discussion on the paper by holmes

2003 Royal Statistical Society 0039–0526/03/52439

The Statistician (2003)52, Part 4, pp. 439–474

50 years of statistics teaching in English schools:some milestones

Peter Holmes

Nottingham Trent University, UK

[Read before The Royal Statistical Society on Wednesday, April 9th, 2003, the President ,Professor P. J. Green, in the Chair ]

Summary. Over the past 50 years the amount of statistics in the English school curriculumhas grown from almost nothing to becoming both a major part of the mathematics taught to all5–16-year-old children and also an integral part of other school subjects. It has also becomewell established as a General Certificate of Education advanced level subject and as part ofmathematics and other subjects at this level.This paper traces the major events that have led tothis penetration of the English school curriculum over this period (generally these comments willalso be true for Wales and Northern Ireland, but not Scotland) and attempts to draw out somelessons that can be learned to make effective future developments in statistical education.

Keywords: Aims of school statistics; Historical development; Lessons to be learned; Projecton Statistical Education; Royal Statistical Society reports on teaching; Teaching statistics inschools

1. Background

Bibby (1986), pages 79–83, identified some moves in the first half of the 20th century to includestatistics in the school curriculum in England. He suggested that Wishart’s paper (Wishart,1939) on some aspects of the teaching of statistics (largely at university level) may be seen as aturning-point in the Royal Statistical Society’s (RSS’s) attitude to statistics and led to the sta-tistical education boom. It is certainly true that following World War 2 in 1945 there was muchactivity in the RSS. E. S. Pearson chaired a committee to report on the teaching of statistics inuniversities and university colleges. This was published as a report of Council (Royal StatisticalSociety, 1947). In it the committee quotes Darwin (1938) as expressing the hope that

‘generations will grow up which have a facility that few of us at present possess in thinking about theworld in the way which quantum theory has shown to be the true one. The inaccuracies and uncertain-ties of the world will be recognised as one of its essential features. Inaccuracy in the world will not beassociated with inaccuracy of thought, and the result will be not only a more sensible view about thethings of ordinary life, but ultimately, as I hope, a fuller and better understanding of the basis of naturalphilosophy.’

The committee goes on to say that they shared that hope and thought that it may ultimately berealized through the teaching of statistics in schools.In the meeting arranged for discussion of this report Wishart (1948) posed the question

‘Should an attempt be made to introduce an elementary course of instruction in statistics into thenormal curriculum of the secondary school?’

Address for correspondence: Peter Holmes, RSS Centre for Statistical Education, Nottingham Trent Univer-sity, Nottingham, NG1 4BU, UK.E-mail: [email protected]

440 P. Holmes

Of three contributors on this point two were very much in favour; one dissented on the groundsthat the secondary school curriculum was overcrowded and, if any time could be spared, hewould rather it went to woodwork, metal-work and drawing. In his summing up of the discus-sion, Wishart referred to the existence of a syllabus for statistics as a subsidiary subject for theHigher School Certificate examination. I have not been able to trace this syllabus nor to findout the number of students who took it.It was at this time that the first steps were taken to include statistics in the school curriculum

for a wider range of pupils. The first syllabuses came in with General Certificate of Education(GCE) ordinary (O-) and advanced (A-) level, which replaced the School Certificate and theHigher School Certificate in 1951.

2. The 1952 report of the Royal Statistical Society

In 1952 the RSS published a report on the teaching of statistics in schools (Royal StatisticalSociety, 1952). The committee was chaired by E. S. Pearson and the report followed closely onthe earlier report on teaching at universities (Royal Statistical Society, 1947).The report was well ahead of its time in considering the teaching of statistics in schools from a

broad educational perspective of what was needed by pupils for their own personal development(section 4). The arguments were summarized in a later paper (Yates, 1968) as follows.

(a) The most fundamental result of a statistical training is that it encourages a habit ofdisciplined thinking about ordinary affairs in terms of quantities (section 7).

(b) The statistician is taught, and the citizen also should learn, to appraise figures critically,to appreciate their fallibility and limitations, and, in particular, to consider the effects ofthe errors with which such figures measure things (section 8).

(c) Students should develop a habit of examining critically and accepting with reserve con-clusions from numerical data drawn by other people (section 10).

(d) Some of the more elementary statistical devices such as charts, averages and percentagesare widely used in the mass media, and the educated citizen should learn to understandthem. Familiarity with social and economic statistical information is a help towards anunderstanding of public affairs (section 11).

(e) Some appreciation of sampling surveys is also needed, since the public is often asked toaccept some result obtained from a sample survey (section 12).

(f) Probability is an idea which has applications, not only in scientific work, but also in dailylife (section 14).

(g) The mathematical specialist will derive satisfaction in finding that statistical theoryapplies concepts and methods that he has learnt in his general mathematics (section25).

I would add to this summary that Royal Statistical Society (1952) also included the followingpoints:

(h) the argument that the development of a balanced and reflective outlook on figures is aslow process, and if it is not begun at school, before the child’s mind begins to crystallize,it may never take place (section 9);

(i) an ability to argue accurately from data is a fundamental need if the person is to be anactive member of society (section 10) and the citizen should learn to appraise figurescritically (section 8).

50 Years of Statistics Teaching 441

It was over 20 years before these ideas were taken seriously for changes in the school curriculumand over 30 years before they received serious implementation.To ensure that statistics at school was taught in practical contexts, the committee recom-

mended that the bulk of its teaching should be done in the contexts of other subjects such asnatural science, geography and history, though they also record that some teachers whom theyhad consulted thought that statistics would only make headway as a separate subject in thetimetable. The committee also recommended that syllabuses be developed for the sixth form(pupils aged 17 and 18 years) related to the GCE A-level that had recently been introduced.In an appendix the committee recorded the way that statistics was included in some GCE

mathematics syllabuses. Of eight examining boards mentioned, four had no statistics at O- orA-level. Three of the remaining four had some statistics in an alternative ordinary syllabus formathematics; three had some statistics in at least one variant of mathematics at A-level.This makes the position look much more encouraging than it actually was. The following

facts are relevant.

(a) When it was first introduced, the GCE O-level was taken by 17% of the relevant schoolpopulation (currently 93% are entered for the General Certificate of Secondary Educa-tion (GCSE) mathematics).

(b) The alternative syllabuses were taken by only a small minority of those taking O-leveland, typically, were those who had stayed at school after 16 years of age.

(c) In these alternative syllabuses the statistics is not the same as that described by the com-mittee in the main body of its report; nor does it have the emphasis that they placed onpractical use.

(d) In all except one of the examples given at alternativeO-level, full marks could be obtainedwithout doing any of the statistics questions; the same was the case for two of the threeA-level syllabuses.

2.1. Some lessons to learn

(a) There is a long way from a group of experts, such as an RSS committee, who decide thatsomething is a good idea to ensuring that it happens in practice.

(b) Good ideas may take a long time to penetrate the public consciousness and to come tofruition.

(c) An overview of the general nature of statistics and its importance for all pupils is neededif a balanced provision of statistics in the school curriculum is to be made.

3. The first introduction at General Certificate of Education advanced level

Following the introduction of some statistics into some A-level GCE mathematics syllabusesdescribed above, a qualification in pure mathematics and statistics was introduced. The firstexamination board to do this was the Northern Universities Joint Matriculation Board. TheAssessment and Qualifications Alliance, which was the successor to the Joint MatriculationBoard, reports that the first examinations for this qualification were in 1961. The statistics pa-per was set as an alternative to theoretical mechanics for students aged 16–19 years who werespecializing in mathematics. The aim was to provide a relevant application of mathematics tothose whose main subject interest was in fields such as geography, psychology and biology. Theother boards followed with their own versions. The Associated Examining Board produced anA-level syllabus in statistics in 1963. The core content was similar to that of the statistics of thepure mathematics with statistics syllabus but the students did not have to study mathematics as

442 P. Holmes

such. These courses have their successors today, though their content and approach has changedsomewhat to become more practical and to take in more of the real needs of the user subjectsoutside mathematics.As far as I can ascertain, the courses were developed by mathematicians and some statisti-

cians from departments of mathematical statistics. No specialists from the user subjects wereinvolved with the development of the syllabus. The syllabuses had to be seen as mathematicallyrespectable since they carried a qualification in mathematics and they were under the authorityof the boards’ mathematics subject committees.They were a brave introduction to elementary mathematical statistics, but they were soon the

subject of criticism. Some of these are referred to in Downton (1968). Holmes (1981a) listed theweaknesses as follows.

(a) The questions tested mathematical principles rather than statistical insight (Barnett(1982) gave a particularly good example).

(b) Little emphasis was given on the practical implications of the final mathematical conclu-sions.

(c) Syllabuses were much less useful than they might be for the user subjects.(d) Neither the syllabuses nor the examinations developed or considered the expertise that

is required in carrying out a long-term investigation.


(a) Involve practical as well as mathematical statisticians in the development of a syllabus.(b) Take advice from the user subjects about the nature of the statistics that they need.(c) There is more to statistics than mathematics—it is essentially a practical subject.(d) Practical statistical skills need to be developed as well as theoretical ones.

4. Primary school developments in the 1960s

4.1. Edith Biggs HMIDuring the 1960s, in the move towards introducing ‘modern’ mathematics into schools, therewas a growth of practical data collection, representation and intuitive inference in primaryschools (pupils aged 6–10 years). This was spearheaded by active promotion by Her Majesty’sInspectors, the most dynamic of whom was Edith Biggs. She wrote the Schools Council’s bul-letin number 1: ‘Mathematics in primary schools’ (Schools Council, 1965), which sold 75000copies in its first year (Fig. 1). There is a chapter on graphical representation in the learning ofmathematics, and most of these graphs are of statistical data. The children were also expectedto collect the data for themselves and to interpret their graphs in real terms. She and othersran many in-service courses for primary school-teachers where there was the same emphasison collecting data for yourself, representing it graphically and drawing (elementary) inferencesfrom the data. Biggs (1971, 1972) give further details.

4.2. Nuffield primary mathematicsThe major curriculum development for primary school mathematics at this time was funded bythe Nuffield Foundation and directed by Geoffrey (later Sir Geoffrey) Matthews. Their mottowas ‘I hear and I forget; I see and I remember; I do and I understand’. To ensure that teacherswere both aware of the philosophy behind their teaching and knew what they were teaching andwhy, they set up a network of teachers’ centres throughout the country.


Fig. 1. Figure taken from Schools Council (1965), page 71 (Crown copyright: reproduced with the permis-sion of the Controller of Her Majesty’s Stationery Office and the Queen’s printer in Scotland)

These centres became a resource for the teachers and a base from which they organized manyin-service courses. They produced many activities for teaching primary mathematics. Activitiesin probability and statistics occurred in several of their books for teachers (NuffieldMathemat-ics Project, 1967). Among other things the book on pictorial representation included a list ofabout 75 topics with indications of what might be appropriate for different ages of pupil. Theyspecifically refer to work connected with geography, science and social surveys. There was alsoan individual book on probability and statistics which was based on practical work (Fig. 2)(Nuffield Mathematics Project, 1969). The text-book for teachers in training by Williams andShuard (1970) gives a good feel for what was happening in primary schools at that time.


(a) Primary school children can learn and enjoy elementary probability and statistics.(b) It is possible to begin to develop early ideas of probability and inference at this age.(c) Pupils learn concepts better if they are introduced in a practical way and calculations are

deferred.(d) Teachers need to be enthused and trained to teach new topics and this takes time and

money.(e) Well-provided local teachers’ centres were a major factor in the success of the Nuffield

Mathematics Project.

