Posted by: Gary Ernest Davis on: November 21, 2010

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A

You described a very disappointing level of achievement among students:

“The students I teach cannot do elementary algebra even in calculus 3. Why then persist with a situation they will not need outside of an academic environment? ”

For those students as you described, their making any realistic progress in physical or mathematical sciences is unimaginable.

To say that something is unimaginable means that you cannot imagine it: It does not mean that no one might ever be able to imagine it.

We have to regularly work around this problem. I am finding better success by letting students use, for example, Maple, to do algebraic and other calculations for them. Their focus of attention then is on other issues, such as what the velocity and acceleration vectors along a space curve are doing, how these relate to the curve, what the curvature of the curve is doing, and how that relates to the shape of the curve, how the arc length from a point relates to the speed along the curve.

These are issues I want students to engage with, and if algebra is holding them up – as it is – then they can use the software to do the calculations for them if they wish. That’s how it is outside of school or college.

1 | Jim Wolper

November 22, 2010 at 12:05 am

This is an appealing presentation, but most of these ideas have been around since the 1980s. I teach an Applied Calculus course for first year students that quite explicitly follows the modeling approach outlined in the talk and replaces most of the hand-calculation with computer or graphing calculator work. Students find it VERY hard, much harder than a tradiational mathematics course, because the demands on their intellectual skills are much BROADER. In fact, for many of them, weak reading and writing skills are more of a challenge than weak calculating skills.

Furthermore, there are many insights into patterns to be learned from hand calculating. Take, for example, minimizing y/x, where y is some function of x, perhaps only given by a graph. A computer program can solve this, but the true insight comes from taking the derivative symbolically, leading to the realization that the minimum occurs where a line through the origin is tangent to the graph. Perhaps someone can come up with an attractive graphical illustration, but that will remove the power of abstraction.

Wolfram’s view of mathematics is too narrow. Concepts from higher algebra are important in applications in coding and cryptography, but his approach deprecates algebra altogether. Following the computational method exclusively will not lead one to classify the line bundles over projective space, but that is in fact an important tool in certain crytographic and coding applications.

Gary Davis

November 22, 2010 at 5:47 am

Jim,

I have to say that I am very much in agreement with Conrad Wolfram. His description of use of technology is pretty much how I teach. I have students do very little by hand. I do get them to do some assignments by hand simply because they say they want to, and that’s fine with me. The reality – as Conrad describes – is that in the real world, outside of school or college academics, they will use computers to do the calculations. The students I teach cannot do elementary algebra even in calculus 3. Why then persist with a situation they will not need outside of an academic environment? Better to emphasize the more important aspect of mathematical activity: the thinking and problem solving aspect.