University of Sydney
Journal of Statistics Education v.3, n.3 (1995)
Copyright (c) 1995 by Sue Gordon, 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: Activity theory; Adult learners; Goals of learners; Experiences; Perceptions of context; Goals of statistics education.
This paper provides examples of students' reflections on learning statistics. The Mathematics Learning Centre, where I teach, offers help to students experiencing difficulty with basic mathematics and statistics courses at university. The excerpts are drawn from surveys or interviews of these and other students studying statistics at the University of Sydney.
Activity theory, which is based on the work of Vygotsky, provides a helpful conceptual model for investigating learning at the university level. From the perspective of activity theory, learning is viewed as a mediated activity in a sociohistorical context. In particular, the way a student monitors and controls the ongoing cognitive activity depends on how that individual reflects on his or her efforts and evaluates success. In Semenov's words, ``Thought must be seen as a cognitive activity that involves the whole person'' (1978, p. 5). Students' interpretations of their learning tasks and the educational goals for their self-development are discussed within this theoretical framework.
1 A mature student gave this reason for giving up her psychology course: ``I was drowning in statistics.''
2 Statistics is often a compulsory unit of university courses such as psychology, business, or health sciences, because it is an important tool for analysing the ``uncertainties and complexities of life and society'' (Mosteller 1989, p. ix). There is, however, often a tension between knowledge as culturally endorsed and the individual's personal appraisal of it (Biggs 1991; Candy, Crebert, and O'Leary 1994) as many teachers of statistics are aware. While no generalisations can be made, an indication of this anomaly was suggested by the results of a survey completed by fifty-two students of the University of Sydney (Gordon, in press). The students were entering second year Psychology, for which statistics is a compulsory and substantial component. One question in the open-ended survey asked the students whether they would have studied statistics given a choice, and why. Their verdicts were overwhelmingly negative with 73% responding ``No'' to the question. This result is consistent with the findings of a number of recent studies (reviewed by Gal and Ginsburg 1994) which show that elementary statistics courses are often viewed with trepidation and hostility in the student culture.
3 The reasons for the decisions reported by these students were classified as positive (i.e., favourable towards statistics) or negative (unfavourable) within two major categories:
Practical Assessment: Perceptions of statistics as useful or not useful for further studies, jobs, societal applications. For example, a positive response in this category was: ``It would probably be useful in whatever job I do.''
Personal Evaluation: Evaluations of statistics in terms of personal considerations such as interest or disinterest in studying it, likes, or dislikes. An example of a negative response in this category was: ``I hated maths at school.''
4 The analysis of the responses showed that the few students who did reply positively perceived statistics as useful. However, the strong and adverse way in which students felt about mathematics dominated the responses of the ``No'' students, and these feelings outweighed any positive assessments of the relevance of statistics to them. These findings suggest that students' perceptions of learning statistics relate to their experiences, their interests, and their goals. Learning is embedded in a social environment.
5 These ideas are developed in this paper. I capture images which describe the learning of statistics as a purposeful and interactive activity that develops in a social context. The following sections of the paper illustrate differing approaches to learning statistics. These approaches reflect students' goals in their societal settings. I describe students' experiences and perceptions of the situations that frame their learning. The article then explores the ways in which students' evaluations and reflections mediate their activities. The concluding sections point out important implications for teaching statistics and ways for educators to relate to learners' perspectives. Directions for further research are indicated.
6 There has been considerable interest in the teaching and learning of statistics in recent years, not only as part of the general growth spurt in mathematics education (Schoenfeld 1994) but as a field of research in its own right. Four international conferences have been devoted to grappling with issues relating to statistical education. (For proceedings of the four International Conferences on Teaching Statistics (ICOTS I, II, III, and IV) see Grey, Holmes, Barnett, and Constable 1983; Davidson and Swift 1987; Vere-Jones 1991; The National Organizing Committee of the Fourth International Conference on Teaching Statistics 1994.) Much topical literature concerns the teaching of statistical concepts to make them meaningful and useful. (See, for example, Green 1994; Hawkins, Jolliffe, and Glickman 1992; Tanur et al. 1989.) What constitutes statistical knowledge and how to enhance the quality of students' learning of it are two of the key problems tackled by educators today (e.g., Anderson and Loynes 1987; Huberty, Dresden, and Bak 1993; Williams 1993). As in other fields, a major challenge currently facing researchers in statistics education is to improve our understanding of learning.