444 P. Holmes

Fig. 2. Page 39 from Nuffield Mathematics Project (1969) (copyright of the Nuffield Foundation and repro-duced with permission)

5. Secondary developments in the 1960s

During the 1960s therewas a revolution in the teaching ofmathematics in the secondary schools.Many groups were active in developing a modern mathematics syllabus and producing text-books for their syllabuses. Groups included theMidlandMathematics Group, the KentMathe-matics Project, the Mathematics in Education and Industry Project and the Inner LondonProject known as ‘SMILE’. The most active of these groups was the School Mathematics Pro-ject (SMP). This was the creation of an enthusiastic group of teachers, mostly working in someof the better-known public schools. The SMP books were the first mathematics texts to includeillustrations and cartoons, which made them more attractive to teachers and pupils (SchoolMathematics Project, 1969). The SMP writers saw probability and statistics as an importantpart of modern mathematics and were concerned to develop a proper understanding of thebasic concepts and techniques. Previously the only statistical technique in the standard mathe-matics course was ‘averages’, which was treated from a very mathematical point of view. TheSMP, and others, introduced a probability and statistics strand into the mathematics coursesfor secondary pupils (aged 11–16 years). The secondary strand tended to emphasize theoret-ical probability and was weak on practical statistics but included many examples using realdata as well as fictitious data. Fig. 3 shows an example where pupils are asked to look at real


Fig. 3. Abstract from page 33 of book 2 of School Mathematics Project (1969) (copyright of the SMP andreproduced with permission)

newspaper articles. It tended to be focused on statistical techniques; some references were madeto the practical uses made of statistics in everyday life but less on statistics in other schoolsubjects.The syllabus included tables, tally charts, bar-charts, the mean, median, mode, range, use of

statistics in newspapers, trends in time series of data on things such as telephones, and pictorialmisrepresentations of data. Probability was introduced as experimental probability and theo-retical probability and went as far as combinations of events by using tree diagrams. There wasa chapter on thinking statistically which looked at common statistical patterns and attemptedto introduce the normal distribution. Book 5 (School Mathematics Project, 1969) ventured onan introduction to the idea of significance.Fig. 4 shows the use of real data in terms of a breakdown of expenditure of tax revenue in

1962, from question 6, and the use of fictitious data from the ‘Richkwick Investment Company’,in question 5. Even with its shortcomings, this first SMP publication was a major step towardsincluding some statistics in the main curriculum for a large number of secondary school pupils.Although it started in public schools it quickly spread into selective state schools and, by theend of the 1960s, books were being produced to cover the lower level Certificate of SecondaryEducation and so the material became part of many secondary modern and comprehensiveschools’ mathematics programmes.


(a) When good enthusiastic teachers are given freedom to experiment, they can change thecurriculum for good and enthuse both their pupils and other teachers.

(b) Plenty of probability and statistics can be done by secondary school pupils.(c) Statistics is more than a set of techniques; probability is more than coins and combinat-

orics.(d) A school needs to decide what is an appropriate education in statistics for all, and how

it can be included.(e) First attempts at introducing new topics into the school curriculum are almost bound to

be flawed. It is best if they can be tried out with small groups of pupils in a context wherequick revision is possible and without penalizing the pupils involved.

(f) Interesting activities are not enough; the topics must link together to form a coherentwhole.

446 P. Holmes

Fig. 4. Abstract from page 37 of book 2 of School Mathematics Project (1969) (copyright of the SMP andreproduced with permission)

6. The 1968 Royal Statistical Society meeting on the ‘Teaching of statisticsin schools’

In 1964 there had been a symposium on the teaching of statistics (Royal Statistical Society,1964) but this had concentrated on the teaching at universities and the training of statisticiansfor government, economics and business, and medical work. It only had incidental implica-tions for school teaching. More importantly for school work, in May 1968 there was an RSSopen meeting at which the RSS Committee on the Teaching of Statistics in Schools, chaired byG. B. Wetherill, presented its interim report (Yates, 1968) and Downton (1968) presented hisresponse paper. Afterwards there was a wide-ranging discussion. The theme of the two paperswas essentially on what should be an appropriate A-level syllabus in statistics. Both the com-mittee and Downton sought to define a course which required only O-level mathematics andlater discussants queried whether this was feasible.The committee referred to a change of view since Royal Statistical Society (1952). Since,

they claimed, the incorporation of statistics teaching into scientific and other subjects was nothappening, it was feasible to introduce statistics as a subject in its own right. After a pass-ing reference to the existence of an O-level in statistics, the rest of the report was on A-level.They suggested that such a course should not have a heavy emphasis on theory; nor should itemphasize computational methods but should

‘make basic concepts clear and . . . will show how these concepts are used in the interpretation ofexperimental and observational data’.

They then gave a course outline for such a syllabus. Downton disagreed with their proposals andconsidered their syllabus to be adding probabilistic experiments to a conventional syllabus. Hisview was that statistical thinking requires high level skills and that an appropriate course couldbe developed based on the different probability distributions as models for practical situationsspread over various fields of application.


The discussion that followed showed a wide range of views from a large number of partic-ipants. Many of the school-teachers were disappointed that so little had been said about theteaching of statistics before A-level.This meeting raised greatly the profile of teaching statistics in schools among RSS Fellows.

The following comment by Downton was repeated by Wetherill in his reply to the discussion:

‘A substantial project is called for to investigate the whole question of teaching statistics in schools’.

This was to have major repercussions in the next few years.

7. The Schools Council Project on Statistical Education, 1975–1980

7.1. The Committee on Statistical EducationLate in 1967, Toby Lewis and Vic Barnett, both Fellows of the RSS who had been workingover the years to help teachers of A-level statistics, gathered a group of people concerned withthe teaching of statistics at school level and formed themselves into an action group called theCommittee on Statistical Education. Most of them were Fellows of the RSS and the group in-cluded not only Vic Barnett and Toby Lewis but also Frank Downton, Dennis Lindley, ArthurOwen HMI and Alan Stephenson of the University of London Schools Examination Board.The group had the encouragement of the RSS and the Institute of Statisticians but was not aformal committee of either body. They looked at the current position of probability and sta-tistics in schools and were not convinced that what was being taught was appropriate. Theiraim was to obtain funding for a major development programme that would consider the roleof probability and statistics in school courses for pupils from the age of 6 to 18 years. In theevent they were successful in obtaining a major grant from a quasi-governmental body calledthe Schools Council to investigate and develop materials for the main secondary school (pupilsaged 11–16 years). This project, the Schools Council Project on Statistical Education (POSE),ran for 5 years, from 1975 to 1980, and had an overall budget of about £220000. It was basedin the Department of Probability and Statistics at the University of Sheffield and directed byPeter Holmes.

7.2. Brief of the Project on Statistical EducationThe aims of the project, as defined by the Schools Council, were the following:

(a) to assess the present situation in statistical education, regarding content, level,motivationand teachers’ attitudes, and to relate these to the position of statistics outside schools;

(b) to survey the needs of teachers, whether they are teaching statistics as a specialist subjector working in a related field, and the implications for both initial and in-service needs;

(c) to devise detailed proposals for implementation of the teaching ideas;(d) to produce teaching materials such as notes for teachers, descriptions of experiments,

work sheets and sets of examples.

The results of the research carried out to cover the first two points of this brief are summarizedin Holmes et al. (1981). The project team found that only between 7% and 10% of all second-ary schools were entering candidates for GCE O-level or Certificate of Secondary Educationsyllabuses in statistics. Of these schools an average of 12.5% and 22% respectively of 16-year-old pupils were entered for these examinations, so statistics as such was taken by only a smallpercentage of the school population.On the positive side the project team found that a growing amount of statistics was being

taught in mathematics lessons and that also much statistics was being taught in other subjects

448 P. Holmes

across the school curriculum. But when the content was considered more carefully the positionwas not as encouraging. Holmes (1981b) noted that the statistics included in mathematics wasthere because it was seen as an element of modern day mathematics, so it was the mathematicaltechniques that were emphasized; there was very little real data or application to real problems.The statistics was included in the other subjects across the curriculum because it was seen asuseful for those subjects and so the difficulties of the concepts being used were not always appre-ciated, and the order in which the statistical ideas were introduced was often not appropriate.For example the team found the use of Spearman’s rank correlation coefficient, together withthe formula

1 − 6∑

d2

n3 − n,

in a geography lesson for 13-year-old pupils.

7.3. Philosophy of the Project on Statistical EducationIn developing their teaching material, the team summarized the aims of school statistics in twoglobal aims.

(a) Children should become aware of and appreciate the role of statistics in society, i.e. theyshould knowabout themany andvaried fields inwhich statistical ideas are used, includingthe place of statistical thinking in other academic subjects.

(b) Children should become aware of, and appreciate, the scope of statistics, i.e. they shouldknow the sort of questions that an intelligent use of statistics can answer, and understandthe power and limitations of statistical thought (Schools Council Project on StatisticalEducation, 1980a).

To put this philosophy into practice the project team developed a large amount of teachingmaterial, which was widely tested in a variety of schools, and published under the title Sta-tistics in Your World (Schools Council Project on Statistical Education, 1980b). The materialcovered a wide range of applications of statistics in society, including, for example, equal pay,smoking and health, quality control, the national census, Premium Bonds and the retail priceindex. Each unit developed statistical techniques in a practical context. Fig. 5 shows a pagefrom ‘Choice or chance’, a unit on probability. Fig. 6 shows a page from ‘Multiplying people’,a unit on population growth. It was the first teaching material in the world aimed at developinga statistical awareness and ability among secondary school pupils through a practical approachusing examples of statistics from everyday life.

7.4. What is statistics?At the time of the project, the prevailing perception of statistics in school mathematics cour-ses, as shown by text-books and syllabuses, was that statistics was a set of techniques such asbar-charts, pie charts, scatter diagrams, the mean, median, mode and interquartile range. Tothe project team this seemed like defining human biology as only the study of the skeleton.As such, these techniques had no life in them and school courses did not give a flavour of theimportance and the ubiquity of statistics in all aspects of life. The project team considered itimportant to have a broad view of statistics so that pupils could gain an insight into the natureof the subject.One way of looking for a definition of statistics is to look in breadth at what statisticians

do. Any definition of statistics must be sufficiently broad to encompass the work of academic


Fig. 5. Abstract from page 12 of Schools Council Project on Statistical Education (1980b) (copyright of QCAEnterprises Ltd and reproduced with permission)

statisticians, applied statisticians in all fields and governmental and official statisticians. Theteam took the following as a working definition.

‘Statistics is a practical subject devoted to obtaining and processing data with a view to making state-ments which often extend beyond the data. These statements are called inferences and take the formof estimates, confidence intervals, significance tests etc. Statistics is concerned with the production ofgood data, and this involves consideration of experimental designs and sample surveys. It has its originin real data and is concerned with the processing of data in the widest of contexts and with a widevariety of applications such as social, administrative, medical, the physical sciences and the biologicalsciences’ (Schools Council Project on Statistical Education, 1980a).

7.5. Why teach statistics to everyone?The POSE team thought it better to consider general educational reasons for incorporatingstatistics into the school curriculum and then to see the implications for what sort of statisticsshould be included and where and how they may best be implemented.Why should we teach statistics to all? Here are the reasons of the Schools Council Project.

(a) Statistics is an integral part of our culture.(b) Statistical thinking is an essential part of numeracy.(c) Exposure to real data can aid personal development and decision-making.

450 P. Holmes

Fig. 6. Abstract from page 15 of Schools Council Project on Statistical Education (1980b) (copyright of QCAEnterprises Ltd and reproduced with permission)

(d) Statistical ideas are widely used at work after school.(e) Early exposure can give sound intuition which can later be formalized.

In this they were in line with the reasons given in Royal Statistical Society (1952).

7.6. Trying to influence the curriculumThematerial that was produced by the Schools Council was highly innovative and was well test-ed in a wide variety of schools with a wide variety of teachers. It was well received by those whoused it. It was not, however, part of the curriculum examined; it did not lead to a qualificationat the end and it did not easily fit into mathematics lessons geared to the existing syllabuses, soit was not widely taken up. At that time there was no national curriculum so the material hadto be marketed and sold on its own merits.


(a) Producing innovative material is very costly in time and people. New ideas must bedeveloped and properly tested to ensure their appropriateness and validity.

