Radical Cheat and Playful Mathematics

One of my favourite professors in university taught a second-year course on solving differential equations. He would bounce into class, silver hair flying, making us feel like it was our lucky day to be studying math on a beautiful day in that stuffy old classroom. I still remember how he would preface the problems we would solve with: “Alright, so that’s the game! Let’s play the game.” And off we would go. After all these years, this phrase stuck with me – admittedly, more than the mathematics I learned during the course. Maybe he was referring to the fact that we were working with “toy problems,” as other professors would say, suggesting that they had little resemblance to the problems mathematicians actually work on day to day; I like to think, though, that he was talking about how mathematics itself can be a form of play.

In his instant-classic lecture, “Mathematics for Human Flourishing” (2017), Francis Su lists play as the first fundamental human desire that mathematics can help cultivate: “Everyone can play. Everyone enjoys play… [And] mathematics makes the mind its playground.” Looking back, the possibility of extending my time in the sandbox is probably what attracted me to pursuing a math degree, and later, to becoming a math teacher. Years later, I’m playing the game with my own students, but as I found out quickly, teaching play is hard work. As Su writes, “It’s actually harder than lecturing, because you have to be ready for almost anything to happen in the classroom.” Brent Davis (1996, p. 222) echoes this sentiment:

“[The teacher] is assigned the task of presenting possibilities and, through attending to students’ responses to these possibilities, opening spaces for play. Such play-fullness is only feasible when one allows for departure from the anticipated … fluidity in the structured … and uncertainty in the known.”

I try hard to open such spaces for my students, choosing tasks that yield themselves to multiple solution paths, encouraging students to pose problems, and exercising a certain freedom in lesson structure that allows us to follow interesting paths that we stumble upon along the way. Of course, there are limits to this freedom, given that we must work within the structure of the curriculum and time constraints. But while structure and constraints may seem antithesis to play, both Su and Davis agree that on the contrary, structure can support playfulness:

There is usually some structure: even babies know ‘peekaboo’ follows a certain pattern, but there is lots of freedom within that structure.
– Su, 2017

Play is not so much an activity as it is an acceptance of uncertainty and a willingness to move … But it is not an abandonment of our quest for structure, order, pattern, and comprehensibility. Quite the opposite, these are the ends of play.
– Davis, 1996, p. 222

Play, then, isn’t about purposelessness or disorder; play is having freedom of movement within a framework of constraints. And sometimes, opportunities for play can arise unexpectedly, even within the most structured of tasks.

These days, we’re playing radicals.

After spending a few days working on evaluating, estimating, and ordering radical numbers, I taught my students how to play Radical War (credit to Andrew Stadel for sharing the idea). The game is played in partners using a  deck that includes a variety of rational and irrational numbers, with values ranging from 1 to 5 (available here). The deck is split between the players. On each turn, both players turn up a card; whoever has the higher card takes both cards into his or her hand. (Special rules apply when both players turn up cards of the same value; full rules available here.) The game is played until one player has won the deck.


As I circulated and listened in on the conversations, I witnessed some great reasoning. The game offered a great, low-risk context to practice estimating and ordering rational and irrational numbers, one that my students seemed to really enjoy. Somewhat ironic, though, that the game itself does not seem to offer a lot of space for play, as Su or Davis would define it. There is plenty of structure given by the rules, but where’s the freedom for movement? With the constraints that are provided, there’s no room for strategy, for deliberating and choosing between possible actions; only luck (or cheating) will determine the winner. Below the surface, then, not much different from a set of drills, with the added twist of competition.

Except that, as anyone who has ever played Monopoly knows, sometimes play is about throwing out the rule book and inventing your own game.

As I circulated, I came upon a group that had abandoned Radical War after a quick victory by one of the players. Each now held their cards fanned out in front of them:

When I asked them if they were going to start round two of Radical War to play best out of three, they sheepishly admitted: “Well… we kind of invented our own game.” And so they told me about Radical Cheat. It’s based on a well-known card game called Cheat (or Bull****, in some circles):

Cards are distributed evenly. On the first turn, a player puts down an Ace card and announces, “one Ace.” If the player does not have an Ace card or would like to get rid of more than one card, he may bluff and play non-Ace cards. If any of the other players do not believe the announcement, they can call “Cheat!”. The person who played the cards must turn them over and show the challenger whether he is bluffing or not. A player who is caught bluffing must pick up the entire pile and add it to his hand. If a challenged player is not bluffing, then the challenger must pick up the discard pile. When the rank to play reaches Kings, it then goes back to Aces and the numbers start again.

