sitting in a courtyard at my alma mater and wanted to jot down a few thoughts. I logged into BetterLesson; they have a nice set-up for uploading and sharing curriculum. I recognized a few names from the few blogs I follow. The site is not incredibly well populated, especially when it comes to people giving critical feedback.

question, then: are there any examples out there of lessons posted that received some good, spontaneous feedback?

I’m also realizing that I have basically two free weeks to work on curriculum and what-have-you before the vacation and move back to New York. So two weeks of dreamy, relaxed planning before the practical demands kick in.

Here’s what I’m thinking/asking myself about this year.

What are the most important mathematical outcomes for this year?

–At first-order of approximation, I would answer that I want all my students to feel like they can solve challenging math problems. This will be because they have practice both with solving familiar and unfamiliar problems, and also that they have some good skills to lean back on. In a sense, I want my students to feel like they’re learning what all high-schoolers should (Algebra skills, graphing, etc) but in a fun and creative way.

What mathematical content do I think is simply THE COOLEST?

–examining infinite quantities, both large and small. Think Zeno’s Paradox. [sequences and series]

–polynomials in generality. How they behave infinitely, how they can be caught and classified (and given proper Latin names?). How they behave near their roots. What this whole “behaving” thing is about.

–quadratics and complex numbers. [thinking about the motivation for imaginary numbers in the context of a motivation for 0, negative numbers, fractions, irrational numbers…. can we just keep going on inventing numbers?]

–exponential growth and its super-polynomiality. using logarithms. (wait do I actually think logs are cool? yes yes.)

–uh, everything else. probability, counting, statistics (gasp), geometric thinking [what is the essential geometry to learn?]

–update: notice that I forgot trig. “Forgot.” There’s a lot of cool stuff in trig, but it’s a beast when it comes right down to it.

–this may be the time to mention something about functions, explicitly.

Ok, looking at a few things, I want to write down a rough order of units/ideas.

Number systems. History, bases. [want a simultaneous problem solving focus…]

Invention of zero, negatives, fractions.

Invention of irrational numbers. Motivation thereof. We’re learning how to reduce radicals and why irrational numbers exist.

Complex numbers as a thing by themselves. Operations on numbers. Meanwhile, there should be a conversation on algebra skills, continuing problem solving.

Uh… quadratics? Don’t overload here by trying to do all of polynomials. Keep it understandable within the framework of having learned about complex numbers.

That might do it for the first semester. There needs to be a project on all this—I’ll keep thinking.

Keep thinking.

BetterLesson seems like the perfect vehicle for sharing but it just doesn’t seem to have the buy-in. The twitter world seems so much more active, but doesn’t offer the repository that BetterLesson could. I have an account but I don’t really check it actively enough.

I’ve always thought that the neatest (and most applicable) thing about complex numbers was their geometry: the fact that multiplication by a complex number models rotation. (Very different from what real numbers can do!) It’s cool in some sense to say you have “solved” a quadratic by writing down a formal expression with an “i” in it, but students rightly don’t get excited by it, because we typically don’t actually *do* anything with the solution. But once one understands the geometry of complex multiplication, one can use it to model rotation, connect to sine and cosine, change hard geometry problems into algebra (although that can make it harder for some students), describe alternating electrical currents, do Fourier series and transforms… now basically none of this is in the standard curriculum, but I wish some of it were, as it connects to how 99% of the world actually uses complex numbers.

Thanks for the comment, and wow, that strikes me as ambitious. But I admit that I’m intrigued. Have you done this in a class, or known anyone who has? I would need applications that don’t require calculus.