Robert Lewis has a great essay on math, as the most misunderstood subject. I couldn't agree more!
In addition to an analogy with sports training similar to one I have here, I especially liked his parable of the hostile party goer. He is confronted by a man complaining that he was forced to memorize the quadratic formula, and yet has never had to use it. Lewis compares this to the absurd notion that we should complain to our first grade teachers that we have not once had to recall the details of the ever popular Dick and Jane books. Why would we need to spend so much time reading the books if the information contained within them can be so easily forgotten? Of course, the answer is that we used the books to learn to read well. Similarly, we practice (even memorize) math to learn to think well.
Lewis has captured exactly the problems most Americans have with mathematics and the ways they misunderstand the subject. But what's next? How can we correct this error for future generations? I suspect it all starts with us: college math professors, especially those of us who get to teach future K-12 teachers. We need to teach with an understanding of what math is, so they will have that understanding, so they in turn can teach in a way that their students appreciate what math is and what it is not. I am not calling for a change in K-12 math curriculum. That is too early to teach the nuances of math. Instead, the teachers must present the mathematics in a way which respects the subject and prepares students to uncover the hidden richness of mathematics as they mature.
Wednesday, December 22, 2010
Tuesday, December 14, 2010
Attendance Issues
At about the midpoint of this last semester, students here at Coastal had a day off (a fall "student holiday"). Not surprisingly, many student took the day before off as well. In my Math 139 classes, I had 15/40 and 11/37 attendance, respectively. It got me thinking about my policy on taking attendance.
I have long held that whether a student wants to come to class is entirely their business. They (or their parents) have paid for college, and if they want to (foolishly) squander their opportunities to learn the material, that is their choice. This would be a fine policy if I did not plan on reviewing material until everyone understands. But of course I do try to ensure that everyone learns the material and this is much more difficult when half the class is a day behind. So even if it appears that the absent students are only hurting themselves, it does effect me as well as the responsible students. Thus encouraging regular attendance is important.
I decided after that class to start sending around a sign-in sheet at the start of every class. I did not change my policy on attendance at all (although the syllabus says I have the option of failing a student who misses too many classes, attendance is not factored into their grade). Attendance shot up. Not to 100% or anything close to it, but considering the course, there was a marked improvement.
A couple of times students would ask if their grade would be effected because they had to miss an upcoming class. I told them not to worry about it. If anyone asked whether I was going to make attendance part of the final grade, I simply didn't answer. It was all done with a wink and a nod, and honestly, I don't think I fooled anyone into thinking that their grade would go down if they missed a class (other than the missed opportunity to learn the material, but that threat had been there from the beginning).
Today I got back my instructor evaluations. In one of my "ways to improve" comments I got:
I have long held that whether a student wants to come to class is entirely their business. They (or their parents) have paid for college, and if they want to (foolishly) squander their opportunities to learn the material, that is their choice. This would be a fine policy if I did not plan on reviewing material until everyone understands. But of course I do try to ensure that everyone learns the material and this is much more difficult when half the class is a day behind. So even if it appears that the absent students are only hurting themselves, it does effect me as well as the responsible students. Thus encouraging regular attendance is important.
I decided after that class to start sending around a sign-in sheet at the start of every class. I did not change my policy on attendance at all (although the syllabus says I have the option of failing a student who misses too many classes, attendance is not factored into their grade). Attendance shot up. Not to 100% or anything close to it, but considering the course, there was a marked improvement.
A couple of times students would ask if their grade would be effected because they had to miss an upcoming class. I told them not to worry about it. If anyone asked whether I was going to make attendance part of the final grade, I simply didn't answer. It was all done with a wink and a nod, and honestly, I don't think I fooled anyone into thinking that their grade would go down if they missed a class (other than the missed opportunity to learn the material, but that threat had been there from the beginning).
Today I got back my instructor evaluations. In one of my "ways to improve" comments I got:
Who knew? The point is, sending around an attendance sheet is almost no work for me, takes almost no time away from class, improves attendance, and is in fact appreciated by the students.
