How Can Average-Ability Students Be Expected to Invent Significant Mathematical Ideas?
The answer to this question stems from the following observations: (a) students do not begin from a state of having no knowledge about the relevant processes and understandings; and (b) model-eliciting activities are fashioned to facilitate certain types of inquiry and development without telling students what to do and how to think. In the middle school curriculum, most of the ideas that are taught have been “covered” during earlier grades, and most of them will be “covered” again in following courses. Few ideas are introduced completely fresh for the first time, and few are introduced for the last time. Furthermore, before students begin to work on a given topic, they usually work on topics that are considered to be prerequisites. So, the next ideas that they are expected to learn are seldom more than extensions or refinements of ideas that have been introduced already. In particular, they usually possess the elements of a language and powerful graphic and symbolic notation systems t
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