Introductory Statistics with a Qualitative Emphasis

by

Michael Bulmer
Department of Mathematics, University of Queensland

 

Abstract

An introductory statistics subject often focuses on developing the necessary quantitative skills for students to be able to work in their discipline. While these quantitative skills are indeed important, at its heart statistics is about the process of scientific investigation. We are currently implementing learning activities which increase the emphasis of the qualitative understanding of statistics and its role. This paper presents some of these activities, together with evaluations and reflections on their effectiveness.

 

Introduction

A service subject in introductory statistics is often seen as providing quantitative skills in a degree program. Indeed, our students in statistics habitually refer to their "maths classes" and "maths assignments". While mathematics and quantitative skill are an important part of statistics, a preoccupation with these aspects can obscure the deeper understandings that students should be seeking. The eminent statistician George Box, quoted by Higgins (1999), captures this well:

Statistics is, or should be, about scientific investigation and how to do it better, but many statisticians believe it is a branch of mathematics... Now I agree that the physicist, the chemist, the engineer, and the statistician can never know too much mathematics, but their objectives should be better physics, better chemistry, better engineering, and in the case of statistics, better scientific investigation. Whether in any given study this implies more or less mathematics is incidental.

We are currently looking at ways to change the quantitative focus in statistics teaching. It is a belief of this study that an introduction to statistics which has an emphasis on the qualitative skills involved in creating and analysing data is at least as useful as one based on quantitative methods. Learning quantitative skills alone is seen as a form of shallow learning, whereas an appreciation for qualitative skills moves towards a deep understanding of statistics.

A second assumption is that students can obtain higher qualitative understanding through physical interaction with statistical scenarios. Traditional statistics teaching involves the presentation of abstract knowledge combined with existing or simple data sets. In recent years this has been augmented with computer simulations to give students a better feel for the role of randomness. However, while both of these techniques are useful and necessary, both are also removed from concrete experience. We have adopted activity work in lectures to provide this concrete level, as well as motivating the need for simulation and theory. This echoes constructivist approaches in other sciences, as described in Dawson (1994).

 

Environment

This study is being performed with the teaching of the first-year subject MS150 Data Analysis and Experimental Design. This is a new subject for 1999, part of the Mathematics Department's move to a unitised course structure.

Currently the focus of first-year statistics teaching in the Department is the large service subject MS113 Elementary Statistics 1, an 8 credit point subject offered to a range of other departments, particularly in the biological and health sciences. With the move to unitisation, this subject is being replaced by a number of new subjects, many of which are more specifically tailored to their target disciplines than the general material in MS113. Most of these replacements are also to be offered in later years of their respective courses. To continue to offer a broad introduction to statistics, particularly for students enrolled in the Bachelor of Arts and combined degrees, MS150 has been introduced as a 12 credit point first-year subject.

MS150 is run with three lectures per week, along with a weekly tutorial hour and a weekly computer laboratory hour. This is an increase of one lecture and half a laboratory per week over MS113. Due to staffing arrangements, two of the lectures and the tutorial are shared with the MS113 students in traditional mode, leaving the focus of this study on the additional lecture and laboratory.

There are no mathematics prerequisites for MS150 meaning that some students can come into the subject with very limited numerical and algebraic skills, while others may be doing tertiary mathematics. The emphasis on qualitative skills plays an additional role here, making it easier to cope with such a diversity of backgrounds.

 

Methods

The goal of improving qualitative understanding has been approached using three methods: the adoption of activities for lectures, the alteration of assessment practice, and the enrichment of laboratory work.

Activities
The third lecture in each week is devoted to activities which highlight some statistical issue. These typically involve carrying out an experiment to collect data and then discussing how it could be analysed. Some examples are given below.

Students are provided with a workbook for them to use during the activities. In it they can record data from their experiments along with comments from the discussion that arises. They are also asked to write a reflective summary of each activity to highlight the statistical issues that arose from it. This follows the general use of student journals presented by Meyers and Jones (1993), encouraging students to create new mental structures by reviewing material from a distance. The journals are also to be used for evaluating the approach.

Activities also aim to encourage cooperative learning, as discussed by Garfield (1993) and Magel (1998). This is felt to improve statistical learning, as well as fostering social and communication skills.

Scheaffer et al. (1996) present materials for an activity-based statistics subject. Many of their ideas for activities have been used, though usually after various modifications for our setting. As an example, one of the first activities used had students pairing up to play a game. A chalk line was drawn near one end of a bench. One student then tried flicking a five-cent piece from the other end with the aim of getting the coin to stop on the line. This is quite difficult - the other student measured the distance from the coin to the line, either positive or negative depending whether it stopped before or after the line. This was repeated 25 times. The students then plotted the data against time and discussed the pattern they observed. This lead to a more general discussion about what you would see if a player was improving, remaining the same, or getting worse. There are two aspects to this, the variability and the centre. These ideas were then related to the application of process control in industry, a topic that would normally not be covered in a first-year course. Finally, a prize was given to the pair whose plot seemed to show the greatest improvement, as voted on by the class.

