A friend of the blog has a thought-provoking long essay on the challenges of teaching students who lack the necessary math skills, but want to get a degree in a math-heavy major.

A friend of the blog has a thought-provoking long essay on the challenges of teaching students who lack the necessary math skills, but want to get a degree in a math-heavy major.

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Thank you!

That essay reminded me of the trauma of having to take real physics (I.e. with calculus) nine years after having last taken calculus and 11 years after last taking algebra. It was hard mostly because of the algebra, but even more so because of the trig. And yeah, I think the professor gave us a break because all of us were premed and taking it because there wasn’t an easier version offered over the summer, and he knew none of us were going to be physicists.

Look, most jobs, even math oriented jobs in consulting and finance, require little more than basic arithmetic. Yet for some reason majors like engineering, business, and economics are all favored by these companies, and they actively look down on people who have degrees in the humanities and social sciences. This is why you end up with the problem of the math illiterate in your classes. People want to be able to get a job when college is over. That’s the primary reason most of them are there. I also can tell you about the innumerable times that some lazy, mediocre, white man-friend of mine condescendingly insinuated I was stupid while he was barely pulling a C in some Econ class he was taking. Of note, all of these men have gone on to be at least moderately successful at the companies that hired them after graduation.

Anyway, this is all to say that while this sounds very frustrating to deal with as a professor, there is a reason this happens. Imagine what it’s like when I am working with an anesthesia resident who is the equivalent. That person is going to graduate and move on to actually kill someone.

Very misleading title. It’s not about teaching math at all, just about kicking people out of a physics major if they have trouble with algebra, which, whatever, there are probably too many physics majors. (My department has a summer program where we teach unemployed physics graduates how to do finance so they can get jobs. It is both popular and expensive.) Seems to me that a better solution would just be to say you have to have a B in calc 2 (or a 5 on the AP exam) to major in physics and then people wouldn’t waste their time in the major.

But I went through all this before the last time this dude posted here, though I didn’t know he was gate-keeping physics at the time. Whatever. The world doesn’t need more physicists, especially not mediocre white dudes. If anon’s department is fine with shrinking their faculty base, go ahead and have an elite program. (Because student enrollment does affect faculty hiring.) That’s their internal decision.

My physics experience in mechanics in high school was terrible because the teacher was a sexist douchebag (he had an article on his office door that argued that feminism had gone too far, made fun of using “he or she” instead of just “he”, explained that he’d had complaints about always hugging women students but didn’t mean anything by it so he was going to keep doing it, and touched my breast once.) I did much better in E&M which was taught by a woman engineer. Even her right hand rule was better. My math skills are and have been excellent.

Omdg, I can assure you that most of my students don’t even know what Goldman Sachs and McKinsey are, let alone that they prefer STEM majors over humanities. We aren’t that kind of school.

Also, my mother-in-law is an anesthesiologist, she has worked at teaching hospitals, and she has similar stories of residents who will be homicidal in private practice. You have my sympathy.

N&M, triaging the bottom 20% does not an elite program make. Also, I’ve never touched a student’s breast (or any other body part aside from handshakes), and I’m not sure how me accepting tuition money from the non-dischargable loans taken out by my weakest male students will right the wrong done to you by that creep. But if the only way to overcome sexism is for me to coddle mediocre men, well, the world is an even stranger place than I thought.

On the subject of privilege, what kind of high school did you attend where the teachers had offices separate from their classrooms and E&M was taught by an engineer? In most public high schools (including mine) the classroom doubled as an office and the desk doubled as a lecture podium. Also, in a disturbingly large number of high schools physics is taught by someone whose degree isn’t in a quantitative STEM field, and only one physics class is offered, not separate mechanics and E&M classes. For someone who loudly displays scars from an inequitable world, I detect more than a trace of privilege in your story.

And if you are in the kind of place where physics grads write large checks to boost their credentials, I doubt your median undergrad (or even summer physics grad) is similar to mine. Try teaching under my conditions before you claim to know how wrong I am. Try spending an hour a week with the individuals whom I am currently seeing in special dedicated office hours (on top of my five regular office hours and four hours of small group tutorial activities) to help with math in my upper division class, who struggled similarly in classes taught by other faculty (including some genuinely warm and compassionate women, and a dude who lets the whole world know how aware he is of privilege and oppression). Try helping them even when they struggle with the alternative assignments that I offered to them and a few other floundering students who need refreshers. Try doing what I do in the setting that I do it in and then maybe I will take your righteous scolding seriously.

Omdg, I can assure you that most of my students don’t even know what Goldman Sachs and McKinsey are, let alone that they prefer STEM majors over humanities. We aren’t that kind of school.

Also, my mother-in-law is an anesthesiologist, she has worked at teaching hospitals, and she has similar stories of residents who will be homicidal in private practice. You have my sympathy.

