## Infinite Geometric Series Through Philosophical Confusion

So, I stole a page from Benjamin Baxter’s playbook and taught a lesson with the aid of comments from the Internet.

Specifically, in Pre-Calculus I was teaching the formula for an infinite geometric sum:
$S=\frac{a_1}{1-r}$
so $a_1$ is the first term and r is the number a term is multiplied by to get the next term.

For example, take .9999…, which is the same thing as
$\frac{9}{10} + \frac {9}{100} + \frac {9}{1000} + ...$

So $a_1=\frac{9}{10}$ and $r=\frac{1}{10}$. Therefore .999… is
$\frac{\frac{9}{10}}{1-\frac{1}{10}}=1$

I looked at the posts from the .999…=1 saga at Polymathematics (check the sidebar) and picked choice excerpts from the comments, i.e.

My first reaction is that I am very worried if you are a maths teacher.

We read them over in class and discussed why the reaction to this problem would be so strong, and why so many people would doubt the mathematics.

One conclusion was that there’s an almost philosophical confusion here; an unwillingness to accept the mathematician’s version of infinity. Last semester I had my Pre-Calculus students write an essay on “do you think infinity exists in real life?” because I wanted them to wrestle the strangeness head-on. The abstract leap to the infinite and infinitesimals is really the crux of Calculus.

### 10 Responses

1. Excellent use of the technique.

Is this a Trig/Pre-Calc class, or just a plain, ol’ Pre-Calc class? I don’t remember much time for philosophy in my high school maths.

2. Our students theoretically take Trig the previous semester, but in practice they only get a quarter worth, so I have to make up ground.

I keep my teaching efficient enough that I can digress when I want to.

3. Thought so. When I went through not too long ago, the courses were combined into a single year.

Calculus was a whole lot more fun, I must say.

4. How did they do with the essays? Was the idea a success or a failure. I’m going to be out of my classroom on paternity leave and need some good projects for them to work. I’d be interested to hear how you set everything up and how you went about evaluating them.

5. The essays went quite well.

I had the students first read the Borges short story “The Book of Sand”. (For my Honors students I cut the story up into 16 parts and we read it out loud while the students were required to keep track of words they didn’t know; for my Pre-Calculus I simply had them read it themselves.) We used that story (which is an exploration of sorts of the ramifications of infinity in real life) as a springboard into our discussion. I used a “talking stick” method (with a thrown ball) where each student provided a reason to be for or against.

I also provided some science terms that the students might not know that could help with research (for example, I wouldn’t expect them to come up with entropy on their own). I did this informally and I think if I do this next year I’ll have a printed list of pointers.

For evaluating I used a rubric our math department has, but it’s not really much different from a regular English department rubric. I did give a heavier weight to the content, and specifically counting the number of arguments provided and their originality of thought.

6. Thanks for the idea! Keep them coming.

7. I cannot BELIEVE I didn’t notice this post earlier…imagine: my blog used as the basis for a class discussion.

Very cool…thanks for the plug.

By the way, a couple of months after you gave the assignment I froze the entries to not allow any more comments; after 2 years, I was still getting them, and none of them provided anything really new.

I hope you can that post or some of my others again someday!

8. Thank *you* for staring the madness in the face. I’d likely have run screaming early on.

It was a popular lesson, I’ll likely repeat it this year.

9. […] source of infinite Internet arguments can also make a source of productive classroom struggle. (I have written about a lesson that included the Internet arguments as part of the […]

10. […] First up in this week’s carnival, we have a practical classroom application involving blogs and the infinite geometric series at NumberWarrior. […]