This activity was very enjoyable for the students because of the active physical dimension, with students deeply engaged in the game and the interaction. The discussion that followed was also more heated than for activities where the students remain seated.

However the activity took longer than planned, not allowing time to look at the data in other ways. It seems likely that any activity with a large "physical" component will require a balancing with less summary discussion. Such issues have been addressed in part by Gnanadeskan et al. (1997). The control of setting, being able to move from the active state to the discussion state, becomes vital in such activities.

A later activity gave the students an introduction to the basic ideas of scientific induction. An automated learning system, described by Bulmer (1996), was used to present a dialogue with someone, who knew nothing, and was learning about the world from an oracle. In the classroom this was done as an audio presentation, using the synthesized voices on the Macintosh, an interesting change from standard audiovisual delivery. The activity started with the following dialogue:

 

Peter is Paul's father. Okay
Peter is male. Okay
Is Peter male? Yes
Is Paul male? Yes, I think so
Is Alice male? Yes, I think so

 

Before the computer was asked each question, the class discussed what the answer might be. After seeing the answer for Paul, everyone agreed on what it would say for Alice. The dialogue then continued, with class suggestions as to what we could say about John to help the computer to decide his sex:

Alice is not male. Okay
Is Paul male? I don't know
Is John male? I don't know
John is Alice's father. Okay
. Is John male? Yes, I think so

 

The students then examined a more complicated scenario:

Mary is Paul's mother, Bob's daughter, Peter's wife, Jill's sister, and Alice's aunt. Bob is Paul's grandfather and George is Alice's grandfather. Peter is Paul's father and John is Alice's father. Jill is Paul's aunt and John's wife. Kelly is Mary's sister. Jenny is George's daughter. Who is Alice's mother?

There are three plausible answers to this question, though students have to ignore any preconceptions to find them all. This aspect was particularly valuable since it reflected the general need for students to be aware of their preconceptions. After some debate, the class discovered two of the candidates for Alice's mother.

It is worthwhile noting that most of the activities can be done without a computer, making their use quite versatile and more socially engaging. It again helps with diversity and equity since there are no technological impediments placed before students.

 

Assessment

The assessment in the course has been changed in two ways. Firstly, the final exam was replaced by project work during semester to make assessment more authentic. This follows the aims outlined by Chance (1997), encouraging the understanding of the statistical process, improving statistical and computer literacy and communication and collaboration skills, establishing a dynamic assessment process, and increasing student interest in statistics. Mackisack (1994) gives an overview of the other benefits of projects. For instance, in addition to providing holistic understanding for students, the students also get an appreciation of the practical issues involved in carrying out experiments and collection data, an outcome also encouraged by Higgins (1999). The data that students generate can later be used to provide genuine examples in lectures.

In our context, the projects typically expand on some issue raised in an activity. For example, the first project this semester built on the process control activity by having the students look at the behaviour of some quantity in their lives that varies over time, such as the hours they sleep each night. (One dedicated student measured the volume of every drink he had over a two-week period.) This project emphasised the identification of variability and bias in measurement and the description of the behaviour above the simple quantitative analysis of the data.

Current assessment practice in mathematics and statistics involves breaking a solution into parts and assigning numerical weights to each part. The final mark is then the simple sum of the marks awarded to each part. By its nature, this is a poor tool for holistic assessment. We want an emphasis on qualitative understanding and communication skills, both hard to quantify in this fashion. To that end our second change in assessment practice has been to grade assignments, laboratories, and projects using a criteria sheet. This has had four dimensions:

The numerical accuracy of their work is still important, but the use of these explicit criteria make it clear that there are other skills in statistics that are equally important.

 

Laboratories

Computer laboratories in a statistics subject serve two roles. They can be used as teaching tools, giving the students the opportunity to explore ideas that are infeasible to examine by hand. But being able to use a statistics package is also a profes

The laboratory exercises for this subject had already been rewritten in 1998. Some of these used simulation to give students a better grasp of the different roles that randomness plays in statistics, while others used statistical software to analyse examples that were more realistic than the ones feasible in lectures. All of the exercises required students to write reports on their analysis.

These have been extended in 1999 by developing web-based tools which the students can use to carry out experiments, adding the important first stage to the process that is usually left out in class work. For example, students had a virtual group of rats for which they could specify diets with varying amounts of calcium and magnesium, the computer returning the resulting blood pressure for that rat. They had to devise an experiment and analyse the results to discover the effect, and any interactions, of the two minerals on blood pressure. Such physical settings are naturally beyond the scope of a statistics course, but this kind of simulation gives students a better feel for the issues involved.

