Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Calculus For The People

geogebra.org

Post reply on HN
Print view View original

71–78 of 78 posts

Re: Calculus For The People

#71

Post by lonelappde »

gregpetrics wrote: post #51
rrss wrote: post #49

Earlier quoted context omitted.

My pages apparently don't align with yours (I see page 46 has only a single exercise), but I don't see where Rudin says anything is "good enough." He states the definition of convergence, meaning that if a sequence satisfies the property then we choose to call it convergent. There is no question of good enough I don't see a claim about an "infinite set of inequalities" - I see an infinite set of I equalities that mus…

Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such that the sequence stays within epsilon of p (the mark) for all indexes larger than N. I know it seems formal because it adheres to a certain structure, but even this is informal at a fundamental level. Specifically, how can we be sure we can perform a countably infinite number of distance measurements in the metric space to be sure the sequ…

Most US schools teach formality in Geometry with 2-column mechanical proofs, following Euclid. Formal and informal belong side by side throughout the curriculum.

Re: Calculus For The People

#72

Post by Retra »

bootlooped wrote: post #65
Retra wrote: post #35

Earlier quoted context omitted.

Learning the meanings of things without learning how to do them leaves you powerless to do anything with that knowledge. It's very easy to walk around saying "we could solve problem X with technique Y", but if you don't actually know how to do Y, then you're just conjecturing fruitlessly. For instance, here you're talking about "memorizing how to manually integrate a 5th degree polynomial" as if that's something anyo…

I'm pretty sure I manually integrated (and differentiated) polynomials that large in my calculus classes. As for the "you don't want to put effort into things", I did put in the effort, I took the classes in question and graduated. I just happen think that particular effort was a waste of time. There's always a deeper or shallower understanding to be had of a subject. Finding what is appropriate for a given task is t…

You don't have to talk about integrating higher degree polynomials, because a polynomial is just a sum of monomials, and you just end up integrating monomials over and over again. If you can integrate a single monomial, you can integrate all polynomials.

That makes your objection seem weak, as though you don't know how to integrate, and you're just imagining that it is hard. That undermines your point because there are actually things in math that are hard, but you'd be woefully unprepared to understand them if you don't see the value in memorizing the absolutely trivial stuff.

It's like a child saying "I don't want to be forced to memorize the shapes of the letters, because that's not what makes a good writer." Does a good writer sit around looking up letter shapes in a diagram all day because they can't be bothered to remember them?

Either way, the optimal solution is the same: do the task over and over again until you are so familiar with it that you can recall it from memory. Memorization is a necessary part of learning.

Re: Calculus For The People

#73

Post by newprint »

lonelappde wrote: post #69
newprint wrote: post #53

Earlier quoted context omitted.

To answer your question, I work in optimization. IMO (International Mathematical Olympiad) are extremely complicated problems that even professional mathematicians often struggle with them. IMO is a level on its own - Gold Standard. Not all Mathematical Olympiads are of the same level of complexity as IMO. Good example are Olympiad caliber problems that are not overkill - Hungarian Problem Books. MO problems are usef…

But industrial mathematics is numerical methods, a rather different beast from the tricky closed form puzzles of Olympiads. I question your claims of causation vs correlation regarding Olympiad work and industrial math and general effort put into basics of mathematical thinking and general mathematical intelligence.

Not entirely true. A lot of optimizations theory and practice are are straight-up inequalities. My claim is very simple: by being able to solve moderately-hard MO problems, you are developing excellent mathematical problem solving skills that let you solve problems in other branches of mathematics or draw interesting connections, because MO problems require ability to solve problem that are at the intersections of different branches of mathematics.

I frequently see topics similar to this one popping-up on HM and a lot of people interested in learning or re-learning math as an adults. Excellent ! So let them learn discreet mathematics by solving problems in algebra, combinatorics and number theory. Once they are comfortable, they can move on to more abstract subjects. My view is very similar to Concrete Mathematics, by Knuth, Patashnik, Graham. Their "concreteness" is very down to earth: combinatorics, number theory and few other things in the mix.

Re: Calculus For The People

#74

Post by falcor84 »

kstenerud wrote: post #14
maxmunzel wrote: post #13

Earlier quoted context omitted.

Actually I think that GeoGebra is by far the most intuitive and for-the-people that a math toolbox can be. When I picked it up in Highschool, I didn’t need to google a thing about it, contrary to the CAS Systems I use nowadays... (Maybe not a fair comparison)

Except that there's no "I don't understand" button where you can tell the author what you don't understand. So it's basically a one-way textbook with no feedback mechanism for the author to discover his assumptions and clarify them.

What other popular resource has such a button?

Re: Calculus For The People

#75

Post by norswap »

Rifled through it, seems to be just the standard stuff.

I was particularly disappointed by:

> The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the Quotient Rule and the Chain Rule, "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives, you need to think a bit like a forensic analyst who studies the site of an explosion to see what sort of bomb was used. We'll discuss this analogy more later when we practice finding antiderivatives.

("Monkey rules" are the derivation rules, this kind of cuteness is a big part of the purported dumbing down)

Anyway, systematically calculating derivatives was always a big sticking point for, as indeed you often need to use multiple rules and it's not quite obvious which chaining of rules will get you there. I was hoping the authors could introduce a systematic algorithm (which no doubts exists but I never bothered looking up - I don't do much integrals day to day) or at least some strong form of intuition that goes beyond "if we did this we'd have something on which we could apply that rule".

Re: Calculus For The People

#76

Post by vibrio »

master_yoda_1 wrote: post #57
vibrio wrote: post #27

Earlier quoted context omitted.

Do you believe school is sufficient?

Yes if you focus and don't waste time. Your whole life won't be sufficient if you can't focus and keep wasting your time in unrelated stuff like this shallow article.

Serially commenting on a shallow article is sort of doubling down on wasted time, no?

Re: Calculus For The People

#77

Post by gregpetrics »

norswap wrote: post #75

Rifled through it, seems to be just the standard stuff. I was particularly disappointed by: > The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the Quotient Rule and the Chain Rule, "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives,…

Good feedback.

I struggled with deciding if I should write activities that illustrate the full algorithm for derivatives and antiderivatives. At this time I left it out, but I do have the materials...

The book was written with a bit of a promise to keep the algebra out, and overdoing it on Monkey Rules (derivatives) and Lucifer's Rules (antiderivatives) breaks that promise. That said, calculating derivatives and antiderivatives is the fundamental algebraic task of a calculus student.

I'm thinking about your feedback right now... and will likely make adjustments in the near future to introduce optional tracks for extra practice on this.

Re: Calculus For The People

#78

Post by norswap »

gregpetrics wrote: post #77
norswap wrote: post #75

Rifled through it, seems to be just the standard stuff. I was particularly disappointed by: > The bad news is that this is a little harder than using the Monkey Rules to calculate derivatives. In some sense the Monkey Rules, particularly the Quotient Rule and the Chain Rule, "blow functions up" when they systematically calculate derivatives. In order to go backwards, and undo the Monkey Rules to find antiderivatives,…

Good feedback. I struggled with deciding if I should write activities that illustrate the full algorithm for derivatives and antiderivatives. At this time I left it out, but I do have the materials... The book was written with a bit of a promise to keep the algebra out, and overdoing it on Monkey Rules (derivatives) and Lucifer's Rules (antiderivatives) breaks that promise. That said, calculating derivatives and anti…

That's great to hear :)
Post reply on HN