Of course, because the cards are non-standard, composed of rational and irrational numbers ranging from 1 to 5, the rules needed to be modified. First and most obviously, the students decided that play would start with 1s rather than Aces and end at 5s rather than Kings. Then, and less obviously, to account for the non-integer numbers in the deck, the students decided that the values on the cards would be rounded to the nearest integer. This way, 1.2 would count as 1, √3 would count as 2, √8 would count as 3, and so on. Simple, and brilliant in its simplicity.

I observed the game play for a few minutes. The element of competition was still present, and students needed to use just as much reasoning as they did in the previous game, because they were required to estimate the values of their cards in order to round them to the nearest integer.  In fact, the stakes for accurate estimates were even higher during Radical Cheat, because players couldn’t rely on their partner to estimate the value of their cards for them—if they made an error, they risked taking the entire discard pile.

I was impressed. Instead of abandoning the game and putting an end to their reasoning about radicals once they got tired of the task, the students demonstrated their ability to play with mathematics, in the sense that Davis describes: “a willingness to move,” without abandoning structure, order, pattern, and comprehensibility. While discarding some of the original constraints (the rules for Radical War), they used the tools they had at their disposal (the cards) to create a new game to play. Note that this form of play is different from playing the game of school. In the school game, which tends to be about learning how to stay within explicit and implicit boundaries, rules are often rigid and not up for negotiation.

But isn’t play what mathematics is all about? As Su (2017) writes, in mathematics,

We play with patterns, and within the structure of certain axioms, we exercise freedom in exploring their consequences, joyful at any truths we find. 

Not to mention that some of the most interesting advancements in mathematics came about when mathematicians pushed against boundaries and discarded seemingly inflexible “rules” to create new games to play.

I used to think that this kind of playful, high-level mathematical activity was next to impossible to attain within the rigid constraints of curriculum. These days, I have a new perspective. As this experience reminded me, playfulness in mathematics is not about abandoning constraints, which can actually serve to spark creative thought by offering something to think about in the first place.  Rather, playfulness involves knowing which rules to keep, and which rules can be broken—say, the rule that exponents must only be positive whole numbers, or that a set of cards must only be used to play Radical War—and giving students the space to break them, and to explore the consequences.

“Alright, so that’s the game! Let’s play.”


Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York, NY: Garland.

Su, F. (2017). Mathematics for human flourishing [Blog post]. Retrieved from mathyawp.wordpress.com/2017/01/08/mathematics-for-human-flourishing/. Audio recording available at  youtu.be/xEtDvc1SWm8


“Ask, and it will be given to you”

It’s February break, which means that I can finally start making a dent in the reading piling up on my bedside table. I kicked things off with Embarrassment by Thomas Newkirk (2017, Heinemann), a book exploring how embarrassment and its associated emotions can get in the way of or even sabotage our students’ and our own learning. (Naturally, there is an entire chapter devoted to “math shame,” with the subheading “Why are we all on the outside looking in?”) It’s an engaging, provocative, and powerful read, with take-aways for teachers and learners of all levels and at all stages in their career. So far (I am halfway through), the most illuminating section for me has been the one entitled “Asking and Receiving.”

Continue reading ““Ask, and it will be given to you””

End-of-semester reflections

I have my students in all of my classes complete reflections about their experience in my class at the end of the semester, which help me reflect on how I can improve my practice as I transition into the next semester. Here are the prompts:

  1. In this class, I felt like I, my ideas, and my questions mattered. (scale of 1-5, from Never to Always)
  2. In this class, I felt challenged. (scale of 1-5, from Never to Always)
  3. In this class, I often felt stressed out. (scale of 1-5, from Never to Always)
  4. The best aspect of this course was…
  5. The worst aspect of this course was…
  6. One surprising or interesting thing that I learned was…
  7. One thing I’d like to tell Mlle V, in all honesty, is…

And here are some highlights from my Grade 9 class. Continue reading “End-of-semester reflections”

Love the questions

There are a lot of quotes (maybe as many as there are teachers) that begin something like this: “The goal of education is…” For example,

“The one real goal of education is to leave a person asking questions.” Max Beerhohm

Undoubtedly, many of these are spoken in a hyperbolic manner, and should be interpreted as such. (Myself, I would be wary of anyone who believed that there are precisely N goals of education, true for all places and at all times.) Nevertheless, this sentiment came to my mind last week, the first week of school. Continue reading “Love the questions”

A small moment to remember

Every being cries out in silence to be read differently. Do not be indifferent to these cries. – Simone Weil

I was substitute teaching today for a group of students I had taught last semester. I noticed that one of the girls had new glasses, and I told her that they looked nice. She replied, surprised, “You’re the only teacher that’s noticed, and you don’t even teach me.”

