Number talks were developed for classroom teachers to engage students in “mental math” by collaboratively grappling with interesting mathematics problems. I was first introduced to the idea of number talks from the book, “Number Talks” by Sherry Parrish. Recently, I had the pleasure of facilitating number talks in 6 third grade classrooms, all at the same school site.These students had not been exposed to many number talks and so were relatively inexperienced in processing math mentally.
After setting up the criteria for how I wanted them to respond, I asked the students to solve the problem “49 + 49”. In all six classes, I watched as almost all of the students attempted to solve the problem using the standard algorithm in their heads. Many of them drew with their fingers on the ground or in the air to help them visualize their moves for solving the problem. When I called for someone to share a solution, in every class the first answer that I received was a description of solving the standard algorithm:
- adding 9 + 9 and getting 18
- taking the 10 from the 18 and “carrying” it to the tens column
- adding 4 + 4 + 1 (the ten from the 18) and getting 9
- So, the answer was 98.
I would then ask if anyone solved the problem a different way (cringing inside as I hoped for any other type of thought process). Interestingly, every class also had two or three students who would raise their hands at this point and explain a different way of solving the problem.
- One student suggested that they had “changed” both 49’s to 50’s. Then they added 50+50, which they explain was much easier, and got 100. But since they had added 2 to the original problem, they then took 2 away to get 98! We decided to call this finding a friendly number.
- Another student explained that they added the tens place and got 80, then added the ones place and got 18, and then they were able to add 18 to 80 more easily and got 98.
As I praised these approaches I saw many third grade eyes widen at the realization that there were other “easier” ways to approach this problem. When I gave a new (second) problem to solve, suddenly there were many students with new approaches to solving the problem without using the standard algorithm. There were still some students who wanted to stick to their original method, but many had adopted the ideas presented in the first problem and had applied them to the second. They were on their way to the realization that there are many paths to solving the same problem, and that decomposing and recomposing numbers is a powerful tool.
In Ernst von Glasersfeld’s book, Radical Constructivism, he states that “when assimilation takes place, a student takes new information and grafts it onto previous schemes adding new understanding to already constructed structures or understandings.” I think the experience above is a prime example of the act of assimilating new information. Those students that had the structure and understanding already in place, were able to take the new idea or way of solving the problems mentally, and apply them to a new problem once they were exposed to the ideas. Some students weren’t there yet. They weren’t able to move from their original understanding…Yet! I believe that as they are exposed to those new ideas repeatedly they will be able to construct and assimilate the new knowledge and have it make sense to them.