Engaging Students in Mathematical Discourse

Engaging Students in Mathematical Discourse
by Chris Siew


An effective mathematics programme is more than just concepts and skills. Problem solving is central to mathematics learning. The development of this ability is dependent on five inter-related components, namely Concepts, Skills, Processes, Attitudes and Metacognition(Mathematics Framework, MOE).
This article highlights the notion that mathematics should be taught in a way that mirrors the nature of the discipline (Lampert 1990), instead of practising concepts and skills in isolation. How can teachers design lessons that create opportunities for students to make conjectures and engage in constructive argument about problems in order to discover important mathematical concepts? What are the attitudes and processes that we need to develop in our students to help them understand the essence of what it means to do mathematics?

Designing the Learning Environment
There is no doubt that accuracy is important in mathematics. However, teachers have to communicate to students that understanding concepts and communicating mathematical ideas are more important than just getting the right answers. Promoting mathematical discourse is a strategy to develop students’ understanding and communication skills in mathematics.
Let us start by examining the kind of structures that need to be in place to facilitate discourse in a math class.

1. Setting expectation for learning – The first step in transforming a traditional classroom to one that promotes discourse is to set the right expectation for learning. Students need to understand that their role in the classroom is not just listening to the teacher and waiting for answers. In the new learning environment, they will be exposed to open-ended tasks and they would need to play a more active role to complete the task. The teacher will assume the role of a facilitator in these activities, who will scaffold and point them to the right direction but not necessary show them the correct answers all the time.
2. Establish classroom relationship – There will also be a shift in relationship between students. Learning will become more social and ideas will be exchanged freely. Their peers will no longer be just someone who sits beside them, they will be their learning partner and collaborator to build each other’s knowledge. The relationship between the student and teacher will also be changing. The teachers will not be the only source of information. Instead, they will direct students to various resources to uncover the key learning points of each topic.
3. Establish rules and management system – Teachers have feedback that conducting collaborative learning activities can be quite chaotic; students are noisy and teachers are not in control. The key to good classroom management is to establish clear rules on the dos and don’ts when students are engaged in learning activities. It is also important to put in place a system to manage conflicts that may arise during interaction among students.

Teacher as facilitator
It is a misconception that in this new learning environment, teachers just present the problem and wait for things to happen. Do you think students will be able to discuss, solve the problem and present the solution automatically? Learning does not happen by chance, it happens by design.
In order to achieve participation and conceptual understanding, teachers have to design lessons that provide opportunities for students to:

1. articulate their understanding,
2. engage in peer-critique and
3. build knowledge collaboratively

To make students thinking visible during discussion and promote engagement, teachers could consider two aspects of teacher discourse: cognitive discourse and motivational discourse. (Stein, 2007)
Cognitive discourse refers to things that teacher says to promote conceptual understanding of mathematics by making their thinking visible. The objectives of teacher discourse are to

1. help students make connection,
2. help them learn from mistakes and
3. stress individual accountability

Motivational discourse refers to things that teacher says to promote students’ engagement and addresses the affective domain. Students need an affective support structure in the classroom that makes it a safe environment that allows them to learn through mistakes and take risks.
The objectives of supportive motivational discourse are to:

1. Promote learning

1. Focus on the process of learning
2. Challenging students
3. Viewing errors as constructive
4. Supporting persistence

• Promote positive emotions

1. Using enthusiasm
2. Using humour
3. Reducing anxiety
4. Addressing emotional needs

• Foster peer collaboration 

1. Building collaboration
2. Emphasizing joint goals/shared responsibilities

Discourse Assessment
Teachers need time to develop competency in using mathematics discourse to advance learning in class. To help track the progress as a class goes through this transformation, Hufferd-Ackles, Fuson, and Sherin (2004) created a framework to describe and evaluate the level of mathematics discourse in the classroom. (Refer to table below)

Components	To compare   lessons from Term 1 to Term 4 based on analysis of lesson observation videos
Questioning	Was there a shift from the teacher as the main questioner to students & teacher as questioners?
Explaining mathematical/ science thinking	Did students increasingly explain & articulate their ideas?
Source of mathematical/ science ideas	Was there a shift from the teacher as the source of all math/science ideas to students’ ideas also influencing the direction of the lessons?
Responsibility for learning	Did students increasingly take responsibility for learning and evaluation of others and self?

Adapted from Hufferd-Ackles, Fuson & Sherin (2004)

Four components are examined – questioning, explanation of mathematical thinking, source of mathematical ideas, and responsibility for learning. Teachers can use this framework to assess the discourse level of the whole class. It can also help them identify individual students who are not actively participating in the discussion so that additional scaffold and support can be provided to encourage this group of learners to be more active in class.
Conclusion
Promoting mathematical discourse in the classroom can help to strengthen students’ processes in mathematical reasoning and their skills in communicating mathematical ideas. As students learn to make and test hypothesis, question, and agree and disagree about problems, they are learning the essence of what it means to do mathematics. As teachers, we need to re-examine our practice and re-think how we can use mathematical discourse to better engage our learners.

References

1	Hufferd-Ackles, K., K. Fuson, and M. Sherin. "Describing Levels and Components of a Math-Talk Learning Community". Journal for Research in Mathmatics Education 35 (March 2004) : 81-116
2	Lampert, M. "When the Problem is not the question and the solution is not the answer: Mathematical Knowing and Teaching." American Education Journal 27 (Spring 1990):29-63
3	Stien, C. "Let's Talk, Promoting mathematical discourse in the classroom."Journal: Mathematics Teacher. Issue: November 2007, Volume 101 , No.4