Statistical Vistas: Perspectives on Purpose and Structure

Eric R. Sowey
University of New South Wales

Journal of Statistics Education v.6, n.2 (1998)

Copyright (c) 1998 by Eric R. Sowey, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.


Key Words: Effective use of overviews in teaching statistics; Long-term learning.

Abstract

A body of research on enhancing the teaching of statistics has been accumulating now for more than fifty years since the pioneering contributions of Wishart (1939) and Hotelling (1940). Yet undergraduates continue to find courses in statistics unappealing. Perhaps this is because their teachers -- even those clear and conscientious in explaining subject-matter detail, and thoughtful in their reading of the statistics education literature -- too commonly fail to open statistical vistas, and thus fail to convey a rich understanding of the purpose and structure of the subject. A vista is inherently a perspective view. This paper shows, with examples, how perspective views can illuminate both purpose and structure. A well-devised perspective on purpose, offered early, can make each topic in the course immediately meaningful. And perspectives on structure, unveiled strategically, can highlight the coherence of statistics. The author's experience over twenty-five years shows that teaching with perspectives can help to produce that ideal -- long-term retention of learning.

1. Introduction

1.1 What Perspectives Reveal

1 My aim is to show how statistics teaching that offers perspectives gives students a strong encouragement to `deep' learning (as defined, for example, by Ramsden 1992, chapter 4). And `deep' learning is usually learning that lasts. Indeed, I argue that having clear perspectives on the subject matter is quite indispensable to long-term retention of learning. Despite its demonstrable importance, this theme is hardly recognised in the past literature on statistics education and, moreover, in the literature on higher education, generally.

2 To see the significance of perspective views, let us look first at two non-statistical instances.

3 A close-up photo of some tiny portion of an ordinary household object (say, the head of a match) reveals mysterious forms. Unless we are also shown a photo of the object in macro-perspective, we shall have great difficulty in identifying what it is or what it could be for. This shows how a perspective view can provide scale and context, and thus let us identify and give purpose to something that is simply unrecognisable in a close-up view.

4 In 1968, Apollo 8 made the first-ever earth-moon transit by a manned spacecraft. The crew took photos of the Earth against the blackness of space, allowing humankind to see the entire disk of our planet for the first time. The indelible impression made by these famous images -- so different from any cartographic projection or aerial photo -- is due as much to their grand perspective as to their powerful romanticism. We see that a perspective view can reveal overall form or structure that is simply unnoticeable in a closer view.

5 Carrying these observations over to our present context, it follows that someone learning statistics without perspective views runs two risks: firstly, failing to find purpose in a mass of detail and, secondly, failing to see an overall structure in the discipline. Would it be surprising, then, if this learner's reaction were negative or hostile to statistics? Hardly! With little sense of the purpose or the structure of what is to be learned, anybody (except unthinking rote learners) would find the task frustrating.

6 Unfortunately, negative or hostile reactions to statistics are not particularly uncommon among college students, as every teacher of the first course knows. Of course, the cause is not necessarily an absence of perspectives. Failure to set an appropriate standard of prerequisite knowledge; rushed, vague or muddled exposition; or absence of real-world illustrations are just a few of the many other factors that can provoke negative reactions. However, some 25 years of observation and discussion with my students have convinced me that the biggest handicap affecting those who are willing to be `deep' learners of statistics is a scarcity of perspectives on the subject matter being studied and on the discipline as a whole. This is particularly true of those students who are, by nature, inclined towards a `big picture' view of the subject matter in anything they study.

7 If teaching too often fails to open vistas, perhaps the situation is saved by the textbooks of statistics? Here the evidence is mixed, though not very reassuring. Two types of perspective seem to be expected in a textbook nowadays (especially one at introductory level). In the Preface a flow chart will provide an overview of chapter linkages, and individual chapters will open with a skeletal overview of the pages that follow. But there is generally little else in the way of perspective.

8 Much more can be done in this regard, as this paper seeks to make clear. In particular, I shall point to the significance of perspectives in helping students to understand not just statistical theory, but statistical practice, as well. All the examples I shall give are of approaches I have used in my own teaching over many years, always with some positive effect. However, I have not written this paper so as to be simply a compendium of examples. Rather, my intention is to offer a systematic and original review of the possibilities of using perspectives, and to illustrate some of these possibilities as a prompt to my readers' creativity in their own spheres of statistics teaching.

