Does mathematics have a place in higher education? 70 points by cottonseed on July 29, 2012 | hide | past | web | favorite | 106 comments

 The future is computers, and computers work through tagged languages and symbolic systems. Everybody needs to understand formal symbolic systems. Math is the best way that I know of to do this.It used to be that to be a composer, first you learned the piano. The piano was thought to give one an appreciation of the nature and nuance of music. From there you could much more easily learn other instruments and then learn how to put them together into something beautiful.Math is the new piano. Learn symbolic manipulation -- and I'd say that means first-year calculus -- and you can work with the kinds of symbolic systems you'll find everywhere else in the modern world.I used to do some tutoring when I was a kid, and I think this level of understanding is achievable by 90% of the population given the right environment. The real question is "why are the structures and environments for education that we create so unable to accomplish this?"Dumbing down the system isn't the answer. It's not like you can dumb down the world to make up for your inability to prepare students for it.
 Why is first year calculus the standard for symbol manipulation? Algebra is also pure symbol manipulation, and the applications of it are more universal, whereas calculus is symbol manipulation over 2 functions, plus algebra; but those two functions have a lot of conceptual baggage that is not nearly as useful.
 I think abstract algebra (group, ring, field) is a better way to teach symbol manipulation. Or even logic. A proof of well-ordering theorem beats integration by parts if your goal is to teach deduction and symbol manipulation.Let's be honest: calculus is taught for its application in engineering, not to teach deduction and symbol manipulation. If that is your goal, there are more suitable maths to teach.
 You need to change the way math classes think of proofs. Every times I have seen a proof in math class, they managed to take a beautiful and elegant argument, and distill it down to the least number of characters that formally proves the statement. Presently, I am doing research work in (applied) chryptography, which as you might imagine involves reading a fair amount of modern math papers. All but one of the papers I read were easier to understand than the textbook proof for Pythagorean's thuerom.
 I agree. More motivations, intuitions, insights behind fewer proofs would be more useful than lots of proofs, if the goal is to teach mathematical thinking, not to teach particular math results.I think proofs are compressed because of lack of margins. Don't laugh! Authors seem to think they need to cover lots of essential results, and if you do motivations, intuitions, insights for all of them, already long math books will be impossible to hold in your hands. In my opinion the obvious solution is to omit most proofs and do far more non-formal discussion around important proofs. Which won't happen because their goal is to teach particular math results, not mathematical thinking, contrary to what they say.
 Because calculus is the first class where functions (e.g. derive, indefinite intergration) take a function as its input, and outputs a function.
 No, all functions do that. Consider the functions f(x)=x^2, and g(x). f(2)=4, but f(g(x))=g(x)^2. A more intuitive example would be abs(x)=|x|. For any numeric input, you get a numeric output, but for a functional input, g(x), you get g`(x) as an output, where g`(x) is a new function simmilar to, but possibly different from, g(x).
 > Everybody needs to understand formal symbolic systems.Everybody does not need to understand this and I dare (with fingers crossed) say a good percentage of those on HN don't either. There needs to be rationalization within the educational system for what is taught, with relevance to education for the sake of knowledge and for how material can be used for future employment prospects & general life skills. That may play to the lowest common denominator or the student mean, but that's largely and unfortunately what public education has become over the past fifty years.
 > Math is the best way that I know of to do this.But is it the best way educators know of to do this? Because at the end of the day, whether math is a good basic skill that pays dividends in logical thinking and symbolic manipulation (dividends beyond what other methods could deliver) is an empirical question. What you say sounds plausible, but theories can be plausible and dead wrong.
 I'm dreaming of a curriculum where students start to program their own computer algebra system in 9th grade (or earlier), say, using Python or Clojure. Starting by teaching it to solve simple equations for x, then proceed over the next 4 years by extending it to know about binomial formulas, solving quadratic euations, perform gaussian elimination, symbolic differentiation and integration, etc. I guess the world isn't ready for that idea, yet ;)
 Automatic differentiation by dual number! Really, after I learned it, I wondered why symbolic differentiation isn't taught this way by default. Good use case of operator overloading, by the way.
