###### February 13, 2007

# Teacher Interventions in Cooperative-Learning Mathematics Classes

By Ding, Meixia; Li, Xiaobao; Piccolo, Diana; Kulm, Gerald

ABSTRACT

Key words: cognitive performance, cooperative-learning mathematics classes, teacher interventions

The potential of cooperative learning to improve students' academic and social performance has been widely recognized (Slavin, 1996; Webb, 1989). In regard to mathematics classrooms, the National Council of Teachers of Mathematics (NCTM; 1989) advocates cooperative learning because "small groups provide a forum in which students ask questions, discuss ideas, make mistakes, learn to listen to others' ideas, offer constructive criticism, and summarize their discoveries in writing" (p. 79). In addition, the NCTM Principles and Standards for School Mathematics (2000) includes communication as one of five process standards. Cooperative learning is an effective way to develop the ability to communicate with others. However, some mathematicians (e.g., Wu, 1997) doubt whether cooperative learning could be used in mathematics classrooms without considering the major purpose of mathematics instruction, which is to help students learn to think mathematically (Schoenfeld, 1988).

Because cooperative learning is used widely in mathematics classrooms, a growing need exists to examine how teachers use this technique in classroom settings. We focus on teacher interventions and use a qualitative methodology to examine whether teachers in cooperative-learning classrooms addressed students' mathematical thinking. Our intent was to help readers become cognizant of possible challenges in cooperative-learning mathematics classrooms and to provide insights for teacher instruction and professional development. Because "the research into the teachers' role in facilitating cognitive and metacognitive gains through cooperative learning is in its infancy, or perhaps childhood" (Meloth & Deering, 1999, p. 254), we contribute to this much-needed research.

Teacher Role in Cooperative Learning

The teacher's role in cooperative learning generally includes (a) specifying objectives, (b) grouping students, (c) explaining tasks, (d) monitoring group work, and (e) evaluating achievement and cooperation (Bettenhausen, 2002). Among those tasks, teacher intervention in monitoring group work is associated directly with students' cognitive performance (Chiu, 2004; Meloth & Deering, 1999).

During group-work monitoring, a teacher is "both an academic expert and a classroom manager" (Johnson & Johnson, 1990, p. 112). However, researchers in earlier studies emphasized the role of classroom manager. For example, Johnson and Johnson (1991) stated,

The teacher monitors the functioning of the learning groups and intervenes to teach collaborative skills and provides task assistance when it is needed. The teacher is more a consultant to promote effective group functioning than a technical expert. Typical statements a teacher may make are, "Check with your group"; "Does anyone in your group know"; "Make sure everyone in your group understands." (p. 61)

Kagan (1985) suggested that teachers should be freed even more in group investigation to allow students to assume responsibility for learning. Teachers typically consult with groups and suggest ideas or possibilities for exploration. Cohen (1991, 1994a) suggested minimizing monitoring to help students become more interdependent, autonomous, and self-directed. Cohen observed that students reduced the amount of cooperation and communication between each other after the teacher intervened. Therefore, Cohen preferred to use the quick- response strategy in which teachers provide brief comments and questions, then move away from the group so that students can continue their discussion.

Johnson and Johnson (1990), Kagan (1985), and Cohen (1991, 1994a) encouraged teachers to monitor the group's on-task behavior and cooperative skills and to provide task assistance when necessary. Therefore, the teachers' role in cooperative-learning classrooms is "more like a consultant who helps improve effective group functioning than an instructor who contributes information or scaffolds students' learning" (Meloth & Deering, 1999, p. 244). The importance of the teachers' role as indicated in these studies was underestimated.

Significance of Teacher Intervention

Johnson and Johnson (1990) suggested that,

Simply placing students in groups and telling them to work together does not in and of itself promote greater understanding of mathematical principles or ability to communicate one's mathematical reasoning to others. There are many ways in which group efforts may go wrong, (p. 104)

One way that group efforts fail is when teachers do not provide necessary help when needed. Meloth and Barbe (1992) examined 180 peer groups and found similar results during most of the group studies. They also reported that 80% of the teachers' monitoring statements were not oriented toward special-task content. Cohen (1994b) emphasized that if teachers do not provide clear and explicit assistance when students need it, students are unlikely to engage in task-specific learning. That lack of teacher assistance was the most important reason for the low achievement of students in group-learning settings (Webb, 1989).

Teacher intervention during peer interactions is also important because teachers need to give students help when (a) no student in the group can answer a question (Hamm & Adams, 2002); (b) students have difficulty communicating with each other, which might cause or reinforce misconceptions in peer interaction (Brodie, 2001); or (c) group members treat one another with authority and no true dialogues exist (Amit &. Fried, 2005). In all those situations, the teacher's intervention is a resource to help students enhance their thinking.

Characteristics of Effective Intervention

What type of teacher intervention is effective in cooperative- learning mathematics classrooms? First, teachers should adapt their help to students' needs (Webb 1989, 1991). Students in Grades 4-8 believed that teachers should provide readily available help to achieve positive group discussions (Ares & Gorrell, 2002). Chiu (2004) explored the Teacher Intervention (TI) model and concluded that if teachers accurately evaluate and adapt their interventions to student needs, they can improve students' problem solving and time on task. second, teachers should focus on cognitive and metacognitive aspects when providing help to students (Deering & Meloth, 1993; Meloth & Deering, 1992, 1999).

