By Sociedade Portuguesa de Matemática

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**Sample text**

T: Could we find a shortcut? Luke suggests reversing the position of a and d, and placing minus signs in front of b and c. T elicits symbolic representation and writes on whiteboard. T: How could we verify this? Ss suggest doing another one. T provides another example; asks students to use ‘Luke’s conjecture’ to write down the hypothetical inverse and check via matrix multiplication. Ss do so; they are convinced the method works. Inverse of ¢a b is ¢ c d d Ϫb ? ¢ ¢ Ϫc a 2 1 inverse 1 Ϫ1 ¢ ¢ ¢ ¢ 1 1 Ϫ1 2 Mathematics-Ch1-9-PAGES 28/6/07 3:56 PM Page 41 Developing mathematical understanding Annotation Interaction Whiteboard Scaffolds Ss’ thinking T gives another example for Ss to try.

Are you getting anywhere with it? Do you really understand what the problem is about? Can you justify that step? Are you convinced that bit is correct? Can you find a counter-example? Have you answered the problem? Have you considered all the cases? Have you checked your solution? Does it look reasonable? Is there another solution? Could you explain your answer to the class? Is there another way to solve the problem? Could you generalise the problem? Can you extend the problem to cover different situations?

In mathematics education, constructivism ‘attends to how actions, observations, patterns and informal experiences can be transformed into stronger and more predictive Mathematics-Ch1-9-PAGES 28/6/07 3:56 PM Page 29 Developing mathematical understanding 29 explanatory ideas through encounters with challenging tasks’ (Confrey & Kazak, 2006, p. 316). In such encounters, cognitive change begins when students experience conflict with their previous ways of knowing and take action to resolve this perturbation.