 f . nice – lalala May 08 '20 at 16:14
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    Right. More generally if $f$ is given by an expression in which $x$ appears several times, and $F(x_1,\dotsc,x_n)$ is defined by replacing each appearance of $x$ in that expression with successive new variables, then $df/dx = \left( \sum_{i=1}^n \partial F/\partial x_i \right)|_{x_1=x_2=\dotsb=x_n=x}$. Once you see it, it pretty much falls out from the "Chain Rule for Paths" in multivariable calculus. But that doesn't take away that it's a really cool idea! – Zach Teitler May 08 '20 at 17:39
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    But how did you mark it? Was there any way to tell if she just tried a guess and was lucky, or did she actually have an insight into multivariable calculus? And in the (maybe more likely) first case, did you still give points for the right answer? Full points even? – Torsten Schoeneberg May 09 '20 at 21:07
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    “The last act is the greatest treason. To do the right deed for the wrong reason.” T.S. Eliot, Murder in the Cathedral – Nicola Ciccoli May 12 '20 at 15:22
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    @Torsten, this happened about 40 years ago, so you'll forgive me if I can't remember how I marked it. – Gerry Myerson Jun 19 '21 at 10:40
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Possibly not what you're looking for, but: the things I've learned while grading are mostly pedagogical, not new mathematical facts (in fact, teaching at a community college as I do, I'm not sure that's ever happened).

One of the main things that sticks with me is this: The rather incredible kaleidoscope of ways that students can misunderstand or be wrong about a thing. Generally the faculty in my department push the thesis that all-multiple-choice testing is fine (even required) for most courses up to calculus, say. The instinct is that it's "obvious" what the common mistakes might be, and these can be covered in a set of 3 or 4 distractor options.

Now, I'm one of the very few instructors (maybe the only one now) who insists on at least a few open-ended questions on any of my tests to see what student work is actually like (and give feedback on it). In doing so, I've discovered a whole lot more "ways to be wrong" then I'd ever imagine, or that could be covered in a multiple-choice test. Looked at from another perspective: any multiple-choice test is an enormous safety net, because it actually rules out the great majority of student responses.

One example: On a college algebra test a few years back, I asked: "Write the equation of the line, with the given properties, in slope-intercept form: Through (-2, 6) and (2, -7)". Out of 40 test submissions, I found there were 26 different unique responses. (!) More specifically: 14 students got the right answer, 2 students duplicated a certain wrong answer, and 24 students each had a unique wrong answer, duplicated by no one else. (Which brings to mind Tolstoy's adage, "All happy families are alike; each unhappy family is unhappy in its own way.")

Second example: For the first time this semester I'm giving programming tests on an actual computer in our lab. (For 20 years I gave programming tests on paper; transitioned to online tests for the COVID pandemic; and found enough advantages that I wanted to keep that as we switched back to in-person teaching.) Coincidentally, the lab has screen-monitoring software that's always on, so without planning it I found myself watching students write code in real-time on a test for the first time ever. I was amazed at how many of my second-semester CS majors couldn't write even basic structures; several were taking many shotgun attempts at simply declaring an array, or couldn't even write a simple for loop, for instance (e.g., mixing up bits of syntax between while, for, and do-while loops, taking as many as 10 minutes of iterations fired at the compiler trying to get it right). One student apparently actually memorized the entire practice test solution, typed that in first (with great difficulty and many compiler errors), and only once that was running tried to modify it to match the actual test question.

Pretty fascinating stuff which I'd have never known if I didn't get to see the students' actual work process.

