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:

2x + 7 = 7

Had the problem been 2x + 9 = 7, they would have easily breezed on through. And yet, when it came to subtracting 7 from both sides in the above equation, many students hit what they felt was a wall: when you subtract 7 from the right side, nothing’s left! On assessments, some wrote “no solution” or simply did not go further. I wasn’t sure how to help these students go beyond this wall, besides repeating that “zero is just another number!” (in hindsight, I should have gone back to the “cover-up” method and asked them “what plus 7 gives you 7?”).

Today, the same question arose in my Grade 10 class in the context of a discussion about prime numbers. (I also posed the question in a Grade 11 class later in the day and received very similar responses, save for that of a very advanced student who brought up the notions of cardinality and measure, which both imply that 0 is a well-defined number.) After debating whether 0 was prime or not, I posed the question: “Is zero even a number?” Here are some of the responses I received from students of all skill levels:

- “No, it’s the
*absence*of number!” (heard from a few students) - “No – with other numbers like 2 or 3, you can multiply them by other numbers and their value changes, but not with zero. It always stays the same.”
- “No, it’s a placeholder – for example, in 10, or 0.1”
- “Google says it’s a number”
- “No, can’t divide by zero”
- “Yes, but it’s a weird number; it’s an exception”
- “No, it’s nothing!”
**“Yes. Like if you have 17 minus 17, that has to be a number, because 17 is a number. And 17-17 is 0, so 0 is a number.”**- “My brain hurts”

I found it really interesting that students echoed many of the same conceptions about zero that humans have held throughout history, such as 0 being simply a placeholder. (On the recommendation of several MTBoSers, I’ve ordered *Zero: The Biography of a Dangerous Idea* to learn more about the historical development of the notion.) The bolded response, though, blew my mind: Offered by a student who normally participates little in class discussions, it used the property that the integers are closed under subtraction to give the most concise, beautiful, compelling argument I have heard from a student so far for why 0 *is* a number. I’ve been thinking about it all day.

I’m not sure that other students found the argument as compelling as I did, though, and the question was left open, to be discussed again tomorrow. (We also dived into the problem of dividing by zero, when one student showed us how 10*0 = 5*0 implied that 10 = 5, if you divide both sides by 0! Another student suggested that something divided by 0 is infinity, which did sound reasonable when we tried dividing 1 by increasingly smaller and smaller numbers… Tonight’s homework question: “Try to figure out what ***** broke by dividing by 0. Either 5 = 10 and all the math we know is wrong, or something isn’t right…”)

Anyway, my question for the #MBToS is: What’s my move here? What might convince students that zero *is* indeed a real number (in the precise mathematical sense, and otherwise)… and how hard should I try to get them there? I’ve seen that confusion about zero has caused trouble when solving equations, so my gut says that this is an important concept to develop. At the same time, I love this ambiguous space we’re occupying, where an idea students have played with since their primary years still seems strange and up for debate. The following remark by Tracy Zager still comes to my mind in these kinds of situations since I read it last year: “Whatever I do, I’m in no rush to define this loveliness away.” Christopher Danielson agrees:

Thoughts? I’d love to hear about similar (or different) experiences and where you took these conversations with your own students.

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Exactly where [a lesson] moves depends on such complex factors as the structures of those present, the context, and what has been anticipated. It may move toward more formulated understandings, if such formulation is relevant to the play space or if it becomes part of a further exploration. It may simply move to other sorts of activities. This, of course, is not to say that we should just allow whatever might happen to happen, thus abandoning our responsibilities as teachers. Rather, it is to say that we cannot make others think the way we think or know what we know, but we can create those openings where we can interactively and jointly move toward deeper understandings of a shared situation.

(Davis, 1996, p. 238-39)

My Grade 9 students are currently working on recognizing, analyzing, graphing, and solving problems involving linear relations. Linear relations lend themselves so naturally to describing patterns à la www.visualpatterns.org, and this is precisely how we got our toes wet in the topic: For several days, my students had been analyzing, extending, and (productively) arguing about a variety of linear and non-linear patterns. The intention of these first few lessons was to have students develop (or, in some cases, refine) an understanding of constant and non-constant change and to connect patterns in pictures to patterns in tables of values.

