When I taught middle school, I always found it interesting that my students could do this task:
Johnny rode 34 miles on Tuesday and on Wednesday he rode 27 miles. How far did he ride over the two days?
Yet they often had no idea what to do when I gave them this task:
Johnny walked ¾ a mile on Tuesday and on Wednesday he walked ⅞ of a mile. How far did he walk over the two days?
The two fractions can be added easily, but I think the reason they were confused is that they saw the fractions and immediately assumed the task was inaccessible. I also think they hadn’t experienced enough types of problems to able to generalize the situation type. It might have been that, instead of focusing on the situation type, they were taught to look for a keyword, or to use some other procedure that did not cause them to recognize the fraction problem as something other than complex and inaccessible.
Recently, we have been doing tasks with second graders that allow them to build concepts that will support multiplication. We had been using one task, but for October we decided to change the context, or situation, of the task for Halloween. Before, we had students building a certain number of towers, each from a certain, varying number of cubes. Then, we would have them hide the towers and make them think about the number of towers they had made, the number of cubes in each tower, and the total number of blocks they used to make all the towers.
We wanted to preserve the concept of making equal groups and then having them think about the groups, the number of items in each group, and the total number of items. So, to do this, we decided to send them “trick or treating.” We got little bags and changed the task from building towers to filling each of the little bags with a certain number of “candies” (little blocks). Then we would have them think about the number of bags, the number of candies in each bag, and the total number of candies.
We saw two positive outcomes. The first was that students were engaged in the task, because it was relevant to them. We also included a story about how they needed to know how much candy they had and how their parents would hide the candy so they wouldn’t eat it all at once. The second positive outcome was that the students had the chance to see the same concept as the towers task in a different situation. In both tasks, the towers and bags of candy, the underlying task is the same: groups, items in each group, and total number of items.
As we move into November, we are thinking ahead to using a new situation that will include groups where the items are connected, like lights on a tree.
Can you think of what other situations could a teacher use? Share your ideas in the comments below!