I was so distracted by never having heard the term “point reflections” before that I forgot to explain in my last post why I think it’s a bad term, and why in particular I think it’s a bad term for middle schoolers.
It’s very common in math that we start out with a very simple concept (like a line in 2-dimensions) and generalize the hell out of it (line in n-dimensions, hyperbolic line, geodesic, affine line over a finite field, projective line, complex line, etc.) in ways that middle schoolers would find confusing. We then end up using the word “line” differently in different subfields of math, and that’s ok. This generalize and rename thing happens again and again at higher levels of math (say integer/algebraic integer vs. rational integer/integer), and this is also fine.
What’s unusual and pernicious is to first introduce a concept in weird generality. Yes it’s great to think about complex reflections, “reflecting about a circle”, affine reflections, etc. when the time is right. But it’s also important to have an intuition for what you’re generalizing, and in middle school that should be reflections about hyper-planes in 1, 2, and 3 dimensions, with almost all attention paid to 2-dimensions. Why? Because reflections in lines in 2-space as a concept is much richer than all order 2 affine transformations. By richer here I mean it makes it easier to think about interesting things. How do you tell if a transformation is a reflection? Thinking about this suggests handedness (intuitive because of mirrors) which is a deep and important property which is accessible to middle schoolers. Thinking about this also subtly primes you for thinking about cosets and their relation to group actions (not so accessible to middle schoolers, but the priming is still there). The fact that every tranformation can be written as a translation and then a rotation or a reflection and then a rotation is a really cool result that introduces key ideas which you could easily think about in early highschool (and again suggests more advanced topics like groupoids). Etc.
Conversely there doesn’t seem to be nearly as much interesting math that’s going to come from thinking of point reflections as a reflection. I could see some of the other order 2 elements leading to interesting questions (in particular “reflecting about a circle” is potentially interesting), but point reflections seem to be a waste.