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.
Wednesday, June 30, 2010
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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