# On the Harvard Consortium Calculus

Oliver Knill

## I. What is Harvard Consortium Calculus?

The "Harvard Consortium Calculus" is an approach to calculus with the following official features [1] which are sometimes called "Harvard Calculus Consortium principles":
1. Mix a graphical, numerical and algebraic approach ("rule of three")
2. Motivate by practical problems ("the way of Archimedes").
3. Chose topics which interact with other disciplines.
4. Formulate open ended word problems.
5. Discourage the mimic template techniques.
6. Use technology to visualize concepts.
7. Prefer plain English over formal descriptions.
It had been promoted in the late nineties [5]. It got its name because some authors of books using the approach taught at Harvard [1] and use this name. The rule of three had been extended later to the "rule of four" [5], the fourth point being "verbally".

## II. What was new?

While none of these "innovations" individually were new, neither pedagogically nor mathematically, it is the combination which can justify its own name.
1. Graphical approaches to calculus predate even Newton. (See Chapter 2 in [8] about Newtons teacher Barrow, who used about 2 drawings for each page in average). Numerical techniques have been used by Archimedes, algebraic approaches have dominated since Descartes.
2. Practical problems have always motivated calculus. Calculus was born on practical problems, like astronomy, probability theory, fluid dynamics or electromagnetism. Almost all formal definitions and procedures evolved from investigation of practical problems.
3. The interaction with other disciplines in moderation has always been done by good teachers. I don't know any any teacher who would not use connections to other fields to make the point.
4. Open ended problems are part of any mathematical research. Open problems have always driven mathematics. Due to the risk of producing frustration, they need to be introduced with care. They are the salt in the dish.
5. Introducing non-template problems is the only way to really understand a subject. Formulating your own sentences in a new language is the only way to learn a language. The trick is to to balance it with more routine problems and drill.
6. Computers have been used in the classroom since they existed. My father has taught middle school math with simple calculators in the early seventies. Graphing calculators became fashion in the eighties and computer algebra systems entered the classrooms in the nineties. The routine use of computer algebra system in the classroom predates Harvard calculus for at least 10 years. The Harvard Calculus Computer algebra effort is therefore an effort to have these benefits survive.
7. Almost all mathematics books use plain English to explain things. As wider the audience, as less formulas and proofs appear. The guiding principle to explain things verbally is rooted in the believe that overuse of formalism is unhelpful. It can be viewed as a remnant of the general counter reaction against the "Bourbaki style" in mathematics which led to efforts of making mathematics more readable and approachable. Plain English has its danger too, one of them being "too chatty" an other one is to be vague.
It is this mix of approaches which defined this particular flavor of calculus.

## III. What benefits are left?

Some benefits:
• The push to include more conceptual problems is a healthy development, but it has to be done in moderation. A valuable remnant of the emphasis on concepts is the use of True/False problems [7] or multiple choice problems to check conceptional understanding. These problems are sometimes non-routine and often difficult and tricky. They are valuable for preparation purposes. They help to find blind spots in the understanding and clear up misunderstandings. They shake up the mind. If they have low weight in the entire exam scheme, they do not contribute much to the usual frustration, teachers and students experience with conceptual problems.
• Spend energy to maintain the use of computer algebra systems keeps the tool alive. When done right, it can produce enormous benefits. Once something is no more new, it is hard to stay motivated using it. The excitement is gone, one can no more brag with innovation. I personally think that computer algebra systems can enhance a great deal any calculus course, but it has to be used with moderation, with low priority and low grade weight up to the extreme that students should be able to avoid them with some penalty. Also, and this is very important, teachers should be able to avoid it if they are not experts in computer algebra systems. We usually have a computer algebra project which can be completed in a few hours and which is fun, not tedious and most of the team of teachers have not to be involved.
• Trying out new ways to teach helps to think about pedagogy and stirs up crusty teaching. It is always good to keep innovating and try new things to keep a course of becoming stale. But this should be done in moderation. Approaches which do not work should be discarded. Being able to correct mistakes is even more important than innovation. It makes sense to innovate one thing, and then evaluate whether it has improved the situation or not. When innovating two or more things, it is hard to assess which part was successful and which part not.
• Open ended problems are healthy and show that mathematics is alive. Mentioning open problems can make things interesting and show that mathematics is a vibrant and living scientific field. Giving open ended problems as homework problems can be problematic however. Such problems have to be produced slowly and carefully and bad problems discarded. Feedback from students is the best guide here.

