Friday, January 26, 2018

Seeking Sequences: What's Next?

Back in November I ran a workshop at the DACTM conference on sequences and series. We worked through and discussed the problems in this document. The main idea was to apply Dan Meyer's "headache-aspirin" approach to sequences and series. That is, we start students with a relatively simple task like finding the next term in a sequence. Then we create intellectual need for a more efficient method by asking for a very distant term (headache). Finally, we help students develop a formula (aspirin).

This problem I stole from Facebook is a great way to introduce sequences:

It's great because it's from social media. If "ordinary" people have spent their free time thinking about this problem, students will surely spend their class time thinking about it.

It's great because in a large group there's always someone who solves it "explicitly" by multiplying and someone who solves it "recursively" by repeatedly adding. So these vocabulary words get introduced naturally.

It's great because, like so many problems on social media, notation is abused badly. It makes the problem inviting. You can put on a show as the angry math teacher yelling that "2 does not equal 6 !!!" and then decree that sequences will always be given a name (as short as possible, only one letter allowed!) and have their terms specified by subscripts. [Or treat sequences as functions whose domain is the counting numbers and use function notation.]

I won't discuss the rest of the problems in detail, but here are two highlights I recall from the workshop.

(1) When finding sums of odd numbers, a lot of people want to solve 161 + 163 + 165 + ... + 777 + 779 + 781 by squaring the difference of the relevant term numbers (391 - 81)^2 instead of taking the difference of the relevant term numbers 391^2 - 80^2. They have the correct intuition to subtract, but they get the order of operations wrong. To answer a question about why this method wouldn't work, we tried things out on a smaller example.

5 + 7 + 9 = 21

The square of the difference method yields (5 - 3)^2 = 2^2 = 4.

The difference of the squares method yields 5^2 - 2^2 = 25 - 4 = 21.

I don't think the square of difference supporters were fully satisfied, and I regret feeling pressed to move on before their confusion was fully resolved. In retrospect, I wish I would have connected it to integrals:
Or drawn a picture:
(Note that the picture also helps show why 5^2 - 3^2 is incorrect whereas the integral analogy is useless for that misconception.)

(2) We came up with a nice example for showing students the efficient method for evaluating a geometric series before developing the formula:
I think the ideal approach would be to have students evaluate a few geometric series by adding, then show them the more efficient method, then have them practice the efficient method, and then finally use the efficient method with no simplification to generate a formula. Thanks to the workshop participants who helped create this example!

Finally, here are some resources that were mentioned during the workshop: