I deduce from your post that you have very little experience in teaching people at the level of beginning algebra. And while one might know geometrically what a parabola is there is a lot one must know before dealing with parabolas algebraically. I suggest that these things appear easy and obvious because you already know them and that you no longer remember what is hard for people learning this stuff for the first time.
I don’t show conic sections in elementary algebra. One typically really mentions the phrase “conic section” is pre-calculus which is 3 courses after elementary algebra. Over the past few centuries the order in which concepts are introduced has been developed. It’s not perfect but one should not be so quick to discount the way things are done without knowledge/experience in presenting these ideas to beginning students.
You've got me. I wish I could delete the comment, or at least edit it.
It's a bit worse than just having no experience teaching algebra, I
didn't have the same experience as most kids trying to learn it. I was
kind of a freak. I was the weird quiet kid in the back of the class who
always knew the answer to every question. (Other kids tended to not like
that, but I'm also very disarming (in person) and so I did alright.) One
year, I was misplaced into a basic geometry class, and the teacher very
kindly let me pick out some calculus textbooks and sit in the back of the
class teaching myself calculus. (Some bureaucratic reason for why I
couldn't transfer, or the calc classes were full, or we didn't have any,
or something, I forget.) Learning math for me feels like remembering
things I always knew.
So, yeah, maybe I should keep my mouth shut when it comes to teaching normal
people how to do math.
Or maybe we should try to figure out what
my brain is doing and how to teach people to do that too?
Maybe I have a
normal brain and I'm just using it differently than most people?
I like that idea better, because then, instead of a freak, I'm a
front-runner. And, in addition, there's hope for a great improvement in
didactic technique and humanity's general "numeracy" level, eh? If we could teach people to math like i do we could compress basic math education (up to calculus) into just a year or so.
"Something self-referential and hopefully unsnarky about how this is a
thread on a how-to-math-better article." ~me, failing at rhetoric.
And for kicks, here's Iconic Math: http://iconicmath.com/ (No
affiliation with me. I'm just trying to end on an upbeat, constructive note.)
I think a big part is that many (most?) children find school – i.e. lectures, textbooks, homework exercises, exams – unmotivating/boring at best, and often extremely stressful/frightening, which means that they aren’t fully focused on it and can easily miss important details. Beyond that, schools often fail to provide meaningful feedback or support when people suffer serious misconceptions or are missing fundamental prerequisite knowledge/skills, which makes it easy for students to fall behind and have great difficulty recovering. Someone who spends the exact same amount of time on academic work but for whatever reason (external help outside school, internal motivation, some insightful introspection, ...) manages to pay closer attention, stay more relaxed, think about things ahead of where the class is expected to be, connect new learning to material learned before and build a better-connected mental map, etc. can end up pulling far ahead.
A whole lot of this has to do with level of preparation before ever arriving at school. Some kids read with their parents for hours every day from age 1–5+, learn to play a variety of strategic games (and games involving basic arithmetic practice), build structures or mechanisms or electronics, practice making art, work through books of logic puzzles, etc. Other kids are left alone and bored without learning materials at a reasonable level, plonked down in front of developmentally inappropriate or just badly produced TV, or handed over to unthoughtful video games.
Then consider how many kids and parents have serious problems at home, with confrontational or even abusive relationships. Pile on work stress, financial stress, poor diet or even hunger, poor sleep, environmental toxins, illness, etc.
I don’t show conic sections in elementary algebra. One typically really mentions the phrase “conic section” is pre-calculus which is 3 courses after elementary algebra. Over the past few centuries the order in which concepts are introduced has been developed. It’s not perfect but one should not be so quick to discount the way things are done without knowledge/experience in presenting these ideas to beginning students.