Introductory Statistics Courses--A New Way of Thinking

Katrina Roiter and Peter Petocz
University of Technology, Sydney, Australia

Journal of Statistics Education v.4, n.2 (1996)

Copyright (c) 1996 by Katrina Roiter and Peter Petocz, 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 authors and advance notification of the editor.


Key Words: Statistics education; Course design; First statistics course.

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.

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
1 = Strongly disagree; 2 = Disagree; 3 = Neutral; 4 = Agree; 5 = Strongly agree.
Q1.   Proofs and derivations of the        1  2  3  4  5
      main results are important.
Q2.   The theory of probability and        1  2  3  4  5
      random variables plays an
      important role in this course.
Q3.   Students are free to study           1  2  3  4  5
      different techniques as long as
      they develop an understanding
      of statistical ideas.
Q4.   One or two open-ended projects       1  2  3  4  5
      are the major teaching/learning
      tools in this course.
Q5.   The main emphasis is on how to       1  2  3  4  5
      plan experiments to collect high
      quality data, and to consider the
      effects of planning on results.
Q6.   Students should be able to critique  1  2  3  4  5
      research papers with respect to
      the design of an experiment.
Q7.   Students need to learn a small       1  2  3  4  5
      basic set of standard statistical
      techniques.
Q8.   Students analyse given sets of       1  2  3  4  5
      data to practice using standard
      techniques.
Q9.   It is important for students to      1  2  3  4  5
      be able to present their reports
      in both written and oral form.
Q10.  Computers will be used as a          1  2  3  4  5
      tool for most applications to
      practical situations.

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
       Possible
       Combinations  
Total  (in any order)  Comment
10     (5,5)           This is considered the most essential
                       approach to this course and will form
                       the foundation of the design.
9      (5,4)           This approach will have a major influence
                       over the course design.
8      (4,4)           This approach is still an important part
                       of the course, with all main aspects
                       considered equally important.
       (5,3)           You may need to consider further how
                       important you feel this approach is.
7      (4,3)           This is considered a minor component
                       in the course design.
       (5,2)           You may need to revise your responses to
                       make this approach more important or to
                       affirm that it is a minor approach.
6      (3,3)           This is considered a minor component
                       in the course design.
       (4,2)           Further consideration may be needed
                       to determine the importance of this
                       approach.
       (5,1)           You may need to revise your responses
                       to make this approach more important or
                       to confirm that it is a minor approach.
5      (3,2)           This is considered a minor approach
                       in the course design.
       (4,1)           Further attention may need to be taken
                       to determine the importance of this 
                       approach.
4      (3,1)           This approach is not deemed to be of
       (2,2)           significant importance.
3      (2,1)           This approach is not important for your
                       course.
2      (1,1)           This approach is not at all important
                       for your course.

Table 3 - Trouble Shooting
Approaches                         Comment

There appear to be no significant  The course needs to be more
approaches in the course.          specific.  Rethink your objectives.

Two approaches appear              Further consideration is needed
to be significant                  to determine if you are trying to
in the course design.              include too much in the course.
                                   Having two different emphases in
                                   this course may dilute the approach.

Three or more approaches           Rethink the objectives of this
appear to be significant.          course.

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.

2. Statistics as analysis of data/as a laboratory subject.

3. Statistics as research design/experimental design.

4. Statistics as a problem-based subject.


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:

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
Question  1  2  3  4  5  6  7  8  9  10
Response  1  2  2  1  2  3  4  5  5  5

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.

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:


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:

C.2 Questionnaire


Table 6 - Results for RESEARCH METHODS (Podiatry)
Question  1  2  3  4  5  6  7  8  9  10
Response  1  1  4  5  2  3  3  2  5  5

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

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%.


References

Blum, W., and Niss, M. (1991), "Modelling, Applications, and Links to Other Subjects -- State, Trends and Issues in Mathematics Instruction," Journal of Educational Studies in Mathematics, 22, 37-68.

Cobb, G. (1993), "Reconsidering Statistics Education: A National Science Foundation Conference," Journal of Statistics Education [Online], 1(1). (http://www.amstat.org/publications/jse/v1n1/cobb.html)

Deming, W. E. (1986), Out of the Crisis, Cambridge: MIT Press.

Gage, N. L., and Berliner, D. C. (1988), Educational Psychology (2nd ed.), Boston: Houghton and Mifflin.

Gal,I., and Ginsberg, L. (1993), "Feeling Certain About Uncertain Things: The Role of Beliefs and Attitudes in Learning Statistics," in Proceedings of the Section on Statistical Education, American Statistical Association, pp. 29-38.

Gourgey, A. (1984), "The Relationship of Misconceptions about Math and Mathematical Self-Concept to Math Anxiety and Statistics Performance," Annual Meeting of the American Educational Research Association, New Orleans.

Harvey, A. L., Plake, B. S., and Wise, S. L. (1985), "The Validity of Six Beliefs About Factors Related to Statistics Achievement," paper presented at the Annual Meeting of the American Educational Research Association, Chicago.

Mosteller, F. (1988), "Broadening the Scope of Statistics and Statistical Education," The American Statistician, 42, 93-99.

Prosser, M. (1994), "Student Learning," presentation to the Statistics Education Workshop, Melbourne.

Ramsden, P. (1992), Learning to Teach in Higher Education, New York: Routledge.

Roberts, D., and Bilderback, E. (1980), "Reliability and Validity of a Statistics Attitude Survey," Educational and Psychological Measurement, 40, 235-238.

Slotnick, H. (1992), "Why They Work," in Proceedings of the Section on Statistical Education, American Statistical Association, pp. 72-77.

Stevens, P. (1982), "Computer Programs to Reduce Math Anxiety," reproduced by EDRS, report submitted to the National Institute of Education, Education Resources Information Center, US Department of Education.

Wise, S. (1987), "The Development and Validation of a Scale Measuring Attitudes Towards Statistics," Educational and Psychological Measurement, 45. 401-405.

Yager, G., and Wilson (1986), "Ten Suggestions on Teaching Research Counselling to Students," Annual Meeting of the North Central Association for Counsellor Education and Supervision, Kansas City.

Zahn, D. (1991), "Getting Started on Quality Improvement in Statistics Education," in Proceedings of the Section on Statistical Education, American Statistical Association, pp. 135-146

----- (1992), "Notes on the Use of Minute Papers in Teaching Statistics Courses," in Proceedings of the Section on Statistical Education, American Statistical Association, pp. 62-68.


Katrina Roiter
Peter Petocz
peterp@maths.uts.edu.au
School of Mathematical Sciences
University of Technology, Sydney
PO Box 123, Broadway NSW 2007
Australia
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