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Jana Kratochvílová

1. Introduction

This paper is focuses on a part of my research. I am interested in the thinking processes involved in solving the mathematical tasks which I have given to 10-11 year old pupils in the United Kingdom. The tasks have a common "element", the triad. I define a triad as three ordered natural numbers (excluding zero) with the third number being the sum of the first two numbers. The tasks with triads were chosen from [9].

The reason why these tasks were chosen is that the tasks are non-standard and therefore the pupils carrying out the tasks cannot use school stereotype answers. A procedure was given to the pupils to generate further triads, the resulting structure produced a pattern. Looking for pattern is the first stage of generalisation which is a part of and an outcome of human thinking. The tasks using the triads are difficult to grasp without a concept of order and an ability in mental arithmetic, which is also a wide field of the research. In addition the structure of triads is a combinatorial structure because they create a tree-graph that appears in many combinatorial and as well as non-combinatorial problems. Finally experience shows that if a task is interesting for pupils they enjoy carrying out the task and discussing it. I found that the tasks I devised with triads answered this purpose.

The 10 to 11 year age group was chosen for the following reasons: It was thought that the level of mathematics needed for the tasks was most appropriate for the average ability of this age group. This age group of pupils moves from Primary to Secondary education at the end of the academic year. A major implication of this is that they move from being taught by generalist teachers for their mathematics in the primary school to specialist teachers of mathematics in the secondary school. We ask ourselves what thinking mechanisms the pupils use before they move to Secondary education.

2. Pilot experiments

The pilot experiments were carried out in a primary school in Derby in the United Kingdom in April and May. Three pilot experiments were done in a quiet library where the experimenter was with three 10-11 year old pupils usually from 9.30 a.m. to 12.00 a.m. The experimenter asked their teacher to choose pupils with average ability for the experiment. Discussion during the tasks was recorded on tape.

2.1. The way of giving the tasks in the pilot experiments

Phase (i) - Introducing the definition of the triad to the pupil:

ˇ Experimenter’s description of a triad – "A triad means three natural numbers (excluding zero), the first two numbers are any numbers, e.g. 9, 6 or 8, 11 or 5, 5. We get the third number by adding together the first two numbers".

ˇ Experimenter gives examples of the triads (e.g. (9,6,15), (8,11,19), (5,5,10)).

ˇ Experimenter gives tasks to the pupil: The first two numbers of the triad are: (3,8, -). The third number is missing. Fill in the number in the third place. The pupil fills in the number.

What is the third number of the following triads (7,5,_), (17,23,_)? The pupil fills in the numbers.

Is the triplet (7,8,12) a triad? The pupil answers the question.

Show why the triplet is not a triad. The pupil shows it.

Phase (ii) - Introducing a procedure to the pupil:

ˇ The experimenter takes a sheet of paper and writes the triad (1,5,6) on line 1. Then the experimenter gives the following procedure:

We put the triad (1,5,6) on line 1. Then we go to line 2 and we get two triads in this way:

1. We take the first number of the triad on line 1, i.e. 1 and put that number in the first place of the new triad on line 2.

- Then we take the third number of the triad on line 1, i.e. 6 and we put that number in the second place of the new triad on line 2.

- Then we add up the first two numbers of the new triad and we get the third number (7) of the first triad on line 2.

2. Then we take the second number of the triad on line 1, i.e. 5 and put the number in the first place of the second new triad on line 2.

- We then take the third number of the triad on line 1, i.e. 6 and put the number in the second place of the new triad on line 2.

- Then we add up the first two numbers of the new triad and we get the third number (11) of the second triad on line 2.

The end point of this phase of the work, i.e. the work the experimenter with the pupil, is Fig. 1a or Fig. 1b. (See Fig. 1a, 1b)

Fig. 1a*

1………………………………………..…(1,5,6)………………………………………….

2………………..…(1,6,7)……………………………...……….(5,6,11)…………………

 

Fig. 1b*

2………………..…(1,6,7)……………………………...……….(5,6,11)…………………

1………………………………………..…(1,5,6)………………………………………….

 

Phase (iii) - The experimenter introduces the tasks to the pupil:

ˇ What will the triads on line 3 look like? What will the triads on line 4 look like?

