Monday, April 24, 2017

Prime Climb Puzzles

Let it be known that I am not a huge fan of math board games. That being established, I have tried on multiple occasions to create one that I like because the undeniable engagement factor is there. One of two things always seems to happen to my attempts:

  • The game does nothing to change how students interact with the mathematics. Rather, it divulges into an attempt to get students to complete drills in order to win points of some type. Here, the math and the game exist as ostensibly separate entities. 
  • The game mechanism does not support flexible mathematics without a plethora of complicated rules. In an attempt to ensure that the first problem does not occur, the game soon balloons out of control until the simplistic spirit of gamification is lost. 
Prime Climb is the first game I've encountered in a long while that avoids both these follies. Also, it has the added benefit of prompting students to work flexibly with the four basic mathematical operations. Often times, games in class are justified loosely on the grounds that they will induce some type of logical thinking. Prime Climb had my students thinking about number strings and composition of numbers within a larger strategy. All of this was tied up with the healthy competition that led one student to declare, "it's like Sorry! for nerds!"

For Canadian audiences, I would peg the mathematics somewhere between a Grade 4 - 6 level. It works with many combinations of the four basic operations, and the elegant board design triggers conversations about prime numbers and factors. It is designed for up to four players, but we played in four teams of three members. Having the groups make communal decisions only made the thinking more audible, adding further value to the experience. 

After we played for a couple days, I introduced my students (who were in Grade 9) to the idea of #PrimeClimbPuzzle because I wanted them to experience a greater creative challenge. The inventor of the game, Daniel Finkel, tweeted a couple of puzzles where a game board situation was imaged and then a question was posed asking the solver to determine what sequence of actions (based on the game rules) led to the provided situation. My class an I went through the two available puzzles and then set out to create our own. 

I am in love with the results. (Which, perhaps is ironic considering the name of Dan's company is called "Math for Love"). Students worked very hard to come up with an interesting hook for the puzzle. On top of that, beta-testing their puzzles furthered the flexible arithmetic that the game initially inspired.

Here are four examples of their puzzles; all eight of their puzzles can be found in this folder. You will have to experience the gameplay to fully understand the puzzles. I recommend you find a way to play the game in your room. 

The "How did we get here?" puzzle.

The "Multiple kills" puzzle.

The "Where did we start?" puzzle.

The "What colour are we?" puzzle.

These puzzles have all been tweeted out using the hashtag #PrimeClimbPuzzle. If you and your students are so inclined, feel free to add to the collection. 


**Thanks to Jenn Brokofsky for providing two copies of Prime Climb for my class to borrow and to Ali Alexander, the photo teacher in my building, for taking the pictures.

Thursday, April 13, 2017

Constraining the Two-Column Proof

There is no dedicated course for geometry in Saskatchewan's secondary curriculum. Instead, the topic is splintered amongst several courses. There are advantages and disadvantages to this, neither of which will be the focus of this post. I just thought that, especially for the non-Canadian crowd, a glimpse of context would be helpful.

The notion of a geometric proof only appears in one course. It is presented as a single unit of study during a Grade 11 course and is preceded by a short unit on the difference between inductive and deductive reasoning. I have taught this course a lot over the past few years, and have always had mixed emotions toward this portion. I love the metacognitive analysis students participate in during the inductive v. deductive reasoning unit. It is a (metric) tonne of fun to teach because it largely entails the completion of games, puzzles, or challenges and a subsequent interrogation of our thinking patterns. This could be my favourite week and a half in the course. After we have experienced the difference between induction and deduction, we spend a couple weeks slogging through angle relationships and parallel lines, triangles, and polygons using the ultimate edifice of deductive reason: The two-column proof.

Let me be clear, I like the two-column proof. It is clear and elegant; its syncopated logical steps appease my brain. However, over the years, I have watched as the emphasis on metacognition slowly fades into an emphasis on rules and their rigid application. 

This year as I was designing a lesson, I tried to design a diagram that would not allow students to use supplementary pairs of angles to move toward a solution. I had noticed this justification emerge several times over the first three days, and I wanted to introduce a greater variety. As I was building the diagram, it hit me:

If I don't want them to use supplementary angles, simply mandate them as off limits. 

It is an example of what complexity thinking (as it has been applied to math education) might call an "enabling constraint". That is, a restriction placed on otherwise virtually limitless possibilities in order to perturb a system's action. "The common feature of enabling constraints is that they are not prescriptive. They don't dictate what must be done. Rather, they are expansive, indicating what might be done, in part by indicating what's not allowed" (Davis, Sumara, & Luce-Kapler, 2015, p. 219). By restricting what can be done, action orients itself to the possible. The divergent paths of deduction that emerged through this simple constraint amazed me. The density of mathematical activity made me kick myself for not thinking of it earlier. 

