Division

Since we talked in-depth about the sub problem last week I wanted to start off with that. I was amazed at how frustrated I was while working on that problem. It's no wonder so many students do not understand fractions or spend a lot of time trying to understand how they work. I'm still at the point like many children where it is much easier for me to think about lots of fractions while drawing pictures. In the fifth grade classroom that I tutor in last Friday they were working with fractions by having fraction bars with each fraction laid out. I thought these were helpful because I was working with some of the students to discover what equivalent fractions were. We worked on taking things like the 1/3 bar and seeing what other bars would line up exactly equal with it, like 2 1/6 bars, or 1/2 and two 1/4 and four 1/8. Students could also explore how as the denominator got larger the size of the bar got smaller.

While I was reading the DMI chapter on division I was trying to pay attention to the different strategies that were being used. The problem like case 24 when the teacher was trying to get students to divide 134 jelly beans among 6 children was just like the division problems I was working on with a student in the fourth grade classroom that I tutor in. I was really amazed at all the different strategies that April came up with to try to solve the problem. Since it took her so many steps I was left thinking the same thing the teacher was at the end, when should we teach students the standard division algorithm? Or should we at all? Is it better for students to struggle and reason through it and then put it into an algorithm or is the algorithm even necessary? I wish I had read through this before working with the 4th grade student in my class on this subject. The teacher had only provided students with these flat counter chips that were rather difficult to work with because it was hard to create piles of them to keep numbers straight. I first tried to have the student model the division out but as the numbers quickly got a lot larger I realized there must be an easier way. The student seemed to sort of grasp dividing by separating into different piles but I'm not sure he completely grasped what he was doing at this most basic step. I think I was leading him too much and not really allowing him to explore the problem on his own. In retrospect I can see this, at the time I was just celebrating that he managed to sit in his desk and work on something for 30 minutes, which was a very large step for this student. I tried to work with him on trying to estimate for some of the larger numbers like April was doing but without a basic knowledge of his multiplication tables this quickly proved to be an impossible task. I worry about solidifying his division knowledge without a solid understanding of multiplication. In Betty's case she was using base 10 blocks with her students. Maybe that might have made this task somewhat less confused because a major part of each problem was sorting out the counters and then recounting and he was often one counter short which was very confusing when trying to arrive at a correct answer. I understood Betty when she was talking about how it was difficult to see the division when she counted out 10 7 sticks and 2 cubes to illustrate 72 divided by 3. I think just looking at that without seeing each one as a single I would not immediately think of my first step for division. I hope after our discussion in class about this issue I will have some more concrete ideas to take back to my 4th grade student next time I tutor him.