This module comprise two LessonSketch experiences, of the similar structure. One deals with Universal statement in Geometry, and the second one deals with an Existential statement in Algebra.

LessonSketch experiences:*What can you infer from this example? - Geometry*
This experience focuses on the role of supportive examples and counterexamples in disproving a universal statement in Geometry.*What can you infer from this example? - Algebra*
This experience focuses on the role of supportive examples and non-confirming examples in proving an existential statement in Algebra. | Teachers' guide:- These two experiences, combined, cover the two possible types of quantified statements: universal (for all) and existential (there exists).
- The experiences highlight the different roles examples play in proving or disproving these two types of statements. Specifically:
- One counterexample is sufficient for proving a universal statement;
- Supportive examples are insufficient for proving a universal statement;
- One supportive example proves an existential statement;
- Non-confirming examples do not disprove an existential statement.
- The summary of these logical aspects can be found in the REP framework (Role of Examples in Proving).
Implementation suggestions: - We recommend doing WCYIFTE - Geometry experience first, since it deals with universal statements, with which PSTs would be more familiar with.
- Follow-up with WCYIFTE - Algebra experience, which deals with existential statements.
- Doing the two experiences in close proximity provides an opportunity to highlight the similarities and differences in the roles of examples in proving/disproving universal and existential statements.
| Other resources:- Buchbinder, O., Ron, G., Zodik, I. & Cook, A.
(2016). What can you infer from this example? Applications of on-line,
rich-media task for enhancing pre-service teachers’ knowledge of the roles of
examples in proving. In A. Leung and J. Bolite-Frant (Eds.),
*Digital Technologies in Designing Mathematics Education Tasks – Potential and Pitfalls*. (pp. 215-235). Springer, Cham.
- Buchbinder, O., & Cook, A. (2018). Examining the mathematical knowledge for teaching of proving in scenarios written by pre-service teachers. In O. Buchbinder & S. Kuntze (Eds.).
*Mathematics Teachers Engaging with Representations of Practice*(pp. 131-154). Springer, Cham.
This chapter analyzes the scenarios written by a group of elementary and secondary PSTs following the enactment of the module in a content course on reasoning and proof. - Buchbinder, O. & Zaslavsky, O. (2009). A
framework for understanding the status of examples in establishing the validity
of mathematical statements. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, C.
(Eds.).
*Proceedings of the 33*. (Vol. 2, pp. 225-232). Thessaloniki, Greece.^{rd}Conference of the International Group for the Psychology of Mathematics Education
This paper outlines the REP (Roles of Examples in Proving) framework. |