1.2 Traditional Statistics Teaching (With Few Perspectives): How It Looks to the Student

9 To be specific, let us consider an introductory (single semester) course in statistics. What topics are found in a traditional syllabus? Textbooks of long standing can provide a reliable guide. The following are some successive chapter headings in a current, U.S.-published, and widely-adopted first-year text in statistics for management and economics, now in its fourth edition. This sequence of chapters has not altered appreciably across the four editions over fifteen years. The particular text is not the issue: there are very many others like it. These chapters together could represent a semester's study.

10 "What statistics is all about; Data collection and sampling theory; Summarising data in tables and graphs; Measures of location and dispersion; Introduction to probability; Discrete probability distributions; Some important discrete distributions; Some useful continuous probability distributions; Sampling theory and some important sampling distributions; Estimating and constructing confidence intervals; Hypothesis testing; Tests of hypothesis involving two populations; Chi-square tests; Analysis of variance; Regression and correlation."

11 What impression does such a syllabus make on the student, as it is traditionally taught -- that is, with a continuous close focus on the subject matter and few (if any) pauses for perspective? Typically, I would say, the student will form a view of statistics rather like this: This is a subject made up mostly of analytical techniques. There seem to be some systematic organising principles underlying these many techniques, but just what those principles might be is elusive. It also seems that the subject contains none of the sorts of contentious issues that one meets in studying, say, political science, which it might be interesting to decide one's position on.

12 Nor is there much escape from heavy abstraction. Probability is defined (in one way) as `relative frequency,' but this can be stated formally only in terms of a limit based on an infinite number of trials! Sampling distributions are, it seems, never constructed in practice, yet they are apparently indispensable to the interpretation of any statistical inference. Estimators are said to be `asymptotically efficient' but we never have `asymptotic' samples. And so on.

13 Further, the material does not appear to have any of its origins in any specific times or places. Names may be mentioned (Bernoulli ... Bayes ... Student ... Poisson ...), but these are all fleeting allusions. One has little idea whether the material is historically young or old, or whether everything that needs to be worked out on a particular topic has already been worked out somewhere. The material just `is.' One is reminded of the tourist who visited Aruba. When asked on his return where Aruba is, he replied: "How should I know? I flew there"!

14 The traditional first course I have just described (yet I would not say caricatured), with its resolutely closely focused presentation, is alas far from obsolete, especially in the realm of service courses.

15 To rescue such courses with such teaching from the motivational vacuum which they offer to students, many effective things can be done (and are being done by thoughtful teachers!), as regards both content and process. Making opportunities for exploratory data analysis (Leinhardt and Wasserman 1979), problem-based learning (Bissell 1975), and activity-based learning -- via individual or group projects (Fillebrown 1994 and Ledolter 1995) or some limited work experience (Jones and Kanji 1980) -- are just a few of the educationally richer and progressively more widely-adopted innovations of the last two decades.

16 What is significant, from the standpoint of this paper, about each of these educationally richer approaches is that it provides some kinds of perspectives in itself. The perspectives implicit in the different approaches are subtle and so are not always recognised by students for what they are. This is not surprising: the main declared merits of the approaches lie elsewhere. It follows that every one of these richer approaches, too, can be enhanced through a more deliberate use of perspective views.

17 In Sowey (1995), I gave a broad analysis of the process issues in statistics education, and emphasised, among other things, the value of teaching with perspectives. I feel some confidence in taking that idea forward here. My confidence has four grounds: teaching with perspectives has self-evident merits in principle; my own use of perspectives has for years motivated very many of my students towards `deep' learning (as they have told me); it is a teaching approach now being advocated in spirit also by others (see Schau and Mattern 1997); and it is adaptable to every kind of course content -- including traditional ones.

18 In the next section (Section 2) I shall look at perspectives over purpose. Such perspectives can provide context for the course and reasons for studying the discipline. They answer the question: "What's the use of statistics?"

19 In Section 3 I shall look at perspectives over structure. These can serve to enhance students' mastery of the content of the course, for structure is the bridge between knowing things and understanding them. These perspectives answer the question: "How does it all hang together?"