Daniel R. Collins
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    The programming example is a horror story. – Steven Gubkin May 13 '22 at 10:37
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    @Steven Gubkin: The programming example has me wondering how they would have handled the only programming course I've taken (PL/1), which was in Fall 1977. After working out how you think to do it (an hour for the first couple of weekly assignments, and 10+ hours for the last 5 or 6 of the semester's weekly assignments) and writing the code lines on scrap paper (after the first assignment you accumulate a lot of computer scrap paper), you walk to the computer lab (if you live on campus or within 2 kilometers), bringing your various attempts (continued) – Dave L Renfro May 13 '22 at 14:20
  • (because what you think is going to work might not work, but maybe you’ll figure out that something else you tried can be used) and of course bringing your text (this was our text) for all the absolutely essential “grammar rules” that, unlike math formulas, you can’t derive or reason out what a missing detail must be. Then you wait for a keypunch station to become unoccupied so that you can type the punch cards, one card for each line of code. There might be a long outside the keypunch room if you go between about 8 AM and 10 PM on a day (continued) – Dave L Renfro May 13 '22 at 14:20
  • when the introductory CS classes have an assignment due, but this wasn’t much of a problem for me as I almost never was there before 10 PM. You’ll spend anywhere from 15 minutes (first assignment) to a couple of hours (last few assignments) typing the cards. Then you go to the drop-off/pick-up desk to deliver your card deck, the top card being one you filled out with your name, instructor’s name, course name, time of submission, and probably your Social Security number (which at that time was what many U.S. colleges used for your student ID number). (continued) – Dave L Renfro May 13 '22 at 14:21
  • Depending on how busy things were (early AM hours had near immediate turn-around times – 10 to 20 minutes), you may have to wait a while until you got the printout back. If you were doing this during usual daytime hours, you could expect 1 to 2 hours, so usually you tried to finish before an upcoming class, submit it, go to your class, then return and hopefully it’ll be ready by then. The first attempt almost never compiled. Thus, you’d get several/very-many compiling errors, and you would have to spend some time going through them on the computer printout sheets (continued) – Dave L Renfro May 13 '22 at 14:21
  • (this is how you accumulate computer printout scrap paper). If it was just a missing semicolon at the end of a line of code, the mainframe’s compiler would fill it in. But often there would be undeclared (or improperly defined) variables, infinite loops that got timed out, and other things that in some cases required careful scrutiny of everything (i.e. usually more an hour’s work). Then you’d repeat the process beginning with typing cards (new cards, replacement cards, etc.). If you’re lucky, it would compile the second time. (continued) – Dave L Renfro May 13 '22 at 14:22
  • But almost always you didn’t get correct results for the trial data/examples you included to test the program (the nature of your trial examples was a significant part of the grade, this being regardless of whether the program worked). I never took another programming course and have essentially stayed away from any coding since (stack exchange mathjax and Latex being about the only exceptions I can think of). One of the things that struck me back then was that I considered these programs very much like writing math proofs, (continued) – Dave L Renfro May 13 '22 at 14:22
  • only they were MUCH HARDER (and I was taking graduate analysis and algebra and topology courses around this time). I never understood how someone could follow the super-restrictive rules and be successful in writing “program proofs” and yet (for some students) have so much trouble with basic upper undergraduate level “math proofs”. – Dave L Renfro May 13 '22 at 14:22
  • There might be a long outside the keypunch room -- Missing word: "... a long line outside ...", in case anyone was confused here. By the way, I normally wouldn't even consider "mathjax and Latex" as being coding, but I've seen them mentioned in this way. – Dave L Renfro May 13 '22 at 14:30
  • I'm distracted by the math not working out. 26 unique answers: 1 correct answer, 1 incorrect answer that was given by multiple students, and 24 unique incorrect answers -- that part works out, but 41 students: 13 correct, 2 duplicating an answer, and 24 uniquely incorrect is only 39. Did 2 students not submit an answer for this question? – shoover May 13 '22 at 15:06
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    @DaveLRenfro I have a theory about how students can succeed in writing programs but have great difficulty writing proofs. If a program is written sloppily, the compiler will say "syntax error" and you can't argue with it. If a proof is written sloppily the professor (or TA) will say something more polite and students can and will argue. – Andreas Blass May 13 '22 at 15:18
  • For those interested, an online history of the CS department I took that 1977 class at is here. It probably didn't help me much that I took the honors version of the class. I'm pretty sure everyone in it had the equivalent of 2 semesters of calculus (a BC academic placement class in high school, or the equivalent somehow), and our first assignment dealt with finding approximations on certain pre-specified intervals of $\sin x$ using its Taylor series expansion. Compared to the later ones involving strings and such, this was quite simple. – Dave L Renfro May 13 '22 at 16:39
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    @shoover: Thanks, and I thought I'd double-checked, too. Per my records, looks like there was one more student who answered correctly, and one who didn't get past copying the point-slope formula, so didn't generate a tally for a response. Fixed it now. – Daniel R. Collins May 13 '22 at 16:39
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Reading the answer posted by Daniel R Collins reminded me of something else I learned while marking student work. Not exactly something mathematical, more something about constructing math exams.

