Interesting, I'm a phd-drop out in computational biology, working as a data science consultant: I use mathematics including multi-dimensional statistics, linear algebra, and calculus everyday. Being self-taught, I'm very self conscience about the math I don't know, but so far, not knowing differential equations doesn't seem to have hurt me. I actually just ran into a problem that uses Hamiltonian dynamics, so maybe I will end up learning differential equations, but it does seem like the course, as taught many places and in the Dover books I own, presents either no useful conceptual insight, like why I learned geometric algebra, or a powerful toolset, like some multi-dimensional statistics.
Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
31–40 of 112 posts
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#32Please 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.
Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later.
One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems.
Maybe Sheldon Axler will do Calculus Done Right.
I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#33Please 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.
Calculus was a lot easier if you paid attention in trigonometry, for me anyway
Of course one of the main texts i learned diff eq from was the Mary Boas book i think she mentioned in that rant, so what do i know? :)
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#34Earlier quoted context omitted.
Your high school didn't teach geometry?
My kids are in high school right now. I was getting my daughter psyched up for geometry, by promising her that she'd get to do proofs. The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class. I was saddened for many rea…
Same deal with my kids in high school, proofs were barely covered.
When I took Geometry in high school about 45 years ago, proofs were essential, and IIRC they dominated the curriculum. And that was good, because it taught logical thinking. Geometry was my favorite course in high school.
This background was incredibly useful to me in both hardware design and in computer programming.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#35Earlier quoted context omitted.
My kids are in high school right now. I was getting my daughter psyched up for geometry, by promising her that she'd get to do proofs. The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class. I was saddened for many rea…
Same deal with my kids in high school, proofs were barely covered. When I took Geometry in high school about 45 years ago, proofs were essential, and IIRC they dominated the curriculum. And that was good, because it taught logical thinking. Geometry was my favorite course in high school. This background was incredibly useful to me in both hardware design and in computer programming.
This was also my experience about 25 years ago. Many classmates were not fond of proofs, so perhaps the move away from them is just giving the customers what they want?
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#36Earlier quoted context omitted.
Your high school didn't teach geometry?
My kids are in high school right now. I was getting my daughter psyched up for geometry, by promising her that she'd get to do proofs. The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class. I was saddened for many rea…
I hated proofs in geometry, which explains why I got A's in all my college math classes except linear algebra, specifically because of those damn proofs.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#37Please 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.
Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick ( EE 120 Linear Systems ) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to l…
I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube.
We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#38Please 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.
Let’s take that up a notch - try both and actually measure the results. It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? So many people (myself included) have stories of, if only I had been exposed to such and such concept in a different way it would have had a much bigger impact. Why not measure multiple aspects? Efficiency of le…
Would the typical ed-school IRB have a problem with this sort of experiment?
Admittedly that is an indictment more of the IRB than of the experiment...
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#39Earlier quoted context omitted.
Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick ( EE 120 Linear Systems ) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to l…
I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube. We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#40Due to an offhand comment from a friend, I started my kid on continuous math, not discrete like they do in school. Went great, until about grade 4 when they actually started to care whether he was aligned with his classmates. Germans are actually pretty intolerant of non-professional pedagogues teaching kids.
But it greatly improved his grasp of elementary math as it matched what he saw in the real world.