Kramarski, Mevarech, and Arami (2002) investigated the effects of cooperative learning with and without metacognitive instruction to solve mathematical tasks of seventh-grade students. They found that students who were taught with metacognitive instruction significantly outperformed their counterparts. Kramareski and Mevarech (2003) further explored the following mathematics teaching methods with 384 students in eighth grade: (a) cooperative learning with and without metacogitive training and (b) individual learning with and without metacogitive training. They concluded that cooperative learning, combined with metacognitive training, was the most effective teaching method.

Finally, teachers should combine teacher and peer resources when intervening with students. For example, Dekker and Elshout-Mohr (2004) compared two kinds of teacher interventions designed to help students. Product-help interventions (p. 44) concerned students' mathematical reasoning and products, such as teachers asking students to explain and justify their work or offering hints and scaffolding their thinking. Process'help interventions (p. 43) addressed peer recourses such as stimulating interaction processes. The authors emphasized that process-help interventions were more effective than were product-help interventions for improving students' mathematical thinking. In summary, effective teacher- intervention strategies used in cooperative-learning mathematics classroom include (a) adapting teacher instruction to students' needs, (b) focusing on cognitive and metacogitive aspects, and (c) combining teacher and peer resources.

Purpose

We examined the ways that teachers use cooperative learning in mathematics classrooms, with a focus on improved student thinking about mathematics. Because some researchers reported that classroom interventions are underestimated and a large gap exists between research and everyday classroom settings about the effects of cooperative learning (Blatchford, Baines, Bassett, Rubie-Da\vies, &. Chowne, 2006), we compared classroom teaching experiences and documented the situations according to two questions: (a) What is the frequency, type, and length of teachers' interventions? (b) What is the quality of teachers' interventions?

Method

We used the qualitative method to explore teacher intervention by focusing on students' cognitive processes (Denzin &. Lincoln, 2000, p. 8); our research questions explored quality, meanings, and interpretations (Janesick, 2000, p. 382). In comparison with the quantitative approach, our method "more easily allows for the discovery of new ideas and unanticipated occurrences" (Jacobs, Kawanaka, &. Stigler, 1999, p. 718). We investigated our questions by using multicase studies (Bogdan & Biklen, 2003; Stake, 2000) and collected data by detailed observation (Michael & de Prez, 2000). We then analyzed, interpreted, and presented 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 allow for sophisticated analysis. Jacobs and colleagues (1999) reported three advantages of this type of data: (a) "relatively 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) and viewable by researchers from diverse cultural and linguistic backgrounds who might bring fresh perspectives to video analysis and examine many facets of the data; and (c) readily watched, coded, and analyzed repeatedly from different dimensions because of data source permanence.

All the authors in this study are members of the Middle School Mathematics Project (MSMP), a 5-year longitudinal study examining how the use of specific research-based instructional strategies in classrooms relates to lasting improvements in student learning. A major role of MSMP members is to observe, transcribe, and analyze classroom videotapes. When the first two authors, who both have 5 years of teaching experience in China, observed the project videos, they found that American teachers tended to use cooperative- learning methods. This type of teaching was totally new and different to Chinese teachers who mainly used direct teaching, with a tendency toward addressing students' mathematical and abstract thinking (Cai, 2001). The different cultural backgrounds of the first two authors allowed them to view American classroom videos with a different perspective. They raised several questions, such as, Why were so many students in cooperative-learning mathematics classes off task? Why did teachers walk around classrooms simply observing whether students agreed with each other? What did students learn in cooperative-learning mathematics classes? The first two authors discussed those questions with their American colleagues- the third and fourth authors. As a result, all the authors decided to examine teacher intervention in cooperative-learning mathematics with the focus on students' mathematical thinking.

We selected six videotaped lessons that examined the same teaching content from the same textbook using only cooperative- learning methods. We chose all of the lessons from the MSMP throughout the 2002-2003 school year. Because the MSMP was not specifically designed for cooperative-learning research, our video data reflected authentic classroom situations in which the teachers used their usual teaching methods. All the videotaped lessons were from 6 sixth-grade teachers, including 5 women and 1 man; all were Caucasian. According to the MSMP teacher database, the teachers came from five schools in three school districts. Three teachers taught in Delaware and 3 teachers taught in Texas. Teaching experience varied from O to more than 25 years.

Among the 6 teachers, 4 teachers had elementary teacher certificates and 2 teachers had middle school teacher certificates. Five teachers had a master's degree and 1 teacher had a bachelor degree. Their professional development training throughout the 2002- 2003 school year varied from 4 to 10 days. From our observation of the videotapes, the classes typically had 18-35 students and lasted approximately 30-50 min. In general, there were three cooperative- learning group sizes: large size (8 students per group), middle size (4-6 students per group), and small size (2-3 students per group). The range of group size produced a good opportunity to study various types of teacher interventions. In addition, in two of the classes, teachers employed cooperative learning immediately after assigning a task; in the other four classes, teachers used individual study for 3-5 min before students began group work. (See Table 1 for detailed teacher and class information.)

Content of Instruction

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

The lesson in this study was about fractions. We asked students to compare 2/3, 3/4, and 6/8, and decide which ones were equivalent to $270 out of $360. We encouraged students to explore by using fraction strips to measure the thermometer that represented the problem in the book. Students had difficulty using and understanding fractions (Hiebert, 1988), so this lesson helped us examine teacher interventions during students' investigation and cooperation. The lesson is reproduced as follows:

At the end of the fourth day of their fundraising campaign, the teachers at Thurgood Marshall School had raised $270 of the $360 they needed to reach their goal. Three of the teachers got into a debate about how they would report their progress.

