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So you want to study mathematics

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Re: So you want to study mathematics

#261

Post by peterhalburt33 »

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’ll admit I’m biased on this issue, but I’m very wary of this type of reasoning. There are plenty of courses that are more useful than calculus, but kids/young adults won’t be convinced to learn a subject purely out of its utility in adult life (same reason I’m against replacing math with a “life skills” class); in g TV fact Calculus was the course that changed my whole college and career trajectory. Up until I took Calculus, I recognized the importance of math, but hated every dull second of it - there was no motivation that appealed to me, and it seemed that math could only be used to solve the most contrived problems (word problems that resulted in linear systems, simple roots of trig or polynomial functions, etc).

The only reason I took calculus is because it was the next math course after pre-calc and I still had a year left in high school. I didn’t even realize that there was a type of math that could be used to exactly calculate quantities related to continuously changing processes, but I was absolutely fascinated by how many real world problems calculus could solve - I realized that the problems given in algebra were contrived out of necessity to resolve in a neat manner. Now I had access to an entirely new vocabulary that allowed me to describe the world as it is, not as it needed to be to neatly fit in a 10th grade word problem.

Until that point I was interested in psychology (and had a vague notion that I’d drop out of school to make a career in music somehow), but I immediately dropped everything to take as many math and physics courses as I could. 14 years later and I work in a very math heavy engineering oriented field.

I know my story is probably atypical, and I have no clue what I’d be doing right now if I hadn’t taken calculus, but it’s one of those things that I look at in hindsight and think that I was that close to giving up on STEM, but for being forced to take Calculus. Instead, I earned my Ph.D. in applied math, and there’s not a day where I don’t use calculus of some kind.

Re: So you want to study mathematics

#262

Post by waste_monk »

brimble wrote: post #213
jackthetab wrote: post #211

Earlier quoted context omitted.

YES! This is exactly my complaint about math. I find the abstract proof-oriented math kinda interesting (although How to Prove It is an exception that I discuss below) but I _really_ want practical real-word applications of these maths. Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that pe…

> Not once have I applied a proof to solve a programming problem. There are people in this very thread insisting that proofs are extremely useful in programming. I dunno if I just picked up the same skills elsewhere (Logic? Philosophy? Just... IDK, thinking and developing an absolute shitload of heuristics through years of experience?) or am entirely missing out and in fact don't have a clue how to program, but I don…

Perhaps by proofs they are referring to formalised / verifiable programming languages such as the SPARK subset of ADA?

Those are indeed very useful for life/safety critical systems such as aerospace, air traffic control, medical devices, etc.

Re: So you want to study mathematics

#264

Post by thoughtfunction »

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…

The problem with this approach is your basically teaching this as a side effect and hoping they're smart enough to actually connect all of that together vs. teaching those concepts directly and being far more understandable.

If you want people to understand that numbers are concepts / descriptions of something potentially infinite, and that it's ok to totally work with non-finite describable numbers without ever really writing out their values, just fucking start with that. I honestly think the start of most undergrad math curriculums should be a logic & foundations class with proofs, sets, number theory and the whole "numbers are more logical concepts, not really specific values" vs. calculus. Then teach calculus, with a big dose of 'what is infinity, really?'.

I think it's actually a great disservice and the hand wave that most calculus math classes do about those exact concepts really fucks over a lot of people. It makes calculus a weed out class because the fundamentals are not explained properly so a good amount of people just go into it as yet another thing they have to ape without real understanding. And for the people who don't work well with things they half understand, they really struggle, like I did. It really made my computer science degree a lot worse, because of the instance of starting all undergrad math with hand wavy calculus, for 3 or 4 classes.

I even wrote blog articles about it, it was probably the worse part of my computer science degree, and if math education was done differently where they didn't handwave, I probably would've had a much better time.

https://www.thoughtfunction.com/2019/11/doing-my-compsci-deg...

https://www.thoughtfunction.com/2019/11/what-i-wished-i-knew...

Re: So you want to study mathematics

#267

Post by peterhalburt33 »

Mimmy wrote: post #6

Going from Strang to D&F seems like a steep jump. The former is an applied textbook for non-mathematicians and the latter is a proof-based text for advanced undergraduate / graduate-level math students. I would suggest working through a proof-based linear algebra book in between to ease the transition. Axler's is a good one. Alternatives include Hoffman and Kunze and the more modern Friedberg, Insel, and Spence.

I’d be hard pressed to recommend D&F to a self study audience. Perhaps it’s just me, but the endless enumeration of examples and counterexamples at the beginnings of most chapters seemed to destroy my intuition rather than reinforce it.

Re: So you want to study mathematics

#268

Post by novok »

bee_rider wrote: post #195
SamReidHughes wrote: post #155

Earlier quoted context omitted.

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.

We already do that in elementary and high school with logarithms, square roots, sin, cos, tan, pi, radians, e, irrational & imaginary numbers, etc. The tables are just built into pocket calculators instead of paper. Even Pythagoras is a bit weird.

Re: So you want to study mathematics

#269

Post by joe_the_user »

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…

My suspicion (again, without the actual background to make this claim with any authority) is that they are dead wrong on point 2—the "real math" parts probably contribute strongly to most folks' dislike of the subject, and the parts the mathematicians didn't like are probably relatively popular among people who don't go on to become mathematicians. This puts point 1 on some shaky ground (though it could still be true and well-justified, for other reasons).

So you're saying you know nothing yet are sure "experts" are wrong, based on no evidence. OK.

Re: So you want to study mathematics

#270

Post by novok »

kjasdfj3 wrote: post #240
wbsss4412 wrote: post #169

Earlier quoted context omitted.

You can certainly make significant headway with some basic combinatorics and basic tables for things like the normal distribution, no calculus needed. This is in fact how most people currently learn statistics.

What happens when you dont have tables? In the land of the blind the guy with one eye walks everyone off a cliff.

Most real life integrals and differential equations are not analytically solvable anyway, so you run into tables like you do with most real life applications of logarithms.

What happens is you learn numerical approximation methods and then recreate the tables with human 'computers', like we used to do before electrical computers came into play. Or create mechanical analog computers like they did in the 1800s or greek times.

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