My class always welcomes conjectures. This is made explicit on the very first day of the semester. This goes for everything from grade nine to grade twelve. As the grades advance, the topics have us venturing into increasingly abstract concepts, but conjectures are *always* honoured.

Certain class structures promote conjecturing more than others. Students offer questions during lectures, but they are often of a surface variety. They notice a pattern that has occurred in three straight examples, or think they have discovered a short-cut. I don’t like using tricks, but if they are “*discovered*” or “*re-invented*” (to borrow a term from Piaget and genetic epistemology), then we use them.

Several posts on this blog have been born out of conjectures offered in class. Many task ideas come from shifts in tasks that we were working on. By far the most profitable, conjecture-rich structure has been that of large whiteboards. There is something about the organization mixed with the rich problems and communication obligations that opens student minds. Regardless of class structure for that day, if a conjecture arises, I get everyone’s attention:

After the conjecture is made, we have a conversation around its feasibility and even vote on its validity. Some groups (or individuals) devote themselves to refutation, while others remain in whole-hearted affirmation. It creates an interesting (yet non-competitive) dynamic. Some students even like having their conjectures disproven and get right back to modifying them to make them stand.

When I feel like I have loosened the curricular pressure, I take my students through explicit problem posing (conjecturing) exercises. We use the process documented explicitly in Brown and Walter’s book. I complete this exercise with a task that is solvable in 5-10 minutes and has many discrete attributes that can be changed. That leaves plenty of time to pose new problems, exchange, attempt solutions, and discuss.

**Side note**: *The Art of Problem Posing* is a must read for any teacher of mathematics at any level. It’s algorithmic encapsulation of the fluid process of posing problems jives well with beginners and is extended easily for experts. Like seriously, stop reading this post and get that book. L-I-F-E C-H-A-N-G-E-R.

On this particular day, I gave my grade 9s the “Poison Pills” problem from Stella’s Stunners. (The initial solution offers avenues into factors and multiples)

- We split pills in half and put in two containers. Take pill from #1, place in top of #2 and then eat from bottom of #2.
- Two players. We used three containers. Take one pill from bottom of all three. Ingest one and place the other two back in whichever column you like. Goal is to be last one alive.
- Switch the bottom and top pill. Then take two from the bottom and put them on top. Eat the next pill.
- Instead of taking two pills, you take three. Eat the third and place the first two back.
- Take three pills and eat the second. Return the other pills if you are alive to do so. Don’t take third pill if you die when you eat second.
- Split the poison pills into six half-pills of poison. You need to eat two of them to die.
- Have 16 pills (12 good, 4 poison) and 12 prisoners. Look if pattern still exists.
- Odd prisoners eat first pill and pass the second one back to the top. Even prisoners pass the first pill and eat the second.
- Add one antedate pill that can make one prisoner invincible to poison.

__Mathematical Intuition__

__Mathematical Complexity__

__Mathematical Ownership__

*solving*. While this is a noble pursuit, I am more interested in resuscitating problems and extending their mathematical lives–problem

*posing*. What can we do to revive or extend this curiosity. A good class, in my books, leaves with more problems than it started.