Something is working

This fall I'm teaching Calculus I for the third time here at Coastal.  Perhaps because I just taught it at the end of the summer, I am finding that it is working particularly well.  I'm not entirely sure why.  Student interaction is good, and I'm sure that is helping.  But beyond that, the lectures seem to be flowing in a way they have not previously.

One possibility is the way I just happen to be presenting the material.  I thought I'd take a moment and record a couple of examples here. 

1. To introduce limits, I started by asking how we might graph $f(x) = \frac{sin(x)}{x}$.  We had previously talked about the graph of $f(x) = \frac{1}{x}$, and what happens at $x = 0$.  We said that while you cannot plug in 0, you can ask what happens near 0.  So we tried that again.  What happens to $f(x)$ as $x$ gets closer and closer to 0.  We made a table, everyone agreed that the $y$-values were getting closer to 1.  Only then did I start using the language "limit."  I gave them the notation as a way to quickly write down what we just did.  This seemed much more natural than giving them a definition of a limit (out of thin air) and then showing them a bunch of examples.
2. For left/right-hand limits, I only introduced them after we discovered the problem in finding limits if the $y$-values approach different values from each side.  I gave them a graph and asked them to find limits in a variety of cases.  We all agreed that at the jump discontinuity, the limit did not exist.  But of course, we can be more descriptive than this.  Coming from one side, the limit does exist.  Then I introduced that notation.
3. For continuity, I drew two graphs, one continuous, and one with a jump discontinuity.  I asked the class what was different about the graphs.  We agreed that the continuous one was continuous and the other not.  I asked what other ways a graph might not be continuous.  We came up with a graph with a removable discontinuity and an infinite discontinuity.  Now, how might we say something about these in terms of limits?  
4. Continuing with continuity, to introduce the difference between continuous at a point vs interval, I covered up the discontinuity and asked if the rest of the graph was continuous.  We agreed it was, so decided that we needed to express continuity at specific points.  This led us to the limit definition of continuity.  This was so much more natural than giving the definition, and then figuring out what it means.

That's all we have done so far.   I'm not sure if this luck will continue.  The common theme seems to be that instead of giving the definition and trying to apply it (which of course has it's place and value), I'm asking them to solve problems, then coming up with the math-way of saying what we are doing. 

I don't know if this is helping the students (we have not had any exams yet) but it definitely feels better to me.   I plan to make an effort to continue using this "technique" for the rest of the class and see how it goes.

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.