I freely admit that my viewpoint is clouded by fluency, but I didn't come up with this today. This is how I've done it since I was still in school and I remember being annoyed by the opaque "just memorize this formula" approach from early on, where the concepts just seemed much clearer. I admit to not being the typical math student. But I think my approach was my competitive advantage, not something that makes my experience inapplicable.

To be clear, you're saying that if you tell a student

    y = x^2 + b*x + c

and you tell them

    x = z + 2

they aren't yet equipped to learn to combine those into

    y = (z+2)^2 + b*(z+2) + c.

Is that right? Why do you teach quadratic equations at this stage? I'd consider those to be much more advanced than simple "replace x with (z+2) everywhere you see it" plus some "alice's house to bob's house to carol's house" problems along a single axis for the concepts.

It seems like you're saying they're taught quadratic equations before they're equipped to poke around with them, which seems to be setting up for the black box/memorize-the-formula version of math.

I remember discovering this for myself when tutoring math in college. It's amazing how difficult it is to explain a lot of "basic" math without using "advanced" math. Trying to solve limits without L'Hospital's rule is excruciating.