I can remember the day well – it was a Friday. My class plan included a discussion of the Mean Value Theorem to be held at the end of class. I really did intend to squeeze it in there, I even had a slide of it to put up on the overhead. Only, things went awry. At Quest, our classes are taught in blocks, which means one course at a time, three hours a day, every day, for three and a half weeks. This particular class was in the latter half of a Calculus 1 course, the second course I had ever taught at Quest. As it was a 3-hour long class, I had given the students a problem at the beginning of class to work on. It was a fairly standard problem in optimization: the students had to determine the best way to build a bridge connecting two points on opposite sides of a river, one some distance down from the other, so that, assuming it is more expensive to build over water than on land, the cost is minimized. As with many such Calculus problems, the beauty is in the set up, and from there it is usually just a matter of applying a few quick and dirty algorithms to get to the answer. For this problem, however, I hadn’t given the students any tools yet – we hadn’t mentioned the words “local extrema”, “critical points”, or “the Extreme Value Theorem” (key concepts for optimization problems) – I had just turned them loose on the problem.

It took them the good part of an hour to get anywhere setting up the problem. Being a mathematician and therefore quite used to symbolic translation and manipulation, I can forget how unnatural an idea it can be for the burgeoning student to assign variables to appropriate quantities, i.e. put an x or y in the “right” place on a picture. Nonetheless, with a lot of collaborative work with each other and a few hints from me, they ended up being able to piece together pictures, derive symbolic relationships, and construct resulting equations. Once various members of the class had written these steps on the board in the front of the room, we were ready at last to leave behind the interpretation of the problem into mathematical terms and get down to the actual solving of the problem. That’s when we hit the second major stumbling block: what to do next.

The next several minutes were spent in open class discussion. A lot of that discussion concerned retracing our steps to what we had done so far. We had a strange looking object on the board with x‘s, numbers and root signs, and it (appropriately) took many of the students several passes to feel somewhat confident that this had anything at all to do with a bridge. Fair enough. After a long while, someone had the idea to draw the graph of the equation we had on the board. With the help of a computer software package, they got the graph on their screens. Then came one of the best moments I have ever had teaching a Calculus course. One of the students, someone who had admitted to actually failing Calculus in high school, said, “Hey, look at the graph. The lowest point looks like the bottom of a parabola, where we know the tangent line is flat.” Wow. What she had so brilliantly and simply observed (just like Fermat had) is the key to the solution of the Optimization Problem, one of the major achievements of Calculus. She had said it as though it were the most obvious thing in the world, her fellow students nodding in agreement. I had never heard a student say such a simple, elegant, self-generated statement about a Calculus result before.

So by the time we actually found the answer – the way to minimize the cost of the bridge – and had summarized our methods and conjectured how to solve similar problems, the full three hour class was over. And I hadn’t gotten to the Mean Value Theorem. Moreover, I knew that the next class was jam-packed, and so this was the only opportunity I had to fit in this topic without it being completely removed from the other topics of the class. How could I let this topic pass? I had made a judgment call – I saw that the students needed to finish off this problem they had started, they needed to put the flag in the top of the mountain they had just bushwhacked their way up. So I had abandoned the Mean Value Theorem and consequently needed to strategize how to fit it in at a later date, or maybe as a problem on an assignment.

Later that day, in my ensuing meditations on what to about the Mean Value Theorem, and in discussions with my colleague Glen (who so often helps me by capturing the awkward phrasing of my insights in his own, beautifully elegant and reflective manner), I realized something. I realized that somewhere, I think in graduate school, I trained myself to teach mathematics as though there were a panel of mathematicians at the back of the room keeping careful track of the mathematical accuracy and consistency of my statements as an instructor. Perhaps it was out of fear of avoiding anything resembling those moments in seminars in grad school when the speaker is suddenly frozen by the penetratingly lethal insight made by a senior audience member noting a devastating mistake. God forbid! No matter how this belief had gotten into my head, I realized that up to that point I had been designing my classes as though they needed to withstand the scrutiny of a fellow Calculus expert, to justify in a kind of legal sense that I had indeed done my duty and presented the mathematical theory of Calculus to the students in its entirety and without omissions. Indeed, I assert that such a perspective is the norm in the mathematical community. The only problem with such an approach is that it is a terrible way to teach.