Take attendance at beginning of semester; I need a reason to come.
Sunday, October 10, 2010
Uniformly bad
A little over half of the courses I have taught over the years have been "uniform" courses. These lower level math courses have multiple sections each semester, so the department has decided to appoint a course coordinator to oversee all the instructors. While there is some variation on how coordinated these courses are, usually it goes as far as common exams and group grading, although most instructors are allowed to write their own quizzes, group work (if any) and assign homework as they see fit. I understand the allure of running courses this way, and there are many arguments in favor of this approach. The problem is that not one of those arguments are for the benefit of the students. Sadly, the more uniform we make our classes, the worse they will be for our students.
Not every college professor is a great teacher. I think we have all run into professors who don't prepare classes well, don't write relevant exams, grade those exams unfairly, and fail to cover all the material the course is supposed to contain. Ideally, uniform courses would correct these problems. They do not. Classes can still be poorly prepared. Exam, unless the coordinator happens to be this bad professor, will be written well, but for students in the bad professor's class, they will not be relevant. Exams will be graded uniformly over all sections, but not uniformly over the exam itself: one page might have a reasonable partial credit policy, while the next be all or nothing. Worst of all, if this poor professor does not cover everything he or she is supposed to in class, then either those students will do badly on that part of the final exam (for not having seen the material ever) or all the sections will drop the material from the exam.
The more uniform the course, the more uniform the level of instruction. Unfortunately, this level drops to near that of the worst teacher in the group. Consider two examples, each occurring this last week. In a class with common exam, graded communally, all sections had to delay returning exams because one professor was too busy to get his pages graded. Another class, again will common exams, although not graded together, a professor decided to let some student take the exam late, resulting in a ban on returning (fully graded) exams to students in other sections. Now in this case, neither professor is the proverbial bad teacher of the previous paragraph: both just happened to find themselves in less than ideal situations. But because of the uniformity of the courses, this non-ideal situation spread to all the sections. Getting exam back quickly is not the most important thing in a college course, but it does benefit students to see their graded exams as soon as possible, as they will then be more likely to look over their mistakes and remember why they made them.
I understand that math departments need to have standards and that universities need to be assessed and accredited. We do not want some student coming out of calculus having just learned the basic derivative rules (without proof) and others getting a full course in real analysis. Some level of coordination is necessary. Here is how I would do it: everyone uses the same textbook, and everyone covers the same sections (with perhaps one or two optional sections, left up to individual instructors, as time permits). Everyone has the same number of exams, worth the same percentage of students' final grades. Exams are graded individually, but the coordinator sends out instructions on the level of partial credit to assign. If the university has a final exam time for all lower level math courses, then there is a common final. For all other exams, instructors are welcome to collaborate, and the coordinator can be the one to write these, but they do not need to do so. That way, if one instructor misses a day and needs to push back his or her exam, there will not be conflicts. All exams should be sent to the coordinator, for approval and record keeping. That is all.
Not every college professor is a great teacher. I think we have all run into professors who don't prepare classes well, don't write relevant exams, grade those exams unfairly, and fail to cover all the material the course is supposed to contain. Ideally, uniform courses would correct these problems. They do not. Classes can still be poorly prepared. Exam, unless the coordinator happens to be this bad professor, will be written well, but for students in the bad professor's class, they will not be relevant. Exams will be graded uniformly over all sections, but not uniformly over the exam itself: one page might have a reasonable partial credit policy, while the next be all or nothing. Worst of all, if this poor professor does not cover everything he or she is supposed to in class, then either those students will do badly on that part of the final exam (for not having seen the material ever) or all the sections will drop the material from the exam.
