COOPERATIVE PROBLEM SOLVING: USING K-W-D-L AS AN ORGANIZATIONAL TECHNIQUE

Cooperative learning continues to prove its effectiveness in many facets of mathematics education. Not only does cooperative learning promote achievement with many levels and types of students (Slavin 1991), but as students work together in groups, communication and interpersonal-relations skills are refined (Greenes, Schulman, and Spungin 1992; AAAS 1989, 1993). Students in small groups are more involved with the subject matter and with one another than they are in whole-group mathematics contexts (Mulryan 1992).

Inherent in cooperative work are such valued processes as clarifying, comparing, and defending ideas as well as the social skills of listening, compromising, and reaching consensus (Rees 1990; Yackel, Cobb, and Wood 1991). Collaborative-group work affords diverse opportunities for engaging students in meaningful discourse (NCTM 1989, 1991). It contributes to a sense of mathematical community as recommended in Everybody Counts (National Research Council 1990).

IMPLEMENTING COOPERATIVE LEARNING IN MATHEMATICS: At the request of teachers in a Professional Development School (PDS) site for the University of Mississippi's teacher-education program, we initiated a cooperative-learning project with fourth-grade teachers and their students. In this rural school district, the teachers had not previously incorporated much organized cooperative learning in their mathematics lessons, and they were eager to find out more about using cooperative-learning strategies effectively. Students in two classrooms at the PDS site participated regularly in cooperative-learning groups for mathematics and other subjects. Students in two other classrooms worked only occasionally in groups. In the classrooms where cooperative learning was a regular practice, students engaged in group problem solving in mathematics two to four times each week. Often cooperative-learning sessions followed large-group introductions to topics. In their groups, the students worked mathematics problems using their textbook materials, exercises from Cooperative Learning Resource Activities (Haubner, Rathmell, and Super 1992) (see fig. 1), material adapted from AIMS (1987) (see fig. 2), real-life situations suggested by their teachers, and materials furnished by university personnel (see fig. 3). We introduced and reviewed several specific problem-solving strategies, such as guess and check, make a chart, and use a picture. In addition, as students initiated, developed, and shared other strategies, these approaches became part of the repertoire of problem-solving strategies that were available. The students worked in groups on problems using problem-solving strategies; they also created and shared similar problems of their own. The students' favorite types were logic problems and open-ended problems developed from everyday-life situations, such as the one shown in figure 4.

K-W-D-L: A TECHNIQUE FOR ORGANIZING AND RECORDING WORK To guide the children's work, we used a modification of Ogle's (1986) K-W-L technique (fig. 5). Originally developed for improving reading comprehension, the technique guides readers through steps that mature readers take as they read expository material. The technique is widely used for reading, but it also holds much potential for use in mathematics investigations. Explanations of K-W-L and the ways it was used for mathematics problem solving follow.

K--WHAT I KNOW In this step, readers brainstorm and discuss what they already know about a topic. The teacher lists their responses and helps the students categorize the pieces of information of which they are already aware. He or she then helps the students identify anything, such as possible misconceptions, that they want to check or clarify as they proceed. For group mathematics problem solving, the "K" step involves students' reading, paraphrasing, and discussing the problem to see what information is provided. It may also include other strategies, such as acting out the problem, drawing pictures, or making a chart so that students begin to understand the problem and recognize what they already know about it.

W--WHAT I WANT TO FIND OUT With the teacher's guidance, students identify areas about which they want to learn. Often they pose questions that have not been answered in the expository text--or raise topics that have not been discussed--and must consult other sources to find their answers and information. For mathematics problem solving, this step may simply involve group agreement on what is being asked--what is the question and what does it mean? The "what I want to find out" step may also involve the students' deciding on a plan to solve the problem. They may agree that they need to find data and then decide on sources of the data. Perhaps they will need to poll or talk with others, make measurements, perform experiments, or consult reference books.

L--WHAT I LEARNED: Ogle's "what I learned step" involves students' reading the text silently and recording their findings. Their responses can be shared in various ways. For example, they can write about the facts they have learned and read their written responses to classmates. This step helps learners refine and expand their thinking about the reading and writing processes.

In mathematics problem solving, the "LL" step requires learners to state and defend their answer or answers and to describe how they went about working on the problem; they can verify their work by letting others check it, or they can speak about the reasonableness of their answers. Groups are also encouraged to reflect on, and write about, any general information that they learned. For example, the students in a group might write and talk about how making a picture was helpful or how they used a guess-and-check strategy.

