In about 1988 the National Science Foundation in the United States made available substantial funding for the development of reform calculus programs. In the first semester of 1993 I taught one of these programs, developed by the Harvard Consortium, at the University of Arizona in Tucson. This paper reports briefly on my experience in the Harvard program, with some additional remarks regarding the calculus reform movement in the USA in general.
The University of Arizona program was essentially a first course in calculus. About half of my students had previously studied some calculus, some successfully in high school, others with less success at university level. I arrived in Tucson with an open mind regarding the virtues of the reform approach; I left convinced that there are lessons to be learnt from it.
The following examples might serve to illustrate the spirit of the Harvard program. The solutions are typical of those produced by my classes; I was impressed by the understanding displayed by students who had never previously studied calculus.
Example 1. Estimate the value of f''(0.3)
for the function given by the table
\matrix
x: & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\
f(x): & 0.10 & 0.21 & 0.33 & 0.46 & 0.63
\endmatrix
Solution. The second derivative is the rate of change of the first
derivative. Using the function values at x = 0.2 and x = 0.3,
f'(0.25) \approx \frac{0.33-0.21}{0.3-0.2} = \frac{0.12}{0.1} = 1.2.
Using the function values at x = 0.3 and x = 0.4,
f'(0.35) \approx \frac{0.46-0.33}{0.4- 0.3} = \frac{0.13}{0.1} = 1.3,
and hence
f''(0.3) \approx \frac{1.3-1.2}{0.35-0.25} = \frac{0.1}{0.1} = 1.
Example 2. A non-fatal infectious disease strikes a small desert island. Susceptible people catch the disease and suffer from it for a month before they recover and are then immune. If N = f(t) describes the number of infected people t days after the first case reaches the island, what (in plain English) does the value of N'(20) tell you?
Solution. Students typically commenced by sketching a graph (a very good way to start) similar to the one below. Their observations at this stage included: the exact shape of the curve for small t isn't important to this question, the first portion of the graph may resemble the logistic curve which they had previously seen, the value of N will climb to a maximum and then fall, N couldn't have reached its maximum by t = 20, N'(t) \rightarrow 0 as t \rightarrow \infty, etc. \midspace{6cm}
Now N'(20) is the slope of the tangent to the curve at t = 20,
and this tangent
is approximated by the chord joining the points on the curve where
t = 20 and
t = 21. So N'(20) is approximately
\frac{N(21)-N(20)}{21-20} = N(21)-N(20),
i.e, N'(20) is an estimate of the number of new cases of the
disease on the 21st day after the initial
outbreak. (Some students used t = 19 and t = 20, or t = 19 and
t = 21, rather than t = 20 and t = 21, as obvious alternatives.)
Example 3. Estimate \int_0^2 \exp(-x^2/2) dx correct to one decimal place. Justify the accuracy you claim.
Solution.
The Harvard program develops the definite integral by using left and right
rectangles, rather than upper and lower rectangles, and restricts
itself to equal
subdivisions. My students had been provided with a TI-81 program which
computes, inter alia , L and R, where L is the sum of the left
rectangles and R is
the sum of the right rectangles. With this notation it follows
readily that if f(x) is monotonic
decreasing over (a,b), then R < \int_a^b f(x) dx < L,
and it follows almost as readily
that if the width of each subdivision is h, then
L-R = h(f(a)-f(b)). \tag*
Example 4. Let f(x) = \exp(x^2), and F(x) be the function such that F'(x) = f(x), with F(0) = 3.
Solution.
I taught two classes - a day class of approximately 25 students and an evening class of approximately 20 students. The day class met in a room which had a computer on every desk running public domain software developed at the University of Arizona specifically for teaching purposes; the evening class met in a normal classroom. In both rooms a computer at the lectern ran the University of Arizona software, projecting the screen onto a whiteboard. All students were required to own a graphics calculator. The TI-81 was recommended; its price was about $100 Australian (cf the current Australian price of about $180 - for further comparison, student fees were some $US15000 per year). Students were advised that they could use any graphics calculator they wished, and that if they used a TI the instructor would help them with it. They were told not to expect help with their calculator if they were not using a TI. In practice this caused no problem: some individual instructors at the University of Arizona produced additional notes for Casio users and made these notes freely available to others, I helped to organise students using specific types of calculator to work together, and my general knowledge and experience enabled me to solve almost all of the problems encountered by users of other brands of calculators. Shortly before I left Tucson, Casio were concerned that TI had such a large share of the market and were trying actively to promote the use of their products.
TI have recently developed the TI-82. Sue Burns, Kings College, London, who had some input into the design of the TI-82, presented a seminar to SUTMEG on the TI-82 and TI-85 at the University of Sydney in August, 1993. The TI-82 seems to be a very suitable tool for teaching the Harvard program.
