John TruranProbability & Statistics-A Tale of Uncertainty

AARE Conference, November 1995Page 1

Australian Association for Research in EducationAnnual Conference - Hobart, November 1995Probability and Statistics in Australian Secondary School Curricula Since the 1960s-a Tale of UncertaintyJohn M TruranDepartment of Education and Department of Pure Mathematics, University of AdelaideAbstractOf all the mathematical topics which entered the school curriculum as part of the "new maths" movement of the 1960s, only probability and statistics ("Chance & Data") remain and their place remains tenuous. This paper examines the forces which led to their introduction into secondary schools in Australia and the ways in which they have been taught, as well as assessing the likely outcome of recent moves which have tried to enshrine the topics in a mathematics curriculum appropriate for all.In the paper it is argued that the utilitarian and humanist reasons adduced for these topics in the northern hemisphere in the 1960s were of less importance in Australia than the pragmatic one that statistics was easily taught in schools with a comprehensive intake. Both topics, although essentially provisional in nature, have since then been presented to students in a deterministic manner. Recent curriculum statements and supporting materials have endeavoured to redress this balance, but it is argued that their neglect of mathematical background and excessively subjective approach make it uncertain that they will be adequately effective.The paper is based mainly on an analysis of curriculum documents and textbooks. It argues that the "colonial echo" model of curriculum development is an inadequate explanation of past events, and proposes a more comprehensive ecological model.IntroductionThe rule is, jam tomorrow and jam yesterday- but never Jam today.1The theme of this Conference is "Directions, Yesterday, Today, Tomorrow". For the teaching of Probability & Statistics in most Australian secondary schools "Yesterday" only started in the late 1960s and "Today" must refer to this testing time when schools try to come to terms with the new demands and philosophies of National Statements and Profiles. "Tomorrow" may already have begun; there is evidence that the topics are already on the wane. Which era managed to provide jam rather than dripping for the loaf labelled "probability & statistics" and how it managed to produce a sufficient to be able to afford such luxuries is the purpose of this paper.This paper is based on work being currently being prepared for a dissertation and is necessarily an abbreviation of the thesis being

developed. Furthermore, it is being prepared for a multi-disciplinary conference whose participants may not have detailed knowledge of mathematics education in Australia. An appropriate approach seems to be to sketch pertinent vignettes which will provide the non-specialist reader with some feeling for the issues and changes involved. References will allow an interested reader to compose a more detailed picture than that provided in the vignettes. I can only apologise if such a reader finds the number of references to my own work to be excessive; given the specialised nature of the topic and the brevity of the paper some such self-referencing is inevitable. Some material referred to are unpublished papers; copies of these are obtainable from the author.Explanatory Notes for the non-Specialist Reader

TerminologyThe term "probability & statistics" needs some explanation. In academic circles "probability" applies numbers to chance events and uses a formal axiomatic system to draw relevant conclusions. "Statistics" is the art and science of collecting and interpreting information. While the two topics are interrelated they remain distinct-probabilistic thinking is deductive, statistical thinking is inductive.2 Until recently the terms were also commonly used with much the same meanings in Australian schools.But since 1991, when the National Statement on Mathematics in Australian Schools 3 was produced the terms "chance" and "data" have come to replace "probability" and "statistics" respectively in schools. Given that they are the only terms use in the two documents Mathematics-Work Samples 4 and Mathematics-A Curriculum Profile for Australian Schools 5 which form the basis for most curriculum planning in Australia at the moment, we must assume that these terms are the received language of Today and will probably remain for Tomorrow. Whether the new terms are synonymous with their predecessors is a matter for debate; I have argued elsewhere that they represent an undesirable simplification.6The term "probability & statistics" is rather a mouthful; some people replace it by the generic term "stochastics".7 While this term is not part of the received language of Today, I shall use it in this paper both for brevity and because it seems to be becoming part of the received language of Tomorrow.The New MathematicsThe term 'new mathematics' has been used with many different meanings. Its most general and least emotive meaning, which will be used here, is simply the new material introduced into western culture schools from as early as 1951, but more commonly from the early 1960s. Its development in the USA and the UK was marked by the establishment of a number of distinct projects, some small, some large, which schools or school systems were free to accept or reject.Some of the new mathematics comprised new topics such as probability, statistics and non-Euclidean ways of approaching geometry. Some

re-examined old topics, emphasising precision of language, symbolism, logic and mathematical structure, with the idea of a 'set' being seen as a unifying feature across the whole subject. Attempts were made to form links between topics previously seen as separate. Some schemes, influenced by educational psychologists like Bruner, emphasised use of the discovery approach.8Of all the topics which entered school mathematics as part of the new mathematics only probability and statistics remain. This makes them particularly valuable topics to examine from the point of view of curriculum change and development.Mathematicians Disagree about Mathematics Non-mathematicians frequently believe mathematics to be a subject not bound by matters of opinion. It often comes as surprise to learn that mathematicians frequently differ in their approaches to problems and that sometimes they may disagree strongly about fundamental ideas.For example, many universities have separate departments of pure and applied mathematics. Pure mathematics deals with theoretical systems and is based on logical principles. Applied mathematics deals with finding theoretical systems which are a good fit with observed data. Both approaches have their value. However, because few people specialise in both fields, different mathematicians have different understandings of what aspects of mathematics are of most importance. Similar disagreements occur within stochastics. Statistical inference always has some degree of uncertainty involved. There is serious disagreement about what the nature of probability is. There is serious disagreement about what forms of probabilistic reasoning are valid. While there is little dispute about the procedures which might be

applied in both probability and statistics (the "pure" aspect of stochastics), there is sometimes significant disagreement about the most appropriate procedures to apply to individual situations (the "applied" aspect of stochastics).9These differences can be relevant to school curricula. Different educational systems tend to emphasise different aspects of mathematics in their secondary courses depending on whether a pure or applied approach is seen to be of greater importance by the leaders of curriculum development.Constructivism or Transmission?It is difficult to write about pedagogy in a way meaningful for some at a time when constructivist theories of learning are in the ascendant. It is difficult to write about "teaching a topic" at a time when some believe this is an inappropriate activity for schools. It is difficult to write about a body of knowledge when all knowledge is seen as relative. It is difficult to write about constructivism when the term embraces so many different views.10 Since what I want to write about is the teaching and learning of stochastics, I must declare a position at this point but without providing a detailed defence.We now know that all knowledge is provisional and we understand better than ever the limitations of "telling" as a form of communication. But

it remains true that western ways of thought have developed many valuable insights which have significant common meaning for many people, as well as proven predictive power and sufficient generality to suggest that they will remain of value in new circumstances. So this paper is written from the assumption that there is a body of knowledge in Western thought called stochastics which is able to be transmitted, and which is of sufficient importance to ordinary people in their everyday lives to make it worth transmitting. Indeed there is some evidence that those who do not share these insights are likely to be disadvantaged or exploited in their own lives. It will, I hope, become clear as I write that this is not some absolutist out-dated diatribe, but rather a flexible accommodation to the need to balance our mortal frames against our seemingly infinite capacity to explore.Theories of Curriculum Development in Australian Mathematics EducationBefore examining specific vignettes it is necessary to set a theoretical framework for this investigation. The full richness of modern analysis of curriculum development has by-passed Australian Mathematics Education. One theory predominates.The Colonial Echo TheoryClements, Ellerton & Grimison11 argue that a 'colonial echo' theory, sometimes conscious, sometimes unconscious, is sufficient to account for major events in mathematics education and for the overall uniformity of Australian schools until the mid-1970s and that it still had some validity in the late 1980s, although by then, they claim, a more mature approach had developed. This "colonial echo" influence is attributed to the assumption that mathematics is culture-free and to many structural and social links with Great Britain. They argue that this has led to school mathematics in Australia becoming both sexist and elitist. Furthermore, they assert that the School Mathematics Project (SMP) and the Nuffield Project, both from England, were the major inspirations for the introduction of the new mathematics into Australia, that even the criticisms of the new mathematics which arose in Australia were copies of criticisms being made overseas, and thatit was the naive willingness of Australian educators to accept untested English ideas in the area of school mathematics which persuaded education authorities, in each Australian state, to commit schools to large-scale, but ill-fated, reconstructions of their primary and secondary mathematics curricula, through the Cuisenaire and 'new Maths' movements.12In an earlier paper13 I have examined these claims and argued that their explanatory value is limited. At that stage I proposed "muddling

through" as a more satisfactory description of the forces acting on the teaching of stochastics in Australia. In the present paper I replace "muddling through" by a more rigorous ecological approach with better explanatory power.A Broad-Spectrum Ecological ModelCrombie has proposed that historical investigation needs to take place concurrently at several different levels.14 One level he calls