As the students began to connect ideas, I looked to develop an activity that gave students an opportunity to apply the generalizations emerging from the phenomena that we were playing with. Melanie Kong must have read my mind, because just as I was thinking about this next lesson, she shared a link to Nat Banting‘s Relation Stations task. In a nutshell, based on their recent experiences with constantly-growing patterns, groups of students are to create their own linear patterns. The three rules Nat provides for this activity are:

- You must use your tiles to model the first three stages of your pattern. (I lifted the restriction of needing to use tiles and allowed students to create patterns on whiteboards.)
- You cannot copy or repeat a pattern we have already seen.
- Your pattern must have a starting point and a constant change.

After the groups create their patterns, the students circulate around the room, drawing the next two stages of the patterns and filling out a table of values for each.

I had been planning a similar activity, but mine didn’t involve students circulating to study each other’s work. I’m glad to have made this change, because I found this to be a great opportunity to simultaneously validate students’ mathematical work, encourage appreciation for their peers’ thinking, and to challenge their own. (I now recognize that I need to provide similar opportunities more often in my classroom.) In their work on this task, my students created some lovely patterns of various degrees of complexity, including the following:

Even more so than I had hoped, the task provided an incredible range of opportunities for discussion that spilled over well into the next day. (What a humbling, rewarding, and fun experience it is to be surprised by students’ creativity.) Notice, for instance, the second pattern, where the number of squares is decreasing rather than increasing. What does Stage 6 look like? Notice also the third pattern, where the relation is either linear or quadratic, depending on what you count:

The main reason for this post, though, was a similarly ambiguous pattern (excuse my finger):

(Yes, I do notice the student doodling in the middle, and yes, I am a very recent convert to one marker per group.)

Here’s my very crude recreation:

I brought this pattern to the next class to use as an opener. The students had had a long weekend, so we needed a refresher on the ideas we had been discussing last week. To be honest, I chose this pattern among the others that the students had created mainly because it was unique in the shapes it used – a combination of triangles and hexagons rather than squares, as every other group had drawn. I did not anticipate what would come to fall out of this pattern.

I set the students to work, with the instructions being to describe the pattern in words, in a table, and in an equation. I had expected that they would count the number of shapes, as in every other pattern we had studied. (I would be curious as to what your first instinct was!) Some of the groups did count the total number of shapes in each step, which is what the creators of this pattern had done. Others counted the number of triangles. One group decided they would count the number of sides/lines, including those on the inside. At first, I worried about the diverging paths the groups were taking, especially because some of those paths led them to non-linear results (this was supposed to be a review of linear relations, after all), and briefly contemplated instructing them to all count the number of shapes. But there was something magical about telling students who had likely waited for instructions for most of their school careers that it was up to *them* to decide where they would take the problem. And so, in Nat’s words, I let the problem drift.

As a group, we first brainstormed the different things we could count in this pattern; my students came up with at least 5. We described the more obvious patterns, after which I set the students to work on describing how the number of lines changed from step to step. (I happily scrapped the next pattern I was going to have the students work on.) What better way to solidify understanding of constant growth (linearity) than by contrasting it with an example of non-constant growth (non-linearity)? A student then mentioned that we could count the number of hexagons, and we argued over whether the pattern 1 (Step 1), 1 (Step 2), 1 (Step 3)… was linear. But then, another student piped up to say that the number of hexagons *was* increasing – indeed, take a look again at the third shape!

Although I hadn’t planned on it, we ended up studying this single pattern, whose depth slowly revealed itself before our eyes, for the greater part of the lesson, and I think we learned as much about Mathematics (its playful side, its emergence from human imagination) as we did about the mathematics (linear relations, constant growth). Looking back, this was one of those rare hours during which it was truly my students who were leading *me* through the curricular landscape (again, Nat’s words, not mine), rather than the other way around. The experience re-minded me of the need to remain flexible in my teaching – to move away from the image of teaching mathematics as a one-way path to an unpredictable and ultimately unknowable landscape, and to always (always) leave space in planning for students, lest we “freeze the body of knowledge that is otherwise dynamic, vibrant, and alive” (Max van Manen, as cited in Davis, 1996, p. 101). As you may have noticed through my references, this lesson brought to life for me several key ideas in Davis’ *Teaching Mathematics: Toward a Sound Alternative *(a work of art or a piece of poetry as much as a book about teaching mathematics). In particular, more than any other lesson so far, the experience has re-minded me that

There is no predicting what ideas will come up, what interests will emerge, what insights will arise. [A] “plan” is thus best thought of as a series of prompts or nudges to encourage movement through a mathematical space. It is not a scheme to be implemented, but a series of possible entry points for teaching action. It is, then, merely a starting place for a continuous process of anticipating; it is more along the lines of a strategy for an as yet unplayed game than an algorithm for reaching a particular destination. It is a way of stepping into the current of a curriculum.

