Separating proofs in low level classes

The semester has gotten off to a busy start, thus the lack of postings.  Anyway, this morning, I was thinking about a student who came to my office hours for some review on derivative rules.  He had forgotten how to take the derivative of $3^x$.  As he flipped through his notes, and eventually found the relevant page, he asked, "Oh right, is that where you sue the frog rule?"  You see, I had joked with my class that I call the logarithm rule for exponents the frog rule because the exponent jumps over the log.  Like a frog.  This of course has nothing to do with the derivative rule for $a^x$, except that when we derived the rule, we needed to use some properties of logarithms (we took the derivative of both sides of $a^x = e^{ln(a^x)}$). 

This got me thinking about a problem that I have noticed before.  Students seem to have a hard time distinguishing between the proof and the result.  This is not as much an issue in a higher level class, where proofs are common.  By then students understand what we are doing.  But in a first semester calculus class, many students are not aware of the important role of proofs in mathematics.  So what to do?

I admit that part of the blame lies with the way I present the material.  I try hard not to make my lecture into a list of theorems each followed by a proof.  Doing so makes the class rather dry, I think.  Plus, not everything we say in calculus comes with a detailed proof.  For the derivative rules, I think it is much nicer to start out with the desire to discover the rule, and then derive it.  That's what I did for $a^x$.  But with students furiously copying down everything I write on the board, the distinction between reason and result can be lost.

Here is my idea: tell students not to take notes.  I want them to see the reason something is true, and I want them to realize that finding this reason is important.  But I don't want them to miss the punchline.  In the end, a student who knows the derivative rules front and back is way better off than a student who has a book full of notes on how the rules were derived.  I think that such a tactic might actually improve understanding of the derivation.  If I tell my students that I just want to them to watch carefully what I am doing, then just maybe they will and see what is going on right then.  When I get to the result, I would tell everyone that this is the thing they should right down and memorize. 

I am sure that this would have improved my evaluations of trigonometry last spring as well.  Many students mentioned that they did not like all the time spent going over why something was true, they just wanted the facts.  I am not about to cut out proofs and derivations, but announcing ahead of time that it is not necessary to copy the whole thing down, might just do the trick.

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.

The problem = equal signs

New research out of Texas A&M suggests that 70% of middle school students don't understand what the "=" sign means.  The article can be found here.  The key point, as pointed out in the article:

The problem is students memorize procedures without fully understanding the mathematics.

It is interesting that the problem runs this deep.  My calculus students have trouble solving velocity problems because they try to memorize the procedure instead of understanding the problem.  Of course, when they forget one step, they are completely lost.  This study suggests that even from the first time students are exposed to math, they are only taught the steps for solving math problems. 

I am not suggesting that we move to a "new math" style of instruction, where answers don't matter, and all the emphasis is on concepts.  However, if students really have this little understanding of even the simplest of math concepts, we are really missing the point of teaching math entirely.

Thinking vs Doing

Word problems are hard for students.  I've never really understood why before.  You read the problem, figure out what it is asking, figure out how to answer the question, and do it.  Usually the mathematics part is not that hard.  So why to students have so much trouble?

One of the reasons might be that many students have never been shown how to think about the problem.  So much of mathematics is about doing. Here are the steps you need to follow to solve an equation.  Here is the process to find the equation of the tangent line.  When you want to find the absolute maximum of the function, first... then... and finally... etc.  Many problems in mathematics are complex and require multiple steps.  When we teach students how to solve these problems, it is very tempting to teach them the algorithm for arriving at the solution.  No doubt there is value in learning how to follow a recipe, but as a method for solving a problem, it can be perilous.  If we simply follow the steps, it is all to easy to miss one without realizing it, resulting in a wrong, or even meaningless answer.

As professional mathematicians, this is rarely a problem, since while applying the problem solving algorithm, we realize why we are performing each step.  If we miss a step, we will notice immediately (most of the time).  But for a student who is first encountering a type of problem, this is near impossible.  They are worried about following the steps, not why the steps matter.  If they would just stop to think about what they are doing, they would realize that something has gone wrong.

Consider this example: in calculus, we ask our students some simple velocity problems.  We give them a formula for the height of an object at time t, ask them to find a formula for velocity, then ask them,

1. What is the maximum height of the object?
2. How fast is the object going when it hits the ground?

This semester, I asked how fast the object was traveling when it was 35 feet above the ground.  A student promptly plugged in 35 for t in the velocity formula.  Why would she do such a thing?  This particular student is usually very bright, and definitely hard working.  I suspect that her thought process went something like this. "Okay, one of these problems.  I need to plug something into an equation.  Here is a number.  Here is equation.  Here we go..."

