a classroom experiment using small-group projects

JMB JOURNAL OF MATHEMATICAL BEHAVIOR, 16 (2), 167-174 ISSN 0364-0213. Copyright 0 1997 Ablex Publishing Corp. All rights of reproduction in any form reserved.

A Classroom Experiment Using Small-Group Projects

JOSEPH F. CONRAD

Solano Community College

It has been suggested that students will learn mathematics more effectively if the traditional classroom lecture format is altered (National Research Council, 1991). One suggested vari- ation is the use of small-group settings. The author taught (Fall 1991) three sections of the intermediate algebra course at Penn State. This course is viewed as a precollege course and credits are not applicable toward any baccalaureate degree requirements. One section was taught in the (author’s) traditional lecture format. The other two sections differed from this in that the students were given group projects and daily quizzes (some of which were in the group format). The purpose of this paper is to report on the outcome of this experiment.

Cooperative learning or small-group learning was first investigated in the 1960s by a variety of researchers. In the succeeding quarter-century many variations of student leam- ing together in groups have been studied (Sharan, 1990; Slavin et al., 1985). Many of these methods have been tried in mathematics curricula. The basic rationale for using group techniques in any setting is given by Davidson (1990, p. 3) as follows: “By setting up learning situations that foster peer interactions, the teacher meets a basic human need for affiliation and uses the peer group constructive force to enhance academic learning.”

In the past several years a new twist to cooperative learning in mathematics has appeared. This is the use of group projects. At this time, done primarily at the college level, a group project approach retains much of the lecture time used in traditional instruction, but gives students the small-group experience as well. The instructor forms groups and assigns problems which complement the material given in class. The students work together, usually outside of class, to produce their results. Cohen et al. (1991) report that their calculus stundents who did projects did no worse than students in traditional sections as measured by pass rates and did better in final exam scores. The Ithaca Project Report (Cohen, Hilbert, Maedi, Robinson, Schwarts, & Seltzer, 1991) does not give any quantitative results, but states that the students gave positive feedback as to their perception of mathematics after taking their group project classes. The experiment reported in this paper deals with apply- ing the group project method to a college algebra class.

DESIGN OF THE EXPERIMENT

The author taught three sections of Math 4, a three-credit intermediate algebra course, at Penn State Altoona during the Fall Semester 199 1. All sections were given two mid-

Direct all correspondence tot Joseph F. Conrad, Solano Community College, 4000 Suisun Valley Road, Suisun City, CA 94585 <[email protected]>.

167

168 CONRAD

semester examinations and a final examination. One section, Class A, was taught in the

instructor’s usual format; namely, there were daily lectures which included a homework assignment that was not collected. The students were given a weekly quiz on Wednesdays which tested the previous week’s material. This class served as a control group. The other two sections, Class B and Class C, also had a daily lecture and home-

work assignment, but they were given two group projects and daily quizzes. The quizzes were of two distinct types which will be discussed below. The projects were given in class instead of a lecture. The students were permitted to work in their groups out-

side of class if they felt it was necessary. (This differs from Cohen et al., 1991, and Hil- bert et al., 1991, where the projects were done strictly outside of class. Both of these reports found that a disadvantage of the project approach was scheduling outside-of- class meetings).

The projects are reproduced in the Appendix.

IMPLEMENTATION OF THE EXPERIMENT

The purpose of this section is to describe how the experiment was carried out and to report observations that were made at that time. In the control class, Class A, the instructor opened each class with a question-and-answer period (generally devoted to doing homework problems) followed by a traditional lecture. On Wednesdays, Class A received a lo- 15 minute quiz at the end of class. The following is an example of such a quiz.

Quiz 3

1. Find each of the following products:

a) (x - 2Y)(x* + 3xy - y*)

c) <X”Y + 3Y)(X2Y - 3Y)

b)(x-3)(2x+ 1)

2. Factor each of the following polynomials completely.

a) 7x2y - 2 lxy3 b)2xZ-6x+&-3z

Note that it is entirely computational and requires only that the students exhibit skills from homework problems and examples done in class by the instructor. The other quizzes were similar in these respects.

In the group-projects classes, Classes B and C, a typical class also started with home-

work questions followed by a lecture. However, each class period closed with a quiz. The majority, 29 of 37, consisted of two questions, one was a computational problem from the previous class period’s homework assignment, the other was a conceptual or noncomputa- tional question from that day’s class. The remainder of the quizzes were done in a group format. The students would join in groups of four (as nearly as possible) and take the quiz together. These groups, as well as those for the projects, were formed by the students, not by the instructor. The following are three quizzes which test roughly the same material as

Quiz 3 given to Class A.