What is the point of the Mean Value Theorem anyway? I mean, I don’t use it in my normal contemplation of Calculus problems, nor do I regard it in the category of the marvels of Calculus. And students never remember it anyway, nor understand why it is ever discussed or forced upon them in a set of  drill problems. Of course I consider the Mean Value Theorem to be important, the way that I consider the carburetor important to the running of my car’s engine. But it isn’t as though I drive my car simply to make the carburetor work, nor is it necessary for a novice driver to know how a carburetor works before they get behind the wheel of a car and get a feel for what driving is. One learns what the Mean Value Theorem is, or what a carburetor does, because one is curious about how the thing actually works or because a problem occurs that organically calls for its understanding. That curiosity or organic thought process develops of its own accord, and not because we preaching instructors try to convince anyone of the value of knowing the legal details of Calculus proofs ahead of time. The Mean Value Theorem was not even a part of the original Calculus of Newton and Leibniz (it was first articulated by Lagrange), so the claim that the Mean Value Theorem is necessary for the learning of Calculus is a fiction, applied in hindsight by mathematicians who are concerned more with logical infallibility than initial comprehensibility. The former is of course an important and worthy goal, but one I think unnecessary in an introductory Calculus class. Time is better spent on having the students meander their ways to the best mathematical vistas, building their sensitivities, and let them go from there.

I learned from class on that Friday that it is actually okay to let mathematics be the messy activity that it is. The students got far more out of clumsily muddling their way through a single thorough problem than they would have from me rushing through the problem for them, only to see an overhead that is more for my conscience than their educational benefit. In the end, I think the sacrifice of the Mean Value Theorem was worth it, as in its stead the students saw the idea behind the solution to the Optimization Problem – a glorious view indeed! If and when the time comes for some of those students to learn about the Mean Value Theorem, I hope it is because their own line of questioning has led them to ask exactly how it is we are certain that a function with a positive derivative is increasing. That kind of subtlety can only be appreciated when one has sensitized themselves to the finer points of mathematics, and for most students that isn’t the first time through an introductory Calculus course.

Next time I teach Calculus, I don’t plan to teach the Mean Value Theorem. I know that puts me at risk of being accused of being incomplete by the committee of mathematicians I imagine in the back of the room, but so be it. In fact, the way I will sort through topics for my next Calculus course will be to ask which topics (i.e. problems) lead to the best mathematical views, not which ones need to be presented to make a complete course (in the mathematically legal sense). In the end, I feel that the few views that the students see from the tops of mathematical mountains of which they themselves have struggled to the top far surpass in satisfaction and wonder the many slideshow photos I could show them of all the mountaintops preclimbed and presented flawlessly in the Calculus album. As my students and I discussed, a Calculus book is a marvelous thing to look at once one has already learned Calculus.

Let me say that the point of this blog entry is not to convince anyone that the Mean Value Theorem should be abolished from the introductory Calculus course. It might very well be part of an excellent course. Instead, I claim that we as teachers provide a better learning environment when we allow students to explore the topics of their classes in the ways they were originally encountered – as mysteries, confusing and tantalizing patterns, and processes of refinement and experiment. One of the best ways to do this is to permit ourselves as teachers to be vulnerable, engage in the process of questioning the subject ourselves, and altering the course as necessary to meet this objective. In my view, teachers should have the freedom to teach a course as they see fit, with their own favorite vistas highlighting the course content, not the predetermined topics produced by curriculum committees. Now, perhaps, I have crossed the line into heresy.

I know that many of the ideas of this entry have been expressed by other educators who have seen and articulated the value of discovery-based mathematical education, including the wonderful writing of one of my mathematical heroes, Paul Halmos. So I don’t presume these thoughts to be original ideas that others should follow; rather, I am interested in how the question Why? impacted both me and my students that day in class. If there is anything I have learned from this experience, it is of the value of questioning my own assumptions, and allowing those questions to be part of the messy, clumsy, and beautiful environment that constitutes a healthy classroom. Such views that are afforded! All of a sudden, Calculus is fun again.