The next day I made a change to the scheduled work period:
  1. I took the diagrams from the textbook questions and put them into presentation slides.
  2. I randomly grouped the class into groups of three and supplied them (as is customary in my room) with a large non-permanent surface and writing supplies. 
  3. I circulated and gave each group a "restriction", thus creating a variety of enabling constraints. 
  4. I projected a new deductive proof task on the board.
  5. Each group completed the problem within their restraints. (If they believed that it was impossible, they needed to supply reasoning as to why). 
  6. Groups visited a neighbouring board and checked the proof for accuracy and validity.
  7. Groups then took on the enabling constraint of that group. 
  8. Returned to Step 4 until the bell rang. 
The restrictions I used are as follows:
  • Cannot use Supplementary Angles
  • Cannot use Alternate Interior Angles
  • Cannot use Corresponding Angles
  • Cannot use Same-Side Interior Angles
  • Cannot use Vertically Opposite Angles
  • Must use Vertically Opposite Angles at least twice
  • Cannot use the fact that angles in a triangle sum to 180 degrees
  • Cannot use the same justification more than once
  • Must "forget" one piece of given information
  • Cannot have a line in the proof that does not deduce an angle required by the task
From a lesson design standpoint, this is a the low-prep-high yield classroom task. I simply used the diagrams provided to me in my resource. From a conceptual standpoint, several nice opportunities arose during the class:
  • Vertically opposite angles are just "double supplementary" angles
    • This is what one student said in the midst of complaining that an adjacent group's restraint was not near as restrictive as theirs. I took the opportunity to pause the classroom hum to ask them to expand on what they meant. Students then began to notice relationships between the justifications. (Corresponding & vertically opposite are just alternate exterior angles, etc.).
  • Students questioned notation
    • They quickly gained a new appreciation for clear communication via notation as they examined classmates' work. It was a nice alternative to the customary lecture on proper proof technique. 
  • Students encountered the notion of unsolvable proofs
    • I did not test to see if each proof was possible before the class. This was intentional. On four occasions, a constraint rendered the task impossible. Rather than critique this as a failure in design, it became a learning opportunity. On three of the four occasions, a neighbouring group joined to help deduce a solution. I reflected afterwards on the sad reality that this may have been the first time that students encountered a problem that was unsolvable. It also gave me a chance to use one of my favourite sayings: "No solution is a solution". 
  • Led nicely into proving that lines are parallel
    • It was much easier to speak about the notion of proving lines parallel with angle relationships once the idea of restriction had been introduced. The process of using special angle relationships to prove lines parallel became one where I "restricted" the use of alternate interior angle, alternate exterior angles, corresponding angles, and same-side interior angles and asked them to prove that at least one of the first three angle relationships resulted in congruency (or same-side interior angles summed to 180 degrees). I had never discussed this topic from a stronger conceptual base.
The whole thing seems oxymoronic at first. How can limiting action actually result in more interpretive possibility? From a systems standpoint, a familiar pattern of action is disturbed and, in doing so, a variety of (perhaps) unanticipated possibilities can then be activated. The job of the teacher is to participate in this possibility--collecting, commentating, and providing more perturbations along the way. A process that is possible even with the structure-heavy two-column proof. 


Davis, B., Sumara, D., & Luce-Kapler, R. (2015). Engaging minds: Changing teaching in complex times (3rd ed.). Mahweh, NJ: Lawrence Erlbaum Associates, Inc.

Saturday, March 25, 2017

Experiencing Scale in Higher Dimensions

A colleague and I have often bemoaned our attempts to teach the concept of scale factor in higher dimensions. A topic that has such beautiful and elegant patterns and symmetries between the scale factors consistently seems to sail directly past the experience of our students. I have tried enacting several tasks with the students including some favourites from the #MTBoS (Mathalicious 1600 Pennsylvania and Giant Gummy Bear). Each time, the thinking during the task seems to dissipate when new problems are offered. It just seems like students have a hard time trusting the immense rate that surface area and volume can grow (or shrink). In the past, I had used digital images of cubes growing after having their dimensions scaled by 2, 3, 4... etc.; students seemed to grasp the pattern yet under-appreciate the girth of 8, 27, 64... etc. times as many cubes. 

For this reason, I went looking for a way to concretely demonstrate these phenomena. I wanted them to feel the weight of increasing volume. 

I settled on a simple construction activity, one that I'm sure has been enacted in several mathematics classrooms. This post isn't as much about a great idea as it is about a reminder that simple designs can sponsor elegant mathematical action. 

