It is the concepts that matter. Students need to learn the underlying ideas of their subject. Knowing the concepts, a student can work his or her way through a forest of ideas to answer a question or produce a result. Not knowing the concepts, a student can only imitate or guess.
Frequently in mathematics, when doing problems or examples, the concepts are obscured by computational difficulties. Machines can be employed to do the computation, so that the student can concentrate on the concepts. This has been a major interest of mine for several years.
The first area where I did this was the application of matrices. Many YSU students take a course where matrices are used to describe problems whose solution involves a system of linear equations. Similar methods are used to work through the optimization procedures of linear programming.
In 1989 I started teaching this material using a primitive computer program on the primitive computers then in Meshel Hall. Since then others are using similar methods in these courses.
Computational difficulties also arise and obscure the underlying concepts in calculus . Only relatively simple functions can be graphed with reasonable ease, yet understanding the graph of a function is a key to calculus. Only relatively simple functions can be integrated with reasonable ease, yet the definite integral is a major concept in calculus to understand and apply.
In 1994 Albert Klein and I started teaching calculus in sections where all students were required to own and use a graphing calculator. We tried to use it so that computational difficulties of secondary importance could become secondary in the students' mind, and the primary concept of the moment could become their primary focus of thought.
In the latter part of the calculus sequence the greater power of the computer, rather than the calculator, is needed. This is apparent when students start to work with three dimensional graphs. In 1995 I started having my calculus 3 and calculus 4 students do some calculus work with the Meshel Hall computers and the program Maple. In addition to graphing, students have used the computer to work through the tedious calculations of the elegant study of curves in space, and in other areas as well.
With the new Mathematics computer laboratory in Cushwa, student work with computers is easier and can be more integrated with the rest of the course. In 1998, I extended the use of Maple into my differential equations course.
What is appropriate use of technology? What should a student be able to do by brain-only power and where should he or she rely on a machine? This is a matter of much debate at all levels of education. There are many horror stories. My opinion tends to be that in the first half of the calculus sequence, machines can help with arithmetic (but not the simplest questions) and graph drawing (but not the simplest graphs), but that the brain should do the symbolic work. In the second half of calculus, the machine can help students out with symbolic work when they get into big "messy" problems (but not for the simplest problems) and with three-dimensional graphs. Others disagree. Some say let the computers do the work from almost the beginning. Others say let the brain do almost all the work to the end. One thing is sure; the brain should do its most effective work to learn the real concept and to solve the real problem of the moment.