teacher interventions in cooperative-learning mathematics classes

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This article was downloaded by: [UTSA Libraries] On: 05 October 2014, At: 05:49 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK The Journal of Educational Research Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/vjer20 Teacher Interventions in Cooperative-Learning Mathematics Classes Meixia Ding a , Xiaobao Li b , Diana Piccolo a & Gerald Kulm a a Texas A & M University b Western Carolina University Published online: 07 Aug 2010. To cite this article: Meixia Ding , Xiaobao Li , Diana Piccolo & Gerald Kulm (2007) Teacher Interventions in Cooperative- Learning Mathematics Classes, The Journal of Educational Research, 100:3, 162-175, DOI: 10.3200/JOER.100.3.162-175 To link to this article: http://dx.doi.org/10.3200/JOER.100.3.162-175 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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Page 1: Teacher Interventions in Cooperative-Learning Mathematics Classes

This article was downloaded by: [UTSA Libraries]On: 05 October 2014, At: 05:49Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

The Journal of Educational ResearchPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/vjer20

Teacher Interventions in Cooperative-LearningMathematics ClassesMeixia Ding a , Xiaobao Li b , Diana Piccolo a & Gerald Kulm aa Texas A & M Universityb Western Carolina UniversityPublished online: 07 Aug 2010.

To cite this article: Meixia Ding , Xiaobao Li , Diana Piccolo & Gerald Kulm (2007) Teacher Interventions in Cooperative-Learning Mathematics Classes, The Journal of Educational Research, 100:3, 162-175, DOI: 10.3200/JOER.100.3.162-175

To link to this article: http://dx.doi.org/10.3200/JOER.100.3.162-175

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Teacher Interventions in Cooperative-Learning Mathematics Classes

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he potential of cooperative learning to improvestudents’ academic and social performance hasbeen widely recognized (Slavin, 1996; Webb,

1989). In regard to mathematics classrooms, the NationalCouncil of Teachers of Mathematics (NCTM; 1989) advo-cates cooperative learning because “small groups provide aforum in which students ask questions, discuss ideas, makemistakes, learn to listen to others’ ideas, offer constructivecriticism, and summarize their discoveries in writing” (p.79). In addition, the NCTM Principles and Standards forSchool Mathematics (2000) includes communication asone of five process standards. Cooperative learning is aneffective way to develop the ability to communicate withothers. However, some mathematicians (e.g., Wu, 1997)doubt whether cooperative learning could be used in math-ematics classrooms without considering the major purposeof mathematics instruction, which is to help students learnto think mathematically (Schoenfeld, 1988).

Because cooperative learning is used widely in mathe-matics classrooms, a growing need exists to examine howteachers use this technique in classroom settings. We focuson teacher interventions and use a qualitative methodolo-gy to examine whether teachers in cooperative-learningclassrooms addressed students’ mathematical thinking. Our

intent was to help readers become cognizant of possiblechallenges in cooperative-learning mathematics classroomsand to provide insights for teacher instruction and profes-sional development. Because “the research into the teach-ers’ role in facilitating cognitive and metacognitive gainsthrough cooperative learning is in its infancy, or perhapschildhood” (Meloth & Deering, 1999, p. 254), we con-tribute to this much-needed research.

Teacher Role in Cooperative Learning

The teacher’s role in cooperative learning generallyincludes (a) specifying objectives, (b) grouping students, (c)explaining tasks, (d) monitoring group work, and (e) evalu-ating achievement and cooperation (Bettenhausen, 2002).Among those tasks, teacher intervention in monitoringgroup work is associated directly with students’ cognitiveperformance (Chiu, 2004; Meloth & Deering, 1999).

During group-work monitoring, a teacher is “both anacademic expert and a classroom manager” (Johnson &Johnson, 1990, p. 112). However, researchers in earlierstudies emphasized the role of classroom manager. Forexample, Johnson and Johnson (1991) stated,

The teacher monitors the functioning of the learninggroups and intervenes to teach collaborative skills and pro-vides task assistance when it is needed. The teacher is morea consultant to promote effective group functioning than atechnical expert. Typical statements a teacher may makeare, “Check with your group”; “Does anyone in your groupknow”; “Make sure everyone in your group understands.”(p. 61)

Kagan (1985) suggested that teachers should be freed evenmore in group investigation to allow students to assumeresponsibility for learning. Teachers typically consult withgroups and suggest ideas or possibilities for exploration.Cohen (1991, 1994a) suggested minimizing monitoring to

Address correspondence to Meixia Ding, Texas A & M University,4232 Harrington Tower, Teaching Learning and Culture, College Sta-tion, TX 77843-4232. (E-mail: [email protected])

Copyright © 2007 Heldref Publications

Teacher Interventions in Cooperative-Learning Mathematics Classes

MEIXIA DING Texas A & M University

XIAOBAO LIWestern Carolina University

ABSTRACT The authors examined the extent to whichteacher interventions focused on students’ mathematicalthinking in naturalistic cooperative-learning mathematicsclassroom settings. The authors also observed 6 videotapesabout the same teaching content using similar curriculumfrom 2 states. They created 2 instruments for coding the qual-ity of teacher intervention length, choice and frequency, andintervention. The results show the differences of teacherinterventions to improve students’ cognitive performance.The authors explained how to balance peer resource and stu-dents’ independent thinking and how to use peer resource toimprove students’ thinking. Finally, the authors suggestdetailed techniques to address students’ thinking, such asidentify, diversify, and deepen their thinking.

Key words: cognitive performance, cooperative-learningmathematics classes, teacher interventions

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DIANA PICCOLOGERALD KULMTexas A & M University

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help students become more interdependent, autonomous,and self-directed. Cohen observed that students reduced theamount of cooperation and communication between eachother after the teacher intervened. Therefore, Cohen pre-ferred to use the quick-response strategy in which teachersprovide brief comments and questions, then move away fromthe group so that students can continue their discussion.

Johnson and Johnson (1990), Kagan (1985), and Cohen(1991, 1994a) encouraged teachers to monitor the group’son-task behavior and cooperative skills and to provide taskassistance when necessary. Therefore, the teachers’ role incooperative-learning classrooms is “more like a consultantwho helps improve effective group functioning than aninstructor who contributes information or scaffolds stu-dents’ learning” (Meloth & Deering, 1999, p. 244). Theimportance of the teachers’ role as indicated in these stud-ies was underestimated.