"historical ecology" which involves "the reconstruction of the physical and biomedical environment and of what people made of it."15 Another level involves examining the "cultural dispositions, habits, motives, opportunities and responses"16 of society and another involves examining the level of scientific thinking available within a society at a given time.Crombie sets historical ecology at the level of nature but his cultural and scientific levels may be seen as operating within a broader ecological framework. The components of these levels are forces which operate within society just as effectively as natural forces. Certainly, the effects and limitations of these forces are less well understood, and probably less predictable, but they remain part of the forces to which an organism or a society responds.The task of the historian is to interpret actions in terms of as wide a set of forces as is feasible and relevant. The position I take in this paper is that the forces operating on the teaching of stochastics may be better understood by considering all three of Crombie's levels united into what I call a "Broad-Spectrum Ecological Approach". Such a model sees society as an organism in which individual members respond to a variety of pressures in ways which are practicable, in the best interests of those individuals, and in many cases in the best interests of society as perceived by those individuals.The Colonial Echo theory belongs within Crombie's second level; it is concerned particularly with cultural dispositions. However, a broad-spectrum ecological approach looks for a wider range of explanations than may be found when working with only cultural evidence.Crombie's three categories suggest some of the other explanations which may become available. But his categories also necessitate seeing the evidence from wider chronological and geographical perspectives than a simple cultural approach requires. Ecologists distinguish "ultimate", "proximate" and "artificial" factors when interpreting biological development.17 For example, breeding behaviour may be described as being brought on by a proximate factor such as increasing hours of daylight or by an ultimate factor such as the desirability for the species of producing young when food and shelter are most likely to be abundant. Proximate causes like daylight may of course be artificially manipulated for animals held in captivity. These terms have been developed for the study of non-thinking organisms and within the contexts of an evolutionary time-scale, but their underlying principles may well have wider applications. Of course, in a study of human history proximate forces may be much more determined by human thought than within classical ecology. Indeed the distinction between proximate and artificial factors may be very difficult to define. However, ultimate forces will tend not to be influenced by human thought and will be similar to those in classical ecology.Why did Stochastics enter the western school curriculum when it did?Within the world today there is a substantial number of competing world views. One of the areas where these world views are most in disagreement is in interpreting the nature of chance events. Hacking18

has described in detail the slow development of the Western scientific interpretation of chance which has taken place mainly over the last five hundred years. Only recently has this interpretation been widely considered suitable for inclusion in school courses for all. A critical question to ask then is why the topic of stochastics has entered the

curriculum at this time.In this paper I examine the issue mainly from a South Australian perspective, but set this regional experience into the wider Australian and international scene. Such an approach may be seen to lack generality. But equally, such an approach may be particularly useful in producing rigorous counter-examples to rebut the over-generalisations of others. If the proximate forces operating within a restricted environment need to be set against the ultimate wider forces then a narrow focus of attention may be appropriate. Since this paper is concerned with Yesterday, Today and Tomorrow it is necessary to look further than merely the early days of the teaching of stochastics. It is necessary to see how well the topic has been received within schools and to assess its viability. So the vignettes I have chosen to illustrate the teaching of stochastics within Australian schools will be grouped under three general headings: Why is Stochastics in the Curriculum?, What is Taught?, and How is it Taught? The paper will conclude by using the answers to these questions as a basis for some crystal-gazing.Why is stochastics in the Australian Curriculum?In the 1960s some changes in Australian mathematics curricula became essential. In primary schools a governmental decision to convert to a decimal currency released large amounts of mathematics time previously devoted to the delights and complexities of British Imperial currency. In secondary schools economic and social circumstances were leading to an expanded and much more comprehensive system, one increasingly suffering from overcrowded schools and a significant shortage of teachers, some trained, some definitely not. The academic curriculum appropriate to secondary education for the intellectually able had, at the very least, to be supplemented by less abstract work. There was also an increasing perceived need for schools to act as significant agents for children's social development and understanding. So new sources of unallocated time and changing social pressures on schools were two obvious proximate forces operating for change. The first of these reasons was, at that time, peculiar to Australia.In 1964 a small national conference was convened by the Australian Council for Educational Research (ACER) to make suggestions about appropriate changes to primary school curricula in the light of the forthcoming change to decimal currency. In 1965 a larger national conference was held under the auspices of United Nations Educational Scientific and Cultural Organisation (UNESCO) to allow secondary and tertiary teachers to reflect on the implications of the new mathematics for secondary schools.Both conferences recommended the teaching of stochastics as appropriate

for schoolchildren. So they may be seen as the symbolic beginning of the "Yesterday" for stochastics teaching in Australia, although some teaching and formal discussions had occurred before that time and certainly influenced the findings. Both conferences tried to provide a comprehensive coverage of what was being done elsewhere in the world. For stochastics at least, such overseas practice was very varied.Two approaches developed in the United States of America (USA). Both were supported by carefully articulated justifications for teaching the topic. Both saw probability as having precedence over statistics. One was a pure approach based on axiomatic thinking,19 the other was an applied approach which was viewed as a critical component of a liberal education for all.20 The British approach was more pragmatic. Initially statistics was seen as more important than probability. Only later was probability seen as useful because of its logical structure,21 and, later still, because of its wide application, one 'that always goes down well in the classroom'.22 In Scotland, with a much higher overall standard of secondary teachers, an entire examinable subject of Probability and Statistics was made available in the final year of secondary schooling.23 Finally, within Continental Europe probability

and statistics were seen as the sciences which completed algebra and topology.24Communication of ideas over long distances at that time was not a serious problem. Australian mathematics educators already knew that there were many models which they might follow or adapt. They knew a lot about these models, though few knew a lot about many of them. In particular, the European approach, initially not available in English, was largely disregarded. Why did Australian educators include stochastics among their recommended new courses? The reasons were poorly articulated and lacked the depth of thinking to be found in overseas projects. Such reasons as were given differed significantly for statistics and for probability.Statistics was by far the more popular. It was practical and concrete. The UNESCO conference was a significant step in the transfer of authority over the syllabus from administrators and academics to classroom teachers. Participants were particularly concerned that syllabuses should be suitable for successful presentation by 'all teachers to all children'.25 They expressed 'general support for the inclusion of probability and statistics] and felt that work could be given at all levels'.26 This opinion seems to have been based on some successful experience of teaching descriptive statistics to less able classes but on little more. Discussion was minimal and superficial.27 The discussion about stochastics in the primary syllabus at the ACER Conference was also minimal. Probability was advocated as an appropriate topic, but without supporting reasons.28 Initial post-conference public documents did not mention probability.29 Yet the official post-conference book, Background in Mathematics ,30 published in 1966, contained clearly articulated reasons for the teaching of probability, reasons which drew heavily on overseas ideas.31 By arguing

that statistics was "the technological brother of probability'32 the links between the two topics were made especially clear. In South Australia, however, the book had little influence: "teachers found it difficult to read'.33There is some evidence from present-day reminiscences that members of the conference saw probability as more important than the initial published documents would suggest. The ACER sponsored a writing team lead by Mr John Izard who produced a card scheme entitled Individual Mathematics Programme to illustrate how the 1964 recommendations might be implemented.34 Although Izard's recollections, made in 1993, are that probability was included, this was not so. Similarly, Professor Philip Hughes, who prepared material for Tasmanian use soon after the conference, has recalled that probability was included in that course, this also does not appear to be the case. In both cases statistics, however, was included, probably because it was concrete and practical.

