### Trying to make precalculus work

My precalculus students wrote a quiz today. On the top of the front page I’d given the instuctions *Solve the following equations for x. Show your work*. If I’d instead typed *Hey, kids! I’m under the impression that you know how to solve quadratic equations, but if you convince me otherwise, I’ll give you ten dollars!* I would have received the exact same thirty papers.

This is the class with the crappy attendance. On Tuesdays, two thirds of my students show up. On Thursdays - quiz days - I get a full house, and then close to half of them jump ship after the break. A few weeks ago, I was willing to believe that a third of my students knew this material well enough that they didn’t need to come to class. Apparently I’m just that naive.

On a related note: a few weeks ago, I was chatting with Dr. Matt about review sheets for math tests. He was ambivalent about them, mentioning (I’m paraphrasing, and possibly misrepresenting, here) that he gives out the sheets one day, and then he goes over them the next class, and he wonders how much good that does. I was thinking about this, and also about a recent post at Learning Curves, during today’s class, when I was fielding questions about the homework. Last class, I’d assigned three word problems; this class, my students asked me to go over…three word problems.

They were difficult questions (relatively speaking), and my students have limited experience setting up word problems, so I wasn’t surprised, nor was I bothered. What did bother me was that when I prompted the students with leading questions - *okay, we are given a perimeter and we need to find an area…what equations should we set up?* I was met with silence. I’d realized it before, but it was only then that I fully saw just how *disengaged* my students were from the material. Math, for them, is a monkey-see-monkey-do affair. I may talk at length about how we need to figure out relationships among variables and translate them into equations, but my students just see how I apply those to this word problem or that one. They can’t apply a method of a problem about a rectangle to a problem about a right-angled triangle.

I started to set up some of them problems, but then realized that I’d tried this before, and it didn’t work; my students had never learned, on a large scale and to my satisfaction, how to set up word problems that differed from the few I’d shown in class. So I decided I was going to try something new. “I’ll go over these next week,” I said, “but I’ll only go over them if I get some feedback about how you tried to set up the problems. It’s okay if you don’t do it correctly; you can learn from false starts. What’s important is that you try.”

I pointed out that I don’t always know how to approach a word problem when I first see it, and that solving word problems isn’t typically something that comes to people all at once. But, I said, it *never* comes to anyone if they don’t try their hands at word problems first.

I know that other instructors of this course have thrown in the towel, and were satisfied with their work if their students were able to answer the same question about maximizing the area of the fence that they’d seen solved in class the week before. I personally don’t see the point in that; if the only problems my students can solve are the ones I did in class, then they can’t do math in any useful sense. From now on, I’ll do a couple of word problems in class, and then assign some that are similar in concept but slightly different in structure. And I’ll only go over those problems in class if I can see that my students have at least attempted to apply my lessons on finding relationships among unknowns. If they haven’t managed to make at least some sense of that, they’re not intellectually mature enough to learn what I’m teaching them, and there’s not much point spending more time on this material; doing word problems with students who can only mimic them isn’t any better than not doing them at all. We’ll see how this experiment goes.

One of the reasons that the Larson/Hostetler/Edwards calculus book is fairly popular with our students is that the text is particularly good at providing examples as templates for exercises and lists of canned formulas where students can find the silver bullet that will slay their problems. The idea behind problem-solving thus devolves to a pattern-matching game (just remember to change the numbers according to the specific exercise) and ends up having almost nothing to do with thinking and strategizing. It’s sad. Even worse, many students will patiently explain to you that you are doing things

wrongif you assign them a problem that cannot be precisely matched to a previously solved example. (God forbid this happen on an exam!)Oh, I get a lot of “It’s not fair! You never showed us how to do this exact problem!” And the pattern-matching technique is so sad, because it’s generally sufficient to get most students A’s in high school. And then, when they get into university…they figure it’s their horrible teacher who isn’t giving them the high marks and reasonable problems they deserve. Argh.

What amazes me is that the offense they take when I give them a

differentword problem suggests that they believe that the reason I’m showing them the problem with the boat rowing with the current and then against the current at 8 km/h and then 4 km/h is so that they will know, forever and always, how to solve the problem with the boat rowing with the current and then against the current at 8 km/h and then 4 km/h, and THAT’S IT.The archives live in a cute little menu somewhere along the side.

What you’re talking about informally in terms of you class is something that I do very explicitly — as nothing is obvious to some students. I set extremely concrete

behavioral(rather than intellectual) objectives.Found the archives, thanks.

Re: behavioural objectives - that’s exactly what I had in mind last night. I wasn’t mad at my students, just frustrated and drained. Their weakness with word problems wasn’t what upset me; what upset me was their FEAR of word prblems and their belief that if they can’t see a solution immediately, then there’s no point in trying. They’re afraid to approach problems, and not approaching problems pretty much guarantees that they’ll never get them.

Another thing I pointed out last night - sometimes it’s a good idea to read the textbook. I hadn’t done a problem that involved having to relate two unknown sides of a triangle to the hypotenuse in class - but there was one in the book! Seemed like none of them had gone back to look. At least I know that this isn’t a math thing - it’s not like students typically do the required readings for their humanities/social sciences classes.

Have you ever done this experiment: take a review sheet with 4 or 6 or N problems, go over it in class, than at test time, present the exact same review sheet?

Word problems are a step up from arithmetic - I think it takes a bit higher level of thought. Arithmetic can be strictly behavioral - I know the addition and multiplication tables, and I can apply the cookbook methods. But word problems require analysis, and that takes a bit of thought and practice.

Maybe you could start with simpler word problems - just to introduce the concept that you can connect arithmetic with words. Something along the lines of “When you double a certain number, you get 20. What’s the number?”

Mike, I did something similar with my weekly quizzes last term: the questions came straight from the homework. This worked reasonably well in my discrete math class, but it was an unmitigated failure in my precalculus class: students who got 10/10 on the quizzes would fail the tests. They were just memorizing how to do the homework questions.

I gave some simpler word problems on the “stuff you should know by now” handout, but there’s a problem with making the questions too easy: if I ask “find a number such that when you double it, you get 20,” most of them will answer “10″ immediately. When I probe further - “how do you know?” they say, “I just do, it’s obvious.” And then when I show them how to set up equations, their eyes glaze over.