“The three-part question was about dividing fractions and used a scenario based on ordering pizza. If a teacher had 31 students in his room and every kid got a fourth of a pizza, how many pizzas need to be ordered?” said Rogers. “In the first part, students had to model division of fractions by either creating a diagram or drawing. In the second part, students were asked to use the standard algorithm to solve the problem. And the third part was to see how they could take that knowledge and apply it to extend their reasoning to explain their solution and why it was a realistic answer.”
McMahan and Rogers discovered that the third part of the question presented a great deal of trouble for a majority of their students. They had difficulty explaining why their answer was reasonable, primarily because many of their answers were not practical at all.
“Instead of thinking backwards and multiplying, they focused on dividing fractions and trying to remember how to do that,” Rogers said. “They didn’t use reasoning or apply any number sense and told us we needed to order 134 pizzas, which is not a reasonable answer.”
Rogers and McMahan realized they were going to have to find a way to teach their students to talk through math problems, to think about the problem before they attempted to solve it and to use number sense to determine if the answer was reasonable.
“We needed to help them make sense of not only the problem, but in connecting the problem to the answer and determining whether or not it’s going to work,” McMahan said.
In the description of their presentation, the math teachers offered to provide “an interactive discussion that shows how giving students number talks can be a powerful tool that math teachers use to increase rigor, encourage critical thinking, promote number sense, and display multiple strategies to solving problems. The focus of this presentation is to give middle school math educators some examples on how to incorporate number talks into their lesson plans to help foster mathematical minds and build a math community.”
It's done just that at HCMS.
Using small groups of mixed ability levels, Rogers and McMahan began leading “number talks” with their students, literally talking through the problem. They started with the pizza question.
“We asked, ‘Why is it that this answer does or does not make sense? Why is it that this is a reasonable answer versus 134 pizzas for 31 kids?’” Rogers said.
Fast forward several months of school and lots of number talks later, and both teachers are seeing results in their classrooms.
“What’s exciting is that at this point in the school year our students are leading those conversations,” McMahan said, pointing out that building the small group dynamic has resulted in improvement for all students. “In the past, the higher achieving student would just work and solve and not even know how to explain how they got the answer. And you’ve have a lower level kid who was embarrassed to ask questions, and the mid-level student just hovered between the two. Now, it’s more natural for all students in a group to engage with each other in that discussion. The lower level kids feel more secure in answering. Sometimes their approach is different, but it’s valid. It’s just a different way of thinking about the problem. It builds confidence in students who may not have had it by providing them with a voice to explore options that other kids at the table may not have considered, regardless of whether they are right or wrong.”
Rogers pointed out that students are much more receptive to considering multiple strategies, instead of subscribing to the idea that there’s only one way to solve a problem. It also initiates the thought process for students, pushing them to think about what they are going to do before they do it. When the answer better connects to the question, students have more “ah hah” moments. Both Rogers and McMahan have seen “drastic improvement” from their students in subsequent extended response questions.
“It helps students understand not just how to get the right answer, but to understand the math reasoning behind the answer and why the answer goes with that problem. You have to be able to apply it – you need to understand the process of it, how to problem-solve,” Rogers said. “We live in a society now where we need problem-solvers, we need people who understand why things work the way they do.”