N&M, triaging the bottom 20% does not an elite program make. Also, I’ve never touched a student’s breast (or any other body part aside from handshakes), and I’m not sure how me accepting tuition money from the non-dischargable loans taken out by my weakest male students will right the wrong done to you by that creep. But if the only way to overcome sexism is for me to coddle mediocre men, well, the world is an even stranger place than I thought.

On the subject of privilege, what kind of high school did you attend where the teachers had offices separate from their classrooms and E&M was taught by an engineer? In most public high schools (including mine) the classroom doubled as an office and the desk doubled as a lecture podium. Also, studies have shown that in a disturbingly large number of high schools physics is taught by someone whose degree isn’t in a quantitative STEM field, and only one physics class is offered, not separate mechanics and E&M classes. For someone who loudly displays scars from an inequitable world, I detect more than a trace of privilege in your story.

And if you are in the kind of place where physics grads write large checks to boost their credentials, I doubt your median undergrad (or even summer physics grad) is similar to mine. Try teaching under my conditions before you claim to know how wrong I am. Try spending time every week with the individuals whom I am currently seeing in special dedicated office hours (on top of my five regular office hours and four hours of small group tutorial activities) to help with math in my upper division class, who struggled similarly in classes taught by other faculty (including some genuinely warm and compassionate women, and a dude who lets the whole world know how aware he is of privilege and oppression). Try helping them even when they struggle with the alternative assignments that I offered to floundering students who need refreshers. Try doing what I do in the setting that I do it in and then maybe I will take your righteous scolding seriously.

@friend – perhaps the problem is that a significant proportion of students where you teach just aren’t very smart? They probably would crash and burn in a writing intensive major as well.

I went to a public magnet high school for Gifted and Talented kids. I think you’re trying to say I’m so privileged that I can’t possibly understand your situation, which… whatever. That’s irrelevant to your argument. Which we have already had the last time you posted about this on this blog. You kept moving goal-posts then too.

It’s just a fact that if you want to increase the cutoff for majors, it becomes a more elite major. Elite is a relative term. Every college and university has elite majors and gut majors. The uni in the town where I grew up (regional state school) had communications as its gut major–the easiest major that took people who had flunked out of other majors. The undergrad I went to had communications as it’s most elite major– you couldn’t even declare it until you were a junior. You get to make these decisions as a faculty, but it is just the truth that administrators try to match faculty sizes with student enrollment. So if you want a more elite major, you will have difficulty justifying replacement hires. We would all like to only have the best and brightest. But there are trade-offs.

PS I haven’t heard Common People in years! Thanks for the ear candy!

“perhaps the problem is that a significant proportion of students where you teach just aren’t very smart?”

Yeah, try saying that in the faculty lounge. I’ll make popcorn. “No, see, everyone can succeed. Grit, growth mindset, etc. Plus, our judgments of who is smart are biased and exclusionary and test scores mean nothing. Etc, etc, etc.”

The last time I tried saying something about kids not being smart my dean heard it and started bragging about allegedly having a low IQ score. I closed my eyes, thought about my outstanding mortgage balance, and refrained from making the obvious joke.

Here is another good song for those who teach at schools higher up in the prestige hierarchy:

Better versions:

I’m pretty sure this is a both/and situation.

Yes, there are students who can’t algebra their way out of a paper bag but who can write a reasonably logical argument in a paper and are better off in majors other than physics.

AND if you look at who walks through the door to college appearing the least capable, it’s mostly students of color, which is not a coincidence. Some of them are too far behind to ever catch up. But some of them just need some extra support for their first couple of years, and have the grit/persistence/whatever to make it work. I’ve seen both, and I can’t always tell who they are from day 1. I’d be willing to bet you can’t either. Telling a student they shouldn’t be a physics major because they are shaky on algebra first semester freshman year unnecessarily discourages some of those students, and even if you apply the discouragement in a race- and gender-neutral fashion looking only at their (current) math skills, the impact is very much non-race-and-gender-neutral due to the social factors at play.

I’m coming around to the fact that there are exceptions. I had an advisee who I eventually realized was not only shaky on algebra, but also struggled with arithmetic and negative numbers as a freshman, and as a junior she still hasn’t caught up (we are a small highly-ranked liberal arts college and we don’t even offer a “college algebra” class — our lowest math is a one-year sequence that does precalc and calc together). I wish I’d caught on sooner and encouraged her to play to her strengths better — she has gotten A’s in social science research methods classes while consistently scraping by with D’s in math and physics. She is smart and determined, but the gap in her math skills is just too large.

On the flip side, you sound just like the physics faculty who discouraged another female student of color in her first year at our school — she almost dropped the physics major, except that our one female faculty member at the time took her under her wing, and she flourished, went through a bridge program, and is now doing a PhD at Harvard. So there.