So it is reasonable to claim that in the mid-1960s Australian educators looked at what changes they might effect and decided that statistics was a potentially workable topic, but were ambivalent about probability. Their reasons were poorly articulated, but tended to be pragmatic. They were certainly not slavish copies of what was happening overseas. Of course, it would have been almost impossible and certainly irresponsible for Australian mathematics syllabuses to have developed in vacuo but the forces acting were very much proximate and local. I argue that ecological forces were of more importance than the cultural ones which Clements et al. have adduced as the guiding principle. Examination what happened in South Australian secondary schools can provide evidence for this argument.The 1965 UNESCO conference provided a focus for the Australian thinking about change which had been being going on for some years.35 South Australian mathematics educators returned from Sydney inspired to effect change and to do so quickly. It was decided in February 1965 to introduce changes on a trial basis immediately.36 This was a change

driven by educators, not by the change of political leadership which occurred soon after. Indeed, it has been claimed that South Australia was the only state in the 1960s which did not introduce the new mathematics by decree.37

The educators wanted to effect change quickly and responsibly. They needed to persuade a creaking Department of Education be persuaded that their plans were, at the least, politically defensible? They decided to use Books 1 - 5 of the English SMP. Was this a colonial echo?The answer must be "No". Facilities for curriculum development in South Australia were at that time pitifully limited. The state had been almost the last to appoint a curriculum officer, and his responsibilities covered both primary and secondary schools. Two earlier recent experiments in mathematics innovation had had sorry histories. One very small project in one secondary school had been

aborted by departmental fiat. 38 One large, high-profile, primary project led by Dr Zoltan Dienes had had a very difficult gestation.39 A trial based on an established course would require less initial resources, was more likely to be politically acceptable, and was likely to have less teething problems than a trial prepared in haste using local resources. The purchase of an established course was almost the only way ahead.But perhaps it might be argued that the purchase of an English text, in preference to an American one, represented a form of colonial echo. There are two reasons for answering "No" to this suggestion as well. Up to 1965 American new mathematics projects were much better known in Australia than British ones. Indeed, there were far more of them. Several had been tried and none had won overwhelming Australian support.40 Both the best American and British material were able to be examined at the UNESCO conference. SMP was believed by many at that conference to be the best material available. This judgement can be shown today to be sound.Most of the material produced by most of the new mathematics projects at that time is collected today in one place in the Library of the University of Bielefeld, Germany. I have examined this collection in its entirety and would support the assertion that SMP was the best course available at the time. Not only was it mathematically sound, but it broke new ground in lively and attractive presentation. It was practically oriented and relatively informal. An added advantage in 1965 was that it was supported in Australia by a publisher who was prepared to work very hard to facilitate its introduction into schools. Even today it is considered by some that SMP was 'way ahead of its time when it was written and is still way ahead of its time. Its writers were perceptive and wrote mathematics books, not time fillers.'41The forces which led to the adoption of SMP were in my view primarily proximate and ecological rather than proximate and cultural. SMP appeared to be the most energy-effective way of introducing sound new ideas without needing extensive time for preparation and trialing. This choice committed South Australia to a course which would contain both statistics and probability. But the decision to include stochastics was a decision by default, rather than by design.It was not long before SMP was found to be unsuitable for Australian comprehensive high schools, and it locally produced materials with much the same subject matter but a rather different approach soon replaced it.42 Planning for a structure of public examinations began at about this time and by 1969 a new mathematics syllabus for public examination at Year 11 was offered which included a pure mathematics approach to probability and some algebra work on included permutations and combinations which it was felt might be applied to probability.43 In 1970 a new Year 12 syllabus ensured that this approach to probability would be seen as an important part of secondary mathematics education for many years to come.44 Statistics, however, was not included at all.

Whatever the reasons for these changes, they did not found their way

into public documents. The decision to teach probability but not statistics in Year 12 must have been a conscious decision. Other decisions were being made in other states, and were known about in South Australia.45 Why was this decision made? No definitive answer can be given, but there is some evidence that E.S. Barnes, professor of Pure Mathematics at the University of Adelaide was particularly influential. He had helped to found the Mathematical Association of South Australia in 1959, chaired the meetings which followed the post-UNESCO conference, and helped to develop the new syllabus for Years 11 and 12. He must have been influential in introducing probability into Mathematics I syllabus at the University of Adelaide in 1969, well ahead of other local tertiary institutions.46 Barnes was a pure mathematician; the South Australian approach to stochastics was decidedly a pure one; in the absence of other evidence it seems reasonable to hypothesise that his influence was crucial. It is clear that the introduction of probability at least was done with little formal or public articulations of suitable reasons. At the upper secondary level the introduction was effected without either prescription or recommendation of a suitable textbook.47 The typical upper secondary mathematics teacher at that time would never have studied formally either probability or statistics, though he or she may have encountered applied statistics in subjects like psychology. Does this mean that the introduction was a mere colonial echo, a mere mirror of what was happening in other parts of the world?48 Connell has talked about English education as being a "selective sieve" for transmitting European ideas to Australia.49 This argument emphasises the cultural forces affecting educational change. But there were others. One was scientific. Mathematical knowledge had developed throughout the world. Barnes, as Professor of Pure Mathematics, had been trained to know where world, rather than merely British thinking had reached. It was his job to ensure that South Australia did not lag behind. The sieve which mediated Barnes' influence was the pure mathematics sieve of scientific knowledge, rather than the cultural one.At the level of scientific knowledge there were very few forces operating in South Australia. There were very few who were interested in school education and with sufficient background knowledge to make judgements about the best mathematics to teach. No textbooks were readily available for the upper secondary school. Barnes was an able academic, a good teacher, amiable, and hard-working. The state did not have the resources to contemplate seriously alternatives to what he was proposing.So I argue here that in South Australia junior secondary schools stochastics was introduced largely by default. In upper secondary schools the influence of one man, a pure mathematician, meant that only the formal aspects of probability were studied. But in both cases the decisions made can be much better understood when limitations of resources and the state of knowledge in society are considered along with cultural influences which may have had some glimmer of colonial echo, but no more than that.

