**Update 12-9-15 **Since pressing publish, I’ve thought of much better titles for this blog. Take your pick:

## The Rhythm Method

or

## Mathematicus Interruptus

—

I returned last night from a thought-provoking PCMI weekend workshop. A+ professional development, friends. Seriously, if you have a chance, go. Huge thanks to Tina Cardone for organizing it.

Each day, we alternated doing lots of math with facilitator Brian Hopkins and reflecting on our practice with facilitators Cal Armstrong and Jennifer Outz. Plus we socialized, shared ideas, and learned from PCMI alumni. It was a great mix and there are many things I could blog about, but I want to focus on some of Brian’s choices that I found surprising in the best way possible. He really has me reflecting on my practice.

**Spoiler alert:** I’m going to introduce some math problems – problems we thought about with other teachers for long stretches of time. After each problem, I’m going to have to give away some of the mathematics to make the point I want to make. If you want a chance to think through the math yourself, pause after each picture. Go play. Then come back. If you keep reading, you’ve been warned. You’re missing out on some of the fun.

**Assurance: **I’m going to be using some math vocabulary, ideas, and representations that were new to me and I spent many hours exploring this weekend. They’re still tentative for me. If you don’t take the time to figure out the math and you don’t know it already, that stuff won’t make sense as you read. Don’t worry about it for these purposes. I’m not trying to recreate PCMI here. I’m trying to make a larger point that you can totally get even if you skim over the math. So skim away, especially in this first bit. Resume close reading after the circular pennies picture.

The first morning, we started with a penny game. (Thanks, Heather, for tweeting these out!)

Find a partner and 9 pennies and play along at home #pcmiTLP pic.twitter.com/eYyUBDMDYg

— Heather Kohn (@heather_kohn) December 5, 2015

We tested what would happen if we had different numbers of pennies or were allowed more moves. We found that multiples of 3 were very important if we were allowed to move 1 or 2 pennies. Multiples of 4 were very important if we were allowed to move 1, 2, or 3 pennies. Brian encouraged us to think about these ideas in terms of modular arithmetic, which we’d been told was going to be a focus of this weekend. My hunch is they chose this topic knowing it was outside of most of our comfort zones. Perfect. It certainly was new to me. I went in having no idea what 2mod3 meant. By the end of the morning, I think lots of us were becoming more comfortable with mod notation and thinking. Brian closed the session with a good laugh over Survivor’s 21 Flags game.

After a reflection session and lunch, Brian passed out this page of primitive Pythagorean Triples. He asked what we noticed and wondered, and encouraged us to play with mods the way we had in the morning.

Make some conjectures on primitive Pythagorean triples: #pcmiTLP pic.twitter.com/4R086kTB5V

— Heather Kohn (@heather_kohn) December 5, 2015

So, what do you notice and wonder? What patterns do you see? What conjectures can you make?

We had all kinds of observations about the ones that were one apart or two apart, different ways to generate subclasses of triples, and what was going on with the evens and odds. Super fun. After the group share out, our group started looking at other group’s conjectures that each triple had a multiple of 3, a multiple of 4, and a multiple of 5. This got us going again with mods, and the squares of mods.

When we stopped, my group was trying out mod8 to see if that helped. I’m still not sure why, but it was fun.

When we arrived the second morning, Brian had a new challenge for us:

Find a partner, 13 pennies, and play along at alone. It's the penny game amped up! #pcmiTLP pic.twitter.com/ERG7XB4U0z

— Heather Kohn (@heather_kohn) December 6, 2015

My partner and I tried and tried to explain this game using mods.

We failed.

The pair next to us was exploring a different technique. They found that the player who forced their opponent to face a symmetric situation won every time, as long as they kept resetting the symmetry. I really encourage you to try this one. It’s lovely and satisfying.

Brian pulled us together and asked what we found. Consensus was that mods were unhelpful, and that symmetry was the more powerful idea here.