It’s a really hard problem, and I think you’re oversimplifying it. It’s possible to do very real (and racially aligned) harm by discouraging students with weak math skills early on (by junior year, OK, fine). If you know a student well enough to know what their strengths are, I think you can maybe VERY gently point out their strengths to them, including if they seem to have a strength that shines through in another area. But I think it is rarely to never the role of a physics faculty member to tell a student in their first, or maybe even second, year that they should not major in physics because their math skills are too weak. By all means, encourage them to build up their math skills, provide resources to help them do that, emphasize how central math is to physics. But don’t be that guy who tells them they can’t do physics. You just don’t know yet.

lyra211, I agree that first semester freshman year is way too early. But I mostly teach sophomores and juniors. And junior year is rather late to change major…

Your points are certainly valid, hence we need a balance. In my institution we are very unbalanced in one direction. Accepting money from people who just aren’t getting it feels less like teaching the lost sheep how the heavens go, and more like accepting 30 pieces of silver. Yes, we can make mistakes if we encourage too many people to switch, but we also make mistakes if we never tell anyone to cut their losses.

Also, everyone is desperate to assume that the students I want to send away are mostly women and under-represented minorities. If they assume that then there’s supposedly moral clarity around encouraging people to continue taking on non-dischargeable loan debt for this. Well, apparently it’s different at your school, but I’ve got a heaping helping of white and Asian men who struggle with math. Can we have some moral clarity about telling them to cut their losses? Or do we have to keep them around because something something equity? I remember a conversation with an Asian guy who was more focused on training for his upcoming weight-lifting competition than developing his math skills, and it showed. Full points for shattering stereotypes, but shattering stereotypes won’t help him graduate.

Finally, the students who can get into a highly selective liberal arts college are on a different level than the students at my decidedly non-elite state school. My students aren’t looking at physics as a way of getting a job offer from an investment bank. So maybe the issues in play are different here.

Friend of blog—to your last point, what do you think the students in your program are after? Why are they enrolled in physics programs to begin with? Presumably you have had some interaction with these students and maybe have some idea of what their needs and ambitions are.

I have a PhD Applied Physics I have worked in industry for nearly a decade. A lot of problems that physicists end up having to solve on the job don’t require much math. These are things like figuring out how to clean glass; figuring out how to maintain a vacuum chamber; interface hardware to software. I actually think that many of these practical skill are not taught well enough in physics programs at universities (undergrad or grad), to the detriment of many things.

Adding to what Lyra said: Every time you tell a kid that they are bad at math, that makes my job exponentially harder. I am a brilliant math teacher, but because of people with fixed mindsets who believe that kids can’t learn math, I have to spend most of my energy in that first stats class removing the math phobia so they can actually learn the material. Almost anybody can get number sense if they’re willing and able to put the time in, but people who have been told they’re bad at something are afraid to put in the effort and get anxious at the idea of just playing with numbers to see what happens.

I had a friend in college who switched from physics to journalism. I asked them why, because their grades were good and they had come out of a math academy high school. They said that every A was excruciating for them and took them way more time (they were dropping and retaking classes) and effort than their peers; this was a person who expected excellence of themselves, and was not failing by any stretch, just not succeeding at the level they considered acceptable. On the other hand, they were fluent in four languages, and said that everything that had to do with languages, writing, and journalism came easily to them. They are still an international journalist, last I checked. I am sure they could’ve gotten their physics degree, but why be miserable when they could be not miserable? They decided the level of excellence they craved was well within their grasp, and required much less effort in some fields than in others. Part of education should ideally be to help students figure out what where their strengths are and what they should specialize in.

I think a lot of the underlying issues in this discussion have to do reframing and not making people make decisions from a defensive crouch (“I’m bad at / failing at something”) but from a position of strength and optimism (“What are things I really enjoy and that come easily to me; how can I find a career that pays well and where I can use these skills?”) Because some majors do require a lot of math and with a high level of proficiency, and everything else is so much easier if you quickly pick up on the math. If the math is a lot of struggle for someone, everything else will be exponentially harder. It’s a kindness to remind the students they have plenty of skills and abilities and do not have to be miserable in their college experience or their careers. (A whole other can of worms is when the student is clearly miserable, but can’t switch majors because their parents are convinced a STEM degree is a path to financial security and won’t pay for college unless it’s in a STEM major or otherwise put pressure on the student.)

One of my pet peeves as a former mathematician and computer scientist is people who misuse “exponentially” to mean “much”. There is a specific technical meaning to the adverb, and it does not refer to just a large increase.

@gasstationwithoutpumps As a former mathematician and (current) economist who teaches math classes, I meant exponentially when I said it. I saw the graph in my head (probably e^x, since that’s the only one I use regularly these days) when I considered the use of the term and decided it was accurate. It is not a linear increase. It shoots up. Possibly it’s hyperbole and it’s only a steep quadratic. But it feels exponential.