What is Taught in School Stochastics classes?YesterdayBarcan has argued that the new curriculum of the 1970s was "antipathetic to the liberal-humanist tradition … and the development in pupils of an organized mastery of knowledge".50 The previous section has shown that the introduction of stochastics into schools was dominated by academics; there is no evidence that the syllabuses developed then were antipathetic to either liberal-humanism or structured knowledge.It is true that the expansion of universities and the proliferation of teachers' colleges in the 1960s produced new founts of wisdom and

authority and opportunities for the changing of traditional structures. The reduction in the number of public examinations provided greater freedom for teachers to develop their own courses. For the first time, mathematics teachers were called on to address, no longer merely issues of "How to teach it", but rather issues of "What to teach, and how to teach it."51 There is little evidence that they did so in any significant way. What was taught may be fairly easily deduced from syllabuses and commonly used textbooks.This is particularly easy to prove in the case of probability in South Australia. For many years the most commonly used textbook contained an incorrect definition of statistical independence which was, moreover, highlighted as something to be taught. The examiners decided that the time had come to scotch this furphy52 and set a question to establish how many schools were teaching the incorrect material. Not only did they establish that many schools were blindly copying the textbook, but they encountered significant opposition from teachers who claimed that marks should not be deducted for reproducing material in a textbook, even if it were mathematically wrong.53So what was taught may easily established by considering textbooks. Two are considered here, both representative of what was commonly used. Clapp et al . (1969) is an example for Year 11 from the first wave of locally produced textbooks and Haese et al. (1984) is an example for Year 10 from the second wave produced about 10 years later. Both deal with probability and statistics in separate chapters, and draw few links between them. Those links which are drawn are technical, not interpretative.Statistics is treated mainly descriptively as a set of techniques to be learned. Clapp et al. do ask some questions which require some critical interpretation of the data. Haese et al. place statistics at the end of the book and include no interpretation.54 In both books probability is also treated in a rigid, deterministic way. Axioms are defined and a calculus of probabilities is developed. Few connections are drawn with real-life events. The influence of the Year 12 publicly examined syllabus may be clearly seen.Blakers, an academic with a strong influence on school mathematics, has observed thata] substantial infusion of old-fashioned Australian conservatism has

resulted in syllabuses and teaching approaches which deviate considerably less from the earlier norms than do many of the overseas counterparts. This has made it relatively easy to accommodate to recent Australian criticisms, some of which have merely been echoes of overseas critics and not always relevant to the Australian situation.55Much more evidence could be adduced, but this will suffice. We may assert that in South Australia probability and statistics were taught as essentially disjoint topics in a deterministic way, with an emphasis on pure mathematics rather than on relevant applications. The content remained liberal-humanist; the practice in general merely dull, certainly not anti-intellectual.The broad-based ecological model is helpful in explaining what happened. In general it was simply beyond the capacities of the teachers to produce markedly original work. "Capacity", of course, is used in its widest sense. Time is difficult to find in any school, and a teacher would be foolish to expend large amounts of time in developing an idea unless there were a good chance that it could be used. Since both secondary and primary schools were dominated by external criteria significant curriculum innovation was unlikely to develop within the ranks. Nor were the ranks sufficiently au fait with the development of mathematical thought to be leaders in bringing this though into the classroom. Lack of resources and limitations of knowledge were important constraints on classroom practice. Ideological issues or the more mature development of the teaching profession were decidedly secondary.

TodayThe issue of what is taught Today is, however, more complex. There have been significant changes in the way school syllabuses have developed. There has been a move to competency-based specifications, rather than knowledge-based ones. There has also been a widening of the mathematics curriculum and stochastics is now officially seen as a desirable activity throughout the primary and secondary school curriculum.The reasons for the increased important of stochastics have been clearly articulated,56 and are similar to the reasons developed in the U.S.A. during the new mathematics movement. The growth in scientific knowledge within the country is starting to have an effect. However, the topic remains unfamiliar to primary school teachers and the competency-based approach is extremely complex and requires the development of many new skills by teachers. Only anecdotal evidence is available for the success of the changes and it seems fair to return a verdict that success is "not proven" at this stage.Neither the competency-based specifications for stochastics in Mathematics-A Curriculum Profile for Australian Schools 57 not the model examples in Mathematics-Work Samples 58 fit easily with academic constructions of the topic.59 There has been a reduction in the content required for secondary students and the links between probability and statistics are kept at the intuitive level.Interest, relevance, equity and success have been the guiding lights

for the changes in the curriculum.60 The ecological forces have had a very strong influence indeed. The influence of scientific thinking has been much reduced. This is of special importance for stochastics because of its indeterminate nature. In the difficult school environment of today a situation where a new, atypical, poorly understood topic is ordered to be taught to all is almost certain to ensure that ecological forces are likely to dominate over cultural or scientific thinking ones.Efforts have been made to provide good models for stochastics teaching and will be discussed below. What is actually happening at the moment in classes is not well documented. What was happening, however, is fairly well known and forms the basis of the final part of this discussion.How is Stochastics Taught?YesterdayThe few published comments relevant to this question would tend to suggest that the answer must be "badly". One quotation from examiners' comments will suffice-"Very poorly done by most. Many answers could only be described as wild lunges at the problem."61 Barcan's observation quoted above about the changes of the 1970s being antipathetic to liberal humanism and organised knowledge may have some validity when applied to the teaching of stochastics even if not to its content .Teaching, especially in secondary schools, is for the most part a very private activity. Sometimes researchers are given the privilege of observing lessons in a systematic way. But researchers' interests tend to be sociological or managerial, rather than pedagogical. In general the conclusions reached have been that mathematics classrooms are authoritarian teacher-centred places where "mathematics is presented as a collection of facts and procedures".62 In general, mathematics educators, particularly those who work from a constructivist perspective, deplore this situation. Vinner, however, has argued that "the traditional classroom … may be the cause of the poor practice that we all denounce, but it may well also be the result of very serious psychological, social and economic reasons."63 Vinner suspects that "some essential elements of the human nature are the cause of current mathematics education practice"64 and doubts that they can be overcome. These remarks, which to my knowledge have not been taken up by others, are of special importance for the teaching of stochastics for two

reasons.Firstly, it is often asserted that teachers teach as they were taught. This may well be true, but it has mainly anecdotal support. Even so, it is of little help for determining how stochastics is taught in schools because few secondary teachers in the 1960s had been taught any stochastics at all and few primary teachers in the 1990s were taught stochastics when they were in primary schools.Secondly, stochastics is an essentially non-deterministic subject. If it is true that the traditional mathematics classroom which emphasises

authority and procedures is so common because of ecological forces founded in human psychology, then the topic is doomed either to failure or emasculation. The latter, it is being argued here, has already happened.In general, the only evidence available to us is the evidence of textbooks. These books probably mirror the practice of teachers of more than average ability. I have elsewhere summarised the approaches of a number of commonly used secondary and tertiary texts.65 These books all follow to a large extent the formal axiomatic approach to probability, and do not include the textbooks of "Today" discussed above. Two generalisations may be made.The first is that there is no consistent approach. Although secondary texts have been being developed over the last thirty years by many writers, usually with one or two revisions, no common way seems to have evolved in the way that occurred with algebra texts in the first part of this century. This is itself would not necessarily be a bad thing, were it not for the importance of the second generalisation. This second generalisation is that most textbooks contain errors of mathematics. Basic principles are skated over, probably with the intention of making the ideas simple, but at the cost of accuracy and, I suspect, clarity. This is not the place to discuss the mathematical errors and fudgings. What is relevant here is how such a situation has developed, and has been allowed to develop, within formal structures which are presumably established to try to eliminate such aberrations.The answer seems to lie in the many difficulties involved in spreading new ideas quickly and widely through a population. We have already seen that changes in South Australia occurred rapidly. The textbooks which were purchased proved unsatisfactory. Locally produced books had to be written in a hurry. There was no time for the authors to develop deep understanding of the topic. The peer review system used by publishers for establishing quality was limited by the limitations of the reviewers. Once the books had been written they acquired a status beyond their achievements. This status tended to be retained even when the weaknesses of the books became apparent. In some cases their authors acquired an authority within the system which may also have helped to maintain that status. New generations of writers tended to work within the approaches set by earlier works.A relatively cheap textbook with abundant exercises proved to be the most efficient way available for communicating new ideas to teachers and students. The critical issues were price and ease of use in the classroom. These are ecological factors, and they are proximate ones. If Vinner is correct, then they may also be ultimate ones. Within stochastics one experiment is currently going on to test this hypothesis.TodaySince the 1970s a small but growing number of mathematics educators have been concerned to effect a radical change in classroom practice. They have attracted large funds for the production of materials suggesting rich activities which will broaden children's understanding of the scope of mathematics and engage them in a wider way than usually