Davis, 1996, p. 127

References

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

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What’s your point, Ilona?

I’m not sure. I guess I just needed to look at those emotions square in the face. Tomorrow, I will try again, hopefully a little wiser. And the next day. And the day after that.

*(Sam assured us that he feels much better now.)

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“What you do speaks so loud that I cannot hear what you say.” ― Ralph Waldo Emerson

Since the first day of school, I’ve been working hard to try to establish a classroom culture where students feel comfortable taking risks, asking questions, sharing and building on each others’ fully-formed and partial ideas, and acknowledging and correcting their mistakes; where all students feel that their contributions and questions are valuable and worthy of consideration. I have tried to do so by pointing out (less often than I should) when a student or a group is exemplifying one of these norms, by waiting (again, less often than I should) after questions and contributions to give more students the time they need formulate and share their ideas, by giving tasks that are accessible to a wide range of students and can be tackled with a variety of strategies, by eliciting and celebrating different solution paths, by highlighting different kinds of mathematical smartness (h/t Ilana Horn)…

And then, I proceeded to potentially sabotage it all with an inexcusable split-second decision.

On Friday, my Grade 9 class was working in small groups on finding different symbolic representations for a variety of images and situations involving fractional amounts. I brought the whole class back together again and invited different groups to share their representations. This was one of the images:

Different students explained why they chose to represent the image as 4/10, 2/5, 6/10, 3/5, 0.4, or 0.6. I carefully recorded their reasoning and their names for all to consider. Then, another student raised his hand and said he had another way of looking at it: “You can also look at it as 0. There are 0 red, or blue, or purple slices colored in. So it could also be 0/10.” I chuckled, acknowledged that this was another way of seeing things, and moved on to the next question. I did not add his explanation to the list, nor his name. For what it’s worth – which is nothing – we were running out of time, and I wanted to, you know, *get to the point.*

“Words cost nothing. Actions can cost everything.” ― Aleksandra Layland

I could have easily forgotten about this incident had the student not came up to me after class and asked, in a hurt voice, “Why didn’t you write my explanation, too?” He explained his thinking again which was, of course, perfectly reasonable. The instructions did not explicitly suggest that students describe how many slices were grey or how many were white. What an interesting, out-of-the-box interpretation! I had *specifically* chosen these images because they were ambiguous, because I wanted students to see that things in mathematics aren’t always black and white… and truly, I did not realize until *after* I wrote the previous line just how ironic it is.

What did my decision to not write the student’s contribution communicate to him and to the other students? That some ideas are, in fact, *not* valuable or worthy of consideration, even if I have prattled on about the contrary point many times since the beginning of the school year. That indeed there *is *a correct answer to a problem that can be interpreted in several ways, and that my apparent curiosity in hearing about different solution strategies and interpretations is just a thinly-veiled attempt to hone in on the *right* one – perhaps, some students had already suspected this, and this incident provided the necessary proof. This split-second decision may very well have discouraged other students from taking a risk and sharing their own divergent line of thinking – and why should they have, if I was just going to blow it off and get back on track to meeting the objective/outcome/target of the lesson?

I fear that I may have done some real damage to the classroom culture and to the students’ conception of mathematics – after all, it’s often the little things that have the biggest impact in teaching. And what if the student in question hadn’t so bravely chosen to confront me about it after the lesson? I may have done it again, and again, and again.

So on Monday, I will start with an apology to my class, as well as a thank you to the student who called me out on my mistake. We will re-consider the problem, and I will listen carefully and record diligently *all* of my students’ contributions, even if it takes up a little too much time and takes us astray from the path that I had anticipated. After all, in Nat Banting’s words,

“Teaching is a fluid movement through a landscape of lived curriculum. Not a mechanical movement through a planned one.”

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- When I see the word ‘mathematics,’ I think of…
- A good experience with mathematics was when…
- A bad experience with mathematics was when…
- This semester, I expect to…

(The students completed the prompts in their new math journals, which I will hopefully get a chance to write about once the routine is more firmly established.)

The idea was to give students the opportunity to reflect honestly on their relationship with and views on mathematics. I plan to repeat the exercise at the end of the semester – it should be interesting and instructive to note any changes in perspective.