Significance of Teacher Intervention

Johnson and Johnson (1990) suggested that,

Simply placing students in groups and telling them to worktogether does not in and of itself promote greater under-standing of mathematical principles or ability to communi-cate one’s mathematical reasoning to others. There aremany ways in which group efforts may go wrong. (p. 104)

One way that group efforts fail is when teachers do not pro-vide necessary help when needed. Meloth and Barbe(1992) examined 180 peer groups and found similar resultsduring most of the group studies. They also reported that80% of the teachers’ monitoring statements were not ori-ented toward special-task content. Cohen (1994b) empha-sized that if teachers do not provide clear and explicit assis-tance when students need it, students are unlikely toengage in task-specific learning. That lack of teacher assis-tance was the most important reason for the low achieve-ment of students in group-learning settings (Webb, 1989).

Teacher intervention during peer interactions is alsoimportant because teachers need to give students helpwhen (a) no student in the group can answer a question(Hamm & Adams, 2002); (b) students have difficulty com-municating with each other, which might cause or rein-force misconceptions in peer interaction (Brodie, 2001); or(c) group members treat one another with authority and notrue dialogues exist (Amit & Fried, 2005). In all those sit-uations, the teacher’s intervention is a resource to help stu-dents enhance their thinking.

Characteristics of Effective Intervention

What type of teacher intervention is effective in coopera-tive-learning mathematics classrooms? First, teachers shouldadapt their help to students’ needs (Webb 1989, 1991). Stu-dents in Grades 4–8 believed that teachers should providereadily available help to achieve positive group discussions(Ares & Gorrell, 2002). Chiu (2004) explored the Teacher

Intervention (TI) model and concluded that if teachersaccurately evaluate and adapt their interventions to studentneeds, they can improve students’ problem solving and timeon task. Second, teachers should focus on cognitive andmetacognitive aspects when providing help to students(Deering & Meloth, 1993; Meloth & Deering, 1992, 1999).

Kramarski, Mevarech, and Arami (2002) investigated theeffects of cooperative learning with and without metacogni-tive instruction to solve mathematical tasks of seventh-grade students. They found that students who were taughtwith metacognitive instruction significantly outperformedtheir counterparts. Kramareski and Mevarech (2003) fur-ther explored the following mathematics teaching methodswith 384 students in eighth grade: (a) cooperative learningwith and without metacogitive training and (b) individuallearning with and without metacogitive training. They con-cluded that cooperative learning, combined with metacog-nitive training, was the most effective teaching method.

Finally, teachers should combine teacher and peerresources when intervening with students. For example,Dekker and Elshout-Mohr (2004) compared two kinds ofteacher interventions designed to help students. Product-help interventions (p. 44) concerned students’ mathematicalreasoning and products, such as teachers asking students toexplain and justify their work or offering hints and scaf-folding their thinking. Process-help interventions (p. 43)addressed peer recourses such as stimulating interactionprocesses. The authors emphasized that process-help inter-ventions were more effective than were product-help inter-ventions for improving students’ mathematical thinking. Insummary, effective teacher-intervention strategies used incooperative-learning mathematics classroom include (a)adapting teacher instruction to students’ needs, (b) focus-ing on cognitive and metacogitive aspects, and (c) com-bining teacher and peer resources.

Purpose

We examined the ways that teachers use cooperativelearning in mathematics classrooms, with a focus onimproved student thinking about mathematics. Becausesome researchers reported that classroom interventions areunderestimated and a large gap exists between research andeveryday classroom settings about the effects of cooperativelearning (Blatchford, Baines, Bassett, Rubie-Davies, &Chowne, 2006), we compared classroom teaching experi-ences and documented the situations according to twoquestions: (a) What is the frequency, type, and length ofteachers’ interventions? (b) What is the quality of teachers’interventions?

Method

We used the qualitative method to explore teacher inter-vention by focusing on students’ cognitive processes (Den-zin & Lincoln, 2000, p. 8); our research questions explored

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quality, meanings, and interpretations (Janesick, 2000, p.382). In comparison with the quantitative approach, ourmethod “more easily allows for the discovery of new ideasand unanticipated occurrences” (Jacobs, Kawanaka, &Stigler, 1999, p. 718). We investigated our questions byusing multicase studies (Bogdan & Biklen, 2003; Stake,2000) and collected data by detailed observation (Michael& de Pérez, 2000). We then analyzed, interpreted, and pre-sented our data with qualitative research resources (Bogdan& Biklen; Denzin & Lincoln; Miles & Huberman, 1994).

Data Source

We chose video data as our primary data source to allowfor sophisticated analysis. Jacobs and colleagues (1999)reported three advantages of this type of data: (a) “rela-tively unfiltered through the eyes of researchers” and“arguably more ‘raw’ than other forms of data” (p. 720); (b)“more versatile than other forms of data” (p. 720) andviewable by researchers from diverse cultural and linguisticbackgrounds who might bring fresh perspectives to videoanalysis and examine many facets of the data; and (c) read-ily watched, coded, and analyzed repeatedly from differentdimensions because of data source permanence.

All the authors in this study are members of the MiddleSchool Mathematics Project (MSMP), a 5-year longitudinalstudy examining how the use of specific research-basedinstructional strategies in classrooms relates to lastingimprovements in student learning. A major role of MSMPmembers is to observe, transcribe, and analyze classroomvideotapes. When the first two authors, who both have 5years of teaching experience in China, observed the projectvideos, they found that American teachers tended to usecooperative-learning methods. This type of teaching wastotally new and different to Chinese teachers who mainlyused direct teaching, with a tendency toward addressing stu-dents’ mathematical and abstract thinking (Cai, 2001). Thedifferent cultural backgrounds of the first two authorsallowed them to view American classroom videos with a dif-ferent perspective. They raised several questions, such as,Why were so many students in cooperative-learning math-ematics classes off task? Why did teachers walk around class-rooms simply observing whether students agreed with eachother? What did students learn in cooperative-learningmathematics classes? The first two authors discussed thosequestions with their American colleagues—the third andfourth authors. As a result, all the authors decided to exam-ine teacher intervention in cooperative-learning mathe-matics with the focus on students’ mathematical thinking.

