Introductory Statistics Courses–A New Way of Thinking

Abstract

This paper presents a framework for the design and analysis of introductory statistics courses. This framework logically precedes the usual process of putting together the syllabus for an introductory statistics course. Four approaches, or paradigms, of statistics teaching are put forward, together with tools for deciding which blend of approaches is most useful in any particular case. These approaches do not correspond to the two traditional schools of thought in statistics education – probability-driven or data-driven – but rather constitute a new approach.

• Statistics education
• Course design
• First statistics course

1. Background

1 Statistics education is currently an active area of research. A review of recent literature shows that it is a dynamic, progressive, and sometimes controversial area. The introductory statistics course is especially important because it may be the only statistics course taken by future users of statistics. It will colour their whole approach and attitude to the subject.

2 It is an interesting paradox that most undergraduate courses in Australia today contain some type of formal statistical training, yet no field seems to be more misunderstood than statistics. When one of the authors was asked to design an introductory statistics course for an undergraduate degree in podiatry, we decided to investigate how such courses were designed. During this process we isolated four main approaches, or paradigms, that are commonly blended to produce introductory statistics courses. We designed a questionnaire to aid users in determining which approaches are best suited to particular situations. We hope that this new method of designing introductory statistics courses will not only prove to be useful, but will provide teachers and lecturers with a simple method for defining and clarifying course objectives. This is especially important for those who are not statisticians, but who need to prepare and teach statistics courses as service subjects.

3 For teachers designing an introductory statistics course, the issue of student anxiety warrants consideration at the outset. The affective domain of learning is sometimes left to chance, but recognising and addressing concerns about anxiety seems to be a vital initial step in the design of introductory statistics courses. Many articles refer to this topic, but there have been few definitive studies conducted in this area. Gourgey (1984), Harvey, Plake, and Wise (1985), Stevens (1982), and Yager and Wilson (1986) are some authors who have investigated this issue in depth. However, current literature on anxiety does little to provide suggestions that can be easily incorporated into daily practice. Ramsden (1992) states that any anxiety students have about a subject will affect the learning styles they exhibit. He goes on to suggest that it is possible for the same student to exhibit ‘deep,’ ‘surface,’ or ‘strategic’ learning styles within the same subject (Prosser 1994). The teaching method implemented should seek to promote the most effective of these styles – deep learning – which will reduce any anxiety that may exist with the student.

4 Another issue raised recently is the effect that students' attitudes or beliefs about a subject have on their anxiety. Gal and Ginsberg (1993) believe that students' preconceived ideas about the nature of statistics can produce anxiety, and they give a critical review of two surveys that are designed to assess student anxiety towards statistics. One is the “Statistics Attitude Survey” by Roberts and Bilderback (1980), and the other is the “Attitudes Towards Statistics” survey by Wise (1987). These studies highlight one approach that teachers can take when assessing students' anxiety. Slotnick (1992) addresses this issue by referring to two needs in Maslow's hierarchy of needs – self esteem and affiliation (Gage and Berliner 1988). He believes that these needs can be directly addressed by the teacher in the classroom. He argues that if students feel that they are valued members of a group and are given tasks that are worthwhile and relevant, then they will feel good about themselves, and their attitude towards the subject will improve.

5 To some extent, these issues can be addressed by use of appropriate teaching methods. Educators have responded to these issues by devising techniques that are sometimes practical, sometimes conceptual, and sometimes philosophical. Whatever the approach, all serve to improve the quality of teaching in statistics. One famous innovation is the “Minute Paper” of Mosteller (1988). This simple and practical idea has produced some extraordinary results. Zahn (1991, 1992) illustrates the effectiveness of this technique in a class of 250 students. Zahn has also extended Joiner's triangle of empowering relationships so that it can be directly addressed within the classroom. The Deming cycle has also been widely used in recent times (Deming 1986). Cobb (1993) discusses the issue of Total Quality Management, and Ramsden (1992) tackles other effective teaching strategies. Addressing these issues from the affective domain is a necessary but not sufficient step in the design of an introductory statistics course. To go further, we need to examine different approaches to the teaching of statistics at the introductory level.

