focus issue: assessment || cooperative problem solving: but what about grading?

COOPERATIVE PROBLEM SOLVING: BUT WHAT ABOUT GRADING?Author(s): Diana Lambdin Kroll, Joanna O. Masingila and Sue Tinsley MauSource: The Arithmetic Teacher, Vol. 39, No. 6, FOCUS ISSUE: Assessment (FEBRUARY 1992),pp. 17-23Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195056 .

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COOPERATIVE PROBLEM SOLVING: BUT

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Diana Lambdin Kroll, Joanna О. Masingila, and Sue Tinsley Mau

your students use cooperative-group work when they are involved in

mathematical investigations and problem solving? If you answered yes, you are in good company. More and more teachers these days are finding that working together helps students become better problem solvers. Do you also grade some of your students' cooperative-problem-solving efforts? If you answered no, you are not atypical. Using cooperative groups for classwork is a lot easier

i

students' comments made during class discussions about their cooperative work, (3) commenting on individual students' write-ups of cooperatively solved problems, and (4) assigning grades.

Note that evaluation and grading are not one and the same thing. Actually at least four reasons can be cited for evaluating students' work: ( 1 ) to make decisions about the content and methods of instruction, (2) to make decisions about classroom climate, (3) to help in communicating what is im-

Four heads are better than one. All these students contribute to the group's solution of a challenging problem.

than assigning grades ror that work, but it can be done - and we think you'll be pleasantly surprised at the results!

The NCTM's Curriculum and Evalua- tion Standards (1989) stresses the impor- tance of using evaluation procedures that match the instructional format. If your students frequently work on problems in small groups, you should also evaluate this

The ideas in this paper were developed during work on a project of the Indiana University Mathematics Education Development Center (MEDC). That project was supported by National Science Foundation Grant NSF TEI-8751478. Any opinions, conclusions, or recommendations expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors would also like to acknowledge the support of John LeBlanc, director of the MEDC, and of Frank Lester and Vânia Santos of the MEDC for their help in conceptualizing evaluationalternatives for use in cooperative class- rooms.

The authors are involved in mathematics education at Indiana University, Bloomington, IN 47405. Kroll is particularly interested in mathematical problem solving, alternative-assessment techniques, and writing to learn mathematics. Masingila has a special interest in cooperative problem solving and ethnomathematics. Mau is interested in the attitudes and beliefs of college students enrolled in remedial mathematics courses.

work. In the words of the standards docu- ment, your evaluation techniques should be aligned with your teaching methods. Many ways can be used to evaluate cooperative work. Among them are ( 1 ) observ- ing as you circulate while students work cooperatively, (2) keeping notes of

portant, and (4) to assign grades (Lester and Kroll 1 99 1 ). Although each of these reasons for evaluation is important, our focus in this article is on evaluation for the purpose of assigning grades. When you gather data for course grades, you probably already use a variety of sources (e.g., individually performed quizzes and tests, individual classwork, homework, classroom observations, individual interviews, students' journals). Our article describes how you can begin to include cooperative problem solving among your sources of grading data.

What Is Cooperative Problem Solving? Before discussing how teachers might grade cooperative problem solving, it is important to be clear about a few defini- tions. When we talk about cooperative work, we envision small groups of two to six students working together to achieve a common goal. The defining feature of a problem situation is that some blockage mustbe experienced by the problem solvers; they do not know at first how to proceed. Thus, in group problem solving,

FEBRUARY 1992 17



the group is confronted with a situation that challenges everyone. They must work together to make sense of the problem; to plan an approach, or several; to try implementing their plan, often revising or replanning in the process; and, eventually, to verify that the solution they reach is appropriate for the problem situation. In cooperative problem solving, all members of the group struggle together to solve a problem that none of them has previously mastered.

