This blog is the fourth in a series of 5. Let's introduce examples and non-examples before we head over to the fifth blog. You can access the first blog here, the second here, third part here, the fourth here.
Providing an example while exploring a concept isn’t out of the ordinary. However, do these examples actually do the job we intended? Can our examples create incidental gaps for understanding to fall through?
In life and in maths, there can be a fine line between something being correct and not quite right. Exploring examples and non-examples provides the when, where, what and why an example works - or the when, where, what and why it doesn’t. When carefully chosen, examples and non-examples highlight relevant and irrelevant characteristics, unpack common mistakes and misconceptions, or reveal variations. For example, think about how a square can be a rectangle but a rectangle cannot be a square. How does a rhombus, a parallelogram, a trapezoid or a quadrilateral fit into this rule?
The distinction between examples and non-examples may seem small when a concept is already understood. But, it can feel more like a chasm when students are still developing boundaries around the concept. In fact, boundary examples are commonly discussed alongside examples and non-examples! Here lies an opportunity to engage in enriching dialogue that moves beyond surface-level understanding, especially when complexities grow, difficulty increases or an unfamiliar perspective is added.
For example, you can help explore solving simultaneous equations by getting students to investigate possible solutions graphically. There are three classes of problems you can surface with students. Straight lines that intersect at one point, straight lines that are parallel, and therefore have no solutions. The third type is when straight lines might look algebraically different, but differ by a factor. This means they will intersect at all points and have infinite solutions.
In this research paper, Bills, Dreyfus, Mason, Tsamir, Watson and Zaslavsky (2006) state “the nature and sequence of examples, non-examples and counterexamples has a critical influence on what opportunities learners are afforded” and, more importantly, how students are interacting within these examples.
“Examples play a crucial role in learning about mathematical concepts, techniques, reasoning, and in the development of mathematical competence. However, learners may not perceive and use examples in the ways intended...especially if underlying generalities and reasoning are not made explicit.”
These examples from Connie Malamed provide some ideas to keep in mind when constructing examples and non-examples, and in this blog post a teacher reflects on their own interesting experience exploring examples and non-examples in. If you’re looking for some more professional reading, these summaries are a great place to start.
After teaching or exploring a concept, have you noticed a particular misconception born from the unclear boundaries or characteristics of an example?
Is there an example where a non-example would create more confusion?
Keep an eye out next fortnight for the final piece of this Small-Group Pedagogy Mini-Series. There will also be a handy video that applies all the explored strategies to a real mini-lesson, including think alouds, checking for understanding and active participation.
Bill, L., Dreyfus, T., Mason, J., Tsamir, P., Watson A. & Zaslavsky, O. (2006). Exemplification in Mathematics Education. In J. Novotna (Ed.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: PME. Retrieved 7th January 2021. from http://mrbartonmaths.com/resourcesnew/8.%20Research/Examples/Exemplification%20in%20Mathematics%20Education.pdf