small group learning in junior high school mathematics

Small Group Learning in Junior High School Mathematics*

Neil DavidsonUniversity of Maryland,

College Park, Maryland 20742

Linda Agreen, Connie DavisMarley Junior High School,

Glen Burnie, Maryland

Are you tired of standing up in front of your class presenting materialeveryday? Would you like a way out of the battle to quiet down the noisystudents and wake up the sleepy ones? If so, you might wish to considerteaching by the "small group discovery method." In this method thestudents join together in small groups to work on mathematicalproblems. You as teacher become the supervisor of the small groups.

In this paper you will find a number of practical suggestions aboutsmall group instruction in junior high. Issues for discussion include thefollowing: What are the basic classroom procedures with small groups?What is the teacher’s role in this classroom organization scheme? Howdoes one form groups that will work effectively on mathematicsproblems? What types of learning materials are most effective forstimulating group problem solving? How do junior high school studentsrespond to the freely structured physical and intellectual environment?

Classroom Procedures and Teacher Functions

In this method the class is divided into small groups, usually with fourmembers apiece, by means described later. Within each group thestudents work together to discuss mathematical ideas and to solveproblems.Group problem solving can be effective only if the members learn to

cooperate with one another. The teacher should promote cooperation bystating some group guidelines such as the following: 1) All students worktogether to get one solution. 2) Everyone must understand how they gotthe solution or answer before the group can go on to the next problem. 3)No person is to be’the boss. 4) Don’t race with other groups to be the firstone finished. 5) Stay in your group. You may change your group onlywith the consent of the teacher. 6) Everyone is to help. 7) EVERYONECOOPERATES!!!A teacher with small groups needs to explain or introduce new

material. Material may be presented to the entire class, as in a traditionalclass, or it may be introduced via a carefully written worksheet, which

*We are most grateful for the many valuable ideas and suggestions contributed by junior high school teachers LindaCraven, Virginia McKenzie, and Al Skierski. ’

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24 School Science and Mathematics

will be discussed later. A t^ird possibility is to present the new material toindividual groups or to a few groups at a time. This can be verysuccessful when there is more than one teacher in the classroom. It isespecially effective if the groups progress at different rates. However, itis difficult to employ with only one teacher present, since groups whichare left to work alone will undoubtedly have questions.

Direction, or guidance, is an important contributor to the success orfailure of small groups. Too much direction may stifle a group and toolittle direction may prevent a group from ever starting. Therefore ateacher must be perceptive enough to know when to give assistance andhow much to give. The quantity of assistance should be limited in orderto provoke the group members to pool their own ideas. Hints from theteacher may be needed if a group is unable to get started, has exhaustedall of its ideas, or has otherwise failed to solve a problem. However, hintsshould only be given if the group is willing to receive them.

It is most important for a teacher of small groups to visit each group asoften as possible to make sure that the students do not become frustratedor bored. Students usually become bored when they do not haveanything to do, do not know what to do, or do not know how to do theproblem. If the teacher is in frequent communication with the groups,these causes of boredom can be eliminated by enrichment problems, ex-planations or directions, or hhits.Group instruction may be initiated in several ways. Many teachers

prefer to "jump right in" because they feel that most students arefamiliar with the concept of groups as a result of informal use forhomework and daily class assignments. Some teachers, however, ease inslowly over a period of several days using group games and listeningexercises. A listening exercise is based on an algorithm or process whichis broken down into several steps. One group member is selected to dothe writing on the board while each of the other members receives a cardwith one step or direction written on it. Each direction is to be read aloudone and only one time. Every group member is to read his/herdirection in turn. The object is to have each student listen and cooperatein order to achieve the correct answer. These listening exercises can beespecially successful with slower students. Another option for initiatinggroup instruction is to have a selected group of students give ademonstration to show how groups should function. This demonstrationshould be well planned and rehearsed if it is to be effective.

