In the latest series of blog posts by the Coordinating Units team, we explored our recent work in the classroom developing multiplicative reasoning with students. One area that we focused on was creating different contexts to present the task. If students can generalize the mathematics and transfer it to different contexts, then they can use it in problem solving instead of memorized procedures.
To begin the school year we used cubes from which the students built towers.
In October, we changed the context of the task, sending them out “trick or treating,” by having them place a predetermined number of candies (small blocks) in bags.
In November, we changed the context yet again, having the students retrieve turkey cut-outs, for their Thanksgiving dinner, with a specified number of feathers on each turkey.
In all the variations, the essence of the task was preserved as the students were asked to make equal groups of items and then asked a series of questions. For example, they were asked to make towers with three cubes, bags of candies with three candies, and turkeys with three feathers, and then asked to determine the number of groups (towers, bags, turkeys), the number of items in each group (three), and the total number of items (cubes, candies, feathers).
Many teachers have experienced students struggling when the context of a concept is changed.
One example of this is:
14 x 10 = 140
So, when multiplying by 10 just add a zero to the end of the number.
Except, an example like this:
1.4 x 10 ≠ 1.40
doesn’t follow this rule even though the problems look the same.
One of the reasons that students struggle is that they are taught to focus in on keywords or formats and then to use a set of prescribed procedures that go along with them. Problems arise when the keywords are not found or the problem is formatted differently. The students see the situation as different and a new set of procedures is needed to solve it. The result is the impression by some students that mathematics is complex with lots of rules and procedures that must be memorized in order to be successful. Teachers want their students to generalize concepts and extrapolate them to situations that have a common underlying mathematical structure even when they are not familiar with the problem. Changing the context gives the student the opportunity to solve problems by understanding the underlying concept instead of memorizing a set of procedures.
In our last teaching session before the Thanksgiving break, one of the students asked us how the task would be changed when we returned. This prompted the other students to start making suggestions such as placing ornaments on trees, presents under trees, bells on a string, and many others. What remained the same in the ideas was that they all could be made into equal groups with a constant number of items in each group. The students have generalized the tasks and realize that the situations contained a common underlying structure of distributing a set of items over a set of groups. They can now produce their own situations using this structure, knowing that the context could change while the math remains constant.