* Ms. Mendoza wanted to announce that the teachers had 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 really the simplest way to describe the teachers' progress.

A. Which of the three teachers do you agree with? Why?

B. How could the teacher you agreed with in Part A prove his or her case?

Procedures

The first author transcribed all of the selected videotapes and recorded and described verbal and nonverbal representations in detail. For example, when a teacher approached a group, how many minutes did she listen to the students' discussion before she intervened? Such information was important because it partially reflected how a teacher used peer resources. After the description, the first two authors parsed the transcribed lesson into small units, each of which represented an action sequence at a particular grain size (Schoenfeld, 1998) or a "sighting" according to teachers' intervention choices and topic changes (American Association for the Advancement of Science [AAAS], 2004). A detailed explanation about sighting is provided in the intervention choice section that follows. On the basis of the videos and video transcriptions, the authors coded and classified teacher intervention length, choice, and frequency in terms of Instrument 1. Moreover, the authors coded and analyzed teacher intervention quality of each sighting by using the categories in Instrument 2.

Instrument 1: Coding Scheme for Teacher intervention Length, Choice, and Frequency

We used Instrument 1 to record teachers' intervention length of each sighting, the preference of intervention choice, and the frequency of a teacher's visits to each group. That type of data was not related directly to teachers' intervention quality, even though it might have affected it (see Appendix). We used one coding sheet to record each visit per group for 1 teacher's intervention. Because a teacher could intervene with different students about various topics during one visit, one coding sheet might include several intervention sightings.

We recorded the start and end times of each intervention sighting. The lengths of each intervention were classified 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 showed whether 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 same group); and (c) whole class (during or right after group intervention, the teacher gave feedback to the whole class). Each teacher interaction with students in any one of the three ways was considered as one intervention sighting (AAAS, 2004). When topics were changed, we recorded additional intervention sightings. We used one coding sheet to record each visit per group for one teacher's intervention. Therefore, we could use the coding sheets to obtain information about a teacher's intervention choice preferences by calculating the percentage of intervention sightings for each choice.

We considered each time that a teacher approached a group and interacted with student(s) as one visit. Because we used one coding sheet to record each visit per group, the sum of these codes provided information about the teacher's frequency of visiting each group. We also recorded the contents used within each intervention for further use. The contents included checking work, providing help, or giving announcements.

Instrument 2; Categories for Teacher Intervention Quality

We focused on teachers' intervention quality. We evaluated each intervention sighting that we recorded with Instrument 1 by using Instrument 2. We created all the categories and indicators on Instrument 2 under the guidelines of NCTM's (2000) Principles and St\andards for School Mathematics. The NCTM teaching principle suggests that "Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well" (p. 10). The five process standards also require teachers to address students' conceptual understanding. Therefore, the basic consideration reflected in our categories is whether teachers' interventions target students' cognitive processes. We adapted the detailed categories and indicators from the AAAS (2004) criteria of teaching quality, which are supported by prior research findings on cooperative learning. The categories are illustrated as follows:

Category I: Teachers' guidance focusing on the learning goal and students' cognitive obstacles (Meloth &. Deering, 1999). Teachers tend to identify students' thinking by asking questions. When students encounter difficulties and cannot figure them out, teacher assistance is the primary resource for enhancing students' thinking (Kramarski et al., 2002). We evaluated that category according to the following four indicators: (a) Is the teacher's guidance related to learning goal? (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 the teacher help students through approaches such as introducing or developing important ideas, helping students relate their own experiences to the mathematical idea, or using alternative representations (AAAS)?

Category II: Promoting student thinking. As facilitators, teachers should provide enough opportunities and create appropriate environments to improve students' thinking (Meloth & Deering, 1999). We used three indicators for this category: (a) When students give a correct answer, does the teacher encourage them to express, justify, interpret or represent their ideas (AAAS, 2004)? (b) When students make mistakes, does the teacher use errors as springboards for inquiry (Borasi, 1994)? (c) Does the teacher encourage students to use different ways to solve the problem (AAAS)?

Category III: Encouraging high-level peer discussion. Even if class instruction begins with individual study, when the phase of cooperative learning begins, teachers should encourage peer interactions, with a focus on elaborating students' thinking (Cohen, 1994b; Dekker & Elshout-Mohr, 2004; Meloth & Deering, 1999). We evaluated Category III by two indicators: (a) Does the teacher ask students to explain their ideas rather than just compare answers with each other (Webb et al., 2002; AAAS, 2004)? (b) When a student needs help, does the teacher respond to questions only after all group members have discussed the problem (Johnson & Johnson, 1990)?

Examples for evaluating intervention quality. We judged teacher intervention quality by examining whether each intervention sighting met each indicator on the instrument with the codes "met,""partially met,""not met," or "not appropriate." For example, some teachers purposely gave students several minutes to do individual tasks before group work. During that period, Category III: Encourage peer discussion would be judged as not appropriate. Also, during group work, some teachers checked students' work by asking, "Are you agreeing?" In that example, without any other guiding or probing, all of the indicators for Categories I and II would not be met; however, the indicator "guiding focus on learning goal" would be partially met. Two detailed examples of our evaluation are shown as follows:

Example 1: The teacher asked the whole class to first explore the new task independently. She then walked around the classroom and came to one student:

Teacher: What did you decide?