ˇ How many triads do you think there will be on line 3, 4, …, 10?

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*In some experiments the line 1 was given up on the paper (Fig. 1a) and in some experiments the line 2 was given down on the paper (Fig. 1b).

ˇ Find the ‘smallest’ triad on line 10. The smallest triad on a line is the triad that has the smallest sum. We get the sum of a triad if we add up the three numbers that make up the triad.

Before you start looking for the smallest triad on line 10 which of the triads (2,5,7) and (3,8,11) is smaller?

ˇ Find the ‘biggest’ triad on line 10. The biggest triad on a line is the triad, which has the biggest sum.

Before you start looking for the biggest triad on line 10 which of the triads (4,6,10) and (5,9,14) is bigger?

(Carrying out and discussing the tasks by the pupils follow.)

2. Two extracts of the pilot experiments

The first extract from experiment 2:

Place: Bishop Lonsdale C. of E. (Aided) School, the library, Derby, UK

Date: 29. 4. 1998

Time: 9.30 – 11.45 a.m.

Pupils: Ben – aged 10 (boy), Stephen – aged 10 (boy), Georgina – aged 10 (girl)

Teacher’s evaluation: Ben, Stephen, Georgina – average ability

Note: Georgina has a new piece of squared paper. She has written the triad on line 1 and then the triads (1,5,7), (5,6,11) on line 2.

Exp. 51: We’ve said we take the first and the third number of the triad on line 1.

Note: The experimenter works with the boys. (2 minutes)

Note: Georgina has corrected the third number that is right. She has written the triad (1,5,6) on line 2 and the triads (1,6,7), (5,6,11), (6,11,17) and (5,11,16).

Exp. 55: Could you have the same triads on two lines? (Pause 10 seconds.)

Exp. 56: What is the first number of the triad on line 1?

Georgina 09: 1. (The experimenter writes the number above the triad on line 2.)

Exp. 57: What is the third number of the triad on line 1?

Georgina 10: 6. (The experimenter writes the number above the triad on line 2.)

Exp. 58: So what is the third number of the triad on line 2?

Georgina 11: 7. (The experimenter writes the number above the triad on line 2.)

Exp. 59: Very good. Take the second number and the third number of the triad on line 1.

Georgina 12: 5 and 6.

Exp. 60: Have you had the triad that has the first number and second number 5 and 6?

Georgina 13: Yes.

Exp. 61: Very good. Try to continue from the line 2.

Note: The experimenter works with the boys. (4 minutes)

Note: Georgina has written the triads (6,7,13), (1,7,8), (5,11), (6,11), (6,17), (5,16) and (6,17) again.

Exp. 73: Each triad has three numbers. (Pause 10 seconds.)

Exp. 74: What do we do with the first two numbers? (Pause 30 seconds.)

Exp. 75: We must add them up. The sum of these numbers is the third number. (Pause 40 seconds.)

Exp. 76: Is (5,11) the right triad?

Georgina 14: 16.

Note: She writes 16 in the triad.

The second extract from experiment 3:

Place: Bishop Lonsdale C. of E. (Aided) School, the library, Derby, UK

Date: 22. 5. 1998

Time: 10.00 – 12.00 a.m.

Pupils: Klaine – aged 10 (boy), Mathew – aged 10 (boy), Claudia – aged 10 (girl)

Teacher’s evaluation: Klaine, Claudia – average ability, Mathew – below average

ability

Experimenter 40: Could you tell me what we get from the triad (1,6,7)?

Klaine 13: 1, 6. (Pause 20 seconds.)

Exp. 41: The one is (1,7,8). But we can get another triad from the triad (1,6,7).

Mathew 15: The triad is …

Exp. 42: Think, Klaine. (Pause 30 seconds.)

Exp. 43: We’ve said that we take the first number and the third number that’s the triad (1,7,8). Then we take the second number and the third number that’s another triad. Listen, we need the second number and the third number that are?

Mathew 16: 13.

Exp. 44: Yes. It is 13 together. But the triad is? (Pause 10 seconds.)

Exp. 45: The triad is (6,7,13). Klaine, please write the triad down on line 3. (6,6,12) is not the right triad. Note: Klaine has written (1,7,8), (11,5,16), (6,6,12), (11,11,22), (6,7,13).