To Ogle's steps we added a "D" step: "What I did." Group members used a recording sheet as they worked problems together. The "What I know" and "What I want to know" steps often helped them understand the problem, plan how to solve it, and evaluate their answers. Their "What I did" narratives and notes helped the students think consciously about the plans and processes they used as they worked together on problems. Our "D" step came third, preceding the "L," or "What I learned," step.

RESULTS We used mathematics problem-solving pretests and posttests for the students in the two sets of classrooms. The tests included versions of two reasoning problems--a two-factor problem and a spatial problem (fig. 6). The children worked in groups using manipulatives as they desired. We scored the groups' work using Charles, Lester, and O'Daffer's (1986) focused holistic scoring scale (fig. 7). The students in the classrooms using cooperative learning scored substantially higher than did the students in the other classrooms. We also compared the problem-solving samples of students in the groups that regularly used cooperative learning in mathematics with samples from the groups that did not. We noticed several qualitative differences. Generally, the responses of the cooperative-learning students were longer and more detailed than those of the other students; perhaps groups accustomed to working together were able to generate fuller descriptions of their reasoning than those who did not regularly work together. In general, the cooperative-group students drew more detailed sketches; on the spatial problem, these students were more likely than the students who did less group work to use designs with more than one layer.

Anecdotal evidence supports increased positive attitudes for students who regularly used cooperative learning with the K-W-D-L technique for mathematics problem solving. The children stated that they enjoyed working together. They expressed growing confidence, interest, and excitement. We heard such statements as "Let's do more!" and "We get it! We can do anything!" The students seemed proud of their growing abilities to solve problems, especially two-factor reasoning problems. As they worked these problems, the children employed various strategies, including drawing pictures (fig. 8), making charts to reflect the two factors (fig. 9), and using guess and check. As they worked in groups, students often remembered to check themselves to ensure that their answers fit the requirements of the problems. The children were generally cooperative and enthusiastic in their work. They learned to come to consensus as needed; often students who did not agree with their group's opinion were encouraged to write their own views and append them to their group's reports.

The teachers in the PDS site remain enthusiastic about cooperative learning for mathematics. They point out such advantages as greater individual involvement and assumption of responsibility by students, more on-task behavior, and the development of group pride and spirit. The teachers state that the use of groups makes mathematics lessons more interesting to both the students and themselves. They use cooperative groups for such activities as mathematics games and homework checks as well as for problem solving. At times, the teachers present certificate-like rewards to groups that work effectively; they observe that the students work well with or without the rewards.

CONCLUSION Having students write about their mathematics problem-solving experiences has been valuable; the process connects mathematics and communication skills and enhances students' reasoning. Using K-W-D-L as a framework to get groups started in organizing and documenting their work has proved workable and effective. Other teachers may choose to implement the technique to help students consider the processes they use as they solve problems together. Educators may want to share the K-W-D-L process with parents and other family members as a structure for helping their children develop study skills and for increasing their academic autonomy.

Added material: Jean Shaw, Martha Chambless, and Debby Chessin are colleagues at the University of Mississippi, University, MS 38677. Vernetta Price and Gayle Beardain teach at South Panola Intermediate School, Batesville, MS 38606. FIGURE 1 Sample of cooperative problems from Cooperative Learning Resource Activities (Haubner, Rathmell, and Super 1992) FIGURE 2 Cooperative-learning activity adapted from AIMS (1987) FIGURE 8 Results from using a drawing-pictures strategy FIGURE 9 Results for the coins problem from using a make-a-chart strategy

BIBLIOGRAPHY

AIMS Education Foundation. Primarily Bears. Fresno, Calif.: AMIS Educational Foundation, 1987.

American Association for the Advancement of Science (AAAS). Benchmarks for Science Literacy. New York: Oxford University Press, 1993.

American Association for the Advancement of Science (AAAS). Project 2061. Washington, D. C.: AAAS, 1989.

Augustine, Dianne K., Kristin D. Gruber, and Lynda R. Harrison. "Cooperation Works!" Educational Leadership 47 (December--January 1990): 4-7.

Charles, Randall, Frank Lester, and Phares O'Daffer. How to Evaluate Progress in Problem Solving. Reston, Va.: National Council of Teachers of Mathematics, 1986.

Greenes, Carole, Linda Schulman, and Rika Spungin. "Stimulating Communication in Mathematics." Arithmetic Teacher 40 (October 1992): 78-82.

Haubner, Mary Ann, Edward Rathmell, and Douglas Super. Cooperative Learning Resource Activities. Boston: Houghton Mifflin Co., 1992.