The more uniform the course, the more uniform the level of instruction. Unfortunately, this level drops to near that of the worst teacher in the group. Consider two examples, each occurring this last week. In a class with common exam, graded communally, all sections had to delay returning exams because one professor was too busy to get his pages graded. Another class, again will common exams, although not graded together, a professor decided to let some student take the exam late, resulting in a ban on returning (fully graded) exams to students in other sections. Now in this case, neither professor is the proverbial bad teacher of the previous paragraph: both just happened to find themselves in less than ideal situations. But because of the uniformity of the courses, this non-ideal situation spread to all the sections. Getting exam back quickly is not the most important thing in a college course, but it does benefit students to see their graded exams as soon as possible, as they will then be more likely to look over their mistakes and remember why they made them.
I understand that math departments need to have standards and that universities need to be assessed and accredited. We do not want some student coming out of calculus having just learned the basic derivative rules (without proof) and others getting a full course in real analysis. Some level of coordination is necessary. Here is how I would do it: everyone uses the same textbook, and everyone covers the same sections (with perhaps one or two optional sections, left up to individual instructors, as time permits). Everyone has the same number of exams, worth the same percentage of students' final grades. Exams are graded individually, but the coordinator sends out instructions on the level of partial credit to assign. If the university has a final exam time for all lower level math courses, then there is a common final. For all other exams, instructors are welcome to collaborate, and the coordinator can be the one to write these, but they do not need to do so. That way, if one instructor misses a day and needs to push back his or her exam, there will not be conflicts. All exams should be sent to the coordinator, for approval and record keeping. That is all.
Wednesday, September 22, 2010
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.
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.
Friday, August 27, 2010
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.
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.
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.
- 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.
- 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.
- 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?
- 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.
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.
Friday, August 13, 2010
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:
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.
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.
Monday, August 2, 2010
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,
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...
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,
- What is the maximum height of the object?
- How fast is the object going when it hits the ground?
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...
Saturday, July 31, 2010
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.
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.
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?"
Wednesday, June 30, 2010
Mathematics as weightlifting
Every math teacher has heard it: "Why do I need to learn this." Like nails on a chalkboard! I find the question especially difficult to answer as a pure mathematician. I enjoy mathematics particularly because it is not something that can be used -- the abstractness is exciting. But students don't want to hear this. Increasingly, students go to college for the purpose of getting a higher paying job, and have little patience for anything they will not need to know in the workforce.
To be clear, there are two very different questions of this type. One is an honest question: "What can this mathematics be used for?" This question deserves an honest and careful answer. The other: "Why do I need to know this if I won't ever use it in my particular job?" It is this latter question I want to answer here. Of course, I realize that often students pose the question as rhetorical, as an exhibition of their frustration. In that case, the following can be used to motivate them to get back to work.
A few years back I was watching a baseball game, and the commentators got to talking about a particular player and how dedicated he was. They said that he spent even more time lifting weights than his teammates did. The whole idea of baseball players weightlifting was perplexing to me: why not spend that time at batting practice, or running sprints? Baseballs are not that heavy, and it's not like once reaching second base the runner must bench-press his bodyweight. But of course upon further reflection, it became clear that while the players do not need to use the particular weightlifting skills they work so many hours on, doing so makes them stronger over all. It makes them better athletes. Similarly, doing mathematics makes (future) scientists better thinkers.
Baseball players are not the only people who lift weights. Most everyone can benefit from weightlifting as it is a great way to stay fit and healthy. Similarly, mathematics is a great way to stay mentally fit and healthy. Like weightlifting, it can be difficult when you first start out, especially if you are not doing it correctly. But once you get the hang of it, not only can you lift more and more weight, but it will become enjoyable. In fact, there are people who enjoy weightlifting so much, they do it professionally.
To be clear, there are two very different questions of this type. One is an honest question: "What can this mathematics be used for?" This question deserves an honest and careful answer. The other: "Why do I need to know this if I won't ever use it in my particular job?" It is this latter question I want to answer here. Of course, I realize that often students pose the question as rhetorical, as an exhibition of their frustration. In that case, the following can be used to motivate them to get back to work.