(b) If philosophy is to be put into practice, teachers need teaching material which embodiesthe philosophy.

(c) To change the teaching in schools requires more than good material; you need to findthe levers in the system and to use them. (At the time the major levers were the exter-nal examinations taken at 16 years of age. Although such examinations in statistics didexist they were not widely taken; nor were they sympathetic to the philosophy behind


the POSE. A syllabus that was more in line with the POSE philosophy was developedshortly afterwards by theNorthern ExaminingAuthority (1988) workingwith the projectdirector as chairman of examiners.)

(d) Major changes of philosophy may take a considerable time to be accepted; a long-termstrategy for acceptance is needed.

8. Changes at advanced level in 1970s and 1980s

8.1. Introduction of project workFollowing the criticisms of the earlier material for 18-year-old pupils, the University of LondonSchools Examination Board introduced an element of practical or project work into its A-levelsyllabus for mathematics and statistics in the mid-1970s. Since this practical and project workwas not assessed directly, it was not taken as seriously as the theoretical aspects of the course.In 1978 the Joint Matriculation Board introduced an A-level in statistics that was designed tohave students work practically through the whole process of doing a statistical investigation andto develop the global skills that are associated with this process. To emphasize the importanceof this practical approach, and to encourage teachers to put the theoretical work in practicalcontexts, the course required a compulsory project which was directly assessable. Although thisfaced some opposition from examiners in mathematics, on the grounds of decreased reliabil-ity of marking, the counter-argument that the assessment was more valid was accepted. Thisparticular syllabus was not like either of the two possibilities suggested by either the RSS 1968report (Yates, 1968) or Downton’s alternative view. It did not, though, require candidates to bestudying mathematics at A-level at the same time. It did not attract large numbers of entries.Teachers coped well with the teaching, but many were put off from trying—the other route ofpure mathematics with statistics, and no project, remained more popular with schools.

8.2. Introduction of advanced supplementary levelsIn 1989 advanced supplementary (AS-) syllabuses in statistics were introduced by most of theGCE boards as part of a move to try to broaden A-level studies. These syllabuses were intend-ed to have half the content of an A-level but to require the same depth of thinking. Many ofthese played down the mathematics side of the subject and increased the emphasis on the designof questionnaires, sample surveys and designed experiments to collect good data. In this theytook more account of the needs of user subjects. The syllabus from Joint Matriculation Board(1989) included an assessable project that was similar to that required for its A-level syllabus instatistics.


(a) If you want project work to be taken seriously you must assess it.(b) If you want project work to be done it must not be an option.(c) It is not easy to set syllabuses that meet the needs of all user subjects—different subjects

require different approaches and different topics.

9. ‘Mathematics counts’: the Cockcroft report

9.1. Influence of the Project on Statistical Education and the Royal Statistical SocietyIn 1978 the Government set up an enquiry into the teaching of mathematics in schools under

452 P. Holmes

the chairmanship of Sir Wilfred Cockcroft (Cockcroft, 1982). The terms of reference for thecommittee were

‘to consider the teaching of mathematics in primary and secondary schools in England andWales, withparticular regard to the mathematics required in further and higher education, employment and adultlife generally, and to make recommendations’.

The committee included probability and statistics as part of their consideration of math-ematics. Their findings on probability and statistics were strongly influenced by the SchoolsCouncil’s POSE and the RSS Education Committee from whom they received evidence (RoyalStatistical Society, 1979). Paragraphs 774–781 dealt specifically with the teaching of probabilityand statistics, though there are other references throughout the report.Two paragraphs in particular reflect the POSE–RSS influence.

‘§775 Statistics is not just a set of techniques, it is an attitude of mind in approaching data. In particularit acknowledges the fact of uncertainty and variability in data collection. It enables people to makedecisions in the face of this uncertainty.’

‘§781 Statistical numeracy requires a feel for numbers, an appreciation of levels of accuracy, the makingof sensible estimates, a common-sense approach to data in supporting an argument, the awareness ofthe variety of interpretation of figures and a judicious understanding of widely used concepts such asmean and percentages. All these are part of everyday living.’

The committee also suggested that each school should have a statistical co-ordinator. Therecently set up Centre for Statistical Education at the University of Sheffield obtained a grantto produce a handbook for such co-ordinators (Holmes and Rouncefield, 1989), but very fewschools took up this idea.

9.2. Influence on Government decisionsIn its turn this report influenced the Government who decided, for the first time in Britishhistory, that there should be a national curriculum. The mathematics content reflected therecommendations of the Cockcroft report.

10. Further developments at Sheffield

Following the success of the POSE and the international reputation that was coming to Shef-field because of this work, several other initiatives to provide resources for school teachers werefunded and based in Sheffield.

10.1. The journal Teaching StatisticsIn 1978 the Teaching Statistics Trust was set up with Joe Gani, Vic Barnett, Peter Holmes andthe treasurers of the RSS and the Institute of Statisticians as Trustees. The aim of the trust wasto promote the teaching of statistics at the school level, particularly by publishing a journalcalled Teaching Statistics. Financial backing came from the RSS, the Institute of Statisticians,the Applied Probability Trust and the International Statistical Institute. The first issue of thejournal was published in January 1979 with Peter Holmes as the first Editor (Fig. 7).It has continued to publish articles of wide applicability with an emphasis on practical help

to pre-university teachers of statistics in all disciplines ever since. It is difficult to quantify theeffect that it has had on teaching, but there is no doubt that Teaching Statistics has raised theprofile of statistics in the school curriculum and encouraged continued thinking on the waysthat it can best be taught.


Fig. 7. Contents page of volume 1, number 1, of Teaching Statistics (reproduced by permission of theTrustees of the Teaching Statistics Trust)

10.2. The first International Conference on Teaching StatisticsIn 1978 the International Statistical Institute’s Education Committee set up a task-force, underthe chairmanship of Lennart Rade, to organize international conferences on teaching statistics.Because of the reputation of the University of Sheffield in this area, and because of the expertisethat the Department of Probability and Statistics had in running conferences, the task-forcedecided to hold the first International Conference on Teaching Statistics in Sheffield in August1982. This conference attracted many teachers and lecturers from the UK, as well as frommorethan 60 other countries, and so raised the profile of teaching statistics and generated even moreinterest in teaching statistics in theUK (Grey et al., 1982). Successive International Conferenceson Teaching Statistics have been held around the world every 4 years.

10.3. The Centre for Statistical EducationThe activities developing from the POSE at the University of Sheffield were carried out underthe umbrella heading of the Centre for Statistical Education. This Centre came into formalexistence as a joint Centre of the University of Sheffield and Sheffield Hallam University inSeptember 1983. The co-chairmen were Vic Barnett of the University of Sheffield and WarrenGilchrist of Sheffield Hallam University. The director was Peter Holmes.Although the Centre was interested in statistical education for all ages, the focus of its activ-

ities was at the school level. It ran several projects aimed at producing material to help teachersto make their teaching more interesting and stimulating.The Statistical Education Project 16–19, funded by the Leverhulme Trust, ran from 1981 to

1984. It conducted a survey of the statistics being used in industry, commerce and Govern-ment by employees starting work at 19 years of age without specialist statistical qualifications(Holmes, 1985) and then set up working parties to produce teaching material for statistics ineconomics (Holmes, 1987), geography, science, business and psychology (Leverhulme Projecton Statistical Education 16–19, 1987).The Department of Education and Science funded a 2-year project to encourage practical

work in teaching A-level statistics. The project officer was Mary Rouncefield and the materialwas published as a text-bookPractical Statistics (Rouncefield andHolmes, 1989a) with a teach-ers’ guide (Rouncefield and Holmes, 1989b). Mary followed this project with a study into therole of the statistics co-ordinator as envisaged by the report of the Cockcroft committee. Thisled to a handbook for use by teachers entitled From Cooperation to Coordination (Holmes andRouncefield, 1989).The Nuffield Foundation funded a project on using databases and spreadsheets to teach

statistics across the curriculum. Mike Hammond was the project officer and the material was

454 P. Holmes

Fig. 8. Activity 18 from Hammond (1990a)

extensively tested in schools and published as a series of worksheets in book form under the titleFifty Things to do with Databases and Spreadsheets (Hammond, 1990a) together with a seriesof handbooks for in-service courses in different subjects (Hammond, 1990b, c, d, e). Fig. 8 givesan example.With the advent of the national curriculum there was a call for practical activities that could

be used to teach the probability and data handling strands. The Universities Funding Coun-cil financed a 1-year project to produce books to meet this need. The project officer was GlynDavieswhowrote the twobooksPracticalDataHandling, bookA, andPracticalDataHandling,book B (Davies, 1993a, b) (Fig. 9).When theUniversity of Sheffield indicated that theywould be closing down theCentre for Sta-

tistical Education at the end of 1995, the RSS decided that this work should continue and estab-lished theRSSCentre for Statistical Education at theUniversity ofNottingham.The two centresoverlapped for one term and then the resources of the Sheffield centre were moved to Notting-ham. In 1999 thismoved toNottinghamTrentUniversity where it currently operates and as partof its portfolio of activities is continuing the Sheffield tradition of being proactive at school level.

10.4. Some lessons to learnIn the context of school statistics, a national Centre for Statistical Education is an importantresource since it can

(a) be the focus for curriculum development and research into statistical education,(b) respond quickly to immediate needs,(c) keep high the profile of statistics teaching both nationally and internationally,(d) give authoritative advice on teaching, curricula and examinations in statistics, and(e) consider developments, such as those in information technology, that have implications

for teaching statistics and make appropriate provision.

11. The first national curriculum in England and Wales

11.1. BackgroundIn July 1987 the Department of Education and Science and the Welsh Office published a con-sultation document for a national curriculum for pupils aged 5–16 years (Department of Edu-


Fig. 9. Activity 62 from Davies (1993a) (reproduced by permission of Hodder Arnold)

cation and Science, 1987). Mathematics was to be one of the foundation subjects and a workinggroup was set up to develop proposals for attainment targets (content) and programmes ofstudy (approach to teaching). The Department of Education and Science and the Welsh Office(Department of Education and Science, 1988) published their proposals for mathematics, basedon the working group’s recommendations, in August 1988 and the revised proposals came intoforce on August 1st, 1989.

11.2. Content and approachThe design of the national curriculum in mathematics reflected both the process of mathemati-cal thinking and the technical content of a mathematical syllabus. Initially the programme wasdescribed under 14 strands—one of which was probability and three were statistics (called datahandling) spanning all ages from 5 to 16 years. In a major revision (Department of Educationand Science, 1991) these 14 attainment targets were reduced to five:

(a) using and applying mathematics;(b) number;(c) algebra;(d) shape and space;(e) data handling;

but the content was essentially unchanged.The technical content was described in great detail under attainment targets with levels 1–8

and exceptional. The programmes of study made little reference to the real life applications ofstatistics or to the use of real data. Non-statutory guidelines for teaching the course were pro-duced (NationalCurriculumCouncil, 1989)which included some excellent ideas such as relatingschool policy across the school curriculum and giving examples of cross-curricular approaches

456 P. Holmes

and activities. Since they were not part of the assessment, they were not taken as seriously asthe main document.

11.3. Assessment and its effectsTo check its effectiveness, the national curriculum included very detailed assessment tests sothat each pupil could be identified as having reached a particular level. Over a couple of yearsor so, the assessment tests so dominated the teaching that the non-statutory guidelines wereeffectively ignored. The emphasis of the assessment was whether the pupils could answer veryclearly defined questions relating to very specifically detailed statements in the attainment tar-gets. Teachers had to make sure that their pupils could answer the test questions, so they taughtto the test. Thismeant that, in statistics, therewas great emphasis on teaching the techniques; anyglobal views of the importance of statistics and the nature of statistical thought were relegatedto secondary importance.


(a) It is possible to include a substantial amount of probability and statistics as part of themain school mathematics curriculum.

(b) If you have statutory and non-statutory parts of a curriculum, the statutory ones willeventually dominate.