 The future is computers,Let's rather hope they're a passing phase, please.
 That is a really pretty analogy. My compliments.
 But is also flawed, because its premise is flawed.
 "School mathematics" is something almost completely different from mathematics. (For an elaboration on this idea, see http://duckduckgo.com/?q=lockharts+lament).Meanwhile, being good at (actual) math is rapidly become a prerequisite for having a bright economic future. School math is worth increasingly little (computers do it better than you). Whereas true math is deeply creative, and computers suck at it.This separation is also why so many programmers say "programming doesn't take math". It doesn't take school math. It rests deeply on true mathematical reasoning.
 My personal anecdote-backed theory is that the lack of proper math teaching (and yes, a little bit of old school drilling) lays the foundation of more advanced "school maths" subjects being perceived as hard: If you struggle with in-your-head multiplication, then the FOIL mnemonic for binomial multiplication is going to be hard, which makes algebra hard. If you can read the language of maths, you can see patterns and understand things in the same way you can when you can read an article.This continues to spill over until it becomes a negative selector, driving a preference for liberal arts type subjects over science and engineering, not by the positive virtues of the former, but by the perceived (and, probably at this point, real) difficulty of the latter - and further augmented by the fallacy that any college education is better than none.We're happy to drill reading in schools, but loath to drill even the simplest maths. Only the most abject failures are allowed to go through school without being able to read a newspaper article at a reasonable speed and give a summary, but if someone breaks down having to do 12 * 9 in his head, it's fine, he's just not very "sciency".Rant over.
 I had to google "FOIL mnemonic" to learn what it is. My reaction would be "why in ???? would you teach your pupils such a trick, given that it breaks down when generalizing to e.g. (x+3y+z)(2x+5y+z)? (anybody who could say "that's just ((x+3y)+z)((2x+5y)+z), so I'll just apply it multiple times would be able the much more general "just add all pairs (something from the left, something from the right; there ar #items on the left times #items on the right such pairs"Do they really teach that somewhere?
 They teach that everywhere. Just graduated high school here, most of my peers still use it. They even taught a special method for three term polynomials. Algebra II is the worst taught class of them all. People spend years after it still learning to factor. But yeah, FOIL is incredibly stupid, I was the only one that refused to call it that, and called it distributing instead (which it was).
 It applies to the period in algebra instruction devoted specifically to quadratic equations. It's a first step towards the concept of "multiply all combinations and combine like terms", taught at a time when solving and factoring quadratic equations are brand new subjects.I don't think FOIL should be given much weight as some sort of eternal mathematical truth, but it's a useful stop on the journey.
 Why is it useful? Isn't it simpler to just say "multiply everything with everything"? This is the way I was taught.
 Same reaction here. It also does not generalize to (x+1)(x+2)(x+3). It especially does not generalize to (x+y)^n, let alone (x+y+z)^n, which I think is more important. I think these kinds of specialized tricks are to be avoided at all cost. Math is about abstraction after all.
 Yes. It's a common teaching method (and not just in math) to teach simplified methods so that people can see them in action, and once students have some experience experimenting with them, delve into the underlying theory. I think it's a perfectly valid teaching method. Students should probably be past the point where they really think "Now I'll use FOIL" by the time they finish a class where it's used as a teaching method. But in any class, there are going to be students who don't really end up grokking the theory.
 But what is simplified about it? It isn't easier than`````` (a plus b) times 'whatever' equals a times 'whatever' plus b times 'whatever' `````` Which you can easily show geometrically, and which helps you generalize to products of sums of three terms, etc.
 What is the underlying theory? I've taken a lot of math courses significantly past the point where I learned FOIL and I don't remember it being explained.
