Tall, Dark, and Mysterious


An equation is a relationship among quantities.

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 9:45 pm.

That’s the sentence that I utter every time I present a word problem on the blackboard in my precalculus class. Then I slowly, slowly outline what quantities we’re working with, which are known, and which are unknown, and how they’re related. Then I work through the problem, and when I’m done, I go over how - and why - we set up the equations that we did.

Today, I graded 30ish tests, and saw that I might as well have spent the above time reinforcing the prevailing view that equations are voodoo amalgamations of letters and numbers, for all the good my attempts did. (Boy, am I ever feeling Rudbeckia Hirta’s frustration today.) Answers to a “rowing with the current/rowing against the current” question revealed that while close to a quarter of my students seem to know that speed=distance/time, about as many are under the impression that speed=time/distance, and many others hold fast to the view that speed=distance*time. I’m sure there’s a bad scifi novel in here somewhere; I’m equally sure that if I asked a random student “if I drive 100 km in 2 hours, what’s my [constant] speed?” they’d reply “50 km/h”. Anyway, this may have happened because I sneakily gave the time in minutes not hours, so the time/distance formula had the advantage of giving a whole number instead of one of those tricky fractions. Another student opted to forgo multiplication and division altogether for this problem, and wrote down the formula “speed of current + 50 minutes = 10 kilometers”. The problem involved the current on the return trip being half of its original value, so some students squished all of the numbers in the original problem together in some haphazard fashion, threw in an x so they’d have something to solve for, and then multiplied the whole thing by one half.

A supply/demand problem had a number (not zero) of students finding that the equilibrium occurred when widgets were sold for negative thirty bucks a pop. No one appeared to bat an eye over this one; they just stated their conclusion and moved onto the next problem. On a question about finding the dimensions of a structure with given area and given amount of fencing, a plurality of students faithfully parrotted the formula that perimeter=2*length+2*width, apparently not noticing (or caring) that the fencing of the figure in question (I provided a diagram) did not surround a rectangle. Nor did it surround a triangle, but many students seemed eager to show off their knowledge of the Pythagorean Theorem.

The class average, despite all that, wasn’t that bad; it’s just that the poorly-done tests were really poorly done, and many of my students seem to have no idea what mathematics is. There was no point in writing comments on some of the papers; so profound is the disconnect between my weakest students and the subject matter.

I have no idea what I’m going to say when I hand these tests back tomorrow. Right now, I’m mostly angry with my students - haven’t they been listening to what I’ve been saying all term? If they were, then they clearly didn’t understand me - why didn’t they ask, in private during office hours if not in the classroom? Much of my frustration is justified - attendance in that class sucks, no one showed up to my office hours during the week before the test - but many of my students are trying. They just don’t have the background right now, and I don’t think I can provide it. But that’s not a productive way to give back the test papers. Students who bombed a test don’t need to be told that I’m disappointed in them; they’re more upset than I am as it is.

Fellow high school/college teachers: what do you say to your students when you hand back a terrible test? I don’t want to upset them, but at the same time - I ain’t scaling these test marks, and the course isn’t going to get any easier.


  1. Have you seen The Miracle Worker? I think of Annie Sullivan often.

    (One of my pre-calc mantras is: What the function does, the inverse undoes.)

    I’m going to write notes on the papers of the students who are trying and who do almost-get it and tell them that the scary speech I’m giving to the class as a whole is not meant for them.

    - Rudbeckia Hirta — 2/10/2005 @ 3:30 am

  2. i remember one particularly bad algebra and geometry test in high school. my teacher had taken the liberty of putting the stack of tests in a big black garbage bag. he brought the bag to class, and pulled them out one by one to give back to us.
    that was how bad they were.

    - haya — 2/10/2005 @ 6:38 am

  3. I had some success last semester with simple “how to be a student” type reminders:

    If you don’t understand something about something I say in class, ask me a question about it, so you can understand.

    You are not the only person confused by math.

    If you are getting lost, it’s probably much better to come to my office hours and talk to me and see if I can help you out about it than for you to hope you’ll get it eventually.

    Surprisingly, sometimes they don’t realize these things, though it’s painfully obvious to us. :)

    (PS: for this “Add ‘em up” thing, do you actually read it or is it an automated thing? Because I SO want to write (for 5+5 =) “5/1 + 5/1 = 10/2″)

    - Moses — 2/10/2005 @ 8:20 am

  4. close to a quarter of my students seem to know that speed=distance/time…

    haha… a friend of mine used this simple computation once at her court date for a speeding ticket, arguing that if she’d been going as fast as the officer said she’d been going, he could not have flagged her down on foot. The prosecuter immediately stood up and registered an objection on the grounds that invoking this physical relationship was “expert testimony.” My friend had to demonstrate for the court that she understood said physical relationship. (which she did to their satisfaction, but still lost and got fined, by the way.)