## IV. About the rule of three

Trying to understand a calculus problem geometrically, algebraically and computationally makes sense because it is good to understand a concept from different point of view. For example, the notion of the derivative of a function of one variable can be understood geometrically as a slope, can be understood through algebraic manipulations like (xn)' = n xn-1 or computationally by taking the limit on a computer. The rule of three is a "meta advise", a special case of "obvious general common sense" that one should use when dealing with a given problem: use all the available tools which come in mind. The rule of three could easily be extended to a "rule of nine". Here are Sofia's "rule of nine" (2004)
 Look at a calculus problem algebraically, analytically, geometrically, historically, graphically, numerically, conceptually, psychologically, as well as experimentally.
For example, to solve the problem to compute the derivative of sin(x) at x=pi/3:
 algebraically we know sin'=cos and cos(pi/3)=1/2 analytically we know sin(x) = x-x^3/3! + x^5/5! - ... and so sin'(x) = 1-x^2/2! + x^4/4!-... = cos(x) so that sin'(pi/3)=1/2. geometrically sin(x) is the height of a right angle triangle with hypotenuse of length 1. The rate of change of this length in dependence of the angle can be seen geometrically. historically the derivative can be derived from Euler's formula exp(i x) = cos(x) + i sin(x) which has the derivative i exp(i x) = i cos(x) - sin(x). Comparing real and complex parts shows cos'=-sin, sin'=cos. graphically draw the graph of sin(x) and determine the slope at x=pi/3 numerically take a small number 1/1000 and compute 1000 sin(pi/3+1/1000)-sin(pi/3)) which gives 0.499567. conceptually since sin(x) increases for increasing for acute angles, the result is positive. psychologically my teacher does not like to assign problems with irrational numbers as answers. The result should be a simple rational number. Because 0 and 1 are out of question, the next reasonable result is 1/2 ...... experimentally here is an esoteric experiment: why not use Fourier series and differentiate that series. To make it interesting, take f(x) = |sin(x)| since sin(x)=|sin(x)| around pi/3 ...

There is a danger when using too many pictures at once: when teaching too many different aspects at once or trying to understand a problem from too many directions too soon, students will by overwhelmed by the complexity and information mass. They will end up knowing less. Its like adding too many ingredients into a dish. Sometimes, it is better to do more with less.

