David Arnold’s discussion of technology in the teaching of Multivariable Calculus was a quick introduction to Matlab, Geometer’s sketch pad and Symbolic toolbox (in Matlab). Much of the talk was also a ‘geewiz’ exposition of some cool things you can do you in Geometer’s sketch pad. I left the discussion not having a good sense for how these technologies might be used to replace (or if they should be used to replace) current curriculum.

Overall, after reviewing Mr. Arnold’s web site, I’m impressed by his extensive use of technology in the teaching of Calculus. He’s even done away with exams and grades exclusively based on performance in labs/quizzes and homework.

The question for me is: is this appropriate. Mr. Arnold has demonstrated that technology can replace a more analytical approach, but should it? For example, you can use the sketch pad to find the arc length of the cycloid without the need of any the analytical tools you traditionally learn in second year calculus. Do you lose anything by orienting yourself to the geometric interpretation in the technology?

One might argue that the analytical tools were invented 100’s of years ago in lieu of tools like the geometer’s sketch pad. After all, the inventor’s of these tools had specific geometric problems in mind (i.e. the orbit of planets) when inventing the Calculus. Thus, its okay to dispense with the analytics to be replaced by high powered technology.

The problem with that argument is that the analytic tools have general application. Not all problems have a geometric interpretation… for example, what picture can you draw that will help you solve calculus problems involving more than 3 variables? The tools are there to help you abstract to these higher dimensions were intuition fails you. Tangentially, some people are more comfortable “pushing symbols” than working with geometry. Personally, I have a hard time imagining shapes in 3 dimensional spaces.

In the end, I applaud Mr. Arnold’s work. My question to him: how does the technology help students get beyond their intuitions of geometry to help equip them with more general analytical tools?

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