Using Cooperative Learning in a Large Introductory Statistics Class

Rhonda C. Magel
North Dakota State University

Journal of Statistics Education v.6, n.3 (1998)

Copyright (c) 1998 by Rhonda C. Magel, 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: Examples of exercises; Team selection.


This article discusses one active learning technique, cooperative learning, that can be used in large classes. This technique requires that students be divided into learning teams. A method for quickly dividing a large class of students into learning teams is presented.  Two examples of cooperative learning exercises used in an introductory statistics class are given. These serve as illustrations of the type of cooperative learning exercises that can be assigned in a large class. In particular, these exercises were used in a class of 195 students. Preliminary findings by the instructor of the advantages of using cooperative learning exercises are discussed.

1. Introduction

1 Research has indicated that students have different learning styles and that their academic performance is related to their learning style and the method of teaching used. If the method of teaching used coheres with their learning style, students tend to do well academically. If the teaching method does not fit their learning style, students tend not to do as well (Higbee, Ginter, and Taylor 1991). Because of the different learning styles of students, faculty are encouraged to use different teaching techniques in their classrooms. One such technique is cooperative learning. Cooperative learning is a form of active learning in which small groups work together on exercises designed to improve learning. As instructors try cooperative learning, however, they frequently run into problems of execution. Many of these problems involve management of the cooperative learning process. This paper discusses one way to overcome a common management problem associated with cooperative learning.

2 Most instructors of large classes resort to the standard lecture format. Studies have indicated that the lecture occupies 80% to 95% of the class time in these classes (Benjamin 1991). The aural-oriented and perhaps print-oriented learner will do well in this type of format, but a student with one of the other learning styles probably will not do as well at processing and retaining the information (Higbee et al. 1991).

3 Benjamin (1991) believes that instructors of large classes often resort to a lecture format because it is easier, and it is safer. There is less that can go wrong when doing a lecture. However, he encourages instructors of these classes to try some active learning techniques.

4 I will start by examining cooperative learning a little more closely. Cooper, McKinney, and Robinson (1991) define cooperative learning as "a structured, systematic instruction strategy in which small groups work together for a common goal" (p. 239). Cooper and Mueck (1990) define six critical features of cooperative learning.

  1. Everyone on a learning team is responsible for the other team members' learning.
  2. Cooperative learning exercises are intended for learning, not for significantly affecting students' grades. Projects done in teams are given little weight. They are thought of as learning experiences. Most of a student's grade comes from individual assignments or tests.
  3. There should be an appropriate assignment of students to teams.
  4. The teacher now serves as a facilitator rather than as an expert dispensing knowledge.
  5. Attention is given to social skills since students must learn to cooperate with each other.
  6. Students gain experience in verbal problem solving.

5 Studies at the K-12 level on cooperative learning indicate many positive advantages over teaching using a more traditional method (Johnson, Maruyama, Johnson, Nelson, and Skon 1981; Slavin 1983). The preliminary results of Cooper and Mueck (1990) on the effects of cooperative learning at the collegiate level also indicate positive findings.

2. Using Cooperative Learning

6 At North Dakota State University we teach several sections per year of an introductory statistics class with the sections ranging in size from 150 to 210. The class is worth three credits and is designed to cover the following material:

  1. An introduction to statistics
  2. Graphing and displaying data
  3. Measures of location and variability
  4. Probability and random variables
  5. Binomial random variables
  6. Normal probabilities
  7. Central limit theorem
  8. Estimation (eight different confidence intervals are introduced)
  9. Hypothesis testing (eleven different hypothesis tests are introduced including the chi-square goodness-of-fit test and chi-square test of independence).

7 Because the class is a prerequisite to other classes on campus, instructors are not given the option of omitting any topics. Most of the students taking this class are in their junior year. Their major areas vary, including business, child development, psychology, agriculture, sociology, biochemistry, zoology, and food and nutrition.

