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So you want to study mathematics

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Re: So you want to study mathematics

#301

Post by dagw »

zozbot234 wrote: post #150
Melatonic wrote: post #95

Earlier quoted context omitted.

Not sure I agree with that - you also have to take into account that calculus is a requirement for many other sciences. I suppose you could just of course make it a pre-req for things like that but I found that in highschool rarely did they go that deep. Physics becomes a hell of a lot easier with basic calculus for example. If anything should be dropped from a highschool level its all of the insane memorization you…

Calculus is also a requirement for statistics and probability. You simply cannot meaningfully talk about the latter without referencing the former.

Yes and no. You have to understand the fundamental concepts of integration, differentiation, differential equations etc. and what they mean and represent. All the time spent learning how to do symbolic integration by hand is however time that could be much better spent.

I also think you could and should start by teaching probability and statistics and using that to introduce calculus as and when it shows up naturally, rather than teaching calculus in the abstract first and then showing the applications much later.

Re: So you want to study mathematics

#302

Post by DocTomoe »

ALittleLight wrote: post #300
DocTomoe wrote: post #292

Earlier quoted context omitted.

When I was younger, I thought like you did. Now that I get older, the value of culture and an understanding of where it came from, how it developed, makes my life just so much richer. That does not necessarily mean that I need to have memorised the whole corpus of Shakespeare, but understanding a reference when it comes around is adding more layers of meaning to other works. This does not only extend to literature. I…

There is a ton of stuff that makes life richer. That doesn't mean we should be teaching it at schools. I can think of movies, games, novels, TV shows, blogs, and music that have had orders of magnitude more impact on my life and thinking than Shakespeare. Should we be teaching any of that stuff at school? I'm willing to believe that some people like Shakespeare, but it's a small minority. You can tell by the number o…

I like to believe that school works as a catalyst here, showing you stuff you wouldn't normally encounter - you have no trouble accessing popular culture, so there's no need to teach you about that (that's being said, one of my more memorable music lessons was the teacher analysing the composition of Pink Floyd's "Shine on you crazy diamond" with us. I can absolutely see Harry Potter becoming an object of academic study and literature teaching in 50-70 years, it is happening with Tolkien already).

On the other hand, less-than-popular culture still is the foundation for our popular culture today - the amount of times "Romeo and Juliet" or "Macbeth" has been adapted, referenced and deconstructed in (popular) media is astonishing, and without knowing the original, you wouldn't fully get the modern references.

You don't need to love Shakespeare (I don't exactly, though I like to see the "Scottish play" every few years) to reap these benefits.

School is not about what people like. If you go by that measure, there's no real need in math, because you will find a lot less people doing calculus for fun, as compared to people who play fantasy football. Of course, you can't really understand the ideas behind fantasy football if you don't know maths, but who cares, it's not popular...

Please note that I did not claim physical labour has no value and one should not know about it. I said that it's relatively easy to pick up, and that it may be more worthwhile to get a professional when the job demands it (a professional, I would like to add, who learned his craft after school, ideally in some kind of apprenticeship). In my country, to get a driver's license, you need to know basic first-aid procedures (and yes, there's a one-day course you need to attend). That doesn't make you a neurosurgeon, and if you have persistent headaches, it's probably better to ask a professional than to depend on me with my first-aid course - or some guy who "learned about surgery techniques" in high school.

Re: So you want to study mathematics

#303

Post by theptip »

joatmon-snoo wrote: post #49
brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs.…

Strong agreement here. I'd go further and say that introductory stats should replace trigonometry too. An understanding of statistics is simply required to function optimally as a citizen in modern society, whereas trigonometry and calculus are niche.

I'm all for giving tasters of as many different "beautiful ideas" as possible in school, but I think we should be elevating practical statistics into the top tier of subjects that we require kids to go through.

Re: So you want to study mathematics

#304

Post by ChrisLomont »

paulpauper wrote: post #204
ChrisLomont wrote: post #131

Earlier quoted context omitted.

At a first glance that book completely fails for an undergrad lin alg course, and looks weak in other areas too. Examples: the words nullity and kernel don't appear, rank of a matrix is not in it, and, well, every topic I can think of for undergrad linear algebra is simply not in the book. It's equally bad for stats: no mention of many common distributions a student would learn for example.

I give a book recommendation and ppl complain, yet no one complains when ppl recommend Khan Academy. Do people think Khan Academy is more rigorous than a textbook?

I'm not complaining about a book recommendation. You claimed this book will cover college-level concepts for linear algebra. It does not even come close to doing that. It's misleading for anyone wanting to learn linear algebra and taking your book recommendation.

