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  1. 1. Cooperative Learning, Mathematical Problem Solving, and Latinos Verónica Galván Carlan, Renée Rubin, and Bobbette M. Morgan The University of Texas at Brownsville and Texas Southmost College Reprint request address: Bobbette M. Morgan, Ed. D. C&I Department, School of Education 80 Fort Brown Brownsville, TX 78520 E-mail:
  2. 2. Abstract Working with fifth grade Latino students, professors engaged students in cooperative activities while solving mathematical problems. Their work was based upon theories of social interdependence, cognitive development, and behavioral learning. Results indicate four changes in student behavior: 1) students became more engaged in problem solving; 2) students moved from a competitive to a cooperative stance; 3) students discovered there were several correct ways of finding a solution; and 4) students code-switched between Spanish and English to ensure everyone in the group understood. Two changes in teacher behavior related to cooperative learning were: 1) the regular classroom teacher moved desks from rows to groups; and 2) the teacher became more aware of the students’ mathematical abilities. Key words: cooperative learning, mathematics, problem-solving
  3. 3. Two challenges currently confronting the mathematics education community are effectively implementing reform-based strategies and increasing the mathematical learning and achievement of Latino students. Mathematics educators are shifting away from traditional classrooms to reform-oriented mathematics classrooms that focus on students actively engaged in mathematical discourse in cooperative settings (NCTM, 2000). Although cooperative learning is widely used and the research supporting this model of teaching is impressive and solid, there is a lack of research on how working in collaborative learning environments affects Latino students’ academic achievement in mathematics (Moschkovich, 1999). The Study The purpose of the study was to conduct research regarding the perceptions of students working in a cooperative learning environment to solve mathematical word problems. The convenient sample consisted of fifth grade Latino students whose ages ranged from 10-12 years old. Participants were primarily first and second-generation immigrants from Mexico, South America, and Cuba who communicated at various levels of proficiency in English and Spanish. Perspectives and Review of the Literature The theoretical framework for this paper centers on cooperative learning. Cooperative learning has its roots in the theories of social interdependence, cognitive development, and behavioral learning. Some research provides exceptionally strong evidence that cooperative learning results in greater effort to achieve, more positive relationships, and greater psychological health than competitive or individualistic learning efforts (Johnson, Johnson, & Holubec, 1994).
  4. 4. Social interdependence theory views cooperation as resulting from positive links of individuals to accomplish a common goal. The Gesalt psychologist Kurt Koffka proposed in the early 1900’s that although groups are dynamic wholes the interdependence among members is variable. Kurt Lewin (1948) stated that interdependence developed from common goals provides the essential essence of a group. This interdependence creates groups that are dynamic wholes. The power of the group is such that a change in any member or subgroup directly changes any other member or subgroup. Within cognitive development theory, cooperation must precede cognitive growth. Cognitive growth springs from the alignment of various perspectives as individuals work to attain common goals. Both Piaget and Vygotsky saw cooperative learning with more able peers and instructors as resulting in cognitive development and intellectual growth (Johnson, et al., 1998). The assumption of behavioral learning theory is that students will work hard on tasks that provide a reward and that students will fail to work on tasks that provide no reward or punishment. Cooperative learning is one strategy that rewards individuals for participation in the group’s effort. A review of the literature on cooperative learning shows that students benefit academically and socially from cooperative, small-group learning (Gillies, 2002). Cooperative learning can produce positive effects on student achievement (Cohen, 1986;
  5. 5. Davidson, 1989; Devries & Slavin, 1978; Johnson & Johnson, 1989; Okebukola, 1985; Reid, 1992; Slavin, 1990). Academic benefits include higher attainments in reading comprehension (Mathes, Fuchs, & Fuchs, 1997) and mathematics (Ross, 1995; Whicker, Nunnery, & Bol, 1997) and enhanced conceptual understanding and achievement in science (Lonning, 1993; Watson, 1991). Social benefits include more on-task behaviors and helping interactions with group members (Burron, James, & Ambrosio, 1993; Gillies & Ashman, 1998; McManus & Gettinger, 1996), higher self-esteem, more friends, more involvement in classroom activities, and improved attitudes toward learning (Lazarowitz, Baird, & Bolden, 1996; Lazarowitz, Hertz-Lazarowitz, & Baird, 1994). Studies on minority students in cooperative settings have increased. Cooperative learning is not simply a matter of grouping students heterogeneously but also in understanding that some groups of students, especially students of color, are more inclined to function better in group settings than individually (Pang & Barba, 1995; Vaughan, 2002). Using cooperative learning groups was found to be a more effective teaching strategy for students of color than for white students in terms of achievement (Slavin & Oickle, 1981). Researchers (Cohen, 1986; Slavin, 1990; Slavin & Oickle, 1981) found that students of color showed greater academic gains in cooperative learning settings than in traditional classrooms, and that cooperative learning strategies improved student performance in mathematics, language arts, science, and social studies (Devries & Slavin, 1978, Okebukola, 1985; Slavin, 1985). A study examining the effects of cooperative learning on mathematics achievement of a group of seventh grade minority students found that students involved in cooperative