Exp. 46: You have from the triad (1,6,7) on line 2 the triads (1,7,8) and (6,7,13) on line 3. Very good. How did you get the triad (11,5,16)?

Klaine 14: 11+5 is 16.

3. Experience gained from the pilot experiments

The pilot experiments were based on the experimenter’s ideas about the way pupils think. However, these ideas did not conform to reality. It was thought that the tasks: What do the triads on line 3 look like? What do the triads on line 4 look like? would be easier than the other tasks and that the focus of difficulties would be in the tasks: Find the smallest triad on line 10. Find the biggest triad on line 10. The answer to the question ‘what do the triads on line 3 or 4 look like?’ requires a knowledge and understanding of the procedure and an ability to follow it. The task to find the smallest triad on line 10 is based on finding the smallest triads on the first four or five lines, finding a pattern and applying the pattern to find the smallest triad on line 10. It was assumed that if the experimenter explained the procedure to the pupils they would be able to follow it. The reality was different. (See examples above.) The focus of difficulties was in the procedure. If the pupils were able to follow the procedure they were mostly able to carry out the other tasks.

There were situations that the experimenter had not expected and therefore the experimenter had not been able to react to the situations. For example, the pupils did not understand the task even after several explanations or their answer to an ‘easy’ question (in the experimenter’s opinion) was not right. It is supposed that it is possible to affront these situations (The other experiments have affirmed it.). For instance, pre-prepared experimenter’s reactions might help the experimenter to be able to use them readily during experiments with the aim to find how the pupils think and not to teach them. But to have pre-prepared reactions does not always mean that the experimenter knows how to react to the situations which arise during experiments. Even though the experimenter has the pre-prepared reactions to the situations it can happen that the experimenter does not always react readily and/or appropriately.

The pilot experiments showed a need to systematise the process of the experiment. A sequence of logical steps was prepared, which it was hoped would help the experimenter as the researcher. In the case of triads the experimenter needs to find out first whether the pupil understands the concept of a triad. Only then can one begin an explanation of the procedure. It is necessary for the experimenter to make sure of the pupil’s understanding of the procedure. After giving another task it is possible to find out by appropriate questioning whether the pupil understands the task. It is hoped that systematised experiments (consisting of logical steps) enables good evidence about the pupil’s solving process to be obtained. However, it must be remembered that if the pupil does not know what a triad means, it is impossible to say anything about the pupil’s concept of order and addition either.

It is thought that not only the systematised experiment but also the standardisation of the experiment might help the experimenter. If the experiment is standardised, it eliminates as many variables as possible. The variables are for example: the number of the pupils in a group for the experiment, the experimenter’s formulation of the tasks and their order, the choice of the experimenter’s questions for the pupils performing the task. By standardisation of the experiment it is hoped that all pupils perform the tasks under similar conditions which will help the experimenter to do comparative analysis and to analyse deeply pupil’s work.

3. The structure of triads

3.1. The concept of triads

In order to understand the structure of triads it is necessary to define what a triad is. Definition: N is supposed to be without zero. An element (a,b,c) of a subset of the Cartesian product N x N x N is said to be a triad if c=a+b.

The triad according the definition means that one of the following situation can happen: a< b, a=b, a>b, but a+b is always in the third place. If aš b then the triads (a,b,a+b) and (b,a,a+b) are different. If a=b then the triads on line 2 made from the triad (a,b,a+b) on line 1 look identical; that is such a triad on line 1 could be represented as (a,a,2a) which leads to the identical triads (a,2a,3a) and (a,2a,3a) on line 2. The both triads on line 2 will produce the same structures and therefore to work with one structure is sufficient. (For the difference see Fig. 2 and Fig. 3)

Fig. 2

1………………………………………...(a,b,(a+b))…...…………………………………..

2……………...(a,(a+b),(2a+b))………...…………………(b,(a+b),(a+2b))……………..

3. (a,(2a+b),(3a+b))((a+b),(2a+b),(3a+2b))…((b,(a+2b),(a+3b))((a+b),(a+2b),(2a+3b)).

 

Fig. 3

1……………………………………….…..(a,a,2a)…...…………………………………...

2………………….(a,2a,3a)………...………………………….(a,2a,3a)……..………….