Evaluating Work on Group Problem Solving Let's examine a grading scheme that can be used when students have been involved in cooperative problem solving in mathematics. This scheme, which involves both group and individual accountability, makes several assumptions: 1 . that the group being graded has had

experience in working together on problem-solving tasks,

2 . that the students have had previous practice in writing out their solutions to problems, and

3. that the students appreciate the impor- tance of ensuring that everyone in their group participates in and understands their group's solution.

The evaluation scheme consists of two phases. In phase 1 , students work in small preassigned groups to solve a problem and to write up a single group solution. In phase 2, students work individually to answer questions about their group

' s solution and to solve several similar problems. Thus, individuals must be able - on their own - to answer questions about the group's solution and to solve extensions of the problem.

The grading of the group's problem- solving efforts is similarly divided into two phases. First, the teacher grades each group's solution. We recommend using some sort of analytic scoring scheme (see our example in table 1). Because all students in the group are assumed to have concurred on the group's solution, all students in the group receive the same score on their solution (e.g., a group's solution might be awarded 1 4 of 1 5 possible points. ) Second, the teacher grades the individual

papers. The individual papers allow the teacher to see which students are able to demonstrate a clear understanding of the problem and of their group's solution. Papers by different individuals may be awarded different scores; for example, if the questions are worth a total of 1 0 points, the four individual papers from a group might receive such scores as 8, 10, 5, and 9.

One possible way to award individuals a grade for their cooperative-problem- solving effort would be total their group's score and their individual score. Thus different students in the same group may receive the same or different grades. For example, if a group paper received 14 (out of 1 5), and individual students in the group received scores of 8, 10, 5, and 9 (out of 1 0), their individual scores would be 22, 24, 1 9, and 23, respectively , from a total of 25 possible points. Under this scheme, each student receives a grade based partly on the group's achievement and partly on his or her individual achievement.

However, if all students in a group receive the same grade - a grade based on the achievement of the group as a whole - often more incentive occurs for individuals to try to ensure that all group members understand the group's solution. Thus, another way to grade cooperative problem solving is to give all group members the

same score. For example, each student could receive the sum of the group's solution score and the average of their individual scores. In the hypothetical forego- ing example, all the students in the group would receive a score of 22 ( 14 + 8, where 8 is the average of the four individual scores: 8, 10, 5, and 9).

Choosinq Problems to Be Graded Careful thought is required to find or develop problems that are appropriate for grading cooperative work. In choosing a problem for grading, several points need to be considered. The problem should be (a) one for which the students possess neither a known answer nor a previously established procedure for finding an answer, (b) neither too difficult nor too easy, (c) interesting to the students and challenging to their curiosity, and (d) one that involves the students in problem-solving behavior (i.e., understanding the problem, making a plan for solving the problem, implementing the plan, and evaluating the solution).

We offer as examples two problems that could be used in grading group-problem- solving efforts. Solutions to all problems discussed are given at the end of the article. For the primary grades, the "school-

^^^^^^^^^1 Analytic scoring scale

Understanding 0: Complete misunderstanding of the problem the problem 3. part of ше problem misunderstood or misinterpreted

6: Complete understanding of the problem

Planning a 0: No attempt, or totally inappropriate plan solution 3. Partially correct plan based on correct interpretation of

part of the problem 6: Plan could lead to a correct solution if implemented

properly

Getting an 0: No answer, or wrong answer based on an inappropriate answer plan

1 : Copying error, computational error, or partial answer for a problem with multiple answers

2: Incorrect answer although this answer follows logically from an incorrect plan

3: Correct answer and correct label for the answer

Adapted from Charles, Lester, and O'Daffer ( 1987. 30)

IB ARITHMETIC TEACHER



picture problem" might be appropriate to read aloud:

When the photographer lined up the three first-grade classes for school pictures, she put 3 students in the first row, 6 students in the second row, and 9 students in the third row. If she continued with the same pattern, how many students would the photographer put in the fifth row?