(c) External assessment moulds the teaching; if you assess atomistically you will obtainatomistic teaching.

(d) If you do not assess global skills and understanding they will not be taught.(e) Integration of statistics teaching across the curriculum requires a whole school strategy.

Statistics must not be seen solely as part of the work of the mathematics department.

12. The current position in the national curriculum

12.1. Changes in the programmes of studyThe current national curriculum in mathematics (Department for Education and Employment,1999) has rationalized the original structure. There is one strand in data handling, which in-cludes probability, another in number and algebra, another in shape and one on mathemat-ical thinking. As well as the technical content, which is still listed under attainment levels,there is a statutory programme of study which indicates how the material is to be taught.The assessment process is meant to be based on these ‘Programmes of study’ which are set atfour key stages. Stages 1 and 2 are for primary schools; 3 and 4 are for secondary school pupils.The RSS Education Committee offered to help with the revision of the programmes of study

for this version of the national curriculum. As a result of this co-operative attitude, two Fel-lows were part of the team that carried out this revision for the Qualifications and CurriculumAuthority (Qualifications and Curriculum Authority, 1999). At key stages 3 and 4 the pro-gramme of study for data handling is clearly written round the statistical process described inthe cycle

Specify the problem

Collect the data

Process and represent

Interpret and discuss


The instructions are that pupils should be taught to carry out every aspect of the handlingdata cycle by using the techniques that are appropriate to the relevant key stage. Pupils areto be taught to communicate mathematically by using diagrams linked to related explanatorytext. They also must make decisions about problem solving strategies to use in their statisticalwork.

12.2. Technical contentThe headings for the detailed syllabus for key stage 3 (pupils aged 11–14 years) are

(a) using and applying handling data,(b) problem solving (includes a description of the cycle), communicating and reasoning,(c) specify the problem and planning,(d) collecting data,(e) processing and representing data, and(f) interpreting and processing results.

A section entitled ‘Breadth of study’ at the end of each key stage is included for all mathemat-ical strands. For key stage 3 it includes the statement that pupils should be taught knowledge,skills and understanding through practical work in which they draw inferences from data andconsider how statistics are used in real life to make informed decisions and that address increas-ingly demanding statistical problems.This is emphasized even further at key stage 4 for the foundation strand (those who may

not be expected to do any more mathematics). These pupils, it says, should be taught aboutthe major ideas of statistics, including the identification of appropriate populations, obtaininga representative sample to draw inferences about populations, different measurement scales,probability as a measure of uncertainty, randomness and variability, an awareness of bias insampling and measuring, and inference and its use in making decisions.Suitably interpreted, this can be considered as a reasonable summary of the major ideas in

statistics and statistical thinking that are appropriate for pupils of this age.In addition there is a section emphasizing the strong links to be made to the use of statistics

in society. Three points made here are that

(a) through problems and investigations pupils should gain insight into how statistics areused in real life to make informed decisions,

(b) pupils should be introduced to important uses of statistics in society and(c) pupils should interpret statistics from society, including index numbers (general index of

retail prices), time series (population growth) and survey data (national census).

12.3. Statistics across the curriculumIt is not only in mathematics that statistics appears in the national curriculum. Holmes (2001a)gave details of the amount of statistical thinking that is required in subjects such as science,geography and history, and across most of the school curriculum. Since these syllabuses wereall developed independently there are discrepancies between the level of statistical competencethat is required in different subjects at the same age. We still have the problems of integratingall this teaching so that the statistics is well taught and the pupils gain an integrated view ofstatistics.

12.4. The present positionStatistics (called data handling, a serious understatement of the ideas that are to be developed) is

458 P. Holmes

now well established as a part of the national curriculum in mathematics. How well it is taughtand learned will depend on the teachers and their resources. The national numeracy strategy,which started in primary schools in 1998 but is now effective in secondary schools, is makinghelp and guidance available (Department for Education and Skills, 2001). Text-books need toincorporate the ideas that are described in the various programmes of study. Crucially the teach-ing will reflect the way that the material is assessed. Officially the assessment should be based onthe programmes of study, and it remains to be seen how much those responsible can abandonthe mentality of assessing individual techniques of the attainment levels and move towards thebroader skills described in the key stages.

13. The present and the future

13.1. The present position at General Certificate of Secondary Education and GeneralCertificate of Education advanced levelThe national curriculum is now the syllabus for GCSE mathematics and, as part of the as-sessment, from 2003 all candidates must carry out a practical project in data handling. Thesemathematics syllabuses are administered in England by the Assessment and Qualifications Al-liance (Assessment and Qualifications Alliance, 2001), Edexcel (Edexcel, 2001) and Oxford,Cambridge and RSA Examinations (Oxford, Cambridge and RSA Examinations, 2001). Thedata handling project carries half the marks that are allocated to the data handling part ofthe syllabus. The general instructions from the Qualifications and Curriculum Authority to theexamining boards require centres to be given the option of choosing their own topics for theseprojects. In fact, all the boards are giving board options, and it is clear that they are applyingpressure on schools to use these options. One of the boards is reported to be constructing alarge, fictitious, database which candidates are expected to use in their projects. This is alien tothe spirit behind using projects, that they should allow candidates to make real inferences inreal contexts. The desire to make assessment manageable is understandable, but there is a realdanger that these projects will become overformalized and so will not encourage or assess theoverall statistical skills that are part of being a statistician. The teaching of data handling inmathematics will depend on the provision of good teaching material for teachers to use. As wasfound in the SchoolsCouncil Project,material that covers the ideas described in the programmesof study is not easy to write. It may be necessary to find resources to provide such material.The Assessment and Qualifications Alliance has made available a GCSE in statistics (Assess-

ment and Qualifications Alliance, 2000a). The syllabus for this is broad and puts the techniquesin contexts. There are, as should be expected, more techniques than are in the data handling partof the national curriculum in mathematics and there is explicit reference to the use of statisticsin real life circumstances. Unfortunately, at the moment, the specimen questions (Assessmentand Qualifications Alliance, 2000b) do not reflect the breadth of the syllabus. They are limitedand unimaginative and do not encourage a discussion of the issues that the statistics raise. Yetit is these practical issues and contexts that make statistics interesting. Most of the questionsconcentrate on techniques and the opportunity has not been taken to widen the type of ques-tion from the fairly narrow sort of question that is asked in mathematics. For example considerspecimen question 5 from the higher paper illustrated in Fig. 10.It is not so much what it does ask as what it does not ask that is at issue. The scenario men-

tions a managing director wanting to assess the effect on sales. This is never referred to againin the question. There is nothing on where the weights for the retail price index have comefrom, whether they are the relevant weights for people buying his product, whether the itemsthat he has chosen are the relevant items and so on. It is not just that these questions are not


5. The managing director of a company is keen to assess the impact that retail prices may have on sales. He therefore obtains, in a summarised form, the following information from the Monthly Digest of Statistics.

Item Group Single Item Index 1995

Single Item Index 1998

Weights

Food 250 291 208Alcoholic Drink 305 310 77Tobacco 358 364 38Housing 308 320 149Fuel and Light 391 402 67

(a) Show that the total weighted index of retail prices for 1995 is 299 (3 marks)(b) Work out the equivalent weighted index of retail prices for 1998. (3 marks)(c) Hence calculate the all item (aggregate) weighted index of retail prices for 1998 using 1995 as the base year. (2 marks)(d) Using your earlier results, if the food group were excluded from the calculations in part (c) what effect would this have had on the resultant index? (2 marks)

Fig. 10. Specimen question from Assessment and Qualifications Alliance (2000b) (reproduced by permis-sion of the Assessment and Qualifications Alliance)

asked here—they never seem to be asked. The statistical and contextual issues, in contrast tothe narrow arithmetical issues, take a distinctly second place. du Feu (2002) analysed the 11questions set on the higher level Assessment and Qualifications Alliance GCSE statistics paperfor summer 2001. He states that, although there were 11 different contexts only one containedreal data (though, arguably, there are three). He found nothing in the paper for the biologists,the geographers, the scientists or the environmentalists.At A- and AS-level we have many modules in statistics but, apart from an AS-level on the

use of mathematics and theMathematics in Education and Industry Project’s AS-level on com-mercial and industrial statistics, they are all part of a mathematics qualification. More of themdo include some practical and project work, but it is still possible to do a substantial part ofstatistics in a mathematics course without doing any practical work. No AS-level in statistics isspecifically designed for user subjects; nor is there one designed for general understanding ofthe nature and role of statistics such as is available in some US colleges for liberal arts students(Moore, 2001). Some of the emphasis in earlier AS-level syllabuses on the collection of data,surveys and design of experiments has been discarded. There is nothing on a deeper reading oftables of data as described in Abramson (1988).

13.2. Have the lessons been learned?Throughout this paper I have tried to draw out lessons to encourage continuing improvementin statistical education. In this section I try to identify which of the lessons have been learnedand to raise points for discussion so that we might see more clearly where to go in the future.Clearly many things have changed for the better over the past 50 years. To a large extent many

of the things that were envisaged in Royal Statistical Society (1952) have been taken on boardand are in place. Statistics is explicitly recognized as a practical and important subject and it isimportant to develop practical as well as theoretical skills. It is firmly part of the primary schoolcurriculum and can be enjoyed there.There are still some worries. Now that we have a national curriculum, howwill things change?

The flowering of ideas in the 1960s and the eventual selection of what was good and the discard-ing of that which did not work required two things. These were an environment where teachers

460 P. Holmes

could experiment with different ideas and a climate where they were empowered so to do. Withthe current emphasis on school assessment and adherence to the national curriculum, teachershave been disempowered and demotivated from experimenting. There needs to be a mechanismwhereby experiments canbemadeon changes in the school curriculumwhich encourage teachersto be active participants and do not penalize the students who are part of the process. This wouldallow continual change and improvement of the national curriculum on the basis of evidence.The lesson that statistics is an interdisciplinary subject has not been learned. The various

subject revisions in the recent review of the national curriculum all took place on the basis ofwhat was important in that subject. So we still have discrepancies across subjects about whattechniques should be included at what level and there is still the danger that there is no properoverview in a school of the received curriculum in statistics.It has been a feature over the years of the RSS reports on the teaching of statistics that teach-

ers need training. In the 1960s and 1970s this was addressed by the growth in practical in-servicecourses and the wide provision through teachers’ centres and support through local authoritysubject advisors. In the 1980s and 1990s funds for these were cut and this support network is notas strong. We still have problems of teachers who are undertrained in statistics and its teaching.The importance of a national Centre for Statistical Education has been accepted by the RSS

and other sponsors, shown by their continuing funding of the Centre at Nottingham TrentUniversity, and this facilitates all the points made in Section 10.4.

