Here's the handout:

But what I wanted to write a note about was the idea of commentary. Mathematics in China followed a bit different path than in other ancient cultures, perhaps because there was more prevalent instruction. Lost is the origin of their ancient text, Nine Chapters on the Mathematical Arts. Instead of the advance coming from a collater, the big jump was Liu Hui writing a commentary on the text. To get the feeling of it, I asked students to solve one or more of the sample problems:

And then to write a commentary on it. This is new territory. So I had them share in groups, and pick someone to share with the class.Chapter 6:12. A good runner can go 100 paces while a poor runner covers 60 paces. The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit. How many paces does it take the good runner before he catches up the poor runner.Chapter 7:1. Certain items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins, and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items.7:18. There are two piles, one containing 9 gold coins and the other 11 silver coins. The two piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins. Find the weight of a silver coin and of a gold coin.

Chapter 8: 1. Top-grade ears of rice, one bundle, medium grade ears of rice, two bundles, low grade ears of rice, one bundle, makes 39 dou. Top-grade ears of rice two bundles, medium grade ears or rice three bundles, low grade ears of rice one bundle makes 34 dou. Top-grade ears if rice one bundle, medium grade ears of rice two bundles, low grade ears of rice three bundles makes 26 dou.

Mostly what they shared was there a solution, so I asked a commentary type prompt after each. Usually I'd be hesitant to put people on the spot, but these are senior students and we've had a pretty open classroom culture so far. I asked about solving with different representations, how you would describe in general the solution method, and an extension question about the mathematics. As they got into those discussions, it occurred to me that this might be a good framework for thinking about writing in our foundations classes. As the students discussed the idea of commentary, they noted that it seemed like a good way to draw attention to the idea of generalization, and a support for student reflection.

So thanks, Liu Hui! We'll see if I can get students writing their own commentaries on the mathematical arts.