20 The objective is always to increase the likelihood that students will retain their learning for the long term -- beyond the semester examination, beyond the graduation ceremony, and into the workplace.

21 This should surely be a fundamental goal of all good education. Yet, judging from its sparse appearance in the statistics education literature, the goal of teaching for long-term retention of learning by students does not appear to be widely supported, even in theory. In practice, much statistics teaching seems to aim little further than getting students through the subject assessment. This is a pity. In Section 4 I summarise the case for keeping the long-term goal always in view.

2. Perspectives on Purpose

2.1 Opening Perspectives: Giving Meaning to the Course

22 A major motivator of learning at university is knowing why one is learning. Of course, the question can be answered at various levels.

23 The line generally taken in textbooks and, surely, the very least that students should be offered runs like this: "doing this subject will give you some knowledge and skill in statistical analysis that will be useful in your intended field of work -- for example, in the following ways ..." If the teacher is willing not to hurry, I want to propose that a rather more ambitious explanation be offered. I shall suppose we are discussing a first course in statistics. More advanced courses would be approached a little differently.

24 My proposal is, first, to specify those fields of knowledge in which the content of the particular statistics course will be applied. These will usually be obvious, since it is widespread practice to have separate statistics courses for separate clusters of disciplines. Thus there will often be one statistics subject for social sciences, one for business studies, one for biological sciences, and so on. The case of a `pure' mathematical statistics subject will be considered presently.

25 Next -- after some very basic statistical terminology has been established -- we need to categorise the main kinds of statistical questions that arise in those fields of application and that fall within the scope of the course. Then, at the outset we take a perspective view of the entire course, describing in non-technical language how each topic in the syllabus is directed to solving one (or more) of the kinds of questions previously identified. In this way, the course, is seen to be a means to practical ends, and its content and purpose are informatively linked.

26 How specifically should one word `the kinds of statistical questions that arise in the fields of application'? Preferably, the wording should be generic to underline the universality of the theme, e.g., "to see if two populations differ in their means," with an added illustration drawn closely from one of the real-world work contexts in which students expect to find themselves, e.g., (for marketing students) "to see if there is a change in the average spending on room service by hotel guests following the extension of daily service hours from 12 to 24."

27 How should an opening perspective be managed for a `pure' mathematical statistics course? Such courses are often condemned as misrepresenting statistics as no more than a branch of mathematics (see, for instance, Kerridge 1976 and Hand 1994). Partial amends can be made by providing contextual illustrations here, as well! Since the students' eventual fields of application may not yet be foreseeable, the teacher is free to select the most striking examples that will accord with students' own interests and experience.

2.2 Later Perspectives: Giving Meaning to the Discipline

28 Every statistics course represents only a slice of the discipline, but it is never a slice with clean edges, so to speak. It has, after all, been lifted from the body of the discipline as a whole and implicitly retains `fibres' on all sides extending back to that body. The teacher should make these fibres visible to students, as occasions arise, and have students look along them so as to get a perspective on the nature of statistics as a discipline, beyond the bounds of the syllabus. Opportunities to offer such perspectives are readily thought of when one is conscious of their value.

29 Here is an example. When studying the properties of estimators and tests for the `classical' linear regression model, students are pleased to discover that all the results are `tidy.' Estimators are optimal in finite samples, test statistics have well-known exact small-sample distributions and many tests are uniformly most powerful against the specified alternatives. It is an opportune moment to explain that these consequences flow from the classical assumptions, not fortuitously, but precisely because the assumptions were chosen to guarantee the consequences. When any of the classical assumptions are relaxed, the `tidy' properties of classical estimators and tests are almost always compromised. Further, the inferential procedures, that have been specially devised for situations where particular assumptions are invalid, generally have only a restricted (e.g., asymptotic) optimality.

30 When students encounter these less-than-ideal results in the general linear model, many tend to think of them as aberrant exceptions. Since their textbooks focus almost exclusively on `tidy' results, this tends to strengthen their conviction that `tidiness' is the rule. Of course, the truth is otherwise. It will be a salutary perspective over the discipline to indicate to students that what seems to them like the rule is, in fact, the exception and vice versa. One constructive context in which to broach the matter is the ANOVA for classical regression, where many optimal results are seen to depend on the partitioning of the total sum of squares with a zero cross-product. "Should we always expect," we may ask provocatively, "that the cross-product will be zero, regardless of estimator and regardless of assumptions?"