  6. 6. learning performed significantly better than students who were not exposed to cooperative learning (Reid, 1992). In a study comparing the effects of cooperative learning to individualistic learning in a racially integrated classroom, Johnson and Johnson (1983) found that cooperative learning experiences resulted in higher academic achievement for minority students. Slavin (1985) examined the effects of Team-Assisted Individualization (TAI), Ability Group Active Teaching (AGAT), and the Missouri Mathematics Program (MMP) on mathematics achievement of third through sixth grade students, using experimental and control groups. He found that TAI, a cooperative learning strategy, had the most significant impact on mathematics achievement. According to Emmer and Gerwels (2002) some research on cooperative learning has addressed instructional components. In a number of studies students have been taught interaction skills, such as how to question or to help each other so that they did not give answers but facilitated each other’s thinking (Fuchs, Fuchs, Kazdan, & Allen, 1999; Gillies & Ashman, 1996, 1998; Nattiv, 1994; Webb, Troper, & Fall, 1995). And, when students are taught such skills, positive outcomes such as increased intrinsic motivation, liking for school, and self-esteem can result (Battistich, Solomon, & Delucchi, 1993). Method The nature of the research study calls for a qualitative method. Utilization of open- ended interviews and written prompts were the most appropriate techniques to capture the essence of how the students perceived engagement in cooperative learningand how this affected their mathematical thinking. The process of video and audio recording of student
  7. 7. interviews, collecting students written responses, and video and audio recording of students engaged in cooperative learning resulted in the utilizationof multiple methods for data gathering. Thus, cross data validity and confidence in the findings were achieved through data triangulation (Patton, 1990). At the end of the research study, participants were asked to respond in writing, in either English or Spanish, to the written prompt asking them to share what they had learned from working cooperatively. Additionally, interview data were video and audio recorded. After clarifying and justifying their mathematical thinking, students were asked questions regarding how working in a cooperative learning environment impacted their ability to solve word problems effectively. Data collection for this qualitative study occurred during one academic semester in an elementary public school in south Texas. Over 95% of the students enrolled qualified for free lunch. The sources of evidence used in this study are the students’ written responses (English and Spanish) and students’ verbal responses that were video and audio recorded. Additionally, audio and video recordings of student dialogues and interactions while participating in a cooperative learning environment were analyzed. Results Analysis of the data for patterns revealed four changes in student behavior and two changes in teacher behavior related to cooperative learning. Students became more motivated, less competitive, more aware of the problem solving process, and developed
  8. 8. language skills in Spanish and English. The regular classroom teacher implemented cooperative learning when the researchers were not present and became more aware of all students’ abilities as they worked in small groups. In this paper direct quotes are the students’ verbatim statements in response to open ended interviews and written prompts. Student names have been changed but reflect the same gender of the student originally making the comment. Students’ spelling and sentence structure have not been changed and demonstrate the fact that they are learning English as a second language. Students became more actively engaged in mathematical problem solving through cooperative learning. In one problem, they had to work out a code before they could solve the problem. Despite challenges decoding the message, they persisted as a team. “We used our brain and our imagination, we worked in a group as a team, we trid to guess the word in the sentences many times unitil get it.” Reluctant learners, who previously did not do their work, began to participate in the problem solving process. José said the hardest problem for him was the “Torn Bus Schedule.” Then he added, “But I had my classimates to help me.” Marcus, who was reluctant to do work on his own, said, “You could do it if a friend or partner is helping you.” Sam, who had previously been retained, wrote, “Yo aprendi que trabajando en equipo podemos resolver probroblemas con las ideas de todos.” (I learned that working in a group we are able to solve the problems with the ideas of everyone.)
