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So you want to study mathematics

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Re: So you want to study mathematics

#191

Post by kevinventullo »

ghufran_syed wrote: post #16

I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to re…

Yes! Combinatorics, graph theory, linear algebra, and even some parts of theory of computation and grad-level abstract algebra and algebraic number theory can be learned without calculus (I think!).

N.b. This is not to say that these are all easier than calculus, and I wouldn’t even really recommend learning say Galois theory before calculus, I’m just saying it seems that one could.

Re: So you want to study mathematics

#193

Post by bitexploder »

exdsq wrote: post #101
potbelly83 wrote: post #85

As a math PhD I have to say the only way you're going to learn mathematics is if you actually have a pressing need to do so. i.e. You have a project at work that needs some math, you have a hobby that needs some math. In this case you just learn what you need. Just learning math for its own sake outside of a University STEM track is just too hard (I wouldn't be able to do it and I've tried).

Correct me if I’m wrong, but I assume you’re looking at this through the wrong lens. As a PhD you’ve learnt something hard at a considerable depth but this isn’t what I, or most people, hope to achieve by learning maths themselves. It’s normally to a far lower depth that’s far more achievable like Calculus I or some aspects of number theory or knowing what all the funky symbols such as Summation mean.

I just want to do basic engineering and physics. Understand some of the higher level models we have to describe our world.

Re: So you want to study mathematics

#194

Post by analog31 »

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.…

I have 2 kids who are in college now. The beauty of math that inspired me to study math (as an undergrad math major) was treated as an aside in K-12 math, and has now been replaced with Grind. I learned to love math thanks to just the enthusiasm of my teachers, and stuff that I did outside of class. Both of my kids got great math scores, good enough to skip college math altogether, which one of them has done.

Today, kids need to be told: Math is nothing like what you learned in high school.

I think K-12 math should be divided into 4 quadrants, taught in a soft of spiral:

1. Arithmetic (symbol manipulation, through algebra and calculus) 2. Computation (numeric and symbolic) 3. Learning from data 4. Theory (sets, proofs, etc.)

Some of these things could be blended with the science curriculum.

Re: So you want to study mathematics

#195

Post by bee_rider »

SamReidHughes wrote: post #155
geebee wrote: post #151

Earlier quoted context omitted.

I'm really wary of this. House wiring would be more useful than Shakespeare. That's not to knock house wiring, or statistics. I'd love to know more about both. But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a fe…

Well, AP statistics covered the area under a probability distribution function and people seemed to understand that -- you look up the answer in a table (or use the TI-83 function). Presumably they'd do the same for a cumulative distribution function.

Highschool is pretty early to teach people that math is too complicated for them to actually evaluate their expressions.

Re: So you want to study mathematics

#196

Post by untake »

Wrong: math is neither hard nor difficult. It is this single belief that deters many people from learning it. Math is all about logic. It is nothing but how to go from A to B. Any person who can reason should be able to be good at math.

Re: So you want to study mathematics

#197

Post by joatmon-snoo »

geebee wrote: post #151
joatmon-snoo wrote: post #49

Earlier quoted context omitted.

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.…

I'm really wary of this. House wiring would be more useful than Shakespeare. That's not to knock house wiring, or statistics. I'd love to know more about both. But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a fe…

Education is a Hard Problem TM because the knowledge of those links isn't useful without sufficient existing knowledge, and you need to introduce things in a way such that the things you hand-wave away are tolerable knowledge gaps. Knowing that a CDF is an integrated PDF isn't useful for someone who doesn't even have a notion of what a PDF is. Describing a state machine that satisfies conditions X Y Z as an Abelian group is similarly not useful for someone who doesn't have an intuition for groups or fields.

My claim is that, given the finite amount of time allotted to mathematics in a secondary school curriculum, it's better to spend that time learning about medians/means, regressions, statistical tests, and so forth, instead of memorizing power laws and derivatives of trigonometric functions.

Re: So you want to study mathematics

#198

Post by fn-mote »

I question just how realistic is is to have "Proofs from the Book" at the start. While the art is wonderful, the background required to read it is not realistically at a lower level than the textbooks listed.

The discussion here has been much more interesting than the actual list, to me. If you wanted to master all of that material, I think a master's program is the way to go, not self-study.

I would be interested in hearing from people who _do_ successfully self-study. What makes it work?

The least interesting (from my point of view) is "already successful in a related field, applied my skills". That would include CS professionals studying maths, I think.

More interesting would be "unable to attend university because of X, did Y and really enjoyed it." Are you a person who completes MOOCs and get something out of them?

Re: So you want to study mathematics

#199

Post by brimble »

LolWolf wrote: post #51
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.…

Sure, here's a fun one ! https://web.stanford.edu/~boyd/vmls/ (I'd even replace Strang's "linear algebra" recommendation with this book.) Imo, proofs are useful in so far as they are enlightening (e.g., the proof that a problem has a minimum is often useful in so far as it tells you how to solve it!) but in many cases they are less so. Math is pretty fun, though, proofs and all, and I'd recommend trying your hand at…

> Math is pretty fun, though, proofs and all, and I'd recommend trying your hand at it as a cool little side hobby! It can often help with "clarity of thought" :) (In many cases, proofs are just one or two lines that tell you something interesting, too, not page-long arguments that are mostly definitions chasing.)

I find proofs and identities very hard to read. I assume it's a bit what having dyslexia feels like. I have to turn them, glyph by glyph, into something more algorithmic to make any sense of them. The only math-for-fun I've ever enjoyed are simple recreational math puzzles—proofs, reading or writing, are torture. I actually enjoyed the parts of math classes that mathematicians insist are bad and are the reason kids don't like math—the parts heavy on memorization and drilling the application of an algorithm—far more than anything that came later. Perhaps not coincidentally, those have also proven to be by far the best bang-for-the-buck of all my time spent in math classes over the years. I use that stuff every day.

Re: So you want to study mathematics

#200

Post by bee_rider »

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.…

I think I'd rather we lose geometry than calculus. Geometry made more sense as an intro to the idea of 'proving stuff' when everybody's dad had a workshop, compass, etc.
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