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.
Saturday, July 31, 2010
Sunday, July 25, 2010
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.
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.
Friday, July 23, 2010
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.
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.
Tuesday, July 20, 2010
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:
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.
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:
- 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.
- 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.
- 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.
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.
Wednesday, July 7, 2010
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:
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.
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.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!)
--Aristotle, Physics VI:9, 239b10
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.
Monday, July 5, 2010
Motivating students motivated by their future career
I stumbled upon a now six month old piece from the New York Times today: Making College 'Relevant'. It discusses the trend, well known to everyone in academia, of colleges and universities catering more and more to students' desires to use college as job training. This is a much graver issue for the liberal arts majors than for the sciences, as a science heavy major is seen as one which will net a high paying job. Mathematics has long straddled the gap between liberal art and science, so even though we might not be hurting for majors, I believe this trend is still worth worrying over.
Even if the top three jobs in the U.S. are mathematics based, mathematics education can suffer from students too concerned with their future earnings. These students are the ones who complain when we show a proof of the mean value theorem, instead of just give them the formula to memorize. They complain that they don't want to learn this or that, because "when will they ever need to know that?" Of course they are missing the point of mathematics if this is their approach.
Luckily there is hope of a compromise. In the article it is reported that a survey by the Association of American Colleges and Universities of employers who hire at least 25% of their workforce from two- or four-year colleges found that 81% asked for better “critical thinking and analytical reasoning skills.” This is kind of mathematics thing. So now we have yet another argument for why students need to know the definition of the derivative: it will build greater critical thinking skills, which in turn will (sigh) make you more money.
Even if the top three jobs in the U.S. are mathematics based, mathematics education can suffer from students too concerned with their future earnings. These students are the ones who complain when we show a proof of the mean value theorem, instead of just give them the formula to memorize. They complain that they don't want to learn this or that, because "when will they ever need to know that?" Of course they are missing the point of mathematics if this is their approach.
Luckily there is hope of a compromise. In the article it is reported that a survey by the Association of American Colleges and Universities of employers who hire at least 25% of their workforce from two- or four-year colleges found that 81% asked for better “critical thinking and analytical reasoning skills.” This is kind of mathematics thing. So now we have yet another argument for why students need to know the definition of the derivative: it will build greater critical thinking skills, which in turn will (sigh) make you more money.
Sunday, July 4, 2010
Mnemonics and Acronyms are BAD (Best to Avoid Discussing)
Please Excuse My Dear Aunt Sally, SOCATOA, All Students Take Calculus, FOIL, ...
It seems math education is riddled with acronyms and mnemonics. Students love them because they afford an easy way to remember what otherwise might be a challengingly complex concept. What is a little more surprising is that many teachers are also very found of these tricks. Last semester I had a student who seemed to have a mnemonic for everything, and she claimed that her high school math teacher showed her dozens of them. What a shame, I thought.
Now I understand that students will not be able to advance very far in mathematics without being able to remember the order of operations, but there must be a better way. Consider the "All Students Take Calculus" example. If you are not familiar with it, this is the trick to remember which trigonometric function is positive in which quadrant. Starting in quadrant I, all three trig functions (sine, cosine, tangent) are positive. In the second quadrant, sine is the only positive one (Sine = S = Students), in the third tangent is the positive one, in the fourth cosine is. Okay, so this definitely works, and I'm sure there are students who know that sine and tangent are negative in quadrant IV but cosine is positive because of this, if for no other reason. But there's the problem: if for no other reason!
Look, let's be honest. Students do not need to know which quadrant tangent is positive in. It is unlikely they will every need that fact on the job, nor be asked about it in a job interview. There are no exclusive math parties where that particular piece of knowledge is required to gain entry. This of course does not mean we shouldn't teach the topic. Determining the sign of a trig function based on the quadrant of its angle is a perfect exercise in understanding the meaning of the trig functions. Students should already know which of x and y are positive in a given quadrant. Students should already know the definition of the trig functions in terms of x and y. Putting these pieces together is exactly why we teach trigonometry at all. Its to develop that kind of thinking.
On the other hand, there are some formulas or concepts which simply need to be memorized. For example, I would not expect my students to be able to derive the quadratic formula each time they need it. If they can think up a little jingle to help them remember which constant belongs where, more power to them. Additionally, I must admit I was surprised to hear that some schools are avoiding mentioning FOIL. I would probably not teach students that method of multiplying two binomials at first, but if they already know the trick, it seems a waste not to use that simply label to remind them what's going on.
So how should we approach mnemonics such as these? Depending on the particular example, my answer might change, but in general, I avoid introducing these tricks completely. I might ask my class how they plan on remembering a particular formula or concept. If a student suggests a mnemonic, I act surprised, as if I have never heard that one before. I then go through it and "check" that it works. This reinforces the original, conceptual basis for the fact, as well as challenges the students to think creatively about the subject. Once the class knows a mnemonic, I tend not to be a hard-ass about it. I would never respond to a student suggesting we FOIL an expression with, "what's FOIL? do you mean doubly distribute?"
It seems math education is riddled with acronyms and mnemonics. Students love them because they afford an easy way to remember what otherwise might be a challengingly complex concept. What is a little more surprising is that many teachers are also very found of these tricks. Last semester I had a student who seemed to have a mnemonic for everything, and she claimed that her high school math teacher showed her dozens of them. What a shame, I thought.
Now I understand that students will not be able to advance very far in mathematics without being able to remember the order of operations, but there must be a better way. Consider the "All Students Take Calculus" example. If you are not familiar with it, this is the trick to remember which trigonometric function is positive in which quadrant. Starting in quadrant I, all three trig functions (sine, cosine, tangent) are positive. In the second quadrant, sine is the only positive one (Sine = S = Students), in the third tangent is the positive one, in the fourth cosine is. Okay, so this definitely works, and I'm sure there are students who know that sine and tangent are negative in quadrant IV but cosine is positive because of this, if for no other reason. But there's the problem: if for no other reason!
Look, let's be honest. Students do not need to know which quadrant tangent is positive in. It is unlikely they will every need that fact on the job, nor be asked about it in a job interview. There are no exclusive math parties where that particular piece of knowledge is required to gain entry. This of course does not mean we shouldn't teach the topic. Determining the sign of a trig function based on the quadrant of its angle is a perfect exercise in understanding the meaning of the trig functions. Students should already know which of x and y are positive in a given quadrant. Students should already know the definition of the trig functions in terms of x and y. Putting these pieces together is exactly why we teach trigonometry at all. Its to develop that kind of thinking.
On the other hand, there are some formulas or concepts which simply need to be memorized. For example, I would not expect my students to be able to derive the quadratic formula each time they need it. If they can think up a little jingle to help them remember which constant belongs where, more power to them. Additionally, I must admit I was surprised to hear that some schools are avoiding mentioning FOIL. I would probably not teach students that method of multiplying two binomials at first, but if they already know the trick, it seems a waste not to use that simply label to remind them what's going on.
So how should we approach mnemonics such as these? Depending on the particular example, my answer might change, but in general, I avoid introducing these tricks completely. I might ask my class how they plan on remembering a particular formula or concept. If a student suggests a mnemonic, I act surprised, as if I have never heard that one before. I then go through it and "check" that it works. This reinforces the original, conceptual basis for the fact, as well as challenges the students to think creatively about the subject. Once the class knows a mnemonic, I tend not to be a hard-ass about it. I would never respond to a student suggesting we FOIL an expression with, "what's FOIL? do you mean doubly distribute?"
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