61–70 of 124 posts
Earlier quoted context omitted.
This is the same progression that I learned, but I am not sure it is the best one. I wonder if there has been any rigorous scientific study on the pedagogical benefits one way or another?
Yeah why don’t we do more a/b testing or controlled testing in education?
Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).
That's the historical progression, not the teaching progression. That's the point. Integrals were conceived over 2000 years ago. Derivatives came Newton-ish. Formalism with limits came last.
Infinitesimal calculus was discarded in late 19th century pedagogy in favor of limits as no rigorous proofs yet existed of its general correctness, whereas such proofs had been formulated for the limits version. It was suspected that while infinitesimals were a nice intuition, they lacked rigor which might lead to unknown errors.
It was eventually rigorously proven in the 1960s that the infinitesimals approach is indeed as complete and correct as the limits version, but by then standard pedagogy had become entrenched in limits.
More on this at https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini... , the modern textbook that re-introduced infinitesimals.
Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).
Integral calculus before differential calculus is how it is done in Apostol's "Calculus". Here are the first several chapter titles to give an idea of what is covered when. Introduction The Concepts of Integral Calculus Some Applications of Integration Continuous Functions Differential Calculus The Relation Between Integration and Differentiation The Logarithm, The Exponential, and The Inverse Trigonometric Functions…
Is this Three Blue One Brown?
I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever mathematics is.
Mythology > MathematicsLet me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…
Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…
literally what they taught me
Heh, ditto. I don't think I got any new info that I didn't get in high school calculus, but it is a well done review of things I hadn't seen in a couple of decades
We spent about 15 minutes (well, ok, one lecture) on series, then a couple weeks on limits, then limits and differential calculus, then the rest of differential calc. (throwing the limit stuff away), then limits and integral calc., and finally the rest of calculus.
Limits are a horrible way to learn calculus if you're trying to understand it at the same time as doing the algebra. :-)
Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…
In my experience, almost everyone out of college who I've met who claims to know calculus.
Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…
I do calculations on Maxwells equations all the time, that's s calculus. Doing CAD, or any finite element calculations - > calculus. Modern engineering would not be possible without calculus. Just because software engineers mostly use linear algebra, doesn't mean that most other technical disciplines are the same.
Was not my question. Nearly everyone uses computers. Computers would not be possible without electronics. How many computer users work with Maxwell's Equations or have even heard of Kirchhoff's circuit laws? A very tiny percentage, undoubtedly.
The reason I asked was the Penn State professor's epiphany that no engineer among the many he asked actually used calculus.