I had decided it wasn't fair to have the "distractors" in a multiple choice exam be answers that a student could come up with by making a simple error such as a sign error, the kind of error for which I would give partial credit if it weren't a multiple choice exam. So I wrote a test where all the distractors were crazy things that no student could possibly come up with.

To my horror, the scores on this test were awful.

I figured out why (I think). If a student makes a simple error, and arrives at an answer that is one of the choices, then the student marks that choice, and moves on to the next question. So, the student gets that one question wrong, but spends no more time on it, loses no confidence, and may well get some of the later questions right.

But if a student makes a simple error, and arrives at an answer that is not one of the choices supplied, then the student goes back over the work, perhaps starts the problem from scratch, makes the same error, or maybe a new one, still doesn't arrive at an answer among the choices supplied, maybe panics, but in any event spends a lot of fruitless time on that one problem, has less time to spend on the rest of the test, and less confidence about tackling the rest of the test.

So I learned the reason why distractors should be "plausible" answers. Maybe everyone else already knew that, but no one ever told me, and I had to learn it the hard way.

Gerry Myerson
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    Great answer. Could you also include an example? I think if the distractors are completely off (for example not even in the right category), one could go for the principle of exclusion. – Jasper May 15 '22 at 08:22
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    @Jasper, that's what I thought, that any student with a bit of insight could see that only one of the choices could possibly be an answer, and then mark that one without even doing any math. But it seems my students didn't approach tests that way. As for specific examples, this was 40 years and four jobs ago, I'm afraid I don't have memories or records of actual examples. – Gerry Myerson May 15 '22 at 10:02
  • Was it like asking for the inverse of a matrix with distractors "42", some integral, and a vector? or was it not "that bad"? – Jasper May 15 '22 at 13:57
  • @Jasper, maybe not quite that bad, maybe more like a question where a moment's thought would show the answer had to be an integer between one and ten, and the distractors were $2000000$, $-3$, and $\sqrt2$. But, really, I don't have access to any notes from way back then, so I'm just guessing. – Gerry Myerson May 16 '22 at 02:03
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    Just go ahead and add this example to the answer, it doesn't matter if it was exactly this one, it is still helpful – Jasper May 16 '22 at 05:23
  • +1 Amazing! (Interestingly, I find that I could pass almost any standard multiple-choice test in mathematics, without looking at any of the questions, by simply taking the intersection of all the options for any answer. This kind of test would at least foil that strategy [not that I think any of my students do that anyway.]) – Daniel R. Collins Aug 01 '22 at 14:59
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I gave an advanced course on Probability that contained some ergodic theory. In exercises, I outlined the usual proof of the equidistribution of $e^{in\theta}$ on the circle, for $\theta/\pi$ irrational. The proof I knew was generalizing equidistribution from indicators of intervals to arbitrary (say, continuous) functions and then using Fourier transform.

