University of Minnesota

Journal of Statistics Education v.1, n.1 (1993)

Copyright (c) 1993 by Joan Garfield, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

** Key Words**: Active learning; Small groups.

Current recommendations for reforming statistics education include the use of cooperative learning activities as a form of active learning to supplement or replace traditional lectures. This paper describes the use of cooperative learning activities in teaching and learning statistics. Different ways of using cooperative learning activities are described along with reasons for implementing this type of instructional method. Characteristics of good activities and guidelines for the use of groups and evaluation of group products are suggested.

1 Cooperative learning is a topic frequently mentioned in conversations about improving education, regardless of the discipline or level of instruction. Some recent definitions of cooperative learning include:

2 An activity involving a small group of learners who work together as a team to solve a problem, complete a task, or accomplish a common goal (Artzt and Newman 1990).

3 The instructional use of small groups so that students work together to maximize their own and each other's learning (Johnson, Johnson, and Smith 1991).

4 A task for group discussion and resolution (if possible), requiring face-to-face interaction, an atmosphere of cooperation and mutual helpfulness, and individual accountability (Davidson 1990).

5 Cooperative learning also falls in the more general category of "collaborative learning," which is described as working in groups of two or more, mutually searching for understanding, solutions, or meanings, or creating a product (Goodsell, Maher, and Tinto 1992).

6 It is important to also consider what cooperative learning is not. According to Johnson et al. (1991), it is not having students sit side-by-side at the same table and talk with each other as they do their individual assignments, having students do a task individually with instructions that those who finish first are to help the slower students, or assigning a report to a group where one student does all of the work and the others put their names on it.

7 Several recent reports urging reform of mathematics and science education in general (e.g., National Council of Teachers of Mathematics 1989, 1991; National Research Council 1989) and statistics education in particular (e.g., Cobb 1992), have described the need for specific changes in teaching. Instead of traditional lectures where teachers "tell" students information that they are to "remember," teachers are encouraged to introduce active-learning activities where students are able to construct knowledge. One way for teachers to incorporate active learning in their classes is to structure opportunities for students to learn together in small groups.

8 The suggestions made in these reports are supported by a growing set of research studies (over 375 studies, according to Johnson et al. 1991) documenting the effectiveness of cooperative learning activities in classrooms. A majority of the published research studies examine cooperative learning activities in elementary and secondary schools, and a subgroup of these studies focus on mathematics classes. The implication of these studies is that the use of small group learning activities leads to better group productivity, improved attitudes, and sometimes, increased achievement ( Garfield, in press).

9 Only a few studies so far have examined the use of cooperative learning activities in college statistics courses. Shaughnessy (1977) found that the use of small groups appeared to help students overcome some misconceptions about probability and enhance student learning of statistics concepts. Dietz (1993) found that a cooperative learning activity on methods of selecting a sample allowed students to "invent" for themselves standard sampling methods, which resulted in better understanding of these methods. Jones (1991) introduced cooperative learning activities in several sections of a statistics course and observed dramatic increases in attendance, class participation, office visits, and student attitudes.

10 Another argument for using cooperative groups relates to the constructivist theory of learning, on which much of the current reform in mathematics and science education is based. This theory describes learning as actively constructing one's own knowledge. Constructivists view students as bringing to the classroom their own ideas, experiences, and beliefs, that affect how they understand and learn new material. Rather than "receiving" material in class as it is "delivered," students restructure the new information to fit into their own cognitive frameworks. In this manner, they actively and individually construct their own knowledge, rather than copying knowledge "transmitted" or "conveyed" to them. A related theory of teaching focuses on developing students' understanding, rather than on rote skill development.

11 Small-group learning activities may be designed to encourage students to construct knowledge as they learn new material, transforming the classroom into a community of learners, actively working together to understand statistics. The role of the teacher changes accordingly from that of "source of information" to "facilitator of learning." Part of this role is to be an ongoing assessor of student learning.

12 As part of the current reform of assessment of student performance, instructors are being encouraged to collect a variety of assessment information from sources other than individual student tests. Cooperative group activities may be structured to provide some rich information for teachers to use in assessing the nature of student learning. While walking around the class and observing students as they work in groups, the instructor is able to hear students express their understanding of what they have learned, which provides instructors with an ongoing, informal assessment of how well students are learning and understanding statistical ideas. Written reports on group activities may be used to assess students' ability to solve a particular problem, apply a skill, demonstrate understanding of an important concept, or use higher-level reasoning skills.

13 A final argument for including cooperative group-learning activities in a statistics class is that businesses are increasingly looking for employees who are able to work collaboratively on projects and to solve problems as a team. Therefore, it is important to give students practice in developing these skills by working cooperatively on a variety of activities. This type of experience will not only build collaborative problem-solving skills, but will also help students learn to respect other viewpoints, other approaches to solving a problem, and other learning styles.

