21–30 of 102 posts
I always thought my calculus 1 & 2 courses focused too much on writing solutions without really understanding concepts. I was pretty lost in calculus 3 and 4.
Yes, it can. The biggest failure in math education is that we spend an inordinate amount of time solving equations without understanding how to apply them to solve real-world problems. Thanks to the calculator we can now spend less time figuring out how to solve x+1=7 and put that time on how to understand when the equation is needed in the real world or how to create an equation that will help solve a real-world pro…
I think a course like that would need to be called something like “Mathematics Appreciation,” in analogy to “Music Appreciation.” I would never hire someone whose only music education was the latter to actually play in a band or write a movie score, and I would never hire someone who didn’t know how to do differentiation or elementary integration by hand to do any work that required an understanding or use of calculu…
Music theory is NOT required to play in a band or write a movie score. Check Gustavo Santaolalla's for an example of a great composer with ZERO musical bg.
I think a course like that would need to be called something like “Mathematics Appreciation,” in analogy to “Music Appreciation.” I would never hire someone whose only music education was the latter to actually play in a band or write a movie score, and I would never hire someone who didn’t know how to do differentiation or elementary integration by hand to do any work that required an understanding or use of calculu…
This analogy doesn't make any sense. Music is not like math and the music industry is not like industries that are related to calculus. Music theory is NOT required to play in a band or write a movie score. Check Gustavo Santaolalla's for an example of a great composer with ZERO musical bg.
That's going to age beautifully
Differentiating by hand is not difficult at all, and graphing a curve and its derivative is a really good way to teach what a derivative actually is.
Integration is harder, obviously, and it is a pain point for sure. Even the idea that an integral is the area under the curve is clearly not trivial to students.
But I don’t see a clear benefit trying to avoid integration. Integration is more difficult, so it should be pointed out as such, but then it’s still really useful to know how to do it…
Earlier quoted context omitted.
This analogy doesn't make any sense. Music is not like math and the music industry is not like industries that are related to calculus. Music theory is NOT required to play in a band or write a movie score. Check Gustavo Santaolalla's for an example of a great composer with ZERO musical bg.
> music is not like math That's going to age beautifully
Edit: inb4, I did read Hofstadter's Gödel, Escher, Bach.
Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do" The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a…
I'm one of those people who find maths really easy. I've also had extensive experience working in mathematics education, from primary to tertiary levels. I've even written policy for our state education department. You are spot on the money. I've seen "progressive" educators who believe that technology, including CAS is the answer to teaching calculus to high school and tertiary students. I think the tech is mostly u…
Earlier quoted context omitted.
> music is not like math That's going to age beautifully
Yeah, my bad, I was mostly referring to "music industry". Music is 100% like math. Music is math, and math is music. Edit: inb4, I did read Hofstadter's Gödel, Escher, Bach.
But yeah, I read TC's post as being less "you can't play music without formal theory" and more "taking a music appreciation class won't (solely) imbue you with the ability to play violin"
...nor would a course which provides a cursory pass over advanced concepts give you applicable (let alone employable) skills without at least a little of the manual labour needed to really feel the intuitions (?)
There probably _are_ some undiscovered clever ways to teach foundations such that you form useful or higher level synaptic links faster - but does seem you still need to then add some brute force to gain adequate fluency..
PS. I don't think that excludes the possibility of Ramanujans or Dizzy Gillespies existing - whether your hard work is unstructured or structured, top-down or bottom up, the key point is - beyond a superficial level there are eventually "no shortcuts"
The question could be changed to a positive. >Can Calculus be taught with differentiating or integrating by hand? And then the answer is yes, as this is how it is done. Now, why would be want to do otherwise? Why would we ever want people to learn less?
>>Can literacy be taught with hitting students with a stick when they err?
>And then the answer is yes, as this is how it is done. (19th century).
>Now, why would be want to do otherwise? Why would we ever want people to learn less?
Maybe you don't learn less by not being hit? Even if my grandparents all had vastly better handwriting than myself. Perhaps I learned some other things not being met with such hostility in my education? I don't know for sure.
At some level of complexity we learn "too hard for me to integrate by hand" and also stop learning how to perform more and more complex integrals. We stop. Where is the most productive place, to maximise useful learning, to stop?
Maybe not being hit with a stick cost me beautiful copperplate handwriting. I feel it was more productive for me to not be hit. I can see someone taking a different view on that with regards themselves. "Never did me any harm" is still a pervasive way of thinking about what others call child abuse. Not many wish they were hit more.
What do you get out of integrating by hand? At each step ratcheting up complexity from y=x, through each increasingly complex function type to where you stopped? Are there diminishing returns? Or do you get past a threshold opening up whole new areas of understanding otherwise not experienced? Something else?
I note anyone can integrate y=x and get 1/2x^2 + c so I don't think avoiding this integration by hand is the point.