7 Researchers are increasingly finding that a Vygotskian perspective provides a helpful lens through which to view the complexities of mathematical learning (Kieran 1994). This approach takes into account social interactions and historical or cultural influences on learning. It aims to understand the roles of language and technology as mediators of meaning (Vygotsky 1962, 1978). Vygotsky's work on cognitive development contributed to the evolution of cognitive psychology (West, Farmer, and Wolff 1991, p. 6). In this framework, not only are the actions of learners important, but so are their goals and how the actions are situated in a socio-historical context.
8 While Vygotsky himself did not develop ``activity'' as a construct, his work laid the foundations of activity theory outlined by his student Leont'ev (1981) and other Russian psychologists. (See Semenov 1978; Davydov and Radzikhovskii 1985.) For activity theorists, the context for all activity is important, and collective external activity precedes individual internal activity. This framework is consistent with theories of situated cognition (Greeno 1991; Lave 1988; Saxe 1991; Scribner 1984) in which a setting is defined as the relationship between an actor and the arena in which s/he acts. Cognitive skills take shape in the course of individual participation in socially organised practices.
9 Leont'ev (1981) analyses activity by referring to three levels of description. The global level identifies the socioculturally defined milieu in which the actions occur. For example, learning may take place in a school context, in a work setting, or during play. The second level is concerned with the actions themselves: how the task is carried out. Actions are defined by the goals directing them. The third level is operational and concerns the tools or resources for carrying out the actions. The word ``activity,'' as used throughout this paper, refers to this framework. For example, a student's statistical activity encompasses her actions within a specific context, her goals (for example, she may be analysing data to arrive at a result or to understand the process underlying the analysis), and the resources available to her, whether technological or her own expertise.
10 Activity theory challenges the notion of the independence of thinking, feeling, and acting, suggesting that there is no division of the process of learning into intellectual or rational aspects and affective elements. Vygotsky describes this dependence:
Thought itself is engendered by motivation, i.e., by our desires and needs, our interests and emotions. Behind every thought there is an affective-volitional tendency, which holds the answer to the last ``why'' in the analysis of thinking. A true and full understanding of another's thought is possible only when we understand its affective-volitional basis. (Vygotsky 1962, p. 150)
11 This perspective complements other conceptual frameworks which emphasise the social and motivational dimensions of learning (for example, Bandura 1986; Eagley and Chaiken 1993; McLeod 1994). Recent research in statistics education (Gal and Ginsburg 1994) confirms the importance of students' attitudes, beliefs, interests, expectations, and motivations to their learning.
12 While there is considerable overlap between activity theory and other theoretical paradigms, and, indeed, many viable teaching strategies follow from more than one theoretical base, I believe the usefulness of activity theory lies in its systemic approach. Activity theory posits a view of learning in which personal experience, goals, subjective perceptions, and socio-historical factors are interwoven. Rather than focussing on separate facets of learning, activity theory implies a commitment to investigating the learning process as a ``dynamic system of meaning'' (Vygotsky 1962, p. 8) in which society and the individual interact and evolve. This view of learning includes notions of growth and diversity. The emphasis on the melding of the individual with society is particularly relevant to the analysis of adult learning, as adults are cognitively fully developed in the biological sense but generate new knowledge through participation in activities that are socially and culturally rooted. Hence, while Vygotsky's theories are pedagogical theories referring mainly to the development of the child, his concepts and the interpretations and ideas of Leont'ev are useful for extending the research on adult development (for example, Candy et al. 1994; Knox 1986; Merriam and Caffarella 1991) by explaining learning through interpersonal interactions.
13 The aim of this paper is to illustrate the theoretical concepts by providing examples of students' statistical activities. The excerpts are from surveys of students entering second year Psychology and from questionnaires and transcribed interviews of eight students who voluntarily attended the Mathematics Learning Centre for assistance with statistics courses. Since the majority of the subjects are female I shall use ``she'' as the general term. All names are fictitious, but the name indicates the gender of the student.
