Lots of discussion in my feeds about this Daily Princetonian piece, “Why won’t anyone teach me math?” by first-year student Abigail Rabieh. She just took Math 202, an intro to linear algebra, and the experience was so lousy she’s done with Princeton math for good. That’s bad!

So what was wrong with Rabieh’s class?

*“Though I passed MAT 202 class just fine, my experience in it was miserable. The way the course was run did not at all set up students to succeed — or even learn math. For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control.*

*This begs the question: what interest does a department have in making it impossible to study? Study materials are given so that students can learn the course material and prepare adequately for the exam. Solution sets are part of this: to properly learn, one needs to be able to identify their mistakes and understand why they are wrong. This struggle was reflected in our exam averages, which were, respectively, in the 50s, the 60s, and the 30s.*

*I am far from the only person who felt this way about my class. MAT 202 has an abysmal rating of 2.71 on princetoncourses.com during the spring 2020-2021 semester. The evaluations on the Office of the Registrar’s website are no better. Students described the course as “disheartening” and said they “lost a lot of respect for the Math department after taking this course.” The advice that came up again and again in many reviews was: “Don’t take this class unless you have to.”*

A lot of math teachers instinctively reacted to this with defensiveness, and I was one of them. After all, what’s so bad here? You hand out practice problems for an exam because you want students to do the problems, not because you want them to read the solutions; the mechanism is that the student works all the problems they can and then asks in office hours or review session about the problems they couldn’t do. I don’t think it’s *bad* to include solutions, but I would never say that not doing so makes it “impossible to study.” Student evals, well — the literature on their unreliability is so vast and I’m so lazy that I’ll only just vaguely gesture at it. And of course, *most* people taking Math 202 are not taking it for intellectual broadening, as Rabieh admirably was; they are taking it because somebody told them they had to. That makes the evaluations impossible to compare with those for a course people take on purpose. And as for those exam scores, well — a median in the 30s is too low, that would mean I’d made the exam much too hard. A median in the 60s, on the other hand, seems fine to me, an indication that I’d written a test with real challenges but which people could mostly do.

But you know what? Our students, especially our first year students, *don’t know that unless we tell them!* A student who got into Princeton, or for that matter a student who got into UW-Madison, has probably never gotten a 60/100 on a test in their entire life. No wonder it’s demoralizing!

What we have here, as they say, is a failure to communicate. Rabieh came away feeling like her teacher didn’t care whether she learned linear algebra. I’m sure that’s not the case. But I think we often don’t *explicitly demonstrate* that care in our classrooms. It makes a difference! We are asking the students to care about our stuff and if we want them to respond, we have to show them that *we care about their stuff*. What do I mean by that, explicitly? I mean that if we think the median score on an exam is going to be in the 60s, we should tell students in advance that we expect that and talk about our reasons for writing the exam that way! I mean that we should ask for student input on how the course is going *before* the semester is over — three weeks in, send out a survey asking for suggestions, and then talk to the class about the feedback you got, showing them you want their input while you can still make use of it. It means that if you teach a crappy lecture one day — it happens! — be open about that the next time and talk about how you intend to present a better one next time. And I feel like these are mostly “things I already do,” which could just be self-flattery on my part, so let me add this: it *might* be that showing students we care could mean making nice prepared slides like the professors in all their non-math classes do, instead of just showing up and writing on the blackboard. (Doing this would be a huge change for me and it exhausts me to think about it — but would it be the right thing to do?)

We don’t really talk about this stuff when we talk about teaching. We mostly talk about *content* and *logistics*; in what order should we present the material, how much should we cover, how many quizzes should we give, what should our grading policy be, should we hand out solution sets for practice problems? That’s how we think about what good teaching is, and that’s how our students think about what good teaching is, and that’s why that’s the language Rabieh reached for in her article when she wanted to explain why she had such a bad time. But I don’t think it’s actually the problem.

I’ll bet her teacher did care. Most of us do. But it often doesn’t show; let’s say it out loud! And strive for a classroom where we’re working as partners towards a goal, not just trying to get to the end without feeling like we’ve failed.

Yeah I plan for median scores in the 60s. And I warn students. And I tell them the grades afterward (like, the A/B divide is at 70, the B/C divide is at 50, …) One thing I try to sell them on is, in this way there’s a lot of room on the upside. If you want to turn around your grade and get a B in this class, just get 60 on the final and you’ll make it! You don’t have to get 115% like you’d need to if our median grades had been 95%.

There’s certainly a “hidden curriculum” problem in STEM that is arguably especially bad in mathematics, but there’s also a weed-out mentality and an innate-talent mentality that is extremely pervasive in R1 math departments (“If you _really_ know the material…”). It’s not much of a surprise that this attitude bleeds through even to well-meaning teaching. And I’d love to know why you think an exam average of 60 in a nonmajors linear algebra course at Princeton is indicative of anything other than ineffective instruction.