13.3. The futureThe place of statistics in the structure of English schools is still not satisfactory. Royal StatisticalSociety (1952) suggested that the place for statistics was with the user subjects. At the time ofthe 1968 report (Yates, 1968) the position had changed to suggesting that statistics should bea subject in its own right in the curriculum. This has not happened in schools. I do not knowof any departments of statistics as such. The de facto position in most schools is that it is themathematics department that takes first responsibility for the teaching of statistics. This has be-come even more the case since the national curriculum in mathematics has included the majorstrand of data handling. Linking across the curriculum is still a problem, and the mathema-ticians’ approach to statistics has its weaknesses as well as strengths. Statistics co-ordinators,as envisaged by Cockcroft, do not exist. How do we link theory with applications and breadthof knowledge about the work of statisticians? How do we best achieve a balanced and relevantstatistical education for all, as envisaged by Royal Statistical Society (1952) and by the POSE?Part of the answer may lie with in-service training.For many years it has been said that the future of school statistics will be influenced by

the growing availability of computers and software. There is no doubt that these facilities arebecoming more available and it is relatively much easier to obtain real data from the Internet.Developments such as the RSSCentre’s project CensusAtSchool (Royal Statistical Society Cen-tre for Statistical Education, 2000) with its large database of pupil responses and wide range ofwork sheets and teachers’ notes have made it much easier for teachers to find good material.The Internet also makes it easier to link across to other subjects in the school curriculum. Itis largely the external assessment that is slowing down this process; examining bodies have notsolved the continuing problem of not being able to assume that all candidates have access toparticular software.It is to be expected that the statistics that is part of the national curriculum in mathematics

should have amathematical bias. Even so, the programmesof study showawell-founded attitudeto the subject. This raises the question of the relationship between GCSE mathematics (with a


data handling strand and a data handling project) and a GCSE in statistics, which presumablyshould bemore rounded and holistic and includemuchmore thanmathematical considerations.At the moment there is a danger that this will be seen as trivial and the whole subject will not beseen as worthwhile by students at a crucial decision time in their lives. The difficulty here appearsto be, not the specification as such, but the assessment. Examiners are largely drawn from abackground in mathematics with its strong emphasis on the reliability of marking. There is amove towards having assessed project work, but this increased validity needs to be extended tothe examinations as well. There is a need for much more holistic assessment with things such asopen-ended questions, discussions of issues, short essays, questions requiring more than simpleor technical answers and items relating the data to a real issue in a context. Only in these wayswill we get the assessment encouraging good statistics teaching. These types of assessment aredone in other subjects; why not in statistics?The price of good statistics teaching is eternal vigilance. If we can encourage teachers, de-

velopers and examiners to incorporate the best of what has been developing over the last 50years then there is some hope. The major need is to maintain a broad vision of the nature ofour subject and its applicability, and continually to apply this vision to the detail of our schoolcourses and their assessment.

Acknowledgements

The idea for this paper came from a short talk given by the author at the Golden JubileeCelebrations of the International Statistical Education Centre in India, and published in theproceedings (Holmes, 2001b). This paper is a very much expanded version of the talk and inwriting it I have had much help from many colleagues, particularly those at the RSS Centrefor Statistical Education. Comments from two referees have helped me to improve the paper. Iam aware that in trying to summarize in one paper all that has happened in school statisticaleducation over 50 years I have had to leave out much detail, and not everyone will agree withmy emphasis on what has happened or be content with some of the omissions. I can only acceptresponsibility for the paper and plead fallibility to those who disagree.

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Discussion on the paper by Holmes

F. R. Jolliffe .University of Kent, Canterbury/It is no mean feat to describe 50 years of statistics teaching in English schools in such detail in one paper,and there is probably no-one more qualified than the author to have done so. As can be seen from themany references to Holmes in the paper, he himself has helped to shape statistics teaching in Englishschools over the more recent decades of the last 50 years. In fact he has made, and continues to make,many contributions to statistical education at all levels and in many countries around the world.

As this Ordinary Meeting is the first and last to be organized through the Society’s Education Section,I feel that it is appropriate to mention some of its history and the contributions that it made to statisticsteaching in schools. In February 1996 the Society’s Council approved the Education Committee’s proposalto form the Section. It was hoped that the Section would appeal to teachers from the Associate Schoolsand Colleges, and the Section’s regulations stated that twoCommittee members were to be appointed fromthose teachers. Recognizing the need to provide meetings for this part of the Society’s membership, theEducation Committee had previously arranged a meeting that took place on June 30th, 1994. About 60people, mainly school-teachers, attended. The first of the two talks at this prenatal meeting of the Sectionwas given by Peter Holmes on ‘Reflecting the statistics in society by the statistics we teach in schools’. Theother talk was given by JimGarbutt, then teaching at a school in York, and was on ‘Motivating secondaryschool pupils in statistics’. The Section’s first meeting was held on June 4th, 1996.

The Section arranged a programme of meetings on statistical education issues at all levels and was alsoresponsible for organizing workshops for school-teachers (previously these had been organized by theEducation Committee). In June 1997 it started to organize workshops for pupils also, with the first a sixthform day in Belfast. Many of these workshops involved practical work and/or information technology.

In December 1999 the Executive Committee established a team to review the Society’s educational pol-icy, and inApril 2001Council accepted the team’s recommendations, including one that the Section shouldbe discontinued. The Section’s last meeting was held on May 1st, 2002. This was a discussion meeting onrecent developments in statistical education in schools and higher education and was held jointly with theRSS Centre for Statistical Education. The meeting started with a presentation on ‘Enhancing statisticalnumeracy in schools for teaching, learning, and assessment’ by Peter Holmes and his colleague DoreenConnor. They considered how an active approach to teaching statistics, using meaningful data collectedfrom and by school-children, can enhance statistical skills to produce a deeper understanding of importantprinciples, and they suggested some methods for evaluating the knowledge gained. The rest of the meetingwas concerned with statistical education at higher levels than school. Thus Peter Holmes was in at boththe start and the end of the Section.

There are important messages under the headings of ‘Some lessons to learn’ in this paper, and many ofthese, sometimes with slight modification of wording, apply to statistics teaching at other levels. I particu-larly like the emphasis that statistics is a practical subject made in Section 3.1 such as ‘involve practical aswell as mathematical statisticians in the development of a syllabus’ and ‘take advice from the user subjectsabout the nature of the statistics that they need’. Sadly, some of the lessons have not yet been learnt, as isnoted in the author’s conclusion.

The place of statistics in English universities is not satisfactory either. In recent years some statisticsdepartments, or even combinedmathematics and statistics departments, have closed. Is this partly because

464 Discussion on the Paper by Holmes

courses given by statistics departments to user departments did not develop syllabuses which took accountof their needs? Were courses too mathematical or taught as a set of techniques with little attention paidto real applications? Is it better if statistics is taught from within the user departments by people who areknowledgeable in statistics instead of by members of a statistics department? It is easy to say that courseswere lost because user departments did not want to give away full time student equivalents for serviceteaching, but would this be so if they had received an excellent service adding value to their graduates? Itwould be interesting to have a paper on 50 years of statistics teaching in English universities.

I should like to end on amore optimistic note. The Society’s Careers Promotion Committee is producingmaterial that is designed to tell school pupils (and others) how useful and exciting statistics is. Look athttp://www.rss.org.uk for details. Perhaps this will help to ensure that our subject has a future foranother 50 years.

I have great pleasure in proposing the vote of thanks.

John Bibby .MatheMagic and University of York/On being offered this onerous task, I was advised that, as Watson to Peter’s Holmes, I may venture totake a critical viewpoint and could even stray from the subject as long as not too many people noticed.For this reason, if for no other, it gives me very great pleasure to second the vote of thanks on thisthought-provoking paper.

The Royal Statistical Society can be justly proud at the role that we have played in enhancing statisticaleducation in the UK (or, more correctly perhaps, England), and via the International Conferences onTeaching Statistics and other endeavours in the wider world. However, as Peter points out, much remainsto be done, some lessons have yet to be learned and pride must not breed complacency.

Peter and I first met in the mid-1970s on an interview bench during the halcyon days of the SchoolsCouncil. We were both short listed in connection with the newly funded Project on Statistical Education.Even then Peter displayed maturity of outlook and a clear focus, whereas I was still in the wishy-washydays of new enthusiasm at the Open University. Needless to say, Peter was appointed, although I was gladto be able to work with him on the project in a very small measure.

Peter has outlined the successes of the project and many other subsequent endeavours which he and hiscolleagues have been involved in. Among these must be counted the dramatically improved position ofstatistics in the school curriculum, and the panoply of publications and projects that are associated withHolmes, Barnett and colleagues.

I am of course delighted that statistics is now firmly rooted in the school curriculum (even if it is oftencalled ‘data handling’) and I acknowledge the role of Peter and his colleagues in achieving this.

However, aswe are discussing history, it behoves us to ask towhat extent these dramatic changes resultedfrom the Schools Council and subsequent projects, and to what extent they were due to exogenous or soci-etal factors.

Undoubtedly themost correct answer is ‘a mixture’. The time was ripe for change; otherwise the SchoolsCouncil and other sources of funding would not have been forthcoming, whatever Vic Barnett’s persuasiveabilities.

But what were the contemporary changes outside these projects—and indeed outside the educationalsystem more generally—that led to statistics’s heightened role?

One thinks of economic supply and demand and the evolving professions, as well as technological andother factors. Also, there was a changing consciousness of the need for numeracy and statistical literacyand for science generally—culminating perhaps in today’s mania where the rule is ‘if it moves, count it’(and the converse, ‘if it is difficult to measure, ignore it’).

Perhaps in his response, Peter could address these environmental issues in somewhat more detail.Peter noticeably does not use the word ‘history’ in his title. Indeed, it is used sparingly throughout the

paper. Thus perhaps he is giving ‘an account’, rather than ‘a history’. (Or perhaps it is indeed ‘his story’,as he personally plays such a prominent role.)

I also note two further restrictions that are implicit in Peter’s title.First, he refers to ‘English schools’—thereby excluding even Scotland, with its distinctly different sta-

tistical and education tradition, from his gaze. This is a lack, for in history as in statistics we often learnmore by shrewd comparisons than we can by examining just a single case. (Another interesting area wouldbe to compare the British colonial tradition with perhaps the French or even the American or Russian.)

Second, by focusing on schools alone, the vast bulk of even the English population is excluded. For,though some of us may have attended school, for many of us learning did not stop there. (Nor indeed didit start there—preschool statistical oracy is an interesting issue to explore.)

Discussion on the Paper by Holmes 465

However, we should not complain about Peter’s narrow focus, for his account provides depth and detailwhere it might lack in breadth.

A more crucial restriction may be the title’s use of the word ‘teaching’. For teaching is not learning. Ineed not elaborate on the many uncharitable things that have been said about the teaching process and itspractitioners, for many of these are untrue—apart from being unsympathetic. However, I would like topresent two basic axioms:

(a) teaching does not necessarily lead to learning;(b) most learning takes place outside the classroom.

In a sense, we are all autodidacts.We learn what wewant to learn—motivation and content are everything,as every good teacher knows. Arguably, this applies especially to elements of literacy, including ‘statisticalliteracy’.

These axioms are not new. The importance of informal learning has long been recognized. However, itis often vastly underrated. This is especially so in the research community, which provides a ton of dataon schooling for every ounce of ideas on informal learning.

The successes of statistics education have been mentioned, and we certainly should celebrate these.However, we may paradoxically be able to learn more by asking ‘What are the failures?’.

For Darwinian evolution teaches us, as do the Taguchi and total quality management approaches toindustrial engineering, that failures may point us to the biggest challenges. By examining ‘failures’ of thepast we may escape Whiggist triumphalism and see more clearly and in a comparative light the historicalfactors which are conducive—and those that are not conducive—to the development of our subject.

I would therefore like to present briefly for your consideration ‘Two great British failures’ in statisticsteaching in England.

The Statistical Society of London’s ‘Statistics in Schools Committee’ (1870)In its pre-Charter days, this Society established a Committee to ‘promote the teaching of statistics inschools’ (Council minutes, July 14th, 1870). This Committee met twice, on July 14th and July 22nd, 1870.William Newmarch and Leoni Levi were its leading personalities. The Royal Geographical Society seemsto have provided the impetus via one of itsmedal schemes—possibly under the influence of FrancisGalton.

However, after its second meeting, the Committee disappeared from view after a lifetime of only 8days—is this a record? The Society’s annual report for that year makes no mention of the Committee’sexistence, and it seems genuinely to have disappeared without trace—‘a veritable Marie Celeste of thestatistical world’ (Bibby (1986), page 79).

2 years later came the second great British failure to which I wish to refer. This was the so-called‘Nightingale Chair’, proposed initially for the University of Oxford, and later (in the 1890s) for the RoyalInstitution. Apart from Florence Nightingale, prime players here includedWilliam Farr, Benjamin Jowettand Francis Galton, with AdolpheQuetelet in a brief role. (For further discussion and original documents,see Pearson (1924) and Bibby (1986), pages 30–43 and 112–147.)

