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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 proof

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've got Mathematics for the Nonmathematician by Kline and that's kinda heading the right way, but what about whole courses of study? More books? It's more of an introduction than a thorough resource or course, and feels like it needs another four or five volumes and a lot more exercises to be really useful. I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.

mjfl 5y ago
The list presented here, calculus, odes + pdes, linear algebra, is essentially that, mathematics for people who just want to use it. It's all undergrad level. There's several layers on top of this - set theory, rigorous probability theory, algebraic geomeetry, topology, that are less useful but interesting to mathematicians.
importantbrian 5y ago
I didn't take any of these classes personally, but I do remember in college there was a whole host of Calculus for Business and Economics type courses that were much more focused on practical application than theory. Maybe picking up some of those textbooks would be the way to go.
susanrigetti 5y ago
If you have a solid background in calculus, I'd recommend Zill's Advanced Engineering Mathematics, which is pretty much basic math for physicists and engineers (aka for people who need to "use it").
- brimble 5y ago
  Heh, I have a long-forgotten-due-to-complete-lack-of-use-for-15-years-straight background in calculus. Solid, it is not. Thanks for the tip, though.
  - susanrigetti 5y ago
    In that case, I recommend starting out with Zill's Precalculus with Calculus Previews and then working through Stewart's Calculus: Early Transcendentals!
    - brimble 5y ago
      I'll check those out, thank you!
jackallis 5y ago
i second this. there should clear distinction between academic math and "real world usage" math.
- Koshkin 5y ago
  But the distinction is generally not as clear as you may think. (1) Much of the mathematics came from real world problems (so, in particular, one may get drawn into some kind of mathematical research and even end up discovering new mathematical facts). (2) Sometimes when applying mathematics one still needs to employ deduction (derive a formula, prove a statement one wants to rely on, etc.).
- psyklic 5y ago
  Having a deep (proof-based) understanding leads to more "real world" insights. So, there may not be a clear distinction.
- Tomis02 5y ago
  Mathematics is a social activity, in the sense that mathematicians create proofs to convince others that their findings are valid. Some proofs are easier to verify than others. Some proofs are "believed to be true" because no one has yet managed to prove them wrong yet. So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented. And just to be clear, there is little to no "real world usage" math without academic math.
  - Koshkin 5y ago
    > create proofs to convince others Well, it is not only about people. You would also want to "convince" Nature (so that the aircraft you designed does not crash right after a takeoff), your own or, often, someone else's wallet (so that money is not spent on nothing).
helpfulclippy 5y ago
I did a degree in applied math. You’d think this would be “math you’ll use,” but the fact is that despite my program having a CS concentration, most of the stuff I did was not really applicable in practice. However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered. I really wish I’d spent a lot more time on proofwriting than say, vector calculus. Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.
- auggierose 5y ago
  Proofs are indeed very practical. For example, I believe Leslie Lamport mentioned somewhere that he only came up with the final version of Paxos once he tried to prove it, and noticed that some condition he assumed wasn't necessary at all.
  - dhosek 5y ago
    The reasoning behind having geometry be the standard high school sophomore math class is that that’s the age where kids would be ready to do proofs. Except that curriculum designers seem to have forgotten this and except in honors classes, most sophomores don’t get taught proofs in geometry and instead get a set of inert rules about shapes that they have no use for.
    - whatshisface 5y ago
      Geometry uses a deduction-only basket of proof techniques that don't prepare students for proofs done afterwards. I would like to see it replaced by elementary number theory which naturally uses a lot of induction/recursion and has some uses for proof by contradiction.
joatmon-snoo 5y ago
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. The vast majority of them, I would assert, are simply pre-med weed-out courses. I still remember freshman year, showing up to the standard intro calc course, and dropping it as quickly as I could in favor of my uni's equivalent of Math 55 (i.e. the hardest u/g intro math course) because of how asinine I found the content...
- antishatter 5y ago
  Statistics converges to calculus at its core. Source: am statistician
- jll29 5y ago
  Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).
  - eftychis 5y ago
    Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at. Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.
LolWolf 5y ago
Sure, here's a fun one ! https://web.stanford.edu/~boyd/vmls/ 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 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.)
- brimble 5y ago
  > 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.
Koshkin 5y ago
> "math for people who just want to use it" It's called "engineering mathematics." (The books by Stroud would be an excellent choice here.)
pphysch 5y ago
> Are there any "math for people who just want to use it" tracks in math pedagogy? Use it for what? That is the question. If you pursue the what, you will inevitably be exposed to genuine ways that mathematics may be employed by it. The academic standard for "learning math" is like "learning programming" by reading the C++ language/STL spec from front to back. No one productively learns programming that way, and even if someone did, they would hardly be well off when faced with a real-world production C++ codebase that follows $BIGCORP's inhouse programming style.
- vharuck 5y ago
  I agree. If anyone wants to study "math in practice," they should pick a science or engineering track they like. Math will be included.
- deleted 5y ago
  [deleted]
k__ 5y ago
I think, the problem is, proofs are what makes math useful. This might sound a bit dense, but the alternative is what 90% of programmers do every day.
- Koshkin 5y ago
  I think one can get far enough by applying known mathematical facts. (Proof: elementary school math is useful.)
  - auggierose 5y ago
    The problem is, usually you need to combine known mathematical facts to solve a problem. Now, how do you know that you combined them properly? Yup. You need to know what a proof is.
    - k__ 5y ago
      This. Applying mathematical facts isn't math, it's calculation at best. That's what most programmers already do. They throw things together in the hope they work without proof they got all edge cases covered. And it works pretty good, because often the edge cases aren't that bad.
dwohnitmok 5y ago
> I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first. The problem is that the answer will depend heavily from person to person and from field to field and often the most sensible answers require a mathematical maturity that creates a chicken-and-egg kind of difficulty. Do you care about engineering? Well you'll need some calculus for that. Do you care about prediction modeling? Well there's some stats you'll need for that. Do you care about finding patterns in the world? Well there's abstract algebra for that. Do you care about reasoning itself? Have fun with mathematical logic. But because humans have different motivations, there's no one-size-fit-all motivation-based approach. This is especially painful for mathematics because I think most people who have learned some amount of pure mathematics will relate heavily to what helpfulclippy says in a sibling comment: "However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?)." The skills of rigorous and abstract thinking that pure mathematics provides is both nearly-universally helpful, but also simultaneously as a result very difficult to motivate. "This will help you think better across everything you do" is lofty-sounding, but generally not a convincing sell unless someone is already curious. But it's true that being able to wrap one's mind around pure abstraction (after rattling off a rigorous definition for an abstract question: Question: "But what is X really?" Answer: "X is just that. No more, no less.") has ramifications for all that one does. And the most painful part of all of this is if you try to start by teaching the wonders of pure mathematics instead of all the messy, boring rote stuff, students' eyes are liable to glaze over even more because of the aforementioned chicken-and-egg issue with mathematical maturity. In this way it's similar to trying to motivate someone to read and write. The key that unlocks that interest for everyone is going to be different and it's very hard to explain the near-universal benefits that reading and writing bring to one's way of thinking (but I can always just have a computer transcribe it or read it aloud to me!) without some inherent curiosity in them.
cptaj 5y ago
I just want a place I can go to weekly for 15min to practice math. It keeps track of what I know and gives me exercises of stuff so I can retain the skills. I've forgotten 90% of the math stuff I learned
- robotresearcher 5y ago
  How about a textbook? At 15 mins a week, a couple of books would last a year.
- Buttons840 5y ago
  Khan Academy. Finish a course, retake the course exam once a year afterwards, fill in knowledge gaps as needed. Doing this should keep all undergraduate math fresh in your mind.
