Henri Poincaré once said: "Mathematics is the art of giving the same name to different things"
I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands.
Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about projectiles, you end up ignoring that quadratic equations are used... heavily... everywhere.
You may see them in chemistry, crypto or nuclear physics. But if your brain only remembers projectiles, it doesn't have the ability to abstract new problems the right way.
As a counterpoint to that, I flat-out despised abstract Math until I got to college precisely because I found it useless. At best I knew I was probably going into engineering in college and I'd been told "engineers use calculus". That was my sole motivation to learn anything about abstract math, and it was pretty bad. It didn't help that the "examples" given in class were mundane toy problems (ex: "how long will this arbitrary tank take to empty?" Well gee I don't know and I don't care. Is this the last tank of water on the island or something?)
Now physics on the other hand, chemistry, computer science, any time I could apply math to a real-world situation got me motivated. I liked Math that could solve actual existent problems, and I learned said math better as a result. The first time I learned about projectile motion I literally started picturing vectors everywhere for the rest of the day.
It also didn't stop me from learning abstraction later in college. In basic signal analysis we learned about the Fourier transform, and then we learned about the Laplace transform and how it was simply a more general Fourier transform (or the reverse if you prefer). Mastering Fourier first gave the Laplace a relatable mental context, it made the abstractions more real.
By contrast I had a Math professor once who tried to teach us theoretical induction and work his way down to examples. I had no idea what the fuck he was talking about until he hit the examples, then everything clicked all the way back up.
Maybe that's just how I'm wired, but I would argue for more and more relevant examples. There are tons of historical uses of math that people would find interesting. Talk about the math used to build the Hoover dam, or telegraphs, fly satellites, or make medicines, or decode Nazi transmissions. And talk about it in real terms with character, if you must simplify for the sake of time then say that's what you're doing. Make people realize they're learning something that has uses beyond Math class or the rarified ivory towers of Math PhDs. That by learning the material they're learning how to do meaningful things. The original, wildly popular SimCity games were basically raw spreadsheets and formulae, but people learned those spreadsheets and formulae because the games had character.
I know to a lot of Math people the abstractions are the "real" part, but most people I know simply don't process the world that way.
This is especially a problem in college. Few math profs seem to enjoy teaching and seem reluctant to ground their subject using concrete illustrations. Without these, there are no tangible points of reference, aside from derivations from known equations. But mere transformations rarely offer insight without adding some sort of context.
I think the crux of the problem in teaching math is the need to illustrate each concept from MULTIPLE perspectives: theoretical, tangible, incremental, graphical, dynamic, etc. It's only by connecting the concepts to manifestations that most of us appreciate the meaning of math. Few are satisfied by disembodied equations or proofs. Most of us need some form of grounding using what we already know to make sense of new concepts. Otherwise math is just castles in the air.
I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands.
Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about projectiles, you end up ignoring that quadratic equations are used... heavily... everywhere.
You may see them in chemistry, crypto or nuclear physics. But if your brain only remembers projectiles, it doesn't have the ability to abstract new problems the right way.