Teaching with Sarcasm

Today we finished the Calc 2 section introducing series. At the beginning of class, we had a lot of confused looks, and questions during class mixed up the (very easy to confuse) sequence with the series. I wanted to find out where we stood in the class, so I gave them a quiz which asked simply,

Using intuition alone, do you think $\displaystyle \sum_{n=1}^\infty \frac{n}{n^3+2}$ converges or diverges? Why?

When I went over the responses, I started a little concerned. But in the end, I had the following results:

12 students had statement something like tail going to 0
3 students though it looked something like a geometric series
5 used the Divergence Test but had an arithmetic mistake to conclude it diverged.
1 had completely wrong math (split the addition in the denominator, mistaking definition of convergence)
3 compared it to the harmonic series (so diverged)
1 liked “convergence” better as a word.

We decided the top three categories, 20 students in all, are in a good way. I grant that $a_n \to 0$ doesn’t guarantee convergence, but intuitively, it’s a good start! Similarly I was happy when they noticed $\frac{1}{n}$ is a small number, like a convergent geometric series. And arithmetic errors are perfectly normal when you first see a topic!

Of the three students who compared it to the harmonic series, two of them wrote to the effect “Well, I think it should converge, but I thought the harmonic series should converge, too” so I had in a sense led them astray by that neat counter-example to the divergence test.

The one person with the snarky response liking “convergence” better as a word is (no surprise to the reader, I’m sure), one of my best students.

By this metric, 20/25 are on the right track. That’s not bad for a first look at series. Tomorrow we learn convergence testing to put their intuition to work!

I’ve been grading a lot of exams lately, so I thought that I’d take a few minutes to write out in words how I grade.

My tests usually have two or three questions per page, with their name only appearing on an attached cover sheet. I start the grading process by flipping over all the tests so I don’t see any names, and shuffling the stack. I then start grading one page at a time, every exam’s last page, then second to last, etc…

For each individual page, I mark all the obviously correct answers and set aside all the questions which require more than a moment or so to think about.

After getting through the whole stack, I compare the two stacks — did most people get it right? Did only a handful of people get it?

At this point, I update the rubric for this page. For example, the question “Expand $\log \frac{27}{12}$ in terms of $\log 3$ and $\log 2$ ” really testing? On last week’s test, many pre-calc students kept missing the negative sign you need to distribute:

$\log \frac{27}{12} = 3\log 3 - ( \log 3 + 2\log 2)$

To me, what I’m really testing is the ability to notice and use all three of the quotient, product, and power rules for logarithms. Distributing the negative is great, but in this context, it’s the least important part. I decided that the students who got almost the right answer could get full marks (while simultaneously thinking, “Damn! I need to work more examples like this in class” and “This question should have been worth more points to account for this…”).

It’s not a perfect system, but so far it’s working for me.

Encountering Series. Good news!

Is that negative sign really that important?