2. The Approaches

6 The genesis of our ideas was an article written by Blum and Niss (1991). We found this article very pertinent to our thoughts about the design of introductory statistics courses. Blum and Niss's article is not specifically related to the area of statistics education, but rather is on the topic of general mathematics instruction. The authors discuss five arguments or theories or approaches to generalised mathematics instruction by considering the design of a course in terms of its purpose, the organisational framework, and the educational histories of the students. The approaches we have isolated are different from those proposed by Blum and Niss in that they are specific to the area of introductory statistics courses at the tertiary level, but the idea presented by them was original and easily adapted to statistics education.

7 The approaches were isolated as a result of an investigation into introductory statistics courses throughout Australian universities. We identified four main approaches: statistics as a branch of mathematics; statistics as analysis of data; statistics as experimental design; and statistics as a problem-based subject. At first glance, the appropriateness of these approaches may not be apparent because many introductory statistics courses in Australia appear to be a blend of them. Only a handful of specialised courses are directly linked with one particular approach. Table 4 in Appendix A displays the approaches. The planning and assessment sections of the table refer to the organisational structure of the course. The content that is included is typical of each particular approach. This does not necessarily mean that the content in each case is exclusive. Indeed, a topic such as confirmatory data analysis could easily appear in any of these approaches. The main differences among the approaches lie in the activities upon which the student learning is based, and are highlighted by the examples given. The same topic is used for the examples for all approaches.

8 In the past, comparisons have been made between these approaches and two well-documented schools of thought in statistics education. One school of thought suggests that statistics education should be probability or model-based, while the other believes that statistics education should be data-driven. Our view of designing introductory statistics courses differs from these schools of thought in two important ways. The first major difference is that proponents of the two schools of thought debate which is better, whereas the approaches we identified appear in no particular order. There is no hierarchal structure suggested here. In our view, different approaches will be more appropriate depending on the particular situation. There is no ‘best’ approach, only the one that best suits each individual case. Another major difference is that the two schools of thought give rise to debates containing conceptual or philosophical viewpoints, while the approaches we have isolated are based on educational theory and are designed to be used in practical situations.

3. How Do We Use the Approaches?

9 Designing an introductory statistics course is not necessarily a linear process. In most cases, the teacher considers issues such as course objectives, availability of resources, and faculty policies simultaneously in an effort to produce a satisfactory outcome. Of course, Table 4 alone is not sufficient for the user to be able to easily identify which approach, or approaches, are most appropriate to a situation. Hence, a questionnaire (Table 1 in Appendix A) was designed to prompt the teacher to think about the course in terms of student learning. As a result, the user will not only be able to identify which approach is most suitable, but also to determine the amount of emphasis that an approach will have in the course. Integrating the questionnaire and the approaches provides an effective mechanism for streamlining the design process by suggesting the following three steps.

3.1 Objectives

10 Having clearly defined goals that the students see as relevant and attainable is the most important aspect of course design. If relevance or attainability of goals is missing, the course will not be effective. As Zahn (1992) asks, “Are you willing to tell the students what results you intend to produce in the course? (Have you even decided for yourself?)” Teachers must also consider what type of skills and knowledge they envisage their students will have as a result of successfully completing this course. To achieve this result, Cobb suggests that we must first ask ourselves, “‘What are the things we do that are most basic to being a statistician?’ We then build a course from those activities…” (1993, paragraph 95).

11 However, educators have traditionally relied on themselves for ideas about course structure and materials. As a result, the design of courses is very often ‘teacher-conception’ dependent. Some educators rely on past experience, while others look at the work of other teachers and use this as a guideline. The approaches and questionnaire presented in this paper are designed to promote discussion about the course and to provide a template for teacher reference. This will allow educators to design courses that will not only take into account the individual requirements of particular situations, but will also provide the theoretical background needed to justify the result.

