Things I Learned While Teaching Undergraduates: Math as a Barrier to Science Edition

Usually I reserve this post title for physics things that I learned while trying to teach them to other people, but since those take a lot of time (and are going to be more scarce than I expected this semester, since I’m teaching very very basic physics) I thought it might be appropriate to use as the title for a little examination of one of the biggest problems my students are having in learning their material.

Now, I understand that the people I’m teaching at the moment aren’t physics majors (most of them are pre-med) but the fact is, a lot of the time when people come by my office hours seeking help on a physics problem, it quickly becomes apparent that they don’t actually know any math. Well, okay, they know some arithmetic, although their multiplication tables are often rusty. But a lot of them simply don’t know any more algebra than your average 8th grader.

I don’t think this is their fault, and this isn’t meant as a rant about them in any way. It’s just plain sad, and it puts me in a difficult position. There’s just no way I can reasonably help these people when they don’t even speak the language I want to use to discuss the problem in. I don’t know how they got through high school and into college (especially something like Stony Brook’s Pre Med program!) without acquiring a little more mathematical proficiency, but clearly they’re not (necessarily) to blame. All I know is, it’s going to make their entire physics class a waste of time.

I’ve come up with two analogies that I think help explain why this is a problem. The first illustrates why I say this will make their physics class into a waste of time: Imagine you were required to take a class in poetry, and you protested that you didn’t want to be a poet. The people forcing you into this unwelcome bit of culture could reasonable claim that they just needed you to be reasonably experienced in appreciating written arts if you wanted to call yourself a graduate of their University. They could tell you they think it’s important that you learn to think abstractly and artistically, no matter what profession you choose. And they’d probably be right, and I’d probably think you should take the class and that it’d probably be good for you.

But then imagine you show up for class on the first day and you discover it is all about German poetry, and that it’s taught mostly in German, and that your only assignments for the course are to recite passages of poetry that you particularly liked. If you don’t speak any German, then this class just became one massive exercise in difficult memorization based on phonetics alone. And at that point I would no longer defend the University’s decision to make you take the course.

On the other hand, the idea of a course taught in another language in some ways still underestimates the difficulty of trying to learn physics in a meaningful way if you don’t know basic algebra. Because at least if you were in a German poetry course, and you came to your TA, a graduate student in German Poetry, he would recognize your insurmountable language barrier and translate the poetry into English for you, so that you could at least get something out of the course. But I don’t have that luxury as a physics TA.

This is partly because I shouldn’t. People should know some basic algebra, because it gives them a systematic way to think about any quantitative question they encounter in life. And particularly if you want to be a doctor, you should be able to think quantitatively. So I feel obliged to keep teaching them in Math/German, since it’s actually part of the value of the course.

But the real problem is that I just can’t. And this is where my second analogy comes in: Solving a physics problem is a lot like choosing a strategy in a sports game. Like football for example. On any given play, you look at the situation on the field, you think about what you’d like the situation to be by the end of the play, and you choose the strategy that you thing gives you the best chance of getting there. Similarly on a physics homework assignment, you’re given a starting point (usually, “lets suppose you know information X and Y) and you have to devise and execute a strategy that gets you somewhere else (like to a place where you know information Z).  Real physics is like this too, just with more interesting questions (you start out with a lot of detectors that can detect everything BUT dark matter, and you have to devise a strategy to use these as a way to look for dark matter).

Not knowing math is like not knowing the rules of football, but still needing to make those coaching decisions. When a kid comes to me, shows me an equation where he has plugged in numbers for every variable except the one he’s supposed to find, and then still tells me he doesn’t know what to do next just because the variable isn’t isolated, it’s like he’s coming to me and saying “I’m on the 20 yard line, it’s the last play of the game, and I’m down by 2 points. I have no idea what I should do here. Can you help me?” I can say “Just kick a field goal!” but they’ll say “Field goal? What is a field goal? Can I get points from that?”