happens in formal academic studies. The activities have been extensively tested within classrooms and found to work. They are carefully structured, with substantial advice to the teacher about how

to conduct the suggested lessons. They emphasise the use of activities, group work, group discussion, and the answering of questions posed by the children. At the same time they remain closely related to traditional content and are moderately teacher-centred. The most extensive materials which covered most of the supper primary and junior secondary curriculum, were produced in 1988 under the title of the Mathematics Curriculum and Teaching Project (MCTP).66 So this approach was easily available before the publication of the competency-based courses. The relationship between these courses and the new teaching material is an issue of its own, and one which has not yet been examined in detail.When the National Statement declared stochastics to be a significant part of the R - 12 curriculum the MCTP authors were well placed to develop appropriate materials in the same vein to assist teachers. Two volumes entitled Chance & Data-Investigations 67 and one entitled Chance & Data-Exploring Real Data 68 were produced. All volumes encouraged an open-ended non-deterministic approach, but they tended to under-play formal understanding of probabilistic ideas. One other book was produced under the auspices of the Australian Association of Mathematics Teachers to provide a specific basis for the in-service training of teachers.69There is very little evidence available to determine how effective these books and their associated materials have been in encouraging the better teaching and learning of stochastics. The special situation in which stochastics finds itself makes this an ideal opportunity to test further Vinner's hypothesis that there are inbuilt psychological reasons for classrooms being as they are and my hypothesis that lasting change is mostly dominated by proximate ecological factors. Tomorrow we may know whether there really was jam today, rather than dripping. But to some extent Tomorrow is already here, so some concluding remarks on this topic are appropriate."Tomorrow"?Whether "Tomorrow" is an appropriate topic for a formal paper to a conference on educational research is a debatable matter. However, if the principles of a broad-based ecological approach are to be of more than academic interest, then they ought to be robust enough to make sensible suggestions about the future. Two will suffice; one is concerned with upper secondary teaching, the other with primary teaching.Stochastics on the Decline in Upper Secondary SchoolsThe story to be reported here is a specifically South Australian story. It has not, to my knowledge, been replicated in other states. The "Topsyist" introduction of probability to the secondary syllabus in South Australia has already been discussed. But By 1985 official documents were claiming that probability was 'a topic essential in

interpreting the results of scientific research and handling much media information' and which would be of value in real life because a student would know how to 'apply it] to problems involving chance events'.70 Students preparing to attend university sat for either Mathematics 1 and 2 or Mathematics 1S, both of which contained a significant study of counting and probability, but no statistics.The topic was not popular. A survey revealed that teachers felt that the survey revealed that Counting & Probability was least enjoyable to teach and very difficult for students. These views were most strongly held by those teaching Mathematics 1S who were more likely to be the teachers less well qualified to teach the topic. "Less well qualified" is perhaps an understatement. One out of six teachers of Year 12 mathematics had only one or two years of study of tertiary mathematics. About 10% of the teachers were teaching the subject because there was not other teacher available.71In the late 1980s the increasing emphasis on applied mathematics in schools led to moves to incorporate a significant amount of statistics

into Year 12 academic mathematics courses. There was significant opposition from some tertiary mathematicians who claimed that statistics was 'not simply a branch of mathematics but a quite separate subject concerned with the logical interpretation of data'72 or because of concern about the lack of confidence in probabilistic ideas which Year 12 students held at that time.73 Others held the view that 'a satisfactory introduction to statistical inference, including a discussion of sampling distributions, can be based almost entirely on the analysis of data sets'.74 This difference of opinion among authorities arose to some extent from different emphases placed on the balance of pure and applied approaches, on new ways of interpreting data which were being developed, and on the views of the purposes of a Year 12 examination in mathematics.In 1989 the relevant curriculum committee of the examining authority proposed that Mathematics 1 and 2 remain in essentially in the same form with counting retained but probability and one other topic omitted 'to allow a more thorough study of a smaller number of topics and an increased range of associated applications'75 and that a new subject, Mathematics 3, be established, 'designed for students wishing to take an applied study in mathematics'76 containing significant study of both probability and statistics.77 It was intended that all three subjects should be 'Publicly Examined Subjects' acceptable to the universities when allocating university places. But until December 1995 Mathematics 3 was unacceptable to two of South Australia's three universities. Their refusal to accept the subject has led to acrimonious debates and an unsuccessful attempt to mount a challenge to their decision in the courts. From January 1996 the subject will be a "Publicly Assessed Subject" acceptable only to the younger two of the three universities, both of whom are experiencing a serious shortage of mathematics and science students.So in 1991, just when the National Statement 78 was creating a special

niche for stochastics throughout the R - 12 curriculum, South Australia removed probability from its year 12 academic curriculum, leaving behind which had been justified back in 1969 as a topic which could be applied to probability. There is some anecdotal evidence that there has been a flow-down effect to junior secondary classes as a result of this decision. Statistics has remained outside the formal studies of the state's upper secondary academic students. The course which has been devised to make these topics available has become a second-rank course which will only be taken by those who do not aspire to attend the University of Adelaide and who do not aspire to be serious mathematical scientists. Stochastics has become a low-prestige subject. What were the forces which led to this decline?Strong arguments for stochastics have been adduced by both pure and applied mathematicians and by practising teachers. But they have not all been talking about the same thing. The division has not been between the teachers and the academics but between, to use rather over-generalised terms, the formalists and the constructivists. Both argue that their approach is valuable, that it has, to use ecological terms, high survival value. If the argument of this paper is correct, neither is related strongly to the criteria for survival which are perceived by classroom teachers. Examiners' Reports for Mathematics 3 contain comments like "There were few very good students who presented for this examination", "in the statistics section, data analysis was handled well, but the probability and inference showed weaknesses", and "candidates were under-prepared in probability".79 The issues have not been resolved by the production of a new course-statistics remains popular, the non-deterministic issues of probability and inference remain much less so. The ecological forces of the reality of the classroom do not seem to have changed.Primary School "Numeracy"A recent development in Australian Education has been some form of

Basic Skills Testing in numeracy among primary school children, usually in Years 3 and 5. The first South Australian tests, held in 1995, were prepared by the New South Wales Department of School Education in association with the South Australian Department of Education and Children's Services. Of the 32 "Aspects of Numeracy" questions administered to Year 3 students, one involved the interpretation of graphical data.80 This question was repeated in the Year 5 test, whose 48 questions contained three other questions concerned with the interpretation of graphs, one question on permutations, and one totally wrong question on the interpretation of tabular information.81 No questions were asked at either level on chance events in any form.It is not unreasonable to deduce from this that the official view in at least two Australian states is that an understanding of chance events is not seen as a critical component of numeracy appropriate to all children. This view conflicts markedly with the view propounded in the official curriculum documents. Two examples will suffice.Misconceptions about chance processes are widespread. Many become

