Shifting from Spectacular Jumps toward Modest Movement

I was in a third grade classroom yesterday, teaching a lesson on ordering representations of multiplication expressions. (See the Tasks for Download section for an example). The students needed to compare representations like eight groups of six, 7 x 8, and 2 x 3 x 6. The first inclination of many students was to solve these expressions and then put the products in order from least to greatest. The teacher and I pushed students to then find an alternative strategy, one that did not involve finding a product. The students really struggled to do this. They could represent eight groups of six as 8 x 6 and they could change 2 x 3 x 6 into 6 x 6, but then they weren’t sure how to compare 7 x 8 and 8 x 6, although they had many interesting and productive discussions about this comparison.

The teacher (who is an amazing, student-centered mathematics teacher) described a future task she might do with the students. She would make one of the factors a really large number (e.g., 8 x 6 compared to 8 x 1,000,000). Her thinking was that students 1) would not be able to calculate the product of the large number and so would need to first turn to other strategies and 2) the really large number would help the students in using factors (rather than products) to compare the expressions.

I also found myself drawn to this revised task. Perhaps if the students worked with these kinds of extreme comparisons (i.e., those with one really large factor), they would then use the same reasoning in situations that involved comparing numbers that were more similar.

But then I realized that while extreme expressions might result in quick and correct comparisons, I was focused on the wrong thing. I was trying to get students to efficiently do something, rather than really valuing the less obvious and less flashy, but really productive mathematical ideas students were debating. In other words, in the task we presented, students were not making huge leaps to new solution strategies, but they were wrestling with important ideas like what the factors in a multiplication expression mean, how to use the commutative property of multiplication, and how great the difference needed to be between factors before you could say anything about differences in products (e.g., can you use comparing factors to determine whether 3 x 15 is greater than 8 x 6?). These were really productive mathematical struggles and the students were quite engaged in these ideas. Yet, it was harder to notice, value, and celebrate these struggles because they did not result in the bright flashbulb of the “Ahha!”, right-answer moment.

This isn’t to say that I might not use an example of an extreme comparison in order to get a student started thinking about comparing factors. However, I am realizing that I need to be less seduced by tasks that result in spectacular jumps to right answers. Instead, I need to continue to celebrate tasks that engage students in modest movements toward essential and exciting, but perhaps less conspicuous mathematical understandings.