Us Math teachers have been talking about lesson plans. “If something is worth learning, then it should be in the lesson plan” is one viewpoint. Substantially changing things on the fly is a sign you haven’t thought hard enough about the learning. Small adjustments to deal with student misconceptions are fine, but don’t diverge for a significant chunk of a lesson.
I disagree. I think that every chance to respond to student’s instinctive thoughts should be grabbed with both hands, as a way of mimicking the process of discovering Math and to pivot around student interest. No matter how hard you think before a lesson you cannot predict what will happen in the heat of the moment. I find that lessons that follow the plan lack zest and joy. When totally unexpected things are happening the students get excited, and I get excited.
Here is what I think is a great example of when not following a plan is great.
 Mr J: Does anybody have any questions about the Pythagorean theorem, before we move on?”
 D: “I just noticed… 5 squared is 12 + 13. 3 squared is 4 + 5”
Note that D did not answer my question at all. I was looking for people who found Pythagoras hard. He just wanted to share an observation.
At this stage, I had two options. Ignore and continue on with the plan, or explore this pattern further. In the end we explored it a tiny bit, then the class voted on which path to go down. They voted unanimously to explore D’s path, which ended up taking the rest of the lesson (30 minutes)
 Mr J: “But 8 squared is 64, and that is not 15 + 17”
 D: “Yes, but 64 is double 15 + 17”
 Mr J: “Why do you suddenly double?!”
 L: “Maybe because the difference between 17 and 15 is 2?”
And now, the game was on. I scribbled a plan on the board:
 Google a list of Pythagorean triples
 Test the conjecture with some more triples
 Prove the conjecture
Students frantically finding the largest triples they could find. Navigating the mathematical internet is a good skill to encourage. Testing, finding counterexamples, realizing their mistakes (“I thought the difference was 25, not 15!”), finding more conjectures (“every triple with smallest number odd contains two consecutive numbers”). G moved to an algebraic statement of the theorem, and noticed it reminded her of something from her Algebra days – the difference of two squares.
A final pattern at the end of the lesson, for triples where the smallest number is even…
 2 lots of 8 is the average of 15 and 17
 3 lots of 12 is the average of 25 and 37
 4 lots of 16 is the average of 63 and 65
What is going on here?
This conjecture is approaching one algorithm for generating triples:
Thoughts:
 Was everyone involved? How do I monitor engagement while also thinking flatout as a mathematician?
 How to persuade my teaching colleagues that this is worthwhile? I am starting from the assumption that the purpose of school Math is to give students a chance to act like Professional Mathematicians. Is that true?
 Payton is currently trying to move towards a model where students in different classes get the same experience. How can that mesh with the idea that lessons should be substantially changed by student conjectures?
By the way, we started this lesson with students reflecting on why they study Math:
Why do I study Math?  If I had to guess, what would I say are Mr Judge’s reasons? 