  - throwamon 5y ago
    I wanted to like Khan Academy, but the pacing of the exercises is so frustrating. They become too easy too fast and then it's an extremely repetitive slog of useless computations. I wish for a service just like them but with many more difficulty levels and less (non-spaced) repetition.
    - lordgrenville 5y ago
      I recently came across brilliant.org, which I think aspires to be this. I didn't stick with it but it might be what you're looking for.
dwrodri 5y ago
Swinging by to plug my personal favorite resource for refreshing my self on Bayesian stats: https://www.youtube.com/playlist?list=PLwJRxp3blEvZ8AKMXOy0fc0cqT61GsKCG https://www.youtube.com/playlist?list=PLwJRxp3blEvZ8AKMXOy0f... I think statistics is by and large the most proportionally underrated subject proportional to its utility. A good command of stats and probability expands your power to use data to reason about answering questions. The channel author, Ben Lambert, has an alternative playlist where he uses some of the techniques taught in this playlist to solve problems in econometrics. However, a lot of what is taught here builds a great foundation for other domains, on everything from bioethics to data journalism to computer vision. Another great channel that focuses a bit more on the machine learning side of things is StatQuest with Josh Starmer: https://www.youtube.com/c/joshstarmer https://www.youtube.com/c/joshstarmer
eftychis 5y ago
Well you can think of it like going to the gym. You don't exactly see people doing squats during their daily life, but you can see the results of people having a good physical core. This needs to be explained further during education and motivated appropriately. We have a short-term utilitarian perspective, and we need to take a step back at times and recall that it takes time and lots of sculpting to transform a wood log to a art piece. As you can jog everyday for fun and/or for the challenge you can also jog to improve your physical health. And not doing proofs is like declaring a guy can weight-lift by just watching videos on youtube and never lifting a weight. Or a guy can "code" without writing a line of code.
konschubert 5y ago
You don’t need proofs, but you still need an intuition for why things work the way they do. Otherwise you will be lost as soon as you leave the textbook territory. Proofs are just one way to build intuition. The best way to learn applied maths and get intuition is an “Introduction to Maths for Physicists” 101 course.
narrator 5y ago
If you just want to learn math for purely practical reasons, Khan Academy (http://www.khanacademy.org http://www.khanacademy.org) is great. It might have added more lessons, but when I went through it, it went up through 1st year college calculus. The thing that makes it VASTLY better than most self study math programs or books is that there are hundreds of exercises that you can do, and see if you got the right answer. If you didn't, it will in most cases explain how to do the problem so you can try again with a completely different problem, so you're not just memorizing the answers. Another thing that makes it great is you can do a little bit a day, start and stop, and come back to it and it will remember your progress and where you left off. Khan is also a gifted teacher. Unlike a lot of math teachers, he has great pronunciation and handwriting and you can watch his lessons as many times as needed.
- macrolime 5y ago
  Where are there exercises? I just tried clicking on some linear algebra topic, but it seems there are just videos https://www.khanacademy.org/math/linear-algebra/alternate-bases https://www.khanacademy.org/math/linear-algebra/alternate-ba...
  - narrator 5y ago
    They don't have exercise for some of the more advanced college level topics. You can see the exercises on the right side of the screen under "practice" for some of the more introductory topics: https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces#vectors https://www.khanacademy.org/math/linear-algebra/vectors-and-...
Siira 5y ago
The proofs can help immensely in remembering theorems and knowing their needed assumptions and application area. Mainstream math education is also already light on difficult proofs in introductory (undergraduate) texts. You can always skip the proofs anyhow.
Jach 5y ago
Yes, all tracks of math involve people using math. Proof writing is a part of use for a lot of it. There are also very many tools within the use of proof writing where a lot of education is around learning a bunch of tools so that you can better "recognize which tool to apply, then apply tool", in a future real-world use where you want or need to prove something. I'm a little snarky, but you have a broken idea of what math is. It's not even your fault. I don't even claim an unbroken idea for myself, I went through public education too, though I do think it's less broken. Somehow compulsory education has managed to get near universal basic literacy, but seems to have failed on whatever equivalent some sibling comments have hinted at exists for math or at least mathematical reasoning. A lot of algebra work taught for junior high can be understood as just a foundation to be able to understand later things (though you can of course use some of it directly as taught without having to learn more for every-day things like some boy scouting activities, or helping with putting together a garden or a fence, or programming -- and some of course is entirely useless). But instead of pushing algebra even earlier, states are instead moving to push it even later. (Let alone trying to spread awareness of even a hint of the subtle divide between more general algebra and analysis that a lot of STEM undergrads don't even really get a whiff of except maybe knowing it's often said to be a thing.) To try and be more helpful, I'll suggest you don't actually want to learn math at all. So don't! At least, not directly. Instead, find something you want to learn more about in science, engineering, or technology/programming, and dig into it until you start hitting the math being used. For many things, especially at the introductory level, it's fundamentally no more complicated than being able to read a junior-high-school level equation. Occasionally you'll need to know about some functions like square root, or sine, or exponentiation, or some other new functions that will be explained (like a dot product) in terms of those things. When you don't understand something, you may need to find an outside reference (or a few) for it, if the book itself doesn't cover it enough or to your liking. Even then, you can often find outside presentations of that thing which are still motivated by the general field and are thus not proof-heavy. However sometimes the best explanation may still be found in a "pure" book just about the thing, and if you can get over whatever problem you have with proofs you can learn to see how they can be used to build your understanding of the thing in smaller pieces, not just as tools to say whether this or that is true or false. In other words, proofs can serve the same function as repetitive problem-solving exercises, and are often given as exercises for that reason. I'm a fan of the Schaum's Outlines series of books just for the sheer amount of exercises available in them, I just wish I had better self-discipline to actually do more exercises. Though they maybe aren't the best resources for a brand-new introduction to something. To give a small example, maybe you're interested in game programming, and eventually want to dive into studying 2D collision detection more specifically so you can implement it yourself instead of using someone else's library, so you might stumble on a copy of "2D Game Collision Detection: An introduction to clashing geometry in games". Its explanation of the dot product comes early (its whole first chapter is on basic 2D vectors), consisting of 2 diagrams and two code examples (the first mostly defining dot_product(), the second using it as part of a new enclosed_angle() function) and some text all over 2.5 pages. It gives things in programming notation instead of mathematical notation, apart from some ² squared symbols occasionally. It gives a few equivalences like a vector's dot product with itself is its length squared, shown as dot_product(v, v) = v.x² + v.y² = length², without proving them, and points you to wikipedia of all places if you want to know more about how that or another detail are true. Why learn it? It's used immediately after in explaining projection, and then later in collision detection functions. Generally that book is structured as: learn the bare minimum of vectors, use them to implement collision detection for lines, circles, and rectangles. I'm not saying this is a great book but it's representative of what you'll find that I think you're really after, which is motivated use of some bits of math. If you don't like that book's treatment of vectors, there are a billion other game programming books that cover the same thing as a sub-detail of their main topic, and maybe even better for you because it'd be grounded in e.g. a graphical application you've already got setup and running to see results rather than a standalone library. Or there's special dedicated math books like "Essential Mathematics for Games and Interactive Applications". Or you can go find dedicated "pure math" books on linear algebra if you want. Or maybe your junior high / high school math education was good enough you can more or less skip most of this and move on to something more interesting, like physically based rendering (https://www.pbr-book.org/ https://www.pbr-book.org/) which also of course has vectors and dot products with brief explanations. Or maybe you don't care at all about game programming, and want to learn about chemical engineering, or economics, or the mechanics of strength and why things don't fall down, or...
joe_the_user 5y ago