We selected six videotaped lessons that examined thesame teaching content from the same textbook using onlycooperative-learning methods. We chose all of the lessonsfrom the MSMP throughout the 2002–2003 school year.Because the MSMP was not specifically designed for coop-erative-learning research, our video data reflected authen-tic classroom situations in which the teachers used their

usual teaching methods. All the videotaped lessons werefrom 6 sixth-grade teachers, including 5 women and 1 man;all were Caucasian. According to the MSMP teacher data-base, the teachers came from five schools in three schooldistricts. Three teachers taught in Delaware and 3 teacherstaught in Texas. Teaching experience varied from 0 to morethan 25 years.

Among the 6 teachers, 4 teachers had elementaryteacher certificates and 2 teachers had middle schoolteacher certificates. Five teachers had a master’s degree and1 teacher had a bachelor degree. Their professional devel-opment training throughout the 2002–2003 school yearvaried from 4 to 10 days. From our observation of thevideotapes, the classes typically had 18–35 students andlasted approximately 30–50 min. In general, there werethree cooperative-learning group sizes: large size (8 studentsper group), middle size (4–6 students per group), and smallsize (2–3 students per group). The range of group size pro-duced a good opportunity to study various types of teacherinterventions. In addition, in two of the classes, teachersemployed cooperative learning immediately after assigninga task; in the other four classes, teachers used individualstudy for 3–5 min before students began group work. (SeeTable 1 for detailed teacher and class information.)

Content of Instruction

The 6 teachers used Connected Mathematics: Bits andPieces I, a sixth-grade text (simply called CMP curriculum).CMP curriculum, which is inquiry and discovery based, is astudent-centered learning textbook (Rivette, Grant, Ludema,& Rickard, 2003). It helps teachers create a supportive envi-ronment, such as a cooperative-learning classroom for stu-dents to investigate interesting problems and to achieve con-ceptual understanding through a real-world context (Lappan,Fey, Fitzgerald, Friel, & Phillips, 1998).

The lesson in this study was about fractions. We askedstudents to compare 2/3, 3/4, and 6/8, and decide whichones were equivalent to $270 out of $360. We encouragedstudents to explore by using fraction strips to measure thethermometer that represented the problem in the book.Students had difficulty using and understanding fractions(Hiebert, 1988), so this lesson helped us examine teacherinterventions during students’ investigation and coopera-tion. The lesson is reproduced as follows:

At the end of the fourth day of their fundraising cam-paign, the teachers at Thurgood Marshall School hadraised $270 of the $360 they needed to reach theirgoal. Three of the teachers got into a debate abouthow they would report their progress.

• Ms. Mendoza wanted to announce that the teachershad it three fourths of the way to their goal.• Mr. Park said that six eighths was a better description.• Ms. Christos suggested that two thirds was reallythe simplest way to describe the teachers’ progress.

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A. Which of the three teachers do you agree with?Why?B. How could the teacher you agreed with in Part Aprove his or her case?

Procedures

The first author transcribed all of the selected videotapesand recorded and described verbal and nonverbal represen-tations in detail. For example, when a teacher approacheda group, how many minutes did she listen to the students’discussion before she intervened? Such information wasimportant because it partially reflected how a teacher usedpeer resources. After the description, the first two authorsparsed the transcribed lesson into small units, each ofwhich represented an action sequence at a particular grainsize (Schoenfeld, 1998) or a “sighting” according to teach-ers’ intervention choices and topic changes (AmericanAssociation for the Advancement of Science [AAAS],2004). A detailed explanation about sighting is provided inthe intervention choice section that follows. On the basisof the videos and video transcriptions, the authors codedand classified teacher intervention length, choice, and fre-quency in terms of Instrument 1. Moreover, the authorscoded and analyzed teacher intervention quality of eachsighting by using the categories in Instrument 2.

Instrument 1: Coding Scheme for Teacher InterventionLength, Choice, and Frequency

We used Instrument 1 to record teachers’ interventionlength of each sighting, the preference of interventionchoice, and the frequency of a teacher’s visits to eachgroup. That type of data was not related directly to teach-ers’ intervention quality, even though it might have affect-ed it (see Appendix). We used one coding sheet to recordeach visit per group for 1 teacher’s intervention. Because ateacher could intervene with different students about vari-

ous topics during one visit, one coding sheet might includeseveral intervention sightings.

We recorded the start and end times of each interven-tion sighting. The lengths of each intervention were classi-fied into six categories: 1 (less than 30 s), 2 (30 s to 1 min),3 (1 min to 2 min), 4 (2 min to 3 min), 5 (3 min to 5 min),and 6 (more than 5 min). Intervention length showedwhether a teacher’s response type was quick or prolonged.

Teachers generally had three choices of interaction: (a)individual; (b) group (two or more students in the samegroup); and (c) whole class (during or right after groupintervention, the teacher gave feedback to the wholeclass). Each teacher interaction with students in any one ofthe three ways was considered as one intervention sighting(AAAS, 2004). When topics were changed, we recordedadditional intervention sightings. We used one codingsheet to record each visit per group for one teacher’s inter-vention. Therefore, we could use the coding sheets toobtain information about a teacher’s intervention choicepreferences by calculating the percentage of interventionsightings for each choice.

We considered each time that a teacher approached agroup and interacted with student(s) as one visit. Becausewe used one coding sheet to record each visit per group, thesum of these codes provided information about theteacher’s frequency of visiting each group. We also record-ed the contents used within each intervention for furtheruse. The contents included checking work, providing help,or giving announcements.

Instrument 2: Categories for Teacher Intervention Quality

We focused on teachers’ intervention quality. We evaluat-ed each intervention sighting that we recorded with Instru-ment 1 by using Instrument 2. We created all the categoriesand indicators on Instrument 2 under the guidelines ofNCTM’s (2000) Principles and Standards for School Mathemat-ics. The NCTM teaching principle suggests that “Effective

TABLE 1. Teacher and Class Information

Teacher information Class informationNo. of years No. of Total Indiv. Coop. No. of Group

Class State Gender teaching Certification Degree PD days time studya studyb groups size

A Delaware F 11–15 Elementary M 4 34 min — 18 min, 51 s 6 4–6B Delaware M 25–30 Elementary M 4 49 min, 50 s — 20 min 6 4–6C Texas F 11–15 Elementary M 4 49 min, 13 s 5 min, 23 s 18 min, 22 s 4 8D Texas F 0–5 Middle School B 10 31 min, 11 s 3 min 4 min 5 4–6E Delaware F 0–5 Middle School M 5.5 37 min 5 min 5 min 5 4–6F Texas F 25–30 Elementary M 5 39 min, 22 s 8 min, 20 s 5 min 9 2–3

Notes. PD = professional development; M = master’s degree; B = bachelor degree. Total time indicates length of class. Dashes signify that the teacherdid not ask students to do individual work at the beginning of class.aIndividual study indicates the length of students’ individual work at the beginning of class (or before the group study). bCooperative study is thelength of students’ group work.