The majority of responses for second two questions were more-or-less predictable in a culture that equates grades with success and intelligence (or lack thereof): e.g., “…when I got 95% on a test” and “…when I got 50% on a test.” A small minority talked about a time when they were able to use mathematics in their daily life, like when they were able to find the best deal at the grocery store; a handful described a time when they solved a challenging problem, others when they had the opportunity to work on a problem with their classmates. In response to the third prompt, a good number of students (especially the older ones) answered “pass the class”; a small number, perhaps thinking that I was looking for a certain type of answer, wrote “listen to the teacher.” One student, IN ALL CAPS, wrote “I expect to… love MATHEMATICS WITH ALL MY HEART”:

Whether he was joking or not, what a challenge to receive!

The students’ responses to the first prompt (“When I see the word mathematics, I think of…”) were particularly interesting and thought-provoking because of their near-uniformity: the vast majority of students completed the statement with words such as “equations,” “numbers,” “problems,” “school,” “a teacher”… In other words, the vast majority of these students associate mathematics with its *products* or with related *objects* – and in particular, with products and objects that are typically imposed upon them, rather than those that students are involved in shaping. Not one student explicitly referred to processes* *or verbs*, *such as asking questions, looking for patterns, making connections, developing logical explanations, generalizing… What does this say about students’ (perceived) relationship with mathematics? Can this relationship be redefined in a culture that assigns higher value to tangible and visible products (answers, grades) than to processes?

I’m still ruminating about all of these responses and how I* *will respond to them through my teaching this semester. I would love to hear your thoughts.

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I offer this simple but impressive trick to future or current substitute teachers, or even as a class starter / warm-up for classroom teachers, but I take no credit for it and am only paying it forward: I was introduced to it by Dr. Egan Chernoff during his Methods in Secondary Mathematics course, who learned it in a class with Dr. Peter Liljedahl, who in turn, came across it in an issue of *Vector* (journal of the British Columbia Association of Mathematics Teachers). Naturally, although I will describe it below, I will not explain how it works… I leave that to you. (I thought about it… but then I remembered Chernoff calling us “answer junkies” and thought better of it.) Afterwards, I’ll reflect a bit on when I’ve found it to be particularly successful and make some guesses as to why.

**Lightning addition
**I start the trick by telling the students that I’m a time traveler. (Of course, you can devise your own story.) “Yes, it’s true. And I’ll prove it to you by adding up a whole bunch of numbers really fast. ‘Cause, see, what I do is go forward in time about a minute, see the answer, then come back and write it on the board before you’re even able to punch in the first number into your calculator. Do you believe me?” (I adapt the story for higher grades so as to avoid the sting of 30 simultaneous eye rolls.)

Have the students take out their calculators. You will also need a student volunteer. Explain that he or she can write any three 4-digit numbers on the board. You won’t look at them while they’re writing, but afterwards, you’ll turn around and write two more. Then, when you say “Go!,” everyone will start adding the numbers. Make sure you tell the students to start putting the numbers into their calculators *only* when you say so – you will all be starting at the same time. Try not to mess up (although this can give students a good hint of what’s going on), prepare for the ensuing amazement.

Here’s an example:

*1. Student writes three 4-digit numbers on the board*

*2. You add two numbers of your own*

*3. When you say “go,” the students start adding the numbers. You, meanwhile, will be able to write the answer within 2-3 seconds:*

Simple. But it *never* fails to amaze. (I once had a Grade 9 student stand up, slowly, and ask: “What* are* you?”)

Of course, I don’t show the trick to the students so that they think I’m a time lord, but because it’s a fun challenge to figure out *how* I was able to do it. Typically, I will do the trick at least twice (often, students will ask themselves to see it several times) to give the students a bit more to go off of. (I’ve included a couple more examples at the end of this post to help *you* recognize the pattern, if necessary.) After this, I’ll simply give the day’s assignment, leaving the examples on the board.

And then, I simply watch. Because without fail, there is *always* (always) at least one student who will stare at the board for a few minutes, then copy down the numbers and try to work it out for him or herself. Sometimes, they will leave it until I give a guiding hint in the middle of class; very often, though, they simply will not be able to leave the problem alone. (When they should be working on Section 4.3, #1-12… oops.) Sometimes, they’ll recognize interesting but non-relevant patterns in the numbers that seem to offer a clue, until I ask whether the pattern holds in all of the examples. By the end of class, someone will have figured it out, and I get them to demonstrate their newfound time-traveling powers in front of the class.