A few years back I was watching a baseball game, and the commentators got to talking about a particular player and how dedicated he was. They said that he spent even more time lifting weights than his teammates did. The whole idea of baseball players weightlifting was perplexing to me: why not spend that time at batting practice, or running sprints? Baseballs are not that heavy, and it's not like once reaching second base the runner must bench-press his bodyweight. But of course upon further reflection, it became clear that while the players do not need to use the particular weightlifting skills they work so many hours on, doing so makes them stronger over all. It makes them better athletes. Similarly, doing mathematics makes (future) scientists better thinkers.
Baseball players are not the only people who lift weights. Most everyone can benefit from weightlifting as it is a great way to stay fit and healthy. Similarly, mathematics is a great way to stay mentally fit and healthy. Like weightlifting, it can be difficult when you first start out, especially if you are not doing it correctly. But once you get the hang of it, not only can you lift more and more weight, but it will become enjoyable. In fact, there are people who enjoy weightlifting so much, they do it professionally.
Tuesday, June 29, 2010
Probability is hard
There is a nice article on the ScienceNews.org site about the Tuesday birthday problem:
I have two children, one of whom is a boy born on a Tuesday. What is the probability that I have two sons?
The article does a very good job of discussing the solution, as well as why it is difficult (and why "Tuesday" has anything to do with the solution). Commenters have pointed out that there is an additional problem: what the meaning of "one of whom" is. It could mean "at least one of whom," which is the strictly correct mathematical interpretation, or "exactly on of whom," which is what most people would read if they are not being careful.
And we wonder why students hate word problems! Problems like this are often offered as examples of how counter-intuitive probability (especially conditional probability) can be. And probability can be counter-intuitive -- just see the Monty Hall Problem. However, problems like these two children birthday paradoxes are often confusing mostly because of the wording. We should be very careful to remove all ambiguity from the question and let the interesting mathematics stand on its own.
I have two children, one of whom is a boy born on a Tuesday. What is the probability that I have two sons?
The article does a very good job of discussing the solution, as well as why it is difficult (and why "Tuesday" has anything to do with the solution). Commenters have pointed out that there is an additional problem: what the meaning of "one of whom" is. It could mean "at least one of whom," which is the strictly correct mathematical interpretation, or "exactly on of whom," which is what most people would read if they are not being careful.
And we wonder why students hate word problems! Problems like this are often offered as examples of how counter-intuitive probability (especially conditional probability) can be. And probability can be counter-intuitive -- just see the Monty Hall Problem. However, problems like these two children birthday paradoxes are often confusing mostly because of the wording. We should be very careful to remove all ambiguity from the question and let the interesting mathematics stand on its own.
Monday, April 19, 2010
Random examples
Some examples are better than others - most of the time. For example, when first teaching the product rule, it is not a good idea to use $xe^x$: since the derivative of $e^x$ is $e^x$, students don't see the form of the product rule explicitly. That said, there are times when the technique being taught are so general, that they would work equally well with any example. In cases like these, I like to construct a random example, with the class's help.
Suppose I wanted to show my students that the Taylor series for any polynomial is simply the polynomial back again. If students have not thought about this yet, the result can be rather surprising (after all, the Taylor series for the other standard examples look nothing like the starting function). What I don't want to do is look down in my notes, carefully copy down a polynomial and start from there. While this would definitely be an example worth sharing, I fear that students would not be impressed. Of course I, the powerful math professor, could come up with an example of a function which is identical to it's Taylor series. Just another example, they would think.
Instead, I write on the board "Find the Taylor series for $f(x) =$" and then dramatically point at a student and demand, "What's your favorite number!?" After regaining his or her composure, the student will say, perhaps, 7. I write $7 x^3 +$ and then repeat with another student. Continuing in this fashion, the class and I together come up with a random polynomial. And wouldn't it be amazing if this random polynomial happened be it's own Taylor polynomial? Surely that cannot be a coincidence.
Random examples like these can be used all over the place, although it is important not to use them as a substitute for a well written lecture. As with any interaction with individual students in class, this technique will keep students alert. Most find it humorous (especially when a student can't remember their favorite number). And most importantly, when used correctly, the random example can drive home the fact that the mathematical technique can be used in any case, not just the special cases the professor has prepared.
