Learning and Mathematics

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Interactive Learning - Brown, Campione, Reeve, Ferrara & Palincsar (1991)

Interactive learning -- learning in which students and their teacher share ideas and take turns leading discussions -- provides students with a model of the way experts work together to learn and understand. Interactive learning also challenges students to develop their own capabilities. Brown, Campione, Reeve, Ferrara, and Palincsar argue that it is necessary to reconsider the traditional roles of teacher and student (where the teacher lectures at the board and students sit passively at their desks, taking notes), and to give serious consideration to the quality of learning possible when classroom learning involves small group work.
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Chapter:

Brown, A. L., Campione, J. C., Reeve, R. A., Ferrara, R. A., & Palincsar, A. S. (1991). Interactive learning and individual understanding: The case of reading and mathematics. In Landsmann, L. T. (Ed.), Culture, schooling, and psychological development (pp. 136-170). Norwood, NJ: Ablex Publishing Co.

Overview:

An alternative to the lecture format, interactive learning provides learners with situations that push the boundaries of their abilities and actively engage them in tasks. It also gives students an opportunity to be learners as they come to master a task (or part of one) and, once they have achieved mastery, to be teachers of those who are still learning. The research indicates that problems that are too difficult at first for students to handle on their own later become problem types they can solve independently when they have first worked on them in a small group setting. This kind of interaction is called reciprocal teaching because both the teacher and the student(s) are teachers and learners.

Reciprocal teaching and scaffolding (in which the 'expert', teacher, or parent decreases assistance and sets up tasks at increasingly challenging levels in response to a child's increased skills and understanding) also enable children to learn a body of coherent, usable, and meaningful knowledge within their zone of proximal development (the distance between what children can do on their own and what they are capable of doing with assistance), and "to develop a repertoire of strategies that will enable them to learn new content on their own" (p. 150).

Direct Quotes (and some comments):

What children can do with the assistance of others is "even more indicative of their mental development than what they can do alone" (Vygotsky, 1978, p. 85).

"The zone of proximal development is the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers" (Vygotsky, 1978, p. 86). [In other words, for every student, there is a range of difficulty he or she is ready to tackle, from what the student knows and can do on his or her own to what can only be done with help. Another way of thinking about it is that the 'ZPD' denotes the space between actual knowledge and potential knowledge. Awareness of a student's ZPD helps a teacher gauge the tasks the student is ready for, the kind of performance to expect, and the kinds of tasks that will help the child reach his or her potential.]

"It is important to note that social settings can provide learning zones for children [students] even in the absence of an explicit instructional goal. Cooperative learning groups provide a learning forum for their members, although the goal is successful problem solution, regardless of individual contributions or the potential for personal development" (p. 137).

In group settings: "Novices may be held responsible for some simple aspect of the task while at the same time they are permitted to observe experts, who serve as models for higher-level participation. In such settings, novices learn about the task at their own rate, in the presence of experts, participating only at a level they are capable of fulfilling at any point in time" (p. 138). [For example, Brown, et al. observed a class in which students worked together with an instructor in small groups. The aim was understanding algebra word problems. Each student in the group took a turn leading the discussion in the group and with the teacher. While the student leader did the problem solving, others acted as supportive critics.]

"Reciprocal teaching was designed to provide a simple introduction to group discussion techniques aimed at understanding and remembering... [Reciprocal teaching is] a procedure that features guided practice in applying simple concrete strategies to the task... [A] teacher and a group of students take turns leading a discussion... The learning leader (adult or child) begins the discussion by asking a question on the main content [problem type] and ends by summarizing the gist [what the problem asks and how the problem solution generated addresses this question]. If there is disagreement, the group rereads and discusses possible question and summary statements until they reach consensus. Questioning provides the impetus to get the discussion going. Summarizing at the end of a period of discussion helps students establish where they are in preparation for tackling a new segment of text" (p. 139).

"The goal [of reciprocal teaching] is joint construction of meaning: The reciprocal nature of the procedure forces student engagement; and teacher modeling provides examples of expert performance" (p. 139).

"... within reciprocal teaching, one member of the group, the adult teacher, does have an explicit instruction goal, and it is part of her responsibility to engage in deliberate scaffolding activities when she works with current discussion leaders in an attempt to improve their level of participation. Thus, reciprocal teaching is both a cooperative learning group jointly negotiating and understanding task and a direct instruction forum wherein the teacher attempts to provide temporary scaffolding to bolster the learning leader's inchoate strategies" (p. 141).

"The idea [of scaffolding] is for the teacher to take control only when needed and to hand over the responsibility to the students whenever they are ready. Through interactions with the supportive teacher, the students are guided to perform at an increasingly challenging level. In response, the teacher gradually fades into the background and acts as a sympathetic coach, leaving the students to handle their own learning. The teacher is always monitoring the discussions, however, and is ready to take control again when understanding fails" (p. 141).

"Reciprocal teaching experience enables the children both to learn a body of coherent, usable knowledge and to develop a repertoire of strategies that will enable them to learn new content [work different types of problems] on their own" (p. 150).

"Students need to practice connecting together their fragmentary knowledge into systems of 'meaningful mathematics'... [Students with this opportunity] argue about the meaning of mathematical expressions and attempt to convince each other of the appropriateness of the algorithms they invent (see Lampert, 1986). They engage in lively discussions about the meaning of what they are doing, and it is these reflective processes that are largely absent from traditional mathematics classes" (p. 151).

"[The strategy criteria of] questioning, summarizing, clarifying, and predicting... were selected [for mathematics] because a byproduct of summarizing what one has just read, asking for clarification, and so on is to force comprehension... [During the process of reciprocal teaching] [t]he four activities became rituals that ensured that a discussion took place and forced comprehension..." (p. 151).

Reflection involves problem extension: "the goal is to make clear that the group has been working on not only a single problem, but also one of a family of related problems. To the extent that they understand the quantities involved in the original problem and the relations among those quantities, they should be able to 'play' with their specific solution and solve related problems. Thus, extension serves to accentuate the need for understanding the problem, as well as providing a way of checking to see if the students do in fact understand the nature of the solution they have generated. This is ... training for transfer [making connections between mathematical concepts in different contexts] which we strongly endorse" (p. 156). [Problem extension can be done, for example, by modifying some quantities in the original problem or by exchanging some of the givens for unknowns.]

"Higher-ability students showed greater degrees of lateral transfer. It appears that academically weak children have particular difficulties applying what they have learned to novel but related situations" (p. 158).

References:

Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305-342.

Vygotsky, L. (1978). Mind in society. Boston: Harvard University Press.

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