Primary School Focus || Doing, Talking and Explaining

Doing, Talking and ExplainingAuthor(s): George BallSource: Mathematics in School, Vol. 23, No. 1, Primary School Focus (Jan., 1994), pp. 2-6Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215064 .

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George Ball, University of Glamorgan

The National Curriculum Orders' demand that the child should be treated as a "mathematical thinker" who is able to reason, communicate and use the mathematics that is taught. The programmes of study for each attainment target make it quite clear that a balance is required between mathematics perceived as a body of knowledge and mathematics as a mode of enquiry. In order to achieve this balance classroom practice requires a practical and exploratory-based element in which doing, talking and explaining play a central role. This element needs to permeate all of mathematics teaching and to be seen as a way of working which places problem solving, investigative activities and practical work at the heart of the curriculum.

In this article I want to report on a series of mathematical activities which I have used to help develop this philosophy. The activities described constitute one day of a ten-day (60 hour) Inset course2 designed to help increase confidence and understanding in primary mathematics teachers.

2

Organisation and Implementation The course, which has been offered to primary teachers in Mid Glamorgan on nine separate occasions, lays great emphasis on the need for "learners" (children and adults) to share, compare and reflect on ideas. A particular focus for the present activities is the need to explain and justify results and not merely to "spot" trends and patterns. The participants at each course, who numbered between 12 and 15 in all instances, were drawn from the whole primary range and were very much a mixed-ability group. As I had planned three separate activities each group was sub- divided into three, and given approximately one hour to investigate an allocated activity. I circulated amongst the sub-groups attempting to act as a facilitator and adopting a supportive role based on the principles set out in my book "Talking and Learning".3

As the activities progressed, the need for participants to develop and communicate a clear understanding of the issues involved was given a sharper focus by extending each group's brief to include a reporting-back requirement. Each group was to give a short (ten-minute) presentation to their peers, describing the activity, the work they had carried out, and the results they had obtained. Each group was also required to justify, by reference to their activity, the validity of their findings. This exercise was designed to focus attention more closely on the three strands apparent in Mal namely: "applications" - "mathematical communication" - "reasoning, logic and proof", or, in my terminology, "doing", "talking" and "explaining".

The Activities The activities were presented in worksheet format and are reproduced in Figures 1, 2 and 3. They were chosen to have the same underlying mathematical structure and

Mathematics in School, January 1994



ODDS AND EVENS

START If you throw an even

S If you throw an odd

number move left. number move ri~ghiI

A B C D

Place a counter on the START position.

Roll a dice.

If it shows an EVEN number move down and to the LEFT.

If it shows an ODD number move down and to the RIGI-FFT.

Continue rolling the dice ard moving your counter until it reaches A, B. C or D.

If you repeat this activity 50 tmes

which of the four channels do you think will be the nmosi used-

Investigate.

Extend the maze to four (or more) rows.

Can you explain your results ?

Fig. 1 Odds and evens

to offer opportunities for a practical, discussion-based approach in a style advocated by Brissenden." Relevant apparatus and materials were readily available for use, including card, gummed paper, scissors, pencils, crayons, etc., as well as Colour Factor and dice. During the developmental period I was careful to allow thinking time

PARTITIONS

You could use Colour Factor or Cuisenaire rods to help with this problem.

The 3-rod can be partitioned (split up) in four different ways:

I i I f

1+2

2+1

3

Inve tigate partitions of different numbers.

Fig. 2 Partitions

and to use open questions which encouraged the participants to explain what they were doing, and/or to consider "what might happen if ...?" All contributions were valued and respected, although I tried not to openly evaluate their quality, or to impose my own thinking on the way things developed. If patterns were spotted the groups were encouraged to explain and justify these by reference to the activity itself. Indeed, in order to set up patterns and provide opportunities for generalisations, the groups were encouraged to view each activity as a special case of a more general family of problems which could be investigated by varying an appropriate parameter, i.e., the number of rows in the maze, the number of items of clothing available for dressing the snowman, or the size of rod to be partitioned. Interested readers may care to explore each of the activities before reading on. The summaries that follow are intended to encourage some doing and explaining on the part of the reader and

SNOWMEN

A snowman can be dressed i, a hat, a scarf, a pair of buttons, a belt, or any combination of these four items.

