### Morbidity and mortality: Term 1

The term is over, and I now know that I’ll be back for another term - though it’s still unclear as to what I’ll be teaching: either a whole lot of statistics, or a calculus/precalculus mix. So it’s time to reflect on what went right and what went wrong last term, so that I can work on improving for next semester, particularly if I teach precalc again.

- I gave weekly quizzes in both of my courses; the quizzes were fifteen minute deals, and the questions were taken directly from the homework. This worked very well for my discrete class, in which quiz marks were strongly correlated with test marks; but it did not work at all for my precalc class. I actually had some precalc students get full marks on their quizzes, and
*then*ask me to explain the questions: it turns out they had just memorized their notes and reproduced the solutions on the quiz - a method that did not prepare them for the tests in the least. The discrete class, though, was one in which hard work translated more directly into success, and so high quiz marks did not mislead my students (or me) about their abilities. - My students in all classes were weaker than I expected. This was more of an obstacle for the precalculus crew than for the discrete crew. So, if I end up teaching precalc again, I will spend the entire first week of classes reviewing grade 9, 10, and 11 math. Good topics to review include:
- Solving linear equations
- The bare basics of setting up word problems - though this is something I’ll have to emphasize later on anyway
- In general, expressing one quantity in terms of another
- Working with rational expressions, including plain old fractions
- Basic properties of exponents

- That said: if I teach precalculus again, I need to explicitly establish what students should know coming into my class. I should prepare a worksheet for this purpose, and tell students that they should be comfortable with every single question on it by the second week of class. I should not be afraid to say to some students who ask me questions in class, “That is a Grade 9 question. You have a lot of catching up to do, and you will not be successful in my class until you get to that level. You can see me during my office hours for help, but I am not going to deal with your question here, because it deals with subject matter that is part of the prerequisite for this course.” This would have saved a lot of time in my weak precalc class; it also would have served to clarify my expectations. For students who are still at a grade 9-10 level by the (relatively late) drop deadline, I should advise them to drop my class and sign up for it again when they have met the prerequisites.
- In general, I should invite students to my office more often. This includes students who hassle me in the ten minutes between classes, asking me questions that can’t legitimately be addressed in that amount of time. Last term I often just tried to give quick answers, which didn’t really address most of the issues.
- I need to be more clear about what does and does not constitute acceptable behaviour in my class. In particular, I should dismiss any student whose cell phone goes off in my class; I certainly warn them enough about that. I should not be afraid to tell wilfully obnoxious students to behave, and I should make clearer earlier in the term to some of my female students that even though it is generally considered socially acceptable for them to giggle about how much they hate math and how much they suck at it, I will not tolerate it in my classroom. (I did make this clear in the last few weeks of classes. They were surprised, but they listened.) I think that this failure on my part played a large role in the downfall of my bad precalculus section, which contained a few strong personalities who succeeded in setting a negative tone for the class.
- I should create worksheets. The text is so horrible that the problems, for the most part, don’t address what I think are the central concepts I want to emphasize. Although I complained about students whining about how the tests were so
*different*from the homework, their complaints were not entirely invalid - particularly as the text had this infuriating tendency to ruin a perfectly good question with instructions such as “and now graph it on your graphing calculator…” - I do, for the most part, defend my decision to give tests that differed somewhat from the homework. I should defend that decision in class, vocally, on a regular basis. Some of my students truly seemed to be under the impression that I had merely forgotten to teach some of the material on the tests - none of which required skills or methods that I hadn’t gone over at some point. I should reach out to those students, to try to show them exactly how the course material related to the test questions - and how to better learn the course material and better prepare for subsequent tests. I thinkI succeeded in this with some students, but (understandably, I’m sure) not with the ones who were indignant about their tests and demanded that I change them.
- I am happy with the way I taught how to set up word problems. I am happy with the way that I emphasized how to set up equations by reading the problem carefully and establishing what quantities we knew, what quantities we did not know, and how they were related. Next term, I’ll spend some more time on this and make that section of the course more interactive, because - truth be told - it’s a far more important skill than most of the other things I teach in that class, like factoring quadratics and graphing parabolas.
- Next term, every time a student asks, “Will (suchandsuch) be on the test?” I should put it on the test. That’ll learn ‘em.

What precalculus text did you use? Was it your choice or had the math dept already chosen it for you?

I used the Larson/Hostetler precalculus text at my community college, my own choice, largely because the math dept has adopted the same authors’ calculus text for our three-semester calc sequence. It’s not a bad book and not too calculator dependent, although the authors are a bit too fond of “this simplifies to” without enough steps to really bridge the gap. (Of course, students never think enough steps are shown.)

I discovered from today’s final exam that my students can find the distance between two points, most can find the distance between a point and a line, and most know (or can recite) that a parabola’s points are equidistant from its focus and its directrix, but damned if they can tell me whether (5, 6) lies on the parabola with focus (7, 9) and directrix 2x + 3y = 15. (It is, in fact, the

vertex.) Unfortunately, because we did no problemsexactlylike this one on the homework, they were thrown for a loss. I should have known.Does your institution have a placement test? I’m assuming they do — otherwise your students would have tried to enroll in calculus straight away instead of taking pre-calc first. If you can get access to the data, you might be able to give the “head’s up” to the clueless.