Suppose I wanted to show my students that the Taylor series for any polynomial is simply the polynomial back again. If students have not thought about this yet, the result can be rather surprising (after all, the Taylor series for the other standard examples look nothing like the starting function). What I don't want to do is look down in my notes, carefully copy down a polynomial and start from there. While this would definitely be an example worth sharing, I fear that students would not be impressed. Of course I, the powerful math professor, could come up with an example of a function which is identical to it's Taylor series. Just another example, they would think.
Instead, I write on the board "Find the Taylor series for $f(x) =$" and then dramatically point at a student and demand, "What's your favorite number!?" After regaining his or her composure, the student will say, perhaps, 7. I write $7 x^3 +$ and then repeat with another student. Continuing in this fashion, the class and I together come up with a random polynomial. And wouldn't it be amazing if this random polynomial happened be it's own Taylor polynomial? Surely that cannot be a coincidence.
Random examples like these can be used all over the place, although it is important not to use them as a substitute for a well written lecture. As with any interaction with individual students in class, this technique will keep students alert. Most find it humorous (especially when a student can't remember their favorite number). And most importantly, when used correctly, the random example can drive home the fact that the mathematical technique can be used in any case, not just the special cases the professor has prepared.
Friday, April 16, 2010
Latex on Blogger
Apparently, I have just enabled latex on Blogger. I did so following the instructions found here. If this is working, then $e^x$ will appear instead of $!$e^x$!$.
Mathematics through puzzles
I love mathematical puzzles. I still remember the first one I ever heard. It was the nine weights puzzle, where you have to find the heavy weight by using a balance scale only two times. I was in forth grade. I remember thinking how clever it was; how simple; how elegant. I can't be sure, but I suspect that puzzle got me on my way to being a mathematician.
Students like these puzzles too. I usually try to give a few of them as extra credit over the course of a semester. What I need though, are some really good ones. I need puzzles that are not only clever, but also remind students about the mathematics we study in class. To keep track of such puzzles, I've started the Math Puzzle Wiki. I would love to find more puzzles, so if anyone has some good leads, please send them along, or add them to the wiki. And of course, feel free to use any of the puzzles you find there.
Students like these puzzles too. I usually try to give a few of them as extra credit over the course of a semester. What I need though, are some really good ones. I need puzzles that are not only clever, but also remind students about the mathematics we study in class. To keep track of such puzzles, I've started the Math Puzzle Wiki. I would love to find more puzzles, so if anyone has some good leads, please send them along, or add them to the wiki. And of course, feel free to use any of the puzzles you find there.
Wednesday, April 14, 2010
Grading is great
This semester I happened to teach three section of Trigonometry, and had an undergraduate grader. He was supposed to grade 15 hours a week (5 hours per class) which meant he could easily grade both homework and quizzes for me. This left me with only exams to grade. Sounds great right?
Turns out, not so much. The grader has done a fine job (although having to get through so many papers meant the students didn't get feedback very quickly). The problem is that I didn't get a chance to grade their work on a regular basis. This in turn held me back from teaching as effectively as I could have.
Apparently I have taken for granted the importance of grading students' work - not because students need to be assessed or get feedback - but because regularly grading allows me to monitor students' progress. Yes, some students ask questions in class, but sadly most do not. Many students will pretend to understand a concept as to not appear ignorant. This semester in particular I have been finding it very difficult to know when I have covered a topic enough so that the majority of my students understand it. I think the reason is, for the first time, I have not been grading the weekly quizzes.
Today I did so for the first time this semester (my grader had a busy week, so I had him just do the homework). It only took me about an hour all together, and going in to tomorrow's lecture, I know that I need to review the polar form of complex numbers, while I should probably not spend much more time discussing the different ways to write vectors.
The point is this: grading, while often a tedious chore, is a great way to ensure that students are getting the most out of course. It is a simple and effective way to take the mathematical pulse of the class. Plus, students appreciate when you get quizzes and exams back quickly, which is easier to do if the grading is done by the professor. From now on, I will grade the quizzes. And I will like it!