established while children are still quite young and are then difficult to overcome. Therefore chance activities should be provided in schools from the earliest stages in order to help students develop more inclusive conceptions.82The achievement of confidence and competence to make sense of and interpret data which have been collected, organised, summarised and represented by others is a major goal for this strand. If they are not to be subject to the kind of exploitation implied by the expression 'you can prove anything with statistics', students need experience in judging the quality and appropriateness of data collection and presentation for answering the questions at hand.83I would argue that the reason for this dichotomy lies more in ultimate forces than in proximate ones. As I wrote at the beginning of this paper, "non-mathematicians frequently believe mathematics to be a subject not bound by matters of opinion". Mathematics is seen to be a black and white subject, free of ambiguities. To the extent that 2 plus 3 is always 5, this is usually true, but there is far more than number truths like this to mathematics. It is also necessary to decide whether addition is an appropriate operation to relate a group of 2 and a group of 3. In most cases this issue could be resolved unambiguously. Choosing the appropriate operation would probably seen by ordinary people as being one facet of numeracy, at least once its importance had been pointed out to them.But statistical thinking is by its very nature indeterminate and ambiguous. Constructing and reading graphs may be determinate, but assessing their meaning always has some element of doubt involved. Calculating probabilities and using tree diagrams may be determinate, but applying these to real life situations inevitably introduces real elements of opinion which cannot be eliminated. Stochastics is a topic where ambiguity is inherent. Of course, this is true in all modelling situations-in physics, for example, equations of motions are presented as absolutes, when everyone knows that they represent a simplified model. The model is sufficiently precise for its imprecision to be unimportant. Such cannot be the case in stochastics.I submit that it is because ambiguous results are not yet seen to be part of the domain of mathematics that the interpretative aspects of stochastics are not seen by our society to be part of the complete education of every child. Graphs are sufficiently ubiquitous and relevant for an understanding of their technical aspects to be seen as important so this aspect of statistics remains part of "education for all". Probability is much more complex, less immediately applicable, and very value-laden, so this is simply neglected.One example, taken from a Minister for Education, must suffice here.I also note in your letter that you believe the new syllabus "encourages children to see maths as a creative project where, in some

situations, answers are not right or wrong, and where there may be several good answers to a question". As a maths graduate myself, I have to say that is not my attitude to maths. I therefore see no conflict

with the introduction of Basic Skills Tests.84Are the forces which influence this position ecological, cultural or scientific? Are they influenced by "colonial echo" or, indeed, any other form of uncritical copying?The forces are clearly not scientific. While there are aspects of stochastics which are still grounds for heated debate, most basic principles of stochastic thinking are well developed and well understood. The relevance of at least some of these principles to everyday life is generally acknowledged. Current scientific thinking has made stochastic ideas potentially accessible to all.The forces are not significantly external. Earlier in his letter the Minister acknowledged his determination to avoid the mistakes which had been made when trying to implement national testing in England & Wales. My judgement of the Minister and his advisers, which is based on moderate to extensive personal knowledge, is that they are making decisions because they see them to be right, not because they have been previously made elsewhere. Such decisions will usually be informed by the experience of others, but it will not be directed by them.It would be possible to argue that the forces are cultural. There is little doubt that most people's memories of mathematics classrooms are of ticks or crosses for right or wrong answers. Most people have had significant exposure to mathematics, many have not enjoyed it, they have constructed mathematics as absolute, and this is what has passed into the culture. It is also true that many potentially creative topics in mathematics are frequently treated in an absolute way. That mathematics can be seen as absolute and non-creative by a responsible and more than competent mathematics graduate is strong evidence that such a view is widely held in society. I would argue that this view is all right as far as it goes, but is somewhat superficial.I would argue that we need to ask why our culture sees mathematics as primarily deterministic, whereas is does not see other basic topics like English in the same way. Within our culture there is a much higher expectation that mathematics is a tool for doing things, rather than a language for examining ideas. Furthermore, the precision required of mathematics is much higher than that required for language. The shop assistant who does not add up correctly will breed bad feelings and will be bad for business. The fact that he or she has limited English need not be a handicap to sustained business success.There are certainly cultural forces influencing our mathematics curriculum. They are based on a limited understanding of mathematics but are nevertheless very real. They arise, not from any colonial echo but from the way in which mathematics is actually used by most people in our society. This dictates to some extent the amount of national resources which they are prepared to put into mathematics education. Unless a strong case can be made for stochastics education for all having high overall survival value, it is unlikely to remain in the syllabus. ConclusionStochastics is a very special and very different topic when compared with others traditionally taught in schools. These differences mean

that less obvious forces operating within education are more likely to be revealed when examining the teaching of stochastics than when looking at the curriculum in a more general way.This paper represents a preliminary attempt to assess the value of a broad-based ecological approach for examining curriculum forces. It provides a richer environment than the colonial echo model, and allows for more detailed assessment of the balance between ecological, cultural and intellectual forces. For stochastics the dominant force appears to have been ecological although much of the rhetoric has been

at the cultural and intellectual level. Recent attempts to change this balance await assessment.For statistics there has sometimes been jam, for probability it has usually been dripping and sometimes dry bread. A sandwich would distribute the jam further, but this product has not yet proved popular. Not only are the topics non-deterministic, but so is their place in the curriculum-this really has been a tale of uncertainty. AcknowledgementsI must thank Professor Philip Hughes, University of Tasmania, Mr John Izard, ACER, Professor John Keeves, Flinders University of South Australia, for allowing me to talk with them about their work. Many libraries and their staffs have helped, especially the Institut für Mathematik, University of Bielefeld. , Members of the Mathematics Education Research Group of Australasia, especially those in the Probability Special Interest Group, have made comments and Dr Andy Begg, University of Waikato, New Zealand has provided hospitality, insights, and access to his department's specialist library. Within the University of Adelaide I thank the staff of the Barr Smith Library, my supervisors, Mr Ian Brice and Associate Professor Paul Scott, for their support, encouragement, comments and patience, Emeritus Professor Alan James, for his clarifications of the complex, and Mrs Kath Truran for reading and commenting on earlier drafts of this paper.ReferencesAbrahamson B. (1969) "Tertiary Mathematics Course in South Australia-First-year Mathematics at Flinders University" The S.A. Mathematics Teacher 1 (1): no paginationAustralian Council for Educational Research (1964) Primary School Mathematics. Report of a Conference of Curriculum Officers of State Education Departments, held in Melbourne 16th - 20th March 1964 to Consider a Desirable Course of Study in Mathematics for Australian Primary Schools, The Manner of Introduction of Decimal Currency into the Course of Study, and Related Matters such as the Place and Value of Structural Aids Hawthorn, Victoria: ACERAustralian Council for Educational Research (1966) Background in MathematicsæA Guide-book to Elementary Mathematics for Teachers in Primary Schools Education Department of Victoria and the ACERAustralian Council for Educational Research (1972) Background in Mathematics æ A Guide-book to Elementary Mathematics for Teachers in Primary Schools Education Department of Victoria and the ACER

Australian Education Council (1991) A National Statement on Mathematics for Australian Schools Carlton, Victoria: Curriculum Corporation for the Australian Education CouncilAustralian Education Council (1994a) Mathematics-A Curriculum Profile for Australian Schools Carlton, Victoria: Curriculum Corporation for the Australian Education CouncilAustralian Education Council (1994b) Mathematics-Work Samples Carlton, Victoria: Curriculum Corporation for the Australian Education CouncilBaker, John R. (1938) "The Evolution of Breeding Seasons" pp. 161 - 177 in de Beer, G.R. (ed.) (1938) Evolution. Essays on Aspects of Evolutionary Biology Presented to Professor E.S. Goodrich on his Seventieth Birthday Oxford, U.K. Clarendon PressBarcan, Alan (1980) A History of Australian Education Melbourne: Oxford University PressBarnes, E.S. (1969) "Tertiary Mathematics Course in South Australia - University of Adelaide" The S.A. Mathematics Teacher 1 (1): no paginationBaxter, J.P. (1972) The Introduction and Development of the New Curricula in Mathematics in Secondary Schools in South Australia 1957 - 1972 M.Ed. Research Essay. Flinders University of South Australia. School of EducationBenz. H.J. (1982) 'Stochastics Teaching Based on Common Sense' pp. 753 - 765 in Grey, D.R., Holmes, P., Barnett, V. & Constable, G.M. (eds)