Other related problems can be made from this problem by (a) changing the context, or setting (e.g., planting flower

seeds instead of lining up students), (b) changing the numbers (e.g. , 4, 8, and 1 2 instead of 3, 6, and 9), (c) changing the number of conditions (e.g., specifying that the rows of students alternate between boys and girls and modifying the question appropriately), (d) reversing given and wanted information (e.g., given that 18 students were put in one row, find the number of the row in which they were placed), and (e) changing some combination of the context, numbers, conditions, and given-wanted information.

Adapting the basic problem in these five

ways allows you to give each group of students in the class a different problem without the difficulty of finding or creat- ing all the problems. Note that some types of modifications will leave the difficulty level of the problem essentially the same, whereas others will result in an easier or harder problem. You will want to choose your method of modification carefully to ensure that it produces a revised problem of the appropriate difficulty level.

A problem more appropriate for middle- grade students might be the "candle problem":

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Two candles of equal length are lighted at the same time. One candle takes 9 hours to burn out, and the other takes 6 hours to burn out. After how much time will the slower-burning candle be exactly twice as long as the faster- burning one?

Using the previously discussed five ways of adapting problems, the teacher could alter the candle problem by (a) changing the context to water draining from tanks instead of candles burning, (b) changing the numbers from 9 and 6 to 12 and 8, (c) changing the number of conditions by having one candle initially be twice as

long as the other, (d) reversing given and wanted information by giving that after 4.5 hours one candle is twice as long as the other and asking how many hours it takes for each candle to burn out, and (e) changing some combination of context, numbers, conditions, and given-wanted information. For an elaborated discussion of this process of adapting a problem, see Teaching Problem Solving: What, Why and How (Charles and Lester 1982).

Here are some other examples of problems that could be used for grading cooperative-problem-solving efforts. For primary-level students who have not yet

been introduced to multiplication, the "parade problem" might be appropriate:

A band marching in a parade had 16 rows with 8 people in each row. How many people were marching in the band?

A problem more suited to the middle grades is the "wrestle-mania problem."

Hulk Hogan entered an arm-wrestling contest along with 7 other wrestlers. Each of the wrestlers had 1 arm- wrestling match against each of the other wrestlers. How many total matches took place by the time the contest was over?

20 ARITHMETIC TEACHER



Gradina Group and Individua) Papers Although finding appropriate problems for grading is not easy, an equally difficult task is deciding how to evaluate group solutions. We suggest using an analytic scoring scale like the one in table 1 . Better to illustrate how this scoring method might be used, we present in figures 1 and 2 two solutions for the "candle problem."

Group A ' s work. Group A demonstrated (see fig. 1) that they understood the relationship between the burning of the two candles (i.e., one candle burned 2/1 8 [ 1/9] of the candle each hour and the other one burned 3/18 [1/6] each hour), so they received 6 points for understanding. Their plan involved making a picture dividing the candles into eighteen units and show- ing the amount of candle left after each hour of burning. Their plan was clear and correctly implemented, so they received 6 point for planning. This group also found and correctly labeled the answer, so they received 3 points for their answer. Thus, our evaluation of group A's work was U-6, P-6, A-3 (overall score-15).

Group B's work. Group В understood (see fig. 2) that the candles were decreas- ing in length at different rates, but they did not understand the relationship between these rates, so they received 3 points for understanding. Since this group tried un- successfully to write an equation and then used the guess-and-check method without being systematic, they received 3 points for planning. No answer was found by this group; as a result group В received 0 points for getting an answer. Their rating, then, was U-3, P-3, A-0 (overall score-6).

To extend the group problem and assess each group member's level of understanding of what their group did, follow-up questions can be designed for each student to answer individually. We recommend including a question to assess each individual's basic understanding of the problem (question 1 in the following), a question giving a problem of similar difficulty to be solved (question 2), and a question giving a problem extension to be solved (question 3). Here are some examples of follow-up questions of these three types for the "school-picture problem."

1 . (2 points) Would the photographer place exactly 7 students in any row?

2.(4 points) Suppose the photographer lined up the three second-grade classes for their picture by placing 5 students in the first row, 10 students in the second row, and 15 students in the third row. If she continued with the same pattern, how many students would the photographer put in the sixth row?

