26 Aug 2005 - 20:01

who died & made calculus king?

From an interesting page by this title at Math Forum:

What causes students to experience difficulty once they are taking calculus?

• Many aspects of algebra and pre-calculus are self-contained. A student's success will not necessarily depend upon his or her proficiency with material in a previous chapter. Calculus, on the other hand, is entirely new material that builds upon itself continuously.

• Calculus students cannot always assimilate the material quickly enough.

• Calculus students can fall behind and find it difficult to catch up.

• Some students have difficulty with the large number of word/application problems.

• Some students do not possess the cognitive flexibility to switch strategies that can be required to solve specific calculus problems.

• Some calculus students have difficulty with active working memory and consequently with manipulating all aspects of a problem without getting lost.

• Some calculus students do not remember all of the necessary material, especially formulas, from algebra and pre-calculus.

• Calculus students are challenged to learn inverse operations in close succession; consequently, they can confuse one type of a problem with its exact opposite.

This list is especially interesting in terms of what cognitive science tells us about learning & remembering.

I mentioned that I need a book that tells me exactly what formulas, concepts, math facts, etc. a person has to know cold in order to take the next course up.

This list reinforces my feeling that we need such a book, along with a workbook that would structure an ongoing sequence of practice.

I think the RUSSIAN MATH book does an amazing job of fending off the last item on the list.

One last note: I suspect this list is probably a good rundown of why students of any age in any level of math have trouble learning it.

cram school

I had an interesting experience yesterday that illustrated the importance of a student having time for math to 'sink in.'

I was trying to teach Christopher the Saxon 8/7 lesson about subtraction of fractions with borrowing (or regrouping).

He couldn't do it at all.

Then his friends Drew & Marc, who've been in Phase 4 since the beginning, came over. Both of them could not only subtract fractions using regrouping, they could do a darn good job explaining what they were doing & why.

They told me they'd learned fraction subtraction with regrouping in 4th grade, back when Christopher was learning basically nothing.

Then they learned it again last year, in 5th grade. I know this, because Christopher & Drew were in the same class by the time Mrs. Woeckener taught the subject.

The difference between Drew & Marc, who've had a year and a half to know what fraction subtraction with regrouping is & how it works, and Christopher, who's had a huge amount of Intensive Math Intervention but didn't learn this topic in the 4th grade, was stark.

He did pick it up almost immediately, after Drew & Marc showed him how to do it. ('Drew and Marc are better teachers than you!') So that's something.

But this is extremely new & fragile knowledge in his head. Drew & Marc have a far sturdier base on which to build.

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Is there any way to tweak wiki and make it display the number of comments after Comments?

This would help keeping track of comments at a glance.

-- CharlesH - 26 Aug 2005

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I've believed for a long time that the way we teach Calculus in this country is absolute rubbish. And I'm talking about teaching it to undergraduates. Then the rubbish got pushed down into the high schools. Now there seems to be a sort of destructive arms race going on among the parents to have their kids take it before any of the other kids so that at least one middle school in Boulder is teaching it in 8th grade.

That's insane.

What's wrong with the way we teach Calculus? Well, we give kids 12 years to, in essence, learn how to do four things and four things only: add, subtract, multiply, and divide. Then we give them typically one semester to learn how to do two entirely new things, differentiate and integrate. These latter operations are just as difficult to learn as adding and subtracting; perhaps more so because they are based on an entirely new philosophical concept called "limits" which, unlike adding and subtracting, has no analog in ordinary life. It makes no sense to spend 12 years learning 4 things and then one semester to learn 2 rather more difficult things.

A great Russian mathematician once dubbed arithmetic "the physics of marbles". There is nothing that one learns in standard school-age math that can't be turned in some sense into a statement about marbles, a statement that can in principle be physically tested or manipulated with marbles.

Not so Calculus.

Calculus was invented by Newton in order to describe the physics of what is now known as Newtonian mechanics. It is difficult if not impossible to understand it thoroughly without simultaneously learning that physics. The two should arguably never have been separated academically. It seems extremely dubious to me that 8th grade students will ever have the solid mathematical maturity necessary for learning the concept of limits. Most undergraduates do not.

There is still controversy among professional mathematicians as to the true meaning of the Real numbers, which is the same to say as the concept of limits as taught in Calculus. There is still substantial disagreement as to the proper formulatoin of the concept of integral.

Finally, in contrast to the standard dogma, I'm rather dubious as to the real utility of calculus. Although all engineers are taught calculus and in my polling are uniformly in favor of its being taught, very few of them ever use it in practice. Most would be perfectly content and productive if they were taught merely a few formulas concerning Fourier transforms. Linear algebra is probably the more useful and tractable subject for the general population of undergraduates. High school students should probably spend an entire year perusing limits from various angles before being taught Calculus in the following year.

-- BernieJohnston - 27 Aug 2005

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Bernie-I'm off on vacation, but I want to talk to you about this when I get back--ok???

-- CatherineJohnson - 27 Aug 2005

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Boy, Charles, I wish I could figure out how to tweak Twiki (ha) to count comments. It's not one of the things I can do easily. I'm definitely thinking over ways to do it, though.

Sorry for the inconvenience. :(

-- CarolynJohnston - 27 Aug 2005

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Actually Bernie, although I have the electrical engineering degree and have never set up and solved a problem using formal calculus in my life, I think learning calculus is useful.

What it gave me was understanding how the directly observable data, its rate of change, and the rate of change of the rate of change are connected. So I could look at a proposal to build tidal electricity generators and immediately know that their output would be dependent on the rate of change of the tide and be about zero at the slack of the tide.

I don't think a few formulas about Fourier equations would be as useful. And I have used fourier transforms in my working career - sometimes it's useful to think in terms of frequencies.

-- TracyW - 23 Jan 2006

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Tracy

What it gave me was understanding how the directly observable data, its rate of change, and the rate of change of the rate of change are connected.

Interesting.

I'm SO looking forward to calculus.

Calculus feels like a HUGE missing chunk in my life....

We'll see.

-- CatherineJohnson - 23 Jan 2006