Encouraging probabilistic thinking

Journol of Computer Assisted Learning (1987) 3. 156-163

Encouraging probabilistic thinking

D. R. Green, University of Technology, Loughborough

Abstract This paper reports on research which indicates the weak understanding of tree diagrams of pupils aged 11 - 16 years and describes a computer program which seeks to overcome this, allowing the pupil to construct and analyse a simulated channel system down which balls may be rolled.

Keywords: Probability; Simulation; BBC microcomputer.

Broadening probability

It is often said that probability is an essential base for the understanding of statistical concepts and that statistical understanding is an important aspect of numeracy. Nevertheless probability is an often neglected topic within school mathematics and there is a tendency to emphasize the analytical (or deterministic, or mathematical) aspects and to play down the experimental (or stochastic, or statistical) aspects. This is reflected in the kinds of question set in examinations. The question below (APU, 1983) is typical;

Item P2 - Age 15 100 discs, numbered from 1 to 100, are placed in a bag and one is drawn out at random. What is the probability that the number on the disc includes the digit 9?

With the availability of microcomputers there is now no excuse, if ever there was one, for perpetuating an ill-balanced approach to a fundamentally important branch of scientific thought. One of the programs produced for the Microelectronics Education Programme (Green, 1986) designed to encourage probabilistic thinking in school pupils is described later in this paper. The idea for this program originated from earlier research which the author carried out at Loughborough which will be described briefly now.

Weighing Up Alternatives

One of the questions of the ‘Chance and Probability Concepts Project’ test (Green, 1982) was that shown in Fig. 1. When devised, this question was expected to be difficult for pupils aged 11 - 14 (Years 1 - 3 of secondary school) but relatively easy for pupils aged 15 - 16 who had met tree diagrams

Invited paper. Correspondence: D. R. Green, University of Technology, Loughborough LEI 3TU.

156

Encouraging probabilistic thinking 157

A robot is put into a maze, which it begins to explore. At each junction the robot is as likely to go down any one path as any other (except it does not go back the way it came). There are eight traps at the ends of the eight paths (see picture). In which trap or traps is the robot most likely to finish up, or are all traps equally likely?

Fig. 1. ‘Robot in Maze’ question from the Chance and Probability Concepts Project.

and multiplication of probabilities. However, the results of large scale testing showed that the item was extremely difficult for all pupils, as indicated in Table 1.

Table 1. Results for ‘Robot in Maze’ question (percentage responses)

School Age Number ‘1 & 2’ ‘ I ’or ‘2’ ‘5-8’ ‘same’ other omit Year lvearsl

2 11-12 640 5 7 22 57 6 3 2 12-13 670 4 5 20 62 6 2 3 13-14 670 7 3 19 62 7 3 4 14-15 540 9 7 16 59 7 3 5 15-16 410 13 4 15 59 5 3

Total 2930 7 5 19 60 6 3

The percentage considering all traps to be ‘equally likely’ remained virtually fixed at around 60% for all ages. Even when accepting as correct a choice of one of the traps 1 or 2 rather than requiring both to be mentioned, the facility only reached 1770, for the oldest pupils.

Interviews conducted with individual pupils indicated that although they had studied tree diagrams they could not see the relevance of that work to the problem posed. The abysmal failure of all Years on this item is in marked contrast to results reported by Fischbein et al. (1967) in a similar but practical setting. A comparison of results is shown in Table 2.

158 D. R. Green

Table 2. Comparison of results for Tree Diagram type problems (percentage responses)

Fischbein et al. (1967) Green (1982)

Age Correct Correct/ Age Correct Correct/ Partly Partly correct correct -

11:03-12:10 58 65 11:l-12:o 5 12 1 2 3 - 13:O 4 9

13:05-14:05 69 69 13:l- 14:O 7 10 14:1-15:0 9 16

It would seem that despite the experience of drawing tree diagrams little understanding of their essential nature is learned, suggesting that a more dynamic and practical approach is called for.

ROLLERBALL

A context in which useful experience may be gained is a series of inter- linking slots (or channels) down which balls may roll into containers (or boxes), reminiscent of fairground games and arcade machines. Two simple examples are shown in Fig. 2. Unfortunately such practical equipment is

Fig. 2. Simple channel designs

difficult to construct, expensive and likely to be unreliable. The microcomputer can provide an alternative which, although not perfect, does have advantages. The package ROLLERBALL, written for the BBC microcomputer, consists of four interlinked programs: 1 DESIGN allows the user to design a channel system which has one or more funnels at the top into which balls can be entered, and connecting paths leading to boxes at the bottom to collect the balls (Fig. 3);


Fig. 3. Constructing a channel system using ROLLERBALL in DESIGN mode.

2 PLAY allows the user to simulate the rolling of the balls down the channels with choice of number of balls, speed of rolling and bias at junctions (Fig. 4);

Fig. 4. Rolling balls down a channel system using ROLLERBALL in PLAY mode.

3 ANALYSE calculates for each collection box in the design the probability of a ball finishing in that box, allowing for multiple funnels and bias (Fig. 5);

160 D. R. Green

Fig. 5. Calculating the probabilities using ROLLERBALL in ANALYSE mode.