14 The use of small-group learning activities appears to benefit students in different ways. These activities often result in students teaching each other, especially when some understand the material better or learn more quickly than others. Those students who take on a "teaching" role often find that teaching someone else leads to their own improved understanding of the material. This result is reinforced by research on peer teaching that suggests that having students teach each other is an extremely effective way to increase student learning (McKeachie, Pintrich, Yi-Guang, and Smith 1986).

15 Just as "two heads are better than one," having students work together in a group activity often results in a higher level of learning and achievement than could be obtained individually. A necessary condition for this to occur is called "positive interdependence," the ability of group members to encourage and facilitate each other's efforts (Johnson et al. 1991). Positive interdependence can be promoted by careful design and monitoring of group activities.

16 Working together with peers encourages comparison of different solutions to statistical problems, problem solving strategies, and ways of understanding particular problems. This allows students to learn first-hand that there is not just one correct way to solve most statistics problems. Small group activities also provide students with opportunities to verbally express their understanding of what they have learned, as opposed to only interacting with material by listening and reading. By having frequent opportunities to practice communicating using the language of statistics they are better able to see where they have not yet mastered the material when they are unable to explain something adequately or communicate effectively with group members. Small-group discussions also allow students to ask and answer more questions than they would be able to in large-group discussions where typically a few students dominate the discussion.

17 Finally, students' achievement motivation is often higher in small-group activities because students feel more positive about being able to complete a task with others than by working individually (Johnson et al. 1991). Working together towards a mutual goal also results in emotional bonding where group members develop positive feelings towards the group and commitment towards working together. This increase in motivation may also lead to improved student attitudes towards the subject and the course.

18 There is not only one correct way in which to use groups, although there are guidelines for the effective use of groups in different types of course settings. The instructor may allow students to self-select groups or groups may be formed by the instructor to be either homogeneous or heterogeneous on particular characteristics (e.g., grouping together all students who received A's on the last quiz, or mixing students with different majors). Johnson et al. (1991) describe several different types of groups, including informal and formal groups.

19 Informal groups are often used to supplement lectures in large classes, and may change everyday. These might consist of "turn to your neighbor" discussions to summarize the answer to a question being discussed, give a reaction to a discussion, or relate new information to past learning. In formal groups, students work with the same students for a longer period of time, sometimes for an entire semester. In these groups students may divide up work to be done on a particular in-class activity, work together to solve a problem or apply a statistical method, or work on long-term projects. Students may also use these groups to review material, complete homework assignments, teach each other information, encourage and support each other, and inform each other about information if a class has been missed.

20 When first introducing a group activity it is useful to establish some rules for students. They should be informed that they are always responsible for their own work but must also be willing to help any group member who asks for help. If they have questions on an activity they should first ask each other, and may ask the instructor only if no one in the group can answer their question. They need to listen carefully to each other and share the leadership, making sure everyone participates and no one dominates.

21 It is also important to establish respect and consideration for all members of the group. One way is to point out that all people learn at different rates and that there are many different ways in which people best learn. It is important that students recognize and accept these differences and be respectful of each other. Students may be told that they will learn statistics more effectively by asking questions, answering questions, helping each other, and analyzing each other's mistakes. This is quite a contrast from the role they may be used to in most college classes where they passively listen to lectures.

22 Finally, students need to be aware that the problems they will be solving may often be solved in different, correct ways. They should be encouraged to try to learn from each other by comparing and explaining different solutions.

23 In order to encourage positive interdependence among group members, students may be assigned to specific roles, which can be rotated each day (Johnson et al. 1991). These roles may help students get started on the activity and also prevent one person from doing all of the work. A "moderator/organizer" is in charge of assigning tasks to the group members, moderating group discussions, overseeing that the assigned task is being carried out, and helping to keep the group on course. A "summarizer's" job is to summarize discussions or group solutions to a problem, so that the "recorder" may write down what the summarizer says. Sometimes it is useful to have a "strategy suggester" or "seeker of alternative methods," who challenges the group to try other methods or explore other ways to solve a problem. A "mistake manager" may ask the group what went wrong and what can be learned from mistakes made. Finally, an "encourager" can be designated to encourage participation from all group members by using probes such as: "What do you think," "Can you add to that?" or by giving positive reinforcement to group members as they contribute to the discussion.