I randomly grouped them into 3s, and assigned each group a total number of linking cubes they were permitted to use. These values ranged from 6 to 9 total cubes. Each group was instructed to combine the cubes to make a shape of their choosing. Now, anyone who has taught middle school needs to know that the irresistible tendency for students to build guns and swords does not dissipate by the 10th and 11th grade. Many, but not all, groups created designs that laid flat on the table. That is, they had a "width" of 1 block. At first I was hesitant about allowing this characteristic to crosscut through each group, but it turned out to be very valuable later on. (see The 2D Builders).

When all the shapes were complete, we defined each dimension of a cube as 1 centimetre. This made "calculating" the dimensions, surface area, and volumes of the arrangements a matter of counting. 

The groups were each asked to determine the surface area and volume of their linking cube arrangements. They kept these values on their group workspace, and then I gave the prompt:

Build a new, larger shape where each of the dimensions is doubled.

After some talk about what counted as a dimension (in which notions like length, width, height, depth, three-dimensional, and two-dimensional all came up), the groups set off to work building their enlargements. It wasn't long until I could see very distinct strategies emerging.

The 2D Builders.
Group builds an enlarged cross by layering two separate cross designs.
These groups all built shapes that were one cube "thick". They constructed their new, enlarged arrangement by doubling the length and width of the shape. Several groups left their work here, and needed to be prompted to double the "thickness" of the shape as well. Generally, I took a subtle approach to this intervention. I would ask them to prove that they had doubled every dimension, and mention, in passing, that the heights still matched--total passive-aggressive teaching move.

The 1-is-8 Builders.
Group builds an enlarged shape by re-enacting the initial construction with larger constituent cubes.
This strategy seemed to emerge from groups with more complicated arrangements. In order to double every dimension, they dissected their original shape into the constituent cubes and doubled each of their dimensions. It wasn't long until the news spread around the room that one cube just becomes eight cubes (a fact that led perfectly into the discussion of volume scale factor). The new, enlarged shapes were then built in an identical fashion as the originals, by constructing new, larger cubes in groups of eight, and then attaching them together.

The Partial Doublers.
A "camel" shape is partially doubled.
A "stairs" shape is partially doubled.
These groups viewed their arrangement in some type of holistic way, as a shape they recognized from the world. (Camel and Stairs shapes imaged above). This naming seemed to cause groups to double some dimensions and overlook others. The camel above has its "hump" fully doubled and its "legs" length doubled, but the width of its "torso", "neck", and "legs" left identical to the original. The stairs above have their height and width doubled, but the height and length of each individual stair is not doubled. These shapes led to the most interesting conversations. 

The Dimension Doublers.
Group builds new, enlarged shape by doubling every edge of the original shape.
These groups did not dissect their original shape, but counted (and often recorded) each dimension. Then they simply doubled each number and re-constructed a shape based on the new blueprint with little reference to the original shape. This abstraction resulted in accurate enlargements. 

When I envisioned the activity, I thought I would spend most of the time focusing on the results of the doubling, but the conversation regarding the various strategies was too irresistible to ignore. After I had groups describe how they went about completing the task, I asked them all to calculate the surface area and volume of the new, enlarged shape. We created a table at the front of the room to record the results of the constructions. 
I then collected the results from the groups. As we went along, each response was recorded without judgement from me. There were times when the emerging pattern fell through and students alerted me to this fact. I would ask for clarification, place a "?" next to the entries in question, and ask the groups to double check their calculations. By the time the last few groups were offering their values, they had become completely predictable. 

The best part about the activity was that students could not believe the size of the new shape after just doubling. They passed them around the group as if to feel the sheer weight of such a small dimensional scale factor. One student commented, "Imagine if we would have tripled the length!"

Opportunity knocks, so you open the door. We spent some time predicting what would happen to the surface area and volume if we tripled each dimension. These were exactly the types of conversations I had hoped to trigger. When I do this again, I think I will have some type of sticker that I will ask groups to use to "count" the surface area. I feel like we spent a lot of time experiencing how much greater the volume grows, but glossed over the growth of the surface area. Having them place a sticker on each face might begin to build an appreciation for the growing surface area as well. 

To reiterate, this idea is not new, nor mine. I would credit it to two things. First, the incessant joy Alex Overwijk gets from playing with linking cubes. (I am often jealous of the uses he finds for them). Second, the recognition that great thinking can emerge from modest tasks and problems, and it is the ability to remain sensitive for those opportunities that can create powerful learning experiences from meager beginnings. 


Sunday, February 19, 2017

Solid Fusing Task

The progression followed by most teachers and resources during the study of surface area and volume is identical. Like a intravenous drip, concepts are released gradually to the patients so as to not overdose them with complexity. Begin with the calculation of 2-dimensional areas, and then proceed to the calculation of surface area of familiar prisms. (I say prisms, so a parallel can be drawn to the common structure for finding the volume of said prisms. That is, [area of base x height]). In this way, surface area is conceptualized as nothing more than a dissection of 3-dimensional solids into the now familiar 2-dimensional shapes. 