14 Fifty-two students starting second year Psychology completed a questionnaire (Gordon, in press). It asked students to write a few lines in response to open-ended questions. These included queries about their attitudes toward studying statistics (described in the Introduction) and their expectations of the statistics course.
15 Most of the comments of the Mathematics Learning Centre students were made by Psychology and Arts degree students interviewed after they had completed a statistics course. Additional material on these students was also obtained from questionnaires: one prior to the start of their courses and one in the second semester. These eight students were a purposeful sample (Merriam 1988) selected for their range of experiences and attributes. Details of the questionnaires and interviews are given in a paper describing case studies of five of the students (Gordon 1993a). As is the case for most students of the Centre, those interviewed were at least 25 years old and had been out of the education system for a number of years. The rising number of older students in many universities means that research on their learning is becoming increasingly relevant and significant (Knox 1986; Merriam and Caffarella 1991). In addition, observations made by mature adult students are insightful and supply valuable material for stimulating reflections on teaching (Belenkey, Clinchy, Goldberger, and Tarule 1986; Slotnick, Pelton, Fuller, and Tabor 1993). I was not involved with the students' formal assessment in any way and had worked extensively with them during the year. The close personal relationship that I developed with the students enabled me to have more access to their thoughts and feelings than is usually possible for someone teaching statistics at university.
16 While the samples cannot be regarded as representative of statistics students in general, I believe that the perceptions and evaluations of these individuals can help educators understand students who study statistics at tertiary level as a service course rather than by choice. This is common in areas such as the social sciences, economics, and para-medical courses. As is often the case for qualitative research, the analysis is interpretive rather than explanatory, acknowledging the complexity of the issues faced; it aims to indicate the dimensions of that complexity, as well as to stimulate reflection and dialogue (Lincoln and Guba 1985; Schoenfeld 1994).
17 Leont'ev (1981, p. 60) proposes that goals are the ``energising function'' of the student's activity. This agrees with research showing that goals act as mediators of university students' performances (Volet and Lawrence 1989; Volet and Chalmers 1992) and that appropriate goals are essential to effective self-regulation, one of the criteria of good learning (Bandura 1986; Biggs 1991). Adult students, in particular, are likely to enter a learning situation with clearly defined goals to which they are strongly committed (Candy et al. 1994; Merriam and Caffarella 1991; Slotnick et al. 1993). In my experience, however, university statistics courses concentrate mainly on the course content with little attention to the students' goals. These are assumed to be irrelevant to students' actions or are simply ignored. The following quotes from three students illustrate that the goals of students are integral to their approaches, furnishing the criteria against which students qualitatively evaluate the outcomes of their learning.
18 Norman (second year Psychology student; grade of Distinction for statistics component):
I have a very pragmatic approach to university, I give them what they want. Arguably if I could guarantee enough knowledge to get ten (full marks) in the stats. tutorial test and the exam. and know that I forgot it all completely afterwards, I'd almost go for that course, 'cause that's what they want. I really do like knowledge for knowledge's sake, but my main motivation is to pass the course.
19 In her survey, Vicki expressed the idea that real psychologists do not do mathematics.
I don't even see the point. In psych why must maths infiltrate itself??? Studies have shown that those who have high maths abilities have low or poor communication & perception skills -- shouldn't psychologists be exceptionally perceptive & able to communicate well? It seems that if there aren't silly numbers to justify things then they aren't plausible in our computer/maths./science promotive society.
20 In contrast Brenda who chose to take a course in statistics said this:
I don't just want to pass, scrape through. I want to understand and know what I'm doing.... If you're going to use it effectively you need to understand it, otherwise, I don't know, it seems pointless to me. If you don't understand what you're doing what's the point wasting your year doing it? ... My whole attitude's totally different from when I was at school. I didn't care when I was at school. I became a Christian a couple of years ago. And so my whole attitude changed and now I guess I've got a goal and now I've got a reason for being alive basically.
I never did well at anything at school and last year the course I did, out of 160 people I came first. That's amazing. That for me is amazing.