(1982) Proceedings of the First International Conference on Teaching Statistics Sheffield, UK: Organising Committee of First International Conference on Teaching StatisticsBlakers, A.L. (1976) 'Mathematics Education in Australia: Past and Present' Australian Mathematics Teacher 32 (5): 147 - 155 & (in 1977) 33 (1): 17 - 22 Brinkworth, Peter (1970) The Introduction of the Primary Mathematics Curriculum into South Australia M.Ed. Qualifying Paper: Flinders University Cambridge Conference on School Mathematics (1963) Goals for School Mathematics Boston: Houghton Mifflin for Educational Services IncorporatedCarroll, Lewis (?) Alice Through the Looking GlassChong, F. (1962) 'Modern Mathematical Programmes in Action' Australian Mathematics Teacher 18(3): 50 - 63Clapp, E.K.H., Hamann, K.M., Lang, I.P., & McDonald, J.A. (1969) Secondary Mathematics Series Mathematics 4 Part Two Adelaide, South Australia: RigbyClements, M.A. (Ken), Grimison, Lindsay A. & Ellerton, Nerida F. (1989) 'Colonialism and School Mathematics in Australia 1788 - 1988' pp. 50 - 78 in Ellerton, Nerida F. & Clements, M.A. (Ken) (1989) School Mathematics: The Challenge to Change Geelong, Victoria: Deakin UniversityClose, R.W. (1962) 'Some Problems of Present-Day Mathematical Education' Education Gazette 15 Jun 62: 154 - 156

College Entrance Examination Board. Commission on Mathematics (1959) Report of the Commission on Mathematics: Program for College Preparatory Mathematics Princeton, NJ: CEEBConnell, W.F. (1993) reshaping Australian Education 1960 - 1985 Hawthorn, Victoria: ACERCrombie, Alistair C. (1994) Styles of Scientific Thinking in the European Tradition London: DuckworthDuncan, A.K. (1969) "Tertiary Mathematics Course in South Australia - Institute of Technology" The S.A. Mathematics Teacher 1 (1): no paginationEllerton, Nerida F. & Clements, M.A. (Ken) (1992) 'Some Pluses and Minuses of Radical Constructivism in Mathematics Education' Mathematics Education Research Journal 4(2): 1 - 22 Félix, Lucienne (1966) Modern Mathematics and the Teacher Cambridge: Cambridge University PressFinlay, Ellen & Lowe, Ian (1993) Mathematics Curriculum and Teaching Program. Chance and Data. Exploring Real Data Carlton, Victoria: Curriculum Development CorporationGigerenzer, Gerd, Swijtink, Zeno, Porter, Theodore, Daston, Lorraine, Beatty, John & Krüger, Lorenz (1989) The Empire of Chance Cambridge, U.K.: Cambridge University PressGray, M.C. (1962) 'The Teaching of Mathematics in South Australia' Australian Mathematics Teacher 18 (3): 71Gregg, Jeff (1995) "The Tensions and Contradictions of the School Mathematics Tradition" Journal for Research in Mathematics Education 26 (5): 442 - 466Hacking, Ian (1975) The Emergence of Probability London: Cambridge University PressHacking, Ian (1990) The Taming of Chance London: Cambridge University PressHaese, R.C., Harris, K.P., Haese, S.H., Webber, B., &Danielsen, F.G. (1984) Mathematics for Year 10 Adelaide, South Australia: Haese & Harris PublicationsImmelmann, K. (1972) 'Erörterungen zur Definition und Anwendbarkeit der Begriffe ‘ultimate factor', ‘proximate factor' und ‘zeitgeber'' Oecologia 9: 259 - 64Izard, John F., Drummond, D.J., Goodger, D.H., Haig, B.D. & Smith, F.L.

(1965, 1970) Individual Mathematics Programme Adelaide, South Australia: RigbyLovitt, Charles & Clarke, Doug (1988a) The Mathematics Curriculum & Teaching Program. Professional Development Package. Activity BankæVolume 1 Canberra, Australia: Curriculum Development CentreLovitt, Charles & Clarke, Doug (1988b) The Mathematics Curriculum & Teaching Program. Professional Development Package. Activity BankæVolume 2 Canberra, Australia: Curriculum Development CentreLovitt, Charles & Lowe, Ian (1993a) Mathematics Curriculum and Teaching Program. Chance and Data. Investigations. Volume 1 Carlton, Victoria: Curriculum Development Corporation

Lovitt, Charles & Lowe, Ian (1993b) Mathematics Curriculum and Teaching Program. Chance and Data. Investigations. Volume 2 Carlton, Victoria: Curriculum Development CorporationMacDonald, Theodore H. (1975) "Needed - A Rethinking of School Mathematics Programs" pp. 32 - 37 in Jeffery, Peter (ed.) (1975) Primary School Mathematics in Australia: 1975: Review and Forecast Report of a Conference Convened by ACER at Ormond College, University of Melbourne, August, 1975. Hawthorn, Victoria: ACERMarch, M.E. (1970) 'Education in the U.K.' Australian Mathematics Teacher 26 (2): 53 - 61 and 26 (3): 103 - 106National Council of Teachers of Mathematics (1961) The Revolution in School Mathematics - A Challenge for Administrators and Teachers Washington DC: NCTMNew South Wales Department of School Education (1995a) Aspects of Numeracy Year 3New South Wales Department of School Education (1995b) Aspects of Numeracy Year 5Owen, John, Johnson, Neville, Clarke, Doug, Lovitt, Charles, & Morony, Will (1988) The Mathematics Curriculum & Teaching Program. Guidelines for Consultants and Curriculum Leaders Canberra, Australia: Curriculum Development CentreSchool Mathematics Project (1965) Teachers' Guide for Book T Cambridge, U.K.: Cambridge University PressSchool Mathematics Project (1970) Teachers' Guide for Book 3 Metric] Cambridge, U.K.: Cambridge University PressSenior Secondary Assessment Board of South Australia (1983) Matriculation Examination 1983. Notes by Examiners. Mathematics 1Shaughnessy, J. Michael (1992) 'Research in Probability and Statistics: Reflections and Directions' pp. 465- 494 in Grouws, D.A. (ed.) (1992) Handbook of Research on Mathematics Teaching and Learning New York: MacmillanShaughnessy, J. Michael & Bergman, Barry (1993) 'Thinking about Uncertainty: Probability and Statistics' pp. 177 - 197 in Wilson, Patricia S. (ed.) (1993) Research Ideas for the Classroom: High School Mathematics New York: MacmillanSumner, Robert J. (1969) The Introduction of Modern Mathematics Courses into the First Four Years of South Australian High Schools Public Examinations Board Streams. M.Ed. Qualifying Paper: Flinders University of South Australia Szendrei, Julianna (1990) "Some Aspects of teaching Stochastics in Hungary" pp. 262 - 283 in Wirszup, Izaak & Stocit, Robert (1990) Developments in School mathematics around the Worls Vol. 2 Reston, Virginia: National Council for the Teaching of MathematicsTruran, John M. (1992) The Development of Children's Understanding of Probability and the Application of Research to Classroom Practice M.Ed. Thesis: University of AdelaideTruran, John M. (1994a) 'Chance & Data Or Probability & Statistics - Are We Oversimplifying?' Möbius 21 (1, 2): 46 - 48Truran, John M. (1994b) 'Curriculum Innovation and the Teaching of Probability in South Australia-Colonial Echo, Mature Development or

