Joyous Math

Michael Pershan recently made this post in response to criticism he received on Twitter. The criticism:

Pershan’s original post talked about microskills, and his follow up made a distinction between “abstract joy” and joy situated in a classroom context.  Pershan puts forth the idea that joy can come from feeling success and confidence in the math classroom, including feeling success over mastering a microskill.

For me, this criticism and his response raise questions I’ve had with the concept of “joyous math”.  Pershan quoted Francis Su, and so I will too:

So if you asked me: why do mathematics?  I would say: mathematics helps people flourish.  Mathematics is for human flourishing.

The idea of bringing joy and a sense of flourishing to children through mathematics is not new.  James Tanton strongly advocates for “joyous” approaches to math education.  His homepage has this stated goal:

The goal of this site is to demonstrate the beauty of mathematics, its wonder and its intellectual playfulness, and to work towards bringing true joy into mathematics learning and mathematics doing for one and all.

The concept of “joy” (better yet, the lack thereof) is sometimes used to criticize the current state of math education.  Possibly the best known criticism of math education comes from Lockhart’s Lament:

In place of meaningful problems, which might lead to a synthesis of diverse ideas, to uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in mathematics, we have instead joyless and redundant exercises, specific to the technique under discussion, and so disconnected from each other and from mathematics as a whole that neither the students nor their teacher have the foggiest idea how or why such a thing might have come up in the first place.

But he’s not alone, to varying degrees.  Here’s a recent book by Alfred Posamentier, who was a teacher in the Bronx and later the dean of math ed at City College in NYC:

Here’s a snippet from (one of) Sunil Singh’s pieces against the current state of math education:

My own personal lament is that the holy trinity of theories — number, graph, and game — are nowhere to be seen in most K to 12 math curricula. Prime numbers are merely a definition. Mathematical mavericks like Martin Gardner, John Conway, Ivan Moscovich, etc. are unknowns. Algebra pops out of a can in high school — with a sequel to boot. There is no organic and seamless bridge between arithmetic and algebra. Teaching algebra as an appendage to teenagers as opposed to teaching it as a circulatory system earlier on is one of the clearest indicators of the mismanagement of mathematics by education. It’s like going to hardware store. Aisle 3, top shelf: nails, washers, and Algebra I. 

I could go on.  But the main issue, as I see it, is an utter lack of joy in the math classroom.  The question then is “What does joyous math look like?”  To many of the mathematicians who advocate for joyous math, it usually takes the form of an elegant or unexpected idea, connection, or result.  Singh cites videos such as these as representing a form of “joyous” math:

I don’t disagree that mathematical ideas like the square-sum problem are mind blowing.  Every time I watch a Numberphile video, I think to myself, “Wow, that’s amazing. I want to show this to my students”.  I’ve gone through James Tanton’s exploding dots videos and read his book Mathematics Galore!  I have the fortune of attending amazing professional development seminars through Math for America (shameless plug!).  I try as much as possible to learn new math, new connections, and new ways of thinking about concepts I thought I already knew.

But here’s my question: will math that makes me feel joy also necessarily make my students feel the same way?  I’ve got 20+ years on my students.  I (pretty much) already know the material that they have yet to learn.  So when I attend a PD where we do familiar math in a cool new way (e.g. clothesline math), I’m stoked because I find myself saying “it makes so much sense now” and “that’s an amazing connection that I had never seen before!”  Moreso, when we’re learning math that is new to me, such as Patrick Honner’s favorite theorem (Varignon’s theorem), I am equally stoked because “this is so elegant compared to what I’ve seen” and “this is so refreshing from the stuff I’m so familiar with”.

But will my 6th grade students, who are fresh to a lot of the curriculum I teach, feel the same kind of joy as I do when I introduce them to clothesline math, exploding dots, visual patterns, and Varignon’s theorem?

Frankly, I’m not sure.  I’ve done some math with them which I consider beautiful, and they don’t seem to be into it.  I thought creating personalized polynomials in Mathematica using Lagrange’s interpolation formula was going to be a hit – I thought wrong.  Or how adding consecutive odd integers starting from 1 always gave a square number because…

img(no excitement whatsoever)

On the other hand, we’ve done math which I don’t consider particularly exciting but which they found really interesting, like using the ladder method to find the gcf of two numbers.

Politicians get a lot of slack for thinking they know about education simply because they’ve been in school.  I worry that teachers fall into the same trap of thinking they know about children simply because they were kids once.  There has been a lot of work done on how children think.  I would be interested in any work that has been done to systematically understand what makes children feel fulfilled.

If making children feel “joy” and “fulfillment” is part of our end game, it can’t come merely from what we adults consider joyous and fulfilling.  I sometimes find myself rolling my eyes at what I, as a 13-year old, once thought made my life complete.  But I don’t regret what once fulfilled me, and I know it’s made me who I am today.  The mathematics we have children do must honor their sense of joy and fulfillment and not merely ours.

That brings me to how I started this post – Pershan’s response to criticism that his way of teaching microskills lacks joy.  I agree with his idea that success can equal joy.  But I also wonder if the question of joy is so context-dependent that it really boils down to his students.  If they feel fulfilled in working through and conquering the micro-challenges he presents to them,  and if they are in fact growing in their mathematical maturity, then perhaps he’s creating joyous math for them and that’s enough.