Although I love to see students in any class working on this problem, what makes me most excited about this trick is that, at least in my experience, it has been most readily picked up in “stretch” or “extended” math courses… i.e., by students who aren’t normally expected to enjoy math, or who don’t think that they’re any good at it. Just this morning I was in such a class (Grade 11), and a group of boys declared that it was now their “sole purpose in life” to figure it out. All throughout the class, they persevered. They struggled. They argued. *And* *then they solved it.* Everyone clapped at the end of the period when one of them demonstrated it on the board.

But why? Why would students who *seem* to have no interest in math spend a class – by their own choice – trying to figure out a silly little addition trick? Well, to start, I don’t believe that there is single a student out there who has absolutely no capacity to get interested in a mathematical problem. The first key, I think, is perplexity, which according to Dan Meyer is the goal of engagement: in his words, the task is engaging because it “induce[s] in the student a perplexed, curious state.” I encourage you to try this in your own classroom one day, just for fun, and experience seeing the shock and wonder on students’ faces. (And notice that although it’s very engaging, the problem isn’t “contextual” or “real world” – I’m not trying to get the students excited by trying to wedge Snapchat or Beyoncé into a math problem. Not that the latter is necessarily a poor strategy, but, as Nat Banting writes, “I would much rather think of classroom materials as either mind numbing or thought provoking. […] A real-world context is the cherry on top.”)

The second key, as I see it, is the task’s accessibility. Although it may not be the best example of one, I do see this trick as belonging in the class of low-floor, high-ceiling tasks (also briefly discussed by Dan Meyer, among many others) – the floor certainly is low, as the majority of even the most struggling high school students understand integer addition and can look for patterns in a set of numbers, and one can think of many extensions to the trick for students who are able to go further (I’ve suggested a few below). It’s easy to understand the problem (essentially, the question is: why did I choose the numbers I chose?), satisfying to figure it out, and even more satisfying for a student who has typically struggled in math class go up to the board and solve what would otherwise be a tedious problem in a matter of seconds, in front of his or her peers. And while I’m not so naive as to think that a student’s confidence (and interest) in math can be dramatically improved within a class period, perhaps a steady drip of these kinds of experiences can help.

I hope you give this little trick a try (and thank @MatthewMaddux – or maybe not, he’ll probably tell me off for “waxing poetically”… do it anyway). Let me know how it goes.

**Extending questions**

- Do the numbers have to have 4 digits?
- Could we alternate writing the numbers? Who has to go first?
- What if the volunteer had written 4 numbers instead of 3? 2? 5? 10?
*n*? - How can the trick be modified so that I know the answer before any numbers are even written on the board? (
*I’ve done this one too, taping the answer under a student’s chair before class . Also very impressive.*)

**Extra examples**

**Postscript
**One day, after staring at the board for a while, a student concluded (in jest) that this was why I was so good at addition:

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The premise is that you are invited to an auction, and given a budget of $10 [I changed the budget to $15 for my students to encourage a bit more risk taking]. Everyone at the auction has the same budget. The participants are all bidding on certain events that may occur when two 6-sided dice are rolled (e.g., both numbers are greater or equal to 5; a single 2 is rolled; both numbers are odd; etc.). After all the events have been auctioned off to the highest bidders, the two dice are rolled 20 times. Each time the event that you purchased occurs, you collect a prize. Bidding always begins at $1 and goes up in increments of $1. You cannot bid against yourself. The order of the events up for auction is known beforehand. If you choose not to spend (some, or all of) your money, the auctioneer will sell you prizes at a cost of $2 per prize after the bidding has ended. **Your task is to get as many prizes as possible.**

A couple of weeks ago, I tried this activity in my math club. Like Nat, I used pattern blocks as currency and brought candy for prizes. Each student received a handout with the description of the events up for auction (you can see Nat’s original handout on his website; I deleted 5 squares because of limited time and because my group is much smaller). I gave the students a couple of minutes to think about which events they wanted to bid on, suggesting that they didn’t share their reasoning at that stage. I didn’t suggest that the students calculate anything, or even mention the word probability; like Nat, I was interested in tapping into their intuition.

Once the auction began, the students bid eagerly, but clearly strategically. (One student did spend a lot of his money on the first square, “both numbers ≥ 5″, which he regretted very shortly after, and there were a few squares that sold for much less than I expected.) Each item opened at $1, and I collected money after each event closed. Bidding went pretty quickly, but still a bit longer than I had hoped, even with a small number of students. (The problem for me is that I only have one lunch hour with these students per week, so time is a precious resource.) After all of the events were sold, the dice were rolled 20 times, and prizes were distributed after each roll. The students kept track of how many times their purchased events were rolled (in the tally section of the handout) so that we could combine the results at the end and discuss the frequencies with which they occurred.