3.2 Questionnaire

12 All members of the design team (ideally several people rather than just one) should fill out the questionnaire in Table 1 separately and use the results as a starting point for discussion. This discussion should centre around the ten statements on the questionnaire relating to introductory statistics courses. Statements 1 through 8 are paired and relate to the four main approaches. Statements 9 and 10 concern the use of computers and communication of results.

13 When responding to the statements, the user must determine the level of importance of various aspects of the course. The items are not designed to be objective; rather, it is hoped that the subjective nature of the statements will promote thought and discussion on each issue.

14 The analysis of the questionnaire is simply a matter of obtaining an aggregate score for each pair of statements. However, these aggregate scores require further consideration; see Table 2. It is possible to obtain a total score that reflects conflicting opinions. For example, a total score of six for a particular approach may indicate two things. It may suggest that the teacher's reaction to that approach is consistent (3,3), in which case the user is prompted to continue. Alternatively, it may indicate mixed ideas ((5,1) or (4,2)). When this occurs, the user is advised to return to the original statements to review the importance of that topic.

15 It is important to maintain an appropriate balance of approaches within an introductory statistics course. If the blend of approaches is not balanced, the course may appear too diluted or confusing for the student. The troubleshooting table (Table 3) is included for cases where there are no clear approaches present, or conversely, too many approaches. In this situation, the user is prompted to return to the initial objectives for clarification.

3.3 Other Considerations

16 Once the objectives and approach are decided, other considerations should be addressed. Issues such as faculty policy, class size and social demographics, choice of text, and timetabling constraints all warrant consideration. However, teachers should endeavour to maintain their original thoughts about the course design, or many of the benefits of the initial stages of the process will be lost. The goal is to accommodate these other concerns into the objectives for the course, rather than allow these issues to dictate the course design.

17 Appendices B and C present case studies showing how our approach was used to design two statistics courses – a one-semester course for mathematics students and a two-semester course for podiatry students.

4. Conclusion

18 Starting with ideas presented by Blum and Niss (1991), we have developed a way of classifying introductory statistics courses that is at the same time theoretically and practically based, together with tools that help to determine which approach is most appropriate in a particular situation. The approaches and questionnaire can logically precede the design of a course by promoting discussion about such issues as structure, use of teaching aids, and assessment of learning outcomes.

Appendix A

Questionnaire and Approaches

The following questionnaire can be used as an aid in the design of introductory statistics courses. It is also useful as a means of critically assessing these same courses. Circle the most appropriate response.

Table 1– The Questionnaire

When designing an introductory statistics course there are three main steps to consider.

Step 1. OBJECTIVES: What do you hope to achieve with this course?

Step 2. QUESTIONNAIRE: Using the questionnaire provided, you will be able to consider your broad objectives while maintaining continuity and clarity in the blend of approaches utilised within the course. Respond to the ten statements, carry out the suggested analysis, and look in the results section (Table 2) for the blend of approaches that you wish to consider.

Step 3. OTHER CONSIDERATIONS: Once the objectives and approach are decided, other considerations should be included in the course design. These include faculty policy, class size and social demographics, choice of text, and timetabling constraints.

Analysis of results: Q1 + Q2 = Approach A Q3 + Q4 = Approach D Q5 + Q6 = Approach C Q7 + Q8 = Approach B

Table 2– Interpreting the results

Table 3– Trouble Shooting

The Questionnaire:

The questionnaire considers the four main approaches to the introductory statistics course that we have identified. By responding to the statements you will be able to isolate which aspects of statistics you wish to emphasise in the course, and give a general indication of which teaching approaches need to be applied. The approaches can be referred to as follows:

A) Statistics as a branch of mathematics

B) Statistics as analysis of data/as a laboratory subject

C) Statistics as research design/experimental design

D) Statistics as a problem-based subject

Table 4– The Approaches

1. Statistics as a branch of mathematics. Planning: Weekly lectures, some tutorials. Typical content: Combinatorics, theory of probability, random variables. Typical activities: Proofs and derivations, advanced mathematical skills. Typical assessment: Mid-semester and end-of-semester exams. Example: The mean of 10 crop yields is 25.3 and the standard deviation is 3.55. What would be the value of s if another crop yield of 28.1 were added to the values?

