Interdisciplinary Math:
A Round-Table Discussion on Calculus Reform

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A Round-Table Discussion with the Client Disciplines

Sheldon P. Gordon, Mathematics, Suffolk Community College

Mathematics: I'd like to welcome all of you to this round-table discussion on the impact of the new calculus courses on the client disciplines. The panelists who have taken the time to share their experience and perspectives with us are:
    Stephen Boyd (Electrical Engineering, Stanford University)
    Tom Daula (Economics, US Military Academy)
    Doyle Daves (Chemistry, RPI)
    David Hanson (Chemistry, SUNY Stony Brook)
    T. J. Mueller (Biology, Harvey Mudd College)
    John Prados (Chemical Engineering, NSF)
    Wayne Roberge (Physics, RPI)
    Mike Ruane (Electrical Engineering, Boston University)
    Clifford Schwartz (Physics, SUNY Stony Brook)
    Angela Stacy (Chemistry, UC Berkeley)

We in the mathematics community are extremely interested in how the changes we are making in calculus are perceived in your fields. Are these changes appropriate to your needs or will they cause severe problems down the road?

Let's begin with the question of how your own disciplines have changed over the years, particularly with regard to the mathematical needs of your students. Anybody want to start?

Electrical Engineering: Let me take a stab at that. There have been tremendous changes in engineering over the last 20 years and these changes are now being reflected in engineering education programs. A lot of the plug-and-chug approach has given way to a much greater emphasis on computing and modeling.

Economics: Modeling -- that's the key word in modern economics.

Electrical Engineering: Exactly. We're now freed of the necessity of looking at just those simple cases that admit of closed form solutions. We can look at more realistic models which include more relevant factors.

Physics: Yes. For example, we always looked at projectile motion with only gravity and then stepped up to include air resistance where the force is proportional either to the square of the velocity or the square root of the velocity. Why? Because those are the cases you can integrate in closed form. But do you really think that either model is the precise truth? In the past, we've been limited by our analytic tools. Now that we have better tools, we in physics are also looking at more realistic situations.

Electrical Engineering: That's exactly the kind of thing I mean. We still start with the fundamental ideas, usually with some form of linearity assumption. However, we quickly move to nonlinear systems with perturbations and then compare the results of linearizing the model to the nonlinear model.

Mathematics: But what does that mean in terms of your courses? Or, for that matter, our courses?

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, engineering students are looking at numerical and graphical representations; less and less are they looking at symbolic methods. In general, the profession of electrical engineering drives the need for qualitative 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 lop- sided in emphasizing symbol moving. When they need analytic representations, we expect them to use some sophisticated computer packages.

Chemical Engineering: That's not only true in electrical engineering, but across all the engineering disciplines. There is far less of the old-fashioned pencil-and-paper mathematical manipulation. In all areas, when students need to evaluate integrals or solve differential equations, they are expected to turn to Mathematica, Maple or MathCad. What is critical is that the students develop a better conceptual understanding of the mathematical ideas and techniques -- What is the effect of changing parameters in a model? What is the behavior of a function, particularly one with several parameters? How do you go between the symbolic representation and the graphical representation of a function? Even something as basic as: How do you distinguish between the dependent and independent variable?

Electrical Engineering: Right! It's the understanding and the applications that count. In terms of what many students are learning today, the traditional symbolic manipulation is totally irrelevant.

Physics: It's exactly the same in physics today. The heavy manipulation is being relegated to Maple or Mathematica. 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: Let me give an example of that based on the fact that the current in a circuit is the derivative of voltage. A common problem in electrical engineering texts is: Here's the graph of the voltage in a circuit; sketch the graph of the current. It's amazing how many students, who have completed a full three semesters of calculus, have no idea where to begin. All they can do is differentiate any conceivable expression, but they don't know what it means. But this problem underlies how an understanding of the mathematics helps in understanding the engineering situation and vice versa.

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. But, we typically require three semesters of calculus for students heading toward organic chemistry, particularly to build up the ability to visualize objects in three dimensions. For students heading toward physical chemistry, we need more, definitely a good course in linear algebra and possibly one in differential equations. However, 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 behavior of its solution. Knowing where to look up integrals seems as effective for most of us as knowing how to integrate. If anything, numerical methods also seem more important now than analytical techniques. Most importantly, students need to bring an understanding of the concepts of calculus.

