Developing a Plan for a Proof - An Analogy

We begin by presenting a strategy game in which there is a key moment before winning where you have already won.  For example, Tic-tac-toe.

Get yourself set up with two possible wins.  Your opponent can only defend one of them, so you win.
But then we have to think of how to get yourself in that position.  There is definitely more than one way to get there.  One way might be more efficient in certain circumstances, but as long as the end goal is achieved, all methods are valid.

This type of thinking will (hopefully) get kids to think about intermediate steps on the way to the end of the proof, but always keeping the end in mind.  And help them to see that you must start with some sort of plan of how you could get to "win".

Hmm, maybe I could even present the discussion with an #anyqs following a heated competition...

I got this idea from a grad class discussion of what another grad student did in collaboration with a geometry teacher.  Bottom line, it's not my own idea, but I can't be sure of where it originated either!

A New Take on Assessment

Consider grading each problem on a test* on a scale from 0 to 4.  Not with a point for each step like I normally do, but more as a rubric ranging from
0  "blank or irrelevant"
1  "did something legit, but far from solution"
2  "going in the right direction, but didn't make it to target"
3  "essentially correct"
4  "completely correct"

I like it because it could be a quick way to assess students' work.  The rubric would be known to them.  They would know how far off from a correct solution they were without me having to necessarily indicate their mistake or misconception.

I don't like it because the overall test grade cannot be the total number of points earned out of total possible points.  This is because some problems are of higher difficulty than others.  So consider classifying the level of difficulty of each problem.  Could I then calculate a weighted average?  But would I weight a novel extension sort of problem twice as heavily as a standard knowledge/procedure-based problem?  Is that what I want the percentage to represent?  The jury is still out, but the ideas have me thinking!

*This post is a reflection on the grading practices of my professor Dr. Bill Martin when he taught at an International Baccalaureate school in India for a semester.