Then one of the students pointed out the following elementary solution. Assume that $I,J$ are half-open intervals on the circle, and $I$ is longer than $J$. Then, you can write $I=I_1\sqcup I_2$, where $I_2$ is a translation of $J$ that follows $I_1$ counterclockwise. Let $n_1$ be the first time $\exp(i\theta n)$ belongs to $I_1$, and $n_2$ is the first time after $n_1$ that it belongs to $J$. Then, $\exp(i(n+n_2)\theta)\in J$ implies $\exp(i(n+n_1)\theta)\in I$, which readily implies $$ \frac{1}{N}\#\{n\leq N:\exp(in\theta)\in J\}=\frac{1}{N}\#\{n_2\leq n\leq N:\exp(in\theta)\in J\}+o(1)\leq \frac{1}{N}\#\{n\leq N:\exp(in\theta)\in I\}+o(1). $$ This means that $\liminf$ of the quantity on the right is greater than $\limsup$ of the quantity on the left. From this and additivity of density the result easily follows.

Kostya_I
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  • You mean for $\theta$ a fixed number irrational w.r.t. $\pi$, yes? (i.e., such that the ratio $\theta/\pi$ be irrational) – Vandermonde May 29 '20 at 18:48
  • @Vandermonde, yes, of course. Thanks! – Kostya_I May 29 '20 at 19:05
  • I could not get this proof to work. There are examples where $exp(i\theta n_2)$ is very close to the right endpoint of $J$, and $\exp(i\theta n_1)$ is very close to the left endpoint of $I_1$. Choose some $n$ such that $\exp(i\theta(n+n_2)$ is very close to the left endpoint of $J$. Then $\exp(i\theta(n+n_1))$ will be outside of $I$. – Yongyi Chen May 18 '22 at 06:09
  • @YongyiChen, I'm no longer sure what I meant two years ago... how about this: let $x=e^{i\theta n_0}$ be the first point hit on $J$. Let $y\in I$ be such that rotating $x$ to $y$ shifts $J$ strictly inside $I$, with an $\epsilon$ of room on both sides. Let $n_1$ be the first time we hit $(y-\epsilon,y+\epsilon)$. Then, $e^{i\theta(n+n_0)}\in J$ implies $e^{i\theta(n+n_1)}\in I$. – Kostya_I May 18 '22 at 21:51
  • @Kostya_I That setup makes more sense. – Yongyi Chen May 19 '22 at 16:08
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An answer I saw a few times while marking a particular question was $$\ln(x+1)=\ln(x)+\ln.$$ I think this explains the 'everything is linear' phenomenon: everything is linear because everything is multiplication.

Jessica B
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Question was:

Evaluate $\int{(\cos^3{x}-\cos^5{x})}\text{d}x$.

Student (desperately weak at math) simply wrote: $\dfrac{1}{3}\sin^3{x}-\dfrac{1}{5}\sin^5{x}+c$.

I thought, "She's using a wrong method, badly"... then I realized that her answer was correct.

It turns out that if you replace the 3 and the 5 (in both the question and answer) with any other numbers, it doesn't work.

Dan
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  • So they'd used Wolfram/somesuch ? – ryang Jul 25 '22 at 16:37
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    @ryang It was an exam question, under normal exam conditions, so no internet. She didn't show any working. I think she just mangled up the power rule together with the integrals of cos and sin, and her answer was correct by coincidence. Either that or it was a stroke of genius. – Dan Jul 25 '22 at 16:44
  • This could be reformulated as a seemingly intimidating homework problem. Very cool. – James S. Cook Jul 25 '22 at 21:39
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This, from an exam I proctored for an absent teacher.

$y=32.5+27.5\cos\frac{\pi}{10}\left(x-8\right)$

Students were given a real life word problem, the movement of a rider on a ferris wheel.

A quick side note, one student came up to me to ask what a ferris wheel was, and I was reminded we often make assumptions about basic knowledge that might not be true. One can get to age 16 and never seen a ferris wheel.

The equation above was the correct equation for the rider height at time x. The next question was to give the rider's height at time x=2. The correct answer was 24.002 which rounds nicely to 24. Multiple students answered -124.42. Because they entered the numbers into their calculator without proper parenthesis.

For me, there are two issues. Students are using calculators from grade school, but not being taught proper use. We can use a bit of time each year to walk kids through the required keystrokes to get good results.

