How much of a difference makes a difference? When my dad taught us to play golf he showed us how being out of alignment by a few inches at the ball would result in being off by 20 yards at the place the ball was going. In teaching, we make decisions about materials we use, questions we ask, the order of assignments we give and, in these and the multitude of other decisions we make, we are always faced with this question: how much of a difference makes a difference?
In working with students in developing fractional understanding, we made a shift in our task to have students take pre-constructed fifths of a whole and use one piece to trace out five pieces to make a new bar. We then asked, “How do the bars compare?” What we were hoping was that they would say, “It’s the same!!” Of course, that didn’t happen.
Instead, they traced using different pieces and when comparing the lengths of the bars they made they noticed they were not exactly the same and so they didn’t have that golden “aha” that we had imagined in our planning. Why? Because there was enough of a difference to make a difference. There were a few differences we needed to make in the way we presented the task.
The first was that the pieces we cut were not perfect squares. When students used the pieces they rotated them and then used a short side. We realized this when one student compared rotated and unrotated pieces and determined they were not the same size. This was important because in previous tasks the focus was on the length of the bars the students had made into fifths, and whether their fifths were too small, too big, or just right. The purpose was to get them to reflect on the size of the fifth they had made. Now they were being asked to recreate the whole with a fifth we had made. There were small differences in the length depending on how they had turned their fifth piece. They began to focus on the small differences in comparison to the original whole and could not see that any piece of the whole could be copied five times to recreate the whole. With our pieces not being exact enough, it caused a problem because students pulled in their thinking from the previous task.
Second, the pieces we used were made of paper and the tracing was not exact enough. The paper bent and was slow to trace and therefore the bars did not come out the same size. See the explanation in the previous paragraph as to the source of this problem.
So how much of a difference makes a difference? In this case, what we saw as “not a difference” made a big difference. The biggest difference is that it failed to meet the fundamental goal of the task.