SMALL GROUP PROJECTS 169

Quiz 6

1. Find the product: (2f + .r)(? + 2ts + s*) 2. What is the first step in factoring polynomials?

Quiz 7

1. Factor the following polynomial: yz + 2y - z, - 2

2. True of False: .X~ = y3 is a prime polynomial

Group Quiz 8

1. Factor the following expression completely.

a3x*y* - a3P + a2x3y* - a*xy4

2. Use the information given on page 45 of the text together with the fact that there are about 900,000 grams in a ton to estimate the number of atoms of hydrogen in the sun. Express your answer in scientific notation with four significant digits.

Notice that the questions are not all computational. Also, question two from the group quiz is not like any problem the students had seen prior to this quiz. The group quizzes were intended to prepare the students for working on the group projects-i.e., to work as a group on a problem which was not familiar.

At this time it is appropriate to state some observations concerning the group quizzes. During the lecture, the students were seated in standard rows. At quiz time, the type of quiz was announces and, if it was a group quiz, the students would arrange themselves into groups. After the first two group quizzes, it was rare for this process to take longer than a minute. Since the students determined their own groups, there were times when they would from a group of six, and be unable to break themselves up without instructor intervention. This, too, was not common and caused little difficulty. One copy of the quiz was then dis- tributed to each group. Most groups would have one student work the problems on the sheet while others watched and commented. Other groups had each student work a problem and then the answers were compared. The accepted solution was then copied on the quiz sheet. Each student signed the group’s quiz sheet and each received the same grade. Inter- estingly, the students often argued over the correct procedure, especially in nonstandard problems. Of course, the correct view did not always win out which led to some “I told you so’s” when graded quizzes were returned.

As the semester progressed, the students tended to form the same groups. This was not discouraged since it allowed the groups to establish dynamics of their own. It also allowed the students to form relationships, mathematical and personal, that extended outside of class. It was not unusual for group members to work together on homework problems.

The personalities of the individuals accepted group performances in many ways. For example, the best student in two of the groups was a nontraditional female student.

170 CONRAD

Despite their abilities, neither of these women were very confident. One was in a group with a forceful, traditional male student who was able on several occasions to persuade his group to adopt his (incorrect) solution rather than the woman’s (correct) solution. (Twice this student asked if she could take the group quizzes by herself because that she would do better. These requests were gently refused and with some encouragement she was better able to defend her positions by the end of the semester.) The other nontraditional female student was in a group with no domineering individual. In fact early in the semester, they would look around at each other saying nothing waiting for someone to take charge. As they recognized the talents of their older colleague, this group looked to her for direction which she ably gave.

Overall, despite the perceptible groan when a group quiz was announced, the students appeared to enjoy working together on the group quizzes. They did not like the fact that the group quizzes were more difficult, hence, the groan. They also did not like having a quiz every class. (It should be noted that the daily quizzes did add considerably to the instructor’s work. There were 39 quizzes to prepare and grade versus 13 in the control class. However, group quizzes were easier to grade in that there were only one-fourth as many of them.)

An entire class was devoted to each project. The students formed their groups and each

student received a copy of the problem. The projects consisted of a real situation in which the students were required to use their abilities to produce some information. The situations were not ones which had been covered in class and the students needed to proceed in a manner that they chose. Each student reported the results of their group’s work in a personal written report due one week after the in-class project day. The grade for the project had two components. First, some parts were graded as a group; in particular, Parts 2 and 3 of Project 1 and Parts 1 and 2 of Project 2. These parts were compared and each person received the minimum grade of those in the group. (This was to insure that the students would indeed work as a group.) The remainder of the project, because of its subjectivity, and presentability was graded individually.

The first project was meant to make the students realize that what makes sense is not always correct and to have them mathematically model a real situation. It took some groups over 15 minutes to understand why $18,000 was not the correct answer. Once they did, many added $18,000 to $90,000, multiplied by .20 and said that $21,600 was the correct amount. Usually, they saw that this was incorrect for the same reason that $18,000 was, but some needed some prodding in the right direction. Some were then convinced that a never- ending process of adding and multiplying was necessary. Eventually, with some constructive hints, all groups found the correct value of $22,500. However, a few groups, some of whom had no difficulty finding the correct value, could not generalize their procedure to answer the final question.

