Designing Learning Environments for Developing Understanding of Geometry and Space by Chazan Daniel. Lehrer Richard

Author:Chazan, Daniel.,Lehrer, Richard.
Language: eng
Format: epub
ISBN: 978-1-136-49058-3
Publisher: Taylor & Francis (CAM)

Summary

Throughout this account, HT and AN were trying to develop a theory of how to make correct predictions. The discrepancies between what they predicted and what they actually found caused them to reflect on their prediction strategies and their conceptions of cube arrays. At first, their enumeration strategies were based on a more primitive spatial structuring of three-dimensional arrays—seeing them in terms of the faces of the prism formed. The students seemed to focus more on numerical strategies than on a deep analysis of the spatial organization of the cubes. However, because their initial spatial structuring led to incorrect predictions, HT and AN refocused their attention on the structure of the cube arrays, which led to a restructuring of their mental models of the arrays. In fact, during their work on Box C, HT and AN seemed to develop a layer structuring of the arrays, a structuring that they verified and refined on subsequent problems.

The interviewer played a dual role in his interactions with HT and AN. Most of his questions were intended to help him understand the students’ current ways of thinking. For instance, to probe whether HT and AN had changed their conception of the cubes in Box B, given their reflection and experience in dealing with Box C, the interviewer requested that the students return to their prediction for B. On Box C, because the interviewer recognized that the students’ use of the phrase “all the way around” might indicate a misconception, he explicitly addressed this terminology with them, hoping that he could understand their use of the term and, if there were a misconception, that the resulting discussion would help the students discover it. Although these actions were meant to probe the students’ understanding, they also served the instructional role of promoting reflection and focusing the students’ attention on relevant aspects of the task. In this sense, the interviewer’s questions were the same questions a constructivist teacher would ask. In fact, to support inquiry-based learning in which students construct personally meaningful concepts, after presenting appropriate tasks, a teacher’s role is threefold: (a) to constantly assess students’ thinking in order to properly guide their activity, (b) to promote student reflection and communication, and (c) (both in interactions with individual students and in moderating class discussions) to focus students’ attention on potentially productive lines of analysis.

Consistent with constructivist accounts of the learning process, the essential components of learning for HT and AN were reflection, cognitive conflict, and abstraction. Reflection and cognitive conflict were promoted by focusing students on predicting the number of cubes in three-dimensional arrays. Errors in predictions, which the students themselves discovered, caused cognitive conflicts or perturbations in the students’ current mental models for arrays. HT and AN attempted to resolve these conflicts by reflecting on the strategies they were using, all the while examining and restructuring their mental models of the arrays. In fact, they moved from an incorrect conception of the arrays, to a period of confusion in which they vacillated between different conceptions, to a viable conception that resolved their confusion.

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