31 Further insights can be offered at the end of the course -- when it is natural to look beyond the course to what else the discipline offers -- by posing questions that reveal how statistical methods can contribute to, for instance, data reduction, data discrimination, and data classification.

32 If these perspectives are well integrated into teaching, students will have a better perception of how the discipline is useful. This can be expected to give them a greater sense of context for the part of it they are studying and a stronger motivation to learn well. If they can also appreciate how the discipline can be useful to them, in their intended professional work, they will have a strong motivation to retain their learning for the longer term.

3. Perspectives on Structure

33 "What's the use of statistics?" is, as we have just seen, a question that should not be skimmed over lightly. Without perspectives on its purpose a course may lack meaning from the outset, leaving students bored and disoriented.

34 "How does it all hang together?" is another question that needs careful attention from the teacher. Without a sure sense of the structure of the course there can be no deep understanding. And then, of course, it is doubtful that there will be long-term retention of learning.

35 To convey a sense of structure is, for the statistics teacher, not a trivial task. Even when the syllabus of the first course is orderly and has been presented in an orderly fashion, one can be disappointed at how little students appreciate of the structure of elementary statistics (let alone of the entire statistics discipline). I can see two reasons for this:

36 First, because -- unless it is problem-based -- the course is not obviously `about' anything. The succession of techniques that are the conventional content of the first course seems, to the student with an inquiring mind, to be a preparation for something that never comes (or at least not before the final examination).

37 Second, because there is more to structure than an orderly presentation of an orderly sequence of topics. Structure is a reflection of the broader notion of coherence of the subject. In Sowey (1991) I identified three dimensions of coherence. These I named theme coherence, pattern coherence, and knowledge coherence.

38 These dimensions of coherence evolve naturally from the metaphor of a tapestry of human knowledge. As in the case of an actual tapestry, there are three `levels' of design: `motifs' (i.e., elemental design units), `patterns' (formed by replicating or slightly varying motifs), and the overall `composition.'

39 In the discipline of statistics, the motifs are themes such as those that form sequential chapters of text books. Theme coherence, then, represents a smooth traverse through a sequential body of theory. Pattern coherence lies in revealing the common elements in the problems and analytical techniques of apparently disparate themes. And knowledge coherence refers to the seamless weaving of statistics into the tapestry of all human knowledge.

40 As these descriptions suggest, theme coherence brings out the continuity of the subject matter, pattern coherence highlights its unity, and knowledge coherence its integrity (i.e., wholeness with its `neighbouring' disciplines).

41 If the goals of statistics education include offering students a rich understanding of structure within the syllabus (as I believe they should), then, clearly, all three of these dimensions of coherence need to be conveyed.

42 Traditional statistics courses (at every level) are seen to aim primarily at theme coherence. It is almost as if teachers judged (what I have called) pattern and knowledge coherence to be of little importance. The truth is probably more prosaic: there is so much to be `covered,' and such limited time, that goals are simply prioritised. Then, consciously or unconsciously, all but theme coherence is pushed to the side. Should we abide by what is traditional, in this way? By no means! If there is not time or opportunity to concentrate properly on pattern or knowledge coherence, let us at least bring out their essence in our teaching. Perspectives offer us a way of doing so.

43 Let us now consider some ways of generating perspectives on the structure and content of the statistics course. I shall look first at theme coherence: here perspectives can complement the traditional close-focus view. Then I shall come to pattern and knowledge coherence: here, as just indicated, perspectives are likely to be almost all that is offered.

3.1 Perspectives on the Continuity of Themes

44 All committed statistics teachers try to bring out theme coherence in their detailed exposition. What is also important, but not always recognised, is the value of injecting frequent perspectives before, during, and after the exposition of detail. I want to distinguish two kinds of perspectives in this context: longitudinal perspectives (`looking along the line of development') and lateral perspectives (`looking from the side at particular topics within a theme-coherent development').

3.1.1 Longitudinal Perspectives

45 The point of longitudinal perspectives is to ensure students always know where they are in the sea of detail. More than simply informing, such perspectives can materially enhance the linkage of theoretical ideas. As well, they have a valuable role in teaching the real-world practice of statistics.