I pointed this out to my class when talking about the problem the next day, and almost all of them agreed that at least one of their math teachers in the past taught them how to do word problems in this way.  To solve the word problem you need to find the equation, and plug in the number, and evaluate.  Quick, don't think about it, just follow the directions.  Now move along.  We cannot blame our students for this poor mathematics upbringing.  I show my students too many algorithms -- it is a fast and easy way to get them to find the right answer, before they forget how the algorithm goes.  In fact, last time I taught this topic, I think I pointed out to my students that questions 1 and 2 above are similar in that you set one of the equations equal to zero, solve for t, then plug that value into the other equation.  Now reflecting back on it, I feel embarrassed to have fallen into the trap.

I am not entirely sure how to fix this problem.  Certainly it is important to, whenever possible, challenge our students to think.  Give them time to struggle with a problem.  Let them figure out the solution without giving them a recipe.  Through practice, the students will come up with their own shortcuts.  This takes a lot of time, and can be frustrating for all parties involved.  But giving them yet another algorithm will only make the problem worse. 

I am reminded of a certain proverb about fish and fishing...

Implicit Differentiation

I recently taught my summer Calculus 1 class about implicit differentiation.  This is usually a difficult subject to teach, as students have trouble understanding why they much include $\frac{dy}{dx}$ whenever they encounter a $y$ term.  I think I have stumbled upon a good way to explain this.

Before saying anything about how to differentiate, I take a few minutes to explain what an implicit function is.  In addition to showing some examples so students see the form of the equations, I point out that even thought the $x$'s and the $y$'s are "all mixed up," $y$ is still a function of $x$.  We just don't know what it is (because it is hard or even impossible to solve for $y$).

Next, I go through an example to see how to differentiate and then solve for $\frac{dy}{dx}$.  When I get to the $y^2$ term, I say that the derivative is $2y$, but because $y$ is a function of $x$, we need to use the chain rule, so we must also multiply by the derivative of $y$, which is $\frac{dy}{dx}$.  I point out that this is a good thing, since we are searching for the derivative of $y$, so one had better show up in our problem.  This, however, is not enough to convince most students.

Hopefully a student asks why we must include the derivative of $y$ (otherwise I ask them to explain in, which usually gets them to ask).  I remind them that $y$ is in fact a function of $x$.  I suggest that maybe $y = \sin(x)$.  Then $y^2 = \sin(x)^2$.  How do you take the derivative of that?  Chain rule.  Okay, what if $y = x^2 + 3$?  Chain rule again.  When writing these two examples up, I will not write out the derivative of the inside, instead leave it is $\sin(x)^\prime$ and $(x^2 + 3)^\prime$.  Then I go back and erase all the "$\sin(x)$" and "$x^2 + 3$" and replace the with $y$'s.  This seems to give students that elusive "ah-ha" moment.  And at least for the next day, they know how to do implicit differentiation.

Derivative of Sine

I recently taught my class the derivative rule for sine and cosine.  These are particularly easy rules to apply, and make for nice examples when doing the product, quotient and chain rules, so I like to introduce them early.  The trouble is that the proofs of the rules are rather complex.  We would never ask students to find the derivative of sin(x) using the limit definition during an exam.  So how much detail should be go into when proving these rules?

In Hughes-Hallett, very little proof is given (it is done in the exercises).  Instead they inspect the graph of sin(x) and use it to sketch a graph of the derivative.  This is a good exercise for the student anyway, and allows them to discover for themselves that the derivative is cos(x).  While this is definitely the way to start, it would be nice to give a slightly more rigorous explanation.

On the other hand, Stewart's calculus text gives quite a rigorous proof for the derivative of sin(x).  Starting with the limit definition, then using the sum formula for sine on sin(x+h), we arrive at $\lim_{h\to 0}\frac{\sin x \cos h + \cos x\sin h - \sin x}{h}$.  This can be regrouped to give $\sin x \lim_{h\to 0}\frac{\cos h - 1}{h} + \cos x \lim_{h\to 0} \frac{\sin h}{h}$.  Now to evaluate those two limits, Stewart takes a page and a half to give a (very nice) geometric argument, using the squeeze theorem.  Last semester, I walked my students through the argument.  It did not go well.  Leaving calculus to look at a geometry problem was just too much.

Luckily, there is another way: look carefully at those two limits.  Do they remind you of anything?  What if instead of the -1, we replaced that with cos(0)?  Or wrote sin(h) - sin(0) in the numerator of the second limit.  That's right, both are simply a derivative evaluated at 0.  The first is the derivative of cos(x) evaluated at 0.  Let's look at the graph of cosine.  What is the slope of the tangent line at x = 0?  Clearly it is 0.  What is the slope of the tangent line to sin(x) at x = 0?  Looks very much like 1 to me.  That, and we already evaluated that limit using approximation, and it also looked like 1.  We are then left with simply cos(x).  And that is not a surprise, since that is what the graph of the derivative looks like.

I know this is not rigorous, but it is convincing.  And more so, it reinforces the definition of the derivative. 

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.