How many different snowmen can you produce?

Fig. 3 Snowmen

consequently become progressively less detailed. If anyone wishes to do some talking, either in relation to the activities themselves, or the Inset course in general, then I would be pleased to communicate with them via the address given above.

Odds and Evens Each of the sub-groups which worked on this activity began with some practical experimentation. Having played the "game" 50 times and discovered B and C to be the more popular end points, most were happy to extend the maze to 4 rows and repeat the activity.

With the new configuration (see Figure 4) the general feeling was that B4, C4 and D4 would be the more likely end points. Symmetry arguments were often given and

3 Mathematics in School, January 1994



One-row maze

Two-row maze

Three-row maze

Four-row maze

START

Al B

A2 B2 C2

As B3 C3 D

B4 C4 D4 E4 A4

Fig. 4 The 'four-row' maze

If you throw an even number move left.

START If you throw an odd number move right.

A3 X B3 - C3 X D3

Fig. 5 Three Routes to B3 which can be denoted by (L, L, R) etc.

everyone thought that there would be "more in the middle", but no-one was initially able to say why this should be so. The empirical approach (replaying the game and counting outcomes) provided reasonably accurate ratios for the underlying theoretical distributions (A3 :B3:C3:D3:=1:3:3:1 and A4:B4: C4:D4:E4= 1:4:6:4:"1) although this was not recognised in all cases.

Asking a group to "show me" how the counter might get to B4 or C4 served to introduce the notion of a route. Coloured pencils provided a useful aid in distinguishing and counting routes, although some groups preferred to work with combinations of "left" and "right". For example, possible routes to B3 are illustrated in Figure 5, and these may be denoted by (L, L, R), (L, R, L) (R, L, L).

Using a "one-row" maze and a "two-row" maze as further special cases provided enough information for pattern-spotting and the impetus for links with Pascal's Triangle as indicated in Figure 6.

One-row maze

Two-row maze

Three-row maze

Four-row maze

Five-row maze

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 1-0 10 5 1

Fig. 6 The number of routes to any given end point (Pascal's Triangle). Summing adjacent entries in any row produces the value of the entry in the intermediate position in the next row. But why should this be so?

Often groups felt they had done enough in identifying the underlying pattern and were happy merely to use their new power to predict results for the "five-row maze" and subsequent cases. However, by continuing to focus attention on questions such as "why should we add adjacent numbers?" and "how do we get to square C4 (say)?"




(see Figure 4) most groups were eventually able to provide a satisfactory reason. Usually the argument was based around the fact that:

To get to C, one would either need to get to B3 and then throw an odd number or get to C3 and then throw an even number.

Since there are 3 ways (routes) of getting to B3 and 3 ways of getting to C3 this means that there must be 6 ways of getting to C4

The experimentation and discussion which had preceded this explanation enabled the majority of participants to accept this argument (and the resulting generalisation for any end point) as almost self-evident once it had been articulated. Such an "understanding" of the underlying mathematical structure was to be a feature of the other activities also, although I strongly suspect that this would not have been the case without the accompanying doing and talking.

Partitions On the face of it, this seems quite a straightforward activity and a little free-play reveals that a doubling rule is in evidence:

Fig. 7 Partitions of a 1, 2 and 3-rod

viz.,

Length of Rod Number of partitions

1 unit 1 2 units 2 3 units 4 4 units 8

Consequently, it is tempting to believe that the investigation is at an end and that the number of partitions of an n-rod (i.e. a rod of length n units) can be found by multiplying 2 by itself (n - 1) times. Although this is the case, the investigation can be easily extended by challenging the group to predict (say) "the total number of partitions of a 7-rod that can be made using just 3 separate rods". Figure 8 shows three ways of performing this task, some systematic way of setting out the blocks (or recording cases) is required if the desired total is to be determined.