Group Formation

Experience shows that math groups with five or more members are toolarge for effective cooperation and participation, except under unusualcircumstances. Groups with any size not exceeding four can be effective,

Small Group Learning 25

but on the average, groups of four work best. Moreover, groups of fourcan split into smaller groups of two on drill and practice material. Theactual number of groups depends on the class size and sometimes on thetopic being studied. Most teachers can readily handle five or six groupson any type of math problem, while some feel that they can handle asmany as eight. In very large classes, an aide is needed to help supervisethe groups; the aide can sometimes be a more advanced student.Formation of the small groups can be done in several ways. The

teacher may select the groups him/herself or allow students to form theirown groups. Friendship is an important criterion, since compatible per-sonalities among the members can result in more cooperation in groupwork. Intelligence level may also be a criterion. Some students feel morecomfortable with a group of their own ability level and working pace.Though groups may be initially formed by the students, if disciplineproblems or other problems arise, placement by the teacher is sometimesnecessary.Homogeneous groups allow both the slower and the faster learners to

move along at their own pace. The brighter students do not need as muchdrill work as the others, nor do they need constant teacher assistance andapproval. However, groups consisting solely of very slow learners aredisasters more often than not.

Heterogeneous groups are sometimes very useful, since they allow thefaster students to help the slower ones. However, not all slow studentslike to receive help and not all faster students are ready and able to giveit. Careful selection of group members should alleviate this difficulty.We have no single "best" answer to the issue of heterogeneous versus

homogeneous grouping. Grouping practices can be varied to meet theneeds in any given learning situation.

In any class there may be a few students who prefer to workindividually. Some of these students are highly intelligent and feel thatothers may hold them back; others are very slow and feel they cannotcontribute to a group. There are also students who simply tend to beloners. In any case students should not be forced to work with others.Absence creates a problem in small groups. Frequent absentees miss

much of the work and other members are often reluctant to take the timeto help them catch up. High absentees can be placed together in groupswith large "paper" enrollments but physically normal size on any givenday.

Students do switch groups. Sometimes, members of a non-productivegroup split themselves up to get better grades. At this age, personalitiesappear to change frequently. A student who cooperates well with hisgroup members one day may find they are his ex-friends the next, andswitch into a group in which he may work more effectively. The frequen-


cy of changing groups decreases as the students learn to work as effectiveteams.The period of time it takes for a group to function well varies accord-

ing to the group. A minimum seems to be three to five classes; thereseems to be no fixed upper bound, but most groups are working well bythe tenth class period. As the students become accustomed to groups,their behavior improves and the classroom noise level drops.

Instructional Material and Pacing

Various types of instructional material can provide a wide range ofactivities for small groups. Textbooks provide ready-made problems fordrill and practice. They can be a source of homework problems forstudents who need individual practice outside their group. Textbooksmay also be used occasionally for example problems and explanations,but the teacher must be aware of the reading level of the students andtheir ability to fully comprehend written instructions.

Worksheets are used most often by teachers in small group instruction.These sheets allow great freedom in construction and they can bedesigned to cover a wide range of material. Worksheets can also providegreat variety in the method of presentation, ranging from programmedinstruction and drill problems to puzzles and games, laboratory work,and guided discovery activities.Worksheets with guided discovery can provide solid understanding of

concepts in a challenging and interesting format. Students are given thechance to think for themselves and develop ideas while being guided to aspecific goal. However, smooth sailing cannot always be expected with adiscovery approach. Writing guided discovery worksheets can be a timeconsuming and difficult job. Not all material can be presented in thismanner and the teacher may be forced to select suitable topics. More-over, many students do not have the patience for discovery learning andlose interest before completing the material. Some students are more in-terested in the answer than the process and get frustrated in having towork out individual steps.The worksheets written and used by classroom teachers tend to contain

problems preceded by a minimum of written material. Instructions areusually given orally in order to insure their full comprehension by thestudents. If new material is introduced on a worksheet, the verbiageshould be as simple and short as possible, and the sheet should usuallyinclude example problems.With concerns about coverage of material in mind, many teachers take

measures to control the pace of their small groups. Some teachers writeworksheets which have very little, if any, discovery work. A betterapproach is to write worksheets with optional discovery problems for the


more capable groups. Also, some teachers specify a reasonable time limitin which the work must be completed. With these measures, coverage ofmaterial tends not to be a serious problem. The intention is that allgroups cover a certain basic amount of material without any sacrifice inunderstanding.

