Another thing that comes to mind about my student is that once he noticed the unifix cubes in front of him, he never looked back! Even though he was more than likely exposed to the standard algorithm to set up multiplication/division problems, he didn't use them unless it was a last resort and the manipulatives were confusing him. If you look at the pictures below, you will see that M is solving a problem with the cubes. In short, the problem says there are 45 students that will be on 3 teams, how many people on each? In terms of a standard algorithm, this is a division problem; 45/3=15. However, M didn't see it that way. He counted out all 45 cubes, then grouped them in to threes and then counted how many groups of three he had. In all actuality, this is a very complex strategy. It appears that he realized that he is trying to see how many times 3 goes in to 45 because he separated the blocks into groups of 3 rather than making three large groups and dispersing the blocks equally. See below...
I think that there are a couple of different ways other students may have solved this problem. The first is by separating the cubes in to 3 different "teams". Count out and put cubes one cube in each group until they are all gone. Once they are all gone, re count (to double check) how many cubes are in each group: 15.
Another way a student could solve this problem is by first counting out all 45 cubes then subtracting 3 from the group until the all the cubes are gone. Once all the cubes are gone, count all the groups. This strategy is very similiar to M's strategy but it is turning the problem into subtraction (and eventually addition) instead of putting them in to groups of three from the beginning.