  9. 9. Students moved from a competitive to a cooperative stance. When the university team first came into the classroom, they did not know about the students’ prior experience with cooperative learning. They asked the students for some rules to follow during the cooperative mathematical problem solving process. Some of the students responded that they should not talk because they were not allowed to talk to one another in the classroom or the cafeteria, only on the playground. Therefore, the researchers knew major changes had occurred when the students talked to one another, wrote about working as a team, and used “we” in their responses. Monica wrote, “We use teamwork. We drew pitchures. We answer qustions. We reread the story. We also reread the qustion.” Rather than competing for the correct answer, they began to share their problem solving ideas and answers. When asked how he figured out a problem, Miguel wrote with the “help of my group.” Kathy wrote, “Four or five minds are smarter than one,” and Carolina described working cooperatively as follows: I could help those people that still do not understand the problem. And that if I share my ideas they can share their ideas too. If we all share our ideas we can work out the problem. And we can put all our minds together and from all of us we can make one big brain. Students were asked what they did when they got stuck on a problem. Prior to working cooperatively, the students mentioned individual methods such as rereading or thinking harder. They also mentioned asking the teacher for help. Initially they believed that
  10. 10. asking another student for help was cheating. Statements from the video recordings revealed that after working collaboratively the perceptions changed. Many of them added that they could ask classmates for help if they got stuck on a problem. In their reflections, the students also mentioned specific skills that they had developed during cooperative learning. These skills included working together without making too much noise, respecting one another, sharing ideas, and negotiating one problem solving process and answer to share with the whole class. Sylvia wrote that learning how to work in groups and solve mysteries is “good for our future too.” Students were asked to work independently first, to share their problem solving processes and answers with one or two other students, and then to share that information with a larger group. At first, students asked each other for their answers. However, they soon began to work with each other on the mathematical problem solving process rather than just seeking the correct answers. They discovered that there are often several correct ways of finding a solution. A good example of students discovering different strategies to solve one problem involved fishing. Part of one problem said that a family of six went fishing and caught a total of ten fish. Everyone caught at least one fish. How many people in the family caught one fish and how many people caught two fish? Students were required to write their own answer, how they got that answer, the answers of other members of the group, and how the other group members got their answers. This process resulted in students understanding that various
  11. 11. strategies can be used to solve this problem. For example, Patricia reported dividing eight by two and two by one to get the answer that four people caught two fish and two people caught one fish each. She then went on to report that “my members drew pictures of fishes and people to understand it better and work out the problem.” Edna reported a different strategy for solving this problem, “I got my answer by subtracting 10 minus 6 and gave me 4 because number 10 was for all together and 6 was… the family.” Griselda reported that she subtracted to get her answer, but her team members added to get the answer. “They looked at the total number of fish and looked at how many people caught one fish and added four to give …ten total fish.” Learning about different strategies went beyond the immediate problem. Students were asked to think about what strategies they used in one problem that might help them with problem solving in the future. Students wrote about using teamwork, finding clues, drawing pictures, and drawing upon prior experiences to help them solve problems in the future. Students’ enthusiasm for problem solving cooperatively was obvious when they pursued the problem further than presented. Although the problem did not ask them how the fish would be divided for dinner, the students became concerned that some people had caught one fish and others had caught two. They discussed ways of solving this inequality. Some used a mathematical solution: Ten fish divided by six people equals one and two-thirds fish each. Others thought more about the family and suggested that some members of the family might be little and eat less or even not like to eat fish at all.
  12. 12. Students’ language developed as they worked together in Spanish and English to solve the problems. The students needed to use general terms, such as “solution” and “strategy;” problem specific terms, such as “down payment” and “rebate;” and technical mathematical language, such as “percent” and “interest rate” during the discussions. They often code- switched between Spanish and English to make sure everyone in the group understood. The students themselves recognized that they were expanding their vocabulary as exemplified when Pablo wrote, “What I like from sharing ideas was that we helped other and I learned new words.” One of the problems contained difficult words, such as sufficient, combustible, and serviceable. However, the students were able to use words they knew in Spanish, such as suficiente, combustible, and servible to understand the English words. In discussing strategies used to solve this problem, Tomas explained that he translated into Spanish. Graciela wrote about doing cooperative learning in English after completing one problem and in Spanish after completing another problem. In English, she wrote that she was able to solve problems “by working in a group using the brain and the imagination and also bay guessing.” On another day Graciela wrote, “Mis compañeros y yo isimos el trabajo en equipo, y asi solosionamos los problemas. Mis compaños y yo nos alludamos y asi garamos que se inificaban las palabras.” (My classmates and I did the work in a team, and that’s how we solved the problems. My classmates and I helped each other so that’s how we got the meaning of the words.)
