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Learner Generated Examples#

See also: teaching-mathematics, some connections with productive-failure

Watson and Shipman (2008)

...learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces.....experience can be oragnised in such a way that shifts of understanding take place as a result of learners' own actions.

Generating personal examples is encouraged by writers of mathematical texts

Example is an instance, illustration, case or element of a mathematical idea, object, process or class.

Perceived obstacles#

  • learner paradox - how can learners construct objects without knowing what to construct?

    However, a basic practice in mathematics is construct unfamiliar objects from what is familiar. Ways to do this include - exercising variation beyond perceived limitations - identifying underlying structures of mathematical representations and re-combining them

    Using practices such as abduction, conjecture without meaning, structure and relationships

  • needing a shift to higher mental functioning. Simply generating and comparing examples are not sufficient for conceptualisation.

    Some of this has to do with going with or across-the-grain. Going with the grain leads to simplistic/missing of underlying structures. But across the grain reveals the internal structure. Which in turn is related to the idea that this work doesn't occur without a level of scaffolding.

  • only for advanced learners

Which leads to three questions - Is it possible for learners to create examples of classes of object they have not met before? - Is it possible for non-advanced students to learn about mathematical objects and relationships by generating their own examples? - If so, what sort of conditions might contribute to learning?

To do#

Read

Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183--194. https://doi.org/10.1007/s10649-008-9143-3

Adler, J., & Pournara, C. (2019). Exemplifying with Variation and Its Development in Mathematics Teacher Education. In International Handbook of Mathematics Teacher Education: Volume 1 (pp. 329--353). Brill. https://doi.org/10.1163/9789004418875_013

References#

Dinkelman, M. O. (2013). Using Learner-Generated Examples to Support Student Understanding of Functions.

Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: learners generating examples. Mahwah: Lawrence Erlbaum Associates.

Watson, A., & Shipman, S. (2008). Using learner generated examples to introduce new concepts. Educational Studies in Mathematics, 69(2), 97--109. https://doi.org/10.1007/s10649-008-9142-4