The Professorship’s aim was

‘for promoting by means of lectures or otherwise the cultivation and improvement of statistical science,and especially its practical applications to social problems’

(Bibby (1986), page 41, Pearson (1924), page 423, and Pearson (1914, 1930)).This was a long-lived proposal, lasting from 1872 until a few weeks before Nightingale’s death. The

coup de grace seems to have been administered by Francis Galton, who suggested that the post shouldbe transferred from Oxford to London and also started to investigate a statistical research programmewhich would spend what was in effect Nightingale’s money. She eventually despaired and withdrew herendowment because she felt that it would ‘only end in endowing some bacillus or microbe, and I do notwish that’ (Cook (1913), volume 2, page 400, and Bibby (1986), page 41).

Commentators have ascribed the fall-out between Galton and Nightingale to the latter’s failure toendorse the newfangled germ theory of disease, preferring the old miasmic theory. However, Kendall(1972), page 141, may be closer to the mark in ascribing its failure to the fact that ‘our senior universitieswere still whispering from their towers the last enhancements of the middle ages’.

In these two great British failures I perceive the following causative factors:

(a) discussions were in terms of structure rather than syllabus;(b) both ventures were ‘supplier led’ (in the case of Nightingale, ‘funder led’).


Client-need was not established nor even considered (although Nightingale argued convincingly thatOxford had a need, because this was where ‘most of our statesmen, and those who later becomeMembersof Parliament, legislators, administrators and holders of executive power carry out their studies’):

(c) in the Nightingale Chair at least, there was a tension between the research function and the peda-gogic function of the professoriate.

If this sounds too contemporary or even Whiggist, let me simply reiterate that history is one step for-wards, several steps back. Learning like epidemics often depends on a ‘tipping point’—or on which side ofthe zero line the infinitesimal " falls. It is often extraordinarily difficult to knowwhatmakes one innovation‘stick’ whereas another fails.

What then do we see for the future? ‘Back to the future’ is where all good histories should start—tosee the future, one must ‘close one’s eyes and wish’. Rewriting history may be frowned on, but can we atleast indulge ourselves a little by creatively considering a range of imaginative histories for what mighttentatively be called ‘statistics 2020’?

Futurology often ascribes undue prominence to technological factors. How will the World Wide Webaffect teaching and learning? What about new computer packages? Will the dynamic visualization tech-niques of DataDesk and Fathom presage new learning methodologies?

However, the future is far more than technology. We must also ask questions about changes in politicalstructure, financial climate, new theories of learning, attitudes to assessment and ownership of learning,etc. At the risk of overdramatizing, will schools still exist in 2020? If so, will we still have teachers?Andwhatwill be their role relative to non-teaching staff, parents, the pupils themselves and indeed new technology?

What about universities?:will they still be there in 2020?Will there be independent statistics departments?These are questions which must be asked, however much they may make us shudder. They shout into

the tunnel far beyond the realm of statistics, and our colleague and friend Professor Adrian Smith will nodoubt come up with some good echoes when his inquiry on post-14-years education reports soon.

My personal worry is that statistics and perhaps mathematics more generally may have developed or bedeveloping to the stage where they no longer require evidential justification—they form part of canonical‘commonsense’ necessity—to paraphrase Huxley, like all truths they ‘began life as heresy, and will end itas superstition’.

To return to Holmes’s paper, in referring to current national curriculum developments and the keystages 3 and 4 strategy (Section 12.1), he uses a Popperian or Tukeyite diagram which he refers to as the‘handling data cycle’. This predates the references given and goes back at least as far as Open Universitycourse DE304, Bibby and Evans (1978) and Bibby (1982) andmultiple works of AlanGraham at the OpenUniversity.

Of the four stages in this cycle, only one is mathematical in nature—the one that Peter calls ‘Processand represent’, but which earlier OpenUniversity versions called ‘Analyse the data’. The other three stagesare substantive in nature. Thus it is no surprise that the iterative data cycle is especially useful in serviceteaching.

Peter notes (Section 13.2) that ‘The lesson that statistics is an interdisciplinary subject has not beenlearned’.

If we are looking for failures, this surely is one that we can learn from—for it is not for want of tryingthat the lesson of interdisciplinarity remains unlearned in Britain.

Others learned early. In the 1860s, discussions of statistics teaching at the International StatisticalCongress were dominated by the requirements of geographers and other applied areas (Bibby, 1986).

A century later, the West African Examinations Council had a surprising innovation in terms of a com-pulsory ‘statistical interpretation’ question which all A-level candidates had to answer. This question wasa compulsory part of the compulsory ‘General studies’ paper, without which one could not gain admissionto a university in any subject. The rationale for this was based largely on a concept of ‘statistics for citi-zenship’ not unlike the rationale adumbrated by this Society in its submission to the ‘Post-14 mathematicsinquiry’ (Royal Statistical Society (2003), paragraph 9.1).

In the mid-1960s I went to Ghana and with my colleague Andy Miller, an economics teacher, wascommandeered to teach statistics to all sixth formers at Accra Academy. They did so well that I waspersuaded by Longman publishers to turn my class notes into a book—and Bibby (1972) was theresult.

Statistics also featured highly on the new ‘Joint schools project’, which I was honoured to co-author.One lesson that stuck with me from these early endeavours was the advantage and joys of team teaching

and of using original sources. If you can persuade somebody from the ‘applied’ department to teach a


service course with you, then this is politically very shrewd (it spreads ownership to parts that other statis-tics courses fail to reach) and can also provide effective ‘idiot proofing’ as well as a good range of appliedperspectives.

This method developed in Ghana followed me to St Andrews. As a result, instead of three first-yearsocial science courses in statistics, there was soon a single course taught jointly by a geographer, a psy-chologist and an economist or statistician (me). Previously, students who studied all three subjects (acommon occurrence in Scotland) had three different methods courses—along with considerable overlapand inconsistencies.

Team teaching is a formidable learning experience for staff as much as for students. Nowhere was thismore apparent for me than at the Open University, where our interfaculty course ‘Statistics in society’involvedmany people, provoked computer overflow problems since up to then nomore than three facultieshad ever collaborated on a course (we had six, but the computerized acronym had to be reduced from‘DEMAST’ to ‘MDST’, pronounced ‘maddest’) and also led to what are best described as ‘interesting’political problems—within the Mathematics Faculty and university wide.

Much to everyone’s surprise, ‘Statistics in society’ appeared on time in 1983. Still more surprisingperhaps, it is running still, 20 years and 15 000 students later.

I think that the key to its success is another aspect of didactic philosophy which must be learned byexperience. This is the maxim ‘Start from your students’—use real life and real perception—as perceivedby the student. Customer-led philosophy beats curriculum led every time.

In the Open University course three main blocks focused on areas of particular interest to mature stu-dents—economics (Labour Force Survey and the retail price index), education (‘How is my child doing?’)and child development. These were reinforced by four television programmes on ‘energy’ (the statistics offuel, cavity wall insulation and renewable power). For other student types, one would choose other focusareas (sport, music, travel—these all have interesting and real statistical questions).

You will notice that I am a great fan of ‘real statistics’—not artificial examples which often stultify avivacious subject and overwhelm it with whimsy. It is time perhaps for a ‘Salters statistics’ course to runparallel to the problem-based Salters chemistry A-level course that was developed at the University ofYork.

However, interdisciplinary teaching does not happen of its own accord. It requires conscious and con-scientious planning. This should include

(a) a base-line assessment of where non-specialist teachers are, in terms of their statistical readiness,(b) training of statistics co-ordinators, who would ideally be specialists in non-mathematical subjects

and(c) structures so that teachers of different disciplines talk to be each other—a rare event in today’s over-

pressed staffrooms.

I am delighted to see that the Education Strategy Group under the leadership of Harvey Goldstein hasrecently taken up some of these issues (mutatis mutandis) with the ‘Post-14 mathematics inquiry’ chairedby Adrian Smith (Royal Statistical Society (2003), paragraphs 3.7 and 5.3). I am particularly pleased thatthey have focused on ‘citizenship’ aspects of statistics and mathematics (paragraph 9.1)—an area that theinquiry’s ‘key questions’ seriously underplays.

In conclusion, I congratulate the speaker, not just on an interesting and useful paper, but also on astunning career of contribution to the subject. In many ways, Peter Holmes’s work has changed lives andhas changed the way that we look at things. I salute him.

The vote of thanks was passed by acclamation.

Toby Lewis .University of East Anglia, Norwich/May I join in congratulating Peter on his important paper?

He asks (Section 7.5) ‘Why teach statistics to everyone?’ Among the answers we read

(a) ‘Statistical thinking is an essential part of numeracy’ (Section 7.5),(b) ‘children should . . . understand . . . the power and limitations of statistical thought’ (Section 7.3)

and(c) ‘statistical ideas are widely used at work after school’ (Section 7.5).

But what are statistical thinking, statistical ideas and statistical thought? There are many views in theliterature. I myself think that the essential idea motivating statistical science is simply that repeats of the


same action or situation may give different results—i.e. uncertainty and randomness. This contrasts withthe deterministic approach which some of the students will be learning in their applied mathematics orphysics. In the paper it eventually appears in Section 9.1, the Cockcroft report:

‘Statistics . . . is an attitude of mind in approaching data. In particular it acknowledges the fact ofuncertainty and variability in data collection.’

In the same spirit I welcome, among the myriad published definitions of statistics, the recent one by Fisher(2001), which is short and to the point:

‘Statistics is the science of managing uncertainty’.

So let students generate their own data and get direct experience of uncertainty and random variation!This would be much more educative for them than having to use in their projects ‘a large fictitious data-base’ (Section 13.1). In my teaching experience I have found simulation an invaluable tool, whether usedby teacher or student.

Finally, I welcome the important Section 4 on teaching at primary level. How right Peter is when hesays that

(a) ‘Primary school children can learn and enjoy elementary probability and statistics’ (Section 4.3,lesson (a)) and

(b) ‘It is possible to begin to develop early ideas of probability and inference at this age’ (Section 4.3,lesson (b)).

I have seen for myself what can be achieved by a gifted teacher. I shall mention only one of her data col-lection, discussion and reporting exercises, the ‘Pedestrian graph’. For this, 9-year-old children observedfrom the pavement and counted the numbers of people crossing the road in a 10-minute period. I recalltwo pleasing entries from their subsequent written reports:

‘Some children went out to do the pedestrians’ (Ian);

‘When we whent it was the 18th of March and we all had something to do. The first day we whent outwe got a lot of children put down’ (Tracey).

Vic Barnett .Nottingham Trent University/I should like to join others in thanking the author for an interesting and informative paper.

One thing which his natural modesty prevents him from telling us is how important was his own teach-ing style and rapport with pupils for persuading them to accept the newer approaches of the ‘Statisticsin your world’ modules in the Project on Statistical Education—indeed sometimes almost hanging on hiswords. I recall the day in the late 1970s when he puzzled a class of 12-year-old children by telling them thatstatistics could reveal which of them smoked behind the bicycle sheds at lunchtimes. A judicious use ofthe (Greenberg) sampling ‘principle of the irrelevant question’, some sampling luck and a few appropriateblushes delivered on his claim and the class was ‘hooked’ on statistics and could not get enough of it.

The author has given us a wide-ranging bibliography. I would just like to extend this a little. In thelate 1970s (again) the International Statistical Institute (ISI) based in the Hague supported several educa-tional task-forces. One, after many years of lobbying through the ISI Education Committee, was chargedwith starting the important series of International Conferences on Teaching Statistics held every 4 years.Another, which I chaired, was concerned with secondary level education, and on its behalf I producedthe wide-ranging review Barnett (1982). This described the situation inmany countries but in particular anopening chapter gave a detailed analysis of the prevailing position in England and Wales (with a supple-ment on Scotland). A similar international resume on university level education was produced by Loynes(1987) on behalf of the ISI Taskforce on Tertiary Level Education.

Harvey Goldstein .Institute of Education, London/I also welcome this important paper and congratulate Peter Holmes not only for an interesting and wide-ranging account but also for all the effort that he personally has contributed to statistical education.