This was the most exciting moment of the weekend for me, pedagogically. The way I thought about it was that Brian disrupted the predictable, pitter-pat routine of math class. Up until this moment, all signs pointed toward mods. I assumed each new activity would build on what we’d done before in this familiar story arc. We’d deepen, add nuance, try new aspects and applications. I did not expect to have Pennies II be totally different than Pennies I, and not involve the handy new tool we’d just learned.

I started scribbling in my notes. When do we ever do this? When do we teach a new tool and then introduce a similar task where this tool is *not* helpful? When do we teach kids the limits of the tool right from the beginning?

What I see in schools is we cue kids to know what tool to use. If we’re two weeks into a unit on fractions and we give them a story problem, the kids figure fractions are involved. If the name of the chapter is “Multiplying Two-Digit Numbers” and it’s written on the bottom of the worksheet, the kids are going to assume they should multiply some 2-digit numbers. If we’ve written an objective about linear equations on the board, kids figure the answer is going to involve linear equations. If my new tool is the hammer that divides fractions, I’m going to use that hammer until my teacher tells me it’s time to switch hammers.

At PCMI, Brian disrupted that process for me. He gave us a new tool, and then gave us a problem where it didn’t apply. Blew my mind. Next up, though, was Pennies III, which he referred to as 11 plus 1, and later told us was called Kayles. It has the same rules as Pennies II, but the pennies start out in this arrangement.

At first, I thought, “Oh, we’re back in the rhythm. I bet this game will give us some amazing synthesis of mods and symmetry. I get it. I see his plan.” Because that’s how math classes normally work, right? I’ve cracked that code.

My table worked and worked and worked at this problem and couldn’t come up with any good rules using mods or symmetry, or anything else for that matter. When we got back together, Brian told us this problem has been around forever and wasn’t analyzed well until computers were invented. It turns out there are no elegant solutions. It’s just a messy, case-by-case thing. There were some groans in the room. Not from me. I was delighted because he’d disrupted the rhythm again. He’d surprised me.

We headed downstairs to set up for the Josephus problem. The rules he played in this version were that Josephus was going to count by 3s and kill whomever he landed on. He was also going to be in the circle. So where should he stand? Which position will be the last one standing?

#pcmiTLP gathers around for the Josephus problem pic.twitter.com/7KKlL93GxU

— Cal Armstrong (@sig225) December 6, 2015

Brian’s shorthand for this problem was “Duck, Duck, Die.” He began killing us off, round and round. With 29 people, player 26 survived. With 20 people, player 13 survived. What was going on here? I heard participants say, “What does this have to do with the pennies?”

I survived the first round of Duck, Duck, Die at #pcmiTLP pic.twitter.com/4T02Bv3FbH

— Jennifer Fairbanks (@HHSmath) December 6, 2015

Over lunch, participants tried out different numbers and Brian synthesized the data.

What do you notice? What do you wonder?

We had lots of questions and observations. Fun. But then we got to the meaty question of can you predict the solution for *any* number? What do you think the answer was? What do you think I thought the answer was?

If he went by typical math class rhythm, the answer would have been some marvelous synthesis of all we’d done. It was the afternoon on the last day. So could he tie together modular arithmetic and symmetry and something wise from Kayles and Josephus? Would he wrap it up in a neat, satisfying bow?

By this point, I had come to expect surprises from Brian, so I really didn’t know what was going to happen. It turns out this problem has to be solved recursively. He had a clean, recursive approach to do it, but people were unsatisfied and wanted a closed formula. He gave us this “big ugly thing.”

I don’t know what it means either. I do know we all cracked up when he revealed the constant. It turns out you have to solve the problem recursively to use the big ugly thing anyway, because that’s how you can calculate κ to enough digits.

Brian talked to us about how not everything in mathematics is solvable in a nice, neat formula. Sometimes this is how it works out.