Physical Set-ups Affecting Small Groups

Small groups can work at the board, on desks or tables, or on carpetedfloors. Teachers who have enough board space for all their groups andhave them working there will find both advantages and problems. Ifthings are organized well, with a set location for each group at the boardand an understanding concerning procedures, there can be two big ad-vantages to board work. First, there is usually good group unity when thestudents work through a problem together at the board. They tend totalk things through as a group and come up with one solution to aproblem. Secondly, group progress and errors can be checked morequickly by the teacher when the groups’ solutions are on the board. Ittakes less time to glance at the board then it does to check the papers ofgroups working at tables, desks or on the floor.Along with these benefits, teachers can run into some problems with

board work. For instance, some students do not feel comfortable work-ing at the board. This can be the result of a lack of experience in workingwith others or a fear of asserting themselves or of making mistakes infront of their peers. There are also students who feel right at home in agroup at the board, but behave in an overly dominant manner and try to"teach" the group. This problem can sometimes be corrected by makingit mandatory that each group member do some of the writing.

Another problem arises with some very conscientious students whowant a copy of every problem and solution. If problems are done on theboard and immediately erased, some students have no work to show theirparents or no problems to study for a test. Teachers can solve thisproblem by passing out copies of the problems and solutions at the endof each week, or by letting students copy correct solutions from theboard.

In general, students at the board work together best on new andunfamiliar material for which they need each other’s help, and worst ondrill and practice work that most could do on their own. Some studentswork particularly well when they have to draw geometric forms on theboard for topics such as plane geometry, perimeter, area, and volume.An alternative to board work is to have the groups work at their seats,

with small tables or flat-topped desks pushed together. Students shouldbe seated close enough together so that everyone can see what is going onand participate.


Putting small groups to work at their seats has both benefits and draw-backs. First of all, control over the class is more easily maintained whenstudents are seated and not moving around. Secondly, some students feelmost comfortable at their seats and work better there than anywhere else.Lastly, students seem to prefer to do routine work such as drill and prac-tice at their seats. On the other hand, unless the furniture permitsstudents to work closely together, group work often turns into individualwork. Also, the check of group progress by the teacher takes longer whenhe has to walk around the room checking papers, rather than glancing atthe board. This last comment also applies to carpeted floors.

Discipline

There are always a few students who are not responsible enough towork in groups. There are rovers who travel from group to group tosocialize or steal a solution. Moreover, some students use the chalk todraw pictures on the board or on other students, or use erasers for throw-ing or creating dust clouds. Some teachers handle these chalk-boarddiscipline problems by making certain students work at their seats.

Occasionally there is a group that will put pressure on another groupto behave, but generally the teacher must devise his/her own methods ofresolving behavior problems. A good approach with which to begin is tostress accepting the responsibilities and freedoms of group work. Severalother methods are listening exercises, as described earlier, and the judi-cious use of grades. Removal of a problem student from a specific groupis possible. If all the problem cases are in the same group, however, thisseparation may backfire. In addition, if some students cannot cope withgroup work, they can be told to work individually. Later, if they show awillingness to cooperate, they can once again work in a group. Anotherapproach is to compromise with the students by allowing "rest later forwork now."Some traditional discipline problems are not present in small groups.

For example, talking and moving around, which were once problems, arenow permitted. Another advantage for the teacher using groups is theability to handle individual discipline problems without disturbing theentire class.

Evaluation of Students

A large variety of evaluation schemes can be used with the small groupmethod. Optional criteria for grading include in-class tests, group tests,group projects, homework, attendance, and group participation andcooperation.A few comments are in order about the various grading schemes. First,

a teacher who feels that tests are necessary can give them on a flexibleschedule to members of each separate group when they are ready. On the


other hand, a teacher can give a test to the whole class on one specificdate. This date should be realistic so that all groups can finish the mate-rial without a mad rush before the test. Excessive time pressure to covermaterial for a test will destroy group cooperation and reduce morale.

If a teacher gives grades for classwork, the grade can be based on at-tendance, participation, and cooperation in the group. The teachershould not grade students on their individual mathematical performanceduring class. Doing so will promote competition among students forgrades, and will greatly hamper the group efforts.

Concluding Remarks

The small group method provides a realistic and practical approach tolearning junior high school mathematics. Virtually all subject matter atthis level can be dealt with at least partly in small groups. Moreover,there is a flexible range of options for working with the groups. Groupscan be used all the time, on specific units, on specific days of the week, orduring portions of any class period.