 What's odd here is the line of demarcation between mathematics (or technical subjects in general) and liberal arts -- two of the quadrivium are arithmetic and geometry. And while we can probably agree that the influential aspects of astrology are no longer worth serious study (except, perhaps, from a historical, psychological or sociological perspective), the periodicity -- the astronomy -- of astrology was also considered part of a well-rounded education. (Music was the fourth element of the quadrivium.)Perhaps it would be best to refer to a completely non-technical education as trivial since it ignores more of the liberal arts than it incorporates.
 This is completely shocking to me. Is it really allowed not to be able to do 12 * 9 in your head after completing school? I am serious. (Since this is not the case in South Korea.)
 Nor around here when you're leaving primary school. So I assume it is a kind of hyperbole used by your parent. At the same time, however, not having any scientific literacy at all seems to be accepted or even glorifies in Western popular culture.Then again, other troublesome behavior is quite accepted as well, like, for example drinking alcohol. How often haven't I heard a parent proudly tell that his or her child just had her/his first beer.Some bad behavior seems just socially and culturally accepted. And changing culture is hard, so, even though there have been programmes stimulating STEM for decades now, there seem to almost no real improvement.
 Well, anecdotally, since that's all I have to rely on, yes. I'm fairly certain that if you were to survey the basic maths skills on an average liberal arts college in most of Europe or the US, you'd get a result that would shock you.
 My experience from the UK is that more kids can do 12 * 9 in their heads at the end of primary school than at the end of secondary. Possibly because by the end of secondary a good percentage of them are too drunk or off their heads on ketamine or methadrone (or whatever it is this week) to either understand the question, or to even understand which way up to hold the bit of paper with the question on it.
 Nicely put. I have increasingly come to the view that computer science really is a branch of applied mathematics. My most intellectually rewarding recent work has been on (programming) projects where I get a chance to use some computational geometry and linear algebra.It goes without saying that almost every non-trivial project requires serious thinking about time and memory complexity costs, and I've been surprised to find theory of computation useful when working even on simple parsers (nothing fancy from compilers classes, just straightforward reading of proprietary formats).
 I started programming when I was 9, mostly taught myself to convert between hex, binary and decimal at about 13. I say mostly out of fairness: high school did spend a day or two on it one year. Needless to say I dropped out in 9th grade; I learned things that would be applicable in the real world better out of school than in. I was making 45K/yr, starting salary, by my mid twenties.Despite that, I'm going to college currently for my degree. I just got an "A" in intermediate algebra, though it was an up hill battle the whole way. The struggle, aside from it being a 6 week summer class, was trying to convince myself that I wasn't wasting my time; that I would learn something applicable to life outside that or the next three requisite classes. Something, that is, besides helping my kids with their math homework.I've come to the firm conclusion that college math exists purely to perpetuate itself, like a virus in our education system; The only job it prepares someone for is to teach the damn class!
 I've found that it is nearly impossible for students to find any math useful until they really understand it. As a simple example, you don't need to know nor understand how to multiply. After all , you could just repeated addition and figure it out. However, there is no need for addition either; you can just count up for a while. point being, the more math you know, the more tools you have. you may never use the quadratic formula outside the classroom, but I use algebra all the time to make my life at both work and home easier. please excuse typing errors as I'm replying from my phone.
 side note: As a programmer I specialized in both data compression and cryptology.
 Math became ludicrously easy for me as soon as I realized how to think about it. It wasn't the same method that they were suggesting in school, and I don't know that my method would work for everyone.
 As someone who still haven't found a way to think about it I would be curious what you ended up with.
 I will be very much interested in learning what method did you use.After dropping from regular education 12 years ago, I have tried to learn mathematics multiple times with moderate success only.Please try to share your methods via a blog post/elaborate comment if you can spare some time.
