In the recent years, statistics educators have been actively rethinking how students learn statistics and how to teach introductory statistics. Furthermore, the current technology continues to open new opportunities for developing innovative teaching strategies. This article presents a paradigm, the PACE approach, for teaching the introductory statistics. PACE stands for projects, activities, cooperative learning using computer and exercises. The approach begins with in-class hands-on activities and cooperative team work. The class lectures are organized to provide the basic concepts and guide students through the activities using team work and computer to help students understand the concepts and problem-solving strategies. Exercises are designed to reinforce the basic concepts and to practice solving real world problems. Projects are self-selected by students under some guidance provided by the instructor. Report writing and oral presentation are emphasized. It is believed that self-selected projects reflect students interest, and hence better motivate them to be active learners. The paradigm of integrating these components together in a structured system motivates students to actively involve with their learning.

Introductory statistics has often been taught as a traditional lecture
and note taking approach. Students tend to think it is a course that fills
with memorizing formulae and finding formulae to fit the problem. Hogg
and et al.(1992) said it well : " *Statistics are seldom designed with
any idea of what it is that students are supposed to be able to do as a
result of having taken the course. .. Statistics is seen as a "subject",
rather than a problem solving tool to be used in the scientific method,
or a useful way to look at the world around us*". In recent years, statistics
educators have been actively rethinking of different ways to deliver the
introductory statistics. Many suggestions have been made. The data-driven
concept is now a commonly accepted approach. A variety of innovative teaching
strategies have been conducted by many statistics educators. Many successful
implementations can be found in the literature. The MAA Notes #26 (edited
by Gordon and Gordon, 1992) collected a list of interesting and useful
ideas and innovative curricula. Cobb(1993) listed and discussed more than
twenty statistical education related projects that were supported by the
National Science Foundation. The Journal of Statistical Education and the
International Statistical Review are two rich resources for issues related
to statistical education.

The PACE approach is a summary of several years of implementing different strategies using the idea of data-driven in teaching introductory statistics. PACE stands for "Projects", "Activities", "Cooperative Learning", and "Exercises". Various hands-on activities and different type of projects were experimented through several years of teaching introductory statistics. They were used either in separate semesters or combined in the same semester. During the early years, because of the limitation of access to computers, most of hands-on activities were mainly used as in-class demonstrations. Different project approaches were experimented in several semesters. The complete implementation of the PACE strategy was conducted in the Fall of 1996. The class size was 28. Students were from departments related to science and technology. Majority were either junior or senior. The introductory course is required. These students usually had precalculus or higher mathematics background with little or no exposure to statistics. Male and female students were about equal.

*"Statistics is the science of data. .... They are numbers with context."*

- People learn better by constructing the knowledge themselves through guided processes.
- Practice and feedback are essential ingredient for sustaining new concepts,
- Active problem-solving in a team work environment promotes active learners.

Each component of the PACE strategy is not new. There have been many
developments that are targeted at a specific component. However, there
are not much discussion of developing a general framework of integrating
several components together. The PACE strategy attempts to provide a structured
framework for integrating projects, hands-on activities that are conducted
cooperatively in a computer classroom environment. The paradigm emphasizes:

- actively engaging students in their learning process,
- closely relating statistics as a scientific tool for solving real world problems,
- providing opportunity for students working as a team,
- actively involving students into report writing and oral presentation.

A traditional lecture has the advantage of being very organized and
giving students many opportunities for reinforcing a concept with exercises.
This important learning techniques may have been discouraged or sacrificed
in many new innovative approaches. The PACE strategy intends to integrate
the new innovative techniques and to maintain the advantage of the traditional
approach of organized structure and reinforcement.

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The complete implementation that I conducted in the Fall of 1996 was given in a computer classroom environment equipped with 28 Macintosh computers, one instructor station and needed software such as M.S. Office and Minitab statistical package. A software, called Look-At-Me, allows every student to connect his/her computer with the instructors station. Students see the instructors screen as well as their own work simultaneously. This allows the instructor and students work together at the same time for the same project. This feature provides a nice setup for instructor to give an organized lecture even during the time when students are working with computers. It also helps the instructor to focus on teaching statistics, rather than, spending time in helping individual student on the computer accidents.

A typical procedure of a PACE teaching strategy begins with a hands-on
in-class activity. Each activity is usually an in-class team project that
is designed to introduce several new concepts and problem-solving strategies.
Figure One shows the flow chart of the guideline for the beginning of the
semester.

