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Making Choices: Part 1. Making Choices ExplicitAuthor(s): Pat Perks and Stephanie PrestageSource: Mathematics in School, Vol. 21, No. 3 (May, 1992), pp. 46-48Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214888 .Accessed: 09/04/2014 15:32

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CHOICES

CHOICES

SMaking SChoices Explicit

by Pat Perks and Stephanie Prestage, School of Education, University of Birmingham

The following scenario is a common one on teaching practice:

Tutor: "What is your lesson today?" PGCE student: "It is an investigation."

Tutor: " Yes but what is your lesson about? PGCE student

(puzzled): "It's an investigation." It is clear that many of our students think that what is

expected is that either a) they teach their pupils how to do investigations or b) they let their pupils do investigations in the spirit of

free exploration.

Either way the link with other aspects of the mathematics curriculum is rarely evident in the planning, execution or evaluation of such lessons. This, despite our best endeav- ours to place the role of investigating and investigations within the whole of the spectrum of the teaching and learning of mathematics. The scenario described is not surprising. It is after all a common interpretation of the role of AT 1 within the curriculum, supported in secondary schools as a consequence of the way that the coursework component of many GCSE syllabuses is defined and assessed.

Although the advent of the National Curriculum has brought about the statutory requirements for the inclusion of both process and content within the mathematics cur- riculum the situation is far from clear. The inclusion of

process aspects into a distinct target has resulted in the possible reinforcement of the perspective that problem solving approaches are separate from, and additional to, the content of the mathematics curriculum. A recent HMI report expresses concern over the lack of evidence of AT 1 in classrooms

Many schools had difficulties with ATs I and 9 [now AT 1] ... because there had been little emphasis on these areas in the past. It was generally considered easier to fill 'content gaps' than to develop a curriculum in which these A Ts underpin the mathematical work done by pupils.l

The Non-Statutory guidance also puts forward the notion of underpining or permeation. Section D3 suggests that:

Using and Applying Mathematics, as represented in Attainment targets 1 and 9 [now AT 1], and the associated elements of study should stretch across and permeate all other work in mathematics providing both the means to and the rationale for the progressive development of knowl- edge, skills and understanding2

How should teachers approach the inclusion of AT 1 into the school curriculum? How should we work with out PGCE students to develop the ideas contained in AT 1? One way is to continue to see these as extra activities to the content curriculum i.e. investigations, practical work,

46 Mathematics in School, May 1992

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problem solving and extended pieces of work. But what about permeation? A model for planning for the inclusion of processes into the curriculum which focuses on math- ematics rather than specific activities, has been developed by an NCC project team (see notes), which included one of the authors, working with a group of teachers in London, Birmingham and Cambridge. It suggests that teaching at all levels should include opportunities for pupils to:

. make and monitor choices * communicate . reason.

The teachers invovled in the project found that through considering these three strands it was possible to address aspects of AT 1 in all their work rather than only in specific activities. They were making more balanced decisions about the learners needs and that it enabled both the teacher and the pupil to be active participants in the classroom.

We were particularly interested in exploring with our PGCE students the notion of making and monitoring choices. Work from the NCC project suggests that this is one aspect that teachers can control and that an increase in choices for the pupil increases also the need for the pupil to draw on aspects of Attainment Target 1 within a given task.

The focus on choice-making raised some interesting issues. In one of the sessions we decided to see how the style and wording of a question might limit or extend choices and analyse the resulting consequences for the classroom. By slightly altering a fairly common text book task we provided alternatives which could involve pupils in different degrees of making choices leaving the teacher with the decision of the appropriate task for a particular lesson. The analysis of the choice making involved also helped:

. to recognise the mathematics hidden in the question,

. to realise the possible stumbling blocks in what seem to be a fairly straightforward question

* and to realise the possibilities available to us as teachers.

We gave the students six versions of similar questions, in the order shown in figure 1.

Version 1

Draw this triangle accurately.

9 cm

NB the diagram is not drawn to scale.

Fig. 1

Version 2 Draw triangles with angles 30., 60., 90. and one side of 9 cm.

Version 3 Draw triangles with angles 30., 60., 90..

Version 4 Draw triangles with angles 30., 60., 90.. Measure and label the length of the shortest side.

Version 5 Draw triangles with one angle of 30.

Version 6 Draw triangles with angles 30., 60., 80..

As they looked at each problem in turn the students were asked to analyse them under three headings:

a) pupil choices, i.e. what a pupil might choose to do, what different interpretations might a pupil make;

b)the possible learning outcomes c) teacher choices.

They were also asked to think about how their analysis changed as the result of looking at the different versions in turn. (In preparation for the session we worked through the activity ourselves, thinking about the choices we could make explicit and how our thinking developed as we looked more closely at the different versions. For example, ver- sion 3 started us thinking about similarity and how this might be developed by changing the wording and asking pupils to measure all sides or whether it would be sufficient to work with version 4.)

The first version can be found in many text books. You might like to consider the pupil choices involved in this question. Initially we could not find any.

An analysis of the choices by the students and discussion of this task produced the following:

Pupil Choices: . Which part to draw first? 9 Whether to use the three angles or to realise that a

checking element is available when 3 angles are given? . How accurately to measure?

Possible Learning Outcomes: . Practice/consolidation of the use of ruler and protractor. * Angle sum of a triangle. * Use of extra data to check.

Teacher Choices: . What aspects to exploit when pupils have tackled the problem?

The recognition of "Which part to draw first" as a pupil choice raised an important point. Whilst it is not usually considered as a choice, it often causes difficulty, because the pupils do not necessarily realise that they have to make a decision.

