Fundamental Constructs in Mathematics Education

Sue Johnston-Wilder, John Mason

• Publisher: Routledge
• Year: 2004
• ISBN:
  9781134338900
• Edition: 1
• Topic: Education
• Subtopic: Education General

Fundamental Constructs in Mathematics Education is a unique sourcebook crafted from classic texts, research papers and books in mathematics education. Linked together by the editors' narrative, the book provides a fascinating examination of, and insight into, key constructs in mathematics education and how they link together. The choice of constructs is based on (some of) the many constructs which have proved fruitful in research and which have informed choices made by teachers.

The book is divided into two parts: learning and teaching. The first part includes views about how people learn - from Plato to Dewey, as well as constructivism, activity theory and French didactiques. The second part includes extracts concerned with initiating, sustaining and bringing to a conclusion learners' work on mathematical tasks.

Fundamental Constructs in Mathematics Education provides access to a wide range of constructs in mathematics education andorients the reader towards important original sources.

Section 1
Activating and analysing learning

This is the first of the two sections making up the body of the text. It consists of extracts from a wide range of authors, addressing questions such as ‘What does it mean to learn mathematics?’ and ‘What is actually learned?’ It is followed by a section on guiding and directing learning.

This section begins with a collection of some of the classic tasks used by researchers to ‘probe’ learners’ understanding. You are encouraged to try these with the learners with whom you work.

The section continues from Chapter 2 with a variety of constructs used in descriptions of what constitutes learning and of conditions which seem to foster and sustain learning. The extracts reveal an ongoing struggle to reach satisfactory definitions of learning, while illustrating a number of different approaches and analyses. The consensus is that learners need to be encouraged and supported in actively taking initiative in their learning. This raises the complex matter of motivation and affect, which attracts researchers to use a wide range of subtly different constructs. If learning involves action, then what learners bring to class to enable them to take initiative and to participate in those actions are natural powers of sense-making. What is learned is the use and extension of those powers applied to mathematical topics. The section ends with constructs which offer a means to expose underlying structure in any mathematical topic.

1 Probing thinking

Introduction

In this chapter we have brought together extracts which include brief descriptions of research tasks which have been designed to probe thinking and which have been taken up in the mathematics education literature and referred to by other workers over the years. The intention of this chapter is to give enough description that the interested reader can try these ‘probes’ with learners of mathematics.

The probes are arranged broadly in order of age of the learners:

• early years (including lower primary);
• middle years;
• later years (secondary and tertiary).

However, many of them are accessible to learners across a wide range of year groups.

Early years

Children’s invention of written arithmetic

I decided to devise a game in which the children’s written representations would serve a clear communicative purpose. The idea for this game arose fairly naturally from my earlier work with boxes and bricks. Young children seemed to be attracted by a closed box containing a number of bricks, and I thought they might be intrigued by the idea of putting a written message on the lid of a box to show how many bricks were inside.

The game centred on four identical tobacco tins, containing different numbers of bricks: usually there were three, two, one and no bricks inside each tin. After letting the child see inside the tins, I shuffled them around, and asked the child to pick out ‘the tin with two bricks in’, ‘the tin with no bricks in’ and so on. At this stage the child had no alternative but to guess. After a few guesses, I interrupted the game with ‘an idea which might help’. I attached a piece of paper to the lid of each tin, gave the child a pen, and suggested that they ‘put something on the paper’ so that they would know how many bricks were inside. The children dealt with each tin in turn, its lid being removed so that they could see inside. When they had finished, the tins were shuffled around again, and the children were asked once more to identify particular tins and see whether their representations had ‘helped them play the game’. The Tins game thus provided not only a clear rationale for making written representations, but also an opportunity to discover what children understood about what they had done.

I carried out a study in which I played the Tins game with twenty-five children, aged 3 years 1 month to 5 years 10 months. Fifteen of the children were in the nursery class and ten children in class 1 of a predominantly middle-class school. Each child was seen individually in a small room away from the classroom … .

There was little doubt about the popularity of the game. The children found the initial guessing-game intriguing and were excited by the idea of making representations with paper and pencil. Several of their comments showed that they were very aware of how this could help them, such as: ‘It’s easy now coz I’ve done some writing.’

There was also little doubt that their representations did in fact help them play the game. Before they made their representations their ability to recognise each tin was at chance level, but afterwards their performance was significantly higher: over two-thirds of the pre-school group and every child in class 1 was able to identify the tins from their representations.

(Hughes, 1986, pp. 64–5)

I was impressed by the children’s ability to recognise their representations, and was curious whether they would still be able to recognise them if some time had elapsed. I therefore returned to the school about a week later and showed each child the tins bearing the representations they had made the previous week. As before, I shuffled the tins, and the children had to guess which tin contained which number of bricks.

The results were striking: the children were just as good at recognising their representations a week later as they had been at the time. Those children … who had made idiosyncratic representations and had been unable to recognise them during the first session, were also unable to recognise them a week later. On the other hand, both Richard and Paul were still able to recognise their idiosyncratic representations a week later, with Richard again spontaneously referring to the ‘tail’ on his representation of zero.

I also used this second visit to the school to find out whether those children who had initially produced any unrecognisable idiosyncratic representations would benefit from the chance to have another go. These children – there were seven, all in the pre-school group – were first asked: ‘Would you like to try it again?’ Most of the children simply responded by saying they couldn’t or wouldn’t think of another way to do it, while those who did try again were no more successful than before. I then suggested the idea of one-to-one correspondence by saying, for example, ‘Why don’t you make two marks on the tin with two bricks in?’ The response to this was immediate. Five of the seven adopted the iconic strategy at once from a single example, generalising without further suggestions to the remaining tins. … The other two children required further examples, but they too eventually adopted the rule. All these children were then able to identify their responses correctly. Thus, by the end of the study, all twenty-five children had produced recognisable sets of responses, with or without prompting.