Don’t try to mathsplain this math nerd. Like, WTF dude.

@xyk There’s a big difference between telling a kid that they can’t do something because they’re bad at math and noticing that they’re not enjoying a major. Those are completely different things, and people with great math skills can still prefer doing something else(!) I don’t think that physics anon guy was arguing that people who don’t enjoy physics should be counseled to switch majors though. That’s a different argument. I’ve got kids who *can* do our major who choose to do a less-technical related major instead because they like managing more than they like economic theory. More power to them.

Tangentially related, this just popped up in my Facebook — a cool quote by Kurt Vonnegut:

When I was 15, I spent a month working on an archeological dig. I was talking to one of the archeologists one day during our lunch break and he asked those kinds of “getting to know you” questions you ask young people: Do you play sports? What’s your favorite subject? And I told him, no I don’t play any sports. I do theater, I’m in choir, I play the violin and piano, I used to take art classes.And he went WOW. That’s amazing! And I said, “Oh no, but I’m not any good at ANY of them.” And he said something then that I will never forget and which absolutely blew my mind because no one had ever said anything like it to me before: “I don’t think being good at things is the point of doing them. I think you’ve got all these wonderful experiences with different skills, and that all teaches you things and makes you an interesting person, no matter how well you do them.”And that honestly changed my life. Because I went from a failure, someone who hadn’t been talented enough at anything to excel, to someone who did things because I enjoyed them. I had been raised in such an achievement-oriented environment, so inundated with the myth of Talent, that I thought it was only worth doing things if you could “Win” at them.

There’s a different between what’s worth doing and what’s worth pursuing as a profession or investing tuition dollars in.

Industry Physicist asks a valid question. I make a point of keeping in touch with alumni and have made some of my classes much more project-oriented, using the sorts of simulation tools they use in their careers. Certainly they don’t use as much math in industry as I use in academic research. Still, the bottom 20% tends to struggle on a number of levels, and some of their struggles carry over to data analysis in lab classes and other areas. I wish for their sake that they’d switch to something else sooner. I completely endorse the notion that one can and should pursue amateur passions without worrying about whether they’ll excel, but that goes to the idea of separating and balancing work and life. What you do for passion or curiosity needn’t be what you do for a living, or what you invest tuition dollars in.

I am also much less likely to counsel a student to reconsider if they are active in extracurriculars that tie in with a STEM career (e.g. robotics club, internships, etc.). If they are scraping by but developing themselves in ways that will compensate, great. But too many of my floundering students have found no strong interest outside the classroom that in some way complements their strengths while compensating for weaknesses. It’s a red flag that they need to reconsider their path.

But everyone just says “No, the world will be a less just place if we ever encourage people to reconsider.” Everyone worries that if kids reconsider something that isn’t doing it for them then we’ll hemorrhage women and non-Asian minorities. Are we here to help students develop themselves, or are students here to help our fields meet certain demographic goals?

As a person who never really “got” math as a kid and who now is going through math with her own kids, can anyone speak to what the best way is to teach math? Here is why I ask. Our district has an accelerated online math option in 5th grade that you have to test into. You can go up to two years ahead in math during that year if you want to. They use a program called ALEKS that uses a mastery model (like Khan Academy). You first take an initial assessment test to see what you already know. Then you briefly learn stuff and then practice it. If you get things right, you get to skip ahead. If you get things wrong, you get more practice. To me, it seems to work infinitely better than how I learned math, with endless homework sheets to do and teachers who weren’t great at explaining things. It also seems to effectively flag what my kid isn’t solid on and make him do it until he gets it. And I should say that his older brother, now in 8th grade, went through this same program and has done well in in-person Algebra I, Geometry, and Algebra II. But I have a colleague who studies math education who says the mastery model is Not Good. I didn’t understand what she thought was better–can anyone comment on that?

@CG– I think the main best way to teach math is with a sense of play. 🙂

The mastery model is currently out of fashion– it will come back. The arguments against the mastery model aren’t that mastery is *bad*– all methods of teaching K-12 math hopefully end in “mastery”. The problem is when it’s mastery via memorization of steps without any push towards understanding. What’s “in” currently is learning different methods of doing the same thing, so my fourth grader has been doing double digit multiplication at least 3 ways instead of just mastering one of them by rote. The idea is that they will get a better understanding of the distributive property and ultimately a better understanding of how numbers work that will make mental math easier and will help algebra to make a lot more sense. So: mastery itself is not bad, mastery without focus on the “why” and “how” is problematic. Mastery by rote is still good, I think for people with developmental delays and it certainly doesn’t hurt as a base for learning the why.