Second, and most concerning. The students graphed the equation. The graphs were beautiful, minimum 5, maximum 60. I struggle to understand how they can so easily get such a result (a negative height!) and not return to the equation to track down their error.

TL:DR

In general, we need to address calculator skills with students. We also need to mindfully teach the skill of checking one's answer.

JTP - Apologise to Monica
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    how did they get the negative number exactly? – Jasper May 16 '22 at 15:30
  • Easy. Put parentheses around cos pi/10. This is what a calculator would give if parentheses are not properly typed in. – JTP - Apologise to Monica May 16 '22 at 15:37
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    +1. That said: I can teach estimating and double-checking basic equations every day to my remedial students and it just comes off as useless blathering to them. (Literally: On the last day I've had students say, "I never got this estimation thing".) Sometimes I think students need to work on a physical construction and have it really break before they might care about this. – Daniel R. Collins May 16 '22 at 16:07
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    I think a lot of students regard "math" as separate from "real world". After all, it is only in "math" that people will buy 300 watermelons and so on. So you get students thinking that if a sweater on 20% sale cost $80, then its original price could perfectly well have been $60, after all 80-20=60, why not; and if they wonder why the original price is higher than the sale price, they probably think, well, it's math, it's not supposed to make sense... – Zach Teitler May 16 '22 at 19:55
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    In regards to the negative height not triggering a sense of "something's wrong", I blame poorly-designed "real life" problems. I recently raised an issue regarding an example exam problem about the behavior of a plucked string. The math all worked out, but the equations described a one-meter string with the elasticity of Silly Putty, "plucked" by pulling the center almost four kilometers to one side. If you want students to sanity-check their answers, you need to consistently provide questions with sane answers. (Negative height just means the wheel is built in a trench.) – Mark May 16 '22 at 21:56
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    Mark - to be clear, multiple students (3) had this incorrect answer. All three had beautiful graphs that had proper min/max. I’m less startled by a subterranean depth than a result at odds with the graph they produced. – JTP - Apologise to Monica May 16 '22 at 22:42
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    About your side note. I’m teaching classes in English, but since I’m not a native speaker a few years ago I asked an English counsellor from the language department to attend one of my classes to give me feedback about my pronunciation and language issues. During that lecture, to illustrate a certain concept, I made reference, as example, to a stagecoach wheel from old western movies. At the end, the counsellor asked me: Are you sure that these students know what a stagecoach is? – Massimo Ortolano May 17 '22 at 19:17
  • @ZachTeitler's comment about students separating "math" from the real world reminded me of another unfortunate separation: "math" as methods of calculating (e.g., methods of integration) vs. conceptual matters (e.g., what is an integral). A student suffering from both of these misconceptions must have a really dismal view of math --- mere calculations disconnected both from reality and from concepts. – Andreas Blass May 18 '22 at 00:14
  • " I struggle to understand how they can so easily get such a result (a negative height!) and not return to the equation to track down their error." They just don't relate one thing to another. These are two completely separate algorithmic tasks for them and they care about the consistency of the results no more than a computer would. The criterion of the success for the latter is that all operations can be carried out and the algorithm terminates in finite time. The fault is partially ours: we often do not teach students but try to program them instead. – fedja Jun 04 '22 at 22:37
  • @fedja - for this specific type of problem, I try to use the real-life aspect of it as much as possible. After the word problem is presented, the first question is to create the graph and then the equation. The graph should be accurate enough to be used as a guide for the answer, if nothing else, to within +/- 15% or so. I try to teach the approach of checking one's work where possible, and to question "does this result make sense?" – JTP - Apologise to Monica Jun 05 '22 at 01:05
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    I saw this when tutoring. Students basically did not know how to properly punch things in, let alone how to punch things in to minimize errors. – DKNguyen Aug 01 '22 at 13:35
  • @DKNguyen - as I ponder this, much of it comes down to PEMDAS, a skill that’s really middle school. HS teachers aren’t spending much time reviewing it as a calculator skill. – JTP - Apologise to Monica Aug 01 '22 at 13:48
  • @JTP-ApologisetoMonica They just don't connect the two together because they have so much coming at them at the same time in a math problem. And even if they do, they still may not punch it in properly because they aren't aware of the calculator's expectations. It's almost like a mini programming task. Way too little use of of brackets. The big trend I find is that students want to enter the entire thing in at once, which tends to make things worse, rather than break it into chunks but that has disincentives of its own like writing things down or planning ahead. – DKNguyen Aug 01 '22 at 13:50
  • What were the units? – Jasper Nov 17 '23 at 22:49
  • The students were instructed to use radians. The pi within the equation should imply radians. – JTP - Apologise to Monica Nov 19 '23 at 14:19
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I once met a student who computed $\sqrt{29}$ to $4$ digits precisions manually. At first I suspected cheating with computer since it's an at home exam. But the student showed me they can use bisection method to compute this so I let it go.