21 The goals reported by these three students reflect quite different social perspectives and values, leading to diverse approaches to the learning of statistics. Norman regarded learning statistics as a commercial transaction in which the aim is to earn the marks. His understanding of learning in the university context is that it is driven by assessment and that other motivations take second place to this. For Vicki, outright resentment at having to study statistics would seem to prevent the task from assuming any meaning. She expresses a perceived conflict between the values of a technological society and her own concern with human interactions. Brenda, however, reported the intention to understand and use the knowledge, an attitude stemming from her acceptance of religious values.
22 These quotes suggest that a student's goals cannot be viewed as isolated and individual but must be understood in the context of institutional or societal forces. Norman's comments attest to an all too often reported paradox: ``Institutions often appear to discourage the very things that, officially, they are committed to foster'' (Biggs 1991, p. 226). Institutions, in turn, are part of communities and cultures. The student is embedded in a hierarchy of systems that are not necessarily congruent. As in Vicki's case, what is perceived to be of cultural worth may be alien to a student's values.
23 The issue raised here is that of freedom of choice for students in their learning. How can the needs of the individual be balanced with the expectations of institutions or the larger society? While there is no simple answer to the dilemma, an initial step is for educators to recognise that the learning of statistics as a meaningful activity must be negotiated, not assumed. For this negotiation to be effective the interaction of individuals, society, and knowledge must be understood. Voigt (1994) considers that to understand the relationships between society and learning we must start by considering the learner's personal experience.
24 Wertsch, in an editorial comment (Semenov 1978, p. 3), states that ``The human problem-solver is viewed as bringing a complex system to a task situation and that this system plays just as important a role in the resulting outcome as does the information from the environment.'' Part of the complex system brought by the learner to the task are her goals. Another aspect that will affect how the individual approaches learning statistics is her experience of statistics or, more commonly, her experience of mathematics. Each individual has a history of learning experiences that may determine how she reacts to learning statistics as a university student.
25 Students' experiences of learning mathematics in the educational setting must be differentiated from their experiences with real life mathematics or statistics. Many adults regard these very differently. (See, for example, Lave, Murtaugh, and de la Rocha 1984; Scribner 1984). Consider the following extract from an interview with a student who, having retired from employment, returned to the education system after many years away from it.
As a hobby for years, I have followed the races. I don't bet very much, but when I do it I am a mathematical punter, I look at the percentages. So I relate to probability. Probability is effectively something that I use. I like the idea of comparing my percentages with those that the bookmaker has on the board, and at that level I have always dealt with statistics.
27 While a positive attitude toward ``real life'' statistics was reported, Ernest indicated a quite different attitude to school mathematics.
Arithmetic I didn't mind at all, but the moment it became more abstract, the moment symbols entered the scene, without a teacher who could relate it to the practical usefulness of it, he lost me. It became an exercise that I couldn't see that it mattered much to me, had relevance to what I wanted to know. I would stare out of the window, it went very much past me.
28 Many students attending the Mathematics Learning Centre have had little formal schooling in mathematics, but their years of working and their life experiences often include the use of mathematics or statistics. These students can regard learning statistics at university with great trepidation, as is evident from the following two excerpts.
29 Alice was faced with the daunting task of studying a statistics course when she had done primary school mathematics only. She said about her forthcoming study:
When I first saw the word statistics in the psychology book I just blacked it out. I thought I'd worry about it when I came to it.
30 Yet for many years Alice had worked as a caterer and had surely had to do quite a lot of mathematics in her work. This is what she said when I asked if she had had any difficulties.
No, but there were two of us doing it and we used to say, okay, you've got two hundred people, you allow a quarter pound of beef (in those days it was a quarter pound) per person so you just multiplied that out, how much steak do you want, and rang the butcher.... No problem at all. But that's really simple stuff!
31 Hettie, who also had little formal schooling, did not regard her life experiences as being helpful to the oncoming formal studies.
I had accounting skills from running a business, so I was not completely innumerate.... In second year, from day one, when we got the handout, I was panicked ... by the algebraic equations, everything. This was what scientists, astrophysicists do, not what I could do.
32 These comments illustrate that mathematics as a tool is meaningful if and only if it is functional. As teachers of statistics we try to guide students to appreciate the power and efficacy of the statistics they learn. The activity framework suggests that no one can persuade a student of the power of a thinking tool or sign system: this discovery depends on her experience of its functionality. As is well documented, students' understandings of statistical concepts are often poor. (See, for example, Garfield and Ahlgren 1988; the sections on students' understanding in Green 1994; Hawkins et al. 1992; and Holmes 1986.) The utility of statistical theory may not always be evident to the learner. If, as is often the case, statistics is taught in a mathematical and theoretical way, the necessary skills for understanding and applying it are put out of the reach of many students.
