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.

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. 

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!