Mulryan, Catherine M. "Student Passivity during Small Groups in Mathematics." Journal of Educational Research 85 (May--June 1992): 261-73.

National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.

National Council of Teachers of Mathematics. Professional Standards for Teaching Mathematics. Reston, Va.: The Council, 1991.

National Research Council. Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D.C.: National Academy Press, 1990.

Ogle, Donna. "K-W-L: A Teaching Model That Develops Active Reading of Expository Text." Reading Teacher 39 (February 1986): 564-70.

Rees, Rebecca D. "Station Break: A Mathematics Game Using Cooperative Learning and Role Playing." Arithmetic Teacher 37 (April 1990): 8-12.

Slavin, Robert E. "Synthesis of Research on Cooperative Learning." Educational Leadership 48 (February 1991): 71-87.

Yackel, Erna, Paul Cobb, and Terry Wood. "Small-Group Interactions as a Source of Learning Opportunities in Second-Grade Mathematics." Journal for Research in Mathematics Education 22 (November 1991): 390-408.

FIGURE 3 Teacher-generated problem Kenny feeds some of the animals on his farm. His mom asks, "Did you feed them all?" Kenny tries to fool his mom and says, "I counted 14 heads of the animals I fed. I counted 32 feet." Kenny's mom thought a while. She knew that Kenny fed some chickens and some horses. She finally figured it out. Can you figure it out too? Reread what Kenny said. See if you can tell how many horses and now many chickens Kenny fed. Show your work.

FIGURE 4 Open-ended, everyday-life problem You want to buy groceries for at least 4 people for 2 meals. You must include each food group in each meal. Use grocery ads from the newspaper. Plan what you will buy to stay within a budget of $20.00. Estimate the costs of the things you want. Discuss how you know your total is close to $20.00. Next use a calculator to find the actual cost. Use the tax table to find and add the tax you will have to pay.

FIGURE 5 Modified K-W-L process K What we know. W What we want to know. D What we did. L What we learned.

FIGURE 6 Sample test items

1. LaToya has 65¢ in her pocket. She has a total of 11 coins. The coins are nickels and dimes. How many of each coin does she have? Figure it out and show your work. Tell how you got the answer. Tell why you think you are right.

2. Mr. Black is making a display of Mother's Day mugs at Panola Variety store. He needs to arrange 36 mugs. The mugs are packaged in gift containers as shown. Design 2 or more displays that Mr. Black might use. Use the cubes or other manipulatives if you wish. Draw your designs. Tell how you worked on the problem.

FIGURE 7 Focused Holistic Scoring Point Scale 0 POINTS These papers have one of the following characteristics:

* They are blank.
* The data in the problem may be simply recopied, but nothing is done with the data or there is work but no apparent understanding of the problem.
* There is an incorrect answer and no other work is shown.
1 POINT These papers have one of the following characteristics:
* There is a start toward finding the solution beyond just copying data that reflects some understanding, but the approach used would not have led to a correct solution.
* An inappropriate strategy is started but not carried out, and there is no evidence that the student turned to another strategy. It appears that the student tried one approach that did not work and then gave up.
* The student tried to reach a subgoal but never did
 
2 POINTS These papers have one of the following characteristics:
* The student used an inappropriate strategy and got an incorrect answer, but the work showed some understanding of the problem.
* An appropriate strategy was used, but-- a) it was not carried out far enough to reach a solution (e.g., there were only 2 entries in an organized list); b) it was implemented incorrectly and thus led to no answer or an incorrect answer.
3 POINTS These papers have one of the following characteristics:
* The student has implemented a solution strategy that could have led to the correct solution, but he or she misunderstood part of the problem or ignored a condition in the problem.
* Appropriate solution strategies were properly applied, but-- a) the student answered the problem incorrectly for no apparent reason; b) the correct numerical part of the answer was given and the answer was not labeled or was labeled incorrectly; c) no answer is given.
* The correct answer is given, and there is some evidence that appropriate solution strategies were selected. However, the implementation of the strategies is not completely clear. 4 POINTS These papers have one of the following characteristics:
* The student made an error in carrying out an appropriate solution strategy. However, this error does not reflect misunderstanding of either the problem or how to implement the strategy, but rather it seems to be a copying or computational error.

* Appropriate strategies were selected and implemented. The correct answer was given in terms of the data in the problem. (From Charles, Lester, and O'Daffer [1986, 35])

SOURCE: Teaching Children Mathematics v3 p482-6 May '97. Reproduced with permission from Teaching Children Mathematics, copyright 1997 by the National Council of Teachers of Mathematics. All rights reserved.