(ibid., pp. 70–2)

Recognising shapes

For this activity, you need a collection of solid shapes (two copies of each). They might be made up from Multilink cubes, or come from a set of prisms and pyramids, they might be packages, etc.

Recognising 3-D shapes: Display one copy of each object, and place the second copy of one of them in a bag or Feely box [so that everyone except the ‘player’ can see]. Now get someone to feel the object and describe what features they are using to identify which of the visible shapes it is.

This is an example of becoming aware of what you are stressing in order to identify something.

Recognising 2-D shapes: Make up a collection of shapes, or a pack of cards with shape drawings … on them, and sort them into groups. Provide a name for each group. Then look at how other people have sorted them, and try to work out the basis of their sorting, providing names for their groups. Then compare notes.

[…]

Revealing shapes: Construct a screen so that you can gradually reveal a large cardboard two-dimensional shape from behind it. Every so often, pause to get learners to discuss all the possible shapes it could be, to make a conjecture, and to say why they think that conjecture might be right.

(Mason, 1990, pp. 9–10)

Covered counting

A collection of objects (bottle tops, cubes, beans, … ) are counted by each and every learner present, and agreement reached as to the number of them (say 12). Eyes are closed, and some of the objects are hidden under a cloth. The remaining objects are clearly visible.

The task is to figure out how many are hidden when there are seven visible and 12 in all. … The child who needs to ‘count all’, and needs perceptual materials, is unable to solve this task as posed. Another child may be able to count on, saying, ‘seven … eight, nine, ten, eleven, twelve’, while putting up one finger with each number word, beginning at eight. She may then notice that five fingers are up and report five as the answer. Another child may proceed in almost the same way but may need to count the fingers that are up to know that five have been counted. Yet another child might say, ‘I know that 8 + 4 is 12 so 7 + 5 must be 12. So there are five hiding.’ This child is using a thinking strategy approach.

(Cobb and Merkel, 1989, quoted in Yackel, 2001, p. 19)

(See Floyd et al., 1982, Block 2 for an equivalent alternative.)

Children making sense

Listening to children, and following their line of thought can be most rewarding. It can also provide insight into the delicate and lengthy process of making sense through acting according to current conjectures and then modifying those conjectures as a result of feedback. In the following extract, a mother and her young child were having lunch when the window cleaner appeared in the garden. The mother went out to the kitchen to get him some water. While she was there she called out to a neighbour, Pamela.

Child: What did Pamela say?

Mother: She’s having to pay everybody else’s bills for the windowcleaner, ‘cause they’re all out.

Child: Why [are] they all out?

Mother: ‘Cause they’re working or something.

Child: Aren’t they silly!

Mother: Well, you have to work to earn money, don’t you?

(Tizard and Hughes, 1984, p. 120)

After an interlude for pudding, the child reverted to the conversation:

3 Child: Umm … she can’t pay everybody’s, er … all the bills to the window cleaner, can she?

4 Mother: No, she can’t pay everybody’s bills … she sometimes pays mine if I’m out.

5 Child: ‘Cause it’s fair.

6 Mother: Mm, it is.

7 Child: Umm, but where does she leave the money?

8 Mother: She doesn’t leave it anywhere, she hands it to the window cleaner, after he’s finished.

9 Child: And then she gives it to us?

10 Mother: No, no, she doesn’t have to pay us.

11 Child: Then the window cleaner gives it to us.

12 Mother: No, we give the window cleaner money, he does work for us and we have to give him money.

13 Child: Why?

14 Mother: Well, because he’s been working for us cleaning our windows. He doesn’t do it for nothing.

15 Child: Why do you have money if you have … if people clean your windows?

16 Mother: Well, the window cleaner needs money doesn’t he?

17 Child: Why?

18 Mother: To buy clothes for his children and food for them to eat.

19 Child: Well, sometimes window cleaners don’t have children.

20 Mother: Quite often they do.

21 Child: And something on his own to eat, and for curtains?

22 Mother: And for paying his gas bills and electricity bill. And for paying for his petrol for his car. All sorts of things you have to pay for, you see. You have to earn money somehow, and he earns it by cleaning other people’s windows, and big shop windows and things.

23 Child: And then the person who got the money gives it to people …

It seems until turn 11 the child was under the impression that the window-cleaner pays the housewives, and not the other way round. In the course of the conversation the relationship between work, money and goods is slowly outlined for her, but it is still unclear from her last remark whether she has really grasped all that has been said.

(ibid., pp. 120–1)

The authors concluded that the conversation of which this extract is part ‘reveals something which is characteristic of the slow and gradual way in which a child’s understanding of an abstract or complex topic is built up’ (ibid., p. 122). It would seem that the child is aware that she has not grasped the full picture and she returns to the conversation repeatedly. In the end the adult closes it down, apparently before the child has fully resolved it to her satisfaction.

Knowing in action, knowing in words

… in the case of a sling made of a ball attached to a string which the child whirls around and then throws into a box, it has been found that the action is performed successfully at age four to five after several tries, but its description is systematically distorted. The action itself is successful: the child releases the ball sideways, its trajectory tangential to the circumference of the circle described while being whirled. But he maintains that he released the ball either in front of the box or at the point of the circumference nearest to it, or even in front of himself, as if the ball pursued a straight line from himself to the box, first passing through the diameter of the circle described by his arm.

The reason is first of all that in ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Sources
• Introduction
• Section 1 Activating and analysing learning
• Section 2 Guiding and directing learning
• Epilogue
• Bibliography