46 Here is an example from a first course in theory. After exposition of the mechanics of least squares trend-line fitting in a bivariate scatter, there is a pause for a new perspective. The following concise wording is intended only to sketch the approach: "What if we now think of the points in the scatter plot as just a sample drawn from some larger population? Let's formalise this idea by defining a model for what we believe the population data generation process to be that produced our scatter plot data. Looking at the irregular pattern of the scatter points prompts us to include a random element into our model. We ought to be able to draw some conclusions about the size of the parameters in this population model from corresponding values in our sample regression. But being inferences from sample to population, i.e., inductive inferences, they may be in error. Perhaps we can limit the risk of error by means of some kind of confidence interval. For this to be feasible, we shall need first to find the statistical distribution of any estimator we devise."

47 Further perspectives may be introduced before presenting (i) the theory of point and interval prediction, (ii) the not-immediately-obvious extension of the regression model to functions nonlinear in variables, but linear in parameters, (iii) the generalisation of simple to multiple regression, and so on. A backward-looking perspective (a `retrospective'?) at the conclusion of the theme would recapitulate the logic and structural coherence of this subject matter, beginning with the intuitive notion of a `line of best fit.'

48 Here is a second example from a higher-level course. In teaching the theory of linear estimation, an initial perspective might indicate that several general principles for the design of point estimators have been devised, and then show how each principle rests on a different conceptual approach to the goal. After presenting (let us say) best linear unbiased estimation, as well as estimation by maximum likelihood, method of moments, and least squares, through particular examples in which they produce unbiased estimators, the teacher might `draw back' from technicalities and ask: "Why this devotion to unbiasedness? Suppose we bring efficiency to the fore? Could it be that we can find a biased estimator more efficient than all the unbiased ones?" In this way, minimum MSE estimation would be unveiled.

49 One reason why such perspectives are, I suspect, not widely used in teaching is that the pedagogical approach in textbooks of statistics is such a poor role-model for inexperienced teachers. In foundation textbooks, as we have seen, an introductory overview of each chapter is now routine, but whether there are further longitudinal perspectives depends on how idiosyncratic the author (or the publisher in a very competitive market!) is prepared to be. Generally, that is not very idiosyncratic at all! Rather than stimulating diversity in expository style, market competition has standarised the product as never before.

50 The best examples of theme-coherent perspectives come, not from elementary textbooks, but from more specialised works. Harris (1994) offers a continual perspective throughout his book on analysis of variance. Similarly, both Gilchrist (1984), who focuses on empirical modelling, and Chatfield (1995), who demonstrates how a practising statistician solves a problem from start to finish, convey the important truth that longitudinal perspective is the natural orientation of the able practitioner.

3.1.2 Lateral Perspectives

51 Perspectives `from the side' enrich understanding of individual topics by putting them in one or another kind of broader context. I have found three kinds of lateral perspective particularly valuable in my teaching. They are the intuitional, the historical, and the methodological.

52 To anticipate (or confirm) some formal statistical proposition by way of an intuitive argument has two constructive effects for the student. It immediately gives the student a feeling of `command' over the result, and it makes the abstractness of the formal demonstration less intellectually intimidating. Both influences can considerably assist `deep' learning.

53 Even studying a counterintuitive result like the central limit theorem can benefit from an intuitive perspective, precisely because the very counterintuitiveness of the proposition will be highlighted, and this will underscore how remarkable it is.

54 What makes for a good intuitive explanation? Three practical guidelines are: to draw analogies with some nonstatistical knowledge the student has; to change the language of the argument from algebra (abstract) to the geometry of a simple diagram (concrete); and to invite the student to go back to first principles, asking "if you had to find a way to do such and such, what logical possibilities can you think of?"

55 This last idea can be illustrated in the context of devising a regression model for a (0,1) dummy dependent variable. Clearly, a function linear in the regressor will not do, and after a little reflection (and perhaps some prompting) the student will hit upon the idea of an S-shaped curve over the range zero to one. Then the next question is: "What kinds of S-shaped mathematical functions can you think of?"