This is not a condemnation of this logical progression; the visualization necessary to dissect prisms is surely an important spatial skill. The trouble for me is in the nature of the problems presented to students. Actually, my problem is that every problem is, in fact, presented to students--ready-formed with a pre-determined solution procedure. Every decision has been made; there is nothing left for students to do. 

Now this is not uncommon throughout school mathematics, but it seems to bother me more with surface area and volume because they present themselves, at least for me, as topics accessible to student reason. Teachers may choose use a variety of physical models, and the concepts can be modelled directly with pictorial representations. They also are concepts that students grapple with at a very young age, making them an area of personal expertise. All these things considered, I find it unfortunate that students are not asked to make a single decision with regards to their manipulation.

The most ridiculous part of the process is the images that appear in the section entitled "Calculating Surface Area and Volume of Composite Figures" (Or whatever your resource decides to name it). Here, we find completely random arrangements of solids, fused in every which way to each other. While some books attach gestalt-like connotations to the fusings (such as the cone and hemisphere that appear to be an ice cream cone), many unabashedly leave them for exactly what they are: Pre-fabricated and essentially random arrangements of solids. 

Building on a common thread that has emerged through my blogging over the years (see here and here for examples), I attempted to design a task that would allow space for students to make decisions about the surface area and volumes of such arrangements of solids. I wanted to create a prompt where there was a larger goal which might allow the concepts of surface area and volume to become relevant during its resolution. In short, I did not attempt to challenge the ridiculous patterns of idealized fusings present in every textbook. Rather, I made them the focal point of a mathematical decision. My intention by doing so was to trigger a deeper mathematical engagement with the topics as students used them to address the problem that emerged as relevant in their active, mathematical decision making. 

The task:
Students are given a set of six solids. It includes a cube; two cylinders; a right, square pyramid; a right cone; and a hemisphere. Rather than provide them with a pre-ordained arrangement of the solids, the task makes the arrangement the key mathematical decision to be made. 

I explain the parameters for a successful fusing. Sides must be fused completely to other sides. Portions of faces cannot 'hang off' or partially fuse. Also, fusings must all be formed between faces. In other words, they could not balance the cube on the dome of the hemisphere, but could fuse one of its faces to the circular underbelly of the hemisphere.

Their task is as follows:

Combine any number of the six solids provided to you to create a shape that has a surface area (in square units) as close as possible to its volume (in cubic units). 

I created this slide to aid in the introduction of the task. 

Anticipated student action:

1) Calculate and then fuse
A logical reaction to a mathematical choice is to gain an understanding for what one is working with. These groups first determine the surface area and volume of each of the six solids and then arrange them in a variety of combinations (usually based on some type of rule). Some choose to fuse the solids that have very similar surface areas and volumes; others choose to 'heal' the shapes with large gaps in surface area and volume through fusing. Either way, they work intimately with both the formulae and the idea of overlapping surface area. 

2) Stacked arrangements
These folk notice that when solids are fused, their surface area is affected but their volume becomes a simple summation. It is then deduced that if they start with a prism that has a greater surface area than volume, they could stack them repeatedly until the values are quite close. Luckily for the teacher, there are plenty of stacking permutations available. Once again, this involves interaction with the calculations as well as a deeper layer of reasoning superimposed through the new decision making required by the task. 

3) Plus-minus arrangements
This strategy uses a calculation of the surface area and volume to develop a third 'statistic' for each solid: Net loss. Here, students calculate the total effect of fusing a solid to an existing arrangement. They do so by subtracting the fused side from the surface area calculation and then comparing it to the solid's volume. If a fusing adds 45 square units to the surface area (taking into account the effect of overlap) and only 35 cubic units to the volume, that fusing has a +10 effect to surface area. Alternatively, some may think of it as a -10 effect to volume. In order to approach surface area and volume parity, solids are chosen to address the current gap of the arrangement. If their fusing currently has 25 more surface area than volume, a solid should be fused that has a negative effect on surface area. That is, would result in a larger gain in volume than surface area, thus narrowing the gap of the arrangement. 

The task is offered with the simple intention of allowing students the space to make mathematical decisions. It shifts the passive digestion of fused solids into an active creation. In doing so, it does not eliminate the opportunity for students to practice calculating the surface area and volume of composite figures. Rather, it couches this execution in a larger strategy. The calculations are deemed necessary to address a problem that emerges through mathematical action. That is, perhaps, the heart of it all. Rather than students being relegated strictly to problem solvers, they are provided opportunities to become decision makers--problem posers. 