3.(4 points) Suppose the photographer lined up some first graders in the same pattern as that in the group problem. If 18 students were placed in one row, what is the number of the row in which they were placed?

Questions that could be used to follow up the "candle problem" are as follows:

1. (2 points) After two hours of burning, how much longer is the slower-burning candle than the faster-burning one?

2. (4 points) Two candles of equal length are lighted at the same time. One candle takes 6 hours to burn out, and the other takes 3 hours to burn out. After how much time will the slower-burning candle be exactly twice as long as the faster-burning one?

3. (4 points) A blue candle is twice as long as a red candle. The blue candle takes 4 hours to burn out, and the red candle takes 6 hours to burn out. After 3.5 hours, how long are each of the candles? The blue candle's length is then what fraction of the red candle's length? Work on the individual problems should

be graded after the group's solution has been assigned point values. Since the individual questions are intended to assess each group member's understanding of what the group did, each individual ' s work must be graded in light of the group's understanding of the original problem. For example, if a group drew a picture that incorrectly matched one hour (rather than 0 hours) with the candles' initial lengths,

group members will likely make the same error in answering their individual questions. Since points have already been deducted on the group's solution for this mistake, points for this same misunderstanding are not deducted again on the individual questions.

Tips for Grading Cooperative Problem Solving in Your Classroom Plan ahead. It's impossible to say too much about this aspect, and, of course, no matter how carefully you plan, some plans will not work quite as expected. Here is a list of several things you need to consider:

1 . Where will you get the problems used for grading? From the textbook? From another teacher? From your district's re- source person, assuming you have such a person? Will they be sufficiently difficult to necessitate a group solution and rich enough for extension questions for the individual part of the examination? It may be useful to start a notebook of interesting problems. Often you can jot down ideas for modifying problems that you find in your textbook or problems you have previously used in class. Perhaps you, along with the other teachers in your school or district, need to start a data bank of problems now. Don't forget that the Arithmetic Teacher and the Mathematics Teacher are excellent sources of problems! A list of several other good sources of problems is provided at the end of this article.

2. If you plan to grade the work of your entire class on the same day, you will probably have five to eight groups working at the same time. Will you use different statements or modifications of the same problem for all groups, or will you choose different problems? If you use restate- ments of the same problem, will the problem arise of groups' overhearing other groups' ideas?

3. What about the room layout? Is sufficient space available for students to work without interfering with other groups? Are sufficient materials (calculators, manipulatives, etc.) on hand for small- group problem solving? If not, can you make the necessary materials or acquire them from another teacher?

FEBRUARY 1992 *f

Evaluation should be aligned with

teaching. ШЕШШШЕШЕШЕШЕШ



4. How will you distribute points? You may want to adapt our guidelines to fit your needs, or you may want to create your own grading scheme. In either approach, plan your point assessments ahead of time and give students some indication of your planned assessment. Waiting until after the problems have been graded can lead to frustration for both you and the students.

Our most important piece of advise is this: Don't evaluate in this way unless you teach in this way. Students don't auto- matically work cooperatively, especially if they 've previously experienced years of instruction focused on individual accom- plishment. Students need time to adjust to group work and to each other. Give each group at least several class periods to work together before you grade their cooperative problem solving. You may even want to "practice grade" the work of cooperative groups once or twice before you actually assign grades for group work. This tactic will give you a chance to hone your grading procedures and will ensure that students understand what type of work you are expecting. For more ideas on using cooperative learning in your mathematics class, consult Artzt and Newman (1990) or Davidson (1990).

Incorporating Grading of Cooperative Problem Solving into Your Overall Evaluation Plan Cooperative problem solving is probably best used as a focus for just one of many types of evaluation in your classroom. Your overall evaluation plan needs to accomplish several goals: to motivate students to learn as much as they can (by participating in a group-solution effort), to give you relevant feedback aboutwhat individual students understand (from the individual portion of the group solution), and to generate data from which you can assign grades (from the combination of the two parts of the cooperative-problem- solving test).