2. Statistics as analysis of data/as a laboratory subject. Planning: Weekly labs with tutorials, class discussion, and group interaction. Typical content: EDA, methods of data collection, hypothesis testing, regression and correlation. Typical activities: Collecting, investigating, and analysing data; confirming hypotheses. Typical assessment: Regular class tests, lab reports, assignments. Example: Collect data on these crops and analyse them. Which crop is significantly better than the others?

3. Statistics as research design/experimental design. Planning: Discussion groups, lab work, group discussion and interaction. Typical content: Analysis of the effect of variables on a response, critical analysis of published papers, understanding regression and ANOVA, interpreting p-values. Typical activities: Designing an experiment and collecting data, interpreting results; theory is discovered rather than presented. Typical assessment: Lab assignments, non-mathematical exams, reports on research papers. Example: What sources of variability could affect the crop yield, and how can we estimate their effects? Discuss the type of research approach used in this research paper and give a critical analysis. 4. Statistics as a problem-based subject.

Planning: Group discussion, project, or consulting work. Typical content: EDA, design of experiments, ANOVA, consultation, and report writing. Typical activities: Students solve problems from their field, working as consultants. Theory is introduced and developed as needed. Typical assessment: Progress reports, essays, final reports, and presentations. Example: A local farmer has asked for a complete analysis of how she can maximise her crop yield.

Appendix B

Case Study 1

Course: Bachelor of Applied Science (Mathematics) Subject: STATISTICS ONE Duration: ONE SEMESTER Hours per week: SIX

B.1 Objectives

Initial discussion produced the following broad objectives:

• Demonstrate an understanding of basic statistical concepts.

• Explain the meaning of common statistical terms.

• Discuss problems associated with collecting data.

• Carry out investigations using summary statistics, graphs, and techniques of exploratory data analysis.

• Make inferences from sample data to populations.

• Use a statistical package.

• Communicate effectively the results of statistical analysis.

• Lay a good foundation for further studies in mathematical and applied statistics.

B.2 Questionnaire

The questionnaire was then introduced to aid in selecting the most appropriate approach to implement. After discussion, the response to each statement was recorded as illustrated below.

Table 5– Results for STATISTICS 1

Analysis of results: Q1 + Q2 = 3 A Q3 + Q4 = 3 D Q5 + Q6 = 5 C Q7 + Q8 = 9 B

The analysis of responses indicated that Approach B was most appropriate to our course objectives – Statistics as analysis of data/as a laboratory subject. The table of approaches provided us with ideas on course structure, content, assessment, and the best method to promote deep learning in the students.

B.3 Other Considerations

Once the approach to the course was decided, it was time to discuss other factors affecting the course. These included the following.

• This course is not a service course, and we know that all students will go on to Statistics 2 – a mathematical statistics course. Therefore, there is little theory, and the emphasis is on developing statistical intuition.

• Class sizes may be as large as 100 students.

• Policy dictates that students should be in the computer labs for a period of two hours per week.

• Faculty Policy covers issues such as assessment procedures.

B.4 Final Design

Further discussions following the suggestions given by the approach of statistics as analysis of data/as a laboratory subject produced the final result. First, a rough outline of the course content was designed using the ideas presented in the table. The discussions then moved to considering the most appropriate method of implementation that would produce the best learning outcomes for the students. As a result, it was determined that the course presentation should follow the format in the table of approaches. The six hours per week were divided into lectures, discussion, tutorials, and experimentation and laboratory work, supported by at least six hours per week of group or individual study. Two hours per week were allocated to the computer laboratory. A set of Minitab worksheets was designed to encourage development of written expression and analytic thought, and to give students experience in actually doing statistics. The assessment also followed the ideas suggested in the table of approaches. The inclusion of two group assignments was considered important for group work skills and organisation. Allowing the students to work in groups of three helped to reduce teacher marking time. The ten Minitab worksheets were to be completed individually and handed in at the end of each laboratory class, with the marks of the best eight to be used in calculating the final mark.