I found new concepts in physics easier to understand after being given a solved example– I was able to relate it to what was explained and then extrapolate to other problems. Some people have no problem starting with the theory and taking it directly to the problems, but I think it takes an awfully good teacher or textbook to not have a little bit of rote mastery be helpful before the actual understanding comes in. Math is a language for many things and it definitely helps it to be fluent before trying to grasp new concepts.

And, to be fair, I thought I really understood the number system after taking number theory — it gave me a real appreciation for how brilliant toddlers and elementary schoolers are just to be able to count and add. But then I took real analysis and looked at it an entirely different way. And then Abstract Algebra in yet another way. There’s always new ways of understanding and appreciating math. And I think starting with a mastery base doesn’t hurt– proving things about the number system is a lot easier when you actually know the rote parts of how to do basic addition/subtraction/multiplication/division and so on.

I love math.

CG, I didn’t go to school in the US so don’t know what mastery paths are, but based on N&M’s comment, it looks like it refers to learning how to do things before necessarily knowing why you do things that way (e.g., learning algorithm for multi-digit multiplication; btw my youngest is doing that now, and he’s got a really good place-value sense, understands fractions and decimal numbers, and still no one seems to want to teach him the multiplication algorithm; I see even my college students do the “box method” where they split, say 325×24 into a box of three rows (300, 20, 5) and two columns (20, 4) and then multiply each row and column and add them up; most of the time, they seem kind of lost without calculators).

Anyway, when I went to school, we’d have lectures and we’d have a book of problems. The books of problems were way more important than the textbooks, which were not good at all. I was lucky to have excellent math teachers throughout (you had to have a degree in math to teach math, or physics/chemistry/geography whatever to teach the corresponding subject in middle and high school; some highschool teachers had MS degrees; all my highschool teachers kicked ass). We never had homework for a grade beyond maybe second grade, the grades were all test based, and we were supposed to practice from the books of problems, which I did, A LOT, because math and physics were always awesome.

My middle kid is doing algebra now, and really likes Khan Academy, and it seems to be pretty great, near as I can tell. His class also uses Kami plus Desmos (graphing tool) and another tool, I forget now what. Teacher has also been sending some packets for practice at home, which are well thought out. My impression is that my kid is doing well and getting the material and getting plenty of practice.

With my middle kid I see that it’s sometimes important to teach him how to do something even if he doesn’t get immediately why, or not fully. Through practice and problems, things fall into place. For example, equations of a line, y-intercept and slope given versus an arbitrary point and slope given. For him, giving him the algorithm relaxed him (“I know how to do it”) and then within a few problems he got why (“Because from a given point I can move left and right using the slope and get more points, including the y-intercept if I want it” ). Lots of practice, lots of slightly different problems, and things fall into place. I think mastery of concepts and mastery of algorithms are not mutually exclusive, and different kids do better being led with one or the other.

(For example, I am sure I was taught the multiplication algorithm by rote, and only the algorithm, but I remember as I practiced it became clear why the algorithm worked.) Anyway, I really like Khan Academy so I don’t really understand the issue your friend in math education has with it. I find it delightful and if it works for your kids, awesome!

Thanks to you both for the replies. I do get what n&m is saying about it being important to understand the “why.” I think what is appealing about Khan and ALEKS is that it feels much more friendly and less daunting than math did when I was a kid. It’s easy to see your progress. I’m actually enjoying going back through it and solidifying some things that I never felt solid on in the first place. For example, every day in 7th grade, I would call my best friend and ask her to re-explain what we had done that day because I just couldn’t understand any of the teacher’s instruction. It probably didn’t help that the boy in front of me sexually harassed me all year, either. I’m glad my kids are having a different, more positive math experience than I did. P.S. I had to google what the multiplication algorithm is (just “long multiplication”, right?). I’m not sure if anyone at school taught my kids that but we taught them because that’s what we know! I do think that for my 2nd grader the method where you think of 196×3 as 200×3 -12 (for example) is helpful because it develops number sense, but I am no expert.

Friend of blog–have you kept in touch with the students who have floundered in the past? Where have they ended up? I have some friends and family members who could fit your description. The ones that come to mind are doing OK now, though it certainly cost people who care about them some heartache. Some of them have pursued trade school or have gone into medicine. Others finally got their act together and are having reasonable success working in physics. Having comfortable backgrounds certainly helped.

As a prospective employer of your students, I suppose I am learning from this thread that I need to not assume competence in algebra from people with physics degrees. Can I work with that? I prefer not to, but I probably can. I’d rather mentor someone in algebra, especially if they know they need to work on it, than mentor someone in how to not be a huge arse-hole in meetings or not be disrespectful to their peers. Of course I would rather have someone who is both competent and respectful to others, but I don’t always get what I want, especially when I am not the hiring manager, and my all-male peers don’t see some of the issues I do. I’ve also been disappointed by the fact that even people who seem to be able to understand some mathematical concepts in isolation have absolutely no clue how to apply those concepts to a real system.