However, to assure that the grades you assign to individuals are valid, you will want to evaluate each student on a variety of tasks, many of which are individual and a few of which are cooperative. For ex-

ample, at the end of a unit you might schedule a test worth 100 points; 25 points to be earned from a group-individual- problem-solving effort, such as the one we've described in this article, and 75 points from a totally individual, more tra- ditional portion of the test. Other grades in a marking period might be obtained from such various other sources as homework, classwork, individual journals, or group projects.

Clearly, teaching according to the vision of the NCTM ' s Curriculum and Evalu- ation Standards (1989) involves changes in how students are evaluated, as well as changes in content and instruction. Assigning grades for cooperative work is just one example. If you would like to be able to grade your students on challenging problems - problems that truly require insight, understanding, and problem- solving skills - the benefits to be gained from having them work in small groups outweigh the difficulties that this new approach may seem to present. Try cooperative problem solving in your classroom soon! We think you'll be glad you did!

Problem Solutions School-picture problem

Group problem: 3, 6, 9, 12, 15 - The photographer would place 1 5 students in the fifth row.

Follow-up problems: 1. 3,6,9 - No, the photographer would

not place exactly 7 students in any row.

2. 5, 10, 15, 20, 25, 30- The photographer would place 30 students in the sixth row.

3. 3, 6, 9, 12, 15, 18- The 18 first- grade students were placed in the sixth row.

Candle problem

Group problem:

After 4.5 hours the slower-burning candle will be exactly twice as long as the faster-burning one. (See group A's solution for a more detailed explanation.)

Follow-up problems: 1. After 2 hours, the slower-burning

candle has burned 4/18, leaving 14/18 of the candle. The faster- burning candle has burned 6/18, leaving 12/18 of the candle. Thus, the slower-burning candle is 2/1 8, or 1/9, longer than the faster-burning candle.

2. Divide the candles into sixths. After 2 hours, the slower-burning candle has burned 2/6, leaving 4/6 of the candle. The faster-burning candle has burned 4/6, leaving 2/6 of the candle. Thus, the slower-burning candle is twice as long as the faster- burning candle after 2 hours.

3 . Divide the red candle into 12/12 and the blue candle into 24/12. After 3.5 hours, the blue candle has burned 21/12, leaving 3/12 of the candle. The red candle has burned 7/12, leaving 5/1 2 of the candle. Thus, the blue candle's length is now 3/5 of the red candle's length.

Parade problem We know that 16 + 16 = 32, and we have four groups of 32; 32 + 32 + 32 + 32 =128. So 128 people were marching in the band.

Wrestle-mania problem We know that 8 wrestlers have 7 matches each; 8 x 7 = 56 matches. We divide by 2 to eliminate duplicating the counting of matches; 56 + 2 = 28. Or, the first person has 7 matches, the second person has 6 additional matches, the third person has 5 additional matches, and so on: 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28. So a total of 28 wrestling matches took place.

Some Good Sources of Problems • Problem-solving Experiences in

Mathematics A series of teachers' sourcebooks, one for each of grades l-8,containing black- line masters for problems that can be solved using a variety of strategies, and useful advice on. the teacher's actions during small-group problem-solving lessons (Charles et al. 1985)

• The Problem Solver: Activities for Learning Problem-solving Strategies Another series of teachers' sourcebooks,

M ARITHMETIC TEACHER



Photo Opportunity The Arithmetic Teacher always has a need for good photographs for use on the cover. Photo- graphs accepted for publication will give credit to the school and to the photographer. The following guidelines should be kept in mind when submitting sets of color slides for con- sideration: (a) The most effective pictures include one or two children; (b) backgrounds should not be too busy; (c) children should be actively engaged in a recognizable non-paper- and-pencil task; (d) the faces of the children should be clear; (e) children of different ethnic backgrounds should be included; and (/) ra- cial, sexual, or ethnic stereotyping should be avoided. Release forms should be completed for each person pictured. Material should be sent to the Director of Publications, NCTM, 1906 Association Drive, Reston, VA 22091.