3………(a,3a,4a)……………(2a,3a,5a)……………(a,3a,4a)…………..(2a,3a,5a)……..

 

3.2. The concept of order in two contexts

The procedure requires the knowledge of order in two contexts.

The first context is based on the fact that a triad (a,b,a+b) consists of three terms, the first term is a, the second term is b and the third term is a+b. All three terms of a triad are in a specific order which is important for the rest of the procedure.

The second context is based on following the procedure in the pupils’ heads. The following description how the pupils follow the procedure might be:

The first triad on line 2 is formed by:

1. Taking the first term of the triad on line 1 and making the first term of the new triad,

2. Taking the third term of the triad on line 1 and making the second term of the new triad,

3. Making the third term of the new triad by adding up the first and second terms of the new triad.

The second triad on line 2 is formed by:

1. Taking the second term of the triad on line 1 and making the first term of the new triad,

2. Taking the third term of the triad on line 1 and making the second term of the new triad,

3. Making the third term of the new triad by adding up the first and second terms of the new triad.

Bold ordinal numbers express the second context of the order. (See Fig. 4)

Fig. 4

1………………………………………...(a,b,(a+b))…...…………………………………..

2……………...(a,(a+b),(2a+b))………...…………………(b,(a+b),(a+2b))……………..

We speak about the first and second and third term of a triad but we also speak about the first triad and second triad and later on about the third and fourth triad on line 3 and so on. We can distinguish between the different contexts of order by looking at the connection of an ordinal number and the word that goes after it. For example, a pupil might say ‘I put the second number of the triad there’ (the first context), or ‘I have already written down the second triad on this line’ (the second context). If new triads on line 3 are made, the pupil needs to think of the two triads on line 2 separately and repeat the previous procedure for each of them. From this point of view the pupils meet a multiplicity of orders and the real need for the concept of order is evident.

3.3. The concept of addition

Addition has three components when applied to the teaching of it to children: concept and skill acquisition and application. (Skill acquisition is when a pupil has an ability to perform a task which might be to recall and use an algorithm correctly for example, or to put numbers into a formula and do the necessary calculations.) Addition is a concept because it is necessary for pupils to understand what is meant by "add this number to that number". One of the simplest ways of defining addition without using numbers is that addition is the putting together of two groups of objects and finding the cardinal number for the totality of the two groups. It is also necessary to interpret and understand not only one but all the many ways in which addition can be expressed. For instance one way is 3 + 4 =, and another ‘what is the sum of three and four’. Addition requires a skill. This means an ability to undertake an algorithm which "performs" the process of addition. This skill needs a complete understanding of the concept of place value as soon as the summation is greater than 9. (See Fig. 5. David does not have the concept of place value and treats the sum as two separate columns of digits.) Using constructivist teaching techniques to give the pupils both understanding and skill, allows the pupils to apply their knowledge in very different situations. If pupils get only one component of addition (mentioned above), for example skill (the process form), then this is formalist teaching in the extreme and can be expressed as – "Addition is the process that must be follow to get the right answer."

Fig. 5

David (9years-old pupil):

There is a rising difficulty of addition involved in following the procedure. At the beginning of the procedure the addition is relatively easy, depending on the initial numbers of the triad on line 1. The answers to the first task - ‘what do the triads on line 3 look like?’ given the triad on line 1 is (1, 5, 6), are four different triads. In one case only the pupils do have to add over ten. To get the third term in each of the triads on line 4 is more difficult than to add the numbers on line 3, because in four cases the sums of the first two terms is over twenty. From the addition of number point of view the task to find the biggest triad on line 10 is the most difficult. As the pupil’s procedure develops from one line to the next line and the biggest triad on each line has the biggest numbers on the line the addition becomes increasingly difficult.

3.4. The structure of triads as objects

The structure of the triads is a branching one like a tree (See Fig.6, see next page.). Each triad produces other two triads that have their own position within the combinatorial structure. There are two aspects of order already mentioned above, one, which relates to the order within the triad and the second which, relates to the order within the procedure. The second aspect of the order defines the combinatorial structure. This dual ordinality makes the undertaking of the whole structure for children both difficult to achieve and even more difficult to visualise.

Fig. 6

4. Analysis of three pilot experiments