56 Historical perspectives give the student a sense of time, place and personalities. Statistical knowledge becomes less `disembodied' if associations are built up in these ways. It repays the teacher to read some of the recent histories of statistics, including Gigerenzer et al. (1989), Porter (1986) and Stigler (1986). From knowing that Fisher's contributions grew out of his agricultural work at Rothamsted, one understands the genesis of terms like `treatments' and `plots' in the theory of experimental design. From reading Plackett (ed.) (1990) one discovers how much more substantial a statistical scholar and insightful a professional man was W. S. Gosset (`Student') than ever one might think from the occasional footnote reference he is accorded in textbooks.

57 Methodological perspective consists in asking questions about the suitability of statistical tools for the declared purposes of the discipline. Effective use of such perspective implies that the purposes of statistics have been made clear -- in itself a fine objective, as I have already emphasised. Many thought-provoking issues can then be raised with students.

58 In a first course, one has, early on, an opportunity to remark on the fundamental contrast between the inductive logic of statistical inference and the deductive logic of (deterministic) mathematical inference. A short overview of the hierarchy of dependable logics -- in order, deduction, induction, and analogy -- makes a natural lateral perspective, and a valuable one! It is quite surprising how many students enter senior subjects in statistics with no clear idea of what the phrase `inductive inference' actually means.

59 Later, one might ask how powerful actually is some UMP test, how useful in small samples is some asymptotically optimal estimator, how robust is some parametric procedure to violations of its assumptions, does statistical significance entail practical significance and vice versa, and so on. The discussions that can follow will be guaranteed to banish forever any initial impression students (mostly abetted by their textbook) may have formed of statistics as a discipline wholly devoid of intellectual fallibility, controversy, or matters for value judgment.

60 In summary, lateral perspectives can make a strong contribution to awakening or stimulating student interest in statistics. But experience shows that their introduction must always be finely judged and low key. Brought in always at a natural point in the exposition, they should be sustained only so long as they hold students' attention. Otherwise students may find them distracting, intrusive, and even irritating, and their effect may then be the reverse of what is intended.

61 In particular, it is unwise to raise methodological critiques too freely with beginning students. Unless students have acquired a fair amount of disciplinary knowledge and some longitudinal perspective first, questions which seem to attack the worth of what they have only recently learned will not be thought very stimulating.

62 What do we find in textbooks? Are they as generally unsupportive of lateral perspectives as they are of longitudinal perspectives? Regrettably, yes: even texts at intermediate or advanced level. Intuitional perspectives appear here and there, historical perspectives rather rarely (Peters (1987) is a striking exception), and provocative methodological perspectives almost never. Educators who seek to inspire students in these ways are, as far as textbook writers are concerned, on their own!

3.2 Perspectives on the Unity of Patterns

63 Like most parts of mathematics, and indeed of science, statistical theory is rich in patterns. Pattern coherence is, as I have said, a focus of teaching that brings out the structural unity of a course by highlighting the existence of such patterns. Unlike theme coherence, pattern coherence is never the principal focus of a course. Yet, it cannot be omitted without limiting students' grasp of structure. The most appropriate compromise is to keep pattern coherence in perspective view throughout the course.

64 Here are several simple examples. The notion of `moments' and the `expectation' notation both unify the arithmetic mean, the variance, and the Pearsonian skewness measure. The concept of `singularity of the moment matrix' in multiple regression encompasses both the undersized sample problem and the regressor collinearity problem. The ANOVA, ANOCOVA, and regression models may all be subsumed under the general linear model.

65 In all these cases, only a short demonstration is required to make the point clear. The intellectual benefits can be substantial. Recognising such patterns allows the student to relate seemingly unconnected themes, and thus to organise the subject matter mentally in a more `compact' way.

66 Making students aware of patterns can have another value: to bring home to them what every inductivist knows, namely, that not all regularities are projectible. The alert beginning student, who learns that the population mean is best estimated by the sample mean, the population variance by the sample variance, and the population proportion by the sample proportion, soon inclines to the view that there is here a universal proposition linking any parameter and its optimal estimator. Alas, explains the teacher, showing the uniform distribution as a counterexample, this is just one of those patterns that are not universal.

67 The scholarly literature has a number of pattern coherent expositions. Del Pino (1989) and Jupp and Mardia (1989) are good examples. But I know of no textbook that systematically points to the patterns in the statistical theory it presents.

