Something is working

This fall I'm teaching Calculus I for the third time here at Coastal.  Perhaps because I just taught it at the end of the summer, I am finding that it is working particularly well.  I'm not entirely sure why.  Student interaction is good, and I'm sure that is helping.  But beyond that, the lectures seem to be flowing in a way they have not previously.

One possibility is the way I just happen to be presenting the material.  I thought I'd take a moment and record a couple of examples here. 

1. To introduce limits, I started by asking how we might graph $f(x) = \frac{sin(x)}{x}$.  We had previously talked about the graph of $f(x) = \frac{1}{x}$, and what happens at $x = 0$.  We said that while you cannot plug in 0, you can ask what happens near 0.  So we tried that again.  What happens to $f(x)$ as $x$ gets closer and closer to 0.  We made a table, everyone agreed that the $y$-values were getting closer to 1.  Only then did I start using the language "limit."  I gave them the notation as a way to quickly write down what we just did.  This seemed much more natural than giving them a definition of a limit (out of thin air) and then showing them a bunch of examples.
2. For left/right-hand limits, I only introduced them after we discovered the problem in finding limits if the $y$-values approach different values from each side.  I gave them a graph and asked them to find limits in a variety of cases.  We all agreed that at the jump discontinuity, the limit did not exist.  But of course, we can be more descriptive than this.  Coming from one side, the limit does exist.  Then I introduced that notation.
3. For continuity, I drew two graphs, one continuous, and one with a jump discontinuity.  I asked the class what was different about the graphs.  We agreed that the continuous one was continuous and the other not.  I asked what other ways a graph might not be continuous.  We came up with a graph with a removable discontinuity and an infinite discontinuity.  Now, how might we say something about these in terms of limits?  
4. Continuing with continuity, to introduce the difference between continuous at a point vs interval, I covered up the discontinuity and asked if the rest of the graph was continuous.  We agreed it was, so decided that we needed to express continuity at specific points.  This led us to the limit definition of continuity.  This was so much more natural than giving the definition, and then figuring out what it means.

That's all we have done so far.   I'm not sure if this luck will continue.  The common theme seems to be that instead of giving the definition and trying to apply it (which of course has it's place and value), I'm asking them to solve problems, then coming up with the math-way of saying what we are doing. 

I don't know if this is helping the students (we have not had any exams yet) but it definitely feels better to me.   I plan to make an effort to continue using this "technique" for the rest of the class and see how it goes.

Product rule vs. trig functions

What is the best first example of the product rue?  Often, books like to use $f(x) = x e^x$.  But this is a horrible first example.  Look at the derivative: $\frac{df}{dx} = e^x + xe^x$.  Can you see the format of the product rule there?  Not in the least.  It would help to use $x^2e^x$, but the real problem is that you do not see the difference between $e^x$ and its derivative.

A much better first example might be $f(x) = x^2\sin x$.  Now $\frac{df}{dx} = 2x\sin x + x^2 \cos x$. Very nice.  You can see exactly how the product rule is used.  But there is a problem: most textbooks do not cover the derivatives of trig functions until the section after the product rule.  The reason for this appears to be the desire to keep all the trigonometric function derivatives in one place.  To get the derivative of tangent, you need the quotient rule, which should definitely be in the same section as the product rule.  What to do?

Whether textbooks like it or not, I think it is worth it to teach the derivatives of sine and cosine first, then the product and quotient rule, and then as an application of the quotient rule, do tangent, and the other three basic trig functions.  This takes a little more forethought, but the benefits clearly outweigh the costs.

The place for proofs

I have been thinking quite a bit recently about the place for proofs in introductory math courses such as trigonometry, calculus, and really anything prior to the "proofs" course.  As a mathematician, I realize the importance of establishing results rigorously.  My students, however, do not.  With the rare exception of the dedicated math major, most students would rather I just tell them the formula, let them memorize it, see a few examples of it in action, and move on.  Finding a balance is no easy task.  There are a few things to keep in mind that can make this challenging task easier.

First, we need to decide on the correct level or rigor in our explanations.  Freshmen will neither appreciate nor understand a complete proof of the mean value theorem, for example.  On the other hand, just stating the mean value theorem makes it unlikely that students will gain an understanding of the concepts contained in the result.  Some explanation is necessary, but that explanation should be used to illustrate what is going on in the theorem, not just to prove that it is true.  In the case of the mean value theorem, this might be to instead talk through a proof or Rolle's theorem (intuitively, there must be a max or min, and that is a place where the derivative is zero) and then maybe show how you can use Rolle's theorem to get the mean value theorem.  Of course, the mean value theorem can also be explained in terms of velocity and in terms of tangent lines, and these, although not proofs, are also important to help students understand the concept.  This is more proof than I would use on other topics, and part of the challenge is that each instance needs to be judged for itself.