 All I can finish today is a rather unhelpful rant. I'm very opinionated and angry about this stuff:http://guscost.com/2012/07/29/mathemantics/But I'll be sure to put together a follow-up instructional blog that skips the vitriol, as nobody should actually have to just believe me. I do think that the very general attitude toward each problem is the only constant feature, and each topic has to be treated as a brand-new exploration, so I'll need to coagulate a topic to start off with. I'm leaning towards degrees of freedom, coordinate systems and the magic of Euler's formula. Sound good?And if not, I'm serious about the last sentence. What topic gave you the most trouble when you tried to learn it?
 I have tried to learn discrete mathematics topic however I have never made progress past Calculus.I had multiple false starts on Linear Algebra, Probability,Graph theory,Set theory, and Logic. Somehow books on these topics tends to run into hundreds of pages with multiple exercises of similar type which kinda encourages me to drop it typically when it starts becoming repetitive.I am not sure how to cross this plateau.
 Well to be honest, I'm not that far along into probably some of those same books myself, and I still get derailed by random curiosities and end up on Wikipedia or ordering a new book. The latest was a tangent into filter theory and digital signal processing, for a real-time audio project. It's hard to anticipate everything I should be able to do but I have some idea of the general order of things to study personally. If I needed to go in depth about Linear Algebra (and I definitely do), first I'd recall whatever I know about vectors and vector spaces, transformations and matrices and systems of equations, and start gathering articles and seeing if I'm missing any critical linked topics. After I get confident enough, I'd probably skip right to some textbook topic that caught my attention, say normed vector spaces, and try to make do with backward-skipping as a last resort.Of course, that's just an artificial way of making the problem harder, basically, so that it's more interesting than safer but repetitive sets of exercises. The best way to be interested, by far, is to actually want to solve some problem. For now, anyway, I heartily recommend looking for problems that these disciplines can solve, and choosing one to solve yourself.
 In my experience Calculus was a tough nut to crack. This was mainly because it is usually taught in a way that forces you to chew lots of preliminary materials which seem completely unmotivated.But obviously, Calculus was not developed the way it is taught. I found historial approaches more motivating. That means either using infinitesimals (17th century approach), or starting from construction of the real number, say Dedekind cuts or Cauchy sequences (19th century approach).
 The original texts are invaluable too. Sometimes ideas don't make as much sense to me after they've been re-explained by a second person.
 If you're having trouble then I'd suggest that you pick up Calculus Made Easy by Silvanus P. Thompson. Best book I've read on the subject by a country mile.
 Here's the follow-up, I tried to explain a topic I'd like to teach, but there should be more pictures and interactive elements too:
 It would be interesting to understand how "school mathematics" displaced "actual mathematics" in our educational system.I suspect that part of explanation has to do with computers. The word "computer" was originally a job title (see, for example the Harvard Computers: http://en.wikipedia.org/wiki/Harvard_Computers). It changed meaning in the middle of the twentieth century when mechanical (and later electronic) computers displaced human computers.It seems like much of "school mathematics" is vocational training to work as a computer. Unfortunately, we don't employ people to do computation anymore because the cheapest and worst computer you can but will be thousands of times more productive at computation than a human being.If our mathematics curriculum was designed to train people to be computers, and that profession no longer exists, it's no wonder that school mathematics seems pointless and disjointed.
 I think "actual mathematics" was used to refer to abstract and/or high level mathematics, the sort a math major might take in college. While "school mathematics" was used to refer to arithmetic and algebra.
 If by "school mathematics" you mean algebra, then I completely disagree with you. Algebra is essential for almost every other mathematical concept, not to mention everyday tasks. Just because a computer can do it easily, doesn't mean that people should rely on them to do simple calculations when they could have easily learned how to do it on their own. That is terribly inconvenient and a waste of resources.
 She makes a good point about algebra students not understanding prerequisites. I worked for two years at a high school where students were having trouble with algebra. Almost invariably this was due to (a) an abysmal grasp of arithmetic; many students couldn't divide or even subtract reliably, leading to insurmountable frustation, and (b) the hand-wavy means by which algebra, and especially its archaic conventions (such as order of operations and elision of the times symbol) were taught.I once solved these problems for a small group of these children to whom I taught algebra from first principles: all grammar was explicit (no elision of parentheses or operators) and we used no numerals. The reduction rules then were able to fit down one side of a whiteboard, and after five one-hour sessions, self-professed math phobes were solving nontrivial equations.