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    Computing square roots by hand used to be taught in secondary school 100+ years ago. The method I learned -- from an 1899 math text -- resembles long division and is not much more complex than that. The method generalizes to cube, fourth, etc. roots. – shoover May 08 '22 at 05:23
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    1897, not 1899. Here's the 1904 edition: https://archive.org/details/arithmeticmensu00schogoog/page/n104/mode/2up – shoover May 13 '22 at 15:11
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    @shoover I was taught the method when I was 11 in middle school, around some 40-odd years ago... No need to go back one century! And I actually used that or series expansion whenever I forgot the calculator at a university exam (something that happened frequently). – Massimo Ortolano May 14 '22 at 11:30
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    I also was taught manual square root finding a little less than 40 years ago. More information about it here. – JRN May 15 '22 at 05:16
  • @JoelReyesNoche This was the algorithm that was taught to me in middle school (I’m sorry that the page is in Italian, but maybe Google translate can help). – Massimo Ortolano May 15 '22 at 09:12
  • @MassimoOrtolano, it seems that the methods we were taught are the same. – JRN May 15 '22 at 13:39
  • In Spain children are still taught how to find square roots by hand (at the beginning of middle school, approximately grade 6-7). The algorithm taught is essentially the same as that linked to by Massimo Ortolano. – Dan Fox May 24 '22 at 07:13
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    @MassimoOrtolano That looks like the same method I learned, only yours is illustrated better. – shoover Jul 25 '22 at 23:04
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An interesting one I saw on a student exam many years ago.

The sequence diverges because the Cauchy criterion is dissatisfied.

Gerald Edgar
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Question was:

"Find the Maclaurin series of $\ln{(2+2x)}$ up to the term in $x^3$.

Student A: $\ln{(2+2x)}=\ln{(2(1+x))}=\ln2+x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-...$

Student B: $\ln{(2+2x)}=\ln{(1+(1+2x))}=(1+2x)-\dfrac{(1+2x)^2}{2}+\dfrac{(1+2x)^3}{3}+...$

Of course, Student A is correct. (Student B has found a Taylor series but not a Maclaurin series.)

But it is interesting to equate the coefficients in the two answers (temporarily ignoring the fact that the two series have different intervals of convergence). We get the following results:

Equate coefficients of $x^1$: $2-2+2-2+2-...=1$, that is, $1-1+1-1+1-...=\dfrac12$

Equate coefficients of $x^2$: $-2+4-6+8-10+...=-\dfrac{1}{2}$, that is, $1-2+3-4+5-...=\dfrac14$

Equate coefficients of $x^3$: $\dfrac43(2-6+12-20+30)$, that is, $1-3+6-10+15-...=\dfrac18$

And so on.

Dan
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  • I once designed a (U.S. based) calculus 2 homework problem based on this idea, where to get things to agree for a particular Maclaurin series term (probably the linear or quadratic term, but I don't remember now) you have to correctly account for all the subsequent contributing coefficients in the non-Maclaurin expansion, and in doing this you'd get an infinite series whose explicit sum can be determined (with hints I provided). – Dave L Renfro Jul 25 '22 at 13:44