The TI-85 had recently reached the market when I was in Tucson. It was examined, but the general impression of the mathematics instructors at the University of Arizona was that it had too much intelligence built in, and so it was rejected. This rejection illustrates an important point that cannot be stressed too strongly. The purpose of the reform program is calculus, not technology. The technology is there solely as a means to an end.
To pursue that point, two alternative approaches to the use of graphics calculators were debated at the Second Annual Conference on the Teaching of Calculus, held at Harvard University in July, 1993: draw the graph first and then analyse what you see (called the Hands-Head approach), or analyse what you expect to see and then draw the graph to confirm your analysis (called Head-Hands). The debate ended abruptly the instant that one participant declared: ``Neither is correct. It must be Head-Hands-Head.''
At no stage in a reform calculus program is a picture on a graphics calculator an acceptable alternative to thinking.
This principle contrasts starkly with what I seem to have heard in Australia in recent years. I know I will be told that I have heard incorrectly, but I am disturbed by the frequency of statements like
In none of these cases have I noticed any significant discussion of what calculus should be taught, or how calculus can be taught better (as opposed merely to being taught differently) using the technology.
``The Rule of Three'' is usually mentioned as the common feature of reform calculus programs, but all of the reform programs I have experienced share a number of common features. These are discussed in this section.
The Rule of Three: All concepts are introduced numerically, graphically, and algebraically. The order in which these approaches are used varies, and all are regarded as important. All students routinely make use of a graphics calculator and/or a computer package as they develop the concepts. Implicit in this approach is that numerical results are perfectly acceptable, but (certainly in the Harvard Consortium approach and I expect in other reform programs) consideration of the accuracy of these numerical results plays an important role, even though there is no formal error analysis in a traditional numerical analysis sense.
Great stress is placed on reading, writing, and modelling of real world situations, both in exercises and in the development of the theory. For example, in the Harvard program the derivative is introduced through the concept of average speed, and the integral is introduced to solve the problem of measuring distance travelled. Problems are frequently taken from the ``real world'', students have to interpret what mathematics is required, the problems are often open ended, and students are expected to write coherent explanations of their solutions.
In keeping with this emphasis, the developers of the Harvard program have talked extensively to the client disciplines, e.g. engineers, biologists, and economists, and have listened to them. At a dinner party the Head of the School of Chemistry at the University of Arizona remarked to me that ``This Harvard Calculus is terrific; it's going to revolutionise the way we teach Chemistry''. These discussions have indicated that two of the major areas wanted by the client disciplines were (i) dealing with functions given in the form of tabular data, and (ii) given f'(x), to investigate the properties of f(x).
Of course, (i) relates to the introduction of concepts numerically, (ii) relates to the graphical approach, and there is ample algebraic treatment at all stages.
Compared with traditional calculus courses, there is a reduced emphasis on technical expertise, with a correspondingly greater stress on problem solving. This is not to say that technical expertise is ignored. The Harvard book devotes some 55 pages of Chapter 4 to developing the rules for differentiation and establishing the derivatives of the usual functions, and some 40 pages of Chapter 6 to techniques of integration (including integration by substitution, integration by parts, and the use of tables of integrals). These chapters also contain several traditional sets of 30 or more exercises to ``evaluate the following'' derivatives or integrals. However, the emphasis is on problem solving, not on computational techniques. As with technology, these computational techniques are unashamedly treated as a means to an end, not as an end in themselves.
Reform calculus courses have removed ``the rigor that was never there''. The view of many in the reform movement is that traditional calculus courses may have claimed to contain a rigorous development of important theorems, but that to a large extent it was there for the consumption of other academics, not for the benefit of beginning calculus students who can rarely come to grips with any rigorous treatment. A participant in the Harvard conference suggested that in a traditional calculus course, ``proof'' means that a statement the student regards as obvious is shown to be a consequence of something that the student doesn't find obvious, usually by means of an argument that the student finds confusing. In reform calculus, ``proof'' is replaced by ``justification'', i.e. by a reasonable argument which leads to a degree of conviction on the part of the student that the result is true. As the student's understanding develops, these arguments can be made more precise.
There has been a strong movement away from ``chalk and talk'', which has been replaced by a great emphasis on class activity (e.g., calculator/computer exploration, discussion of problems), group work, etc. This is a natural consequence of the reform approach, because of both the open-ended nature of so many of the problems, and the stress placed on communication.
A result of the problem-solving emphasis and the class activity approach is that the Harvard program doesn't try to do as much as a traditional course; the aim is to teach less, but to teach it better. I've heard this objective quoted so frequently in regard to traditional courses that it's almost a cliche, but in the end most programs seem to succumb to a perceived need to cover a certain amount of material. The Harvard program genuinely tries to move more slowly and to develop understanding.