24 It is recommended that the instructor carefully read one of the excellent resources on cooperative or collaborative learning in higher education (e.g., Goodsell et al. 1992; Johnson et al. 1991) before incorporating cooperative groups in a statistics class. These resources provide complete information on structuring and monitoring groups, and developing and evaluating group activities. Based on the models of cooperative learning in these references, cooperative group activities for a particular statistics class can be developed. These activities might include:

- 25 Having groups individually solve a problem and then compare their solutions (e.g., homework problems or problems from the textbook requiring particular skills).
- 26 Having groups discuss a concept or procedure, or compare different concepts or procedures (e.g., discuss the steps involved in testing an hypothesis, or compare the advantages and disadvantages of using the mean, median, and mode to summarize a data set).
- 27 Giving groups a data set to analyze and to discuss, followed by a written report of what they have learned about the data (e.g., data sets from the Quantitative Literacy materials or data generated in class such as estimating the distribution of the number of raisins in a small box).
- 28 Having each group collaborate on a large project involving collecting, analyzing, and interpreting data. Groups may meet in and/or outside of class to work on these projects, and may present the results in a written report and/or an oral in-class presentation.
- 29 Using groups as a way to learn new material. The jigsaw method can be used, where students are assigned temporarily to new groups, and each new group learns something new, such as a different type of plot. Then, students return to their original groups and teach each other the material they just learned.
- 30 Having groups compare their ideas about chance phenomena, and then generate or simulate data to test their beliefs (e.g., distributions of heads and tails when coins are tossed, or the best strategy to choose in the Monte Hall problem).

31 The three articles mentioned previously on using cooperative groups in college statistics courses (Dietz 1993; Jones 1991; Shaughnessy 1977) provide more detailed descriptions of particular cooperative group activities. Samples of other group activities are included in the appendix.

32 Although cooperative group activities can be used in many different ways, it is important to consider characteristics of good activities in designing activities for a statistics class. Activities should require that all members of the group be involved, and not allow just one or two students to do the work while the others observe them. The instructions for the activity should be made very clear so that students do not spend time trying to figure out what it is they are supposed to do, or take a wrong path because of misunderstanding the activity.

33 Students should know in advance that they are accountable for the end product, both individually and as a group. This may result in individual write-ups or contributions to a group product, where students may be asked to evaluate the extent of their individual contributions. There should also be some assessment of the results of a group activity so that students receive feedback and learn from any mistakes made.

34 It is important that group activities conclude with some type of summary of what students have learned. Students may be asked to turn in their individual work or to write one group summary of the results of the activity. Grades or points may be assigned in different ways. If students work together but turn in separate reports, these may be rated individually and then a group score based on the average assigned as well. If only a group score is assigned to a group product, students may be asked to volunteer the percentage of their contribution, and that may be used to determine their share of the group points. Or, a group score may be assigned and everyone receives that score. Methods will vary based on the types of projects and students.

35 In addition to having the instructor evaluate a group product, students should be encouraged to assess their own group product, as well as how effectively their group worked together. Sometimes products may be exchanged between groups so that students may critique each other's work. Students may also participate in a group quiz, where they work together to solve problems on the quiz. This method is particularly effective if it takes place after students have individually taken the same quiz.

36 Despite the recommendations and encouragement for using groups, there are still concerns expressed by statistics instructors who are either contemplating using groups or have tried it and have had some negative reactions.

37 Some instructors feel uncomfortable losing their role of being on center stage, performing in front of appreciative students. In using groups, the teacher's role in class is more in the background, where they may observe, listen, and assist students only as needed. Instead of elegantly demonstrating the solution of problems or proofs in front of the students, instructors will often step back and watch the students struggle through these same problems. They also take on the role of questioners, asking members of the groups about their conclusions or solutions to problems, asking them to justify what they did and why.

38 Instructors may be discouraged by students who resist an activity that appears challenging and difficult, forces them to think, and does not allow them to be passive learners, because students are used to sitting in lectures where they are not required to talk, solve problems, or struggle with learning new material. Students may want the teacher to do more explaining, and telling them the right answers, rather than struggle with a problem themselves. Some students may prefer to work alone, and resist being forced to work in a group. Sometimes this concern is related to the issue of grading fairness; students may feel it is unfair to give one grade to the entire group rather than separate grades to individual students, especially if students do not contribute equally. By adopting a grading policy where the amount of individual student contributions are used to assign points may alleviate this concern.

39 Sometimes group activities are an instant success, and the instructor is encouraged to continue using groups in class. Other times concerns such as the ones listed above arise. Goodsell et al. (1992) have several helpful suggestions for overcoming problems with students in using groups. As teachers develop more experience in designing and managing group activities, and as students become more accustomed to learning and working together, these problems usually disappear.

40 Cooperative group learning includes a wide variety of activities that may be implemented in several different ways in a college statistics class. These activities offer ways for students to become more involved in learning and to develop improved skills in working with others. The strong support of research and the recommendations from recent reports urging educational reform should encourage more instructors to introduce cooperative group activities in their classes. Perhaps as more statistics faculty begin to experiment with the use of small-groups and to evaluate their effectiveness in improving student learning, we will be able to develop a core of research to inform us as to the best types of activities to use in helping students learn particular statistical concepts.

