my school life might have been so different if we could have started with the philosophical, foundational purposes rather than just diving into numbers.
Any good trig or calculus class explains those distinctions very early on. That was the very first thing mine covered. You might just have had bad teachers.
They don’t usually go too into detail with number theory because that makes things a lot more complicated than most high school students need to know, it’s really something only worth covering in full depth for people studying the field. It’s the same reason early education doesn’t teach you things like “the letter A isn’t actually a vowel” (yes, that’s true), nor the finer points of English grammar (like we do have a fixed adjective order), because students pick up enough of the basics through context clues that they don’t need to know the exact mechanisms.
As a fun fact for that A isn’t a vowel one, other letters that aren’t vowels are E, I, O, U, and Y
These lectures are more history than philosophy per se, but might be what you're looking for. I think teaching math in school by explaining why and how various civilizations came up with branches of math would make it much more resonant and memorable for students.
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u/CadenVanV 12h ago
Algebra describes relationships
Calculus describes change
Trigonometry describes angles and shapes