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.

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