## 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.

NatBanting

**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.

NatBanting

References
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.

NatBanting

## Sunday, February 19, 2017

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.

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.

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.

Summary:
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.