The first step is to provide an environment in which students feel comfortable about presenting their ideas, knowing that they will be listened to and that they will receive useful feedback. The smallgroup structure of ALiM groups is an excellent setting for students to participate in mathematical conversations.
The next step is to lead students along the path towards mathematical discourse – learning how to engage in mathematical argumentation.
Quality reasoning and clear communication do not always go hand in hand. Some students struggle to put their thoughts into words, even if their reasoning is sound. Revoicing is a useful way to clarify a student’s response.
In simple terms, revoicing involves restating what someone has said in your own words. An effective teacher can use this to model correct mathematical language.
In some instances, a teacher may understand what a student is trying to say but the rest of the group do not. Revoicing can be used to clarify a student’s idea so that other students can engage with it. The idea can then be used as the basis for pair, smallgroup, or class discussion.
After revoicing, it is important to check with the student that their idea has been correctly captured.
For example: A student is exploring fractions. The teacher gives them two same-sized rectangles, one showing three thirds and the other showing four quarters. The teacher asks the student what they notice about the shares.
Student: This one is bigger [points to one third] because there are less bits, and this one is smaller [points to one quarter] because there are more bits.
Teacher: Let me see if I understand. This shape is shared into three parts – I can see three thirds here. This one is shared into four parts – I can see four quarters. When I compare the shares, one third is bigger than one quarter. That’s because if we divide something into lots of parts, the parts get smaller. Is that what you meant?
Revoicing also creates time for other students to think about an idea and how it relates to their own ideas or to previous activities.
Increase wait time
A simple way to improve the quality of teacher and student discourse is to extend wait time after posing a question. Wait time allows students to think about and formulate their responses. Making time for this indicates that you are genuinely interested in what your students have to say. Interestingly, increasing wait time can also increase the quality of questions that teachers ask.
Lengthening the time provided for thinking helps students to understand that uncertainty is a normal part of problem solving.
Use partner talk
Some students lack the confidence to share their ideas with a group. Talking to a neighbour gives students a chance to practise putting their thoughts into words. Partner talk also gives students an opportunity to quietly ask their classmate a question.
Think, pair, share
Partner talk time can be extended by using think, pair, share.
Note that teachers need to be active listeners when students are sharing their ideas.
Use question cards
Students who are reluctant to share their own ideas can increase their level of participation by asking questions. Giving students a store of questions to use can enliven class discussions. One of the most important questions in the classroom is “Why do you think that?”
Create a framework for interaction
Seating arrangements can affect how readily students share their ideas. Encourage students working in pairs to face each other. Small groups often function better when they are sitting in a circle. Choose whether to be part of the circle or to sit or stand within earshot. Sometimes removing yourself to the periphery frees up students to discuss their ideas more openly.
Clarify the process you want students to follow. For example:
I want you to explain to the people in your group how you think you are going to work it out. Then I want you to ask if they understand what you are on about and let them ask you questions. Remember, in the end, you all need to be able to explain how your group did it. Think of questions you might be asked and try them out.
Mathematics BES, page 73
Co-construct a flowchart showing a process that students can follow as they discuss their ideas. Sentence starters are a useful way to get students thinking:
Promote mathematical discourse
When students support their ideas with examples, look for and describe patterns, or make predictions, they are using mathematical thinking. When students share this thinking with others or respond to the thinking of their peers, they are participating in mathematical discourse. ALiM groups provide a safe setting for students to practise this.
You can invite a student to justify their ideas by simply asking “How do you know?” when they offer an answer. By explaining their reasoning, students often cement their understanding of a concept or, in the process, recognise flaws in their own logic.
Explaining, clarifying, and justifying ideas are an essential part of mathematical argumentation, in which ideas are shared, debated, and developed in order to reach a shared understanding. Research shows that young children can learn how to participate in mathematical argumentation. Andriessen (2006) calls this process “arguing to learn”.
Students may struggle to challenge another group member’s thinking, but a teacher can model how to do this. Counterexamples are an excellent way to challenge student thinking. For example, in the Revoicing example on page 2, the student is given two equally sized rectangles. The student understands that increasing a number of shares decreases the size of each share. The teacher could then present a third, significantly larger rectangle that is divided into four shares, or a third rectangle divided into three uneven shares, to explore the limits of the student’s understanding of thirds and quarters.