A reform calculus program requires a different mode of thought from the traditional: the student (and the instructor) must escape from a mindless use of formulae, regard graphical techniques as a natural approach to many problems, learn to communicate, etc. A reform calculus program cannot be grafted onto a traditional course just by introducing some graphical work, using a computer, etc. Using diagrams, sketching graphs, making calculations, etc, don't constitute a reform calculus program unless the student instinctively uses these methods without prompting whenever it is appropriate.
Two examples will illustrate this distinction. One of several programs I issued to my class allowed the student to provide a function f(x), an initial point x_0, and a step-size h, and provided as output the value of (f(x_0 + h)-f(x_0))/h. Students used this program freely at various stages throughout the course to estimate derivatives at a specific point by taking successively smaller values of h (of their own choosing). When I set a test question early in the course to estimate the derivative of x^x at x = 3, I was pleasantly surprised by the understanding of the definition of derivative that they displayed. By contrast, a colleague at Macquarie showed his students the same approach, and set them exercises to do on their own calculators at home; he was quite disappointed with the performance of his class when he asked them to estimate the derivative of 6^x at x = 2. To my mind, this difference in outcomes highlights the contrast between ``teaching'' students these ideas as opposed to allowing them to develop the ideas as an integral part of the entire course.
The ease with which my students coped with the concept of a limit provides a further illustration. We started by solving x^2-7x+10 = 0 iteratively, and noted that the calculator was ``trying to tell us'' the number 2; after a few weeks of referring to ``the number the calculator is trying to tell us'' I found it natural to introduce a shorthand word, ``limit'', to replace this phrase; then, after using the word ``limit'' for a few weeks we discussed the ``proper'' meaning of the word. Since this was a first course in reform calculus we did not take the extra step of formalising the \epsilon-\delta definition in class, but discussion with individual students suggested that this would be a reasonably easy step to take, since by this stage they understood the concept so clearly.
Based on my obviously limited experience, I felt that students' understanding of calculus was significantly improved by the reform calculus approach.
The following reports from the University of Arizona might be weighed against the fact that the faculty members concerned are certainly missionaries for calculus reform, and perhaps also by the high quality of teaching that I observed there; however, they accord with my own impressions of the program.
I saw graphic evidence of this benefit in the Mathematics Tutoring Room. Students from any first year class were encouraged to visit the tutoring room between 12:00 and 2:00 Mon-Fri, and all faculty were required to spend one of their rostered office hours in the tutoring room. After the first few weeks I found I could predict accurately from their first words to me whether students were taking traditional calculus or reform calculus: a student from a traditional class almost invariably commenced by saying ``I can't do (this question)''
or ``How do I do (this problem)?''.
A student taking a reform stream nearly always commenced with something like ``I've reached (this stage) and now I'm stuck. I've tried (such and such an approach), but it didn't work. What might I try next?''
In fact, teaching in this environment was so enjoyable that I chose to spend several extra hours above my scheduled load in the tutoring room - this was an extra benefit to the teacher that I hadn't expected!
Since my return to Macquarie University, the Harvard Consortium book has been adopted as the text for our first year introductory unit (which bridges the gap between HSC 2-Unit Mathematics and our mainstream first year units), and the material has been taught somewhat in the spirit of the reform program. However, the use of a programmable graphics calculator as an integral part of the course is essential if the true spirit of the Harvard Program is to be met, so Macquarie's adoption of the program has been only partial.
A number of issues must be resolved before a reform program can be introduced in full. These include the following.
The first course textbook, Calculus by Deborah Hughes-Hallett, Andrew Gleason, et al. is currently available from the publishers, and a draft edition of the second book, Multivariable Calculus by Bill McCallum et al. is also available in Australia. This second book was to be used at the University of Arizona in 1994.
The University of Arizona software is a suite of public domain programs developed specifically for teaching purposes. The 12 megabytes of programs contain five sets of ``Are You Ready?'' tests (oriented to the U of A courses), ten ``Slide Shows'' of images that would be difficult or impossible to draw on the board, and twenty five ``Toolkits''. These toolkits are the heart of the software; they include programs for graphing and/or fitting polynomials, complex numbers, numerical integration, Venn diagrams, graphs of implicit functions, Fourier series, Taylor series, solution of differential equations, a comprehensive linear algebra package, etc. All programs are comprehensively documented and have been designed to be easy to use even for students who are unfamiliar with computers, and many of them have excellent sets of problems attached as ``projects''. The software is available (free of charge) from University of Arizona via ftp .
I would be happy to provide any further information I can to anyone who wants it.
\ending{Mathematics Department} {Macquarie University NSW 2109} {ted@mpce.mq.edu.au}