Mathematics for Curriculum Leaders

Bill Rawson

• Publisher: Routledge
• Year: 2003
• Print ISBN: 9781138442276
• eBook ISBN: 9781134849413
• Topic: Bildung
• Subtopic: Bildung Allgemein

Mathematics for Curriculum Leaders involves teachers in a deliberate enquiry into the nature of understanding in mathematics and the ideas underlying its teaching and learning. Helping children with the language of mathematics is shown to play an important part in mathematics teaching. The pack is divided into 7 units drawing upon the demands of the National Curriculum and providing activities to support children in their attempts to report their thinking. Sensitive collection and interpretation of this information in order to guide action is an essential feature of each unit.

UNIT 1

TALKING MATHEMATICS

LEADER’S GUIDELINES

This Unit identifies discussion as an important element of mathematics learning. It establishes the principle of encouraging talk about mathematics as a means of enhancing thinking about mathematical ideas. Implicit in each of the ensuing Units is the high profile given over to talking and listening.

Briefing Paper

You should distribute copies of this at the end of the Stimulus Activities. It contains recommendations for further reading on this subject.

Stimulus Activities

Three hours should give you sufficient time to work through the activities and consider the roles of both teacher and pupil during mathematical discussion.

1. Sharing recordings of mathematical discussions. Before this session, participants should make recordings of either a teacher-pupil or pupil-pupil discussion in mathematics. They should prepare a short transcript. Distribute copies of Stimulus Activity 1 well before the session in order to help each participant in their preparation and analysis of their transcripts.

2. Mathematical terms. Keep the group together for this activity and write down on a large sheet of paper the mathematical terms they use.

3. Fishbowl. Have ready a supply of pens and paper, geometric shapes, building blocks, measuring instruments. Participants work in groups of three. One observes the other two working on a problem they have chosen from the list. Depending on the time available, participants can repeat the process by swapping roles so that someone else can observe while the other two work on another problem.

Classroom Activities

Participants should be given around 2 weeks to arrange for small groups of children to work on some of the classroom challenges.

1. Number stories.

2. Which is heavier?

3. Triangles on a 3×3 pinboard.

4. Behind the ‘wall’.

5. Between 8 and 10.

6. Plan a day.

Review Session

About 3 hours should be assigned for this session.

1. Reporting on classroom activities and considering classroom roles.

2. Advantages and disadvantages of encouraging discussion.

3. Brainstorming further points and activities.

BRIEFING PAPER

Most teachers in primary schools would like to think that their classroom is a place where children are encouraged to handle, examine, explore, manipulate and talk. Indeed, HMI report that:

The predominant way of organising effective mathematics work in the primary classrooms was within small groups. Working in this way enabled the teacher to promote discussion as an important element of mathematical learning. (DES 1989:20)

Being encouraged to work collaboratively in this way on a shared mathematical task provides children with opportunities for posing questions, testing ideas, removing the fear of taking risks and supporting one another as they develop in understanding. However, conversations during mathematics lessons can easily turn to a form of exposition where the teacher instigates the question, prompts the child to elicit a response, and then supplies some comment which indicates an evaluation of that response. Often quite unintentionally, adopting this kind of teaching style can discourage children from contributing to classroom discussion.

While working alongside groups of children, you may soon become aware of the great amount of off-task conversation. Does this indicate a need to examine both the task and the group composition? A closer examination of children’s talk during collaborative work often reveals the ‘cut and thrust’ of statements uttered, solutions offered, accepted, ignored and often left unchallenged. This can be disconcerting and can provide you with sufficient evidence to convince you to change your teaching style!

Mathematics in the National Curriculum (DES 1991) identifies in Attainment Target 1 the important role of talk which is further supported by Mathematics: Non-Statutory Guidance (NCC 1989). This Unit aims to provide you with opportunities for analysing various classroom activities that involve mediating learning through conversation. Further reading on discussion-based teaching is recommended, and details will be found in the References at the end of this Briefing Paper.

Children Talking

Analysing conversations between children while working on mathematical problem solving reveals how they apply numerous ‘verbal strategies’. Some of these are identified in the following extracts taken from transcripts:

1. Here is an example of a breakthrough in understanding by a nine-year-old. No doubt she has worked on many activities dealing with ¼+¼=½ in her school life, yet in this specific context she now appears to have made a connection:

Pupil

: If I put half with this half, it makes one big square. This quarter joined to this. Eh, does a quarter and a quarter make a half? Two quarters make a half.

2. Examining the structure of a situation is an essential part of the problem-solving process. Questioning and mulling over the problem is a feature of this section of the dialogue:

Pupil

: Was there six between the next stage?

3. Causal links between parts of an experience are justified as a child makes deductions. On the basis of these, the child becomes satisfied with the work that has been carried out so far:

Pupil

: Ten of these must equal forty-one because if you add four each time…

4. In order to ensure that successful and reasonable outcomes are possible, children apply a system of checking and modification. This is captured in the following extract:

Pupil 1	: Let’s work it out again. Let’s work out eight. Four made seventeen, five equals twenty one…
Pupil 2	: But that can’t be thirty-three. It must be thirty-two. You take another one off that. Remember?

5. Having to explain the results of your efforts to someone else is an excellent way of clarifying your own thinking:

Pupil

: We’ve just worked out four. If you times that by two, that’s eighteen. Take one, that’s seventeen. So you times that by two and take one.

6. As you observe groups working together on a problem, a wide range of abilities among those involved soon becomes apparent. Insights for one child may be of no value for another at that moment. On such occasions, these insights are often ignored. Four children were involved in identifying the shapes within shapes. Three were counting hexagons and triangles while the fourth noticed:

Pupil

: A hexagon looks like a box. When I am close, I see a box. When I am far away, I don’t see a box.

This information was ignored by the other children.

7. Placing children in groups provides examples of interesting aspects of ‘collaborative discussion’. It appeared that these two boys were working independently, following their own lines of thought throughout the problem. A look over their comments, however, for the entire problem-solving session does show how their comments formed an interwoven whole and appear eventually to have influenced each other’s thinking:

Pupil 1	: So that would equal eight.
Pupil 2	: Would it work for odd numbers, though?
Pupil 1	: Forty-five can’t be right.
Pupil 2	: They’re all odd numbers, though.

8. The mathematical terms that are used can often be difficult to understand. Fortunately, on this occasion the child asked for an explanation:

Teacher	: We are going to do area today. I want you to cover this surface with these shapes.
Pupil	: What’s surface?

Strategies

Listening to oneself after recording a session often reveals how easy it is to slip into a teaching style that produces the familiar pattern: question by the teacher—response from the pupil—comment by the teacher—next question…. You need, therefore, to make a conscious effort, while in the ‘heat of the moment’ of discussion, to refrain from immediate intervention and think carefully about the kind of interaction that is taking place. Opportunities for mediating learning through structured conversation are characterized, for example, when a ‘Well done’ is changed to ‘Tell me how you worked that out’. ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Introduction to the Primary Inset Series
• Introduction to Mathematics for Curriculum Leaders
• Unit 1 Talking Mathematics
• Unit 2 Developing Mathematical Ideas
• Unit 3 Applying Mathematics
• Unit 4 Mathematical Problem Solving
• Unit 5 Children’s Understanding of Mathematics
• Unit 6 Calculating in Mathematics
• Unit 7 Representing Mathematics