I think what I am inclined to do is establishing the notion of convention in classroom, so whenever I inform the arbitrary, students know that what I'm telling them there are conventions in mathematics and they are expected to know it, remember it, and use it. Similarly when teaching something necessary, I would like to prompt them that those things are deducible. However I don't think it is realistic to not "give" students any necessary and have them com to know it by themselves.
I feel like I just wrote something really boring above. I think I totally agree that the awareness of what is arbitrary and what is necessary is very important for both teachers and students. But I am not sure how can I encourage that awareness effectively. Maybe incorporating some visual aid? (e.g. box all the arbitrary using red throughout the materials or in-class writing)
I think everything just comes down to how can we help students to understand the math. Since even when students know the presence of necessity in what we teach, they often end-up thinking "but memorising is just easier for me."
You’re really thinking about how to make the arbitrary/necessary distinction visible for students, and I like the honesty in how you reflect on what feels realistic in a classroom. Your idea about marking conventions visually shows good instinct for practical tools. I’d love to see you anchor this in one small moment from your own teaching where you noticed students leaning on memorizing.
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