Tuesday, November 1, 2016

Real-World: An Attack on "Relevance"

**deep breath**

Last week, I caught myself saying something to a pre-service teacher as we planned a Grade 1 lesson for the making of 10s. I asked her, 
"Why would the students need to know how to make up 10s?"

When she was auspiciously silent, I filled the space with a statement said entirely tongue-in-cheek. It was only upon reflection, that I kicked myself for not being able to shut up and allow her to think. I said,

"...because my job is to convince teenagers they need logarithms, and that is much more difficult."

Now, aside from the unwarranted attack on logarithms and the ridiculous and demeaning insinuation that a high school mathematics teacher somehow has drawn a short straw, this statement lurched the idea of relevance to the forefront of my mind. I have a special place reserved for the debate of relevance, repressed deep into the damp catacombs of my consciousness. There is good reason for this, because every time it creeps up, long-winded and ranty blog posts are written. 

**deep breath**

So what is my problem with relevance and math's application to the "real world"? (pause to allow the shudders down my spine to digress). Well, in a nutshell, I think that justifying the study of mathematics through a claim that we are somehow doing kids a solid by preparing them for the real world a) ignores the fact that for students, school is their real world, and b) is an extension of a myriad of things us teachers do just to appear "teacher-y". 

I lump finding strained applications to the "outside world" in the same category as creating rubrics with vague language gradations so we can somehow move closer to understanding the difference between "often" and "frequently". These are hairs that never need to be split, because in doing so, we cheapen the activity which we are trying to understand. In the case of mathematics, grasping at some semblance of relevance is insulting. 

Here, I would like you to image a well-reasoned exposition of all the possible justifications for teaching math. Math as beautiful, math as employable, math as a pursuit of pure reason, etc. 
You can imagine my surprise while I was listening to a podcast during a morning walk and heard mathematician Edmund Harris say:

"In a massive intellectual land grab, I claim mathematics is everything that you can think about without reference to the real world."


You are trying to tell me that the very thing that defines mathematics is the only thing we, as teachers, try to use to get kids to learn it? This could stand as the largest piece of educational irony in history. 

Here is my alternative; I believe it provides a much steadier footing for mathematics education as well as a productive pedagogy therein. What if we stopped thinking of math as tool only useful for other things, and began to think of it as a discipline that a) has self-contained beauty and utiliy and b) has the sneaky tendency of becoming relevant to the world around us? Why do we need to tether worth to application? Can't application be some sort of serendipitous occurrence? A cherry on top--so to speak?

Skeptic: Fine then, but if we lose the "real-world", how will we motivate students?

Me: Glad you asked. 

First, real-world is a crappy motivator. Again, that is teachers being teacher-y. We are creating a reason why we think this would be worthwhile, and projecting it. 
Second, if we forget about relevance and "real-world", the project of schooling stops being one of spoon-feeding isolated topics, each with some white-washed and over-simplified connection to the natural world. The project of teaching mathematics is creating the conditions to harness the inescapable human tendency to make meaning. Our job isn't to tell why something is relevant, it is to open a space for which the problems become relevant. 

Skeptic: Wow... how romantic.

Me: Isn't it, though?

Think about your "best" or "favourite" lesson. Seriously interrogate the relevance it held. Here are two examples of low-hanging fruit from the internet. First, Barbie Bungee is an ageless task where students are asked to determine how many elastics will allow a Barbie to safely bungee jump off of a certain height. This task occurs in the real-world, but I am doubtful that a student's full-time profession will fall at the juncture of extreme sports and children's toys. 
Second, Marble Slides (from Desmos) are amazing. I have witnessed student forays into uncharted territory of engagement. Sure, they evoke the principles of physics, but, again, I don't recall seeing too many job postings or graduate degree programs being offered in marble rolling. 

The point is not to attack these lessons, but rather point to what makes them great. They are so irrelevant that students find themselves lost in their contexts. They afford them a chance to return to their roots as natural mathematicians and entertain their irresistible urge to make meaning. 

Please stop looking to embed math class in reality. I'm begging you. The task of teaching mathematics is actually much more difficult than that. We need to focus on creating experiences where students can suspend reality and simply be lost in a state of mathematical vertigo for a while. 


Monday, August 22, 2016

100 Rolls Task

Most probability resources contain a familiar type of question: the two-dice probability distribution problem. 

Often times, it is accompanied with questions concerning the sums of the faces that appear on each dice. 

For example:

Roll two fair, 6-sided dice. What possible sums can be made by adding the faces together?
What is the probability that:
a) the sum is 6
b) the sum is a multiple of 4

c) the sum is greater than 15?