8 Students taking the introductory statistics class attend a large lecture portion twice a week, meeting 50 minutes each time. All of the students in a particular section of this class meet in the large lecture. One section is split into five smaller ones once a week for students to attend computer labs and/or take quizzes. A faculty member teaches the large lecture portion for a section of this class. Graduate teaching assistants conduct the computer labs and/or monitor short quizzes. Exams are given in the large lecture portion. All of the comments that follow refer to activities in the large lecture portion.

9 I have taught many sections of this introductory statistics class during the time I have spent at North Dakota State University. During one spring semester, I had a section of this class consisting of 195 students. It was during this spring that I decided to experiment with using cooperative learning.

10 Prior to this time, I had been using an active learning technique in my classroom. Approximately 25 minutes of every large lecture portion class was spent lecturing. For the remainder of the class, every student was given a worksheet to complete. The worksheet was to be turned in at the end of the class period. Students could work alone or with other students of their choosing, but each student needed to complete his or her own worksheet. Some of the worksheets used are given in Magel (1996).

11 When using this active learning technique, I was originally concerned about whether I could cover all the material that I had covered when I had used a standard lecture format. It turned out this was not a problem. Everything did end up getting covered. The average exam scores of the students also, if anything, improved. Further details are given in Magel (1996).

12 Overall, I was fairly happy with this technique, but I had noted at times students who had poor quantitative reasoning skills would work with other students of the same type. I spent some time helping these groups, but they were not benefiting as fully from the classroom experience as I felt they would if they worked with students with more highly developed quantitative reasoning skills.

13 It is because of this observation that I decided to experiment with using cooperative learning in the class. One problem that I initially faced in doing this was how to quickly divide the students into learning teams. I felt that it would be best to randomly assign students to teams since I did not know the personalities and abilities of the students well enough (there were 195 of them) to figure out which students would work well together. I also wanted to vary the teams each time, and I needed a quick way to do it.

14 The class that I was teaching was given in a standard auditorium consisting of three sections (left, middle, and right). After examining the layout of the auditorium, I decided to have learning teams assemble in 13 rows of each section. Rectangles were cut from 13 different colors of construction paper and marked with one of the numbers "1", "2", and "3" ahead of time. The number on each card represented a section of the auditorium with "1" = left, "2" = middle, and "3" = right. The color represented a row in that section. Five cards of each combination were made. This was the maximum size of each team.

15 The cards were shuffled and given to the students as they came into the classroom on the day that a cooperative learning exercise was to be used. It should actually be noted that the auditorium I was using had three different doors, and so I had to do the best that I could at passing out cards. I made sure that every student did have a card before we started. Students originally sat wherever they wanted to until it was explained what we were going to do in class that day, and how we were going to split up into groups. Students were then to go to the section designated by the number on their cards and then to the row of that section designated by the color they were holding. The colors that were to represent the rows were placed on an overhead transparency so that students could determine which row corresponded to their color. For example, students holding blue cards would go to the first row of their assigned section. Students holding tan cards would go to the second row, purple cards to the third row, and so on.

16 The first time that I used this method for dividing students into teams, it took approximately two to three minutes for everyone to get to their appropriate places. Different teams were formed by the same method the next time I used a cooperative learning exercise. The process of dividing students into teams took less than two minutes the second time.

17 Not every auditorium will be in exactly the same arrangement as the one I was using. For example, I have been in some auditoriums with one large section and one small section. In this case, the large section could possibly be divided in half by using a seat number. The instructor will have to adapt the above technique to fit his or her classroom conditions. The point here is that we can take an auditorium designed for lecturing and use it to do small group cooperative activities.

18 Two examples of cooperative learning exercises that I gave in my class follow. The first exercise (Example 1) was given on Valentine's Day. It was an effort to get the students to understand and reinforce the concept of a discrete random variable, its probability distribution, and the mean and variance of a random variable. It was also given to further reinforce the idea of sampling with replacement versus sampling without replacement.