I found the article frustrating for the reasons you said, but partly that’s the nature of op-eds where they’re too short to really communicate much. But I think a key point in teaching is that you should do things the way students want unless there’s a really good reason for it. One reason for this is that it buys you some good will to not do what they want when you really do think it’s important. In the case of solution sets I do think there’s a really good reason not to post solutions at least to some of the practice problems. It’s just too tempting to look at the answer, and eventually you need to start learning how to check things yourself. And if students want problems with solutions, usually their book already has them. By contrast, I think there’s just no good reason to have exams with medians that are lower than students expect. It frustrates them, and I don’t think there’s a good pedagogical reason for it, because exams settings are not good settings for challenging students because of the stress and time pressure. But basically I think it’d be good for teachers to ask themselves more often “is this really the hill I want to die on”?

1. We overestimate the importance of caring. there are very good (or at least very popular) teachers that do not care. They are just very good at manipulating the students. There are lots of teachers that care too much, which make them do too much.

2. Lots of mathematicians see the world as little children, they believe everyone else has the same knowledge and logical abilities they do. Therefore, they do not explain things that they see as trivial. You can see this is in research, but also in teaching. I think it is extremely important to explain to students why you are doing something. If you do that, they will be willing to accept non-standard things.

3. It is very important to manage expectations with students. They need to know what you expect of them and what they should expect of you, This minimizes panic.

4. I provide solutions to all homework problems, because I think it is very important for students to see not necessarily the solution, but how to write it properly. However, there is limit to how much I can provide and in some stage students should learn to check their own solution. For many years as department we did not provide solutions to old exams. This enables you to use variants of old exams questions without people trying to learn by heart old solutions and then repeat them. Sometimes this phenomenon is especially bad when they do not notice a change in a new question compared to an old one.

I suspect there’s an additional communication problem here, which is that the student doesn’t seem to know what the goals of the course are. She says “the priority of the class should be…”, but I don’t get the impression she knows what the priority of the class is! Making this clear would not only tell her what she’s supposed to be focusing on, it would also make it easier for the professor to say things like “I am doing X [not releasing solutions?] because Y, and Y is related to learning objective Z.”

To further complicate things, this appears to be a large coordinated course, and in that setting, there’s plenty of variability in the extent to which even the individual *instructors* know what those learning goals are. And being a service course, even the department is probably juggling the goals that are given to it by other departments, which may be poorly articulated, orthogonal to each other, or even directly contradictory.

And linear algebra has its own unique set of issues, given the spectrum of different ways to teach it, from almost-all-computation to almost an intro-to-proof style. If you go close to the latter route, then there will always be pushback from students no matter how well you communicate what you want them to learn from it. I can’t tell, from reading between the lines, where this course falls on the spectrum.

my reaction was 1) you do all the problems in study group. the ones nobody gets (and can’t solve with Matlab/Octave or python), should talk to the TA.

2) Linear algebra is a weird class for a history major to take, unless they are going to do some weird data science PCA history project. Frankly, if linear algebra turns a history major off of math, it’s normal. Calculus, stats, probability are more applicable. Somebody should have told you, don’t take that.

3) sometimes there’s a bit of an advising gap, students need a bit more mentorship than is available. I did a not extremely math-heavy econ major, worked in an industry economics department, applied to grad school, got told to go away and not come back until I aced upper level differential equations. Somebody should have told me to take linear algebra etc., I ended up doing ULAFF / Khan Academy, which actually do want to teach people math more than your typical undergrad courses

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The post title reminds me of the 1980s band “Y Kant Tori Read.”

As a former student of both STEM and liberal arts, this was my reaction to that piece, as well.

When I was in college (not Princeton but another highly ranked and competitive school), it was not unusual for me to take a history test where professor said he was disappointed that the median was only 75, and then take an engineering test where the median was 25 (also out of 100) and the professor didn’t find the low grades to be noteworthy.

I don’t know if it’s inherent to the subjects, or cultural differences between departments, or that liberal arts professors have smaller departments with fewer prereqs and they’re trying to attract more non-majors to take their courses as electives, or what — but test grades across departments are not generally comparable.

The opinion sections of college newspapers are of widely varying quality. I wouldn’t worry about what first-year students think of college. It could have been any course thrown under the bus. If they think it’s tough when they don’t get the answers handed to them, just wait until they get into the real world!

Princeton’s math department is consistently rated (by non-freshmen) as one of the best in the world.

Having misbehaved for most grades because of boredom, almost 57 years ago I found myself in an honors calculus class, emphasis on theory/theorems/etc. Not being able to connect it to anything, I was alienated. Forward some years to a graduate course in “applied linear analysis” I found it very interesting, connected to a multitude of real problems (similar courses in Fourier and Complex Analysis). I suggest it isn’t simply having solutions for some students, it is about seeing the connections to the real world. I understand some people find beauty in math, and elegant connection like those described in “Shape”. However, many of us wish to understand the power of mathematics and how it can be used to solve problems in science or engineering.

The main complaint in the article seems to be the lack of worked solutions, apparently it’s impossible to learn without them. Maybe, but providing worked solutions to everything often leads to mindless imitation, without understanding. Even worse, when the imitator gets it wrong, they have no idea about why they got it wrong.

The other striking thing in the article is the absence of any reference to a library. We’re a long way from the medieval university where the professor was the one with the book. If you want to learn linear algebra or any other subject, is it really so hard to seek out alternative sources of information? Schaum Outline Series, anyone?

Many years after I had a similar experience at MIT I read Hadamard, Jacques (1949), “The Psychology of Invention in the Mathematical Field”, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954 and began to understand why math education failed me.