I think the obsession with this specific subdomain of probability questions stems from the elegant way in which a table of outcomes (pictured below) leads to a counting of favourable and total events. 
The sums of two dice neatly organized into a table.
Shifting the problem ever so slightly offers a new sub-domain of question concerning the product of the two dice. A typical problem might look like:

Roll two fair, 6-sided dice. What possible products can be made by multiplying the faces together?
What is the probability that:
a) the product is 6
b) the product is a multiple of 4
c) the product is greater than 15?

Note the small shift necessary to surround the problem with the same structure of a table of outcomes (pictured below). I am assuming that most teachers of mathematics could construct numerous additional sub-questions of the same variety. The question grows in size, but remains one that requires simple counting.
The products of two dice neatly organized into a table.
I believe these types of questions are a direct result of math teaching's infatuation with tidiness, especially within the content domain of probability. This is unfortunate, because probability offers accessible forays into imagination and allows intuition to play the dual, contradictory role of guiding light and devil's advocate. In other words, posing an elegant probability problem leaves the solver balancing their wonder (driven by the question, "what if?") with their rationality (driven by the assertion of, "that can't be!") 

In an effort to make this domain of two-dice probability distribution problems more interesting, I flipped* them.

I rolled a pair of dice 100 times. I multiplied the result from each die after every roll and recorded them in this table.

What numbers appeared on the faces of each die?

[Sidenote Rant:

It should be noted that opening up a problem to divergent and emergent action does not de-value the elegant organizations afforded to mathematics. Many think that rich, open tasks are just a rejection of neat and orderly work. However, open tasks signal a rejection of the notion that interaction with a problem should be neat and orderly, and, by extension, that learning mathematics is the process of achieving lockstep with an ideal way of acting. The learning of mathematics is a holistic, messy, and convoluted experience filled with triumphs, dead ends, and ways of representing action. In order to harness these movements, the problems need to allow space for them. The structure of the table of outcomes may well emerge as useful to students working on this problem. In fact, as you will see below, it was eventually encouraged, but not legislated. 

End Rant.]

As a frame of reference for the students, I offer them the following image of 100 products from two standard dice. 

The problem posed is intentionally vague. I want students to develop the culture of precision, and demand that the problems I present meet the same requirements. I expect students to ask a myriad of important clarifying questions:

  • Are both dice identical? 
  • Can a number appear on a die more than once? 
  • How many faces are on each die? 
  • Can the faces have fractions or decimals?
  • etc. 

In this case, I refine my parameters to two dice, each with six sides. The dice do not have to be identical, and all faces from both dice feature whole numbers. A number can appear more than once on a single die, and that will be reflected in how many times certain products are created. 

Part way through their action with the problem, I provide them with two handouts. This helps them structure their work (if necessary), and provides them an artefact to hand in to me when the time comes. 

The first is a sheet with a completed table of products from two normal dice (similar to the first image in this post) and the corresponding nets of the dice that would create the distribution. (download .pdf here). The second is an identical sheet with the table and dice nets left blank for them to work with when creating the dice that resulted in this new distribution. (download .pdf here). 

I give each group a single copy of each handout (to eventually hand in) and have them do the majority of their work on a shared whiteboard space. I find the communal (and non-permanent) workspace to encourage a greater density of neighbour interactions.

Anticipated student action:

I've chosen to organize anticipations of student action around possible questions that 1) groups may pose during the course of action or 2) teachers may pose to trigger student action or perturb student reasoning. 

(1) Will all possible combinations of rolls occur in the 100 trials?
This question gets at the very heart of probability. I guess, in theory, the answer is: No, not every possible pairing must have been rolled in the 100 trials. But there are three caveats to this: First, every product must be possible. That means that all products must have at least one pair of factors that exist on opposite dice. Second, how many products are there? If there are less than 36, that doesn't mean that some did not occur. It could mean that some have multiple factor pairings that produce them. Third, each outcome has a 1 in 36 chance in happening. Even if each product were unique, there would still be a good chance that each occurred in 100 rolls.

(2) Does where you place the factors make any difference?
We assumed the dice were fair so where you place the numbers on a single die will not matter. However, switching numbers between dice will change the possible outcomes. Keep in mind that each product needs to have one factor on each die. 

(3) How often will certain outcomes occur?
My guess is students will set up some type of table of results and compare the experimental results with a dynamic theoretical probability as they build their dice. They will feel comfortable when their theoretical calculations closely match the experimental results. If they set up a system where "6" should be rolled 5 times in 36, but only appears 7 times in the 100 trials, are they wrong? What is the margin for error? What if theoretically rare outcomes actually appear more often than theoretically common ones in the trial? Should we be switching numbers, or assuming that the trial resulted in a deviance from expectation? 