TonyB: Here’s the lowdown on the precalculus piece of crap I used. I didn’t choose it, and I don’t even know how much influence the department head had, as he seemed appalled by the book and said he planned to replace it next year. I’ll pass on your suggestion to him (he makes the final decisions, but he listens to what his instructors say). “Not showing enough steps” is a problem that I, as a teacher, can work around reasonably well. The text being disorganized and ineffective - which mine was - is a lot harder to deal with.

I’m not surprised by the focus/directrix results. (We don’t even cover that stuff in my class; I’d lose 90% of my students if I did.) A few months ago, I did a bunch of standard rowing with the current/against the current problems: eg, I can row 10 km/h with the current, and 6 km/h aganist the current. What is my speed in still water, and what is the speed of the current? On the test, I gave a similar problem, but on the way back, the current had died down to half its original speed. One girl called me over during the test and helpfully informed me - as though this was an oversight on my part - that I had never done a problem where the current was HALF.

Rudbeckia Hirta - the prereq at my school for taking precalc is “at least C+ in grade 11 math, or at least a pass in grade 12 math.” I wish my school also had its own placement test, because getting a C+ (or even an A) in high school math is, I’ve learned, completely meaningless. The problem is that if a student, for instance, can’t add fractions, then there’s nowhere for them to go: my university doesn’t offer remedial courses at that level (nor is it its job to), and if my students didn’t learn fractions in high school the first time around, they’re likely not going to learn them the second time around.

But regarding “if they met the requirements, they’d just take calculus right away”: I had one student who apparently DID meet the requirements, but was lost in calc, so she joined my class. Her grade on her first test was less than 20%, and she soon dropped out. Since she met the calculus requirements, that means that she had supposedly, at some point, gotten a mark of C+ or higher on THE VERY MATERIAL I TEACH IN PRECALC. The mind boggles.

“if my students didnâ€™t learn fractions in high school the first time around…”

the first time around wasn’t in elementry school?!?

We try to enforce prerequisites and provide placement services, but the process is leaky. Some of our students use the on-line self-assessment tool we have and either lie or manage to fool themselves into believing they know more than they do. The tests in our assessment center are better placement tools, but not everyone takes them. We math instructors make a concerted effort during the first week of each semester to check transcripts or collect prerequisite verification slips from each student, but when a counselor provides a student with a verification slip and and merely checks the Interview box (as opposed to Transcript or Placement Test), it’s anyone’s bet as to how prepared the student really is. (Math teachers sure don’t want to do the job of counselors, but it appears some counselors aren’t too crazy about doing it either.) The delightful upshot of all of this is that the first part of each semester is a weeding-out process as students who thought they were going to leap-frog ahead actually discover they’re going to fall behind (and, of course, it’s the teacher’s fault for making the class too hard).

I shouldn’t let all math instructors off the hook, though. My colleagues and I are good guys and gals, really, but we naturally have a wide range of opinions about our mission. I happen to think we’re trying to instill at least technical competence in mathematics at levels appropriate to ensuring that students who go on to subsequent courses, in math or otherwise, don’t fall on their faces. A couple of my colleagues will instead assert that our sacred mission is to ensure that students succeed (sounds good), but interpret that to mean that students pass the current class (to hell with what happens later) even if chapters have to be dropped from the syllabus and tests have to be retaken (”Oh, did you flunk the exam? Here’s a makeup test with the exact same problems.”) until everyone ekes out a C. Arrrggghhh!

Perhaps this is a natural consequence of our two-fold nature as a community college: We both back-fill those basic math courses that students blew off in high school (we politely call our algebra courses “developmental” rather than “remedial” to minimize embarrassment) and prepare students with calculus that transfers to four-year colleges. (We are a major feeder campus for the Cal State University system and the University of California.) If all you want to do is get the students out the door with a two-year college Associate of Arts degree, then perhaps you can dumb down the curriculum without severe consequences (except to make the AA no better than a high school diploma). If, however, you want to maintain a student’s options to continue his or her education, you’re effectively foreclosing that option whenever you give credit for courses of minimal content. The students have credit for it, but don’t know the material, and this will come out in an awful way when they enroll in a UC campus as a transfer student and find out that they don’t know beans. It’s not a dilemma to a hardcase like me, but to some community college instructors it’s a choice between mastery and “success lite”.

By the way, although I’ve taught precalculus several times, this class still bemuses me. I never took it, going straight from high-school trig to college calculus in my younger days. The growth of such prep courses must be a sign of our failure to get students to understand what they’re doing, which is why intermediate algebra repeats the content of beginning algebra and why precalculus recapitulates all of algebra, with a repetition of trig thrown in for good measure. Perhaps you’ve heard of pre-algebra, which we teach as a transition from arithmetic to algebra. I get to teach it next semester for the first time. Oh, goody. Despite my reservations about the course, I’m going to use it as an opportunity to get my students ready for algebra, helping them to overcome what must be gaping holes in their preparation. Should be educational for me, too.

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