    I guess it is somehow special to understand that speed=distance/time :P

    - Jen — 2/10/2005 @ 9:51 am

  5. Have you gone through unit analysis - the thing that says, in d=rt, for example, that “km = km/h * h” (it works better with the long horizontal for the division sign). “Then the ‘h’s cancel and we’re left with ‘km’ on both sides.”

    And if someone writes “r = dt”, we can see right away that the units don’t work out - mi/h is not mi*h.

    And you get across the concept of operaing without numbers at all.

    You can even do a longer calculation like “if gas costs $2 a gallon, and your car gets 20 miles per gallon, what does it cost in cents per mile?” ($/G * G/mi * cts /$ = cts/G

    (Sorry, we still use those antiquated units down here.)

    Another thing you might do is have dated lecture note outlines for each topic. Then, after the test, you pull out the outline and say, “we covered problem nr 1 on this day, like this”.

    - Mike — 2/10/2005 @ 11:14 am

  6. I wonder what the correlation is between those students who can parse a question into subject, verb, object, etc. and those who can write an equation corresponding to the question. Probably somewhere near 0.99 I suppose.

    - Dorn — 2/10/2005 @ 1:15 pm

  7. Try to turn it into constructive feedback for your students. Can you suggest some place where they might get additional help? Is there any tutoring available? Perhaps suggest that they form study groups. Like you said, the course isn’t going to get any easier.

    - EdWonk — 2/10/2005 @ 3:42 pm

  8. Thank you, everyone, for your suggestions. They’re both encouraging and discouraging: encouraging, because I already follow most of them, so I’ve been doing things pretty much right; but discouraging because even though I’ve been doing everything right, the class is such a disaster anyway.

    RH, I’ve never seen The Miracle Worker, though back in grade school I did a project on Helen Keller, so I have a decent idea of what it would be about. My movie inspiration is Stand and Deliver, but it’s no coincidence that right now, the only line I can remember from it it “You can’t teach logarithms to illiterates.”

    Jen - wow, that’s funny. I mean sad. I mean sad and funny.

    Dorn - actually, I’d put the probability at much lower, as the majority of my students who actually got the word problems did not speak English as a first language. Ironically, it’s the unilingual anglophones who complain about not understanding what the word problems are asking.

    (Moses - the Add ‘em up is a spam blocker, which, conveniently enough, also guards against the possibility that I’d ever get blog comments from half of my students.)

    Mike - I’ve done some unit analysis, though I pretty much glossed over it this term. It’s a very useful way of looking at problems, I agree, but it seems that no matter what I do, my students just zero in ont ehe aspect of my lesson that pertains to the particular question I’m solving on the blackboard. Which is why I don’t want to implement your suggestion about the dated class outlines: I very deliberately give questions that use the same skills as the ones developed in class, but that do not follow the template of the class problems. If they can’t deal with problems that aren’t word-for-word identical to the ones I did in class, then they can’t do math, period.

    Which, by the way, is the reason I haven’t yet replied to this one student’s email:

    Dear [Teacher],

    I found the mid-term quite difficult, although I studied the text, and accessed the tutorial centre somewhat.

    I will attend the tutorial centre more but am wondering if you have any extra math sheets with questions similar to what may be on tests. I found it more challenging that the text.

    Perhaps when I see how I did on the test and focus on the weaknesses noticed it may improve but if you have any extra math sheets it would be much appreciated.

    Thank you,

    Argh. This student was one of the ones who submitted a jumbled mess of letters and numbers , multiplied by 1/2, in the rowing question. The one with the area/fencing, she presented the quadratic formula (why?) and left the rest of the page blank. I assign 30-50 practice questions every week; the problem isn’t that this student doesn’t have enough to work on, it’s that she’s locked into this mentality that she just needs to memorize MORE types of problems. I could give her worksheets, but they’d still be different from the test questions, because I’m testing my students in math, not in pattern-matching or in mimicry.

    - Moebius Stripper — 2/10/2005 @ 4:04 pm

  9. There are people who simply cannot “get” math. I refuse to believe they were born this way, but they grow into it. I’m a grad student in geology, a field far enough removed from my baccelaureate studies that I’ve had to take many undergraduate classes. In those classes I’ve met many students who are okay (not great) with math until they realize they’re doing math, when they freeze up and their brains refuse to work. Mind you, these are future scientists (admittedly not at a tier 1 institution, but still…)

    - Karen — 2/11/2005 @ 12:25 am

  10. In my grad complex analysis class, the professor came in and said, half jokingly ” I tried grading your tests, but I only got halfway through and realized I was too depressed to go on. I dont know why I even taught you this, but at least I have my research to keep my happy.. ” ( the professor does have very high standard) He then proceeded to treat us like mathematical babies the rest of the lectures, proving extremely obvious statements, etc.