Fig. 8 Three ways of constructing a 7-rod using exactly three rods

One possibility is to write:

1 -rod (1)

2-rod (2) (1,1)

3-rod (3) (2,1)(1,2) (1,1,1)

4-rod (4) (3,1)(2,2)(1,3) (2,1,1)(1,2,1)(1,1,2) (1,1,1,1)

Fig. 9 Systematic method of recording possible partitions of an n-rod

where (2) indicates that a 2-rod has been made with just one rod, and (2,1,1) indicates that a 4-rod has been made with three separate rods (a 2, a 1, and a 1-rod).

It is now a small step to identify the number of ways of making an n-rod using one rod, two rods, three rods ....

Size of rod Number of ways

1 -rod 1

2-rod 1 1

3-rod 1 2 1

4-rod 1 3 3 1

5-rod 1 4 6 4 1

Fig. 10 Number of possible partitions of an n-rod

Clearly, Pascal's triangle describes the underlying structure once again. However, it is a much more demanding task to explain why this structure is in evidence. To do this it is necessary to interpret clearly the significance of individual entries in the 'triangle'. (See Figure 11) The 'boxed' entry of 15 (in the seventh row of Pascal's Triangle) represents the number of ways of making a 7-rod with three separate rods.

Since a 7-rod can be made from a 6-rod by adding a 1-rod, or from a 5-rod by adding a 2-rod, or ..., it follows that this number can be found by summing:

the number of ways of making a 6-rod with two separate rods, the number of ways of making a 5-rod with two separate rods, the number of ways of making a 4-rod with two separate rods, the number of ways of making a 3-rod with two separate rods, the number of ways of making a 2-rod with two separate rods.

Notice that the addition pattern generated is indicated in Figure 11 and is not the more apparent and generally stated "rule" for Pascal's Triangle.




Fig. 11 Significance of individual entries in terms of partitions

1

1 1

1 2 1

1 3 3 1

4 6 4 1

5 10 10 5 1

15 20 15 6 1

1

1

1 ~ 6

Snowmen Dressing showmen with one, two three, ... items of clothing produces equivalent results to those shown in Figure 6. In this case however, the horizontal numbers relate to the number of different snowmen that can be made using 0, 1, 2, ... of the available items of clothing. Thus the boxed entry in the fifth row of Figure 6 can be interpreted as the number of different snowmen that can be made using just two of five available items of clothing. Can you explain, in the context of this particular investigation, why it is that the 4 and the 6 in the preceding line of Pascal's Triangle should be added?

The Snowmen activity is referred to by PrIME' and the work of a group of bright seven year olds is reported in some detail. I find it particularly useful to refer teachers to this case-study (and similar ones which relate to Partitions and Odds and Evens) after they have investigated such problems themselves. Indeed, the intro- duction of case-study material during the reporting-back phase has proved extremely valuable in revealing parallel lines of thought and stages of development within teacher and pupil work. The discussion that arises also provides opportunities for teachers to consider how they might introduce, or modify, an activity themselves, and to reflect upon the types of teacher-pupil and pupil-pupil interaction which are in evidence.

Concluding Remarks Although the main thrust of this article may appear to centre around three distinct investigations, the underlying philosophy, and one which is made explicit during the full Inset course, is the need for pupils (and teachers) to work in an investigatory manner. This does not mean using investigations and real-world problems as 'add-ons' to more traditional classroom practices, but requires that the doing, talking, explaining ethos becomes part of all mathematics teaching. Later elements of the course focus attention on how this can be achieved within the main content areas of number, algebra, shape and space and data handling. M

References 1. Mathematics in the National Curriculum (1991) The Department of

Education and Science. HMSO. 2. Primary Mathematics Coordinators Course, University of Glamorgan/

Mid Glamorgan LEA. 3. Ball, G. (1990) Talking and learning. Blackwell. 4. Brissenden, T. (1988) Talking About Mathematics. Blackwell. 5. PrIME. Children, Mathematics and Learning, p. A19. Simon and

Schuster (1990).

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Primary School Focus || Doing, Talking and Explaining

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