Math education has become torturously miserable in the US by moving extraordinarily slowly. You have multiple years of working on only slightly more complex equations and concepts and naturally people get sick of it. You have generations of teachers who barely know math and view it as a punishment, teaching kids and instilling the same views in them. And then you have outsider still saying "can't we have condense it and simplify it further so we won't have to learn all these useless abstractions" and the curriculum bends further this way. But these actual situation of math is that not understanding what's happening is the thing makes it an empty and unpleasant activity. Edit: Also, yeah, 90%-99% of math can be accomplished with some math software. It's just for the remaining small percentage of stuff you need some understanding and for a small percentage of that you need lots of understanding. So most of this seems useless but 99% correct is actually not enough in some significant number of technical situations, etc.
hardwaregeek 5y ago
I dunno, at that point, are you really doing math? Math is proof. Now, granted, nobody expects you to know exactly how everything is proved. But there is an expectation that you can prove most of the stuff. Otherwise, it's very easy to stray from the truth into the plausible-sounding, but incorrect. That said, you're absolutely correct that more justification and motivation is important. So much of math can be taught with problems from physics, computer science, etc. Perhaps a good book for you would be Concrete Mathematics by Knuth? I haven't read it but people swear by it.
- analog31 5y ago
  We're now at a point where college math departments have to offer an "introduction to proofs" for students who have never seen one. Even high school geometry is now taught with few or no proofs.
pnin 5y ago
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't see how you can be an intelligent tool user without at least a little curiosity about how your tool functions. Maybe you can apply your tool, even be highly effective, in certain instances. But this is inherently brittle knowledge. When the parameters of your problem change and you don't understand your tool well enough to adapt, you're lost. "Math for people who just want to use it" is very broad. What do you want to use it for? Physics, biology, chemistry, computer science? Sociology? Economics? There might be some shared stuff, but for all of these disciplines there is a vast space of mathematics that might be relevant. I think Eliezer Yudkowsky's idea of a book (series) covering "The Simple Math of Everything" is fantastic. I would love to read that book. https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple-math-of-everything https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple...
- brimble 5y ago
  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 math. 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).
  - syntheweave 5y ago
    I think it's a kind of rope that frays on both ends for systemic reasons: 1. Students in math courses and their parents grow to prefer(through the overall institutional constraints) to have a simple exercise that guarantees them credit - while actually doing math is a matter of crossing the Rubicon into tough puzzle-solving, and it needs some guidance for unexceptionable students to start enjoying. 2. Math teachers, particularly in the lower grades where qualifications are lower, have a harder time teaching concepts than they do exercises. And they are also incentivized to hand out a grade, preferably one that satisfies the parents. So no matter how the high level is set up, everything converges into giving the kids a worksheet to "plug 'n chug." Which is just a confusing, badly paced grind, and therefore an easy reason to hate math. Either you get it completely and are just sitting there chugging through the problem set, or you have no idea what's going on and it's due tomorrow so your grade rests on something you feel defeated by. I actually think that for the parts that are currently treated as rote memorization work, the curriculum should lean into it and treat it like learning the alphabet, with worksheets where you literally fill in the dotted lines repetitively; hand them out to everyone as a portion of the homework. And then the logic and critical thinking aspects need to proceed like a philosophy course, with interaction through a step by step process, not "get the answer in the back of the book". This element is something I've long thought could be automated in some degree with computer systems that let you play with the concepts, and therefore correct your thinking.
analog31 5y ago
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.
screye 5y ago
This is more of a university perspective, but...... > Are there any "math for people who just want to use it" tracks in math pedagogy It's called Industrial Engineering /s (but only kind of) It is tame as compared to a pure math or physics B.S. But, you pretty much cover the gamut of every math tool that is useful in the real world. From stats, supply chain, search, calculus to combinatorics and so on. Each concept is squarely grounded in a field where it is used. I was surprised at how well my mechE (usually the closest undergrad degree to Industrial) degree prepared me for applied math and ML coursework for my CS masters. In comparison, a lot of CS undergrad peers struggled in those courses.
jackthetab 5y ago
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 perspective in many other fields. Not once have I applied a proof to solve a programming problem. I've thought of publishing, i.e.blogging, examples that I've come across but that would just be a mish-mash of stuff I've read elsewhere with no overarching theme/framework. Besides, someone else must have done this, no? EDITED: Used the correct book title.
- brimble 5y ago
  > 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't see it (outside some rare niches where it probably is useful—coq exists, after all). Sure, the word "exhaustive" can apply both to accounting for all (reasonably) possible problems in a block of code, and also to proofs, but the former doesn't feel at all like working on proofs, to me, to pick just one example (and some posts have seemed to imply that accounting for e.g. edge cases is exactly one case in which experience with proofs come in handy, but man, they feel like very different and barely-related activities to me).
  - Jhsto 5y ago
    In a way, writing types is proof-esque. It establishes some basic correctness guarantees regardless of the programming language. Some concepts like refinement types make it seem more "mathy", by requiring quantifiers within programs: https://ucsd-progsys.github.io/liquidhaskell/blogposts/2019-10-20-why-types.lhs https://ucsd-progsys.github.io/liquidhaskell/blogposts/2019-... Someone without any background in the subject would probably find Dafny interesting.
  - 5y ago
kovac 5y ago
It depends on which department you take math course in. Any course offered in a math department, even statistics and probability will be treated formally there because that's what mathematics is. What you probably need is to take math courses from an applied mathematics department like physics or engineering. But math is a broad field so you're going to have to pick specific courses. For example, partial differential equations are quite common. But if you learn it in a math department it'll have proofs and full rigour just like a course in set theory. While some techniques for solving them will be covered, we mostly study the underlying mathematical structures, why certain techniques work, etc. If you take it in a physics department they'll teach you techniques and numerical methods to solve a certain class of problems like heat equations or fluid dynamics. But learning through direct applications will inevitably limit you to those techniques while learning things in full generality tends to make it easier to pick up specific techniques when needed.
j2kun 5y ago
I'm actually currently writing a book with exactly this emphasis! The working title is "Practical Math for Programmers," and the idea is to build a collection of 60-75, 3-page long, _compelling_ demonstrations of mathematics used in production settings, biased toward stuff a generalist programmer might find useful. Not going into proofs or foundations, but providing lots of references to further reading. I'm aspiring for it to be like a Hacker's Delight, or Programming Gems, but just for math that is genuinely useful. Read more here: https://buttondown.email/j2kun/archive/a-week-of-book-writing/ https://buttondown.email/j2kun/archive/a-week-of-book-writin... Sign up to hear when it's released here: https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e97df95d3a140b1a3f&id=fbe11f6b59 https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e9...
andi999 5y ago
Theoretical physicists use probably more math than other subjects. Generally the litarature to use math is called 'mathematical methods for physicists'. The book by Arfken is considered standard by many. Did you mean something like that?
eternityforest 5y ago