I can’t tell you how satisfying I found it that there wasn’t some “satisfying” synthesis in the end. I mean, I love making connections. I love thinking relationally. But not everything connects nicely. In math class, we often create an artificial story arc where one thing leads to the next and then the next in this nice flow. We leave out the part of the story where mathematicians struggled for hundreds of years between these ideas we’ve connected in 45 minutes. We leave out the part of the story where mathematicians aren’t told what tool to use when they attempt a problem. They might try this or that. Maybe mods are helpful? Nope. Symmetry? Nope. Should I try to graph something here? Maybe that will be illuminating?

What I’m thinking about most, though, is that we teach in this familiar rhythm – this unit is factoring and every question will be solved using factoring – almost all year long. And then we’re frustrated when kids don’t “transfer” what we’ve taught them in novel situations. “We covered that!” we yell, when we see they didn’t get the factoring question right on the big state test because they didn’t recognize it as a situation where factoring would be helpful. We shouldn’t be at all surprised. We’ve been cuing them all along. The tool of the week is tables, or skip counting, or measurement. Attention everyone, the new tool of the week is negative numbers, or symmetry, or finding the intercepts. Without the cuing, where are they?

Some of this is for good reasons. It’s important to dig into ideas with depth and connect them. I’m not saying to jump around willy nilly. But I wonder if we can learn something from Brian? Next time you teach a tool, how about giving students a chance to figure out when it *doesn’t *help as soon as they’ve started figuring out when it *does*? That would give them a fighting chance of deciding when it might be helpful in novel situations. Brian’s way to help us think about the usefulness and limitations of our new gadget was to give us seemingly related problems that were actually quite different, mathematically.

I’d love to put our heads together and think about content examples in the comments. Pick a concept. How could we use Brian’s technique?

I keep thinking about a snippet of an Ira Glass talk from 1998 that has always stayed with me. He was talking about the shortcomings of the rhythm of a typical National Public Radio news story, which led him to create *This American Life*:

And there was something dull about the rhythm, to me as a radio producer, where every story was set up so there was a little bit of script and then you’d hear a quote, and some script and then some quote. And radio, you know, functions a lot like music, even though it’s speech. It had this very predictable rhythm.

And we never get to know any of the characters, enough to feel anything or empathize in any way or to be amused or to feel angry or to be surprised.

This weekend, I felt things. By disrupting the typical rhythm of math class, Brian gave me a chance to feel amused and confused, surprised and delighted.

It’s really worth thinking about how we can do that for our students.

This got me thinking ! In particular about factorisation.

What about this, and variations:

“What is the probability that a quadratic equation x^2 + ax + b = 0 has whole number valued roots if the values of a and b are taken from the set {1,2,3,4} ?”

Any bets on how confused kids will if they attack w/ the quadratic formula? “Hmm, what’s ‘b’, what’s ‘a’, what’s ‘c’ again?”

Why stop at 4? I would go for “what is the probability that the roots of x^2+ax + b = 0 have whole number valued roots if a and b are randomly chosen digits (0..9)?”

Since I like to tease students, I would definitely remind them to consider the case that only one root is a whole number and the other is something else.

I can’t tell if you reached the big payoff for Pythagorean triples – that they can all be generated via some simple functions.

https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple

Not to worry. Nothing publishable here! LOL. We were playing around with how to, for a given odd number, generate two other sides that were 1 apart, e.g., (5,12,13), (7,24,25)… We were working from the observation that 5^2 = 12 + 13, 7^2 = 24 + 25, and so on. Then we were messing around with the triples that started with an even number and had two other sides that were 2 apart, e.g., (8,15,17), (12,35,37)… There were some interesting patterns there too. It was such an open task that there were many directions we could head in, and this was ours. Really fun and new to us, but certainly not new to mathematics.

Really cool that you had this experience and the right mindset for it. I wonder how much of this other participants got (I’m thinking about the groans you mentioned with Kayles and the Josephus recursion)?

One easy way to disrupt the rhythm a bit is to pre-plan some questions/problems/challenges from a past unit a couple weeks (or months?) later, in the middle of a new module. Alternatively, grab a random NRICH activity or math contest problem without pre-selecting it for compatibility with the current topic?