Second, and the idea I've been spending most of the time thinking about lately, is where in the lecture to place the proof, whatever level of detail that might entail.  There seem to be three basic ways to go:

1. The classic: you state the result, then ask why it is true.  Then give an argument to try to convince everyone that the result holds.
2. The quest: you state what sort of result you are looking for.  For example, you want to find a derivative rule to help you take the derivative of a product of two functions.  You go through the "investigation" and derive the rule.
3. The sneak: you do not say what you are looking for, or even that you are looking for anything at all.  Instead, you say that you want to "play around" with these formulas and see what you get.  Then you "stumble" upon some nice formula, and put a box around it.

For one reason or another, I have been using option 3 recently.  I don't know why I have fallen into this: maybe a book or lecture I saw used it and I picked it up.  In any event, I think it is a mistake.  A healthy mix of styles 1 and 2 are appropriate, but the third option should really be avoided.  At first glance, it appears that it would be a reasonable way to sneak in proofs under the objection radar of the students.  But students are much more savvy than that.  As soon as you put the box around the derived formula, they realize they've been had and respond with the likes of, "you mean the last five minutes were not the important part?"

I can't blame them for thinking this.  The "playing around" part often entails a lot of algebra.  Students have a tough time keeping up with the notes, let alone understanding it.  By the time I get to the formula they may be two or three lines behind, and miss my comments about the whole point of the exercise. 

Options 1 and 2 both have their merits, and using both is probably the best way to keep the class exciting.  In general, I think it is a good idea to state the result first when you can - giving your students a rough understanding of a proof is useless if they don't know what it is the proof establishes.  However, we have all been in a lecture where it was nothing but statement proof repeat and that can be very dull.  Additionally, searching for an answer highlights an important aspect of problem solving: we want students to try different things when they get stuck, so modeling this behavior has benefits beyond that of including proofs.  Similarly, option 1 impresses the importance of critical thinking.  We want students to check their answers, or even ask themselves if their answer makes sense.  This is what we are doing when we ask why the theorem might be true.

While many students might not be overly enthusiastic about seeing the proof of various results, I have found that if I am upfront with them about it, they will usually listen.  Maybe they will not all take notes when the proof begins, but then I would rather they sit and listen and think than mindlessly copy every line.

The "Zeno's Paradoxes and Calculus" Paradox

One of my fondest memories of taking freshman calculus was the brief discussion of Zeno's paradoxes. For anyone unfamiliar, the particular one I remember is the Dichotomy paradox:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

--Aristotle, Physics VI:9, 239b10

That is, if you are walking towards the wall, first you must travel half way there.  Then you must travel half way from there to the wall, then half way again, and so on.  Thus you will never reach the wall!  (Actually, this is backwards from the traditional reading of the paradox: before you travel half way there, you must first travel half way to that half way point, and before that, half way to there, so in fact you never start moving at all!)

This paradox is often used as an example of a great mystery that calculus can help us solve.  I so wish that were the case.

I have given my students this example in my own calculus courses.  It always goes over very well.  When first describing the situation, I ask given all this whether I will ever reach the wall, to which most students say that I will not.  "Great," I say, and proceed to walk straight into the wall.  It is a fun activity that engages students, and is related to mathematics.  Except that it is not related to mathematics.

The usual explanation of the paradoxes using calculus is to show that the geometric series with ration 1/2 converges.  But this is not what is perplexing about Zeno's paradox.  In fact, doing the math behind this series is much more complicated than just looking at a picture of a line divided first in half, then the next part in half again, and so on.  Clearly the sum of 1/2, 1/4, 1/8, ... is 1.  The fact that mathematicians have been able to develop a the notions of limit and infinite series to a level of precision which agrees with our intuition is remarkable, yes, but the result is not surprising.  This explanation acts as if the perplexing thing about Zeno's paradox is that the result of traveling these half distances is just the whole distance, and not an infinite distance.  After all, at first glance, adding up an infinite number of things should not give you something finite.

Perhaps better would be to use the geometric series to represent time.  Say you walk one mile at one mile per hour.  In half an hour, you have walked half a mile.  Then you need 1/4 of an hour to get through the next 1/4 of a mile.  Then you need 1/8 of an hour to go the next bit, then 1/16 of an hour, and so on.  You add up all these times, you get 1 hour, and you have traveled 1 mile (again by adding up all those distances).  Alright, so this definitely is convincing.  I am now sure that I will reach my goal in a finite amount of time.  Of course, I knew that already: I walked right into the wall.  Anyway, Aristotle even gave that explanation, and he didn't know any calculus.

The reason Zeno's paradox is compelling is that it requires you accomplish an infinite number of steps.  Not an infinite number of steps in 1 hour, but an infinite number of steps at all.  I can see that this particular infinite sum is a finite number, but what I cannot see is that I would be able to ever arrive at that number by physically entering infinitely many terms into my calculator (even if I could do so at an ever increasing rate).  This seems like a problem for physics, not mathematics. 

And yet, Zeno's paradox is such a great teaching tool.  If only there were a way to use it that did it justice.