Questions and Answers

The following dialogue is a slightly paraphrased composite of several email discussions between a skeptic (asking the questions) and a technology advocate (supplying the answers).

Question: Do you think that it is possible to master calculus (or other mathematics) without putting in long hours of toil?
Answer: First of all, we should not confuse a student's grinding through template activities with "mastering calculus."

Question: But aren't you saying that clicking buttons, or pulling down menus will replace thinking about calculus concepts?
Answer: People used to complain about "cookbook" approaches to calculus, when the sliderule was the only calculating device available. We are not trying to create a "clickbook" approach.

Question: Still, you seem to be saying that technology will take the hard work out of learning calculus.
Answer: No. We agree that mathematics is not "easy." The fallacy that mathematics is something that comes easily to people with a certain "gift" is definitely not one we want to support. Mastery of calculus is hard work, but it doesn't have to be centered on tedious template activities. That's why Calculus WIZ was invented. Calculus WIZ will do almost all the students' homework in the traditional Calc 1 & 2 courses--solving the problems in detail and typesetting the solution, ready to hand in.

Question: I and many of my colleagues feel that routine drill problems are an essential tool for elaborating, reinforcing, and illustrating points made in lecture, and that students who are never exposed to a large number of such problems are often cheated of any real opportunity to develop their skills before they are faced with an exam--which they will in all likelihood flunk, if they have done all their homework using a computer tool.
Answer: We have some evidence that technology does not impair student learning.
The BYU Outcomes Study [15] compared the grade-point averages of students taking courses in a traditional format with students using the Harvard Consortium materials and students using the Mathematica-based course "Calculus, the Language of Change" [C:TLC]. Results for seven different courses are shown below:

Physics 221 (with Calculus prerequisite)
     C:TLC     Harvard     Traditional
      3.32         3.09           2.99
          
Physics 122 (Calculus co-requisite, second semester)
     C:TLC     Harvard     Traditional
     2.97         2.93          2.91  

Theory of Analysis 315
     C:TLC     Harvard     Traditional
      3.16         2.44           2.72
      
Linear Algebra 343
     C:TLC     Harvard     Traditional
      2.96         2.80           2.74
      
Multivariable Calculus 215
     C:TLC     Harvard     Traditional
      3.33         2.99           3.02
      
Advanced Engineering Math 312
     C:TLC     Harvard     Traditional
      2.93         2.86           2.959
      
Statistics 321
     C:TLC     Harvard     Traditional
      3.33         2.99           3.02

Question: These are interesting (and rather surprising) outcomes to me. So, do I understand then, that your goal is to train students in using calculus with a computer-based aid such as Mathematica to attack challenging and realistic applied problems?
Answer: That's right.

Question: What about the whole range of clever analytic tricks, for example to find , which I think it is still important to expose students to?
Answer: What makes more interesting than ? They both use the same two functions. Most of the "techniques" of integration are not interesting by any stretch of anyone's considered imagination. If we decide that symbolic integration should be studied, our students would be better off studying the algorithms used by symbolic algebra programs, rather than learning how people found integrals before there were computers.

Question: I am willing to consider that you might be right about this. I still think that students should have the experience of studying a coherent and successful mathematical theory, of which calculus is one of the best examples. And I think that for scientists and engineers in the next century, training in real and complex analysis is going to be even more important than it is already. But when I see students trying to compute an average by entering a + b/2 into their calculators, I wonder if technology is limiting their understanding. Clearly such students are a menace if they are using a CAD program to design an airplane.
Answer: My goal is to attract and train future scientists. Will they solve problems with computers, or should I restrict their training to contrived 19th century paper-and-pencil problems that come out in integers? Using Mathematica enriches the course, makes concepts clearer, allows for solution of more problems that are interesting to students, and students are learning to use a very powerful new tool.