The concepts of "arbitrary" and "necessary" are central to understanding how students engage with mathematics. Hewitt discusses how certain mathematical concepts can seem arbitrary to students if they do not see the inherent necessity behind them.
For my lesson, instead of presenting mathematics as a set of rules to be memorized, I would encourage exploration and discovery. This might involve presenting students with a problem before teaching them the method, so that they can see for themselves the necessity of the mathematical concept. This is also what one of my SAs does in her class. I really like this idea.
The language I use can make a concept arbitrary or necessary. I would be mindful of how I introduce terms and symbols, making sure they have a clear purpose and are connected to students' prior knowledge. This is also important for a class with a large number of ELL students.
I would encourage students to discuss their understanding of why certain procedures are used in mathematics. This might help them move from seeing concepts as arbitrary to seeing them as necessary.
Thanks for this interesting and thoughtful reflection, Sally. It's interesting to think about the ways that necessary concepts might appear to be arbitrary to students if they're not given a chance to develop concepts for themselves!
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