My ideas about Calculus reform are based on three considerations.
My view of the development of Calculus over the last few decades.
Two or three decades ago, university faculty began a gradual de-emphasis of the analytic aspects of Calculus, i.e. the epsilon and delta proofs and the art of estimation. By the time I became a professor, Calculus courses consisted essentially of drill in computing derivatives and integrals exactly, with an occasional toy application to graphing and optimization. Of course, such courses are neither interesting to teach nor to take and are not particularly relevant to engineering. So, the pressure to "reform" calculus steadily grew all the while.
A few years after I became a professor, symbolic manipulation software like MAPLE and MATHEMATICA became widely available. These programs promised to take over the tedious aspects of computing derivatives and integrals, theoretically freeing the students to learn mathematics, and consequently they became the centerpieces of a new reform movement. Many faculty members embraced their use whole-heartedly, going so far as to spend considerable time teaching the use of the software in class at the expense of teaching mathematics.
But the promise of a reform based on such software failed because the mathematical content of the course had already been removed years before. Once the drill in computation was removed, Calculus was completely stripped of its heart. In fact, most students passing through a reformed Calculus course neither understand the basic concepts of analysis and Calculus nor can compute derivatives and integrals with proficiency. Anyone teaching third and fourth semester courses is confronted with this fact as the failure rate is very high considering that we presumably fail out the incompetent students during the first two semesters which have even higher fail rates. In the vast majority of cases, students fail because they cannot compute the simplest integrals or derivatives and have no understanding of the limit and derivative despite having passed the previous semesters.
It is bitter to admit that we "reformed" Calculus in the wrong direction after spending so much energy on it over the last few years. And it is perfectly understandable that we might peevishly decide to get out of the business of teaching first year Calculus altogether. This is even more tempting when we consider that a slight majority of our students have some prior exposure to computational aspects of Calculus before university.
However, this is a responsible decision only if we accept the notion that Calculus is about computing derivatives and integrals exactly, because that is the only thing that is taught in High School. (Actually, even this rationale stands on weak ground since the high percentages of students that fail in all four semesters of Calculus belie any claim that students have mastered even the computational aspects of Calculus in High School.) Dropping elementary Calculus is downright irresponsible if we consider Calculus from an old-fashioned point of view as the first course in analysis given to engineering and science students. Only mathematicians are qualified to teach this level of mathematics.
My roots in constructive mathematics.
L. Bers exposed me to the constructionist point of view when I first began to study analysis. He felt this was the best way to teach, as well as do, analysis, and I inherited this philosophy. This is a large part of the reason that I moved into scientific computing since numerical analysis is nothing more than the practical implementation and use of constructionist tools. Working in numerical analysis allows me to put my hands on quantities that otherwise remain mathematical abstractions that are difficult for me to comprehend. I believe many students share my feelings about this.
Problems inherent to teaching mathematics to engineering students
Mathematicians have the tendency to view mathematical education from the point of view of educating more mathematicians. However, educating a student that loves abstract mathematics is a much different problem than educating an engineering student, no matter how talented at mathematics. The difference is one of motivation. I believe that one of the main hurdles to learning a difficult concept or technique is in fact motivation. If a difficult idea is left unmotivated, it raises a barrier for the understanding and makes it impossible to place the idea into new contexts. Consider the analog of learning to cook. A cookbook that simply gives ingredients and recipes is a disaster for a novice trying to learn how to cook. With no explanation of why certain steps are performed and the timing, an average reader can only follow the instructions exactly. If that is not possible - if the cook makes some seemingly innocent variation or when something goes wrong or some ingredient is missing - a ruined dish and frustration inevitably follows. A good book explains the how and why of techniques so that an average reader ends up with the ability to concoct a good meal with the ingredients at hand in any strange kitchen.
In our natural tendency to aim our courses at the converted, we forget that analysis is rooted in the practical. Calculus was invented by people as a tool to solve scientific and engineering problems. The best way to teach Calculus to an engineering student is to go back to these roots and bring out the concepts in context of mathematical modeling of physical phenomena and solving mathematical models in order to predict physical phenomena. This provides a natural motivation for the concepts of analysis: convergence, limits, derivatives, integrals, and so on. The engineering student literally passes by the mathematical difficulties because they become focussed on practical difficulties in the same way that it is much easier to walk along a narrow rock ledge a hundred feet above the ground if one is focused on climbing a sheer face above the ledge.
I am assuming that the reader agrees with the view that a modern engineer needs an understanding of the rudiments of analysis. The prevalent use of computers to solve mathematical problems, the sophistication of modern mathematical models of physical phenomena, and the techniques needed to solve them all mark the importance of mathematics to engineers. So do the higher salaries commanded by students who demonstrate a proficiency in mathematics.
Proposal for Calculus Reform
"Reform" is a misnomer because my proposal is quite "retro" in that I believe we should return to teaching analysis, as was the goal in the decades before the modern reform era. There are new aspects to my proposal, primarily in the way in which some of the standard material is presented and in the way that mathematical modeling and solving mathematical models are used to motivate the mathematics. But despite these concessions to modern times, my proposed syllabus most closely resembles what would have been taught in Calculus courses before any of us were students. The material passes from the analysis of numbers, with the highlight of the construction of the real numbers as solutions of nonlinear equations, to the analysis of transcendental functions, as solutions of differential equations. The material is distinguished principally by the following characteristics:
- Motivate the mathematics in the context of models and explain where the models come from.