33 Goals and experiences are two factors that enable students to make meaning of their learning. The third factor that contributes to the student's activity is that of the perceived context of the learning. While goals may direct actions, implementation of intentions is strongly influenced by situational factors such as constraints, social expectations, and opportunity. Leont'ev (1981) proposes that any learning task is framed by its setting, a system with its own set of values and practices. Some of these are unique to the educational setting. Where but in a university, for example, would you get a large number of students sitting in a lecture theatre writing busily, whether or not they understand what it is they are writing? As Alice put it:
I was sitting watching him go through the overheads of our notes and if he said ``this is very important,'' I would underline it and if he said ``don't worry about these pages,'' then I wouldn't worry about those pages. I assumed the man knew what he was doing because he wrote the notes.
34 The problems individuals have to solve are structured by the bounds of this system; individuals become socialised by this system at the same time they act within it. Culture plays an important part. One student who had completed her schooling in Vietnam found studying in Australia very different from what she was accustomed to.
There is a big difference between the mathematics I did at high school and this course. Everything is new for me ... there is a lot of things different from how I learnt at my high school. A different way of thinking. I have to remember not only the rules but how to apply the rules.... But it is hard to learn to solve problems because in my country the way to learn is just to memorise by heart. We had no chance to practice to think for ourself. Just to memorise what the teacher said to us.35 Diametrically opposing this view, Ernest, commenting on his Australian university education said this:
All they can possibly say in defense of their way of teaching is that the intellectually curious student may still find it a useful base to depart from and to develop, evolve from. What they overlook is that a lot of the people who are not so interested, who do not have this native interest in the subject, they will not get it because of the way they teach. If they taught it in a more interesting way, if they taught people to think about it, rather than to repeat it, people might think about the importance of these questions that we look at.
We do nothing to arouse intellectual interest apart from making people exam smart and certainly with young students, all they talk about is what is needed for the exam.
36 The perceived context of an activity determines how it is construed. If students view learning statistics as an enterprise for gaining examination marks then the process of learning is directed towards that end, and understanding what they learn is irrelevant. Ernest's comments indicate that for him learning for examinations is what Leont'ev (1981, p. 253) calls ``alienated labour,'' where statistical knowledge viewed as a commodity evaluated through assessment is ``foreign'' to his sense of its relevance or worth.
37 Statistics courses, in common with other undergraduate courses, can suffer from the ``lamentable lapses'' of an overloaded curriculum, imposing too much detail at too advanced a level, failing to connect learning with the world of practice, and using forms of teaching and assessment that encourage rote learning (Candy et al. 1994, pp. 84-89). To promote the ideals of education in the university setting, statistical teaching must overcome these lapses. Hawkins et al. (1992) and Garfield (1994) suggest teaching methods and assessment practices that surmount many of these problems.
38 Perhaps the most important factor in students' regulating of their learning is their own evaluation of it. As Leont'ev (1981, p. 126) explains: ``Meaning mediates man's reflection of the world.'' That is, individuals' experiences of a socially organised practice are filtered through their awareness of its purpose and significance. Clearly for all students the examination mark is important. However, as indicated by the above example, not all students evaluate their learning entirely by this quantitative level. In fact, the responses of the 52 Psychology students that I surveyed indicated that many students do not even think it is desirable to be good at any mathematical subject. One of these students painted the following unflattering portrait of people who are good at mathematics:
They probably make up the questions to certain answers & lie in bed at night with thoughts of Pascal, Pythagoras, Newton and Jim Coroneos having disgusting orgies with live badgers and calves liver.
39 Another reported a reluctance to be part of the perceived elitism of the ``culture'' of mathematicians:
Maths is an exercise in agony, because the people who teach it make one feel as though maths belongs in a higher plane of evolution. Even though the number system is for everyone, and the concepts are there for everyone, the feeling (especially if you are doing pass options) that you do not deserve to know anything runs rampant.... Maths, in short, is a lofty pain and a real headache to study.