The great thing about this activity is the conversations that happen afterwards. Who “won”? Was there an event that occurred more often than expected? Less often? Did someone pay too much? Too little? Is there a square you regret paying for? (The student who spent much of his money on “both numbers ≥ 5″ certainly did.) Which events would you bid on next time?

These students were more comfortable with the basic concepts of probability than I expected, so most of their arguments were mathematical (“Well, there are *x* different ways of getting…”). We discussed the difference between theoretical probability and experimental probability – or rather, why there *was *a difference between the two. (I was surprised when one of my students brought up the Gambler’s Fallacy in her explanation – she’s clearly been doing some extra-curricular reading!) Unfortunately, the discussion didn’t last as long as I had hoped, and I didn’t have the luxury of carrying it over to the next day. Because my students loved the activity, and because I wanted to squeeze more mathematics out of it, I decided I would do it again – but with a twist.

It was clear to me that the students were very comfortable with the notion of independence and with calculating the probability of independent events. So, to make the task more challenging, I decided to switch from dice to cards. The premise of the auction is exactly the same (see above) – however, instead of two dice being rolled on each turn, two cards are drawn from two separate decks. The decks are *not* shuffled together (although this is certainly a valid variation), and the cards are *not* replaced. (Of course, you can also introduce dependence with dice, which would necessitate that you keep track of previous rolls. Creating suitable, not-too-unlikely events is a bit more challenging of a task in this case.)

I introduced the idea to my students last week. After we had a discussion about what was different about the two auctions, they suggested a few possible events that could be auctioned off (which had probabilities strictly between 0 and 1), and we had some great conversations about the probabilities of these events on the first draw, on the second draw, and so on. Since some of these events involved sums and products, we decided on a few rules for the Ace, the face cards and the Joker: in particular, the Ace has a value of 1, the face cards (Jack, Queen, and King) all have a value of 10 (to simplify the calculations of sums and products), and the Joker has a value of 0. You and your students may come up with different rules for these cards, which would change the probabilities for certain events – another interesting topic of conversation!

Again, time was not on our side, so I took the students’ suggestions but needed to add a few events myself. I wanted to make sure that the board was “balanced” in that there weren’t too many very likely and very unlikely events, so a friend and I played around (at a pub, over drinks – is there a better way to do math?) with a lot of different possibilities until we got a well-rounded board. To simplify our calculations, we only considered probabilities for the *first* draw, but of course, you would want to talk with your students (as I will with mine) about what happens in subsequent draws. The calculations do get unwieldy, though, so if you are interested in finding exact probabilities (e.g., for the first 1, 2, or 5 draws), I would choose only a few events from the board to focus on. There are several on here that could generate some great discussions and interesting solution methods (e.g., sum is odd; product less than 10, sum less than or equal to 9). For students new to probability, however, I imagine this being used as an introduction activity to the notion of dependence, leading to interesting discussions and giving just a taste of the calculations involved.

Below is the board that I will be using with my students (link to handout here). I have narrowed it down to 16 events and will be making only 15 draws because of limited time and a small number of students; for a larger group, you will want to add events to give everyone the opportunity to participate. As I suggested above, the activity is infinitely adaptable – not only in the events that you (and your students) choose, but also the rules that you assign to the various cards, how/whether you shuffle them, how many decks you use… If you have any suggestions for adaptations, comment below or on Nat’s original post (and do check out his other tasks, if you haven’t already – his blog is a treasure trove of interesting activities and projects).

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Like many in the mathematics education community, I had little interest in and and even less patience for what a *political science* professor who once taught a numeracy course to undergraduates had to say about the ills of secondary mathematics education (*the audacity*!). And I still stand by much of what I said: e.g., I still think that comparing *the entirety of the field of mathematics* to chess and crossword puzzles is rather petty and small-minded. Nor do I believe that mathematics doesn’t have the capacity to “move minds toward controversial terrain.” Of course, I was not the only one provoked by Hacker: Educators, mathematicians, and others more knowledgeable and experienced than I have written extended, thoughtful critiques of his arguments, and I will not repeat them here.

