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: bmorgan@utb.edu

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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