As a taxpayer, I’d rather pay taxes to support students retaking classes floundering in physics class than to pay for them to be in jail or in drug rehab. This seems like a dumb choice, but if that’s where we are as a society then I know which version I prefer. It also doesn’t sound like you have the power to create better choices beyond trying to communicate your concerns to your dean in a way that she is more likely to understand correctly. As a pragmatic point, “only encourage student away from physics if they are white and male” seems like an ok practical guideline to address the concerns of commenters in this blog.

Surely there are more choices in life than “Get inflated C- in physics” or “Smoke crack and end up in jail and/or rehab.” If not then we have a much bigger sociopolitical problem than the demographics of physics grads. We need to look broadly at options and paths in life if we really care about a fair society.

One might not need algebra to experiment with different glass cleaning processes or interface a camera and automate a data acquisition process. (At least if key software for signal processing available in canned form.) OTOH, if someone wanted to look at that data and extract certain features, learning the relevant algorithms well enough to code them might require some math. If someone wanted to do some statistical analysis on the output then you need math, or atleastsome facility with abstraction to translate scenarios into code. The person who can’t do math will hit certain obstacles when the team needs some sort of analysis or coding that isn’t already canned.

And not all of the weak math students are shy women and minorities whose impostor syndrome makes them so deferential that you’ll need to mentor them in confidence rather than decency. Some are dilettante dudes who are convinced that they could do fundamental physics because they get, like, the ideas, man, and they don’t like people who discourage their creativity. Einstein failed math class, according to urban legends, and this mean prof says these students are also bad at math, so they must be like Einstein, right? Yeah, I’ve had a few of those. Einstein may have needed help with differential geometry, but these dudes need help with freshman calc. I’ve had some of them utter things so profoundly ignorant about math, in the middle of class, that there were math majors (women, FWIW) with dropping jaws. Yes, really.

I want a culture where those dudes learn a bit shame.

The dudes who are good at math actually tend to be less vocal and more interested in applied physics. The dudes who are bad at math (some pale, some not) have excuses and more “pure” interests.

Finally, if the only weak students who get filtered are white (and maybe Asian) men, that will have certain downstream effects. It might shift the raw numbers in more progressive directions, but other outcomes will shift in different directions simply because one group got better quality control. And when people notice what’s going on the resulting perceptions will not further the cause of equity.

BTW, I would not be averse to offering a track that has few to no theory classes beyond the sophomore level. I would be fine if there were a track where, beyond the sophomore level, the classes were on laboratory methods, use of simulation software (as opposed to developing and implementing the algorithms yourself), more applied topics taught in qualitative ways (e.g. solid-state physics taught from a standpoint of “Here are properties of semiconductors” rather than “Let’s derive Bloch’s Theorem and use it to solve different models”), etc.

However, let’s not kid ourselves: There would still be a place for a more quantitative track, and not just to prepare people for grad school, but to prepare people for the kind of industry jobs where you actually do need to know a lot about EM waves, Fourier transforms, numerical algorithms, statistics, etc. And that more quantitative track would have to have some serious math requirements, and there would be some students whom we would have to say “No” to. All of the debates over whether it’s right to tell someone “No”, whether it’s right to steer someone away, those debates would still unfold. Replace “Teaching physics to the mathless” with “Teaching this particular physics track to the mathless.”

Mind you, I’m still in favor of it, because I think offering people practical options is a good thing regardless of how much or how little math they know. I think some people who are weak at math might choose that track, for obvious reasons, but some strong math students might frankly like the practical track for their own reasons, and that’s great too.

Well, I think we do have bigger societal issues than the demographics of physics majors. However, neither you nor I are in a position to professionally influence those issues.

I completely agree about wanting a culture where dudes learn a bit of shame. I’m not sure how to achieve it without needlessly demoralizing less clueless and also more capable people who have a more highly cultivated sense of shame. When I was in college, I would feel great shame and get very demoralized when I heard comments from instructors that were, in retrospect, directed at clueless entitled white dudes and not at me. I sympathize with the people making those comments and understand where they are coming from, especially with some hindsight, but it definitely impeded my learning to have to hear them.

I agree that there would always be a place for a more quantitative track even if you were to offer a less quantitative one. It sounds like it would be more work for you because you would have to offer it, but on the other hand those students stay in your department and you get to keep their enrollment and tuition dollars. It seems like that changes your problem from “lets send these students away from this major and make them someone else’s problem” to “lets create objective prerequisites for the quantitative track.” And I think that would change the moral and ethical aspect of some of the discussions in this thread. As a prospective employer, I would probably eventually figure out that XY program has a quantitative and non-quantitative track, and adjust hiring practices accordingly. I might even prefer students from the qualitative track, depending on what problems I was hiring them to solve.

