My Number Theory Class :)

What do I want to teach my students next year?

I will have the most amazing class. About 8 - 10 kids, 7th and 8th graders, who are all avid math learners, who have excelled beyond their years. They ask good questions, constantly draw connections, love to tackle new problems, they are willing to work through frustration when problems take a long time to solve, and they tend to justify their reasoning. Many know how to write proofs. All have completed Algebra I, several have completed Algebra II and Precalculus. I have worked with most of them before, and I like them, and I think they like me. Yeah, insanely lucky.

So, what is there to teach them? Content-wise, plenty. The whole world of math is at their feet. My intension is to spend most of the year on Number Theory. Mostly because Number Theory is awesome, fundamental, and deep, and also because the textbook I have is awesome and readable and funny and engaging.

But in terms of enduring understandings, and in terms of real take-away lessons about academics and about math: what do I want to teach? I want to have a few ideas to start the year with, so I can set a direction for the year.

What about the art of questioning? Have they ever looked at a situation in math class and come up with their own questions? Probably not much. I've tried it a bit, and only gotten so far with them, and then let it go. That's something that mathematicians do that they have not yet internalized.

The problem is, I'm not used to teaching this skill. It is SO much easier to pose a question than to just present a situation and leave the questions to the kids. My teachers did not model this for me! And I left grad school at just about the point when I'd be expected to come up with my own questions.  I can do it (I mean, that's why I write such challenging tests and fun worksheets!), but to teach questioning, that would be new.

Hmm, what if I start with the sequence that came up on Twitter today ...

1, 11, 21, 1211, 111221, ...

It's a really annoying pattern, especially if you like math. You can run yourself ragged trying to find an answer, but (SPOILER ALERT) you just read it aloud: one 1; two 1's; one 2, one 1; etc. It's not really math. Blah.

But you can ask math questions about it. What is the largest digit that appears? Any number without a 1? What proportion of the digits are 1's in the long run? Does the pattern ever repeat? What about this variation:

1, 11, 21, 1112, 3112, 211213, 312213, ...

How are they different? How do the properties of the two sequences compare?

The point is, I can start with this annoying non-math sequence, and then ask the students to ask math questions about it. That might be a great place to start this journey together. We need to take time to ask more questions as we go, I need to remember to give them room to ask questions. And I need to give them situations that are worth asking questions about. Games. Stories. Pictures.

Okay, one goal set. What is next?

I can teach them revision. That is a skill that they will need for life, how to go back and make your work better, more approachable to an outsider, and more aesthetically pleasing to another mathematician. It's something you should do with all writing, and math at this level is writing. One of my favorite things about grad school was the way that people would not be satisfied with answering a question - they'd want a GOOD answer. A beautiful answer. I think these kids already have the sensibility - they will argue over whose method is the better one, which is the essence here. But I want to connect it to the way they write up their work. To pick a problem to go back and rewrite, and write it more clearly the second time. How do you get kids to value that work? I should talk with the English teachers about that, they must do it all the time.


Okay, two goals, not bad for one blog entry! Here I am asking the big questions. I'll come back and revise my thinking about it later in the summer.

Cheers!