Not pretending to be a hero. There have undoubtedly been countless occasions that I’ve been blind when I should have seen. Just tying a knot in my mind to notice, amidst the chaos, the students I’m not seeing in my own classroom.

Reframing a classroom culture problem

tl;dr: Sometimes, a “problem” in the classroom is really a symptom of a bigger issue.

Here’s the backstory. Lately, I’ve been struggling with a lack of engagement in whole-group discussions after small-group work. I was growing increasingly frustrated that the majority of the students – especially the students who had a good grasp on the concepts – didn’t volunteer to share their thinking, and that they had no qualms about chatting with their friends when a student (or me) did choose to put forth a question or share an idea. Increasingly, I was feeling like I was losing them, and searched for explanations ranging from summer being around the corner, to deficiencies on the part of my students (kids these days!), to the possibility that I am a terrible teacher for whom there is, frankly, no hope. (During a recent conversation, Jimmy Pai described teaching as a balancing game between overconfidence and despair, and these days I’ve been leaning heavily to the right.) Continue reading “Reframing a classroom culture problem”

On balance scales and trusting your students

At some point last semester, I remember feeling apprehensive about using balance scales as a model to introduce linear equations. I think I was worried about students’ ability to move beyond the model and consider things like negative and decimal coefficients. Would they be lost when they no longer had their crutch?

I should have trusted them.

Nat Banting has written about balance scales here. Although the metaphor of balance scales is referred to often in math classrooms – during the linear equations unit, students must hear the mantra “whatever you do to one side, you do to the other” at least 1000 times a class – the model really becomes powerful when students actually work with balance scale problems, and that they do so right from the start. Nat writes:

My Grade 9 students don’t see an equation for the first two weeks of their unit of solving linear equations. That is because I think students get all bogged down in the notation, and lose their problem solving intuition.

I decided to follow Nat’s advice. For a few days, my students worked through sequences of problems such as these, increasing in difficulty, where the goal is to find the “weight” of the square: Continue reading “On balance scales and trusting your students”

The kids are alright

Quelle est une chose qui devrait être interdite à l’école ?

I often start French class with a question. The students reflect and discuss in their table groups, then share their responses with the rest of the class during the whole-group discussion. Yesterday, I opened with the above. In English: “What is one thing that should be banned at school?”

“Littering!” “Homework!” “Vaping in the bathrooms!”, a few students volunteered (in French). Then, a student offered, “Cell phones!” I’m sure you can imagine the disbelief and anger that this response evoked from a class of Grade 9 students, the majority of whom have an iPhone permanently glued into their hand and probably could not imagine a time without them. Once the others died down, I gave the student the floor, who argued with sincerity: “Cell phones are a huge distraction. We’re paid to be here by tax payers, which means we have a responsibility to learn and not waste time on our phones.” Once again, the other students erupted. It took a minute to calm them down.

Finally, a student responded bitterly: “If we’re paid to be here, where’s my money?”

I knew at that moment that French class, strictly speaking, was over. This was going to be something different, and I liked where we were going. Continue reading “The kids are alright”

The absence of number

One of my favorite conversations to have with my students (Grades 9-11) has quickly become centered around the concept of zero.

It first arose naturally last semester, when a Grade 9 student blurted out: “But 0’s not a number, it’s the absence of number!” (I’ve forgotten the context of this remark.) I remember a few other students laughing at this, and I know that this was unfortunately a negative formative experience for her, because she wrote about it in her end-of-year reflection. And yet, when I prodded further at a later point in time, I found that many other students, if not most, held the same belief. We had a whole-group conversation about it, talked about how 0 behaves in a similar way as other numbers in many ways (you can add it, subtract it, multiply it…), but still, I think a lot of students held on to this conception. This was especially clear when the students were solving equations and came to a problem such as this: Continue reading “The absence of number”