Student: 3/4

Teacher: 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' group work. The teacher checked the student's thinking by asking him several questions (Table 2 shows the evaluation of this sighting).

Example 2: During group work, Teacher B came to a group. He observed the students folding fraction strips and listened to 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. Go to 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 thermometer 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. The teacher guided the students because the whole group encountered difficulty (Table 3 shows the evaluation of this sighting).

After completing the coding, we counted the number of met, partially met, and not met sightings for each indicator. (We did not tabulate the inappropriate sightings.) For example, if there were 7 met, 6 partially met (considered as 1/2 6 = 3 met), 2 not met, and 3 not appropriate, the calculation for that percentage would be (7 + 1/2 6)/(7 + 6 + 2) = 67%. That means, 67% of the teacher's intervention sightings met the indicator.

This study is part of the MSMP. All authors had been trained for video transcription, coding, and analysis by AAAS experts for approximately 2 years. There were two types of training. One type of training was short term; AAAS experts came to the MSMP to train the coders for 2 days. The coders also traveled to Washington, D.C. for several days of training. Another type of training was long term; AAAS experts joined the MSMP meeting on video coding and analysis by telephone every 2 weeks. After 2 years of professional training, the MSMP members were skilled at video transcription, coding, and analysis. To improve reliability, the first two authors coded and evaluated each intervention sighting independently, then compared the outcomes with each other. The consistency of evaluation reached 90%. Each inconsistency of evaluation was reconsidered or discussed with the other authors until an agreement was reached.

Results

Differences of Intervention Length, Choice, and Frequency

Intervention length. Figure 1 shows teachers' length of interventions. For example, Teacher A had 21 intervention sightings that were less than 30 s. Teachers B and C each had a long intervention sighting lasting more than 5 min.

Table 4 shows the teachers' response types according to their intervention length. For interventions shorter than 1 min, there were more than 90% sightings in Classes A, D, E, and F versus less than 50% in Classes B and C; for interventions longer than 2 min, there were zero interventions in Classes A, D, E, and F versus 40% interventions in Classes B and C. Thus, interventions in Classes A, D, E, and F were quick responses, whereas in Classes B and C they were prolonged responses.

Intervention choice. Figure 2 shows that all teachers had lower intervention frequencies with whole classes than with individuals and groups. Teacher A had the highest frequency for intervention with individuals, whereas Teacher B never interacted with individuals. All teachers had several interventions with groups.

Because many mathematics classes combined individual study and group work within a lesson, we closely examined teacher intervention choices (whole class was not considered here) from different periods. Table 5 shows teachers' intervention choice preferences. Teachers A and B did not ask students to work individually during the mathematics lessons; all their interventions involved only group work. During group work, Teacher B intervened with whole groups (87.5% with group), whereas Teacher A intervened with individuals within the group (73% with individual). Teachers C, D, E, and F combined individual and cooperative study techniques. During individual study periods, Teachers C and E intervened with individuals (38.5% and 47.4%, respectively), more than did Teachers D and F (30% and 22.2%, respectively). During group-work periods, Teachers D and F intervened with groups (60% and 50%, respectively) more than did Teachers C and E (30.8% and 31.6%, respectively).

In general, interventions with individuals were more likely to occur in individual study periods, whereas interventions with groups occurred during the group study. Although Teachers A, C, and E interacted with individuals, Teacher A interacted only with students during group work. In contrast, Teachers B, D, and F interacted with groups; however, Teacher B interacted only with groups.

Intervention frequency. Table 6 illustrates the teacher intervention frequencies for each group. The teacher visited each group in the six classes at least once, with the exception of one group in Class F. Students in Class F were clustered into nine groups, which may have contributed to the teacher's inability to visit with all groups. According to the percentage of groups that were visited, we classified teacher intervention frequencies into one of three categories: high, middle, or low. For example, in Class A, the teacher visited four groups twice, and another two groups four times. Thus, all groups in Class A were visited at least two times, resulting in a high intervention frequency for Teacher A. In Class B, the teacher visited five of the six gr\oups only once. Therefore, only 17% of the groups were visited at least two times, resulting in a low intervention frequency for Teacher B. In Classes C, D, E, and F, the teacher intervention frequencies were inconsistent. Some groups were visited as many as three or four times, whereas other groups were visited once or not at all. On average, teacher interventions in the four classes could be classified as middle frequency because the percentages were from 33% to 60%.

Differences of Intervention Quality

Table 7 shows the percentage of intervention sightings that each teacher met for each indicator. The percentage equal to or higher than 70% was highlighted. The pattern in Table 7 shows that, overall, the teacher intervention quality in Classes A, C, and E was higher than in Classes B, D, and F. Across the categories and indicators, we found generally that teachers were not skillful at guiding students with scaffolding questions or through multiple approaches. Moreover, teachers did not promote students' thinking by capitalizing on errors or by encouraging them to solve problems in different ways. Finally, teachers were not proficient at encouraging peers to elaborate on their ideas to each other.

Focusing on teaming goal and cognitive obstacles. Teachers A, C, and E focused better on learning goals and cognitive obstacles than did Teachers B, D and F. Teachers A and E used scaffolding-type questions better than did Teacher C when they guided their students. The following intervention 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 cognitive obstacles with simple scaffolding questions. Similarly, Teacher E checked students' ideas by asking "What did you decide?""Why did you decide Ms. Mendoza?" and "How did you decide Ms. Mendoza?" However, the questions asked by Teacher C were not as scaffolded, even though the length of her intervention was prolonged.