The second project was designed to teach them about monthly payments, to force the students to read the text and decide which formula was appropriate (we had not discussed this topic in class), to see if they could use their calculators properly, and to present them with an open-ended problem. Despite the text, two groups found their payments by assum- ing simple interest and then dividing the total by the number of months. Many groups had difficulty executing the complicated monthly payment formula on their calculators. Some realized this when their answers were nonsensical, but most cross-checked their values with other groups and observed discrepancies. Because none of the options gives a monthly


payment of under $175, Question 3 yielded some interesting results. Some students

explained how they could adjust their budgets to increase income or decrease spending, but

few said that they would have to look for a different car.

The students found the group projects instructive. They enjoyed the fact these exercises

seemed to be real, but they did find them more difficult than the normal word problems in

the book. As with the quizzes, they worked well together and would sometimes get into

heated discussions about the correct way of doing a problem.

RESULTS OF THE EXPERIMENT

Quantitative results were produced in the following way. Prior to registering for their first

semester, such student at Penn State is required to take placement exams. The scores for

those students who took these exams immediately prior to Fall 1991 served as a basis for

comparison. The final examinations, which were essentially identical for all sections, were

used to assess achievement. They were standard computational examination taken individ-

ually. (Any group format for the examinations was rejected due to the service nature of this

course. It was felt that the students needed to be able to perform the requisite skills in the

same atmosphere which their (academic) futures would present to them.) The means of

these data are reported in Table 1. (Note that Class B and Class C received the group format

while Class A received the standard format.) Class B + C was found by combining Classes

B and C into a single treatment.

It appears that, despite similar pre-registration score means, the final exam mean in

Class A is much better than in Classes B or C. These test statistics were analyzed in two

ways. First, a two-sample t-test was performed which compared each of the three statistics

for the combined Class B + C versus the control Class A. Also, a one-way analysis of vari-

ance was performed comparing the test statistics for all three classes separately. The p-val-

ues of these tests are given in Table 2.

Both types of analysis indicated no significant difference (p < .06) in pre-registration

scores between the control group, Class A, and the classes which used the group format,

Classes B and C. Both analyses also showed a significant difference in final exam scores

@ < .06) with the control outperforming the group-project classes. (The implications of this

will be discussed.)

TABLE 1. Test Statistic Means

Pre-Registration Class No. at Final No. in Sample Score Mean Final Mean

A 20 17 17.29 112.7 B 29 25 17.52 94.8 C 28 15 15.73 88.7

B+C 57 40 16.85 92.5

Note: “No. at Final” gives the total number of students who took the final. “No. in Sample” gives the num-

ber for whom appropriate pre-registration scores were available. Class B and Class C used the group format; Class

B + C combines them into one treatment.

172 CONRAD

TABLE 2. Analysis of Variance.

Pre-registration scores

Two-sample t-test

.70

One-way analysis of variance

,353 Final exam scores ,004 ,057

Nofec This table gives the p-values associated with each of the indicated statistical

tests. The two-sample t-test compared the combined group format classes with the control

and the one-way analysis of variance compared the three classes separately.

ANALYSIS OF RESULTS

Recall (Table 2) that the statistics indicated that there was no significant difference

between the pre-registration scores of the classes. Also, recall that there was a significantly better performance on the final exam by the class that did not receive the group format.

Consequently, it can be concluded that, tested achievement was significantly less on the part of those who received the group format.

Should we blame the group-project approach for this disappointing outcome? There are

several possible factors that could have contributed to it. First, the size of the classes could have had an effect. Class A had approximately ten fewer students (abut 20 versus about 30)

throughout the semester than either Class B or Class C. It is generally accepted that smaller

classes lead to improved performance. However, the group format should have mitigated against this being a major factor.

A second observation that could help explain this outcome is the daily quizzes would not have required regular review on the part of the students. In other words, those who had

weekly quizzes were forced to review material from the previous week while preparing for

a particular quiz. Having gone through a review process such as this would have been an aid at exam time when a review of five weeks (for a mid-semester exam) or 15 weeks (for

the final) was necessary. On the other hand, those who had daily quizzes only needed to

concern themselves with the previous day’s class; they never had a chance to practice

reviewing for their exams, so to speak.