I would be cautious about introducing a case where a tool doesn’t help immediately after introducing the tool. My concern is that there is a stage where the learners only have a tenuous understanding and I’m not sure the switch up would help. On the other hand, there is always some value in compare/contrast activities. For example, both differentiation and factoring can tell us interesting things about polynomials. What are some things we might ask where differentiation is more useful? Where factoring is more useful? In those cases, does the sub-optimal tool still tell us something or just lead us astray? Are there case where using both is more helpful?

other misc comments:

– it was physically painful for me to watch the survivor 21 flags video, as each team made suboptimal moves. I’ve played NIM variations with young kids (3rd grade and younger) several times and they consistently seem to get the strategy very quickly, so I was surprised that the survivors struggled so much. I will watch this vid with my little ones and see what they think.

– Pennies I: it might be interesting to go back to this and work out why symmetry isn’t helpful for this game. can’t the first player just play in such a way that the arrangement is always symmetrical? Actually, they can, so why doesn’t that win for them?

– Kayles: can’t the first player just take the penny second from the left, then play the symmetry strategy from Pennies II? I guess maybe I’m missing something about the generalized form of the game.

Here is another idea about “messy/case-by-case” games or problems that don’t have closed solutions: most interesting things are actually like this and the tools/techniques for attacking them are still really cool. Examples abound; chess is likely the most familiar example.

Fantastic comments, Joshua, as always. I love your thoughts about productive disruptions. Two things come to mind about comparing different strategies and evaluating their strengths and shortcomings:

-One is the “Compare and Connect” strategy in Intentional Talk by Elham Kazemi and Allison Hintz. Have you read it? It’s so good. They talk about choosing two representations/methods/strategies and putting them side by side for comparison and discussion. Lots of time the conversation is “Can we see the _____ from this representation in this representation?” Great. But other times we head into which method is best for what types of situations discussions?

-Two is Chris Luzniak’s work on making math debatable. He’ll give give kids a problem with some ambiguity and have them debate which is the BEST or WORST way to go about it. https://www.bigmarker.com/GlobalMathDept/14Oct2014

In both cases and what you’re suggesting, there has to be an intentional decision on the part of the teacher to invest time in this bigger thinking about strengths and limitations of the different tools, how do they relate and not, when might each be a good choice? With all the pressure to teach the next thing all the time, this decision is not obvious. Huge payoff though.

As for Survivor 21, I think that’s fascinating. So what are the conditions that made it hard for these players to be able to figure it out? I’d argue there were two big ones. 1) They were evaluated on their first exposure. They had no opportunities for exploratory play to get a sense. 2) It was high stakes and timed, right? So, if a student faces new-to-them mathematics on a timed, high-stakes test, can we realistically expect them to take the time to go back and figure it out from first principles? Or is this the more likely outcome?

Finally, play Kayles. We tried the strategy your suggesting. There is still a way for Player 1 to force Player 2 to face a different symmetric scenario in which Player 1 wins. But it’s not guaranteed. Have fun!

Thanks for the reading/reference pointers!

Ah, just realized that I missed a key point about

whohas a winning strategy in Pennies II that gave me the wrong idea about Kayles.Will write up my notes from showing my kids the survivor video and playing some of these games with them. Suffice it to say my original judgment was overly harsh.

I loved your article! I tutor math students and right now they’re deeply involved in factoring higher degree polynomials so one thing that came to my mind is: after they’ve been taught the basics of differences of squares, differences/sums of cubes, quadratics and grouping…there area some great problems like x^4 + 11x^2 + 36 where they need to brainstorm how to rewrite it so that it’s a perfect square trinomial and difference of squares x^4 + 11x^2 +x^2 + 36 – x^2 which can then be factored. I’m not sure if that’s “disruptive” enough, but it sure seems to disrupt my students!!

Thanks, Michelle!

Here’s a geometrical one, with many ways of dealing with it, one of which is a one liner:

Given two points A and B find the point 3/8ths of the way from A to B