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Projects provide students opportunities to actually conduct a study themselves and learn how to write a scientific report and presentation. There are various ways to conduct projects in an introductory statistics course. They can be open-ended projects or using real data sets for statistical analysis. They can be designed by the instructor or selected by students themselves. Projects designed by the instructor are usually more structured. However, it may not be relevant to students interest. As a consequence, the level of enthusiasm is not easy to maintain. Self-selected projects, although not as structured, they tend to make the learning process much more interesting throughout. Students tend to be more enthusiastic in solving their own projects. As a result, students have shown higher motivation and interest in learning how statistics can be applied to solve their selected projects. Fillebrown (1994) reported a similar experience of implementing selected-project approach in the elementary statistics course.

It is critical that students' projects are guided through out the semester. To make the projects work smoothly, the guidelines are important. These guidelines include

- how the project should be conducted,
- where to find the related information or data sets,
- how the project will be graded, and
- how a report should be prepared.

In an introductory statistics course, typical types of problems include one sample, two independent sample, paired sample , categorical , and correlation and regression problems. Using the PACE strategy, the concepts, sampling design, and characteristics of these problems are introduced as part of descriptive and exploratory analysis. It is appropriate to introduce planning, project design and data collection in the early stage when exploratory analysis is introduced.

Based on the experience, students usually look for data or projects which they are familiar with and are interested in. Students from health related areas find projects related to health; while students in Geography would be interested in weather and environment issues. Many students look for data related to their campus life or sports. I also notice that more and more students search through internet to find interesting data for their projects.

Three reports were required from the projects.

- First report: students are asked to describe the data they obtained by describing

- where they find the data,
- how the data are collected,
- how the variables are matched with the type of data discussed in class as two independent sample data, paired data, regression and correlation data and categorical data.
- What questions they are interested to investigate.

- The second report: students are asked to analyze the data using descriptive and graphical tools. A computer package with graphical capability is necessary. In this course, we use Minitab. In addition to the analysis, students also learn how to integrate computer output, graphs and spreadsheets into their report.
- The third report: students are asked to complete their final analysis and report using appropriate statistical methods for their project.

The use of hands-on activities in class has been developed and advocated by statistics educators for years. Schaeffer, Gnanadesikan, Watkins and Witmer (1996), Tanner and Wardrop (1992), etc. are among the most active educators who develop and promote the activity-based curriculum. The use of hand-on activities is an important part of the PACE strategy. Students conduct hands-on activities in class guided by the instructor and collectively enter the data into computer. Two students are paired to do the computer work and write a brief report for each in-class hands-on activity. Students are required to integrate the computer results and graphs into their reports. The emphasis is on the entire process of problem-solving. The pairing can be different from activity to activity. Since most of the activities for each in-class project are completed during the class time period, the typical problem of finding extra time to work on a project is no longer a serious problem. Most teams either come to the computer classroom one hour earlier or stay a little late to finish their reports. Some hands-on activities that I have used include

- How far are students away from home to the university?
- Is your pulse rate increases significantly after one minutes of exercise?
- Can your palm size predict your height well?
- Which party would you vote for if the election is held today?
- How well can you draw a line segment of one inch?
- Counting M&M.
- Roulette game - what is the chance of winning?
- Baseball hitting rate simulation.
- Lets estimate the number of raisins.
- Do you believe the mathematics background of current CMU students is really worse than that twenty years ago?

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**3.3 Class Lectures involving cooperative learning and computer**

An organized lecture is much easier to conduct in the traditional lecture-note taken environment. A lecture which uses cooperative groups and is conducted in a computer class room environment is not designed for a typical organized lecturing. It is nature that the lecture is a guided learning process, where students are often conducting activities, discussing the problem-solving and working on computer. It is, therefore, very important to prepare a set of guideline sheets for each activity:

- An activity sheet that describe the activity to be conduct and the related questioned to be addressed.
- A worksheet for data recording and key computer commands for data analysis.
- A handout that demonstrate the new concepts and the related concepts applied for the activities.
- A work problem sheet that lists skilled problems for in-class reinforcement of new concepts.

The PACE paradigm considers class lecture as one of the major components. The interactive nature between students and the instructor is a key ingredient for the success of this paradigm. Although the instructor is placed more as a facilitator and a motivator, the interpretation and demonstration of subtle concepts are essential to help students understand the concepts and ease their mathematics anxiety. If students understand the concepts and work on basic skills, then, activities can be much more creative and generates enthusiastic and active participation. Otherwise, an activity is easily to be only a fun game. The sequence of activity, lecturing and working problems is interchangeable, depending on the concepts and skill involved. Activities conducted now may be used later after appropriate concepts are introduced, and vise versa.

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**3.4 Exercises**

Traditional exercises often are small artificial problems that are aimed for a specific concept. Although these skilled problems are important, they do not provide students a complete picture of statistical thinking and problem-solving. With the availability of computer technology and real world data that can be easily obtained from various resources such as the Journal of Statistical Education and many other internet resources, it is no longer difficult to design more open-ended data analysis and require writing reports and presentations. Through the years, the four major resources where I find small projects and data for lab problems are:

- faculty research projects that I consulted with in my statistical consulting service.
- data and problems from the internet, in particular, the Journal of Statistical Education.
- news papers, in particular, the USA Today and Wall Street Journal.
- projects and data that were collected by students from previous semesters.