Among Teachers B, D, and F, Teacher B recognized students' difficulties and tried to focus on them. However, when he guided students, he always repeated the same sentence "This is your start line, this is your finish line" or "Start line is here, correct? Finish line is here, correct?" He did not scaffold his questions and did not often use multiple approaches to guide students. Therefore, even though Teacher B spent prolonged time with each group, he did not move students' thinking ahead; most students were still confused by the end of the class. Teachers D and F checked group work and asked only simple questions, such as whether students agreed or disagreed. These questions did not allow Teachers D and F to identify students' cognitive obstacles and to further guide them.

Promoting student thinking. Teachers A, C, and E promoted student thinking better than did Teachers B, D, and F. Even though Teachers A, C, and E asked students to explain their ideas, only Teacher A effectively used errors as springboards for students' inquiry. In the lesson on promoting student thinking, there was a common mistake that occurred in several classes. When students explained why 3/4 was equal to 6/8, they understood that 3 2 was 6 and 4 2 was 8. However, some of them made mistakes in their verbal representations, such as doubling 3/4, or 6/8 divided by 2. In addition, some of the students made mistakes in written form, such as 3/4 2 = 6/8 or 3/4 + 3/4 = 6/8. Although Teachers A, C, and E noticed the type of error, they dealt with it differently. For example, 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 you think these are the same thing?" When there was no response, Teacher C said, "Ok, this is actually what we are 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 understood the procedure, she moved toward another topic. In contrast, Teacher A insisted on requiring students to inquire about the mistake: doubling 3/4. The following is an 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 multiplying 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 and multiply 4. So what are you doing?

Student 3: 3/4 is 6/8.

Teacher : 1 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? Think about it. I'll come back. Talk to your table. Because you say you are doubling, I want you to think about what you are doing.

Teacher A grasped students' verbal mistake and continued to ask students to consider their actions when they said that they were doubling fractions because their error was the same as with the written form, 3/4 2 = 6/8. She expected the students to figure out that 3/4 was multiplied by 2/2, which equals 1, rather than by 2. Therefore, even when 1 student in the group discovered that if 3/4 were multiplied by 2, the answer would be 1.5, not the same value as 3/4, the teacher prompted students to continue their discussion (detailed in Ding & Li, 2006)

Teachers D and F did not check students' thinking during intervention, thus, they had few opportunities to find students' mistakes and to promote their thinking. Teacher B generally had a negative attitude toward students' errors. For example, when the teaching assistant told him that one of the groups had made mistakes and needed his help, the teacher showed his disappointment and said, "Oh! No, they don't need help!" Therefore, repeating errors, rather than promoting student thinking, commonly occurred in Class B.

All teachers except Teacher C scored very low on their incentive to encourage students to use different ways to solve problems. Teachers were generally satisfied with one solution in a group and asked students to share their answers in the report phase at the end of the class.

Encouraging high-level peer discussion. Because the classes used cooperative learning, all teachers put students in small groups and allowed them to talk with each other. However, Teacher A, who was good at guiding and promoting students' thinking, scored lowest within the two indicators of the category "Encouraging high-level peer discussion." In Class A, only one group out of six groups had a peer discussion (discussions occurred in only 19 interventions). Teacher A did not encourage the groups to discuss problems with each other, but tended to respond immediately to students seeking help. When students in Class A needed assistance, they tended to seek the teacher rather than their peers. For instance, one student wanted to ensure that his solution was correct. He continued to raise his hand for more than 3 min because the teacher had left to interact with other groups. When she tried to give a student help, she never checked whether the student had first asked for peer help. Also, a boy in Class A asked for the teacher's help even though a girl in the same group was just praised by the teacher for her solution to the same problem. The teacher came to the group, giving the boy immediate assistance without any suggestion such as, "Why not discuss this with Cathy?" or "Cathy, can you explain this to him?"

In general, Teachers B, D1 and F walked around their classrooms and listened to groups talk before they intervened in them. However, they did not often ask students to explain their thinking to each other; Teachers D and F asked groups only to compare their answers. Teachers C and E used peer resource more than did the other 4 teachers.

Discussion

In cooperative learning, teachers' intervention frequency, length, and choice may influence the quality of the intervention. Regardless of teachers' intervention types, they need to address students' mathematical thinking and understanding. Concerning the length of teacher interventions, researchers suggested two different strategies; the first was Cohen's (1991, 1994a) quick response type. In contrast, there was the prolonged response type; Brown and Palincsar (1989) examined a program that implemented "guided cooperative learning-reciprocal teaching." They recommended that teachers help groups through modeling strategies and steering discussions. The prolonged monitoring 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 intervention. However, those two teachers displayed different intervention qualities of guiding or promoting student thinking. Although Teachers A, D, E, and F used quick interventions, their effects on students' thinking varied.

Thus, there are no absolute rules regarding intervention length. Using quick or prolonged response types depends on group situations. In addition, as demonstrated by Teacher B, those teachers who spend too much time with certain groups may ignore other groups and rush to finish the teaching task. Likewise, spending too little time with groups may not give teachers adequate time to identifyand develop students' thinking abilities. Therefore, responses that are too quick or too prolonged can have negative consequences for promoting student learning.

