it appears that many basic statistical concepts have

been acquired already. This not only prevented the

order of acquisition of skills being distinguished but also rendered any investigation of the age of

acquisition difficult. Similarly at the upper end, in

some areas where there was only a small number acquiring the skill it was difficult to make order

distinctions. At one point it was considered giving

only older subjects questions at the higher level over all five topic areas and conversely at the lower end. This would have enabled more questions to be set on each skill and a more detailed Inclusion Analysis to

take place. However in view of the wide range of

abilities, particularly at the lower age range, this would have hampered the ability to look at the age of acquisition of skills.

The initial testing had proved extremely valuable in removing questions from the tests which were

inappropiate. The final tests on the whole gave very

few problems in their design. The lengths gone to make

the tests as visual and 'user friendly' as possible seemed from observation to have given subjects a high degree of motivation and encouragement.

In examining the order of acquisition of skills it was felt that some measure of success was achieved.

After some re-arrangement of initial skills the final

hierarchies went largely uncontradicted. Whilst it was

not possible to confirm the existence of some of the precedents many came out highly significant. Various key elements emerged in individual topic a r eas:-

(i) Sorting & Grouping Skills - The ability to sort data into categories is clearly a skill acquired early

in a child's development. Even to the extent of using

different forms of data in frequency tables this

presents little problem to children by the early teens. Although it was not possible to tell whether the skill of compiling frequency tables is possible without

necessarily being able to interpret the results, the latter still seems to be acquired at an early age.

This is however an area that all subjects are likely to

have practised at Primary level. In order to progress

further in this area, effectively from putting data into given categories to making up their own

categories, subjects need an understanding of the

nature of more precise groupings. Only those in the

oldest age ranges, with presumably greater experience, were able to understand the precise group definitions required and the implications of different groupings in order to set up and use their own tables competently.

(ii)Measures of Location:-This was the most complex of the hierarchies and although many precise conclusions of precedents were not possible, a skeleton pattern

emerged. The earliest skills centred around finding measures where the answer was a whole number and, in

particular, one of the data set. The mean is rather a

more difficult concept which seems to be understood in

the context of its balancing properties. One of the

biggest stumbling blocks in all strands appears to be the using of an answer which is 'non-real' i.e. either where the answer is not one of the data set or, even

more difficult, not a possible concrete example. The

classic example being the difficulty of understanding what is meant by 'the average size of family is 2.2

children'. Finding the values of means and medians

from frequency tables and weighted means involved much

more complex skills and few could complete these. What

is perhaps of more concern is the lack of understanding of the limitations of the various measures by even the

oldest of subjects. This leads to the conclusion that

even at the end of many years of experience of the

measures most people have only an algorithmic knowledge of measures of location.

(iii) Distribution & Dispersion;- This was a more

limited area than the others and the analysis was less

conclusive. Many subjects at the lower and middle age

ranges have a poor concept of the overall idea of

spread. Even the idea of a simple range seems to be

only partially understood by many. As age progresses

the concept of spread becomes more likely to be related to the idea of centrality than homogeneity amongst the

data. The more difficult ideas of quartiles and

presumbably other measures of dispersion are apparently only attainable by the oldest subjects.

(ivlBivariate D a t a ;- This was the most involved hierarchy as there were in it strands between which

there was little direct connection. Much of the

initial hierarchy in this area was radically altered and of all the areas this perhaps gave the most

enlightening insights into the way concepts in the

topic are understood. The topic was also of great

interest as many of the ideas here are not ones which the younger ages would have met in formal teaching.

The basic idea of 'correlation' is not a difficult one for even the younger subjects.

Suprisingly, they even seemed to cope well with the

idea of negative correlation. The use of bivariate

frequency tables with qualitative and discrete data were well understood by a majority and enabled fairly quick progression onto dealing with the grouped

situation. Despite this there seemed to be more

difficulty in understanding what these tables actually

showed in terms of variable relationships. The nature

of this difficulty is clearly something worthy of

further investigation. Plotting paired data on a

scattergram rarely proved difficult, though many found obtaining a straight line from this rather harder. Without having developed these skills subjects seem

unable to progress to more sophisticated judgements about correlation.

(v) Sampling Skills;- After the considerable

simplification of the structure this was the most

strongly supported hierarchy. This perhaps suggests

that simpler hierarchies in other areas might have been

useful. The 'pinnacle' of this hierarchy was in a

crude way to understand the basic principle of a

confidence interval

,

probably close to the true value. The series of skills

needed to acquire this was strongly supported by

analysis. The increasing understanding of the skills

with age was also most marked with only a few of the

oldest subjects obtaining full understanding. This an

area rarely dealt with in school curricula and possibly shows the clearest example of skills acquired through everyday experience.

Whilst the precise linkages might have been difficult to analyse in some areas there is

nevertheless strong support for the existence of three

levels of concepts. Whilst these may not concur in the

five topic areas consistently with age, the general

pattern of development seems consistent. Indeed one of

the difficulties in analysing the precise linkages was due to the fact that in most areas all the subjects had largely acquired the Level I skills and only the most

structure used by Green[1982], Fischbein[1975] and others, as outlined in Chapter 2, of the three stages of development seems consistent with the results

C h a p t e r 7

Implications for Curriculum Designers and Teachers

7.1 Background

At the outset of this project it was hoped to gain a greater knowledge of the way in which statistical understanding developed amongst 12 - 18 year olds. What were the preconceptions they brought in from

earlier experiences, what were the stumbling blocks to understanding and what were the limitations of

understanding for children at different ages? Although many questions remain unanswered, the research has

nevertheless given many insights into the development of statistical concepts.

Statistics as a taught subject in the UK has been an area of greatly increased interest in general

mathematics teaching, in 'user' subjects and as a

subject in its own right. The Cockcroft Report in the

1980s had considerable affect on mathematics syllabi for pre-16 courses and the Schools Council Project on Statistical Education in its turn influenced the

compilers of the Cockcroft Report in their strong

recommendations to promote the teaching of Statistics. Not only did this lead to the large Data Handling

section in the National Curriculum (1990) but also led to recommendations on teaching styles

11 Statistics is essentially a practical subject