Cooperative learning has been championed by many advocates. It was designed and implemented in order to develop social strategies and social attitudes in students, and to improve social relations within and between groups. In addition, there is a large cluster of cooperative learning models aimed at cognitive development e.g. in mathematics. Sometimes cooperative learning is directed at both the social and the cognitive side of human development (Gillies, Ashman & Terwel, 2008).
The central questions to ponder upon are:
• Should knowledge be provided or generated in mathematics education?
• What do we know about the feasibility of cooperative learning in mathematics education?
• What are the underlying mechanisms of cooperative learning in mathematics education?
• What kind of effects may we expect from learning in cooperative groups in mathematics?
• What are the criteria for curriculum materials and assignments for learning in
cooperative groups in mathematics education?
The purposes and aims of cooperative learning need to be elaborated within certain domains of study. Cooperative learning is not a technique for its own sake but needs content in order to be useful. The specific content or subject matter is not a result of arbitrary choice, without any consequences for the design of a curriculum in which cooperative learning takes place. Content has its own characteristics, which can be utilized in the designing process and in the classroom in order to facilitate the development of thinking as a human activity. Mathematics education, for example, offers specific opportunities for cooperative learning with this purpose in view. To put it differently and to make the general idea more specific, the content of mathematics allows for specific models of cooperative learning in order to accommodate individual differences between students. Mathematical problems can be situated in real-life contexts and designed in such a way that solutions can be reached along different routes and at different levels. This makes cooperative learning in mathematics different from cooperative learning in other domains, such as languages and world orientation. Each domain has their own opportunities for teaching and learning with regard to individual differences among students.
Purpose, Organization and content may be summed up in the following composite question: Should all students pursue the same purposes and content or should different programs be offered to different categories of students? My own position has always been that a common curriculum should be offered to all. I am inspired by Hans Freudenthal (1991) who proposed ‘mathematics for all’ in the context of comprehensive education for students between the ages of 12-16 (see also Gravemeijer & Terwel, 2000). However we should recognize large differences between students, especially in domains like mathematics and languages. Therefore, a common curriculum should always been accompanied by opportunities for enrichment, remediation and choice. The question is: how to implement this innovation in the classroom? Could cooperative learning offer a solution?
There are many instructional approaches. Most of them can be categorized in the dichotomy ‘providing versus generating’. Important question is: Should knowledge be provided or generated in mathematics education? In our research, several research projects are about representations in mathematics. By representations one can think of drawings, graphs, verbal descriptions, concepts, symbols, algebraic formula’s, designs and proto-types. Point of departure was one of the major questions in learning theory and curriculum design: Are representations to be provided or generated? I have tried to overcome this dichotomy by searching for a third way. First I have designed an instructional model for cooperative learning and adaptive instruction for students between the ages of 12-16 (the so called AGO-model). Second by designing an instructional approach for primary mathematics called ‘guided co-construction’.
Cooperative Learning and Adaptive Instruction: AGO-model
The AGO-model is a whole-class model for cooperative learning that allows for student diversity through situational remediation and enrichment within small groups. The AGO-model consists of the following stages:
1. Whole-class introduction of a mathematics topic in real-life contexts;
2. Small-group cooperation in heterogeneous groups of four students;
3. Teacher assessments: diagnostic test and observations;
4. Alternative learning paths depending on assessments consisting of two different modes of activity:
a) Individual work at individual pace and level (enrichment), in heterogeneous groups with the possibility of consulting other students, or;
b) Opportunity to work in a remedial group (scaffolding) under direct guidance and supervision of the teacher;
5. Individual work at own level in heterogeneous groups with possibilities for students to help each other;
6. Whole-class reflection and evaluation of the topic;
7. Final test.
The model provides for diagnostic procedures and special instruction and guidance by the teacher in a small remedial group for low-achieving students. This cycle is extended through a series of lessons (units) over for example three to five weeks, preferably in extended units of uninterrupted instructional time.
In a pretest-posttest control-group experiment, the AGO-approach was put to the test (Terwel, Herfs, Mertens, & Perrenet, 1994). Students in the experimental (AGO) condition outperformed their counterparts in the control group (N=582). In this project an effect size of .68 was found. In addition a significant effect of class composition was found. Students in classes with a higher mean ability outperformed their counterparts in classes with a lower mean, after controlling for initial individual differences in mathematical ability. Indications were found that low-achieving students profited less from learning in small groups than high-achieving students.
Guided co-construction of mathematics
In our curriculum design and research projects we are inspired by many authors (Dewey, 1902, Brown & Palincsar, 1989; Freudenthal, 1991; Mercer, 1995; Hardman, 2008). Point of departure was Dewey’s famous adage about how knowledge construction should progress:
“It is continuous reconstruction, moving from the child’s present experience out into that represented by the organized bodies of truth that we call studies” (Dewey, 1902). Freudenthal’s main concept is guided reinvention of mathematics. Freudenthal refers to the guidance of the teacher in reinventing mathematics as a human activity. Cooperative learning in small groups of four is a central part of Freudenthal’s instructional approach. Our description of the instructional approach Guided co-construction of mathematics entails the following three core elements.
1. ‘Guided’ refers to the explicit role of the teacher for whole-class instruction and the scaffolding of students either in groups or individually.
2. ‘Co-’ refers to cooperative learning as an essential component of mathematics as a social, human activity and a cultural tool. In contrast to mathematics as a closed system to be transmitted to students.
3. ‘Construction’ refers to the recognition and construction of concepts, models, symbols by students on the basis of their prior knowledge and experiences.
Taken together, these elements imply that teachers facilitate the understanding of mathematics by presenting concepts, models, symbols etc. but also elicit and scaffold contributions and constructions from students within a meaningful (real life) context. In this interactive process, the differences between students are actually called upon and mathematics is not only prescribed ahead of time but also created by the students and teacher as they interact and move along. And such a process is also called co-elaboration, co-construction or the guided reinvention of mathematics (Brown & Palincsar, 1989; Dewey, 1943; Freudenthal, 1991).
In one of my research projects on ‘guided co-construction’ I found promising results (Terwel, Van Oers, Van Dijk & Van den Eeden (2009). My research question was: With regard to transfer, is it better to provide pupils with ready-made representations or is it more effective to scaffold pupils’ thinking in the process of generating their own representations with the help of peers and under the guidance of a teacher in a process of guided co-construction? The sample comprises 10 classes and 239 Grade 5 primary school students, age 10–11 years. A pretest-posttest control group research design was used. In the experimental condition, pupils were taught to construct representations collaboratively as a tool in the learning of percentages and graphs. Children in the experimental condition outperformed control children on the posttest and transfer test. Both high- and low-achieving pupils profited from the intervention. This study shows that children who learn to design are in a better position to understand representations like pictures, graphs, and models. They are more successful in solving new, complex mathematical problems.
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