### Blah blah asymptote blah blah intercept

Three of my four classes are full of students who seem to be getting at least something out of my efforts, but lately I can’t think about my early afternoon precalculus class without being reminded of this Far Side cartoon:

For whatever reason, the students who can’t solve linear equations, the students who can’t add fractions, and the students who can’t even come close to formulating an equation from a word problem, all ended up in this section. A typical Q&A session in that class goes something like this:

Student: (after I’ve graphed one function on the board) So basically, when you’re graphing a function, it goes positive-negative-positive-negative, and has two vertical asymptotes.

Me: (not knowing where to begin) Well, no, not really: in the case of the example on the board, yes, but that’s because of the factors of the two polynomials in the quotient. (expands on this a bit, explaining where the zeroes and intercepts came from, again) You need to look at the factors of the numerator and denominator, and take test points in all of the intervals that you obtain from the critical points. (does example on blackboard) Does that make sense?

Student: So basically what you’re saying is, yes, in general there’s two vertical asymptotes and the graph goes positive-negative-positive-negative.

Most of these students would benefit from a good, solid, *grade nine* math class, and I’m at a loss over what to do with a university precalculus class full of them.

It could be worse. Have you seen the corresponding Far Side cartoon about cats?

I always feel sorry for you when I read about your precalc class. Really, there is absolutely no excuse for a student being allowed to enter university and not being aware of what a ‘function’ is in the context of mathematics. At least you know which class to avoid teaching in the future :-)

If it makes you feel better, I TA business calculus. Last week I covered for the prof and taught partial derivatives and level curves. A student came to my office later with a couple of questions (a girl who asked me why (e^r)^1 = (e^r) (my answer was, ‘There are may reasons, we can view (e^r)^1 as e^(r*1) = e^r. Also, we could just just by definition that a number raised to the first power is itself”. Response from student, “so it sis something I should just write down on my cheat sheet”)) and asked me about partial derviatives.

We started slowly, with x^2 + y^2 and then x^2 + y^2 + 4xy and then I gave her the example xe^y expecting to trip her up slightly. So I went through the entire derivative process showing her to hold things constant etc and she was like yeah I get this. My next example was x ln(y). DO you think that she took what she learned from the previous example… of course not… We had to start over.

Also, another amusing ancedote from a quiz. We asked the astudents to determine how much money a person has in the retirement plan at age 65 assuming the person invests 2000$ a year between the ages of 20 and 39 under some continuous compounded interest scheme. One answer was $1.07835678987x10^75. 9 (m not joking)

I wrote on the quiz that the (American) national debt was about $10^13 and that the # of particles in the universe was estimated at around 10^78. Do you think the student actually thought the answer was correct?

Rohan - thanks. You know, I don’t mind teaching weak students. What I can’t do - and what few people can - is teach material to students who are five or so grade levels behind. In my weak precalc section, I have a handful of students who interrupt me during explanations to ask “how did you get from the second line to the third”, when the second line was something like 2(x+3) and the third like was 2x+6. And God help us if there are any

fractions.Marc - man, I hear ya. Regarding your “does you answer [to the money question] make sense” - that’s something I’ve been trying to drive home to my morning class, in which we do financial math; it’s gone way over most students’ heads. For instance, I’ll ask how much one has to invest each month in an annuity at 9%/a compounded monthly, if one wants to save $1200 by the end of the year. I’ll say “well, let’s see - if there were NO interest, it’d be $100/month - so we should expect our monthly payments to be LESS than $100.” It’s lost on a lot of students, who get completely wrong answers on quizzes and hand them in ten minutes early, when they could have easily checked their work and seen that they’re answer was an order of magnitude too small.

One of the most disturbing aspects of my students’ weakness is their general inability to reason quantitatively. If they can’t graph a parabola, fine, we’ll all live; but if they can’t ask themselves things like “what is a reasonable maximum for this area to be” - that just indicates to me that quantities are TOTALLY ALIEN to them. sigh.

I don’t teach, so I haven’t tried this myself, though I have read several, separate reports from teachers who say that it has worked well with students in high school and in college. It’s worth considering.

Make part of the score on any assessment item tied to an analysis of whether or not the answer is correct or reasonable. For example, on a 4-point problem, the student writes whether s/he thinks the provided answer is worth 1, 2, 3, or 4 points. If that estimate is correct, it’s worth a point (making the problem actually worth 5 points.)

The idea is to give the student credit for actually analyzing whether an answer could be correct. The reports that I’ve read (and, no, I don’t have references at hand) claim that at least some of the students learn the value of reviewing their answers for meaning in the context of a given problem.

Lisa, I do something similar: If a student works through, say, a maximization problem and concludes “therefore, the maximum area is negative forty-five”, they’ll get at most 4/6 on the question. If they write, “I’m getting a maximum of negative forty-five, which can’t possibly be right, because the area has to be a positive number”, then they’ll get 5/6.

(Vaguely related: I recently administered a test on permutations and combinations, and some of the questions involved very large numbers. One student didn’t have a scientific calculator - which was the reason he gave for leaving an answer in the form “308915776*10000″. I docked a mark for that. If that’s what it takes for him to learn how to

multiply by powers of ten without a calculator, then so be it.)I force my student to explain how they test the answere they get to the problems I give on tests. On a typical 3 point problem i can give up to 2 point for a good way of testing the answere.