Learning and Mathematics

Small group interactions - Yackel, Cobb, & Wood (1991) Yackel, Cobb, and Wood, of Purdue University Calumet, Vanderbilt University, and Purdue University, respectively, describe an experiment in which all instruction in a second grade mathematics classroom was replaced by small group problem-solving strategies for an entire school year. Children were grouped in pairs and spent half of each math period working together to solve math problems. The other half of the period was used for whole group discussions of the paired activities. Work was not graded, nor was there a set amount of work to be completed; rather, the teacher gave report card grades based on her knowledge of the children, and the students were allowed to spend as much time as they needed to discuss each problem and arrive at a solution. The goal of the project was to foster collaborative learning and build conflict resolution skills, as well as to provide for other learning opportunities that do not arise in more traditional classrooms. Article: Yackel, E., Cobb, P., & Wood, T. (1991) Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22 (5), 390-408. Quotes and Comments: "The salient features of the project for study are these. First, instruction was provided by the regular classroom teacher in the regular classroom setting. Second, the duration of the study was an entire school year. Third, a cooperative learning approach was used for all aspects of second-grade mathematics instruction including computational activities, usually thought of as skill and practice activities, and other traditional second-grade topics such as time, money, and measurement. This third feature is especially important, since small-group problem solving in mathematics education has come to mean that students work together on word problems or nonroutine problems. Our approach, however, was to develop instructional activities so they were likely to give rise to problematic situations for the children, rather than be treated as exercises for practice or drill... Finally, the classroom norms for cooperation were mutually constructed in the context of working on mathematics activities and in the absence of an external reward system for either individuals or groups" (390-1). "We set out initially to form pairs of a variety of types to gain information about influences that might be relevant in developing productive working relationships. Accordingly, we formed pairs that were homogeneous with respect to mathematical development and pairs that were heterogeneous in this regard, and we were careful to include male-male, female-female, and male-female pairings" (392). Further on in the article, the authors evaluate the results of the homogeneous and heterogeneous ability groupings: "[I]n groups where children were of different mathematical conceptual levels the teacher sometimes modified the obligation that they should reach a consensus about solution methods because she realized that this was an impossibility, given the disparity in their mathematical understandings. Instead, the children had to agree only about an answer, and it was acceptable for the more advanced child to construct a solution method that the other could not understand provided the weaker child had also constructed a solution. Although this appears to suggest that children of different mathematical conceptual levels should not be partners, there are other factors that make such partnerships viable. For example, one such pair in the project classroom formed a productive working relationship because the weaker child continually drew his partner back into active involvement... The partnership proved mutually beneficial because of the social relationship that developed between the two children" (400). "...[T]eaching through problem solving acknowledges that problems arise for students as they attempt to achieve their goals in the classroom. In this approach students are seen as the best judges of what they find problematic and they are encouraged to construct solutions that they find acceptable, given their current ways of knowing. The situations that children find problematic take a variety of forms and can include resolving obstacles or contradictions that arise when they attempt to make sense of a situation in terms of their current concepts and procedures, accounting for a surprise outcome (particularly when two alternative procedures lead to the same result), verbalizing their mathematical thinking, explaining or justifying a solution, resolving conflicting points of view, developing a framework that accommodates alternative solution methods, and formulating an explanation to clarify another child's solution attempt" (394-5). Thus the authors believe that social situations are crucial in mathematics instruction, especially with regard to allowing the child to construct and modify his or her own mathematical knowledge. "In the project classroom, the teacher initiated and guided the mutual construction of a variety of social norms as explained below. These included the following: that students cooperate to solve problems, that meaningful activity is valued over correct answers, that persistence on a personally challenging problem is more important than completing a large number of activities, and that partners should reach consensus as they work on activities... In addition to norms for social cooperation, there were classroom norms for individual activity. These included the following: that children figure out solutions that are meaningful to them, that they explain their solution methods to their partner, and that they try to make sense of their partner's problem-solving attempts" (397-8). On pp. 398-99, the authors discuss two instructional methods used by the teacher to create and reconfigure these norms. In some cases she would use an incident that had occurred in the classroom to illustrate a general class expectation, in others she would explicitly introduce a norm to the whole class - drawing the children into a discussion of the particular obligation and the reasons for its existence. "Children engage in two types of problem solving as they work together in small groups. On the one hand, they attempt to solve their mathematical problems, and on the other hand, they have to solve the problem of working productively together, as discussed above. Once social problems have been temporarily resolved, the interactions that take place give rise to opportunities for learning that result directly from the interactions. As the children work together and strive to communicate, opportunities arise naturally for them to verbalize their thinking, explain or justify their solutions, and ask for clarifications. Further, attempts to resolve conflicts lead to both the opportunity to reconceptualize a problem and thus construct a framework for another solution method, and the opportunity to analyze an erroneous solution method and provide a clarifying explanation. These types of discussion-based learning opportunities do not typically occur in traditional classroom settings and are qualitatively different from the opportunities that do arise in traditional settings where children typically do not engage in mathematical discussions with each other but work individually to complete many exercises by repeating a method demonstrated by the teacher" (401-2). - summary by Jane Ehrenfeld

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