How much calculation is necessary in a calculus course? After all, that's what the word means: a system of calculation. We want our students to have good “symbol sense” -- but do we know what that means?
This was not much of an issue with the traditional course because the whole course was about calculation. We assumed -- with no theoretical or empirical justification -- that students could learn calculus by watching us calculate, and that we could learn what they knew by watching them calculate. Our job was to find the best examples to show them and then to test them with examples that would surely show whether they “understood” or not.
Thinking about who our students are and how they relate to mathematics, we have found it much more productive to embed the need for calculations in contexts that students find meaningful. When we start with a problem students might actually care about and find that we need a new technique to solve the problem, there is a motivation for learning that new technique. “You're going to need this later” never was much motivation, especially if it turned out not to be true.
On the other hand, there is some justification for a modest amount of unmotivated and uncontextualized computation -- if it's not too difficult -- because students whose entire mathematical experience has been symbol-pushing find a comfort level in doing familiar activities.
Apart from making students feel comfortable, we think that some amount of symbol manipulation is necessary in order to understand what machines do for us and to monitor the reasonableness of machine-generated answers. We are convinced that will continue to be true even in environments where essentially all calculations are done by machine. Lacking any theoretical base for deciding how much is enough, we continue to address this problem empirically. However, we have found through our use of gateway tests that most students can learn appropriate symbolic skills on their own and from each other, with relatively little class time or text space devoted to this.
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Send comments to the author <das@math.duke.edu>Last modified: May 17, 1997