# Mathematical communication

Effective teachers facilitate classroom dialogue that is focused on mathematical argumentation.

#### Guidance for effective practice:

1. Employ, and encourage ākonga to employ, talk moves to orchestrate learning productive conversations in group settings.
Caroline believes in the power of Talk Moves for empowering students to interact effectively in collaborative groups. As a pre-service teacher Caroline frequently participated in role playing where Talk Moves were practiced. Caroline also ran a professional development session for her colleagues on Talk Moves using the module Introduction to Talk Moves in PLD 360.
To prompt her students, she displays posters of the main talk moves around the room, with examples:
• Revoicing (Repeating what another person said, possibly clarifying their language)
• Providing wait time (Given a responder adequate time to construct their answer)
• Adding on (Building on the ideas of another person)
• Explaining (Clarifying the ideas of another person)
• Comparing (Comparing the different ideas of two or more people)
• Applying (Using someone else’s idea to solve a new problem)
• Justifying (Showing why an idea or strategy is correct)
Caroline knows that Talk Moves take regular practice to become habits. In the first few weeks she points out to students whenever she uses the moves herself and prompts her students when they are working in groups. During whole class summaries at the end of lessons she asks students for examples of how they used the moves.
She also provides a chart of question starters to help her students begin constructive responses to their classmates.

Source: Makar, Bakker, & Ben-Zvi (2015).

2. Set clear expectations about what constitutes a mathematical argument.
Page’s year 3 class are investigating what happens when odd and even numbers are added and subtracted. They begin by deciding what is meant by odd and even numbers. Using tens frames they decide which numbers of students can be organized into pairs with ‘no odd person left out’. The tens frames for even numbers, 2, 4, 6, 8, 10, are grouped together, and the left-over frames are discussed.

Her students conclude that an odd number of students have ‘an odd person left over’ after the pairing is done.  The class investigate the whole numbers from 11 to 20, and correctly classify the numbers as even or odd. Using a Hundreds board the class highlight the even numbers and notice that the numbers exist on columns. Other numbers greater than 20 are investigated to establish if the column pattern continues.

Page poses this problem: “If I add two even numbers, or two odd numbers, is the sum always even or odd. Why?”
Students investigate the question in pairs using blank tens frames, counters, and cubes. They record their thinking for the sharing session.
Page: Charley and Rynan, what did you find out?
Rynan: We tried 4 + 6, 2 + 10, and 8 + 6. The answers were 10, 12, and 14.
Page: What kinds of numbers are 2, 4, 6, 8 and 10?
Charley: They are all even and the answers are even.
Page: That’s interesting. Did other people find that?

Many of the pairs produced examples. Page lists the examples as ‘even sums’ and ‘odd sums’ referring to the answer.
Page: What do you notice?
Leonard: All the answers are even. You always get an even answer.
Page: Could we get some odd answers? How?
Several pairs have odd answers because they added even numbers to odd numbers by mistake, e.g., 3 + 4 = 7, 6 + 5 = 11, 1 + 8 = 9. Page lists the equations in the Odd Sums column.
Page: Now I want you to go back and figure out why these things always happen. Why is it that you get an even answer if both addends are odd? Why is it that you get an odd answer when you add an even number to an odd number?

Page is aware that noticing pattern in examples is a first step to generalization. However, she wants her students to provide a justification of why these patterns occur. She expects students to go beyond specific numbers to the structure of any even or odd number.