One important distinction between the pre-1990 and post-1990 period has been the move from a largelydecentralized school curriculum and assessment system to one controlled by central Government. This


implies that innovations are more likely to be successful if initiated from the top down, and this poses par-ticular problems for statistical educationwhere, as Peter points out, themost successful developments havearisen from enthusing individual teachers and educators at the chalk face. Furthermore, the imposition ofexternal targets and regular key stage, easily measured, assessments makes the development of practicalproject work more difficult. This is especially so in statistics which ideally should be interdisciplinary andlong term.

I would also like to emphasize the role of developments in information technology which make possiblethe statistical analysis of large scale real data sets. The RSS Centre for Statistical Education has madenotable contributions in this area with its CensusAtSchool project and its new collaboration with theOffice for National Statistics to exploit national data sets for teaching. Alongside this I would want to seeways in which pupils could be helped to generate their own large scale data sets, e.g. by discovering andinterrogating existing databases. The role of data analysis software for schools is also an issue that requiressome thought. The use of existing packages, such as spreadsheets, may be convenient but is not necessarilyoptimum for both carrying out data analysis and enabling pupils to learn statistics. Developing suitablesoftware tools would seem to be an area where some international collaboration could be useful.

Peter mentioned the idea of a statistical learning co-ordinator in each school. The Royal Statistical Soci-ety’s Education Strategy Group, in its evidence to the Smith inquiry into mathematics teaching, extendedthis idea to a co-ordinator in initial teacher training and for continuing professional development: such aperson does not have to be a mathematician. If such a proposal is taken up it could then form the basisfor an extension of the co-ordinator idea into schools themselves, with all the benefits that can bring.

Finally, it is worthmentioning that statistical literacy has an important rolewithin citizenship education.Given the importance that is currently attached to citizenship education, this seems a promising route tofollow, developing materials and modules. I would be very interested in Peter’s views on this.

Denise Lievesley .United Nations Educational, Scientific and Cultural Organisation, Montreal/I congratulate Peter Holmes not only on an interesting and thought-provoking paper but also on thesignificant influence that he has had on the quality of statistics education, particularly in the UK, throughhis distinguished career and his devotion to the communication of statistical thinking.

PeterHolmesmentioned the role that he and others played in establishing the International Conferenceson Teaching Statistics and I wish to underscore the value of these conferences as a forum to share expe-riences of statistics teaching and to explore the effect of different national policies. In the United NationsEducational, Scientific and Cultural Organisation we are involved in cross-national studies of education.Such comparisons can be extremely valuable as long as the users are willing to be open to new ideas andto learn. Unfortunately, as Peter indicates, evaluation can be a double-edged sword. Some countries aredefensive and poor comparative results lead them to encourage ‘teaching to the test’. Other countries arekeen to understand how their policy and performance can be improved from learning from others’ expe-riences. For example it would be very instructive to understand how Finland and many Asian countriesachieve the high levels of performance in numeracy that are shown in the ‘Programme of internationalstudent assessment’.

I would have found it interesting to hear Peter Holmes’s opinion on the benefits of specialist teacherscompared with equivalent resources being used to provide statistical training to teachers of mathematicsor other subjects. Also to what extent can statisticians who are outside the teaching profession supportand provide applied material of use to those who have the direct responsibility for statistics education?Can applied statisticians spend some time in schools helping teachers? Can the Royal Statistical Societypromote such interchange?

Finally I wish to express my disappointment at the lack of statistical content in the various initiativeswhich are taking place worldwide to define and measure life skills or ‘competences’. Statistical under-standing and skills must be a critical part of a modern citizen’s capabilities yet the statistical community isnot playing the advocacy role that is needed to ensure that they are included in appropriate assessments.

Pamela Morris .Cambridge/With reference to statistical thinking as mentioned by Toby Lewis, the Royal Statistical Society used toaward a prize each year to the candidate whose course work in the AS-statistics examination of the Oxfordand Cambridge Schools Examination Board best illustrated statistical thinking. AS denoted ‘advancedsupplementary’ then, not ‘advanced subsidiary’ as now.

The development of the pocket calculator in the late 1970s eased statistical calculations, thus helpingpractical work.


The quantitative revolution in geography began in 1958, accelerated into the 1960s and the 1970s, andthen became integrated into certain areas of the subject. Statistical techniques became applied to thesubject in some schools.

In 1965 ‘Mathematics in action: statistics’ was produced as a series of 10 television programmes by theBritish Broadcasting Corporation with an associated course. Initial topics, as may be recalled, concernedthe estimation of the number of squirrels in a wood, and fish of a certain species in a pond.

The following contributions were received in writing after the meeting.

Neville Davies .Nottingham Trent University/In primary and secondary schools evidence for motivating pupils in learning statistics by using real datacomes from the international CensusAtSchool project (Connor and Davies, 2002; Connor and Holmes,2002; Davies et al., 2003). It involves collecting real data from and about pupils, their environment etc.,and creating a range of learning and teaching resources. Several million children have taken part in theUK, South Africa and Australia, and soon even more will be participating in New Zealand and Canada.

In higher education the integration of real and publicly available data into learning and teaching wasa key recommendation of a report funded by the UK Joint Information Systems Committee (Rice et al.,2001).

In the working population the application of even the simplest statistical techniques has been shownto be poor by Hoyles et al. (2002). A key observation is that a real data and work-related approach arecrucial for improving statistical numeracy of employees.

It is time to take an integrated approach to statistical education throughout life, using real data.

(a) We should involve all students from an early age in the information and communication technologydata handling and statistics cycle of design, collection, presentation, interpretation and decision-making with real data which they collect and produce. In UK schools the structure and syllabuscontent are already in place through the data handling components of many national curriculumsubjects, but the approach in practice within and between subjects is not connected. Using relevantand real data can ensure that statistical skills are learned in a non-threatening way and conclusionsthat students draw are more meaningful.

(b) All students who enter higher education should study to some minimum level of statistical numer-acy. This has been proposed before, but not in the context of relating it to earlier studies, using realdata, and learning skills that will be relevant in the workplace.

(c) In the workplace we should conduct a needs analysis and, where necessary, roll out nationwideprogrammes of information and communication technology statistical numeracy training designedto reflect work and business data and contexts. Such training programmes should lead to improvedstatistical numeracy, better evidence-based decision-making and improved productivity.

(d) Community solicitors of data and statistics should be established, with the purpose of providing aone-stop advisory andmentoring service for individuals, businesses and all who need advice on dataand statistical analyses. These services could, for example, be linked to existingGovernment-fundedregional development agencies.

Implementing these proposals will require planning and a large amount of resources, and, of course, manyinspired people like Peter Holmes to inspire others to champion the cause.

Chris du Feu .Beckingham/In Section 8.8, Peter notes that project work must be assessed if it is to be taken seriously and in Section11.4 that external assessment moulds teaching.

A problem in our current climate is that assessments of project work often do not mould in the rightdirection. They demand that candidates merely demonstrate a grasp of a selection of techniques withinthe syllabus. This results in projects which are little more than collections of calculations and charts whichmay gain points but show no underlying statistical coherence. The pressure of time in schools often meansthat the only project which students do is the assessed project, thereby reducing its value as a teaching orlearning experience.

There is an increasinguptakeofGeneralCertificate of SecondaryEducation (GCSE) statistics in schools.A second board (Edexcel) is preparing aGCSE statistics syllabus. This is, apparently, good news.However,the pressure on schools, teachers and pupils is to enter more examinations because they are all judgedby examination success. GCSE statistics is often presented as a course in which able pupils gain an extra


GCSE for only a little extra work above GCSE mathematics and one where less able pupils have somechances of gaining a mathematically related GCSE qualification. The message that pupils receive is thatthe only value of statistics (which is obviously a trivially easy and worthless subject) is to gain an extraGCSE.

Examination boards have an interest in providing a syllabus which will be suitable in schools that areconcerned about league tables. Such a syllabus will be as easy as possible within the constraints set bythe Qualifications and Curriculum Authority. The trend for all syllabuses to have an exactly matchingtext-book will further damage the value of the qualification.

One solution is for the Royal Statistical Society to provide a GCSE statistics qualification—no otherbody in the country is better placed to produce a suitable syllabus. The problem of running the course isnot insurmountable. Oxford, Cambridge and RSA (currently the only board without a GCSE statisticssyllabus) runs GCSE and A-level examinations on behalf of groups such as theMathematics in Educationand Industry Schools Project. Why should a similar arrangement not work for a GCSE (or AS- or A-level)statistics qualification with the Society?

In such a qualification, it is to be hoped that the project work assessment would not demand that any ofthe techniques on the syllabus be demonstrated. All it would require was that the work contained statisticalmaterial of at least comparable technical demands of the syllabus and that the whole work demonstrateda sound application of statistics to a real problem in line with the data handling cycle approach in thenational curriculum.

D. V. Lindley .Minehead/This is a splendid paper and the author has done us all a great service in describing the problems andsolutions in the teaching of statistics in schools. I findmyself in general agreement with the ideas presentedand my comment concerns a matter of emphasis. A major task of an educator is to help the pupils tothink for themselves and, in particular, to think about uncertainty, since uncertainty seriously affects allof us. Now uncertainty must be described by probability and therefore everyone needs an introduction tothat calculus. It is this essential emphasis on probability that I find lacking in the paper. Data are vitalingredients of statistical thinking but can only be evaluated in terms of probability. Of course, even profes-sional statisticians do not always appreciate the role of probability, for example, when they use confidenceintervals, which are not expressions of uncertainty about parameters that combine according to the rulesof the probability calculus. A large and popular US text on statistics has little on probability and hardlymentions conditional probability, so students who are brought up on this text will not be able to think,only to manipulate.

Even when probability is included it can be done poorly. For example, 1/6 is the answer expected inFig. 5. But suppose that the taster is a Kenyan whomay be sure to classify correctly their own coffee but beat a loss with the other brands; the probability of their getting all correct is 1

2 . Probability is an expressionof your uncertainty and the assignment of the classical values, as used in Fig. 5, may be inappropriate.Probability is the language that we must all use in expressing our view of this uncertain world.

Margaret Rangecroft .Sheffield Hallam University/The only time that my history teacher smiled at me was when I told her that I would be dropping historyO-level. At the time we were studying the NapoleonicWars and I simply could not remember the long listsof alliances and dates. I have never been good at rote learning; I need to see the logic behind what I amlearning and at that time there seemed no logic in the type of history that I was being taught. Thankfullysince then I have realized that much of the logic behind learning history is to see what leads to success andwhat to failure.

For me, therefore, the great strength of this paper is the inclusion of sections entitled ‘Some lessons tolearn’. The discussion of the history of statistics teaching in English schools is well written and interestingbut the real force of the message is in the discussion of how that impinges on how we can learn from whathas gone before and how it should influence how we proceed. This raises this paper from being a compre-hensive and careful discussion of the past to a major contribution to the discussion of future policy. It setsthe scene for the next stage in the important role of the Royal Statistical Society in promoting statisticseducation.

Very important issues are raised in Section 13.3 to which I would like to add one more: the trainingof teachers. No matter how much we improve syllabuses and resources, statistics stands little chance ofengaging pupils unless it is delivered by enthusiastic and knowledgeable teachers. There is a need for bettertraining at both pre-service and in-service levels. The RSS Centre for Statistical Education together with


individual members of the Society run workshops and short courses, particularly in A-level statistics. Hereat Sheffield Hallam University we continue to run a post-graduate course, the content of which PeterHolmes has influenced in no small degree. Unfortunately all these courses are influencing such small num-bers of teachers that their effects are but a drop in the ocean. Alongside the very necessary developmentin syllabuses and assessment we need to capture the hearts of teachers and through them the pupils.

Gordon Skipworth .Edexcel, London/Although this paper contained much of interest, most of which could be accepted, there are several pointsto note about it. As is to be expected it is written in academic style from the viewpoint of the author andomits some important facts. Edexcel has been involved with A-level course work for many years. This hasentailed students carrying out individual pieces of course work on their own ‘real world’ topic—a termfavoured by the Qualifications and Curriculum Authority and specified in the assessment objectives thatare laid down in the specification. It is likely that the author has never seen a piece of good statistics coursework by candidates from this awarding body.