I am literally working on exactly that, as an project I'm calling Math: ELI5, all open source on GitHub: https://github.com/EternityForest/AnyoneCanDoIt/blob/master/MathELI5.md https://github.com/EternityForest/AnyoneCanDoIt/blob/master/... The catch is that I myself don't actually understand this well. It's just "What I learned googling stuff while working as a programmer. I've had help reviewing it but there could be errors. I'm trying to cover all areas of math that a nontechnical person, or typical non math focused programmer would need to know, but I treat actually doing any of it by hand on paper as an arcane thing for the really dedicated, so there's not really any excercises. Instead of actually teaching a real understanding of math, which I can't do, because I don't know it well, I just explain what people who do understand it use it for and why you might want to go actually learn it. I also have any historical math related stuff that I find interesting. I can't tell you how to derive or prove Euler's equation , but you don't need to know math to understand the emotional impact from a humanities perspective, and be amazed that all those constants fit together like that, and that someone could discover it. Ultimately, I think traditional math education has it totally right. My life would be so much better if I knew it, because there's jobs that seem to require exactly what they tech in math class. It's not directly useful for non STEM types, but the idea seems to be give everyone a head start since so many do want STEM jobs. I think you really do have to get to the being able to do proofs level to make use of it in the real innovative applications. The common everyday applications math people like to cite can usually have a dedicated software package. It's not like we still need to add two numbers on paper. If you want to build something, we have RealThunder's FreeCAD. Excel's Goal Seek and CAS systems do the stuff people say we will use algebra for. But if your in tech eventually you run into the wall and need to do something like a Kalman filter or calculate stresses in a bridge and you're screwed.
tshaddox 5y ago
Proofs certainly aren’t the only aspect of what one might call “understanding mathematics.” Aren’t proofs generally limited to very specific mathematical courses (at least until you get to the cutting edge where new results inherently require proofs)?
PennRobotics 5y ago
Street Fighting Mathematics. It's still a bit much to absorb, but you can't really ever say the author didn't attempt to describe each lesson in real world terms. You memorize or learn to graphically/logically derive a few things, like how log(1+x)≈x for small x or how you can sorta guess for most quantities or plots. (One book example figures the data capacity of a CD if you know its music storage using informed guessing of each conversion factor from seconds of music to bits.) As another example: powers of cos x from -pi/2 to pi/2... you can sketch cos x and then roughly sketch the second power, and then it's clear that the more times you do that, you get a bit more of a bell curve. One in the middle will always stay one in the middle, but you get less area with each iteration as the rest of the function approaches zero. If you wanted an integral, you eyeball where the plot is at 0.5, draw a full-height rectangle from the left 0.5 value and the right 0.5 value. Because the height is 1, the width is pretty close to the true area. You can decide at this point---if you want more precision---to visually guess if the tails or the main part of the function should have more area and adjust your answer. There's an Open Access PDF at https://mitpress.mit.edu/books/street-fighting-mathematics https://mitpress.mit.edu/books/street-fighting-mathematics There were lecture videos on MIT's Open Learning Library (and somewhere else before that), but they're currently down. The class site is https://ocw.mit.edu/courses/mathematics/18-098-street-fighting-mathematics-january-iap-2008/index.htm https://ocw.mit.edu/courses/mathematics/18-098-street-fighti... as well as http://web.mit.edu/18.098/ http://web.mit.edu/18.098/
- PennRobotics 5y ago
  The audio CD thing is pretty clever. Even if you don't know important factors like the maximum frequency, you can get a great guess based on what you already know... like knowing Freddie Mercury could sing four octaves, starting from probably somewhere above "transformer hum sound". You'd have to know each octave doubles in frequency. Side quest: When you play the bugle, the played frequency increases or decreases by MULTIPLES of the base frequency---NOT powers of 2. Suppose this base frequency is 250 Hz. There is an octave from 250 to 500, but there's a note between the octave from 500 to 1000 at 750 Hz, and a few notes between 1000 and 2000 Hz, which is the part of the musical scale something like Reveille is played. If Reveille jumped from octave to octave, it would just sound like the intro to Justin Hawkin's cover of This Town Ain't Big Enough. So, if you know transformer hum is 50 or 60 Hz and Queen's frontman starts his singing at 100 Hz, then he can sing up to 1600 Hz, or four octaves. Mentally recalling what his falsetto sounds like, you can imagine a really high-pitched guitar solo an octave above this, and you can still imagine what an octave above that would sound like. (Maybe you're getting close to dog whistle territory in your imagination.) This, then, is 6400 Hz you are imagining. The top of each sound wave to the top of the next is 6400 Hz. To record this, you'd need the top AND bottom of each sound wave, because the speaker cone moving from maximum to minimum displacement is how the sound is made. If you want to make sure you aren't accidentally recording the middle (zero crossing) of each wave, you can even take three or four or five samples per sound wave instead of two. It's a lot of thought, but you can reasonably decide that 25000 Hz is a good sampling rate for capturing much of the range of human hearing. Going too far beyond that, you're wasting storage space. A CD holds a bit more than an hour of music, or 3600 seconds. If you've listened to Dire Straits, Eagles, Cyndi Lauper, Metallica, David Bowie, Led Zeppelin, ELP, or nearly any other band, you're probably aware the recordings have independent left and right channels. Finally, each sample is going to be somewhere between "speaker fully retracted" and "speaker fully extended". With 5 bits, this gives 16 "stops" from the middle point to fully extended. But we know that music can get really quiet when it fades out, and a lot of volume knobs can go from zero to thirty and sometimes higher. When you have the volume at one, you can still tell the difference between loud parts and quiet parts, so you'd need an extra 5 bits just to get good dynamic range at loudest and quietest volume settings, or 10 bits. What happens when you double this? If you have 20 bits, you are probably close to wasting bits. You have a million places where the speaker coils can move to. For a speaker that moves a few millimeters, this means 20-bit resolution allows steps of a few nanometers. This is the scale of computer chips and color wavelengths. If you took the color blue and shifted its wavelength by a few nanometers, it would still be practically the same shade of blue! Without knowing about bit depth, you can reasonably assume 16 bits is good because it's a power of two and will give a lot of dynamic range. 8 would be too low. 32 is just wasteful. With 32 bits, a speaker capable of moving 1 cm end-to-end will have 10 carbon atom diameters of linear resolution. The ears are impressive, but I don't know they can differentiate the air displacement of (speaker cone area) x (ten carbon atoms). Even having 0 to 100 on the volume knob, this leaves 25 bits of range at each volume setting. This is audiophile (and arguably, snake oil) territory. So then, you can say 3600 seconds is pretty close to 3000 seconds, 2 channels is close to 3, 16 bits is close to 10, 25000 Hz is close to 30000 Hz... 3 x 3 x 3 x 10 x 1000 x 10000 ≈ 3,000,000,000. Since a byte has about 10 bits, divide by ten, and this yields a first approximation---based on logic reasoning of what we know---of 300 MB. It's wrong, but it's not "very" wrong. (It's off by a factor of two, not a factor of ten! Not bad for 4 rounded, intermediate conversion terms...) (The idea is to round each term to a value starting with 1 or 3, because multiplying 3 and 3 is close to 10. The reason 2 is close to 3: 10^(1/2) = 3.16. This states that a good midpoint of 1 and 10 is 3.16, because if you square each term, you get: 1, 10, 100. Now, 10^(1/4) = 1.78. This means that any value less than 1.78 would be closer to 1 after squaring, and any value higher will be closer to 10.) You can even take the analysis further and back-calculate things like how fast the CD might spin by guessing the track width and bit area, how long a track skip would be, whether the size limitation of the CD is due to optical or material properties, how far the laser would need to be to converge at one bit while being close enough that any deviation in the surface flatness doesn't send the return beam away from the sensor, etc. (This is all the info you'd probably use to begin the approximation if you weren't aware an audio CD holds an hour of music, like if you were asked in 1975 to "back of envelope" whether a compact, non-contact, vinyl-like, LP-length recording medium was possible.)
Aperocky 5y ago
There's even a package for that, it's called numpy.
crispyambulance 5y ago
> Are there any "math for people who just want to use it" tracks in math pedagogy? If you're talking about college level, "math for people who just want to use it", is basically all it is (outside of math departments and perhaps outlier curriculums in some elite places) and that's a problem. Learning just the applications without picking up the theorems and without a true understanding of the concepts makes more advanced work in whatever discipline one chooses more difficult. Why? Because you'll have to follow mathematical arguments and it's so much easier to do that when you got the background to fall back on. I think students before college need more focus on mastering the basics. They're rushed so much, and tragically, math is very cumulative. Any one who has tutored before will notice that it's disturbing how many people don't really understand how to manipulate fractions as they enter a calculus course.
xtiansimon 5y ago
History of mathematics is a cool and relatable subject in so far as you can imagine the problems of ancient Egyptian worker paid in bread, and land survey/architecture. problems to Nazi code breaking, yeah?

