The other day in Algebra 1, I was talking about exponents and their connection to orders of magnitude and fractal structure, which I see as the fundamental concept here. Consider the powers of 3. Naturally, 3 is a triangle. Then 9 is simply a triangle of triangles – a “meta-triangle.” By the same thinking, 27 is a triangle of triangles of triangles. If you’re thinking like we were, maybe you’d call 27 a triangle of meta-triangles. Or is it a meta-triangle of triangles? Amazingly, it’s both!!!

(*Is it fair to call 27 a meta-meta-triangle? Math up these comments.)*

Anyhow, my students made the connection to Sierpinski’s triangle, and the whole “meta” conversation brought up Inception, video feedback, and other cultural references with which they are familiar, so the whole thing really connected to things they know. The exponent tells you how many levels deep the “inception” goes. Of course I talked about how 3^4=1x3x3x3x3, but these patterns are too rich to leave boiled down into something that simple.I hope you don’t think that’s too abstract or confusing for 8th graders. These ideas *are* somewhat mind-boggling or perplexing, as Dan Meyer might say. They’re not impossibly difficult. They’re just weirdly fascinating and challenging. Ideas like that (and damn good problems) draw learners into the math world. The point here is that we had a chance to struggle with instantiated concepts, and by wrestling with things like meta-meta-triangles and by arguing about them, we did some math together. (Opinion: A mathematical conversation *is* math. Agree?)

If I tried to wrap the material up into a perfect little package that could be quickly delivered and easily understood or memorized, then there wouldn’t be any mathematics left at all.

* * *

The base ten number system is built on exactly these ideas, and scientific notation exploits that beautifully, using the exponent to indicate the level of depth. This was where I hoped to lead the class, and I had some (slightly dry) worksheets at the ready, so I handed them out.

While they were working, I pulled up an AMAZING visualization called Scale of the Universe 2. (If you’ve never seen this before, I wouldn’t blame you if you stopped reading and just played with that for 40 minutes.) I was happy to report that it was designed and programmed by some high school kids in California, but I was even happier that the room started to fill with energy. I narrated our little viewing a bit, but I could hardly get a word in between their questions! *Their* questions. A marathon is *that* big? How many central parks is Angel Falls? What’s “total human height?”

I answered a few, which was almost too much fun, and then we started researching. We pulled up wikipedia and google and found the most amazing information. Even I learned an incredible amount, and for the rest of the period we were tapped into the magic of curiosity and learning in the information age. Sometimes we would do computations in our head, sometimes on the board, and sometimes we would just pull up WolframAlpha and had the computer solve it for us. Arithmetic, paper, and pencil have their place, no doubt, but using a great tool isn’t a sin. It’s a virtue. Especially today, with the most incredible tools readily available.

I didn’t care that we stopped working on the scientific notation problems. We were using scientific notation right then to understand our world. It was OK that the sheets didn’t get filled in. We could do that later. *This *was going to leave a mark. I went next door, grabbed 10 netbooks, and put them in their hands. “Keep going. Whatever questions you have, answer them.”

I did it again with my 10th graders and again with my 5th graders. Amazingly, the Algebra 2 kids would ask me questions and not even realize they were staring at the screen of the most powerful information tool ever available. “Google that!” All of a sudden this was a lesson in using the internet. Whatever you find, whatever questions you answer, record it. Tomorrow we’ll share. “Where? You could use Google Docs…”

If you believe that school is where we equip students for the world, is there any doubt they need these skills and tools in their kit?

* * *

I’m telling this story for three reasons:

i) That’s cool math up there. Exponents and fractals are amazing, and they actually connect to my recent post, quite possibly the coolest problem I’ve ever come up with.

ii) To show how computers can help us meet the enormous demands of true interest. 100 years ago, when our school system was designed, I think this was literally impossible. Schools today, however, have an amazing opportunity if they can shift their mindset.

iii) I love when a class spins on a dime in the most unpredictable ways. It tells my students are awake. It tells me they are conscious and active, processing and guiding our work. I love it. I think of the transfer of power (not just information and technique) as a central goal of my teaching, so when stuff like Scale of the Universe 2 falls into my lap, my job can get very easy.

I’m so thankful to be at a school where we can follow our whims and simply study whatever is most fascinating.

Awesome. Any way to extend this to negative exponents?

They’re already there if you think in reverse. Looking at 9 for example, one meta-triangle is 3^2 points. One point is 3^-2 meta-triangles.

“A triangle of meta-triangles. Or is it a meta-triangle of triangles? Amazingly, it’s both!”

Well, thats t(m(t)) = m(t(t))

Oh wait, I’m using t as both a function and a base case argument. Try:

t(m(t(1))) = m(t(t(1)))

Which would seem to imply that t and m are the same function, and that all that matters is how deep you nest them, and viola lambda calculus.

And that was before I read the rest of the post. I managed to only lose 10 minutes to the scale of the universe, but boy am I jealous of your kids.

I like your Algebraic, function argument!

Although, t(m(t(1)))=m(t(t(1))) only implies that t and m commute on the value of t(1), right? They are not necessarily the same function. In fact, it doesn’t feel right to me to say they both are functions at all. I mean what does t(1) mean? What’s t(2)? What are the allowed inputs for m? Because you have it taking in t(1), but also t(t(1)), so does t output numbers?

Oh math! You’re so silly sometimes.

I have my own thoughts on the meta-meta-triangle question, but I’m hoping someone else will post, so I’m waiting to share.

Thanks soooo much for commenting, Max.

Mind, blown.

I am going to go out on a limb here, but I think this is on of the greatest posts I have read about utter engagement, starting with pure math.

“When are we ever going to use meta-triangles in the real world?!” Not even a grain of thought in your students mind at the time, but very simply could have become one had you let them work on worksheets any longer. Instead you gave them the chance to see where the abstract philosophical leads into the real. I am in awe, Thanks for this.

Thanks so much for the mega compliment!

I don’t want to make it sound too perfect, because it was still a day of teaching, like every other. It had hiccups and pauses and moments where I didn’t know if I was doing the right thing. But it all feels right, in terms of what I think makes for good teaching.

Thanks, Timon.

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I think this is brilliant, Paul! Thanks for sharing this knockout lesson that took off on wondrous tangents, I also really appreciate your mentioning that you did this with both your 10th and 5th graders. Now, I think THAT is a mark of a great math lesson, speaks to its adaptability, flexibility, freedom for kids to think and explore!!!!!!!!

Like you, I’m also very “thankful to be at a school where we can follow our whims and simply study whatever is most fascinating.” I couldn’t ask for a better principal and a better superintendent (one school K-8 district), so I don’t take their trust and support in me for granted. We try to do what is best for our students, and when admin is on board with this, it really helps.

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