#11. Discussions about mathematical reasoning: Accountable Talk® by students of all ages

Posted on August 5th, 2014

From the Institute for Learning, this is Victoria Bill.

People often wonder, are Accountable Talk discussions valuable for students learning any level of mathematics? Are young children capable of engaging in Accountable Talk discussions? The answer to both questions is yes. But don't take my word for it; listen to the details of student discussions.

Let's start in a middle school classroom. A group of students is solving a task. They have been told that there are a total of 20 houses, some blue and some red, and the ratio of red to blue houses is 2 to 3. Their task is to determine how many of the houses are blue. One student claims there are 10 blue and 10 red houses. Another student says, "It doesn't say that half are red and half are blue, it says the ratio of red houses to blue houses is 2 to 3, so you need more blue houses."

A third student reaches for counters and makes a visual display of the 2 to 3 relationship with 2 red counters and 3 blue ones. The student starts to iterate the display of the 2 to 3 relationship. You can hear several students saying, "I see it now." The student who originally said 10 red and 10 blue creates a table on a sheet of paper and recites, "2 is to 3 as 4 is to 6 as 6 is to 9 as 8 is to 12..." Suddenly, another student says, "STOP! You have 20 houses now. I see 8 red and 12 blue. Altogether there are 20 houses." Another student claims, "8 is to 12 is the same as 2 is to 3." When pressed to say in what way, the student says, "2 is to 3 is still 8 is to 12, but four iterations of the 2 to 3," and she points to the four iterations of 2 to 3 in the counters as evidence to support her reasoning.

Through an Accountable Talk discussion, the students have used each other’s thinking about the task, including the representations, and they have made sense of the problem.

These middle school students sound pretty sophisticated. They know how to manage their learning, which tools to draw on, and how they can build on each other's learning. Can this happen in a kindergarten classroom?

Let's look into a kindergarten classroom where students are attempting to determine the total number of donuts if there are 3 chocolate and 4 banana donuts. Making you hungry? One student displays 3 counters and 4 more counters and writes 3 + 4. Whereas, a second student displays 4 counters and then 3 more counters and writes 4 + 3. So now we have 3 + 4 and 4 + 3 written on the board. Immediately, students claim, "4 + 3 is more than 3 + 4." The teacher says, "Who agrees or disagrees?" and encourages the students to turn and talk. Many students immediately recalculate the 3 + 4 and the 4 + 3.

Throughout the room we hear students claiming "I got 7 for 3 + 4" and "I got 7 for 4 + 3". Suddenly, the student who originally claimed that 4 + 3 was greater than 3 + 4 says, "Ah, I get it. I have 3 + 4, and she has 4 + 3, and we have the same amount. She just has 4 donuts first, and I have 4 donuts second." The teacher immediately asks, "Who understands what she is claiming?" Several students point to the donuts and explain that 4 + 3 is 7 and 3 + 4 is 7, and they both show the same amount. The teacher records 3 + 4 = 4 + 3 and says, "Is this statement true?" Many shouts of "Yes!" can be heard throughout the room. "It's just the number of donuts. It doesn't matter where you put them." These kindergarten students made sense of the problem and shared their mathematical reasoning in the same way that the middle school students did.

So, what is the value of learning mathematics in this way, by making sense of tasks and thinking through them with others? By making sense of problems and talking with others, students express their own thinking and hear that of others. They come to understand what happens mathematically and why the system is working the way it is in mathematics. The Common Core Standards refers to this as the structure of mathematics. I'll leave you with this final thought…In our middle school and kindergarten examples, the teachers played a critical role. What was it?

Victoria Bill

Victoria Bill is the Chair of the Institute for Learning's mathematics Disciplinary Literacy Team.

More information on Victoria Bill.