 

Evaluation

The new aspects of the subject are being evaluated in a number of ways. In addition to standard questionnaires on student attitudes, students have also been asked to keep journals of their activity work and to write reflections on each activity. Primarily introduced as a learning aid, these reflective journals will be borrowed from students at the end of semester to obtain an impression of the types of knowledge they are taking from the activities. The lecturer and tutor are also keeping reflective journals on the activities and laboratories, respectively, again serving the dual role of learning aids (for the staff) and evaluation tools.

At the time of writing, the evaluations have not yet been collected and analysed. However, one crude comparison comes from the mid-semester test, a multiple-choice test sat in common by all MS150 and MS113 students. Of the 346 MS113 students, only 1 obtained full marks on the test, whereas 3 out of 14 MS150 students obtained full marks. There are many other factors to consider in this outcome, but it is nevertheless encouraging as it suggests that the practical work in activities, laboratories, and projects is transferring to understanding of the traditional material.

 

Discussion

The informal feedback from students and the tutor has so far been very positive. However, there are a number of outstanding issues that are worth discussing.

The 1999 enrolment in MS150 has only been 14 students (due to an administrative error which left MS113 still available to general students). This raises the concern of scalability, since the full audience for introductory statistics at this university is in the thousands. A few of the activities, particularly the physical ones, would become a logistical problem for significantly larger classes, but many of the other activities would benefit from larger sample sizes. There are no problems with the laboratory exercises since these are available flexibly on the web and only require software access to complete. As expected, the most difficult approach for large classes is project work. Giving quality feedback on large numbers of detailed reports is extremely time consuming. Some of the activities and limited project work will be trialed on a larger group in 2000 to determine if it is feasible to scale this approach.

Whenever a new teaching mode is considered, such as the use of activities for improving statistical understanding, the immediate question arises as to whether the students have the necessary learning skills to match the new mode. This is certainly the case with activities since they are a departure from traditional teaching (though they may in fact be closer to the secondary school experience that the first-year students have come from).

Related to the issue of learning skills, it is important to ensure that students receive an equitable learning experience from the new teaching method. Of concern is whether or not students who feel very comfortable with the traditional lecture setting will be able to get as much from the activity-based setting as other students. There is obvious potential for alienation, particularly if the activities are considered too "childish". This will be evaluated through the questionnaires on attitude.

Velleman and Moore (1996) highlight the concern that any approach to student-centred learning must achieve a balance between student choice in what they do and a structured presentation of the discipline. The use of guided activities should manage this, but again it will be addressed in the evaluation.

At the first meeting with students, they were asked whether they would be happy to do project work during semester rather than have a final exam. In choosing the project work, they now have weekly assignments, weekly laboratory reports, weekly journal reflections, and several projects to be assessed. Though all of this is formative, there is a concern that the assessment burden may be too high.

Finally, the successful use of activities seems very dependent on the parameters of the teaching space available. The activities in this semester have mostly been successful despite the current space being relatively cramped. It is hoped to try a different style venue in 2000 to determine if it has any effect on student perceptions of the activities.

Overall, pending final evaluations, the use of activities and projects seems to have been successful for the students and, just as importantly, has also made the teaching of the subject very enjoyable and rewarding.

 

References

Bulmer M. (1996) Inductive Theories from Equational Systems, Information, Statistics and Induction in Science, ISIS'96 (Dowe D.L., Korb K.B., and Oliver J.J., eds.). World Scientific. pp. 260-268.

Chance B.L. (1997) Experiences with Authentic Assessment Techniques in an Introductory Statistics Course, Journal of Statistics Education, 5 (3) .

Dawson C. (1994) Students' prior knowledge and teaching approaches: transmissive and constructivist teaching, Science Teaching in the Secondary School. London: Longman. pp. 47-55.

Garfield J.B. and Gal I. (1999) Assessment and statistics education: current challenges and directions, International Statistical Review, 67 (1), pp. 1-12.

Gnanadeskan M., Scheaffer R.L., Watkins A.E., and Witmer J.A. (1997) An Activity-Based Statistics Course, Journal of Statistics Education, 5 (2) .

Higgins J.J. (1999) Nonmathematical Statistics: A New Direction for the Undergraduate Discipline, The American Statistician, 53 (1), pp. 1-6.

Mackisack M. (1994) What is the Use of Experiments Conducted by Statistics Students?, Journal of Statistics Education, 2 (1) .

Magel R.C. (1998) Using Cooperative Learning in a Large Introductory Statistics Class, Journal of Statistics Education, 6 (3)

Meyers C. and Jones T.B. (1993) Promoting Active Learning: Strategies for the College Classroom. San Francisco: Jossey-Bass.

Scheaffer R.L., Gnanadesikan M., Watkins A., and Witmer J.A. (1996) Activity-Based Statistics. Springer.

Velleman P.F. and Moore D.S. (1996) Multimedia for teaching statistics: promises and faults, The American Statistician, 50 (3), pp. 217-225.

 

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