19 Each learning team was given an envelope containing a mixture of red, white, and pink M&M's. Students were then required to work on the following sheet (Example 1). They could get help from other team members. Teams could also ask questions of me. Members of the team were required to check and discuss answers with each other before turning in the worksheet. No member was to turn in his or her worksheet until all members were done. One downfall of this process is that it is hard to monitor in a large class to what extent the first critical feature of cooperative learning, namely everyone on a learning team is responsible for the other team members' learning, is met. This was certainly encouraged by telling students that answers should be discussed and checked with other team members, and by not allowing students to turn in worksheets early. I also required each student to list the names of the other team members on their sheet.

20 Many teams had questions getting started with this exercise. I gave all the teams a hint after letting students think about this for a few minutes. This was done to keep the process moving. I am pointing out this particular part because it is the instructor's job in Cooperative Learning to serve as the facilitator. If problems arise in doing the exercise, adjustments must be made. At the end of the 50-minute class period, approximately half of the teams had completed the exercise. The other half asked to take it home to complete it, and to bring it back the next time. This exercise was not to be graded for accuracy and every student who made a reasonable attempt to complete it was awarded five points (even if they didn't complete it since we were under a time constraint). It was a good sign that students wanted to complete it.

21 No lecture or examples were given during this class period. Random variables, probability distributions, and the mean and variance of a random variable had been defined earlier. Instead of doing additional examples in class, I let students work on their own. This goes along with Cobb's (1991) proposal that we as instructors should do less lecturing.


STAT 330
NAME: ______________________________

Happy Valentine's Day

1. Count the number of M&M's in the envelope.

Number of Pink = ____
Number of Red = ____
Number of White = ____

Suppose three M&M's are randomly drawn from the envelope one at a time and without replacement.

(a) Write down the sample space.
(b) Let X = the number of pink M&M's drawn. What values could X assume?
(c) Find the probability function for X.
(d) Find the E(X). Write a brief interpretation of this value.
(e) Find the standard deviation of X.
(f) Graph the probability histogram of X.
(g) Check and discuss answers with other team members before going on to Number 2.

2. Suppose now that a random sample of size three is drawn with replacement. Let X = the number of pink M&M's drawn.

(a) Find the probability function of X.
(b) Find E(X) and the standard deviation of X.
(c) Graph the probability histogram of X.
(d) Check and discuss answers with other team members.

22 A second example of a cooperative learning exercise (Example 2) was given to help students understand the concept of estimating the difference between two population proportions. Students had been given the formula for the confidence interval for p1 - p2, but examples had not been worked. Each team was given two sets of cards. The first set of cards was to represent a population of people given treatment A to fight a particular type of cancer. The students were to estimate the proportion of people from this population who survived at least five years after receiving treatment A, p1. Each card had either a "0" or a "1" on it with "0" indicating that the person died before the end of the five years, and a "1" indicating that the person selected had survived at least five years. 


STAT 330
NAME: ___________________________
DATE: ___________________________


1. Assume the cards you are given represent a population of people given treatment A to fight a particular type of cancer. This population is very large. We want to estimate the proportion of people who survived at least five years, p1. We will do this by randomly drawing a sample of size 20 from this population. Shuffle the cards, draw 1 card, replace it, shuffle the cards, draw another card, replace it, etc. A "1" indicates that the person selected has survived at least five years. A "0" indicates that the person selected has died before the end of the five years.

A point estimate for p1 is:

$\hat{p}_1 = \frac{x}{n}$,

where in this case x is the number of people in the sample who survived at least five years.

(a) Write down your point estimate of p1.
(b) Find a 90% confidence interval for p1.
(c) Examine your cards. What is the true value of p1?
(d) Is the value of p1 in your confidence interval?
(e) Discuss results with members of your team.

2. The second deck of cards you received represents a population of people given treatment B to fight the same type of cancer. Randomly draw a sample of size 20 from this population and estimate the proportion of people receiving treatment B who survive at least five years, p2.