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mathematics teaching requires understanding what studentsknow and need to learn and then challenging and support-ing them to learn it well” (p. 10). The five process standardsalso require teachers to address students’ conceptual under-standing. Therefore, the basic consideration reflected in ourcategories is whether teachers’ interventions target students’cognitive processes. We adapted the detailed categories andindicators from the AAAS (2004) criteria of teaching qual-ity, which are supported by prior research findings on coop-erative learning. The categories are illustrated as follows:

Category I: Teachers’ guidance focusing on the learning goaland students’ cognitive obstacles (Meloth & Deering, 1999).Teachers tend to identify students’ thinking by asking ques-tions. When students encounter difficulties and cannot fig-ure them out, teacher assistance is the primary resource forenhancing students’ thinking (Kramarski et al., 2002). Weevaluated that category according to the following fourindicators: (a) Is the teacher’s guidance related to learninggoal? (b) Is the teacher’s guidance addressing the students’current thinking and their cognitive obstacles (Meloth &Deering, 1999; Webb, Farivar, & Mastergeorge, 2002)? (c)Does the teacher provide scaffolding questions or tasks(Mercer & Fisher, 1998; AAAS, 2004)? (d) Does theteacher help students through approaches such as intro-ducing or developing important ideas, helping studentsrelate their own experiences to the mathematical idea, orusing alternative representations (AAAS)?

Category II: Promoting student thinking. As facilitators,teachers should provide enough opportunities and createappropriate environments to improve students’ thinking(Meloth & Deering, 1999). We used three indicators forthis category: (a) When students give a correct answer,does the teacher encourage them to express, justify, inter-pret or represent their ideas (AAAS, 2004)? (b) When stu-dents make mistakes, does the teacher use errors as spring-boards for inquiry (Borasi, 1994)? (c) Does the teacherencourage students to use different ways to solve the prob-lem (AAAS)?

Category III: Encouraging high–level peer discussion. Even ifclass instruction begins with individual study, when thephase of cooperative learning begins, teachers shouldencourage peer interactions, with a focus on elaboratingstudents’ thinking (Cohen, 1994b; Dekker & Elshout-Mohr, 2004; Meloth & Deering, 1999). We evaluated Cat-egory III by two indicators: (a) Does the teacher ask stu-dents to explain their ideas rather than just compareanswers with each other (Webb et al., 2002; AAAS, 2004)?(b) When a student needs help, does the teacher respondto questions only after all group members have discussedthe problem (Johnson & Johnson, 1990)?

Examples for evaluating intervention quality. We judgedteacher intervention quality by examining whether eachintervention sighting met each indicator on the instrumentwith the codes “met,” “partially met,” “not met,” or “not

appropriate.” For example, some teachers purposely gavestudents several minutes to do individual tasks before groupwork. During that period, Category III: Encourage peer dis-cussion would be judged as not appropriate. Also, duringgroup work, some teachers checked students’ work by ask-ing, “Are you agreeing?” In that example, without anyother guiding or probing, all of the indicators for CategoriesI and II would not be met; however, the indicator “guidingfocus on learning goal” would be partially met. Twodetailed examples of our evaluation are shown as follows:

Example 1: The teacher asked the whole class to firstexplore the new task independently. She then walkedaround the classroom and came to one student:

Teacher: What did you decide?Student: 3/4Teacher: Why did you decide Ms. Mendoza 3/4 was the best?Student: Because it is the simplest form of the fraction.Teacher: How did you decide 3/4 was the simplest form?Student: (Inaudible) Teacher: Ok, good. Thank you. Good job. (Left)

That is a sighting that occurred before students’ groupwork. The teacher checked the student’s thinking by ask-ing him several questions (Table 2 shows the evaluation ofthis sighting).

Example 2: During group work, Teacher B came to a group.He observed the students folding fraction strips and listenedto their discussion.

(Start time 32:21– )Student 1: I think it is 3/4.Student 2: This teacher said 2/3.Student 1: (After awhile) I think it is 2/3.Student 2: (After folding a paper) Not 3/4, but 2/3. Student 3: 2/3.Student 1: Yes, 2/3. It’s 2/3. Student 4: Ok, 2/3.(Time 33:07– )Teacher: (After listening to students’ discussion for about

50 seconds) Wait a minute. Wait a minute. Goto the 2/3 again. Put 2/3 up.

(All students in the group took out the fraction strips)Teacher: (Pointed to the fraction strip and the thermome-

ter on the book) Now, we got the start line.That’s the finish line. Does the strip fit?

Student 4: Oh!! Student 2: (Inaudible)Teacher: But it has to make sense. It has to fit. Yours does

not make sense to me. (Went away)(End time: 33:34)

This sighting happened during students’ group work. Theteacher guided the students because the whole groupencountered difficulty (Table 3 shows the evaluation of thissighting).

After completing the coding, we counted the number ofmet, partially met, and not met sightings for each indicator.(We did not tabulate the inappropriate sightings.) Forexample, if there were 7 met, 6 partially met (considered as1/2 × 6 = 3 met), 2 not met, and 3 not appropriate, the cal-

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culation for that percentage would be (7 + 1/2 × 6) / (7 +6 + 2) = 67%. That means, 67% of the teacher’s interven-tion sightings met the indicator.

This study is part of the MSMP. All authors had beentrained for video transcription, coding, and analysis byAAAS experts for approximately 2 years. There were two

TABLE 3. Sample Evaluation for One Sighting During Group Work

Category Evaluation Comment

I (Guiding focus)1. Learning goal Met The students were exploring the new task and the teacher’s

help was related to it.2. Cognitive obstacle Partially met The teacher identified that students were confused by how to use

fraction strips. However, he just pointed out the “start line” and “finish line” rather than ensuring that students knew whythe finish line was important and what the relationship was between the thermometer and the fraction strip.

3. Scaffolding questions Not met The same comment stated above.4. Multiple approaches Met The teacher used multiple representations during his guiding.

The teacher guided the students to look at the thermometer,which represented the fundraising process in the textbook.