## V. What were the mistakes?

While not advertised, the "Harvard Consortium Calculus approach" came also with changes which were considered mistakes or pitfalls: here are the most important ones (see [2] for an other list):
1. Incorrect or imprecise definitions.
2. Important mathematical topics were left out.
3. Overuse of computer algebra system dragging resources.
4. Avoiding routine problems makes the subject hard.
5. Overuse of applications leading to complex and complicated texts.
6. Obscure motivating examples which confuse.
7. Too much numerics. Numerical concepts are hard.
8. Too much complexity. Problems embedded into too long stories.
9. The need to motivate every result experimentally.
More details:
1. While using incorrect definitions is a "criminal offense" to a mathematician, it produces also problems for non-mathematicians. Good and precise definitions are always clearer and easier to understand and do not lead to problems later on. This does not mean that the definition must become too formal. Examples:
• Definition of a "vector" as an object with direction and length. This vague definition even appeared in Britannica: it defines an object by introducing two non-defined quantities "direction" and "length". It does not work for all vectors (the zero vector) and is so general that it also applies to other objects like vectors in a larger dimensional space, a "walking man" or a "movie".
• Double equivalent definitions, where the equivalence is later proven. Example: Define v.w = |v| |w| cos(alpha) = u1 v1 + u2 v2 + u3 v3. Hidden in the definition is a theorem as well as a new definition, the "angle".
• Negative r values for polar or cylindrical coordinates. While convenient for drawing polar graphs, it produces ambiguities and difficulties for example, when the topic of changing variables in integration appears. It produces problems even when dealing with spherical coordinates.
2. Here is an example of a neglected topic: when focusing on parametrized surfaces r(u,v) = (x(u,v),y(u,v),z(u,v)) in the case of graphs r(u,v) = (u,v,f(u,v)) only, one works only with a limiting situation. It even does not include important cases like the sphere or surfaces of revolution. Parametrization might be a harder concept to master, but it is worth the effort.
3. Teaching with technology is challenging for teachers. Even when using a computer algebra system on a daily basis, it can happen that something goes wrong during a presentation. Teaching with technology is always a risk. For students, using too much computer algebra systems during homework prevents to practice and makes the student dependent on technology. It is a matter of finding the right balance.
4. This is serious and often a typical mistake, especially by experts: difficult exceptional cases and examples are presented in a class because they appear more interesting to them. It is justified by the fact to promote understanding. But it produces insecurity. Every piano player, or beginner in a new language knows that routine practice is an important part of training. It needs routine to get to the level of conceptional understanding. Repetitive practice is the soil, on which conceptual insight can grow.
5. When teaching an application from physics for example, one has the difficulty that both topics, mathematics and the subject physics appears. This makes things more interesting, but it produces complexity. If the teacher does not know the other subject well enough, it is a guaranteed to become a disaster. Designing good problems with applications is an art and each case has to be "field tested" to prove its value. Especially here, what works for one teacher might not do it for an other teacher.
6. It can be challenging to produce motivating problems. So called hook-up or Hatsuma problems are problems which lead naturally to a subject. While some are great, it can also happen that such starters are painful. I have seen practice lectures in single variable calculus, where a motivator was so obscure that I had no idea what subject it belongs to. Again, it is not so much the approach, but the limitations of the teacher which can make this challenging.
7. Numerics can be hard. It needs a course on numerical analysis to appreciate this. It can also be impractical. The obsessive use of Riemann sums for example can frustrate students. While it is helpful and pivotal to know Riemann sums (it helps how to set up integrals both in one and higher dimensions), there is a point in any course, where solving things directly is more convenient. Mathematics is about elegance. The most simple solution wins. Every math problem should be an advertisement, how powerful it is. Mathematical problems should be solved eventually in the most efficient way. Otherwise it alienates.
8. Motivating every example from other disciplines can be artificial up to the point where it is painful. It can be challenging to produce good and clear problems which involve other disciplines. It is not easy to write original applied problems which do not have some ambiguity. Even in a team of teachers which double check everything and if armies of publishers and teachers proofread it, pitfalls can remain. Example: there is an example in Stewart's popular multivariable course, where the answer depends on whether the earth gravitational constant is chosen to be 10 or 9.98. [3] It is a typical pitfall of an applied problem and hard to catch.
9. Motivating results by experiments is the researchers dream, but it can be frustrating in pedagogy. A real life example [4]: a teacher wants to guide his students to Pythagoras theorem and let students draw random right angle triangles to find relations between the side lengths. He gives a hint to pay attention to a2,b2 and c2. The class is busy for an hour and collects data. At the end, a student presents the findings of the group in front of the class:
 "In our experiments, we found the following general rule: while a2+b2 - c2 was always close to zero, it was never zero!"

## References:

[1] Hughes-Hallet et all, Calculus , Alternate Version, 2nd Edition
[2] http://mathematicallycorrect.com/hc.htm
[3] Stewart Calculus, problem 26 of section 10.4, one has to use g = 3.2808399 * 9.80665 = 32.174 ft/sec2 Using g=10 m/sec2 gives an other result.
[4] Personal communication, Daniel Goroff, 2001
[5] About the Calculus Cosortium (Wiley)
[6] NSF grant
[7] Examples of Robert Curtis 1992
[8] V.I. Arnold, "Huygens and Barrow, Newton and Hooke", Birkhaeuser, 1990.

 first Draft: June 19, 2002, updated April 8, 2004, January 20, 2009 (added styles) and February 14, 2009 (corrected some english and added [8]).