But this could not refer to any of us, could it?
40 In contrast, however, Sandra said this of her statistics course:
While a lot of course material is not relevant for counselling, a lot of the attitudes that you learn are important. A sense of professionalism in your approach, not a knee-jerk reaction to things, but sitting back and assessing it. The course teaches you to take many theories in and assess the different theories ... not what is right and wrong but hold many different theories in mind at once. And sit back a bit more. People who haven't had that training ... they have more ... a knee jerk reaction.
By the end of the year I thought, it doesn't really matter how I go in this exam, I'm not going to let the exam mark dictate to me my knowledge, because I knew I had a better grasp at the end of the year.
For this exceptional student, at least, evaluation was in terms of the quality of learning she felt she had achieved.
41 These excerpts suggest that knowing statistics includes perceptions of its value. The social interests and preoccupations of a culture are not automatically assimilated by an individual, but are monitored in terms of her own value system. Shared interactions trigger differing insights and feelings in participants. In social cognitive theory, evaluative processes are referred to as self-regulation, by which students observe, judge, and react to their perceived progress towards goal attainment (Bandura 1986, p. 337). Consistent with this view, activity theory posits that monitoring processes, which have their roots in the social world, mediate students' experiences, guiding thought and action and are the means to self-realisation (Semenov 1978).
42 The examples in the preceding sections are perhaps extreme, but some illustrate a common problem. Not only is there a wide gap between the knowledge of educators and students, but, less commonly acknowledged, teachers and students often have different perspectives and expectations. Most of us have to be regarded as success stories in terms of our statistics education or why would we be teaching it? This means that there is a gap between our perceptions of learning statistics and those of our sometimes reluctant students. The issue for consideration is this: how do we try to bridge that gap? How can we understand what it is like for our students to learn statistics, and how can we provide educational activities appropriate for students' needs?
43 Slotnick et al. (1993) suggest that both cognitive and identity development impact students' learning. These are separate but related areas of adult development. Cognitive development describes changes in people's thinking and problem solving, while identity development is about changes in how people feel about themselves and in how they relate to others and to the environment. The stages reached in both these areas of growth will affect how students interact with others, how they cope with the courses, what commitments they make, and hence how they approach their learning. To provide educational activities appropriate for students' needs, the statistics educator should address both these areas of development. In answering a student's questions, an instructor not only helps that person advance her skills in reasoning and logic and provides a role model for dealing with difficult issues, but also ``validates'' that person, that is, confirms her identity (Slotnick et al. 1993, p. 18).
44 What works for me in trying to view learning statistics from the student's perspective is to put myself into the learner's shoes -- a novice, but an adult. I took up ice skating a few years ago. There are similarities in the way I think of learning to skate and some of my students' ways of relating to mathematics (cf. Gordon 1993b). Firstly, I did not learn the skills as a child when I was at school, and many adult students, either through lack of opportunity or interest, did not learn much mathematics at school. Secondly, I am aware that I have no natural talent for the sport. I am sadly deficient in the 3 C's: Coordination, Composure and Courage. Many adults perceive themselves to lack innate mathematical ability. While the risks of learning to ice skate are fairly unsubtle, the risks in learning mathematics relate to the loss of self esteem associated with not succeeding.
45 There are three principles derived from the ice skating analogy that I find useful to guide my teaching. Firstly, we must create a supportive environment. Students cannot learn if they feel threatened and highly anxious. It must be safe to take risks. We all try to encourage discussion and questions, but there is more to a supportive environment than that. One has to ensure that students are supportive too. Older students have reported encountering younger students who are hostile to the type of questions they ask, particularly if they are of the ``why'' kind instead of the ``how to'' kind. A student explained to me that ``it doesn't seem to be the done thing.'' The activity framework highlights the need to recognise that the classroom is not just a place where instruction is received, but a social structure in which students' actions form. The statistics classroom can be structured so that students can safely take the initiative, discuss things with each other, do the teaching. For example, cooperative learning activities (Garfield 1993) can help students advance both intellectually and psychologically.