These resources appear to be related to currents and students interest well. Internet is one of the richest source for activities, case studies and interesting data sets. The following World Wide Web pages will link to most of statistics resources available on internet:

StatLib (http://www.stat.cmu.edu/),

Statistics Education Links (http://www2.ncsu.edu/pams/stat/stated/staedlinks.html),

WWW Virtual Library-Statistics (http://www.stat.ufl.edu/vlib/statistics.html),

Chance Database (http://www.geom.umn.edu/locate/chance/).

In the PACE approach, both skilled exercises and problem-solving exercises are required. For an introductory statistics course, computer lab projects are important exercises for reinforcement of both concepts and problem solving strategies. Without reinforcement, a large proportion of students may not retain much of what they learned from activities and lectures.

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**3.5 The assessment of students learning**

In addition to the typical quizzes, tests and exercises, the PACE approach adds two more assessment components. One is the team work report for each in-class activity and the other is student-self selected projects. The self-selected projects are evaluated in three stages. The first stage is the problem description, sampling and data collection. The second stage is the exploratory analysis. The third stage is the final report that answers the problem defined at the first stage. It can be overwhelmed by having all of these assessment. It is necessary to balance the amount of exercises, quizzes, team work reports and self-selected projects. One rule that I used is that there is no more than two types of assessments per week. For most of the weeks, there is usually one assessment per week.

Conducting projects is one of the effective activities for students to gain an overall picture of what they learn. However, it can be overwhelmed for both students and instructor. I find it is particularly difficult to ask students forming teams to work on off-class projects. Finding schedule to meet each other among team members seems extremely difficult. As a consequence, most of groups did not do their projects properly. The self-selected projects, self-paired teams and three stages of evaluation, thus far, work the best in my experience.

Although it is easy to access computers at our university, most of students
coming to the introductory statistics course had little background of statistical
software or knowing how to integrate graphics and spreadsheets into reports.
It can be very time consuming for students to be familiar with computer
and for the instructor to spend extra efforts on this matter. In order
to minimize students frustration and anxiety, a simplified computer manual
will be necessary.

It is common that there are many hurdles to overcome in order to make
a new implementation smooth. There is no exception for implementing the
PACE approach. The most important factor for the success is the preparation.
This article shares my experience of implementing the PACE approach and
discusses what preparations are needed. There are many works to be done.
One area is the assessment of the model.

Computer technology is a necessary tool in the PACE model. However,
its purpose is mainly to easy the computational process. Graphing calculator
has similar capability, and can provide similar functions. The cost and
portability of graphing calculators seem to make it more suitable for large
class sizes.

The PACE model summarizes many years of my experience in teaching the introductory statistics. The successful implementation is very encouraging. Continuous refinement and improvement are essential. The assessment of the model and how it meets the learning theoretical framework are currently under investigation.

This work was supported by the educational technology grant from Central Michigan University and by an NSF/ILI mobile computer lab grant in 1996. The author is grateful for these supports.

Cobb, G. (1993) Reconsidering Statistics Education: A National Science Foundation Conference. J. of Statist. Edu., V1,n1.

Dietz, E.J. (1993) A cooperative learning activity on the methods of selecting a sample. American Statist. 104-107.

Fillebrown, S. (1994) Using projects in an elementary statistics course for non-science majors. J. of Statist. Edu., V2,n2 .

Garfiled, J. (1993) Teaching statistics using small-group cooperative learning. J. of Statist. Edu., V1,n1.

Garfield, J. (1995) How students learn statistics. Internat. J. of Statist. Review, 25-34.

Gordon, F. and Gordon, S. (1992) *Statistics for the Twenty-First
Century*. MAA Notes # 26.

Hillmer, S.C. (1996) A problem-solving approach to teaching business statistics. American Statist. 249-256.

Hogg, R.V., et al (1992) Towards lean and lively courses in statistics. MAA Notes # 26, 3-13.

Keeler, C. and Steinhorst, R. (1995) Using small group to promote active learning in the introductory statistics course : A report from the field. J. of Statist. Edu., V3, n2.

Moore, D. (1992) Teaching statistics as a respectable subject. MAA Notes #26, 14-25.

Sowey, E. (1995) Teaching statistics: Making it mamorable. J. of Statist. Edu. V3,n2.

Scheaffer, R. L., Gnanadesikan, M., Watkins,A. & Witmer, J. (1996)
*Activity-Based Statistics: Instructors Resources*, New York, Springer.

Tanner, W. and Wardrop, R. (1992) Hands-on activities in introductory statistics. MAA Notes #26, 122-128.