In our experience there have been no parental complaints aboutchildren being involved in small groups. To date, administrative com-ments from the schools involved have been favorable.What are some of the disadvantages of learning in small groups in

junior high? First, coverage of material may be a problem for someteachers who are inexperienced with the small group method. Anotherdisadvantage is the initial difficulty in forming effective working groups.Success of the method requires cooperation that is sometimes hard toachieve, and absenteeism can be a persistent problem. Success of themethod also depends heavily on the quality of the written materials. Ateacher who writes his own materials will find this a difficult job whichrequires much preparation time. In addition, the teacher in a large classcannot provide enough help to all the groups, and an aide is needed.Discipline problems can be a major hurdle to overcome. Finally, a smallpercentage of students cannot or will not work in a group, and theyshould not be forced to do so.We conclude by stating some of the advantages of learning by the

small group method in junior high. Students learn mathematics byactively working at a comfortable pace. At the same time they learn tocooperate with others and to talk about mathematics. The classroomatmosphere tends to be relaxed and informal, help is readily available,questions are freely asked and answered, and even the shy student finds iteasy to be involved. Traditional "discipline" problems of talking andmoving around are eliminated by definition. Students tend to becomefriends with their group members, and the teacher-student relationshiptends to be more relaxed and pleasant than in a traditional approach. In


addition, many students maintain a high level of interest in the smallgroup activities. Many students, including some slow learners, like mathmore�or at least hate it less�than in teacher-directed approaches.Finally, the available experimental evidence consistently shows that thestudents learn as much mathematics in a small group approach as inother approaches, and sometimes even more.

REFERENCES

BIGGS, EDITH E., and JAMES R. MACLEAN, Freedom to Learn: An Active Learning, Ap-proach to Mathematics, Addison-Wesley (Canada) Ltd., 1969.

DAVIDSON, DENNIS. "Learning Mathematics in a Group Situation." The MathematicsTeacher, February 1974, pp. 101-106.

DAVIDSON, NEIL A. The Small Group-Discovery Method of Mathematics Instruction asApplied in Calculus. Doctoral dissertation, University of Wisconsin, 1970. TechnicalReport no. 168, Wisconsin Research and Development Center for Cognitive Learning,Madison, Wisconsin.

DAVIDSON, NEIL A. "The Small Group-Discovery Method as Applied in Calculus Instruc-tion." American Mathematical Monthly, August-September, 1971, pp. 789-791.

DAVIDSON, NEIL, RONALD MCKEEN, THEODORE EISENBERG, "Curriculum Constructionwith Student Input," The Mathematics Teacher, March 1973, pp. 271-275.

MCKEEN, RONALD and NEIL DAVIDSON, "An Alternative to Individual Instruction inMathematics," American Mathematical Monthly, December 1975, pp. 1006-1009.

STEIN, SHERMAN and CALVIN CRABI.LL, Elementary Algebra: A Guided Inquiry, HoughtonMifflinCo., 1972.

CLEAR AIR TURBULENCE JUST AHEAD

Clear air turbulence, sometimes dangerous, may be reliably detected by infra-red water vapor measurements made available in the aircraft cockpit usually intime to take necessary precautions. Passengers can be buckled up, air speed re-duced, or the area of hazard successfully avoided. For 51 test situations, signalanomalies from 4 to 10 minutes in advance of subsequent clear air turbulence en-counters were noted 80 percent of the time, while 12 percent were false alarmsand 8 percent were encounters with no warning. The high percentage of success-ful and timely warnings of impending clear air turbulence provided by the infra-red device should make it a valuable and welcome addition to aircraft onboardinstrumentation, especially that of large, fast, high-flying aircraft in both sub-sonic and, hopefully also, supersonic flight.

STUDYING IONS

Last year scientists identified for the first time the infrared spectrum of the hy-drogen molecular ion. Now under a two-year National Scieence Foundationgrant University of Arizona researchers will continue the work that led to thatdiscovery. Primarily they are studying atomic systems with three particles�verysmall atoms and molecules. They think their work will help them make very pre-cise measurements of certain ions and perhaps lead the way to theories thatwould predict certain constants in nature, such as the ratio of the proton mass tothe electron mass. They also plan to study the helium atom in states of high exci-tation. A highly excited helium atom may be 10,000 or more times larger thanone not excited. Atoms in highly excited states are sometines found in the upperatmosphere and in the interstellar medium. Another area the scientists will studyis the interaction of helium and neon atoms that occurs in helium-neon lasers,the most common laser.

small group learning in junior high school mathematics

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