Physics: The changes I see happening in physics go beyond that kind of thing to the development of a very different learning environment, one that makes the student a far more active participant in the learning process. Research has shown that the traditional lecture course in physics is only minimally effective in stimulating learning for today's students. We are attempting to replace it with different settings based on collaborative learning, cooperative activities, and interactive learning. Some of us call it a workshop environment. It appears to be very effective as an educational approach, but also better prepares students for the workplace. We believe that students are unlikely to find satisfying employment in physics without effective communication, cooperation and leadership skills, none of which is fostered in the passive learning environment.

Chemical Engineering: Yes! That's exactly the cornerstone of the changes happening across the engineering disciplines today. The practice of engineering focuses much more heavily on some of the softer skills that we have ignored in the past -- things like communication, teamwork, and the need to examine ill-posed problems. Engineers do not work on projects individually; they are always part of teams, and the teams typically involve individuals from a number of different disciplines. Also, companies can no longer afford the luxury of graduating engineers basically serving a one- or two-year apprenticeship as they learn how to be practicing engineers. The employers want to hire people who are able to contribute almost from day one.

The teaching of engineering is rapidly changing to reflect this paradigm with a more integrated approach combining engineering, science and mathematics done in context. We are attempting to develop a far greater degree of teamwork, communication, and broader information about associated fields. For instance, we would like to see students of engineering learn many of their mathematical skills in the context of solving sophisticated problems. This involves a lot of team teaching, for example. There are a number of pilot programs underway to implement these ideas and they are quite successful -- at one location, graduation rates for engineering majors have increased by about 50%.

Chemistry: Many of the same themes are being discussed in the chemistry community. Mathematical modeling is becoming increasingly important as commercial software becomes widely available. Differential equations play a big role here, especially in kinetics and quantum mechanics. But the new pedagogy will also entail more student involvement and less lecturing. As in engineering, projects in chemistry are done by teams, not by individuals, so it is particularly important that we place more emphasis on collaborative work in our introductory courses. Of course, chemists are conservative by nature, but I expect that they will fall in line when they see the positive results.

Biology: I think that people in all fields are beginning to realize that the traditional lecture approach is great for covering a lot of material quickly, but is even better at putting the students to sleep. It is critical that we get them actively involved in the learning process. When we do that, they will learn far more and it will stick better. In fact, we must refocus our own perspectives to emphasize learning, not teaching. The professor must become more of a facilitator or learning director than the leader. And that's not easy.

Physics: But, to do any of that means that we must accept the principle of covering less material, but looking at the core topics in more detail and in greater depth. There are some wonderful traditional texts out there, but they are far too encyclopedic to serve as introductory textbooks. You have to go through them so quickly that most students come out with very little understanding of the fundamental concepts of physics. The key is to focus on how much you learn, not how much you cover.

Biology: I couldn't agree more. We should be worried much more about concept than about content. It's not so terrible if we cover less provided that our students learn how to learn.

Physics: There is another aspect to our philosophy of having students learn physics in a workshop atmosphere. We find that the application of the concepts immediately after they have been introduced forces the students to confront weak spots in their comprehension and ask the instructor for help. It gets very hard to simply hide in the back of the lecture hall.

Mathematics: I find it amazing how parallel the thinking appears to be in all of your fields to what we have been saying and doing in improving the mathematics curriculum. Are there any other changes taking place in your fields that relate to mathematics?

Biology: More and more, we are looking at using simulations of biological processes and systems. For instance, they might be on population growth or physiological processes or environmental issues. The advantage to using simulations instead of the actual processes is that they are faster, safer and far less expensive. When the students get involved in using simulations, we want to change the focus and have them begin to question how appropriate the model is: Does it truly model the situation under study? What happens when you vary the parameters in the model? We want them to question the basic assumptions in the model, if it seems appropriate, and to change the model to build a more sophisticated one.

But, when they get into issues such as that, they must come to a deep understanding simultaneously of the biological processes and the modeling process, and the latter requires understanding the mathematics used in developing the model. Typically, that is a differential equations model. Also, we are often looking at discrete processes, so there is a considerable emphasis on difference equations. Many models require a knowledge of linear algebra and certainly some understanding ofbasic applied statistics.

Economics: The same is true in economics. In fact, the majority of the models we use are discrete, typically based on difference equations. They also tend to involve a stochastic component, so that students need an introduction to probabilistic reasoning early in their studies.

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. For instance, we might think of this in the framework of mathematical modeling -- look at the behavior of a real system and then abstract the general rules. The key in this process is finding the limitations of the model. For example, electronic components are usually rated according to a range of values under which they behave linearly; but what are the limits under which the linearity assumption breaks down and how do the components behave outside that interval?