I’m only half joking when I say that we need two tracks. Most women and selected men get the supportive track. Most men and a few princesses get the “Seriously, you won’t be doing string theory if you can’t even do freshman calculus” track.

I’m not joking at all. I think two tracks is a great idea. I’m not senior enough in my career to be able to offer industry money to support such a thing, but I strongly suspect industry money might be available. My company takes outreach to universities pretty seriously and they are definitely not alone. I know there was some industry money provided to my PhD program, which has a pretty applied bent, though I was not interested in the details at the time.

The comment about semiconductors hit a nerve for me that’s sort of the opposite of the complaint in the essay. I have taken several solid state physics classes at the undergrad and grad level, I have derived Bloch’s theorem many times in classes. I still don’t understand why it’s important, and what laws of physics would be different if Bloch’s theorem were not true. I’ve read Kittel, Ashcroft and Mermin, and the wikipedia page on Bloch’s theorem. My deficiencies in this regard have nothing to do with innumeracy, but they are still real. This is just an aside rant and not intended to adress the issue of teaching physics non-quantitatively, but if anyone has an explanation or link to a good explanation about Bloch’s theorem, I would definitely read it.

The two tracks in my immediately prior post are different from (and less serious than) the ones referenced a few posts prior. I’m quite serious about the earlier proposal for quantitative and practical tracks (for lack of better terms). The more recent post is a sort of joke about how to teach the quantitative track: Men in one room being yelled at for not knowing the basics yet thinking they’re cut out for string theory, women in the other room getting more humane treatment.

Industry Physicist: I doubt I will be able to do better than the holy solid-state trinity of Ashcroft and Mermin, Kittel, and Wikipedia, but here’s how I think about the Bloch theorem and some of how I introduce it to my students. Some of what follows will be mathy, but some of it won’t. Maybe something resonates.

Basically, the Bloch theorem does for spatially periodic systems (systems with an underlying Bravais lattice in real space) what physicists routinely do for all sorts of other systems: use symmetry operations to figure out what the relevant quantum numbers are for the system and thereby classify the eigenstates and eigenvalues of the system’s Hamiltonian.

For example, you know how whenever you deal with a diatomic molecule, you talk about symmetric and antisymmetric electronic states? Also whenever you deal with a potential that’s symmetric (e.g., particle in a box)? Or how, when you have a cylindrically symmetric system, you immediately know the eigenstates will have a term exp(i*m*phi), where m is an integer? Or how, when you have an unbounded system in a certain direction, say, a free particle moving in 1D, you expect the states to look like ~exp(ikx)?

These are all manifestations of a handful of overarching ideas. Bear with me.

(Math alert) If a system is invariant under all elements of a Lie group (e.g., all rotations about an axis form the SO(2) group), the operators that are the Lie group’s generators (e.g., the generator of all rotations around z is Lz, the z component of angular momentum) will be a constant of motion, and the eigenvalues of the generators (e.g., Lz’s eigenvalues are any interger m times hbar) are good quantum numbers. (This follows from Noether’s Theorem.) Another example is translation, its generator the momentum operator, and its eigenvalues (any vector in 3D times hbar) are good quantum numbers for the free-particle problem.

But, discrete groups are very good at helping us classify states and energies, too! That diatomic molecule? Has (sub)group of symmetry C2, which contains the reflection from the point in the middle of the molecule. The so-called irreducible representations of this group represent the reflection operation as either 1 or -1; that’s what symmetric and antisymmetric come from. A triatomic molecule? Wavefunctions are such that, when you perform a rotation by 2pi/3, they will spit out a phase factor that looks like 1, exp(i2pi/3), or exp(-i 2pi/3). Note how performing a symmetry operation spits out a phase factor (1 or -1; 1, exp(i2pi/3), or exp(-i 2pi/3))? These are so-called one-dimensional unitary irreducible representations of the corresponding symmetry (sub)groups.

You know how spherical harmonics show up when you solve the eigenvalue problem of the hydrogen-atom Hamiltonian? Well, it turns out spherical harmonics represent the group of rotations in 3D (so called SO(3) group; it’s a Lie group, its generator is the angular momentum) in the space of square-integrable functions on a sphere. It’s the group of rotations — the Hamiltonian symmetry group for the hydrogen atom — that gives the beloved wavefunctions Y_l^m and and the quantum numbers l and m.

Are you still awake? Just barely? 🙂 Hang in there!

If you don’t care about this stuff, bottom line is that knowing about a spatial symmetry of a physical system is really important if you want to solve the eigenvalue problem of its Hamiltonian. Symmetry gives rise to relevant quantum numbers.

Where does the Bloch theorem come to play?