B.5 Final Result

Textbook:

The text selected was one that was most conducive to our approach: Moore, D. S., and McCabe, G. P. (1993), Introduction to the Practice of Statistics (2nd ed.), New York: Freeman.

Topics Selected:

1. LOOKING AT DATA: Distributions; relationships

2. PRODUCING DATA

3. PROBABILITY: Probability rules; probability calculations; from probability to inference

4. INFERENCE: Introduction; inference for distributions; count data; regression; analysis of variance; distribution-free inference

Assessment:

• Two assignments (20% total). Students work in groups of three.

• Ten laboratory worksheets (20% total).

• Final examination (60%).

Appendix C

Case Study 2

Course: Diploma of Podiatry Subject: RESEARCH METHODS 1 AND 2 Duration: TWO SEMESTERS (36 Weeks) Hours per week: THREE

C.1 Objectives

Initial discussion with representatives of this department produced the following broad objectives:

• Develop an understanding of statistical reasoning and use this to critically analyse current publications.

• Introduce an appropriate set of statistical concepts and techniques that provide a basis for effective understanding and use of statistical methods.

• Carry out investigations using summary statistics, graphs, and techniques of exploratory data analysis.

• Make inferences from sample data to populations.

• Use a statistical computer package.

• Be able to communicate clearly the results of statistical analysis in a variety of different media.

C.2 Questionnaire

Table 6– Results for RESEARCH METHODS (Podiatry)

Analysis of results: Q1 + Q2 = 2 A Q3 + Q4 = 9 D Q5 + Q6 = 5 C Q7 + Q8 = 5 B

The analysis of results indicated that Approach D was most appropriate to the course objectives – Statistics as a problem-based subject. The approach provided us with ideas on course structure, content, assessment, and the best methods for promoting deep learning in the students.

C.3 Other Considerations

• This is a service course. The representatives expressed concern as to how this course may affect students' commitments to other subjects.

• The class size is approximately 20 students.

• Faculty policy dictates that students must attend at least 80% of lectures to obtain a passing grade.

C.4 Final Design

The identification of Approach D causes certain topics and methods to predominate in the course design. As this is a service course, some content comes from the client, and we fill in the details and the gaps. In this course, the emphasis was on using current publications to highlight the use of statistics as a tool within podiatry. It was stressed that students should develop an appreciation of statistical techniques through writing, understanding, and interpreting results. First, the selection of content was discussed following the ideas presented in the table. It was decided that the course would consist of one major project, where the students would work in groups. This would serve to develop students' analytic thought, appreciation of statistics, and communication skills. Classes would be held entirely in a computer laboratory that had access to teaching facilities. Introductory techniques would initially be discussed within the class, and then students would be free to select from a composite of current publications, data sets, and problems to work through. As a result, classes would be informal and would consist of general discussions, small group discussions, and individual work. Assessment would consist of students' presentation of problems from publications and regular progress reports on the major group project, both in written and oral form. The exam would be held before the end of semester, so as to minimise the work load on students at the end of semester.

C.5 Final Result

Textbook:

No single textbook was found that was appropriate. However, a detailed reference list was constructed.

Course Planning:

Semester 1: Introduction; introductory problems (each student given a problem highlighting a different statistical approach); presentation of solutions to problems (during which the basic statistical theory will be discussed); theory exam; selection of project; data collection.

Semester 2: Revision of semester 1 material; more specific statistical techniques; further problems and presentation of results; theory exam; presentation of projects.

Assessment:

Students will be given an aggregate mark after the completion of the second semester. Students must successfully complete each piece of assessment to obtain an overall pass in the subject.

Semester 1: Exam 30%, problem presentation 10%, project report 60%.

Semester 2: Exam 25%, problem presentation 10%, project report 65%.