Teachers' intervention frequency, a factor related to intervention length, is important for the quality of teacher intervention. Chiu (2004) suggested that teachers revisit groups because students have a tendency to move off task after the teachers leave the students. Webb (1989, 1991) also suggested that, after providing necessary assistance, the teacher should leave the group and give students an opportunity to use the explanation that he or she provided. Later, the teacher could return to the group and assess how the student(s) used the explanation (Webb et al., 2002). Our study supports the prior research findings. For example, the infrequent visits of Teacher B to each group stemming from his overly prolonged responses prevented his checking students' understanding of his guidance, resulting in the students being off task.

Teachers in cooperative-learning mathematics classes have three intervention choices: individual students, small groups, and whole class. Regardless of their intervention choices, teachers should address students' mathematical understanding. When teachers intervene with a student, they need to consider how to use peer dynamics to foster his or her thinking. When teachers intervene with a group, they must consider individual students' thinking, such as whether they comprehend the lesson. Moreover, if all classroom groups have the same types of questions or confusion about mathematics concepts, teacher feedback to the whole class might promote students' thinking.

Teachers who intervened with individual students had more opportunity to check their independent thinking, although some of the students did not use peer resources effectively. In contrast, teachers who intervened with groups depended more on peer interaction than did other teachers, but some of them neglected students' individual thinking. As a result, techniques to balance peer resource and independent thinking and to use peer resource to improve students' mathematical understanding raised challenges for the teachers' interventions.

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

Two problems related to using peer resource were noteworthy: 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 discuss ideas with each other, preferring instead to give immediate answers. Thus, teachers are often so busy interacting with many individual students that they neglect peer interaction. Teachers should encourage peer discussion and spend more time with groups that require help, such as groups in which all members are confused or cannot reach an agreement. Therefore, encouraging peer resources enables teachers to deal with three intervention choices simultaneously and to keep more students on task-related thinking, thus greatly improving the efficiency of cooperative learning.

Second, although teachers should encourage peer interaction, they should not be too dependent on it. Simply asking students to compare their answers with each other is being too dependent on peer resources because students' independent thinking is not verified. Sometimes, lower level students tend to view higher level students as authorities and follow them without understanding the logic behind their answers (Amit & Fried, 2005). Thus, if the higher level students make mistakes, then the whole group would 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 agreed with her, "Yes, 2/3. It's 2/3." The teacher, who had been listening to the group discussion, recognized the mistake and provided help. If he had just checked the group by asking "agreeing or disagreeing" type questions, he would have never known the actual situation about the students' thinking.

There are two ways to balance peer resource and students' independent thinking. First, teachers could encourage students to elaborate their thinking. Sawyer (2004) suggested using the creative teaching strategy, which emphasizes the importance of students' active participation, including exploratory discourse and elaboration. Veenman, Denessen, Akker, and Rijt (2005) found that high-level elaborations of sixth-grade students were related positively to their mathematical achievement. Therefore, teachers could use both strategies: "Encourage explaining to each other" (use peer resource effectively) and "Encourage explaining to the teacher" (not be too dependent on peer resource) to balance the relationship between peer resource and students' independent thinking. second, teachers could combine cooperative learning with individual study. Some teachers in our study gave students 3-5 min to work individually before group work. Working individually before cooperative learning occurs could provide groups with mathematical content for discussion. Lower level students would have time to think about their problem so that they would not have to rely on their peers for help. In addition, higher level students would have time to think in multiple ways to improve their thinking.

In general, teachers in cooperative-learning mathematics classes could use peer and individual resources to improve students' mathematical thinking. The differences in teacher intervention quality scores show that not all teachers were good at cultivating students' independent thinking. The high-quality interventions of some teachers highlighted ways to address students' thinking, that is, analyze, diversify, and deepen their thinking.

Analyze students' thinking means that teachers should first identify students' cognitive obstacles, then focus on these obstacles to guide the students. Some teachers in this study identified students' ideas while scaffolding questions and made instructional decisions according to students' thinking, as suggested by Timmerman (2004) when he described "structured interviews." Chamberlin (2005) suggested four interaction patterns such as, "terminology-search interaction pattern" and "question- initiated interaction pattern" to help teachers meet the challenges of interpreting students' thinking. If teachers only check the agreement of group answers or repeat the same leading questions, they will not be able to identify students' cognitive difficulties. In contrast, if teachers scaffold their questions while guiding students, they can guide them on the basis of their mathematical thinking.

Teachers could diversify students' thinking in two ways. First, teachers guide students through multiple approaches. Throughout the six videos, some teachers used multiple representations such as fraction strips, pictures, number lines, or manipulatives. Teachers also linked students' current cognitive obstacles with the previous knowledge such as, "remember before, we folded it to make it work." Other teachers linked students' ideas with real-life experiences such as, asking about sharing candy bars. second, when teachers monitor group work, they should encourage students to solve problems in various ways. For example, if a group proved their problem solving by using fraction strips, the teacher could encourage them to try other approaches and also praise them for their first solution.

A major requirement for teacher intervention is to deepen students' thinking. Cooperative learning is a process for students to discover and construct their knowledge. Thus, student errors were a common and natural thing to occur during group study. Teachers should view errors as opportunities for inquiry. Teachers can often encourage students, even when the students make mistakes by saying, "I like the way you play with numbers" or "You are on the right track." Moreover, teachers should capitalize on errors as springboards for inquiry rather than only using errors as a diagnosis and remediation of students' knowledge weaknesses. For example, regarding the common error 3/4 x 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 students may not only have understood this concept but also could have improved their understanding of the equal sign, a critical notion for later algebra study (Falkner, Levi, & Carpenter, 1999; MacGregor, 1999; NCTM, 2000; Senzludlow &. Walgamuth, 1998).

