Tall, Dark, and Mysterious

5/31/2005

On teaching college students what they should already know

File under: Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 6:41 pm.

Rudbeckia Hirta succinctly explains that if you can’t do algebra, then you can’t take calculus:

Due to reasons beyond my understanding, high school math and college math are completely unaligned. The K-12 system sends us students whose knowledge is a mile wide and an inch deep: we get students who are shaky at algebra, frightened of fractions, and unsure of how to find the areas of basic plane figures (and completely unable to accept the idea that it is a reasonable request to ask them to solve non-standard problems where the method of solution is not immediately obvious), but they have been exposed to matrix arithmetic, computations from polynomial calculus and other supposedly “advanced” procedures. You would think that the “college prep” track would prepare students for college, but it doesn’t. Recently I read somewhere (maybe in Focus?) that there is more calculus taught in high schools than there is at colleges.

See also: mathematics by pattern matching, word problems, et al. So much of this problem can be traced to the fact that students do not understand that an equation is a relationship among quantities. Each equation they see is a concept unto itself, to be memorized and applied to the word problem on the test that looks like the word problem that I did on the blackboard and that used a similar equation. If the word problem on the test doesn’t look like any of the problems I did in class, then that question is “totally unfair”. Most students will leave the question blank, or solve a completely different problem (one with a ready-made equation) in the space provided.

This New York Times op-ed is about teaching freshman English to illiterate college students, but I’m sure that anyone teaching freshman math to innumerate college students can find plenty to relate to. I’m not sure I’m willing to buy into, wholesale, the author’s belief that content should be ignored in favour of form, but I can’t argue with success:

On the first day of my freshman writing class I give the students this assignment: You will be divided into groups and by the end of the semester each group will be expected to have created its own language, complete with a syntax, a lexicon, a text, rules for translating the text and strategies for teaching your language to fellow students.

…14 weeks later - and this happens every time - each group has produced a language of incredible sophistication and precision.

How is this near miracle accomplished? The short answer is that over the semester the students come to understand a single proposition: A sentence is a structure of logical relationships. In its bare form, this proposition is hardly edifying, which is why I immediately supplement it with a simple exercise. “Here,” I say, “are five words randomly chosen; turn them into a sentence.” (The first time I did this the words were coffee, should, book, garbage and quickly.) In no time at all I am presented with 20 sentences, all perfectly coherent and all quite different. Then comes the hard part. “What is it,” I ask, “that you did? What did it take to turn a random list of words into a sentence?” A lot of fumbling and stumbling and false starts follow, but finally someone says, “I put the words into a relationship with one another.”

An equation is a relationship among quantities. How many times have I tried, and failed, to get this idea across? Imagine getting students to realize it for themselves! Unfortunately, many students lack the intuitive ideas about math needed to know, even subconsciously and with the help of leading questions, that equations are anything other than a jumble of letters, numbers, and symbols. (Even though I spent twenty minutes showing how we could use the definition of a circle and the Pythagorean Theorem to derive the equation of a circle in the Cartesian plane, nearly all of my students were angry that I wouldn’t provide the formula on the test. I mentioned, incorrectly, that they could derive the formula themselves, on the test, if they needed to; of course, I was wrong.) But I would gladly sacrifice 80% of the poorly-learned content in a first-year college math course if I could instead effect a solid understanding of what equations really meant - and how to get one from a sentence or two of information. I wonder if Fish’s lesson could be adapted to the first-year math classroom.

At Critical Mass, where I found the NYT piece, Erin O’Connor isn’t optimistic even about applying it to the English classroom:

[M]ost university composition courses are taught by graduate students who are a) not necessarily good writers themselves, and b) often more interested in using the composition classroom to practice teaching the content they hope to teach as non-composition teaching English professors, and you’ve got a situation in which the Fish vision, regardless of its merits, is pure pipedream.

Ditto. No one teaches precalculus if they can avoid it; at Island U, it got passed from temps to new faculty to the department head, who teaches the courses that no one else will teach. Everyone says that the precalculus course needs to be completely revamped, but given the option, they’d rather take a less thankless teaching assignment than put forth the effort to revamp it. And around and around we go.

Gather ye round and swoon over my (ex) department head.

File under: Righteous Indignation, Those Who Can't. Posted by Moebius Stripper at 8:18 am.

The second semester of the year has ended, this time for real: this brat wrote her final exam last week. Finally. Department Head handled the whole affair, from arranging the exam date to grading the paper, and MLIHASIM got the C that she needed. So, good on her, I guess. Not content, however, to leave the course with her honour even vaguely intact, she penned a valedictory email to Department Head, thanking him for supervising and grading her exam, which she’s sure was a huge burden for him, but what could she do? - if it were up to her, she pointed out, she wouldn’t have had to write that exam at all. She informed him that for the most part she “enjoyed having [me] as a teacher”, and that I was “good at explaining the basics”, but that my “tests and exam were significantly harder than the homework” and in fact contained some “questions that we never did in class.”* Since she hadn’t done math in years and years, she had a tutor “show [her] how to do all of the homework problems” and yet she still found the exam “very hard”. Oh, and she talked to some other students and they agreed with her, and would Department Head “please keep this sort of thing in mind” the next time he hires faculty? If this sounds familiar, it is: she aired precisely this grievance (minus the hiring advice) every single goddamned time we met outside of class, not to mention several times over email.

Department Head forwarded me this note. He also, God bless him, forwarded me his reply, which was basically all,

Dear MLIHASIM,

I’m glad you enjoyed Moebius Stripper as an instructor. As for your other remarks, I’m afraid that your expectations of a college-level math course are incompatible with reality. MS’ exam was no harder than the ones I give when I teach this class with the same text (and hence with similar homework). Math is not about memorization; in fact, mastering it requires that you be able to apply the concepts you learned to new problems. That you did not learn to do this in spite of the effort you put into this course indicates that a C was an appropriate, if not generous, mark for you to achieve. MS taught this course exactly as it should be taught, and exactly as I would teach it - though I don’t think I am as patient as she is! Speaking of which, your hiring advice is rather moot, as I’m the one who will be teaching this course - as well as the follow-up - next semester. Say - I guess that means I’ll have you in my class! See you next term and have a good summer, and I look forward to seeing you in the fall.

Dealing with students who think that they should be allowed to dictate the terms under which they learn (or fail to learn) the subject is frustrating. I can’t imagine how much more frustrating it would be if those students had the support of my boss.

* However, some of the exam questions were actually identical to questions on the review sheet - which MLIHASIM had actually told me she wasn’t going to do, because it too was too hard.