mjfl 5y ago

The list presented here, calculus, odes + pdes, linear algebra, is essentially that, mathematics for people who just want to use it. It's all undergrad level. There's several layers on top of this - set theory, rigorous probability theory, algebraic geomeetry, topology, that are less useful but interesting to mathematicians.

importantbrian 5y ago

I didn't take any of these classes personally, but I do remember in college there was a whole host of Calculus for Business and Economics type courses that were much more focused on practical application than theory. Maybe picking up some of those textbooks would be the way to go.

susanrigetti 5y ago

If you have a solid background in calculus, I'd recommend Zill's Advanced Engineering Mathematics, which is pretty much basic math for physicists and engineers (aka for people who need to "use it").

brimble 5y ago
Heh, I have a long-forgotten-due-to-complete-lack-of-use-for-15-years-straight background in calculus. Solid, it is not. Thanks for the tip, though.
- susanrigetti 5y ago
  In that case, I recommend starting out with Zill's Precalculus with Calculus Previews and then working through Stewart's Calculus: Early Transcendentals!
  - brimble 5y ago
    I'll check those out, thank you!

jackallis 5y ago

i second this. there should clear distinction between academic math and "real world usage" math.

Koshkin 5y ago
But the distinction is generally not as clear as you may think. (1) Much of the mathematics came from real world problems (so, in particular, one may get drawn into some kind of mathematical research and even end up discovering new mathematical facts). (2) Sometimes when applying mathematics one still needs to employ deduction (derive a formula, prove a statement one wants to rely on, etc.).
psyklic 5y ago
Having a deep (proof-based) understanding leads to more "real world" insights. So, there may not be a clear distinction.
Tomis02 5y ago
Mathematics is a social activity, in the sense that mathematicians create proofs to convince others that their findings are valid. Some proofs are easier to verify than others. Some proofs are "believed to be true" because no one has yet managed to prove them wrong yet. So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented. And just to be clear, there is little to no "real world usage" math without academic math.
- Koshkin 5y ago
  > create proofs to convince others Well, it is not only about people. You would also want to "convince" Nature (so that the aircraft you designed does not crash right after a takeoff), your own or, often, someone else's wallet (so that money is not spent on nothing).

helpfulclippy 5y ago

I did a degree in applied math. You’d think this would be “math you’ll use,” but the fact is that despite my program having a CS concentration, most of the stuff I did was not really applicable in practice. However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered. I really wish I’d spent a lot more time on proofwriting than say, vector calculus. Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.

auggierose 5y ago
Proofs are indeed very practical. For example, I believe Leslie Lamport mentioned somewhere that he only came up with the final version of Paxos once he tried to prove it, and noticed that some condition he assumed wasn't necessary at all.
- dhosek 5y ago
  The reasoning behind having geometry be the standard high school sophomore math class is that that’s the age where kids would be ready to do proofs. Except that curriculum designers seem to have forgotten this and except in honors classes, most sophomores don’t get taught proofs in geometry and instead get a set of inert rules about shapes that they have no use for.
  - whatshisface 5y ago
    Geometry uses a deduction-only basket of proof techniques that don't prepare students for proofs done afterwards. I would like to see it replaced by elementary number theory which naturally uses a lot of induction/recursion and has some uses for proof by contradiction.

joatmon-snoo 5y ago

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. The vast majority of them, I would assert, are simply pre-med weed-out courses. I still remember freshman year, showing up to the standard intro calc course, and dropping it as quickly as I could in favor of my uni's equivalent of Math 55 (i.e. the hardest u/g intro math course) because of how asinine I found the content...

antishatter 5y ago
Statistics converges to calculus at its core. Source: am statistician
jll29 5y ago
Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).
- eftychis 5y ago
  Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at. Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.

LolWolf 5y ago

Sure, here's a fun one ! https://web.stanford.edu/~boyd/vmls/ 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 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.)

brimble 5y ago
> 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.

Koshkin 5y ago

> "math for people who just want to use it" It's called "engineering mathematics." (The books by Stroud would be an excellent choice here.)

pphysch 5y ago

> Are there any "math for people who just want to use it" tracks in math pedagogy? Use it for what? That is the question. If you pursue the what, you will inevitably be exposed to genuine ways that mathematics may be employed by it. The academic standard for "learning math" is like "learning programming" by reading the C++ language/STL spec from front to back. No one productively learns programming that way, and even if someone did, they would hardly be well off when faced with a real-world production C++ codebase that follows $BIGCORP's inhouse programming style.

vharuck 5y ago
I agree. If anyone wants to study "math in practice," they should pick a science or engineering track they like. Math will be included.
deleted 5y ago
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k__ 5y ago

I think, the problem is, proofs are what makes math useful. This might sound a bit dense, but the alternative is what 90% of programmers do every day.

Koshkin 5y ago
I think one can get far enough by applying known mathematical facts. (Proof: elementary school math is useful.)
- auggierose 5y ago
  The problem is, usually you need to combine known mathematical facts to solve a problem. Now, how do you know that you combined them properly? Yup. You need to know what a proof is.
  - k__ 5y ago
    This. Applying mathematical facts isn't math, it's calculation at best. That's what most programmers already do. They throw things together in the hope they work without proof they got all edge cases covered. And it works pretty good, because often the edge cases aren't that bad.

dwohnitmok 5y ago

> I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first. The problem is that the answer will depend heavily from person to person and from field to field and often the most sensible answers require a mathematical maturity that creates a chicken-and-egg kind of difficulty. Do you care about engineering? Well you'll need some calculus for that. Do you care about prediction modeling? Well there's some stats you'll need for that. Do you care about finding patterns in the world? Well there's abstract algebra for that. Do you care about reasoning itself? Have fun with mathematical logic. But because humans have different motivations, there's no one-size-fit-all motivation-based approach. This is especially painful for mathematics because I think most people who have learned some amount of pure mathematics will relate heavily to what helpfulclippy says in a sibling comment: "However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?)." The skills of rigorous and abstract thinking that pure mathematics provides is both nearly-universally helpful, but also simultaneously as a result very difficult to motivate. "This will help you think better across everything you do" is lofty-sounding, but generally not a convincing sell unless someone is already curious. But it's true that being able to wrap one's mind around pure abstraction (after rattling off a rigorous definition for an abstract question: Question: "But what is X really?" Answer: "X is just that. No more, no less.") has ramifications for all that one does. And the most painful part of all of this is if you try to start by teaching the wonders of pure mathematics instead of all the messy, boring rote stuff, students' eyes are liable to glaze over even more because of the aforementioned chicken-and-egg issue with mathematical maturity. In this way it's similar to trying to motivate someone to read and write. The key that unlocks that interest for everyone is going to be different and it's very hard to explain the near-universal benefits that reading and writing bring to one's way of thinking (but I can always just have a computer transcribe it or read it aloud to me!) without some inherent curiosity in them.

cptaj 5y ago

I just want a place I can go to weekly for 15min to practice math. It keeps track of what I know and gives me exercises of stuff so I can retain the skills. I've forgotten 90% of the math stuff I learned

robotresearcher 5y ago
How about a textbook? At 15 mins a week, a couple of books would last a year.
Buttons840 5y ago
Khan Academy. Finish a course, retake the course exam once a year afterwards, fill in knowledge gaps as needed. Doing this should keep all undergraduate math fresh in your mind.
- throwamon 5y ago
  I wanted to like Khan Academy, but the pacing of the exercises is so frustrating. They become too easy too fast and then it's an extremely repetitive slog of useless computations. I wish for a service just like them but with many more difficulty levels and less (non-spaced) repetition.
  - lordgrenville 5y ago
    I recently came across brilliant.org, which I think aspires to be this. I didn't stick with it but it might be what you're looking for.