Turns out, not so much. The grader has done a fine job (although having to get through so many papers meant the students didn't get feedback very quickly). The problem is that I didn't get a chance to grade their work on a regular basis. This in turn held me back from teaching as effectively as I could have.
Apparently I have taken for granted the importance of grading students' work - not because students need to be assessed or get feedback - but because regularly grading allows me to monitor students' progress. Yes, some students ask questions in class, but sadly most do not. Many students will pretend to understand a concept as to not appear ignorant. This semester in particular I have been finding it very difficult to know when I have covered a topic enough so that the majority of my students understand it. I think the reason is, for the first time, I have not been grading the weekly quizzes.
Today I did so for the first time this semester (my grader had a busy week, so I had him just do the homework). It only took me about an hour all together, and going in to tomorrow's lecture, I know that I need to review the polar form of complex numbers, while I should probably not spend much more time discussing the different ways to write vectors.
The point is this: grading, while often a tedious chore, is a great way to ensure that students are getting the most out of course. It is a simple and effective way to take the mathematical pulse of the class. Plus, students appreciate when you get quizzes and exams back quickly, which is easier to do if the grading is done by the professor. From now on, I will grade the quizzes. And I will like it!
Wednesday, January 13, 2010
About Math for Profs
Welcome to my new blog. Since entering graduate school in Fall 2004, I have taught a variety of math courses at the undergraduate level. These five and a half years of teaching by no means make me an expert on the subject of teaching mathematics, but then that is part of the reason I wanted to start this blog. I really really like teaching. It is the reason I went to grad school in the first place. I love being in front of a class. I love watching students come to realize that mathematics can be attainable, even enjoyable. But most of all, I love figuring out how best to convey the mathematics to my students.
I like teaching math because I like math. There are some college professors who view teaching as a necessary evil -- an activity that pays the bills so they can spend time doing mathematics. I do not take this view. For me, solving the puzzle of how best to teach a particular topic is just as enjoyable as solving the puzzles of mathematics proper.
Plenty has been written on how to be an effective teacher. Unfortunately, most of this advise centers on general techniques of conveying information: speak and write clearly, ask engaging questions, encourage discussion, be well organized, etc. Of course this is important. But what I want to think about and share in this blog is much more specific. I want to know the best way to teach implicit differentiation. What is the best example to answer a student when she asks why we need to learn about sequences and series? What test questions should be avoided because they either are ambiguous or are mathematically unsound? In other words, what is the best mathematical content for excellence in mathematics teaching.
In the months to come, I hope to have the time to explore some of these topics. This semester I am teaching a few sections of Trigonometry, as well as a Discrete class for future Middle School teachers. I suspect most posts will have to do with these subjects, but when I think of anything from other courses, I'll include those as well. Oh, and if anyone actually finds this blog and reads it, I would love your feedback and input. Let's get started...
I like teaching math because I like math. There are some college professors who view teaching as a necessary evil -- an activity that pays the bills so they can spend time doing mathematics. I do not take this view. For me, solving the puzzle of how best to teach a particular topic is just as enjoyable as solving the puzzles of mathematics proper.
Plenty has been written on how to be an effective teacher. Unfortunately, most of this advise centers on general techniques of conveying information: speak and write clearly, ask engaging questions, encourage discussion, be well organized, etc. Of course this is important. But what I want to think about and share in this blog is much more specific. I want to know the best way to teach implicit differentiation. What is the best example to answer a student when she asks why we need to learn about sequences and series? What test questions should be avoided because they either are ambiguous or are mathematically unsound? In other words, what is the best mathematical content for excellence in mathematics teaching.
In the months to come, I hope to have the time to explore some of these topics. This semester I am teaching a few sections of Trigonometry, as well as a Discrete class for future Middle School teachers. I suspect most posts will have to do with these subjects, but when I think of anything from other courses, I'll include those as well. Oh, and if anyone actually finds this blog and reads it, I would love your feedback and input. Let's get started...
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