I took two courses in ODE. I got an A in the first semester. I still have no idea what they are or what's going on. It's all just relationships and patterns to me--but with no intuition or understanding behind them. It never "sunk" in like Calculus did where "Area under the curve" and "tangent line" are super obvious in retrospect and immediately applicable to your daily life. "What's velocity? The change in distance over a unit of time." Done. Presto.
Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
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Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#102I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were stu…
>The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure. This is a prime example of what I believe is wrong with traditional education (i'm not judging your teaching ability o…
I'm in the same boat as you. 25 and starting my career, severely limited by my poor performance in school.
Sundays are study days from now on. Hopefully I can catch up.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#103Earlier quoted context omitted.
>The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure. This is a prime example of what I believe is wrong with traditional education (i'm not judging your teaching ability o…
I'm in the same boat as you. 25 and starting my career, severely limited by my poor performance in school. Sundays are study days from now on. Hopefully I can catch up.
I think it's widely accepted that if you are in tech, your success is far more dependent on your actual ability, and schools haven't ever exactly been able to help with tech careers so far.
It's only University level certifications that make any difference for your prospective employers, but that does not necessarily guarantee the skills you need as an engineer (if that's what you are aiming for), and many businesses recognise this.
I got into full time programming job at around your age after a variety of education routes in sciences and arts (no comp-sci) and variety of crappy jobs.
No one seemed to be particularly interested in my lack of certifications when it came to interview time (even though they all have them on the job app). Once I could talk shop and show some small pieces of work and my potential could be recognised I was hireable.
I'm not saying it's ever easy to get a foot in the door, but if you focus on the work, your ability will show through in the interview process (if they are not simply a large corporation hiring through box ticking criteria, probably a bad place to start out when you are a junior anyway).
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#104Earlier quoted context omitted.
>It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? Yes and...of course it is done and tried. The problem is that most parents (especially those whose kids are most likely not to have the support to get past these struggles) are not exactly ecstatic for their children to be used as experiments. They want concrete answers, fixes, sol…
That’s helpful background, thank you. A lot of it makes sense but, hear me out, I don’t believe you’re point on difficulty or definitions has much to do with the the most frustrating parts you mention. Research is always difficult, in any field. You don’t get a PhD for courses, when you’re expected to have insights that no one ever has had before, that’s just a hard thing. Same with designing research. It’s so diffic…
All excellent points - I would opine though that education has a particular difficulty in that unless one trusts a standardized test every year, the experiment needs to run for a very long time by the standards of human trials.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#105Earlier quoted context omitted.
I'm in the same boat as you. 25 and starting my career, severely limited by my poor performance in school. Sundays are study days from now on. Hopefully I can catch up.
I think it's widely accepted that if you are in tech, your success is far more dependent on your actual ability, and schools haven't ever exactly been able to help with tech careers so far. It's only University level certifications that make any difference for your prospective employers, but that does not necessarily guarantee the skills you need as an engineer (if that's what you are aiming for), and many businesses…
[Edit] I'm presuming your path is programming in some form... of course there are other areas of tech where you can't really get away from formal education and certification, e.g no company is going to let you near their ASIC design until you have some sort of qualification in electrical engineering + basic comp sci. If you want to just to coding generally however then my previous post applies.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#106Please 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.
Most students will benefit from already understanding calculus, when they are learning trigonometry.
The argument cuts both ways: not only can one strongly benefit from a basic understanding of trigonometry when learning calculus (how the derivatives of sin/cos/etc all related to one another) - it really isn't possible to get far with basic integration techniques without learning the fundamental trig-based substitutions. And ultimately you can use any set of subjects, in just about any order. What matters is the thought processes, and that what you're really trying to teach is not a set of facts or techniques - but the underlying mathematical essence of these techniques.
So at the end of the day, what it really comes to is: "It doesn't matter so much what you teach, but how you teach".
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#107Please 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…
Calculus. Surely any child who has ever ran, or ever seen a hill, knows about velocity and slopes - concrete instances of the abstract concept.
But I would say that that's not math, any more than speaking means you know grammar, or digesting means you know biochemistry. Math is formalization.
For calculus, it's slopes at a point without dividing by zero (using the cheat of limits).
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#108Earlier quoted context omitted.
More like `from a computational point of view.' Analytically, integration is much more difficult and richer than differentiation!
Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that. Note the key word there is a function not a function with a closed form that's a tiny subset.
"Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that."
The OP said the opposite, that differentiation is harder 'more finicky.' I agree that the concept of integration is much richer.
Also, I didn't mean 'closed form solution' when I said 'analytic.' I also didn't mean 'analytic functions.' I meant that the analytic machinery you have to develop in order to have a theory of integration is far richer than for differentiation - i.e, proving the multivariate change of variable theorem.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#109Earlier quoted context omitted.
Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that. Note the key word there is a function not a function with a closed form that's a tiny subset.
"Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that." The OP said the opposite, that differentiation is harder 'more finicky.' I agree that the concept of integration is much richer. Also, I didn't mean 'closed form solution' when I said 'analytic.' I also didn't mean 'analytic functions.' I meant that the analytic machinery you have to develop in order to ha…
Thanks for clarifying what you meant by 'analytic'.
To me, 'harder, more finicky' means exactly that it is of a more constrained scope, so I don't think I interpreted OP wrong.
Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]
#110Interesting, 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…
So how do you reason with abundance of molecules? mRNA and protein can be modelled from simple linear differential equation