    The guy is very funny and an excellent teacher. Needless to say we all did much better on the final after that. I think students forget that professors sometimes take tehir kids
    grades personally as a sign as to the quality of their teaching. All though I did well on the midterm, I worked that much harder not to let him down again.

    - Marc — 2/11/2005 @ 7:19 am

  11. I went to a talk at NCTM last year given by Bill Handlin. He was talking about his work with high school students who have failed Alg. 1 at least 3 times. (And apparently, he’s having pretty good success with them.) One of his major goals is to get the students to realize that they’re incapable of doing math *because* their negative reactions cause their brains to shut down. He starts by making them write, on homework and tests, a + or - next to each problem based on whether their first thought on seeing the problem was positive or negative. Next they have to write down a brief explanation why they reacted that way: Ugh. FRACTIONS or Yay, I know this one or Sue’s my girlfriend’s name. Eventually, he makes them keep writing down reactions until they can find a positive one. It seems that once they can find a positive reaction (I see how to get rid of the fractions/this is like that other problem/I understand this part of it) they are in a better frame of mind to try to solve it.

    Seems to me that college students would bristle at being forced to do this exercise, but it’s such a good lesson: you can’t solve problems if you won’t think about them.

    - lisa — 2/11/2005 @ 12:27 pm

  12. Marc - I can see that working in an upper-year course for majors, but I assure you that my students would LOVE it if I babied them. Some of my students are lazy, but a lot of them CANNOT do math at even the junior high school level. They’d appreciate it if I took three weeks to teach them fractions. In a way, I’d like to do that as well; at least they’d learn them. But I can’t, and besides - this is university.

    Lisa - I’m intrigued, largely because that method seems so hokey that I can’t imagine it working. But if it did, hey, I’m not about to argue with success…though I can’t imagine it working with my more jaded adult students. As for “you can’t solve problems if you don’t think about them” - I’ve actually told my students as much, and have told them that I will not just show them how to do the problems if I don’t see that they’ve at least tried.

    - Moebius Stripper — 2/11/2005 @ 6:40 pm

  13. One of my most horrible experiences was with a student who absolutely would not listen to me. She was very upset because she was flunking the quizzes and exams even though she was doing the homework. When I asked about the homework, it turned out she was doing a lot more than I assigned. In discussing the matter with her, I learned that she persisted heroically in the face of failure, doing more and more problems even though she was not getting any of them right! “You need to do fewer problems and do them more carefully,” I said. “No! I need to do more!” she replied. Although she was, if anything, very deferential toward me in other respects, she would not take my advice to slow down and think about what she was doing, to work in concert with other students, to allow me or a tutor to help her go step by step. No! She had to do more problems until some miracle happened and she figured out what to do. The miracle never occurred and she kept practicing how to do problems incorrectly unless she dropped the class in a fit of deep despair.

    I feel less bad about other students who fail: those who aren’t motivated to try. I will make an effort to get them involved, but if they deem it to be too much trouble to invest some time and thought in math, I will feel no guilt in turning my attention where it seems to do some good. Still, you always wonder if you missed something that could have sparked their interest. Triage is less bloody in the classroom than it is at a disaster scene, but it’s still painful.

    - TonyB — 2/13/2005 @ 11:12 am

  14. “speed of current + 50 minutes = 10 kilometers”
    This may be a problem of notation - using add and equals signs in an informal way, and then getting confused?

    Regarding the formula for speed… a good way of remembering this is to ask something like..

    Q. How do we measure SPEED - what units?
    A. Kilometers per hour (or similar, hopefully)

    and then note that PER means DIVIDE…
    so kilometers per hour are kilometers divided by hours…
    (getting students think of it this way may also prevent problems converting units)
    in other words, in general, speed (kph) = distance (km) / time (h).

    Good luck!

    - Ronald — 2/13/2005 @ 1:06 pm

  15. >so hokey that I can’t imagine it working

    He started the talk by putting up a question about football (Name three things that a can do to block the runner.) My first thought was “It’s about football. I don’t even want to read the problem.” Once I calmed down and *read* the problem, I realized that I actually could answer it based on the name of the position and my minimal understanding of sports. I haven’t had the opportunity to try it out on students ‘cause I’m not teaching, but I’ve found that the mental version works for me.