dwrodri 5y ago

Swinging by to plug my personal favorite resource for refreshing my self on Bayesian stats: https://www.youtube.com/playlist?list=PLwJRxp3blEvZ8AKMXOy0fc0cqT61GsKCG https://www.youtube.com/playlist?list=PLwJRxp3blEvZ8AKMXOy0f... I think statistics is by and large the most proportionally underrated subject proportional to its utility. A good command of stats and probability expands your power to use data to reason about answering questions. The channel author, Ben Lambert, has an alternative playlist where he uses some of the techniques taught in this playlist to solve problems in econometrics. However, a lot of what is taught here builds a great foundation for other domains, on everything from bioethics to data journalism to computer vision. Another great channel that focuses a bit more on the machine learning side of things is StatQuest with Josh Starmer: https://www.youtube.com/c/joshstarmer https://www.youtube.com/c/joshstarmer

eftychis 5y ago

Well you can think of it like going to the gym. You don't exactly see people doing squats during their daily life, but you can see the results of people having a good physical core. This needs to be explained further during education and motivated appropriately. We have a short-term utilitarian perspective, and we need to take a step back at times and recall that it takes time and lots of sculpting to transform a wood log to a art piece. As you can jog everyday for fun and/or for the challenge you can also jog to improve your physical health. And not doing proofs is like declaring a guy can weight-lift by just watching videos on youtube and never lifting a weight. Or a guy can "code" without writing a line of code.

konschubert 5y ago

You don’t need proofs, but you still need an intuition for why things work the way they do. Otherwise you will be lost as soon as you leave the textbook territory. Proofs are just one way to build intuition. The best way to learn applied maths and get intuition is an “Introduction to Maths for Physicists” 101 course.

narrator 5y ago

If you just want to learn math for purely practical reasons, Khan Academy (http://www.khanacademy.org http://www.khanacademy.org) is great. It might have added more lessons, but when I went through it, it went up through 1st year college calculus. The thing that makes it VASTLY better than most self study math programs or books is that there are hundreds of exercises that you can do, and see if you got the right answer. If you didn't, it will in most cases explain how to do the problem so you can try again with a completely different problem, so you're not just memorizing the answers. Another thing that makes it great is you can do a little bit a day, start and stop, and come back to it and it will remember your progress and where you left off. Khan is also a gifted teacher. Unlike a lot of math teachers, he has great pronunciation and handwriting and you can watch his lessons as many times as needed.

macrolime 5y ago
Where are there exercises? I just tried clicking on some linear algebra topic, but it seems there are just videos https://www.khanacademy.org/math/linear-algebra/alternate-bases https://www.khanacademy.org/math/linear-algebra/alternate-ba...
- narrator 5y ago
  They don't have exercise for some of the more advanced college level topics. You can see the exercises on the right side of the screen under "practice" for some of the more introductory topics: https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces#vectors https://www.khanacademy.org/math/linear-algebra/vectors-and-...

Siira 5y ago

The proofs can help immensely in remembering theorems and knowing their needed assumptions and application area. Mainstream math education is also already light on difficult proofs in introductory (undergraduate) texts. You can always skip the proofs anyhow.

Jach 5y ago

Yes, all tracks of math involve people using math. Proof writing is a part of use for a lot of it. There are also very many tools within the use of proof writing where a lot of education is around learning a bunch of tools so that you can better "recognize which tool to apply, then apply tool", in a future real-world use where you want or need to prove something. I'm a little snarky, but you have a broken idea of what math is. It's not even your fault. I don't even claim an unbroken idea for myself, I went through public education too, though I do think it's less broken. Somehow compulsory education has managed to get near universal basic literacy, but seems to have failed on whatever equivalent some sibling comments have hinted at exists for math or at least mathematical reasoning. A lot of algebra work taught for junior high can be understood as just a foundation to be able to understand later things (though you can of course use some of it directly as taught without having to learn more for every-day things like some boy scouting activities, or helping with putting together a garden or a fence, or programming -- and some of course is entirely useless). But instead of pushing algebra even earlier, states are instead moving to push it even later. (Let alone trying to spread awareness of even a hint of the subtle divide between more general algebra and analysis that a lot of STEM undergrads don't even really get a whiff of except maybe knowing it's often said to be a thing.) To try and be more helpful, I'll suggest you don't actually want to learn math at all. So don't! At least, not directly. Instead, find something you want to learn more about in science, engineering, or technology/programming, and dig into it until you start hitting the math being used. For many things, especially at the introductory level, it's fundamentally no more complicated than being able to read a junior-high-school level equation. Occasionally you'll need to know about some functions like square root, or sine, or exponentiation, or some other new functions that will be explained (like a dot product) in terms of those things. When you don't understand something, you may need to find an outside reference (or a few) for it, if the book itself doesn't cover it enough or to your liking. Even then, you can often find outside presentations of that thing which are still motivated by the general field and are thus not proof-heavy. However sometimes the best explanation may still be found in a "pure" book just about the thing, and if you can get over whatever problem you have with proofs you can learn to see how they can be used to build your understanding of the thing in smaller pieces, not just as tools to say whether this or that is true or false. In other words, proofs can serve the same function as repetitive problem-solving exercises, and are often given as exercises for that reason. I'm a fan of the Schaum's Outlines series of books just for the sheer amount of exercises available in them, I just wish I had better self-discipline to actually do more exercises. Though they maybe aren't the best resources for a brand-new introduction to something. To give a small example, maybe you're interested in game programming, and eventually want to dive into studying 2D collision detection more specifically so you can implement it yourself instead of using someone else's library, so you might stumble on a copy of "2D Game Collision Detection: An introduction to clashing geometry in games". Its explanation of the dot product comes early (its whole first chapter is on basic 2D vectors), consisting of 2 diagrams and two code examples (the first mostly defining dot_product(), the second using it as part of a new enclosed_angle() function) and some text all over 2.5 pages. It gives things in programming notation instead of mathematical notation, apart from some ² squared symbols occasionally. It gives a few equivalences like a vector's dot product with itself is its length squared, shown as dot_product(v, v) = v.x² + v.y² = length², without proving them, and points you to wikipedia of all places if you want to know more about how that or another detail are true. Why learn it? It's used immediately after in explaining projection, and then later in collision detection functions. Generally that book is structured as: learn the bare minimum of vectors, use them to implement collision detection for lines, circles, and rectangles. I'm not saying this is a great book but it's representative of what you'll find that I think you're really after, which is motivated use of some bits of math. If you don't like that book's treatment of vectors, there are a billion other game programming books that cover the same thing as a sub-detail of their main topic, and maybe even better for you because it'd be grounded in e.g. a graphical application you've already got setup and running to see results rather than a standalone library. Or there's special dedicated math books like "Essential Mathematics for Games and Interactive Applications". Or you can go find dedicated "pure math" books on linear algebra if you want. Or maybe your junior high / high school math education was good enough you can more or less skip most of this and move on to something more interesting, like physically based rendering (https://www.pbr-book.org/ https://www.pbr-book.org/) which also of course has vectors and dot products with brief explanations. Or maybe you don't care at all about game programming, and want to learn about chemical engineering, or economics, or the mechanics of strength and why things don't fall down, or...

