Classroom Interaction

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their interpretation is only valid for special cases. However, the geometrical meaning of this property is related to the cosine formula learned in the Compulsory Part. If some groups can find that the property is the vector version of the cosine formula, they can be invited to explain how they arrived at this geometrical meaning.

If none of the groups can arrive at the actual meaning, the teacher may guide them to find it out by giving prompts. Some well-constructed prompts (or scaffolds), such as asking them to draw various types of triangles and find clues to connect |a−b|, a⋅b, |a| and |b| with the triangles drawn, may be provided. The co-construction approach is adopted here.

After understanding the geometrical meaning, the result can be derived by applying the cosine formula learned in the Compulsory Part. The groups are further asked to explore alternative proofs. Here, the inquiry approach is employed. The groups may not think of proving this property with |x|² = x⋅x directly. The teacher may give some hints to guide them. In this case, the teacher and the students are co-constructing knowledge. If the students still cannot prove this property, the teacher can demonstrate the proof on the board using the direct instruction approach. Whatever methods the students use, they are invited to explain their proofs to the class. During the explanation, the teacher and students may raise questions and query the reasoning.

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quadratic equations?”

 Be patient and use silence to encourage reflection

There should be pauses after questions to encourage student responses. The teacher should wait for students’ replies, and resist the temptation to break the silence by answering the questions for them.

 Arrange and use classroom space to encourage interaction

Putting students into small groups or arranging them to sit face to face in a circle may promote discussion and interaction among them.

 Create a friendly environment

The teacher should welcome students’ ideas and consider all their views without rushing into making judgments. He/She should also focus discussion on ideas rather than on judging people.

How teachers view and handle interaction is a key factor in determining the effectiveness of classroom interaction. Making use of effective questioning techniques, providing an encouraging classroom environment, scaffolding and providing appropriate feedback are important for maintaining the sort of interaction that leads to learning.

(a) Questioning

Appropriate questioning practices not only elicit students’ thinking but also enhance their understanding. The questions the teacher asks can take various forms.

Sometimes they may be simple, lower-order questions, for which students may have an immediate answer, to check whether they have learned factual information. At other times the teacher has to ask more open questions to which there may not be any one or simple answer. For example, it is helpful for the teacher to ask: “Can you explain how you got this answer?” or “Could you make use of a diagram to help with your explanation?” Instead of merely giving correct answers, the teacher has to be a good listener and give students sufficient time and opportunities to build up their understanding of mathematics through contextualised problems that make sense to them. This allows the teacher to gather useful information on students’ progress in understanding mathematical ideas.

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Tips for questioning in the classroom

 Pause (giving students time to think) after asking a question.

 Avoid asking questions which require “yes” or “no” answers.

 Follow up student responses by asking “why” or by passing the question to the class or another student for a reaction.

 Limit the use of questions that rely almost completely on memory.

 Avoid directing a question to students for disciplinary reasons.

 Avoid asking questions that contain the answer.

 Do not call on a particular student before asking a question.

 Ask open-ended questions.

 Do not label the degree of difficulty of a question.

 Avoid asking for choral responses from the whole group.

(b) Scaffolding

One of the teacher’s roles has been conceptualised as “scaffolding”. Scaffolding is the support that a teacher provides in helping students learn how to perform a task that they cannot yet manage on their own. In Mathematics, teachers’ scaffolding practices can be categorised into three levels. Level 1 scaffolds refer to those prompts and stimuli that already exist in the environment, either as a result of deliberate planning or by chance, and which serve to support students’ learning in Mathematics. These may include, for instance, a poster displaying useful formulae, and computer manipulatives such as dynamic geometry software and self-correcting mathematical games. Although the teacher’s immediate involvement may be low, the level of support can be high depending on the time and effort he/she has expended in finding out which displays, tasks and materials are available. Level 2 scaffolds include direct interactions between teachers and students focusing on the specific tasks in hand. Practices that require students to be involved more actively in the learning process are also included in scaffolding at this level. The strategies adopted may vary from direct instruction, showing and telling, to more collaborative meaning-making. Level 3 scaffolds aim to make connections between students’

prior knowledge and experience and the new mathematics topics to be learned.

Students are likely to engage in longer, more meaningful discussions, and meanings are more easily shared when each student engages in the communal act of making mathematical meaning. The two Further Learning Units – “Further Applications”

and “Inquiry and Investigation” – aim at providing teachers with a platform for

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providing scaffolding at this level.

It is worth noting that the approach of encouraging student discussion is heavily dependent on establishing mutual trust between the teacher and the students. On the one hand, the teacher has to trust the students to persist in attempting to solve mathematical problems so that they can feel free to talk about their solutions and describe how they reached them. On the other hand, the students must trust the teacher to respect their efforts and give them opportunities to discuss their understanding and explain their attempts to solve the problems.

Criteria for choosing a mathematical problem

A good mathematical problem:

 has important, useful mathematics embedded in it;

 can be approached in several ways, using different solution strategies;

 has various solutions or allows different decisions or positions to be taken and defended;

 encourages student engagement and discourse;

 requires higher-order thinking and problem-solving skills;

 contributes to students’ conceptual development;

 has connections with other important mathematical ideas;

 promotes the skilful use of mathematics;

 provides an opportunity to practise important skills; and

 creates an opportunity for the teacher to assess what his/her students are learning and where they are experiencing difficulties.