This term's elementary algebra class is one of the weakest collections of math students I have ever had the experience of teaching. (This is the same five-day-a-week class that contained the student who thought we met only on Tuesdays and Thursdays.) Only five students (out of 23) passed the exam on the equations and slopes of straight lines. It was that big a disaster.
Algebra is one of those subjects that can be learned in at least two completely distinct ways. While I prefer conceptual learning, I will settle when necessary for mechanical. Many of algebra's techniques can be learned by rote and applied mechanically. Anyone capable of tying shoelaces should be able to master a short progression of steps and apply it in specific circumstances. It's a fall-back position, but I thought it was a robust one—especially when fortified by my deliberate policy of allowing students to maintain and use a notecard of formulas, instead of emphasizing memorization.
Shows what I know. Our most recent chapter was a paean to the beloved slope-intercept form for the equation of a line. The equation y = mx + b is a delightful mathematical construct, manifesting in all their glory the slope m and the y-intercept b. Yet when I asked my students to find the slope of the line 2x − 3y = 18, did they remember how time and again we had solved for y so as to pluck the slope from in front of x in the resulting equation? They did not.
No, some of them began to painstakingly grind out points on the line so that they could use the “rise over run” formula for slope. That shouldn't have been too bad, since I gave them a line with simple axis intercepts, but they spurned those, too. One student plugged in x = 1 and discovered to her dismay that the corresponding value of y was −16/3. She would have to stick fractions into the ratio for slope and then simplify the result. My students hate fractions, probably because they worship the decimal approximations that tumble out of their calculators. (Many modern calculators can do fractional arithmetic, but most of my students clearly don't know how to use that feature. Probably just as well.)
Once again, the students who pick a clumsy but correct technique lack the chops to persevere to a successful conclusion. The ones who could have pulled off the calculations never had to, because they (all five of them) chose more efficient approaches to the problem.
Lost beyond redemption, one student came up to me to ask a clarifying question. I encourage them to see me when they're stuck, even during exams and quizzes. If they can answer the questions I ask, they can usually do the problem on which they've been stuck. It's a Socratic kind of thing. In this case, my student was so anxious she could not remember the slope-intercept form that we had been working on so hard. It was not even on her notecard of formulas. (When students follow my directions and carefully annotate the formulas on their notecards with reasons to use each one, they often remember the formulas and their applications without needing to look at the card during an exam. They've accidentally learned the techniques.) We quickly came to an impasse, so she handed in her paper and returned to her desk, hyper-ventilating and shaking.
I don't know what to do, but I think a lot of one-on-one student-teacher conferences are coming up. We're moving from simple linear equations to a chapter on solutions of linear systems. The dean won't be pleased if I salvage no more than five students out of my entire enrollment. And I sure won't be pleased either. And what are my students thinking?
A big part of it is probably test anxiety, since several students who got D's and F's are much more successful in classroom discussions. They ask intelligent questions and seem to be following what's going on. Most of them have modest success on single-problem quizzes that I frequently give, but multiple-problem exercises or four-page exams strike terror into their hearts. I'll have to talk many of them into getting the test-anxiety counseling that we offer our students. Clearly they need it.
One algebra student has sent me a request, but I don't think I'll be able to honor it:
Professor Z:I'm quite certain that I am not permitted to pummel my students with erasers, even for pedagogical purposes. It may be that this student is still working out old classroom traumas, but what's going on with all of her classmates?
I have a request. Right before I start taking my next test would you please take the chalk filled eraser and whack me upside the head with it in hopes it will knock the “idiot” out of me? Remember my first assignment in which I informed you of the Algebra for Idiots book? You said you felt “we” could get me above idiot stage. I am so frustrated with myself for this last test as I truly did know how to do the problem. I think it's psychological due to a bad math teacher in the 4th grade, made me cry and called me stupid. His name was Mr. Xxxxx. I appreciate that you haven't done that. All joking aside...
Time for me to get busy finding out.