Once we had stimulated the creativity of our students, they recognised issues which had never occurred to us. For example, "What if the pupils are not allowed to use protractors?" The focus for the learning could now change to the use of constructions or trigonometry, depending on whether pairs of compasses are permitted or not. The provision or restriction of apparatus can change the type of mathematics being considered. This prompted the ques- tion "What if the pupils were asked to draw this on a

Mathematics in School, May 1992 47

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computer using LOGO?" As a teacher do I need to change the 9 cm to 90 units? Or would I like the pupils to explore how close they can get to 9 cm? Is the concept of 'triangle' different when drawing the shape with pencil and paper and drawing it with LOGO? (Leron 1989). The focus for learning may, for example, change to exploring external angles. By approaching a fairly standard text book question in this way two new possibilities have become avialable which may not have occurred to us in any other form of planning.

The next version offers a similar task without a picture.

Version 2

Draw triangles with angles 30., 60., 90. and one side of 9 cm.

Fig. 2

Pupil Choices: * What does the triangle look like? Any horizontal or

vertical lines? Which part to draw first? Whether to use the three angles or to realise that a checking element is available when 3 angles are given? How accurately to measure? How many triangles need to be drawn?

Possible Learning Outcomes: Practice/consolidation of the use of ruler and protractor. Angle sum of a triangle. Use of extra data to check. Essentially only three different triangles.

* Lengths of sides and the position of the shortest side?

Teacher Choices: * What aspects to exploit when pupils have tackled the

problem? * Does the wording of the activity need changing? * Should the pupils be asked to measure and label each

side in order to highlight the last two foci for learning?

The small alterations made to the original question appear to have extended the task considerably. Some of the decisions pupils have to make may now be more explicit to them. For example, they need to decide how many triangles should be drawn. The task for the teacher is to help them to become choice-makers, not by telling them the answer to "How many?" but by encouraging them to make their own decisions. The possible choice-making was extended even further by our students who noticed the ambiguity in the wording of the question. We as the writers of the task 'knew' what the words meant and had not expected the possible choice shown in figure 3.

gcm 9cm 9cm

Fig. 3

The third version offers less information to the pupils.

Version 3

Draw triangles with angles 30., 60., 90..

Pupil Choices: * What does the triangle look like?

Are there any horizontal or vertical lines? Which part to draw first. Whether to use the three angles or to realise that a checking element is available when 3 angles are given. How accurately to measure.

* How many triangles need to be drawn. Do I draw a triangle with an angle of 30. and the other two angles can be anything I like?

Possible Learning: Outcomes: Practice/consolidation of the use of ruler and protractor.

* Angle sum of a triangle. Use of extra data to check.

* Specialising - drawing particular cases. * Generalising - notions of similarity.

Teacher Choices: What aspects to exploit when pupils have tackled the problem. Does the wording of the activity need changing? Should the pupils be asked to measure and label each side in order to highlight the last two foci for learning? Or measure and label the shortest side? Would these exploit different aspects?

Again a very slight alteration to the wording has extended the pupils choices and the richness of the activity. Children may recognise that each triangle is related to all the others drawn and come to recognise the existence of the "one" triangle which represents the solution set. The act of practising the drawing of triangles offers special cases. The process of specialising permeates the activity and might enable the pupils to generalise about similarity.

Each of the versions considered offered different possibil- ities and also practice in recognising these. The more we analysed the more we were able to analyse and the more the opportunities for teaching and learning become explicit. Analysis of later versions enabled us to find new possibil- ities in earlier versions. Any task has inherent within it pupil choices and, as we have shown, small alterations may extend or limit those choices. As the chocies were extended so aspects of attainment target 1 with attainment target 4 became explicit. (Compare the process and content out- comes for versions 1 and 3). Once the choices have been identified, we as teachers may offer many of these or just a few depending on the objectives of the lesson. The issue of how we help the pupils to recognise their role in choice making is the starting point for another article.

By analysing these differing questions in terms of the choice-making it is possible to have a way of looking at what mathematics might happen and also open up our minds to other possibilities. No longer do we have the problem of searching for new tasks since by adapting old ones the resources become endless. As one of our students asked:

What happens if these triangles are drawn on a balloon?

Note The NCC project, "Using and Applying Mathematics" started in January 1990 and finished in July 1991. The project director was Margaret Brown (King's, London) and the rest of the team Mike Askew (King's, London), Angela Walsh (Institute of Education, Cambridge) and Stephanie Prestage (Birmingham). The results of the pro- ject have yet to be published. References 1. HMI (1990) Mathematics, Key Stage 1 and 3, HMSO, London. 2. DES (1989) Mathematics in the National Curriculum, HMSO, London.

48 Mathematics in School, May 1992

Article Contentsp. 46p. 47p. 48

Issue Table of ContentsMathematics in School, Vol. 21, No. 3 (May, 1992), pp. 1-52Front MatterEditorial [p. 1-1]Curve Stitching in LOGO [pp. 2-7]Mathematics Homework on a Micro [pp. 8-11]What If...? 10. Bananas [p. 11-11]It's Life but Not as We Know It [pp. 12-15]Let's Twist [pp. 16-18]Using Video in the Teaching of Mathematics [pp. 19-21]Puzzles, Pastimes, Problems [pp. 22-23, 40-41]Welcome 1992! [p. 24-24]Mathematics in a Matchbox. Part 1 [pp. 25-28]On Rectangles in Rectangles [p. 29-29]Force Five in the Forties: Numbers out of the Air [pp. 30-33]Girls into Mathematics: An LEA Based INSET Initiative [pp. 34-35]Back to the Future: 10 Investigations Based on the Puzzles of H. E. Dudeney and...