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Teaching and Learning by Example


"The mathematical problems, tasks, demonstrations, and exercises that teachers and students engage with in classrooms are, in general, specific instantiations of general principles. Indeed, the usual purpose of such examples is to illustrate those principles and thus facilitate their learning. With this in mind, it is clearly important for teachers to be able to choose or design suitable examples, to recognise what is offered (or afforded) by particular examples, and to know how to adapt an already existing example to better suit an intended purpose. Although writers of textbooks and other teaching resources also need these skills, it is ultimately the teacher who puts the examples to work in the classroom. Teachers' choice and use of examples is indicative of their pedagogical content knowledge (PCK)-- the complex amalgam of mathematical and pedagogical knowledge fundamental to teaching and learning--and reflects their understanding of the mathematics to be taught and how students can be helped to learn it. This paper examines some of the issues associated with example use and how it is informed by and can inform us about PCK."