01 Nov 2005 - 18:15

Rudbeckia Hirta on calculus

...answering these questions:

In terms of teaching calculus to college freshmen, it would be easier if they had NOT taken calculus in high school. Or, rather, it would be best if the students who took calculus in high school all—or almost all—scored well enough on standardized tests so as to not retake calculus in college. It's the retaking that sucks (and the "wasted" year).

Why do you say this?

Are these kids hard to teach....or are you saying they're wasting their time re-taking calculus....??

And do you think it would be better to take neither of the AP calculus courses?

If so, is there a math course you'd prefer these kids take senior year?

The first issue is one that we read here all the time: teaching by "exposure" versus teaching to mastery. If you want the students to know calculus (not to have merely "taken" calculus), then you want them to have been in a class that expects mastery. Some high school classes do, and some high school classes don't; some college classes do, and some college classes don't. But most importantly why spend two years on a task that most students are capable of doing in one?

(I believe it took Newton two years to INVENT calculus. Of course he was working on it full time because everything was closed due to some plague.)

The other problem with re-teaching is that the students think they already know everything. This leads to a few common problems: the students think they know everything already, so they are reluctant to put effort into learning (coming to class, doing homework)—instead trying to get by on what they already know. If they took a sub-optimal high school calculus class, the teacher may have treated the foundational material (which is very abstract and difficult for students) as "unimportant"; the students often pick up on this attitude. During certain parts of the class, the point of the lesson is to understand a certain idea (the definition of the derivative and its connection to slope), and students who have already taken calculus will instead choose to (incorrectly) answer the question by using an easier calculation taught later in the course. (It's not that I'm against "short cuts"; the point of the lesson is to understand what's going on behind the scenes of the shorter calculation.)

In terms of the AP calculus classes, I really like the BC calculus. (Biased I am, as I took BC calculus myself during the 1989-90 school year at Niskayuna High School in Niskayuna, NY.) There are very few circumstances where I would recommend AB calculus. There are some obvious ones (teacher, schedule logistics, etc.) but aside from that, the only other time I would recommend AB over BC would be for a student who has struggled greatly in precalc, who is planning on studying the humanities in college, and who is planning on attending a college where part of the gen-ed requirements can be fulfilled by scoring well on the AB calculus exam. However, a BC calculus course that prepares students to take the BC exam is a fine opportunity.

In terms of 12th grade math offerings, that would be a fairly place-dependent recommendation. If the school has a corps of students who finish 11th grade ready for a real calculus course, then something like BC calc would be a canonical recommendation. Otherwise it then becomes an issue of looking at WHY aren't there students ready for 12th grade calculus (school too small for critical mass? ineffective programs? something else?) and making decisions based on that. There is a lot of interesting mathematics that can be done at the high school level, and the "right" course will depend on both the school and the students.

people hate learning things all over again

The other problem with re-teaching is that the students think they already know everything. This leads to a few common problems: the students think they know everything already, so they are reluctant to put effort into learning (coming to class, doing homework)—instead trying to get by on what they already know.

As far as I can tell, this is a huge, 'foundational' problem at all levels.

My Singapore Math kids are in open revolt at having to learn to do bar models when they can already solve a problem doing simple calculations (i.e. subtracting). Christopher, too, runs amok whenever I ask him to 'back up' a step.

My neighbor and I were talking about this one day, about why it's so aversive—and aversive is the word—to 'go back to the beginning,' or to 'do things more than one way.'

I came up with a theory, which I have now forgotten.

It will come back to me.

discovery ≠ memory

That's something I've been meaning to point out.

I've seen constructivist educators claim that discovery is a memory aide. Something you discover for yourself you remember better.

I've also seen cognitive scientists say this is rubbish, and I can tell you it's rubbish without recourse to PubMed.

Nonfiction writers are constantly forgetting their discoveries (which we call 'ideas'); this is why writers carry notebooks and pencils around; this is why some writers will actually get up out of bed in the middle of the night to write down middle-of-the-night discoveries on a piece of paper.

If you don't write your ideas down, they're gone.

As a matter of fact, new ideas are far less sturdy than old ideas, precisely because they are new. They haven't been rehearsed, by definition.

what's the code for the 'does not equal' sign?

Remember: it pays to ask a question!

never mind

I found it here

Back to main page.

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When you forget the code for the "does not equal" sign, you can always use "!=". That's the symbol used in c programming, and at least the geeks out here will know what you're talking about...

-- StephanieO - 01 Nov 2005

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I know that two years ago I spent a great deal of time figuring out the calculations and assumptions behind a figure for my then job. I now can't remember on anything but the highest level what they were or what was so difficult about grasping what was going on.

And further back, at engineering school I developed an algorithm for designing crossword puzzles that involved a trick I remember being rather pleased with myself over. I just can't recall the trick.

-- TracyW - 01 Nov 2005

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And further back, at engineering school I developed an algorithm for designing crossword puzzles that involved a trick I remember being rather pleased with myself over. I just can't recall the trick.

Yup.

I've probably forgotten as many good ideas as I've remembered.

-- CatherineJohnson - 01 Nov 2005

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And look what happened when Fermat didn't write down the proof of his famous theorem because the margin was too small.

-- BarryGarelick - 02 Nov 2005