Cooperative learning is a good teaching method in mathematics classrooms, but it is also a complex system (see Figure 3). Whatever teachers' intervention choices and strategies are, they should focus on the fundamental goal, which is to address students' mathematical thinking and to help them think mathematically.

Conclusion

We found that the length, frequency, and choices of teacher interventions somewhat influence intervention quality; we suggest that regardless of teachers' intervention types, they should address students' mathematical thinking. Moreover, our findings of teacher intervention differences allow teachers to recognize the challenge of balancing peer resource and students' independent thinking. We suggest that teachers use peer resource to help students think mathematically. Finally, our findings of the higher quality interventions such as helping students overcome cognitive obstacles by using simple scaffolding questions provide teachers with insightful ways to address students' thinking.

We also formed insights for teacher professional development \in cooperative learning. During teacher training, it is not enough for teacher educators to provide mathematics teachers with strategies for classroom management and group function. Educators also need to cultivate teacher beliefs of using cooperative learning to improve students' mathematical thinking and to provide teachers with techniques to effectively address student thinking. Researchers should focus on (a) ways to improve cooperative-learning effects by addressing students' mathematical thinking and (b) techniques to improve students' mathematical thinking by taking advantage of cooperative learning. The underlying reasons for teacher intervention differences when addressing student thinking and the fundamental mechanism of high-quality teacher interventions also need further study.

REFERENCES

American Association for the Advancement of Science. (2004). Instructional criteria. Retrieved February 11, 2006, from http:// www.project 2061.org/publications/textbook/algebra/summary/ criteria.htm

Amit, M., & Fried, M. N. (2005). Authority and authority relations in mathematics education: A view from an 8th-grade classroom. Educational Studies in Mathematics, 58, 145-168.

Ares, N., & Gorrell, J. (2002). Middle school students' understanding of meaningful learning and engaging classroom activities. Journal of Research in Childhood Education, 16, 263- 277.

Bettenhausen, S. (2002). Students as teachers. Kappa Delta Pi Record, 38, 188-190.

Blatchford, P., Baines, E., Basse, P., Rubie-Davies, C., & Chowne, A. (2006, April). Improving the effectiveness of pupil group work: Effects on pupil-pupil interactions, teacher-pupil interactions and classroom engagement. Paper presented at the annual meeting of the American Educational Research Association, San Francisco.

Bogdan, R. C., Biklen, S. K. (2003). Qualitative research for education: An introduction to theory and methods. Boston: Allyn & Bacon.

Borasi, R (1994). Capitalizing on errors as "springboards for inquiry": A teaching experiment. Journal for Research in Mathematics Education, 25, 166-208.

Brodie, K. (2001). Teacher intervention in small-group work. For the Learning of Mathematics, 20(1), 9-16.

Brown, A. L., &. Palincsar, A. S. (1989). Guided, cooperative learning and individual knowledge acquisition. In L. B. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Gloser (pp. 393-451). Hillsdale, NJ: Erlbaum.

Cai, J. (2001). Improving mathematics learning: Lessons form crossnational studies of Chinese and U.S. students. Phi Delta Kappan, 82, 400-402.

Chamberlin, M. T. (2005). Teachers' discussions of students' thinking: Meeting the challenge of attending to students' thinking. Journal of Mathematics Teacher Education, 8, 141-170.

Chiu, M. M. (2004). Adapting teacher interventions to students' needs during cooperative learning: How to improve student problem solving and time on-task. American Educational Research Journal, 41, 365-399.

Cohen, E. G. (1991, April). Classroom management and complex instruction. Paper presented at the annual meeting of the American Educational Research Association, Chicago.

Cohen, E. G. ( 1994a). Designing group work: Strategies for the heterogeneous classroom (2nd ed.). New York: Teachers College Press.

Cohen, E. G. (1994b). Restructuring the classroom: Conditions for productive small groups. Review of Educational Research, 64, 1-36.

Deering, P. D., & Meloth, M. S. (1993). A descriptive study of naturally occurring discussion in cooperative learning groups. Journal of Classroom interaction, 28(2), 7-13

Dekker, R., & Elshout-Mohr, M. (2004). Teacher interventions aimed at mathematical level raising during collaborative learning. Educational Studies in Mathematics, 56, 39-65.

Denzin, N. K., & Lincoln, Y. S. (Eds.). (2000). Handbook of qualitative research (2nd ed.). Thousand Oaks, CA: Sage.

Denzin, N. K., &. Lincoln, Y. S. (2000). Introduction: The discipline and practice of qualitative research. In N. K. Denzin, &. Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd d., pp. 1-28). Thousand Oaks, CA: Sage.

Ding, M., & Li, X. (2006, February). Students' errors in transitions from verbal to symbolic understanding. Paper presented at the 33rd annual meeting of the Research Council on Mathematic Learning, University of Nevada, Las Vegas.

Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-236.

Hamm, M., &. Adams, D. (2002). Collaborative inquiry: Working toward shared goals. Kappa Delta Pi Record, 38, 115-118.

Hiebert, J. (1988). A theory of developing competence with written mathematics symbols. Educational Studies in Mathematics, 19, 333-355.

Jacobs, J. K., Kawanaka, T. K., & Stigler, J. W. (1999). Integrating qualitative and quantitative approaches to the analysis of video data on classroom teaching. International Journal of Educational Research 31, 717-724.

