Teaching Pythagoras Theorem in China – a case study

How a Chinese teacher improved classroom teaching in Teaching Research Group: a case study on Pythagoras theorem teaching in Shanghai by Yudong Yang published by ZDM Mathematics Education (2009) 41:279–296 DOI 10.1007/s11858-009-0171-y.

In China, a school-based teaching research system was built since 1952 and Teaching Research Group (TRG) exists in every school. In the paper, a teacher’s three lessons and the changes in each lesson were described, which might show a track of how lessons were continuously developed in TRG. The Mathematical Tasks Framework, The Task Analysis Guide, and Factors Associated with the Maintenance and the Decline of High-level Cognitive Demands developed in the Quantitative Under- standing: Amplifying Student Achievement and Reasoning project, were employed in this study. Based on the perspective of Mathematical Task Analysis, changes of three lessons were described and the author provided a snapshot for understanding how a Chinese teacher gradually improved his/her lessons in TRG activities.

Abstraction and Algebra

Just how essential abstraction and algebra are? Here’s what Caleb Gattegno says on the importance of abstraction and algebra: 

Nobody has ever been able to reach the concrete. The concrete is so ‘abstract’ that nobody can reach it. We can only function because of abstraction. Abstraction makes life easy, makes it possible. Words, language have been created by man, so that it does not matter what any reader  evokes in his mind when he reads the word red, so long as when we are confronted with a situation, we are using the same word even for different impressions. Language is conveniently vague so that the word car, for example, could cover all cars, not just one. So anyone who has learn to speak,  demonstrate that he can use classes. There are no words without concepts. …

Therefore how can we deny that children are already masters of abstraction, especially the algebra of classes, as soon as they use concepts, as soon as they use language, and that they of course bring this mastery and the algebra of classes with them when they come to school. […]

… The essential point is this: that algebra is an attribute, an essential power, of the mind. Not just of mathematics only. Without algebra we would be dead, or if we have survived so far, it is partly thanks to algebra – to our understanding of classes, transformations, and the rest …

(Gattegno, 1970, pp 23-5)

I think the ‘algebra of classes’ is a bit of a stretch but I completely agree that our young learners are very much capable of making abstraction, and yes, to engage in algebraic thinking.

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