joe_the_user 5y ago

Math education has become torturously miserable in the US by moving extraordinarily slowly. You have multiple years of working on only slightly more complex equations and concepts and naturally people get sick of it. You have generations of teachers who barely know math and view it as a punishment, teaching kids and instilling the same views in them. And then you have outsider still saying "can't we have condense it and simplify it further so we won't have to learn all these useless abstractions" and the curriculum bends further this way. But these actual situation of math is that not understanding what's happening is the thing makes it an empty and unpleasant activity. Edit: Also, yeah, 90%-99% of math can be accomplished with some math software. It's just for the remaining small percentage of stuff you need some understanding and for a small percentage of that you need lots of understanding. So most of this seems useless but 99% correct is actually not enough in some significant number of technical situations, etc.

hardwaregeek 5y ago

I dunno, at that point, are you really doing math? Math is proof. Now, granted, nobody expects you to know exactly how everything is proved. But there is an expectation that you can prove most of the stuff. Otherwise, it's very easy to stray from the truth into the plausible-sounding, but incorrect. That said, you're absolutely correct that more justification and motivation is important. So much of math can be taught with problems from physics, computer science, etc. Perhaps a good book for you would be Concrete Mathematics by Knuth? I haven't read it but people swear by it.

analog31 5y ago
We're now at a point where college math departments have to offer an "introduction to proofs" for students who have never seen one. Even high school geometry is now taught with few or no proofs.

pnin 5y ago

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't see how you can be an intelligent tool user without at least a little curiosity about how your tool functions. Maybe you can apply your tool, even be highly effective, in certain instances. But this is inherently brittle knowledge. When the parameters of your problem change and you don't understand your tool well enough to adapt, you're lost. "Math for people who just want to use it" is very broad. What do you want to use it for? Physics, biology, chemistry, computer science? Sociology? Economics? There might be some shared stuff, but for all of these disciplines there is a vast space of mathematics that might be relevant. I think Eliezer Yudkowsky's idea of a book (series) covering "The Simple Math of Everything" is fantastic. I would love to read that book. https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple-math-of-everything https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple...

brimble 5y ago
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 math. 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).
- syntheweave 5y ago
  I think it's a kind of rope that frays on both ends for systemic reasons: 1. Students in math courses and their parents grow to prefer(through the overall institutional constraints) to have a simple exercise that guarantees them credit - while actually doing math is a matter of crossing the Rubicon into tough puzzle-solving, and it needs some guidance for unexceptionable students to start enjoying. 2. Math teachers, particularly in the lower grades where qualifications are lower, have a harder time teaching concepts than they do exercises. And they are also incentivized to hand out a grade, preferably one that satisfies the parents. So no matter how the high level is set up, everything converges into giving the kids a worksheet to "plug 'n chug." Which is just a confusing, badly paced grind, and therefore an easy reason to hate math. Either you get it completely and are just sitting there chugging through the problem set, or you have no idea what's going on and it's due tomorrow so your grade rests on something you feel defeated by. I actually think that for the parts that are currently treated as rote memorization work, the curriculum should lean into it and treat it like learning the alphabet, with worksheets where you literally fill in the dotted lines repetitively; hand them out to everyone as a portion of the homework. And then the logic and critical thinking aspects need to proceed like a philosophy course, with interaction through a step by step process, not "get the answer in the back of the book". This element is something I've long thought could be automated in some degree with computer systems that let you play with the concepts, and therefore correct your thinking.

analog31 5y ago

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.

screye 5y ago

This is more of a university perspective, but...... > Are there any "math for people who just want to use it" tracks in math pedagogy It's called Industrial Engineering /s (but only kind of) It is tame as compared to a pure math or physics B.S. But, you pretty much cover the gamut of every math tool that is useful in the real world. From stats, supply chain, search, calculus to combinatorics and so on. Each concept is squarely grounded in a field where it is used. I was surprised at how well my mechE (usually the closest undergrad degree to Industrial) degree prepared me for applied math and ML coursework for my CS masters. In comparison, a lot of CS undergrad peers struggled in those courses.

jackthetab 5y ago

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 perspective in many other fields. Not once have I applied a proof to solve a programming problem. I've thought of publishing, i.e.blogging, examples that I've come across but that would just be a mish-mash of stuff I've read elsewhere with no overarching theme/framework. Besides, someone else must have done this, no? EDITED: Used the correct book title.

brimble 5y ago
> 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't see it (outside some rare niches where it probably is useful—coq exists, after all). Sure, the word "exhaustive" can apply both to accounting for all (reasonably) possible problems in a block of code, and also to proofs, but the former doesn't feel at all like working on proofs, to me, to pick just one example (and some posts have seemed to imply that accounting for e.g. edge cases is exactly one case in which experience with proofs come in handy, but man, they feel like very different and barely-related activities to me).
- Jhsto 5y ago
  In a way, writing types is proof-esque. It establishes some basic correctness guarantees regardless of the programming language. Some concepts like refinement types make it seem more "mathy", by requiring quantifiers within programs: https://ucsd-progsys.github.io/liquidhaskell/blogposts/2019-10-20-why-types.lhs https://ucsd-progsys.github.io/liquidhaskell/blogposts/2019-... Someone without any background in the subject would probably find Dafny interesting.
- 5y ago

kovac 5y ago

It depends on which department you take math course in. Any course offered in a math department, even statistics and probability will be treated formally there because that's what mathematics is. What you probably need is to take math courses from an applied mathematics department like physics or engineering. But math is a broad field so you're going to have to pick specific courses. For example, partial differential equations are quite common. But if you learn it in a math department it'll have proofs and full rigour just like a course in set theory. While some techniques for solving them will be covered, we mostly study the underlying mathematical structures, why certain techniques work, etc. If you take it in a physics department they'll teach you techniques and numerical methods to solve a certain class of problems like heat equations or fluid dynamics. But learning through direct applications will inevitably limit you to those techniques while learning things in full generality tends to make it easier to pick up specific techniques when needed.

j2kun 5y ago

I'm actually currently writing a book with exactly this emphasis! The working title is "Practical Math for Programmers," and the idea is to build a collection of 60-75, 3-page long, _compelling_ demonstrations of mathematics used in production settings, biased toward stuff a generalist programmer might find useful. Not going into proofs or foundations, but providing lots of references to further reading. I'm aspiring for it to be like a Hacker's Delight, or Programming Gems, but just for math that is genuinely useful. Read more here: https://buttondown.email/j2kun/archive/a-week-of-book-writing/ https://buttondown.email/j2kun/archive/a-week-of-book-writin... Sign up to hear when it's released here: https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e97df95d3a140b1a3f&id=fbe11f6b59 https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e9...

andi999 5y ago

Theoretical physicists use probably more math than other subjects. Generally the litarature to use math is called 'mathematical methods for physicists'. The book by Arfken is considered standard by many. Did you mean something like that?

eternityforest 5y ago

I am literally working on exactly that, as an project I'm calling Math: ELI5, all open source on GitHub: https://github.com/EternityForest/AnyoneCanDoIt/blob/master/MathELI5.md https://github.com/EternityForest/AnyoneCanDoIt/blob/master/... The catch is that I myself don't actually understand this well. It's just "What I learned googling stuff while working as a programmer. I've had help reviewing it but there could be errors. I'm trying to cover all areas of math that a nontechnical person, or typical non math focused programmer would need to know, but I treat actually doing any of it by hand on paper as an arcane thing for the really dedicated, so there's not really any excercises. Instead of actually teaching a real understanding of math, which I can't do, because I don't know it well, I just explain what people who do understand it use it for and why you might want to go actually learn it. I also have any historical math related stuff that I find interesting. I can't tell you how to derive or prove Euler's equation , but you don't need to know math to understand the emotional impact from a humanities perspective, and be amazed that all those constants fit together like that, and that someone could discover it. Ultimately, I think traditional math education has it totally right. My life would be so much better if I knew it, because there's jobs that seem to require exactly what they tech in math class. It's not directly useful for non STEM types, but the idea seems to be give everyone a head start since so many do want STEM jobs. I think you really do have to get to the being able to do proofs level to make use of it in the real innovative applications. The common everyday applications math people like to cite can usually have a dedicated software package. It's not like we still need to add two numbers on paper. If you want to build something, we have RealThunder's FreeCAD. Excel's Goal Seek and CAS systems do the stuff people say we will use algebra for. But if your in tech eventually you run into the wall and need to do something like a Kalman filter or calculate stresses in a bridge and you're screwed.

tshaddox 5y ago

Proofs certainly aren’t the only aspect of what one might call “understanding mathematics.” Aren’t proofs generally limited to very specific mathematical courses (at least until you get to the cutting edge where new results inherently require proofs)?