Another factor that should be included in this discussion is that the final examination was given in the traditional format. In other words, the exam was composed of standard

computational problems similar to those done in lecture and on the homework. This, if Class B or C students gained in any other respect, it was not measured. The question of student evaluation is a difficult one. Cohen et al. (1991, p. 18) state, “the positive effects of

the program that we are seeking may require very different measures of success [than traditional exams.]” They administered surveys in addition to analyzing test scores and pass- fail rates. The surveys found that the organizational aspects of the course presented more

difficulties than the mathematics. Interestingly, the surveys indicated that there was a “higher acceptance of the project approach.. exhibited by more advanced students (Cohen et al., 1991, p. 25),” which these students were not.

In conclusion, this experiment did not produce a success for the group-projects approach. It is clear that more research on the efficacy of small-group learning at the college level is desirable. For example, an attempt could be made to measure student abilities

with non-computational problems or to measure attitudinal changes brought about by this


approach. This research should be conducted to analyze both the well-established cooper-

ative learning techniques and the group-project approach. With the trend in business

toward a TQM, emphasizing the team concept, it should be interesting to follow students

who have been exposed to group-project learning environments into their business careers.

APPENDIX

Project 1

Suppose your father is the chairman of the Budget Committee of a church. He is new to

the job and asks for guidelines for next year’s budget. He is told that the only hard part

of the job is to make sure that 20% of the total church budget is sent to overseas mis-

sions. His committee meets and decides that the church budget without overseas, mis-

sions will be $90,000. At home, he tells you that his job is easy. All he has to do is to

take 20% of $90,000, which he quickly finds is $18,000 and let that be the overseas mis-

sion budget. Due to your Math 4 background, you realize that he is wrong and you

(respectfully, as always) point that out to him. He replies, “I don’t see anything wrong,

why am I wrong?’

PART 1: Why is $18,000 the wrong amount for the overseas missions budget?

After you explain what is wrong with what he did, your father says, “Well, if you’re so

smart, what should the amount be ?”

PART 2: What should the amount be?

Your father must present his committee’s proposal at a church meeting. He is afraid that

someone will want to change the $90,000 figure and maybe even the 20% overseas mis-

sions budget goal. Obviously, either one of these two changes would change the proper

overseas missions amount. He asks you to come up with a formula that would give the

proper overseas missions amount, M, given B, the total budget without overseas missions

and P the percent of the total church budget devoted to overseas missions.

PART 3: What is the proper formula?

Project 2

Suppose you are interested in buying a car. You know your finances will not permit a new

car, but you hope to get a late-model used one. After shopping around you find the perfect

vehicle and deal your way to a bottom-line figure of $9,900. Now comes the fun part-

financing. You have $2,500 in savings, so you can’t pay cash, but you can put some money

down. The dealer is happy to talk a four year loan, but the interest rate seems high at 13.5%.

You check with two local banks and they give you rates (again over four years) of 11.5%

and 12.0%, respectively.

174 CONRAD

PART 1: Decide what you can spend out of your savings as a down-payment. (This is up to you, but remember it is not wise to spend all your savings.) Using this figure, compute the monthly payments for each of the interest rates. (See page 225 in the textbook.)

PART 2: For each of the options, find the total amount of money you would pay over the

life of the loan.

PART 3: You know your budget could not take monthly payments over $175.00, what would you do? (You REALLY want this car!)

REFERENCES

Cohen, Marcus, Gaughan, Edward D., Knoebel, Arthur, Kurtz, Douglas S., & Pengelledy, David. (1991). Student research projects in calculus, MAA Spectrum Series.

Davidson, Neil (Ed.) (1990). Cooperative learning in mathematics: A handbook for teachers. New York: Addison-Wesley.

Hilbert, Steve, Maedi, John, Robinson, Eric, Schwartz, Diane, & Seltzer, Stan. (1991). The Ithaca

college caZcuZus project report. Presented at the Baltimore Joint Meetings of the AMS-MAA. National Research Council. (199 1). Moving beyond m_yths: Revitalizing undergraduate mathematics.

Washington, D.C.: National Academy Press. Sharan, Schlomo (Ed.) (1990). Cooperative learning theory and research. New York: Praeger. Slavin, R., Sharan, Schlomo, Kagan, S., Lazarowitz, R.H., Webb, C. & Schmuck, R. (Eds.) (1985).

Learning to cooperate, cooperating to learn. New York: Plenum.

a classroom experiment using small-group projects

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