(a) Write down your point estimate of p2.
(b) Write down your point estimate of p1 - p2.
(c) Find a 90% confidence interval for p1 - p2.
(d) Examine your confidence interval for p1 - p2. Can you conclude one treatment is better than the other? Why or why not?
(e) Look at your second set of cards and write down the true value of p1 - p2.
(f) Is the true value of p1 - p2 in your confidence interval?
(g) Discuss results with members of your team.

23 Students were to randomly select twenty cards, one at a time, always replacing the card they drew and reshuffling the cards before making the next draw. One team member was responsible for holding and reshuffling the cards. All other team members were to be given an opportunity to draw cards. The cards were to represent a very large population and thus the probability of drawing a "1" always remained the same.

24 After calculating a point estimate and a 90% confidence interval for p1, the students were to examine their cards and write down the true value of p1. Each group had 10 total cards. If a "1" appeared on three of them, their cards represented a population in which p1 = 3/10 = 0.3. This value varied from team to team. The students were then asked whether the value of p1 was in their confidence interval. This exercise was an attempt to get students to understand the concept of estimation. Members of each team were supposed to check their answers with each other before proceeding to the next part. This was done in order to try and meet the first critical feature of cooperative learning, that everyone on a learning team is responsible for the other team members' learning. As stated earlier, this feature of cooperative learning is hard to monitor in this setting. The best that I felt I could do was to try to encourage it.

25 The second set of cards each team was given was to represent a population of people given treatment B to fight the same type of cancer. Students were to draw a random sample of size 20 as before. They were then to use their estimate of p2, and of p1 in the first part, to derive a 90% confidence interval for p1 - p2. Next, they were to look at their cards and find the true value of p2, then of p1 - p2, and determine if the value of p1 - p2 was in their confidence interval. This was again to get students to understand what p1 - p2 measured and the idea of estimating this quantity by a confidence interval. Student teams were again supposed to discuss and check their answers.

26 It should be noted that throughout the semester, I interspersed cooperative learning exercises with the active learning technique that I had been using earlier. That is, I continued at times to let students do worksheets either by themselves or with one or two other students. The exercises on these worksheets did not require one to interact with other students in order to complete them. I used both techniques in this class, because some students felt more comfortable choosing the students that they wanted to work with (or in some cases alone), but at times I wanted to mix students up and force them to interact with other students. I wanted to make sure that the weaker students participated in some interactions and did not always work with other weaker students.

3. Results

27 Three exams are given in my class during the course of one semester. Eighty percent of a student's grade comes from these exams. The cooperative learning exercises that students did this spring semester were based on material covered on the first and third exams. For each of those exams, the cooperative learning exercises used more than half the concepts covered. After the class was over for this spring, I decided to go back and compare the students' performance in this class with the students' performance in the class that I had taught during the previous spring semester. I thought it would be better to use the previous spring semester's class (rather than fall, as originally planned) to keep the students as homogeneous as possible. Some departments encourage their student majors to take this class during a particular semester (spring or fall).

28 The three exams given in both classes were not the same, although they did cover the same material, and I tried to make them of the same level of difficulty. It would be virtually impossible to keep the three exams exactly the same and have any meaningful interpretation of the comparisons of the two sets of scores, because students in the earlier class would give information to students in the later class. Preliminary findings indicated that, on average, students' scores improved when cooperative learning exercises were used. I would hesitate, however, to use these findings to say that more learning had taken place. These tests were not standardized.

29 Exam Number 1 covers topics 1 through 4 as mentioned earlier. In the class in which cooperative learning exercises were not used, 8.28% of the students' scores on this exam were below 60, and 8.92% of the scores were in the 60's. When cooperative learning exercises were used, only 2.03% of the students' scores were below 60, and 5.08% were in the 60's. This gives some indication that weaker students may benefit. Similarly, on Exam Number 3, there was a lower percentage of scores below 70 when cooperative learning exercises were used. This exam covers topics 8 and 9 as mentioned earlier. When cooperative learning exercises were not used, 28.76% of the students scored below 60, and 18.95% scored in the 60's. This changed to 20% scoring below 60 and 13.51% scoring in the 60's when the exercises were used.