When I start talking about crystals with my students, I draw a 2D lattice and we admire its periodicity. I ask them yo imagine they know the eigenfunctions of a Hamiltonian of an electron moving through this lattice, and remind them that eigenfunctions are states particularly well attuned to the system’s geometry. Then I ask them to imagine an electron described by one of the eigenstates of the Hamiltonian. Next, I ask what they think should hold about the probability density of finding (“fishing out”) an electron at some point A, and at another point B that can be obtained from A by a lattice-vector translation. They all agree that the probability density (being in principle measurable) should be the same at A and B. I say that means the actual wavefunctions at A and B must differ by at most a phase factor; they agree. Then we show that, if you go from A to B then to C, exp(i*phase(A-B))*exp(i*phase(B-C))=exp(i*phase(A-C)).

This way, before we even start deriving the Bloch theorem, I’ve already introduced (without calling it that) the notions that, under symmetry operations, eigenstates of a Hamiltonian must behave according to 1D representations of the symmetry group (basically, perform translation, spit out a phase factor).

Translation groups are commutative (Abelian) which means they have only 1D irreducible representations, and guess what? The irreducible representations of the translation group are numbered by the k’s from the first Brillouin zone! Isn’t that wild?

If you want to be formal, the Bloch theorem means the following: You are given a crystalline Hamiltonian that’s invariant under a group of translations depicted by a real-space Bravais lattice. Then you are guaranteed that wavevectors from the 1st Brillouin zone (interior of the Wigner-Seitz primitive cell in reciprocal space) enumerate the irreducible representations of this group of symmetries. These wavevectors will be good quantum numbers helping enumerate energies and the associated eigenstates of your Hamiltonian. Furthermore, each eigenstate will transform under symmetry operations according to the appropriate 1D irreducible representation. In other words, I take an eigenstate of a crystalline Hamiltonian, and perform translation, out pops a phase factor, and that phase factor looks like exp(i*k_1stBZ*translation_vector).

If you want to be less formal, the Bloch theorem helps you relate the existence of an underlying crystalline symmetry group (a real-space lattice) to the quantum numbers arising from symmetry (vectors from the first Brillouin zone, a very special part inside reciprocal space). These quantum numbers enumerate your eigenfunctions and your energies for any particle moving through such a system (not just electrons). The Bloch theorem also explicitly shows a general form of the eigenfunction (must be of form exp(ikx)*something periodic if it is to spit out a phase factor exp(ikR) upon translation by R) and helps showcase its connection to the quantum number. If the Bloch theorem didn’t hold, the system wouldn’t be a periodic crystal, wouldn’t have bands and gaps, and would have very different physics.

Thank you, that was wonderful. The analogy to symmetry in atomic and molecular orbitals was particularly helpful since between chemistry and the hydrogen atom from quantum mechanics, orbital symmetry is quite familiar to me. I guess the point with Bloch’s theorem is that it specifies the type of symmetry that wave functions in crystalline solids have. Is that right? So if we are trying to solve for the electronic wave function for a system of interest, we could use Bloch’s theorem to determine that some functions cannot be wave functions of electrons in a crystalline solid because they don’t satisfy the requirement imposed by the theorem.

So to take an extreme example, the solution for the wave functions of a hydrogen atom would not be a solution for a crystalline lattice because the electronic wave functions of a hydrogen atom don’t satisfy Bloch’s theorem. The underlying reason is that the symmetry of the systems is different, but Bloch’s theorem spells out the mathematical implications.

Is that right?

Now I need to learn about what irreducible representations are. I own a copy of Cottons “chemical applications of group theory,” but haven’t read it seriously (I was a chemistry major in college before switching to physics. )

Thanks again. I will be back to read this explanation again and I definitely feel enlightened!

“So if we are trying to solve for the electronic wave function for a system of interest, we could use Bloch’s theorem to determine that some functions cannot be wave functions of electrons in a crystalline solid because they don’t satisfy the requirement imposed by the theorem.”

That’s correct!

You might also like this book by Michael Tinkham, “Group Theory and Quantum Mechanics”:

Have fun!

Noether’s Theorem is the most beautiful thing in physics. It’s even more beautiful than fractals in chaotic systems or multi-color fluorescence microscopy images.

Ok ok, I read a bit of the book and I think I understand now that a reducible representation is one that can be separated into two or more representations, like spin and orbital wave functions in simplified systems where spin-orbit coupling terms are neglected. And irreducible representations can’t be reduced further–like we can’t separate the three spatial dimensions int he hydrogen atom because it has spherical symmetry. Thank you for the suggestion! I bought the kindle version and might also buy the paper copy so I can write on it.

I think the most beautiful thing in physics is self-phase modulation leading to super continuum generation. One wavelength goes in, rainbow goes out. It was asserted on many occasions that symmetry has to do with whether a material (e.g. sapphire, but not BBO) supports this process or not, so I think Noether’s theorem ties in, and I would like to understand how it ties in.