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.