PennRobotics 5y ago

Street Fighting Mathematics. It's still a bit much to absorb, but you can't really ever say the author didn't attempt to describe each lesson in real world terms. You memorize or learn to graphically/logically derive a few things, like how log(1+x)≈x for small x or how you can sorta guess for most quantities or plots. (One book example figures the data capacity of a CD if you know its music storage using informed guessing of each conversion factor from seconds of music to bits.) As another example: powers of cos x from -pi/2 to pi/2... you can sketch cos x and then roughly sketch the second power, and then it's clear that the more times you do that, you get a bit more of a bell curve. One in the middle will always stay one in the middle, but you get less area with each iteration as the rest of the function approaches zero. If you wanted an integral, you eyeball where the plot is at 0.5, draw a full-height rectangle from the left 0.5 value and the right 0.5 value. Because the height is 1, the width is pretty close to the true area. You can decide at this point---if you want more precision---to visually guess if the tails or the main part of the function should have more area and adjust your answer. There's an Open Access PDF at https://mitpress.mit.edu/books/street-fighting-mathematics https://mitpress.mit.edu/books/street-fighting-mathematics There were lecture videos on MIT's Open Learning Library (and somewhere else before that), but they're currently down. The class site is https://ocw.mit.edu/courses/mathematics/18-098-street-fighting-mathematics-january-iap-2008/index.htm https://ocw.mit.edu/courses/mathematics/18-098-street-fighti... as well as http://web.mit.edu/18.098/ http://web.mit.edu/18.098/

PennRobotics 5y ago
The audio CD thing is pretty clever. Even if you don't know important factors like the maximum frequency, you can get a great guess based on what you already know... like knowing Freddie Mercury could sing four octaves, starting from probably somewhere above "transformer hum sound". You'd have to know each octave doubles in frequency. Side quest: When you play the bugle, the played frequency increases or decreases by MULTIPLES of the base frequency---NOT powers of 2. Suppose this base frequency is 250 Hz. There is an octave from 250 to 500, but there's a note between the octave from 500 to 1000 at 750 Hz, and a few notes between 1000 and 2000 Hz, which is the part of the musical scale something like Reveille is played. If Reveille jumped from octave to octave, it would just sound like the intro to Justin Hawkin's cover of This Town Ain't Big Enough. So, if you know transformer hum is 50 or 60 Hz and Queen's frontman starts his singing at 100 Hz, then he can sing up to 1600 Hz, or four octaves. Mentally recalling what his falsetto sounds like, you can imagine a really high-pitched guitar solo an octave above this, and you can still imagine what an octave above that would sound like. (Maybe you're getting close to dog whistle territory in your imagination.) This, then, is 6400 Hz you are imagining. The top of each sound wave to the top of the next is 6400 Hz. To record this, you'd need the top AND bottom of each sound wave, because the speaker cone moving from maximum to minimum displacement is how the sound is made. If you want to make sure you aren't accidentally recording the middle (zero crossing) of each wave, you can even take three or four or five samples per sound wave instead of two. It's a lot of thought, but you can reasonably decide that 25000 Hz is a good sampling rate for capturing much of the range of human hearing. Going too far beyond that, you're wasting storage space. A CD holds a bit more than an hour of music, or 3600 seconds. If you've listened to Dire Straits, Eagles, Cyndi Lauper, Metallica, David Bowie, Led Zeppelin, ELP, or nearly any other band, you're probably aware the recordings have independent left and right channels. Finally, each sample is going to be somewhere between "speaker fully retracted" and "speaker fully extended". With 5 bits, this gives 16 "stops" from the middle point to fully extended. But we know that music can get really quiet when it fades out, and a lot of volume knobs can go from zero to thirty and sometimes higher. When you have the volume at one, you can still tell the difference between loud parts and quiet parts, so you'd need an extra 5 bits just to get good dynamic range at loudest and quietest volume settings, or 10 bits. What happens when you double this? If you have 20 bits, you are probably close to wasting bits. You have a million places where the speaker coils can move to. For a speaker that moves a few millimeters, this means 20-bit resolution allows steps of a few nanometers. This is the scale of computer chips and color wavelengths. If you took the color blue and shifted its wavelength by a few nanometers, it would still be practically the same shade of blue! Without knowing about bit depth, you can reasonably assume 16 bits is good because it's a power of two and will give a lot of dynamic range. 8 would be too low. 32 is just wasteful. With 32 bits, a speaker capable of moving 1 cm end-to-end will have 10 carbon atom diameters of linear resolution. The ears are impressive, but I don't know they can differentiate the air displacement of (speaker cone area) x (ten carbon atoms). Even having 0 to 100 on the volume knob, this leaves 25 bits of range at each volume setting. This is audiophile (and arguably, snake oil) territory. So then, you can say 3600 seconds is pretty close to 3000 seconds, 2 channels is close to 3, 16 bits is close to 10, 25000 Hz is close to 30000 Hz... 3 x 3 x 3 x 10 x 1000 x 10000 ≈ 3,000,000,000. Since a byte has about 10 bits, divide by ten, and this yields a first approximation---based on logic reasoning of what we know---of 300 MB. It's wrong, but it's not "very" wrong. (It's off by a factor of two, not a factor of ten! Not bad for 4 rounded, intermediate conversion terms...) (The idea is to round each term to a value starting with 1 or 3, because multiplying 3 and 3 is close to 10. The reason 2 is close to 3: 10^(1/2) = 3.16. This states that a good midpoint of 1 and 10 is 3.16, because if you square each term, you get: 1, 10, 100. Now, 10^(1/4) = 1.78. This means that any value less than 1.78 would be closer to 1 after squaring, and any value higher will be closer to 10.) You can even take the analysis further and back-calculate things like how fast the CD might spin by guessing the track width and bit area, how long a track skip would be, whether the size limitation of the CD is due to optical or material properties, how far the laser would need to be to converge at one bit while being close enough that any deviation in the surface flatness doesn't send the return beam away from the sensor, etc. (This is all the info you'd probably use to begin the approximation if you weren't aware an audio CD holds an hour of music, like if you were asked in 1975 to "back of envelope" whether a compact, non-contact, vinyl-like, LP-length recording medium was possible.)

Aperocky 5y ago

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

crispyambulance 5y ago

> Are there any "math for people who just want to use it" tracks in math pedagogy? If you're talking about college level, "math for people who just want to use it", is basically all it is (outside of math departments and perhaps outlier curriculums in some elite places) and that's a problem. Learning just the applications without picking up the theorems and without a true understanding of the concepts makes more advanced work in whatever discipline one chooses more difficult. Why? Because you'll have to follow mathematical arguments and it's so much easier to do that when you got the background to fall back on. I think students before college need more focus on mastering the basics. They're rushed so much, and tragically, math is very cumulative. Any one who has tutored before will notice that it's disturbing how many people don't really understand how to manipulate fractions as they enter a calculus course.

xtiansimon 5y ago

History of mathematics is a cool and relatable subject in so far as you can imagine the problems of ancient Egyptian worker paid in bread, and land survey/architecture. problems to Nazi code breaking, yeah?