This is worse than a course being taught in German because there are things I just can’t translate. I can explain to you what the terms in an equation represent physically, but it’s unreasonable to think I can explain the physical significance of something like factoring an expression so that you can solve for the desired variable. “Factoring” is just one of those things that you need to know you’re allowed to do when you’re designing a physics strategy. It needs to be in your playbook just like a football coach needs to know that there’s such a thing as a “handoff.” If you don’t know, there’s nothing I can do to “translate” the problem into words that make sense to you. Now, I can teach you what factoring is, but that’s not my job. That’s something that was supposed to happen during several weeks of your 8th grade math class. I can’t do it in 5 minutes as a little aside from describing how to solve for the tension in an Atwood machine. It would be like asking me to tell you what a “down” was while I was trying to describe to you the advantages of running a Nickel defense when your opponent is facing a third and long.

I don’t know what to do about this. As I said, I don’t blame the students, and for the moment I’m still trying to explain the math to them whenever possible. It’s also worth noting that this is only true for a small portion (maybe 10 percent ?) of the students who come to see me. But for those 10 percent, I can tell it’s just not working. I end up having to walk them through every step of the algebra, and, try as I might to explain it in a memorable way, I’m left with the impression that they’ll only be able to use the trick I’ve taught them again if it shows up in exactly the same context the next time they need it. Which of course it won’t.

Part of me says that these kids should be allowed to realize that they are not meant for a career that involves quantitative reasoning, and that I shouldn’t spend half an hour reminding them how square roots work if it comes at the expense of someone I could really help. But part of me has no idea how much math doctors and the various other people in the medical field actually have to do, so maybe I’m being ridiculous. Either way, one thing’s clear: if we’re going to require people to take hard-science courses outside of their immediate field of study, we have to start teaching them more math. Otherwise, the entire thing is a waste of my time and, more importantly, theirs.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

8 Responses to Things I Learned While Teaching Undergraduates: Math as a Barrier to Science Edition

  1. S says:

    Those analogies were really entertaining.

    I wonder if teaching math is analogous to building gears with tolerances. People learn at different rates. The number of times they have to be exposed to a new concept is kind of like the number of teeth their gear has around its edge. Their overall patience is kind of like the radial width of the gear. The teacher forces the material at a set rate for the students in their classroom, and mechanical strain deepens the teeth every time it’s gone over. For students with little patience, the gear eventually breaks and they hate math. For students who get everything the first or second time through, the repetition wears on the gear, but potentially deepens their understanding.

    It’s got to be difficult to teach people math because you have to strike a really delicate balance in both accommodating everyone and simultaneously covering the material at the prescribed rate.

  2. Moominmamma says:

    As someone who is not math-oriented, I have always found this a particularly interesting question. I am personally of the belief that certain brains just can’t do higher-order math, and that includes what you would call “simple” algebra. At least, young brains can’t. My experience has led me to believe that perhaps there is a level of brain maturity that needs to occur before certain concepts can be understood, and that in many cases this doesn’t happen until well into the twenties or even thirties. We know from brain research that portions of our brains don’t mature until our mid-twenties, so maybe the thirties is hyperbole on my part, but from my own experience, I believe that to be true. Or, perhaps, it takes some of us into our thirties or forties to see the value in this kind of math and then take the time to learn how to do it. Now, it does seem reasonable to require pre-med students to be able to do this. I would hope my doctor could do it. However, I , like you, really don’t know how much doctors need to use this skill in their everyday work, or how much their ability to do it enhances their skills as a diagnostician or healer. I would assume researchers need this skill to a greater degree than practicing physicians or surgeons.

    I am curious as to whether the students even remember being exposed to these concepts. Do they have an “aha” moment when they say, “Oh yeah, I sort of remember doing this in school”? Or does it seem completely alien to them?

    I, too, enjoyed your analogies. S’s analogy is going to require a little more thought on my part before I can decide what I think about it. Thanks to you both for the brain food for this evening!

  3. Moominmamma says:

    I keep thinking about this. I have a couple of questions. Are the students who come to you without adequate math skills the same ones who bring reams of paper over which they have spread their calculations? Perhaps they’ve never learned to be systematic, which I think represents a level of rigor most teachers are reluctant to engage in these days.