46 Secondly, guidance is important. Mathematics has been around for a long time, and many brilliant people of diverse cultures have contributed to its growth. One cannot expect students spontaneously to make all these leaps and discoveries for themselves. Vygotsky (1978) posits that learning is a process of enculturation whereby the novice is guided towards expertise. However, the type of guidance is critical. If the teacher is the only one in the classroom who is being creative and thoughtful, and students are expected merely to react to her or him, then the guidance will not succeed in assisting students to become independent and confident learners. We can learn to guide students by studying the ``artistry of good coaching'' (Schon 1987, p. 17) whereby aspiring practitioners acquire new skills or insights by being helped to build on what they already understand and know how to do.
47 This brings us to the third principle that I find useful: teaching needs to build on the personal experience of the learner. In trying to connect abstract statistical concepts with personal experiences, analogies, similes, and metaphors may be useful instructional tools. West, Farmer, and Wolff (1991) describe the use of metaphor as one of the teaching strategies that enable students to apply prior knowledge to new knowledge. In summarising the research on how metaphor and related devices assist cognition, they propose that appropriate metaphors are effective bridging mechanisms that aid long term recall and can often convey more cognitive and affective meaning than literal language. The power of these devices often lies in their richness in imagery and ``top down'' processing (whole to part rather than part to whole) allowing the creation of an organised set of similarities in the minds of students. From a Vygotskian point of view (Vygotsky 1962) language is used by learners to direct their thinking, enabling self-control and mastery of the environment.
48 This suggests that students' understanding of statistical concepts may be enhanced by appropriate use of metaphor and analogy. For example, a basic concept in inferential statistics is that of the null hypothesis or hypothesis of ``no effect.'' Students studying elementary statistics at university are often puzzled as to why an experimenter would set up a hypothesis which asserts that what has been observed is merely chance variation. To explain to students that this way of acting puts the onus on the experimenter to provide *evidence* that something (other than chance) is at work, I use an analogy to the ``innocent until proved guilty'' assumption that our courts of law use. It should be noted that this analogy will enhance a student's understanding only if she is aware of the procedures of this system of law.
49 The use of images and metaphors can also enrich the classroom environment. I have found that the following metaphor is appreciated by students who experience anxiety because what they learnt a week ago, or two weeks ago, has been forgotten. It is to ask students to think of their learning of statistics as building a road.
It's a wonderful road, it will take you to places you did not think you could reach. But when you have constructed one bit of road you cannot sit back and think ``Oh, that's a great piece of road!'' and stop at that. Each bit leads you on, shows the direction to go, opens the opportunity for more road to be built. And furthermore, the part of the road that you built a few weeks ago, that you thought you were finished with, is going to develop pot holes the instant you turn your back on it.
This is not to be construed as failure on your part, this is not inadequacy. This is just part of road building. This is what learning statistics is about; go back and repair, go on and build, go back and repair.
50 I find this useful because when I start a new section in statistics, say linear regression, I want to relate it to something they covered a while back. I ask ``Do you remember the analysis of variance?'' and look round to see if they do. Eventually someone says ``Oh, oh, pot hole'' and everyone laughs. The tension breaks: it is acceptable to admit that you have forgotten the analysis of whatsisname.
51 Because of the differences between the backgrounds, knowledge, and life experiences of educators and students, the meaning of classroom tasks must be negotiated. Metaphors are one way of enhancing communication. They may be a particularly pertinent strategy for doing so when students are more comfortable with the traditions and culture of the arts and humanities than the technology and logic of scientists. Metaphors and analogies may help to decrease the distance between students and statistics educators and improve students' active participation in statistical education.
52 The examples of students' reflections given in this paper suggest that to teach statistics to adult and possibly reluctant learners -- and be understood in our teaching -- we must first understand them. Before we can appreciate the students' frames of reference we need to acknowledge their experiences. We need to be aware of their beliefs so that we can communicate effectively. To know what meaning learning statistics has for them, we need to realise what their values are. In addition, we need to clarify our own goals as educators. Do we want to train students or educate them? What do we define as statistical education? According to Davydov and Markova, for any educational activity to develop ``it is necessary to ascertain and create conditions that will enable activity to acquire personal meaning, to become a source of the person's self-development....'' (Davydov and Markova 1983, p. 57). Davydov and Markova were referring to activity for schoolchildren, but their perspective is in accord with notions of adult development (Knox 1986; Slotnick et al. 1993) and the concept of lifelong learners as described by Candy et al. (1994).