Three Sample Group Activities

Each group of students is given some different data sets (e.g., prices of running shoes, fat content of fast foods, Olympic medals, temperatures for a month). These may either be plots of data or sets of numbers that students can use to first construct simple plots. They are given the following instructions:

- In your group, discuss each of the four measures of center you read about in Chapter 4. Make sure that everyone understands what each measure is and how it is calculated.
- Discuss the advantages and disadvantages of using each of the four measures to summarize a data set.
- For each of the distributed data sets, determine which measure of center would be most appropriate as a single number summary and why.
- Turn in one written summary of your discussion. Be sure to include a description of each measure and how it is calculated, advantages and disadvantages of each measure, and a discussion of which measure of center is most appropriate to use in representing each data set and why.

This activity was developed by the NSF-funded CHANCE project. For this activity you will need some large bottles of Coke and Pepsi and many small paper cups. Students are asked to select one person in their group who they think can distinguish between Coke and Pepsi. They are asked to design and conduct an experiment to determine how well this person can actually distinguish between the two types of drinks. After the experiment is completed, groups share their methods and results in a class discussion. Characteristics of good experiments are highlighted and students are asked to consider which results they most trust.

This activity is designed to help students empirically test the wisdom of their intuitive ideas about chance events. They are given the following scenario:

Suppose you're on the game show, Let's Make a Deal, and you're given the choice of three doors. Behind one door is a car; behind the other two doors are goats. You pick a door, say Door Number 1, and the host, who knows what's behind the doors, opens another door, say Door Number 3, which has a goat. He then says to you, "Do you want to pick Door Number 2?" Is it to your advantage to switch your choice?

- Discuss in your group whether you should switch or stay with your original choice, or whether it makes no difference. What assumptions does your answer depend on? What other assumptions might you make and how would they affect your answer? Summarize your discussion in a paragraph.
- Carry out an experiment to test your decision. For example, one person can serve as the host, Monte Hall, and the other group members as contestants. The host takes three cards, each representing a door. On the back of each is a goat (on 2 cards) or a car. The host lays out the three cards, blank side up, making sure he/she knows which one has the car on the other side. After each game, the cards are rearranged and the game is played again, keeping track of the strategy used (switch or stay) and the result.
- Which strategy is better? Summarize the data used to support your decision. Determine the probabilities of winning if you switch doors and if you stay with the original choice of doors.

Artzt, A. and Newman, C. (1990), How to Use Cooperative Learning in the Mathematics Class, Reston, VA: National Council of Teachers of Mathematics.

Cobb, George (1992), "Teaching Statistics," in Heeding the Call for Change: Suggestions for Curricular Action, ed. L. Steen, MAA Notes, No. 22.

Davidson, N. (ed.) (1990), Cooperative Learning in Mathematics: A Handbook for Teachers, Menlo Park: Addison Wesley.

Dietz, E. J. (1993), "A Cooperative Learning Activity on Methods of Selecting a Sample," The American Statistician, 47, 104-108.

Garfield, J. (in press), "How Students Learn Statistics," International Statistical Review.

Goodsell, A., Maher, M., and Tinto, V. (1992), Collaborative Learning: A Sourcebook for Higher Education, University Park, PA: National Center on Postsecondary Teaching, Learning and Assessment.

Johnson, D., Johnson, R., and Smith, K. (1991), Cooperative Learning: Increasing College Faculty Instructional Productivity, ASHE-ERIC Higher Education Report No. 4, Washington, DC: The George Washington University.

Jones, L. (1991), "Using Cooperative Learning to Teach Statistics," Research Report Number 91-2, The L.L. Thurstone Psychometric Laboratory, University of North Carolina.

McKeachie, W., Pintrich, P., Yi-Guang, L., and Smith, D. (1986), Teaching and Learning in the College Classroom: A Review of the Research Literature, Ann Arbor: Regents of the University of Michigan.

National Council of Teachers of Mathematics (1989), Curriculum and Evaluation Standards for School Mathematics, Reston, VA: Author.

National Council of Teachers of Mathematics (1991), Professional Standards for Teaching Mathematics, Reston, VA: Author.

National Research Council (1989), Everybody Counts: A Report to the Nation on the Future of Mathematics Education, Washington, DC: National Academy Press.

Shaughnessy, J. M. (1977), "Misconceptions of Probability: An Experiment With a Small-Group Activity-Based Model Building Approach to Introductory Probability at the College Level," Educational Studies in Mathematics, 8, 285-316.

Joan Garfield

The General College

University of Minnesota

140 Appleby Hall

128 Pleasant St. S.E.

Minneapolis, MN 55455

jbg@vx.cis.umn.edu

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