II (Promoting thinking)1. Encouraged explanation to teacher Not met The teacher did not ask students to explain their ideas.2. Used error as springboard Not met Although the teacher found the students made mistakes, he

just asked them a simple leading question, “So does it fit?”3. Encouraged different solutions Not met This teacher did not ask students to explore with different

tools or to work out the problem through different ways.III (Encouraging peer discussion)

1. Encouraged explaining to each other Met The group of students continued with the discussion after being encouraged by the teacher.

2. Provided response after group discussion Met The teacher stood by the group, observing students folding papers and listening to students’ discussion for about 50 s.He gave them help only when he found that the whole group made mistakes.

Notes. Met signifies that the teacher’s interventions during this sighting met the indicator of corresponding category. Partially met means the teacher’sinterventions during this sighting only partially met the indicator of corresponding category. Not met means the teacher’s interventions during thissighting did not meet the indicator of corresponding category.

TABLE 2. Sample Evaluation for One Sighting Before Group Work

Category Evaluation Comment

I (Guiding focus)1. Learning goal Met The teacher’s questions were related to the new task.2. Cognitive obstacle Not appropriate The student did not encounter any difficulty.3. Scaffolding questions Met What, why, and how were hierarchical questions, according

to students’ thinking.4. Multiple approaches used Not appropriate The student did not encounter any difficulty.

II (Promoting thinking)1. Encouraged explanations Met The teacher asked the student to explain why and how.2. Used error as springboard Not appropriate The student did not encounter any difficulty.3. Encouraged different solutions Not met This teacher just praised the student’s solution and then left.

III (Encouraging peer discussion)1. Encouraged explanations to each other Not appropriate The teacher deliberately asked the class to do individual

work during this period.2. Provided response after group discussion Not appropriate The same comment stated above.

Notes. Met signifies that the teacher’s interventions during this sighting met the indicator of corresponding category. Not appropriate means thisindicator is not appropriate for evaluating the teacher’s interventions during this sighting. Not met means the teacher’s interventions during thissighting did not meet the indicator of correspnding category.

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types of training. One type of training was short term;AAAS experts came to the MSMP to train the coders for2 days. The coders also traveled to Washington, D.C. forseveral days of training. Another type of training was longterm; AAAS experts joined the MSMP meeting on videocoding and analysis by telephone every 2 weeks. After 2years of professional training, the MSMP members wereskilled at video transcription, coding, and analysis. Toimprove reliability, the first two authors coded and evalu-ated each intervention sighting independently, then com-pared the outcomes with each other. The consistency ofevaluation reached 90%. Each inconsistency of evaluationwas reconsidered or discussed with the other authors untilan agreement was reached.

Results

Differences of Intervention Length, Choice, and Frequency

Intervention length. Figure 1 shows teachers’ length ofinterventions. For example, Teacher A had 21 interven-tion sightings that were less than 30 s. Teachers B and Ceach had a long intervention sighting lasting more than5 min.

Table 4 shows the teachers’ response types according totheir intervention length. For interventions shorter than 1min, there were more than 90% sightings in Classes A, D,E, and F versus less than 50% in Classes B and C; for inter-ventions longer than 2 min, there were zero interventionsin Classes A, D, E, and F versus 40% interventions inClasses B and C. Thus, interventions in Classes A, D, E,and F were quick responses, whereas in Classes B and Cthey were prolonged responses.

Teacher

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FIGURE 1. Teacher’s intervention length. 30’’ = 30 s; 1’ = 1 min; 2’ = 2 min; 3’ = 3 min;5’ = 5 min.

< 30”

A B C D E F

20

15

10

5

030”–1’ 1’–2’ 2’–3’ 3’–5’ > 5’

Intervention Length

TABLE 4. Teacher’s Response Type (in Percentages)

Teacher < 1 min > 2 min

Quick interventionA 93.5 0D 100 0E 94.5 0F 100 0

Prolonged interventionB 28 42.9C 46 36

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Intervention choice. Figure 2 shows that all teachers hadlower intervention frequencies with whole classes thanwith individuals and groups. Teacher A had the highest fre-quency for intervention with individuals, whereas TeacherB never interacted with individuals. All teachers had sev-eral interventions with groups.

Because many mathematics classes combined individualstudy and group work within a lesson, we closely examinedteacher intervention choices (whole class was not consid-ered here) from different periods. Table 5 shows teachers’intervention choice preferences. Teachers A and B did notask students to work individually during the mathematicslessons; all their interventions involved only group work.During group work, Teacher B intervened with whole

groups (87.5% with group), whereas Teacher A intervenedwith individuals within the group (73% with individual).Teachers C, D, E, and F combined individual and coopera-tive study techniques. During individual study periods,Teachers C and E intervened with individuals (38.5% and47.4%, respectively), more than did Teachers D and F(30% and 22.2%, respectively). During group-work peri-ods, Teachers D and F intervened with groups (60% and50%, respectively) more than did Teachers C and E (30.8%and 31.6%, respectively).

In general, interventions with individuals were more like-ly to occur in individual study periods, whereas interven-tions with groups occurred during the group study. AlthoughTeachers A, C, and E interacted with individuals, Teacher

Teacher

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FIGURE 2. Teacher’s intervention choice.

Individual

A B C D E F

20

15

10

5

0Group Whole class

Intervention Choice

TABLE 5. Teacher Intervention Choice Preference (in Percentages)

Individual study period Group study periodTeacher Individual Group Individual Group

A 0 0 73 20C 38.5 0 15.4 30.8E 47.4 0 15.8 31.6B 0 0 0 87.5D 30 0 0 60F 22.2 0 0 50

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A interacted only with students during group work. In con-trast, Teachers B, D, and F interacted with groups; however,Teacher B interacted only with groups.

Intervention frequency. Table 6 illustrates the teacherintervention frequencies for each group. The teacher visit-ed each group in the six classes at least once, with theexception of one group in Class F. Students in Class F wereclustered into nine groups, which may have contributed tothe teacher’s inability to visit with all groups. According tothe percentage of groups that were visited, we classifiedteacher intervention frequencies into one of three cate-gories: high, middle, or low. For example, in Class A, theteacher visited four groups twice, and another two groupsfour times. Thus, all groups in Class A were visited at leasttwo times, resulting in a high intervention frequency forTeacher A. In Class B, the teacher visited five of the sixgroups only once. Therefore, only 17% of the groups werevisited at least two times, resulting in a low interventionfrequency for Teacher B. In Classes C, D, E, and F, theteacher intervention frequencies were inconsistent. Some

groups were visited as many as three or four times, whereasother groups were visited once or not at all. On average,teacher interventions in the four classes could be classifiedas middle frequency because the percentages were from33% to 60%.

