What Is the Thinking We Want? The Dynamics of Change

As most calculus instructors know, the majority of their students do not become mathematicians, but apply their mathematical knowledge to various engineering, scientific, or other professional fields--the client disciplines. These disciplines display a converging set of opinions regarding mathematical preparation, as illustrated in the following series of quotations [19].

Chemistry: "I'm a bit embarrassed to admit that I haven't evaluated an integral in 40 years! That kind of thing is not really what many of us need calculus for in chemistry... what seems to be important is for students to understand the meaning and application of derivatives and integrals, how to set up a differential equation and interpret the behaviour of its solution...If anything, numerical methods also seem more important now than analytical techniques."

Physics: "What we want is for students to bring a basic understanding of fundamental concepts of calculus into their physics courses. Right now, they are very good at taking the derivative mechanically, but have little idea of what the derivative tells them."

Electrical Engineering: "Our students must be able to interpret the behavior of solutions based on graphical output. They must develop a much better understanding of the function concept. More and more, engineeering students are looking at numerical and graphical representation and less and less are they looking at symbolic methods. In fact, the message I continually have to give some of my students is that the vast majority of things do not have algebraic formulas. Their calculus training was just too lopsided in emphasizing symbol moving. When they need analytic representations, we expect them to use sophisticated computer packages."

Chemical Engineering: "In the traditional educational approach, we in engineering tended to begin with very general, abstract principles and eventually worked our way down to specific applications. We now realize that this is not the best approach for most students. They are better served by starting with a series of down-to-earth examples and then generalizing to discover the fundamental principles."

Biology: "Computer simulations are an extraordinary tool for involving students in a problem-solving environment. It encourages them to interact at a much deeper level of involvement. Perhaps more importantly, it opens up doorways to them. A textbook approach is very narrowly focused, with the author directing the reader along the prescribed course in a totally linear fashion."

A composite student emerges from these descriptions: one who understands visual representations, is comfortable with computer applications, understands concepts of calculus, and is able to explore models and simulations.

An accompanying new theory of instruction also emerges. Traditional calculus lectures tended to present the subject in the same style as a mathematics research paper, with all details carefully worked out in advance. Textbooks also reflected this practice: "...the terms involving and can each be made smaller than epsilon/3, say, by restricting....". [20]. The expression epsilon/3 is suddenly just there. With some reflection, the student can see that "this works," and a few students will intuitively see this. But many students will be troubled, partly because epsilon/4, say, might also work, if not so neatly.

New approaches to instruction are designed to get students actively involved in their own learning. This involvement is often expressed in statements such as "students construct their own knowledge within zones of expert knowledge." A key to this construction is experimentation: students should be given exercises that reflect the empirical as well as the deductive side of mathematics. The project-based instruction pioneered by Davis, Uhl, Porta, Stroyan, and others puts this belief into practice.

Within the new calculus pedagogy these ideas are expressed in statements specific to mathematics. Students need to learn to read mathematics and express their own mathematical ideas in writing, and, mathematics instructors need to consider variations in the ways students learn. But these ideas are special cases of more general beliefs about learning and higher-order thinking. Lauren Resnick has written an excellent description of the characteristics of higher-order thinking. For example, "Higher-order thinking in general can be characterized as nonalgorithmic, complex, involving uncertainty. Higher-order thinking involving self-regulation of the thinking process, rather than someone else 'calling the plays', involves imposing meaning and finding structure in apparent disorder, and higher order thinking is effortful. There is considerable mental work involved..." [21].

The client disciplines outline a new approach based on applications of mathematics. It is important to point out that the impact of the computer is also significant in research mathematics. Here is another round-table discussion, in this instance between recipients of the Fields Medal in Mathematics [21]:

Smale: I think that computers help a lot in the day-to-day practice of experimental mathematics, but there is a much deeper effect of computers on mathematics, which is changing the whole structure of mathematics. Many of you heard me say recently that the greatest problem in mathematics in the last half of the twentieth century comes from the computer. It is the problem "Is NP not equal to P?" It is creating a whole different way of thinking about every subject in mathematics. It is about the way we do mathematics, about the emphasis on the algorithms.

Jones: How do we know that our computer program is working right? This is very important. One of the main things here is actually to be able to see what is happening. So, in fact, a very important part of this computer project is to have good graphics, so we can control and see that everything is not going wild. Thus, I think there are some very interesting problems coming up from simulation, as to whether what the computer is doing actually has anything to do with the realities.

The implication is that mathematics students should be introduced to computing concepts early on, so that they will understand the role of the computer in carrying out the mathematical process. The computer is more than a tool for learning calculus--the computer is an important subject in itself.