4 LIST provides details of the channel designs which the teacher or pupils have created and saved on disk.

Using ROLLERBALL

There are several ways in which ROLLERBALL can be used effectively in the class, allowing varying degrees of freedom for the pupil: 1 The teacher prepares channel systems using the DESIGN mode which are stored on disc and the pupils are given instructions to load a design and then explore its properties by means of the PLAY mode. Pencil-and-paper investigation to try to model the behaviour of the system would then be encouraged before instruction to proceed to A N A L Y S E mode which provides the theoretical probabilities of different outcomes. 2 The teacher prepares worksheets depicting designs which the pupils construct for themselves before undertaking investigations such as outlined in 1. A sample worksheet and datasheet are shown in Figs 6 and 7. There is reason to prefer this approach if time permits since it involves more pupil participation and allows some individuality interpreting the designs given on paper. This interpretation requires deciding which are the essential features (junctions, funnels, tops, order of connection) and which are irrelevant (lengths of channels, position on screen, vertical or diagonal direction). If attention is drawn to these aspects valuable incidental experience is provided. 3 The most liberal approach is when pupils are allowed free rein to design, construct and investigate their own channel systems. The accurate recording of the design and of any bias employed and the numbers and percentages of the outcomes are particularly important in such cases.

Recording results is not to be considered an end in itself but rather as enabling meaningful comparisons, interpretations and deductions to be


1 Using option 3, ‘Call Design Program’ design each of the following shapes in turn and use the designs to roll 100 balls down each.

Choose the ‘Random’ option and then the ‘Fair’ option so that the balls are just as likely to go Left as Right at each junction. Record your results on the Rollerball Results Sheet.

U A A B

U 4 A B

A B

2 (a) (i)

(ii) Can you explain why? Which shapes (if any) gave many more balls in one box than the other? Can you explain why?

Try to predict what will happen: when 100 balls are rolled down each of the designs below, when the ‘Fair’ option is selected.

Which shapes (if any) gave about equal numbers of balls in boxes A and B?

(b)

3 (a)

A B

u .$ uu A B C D

U ii A B C

(b)

(c)

Design each of the shapes in Question 3a and roll 100 balls down each. Compare the results with your prediction. Write comments about each set of results and your predictions.

4 Repeat Question 3 when the ‘Biased’ option is selected. First choose the bias to be 25% going Left, then choose the bias to be 75% going Left.

5 Repeat Question 4 selecting various other biases. Be sure to write down comments about your predictions and the actual results.

6 Make up your own design. Predict how many balls will end up in each box with 100 balls rolled, when the ‘Fair’ option is chosen, and say why. Then check your prediction and comment on it.

7 Repeat Question 6 choosing the ‘Biased’ option for various biases.

Fig. 6. Worksheet for use with ROLLERBALL.

162 D. R. Green

ROLLERBALL DATA SHEET 1

Fig. 7. Datasheet for use with ROLLERBALL.

made. The fact that repeated trials generally give different (but not dissimilar) results can be readily perceived and should be fully discussed and explored as an important feature of such simulations. The advantage of using the computer’s pseudo-random number generator, rather than ‘mother nature’, is that it does allow the teacher to repeat a particular run of the simulation if he so wishes. However the whole business of generating numbers by computer needs to be carefully explained to pupils if confi- dence is to be maintained. Essentially it comes down to being prepared to discuss the modelling process, and the teacher must be willing to allow pupils to investigate the validity of the model for themselves, based on agreed criteria.

Statistical software

Quite a few ‘statistics’ programs are now available for microcomputers but many of these are convergent teaching programs, aimed at mastering some particular skill, and others are tools for mechanical calculation. Undoubt- edly such programs have their place but neither of these types makes full use of the inherent flexibility of the computer, and by their nature they constrain the user. ROLLERBALL is an attempt to make available to the user a flexible exploratory tool conducive to investigational work. It will come as no surprise to learn that ROLLERBALL is extremely popular in the class- room because it offers the opportunity for meaningful pupil-centred game- orientated work, and its appeal extends to the teacher who is aware of its educational value for group or individual work. ROLLERBALL can lead to extremely important statistical work but inevitably it requires much of the teacher if full use is to be made of the opportunity. Rather than replacing the teacher this kind of program makes his presence and expertise even more vital.


There is no doubt that the current restructuring of the English school examination system at age 16 will lead to demand for many more programs lof this type for mathematical and scientific subjects. Who will produce them?

References

Assessment of Performance Unit (1983) A Review of Monitoring in Mathematics

Fischbein, E., Pampu, I. & Minzat, I. (1967) L'intuition probabiliste chez l'enfant.

Green, D. R. (1982) Probability Concepts in 11-16 year old pupils. CAMET,

Green, D. R. (1986) Probability and Statistics Programs for the BBC micro. Capital

1978 - 1982, Part 1, Department of Education and Science.

Enfance, 2, 193-208.

Loughborough University.

Media, London.

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Encouraging probabilistic thinking