Re: So you want to study mathematics

#305

Post by ChrisLomont »

wizzwizz4 wrote: post #140
ChrisLomont wrote: post #131

Earlier quoted context omitted.

At a first glance that book completely fails for an undergrad lin alg course, and looks weak in other areas too. Examples: the words nullity and kernel don't appear, rank of a matrix is not in it, and, well, every topic I can think of for undergrad linear algebra is simply not in the book. It's equally bad for stats: no mention of many common distributions a student would learn for example.

Is that such a bad thing? I'm half-convinced that the “rank of a matrix” is just an artefact of a particular algorithm for inverting matrices. And distributions aren't everything; I can look up any distribution I want on Wikipedia, just as soon as I need it, but a proper foundation in what statistics means is much harder to come by. (I have enough of a foundation to know when it's being taught very wrong, but not eno…

>I'm half-convinced that the “rank of a matrix” is just an artefact ...

The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space". It becomes more fundamental if the person goes on to study deeper math. It tells you how to compute the size of a basis for the target space. The uses go on and on.

>And distributions aren't everything; I can look up any distribution I want on Wikipedia

Yes, you can look up anything on wikipedia, but not in this book, which is why this book will not teach you the things the OP claimed it would.

>a proper foundation in what statistics means is much harder to come by

It's very hard to get that proper foundation from a book that does not cover those foundations. One is better served by using a proper book with a consistent and well laid out presentation of the needed concepts. Saying one can look fundamental stuff up elsewhere means the book is lacking.

Re: So you want to study mathematics

#306

Post by dr_dshiv »

openfuture wrote: post #295
BeetleB wrote: post #233

Earlier quoted context omitted.

> Why is it that every time any subject about mathematics comes up there is always a complaint about notation? The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.

Yep, blaming your tools is what the incompetent do.

Blaming the user is also what the incompetent do.

Re: So you want to study mathematics

#307

Post by tcmart14 »

brimble wrote: post #209
pnin wrote: post #176

Earlier quoted context omitted.

I have some sympathy for this sentiment, and there is no doubt that the producers of mathematics could and probably should spend more time making life easier for the users and consumers of mathematics. This is true even /within/ the discipline, for very theoretical stuff. In the end this takes vastly more work than people expect. That's not an excuse, but it's true. However, I also have a fundamental objection. I don…

I do not have the right background to be an authority on this, but from going through school myself and paying a fair amount of attention to math reformers over the years, it seems like they usually believe: 1) The problem with math in school is that there's not enough "real math". 2) Relatedly, insufficient exposure to "real math" in compulsory schooling is also (a major part of) why people think they don't like mat…

No expert either, but I can speak as a non-traditional student (didn't go to college right out of high school join the Navy and started a family first) about to graduate with a degree in engineering. I was decent in math in grade school but did not like it. I didn't learn to actually enjoy math until my college calculus classes. And it goes back to the real math you are referring to. Looking back, Algebra in highschool just felt like, "memorize this type of problem and the steps," but didn't do anything to build intuition in actually understanding the why. Then calculus comes and I felt like that gave me the 'why.'

Re: So you want to study mathematics

#308

Post by pnin »

wanderingmind wrote: post #230
pnin wrote: post #176

Earlier quoted context omitted.

I have some sympathy for this sentiment, and there is no doubt that the producers of mathematics could and probably should spend more time making life easier for the users and consumers of mathematics. This is true even /within/ the discipline, for very theoretical stuff. In the end this takes vastly more work than people expect. That's not an excuse, but it's true. However, I also have a fundamental objection. I don…

Its like how people use tools, like gears, knowing when to use what type of gears without needing to understand the microscopic material structure that provides its strength, or the mathematical definition of it's kinematics.

I think if you use gears, it would be well to know some Newtonian mechanics. The material structure might not be relevant, but in mathematics you also don't need to go down to the foundational level for every problem. You don't need to go deep into set theory to understand the proofs in calculus (though it is true that Cantor's investigations into set theory started from the question of convergence of certain infinite series, which is a calculus problem).

Re: So you want to study mathematics

#309

Post by Aperocky »

brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

There's even a package for that, it's called numpy.

Re: So you want to study mathematics

#310

Post by trosi »

It's true that a standard undergraduate curriculum in mathematics will contain a lot of analysis (including calculus), a lot of ODEs/PDEs and some algebra. But if I wanted to get someone interested in math I would also point them towards some of the more fun (and less known stuff), such as combinatorics, probability, topology, differential geometry and number theory. Some of these are also much more applicable today than, say, complex analysis.
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