I think what I would want to see next is whether these students can correctly use the distributive property when solving equations/simplifying expressions. I would also want to know if, given an example of the distributive property, they can explain why it makes sense to change the expression in that way. With a question like this, a student can get stuck just by not knowing the name “distributive property,” and if that happens I have no idea whether they can use the distributive property, or, more importantly, whether they understand it. For example, the last student seems to be accurately describing the commutative property, but I know next to nothing about what he knows about the distributive property from that answer.

Some of the others seem more informative: it seems as though many of the students associate distributive property at least with the visual form of some number next to two things inside parenthesis. I’m particularly worried about the students who distributed multiplication over multiplication- that makes me think they see the form, but don’t have the understanding.

I have a couple of ideas about how to teach the distributive property. I’ve seen teachers have success with algebra tiles. For example, using the first student’s idea, we could make 3(x+2) with tiles. That means we need an x tile and 2 unit tiles. And then we need three groups of that in total. Then, the students can count the tiles, and see that it comes out to 3x + 6, just as if you had multiplied the three by both. If they’re thinking of multiplication as making groups of things, then you can generalize from there (not that I would express it in these terms, but just for clarity): if you have x groups of (y+z) then you have to have x groups of y and x groups of z, every time, no matter what x, y, and z are. (Although multiplication as grouping breaks down pretty easily: what does it mean to have pi, or -7 groups of something?) Personally, I really like using multiplication as an area model to show this: if you have a rectangle with one side (x+2) and one side 3, then that has the same area as two rectangles, one with sides x and 3 and the other with sides 2 and 3. That’s very visual, and I think it’s easy to see that that picture would be true no matter what labels you put on there (if positive- this model isn’t perfect either). The example the second student used might help students develop an intuitive sense that this should work, because you can actually add 10 and 2, so you can show that 3(10+2) = 3(12) = 36 = 30 + 6 = 3*10 + 3*2. I would also show examples of it not working for multiplication, because I think often we only show students things that do work, and then they apply those ideas to situations where it does not make sense. One final idea I had is that some students may already use the distributive property while trying to multiply numbers- for instance if they are trying to find 9*13 and they know 9*12= 108, then they may add 9 to 108 to get 117. That’s in a way the same as saying 9(13)=9(12+1)=9*12+9*1. I doubt they think of it in those terms, but any way we can connect a new concept to something they already know is valuable, I think.

I have no idea what the third student is thinking.