Economics: That sounds remarkably similar to what is happening in modern economics and finance. Our primary objective is to develop mathematical models, but the key is in understanding the assumptions on which the model is based. Questions that students must come to grips with are the sensitivity of the results to changes in the underlying assumptions, what the limitations are of the models, and identifying conditions under which certain types of behavior are possible.

Biology: What I see is the need for students to be prepared to face unknown problems. Too often, students, even very good students, approach every new situation with fear because their previous experience has been very narrow. I'd love to see them willing to just dive in and try to formulate a differential equation to describe a process and then examine the behavior of the solution to see whether it reasonably describes that process.

Mathematics: But how do you see getting to that stage?

Chemical Engineering: One of the major trends we see in engineering is the development of multimedia courseware. That has the potential to do some truly wonderful things in that direction.

Biology: Yes. 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 -- the author directs the reader along the prescribed course in a totally linear fashion. A lecture approach tends to do more of the same. The teacher leads the discussion, highlighting the points he or she thinks are important, and diverting the students away from questions that they may feel are significant; often this occurs because the instructor may feel ignorant of the side issues. But a true multimedia environment allows the students to go off the primary path to find the answers to the often unexpected questions that arise in their own minds.

Physics: That's exactly the kind of thing we are doing in physics as well. It allows us to get into real-world issues that you can't get into with purely analytic methods. Good computer courseware provides an interface between the students and physical experiments by collecting data, analyzing the data, allowing the students to visualize the behavior of the variables, and looking at the effects of changes of parameters. As an example, someone already mentioned how current is the derivative of voltage. But any measurement of the voltage includes a certain amount of "noise"; when you differentiate that quantity, the noise gets amplified. Students must understand this as a general principle and that can best be conveyed to them via computer simulations to see the effects of different assumptions about the noise.

Chemical Engineering: Another interesting thing we have observed is that many females tend to come into engineering classes with a less well developed ability for three dimensional visualization. We have found that the use of graphical technology is very effective as a means for remediating this problem.

Mathematics: To what extent are the ideas you have been discussing actually being implemented in your fields?

Electrical Engineering: Obviously, there is a broad spectrum of people in every discipline. As someone mentioned above, chemists tend to be conservative. There are conservatives in all fields, except possibly in math where you have made such extensive changes across the discipline. In electrical engineering, most of us are moving in the directions I mentioned before, though I might be considered somewhat of an outlier. Nevertheless, electrical engineering is changing rapidly and the mathematics you teach is clearly there to reinforce the engineering ideas.

Chemistry: The need to reform engineering is becoming very widely accepted and momentum for change is truly moving rapidly. The NSF has funded the creation of a group of what are called "engineering coalitions" which are designed to develop and implement engineering programs based on the comments we have made. The institutions involved in these coalitions enroll approximately one-third of all engineering students in the country. Further, the math departments at virtually all of these coalition schools are heavily into calculus reform activities and we have not heard any negative comments from the engineering and related departments at any of them. In fact, ABET (Accreditation Board for Engineering and Technology) visitation teams have typically reported very favorably on the calculus activities. Also, when you speak to many of the engineering faculty, a theme that comes through regularly is that they are pleased to hear that the mathematicians are also concerned with their students.

Economics: In economics, we seem to be heading toward a two-tier system. The best schools have moved very far in the directions I discussed above; the focus in their courses is highly mathematical with tremendous emphasis on modeling situations. Many other schools are still giving relatively non-mathematical approaches to the field and there is a great potential that the students coming out of such programs will find themselves locked out of the job market, at least in terms of the best jobs, and they will find themselves locked out of top graduate schools. But because the world is becoming ever more difficult, and the economic problems we face become comparably more complicated, we will see pressures to upgrade all our offerings. Also, remember that there are many more economics majors than there are engineering majors, so there are some major implications for the mathematics community.

Physics: There are a number of projects designed to implement these approaches to introductory physics and their preliminary results have been very positive. Many other institutions are looking into these projects with the intent of modeling them, so we think that these ideas will spread very quickly.

Mathematics: Most of what you have been saying about the curricular activities in your different fields is very new to me as a mathematician. I am not really aware of the extensive changes taking place in these areas. Is the reverse true? Are you aware of what has been happening in mathematics in terms of changing the calculus and related curriculum?