 Interesting! This same problem occurs in calculus as well. Leibniz's notation is pretty offensive, for example: in basic calculus, d/dx is an operator. In differential equations, you suddenly begin treating dx as a separable algebraic term â€” and way too many teachers suck at explaining why this is suddenly permissible.
 I recommend Elementary Calculus: An Infinitesimal Approach to those interested in calculus. It treats dx as a term from the start, and matches my (and I suspect most people's) intuition of the subject much better than epsilon-delta. And it is actually completely rigorous, not hand-waving infinitesimals.
 The book is available as a free online pdf:http://www.math.wisc.edu/~keisler/calc.htmlChecking it out now.
 My high school has a relatively progressive math program, and we never learned d/dx as a function. We began calculus with limits, then started solving Lim[dx->0](dy/dx), then we started using d/dx as a shorthand so we did not need to pull a nested expression out to take its derivative. I never thought to think of d/dx as a function until I started doing functional programming.
 +1 on explicit grammar. In my recollection, that was the biggest source of frustration in studying math subjects: decoding the actual meaning of a math text fragment and separating random elisions from errors.
 It's funny how in grade school the multiplication symbol is omitted, only to be (more confusingly replaced) by a period later on when the expressions get complicated.I suspect it's the lack of arithmetic skill more than the conventions that are the heart of the problem. I think kids should be required to learn to do mental arithmetic before they're allowed to use calculators, but this may be impractical.
 I think kids should be required to learn X before they're allowed to do Y,Clearly you were never a kid.
 Why is it impractical? (It's the way we do it in South Korea.)
 In the United States denying children access to things seems to be quite impractical :-)I also worry that smart kids will work out that calculators are ubiquitous before they force themselves to become proficient at mental arithmetic (e.g. Google and Spotlight are both far more powerful calculators than anything I had access to as a student).
 What a marvelous stupid idea.Math and Applied Math is more than numbers and math for the sake of math. It is a way to bring your level of abstract thinking higher - much the same way learning languages does.Math is so much more than algebra and calculus. It will teach you pattern recognition and deduction - two highly sought after skills in engineering - but can be applied to any profession.If anything I believe we should bump up the level of math that high school children learn.
 You seem to imply that actual mathematical content is rather less important. Then why stick with algebra and calculus?Frankly, I agree with Hacker: quadratic equations are boring. Teaching number theory will do better job of teaching pattern recognition and deduction. More Fermat's little theorem, more Chinese remainder theorem, less quadratic equations. I found undergraduate number theory to be far easier and more fun than calculus, and you need no calculus to start on number theory.
 > You seem to imply that actual mathematical content is rather less importantNo. The point was that it is much more than what you call the content.
 "Nobody ever brags about not knowing how to read, but people brag all the time about not knowing how to do math."This is not just math, it's virtually every subject you learned in school. Imagine the following at a cocktail party:"I had to read Joyce, Whitman, and Hawthorne, but I never got the point of any of them.""I took a lot of history, it was a bunch of dates which I could never remember.""I took three years of Spanish, but I don't remember a word."Et cetera.
 All perfectly understandable. It's called "the path less taken."A lot of people just don't click with formalized education at all in their early years. They just couldn't throw themselves behind all those authors and facts they're "supposed to" be up on. Some of them are too busy learning informally by dealing with the world around them -- i.e. with things like nature, work, and you know, other people.Then slowly grow to form their own view of the world, and seek their own path to obtaining a more structured comprehension of it -- which might well mean starting (on their own volition) formalized education sometime in their 20s or 30s -- or perhaps not.A great many who choose this route, or have it forced upon them may fail or flounder -- but a great many also don't, and end up with accomplishments for others to read about or hear about in those books, classes and received intuition that some people think are the only valid starting points to figuring out how to make a dent in the universe.