Muddling Through?' pp. 632 - 640 in Bell, Gary, Wright, Bob, Leeson,

Neville & Geake, John (eds) (1994) Challenges in Mathematics Education: Constraints on Construction Proceedings of the Seventeenth Annual Conference of the Mathematics Education Research Group of Australasia. No place of Publication: MERGA Truran, John M. & Kathleen M. Truran (1994) 'Chance & Data and the Profiles in Secondary Schools' pp. 132 - 134 in Beesey, Cathy & Rasmussen, Duncan (eds) (1994) Mathematics without Limits Proceedings of the Thirty-First Annual Conference of The Mathematical Association of Victoria. Melbourne, Victoria: Mathematical Association of VictoriaTruran, Kathleen M. (1993) Report of a P.E.L. Investigation on the Impact of the National Curriculum of England and Wales January - April 1993 Adelaide, South Australia: University of South Australia, School of Education, Salisbury CampusTruran, Kathleen M. & Truran, John M. (1994) 'Chance & Data and the Profiles in Primary Schools' pp. 126 - 131 in Beesey, Cathy & Rasmussen, Duncan (eds) (1994) Mathematics without Limits Proceedings of the Thirty-First Annual Conference of The Mathematical Association of Victoria. Melbourne, Victoria: Mathematical Association of VictoriaUnited Nations Educational, Scientific and Cultural Organization (1965) Mathematics in Australian Schools Sydney, New South Wales: UNESCOVinner, Shlomo (1994) "Traditional Mathematics Classrooms-Some Seemingly Unavoidable Features" pp. 353 - 360 in da Ponte, João Pedro & Matos, João Filipe (eds) (1994) Proceedings of the Eighteenth International Conference for the Psychology of Mathematics Education Volume IV Lisbon, Portugal, Program Committee of the 18th PME ConferenceUniversity of Adelaide. Public Examinations Board. (1968) Manual of the Public Examinations Board 1969 Part I Syllabuses for the Year 1969 Adelaide, South Australia: Griffin Pressvon Glaserfeld, Ernst (ed.) (1991) Radical Constructivism in Mathematics Education Dordrecht: KluwerWatson, Jane M. (1994) Australian Maths Works. Teaching and Learning Chance and Data. Adelaide: Australian Association of Mathematics Teachers.John TruranPO Box 157GoodwoodSouth Australia 5034(08) 373 0490jtruran@arts.adelaide.edu.au1Carroll (?, ch. 3))2I am grateful to Dr Alan James, Emeritus Professor of Statistics, University of Adelaide, for pointing out this distinction.3Australian Education Council (1991)4Australian Education Council (1994a)5Australian Education Council (1994b)6Truran (1994a)

7E.g. Benz (1982); Szendrei (1990); Shaughnessy (1992, p. 466); Shaughnessy & Bergman (1993, p. 178). The list longer than some have asserted, although it is true that this meaning hass not yetbeen listed in standard dictionaries.8NCTM (1961, pp. 21 - 27; p. 79)9E.g. , vide Gigerenzer et al. (1989)10Ellerton & Clements (1992); von Glaserfeld (1991)11Clements, Grimison & Ellerton (1989, pp. 51 - 68)12Clements et al. (1989, p. 68)13Truran (1994b)14Crombie (1994, vol. I, pp. 63 - 69)15Crombie (1994, vol. I, p. 63)16Crombie (1994, vol. I, p. 64)17Baker (1938); Immelmann (1972)18Hacking (1975; 1990)

19College Entrance Examination Board (1959b, pp. 29 - 32)20Cambridge Conference on School Mathematics (1963, pp. 70 - 71)21School Mathematics Project (1965, p. 82)22School Mathematics Project (1970, p. 1)23March (1970)24Félix (1966)25UNESCO (1965, p. 10)26UNESCO (1965, p. 21)27Blakers (1976, p. 147)28ACER (1964, Annexure 3)29ACER (1964, Annexure 10); Education Gazette 1 Jun 1964, pp. 187 - 18830ACER (1966) and lightly revised in 1972. Page references are to the 1972 edition.31Keeves (pers. comm.)32ACER (1972, p. 185)33Brinkworth (1970, p. 39, text and footnote). Much of the book had been checked and corrected by the eleven-year old niece of the author! (Keeves, pers. comm.)34Izard et al. (1965, 1970)35E.g. , Chong(1962); Close (1962); Gray (1962)36Baxter (1972)37MacDonald (1975, p. 37)38Sumner (1969)39Brinkworth (1970)40Connell (1994) writes from an ACER perspective and understates the natue and reange of experimentation which occurred.41Truran, K. (1993)42Secondary Mathematics Series (SMS), e.g. , Clapp et al. 2(1969)43Univesity of Adelaide. Public Examinations Board (1968, p. 41)44Univesity of Adelaide. Public Examinations Board (1968, pp. 106 - 108)45Reading List for Mathematics Teachers Recommended by the Mathematical

Association of South Australia, March 1967 46Abrahamson (1969); Barnes (1969); Duncan (1969) 47Univesity of Adelaide. Public Examinations Board (1968, p. 40)48Letter to author from Professor MA Clements 20 Jan 1994.49Connell (1993, p. 140)50Barcan )1980, p. 344)51Editorial The S.A. Mathematics Teacher 1(1): no pagination52Mr Ross Frick, University of South Australia (pers. comm., December 1994)53Truran (1992, p. 99)54What they refer to as "interpretation" is only graphical reading skills.55Blakers (1976, p. 17)56Australian Education Council (1991, pp. 163 - 164)57Australian Education Council (1994b)58Australian Education Council (1994a)59Truran, J. & K. (1994); Truran, K. & J. (1994)60Australian Education Council (1991, pp. 1 - 39)61Senior Secondary Assessment Board of South Australia (1983)62Gregg (1995, p. 443)63Vinner (1994, p. 354)64Vinner (1994, p. 354)65Truran (1992, pp. 98 - 138)66Lovitt& Clarke (1988a, 1988b) Owen et al. (1988)67Lovitt & Lowe (1993a, 1993b)68Finlay & Lowe (1993)69 Watson (1994)70Subject Guide produced by the Senior Secondary Assessment Board of South Australia about 1985.

71Unpublished Evaluation Report produced by the Senior Secondary Assessment Board of South Australia about 1987.72This view was expressed by a member of the University of Adelaide in late 1985 in correspondence with the Mathematics Curriculum Area Committee.73This view was expressed by a member of Flinders University of South Australia in early 1986 in correspondence with the Mathematics Curriculum Area Committee.74This view was expressed by a member of the South Australian Institute of Technology in early 1986 in correspondence with the Mathematics Curriculum Area Committee.75Recommendation 16 of Mathematics Curriculum Area Committee to Senior Secondary Assessment Board of South Australia 13 Feb 1989.76Recommendation 1 of Mathematics Curriculum Area Committee to Senior Secondary Assessment Board of South Australia 13 Feb 1989.77The name of this new topic has changed many times; 'Mathematics 3' is used for convenience.78Austalian Education Council (1991)79Undated MS headed "Chielf Examiner's Notes Quantitative Methods",

probably 1993 from the Senior Secondary assessment Board of Souith Australia80New South Wales Department of School Education (1995a)81New South Wales Department of School Education (1995B)82Austalian Education Council (1991, p. 163)83Austalian Education Council (1991, p. 164)84Letter from the Hon. Robert Lucas, Minister for Education and Children's Services, South Australia to J.M. Truran, 7 Nov 95