The remarks about the present state of statistics teaching and examining are idealistic and do not takeinto account the aptitude and abilities of the students or the time that is available to the teachers. Contextualissues are very difficult for students of General Certificate of Secondary Education andGeneral Certificateof Education statistics and although it is desirable to teach statistics in context it is not necessarily realistic.

Having quoted in the paper that ‘It enables people tomake decisions in the face of uncertainty’, studentsneed to be taught the techniques to enable them to make such decisions. They also need time to practicethe skills of decision-making. How long did it take the author and other professional statisticians to learnthese skills? Many years is the likely answer. Part of the ‘fun’ of statistics is this uncertainty and studentsfind it difficult at this stage of their lives to accept that there is not just one correct answer but that theremay be several acceptable answers. Evidence suggests that teachers do not have this time. It is doubtfulwhether all teachers would share the author’s views.

The idea of a Department of Statistics in a school is very idealistic. Perhaps a Department of Mathe-matics and Statistics with a statistician as Head and responsible for interdepartmental liaison would bemore realistic.

The evidence from the many thousands of candidates who sit the statistics examinations and submitcourse work for Edexcel is perhaps one of the milestones in the teaching of statistics over the past 50 years.

The author replied later, in writing, as follows.

I would like to thank all who have contributed to the debate in my paper and I am grateful for all theencouraging comments.

It is good to acknowledge, with Flavia Jolliffe, the active role taken by the Royal Statistical Society’s(RSS’s) Education Committee and the Education Section in developing the Associate Schools scheme andthe various workshops for teachers and pupils. I hope that this will continue under the new structure ofthe Education Strategy Group. She also reflects on the teaching of statistics in universities. This was notpart of my paper but it is an important topic for the future of the subject and profession.

John Bibby is correct in suggesting that my paper is an account rather than a history. He encourages meto address some of these issues. Harvey Goldstein makes an allied point when he describes the distinctionbetween the pre-1990 and post-1990 period—though I would date the change as starting several yearsearlier. I am not a historian and am not particularly competent to weigh up the effect of exogenous andsocietal factors. It is probably no coincidence, though, that the creative bottom-up development was inthe 1960s where it can be seen as part of general creativity and experimentation. Certainly the greater cen-tralization was part of the 1980s political programme of the Thatcher era, with suspicion of professionals,including all parts of education, and the move to greater central control under the guise of accountability.It is useful to compare the way that statistics in schools has developed in different countries. A snapshotof the position in Scotland in the late 1970s is in Barnett (1982) along with a reference to some of theother traditions that John Bibby mentions. A follow-up to this, for European countries, is given in Holmes(1994).

John Bibby also rightly says that we should learn from our mistakes and gives two examples from thepast. The failure to learn the implications that statistics is an interdisciplinary subject is picked up by sev-eral contributors. At school level this is both a structural and a personnel problem. I agree with GordonSkipworth that it is not practical to consider having departments of statistics in schools. The mathematicsdepartment is the obvious place to base someone with responsibility for ensuring appropriate teaching


across the curriculum of what is an essentially numerate subject. This idea of a statistical co-ordinator, alsoreferred to by Harvey Goldstein, was part of the report of the Cockcroft Committee (Cockcroft, 1982). Ifit is to happen in any school it needs support from the very heart of the school administration. The teamteaching solution of the Open University does not easily translate to schools. From this point of view themathematics department is not ideal; in most schools it is the department that is least involved with inter-departmental teaching. For example, although several contributors have rightly stressed the importanceof statistics as part of citizenship, people running these courses find it difficult to involve mathematicsdepartments. Interdisciplinarity is also a personnel problem in that there are very few teachers with asufficiently strong and broad background to appreciate the role of statistics across the curriculum and insociety and how all these can link together in a pupil’s complete educational experience. Neville Daviesreminds us of how useful a set of real data can be, as the CensusAtSchool data are, in encouraging teachersfrom different disciplines to look for real insights through statistical analysis. Such real data that originatewith the pupils also engage them more fully.

This brings us to the issue of teacher training, raised by Denise Lievesley, Harvey Goldstein and Mar-garet Rangecroft. It is crucial to the teaching and learning of the subject. Sheffield Hallam Universityhas a well-deserved reputation in this field, but major changes will only come if appropriate experiencesare incorporated into initial teacher training courses and in-service courses. Maybe we can learn from theUSA on the first of these. A joint working group of the American Statistical Association and the NationalCouncil of Teachers of Mathematics is currently working on proposals and material that should be incor-porated into all initial teacher training courses for mathematics teachers. We ought also to be willing tohelp with the programme of continuing professional development that is being proposed by the AdvisoryCommittee on Mathematics Education and accepted in principle by the Secretary of State for Education.Denise asks my opinion on the benefits of specialist teachers compared with statistical training of teachersof mathematics or other subjects. The more teachers with specialist statistical background we can put intoschools the better. But the reality is that very few specialist statisticians choose this career path, and thosewho do must be willing to teach another subject as well. I think that the best use of what resources we havewould most usefully be put into improving the statistical understanding of mathematics teachers. Thoseoutside the teaching profession can help. One model for this can be seen in the co-operation betweenPfizer and the Kent Local Education Authority on agreed development projects. Partnerships are usuallywelcomed by schools. Some RSS Local Groups are already involved in on-going co-operative ventureswith local schools. More of this would be helpful.

Toby Lewis and Dennis Lindley both raise the problem of the essential nature of statistics as a subject.Both remind us of the essential place of uncertainty at the heart of the statistical process and hence theuse of probability to measure this uncertainty. There is a large part of statistics for which this is true andit has been the major strand of school statistics in England since the 1960s. There may not be as muchprobability in the national curriculum as Dennis would like, but some is there. There are problems oflinking probability and data at the earliest stages, but it can be done as some of the booklets from theProject on Statistical Education show. The project itself included subjective probability as a strand andhad examples at the top level of using the simplest form of Bayes’s theorem in considering the practicalconsequences of the breathalyser test. It is this approach to statistics which is embodied in A-level mathe-matics and statistics courses. But, if we consider statistics as what statisticians do, there is much that doesnot come under this banner. In a recent editorial in RSS News, Ray Thomas introduced the two terms‘pstatistics’ and ‘mstatistics’ to distinguish (Thomas, 2003). The examples given by John Bibby at the endof his contribution to this debate require more thinking on what data to collect, how the data are to becollected and what inferences we can draw from these data about how they came to have the shape thatthey have, starting to identify some possible underlying causes and allowing for possible confounding.In many cases the data are not samples from which we draw inferences about the population—they arethe population. An example currently in the news is foundation hospitals. Newspaper headlines mademuch of the fact that the first group of foundation hospitals, supposedly the best, had higher death-ratesthan others that were not in the list. No reference was made to the possible confounding effect of bet-ter hospitals having to treat a larger proportion of very ill patients. There are statistical issues of whatto measure and the practical consequences in the system of choosing some measures rather than others.Of course there is also uncertainty in these figures, but the main concern is the interpretation of the leveland interrelationships rather than the inherent variability in collecting the data.

Gordon Skipworth reminds us of the large number of A-level projects that have been done for examiningboards over the years. I pointed out the pioneering role of the University of London Schools ExaminationBoard (one of Edexcel’s predecessors) in Section 8.1. At the same time that board also had an AO-level in


statistics with a compulsory question on interpreting in a context a table from official statistics. This wasa task that the candidates learned to do, which is evidence against some of the other points that he raises.Denise Lievesley and Chris du Feu also raise the issue of the influential nature of assessment. It can be adouble-edged sword and there are stultifying effects from teaching to the test. I agree with Chris that weneed to get assessment right if we are to encourage good learning and a correct appreciation of the impor-tance of the subject. With the current inquiry into mathematics teaching post 14 years and the possibilityof a move to a baccalaureate-type system it is probably not the right time to proceed with an RSS GeneralCertificate of Secondary Education in statistics to go along with the other RSS qualifications; but this isan idea that would be worth following up after things have settled down.

Gordon Skipworth accuses me of being idealistic in my remarks about the present state of statisticsteaching and examining. I disagree. Most of my remarks are factual. My own teaching and examiningexperiences, as well as communications with teachers, show that pupils and students enjoy and can discusscontextual issues and that this gives purpose to learning techniques. There are many places in the schoolcurriculum where students accept that there is more than one correct answer—discussion on politics, reli-gion, citizenship issues, lessons from history etc. It is of the essence of inference that there are alternativepossible answers. For contrary views to his, even within the discussion on my paper, I refer him to thecontributions of Bibby, Davies, Goldstein and du Feu.

References in the discussion

Barnett, V. (1982) (ed.) Teaching Statistics in Schools throughout the World. Voorburg: International StatisticalInstitute.

Bibby, J. (1972) Living Statistics: an Introductory Text for West African Students. London: Longman.Bibby, J. (1982) Teaching statistics to non-statisticians—Open University course. In Proc. 1st Int. Conf. TeachingStatistics (eds D. R. Grey, P. Holmes, V. Barnett and G.M. Constable), vol. 1, pp. 238–249. Sheffield: StatisticsTeaching Trust.

Bibby, J. (1986) HOTS—Notes towards a History of Teaching Statistics. Edinburgh: Bibby.Bibby, J. and Evans, J. (1978) Teaching Statistics at The Open University, and Other Problems. In Proc. NeueFormen des Lehrens und Lernens in Hochschulalltag. Berlin, 1977. Dortmund: Arbeitsgemeinschaft fur Hoch-schuldidaktik.

Cockcroft, W. H. (1982) Mathematics Counts: Report of the Committee of Enquiry into the Teaching of Mathe-matics. London: Her Majesty’s Stationery Office.

Connor,D. andDavies, N. (2002)An international resource for learning and teaching.Teachng Statist., 24, 59–61.Connor, D. and Holmes, P. (2002) Classroom and worksheet activities across the curriculum. Teachng Statist.,

24, 55–58.Cook, E. (1913) The Life of Florence Nightingale. London: Macmillan.Davies, N., Connor, D. and Spencer, N. M. (2003) An international project for the development of data handlingskills of teachers and pupils. J. Appl. Math. Decsn Sci., 5, 1–11.

Fisher, N. I. (2001) Crucial issues for statistics in the next two decades. Int. Statist. Rev., 69, 3–4.Holmes, P. (1994) Teaching statistics at school level in some European countries. In Proc. 1st Scientific Meet.International Association for Statistical Education (eds L. Brunelli and G. Cicchitelli), pp. 3–11. Perugia:University of Perugia.

Hoyles, C., Wolf, A., Molyneux-Hodgson, S. and Kent, P. (2002) Mathematical skills in the workplace.Report. Science, Technology and Mathematics Council and Institute of Education, London. (Available fromwww.stmc.org.uk/pdf/maths-skills-work-final.pdf.)

Kendall, M. G. (1972) Measurement in the study of society. In Man and the Social Sciences (ed. W. A. Roson).London: Allen and Unwin.

Loynes, R. M. (1987) (ed.) The Training of Statisticians round the World. Voorburg: International StatisticalInstitute.

Pearson, K. (1914) The Life, Letters and Labours of Francis Galton, vol. 1. Cambridge: Cambridge UniversityPress.



Rice, R., Burnhill, P., Wright, M. and Townsend, S. (2001) An enquiry into the use of numeric data in learn-ing and teaching. Report. University of Edinburgh Data Library, Edinburgh. (Available from http://datalib.ed.ac.uk/projects/datateach./html.)

Royal Statistical Society (2003) Post-14 mathematics inquiry: submission from the Royal Statistical Society.Royal Statistical Society, London.

Thomas, R. (2003) Pstatistics and mstatistics. RSS News, 30, no. 9, 1–3.

discussion on the paper by holmes

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