Differences of Intervention Quality

Table 7 shows the percentage of intervention sightingsthat each teacher met for each indicator. The percentageequal to or higher than 70% was highlighted. The pattern inTable 7 shows that, overall, the teacher intervention qualityin Classes A, C, and E was higher than in Classes B, D, andF. Across the categories and indicators, we found generallythat teachers were not skillful at guiding students with scaf-folding questions or through multiple approaches. Moreover,teachers did not promote students’ thinking by capitalizingon errors or by encouraging them to solve problems in dif-ferent ways. Finally, teachers were not proficient at encour-aging peers to elaborate on their ideas to each other.

TABLE 6. Teacher’s Intervention Frequency for Each Group (in Percentages)

Groups GroupsGroup visited visited

Class 1 2 3 4 5 6 7 8 9 ≤ 1 time ≥ 2 times

A 4 2 4 2 4 4 — — — 0 100B 2 1 1 1 1 1 — — — 83 17C 3 1 3 1 — — — — — 50 50D 1 3 1 2 1 — — — — 60 40E 4 3 3 1 1 — — — — 40 60F 3 1 3 0 1 2 1 1 1 67 33

Note. Dashes indicate that the group did not exist.

TABLE 7. Teacher’s Intervention Quality: Intervention Sightings Met for Each Indicator (inPercentages)

Category A B C D E F

I (Guiding focus)1. Learning goal 100 100 100 70 100 65.22. Cognitive obstacle 87 63 92 33 70 363. Scaffolding questions 71 25 42 20 70 04. Multiple approaches 79 40 77 25 43 18

II (Promoting thinking)1. Encouraged explaining to teacher 73 63 71 40 81 552. Used error as springboard 68 13 42 10 50 03. Encouraged different solutions 50 50 75 0 33 44

III (Encouraging peer discussion)1. Encouraged explanation to each other 12 50 100 25 100 632. Provided response after group discussion 15 83 67 100 100 100

Note. Percentages ≥ 70% are highlighted in boldface type.

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Focusing on learning goal and cognitive obstacles. TeachersA, C, and E focused better on learning goals and cognitiveobstacles than did Teachers B, D and F. Teachers A and Eused scaffolding-type questions better than did Teacher Cwhen they guided their students. The following interven-tion sighting was from Teacher A:

Teacher: Hey, which one you try first?Student: 2/3 . . .Teacher: Oh, good point. Think, where is the goal?Student: (Point out)Teacher: So what’s wrong with the fraction strip?Student: Too short, it doesn’t work.Teacher: So where you can make it work?Student: Try to find the length that fit into.Teacher: Ok, you can try that.

Teacher A successfully helped the student overcome cogni-tive obstacles with simple scaffolding questions. Similarly,Teacher E checked students’ ideas by asking “What did youdecide?” “Why did you decide Ms. Mendoza?” and “Howdid you decide Ms. Mendoza?” However, the questionsasked by Teacher C were not as scaffolded, even though thelength of her intervention was prolonged.

Among Teachers B, D, and F, Teacher B recognized stu-dents’ difficulties and tried to focus on them. However,when he guided students, he always repeated the same sen-tence “This is your start line, this is your finish line” or“Start line is here, correct? Finish line is here, correct?” Hedid not scaffold his questions and did not often use multipleapproaches to guide students. Therefore, even thoughTeacher B spent prolonged time with each group, he did notmove students’ thinking ahead; most students were stillconfused by the end of the class. Teachers D and F checkedgroup work and asked only simple questions, such aswhether students agreed or disagreed. These questions didnot allow Teachers D and F to identify students’ cognitiveobstacles and to further guide them.

Promoting student thinking. Teachers A, C, and E pro-moted student thinking better than did Teachers B, D,and F. Even though Teachers A, C, and E asked studentsto explain their ideas, only Teacher A effectively usederrors as springboards for students’ inquiry. In the lessonon promoting student thinking, there was a common mis-take that occurred in several classes. When studentsexplained why 3/4 was equal to 6/8, they understood that3 × 2 was 6 and 4 × 2 was 8. However, some of them mademistakes in their verbal representations, such as doubling3/4, or 6/8 divided by 2. In addition, some of the studentsmade mistakes in written form, such as 3/4 × 2 = 6/8 or3/4 + 3/4 = 6/8. Although Teachers A, C, and E noticedthe type of error, they dealt with it differently. For exam-ple, in Class C, students wrote down 3/4 × 2 = 6/8 and 3× 2 / 4 × 2 = 6/8, and said that they were the same.Teacher C asked the whole class, “How many of youthink these are the same thing?” When there was noresponse, Teacher C said, “Ok, this is actually what weare going to learn later.” In Class E, when a student said“divided by 2,” Teacher E asked the student to write down

what he meant. When she found that the student under-stood the procedure, she moved toward another topic. Incontrast, Teacher A insisted on requiring students toinquire about the mistake: doubling 3/4. The following isan example:

Student 1: 6/8 equals to 3/4.Teacher: How do you know that? Student 1: To double it.Student 2: 3 × 2 is 6.Teacher: You are not doubling, what are you doing?Student 1: 3 × 2 is 6, and 4 × 2 is 8.Teacher: But that is exactly multiplying by number 2?Student 2: That’s a common denominator.Teacher: Wait, wait and think. Listen to my questions

before you give me an answer. He says he is mul-tiplying by 2. Are you really multiplying 3/4 by 2?

Student 3: No. If you multiply 3/4 by 2, you get 1.5.Teacher: Right. So what are you really doing? Because you

are right when you say you multiply three andmultiply 4. So what are you doing?

Student 3: 3/4 is 6/8.Teacher : I know, but we are trying to figure out why.Student 2: They were just changed numbers. They are the

same fractions.Teacher: Yes, they are equivalent fractions, you are right.

But how are we changing the numbers? Thinkabout it. I’ll come back. Talk to your table.Because you say you are doubling, I want you tothink about what you are doing.