    From the fog of my distant past, I cannot recall being taught 8th grade algebra in any way that made me realize that the “variables” in my equation could in any way relate to any actual physical phenomenon. Was this the case with you? I will question our math specialist at school about this, but it seems to me that perhaps we are introducing these concepts in entirely too abstract a way. Is it possible to use these equations when introducing basic physics in eighth grade? Is this done? I have absolutely no recollection of anything from my eighth grade physical science class besides “the distillation of wood” experiment. Seriously. How pathetic is that?

    • Colin West says:

      :-) I’m not ignoring you, but I have to go now. I’ve thought about this a bit too and in principle I think you’re right that a lack of rigor is partly to blame.

      Which is also why I have to go be more rigorous on my homework and not give in to the temptation to blog about things all night!

  4. Evan West says:

    “Trying is the first step towards failure” believe it or not that’s the first thing I found when looking for Simpson’s pie quotes to match your picture, but then it occurred to me it really matched what I was gonna say anyway.

    The worst thing you could do as a teacher, and someone they should look up to is tell them despite their best efforts things aren’t going to work out for them, and that they’re going to have to completely reorganize their lives, which is in effect what you would be saying if you told them they’re not meant for the career they’re in College studying for.

    some people, like myself, recognize we’re not cut out for math and science. And most of those people have experienced the opposite spectrum (whoops thats a science word isn’t it ;) and won’t push their luck on these types of things if they don’t understand it. If you have people coming to you to consistently, no matter how trivial or obnoxious their questions might seem or how annoying its that they take up so much time, its been my general observation that they have probably already made the personal decision that the other fields aren’t for them and they would rather go down trying in the one that their most passionate about, than simply have someone tell them they should call it quits cause its not for them.

    • Colin West says:

      Ha! What an ironic quote to be associated with that. I was really hoping to find a picture I’d seen before with Mrs. Krabapple in front of the chalkboard with a glaring arithmetic error written behind her, but I couldn’t find it, and by then I was hooked on the Simpsons theme, so I went with this.

      I hope you don’t think I would ever take it upon myself to tell someone they’re not cut out to pursue their chosen career although sometimes I can’t tell whether they’ve chosen it or they’re parents have chosen it for them. Either way, however, I that’s clearly not my place, and as long as they’re willing to put in the effort, I’ll certainly try to the best of my ability to help get them there. I guess I was just musing more generally about how the education process should work: when people are really small, it’s obvious it should just be nurturing and supportive, and try to teach people anything they need or want to know. And at extreme ends, like a med school, it’s obvious that instead of being as nurturing, it should be a test of one’s aptitude to some extent, since we can’t have a board-certified brain surgeon who doesn’t know how a brain works. The tricky part is what happens in between: even if I don’t directly discourage anyone from pursuing medicine, if I end up giving them a poor grade because they couldn’t pass the exams, I’ve indirectly done just that as it will hamper their med-school applications. That doesn’t seem nice, but then again, it seems like it would be wrong to give the schools they’re applying to the impression that they know more math and physics than they really do.

      This leads me to a second point: I also didn’t mean to say that I hold these kids in any way accountable for the fact that they seem ill-prepared to do the math that’s being asked of them. I hope it didn’t come across that way. As a matter of fact, I just feel terrible for them, because they’re being asked to understand all these complex physics ideas that are being taught in a “language” that they don’t speak to begin with. That must be really unpleasant.

      And a lot of the time, as both you and mom have pointed out, it just seems like they aren’t “wired” the right way to really get math, and obviously that’s not their fault. But more often, I sometimes think that some previous teacher must have done them a disservice by either not teaching the material well or letting them pass when they didn’t deserve to, because it just doesn’t seem like they should be able to take a class that has “college algebra” as a pre-req. if they tell me they didn’t understand the content of that class.