Janesick, V. J. (2000). The choreography of qualitative research design: Minutes, improvisations, and crystallization. In N. K. Denzin, &. Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd d., pp. 379-399). Thousand Oaks, CA: Sage.

Johnson, D. W., & Johnson, R. T. (1990). Using cooperative learning in math. In N. Davidson (Ed.), Cooperative learning in mathematics-A handbook for teachers (pp. 103-124). New York: Addison- Wesley.

Johnson, D. W., & Johnson, R. T. (1991). Learning together and alone: Cooperative, competitive, and individualistic learning (3rd ed.). Englewood Cliffs, NJ: Allyn & Bacon.

Kagan, S. (1985). Dimensions of cooperative classroom structures. In R. Slavin, S. Sharan, S. Kagan, R. H. Lazarowitz, C. Webb, &. R. Schmuck (Eds.), Learning to cooperate, cooperating to leam (pp. 67- 102). New York and London: Plenum Press.

Kramarski, B., Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effects of cooperative learning and metacognitive training. American Educational Research Journal, 40, 281-310.

Kramarski, B., Mevarech, Z. R., & Arami, M. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks. Educational Studies in Mathematics, 49, 225-250.

Lappan, G., Fey, J. T, Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (Eds). (1998). Connected mathematics: Bits and pieces I, Understanding Rational Numbers (Teacher's edition). Menlo Park, CA: Dale Seymour.

MacGregor, M. (1999). How students interpret equations: Intuition versus taught procedures. In H. Steinbring, M. G. B. Bussi, & A. Sierpinska (Eds.), Language and communication in the mathematics classroom (pp. 262-270). Reston, VA: National Council of Teachers of Mathematics.

Meloth, M. S. (1991). Enhancing literacy through cooperative learning. In E. Hiebert (Ed.), Literacy for a diverse society: Perspectives, practices, and policies (pp. 172-183). New York: Teachers College Press.

Meloth, M. S., & Barbe, J. (1992, April). The relationship between instruction and cooperative peer-group discussions about reading comprehension strategies. Paper presented at the annual meeting of the American Educational Research Association, San Francisco.

Meloth, M. S., & Deering, P. D. (1992). The effects of two cooperative conditions on peer-group discussions, reading comprehension, and metacognition. Contemporary Educational Psychology, 17, 175-193.

Meloth, M. S., & Deering, P. D. (1999). The role of the teacher in promoting cognitive processing during collaborative learning. In A. M.

O'Donnell & A. King (Eds.), Cognitive perspectives on peer learning (pp. 235-255). Mahwah, NJ: Erlbaum.

Michael, V. A., & de Prez, K. A. M. (2000). Rethinking observation: From method to context. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 673-702). Thousand Oaks, CA: Sage.

Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis. Thousand Oaks, CA: Sage.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Rivette, K., Grant, Y., Ludema, H., & Rickard, A. (2003). Connected mathematics project. Retrieved March 30, 2006, from http:/ /www.math. msu.edu/cmp/shareware/CMP%20Booklet%20FINAL.pdf

Senz-ludlow, A., & Walgamuth, C. (1998). Third grader's interpretations of equality and the equal symbol. Educational Studies in Mathematics, 35, 153-187.

Sawyer, R. K. (2004). Creative teaching: Collaborative discussion as disciplined improvisation. Educational Research, 33(2), 12-20.

Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of "well-taught" mathematics courses. Educational Psychologist, 23, 145-166.

Schoenfeld, A. H. ( 1998). Toward a theory of teaching-in- context. Issues in Education, 4, 1-94.

Slavin, R. E. (1996). Research on cooperative learning and achievement: What we know, what we need to know. Contemporary Educational Psychology, 21, 43-69.

Stake, R. E. (2000). Case studies. In N. K. Denzin, & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 435- 454). Thousand Oaks, CA: Sage.

Timmerman, M. A. (2004). The influences of three interventions on prospective elementary teachers' beliefs about the knowledge base needed for teaching mathematics. School Science and Mathematics, 104, 369-382.

Veenman, S., Denessen, E., Akker, A. V. D., & Rijt, J. V. D. (2005). Effects of a cooperative learning program on the elaborations of students during help seeking and help giving. American Educational Research Journal, 42, 115-151.

Webb, N. M. (1989). Peer interaction and learning in small groups. International Journal of Educational Research, 13, 21-40.

Webb, N. M. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22, 366-389.

Webb, N. M., Farivar,S. H., & Mastergeorge, A. M. (2002). Productive helping in cooperative groups. Theory into practice, 41, 13-20.

Wu, H. (1997). The mathematics education reform: Why you should be concerned and what you can do. The American Mathematical Monthly, 107, 946-954.

MEIXIA DING

Texas A & M University

XIAOBAO LI

Western Carolina University

DIANA PICCOLO

GERALD KULM

Texas A & M University

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

Cofryright 2007 Heldref Publications

MEIXIA DING is a doctoral student in mathematics education, Texas A & M University. Her research interests include cooperative learning and teacher decision making. XIAOBAO LI is an assistant professor in mathematics education, Western Carolina University. His research includes error analysis and curriculum study. DIANA PICCOLO is a doctoral student in mathematics education, Texas A & M University. She is interested in pedagogical content knowledge development between student teachers and interns. GERALD KULM is the Curtis D. Robert Endowed Professor of mathematics education. His research includes middle grades teaching and learning, mathematics assessment, and teacher knowledge.

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