ORIGINAL ARTICLE Qualities of examples in learning and teaching Anne Watson • Helen Chick Accepted: 5 December 2010 / Published online: 16 December 2010 Ó FIZ Karlsruhe 2010 Abstract In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theo- rising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of ‘‘didactic object’’ (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Nether- lands, pp 191–212, 2002), we talk about ‘‘didacticising’’ an example and observe that the nature of students’ engage- ment is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root ‘‘didact’’. We use the verb ‘‘didacticise’’ but without implying any connection to particular theoretical approa- ches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities. Keywords Examples  Didactic object  Generalisation  Learning from examples 1 Examples in learning and teaching 1.1 The relations between examples and mathematics for learners In her seminal paper, Rissland Michener (1978) examined the role played by examples in understanding mathematics. She described examples as ‘‘illustrative material’’ (p. 362) and highlighted an important dual relation: that examples can be constructed from results and concepts, and in turn examples can motivate concepts and results. Borrowing from Freudenthal’s definition of models-of and models-for (Freudenthal, 1975; cited in van den Heuvel-Panhuizen, 2003), we might view the nature of examples in Rissland Michener’s dual relationship as examples-of—in which the examples are specific instantiations of a previously defined class—and examples-for—in which the examples are the genesis for identifying an as-yet-uncharacterised class. Rissland Michener delineated different roles that examples can play in understanding mathematics. Start-up examples motivate definitions and build a sense of what is going on; reference examples are ‘‘standard cases’’ that link concepts and results, and are returned to again and again; model examples indicate generic cases and can be copied or used to generate specific instances; and, finally, counter-examples sharpen distinctions between, and definitions of, concepts. Lakatos goes further and suggests that counter-examples have historically generated inquiry into new classes of objects (1976), while Goldenberg and Mason (2008) high- light that the difference between example and counter- example depends on one’s attention or emphasis. If attention and emphasis are relevant, then whether an example is ‘‘of’’ some class or actions that are already familiar or ‘‘for’’ the construction of something new depends on the person undertaking the mathematical activity. Thus, a A. Watson (&) Department of Education, University of Oxford, 15 Norham Gardens, Oxford OX2 6PY, UK e-mail: Anne.watson@education.ox.ac.uk H. Chick Melbourne Graduate School of Education, University of Melbourne, Melbourne, Australia 123 ZDM Mathematics Education (2011) 43:283–294 DOI 10.1007/s11858-010-0301-6