Teacher A grasped students’ verbal mistake and continuedto ask students to consider their actions when they saidthat they were doubling fractions because their error wasthe same as with the written form, 3/4 × 2 = 6/8. Sheexpected the students to figure out that 3/4 was multipliedby 2/2, which equals 1, rather than by 2. Therefore, evenwhen 1 student in the group discovered that if 3/4 weremultiplied by 2, the answer would be 1.5, not the samevalue as 3/4, the teacher prompted students to continuetheir discussion (detailed in Ding & Li, 2006)

Teachers D and F did not check students’ thinking dur-ing intervention, thus, they had few opportunities to findstudents’ mistakes and to promote their thinking. TeacherB generally had a negative attitude toward students’ errors.For example, when the teaching assistant told him that oneof the groups had made mistakes and needed his help, theteacher showed his disappointment and said, “Oh! No,they don’t need help!” Therefore, repeating errors, ratherthan promoting student thinking, commonly occurred inClass B.

All teachers except Teacher C scored very low on theirincentive to encourage students to use different ways tosolve problems. Teachers were generally satisfied with onesolution in a group and asked students to share theiranswers in the report phase at the end of the class.

Encouraging high-level peer discussion. Because the classesused cooperative learning, all teachers put students in smallgroups and allowed them to talk with each other. Howev-er, Teacher A, who was good at guiding and promoting stu-dents’ thinking, scored lowest within the two indicators of

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the category “Encouraging high-level peer discussion.” InClass A, only one group out of six groups had a peer dis-cussion (discussions occurred in only 19 interventions).Teacher A did not encourage the groups to discuss prob-lems with each other, but tended to respond immediately tostudents seeking help. When students in Class A neededassistance, they tended to seek the teacher rather thantheir peers. For instance, one student wanted to ensure thathis solution was correct. He continued to raise his hand formore than 3 min because the teacher had left to interactwith other groups. When she tried to give a student help,she never checked whether the student had first asked forpeer help. Also, a boy in Class A asked for the teacher’shelp even though a girl in the same group was just praisedby the teacher for her solution to the same problem. Theteacher came to the group, giving the boy immediate assis-tance without any suggestion such as, “Why not discuss thiswith Cathy?” or “Cathy, can you explain this to him?”

In general, Teachers B, D, and F walked around theirclassrooms and listened to groups talk before they inter-vened in them. However, they did not often ask students toexplain their thinking to each other; Teachers D and Fasked groups only to compare their answers. Teachers C andE used peer resource more than did the other 4 teachers.

Discussion

In cooperative learning, teachers’ intervention frequen-cy, length, and choice may influence the quality of theintervention. Regardless of teachers’ intervention types,they need to address students’ mathematical thinking andunderstanding. Concerning the length of teacher interven-tions, researchers suggested two different strategies; the firstwas Cohen’s (1991, 1994a) quick response type. In con-trast, there was the prolonged response type; Brown andPalincsar (1989) examined a program that implemented“guided cooperative learning—reciprocal teaching.” Theyrecommended that teachers help groups through modelingstrategies and steering discussions. The prolonged monitor-ing resulted in a substantial impact on student learning(Meloth, 1991; Meloth & Barbe, 1992; Meloth & Deering,1999). In our study, Teachers B and C used prolonged inter-vention. However, those two teachers displayed differentintervention qualities of guiding or promoting studentthinking. Although Teachers A, D, E, and F used quickinterventions, their effects on students’ thinking varied.

Thus, there are no absolute rules regarding interventionlength. Using quick or prolonged response types dependson group situations. In addition, as demonstrated byTeacher B, those teachers who spend too much time withcertain groups may ignore other groups and rush to finishthe teaching task. Likewise, spending too little time withgroups may not give teachers adequate time to identify anddevelop students’ thinking abilities. Therefore, responsesthat are too quick or too prolonged can have negative con-sequences for promoting student learning.

Teachers’ intervention frequency, a factor related tointervention length, is important for the quality of teacherintervention. Chiu (2004) suggested that teachers revisitgroups because students have a tendency to move off taskafter the teachers leave the students. Webb (1989, 1991)also suggested that, after providing necessary assistance, theteacher should leave the group and give students an oppor-tunity to use the explanation that he or she provided. Later,the teacher could return to the group and assess how thestudent(s) used the explanation (Webb et al., 2002). Ourstudy supports the prior research findings. For example, theinfrequent visits of Teacher B to each group stemming fromhis overly prolonged responses prevented his checking stu-dents’ understanding of his guidance, resulting in the stu-dents being off task.

Teachers in cooperative-learning mathematics classeshave three intervention choices: individual students, smallgroups, and whole class. Regardless of their interventionchoices, teachers should address students’ mathematicalunderstanding. When teachers intervene with a student,they need to consider how to use peer dynamics to fosterhis or her thinking. When teachers intervene with a group,they must consider individual students’ thinking, such aswhether they comprehend the lesson. Moreover, if all class-room groups have the same types of questions or confusionabout mathematics concepts, teacher feedback to thewhole class might promote students’ thinking.

Teachers who intervened with individual students hadmore opportunity to check their independent thinking,although some of the students did not use peer resourceseffectively. In contrast, teachers who intervened withgroups depended more on peer interaction than did otherteachers, but some of them neglected students’ individualthinking. As a result, techniques to balance peer resourceand independent thinking and to use peer resource toimprove students’ mathematical understanding raised chal-lenges for the teachers’ interventions.

One of the main purposes of mathematics instruction isto help students think mathematically (Schoenfeld, 1988).In cooperative-learning classrooms, even though studentsare grouped, teacher intervention should not focus only ongroup function but also on students’ cognition (Meloth &Deering, 1999). That is, teachers should use peer resourcesto foster students’ mathematical thinking.

Two problems related to using peer resource were note-worthy: First, teachers should use peer resources effectively.Some teachers are good at guiding and promoting students’thinking, but they may not often encourage students to dis-cuss ideas with each other, preferring instead to give imme-diate answers. Thus, teachers are often so busy interactingwith many individual students that they neglect peer inter-action. Teachers should encourage peer discussion andspend more time with groups that require help, such asgroups in which all members are confused or cannot reachan agreement. Therefore, encouraging peer resourcesenables teachers to deal with three intervention choices

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simultaneously and to keep more students on task-relatedthinking, thus greatly improving the efficiency of coopera-tive learning.