3.3 Perspectives on the Integration of Knowledge

68 Knowledge coherence in teaching traces the integral links between statistics and its `neighbouring' disciplines, i.e., those from which its techniques derive (mathematics and logic) and those in which its techniques are applied (list here all the sciences, technologies, and humanities!). Two further sets of issues, that impinge constantly on real-world statistical practice, belong with the notion of `integration' also. They are generally grouped under the headings `data quality' (e.g., accuracy, continuity, and completeness of data collection by official agencies, and accessibility of databases), and `statistical consultancy' (e.g., communication problems with non-statisticians and ethical aspects of statistical practice).

69 Which of these links will be pursued at length in statistics education? Often the answer is `very few.' This is not usually for want of good intentions. Rather, it is shortage of time that inhibits the initiative.

70 To the extent that knowledge coherence is pursued, its direction will depend entirely on the aims of the course. Courses in pure theory sometimes look back to origins (the analysis of gambling games, the medical struggle against epidemics, the challenge of designing a biological taxonomy).

71 Courses keyed to some discipline of application will refine classical methods and develop new ones to tackle the practical statistical issues in that discipline and to accommodate the kinds of data to be found there. Thus, statistics for psychologists will refine and develop topics in experimental design and ANOVA; statistics for economists will emphasise extensions to the regression model that cope effectively with nonexperimental data; statistics for medical researchers will elaborate on the design of randomised clinical trials, and so on. But where the course is run purely as a service course (e.g., statistics for psychologists) and is not taught by a subject-matter specialist (a psychologist), it is doubtful whether much attention will be given to integrating the ways a statistician and a psychologist form their judgments on quantitative issues -- a process so well demonstrated by Everitt and Hay (1992) -- which is yet another facet of knowledge coherence.

72 Of course, where knowledge coherence is emphasised, then longitudinal perspectives are as relevant as they are for theme coherence (see Section 3.1.1, above).

73 What about issues of `data quality' and `statistical consultancy'? These are broad matters of professional importance. Not to understand them will be to have a blinkered view of statistics as a practical discipline. But, here too, the narrow focus on formal technique in the traditional statistics course, and the shortage of time, tend to push these broader matters to the side. Given that these are matters that should certainly not to be neglected in a good statistical education, then maintaining a perspective is once again the answer. Students are alerted early to the data quality and statistical consultancy issues most relevant to the purpose of their course. Then, as the course progresses, its topics are presented with a data quality perspective or a consultancy perspective, as relevant.

74 Thus, for example, where the estimated size of some parameter is of vital significance in a modelling exercise, the risk of measurement error in the data used in estimation would be remarked on and the idea of doing a model sensitivity analysis suggested. Similarly, in testing, the critical point that failing to reject a null hypothesis does not necessarily mean accepting that hypothesis would be explained in the context of the statistician's responsibility to properly advise the client.

4. Perspectives Enhance Persistence of Learning

75 In this paper, I have argued for the cultivation of perspective views in teaching statistics. The immediate value of such an approach is in ensuring that students appreciate clearly `what it's all for' and understand firmly `how it all hangs together.' The combination of understanding and motivation that results has a valuable consequence, for what is better understood and better motivated is better learned and better retained.

76 I have categorised perspectives by their objective, i.e., whether they illuminate the purpose or the structure of statistics. The explorations of this paper show that they can also be classified in a different way. Some (e.g., longitudinal perspectives) will be brought into play systematically, while others (e.g., lateral perspectives) will be introduced more sporadically -- and perhaps spontaneously in the dynamic of teaching.

77 Whatever the perspectives, it is wise to introduce them naturally and in a low key fashion. They need not take up a great deal of time. They are there to enrich the value of the course to the student, not to intrude on its coherent flow, and not (it needs to be said) to give an opening to the teacher to impress with his/her broad erudition.

78 If all goes well, students will pass their assessments in statistics year by year. Attuned to and informed by perspective views, they will very likely find that they have also retained the fundamentals of what they learned in previous years. Then their years of study will represent not just the price of a paper qualification but also something cumulative: a statistical education.

Acknowledgments

In an earlier version, this was an invited paper at SISC-96, the Sydney International Statistical Congress, July 1996. I am indebted to three anonymous referees for their perceptive and helpful critiques.


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Eric R. Sowey
Department of Econometrics
University of NSW
Sydney, NSW, Australia 2052

E.Sowey@unsw.edu.au


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