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Who you callin’ crazy now?

Rudbeckia Hirta of Learning Curves writes about her latest crop of college math students. It’s not pretty.

First the Gen-Ed kids:

Unfortunately for my gen-ed class, many of them cannot do algebra at all. Again, these students are bright and interested. Good students; they seem to like the class. Almost all of them hard-working and with excellent attitudes. … If these students were as serious about their high school educations as they are about their college educations, their lack of algebra skills cannot be entirely their faults.

And yet, the most missed question on the test (tripping up between a quarter and a third of the class) was missed because of algebra. They can set up the equation, but they can't solve for x.

No surprises here. They know almost no algebra. If you look at the sample exam questions given, you’ll see that she’s testing very basic algebra. Of course, the calculus students know more algebra. But do they know enough?

Allow me to point out that almost all of the calculus students can do algebra; their mathematical problems are fairly minor. Some may have to work at it more than they should, and we have had some parentheses issues, but, overall, the calculus students can use the distributive law and do not do stupid things with radicals and exponents. They are wary of rational expressions and a bit shaky on trig identities, but they are as skillful and well prepared as any students that one might recall from some mythical glory days back when freshmen could do algebra.

So their algebra skills aren’t exactly carved in stone. And, how much more of their algebraic knowledge is still at the inflexible stage? The wheels on the train are getting wobbly. Will the train stay on the tracks all through Calc I? And then II? We’ll find out in Physics.

-- KDeRosa - 30 Oct 2005


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Since I tutor at a community college, I can tell you that the wheels come off for using the chain rule for finding the derivative and related rates problems.

The chain rule poses problems because many end up with an expression with two to three terms. One of them may have a radical in the denominator. Now, the students have to get a common denominator that involves a radical.

In teaching related rates, many professors start with geometry. For example, if the volume of a sphere is increasing at a certain rate, what is the rate of increase of the radius. This involves doing algebra with the pi intact. It totally throws them off.

-- AnneDwyer - 30 Oct 2005

Related rates problems were the bane of both Bernie's and my existences. We were talking a while ago about hitting walls? B and I both hit the walls at related rates. And then, later, they fell into place, basically without further effort. And the problem wasn't algebra.

I think what we got stuck on was that the components of formulas like V=4piR^3/3 were suddenly to be thought of as functions of some third hidden variable (usually time), rather than as placeholders for numbers to be plugged in (as they had always been before).

This is quite a cognitive leap, and it happens early in calc 1. A while later, you've had a lot more practice thinking of formerly static variables as functions, so the problem resolves itself. In the meantime, if you have procedural knowledge, you're doing well.

Even implicit differentiation is not as bad as related rates, because there is no 'hidden third variable' involved.

-- CarolynJohnston - 30 Oct 2005

Hey, we have to blogroll Rudbeckia Hirta! She's a knitter and a math person (Catherine is a hardcore knitter too).

-- CarolynJohnston - 30 Oct 2005

I googled chain rule and algebra and struck math mistake gold:

Most of the mistakes that occur [in calculus]repeatedly involve algebra, rather than calculus. They can be avoided by being careful and checking your work. Others involve common misunderstandings about various aspects of calculus.

More vindication. The chain rule error is number 10 on this list.

-- KDeRosa - 30 Oct 2005

I was going to say that!

You have to get Rudbeckia Hirta up!

-- CatherineJohnson - 30 Oct 2005

Here's another take on the subject of 'why is calculus hard?'

Who Died and Made Calculus King?

Also, here's the post I wrote about it last summer, I think: calculus in high school

-- CatherineJohnson - 30 Oct 2005

The site Ken found is apparently based on How to Ace Calculus which seems to be a terrific book (meaning, terrific word of mouth).

-- CatherineJohnson - 30 Oct 2005

Here's Bernie's post on calculus

-- CatherineJohnson - 30 Oct 2005

This is interesting:

Students who wish to apply to highly competitive colleges are likely to opt for one of the calculus choices. College guidance counselors and college admissions officers both imply, if not explicitly state, that calculus is a requirement for acceptance at such schools.

I'd put money on it that the exclusive schools are looking for AP calculus. Ed doesn't want to believe this, but I've now seen it cited in two separate places, and my best friend in CA says it's true (her kids have just gone through admissions process).

I love this, too:

Very diligent and committed students who have achieved mathematical success through their own hard work and/or excellent teaching in previous math courses will want to attempt calculus. These students, however, may not be intuitive mathematical thinkers and may have experienced some difficulties with math instruction in the past.

A final group of students appears to end up in a calculus course because a some scheduling constraint that is beyond their control. A few years ago two of my students were "promoted" from a standard Algebra II course directly into Honors Calculus because the school did not have enough students to warrant a pre-calculus course. They succeeded - only because of a wonderful teacher who was committed to filling in all of the gaps in their mathematical instruction.

-- CatherineJohnson - 30 Oct 2005

Listmania of calculus books

-- CatherineJohnson - 30 Oct 2005

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