Second, although teachers should encourage peer inter-action, they should not be too dependent on it. Simply ask-ing students to compare their answers with each other isbeing too dependent on peer resources because students’independent thinking is not verified. Sometimes, lowerlevel students tend to view higher level students as author-ities and follow them without understanding the logicbehind their answers (Amit & Fried, 2005). Thus, if thehigher level students make mistakes, then the whole groupwould make the same mistakes. In the example of Class B,when one student in a group loudly announced her finding,“Not 3/4, but 2/3,” all other students in this group agreedwith her, “Yes, 2/3. It’s 2/3.” The teacher, who had been lis-tening to the group discussion, recognized the mistake andprovided help. If he had just checked the group by asking“agreeing or disagreeing” type questions, he would havenever known the actual situation about the students’thinking.

There are two ways to balance peer resource and stu-dents’ independent thinking. First, teachers could encour-age students to elaborate their thinking. Sawyer (2004)suggested using the creative teaching strategy, whichemphasizes the importance of students’ active participa-tion, including exploratory discourse and elaboration.Veenman, Denessen, Akker, and Rijt (2005) found thathigh-level elaborations of sixth-grade students were relatedpositively to their mathematical achievement. Therefore,teachers could use both strategies: “Encourage explainingto each other” (use peer resource effectively) and “Encour-age explaining to the teacher” (not be too dependent onpeer resource) to balance the relationship between peerresource and students’ independent thinking. Second,teachers could combine cooperative learning with individ-ual study. Some teachers in our study gave students 3–5 minto work individually before group work. Working individu-ally before cooperative learning occurs could providegroups with mathematical content for discussion. Lowerlevel students would have time to think about their prob-lem so that they would not have to rely on their peers forhelp. In addition, higher level students would have time tothink in multiple ways to improve their thinking.

In general, teachers in cooperative-learning mathemat-ics classes could use peer and individual resources toimprove students’ mathematical thinking. The differencesin teacher intervention quality scores show that not allteachers were good at cultivating students’ independentthinking. The high-quality interventions of some teachershighlighted ways to address students’ thinking, that is, ana-lyze, diversify, and deepen their thinking.

Analyze students’ thinking means that teachers should firstidentify students’ cognitive obstacles, then focus on theseobstacles to guide the students. Some teachers in this studyidentified students’ ideas while scaffolding questions and

made instructional decisions according to students’ think-ing, as suggested by Timmerman (2004) when he described“structured interviews.” Chamberlin (2005) suggested fourinteraction patterns such as, “terminology-search interac-tion pattern” and “question-initiated interaction pattern”to help teachers meet the challenges of interpreting stu-dents’ thinking. If teachers only check the agreement ofgroup answers or repeat the same leading questions, theywill not be able to identify students’ cognitive difficulties.In contrast, if teachers scaffold their questions while guid-ing students, they can guide them on the basis of theirmathematical thinking.

Teachers could diversify students’ thinking in two ways.First, teachers guide students through multiple approaches.Throughout the six videos, some teachers used multiplerepresentations such as fraction strips, pictures, numberlines, or manipulatives. Teachers also linked students’ cur-rent cognitive obstacles with the previous knowledge suchas, “remember before, we folded it to make it work.” Otherteachers linked students’ ideas with real-life experiencessuch as, asking about sharing candy bars. Second, whenteachers monitor group work, they should encourage stu-dents to solve problems in various ways. For example, if agroup proved their problem solving by using fraction strips,the teacher could encourage them to try other approachesand also praise them for their first solution.

A major requirement for teacher intervention is to deep-en students’ thinking. Cooperative learning is a process forstudents to discover and construct their knowledge. Thus,student errors were a common and natural thing to occurduring group study. Teachers should view errors as opportu-nities for inquiry. Teachers can often encourage students,even when the students make mistakes by saying, “I likethe way you play with numbers” or “You are on the righttrack.” Moreover, teachers should capitalize on errors asspringboards for inquiry rather than only using errors as adiagnosis and remediation of students’ knowledge weak-nesses. For example, regarding the common error 3/4 × 2 =6/8, if teachers had grasped the notion of “equality” or“equal sign” to guide students to check 3/4 and 6/8, the stu-dents may not only have understood this concept but alsocould have improved their understanding of the equal sign,a critical notion for later algebra study (Falkner, Levi, &Carpenter, 1999; MacGregor, 1999; NCTM, 2000; Sáenz-ludlow & Walgamuth, 1998).

Cooperative learning is a good teaching method inmathematics classrooms, but it is also a complex system(see Figure 3). Whatever teachers’ intervention choicesand strategies are, they should focus on the fundamentalgoal, which is to address students’ mathematical thinkingand to help them think mathematically.

Conclusion

We found that the length, frequency, and choices ofteacher interventions somewhat influence intervention

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quality; we suggest that regardless of teachers’ interventiontypes, they should address students’ mathematical thinking.Moreover, our findings of teacher intervention differencesallow teachers to recognize the challenge of balancing peerresource and students’ independent thinking. We suggestthat teachers use peer resource to help students thinkmathematically. Finally, our findings of the higher qualityinterventions such as helping students overcome cognitiveobstacles by using simple scaffolding questions provideteachers with insightful ways to address students’ thinking.

We also formed insights for teacher professional devel-opment in cooperative learning. During teacher training, itis not enough for teacher educators to provide mathemat-ics teachers with strategies for classroom management andgroup function. Educators also need to cultivate teacherbeliefs of using cooperative learning to improve students’mathematical thinking and to provide teachers with tech-niques to effectively address student thinking. Researchersshould focus on (a) ways to improve cooperative-learningeffects by addressing students’ mathematical thinking and(b) techniques to improve students’ mathematical thinkingby taking advantage of cooperative learning. The underly-ing reasons for teacher intervention differences whenaddressing student thinking and the fundamental mecha-nism of high-quality teacher interventions also need fur-ther study.

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Students’ mathematical thinking

FIGURE 3. Teacher interventions and students’ mathe-matical thinking.

Individual Group(Peer resource)

Whole class

Teacher interventions in cooperative-learning mathematics classes

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APPENDIXCoding Scheme

Class ————————————— Group —————————————

Intervention choice Time (from __ to __) Response type Content

IndividualABCDEF

Whole groupWhole class

Note. Response types: 1, 2, 3, 4, 5 and 6 denote less than 30 s, 30 s to 1 min, 1 min to 2 min, 2 min to 3 min,3 min to 5 min, and more than 5 min, respectively.

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