Saturday, February 17, 2007

Crossed lines

A tale of math panic

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 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...
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?

Time for me to get busy finding out.


Andrew - Australia said...

I emphasize with the students having more frequently than I like sat in exam feeling absolutely sick to the stomach because I have no idea what is being asked for!

I recall my high-school matriculation exam which had a 20% question that started from the volume of a solid fo rotation (a simple circle in this case). I could do all the other bits, but I had totally forgotten the formula therefore, I couldn't get to the other bits. I did the rest of the exam and then spent 45 minutes trying to get the formula from first principals.

After the exam I asked one of my fellow students. "4/3 pi r cubed" cam the answer. I've forgottem most of my maths, but I can still remember that one.

Maybe your students will have y=mx+b etched on their frontal lobes from now on.

BTW do all your students look like the one in the illustration? I might be tempted to re-learn differential calculus...

Anonymous said...

Yikes. This is a college class? [Although thinking back, I did take a college class where the textbook was Basic Algebra I.]

Anonymous said...

For a moment I thought you were talking about our class. I've enrolled back into college as a freshman. I've been out of school for more than a decade, but I managed to get placed directly into the calculus class. I along with most of our class totally bombed on the review chapter that is really just all about algebra and linear equations. The only good thing is that we chose to be rated on a curve (is that a pun?) It has taken me some time to memorize the different formulas (point-slope, rise-run, eh..whatever the third was) and to actually remember when to use each. I think the last time we did this in school we used graphing calculators so it's definitely been a struggle; we don't use calculators at all now.

David Harmon said...

Sounds like in many cases the anxiety is the problem. Remember, anxiety and panic shut down the higher mental functions....

Heather said...

Minor thread hijack here, have you gotten a look at that video about math texts in Washington state yet?

Zeno said...

Yes, Heather, I've seen it and pondered it. Haven't been able to get my hands on the texts denounced in the video yet (don't want to actually buy them, of course). And I've had too many course preps this term to find time to do some proper research on the topic for a blog post. The video link is bookmarked in my Halfway There slush file for later reference, so I haven't forgotten it. Just not gotten to it.

Thanks for the reminder. Maybe during spring break.

Rick @ shrimp and grits said...

I suppose that this sort of thing is why I have so much trouble with students being able to find a Beer's Law constant in my freshman chem class.

(Beer's Law - for the non-chemists - says that the amount of light absorbed by a compound is directly proportional to the concentration of the compound.)

They use Excel to plot their absorbance and concentration data - which generates a nice straight line. I tell them "Okay, the Beer's Law constant is the slope of that line you just plotted." More often than not, they look at me as if I've just told them that the constant is wkoj fasdkjfdskjhfdsa kjsdahf sajf hdsef akjh.

Pareto said...

I learned y = mx + b when I was 12 or 13; I'm probably more mathematically inclined than average, but are people not learning this by the time they graduate? And the concept of the slope is so integral (har har) that it's shocking to find that many people haven't properly understood it.

Peter Lund said...

How old are your students?

Please tell me they aren't in college!

Zeno said...

Oh, yes, these are college students. My school is a community college, which means we offer the usual postsecondary courses for college freshmen and sophomores, but we also have the complete gamut of earlier courses (at least as far as math is concerned). Many of our students carefully avoided all math classes in high school (or flunked them) and now are having to make up for lost ground. Some are adults who didn't go to high school at all.

As a result, our courses in computational math (that's arithmetic), pre-algebra, and algebra are full of people who are selected for their fear and avoidance of math. It's usually not as bad, though, as this semester's elementary algebra class, which is epic in its inability to cope with equations of lines.

TheBrummell said...

...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?

I'm glad I wasn't drinking coffee when I read that. Laugh-out-loud funny.

I'm quite certain that I am not permitted to pummel my students with erasers, even for pedagogical purposes.

But she asked for it!

OK, now I feel this comment slipping towards somewhere less pleasant. Oh well. I hope your students improve their skills, and pass your class.

Jonathan said...

It's not just you (but I am sure you know this). I teach a "workshop," a preparation class for college algebra. The students handle short quizzes, but I am concerned about our first exam. I teach both the procedural and the conceptual side, hoping that the concept will help the procedures stick. And they do, to a point.

The hardest part is what you call the "self-selection." The weakest math students put it off until their final year. So I see them not only weak, but also with 3 or 4 years of absolutely no work in mathematics at all

Peter Lund said...

But... how do they get into college when they are /so/ unqualified?

I mean, this is not just discalculia and anxiety...

Zeno said...

California community colleges have open admissions. Any adult or high school graduate can attend. It's part of our mission to provide instruction at any level up to and including the sophomore year of college (and our college units transfer to the University of California, the California State University, Stanford, Caltech, etc., etc.), but that leaves a lot of open space down below. We teach non-college-credit courses like arithmetic because there's a demand for it -- people who never learned the most basic math. (Some skipped school. Some flunked all their math courses.) Those who decide they want to learn the basics can't go to school with the children, so they come to us.

Hence they're all welcome, although I strongly suspect in the case of my algebra class that several of them must have slipped through the cracks of our overworked counseling office. Perhaps they got someone to sign placement slips for a course they're certainly not ready to take. We check pre-requisites (which for algebra is pre-algebra or a placement test), but that part of our system is definitely not working properly. I will be lucky to save half this class.

Anonymous said...

Ah yes, the curse of the pocket calculators...

In Introductioon to matrices class, there was a student relied on one a bit too much.

Step one: Divide the matrix by 6. One of the terms was 1.

Step two: multiply the matrix by 3.

Output: 0.48, since 1/6 = 0.16, and 0.16 * 3 = 0.48.

And it had to be correct, because the calculator said so!

That made calculating inverses and interesting challenge.