Making Choices, Part 3: Choices, Constraints and ControlAuthor(s): Pat Perks and Stephanie PrestageSource: Mathematics in School, Vol. 21, No. 5 (Nov., 1992), pp. 44-45Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214941 .

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MAKIN CHOICES

PART 3

Choices, Constraints and

Control

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

In the Southern Examining Group's criteria for coursework assessment, one of the factors by which pupils are judged to be of a higher standard is their ability to extend a task and to see alternative pathways. Similarly, the National Curriculum ATI level 2 states that pupils should be able to:

Respond to the question "What would happen if ...?"

Level 6 talks about pupils "Posing own questions..." though the level 4 statement "identify and obtain infor- mation..." and level 2 "select materials needed..." imply that the pupils should take an increasing degree of responsi- bility for mathematical tasks and not wait for level 7 before being expected to:

Follow new lines of enquiry when investigating within mathematics itself ...

For both pupils and teachers the notion of being able to change and extend tasks is often one that appears quite difficult unless some method of practising how to change and extend can be found. One method is what has come to be called "What if not ..." as a shorthand for the activities described very clearly by Brown and Walters in their book "The Art of Problem Posing".

Having worked with activities described in Brown and Walters in the past and having found them helpful, it seemed natural to offer this method to the PGCE students. Initially our intention was to provide them with a way of extending tasks to create new ones but other things hap- pened along the way. The first task analysed was an old favourite, the elephant in the bun house.

The elephant has a house, which has a number of rooms each room having door(s) to other rooms. In those rooms there are buns.

entrance.

exit

The elephant can only come into the house through the entrance and leave by the exit. The elephant never goes into a room unless there are buns in the room Once in a room, the elephant eats all the buns.

How many buns does the elephant eat?

* Does the elephant have to go into all the rooms? * Is the elephant really hungry? Is it trying to eat

as may buns as possible? * Does the elephant have to come out of the house? * Can it leave some buns behind?

Some of these questions can be answered by rereading the problem, but some questions begin to illustrate that choices can and have to be made.

In order to consider the possible choices you might have available for changing this problem, the task itself needs to be analysed. The students were asked to write down all the things they noticed about the wording of the problem, the picture and the rules, in order to isolate all the individual components which make up the complete task. A list began to emerge:

44 Mathematics in School, November 1992

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* the house has 8 rooms * each room is square 9 there is only one exit * there is only one entrance * the number of buns goes 1, 2, 3, 4 ... * there are a different number of buns in each room * some rooms have 2 doors others have 3 * the rooms are arranged as a two by four array * every room is the same size * all the doors are open

As this list was being collected from the students other items appeared that did not strictly belong to the above:

* Does the elephant have to enter every room? * Is the elephant on a diet? * Can the elephant leave some buns behind for

later?

In searching for the words to identify the differences between some of these statements and the ones that belonged to the original list, the word choices was used frequently but a new word appeared - constraint. For example it is a constraint of the rules being applied correctly that the elephant cannot enter every room. Whether the animal is on a diet could be seen as a choice or a constraint. Interpreted as a choice, the elephant still obeys the rules and scoffs everything in a room, but enters the rooms in a particular order so as to eat the least number of buns. Interpreted as a constraint, the elephant cannot be on a diet because the rules say that the elephant has to eat all the buns in a room and cannot control the intake of buns sensibly.

The use of the word constraint in describing attributes of a problem seemed to be very useful. It is in acknowledg- ing the constraints and obeying them that enables pupils to arrive at similar solutions. If choice-making were to be completely free, solutions would bear no resemblance to the original questions. In the choices that a teacher makes, the constraints become the most identifiable source of the control that a teacher has over the activities that the pupils are doing. In giving particular rules the question can be tightly structured. In recognising the rules, pupils accept the task as set and the constraints this places upon their choice-making.

For example, in offering the task to the pupils by restating the question as:

What is the maximum number of buns the elephant can eat?

they are directed to finding the answer to a particular question, the pupils do not have to make decisions about what the answer should look like. The teacher is then free to extend or change the task in a particular way. The next questions posed might be:

Supposing the next day each room had an extra bun, what would the answer be then? What if there were two extra buns in each room or three or four or ...?

In imposing a particular constraint and extending the task in this way the focus of the activity is shifted to considering strategies for finding the totals of sets of numbers of the following form:

1, 3, 4, 5, 6, 7, 8 2, 4, 5, 6, 7, 8, 9 3, 5, 6, 7, 8, 9, 10 etc

If, on the other hand, the focus of the activity is to find the sum of the numbers from 1 to 8, 2 to 9, or 3 to 10 then the diagram needs to be altered. So, by fixing the constraints the teacher can control the task quite tightly.

The task could be changed by allowing the doors to be open or closed and changing the eating rule from that of the elephant eating all the buns in a room to eating at least one bun in every room visited. If the question

How many buns does the elephant eat?

is posed, there are many possible solutions. For example, the least number of buns which can be eaten and allows the elephant to get out of the house, is 5. This also allows there to be 6 doors closed. The elephant can do this in four ways. Is it possible for the elephant to eat 6 buns? What are the strategies for finding the minimum to the maximum number of buns? The methods of solution and the organisation of planning remain as choices of pupil?

The task becomes more open and possily more diverse in its solutions by releasing constraints and extending pupil choices. Analysing aspects of a problem in terms of choices and constraints provides both a focus for changing the task and a planning tool for the way a task might be controlled. It is in balancing the choices and constraints that the teacher can plan for the freedom or otherwise of their pupils' involvement. The structure of a lesson is planned by the teacher. Planning requires tools. From looking at the elements generated by the elephant problem a good process appears to be:

* analyse the possible choice making in the task * choose the level of control of the task * introduce or remove constraints to limit or extend

pupil choices to support the control.

Reference Brown, S. I. and Walter, M. I. (1983) The Art of Problem Posing. The

Franklin Institute Press.

New Thinking about

the Nature of Mathematics

a collection of non-technical papers by Philip Davis, Ray Monk,

Warwick Sawyer, Paul Ernest, David Henley, Eric Blaire

Edited by Chris Ormell

ii + 84 pp. ISBN 0 907669 20 4 Paperback, approximately B5 size

Publication: November 1992

available from MAG-EDU, University of East Anglia,

Norwich NR4 7TJ England x5 post free to addresses in the UK. Add x1 for EC and x2 for non-EC

Cheques, drafts in Sterling payable to MAG-Ashby

Mathematics in School, November 1992 45

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