But – and this is where you may disagree – I think that Hacker has raised a crucial question: Namely, why? Why quadratic equations, why algebra, why math? He’s not the first to ask, and not the first to suggest that mathematics isn’t as widely used in daily life as some would believe. Paul Lockhart, for instance (a favorite among mathematics educators), lamented in his *Lament* (2002) that most people are “under the gross misconception that mathematics is somehow useful to society” (p. 6), criticizing perhaps even *more* vehemently than Hacker the current state of mathematics education (which he called “excruciatingly boring” and “disturbingly fractured,” “a senseless bouillabaisse of disconnected topics,” among other descriptions that I probably shouldn’t print here [see, e.g., p. 25]). Of course, Lockhart is a mathematician, which certainly gives more cred(ibility) to his critiques; moreover, he wasn’t interested in getting mathematics out of schools, but rather in having educators recognize and teach mathematics as an art. One certainly doesn’t get the sense that Hacker sees anything particularly beautiful about the subject. The point, however, is that many of the arguments that Hacker has put forward are not new – the purpose and the goals of mathematics education have been debated throughout human history, and Hacker is simply the latest to join the line (for more recent essays on the topic, see, e.g., O’Brien, 1973; Dudley, 1987, 1997; Smith, 1989).

The Question is an uncomfortable one to ponder over. It raises other, equally uncomfortable questions, such as: What, then, is *my* purpose as a mathematics teacher? These are not easy questions to answer, and ~~perhaps~~ most likely, definitive answers simply do not exist (certainly, they won’t be found in the back of the book). Nevertheless, some bold educators have recently chosen to tackle the Question head-on, whether as a direct or indirect response to Hacker: see, e.g., Ben Orlin’s Why You Need Math and Karim Kai Ani’s On Purpose. (In fact, even *I’ve* even tried to answer it, albeit several months ago, before the release of Hacker’s book made waves – I did so in response to my students’ interrogations, who [as teenagers and children often do] did not shy away from asking difficult and controversial questions. Children certainly have a knack for getting at the heart of an issue – I think they’re far more philosophical than many would believe.) Others, however, have focused on other flaws in Hacker’s arguments, such as his (possible) misunderstanding of the nature of the number pi – and here, I will be bold and say that I think we may be missing the point.

I know that the Question (i.e., Why math?) may seem like a threat. I certainly felt threatened by Andrew Hacker, and I suspect that many other educators did too. Perhaps his critique was not especially well-constructed, and maybe there’s good reason to be suspicious of a political science professor claiming to be an expert in the teaching and learning of school mathematics. He certainly should have been more explicit and humble about his lack of expertise. But the Question still stands, and I think that it’s critical that we examine it honestly, with open minds and without fear, because the way in which we answer it (personally and collectively) has a tremendous impact on curricula, on the decisions that we make in the classroom, on our students’ learning and enjoyment of the subject, and on their future. Although we ~~might~~ will disagree about the purpose and the goals of teaching mathematics, we can start by recognizing that, at the very least, most of us agree that mathematics is valuable, and that (to some extent or other) it is worth learning. Even Hacker would sign off on that.

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Last week, Dan Meyer wrote a brief reflection on Ed Beagle’s First and Second Laws of Mathematics Education:

- The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
- Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

The second law particularly resonated with me, a soon-to-be teacher. The more I learn about mathematics education, the more I realize that there is still so much to learn, and that anyone who says it’s simple is selling you something (Dan Meyer). My to-read list is growing longer and longer, even as I realize more and more fully that what matters most is not what I read, but what I *do *at the ground level with my students. (Side note: Last week I also began my foray into John Mason’s work – thanks, Danny Brown.)

At the same time, I worry that the talk that Dan linked to is too easily misinterpreted, especially if the date (1971) gets lost in translation. In the article, Begle provides a long list examples of beliefs about mathematics education that have proved to be false and innovations that have been found to be ineffective – almost enough to make a mathematics teacher, especially one on the cusp of beginning her career, throw her hands up in the air and give in to the kind of extreme relativism or skepticism that leads to complacency. In other words, as Raymond Johnson wrote, “the kind of attitude that verges on fatalism: ‘Nobody seems to know the ‘best’ way, so who cares if I just go with my way.'”

Now, I’m certainly not about to throw in the towel. I have certain beliefs about the teaching and learning of mathematics that aren’t shaken every time I see a meme about the supposed failure of Common Core, though I am open to reevaluating these beliefs should research and experience suggest it necessary. I know (or, at least, I feel) that Begle’s goal was to arouse some necessary humility in the field of mathematics education and to spur on further research. I agree with Dan Meyer that Begle’s words are still very relevant today, and that they may even offer a sort of comfort (though, again, not the sort of comfort that leads us to complacency).

