Yesterday in my Algebra II class I went a bit off the deep end in terms of tangents. We were studying complex numbers, and the day before, I had shown them Schrodinger’s Equation (which has i in it). One of my students wanted to know what it was, so I naturally told them.
NOT.
I had this student research it and come to class and explain what he could suss out. He did, in a really funny way, and it was totes good times.
When he got to Schrod’s cat, however, his explanation fell a bit short, so I had him go back an re-research it. We’ll see how that goes.
That was DIVERSION 1.
Then when we were discussing how all our plethora of numbers fit together, we produced
and of course, of course, a student raises his hand and asks “where does infinity go?”
Because the answer to that takes a long time to properly address, and I wanted to move on without spending 40 minutes on that question, I talked about how infinity was, for us, not a number but rather an idea of unboundedness. The game of “the highest you can go, I can go higher.” And of course, I gave them a teaser about the levels of infinity… I didn’t explain it, but I talked them through the basic conclusion (that the real numbers are a different level of infinity than the integers), and hopefully blew their minds by saying that the positive integers are the same level of infinity than the negative integers.
That was DIVERSION 2.
Finally, I concluded by explaining the broad notion of fractals and how each point of a fractal is painted black or a color… and what being painted black or a color means. We did a few examples on our calculators (of evaluating points, and decided whether to color it black or a color).
And that was DIVERSION 3.
I didn’t get to start completing the square, as I had hoped. But you know, the kids will remember this more, and it will raise their general appreciation and wonderment of mathematics… more than completing the square anyhow.
Continuous Everywhere but Differentiable Nowhere 
Diversions or real interests being explored – as an adult – I choose what I learn – I did Chinese for five years and have studied sign language and am currently working through and enjoying Visual Complex Analysis by Tristam Needham. All this is good for their general education and their sense of interest.
The diagram of the real numbers and the imaginary numbers and the complex numbers could be linked to their ideas that numbers occur in a number line – then they you can start to think of the real numbers as being a thin and not very common type of number and similarly with the imaginary and all the rest of the complex space is complex numbers.
Can I say that the language is truly irritating as complex numbers are not complex and imaginary numbers aren’t imaginary any more than real numbers are real.
I *hate* the titles “real” and “imaginary” — agreed. BARF. I really try to get kids to understand that “imaginary” doesn’t mean fake. It’s not easy. I don’t always succeed.
*lol* This isn’t a diversion from math — this is math! Math the way it should be (reminds me of Vi Hart’s playful applications of graph theory in doodling). Really, math the way it actually is done by people who use it outside of schools — as a process of inquiry, testing hypotheses, and finding the joy in disproving our models by finding counterexamples (you think your number model is so good? Then where does infinity go, huh? Hah!). Way to go! Can we have a high school math credit full of this stuff?
I’m with you. How much fun would it be for both students AND teachers to have a class of these “odds and ends” sort of things that pop up in math all the time, but usually too late for many students to get interested in it?
A lot of these topics could at least be introduced to students on almost any level. Knot theory is playing with string; fractals can get started with messing with a computer program; infinity is a crazy and interesting subject on its own; etc.
“What do you do for math research?” is one of the most common questions I got in grad school and one I didn’t even understand myself until my junior year of college majoring in math. I think it would be an eye-opener to many students and show them the “beauty of the grand canyon” without our “pushing them off.”
Yeah this is some of the fun stuff. At least for us math geeks. :)
I do think students find it mind-blowing if they really understand how weird infinity is so that’s a lesson totally worth teaching.
A saying that I’ve seen in many places (but I first heard it attributed to Dan Teague): “Teach every class as if it were their last.” So far I succeed in that about 1% of the time. Sounds like you succeeded with that one.
Oh, I love Dan Teague AND that quote. I can’t live by it… but it’s nice and aspirational.
What was nice is that I sent an email asking for feedback on the class, not writing the email in such a way that I let on what I wanted to hear. All the responses I got were positive and enthusiastic about going down tangent road when the opportunity presents itself.
I wouldn’t blame the kid for getting the cat wrong—95% or more of the popular presentations I’ve seen get it all wrong anyway, claiming it has something to do with human cognition.
It does sound like you teach a proper math class, not like the one my son just dropped (details in http://gasstationwithoutpumps.wordpress.com/2011/02/02/trig-and-anal-geo/ ).
As the student does some more research on Schrodinger’s Cat, perhaps he/she may come across these clips from one of my favorite shows.
Hmm. Not sure why the first one showed up as a link, and the second as the embedded one that won’t play.
Yay for Diversion 2! There must have been a moment of perfect synchronicity in the math-teaching cosmos this week, because we this same moment on Monday, when we were working on end behavior in graphing polynomials on Monday. My Algebra 2 kids got totally excited about the idea of infinity as an utter absence of all boundedness. They wrote for ten minutes straight in their math journals on their relationships with it and ideas about it. I projected an M.C. Escher lithograph of one hand drawing another on the wall and it totally blew their minds.
– Elizabeth (aka @cheesemonkeysf on Twitter)
Am I wrong to treat infinity, as it arises, as a cardinality rather than a number? Kids seem to at least accept that, and some of them kind of get it.
Jonathan