11–20 of 112 posts
Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science. Vested interests dominate every nook and cranny of our society, even the society of mathematicians. Truly we live in a decadent age. With this much fuel piled up, who w…
I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physics) classes would leave me with very uneasy feeling that something crucial is left unmentioned, something important is swept under the rug and something important is hidden for whatever reason. Turns out I was wanting for proofs, but couldn't articulate it - I…
Your high school didn't teach geometry?
Earlier quoted context omitted.
Your high school didn't teach geometry?
Geometry proofs aren't real proofs.
Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.
Interestingly, children as young as 5 show an aptitude for understanding overarching concepts of calculus.[1] This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children…
And I love that article. She really captures the damage that my early math education did (which I've been working the last year to overcome).
"Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon... or memorizing multiplication tables as individual facts rather than patterns."
Earlier quoted context omitted.
Your high school didn't teach geometry?
Geometry proofs aren't real proofs.
Earlier quoted context omitted.
I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a…
Earlier quoted context omitted.
I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a…
Earlier quoted context omitted.
Your high school didn't teach geometry?
Geometry proofs aren't real proofs.
Earlier quoted context omitted.
Geometry proofs aren't real proofs.
Why do you believe this? I think they are. They provide a nice example of using logic and axioms. I don't think one will be able to have a proof based course other than geometry before calculus. Students just aren't mathematically mature enough for that.
Whereas in calculus and algebra and analysis and number theory, the proofs often had different paths to prove them or different constructions/descriptions and the things we proved felt substantial.
Even worse is when you have all the proof based and concept oriented course and are tested on trickeries on exams.