What I am wondering, however, is this: What *have* we learned since Begle’s talk? Raymond Johnson suggests that we’ve made enormous progress, and I’m inclined to believe that this is true. Luckily, Danny Brown recently provided a list of key ideas he feels are important to teaching mathematics, and I’m very much looking forward to digging into it (my to-read list grew rather substantially after this post). I’m hoping other math educators could offer their own perspective:

**What do you feel are one or two key understandings about the teaching and learning of mathematics that we’ve gained since Begle’s talk in 1971? (Bonus: Where is our understanding still lacking?)**

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**Last cookie** is a game for two players. In this game, a number of cookies are divided between two cookie jars, and each jar has at least one cookie. Each player can take cookie in one of two ways:

- By taking any number they like from just one jar or
- By taking the same amount from both jar.

The winner is the person who takes the last cookie(s).

Some of you may recognize this as Wythoff’s game; on the NRICH website, which has a great online version of the game, it’s called Last Biscuit (I modified the name for a Canadian audience). If you aren’t familiar with the problem, I highly encourage you to give it a go – very easy to understand and play, but the optimal strategy is not particularly obvious.

I assigned this game as homeplay (get it?? because it’s a *game*… terrible, I know). Then, the other day, we picked it up again after about a week of rumination.

To my surprise, one of the students *had* come up with the optimal strategy, which (spoiler alert) revolves around “cold piles” and “hot piles” (credit to Wikipedia for the terms): when the piles are “cold” (e.g., 1 cookie and 2 cookies), the player whose turn it is to draw will* lose* with best play; conversely, when the piles are “hot” (e.g., 1 cookie and 3 cookies), the player whose turn it is to draw will *win* with best play.

I let the students play the game for a bit in pairs to test and refine their strategies; however, there seemed to be a roadblock that prevented productive exploration: because one of the students had already *found* the optimal strategy (and was* very* vocal about it), the students seemed to feel that the work was done – even though this strategy was not universally understood. Why were the pairs 1 and 2, 3 and 5, and 4 and 7 special? Because a (peer) authority said they were. Case closed.

Luckily, I had something in my back pocket. Having done some research about Wythoff’s game, I came across this article, which shows that it is isomorphic to a came called “Cornering the Queen”. (This game is played with a chessboard and the Queen piece. The Queen is initially placed in the far right column or in the top row of the chessboard, and can only move south, west, or southwest. The player who gets the queen to the lower left corner is the winner.) Not mentioning the connection, I gave the students some paper chessboards and let them play in pairs, keeping an eye out for optimal strategies.

This turned out to be far more productive – and if you compare the intermediate work of one group below to the images in the article cited above, you will see that the students were hitting the main ideas. Eventually, *all *of the groups found the first few “cold positions” (actually, they called them “dead zones”) and were able to make the connection between Cornering the Queen and Last Cookie. They were even able to extend the chessboard to find *more *dead zones. Brilliant! A happy ending! Except…

Even when *everyone else* had found the pairs, the student who had found them first through Last Cookie (and was therefore quickly able to apply them to Cornering the Queen) still felt that he “owned” the solution: “They’re *my* numbers!” he repeated – jokingly, but with a slight air of resentment. He found them *first,* after all. “Nobody *owns* numbers,” I replied politely, but this did not seem to help. I think that some of the students left feeling that they somehow had less of a claim on the solution, even though they had all worked so hard to find it.

But, in today’s day and age, don’t many teachers *want* students to “own” their knowledge? I think that many of us want to get away from the scenario where it is the *teacher* who owns the knowledge, doling it out to passive recipients who are expected to gulp it all down, no questions asked. I certainly do. So what was wrong with this situation? I’m trying to pin it down, and I think that perhaps, what we may actually be looking for is a sense of *community *ownership of knowledge among students in our math classrooms. This was certainly missing in the situation I described. So how do we build it?

I think I’ll need to refer back to *Strength In Numbers, *Ilana Horn’s excellent book on community learning, for some ideas. However, I would also love to hear what you think. Have you had similar experiences? Do you strive to foster individual or community ownership of knowledge in your math classroom? Perhaps you have a different perspective on the issue here? Feedback would be very much appreciated. (In other words: help!)

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