30 Some students indicated on evaluation forms at the end of the semester that they liked doing the cooperative learning exercises. Other students commented that they would rather choose their own teams.

31 The students in the class appeared to be actively involved in doing both of the cooperative learning exercises mentioned in this paper. Every group had extensive discussion during the exercises. Students' discussions in class were on the exercise. In some teams, members were debating which of two methods was the correct way to solve the problem. I did get asked after awhile which way was correct. In other teams, a member was explaining to another member how to go about solving a particular problem.

4. Conclusions

32 Cooperative learning exercises can be used in a large class, at least one of size 195. They do require planning ahead of time. Dietz (1993), who used a cooperative learning exercise to teach sampling in a statistics class of size 41, has also pointed out that planning is crucial. The instructor must be willing to try something different, not worry that something may go wrong, and be ready to make adjustments to the exercise in class. The instructor may have to experiment with different ideas and find out what works best for his or her situation.

33 It has been my experience that a majority of the students enjoyed doing cooperative learning activities (or perhaps just having a variety of different activities). Some students did indicate they would have preferred to select their own teams. Dietz's (1993) findings also support this. However, in order to achieve my purpose of not having some of the students with weaker quantitative reasoning skills working with students of the same type, I needed to be the one who assigned teams. What I felt was a good compromise was doing approximately four cooperative learning exercises in class for which there were randomly assigned teams, and also continuing with the other active learning technique which allowed students to work alone or with other students of their choosing.

34 The random method of assigning students to teams might not always give the best team formations. Weaker students could still end up with weaker students, but it was the best I could hope for since it would be hard to know all of the students' abilities. I also randomly assigned students to new teams each time a cooperative learning exercise was to be done. If a student was on a team that did not work well together the first time, the student would at least have a chance of being on a team that worked well together for the next exercise.

35 Preliminary findings based on comparing exam scores from this spring semester's class with the class from the previous spring semester indicate that there was a significant increase on the average exam score when cooperative learning exercises were used. A smaller percentage of students also scored below 70. There were too many other factors involved to claim that cooperative learning exercises were responsible for this, but these findings merit further investigation.

36 I do think that what is important is to give students a variety of experiences in class. This keeps things interesting and should help make learning statistics fun!


Benjamin, L. T., Jr. (1991), "Personalization of Active Learning in the Large Introductory Psychology Class," Teaching of Psychology, 18(2), 68-74.

Cobb, G. (1991), "Teaching Statistics: More Data, Less Lecturing," UME Trends, 3.

Cooper, J., and Mueck, R. (1990), "Student Involvement in Learning: Cooperative Learning and College Instruction," Journal on Excellence in College Teaching, 1, 68-76.

Cooper, J., McKinney, M., and Robinson, P. (1991), "Cooperative/Collaborative Learning: Part II," The Journal of Staff, Program and Organizational Development, 9(4), 239-52.

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

Johnson, D. W., Maruyama, G., Johnson, R. T., Nelson, D., and Skon, L. (1981), "Effect of Cooperative, Competitive, and Individualistic Goal Structures on Achievement: A Meta-Analysis," Psychological Bulletin, 89, 47-62.

Higbee, J. L., Ginter, E. J., and Taylor, W. D. (1991), "Enhancing Academic Performance: Seven Perceptual Styles of Learning," Research Teaching in Developmental Education, 7(2), 5-10.

Magel, R. C. (1996), "Increasing Student Participation in Large Introductory Statistics Classes," The American Statistician, 50, 51-56.

Slavin, R. E. (1983), "When Does Cooperative Learning Increase Student Achievement?," Psychological Bulletin, 94, 429-445.

Rhonda C. Magel
Department of Statistics
North Dakota State University
Fargo, ND 58105

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