- Discuss sequences and limits primarily from the point of view of Cauchy sequences, which are very practical. For example, the standard engineering way to test convergence is to test the Cauchy condition on a couple of elements of a sequence.
- Emphasize Lipschitz continuity over the standard weaker definition. This fits the idea of trying to understand nonlinear functions by replacing them by linear approximations and it removes several technical difficulties.
- Define the derivative through the linearization of a function, again fitting the notion of approximating a nonlinear function by a linear function. This also serves as an introduction to asymptotic analysis.
- Define integration as a procedure for computing an approximate solution of a differential equation. The mathematical highpoint is the proof that numerical solution produced by the rectangle rule generates a Cauchy sequence that converges to the solution of the differential equation.
- Use constructive proofs and apply theorems to the study of specific mathematical models. This naturally leads to problems involving implementation of proofs on the computer, which means that students really understand the proofs in a way that is not possible simply by reading. This also gives the students experience in treating mathematics experimentally, which is important training for engineers.
- Consistently develop techniques for solving nonlinear equations, with chapters on the Bisection Algorithm, Fixed Point Theory, and Newton's method.
- Develop the notion of passing to a continuum limit as a step in model derivation. .
The third and fourth semesters should be devoted to linear algebra and analysis in several variables, i.e. to discussing the difficulties that arise in higher dimensions. The linear algebra course should be modern: emphasizing the structure of vector spaces with concrete examples of vectors spaces and the factorization of transformations. For engineering purposes, vector spaces of piecewise polynomials and trigonometric polynomials provide an important class of examples while the PLU and diagonalization of S.P.D. matrices are the first important factorizations to teach, with QR and SVD not far behind. Applications to solving large systems, differential equations, least squares, image compression and so on would be developed along with the mathematics and the students would have to solve real life problems.
The fourth semester would develop analysis in several variables. With the proper development in one dimension, much of the analysis in one variable carries over immediately to higher dimensions. This means the course would be able to skip over many technical details of proofs, allowing a focus on the specific issues rising in several variables, such as the variations of integration by parts formulas, etc., and a derivation of the basic models in physics: electromagnetism, fluids, etc. This fourth semester can therefore be closely connected to the physics courses for example, with the students getting their first exposure to the mathematical models that they will spend the rest of their careers solving.
This approach has been tested in Georgia Tech in the Honors Course and in my opinion this was an overwhelming success. The honors students passed at least the AB version of the Advanced Placement exam and their abilities placed them at the level of the A and B students in the typical classes. In all classes, the students performed way beyond my expectations on the mixture of group projects, problem sets, and exams that I assigned. The retention rate after the first semester was very high, around %98, while the students requested that I teach two more classes (Linear Algebra and Multivariable Calculus) after the standard two semester sequence finished. The retention rate here was still very high, around %80, with many of the students taking one or both of these classes as electives. I can distribute copies of the problem sets, exams, and projects if you want to see the level of mathematical maturity reached by the students. One of the highlights was a project in which the students read a constructive proof of the Mean Value Theorem, which is roughly 3 pages long and uses a sort of iterated Bisection search, on their own and wrote programs that implemented the proof and produced the point in the Mean Value Theorem. The same material is now being tested on a select group of engineering students at Chalmers University of Technology in Sweden.
Of course there are difficulties with this approach. Much of the material has to be developed, though there are books that could be used in the interim (like Linear Algebra Done Right by Axler, Vector Calculus by Mattthews, Vector Calculus by Marsden and Tromba). This material must really be taught, you can't just lecture and be sure that most students will understand. The weaker students in the normal sequence will have to be treated very carefully, it might require offering more lecture hours and so on. But the gain is that we finally get a Calculus course that is fun to teach and to take, and we would be training our students to become engineers. I also believe that we would attract a steady stream of mathematics majors as students find out that mathematics is both interesting and useful.
Syllabus for the first and second semesters
Mathematical modeling, integers, induction, rational numbers, infinite decimal expansions, functions, polynomials, Lipschitz continuity, sequences, real numbers, the Bisection Algorithm, Fixed Point Theory, the linearization of a function, the derivative, the Mean Value Theorem, modeling with differential equations, antiderivatives, integration, transcendental functions, inverse functions, Newton's method, optimization problems, polynomial approximations, applications of the integral, a general initial value problem.
Syllabus for the third semester
Vector spaces, linear combinations, linear dependence and independence, span, subspaces and direct sums, bases and the direct sum theorem, dimension, norms and inner products, orthogonality and linear independence, orthonormal basis, Gram-Schmidt algorithm, linear transformations, null space, range, 1-1 and onto transformations, the matrix of a transformation, change of basis, the inverse problem, elementary transformations and permutations the PLU decomposition, complex vector spaces, eigenvalues, invariant subspaces, diagonalization and triangularization, as time permits: transformations on inner-product spaces, self-adjoint and normal transformations, the Spectral Theorem, positive definite transformations, the SVD decomposition
Syllabus for the fourth semester
Vectors, vector fields, line, surface, and volume integrals, partial derivatives, gradient, divergence, curl, the suffix notation, Taylor's theorem, integral theorems, curvilinear coordinates, tensors, derivation of the basic equations in mechanics (heat transfer, electromagnetism, continuum mechanics and the stress tensor, solid and fluid mechanics), Newton's method, optimization problems