    - lisa — 2/14/2005 @ 10:16 am

  16. […] es — Moebius Stripper @ 11:07 am

    Following the emotional turmoil that was grading the precalculus tests, I decided that […]

  17. The Carnival Of Education: Week 2
    Here is a variety of interesting and informative posts from around the EduSphere (and a few from the Larger ‘Sphere) that have been submitted by various authors and readers. We think that they represent a great variety of…

    - The Education Wonks — 2/16/2005 @ 12:16 am

  18. I’m a little late to the discussion, linking in from the Carnival of Education. I just want to vehemently agree with the comment above from Mike. TEACH DIMENSIONAL ANALYSIS! One of the things I liked about my daughter’s third grade Saxon math was that they always had to include units in their answers. But then, when they got to multiplication, suddenly the units got dropped. An example problem might be:

    There are six students. Each student has 5 crayons. How many crayons are there altogether?

    The answer they wanted was:

    6 * 5 crayons = 30 crayons.

    But this drops the ball! The right answer should be:

    6 students * 5 crayons/student = 30 crayons.

    The unit student in the numerator of the first factor cancels the unit student in the denominator of the second factor, leaving crayons as the unit for the product.

    Sure, it involves an early discussion of numerators and denominators, but learning to handle the dimensions correctly as students are first learning multiplilcation story problems will help immensely when students are trying to do your rate and distance problems. The dimensions deliver the right answer on a silver platter.

    Dan K.

    - Dan K. — 2/16/2005 @ 7:45 pm

  19. Believe me, I’m a big fan of dimensional analysis. But, Dan K, the answer to why I haven’t emphasized it more is in your comment - your daughter’s third grade math class did this. My students are college students. They’re missing ten years’ worth of math background, and there is no way I can provide it in three months. I’d like them to know dimensional analysis, so I can spend some time teaching that. I’d also like them to know how to add fractions, which would take me a little while to teach. I’d also like them to know how to various geometric formulas - perimeter of a rectangle, area of a circle, volume of a pyramid. They should have learned these years ago, but they didn’t - or, they did, but it didn’t take. I’d like them to be able to translate simple statements such as “five more than double a number is 11″, but they can’t.

    I have three months to cover four chapters of a textbook. If I provide clear explanations for everything in those four chapters, then that takes the entire term. I try to segue into lessons on prerequisite material when I can, but time is limited; if I taught them everything I thought was useful for this course, I wouldn’t get to the course. Maybe I should make some time for dimensional analysis, but I’m not sure it’s that much more important than any number of other topics that aren’t explicitly in the curriculum but that would help them understand the material better.

    - Moebius Stripper — 2/16/2005 @ 9:07 pm

  20. Again, a late addition… (I do love your blog!)
    You may already do this, but have you considered offering a list of “must knows” at the beginning of the semester, ideally in the form of a self-test followed by a list of the concepts tested on the self-test, and telling students that if they can’t do the test or don’t remember it, then they need to catch up? (I saw a psychology stats prof do this with decent success - she also held two tutorials on the background material, to help people remember the concepts they might’ve forgotten, but you could also simply refer people to the tutoring centre.)
    Obviously, this will only help those who want to succeed, but at the beginning of the semester many students do (after all, they odn’t have much work to do yet, haven’t been beaten down by midterms etc), and at least those who want to succeed would have a fighting chance.

    These are all, y’know, hypothetical suggestions. What with the unemployment ‘n all. But hey, why not…

    - parodie — 6/1/2005 @ 6:46 am

  21. I actually did give a “things you should know by now” handout at the beginning of the course, inspired by the group of students last year who, three months into the term, persisted in delaying the lesson while they grappled with the fact that 2/3+4/5 is not equal to 6/8. I told students to go over it, this week, and come to me or the tutorial centre immediately for help. I told them that if they would not be successful in my class if they couldn’t do the questions on the handout.

    A few weeks before the final, one student came to me for help with word problems. I asked her if she’d tried the simpler word problems on the handout. She hadn’t, she said; she’d done the other problems, but she didn’t know how to do the word problems, so she skipped them.

    It’s not math these students are struggling with…it’s the idea of taking a class.

    If I teach this course again, I will give something like the handout, but as a timed test. (Someone else made a similar suggestion in the comments here awhile ago.) I will grade the test, but not count it toward the student’s mark. If the student gets below a certain score, I’ll tell them to either a) get lots of help, soon, b) drop the course, or c) prepare themselves emotionally not to do very well. I think that this might drive home the message that a simple “here’s a worksheet you might want to look at” doesn’t get across very effectively. But if I return their work with a “11/20″ in red ink, maybe they’ll figure it out. Maybe.

    - Moebius Stripper — 6/1/2005 @ 10:06 am

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