I think the beauty of the problem is in this question. Arguments are rooted in theoretical mathematics (which are most likely emphasized in the curricular outcomes), yet guided by experimental probability. In the middle space between the two exists mathematical intuition.

(4) What do prime numbers tell us?
Well, a prime number has two factors, so we know those two factors must exist on opposite dice. Also, one of those factors needs to be a "1". The case of "1" is interesting. How can we know that a "1" won't exist on both dice? If we know a single prime product exists, can we be sure that others must exist? 

(5) What do products with only four factors tell us?
Every product will have "1" and itself as factors, but what if there are only two other options? Take the example of "27". We are fairly certain that "1" and "27" didn't multiply to achieve "27" because if "27" existed on a face of a die, we would (likely) see other multiples of "27" in the results. There is one (54), but it seems unlikely that either the other dice is dominated by 2s and 1s, or the "27" just wasn't rolled in conjunction with the other options. It leaves "9" and "3" as excellent options for dice sides. 

(6) What do perfect squares tell us?
The knee jerk reaction here is to assume that a perfect square is created by the multiplication of identical factors. There is something attractive about that symmetry. It is not true in the case of "4" and "9", and gets even more convoluted in the case of "36". This assumption, of course, ignores the possibility of pairing the perfect square with a "1". It is interesting to note that a "4" must be constructed with two 2s if there is no "1" / "4" pairing possible. The same goes for the product of "9". The existence and placement of a "1" gains importance yet again in this case. 


If your goal is to study factors and products, allow the probability to slink into the background and remain governed by intuition. Focus on the composition of the numbers and the characteristics of primes, perfect squares, and common factors. 

If your goal is to study probability, use the factors as a vehicle to have conversations of likelihood. Encourage formality to justify conjectures of probability, and live in the very real tension between experimental and theoretical probability. 

The task doesn't throw out the representations of classical probability or the theorems of factors and products. Rather, it opens a space for students to act on both topics in ways that emerge as necessary at the time. 

While I don't tend to give out solutions to problems, I feel like teachers are more likely to use this task if they know the composition of the dice that created the 100 trials. (Yes, I created the dice and actually rolled them 100 times).  

Dice 1: 2, 3, 1, 6, 4, 6
Dice 2: 9, 10, 5, 3, 2, 2

My suggestion is to never reveal this composition to your students. I know you, as a teacher, probably have this weird "master of math" vibe and most likely feel incredibly insecure in the uncertainty of all of this--but your students haven't earned the right to be this unnaturally uptight about their work yet. 

As far as classroom assessment, I like to collect viable solutions complete with justification. I attempt to focus on the process of establishing the mathematics, and a grand reveal of the right answer bankrupts that immediately. 

Happy rolling. 


*tongue-in-cheek reference to the idea of "flipped classrooms" of which I remain extremely skeptical.

Wednesday, June 29, 2016

TDC Math Fair 2016: A Summary

On June 15th, my Grade 9 class and I hosted our second annual math fair. What started out as a small idea has grown into a capstone event of their semester. This year, we had 330 elementary school students visit our building to take part in the fair's activities. Several people (following the hashtag #TDCMathFair2016) commented that they would like to do similar things with their student transitions. This post details the rationale behind the event, how we structured it, what stations we had, and feedback/advice from our exploits.

I pursued this opportunity with a two-pronged focus. First, I wanted to showcase a mode of teaching mathematics that relies heavily on student collaboration, conjecture, and re-calibration to a problem situation. I wanted students to experience real mathematical choice after encountering a task that is not immediately solvable. Through their actions, they expand the set of information they have and unlock further opportunities for action. In doing so, they activate curricular understandings. We designed or adapted classroom activities to fit the scale of the event, but their classroom viability remained at the heart of the event. I didn't want the fair to be something that was only possible outside of the classroom walls; I wanted to emphasize to all involved that mathematics has an active character and we can carve a curriculum out of that action.

Second, I wanted an event where school communities would have a chance to network. The transition from middle school to high school is an onerous one filled with social and academic challenges. I wanted future students to meet future peers and teachers as well as become familiar with the building. This is a very pragmatic concern. On top of that, I wanted the opportunity to communicate with the elementary school teachers of my future students. Very rarely do we get a chance to talk. This was one way where I could plan an event for their kids and open lines of communication for future interactions.

The day was built around groups of students rotating through four stations. Each group of elementary students (7-9 students/group) was paired with two of my high school volunteers. The older students acted as guides throughout the event. This was a good ratio; there was hardly a time when a group of elementary students were left alone. There was always someone there to support or extend thinking. I also had a teacher at each station to spearhead the activity. These were pre-service teachers or teachers who just received their certification. That meant that they were available during the school day.