Authors Sought for 1994 Yearbook The Council's yearbook for 1994 is tentatively titled Professional Development for Mathematics Teachers. The Editorial Panel, headed by Douglas Aichele, is seeking papers relating to the following categories: • Professional-development issues and perspectives

• Initial preparation of mathematics teachers • Professional development of practicing mathematics teachers

• Roles and responsibilities of providers of professional development

• Resources and delivery systems • Professional-development programs and practices that work

Initial manuscripts are due by 1 March 1992. Complete information regarding these categories is available from Arthur F. Coxford, General Editor, 1228 I School of Education, University of Michigan, Ann Arbor, MI 48109-1259. The guidelines can be obtained by writing or calling 313/764-8420 during working hours.

In NCTM Journals Readers of the Arithmetic Teacher might enjoy the following articles in this month's Mathematics Teacher: • " Soundoff: A Computer for All Students,"

Franklin Demana and Bert K. Waits • " Tips for Beginners: Summary Cards for

Tests," Edwin L. Clopton • " Activities: Mathematical Connections

with a Spirograph," Alfinio Flores

FEBRUARY 1992 23

one for each of grades 1-8, with black- line masters and teachers' notes (Hoogeboom et al. 1987, 1988)

• Make It Simpler A book of black-line masters for problems for middle-grade students, with accompanying teaching sugges- tions on using cooperative problem solving in the classroom (Meyer and Sallee 1983)

• A Sourcebook for Teaching Problem Solving or Problem Solving: A Hand- book for Teachers

Two books, each of which contains problems for all grades, along with notes about teaching problem solving (Krulik and Rudnick 1984, 1987)

References

Artzt, Alice F., and Claire M. Newman. How to Use Cooperative Learning in the Mathematics Class- room. Reston, Va.: National Council of Teachers of Mathematics, 1990.

Charles, R., G. Gallagher, D. Garner, F. Lester, L. Martin, R. Mason, E. Moffatt, J. Nofsinger, and С White. Problem-solving Experiences in Mathemat-

ics. Teacher sourcebooks for grades 1-8. Menlo Park, Calif.: Addison- Wesley Publishing Co., 1 985.

Charles, Randall, and Frank K. Lester, Jr. Teaching Problem Solving: What, Why and How. Palo Alto, Calif.: Dale Seymour Publications, 1982.

Charles, Randall, Frank Lester, and Phares O'Daffer. How to Evaluate Progress in Problem Solving. Reston, Va.: National Council of Teachers of Math- ematics, 1987.

Davidson, Neil, ed. Cooperative Learning in Math- ematics: A Handbook for Teachers. New York: Addison-Wesley Publishing Co., 1990.

Hoogeboom, Shirley, Judy Goodnow, Mark Stephens, Gloria Moretti, and Alissa Scanlin. The Problem Solver (Grades 1-8): Activities for Learning Problem-solving Strategies. Sunnyvale, Calif.: Creative Publications, 1987 and 1988.

Krulik, Stephen, and Jesse A. Rudnick. Problem Solving: A Handbook for Teachers. 2d ed. Boston: Allyn& Bacon, 1987.

. A Sourcebook for Teaching Problem Solving. Boston: Allyn & Bacon, 1984.

Lester, Frank K., Jr., and Diana Lambdin Kroll. "Implementing the Standards: Evaluation: A New Vision." Mathematics Teacher 84 (April 1991): 276-84.

Meyer, Carol, and Tom Sallee. Make It Simpler: A Practical Guide to Problem Solving in Mathemat- ics. Menlo Park, Calif.: Addison-Wesley Publish- ing Co., 1983.

National Council of Teachers of Mathematics, Commission on Standards for School Mathemat- ics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. Щ

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focus issue: assessment || cooperative problem solving: but what about grading?

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