This is an account of the second lesson with upper primary (ages 8-11). In the first lesson, we gained familiarity with the idea that we can form equations with the rods, and that these can represent numerical calculations once we give a value to one of the rods. In what follows, I chose to set dark green equal to 1.
In this lesson, I toyed with the idea of presenting this set of tasks. In the event, I decided not to. Primarily, I was not happy with the second two tasks in which I set a series of questions; after a couple of conversations with others, I chose to invite the children to create their own questions, for others to work on. It turned out I did not to give any written instruction, and it was only necessary to present an example of what I wanted them to do on the board.
@MichaelOllerton also suggested the important idea that I constrain the number and colour of rods that I gave out. In the end I decided to give two of each colour, up to dark green, for each pair of children. This constraint was important, in an activity in which there is already considerable freedom.
The first problem I wrote on the board was 2/3 + 1/6 = ?. I chose Milos, and he gave the answer 5/6. I asked him how he knew the answer was 5/6. He explained: "Two thirds, or four sixths, is the same as two reds, or one pink, and one sixth is a white. If you put them together it's the same as a yellow, which is five sixths." Much nodding of heads.
The second problem I wrote on the board was 7/6 - 4/6 = ?. I chose Caitlin, she gave the answer 3/6. I asked her how she knew. She said: "I just did seven take away four, which is three, so the answer is three sixths." More nodding.
The third problem I wrote was 2 - 2/3 = ?. I chose Sarah, who gave the answer 1 1/3. I asked her how she knew, she said: "I just took two thirds away from two." I asked her if she could elaborate, but she couldn't, in that moment. I asked if anyone else wanted to share their method, and Violet offered this explanation: "Well, it's basically twelve take away four. Two is twelve (whites), and two thirds is four (whites), and twelve take away four is eight, so eight sixths." I wrote 8/6 on the board, next to Sarah's 1 1/3 and asked: "Are these the same?" Nodding. A sense of "Yes, of course they are."
There seemed to me to be real evidence of understanding.
There was evidence, for example in Milos' response, of visualising the fractions as (algebraic?) objects that could substituted for others, or be added or subtracted. This will be useful later.
There was evidence, for example in Violet's response, of the children informally constructing the algorithm for adding and subtracting, without the language of common denominators.
They were developing a sense of what adding and subtracting fractions is about, switching between enactive, iconic and symbolic modes of representation. The algorithms will follow.