53 In particular, with respect to the task of teaching statistics at university, what sort of self-development could students achieve? There are, I believe, three areas of development to aim for: intellectual development, the development of students' conceptions of statistics and approaches to learning it, and their personal growth.
54 Intellectual development is characterised by an improved capacity for abstract thinking, better methods of learning, and conscious control over the processes of learning. Educational activity in this context would be made up not only by those actions aimed at the mastery of knowledge, skills, and technical abilities, but also by those directed at enhancing mental capacities such as the ability to reflect, to understand the connections between statistical concepts, to see ahead, and to generalise.
55 It could also be expected that educational activity would result in students developing their conceptions of what statistics is. Research that colleagues and I conducted at Sydney University shows that currently many students of mathematics enter university believing mathematics to be a fragmented collection of rules, formulae, and algorithms, learnt in much the same mechanical way as one would learn touch typing (Crawford, Gordon, Nicholas, and Prosser 1994). We surely aim to teach students to view statistics as integrated and structured knowledge that enhances their understanding of the world.
56 Finally, and perhaps most importantly, an area in which an adult learner can develop by studying a statistics course is that of personal growth. Students who succeed in overcoming their reluctance to tackle mathematics-based courses, who conquer long standing difficulties with mathematics or a severe lack of confidence in their abilities to do mathematics, often report feelings of achievement such as are reported below by two statistics students.
I'm glad that I'm doing maths, 'cause for me that has a lot of value. Maths has kudos for me. I found that I actually knew more than some other people. Instead of being the one who knew the least. And that gave me an enormous sense of power. Or a bit of power anyway, that I could actually know more than somebody else.
58 Summing up her feelings, Sandra, after successfully completing her statistics course, reported that:
All my life it felt like I had this dark secret: that I felt really stupid about this area. I'd cover it up so no one would know. This really feels like growing up.
As the above two reports indicate, students can find learning statistics empowering.
59 The theoretical perspectives of Vygotsky and Leont'ev posit that society and individuals change and evolve together. It is through educated and thinking individuals that societies are enriched. Individuals, in turn, benefit from the progress of their communities. Statistics education has an increasingly important part to play in this dynamic relationship. Technological advances mean that people encounter a rapidly growing mass of information during their lives. There is, therefore, a growing need for statistically educated people who can assess information critically and use it effectively for their own benefit and that of their communities.
60 Learning, viewed through the lens of activity theory, is a dynamic, interactive process mediated by the social institutions in which it occurs and the position of the individual in these (Leont'ev 1981). From a pragmatic point of view, activity theory shifts the task of the statistics teacher from a preoccupation with how best to present information to a focus on what students are actually doing and what goals and experiences underlie their actions. It points to the need to look for the links between the social environment and the individual, to be aware of and responsive to individual diversity, and to regard affective elements as an integral rather than peripheral part of learning.
61 This paper has explored students' reflections on learning statistics from the perspective of activity theory. Research is needed on students' statistical activities in various contexts. As Hawkins et al. (1992, p. 99) point out, investigations tend to focus on the methods used to teach statistics rather than the ways students learn it. What are the different ways that students perceive their learning tasks? How do the conditions of learning affect students' goals and actions? Most importantly, what meanings do students use to regulate the information received? Attention to these questions is essential for ensuring that students' activities reflect the ideals of university education.
62 From a Vygotskian viewpoint, the aim of education is to increase the control and responsibility of learners. As Voigt (1994, p. 191) expresses it: ``There is the hope that if we take care of the quality of the negotiation of mathematical meaning we could improve the culture of the mathematics classroom as well as the education of the `competent layman'.'' The challenge for statistics educators is to find a way to communicate that enables students to view statistics as meaningful and useful knowledge that promotes their development and helps them tackle the complex issues of modern society. In these tasks there is much work to be done.
Earlier versions of this paper were presented at the Post Secondary Mini-Conference of the Mathematics Association of New South Wales (Sydney, April 1994) and the Australian Bridging Mathematics Network Conference (Sydney, July 1994). I wish to thank the reviewers of this paper for their helpful criticisms and suggestions.
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