Chemistry: The mathematicians have kept calculus reform a secret. They need to come talk to us about the interrelationships between what they are doing in calculus and our courses. Only a few months ago, someone mentioned to me that we should be making use of the graphing calculators that all calculus students purchased and learned how to use. But we were totally unaware of this. Mathematicians seem to be isolated and not concerned about the rest of the world. There is a big difference if I go to a mathematician and initiate a conversation about incorporating math into chemistry compared to if that person comes to me.

Electrical Engineering: I agree completely. Engineers in general do not know about calculus reform activities. For example, I learned about what was happening because my own son took one of the reform courses; if not for that, I would probably still be in the dark.

Mathematics: Then what do you suggest that we do?

Chemistry: Come over and talk to us. From our discussions here, it is obvious that there is a tremendous degree of commonality of philosophy; let's share it for the good of all.

Chemical Engineering: That's fine on a local scale. However, the mathematicians should be more active globally at informing us of what is happening in calculus. We in engineering have a variety of publications devoted to engineering education in each of the subspecialties; write some articles describing your activities, your philosophies, and your goals in calculus reform. They will be very welcome. You might also want to look at them yourselves to get a better feel for what is happening in the reform projects in engineering.

Physics: I agree completely. There is a journal in physics also dedicated to education and articles describing the changes in calculus would be extremely welcome. Also, come to some of our conferences and give presentations; show us what you are doing.

Electrical Engineering: You should also come to some of the national engineering education meetings. You will be very welcome there. And, maybe you should invite us to come to your meetings both to let you know what we are doing and to let us find out more detail about what you are doing.

Mathematics: Thank you all for your comments and insight. It has been a very informative and eye-opening experience. I hope that the discussion we have conducted here will be the start of an on-going dialogue among all our disciplines.

Acknowledgment

The above discussion was culled by the author from a series of individual interviews with and talks by the individuals listed at the beginning of this article. The author is extremely appreciative of the time and cooperation extended by these colleagues, and wishes to thank them for their interest and concern for the education of their own students and their willingness to assist us in the mathematics community.

Part IV Connections

Calculus does not exist in a vacuum. By its unique nature as one of the greatest and most powerful intellectual developments, calculus is inextricably connected to a variety of other disciplines and other portions of the mathematics curriculum. It is a primary mathematical tool for physics, engineering, and chemistry; it is almost as valuable a tool for modern economics and biology, as well as many other fields. In the high schools and two-year colleges, calculus is often regarded as the culmination of all previous mathematical experiences. In the universities, it is the gateway through which students must pass to more advanced courses in mathematics and in the client disciplines. If we are to talk intelligently about changing calculus, then we must also consider all of the other areas to which calculus is connected.

If significant changes are to occur in calculus involving different skills, different expectations, different experiences, and different learning environments, then we should want our students to come into calculus from a comparably different preparatory experience. Such courses are currently being developed and implemented. We refer the interested reader to Preparing for the New Calculus, edited by Anita Solow (MAA Notes, Number 36) for descriptions of several such efforts and discussions of the objectives and philosophies for such new preparatory courses. Of course, many students come to college ready for calculus, having taken all their preparatory mathematics in high school. But, the high school mathematics curriculum is changing rapidly, in large measure due to the NCTM Standards. In his article for this section, John Dossey describes the status of the reform movement in the secondary schools and discusses how it connects to college calculus, both traditional and reform. In addition, the AP calculus exam is currently being revised to reflect many of the ideas common to the new calculus courses at the college level. This is discussed in Part III of this volume.

Furthermore, once the calculus experience changes, it becomes essential that the courses that follow calculus be rethought as well. Many of the new calculus projects have incorporated significant introductions to differential equations, which has major implications for the first course in differential equations. A separate reform movement has been working to develop modern approaches to these courses, as described by Bob Borrelli and Courtney Coleman in their article in this section. Similarly, there are major implications for the upper division mathematics offerings. David Carlson and Wayne Roberts discuss that issue in their article.

Another issue connected to calculus reform is the new GRE mathematics examination that will be given beginning in 1997 to all students going on to graduate programs in any of the quantitative disciplines including mathematics, the physical and biological sciences, engineering, computer science, economics and business. An article discussing this test, along with some sample questions that reflect the spirit of the new calculus, is included in Part III.

Finally, no discussion of the connections to calculus is complete without considering how the client disciplines react to the changes, because other departments depend so heavily on calculus. Simultaneously, major changes are happening in most of the other curricula. What implications do they have for us in mathematics? Both of these issues are addressed in an article presenting a round-table discussion among some of the leading educators from most of the disciplines that use calculus.

Sheldon P. Gordon, Suffolk Community College

 

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