[pointedset]

I think your quote from Gallian doesn't address syzarian's example, actually.

The two properties you quoted are about the fact the distributivity works whether you're multiplying on the left or on the right. That's one possible left-right confusion, but I would argue most weak students believe it holds even when it doesn't, so in a sense they're too permissive in their reasoning.

However, syzarian's example is about a different left-right confusion: whether you can read an equality both forwards and backwards. I've seen this confusion in students: they will readily believe that you can distribute a factor over a sum (going forward), but be very skeptical about the act of factoring out (going backward), even though it's justified by the same equation. In this case, the students aren't permissive enough in their reasoning.

This is the kind of misconception (that equality has a "direction") that's much easier to suss out with an in-person interaction, whether it's with a teacher or other students.

[rrrrrrrrrrrr2]

Oh. Wow, you are right. I honestly just read the definition in Gallian wrong, and then, even as I was typing it, I swear I thought it said "ba + bc = b(a + c)". What a crazy experience of confirmation bias by me.

I obviously agree that a human is extremely effective at explaining that "equals" is symmetric and not the same as "implies". I'm just arguing that the task also can be done in prose to similar, if inferior, effect.

I did agree however that because math has so many gotchas like this, then no textbook will ever disclaim all potential sources of confusion like the example by GGP. And so our discussion boils down to the tradeoff between the price and time investment of taking a class vs. the increased difficulty of self-studying. And how different people assign value differently to each side of the tradeoff, which I claim is the root of our disagreement.