The class and I worked out which groups would rotate to which stations during the four time slots. Each group leader received a rotation sheet and followed it. This worked very well; it allowed us to keep numbers consistent at each station throughout the day. We had 18 groups come in the morning and 20 in the afternoon. It was a full day for the leaders.

I wanted to have an outlet for the leaders if a group of students was operating well below grade level or had exhausted the possibilities at the station. We provided this in two ways. First, there was an estimation station set up where students were asked to find a time to provide an estimate of the number of balls in a large cube I found in the school's storeroom.

Second, we set up a free play symmetry station with a collection of tiles from Christopher Danielson and TMWYK. If there was time to burn, we sent kids there. Only problem was, it was tough to get them to leave. If you are planning an event with this many students, I would recommend you have these safety valves worked into the day.

The Stations:
There were four stations that students rotated through. Each lasted 30 minutes with a 15 minute break between stations two and three for snacks and estimations.

Sweet 16
This activity focused on balancing multiple equality statements. It contains a very low floor. Students can deduce viable options to some of the equality statements. It also has a very high ceiling. To balance one of these, it takes inductive leaps of faith and several revisions. I adapted puzzles from this book into a classroom resource published here. The ten puzzles in my resource are the ten that students worked on at the fair. Students worked on 2ft by 2ft whiteboards with the 16 Boxes template on them.

Here, students played a game of Tic-Tac-Toe where each space in the large grid is filled with another game of Tic-Tac-Toe. The rules are explained here, but we made a single adaptation. If you send your opponent to a board that has already been won, they do not get to go anywhere they like. Instead, they must play in the already won square with no chance of winning it back. In this variant, it is an advantage to send your opponent to a previously conquered square because, while they still get to dictate your next play, they place a meaningless mark. As students approached mastery of our version, we introduced the rule change to keep them on their toes.

Spa-Ghet-Rekt   (the kids named it)
This was an adaptation of Alex Overwijk's lesson. The students were presented with a pile of pennies, more spaghetti than they could ever need, and a cup that had been fashioned into a hanging basket with paperclips. They were asked to simply test how many pennies a piece of spaghetti could support. After this, they were asked to estimate how many two pieces of spaghetti would support, what about three, etc. You get the point. The whole time, we were entering their data into a communal Desmos graph and asking students for their estimations and reasoning before mindlessly piling pennies into baskets.

About half way through, we told the kids that there 184 pennies weighed one pound and asked them to estimate how many pieces of spaghetti they would need to hold a 2.5 pound weight. When they gave their reasoning, we went to the crash mats and tested it. Tension ran high as pieces of spaghetti were removed one at a time to reach a breaking point. This was the highlight for many students. We harnessed mathematical intuition, extrapolation, graphical literacy, and even some conversations on gravity, force, and potential energy.

Word to the wise: this makes a mess. We went through over 20 kilograms of spaghetti in the four hours of running the activity. Our kids did a fantastic job cleaning up, but get the okay from your caretakers before including this station.

Dark Room Escape
A colleague took the lead on designing a dark room escape in a separate room that could accommodate 40 students at a time. He ended up splitting them into four quadrants; each quadrant needed to solve a network of puzzles to unlock their part of the final challenge. If all four succeeded, the group won a prize.

The challenges were very intricate and well thought through. It took a lot of work, but we wanted to have a signature station that students would remember and look forward to next year. We are planning on using some more budget to get more materials to expand the dark room for the 2017 edition of the fair.

The response to the fair was overwhelmingly positive. It was really cool to have an entire educational community pull together for a day. Teachers commented that is was really good to see the consultants, coordinators, superintendents, and trustees in the building and interacting with students. We really were all together as a division devoted to mathematics.

The local newspaper was there and published a short article the next day. My students commented that they were getting texts from friends who attended still thinking about the stations. One student said that she lives on the same street as a elementary participant. As she walked into her house, she overheard the student telling her mom her strategy on Spa-Ghet-Rekt.

Several students attended for the second time and commented that it was "just as much fun as last year" and they "wish they could come back again". I just smiled and told them that they could, but they would be group leaders. I found out later that one school even included pictures of the event in their Grade 8 graduation video. 

The Saskatchewan Mathematics Teachers' Society periodical The Variable published a recap article from the perspective of a teacher volunteer as well as one of my Grade 9 students. 

Perhaps the coolest thing to come from this was a phone call from a local woman who read the article in the paper. She talked to me for ten minutes about her 10-year-old great-grandson who loves mathematics. She was just calling to chat about how mathematics can challenge "youngsters" (her word). We finished our conversation with her saying, "It's just so good for this old lady to see kids still having fun in such an awfully serious world". After hanging up, I sat in silence for a few moments in my classroom. At that moment, I knew we had achieved our goals.