Introduction
‘Doing maths’ is looking a little different these days in some classrooms, as more collaborative approaches to solving mathematics problems are being encouraged.
In some South Auckland schools, in particular, this development of “communities of mathematical inquiry” is being led by Dr Bobbie Hunter, a researcher, academic and New Zealand-born Cook Islander, whose focus is on better meeting the needs of all students and of Pasifika learners in particular.
The roles and responsibilities of teachers and students are changing as the patterns of communication and participation have students taking more responsibility for active listening and sense-making. In a lesson, the solutions to problems are discussed, negotiated and constructed in a collective way. Learning conversations include all students, and everyone feels that their contribution is valued. Students feel that everyone succeeds when the group succeeds.
This video briefly explores the components of such a lesson. We’ll look at the mathematics problems themselves, the way in which these are ‘launched’ in the lesson before groups work together on constructing solutions, we’ll see the collective sharing of these and the way in which the teacher connects group learning to important big mathematical ideas. Finally, we’ll identify the underlying communication and participation dynamics that are essential to a successful lesson.
The problem
The problem itself is carefully chosen and crafted.
When I write a number problem, for example, my starting place is just what’s happening in my room and in the world of my students. I think hard about the numbers I want the students to work with, and I ensure that the solution can be reached using a number of different strategies or approaches. They’ve got to be able to come at the problem in a number of ways.
The problems I write for the kids must be both achievable and sufficiently challenging. But I need to be really clear myself about the big idea I want to connect to. So knowing the number of property that I want my kids to learn about is a useful starting place. Generalising is such an important part of mathematics learning.
As I create a problem, I’ve got to really know my students and know my maths! I know which strategies I want them to explore, but I also need to have anticipated, ahead of time, all the possible approaches the students might take. I also need to recognise the possible errors they might make along the way so I can use these to unpack the students' thinking and to take them forward.
We have heard how a problem should:
- be culturally appropriate;
- be related to a current experience or a context that is familiar;
- challenge the students’ thinking;
- have multiple entry and exit points;
- be one in which a number of approaches or strategies can be used;
- lead to the big idea and generalisation;
- be fully explored by the teacher before the lesson, so that all outcomes and possible errors are anticipated.
The learning environment
The kind of mathematics problems teachers give their students is really important. So are the cooperative and collaborative learning behaviours in the classroom!
For everyone to be able to contribute their maths ideas to a collective solution and feel that their input is valued, teachers themselves must rethink their role and reconsider the ‘power base’ in their classroom.
Classroom behaviours and cues, attitudes and beliefs about what is important, and the implicit and explicit rules that govern the interactions in a lesson, need to be talked about openly. Changing classroom norms, and the way people communicate and participate, take very careful scaffolding. And it takes time!
Launching the problem
In order for each student to actively contribute to an agreed group solution for a given problem, all students need to understand what is being asked of them. They need to be given time to get to know and interpret the question.
Giving class time to understanding the problem may involve the teacher in making time to wait for a student to clarify their thinking before they contribute their idea, echoing an interpretation to check for meaning, or asking a student to repeat what another student has said.
Group work and teacher role
Groups are comprised of students of a range of abilities. The onus on any student in the group is to ensure that all other group members understand the idea they have shared, and also to ask questions if they do not fully understand the idea of another student. All students must take responsibility for reasoned sense-making.
As the teacher roves between groups, their focus is on the group process and dynamics. The teacher’s role is to prompt individual and group responsibility for a collective understanding of and negotiated agreement on a problem solution. It is not their role to simply give clues to find the right answer.
The teacher may prompt the group to consider how they will support one of their members to present their agreed strategy and solution to the whole class. The group must be also prepared to answer questions and respond to challenges from class members, and explain their mathematical reasoning.
The teacher is also actively seeking the group whose strategies and solutions will when shared, most effectively lead to new learning for the whole class. It takes astute noticing, and sound mathematical knowledge of a well-constructed problem, for the roving teacher to identify the group, or groups, whose solutions will best connect with the important big mathematical idea.
Plenary (sharing)
It takes skilful orchestration on the part of the teacher to ‘conduct’ the sharing session in which representations, explanations, reasoning and justification play a key part.
As students explore and explain differences and similarities in approaches and representations, they revise and extend their thinking. The questioning and argumentation that the teacher encourages, further deepens the understanding of all students.
Making connections
When the problem was being written, the important mathematical idea that this problem would lead to was clearly identified by the teacher. The nub of the lesson is at this point when the teacher makes this connection explicit for the students, and the important generalization is made.
Participation and communication at work
Throughout the lesson, a finely tuned synthesis of participation and communication structures are at work to create a fully inclusive community of mathematics ‘inquirers’.
I’ve had to work really hard on developing active listening with my kids and having them understand how important it is to ask questions if they don’t understand. My message to the kids is really about actively taking responsibility for their own learning, and for their classmates’ learning too.
It’s taken time for my students to get the whole idea of supporting each other so that everyone understands and shares in the success of the group. It's a big shift to have them thinking about clearly communicating an idea so that everyone gets it.
I’ve worked hard to have the children understand and feel comfortable using mathematical language and terms. I’ve also had to work hard to clearly understand number properties myself so that I can write problems well, and then make the connection for the children.
I’m really pleased that my kids are now learning to take a risk and will disagree or challenge someone if they’re thinking differently. They’ve had to feel safe, and that takes time. We’re now just starting to get some really good conversations about mathematics in a problem, and it’s really exciting.
Conclusion
In a classroom in which mathematical inquiry and learning is the responsibility of everyone in the class community, the teacher will often be heard to say to their students, “Remember, for a group to be successful, you all need to be asking a lot of questions”.
In fact, this deceptively simple reminder may disguise the complexity of the factors that the teacher has skillfully blended to create a safe, inclusive environment in which all the students can actively contribute to the collective success of mathematics.
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