I’m hard at work on the post I promised over a month ago on the history of algebra, which turned into a whole research project. In the meantime, a shout out:

I met Sam Coskey and Avital Oliver on a Chinatown bus from New York to Boston. All three of us were taking the day to go visit Bob and Ellen Kaplan’s math circle in Cambridge, MA. Avital and I had independently read *Out of the Labyrinth* and were excited enough about it to contact the Kaplans and ask to visit.

There I was, on the bus with two strangers, united by an interest in math education taking place outside the timeline and curricular demands of school. What was it like?

Avital is not professionally an educator but he is as intense and passionate about math education as anyone I know. He and Sam met in the math department at Rutgers. I am a little vague on the details of the story but after Sam finished his PhD they were both in NYC and started up The School of Mathematics: Avital rented an artist studio in a basement in Fort Greene, Brooklyn, they outfitted it as a (very alt/boho-feeling) classroom, and started holding free math classes on Saturday mornings for all comers. As I described before, these sessions always start with a simple, usually elementary-level question. Then they go wherever the participants take them. Sam and Avital’s roles are felt more than seen: every time I’ve been present, the conversation has magically turned into a deep exploration of important content (why do triangles with congruent angles have proportional sides? how would you write down numbers if you forgot the usual way? why do parallelograms with equal base and height have equal area? why is multiplication associative?), without being visibly directed almost at all.

Anyway, on the bus, after hearing what they were up to, I found myself in an intense conversation with Avital about the nature of mathematics and the purpose of math education. I had spent the prior year and a half rediscovering my own relationship with math by working page-by-page and problem-by-problem through Michael Artin’s *Algebra* textbook (which is awesome, btw). I was having my mind completely blown by algebra and was really excited about it. I had also started to be certain that I wanted my professional life to be concerned with giving students the opportunity to have an experience akin to the experience I was having. I saw that experience as being defined by rigorous but creative play in an imaginative wonderland (the mathematical universe). I described this as my goal. Enticing students into this wonderland by setting them up to see some really intriguing patterns, images, and connections, was my plan to accomplish it.

Avital said, “that’s not mathematics.” What was it then? “Some sort of pied-piper, luring-them-in-with-candy typed thing.” (I’m paraphrasing since I can’t remember exactly, but this is the idea.) To Avital, it’s not mathematics unless you’re trying to answer a question you already had. You’re not *teaching* mathematics unless you’re working with people to build tools to find answers they already wanted. (I’m sure you’ll recognize the spirit of this point of view in the work of Dan Meyer and Shawn Cornally, but Avital is the most uncompromising purveyor of it I’ve come across.)

It’s not like I was converted on the spot – oh, never mind, yeah, I’ll forget my whole plan and start over. But it’s been deeply valuable to me over and over again to put to myself “Avital’s challenge”: say an idea is beautiful and I want you to see this. What question could we entertain that would make this idea necessary? And why would you want to answer this question? How could I get you interested in this question? And, if the notion of you being interested feels at all strained, maybe this is not the idea I should be leading you to.

I think with that conversation Avital has won my personal “most influential conversation with a stranger on a bus” prize.

Avital moved to Israel last summer, and Sam took over leading the School of Math, which he does really beautifully. Sam is leaving NYC as well at the end of this academic year. The future of the School is unclear. So, if you live in NYC, you might want to check it out before the end of May. Classes are on Saturdays and start at 11. This coming saturday, Avital is briefly back in town from Israel, so this might be a good one to be there for.

Just started reading recently, and am enjoying it. This post, though… well, let’s say it challenged me.

Is art “necessary”? What about music? What problems do they solve?

I love math primarily for its infinite, intricate, often surprising beauty (and I’m guessing you feel similarly). So this idea of motivation–while of course it is important, and I eat up Dan Meyer’s stuff like candy–always dismays me a little. Am I allowed to try to show my students some of that beauty? And if we say that appreciation of the aesthetic aspects of math might (hopefully) come along once we’ve “hooked” them with the well-manufactured “necessity” of various problems and concepts, isn’t that subject to Avital’s “pied piper” criticism?

Keep it up, I’m really loving reading all these math-ed blogs and thinking through this stuff.

I love this comment because it reminds me of how I felt when I had the conversation with Avital in the first place.

I loved his passion and I respected his vehemence but at the time I didn’t agree. I thought (like you are thinking) that granting a central place to the role of seeking an answer to a question somebody already has meant I had to give up the idea that the aesthetics of mathematics are a legitimate driving force for math education. I felt sure (still do) that these aesthetics are an important part of what I’m trying to teach.

I still don’t know if I “agree” – I mean I would never be as uncompromising on this point as Avital is. But this conversation has shaped my thinking in ways I would never have foreseen at the time. In particular, let’s zero in on the case where the aesthetics are what we’re going for. Let’s temporarily forget all other purposes, and say that the point is for students to see and appreciate the beauty of a particular mathematical idea.