      I guess the point is twofold: first, I agree that everyone should pursue whatever they’re interested in, with the understanding that if it’s not something they’re inherently good at, they’ll have to work hard to get there. Secondly, I don’t fault the students if they haven’t gotten the proper background, but it puts me in a difficult place where they need me to spend a whole hour explaining to them how to add fractions, when there are 6 other people waiting to ask me for help. Whatever I do, I end up feeling unfair to someone.

      So I definitely agree about what I shouldn’t do. Any thoughts about what I should ? Do I owe these people as much time as it takes to get them caught up? And if not, how do I know when it’s okay to say “You’ll have to try to learn this on your own; I have to go help this next person now”?

  5. Moominmamma says:

    I have a suggestion or two as to what you should (or could) do. First, no, I don’t believe you owe the students as much time as it takes to get them caught up. They are young adults who need to learn to take responsibility for their own learning. When a course requires a prerequisite, they need to be able handle the material from the prerequisite course.

    That said, you are a nice guy, and as a good teacher you wish to help your students as much as possible, and face-to-face it is difficult to “turn down” a needy person. I suggest, if you teach this course again, that you give all students in your lab sections a pre-assessment on the day of your first lab class. This would consist of a short (3 problems?) algebra test that you could devise so that it encompassed many of the skills with which you currently see students having the most problems.-factoring, adding fractions, etc. After examining the responses you would be able to discern who would have difficulty with the required math. the next step would be to gather these students and let them know that it looks like they aren’t “fluent” in algebra and that this will (in your experience) cause them problems as they try to do the lab work. Then you let them know what support systems are available for them. If you are so inclined, you could offer to run a one-hour tutorial session weekly (or as needed) for all of them as a group. You could suggest that they pool their resources and hire a tutor for a weekly or more often tutoring session. They could form their own study group with a sympathetic and math-knowledgeable friend. Look up what the student support center run by the university has available and make those services known to them. Give them the name of learning support centers in town (there might even be several of these commercial entities in the vicinity). Give them a list of homework help and tutoring sites available on the web ( I just saw something on TV a few days ago about the vast and burgeoning online-tutorial business that exists in India for American students). You could actually hand them a piece of paper with suggestions and lists (disavowing all legal responsibility-this could actually be a problem. You might just do better to hand them the list of “in theory” places, instead of actual names or companies). A paper like this would lend gravity to the situation. Repeat the info. on a website. Follow up with the students at the next class to see if they have a plan in place. Be entirely sympathetic, but continue to let them know that they need to come up with a solution. I would also set a rule regarding my office hours: I’m willing to help anybody with basic math problems as long as there is not somebody else waiting outside my office for help. If another student shows up, you have only x many minutes of time. You are welcome to wait or come back after the other students leave.

    Since you are already halfway through the semester, it is too late for the pre-assessment, but you could suggest other avenues for help with math problems, and you could institute the office hours time restrictions rule.

    Another idea is to find some brief tutorials online for the kinds of skills/knowledge the kids are missing ( e.g., factoring) and have them printed (if allowed by copyright) or have the websites url’s printed so you can just hand them some sheets of paper and ask them to look at that first, and come back if they still don’t understand.

    Obviously I have no idea whether the kind of math I am suggesting meets your needs. I would focus on letting kids know as early as possible (through pre-assessment) if it looks like they might have trouble, making resources other than yourself available to them, and setting restrictions on your time if other students’ needs are being harmed by the low-math kids. If Evan is right (and I think he had a good point), the students who are still there despite knowing that they have problems with math might be very highly motivated to deal with the problem, if you just give them options and let them know that you can only provide so much support. This may come off sounding kind of hard-core or mean, and I don’t mean for it to at all. It should be dealt with up front (preferably at the beginning of the class): Some of you are going to have trouble with the math you need for this class, here are some ways you can help yourself and each other with that, I’m happy to help but my time is limited and other students need help too, so here are my rules for how we handle the basic algebra questions during office hours.

  6. Colin West says:

    These are such excellent suggestions.

    Remind me again why you’re not designing curriculum someplace?

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