The topic of the latest AIMS Center colloquium was “What every student needs to know for multiplication” (Video Archive)(Resources). This presentation highlighted the work that we have been doing around understanding how students develop multiplicative reasoning. One of the things we are doing is implementing a task with second and third grade students called the Towers Task. In this task, students work in pairs, one partner is the sender and the other is the bringer. We have bins full of connecting cubes located across the room. The sender is given a card containing two values: first, an amount of towers, and second, the number of cubes in each tower (example – 4 towers with 3 cubes in each tower). The sender then “sends” the bringer across the room to the bins to build and bring back towers, one tower at a time, until they have made the correct number of towers. By having the student build one tower one at a time, they have time to reflect as the bring it back and see the “3-cube-tower” they built as something that is countable. After the bringer has built the towers, the sender hides the towers and asks three questions:
- How many towers did you make?
- How many cubes are in each tower?
- How many cubes are in all the towers?
Students who are just developing their multiplicative reasoning have trouble keeping track of the towers and cubes, especially with the last question. A hint that we encourage the sender to give the bringer is to ask them if they can imagine the towers and cubes. With some students, we have seen that, once given this hint, they are able to imagine the towers and tap or point to each individual cube and count. If they are still struggling, the sender can uncover one tower at a time and, if needed, all towers can be shown and the bringer can count the cubes. After the bringer answers all three questions, they draw a picture of the towers or a representation of how they knew the total amount of cubes in all the towers. The sender goes and puts the connecting cubes back in the bin and then it’s time for the partners to switch roles. Please view the video clip to see Towers Task in action.
By engaging in the activity, students have the experience of building a composite unit and figuring out a way to keep track of counting the items within the composite unit as well as the composite unit itself. We have seen that some students go from depending on counting the cubes on the imagined towers to being able to keep track of the amount of towers on one hand while counting the number of cubes on the other hand. What is the next step when child is able to do this? Stayed tuned for Part 2 of Tower Tasks where we will describe what comes next.