  13. 13. After observing the researchers implementing cooperative learning during mathematics, the regular classroom teacher moved the desks from rows to groups and began implementing cooperative learning as well. She put posters on the wall explaining the various roles students would take in their groups. The teacher also began encouraging conversation among students rather than encouraging them to remain silent at all times. The classroom teacher also became more aware of students’ abilities when they worked in small groups. By its very nature, cooperative learning requires students to clarify and justify their ideas verbally. By listening to the students talking within their group and reporting to the whole class, the teacher became more cognizant of the students’ mathematical thinking processes. The thinking that goes on during mathematical problem solving is often hidden when students are working silently and individually but becomes apparent when they discuss these processes aloud with other students during cooperative learning. Discussion This study brings forth information about collaboration between professors from a university school of education in south Texas and a public school to teach mathematical skills to a fifth grade class of English language learners weekly over a semester. Additionally, this qualitative research study begins to address the limited amount of research regarding Latino students engaged in mathematical problem solving in a cooperative learning setting. The results of this study will contribute to the body of knowledge about collaboration of university professors with public schools and teachers, cooperative learning strategies use
  14. 14. with English language learners, and fifth grade English language learners’ perceptions about participation in cooperative learning structured groups. This research lends support to the findings in an article by Calderón, Hertz- Lazarowitz, and Slavin (1998): Research on second-language learning has shown that for students to reach high levels of proficiency, they must engage in a great deal of oral interaction, jointly negotiating meaning and solving problems (e. g., Krashen, 1985; Losey, 1995). Cooperative learning routinely provides opportunities for students to work together to construct meaning and share understandings (Durán & Szymanski, 1993; Prado-Olmos, Smith & Szymanski, 1993) (p. 154). Calderón and colleagues expressed that there currently exists a limited amount of research regarding cooperative learning and English language learners. …because cooperative learning engages students in frequent cognitively complex interactions around the solution of real problems and because of its demonstrated achievement effects in both monolingual and ESL settings, the effectiveness of cooperative learning for bilingual education is often assumed. This is the first study of cooperative learning in U.S. bilingual classes, and the results generally support this expectation (p. 163). Having students work together can increase students’ vocabulary and concept development in both language arts and mathematics. Research on cooperative and
  15. 15. collaborative learning supports the cognitive and affective benefits of collaborative learning (Cohen, 1994; Johnson & Johnson, 1985; Slavin, 1988). In addition, researchers (Carlan & Rubin, 2003; Rubin, 2002) found special benefits for bilingual learners. Latino students code-switched between Spanish and English to make sure everyone in the group understood. Some students may understand the concepts but lack the English vocabulary to express it while others may have the necessary English vocabulary but lack the concepts. By working and talking together, they can develop the necessary vocabulary and concepts in both languages. In addition, students revalue their own abilities when they work together on meaningful problems or projects (Goodman, 1996). Students become more engaged in the learning and make new connections between the two languages (Rubin, 2002). A summary of the results of this study indicate that: 1) Students became more actively engaged in mathematical problem solving through cooperative learning. Reluctant learners, who previously did not do their work, began to participate in the problem solving process. 2) Students moved from a competitive to a cooperative stance. Rather than competing for the correct answer, they began to share their problem solving ideas and answers. As one student wrote, “Four or five minds are smarter than one.”
  16. 16. 3) At first, students asked each other for their answers. However, they soon began to work with each other on the mathematical problem solving process rather than seeking the correct answers. They discovered that there are often several correct ways of finding a solution. 4) Students’ language developed as they worked together in Spanish and English to solve the problems. The students needed to use general terms, problem specific terms, and technical mathematical language during the discussions. They often code-switched between Spanish and English to make sure everyone in the group understood. 5) After observing the researchers implementing cooperative learning during mathematics, the regular classroom teacher moved the desks from rows to groups and began implementing cooperative learning as well. 6) The classroom teacher also became more aware of students’ abilities when they worked in small groups. Some students who did not normally participate in whole group activities were actively involved in small group work.
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