Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
1–10 of 112 posts
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#2Please 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.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#3Please 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.
I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#4Please 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.
I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
Especially that darn three-body problem. (Not poking fun, just agreeing, it's genuinely tough.)
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#5Please 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 can handle the rigor. For now, let them play with it. That makes math a lot more fun and less draining, esoteric, impenetrable.
[1] https://www.theatlantic.com/education/archive/2014/03/5-year...
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#6I 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 just sensed that something was off :)
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#7I 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?
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#8Why 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 will be surprised by the conflagration?
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#9Please 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.
I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#10Earlier quoted context omitted.
I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
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 solved problem, differentiating even a reasonably smooth function numerically is much harder. If there is noise in your function, it doesn't matter so much for integrating, but can completely break the derivative.