How is this goal best accomplished?What I’ve found myself noticing in the last year (since the conversation with Avital on the bus) is that the times I’ve felt that I’ve accomplished this goal most satisfyingly are all occasions on which my students hit upon (or were shown) the idea

while they were trying to answer a question that was separate from the idea, that the idea in question helped elucidate.In contrast, on occasions where I’ve led students to a pattern or concept outside of the context of them trying to answer a separate question, there has usually been something unsatisfying-feeling about the moment of encounter with the idea. For example, I used to start off my Algebra I class with a lesson on the Pascal’s Triangle. I would show the kids how it was constructed, make them do some computations, and then tell them to look for patterns, maybe with a few hints. What killed me is that

they would find them, and then not be impressed. Here I am, over here, thinking it’sso awesomethat the sums of the rows are powers of 2, that the pattern of evens and odds makes this awesome visual pattern (the sierpinski gasket), etc., and the kids think these things are, at best, marginally cool for about 45 seconds.All this is just to say that I don’t take the insight I got out of that conversation to invalidate sharing the aesthetics of math with students. I think it’s really important for teachers to have their classes be impassioned by what impassions them personally, and if for you (as for me) mathematical aesthetics are key to that, then that needs to be at the heart of your teaching. What it does is give a (totally helpful, in my experience) heuristic that can be used to make mathematical encounters more powerful. (Including encounters with the aesthetics of math.)

One more (related, hopefully helpful) thing: I don’t treat my memory of that bus conversation as a claim about mathematics and math education to either agree or disagree with. I treat it as one among several standards against which I measure my pedagogical ideas, when I want to. I’ve found many times that it has made them better. Other times, I didn’t feel like engaging with it and that didn’t mean the lessons weren’t good too.

I’m so sad! After your last post, I read everything on their site, and wished I could visit. I hope it somehow continues…

Somewhere along the way I started mentally dividing math up into applied math and theoretical (or pure) math. Applied math is using math to answer a question, includes the math used in engineering, science, and business. I would guess that Avital considers “real mathematics” to be applied math. Theoretical math doesn’t have an application; it’s done for its own sake. When Matt E calls math “an art”, I think he’s referring to theoretical math.

I think the debate over which of these is “real math” is missing the point. They are both math, in much the same writing a short story and writing a lab report are both writing. Furthermore, the same idea can reach into both sides of mathematics. For example, Euler’s equation (e^(i*pi)=-1) is beautiful in a theoretical mathematics way, but the same math is used constantly in engineering any alternating current (AC) circuit.

Some students (and some teachers) are going to prefer one to the other, but they are both valuable in different ways. Can’t we teach our students about both the usefulness and beauty of math? Wouldn’t that be giving them a better sense of what math is?

There’s a miscommunication here. There was no discussion about pure vs. applied math. There’s an important (though subtle) distinction that Avital made in the original conversation, that I now see I should have included in my post, between math being “motivated” vs. “applied.” Avital has no beef with pure math.

What he favors is

motivatedideas vs. ideas that are not motivated. The motivation doesn’t need to come from outside math – it just needs to come from outside whatever the idea itself is. Math can be its own motivation and yet each individual idea may be generated by the need to answer a question about some other idea.An example I recall Avital using in the original conversation is the idea of negative numbers. Mathematicians were treating negative numbers dubiously as late as the 16th, possibly the 17th century. According to Avital, what made them finally gain wide and full acceptance in the mathematical community was that they were seen to unify all the different cases in the general solution to quadratic, cubic, etc. equations. For example, from antiquity to the 16th century, quadratics were seen as 3 distinct types:

x^2 + ax = N

x^2 + N = ax

x^2 = ax + N

Fully admitting negatives to the table allowed all three cases to be unified into one:

x^2 + ax + N = 0

and handled with a single formula.

The point is, negative numbers gave new insight into questions about equations that people were already asking (totally independent of negatives), and this is what led them to be fully accepted.

By the way, I (Ben, not Avital) totally agree that different folks will relate to different parts of math and in the final analysis there’s no need to be dogmatic about any one view of the practice. That said, I’ve found Avital’s point of view very useful to think about while lesson planning. Making myself try to answer to it makes my lessons better.

(Yo, Avital, if you’re reading this, I’m waiting for you to comment. Here I am, representing your point of view to people I don’t know, on the basis of a single conversation we had 15 months ago! Feel free to correct me completely.)

That’s fascinating. I’m not sure I agree with the big idea here… but it’s certainly a Big Idea, and it is a conversation well worth having.

And even if I don’t end up agreeing, I don’t know that I disagree either… it sounds like it leads to an engaging approach to getting others to do mathematics, and that’s certainly worthwhile.

Jonathan

Yes, exactly. This is just how it’s played out for me – I don’t exactly agree, I don’t disagree, I just use this idea to help me plan teaching experiences and it makes them better…