 You're right, but he's claiming this at a lower level. I imagine it's more embarrassing to have to ask, "What does this say?" than to have to ask, "How much change should I receive?"
 My dream educational system destroys the concept of "subjects" - no more siloing knowledge within artificial boundaries. Students think things like "Algebra is Math and I hate Math so I hate Algebra" or "Math is my worst subject." It is an easy label to associate their failures with.An example of learning without boundaries:One day, I decide I want to play the guitar. I get my hands on one and fiddle around. Perhaps a brief history on the instrument will give me some better context as to why the instrument is shaped as it is, and what all the features are. An exposure to famous guitarists and their music improves my appreciation of music and gives me context to work with. As I continue to struggle with notes and basic music theory, it may be a good time to take a brief detour into the realm of physics - a lesson in simple harmonics. If interested, I can dive further into the required algebra relevant to harmonic equations. Hmm... maybe learning to use a computer program can help me figure out how messing with these waves changes the sound...And so it goes. A familiar story to most hackers, but sadly an approach that most educational systems do not encourage.Is this exploratory form of learning superior? My theory is that the associations formed between nodes of knowledge are necessary for any sort of actual deep learning to occur. By sandboxing subjects, the standard school curriculum limits the possible associations that students think are allowed, thus limiting the probability of successful links between nodes. Even an above average student can graduate from college, and be left with nothing but a bunch of barely reachable random islands of knowledge, floating further out to sea every day.In other words: school sucks, learning rules.
 I really love that idea. I think a lot of students which are bored by school (and/or feel left behind) would regain their natural curiosity in such an environment. I think the technical possibilities we have now make much more individual approaches to learning possible and computers can help with this cross-subject approach.Based on your example they could then go on to construct some instrument (say, a flute) by writing a little program and print the instrument with a 3d printer or have a rough version cut by a laser cutter. The question is: how to integrate ideas like that in the current school system? ...
 I can see an immersive virtual world being amazing for encouraging curiosity. This world, the information stored within it, and the humans who frequent it, would attempt to encapsulate all human knowledge. Sound familiar?We already have such a world - it is the Web. And we know it is a mess, a time sink, and a minefield. However, with focused effort, it does function as an amazing learning environment.Unfortunately, it is not a suitable alternative to school yet. Telling kids to drop out of school and hit the 'Net instead is most likely a bad idea.Perhaps the future role of Teachers, will be to function as Guides in this virtual world of information. Most awesome field trip ever!
 So you aspire to be a Renaissance Man?
 I guess you could put it that way.I see in his description more of a tech tree. It'd be more akin to what you'd see in Diablo 2, but expanded by a 1000. The tech tree method would also explain prerequisites. Ideally, if you can prove you have the knowledge of a node, you could pass through it and continue learning.No clue how we would go from factory farmed schools to that though. It's another educational pie-in-the-sky thought.
 A tree is a great analogy for the model I am talking about.I think refactoring current educational systems - let's call them assembly lines - into this model is very difficult, and perhaps not worth the energy. A traditional master/apprentice system might be a better base to build on top of. A master can provide the ad hoc guidance that is helpful during free exploration. With the Internet, a distributed master/apprentice system can become a reality.
 Nobody seems to understand what college-level math is. This isn't balancing checkbooks, folks. I loved being a math major, and I have trouble doing simple math. College-level mathematics is about as far removed from the "real world" as you can get. We convince ourselves that this learning is valuable because those who have certificates saying they've learned it get better-paying jobs -- because we've convinced ourselves that it's valuable.Don't get me wrong - math is awesome, and everyone, especially young people, should have the opportunity to explore it. Perhaps it's helpful to think of math as a cultural activity, which is mostly removed from practical concerns.
 I don't think she is talking about Math majors. She is talking about some requirements that other majors have to complete as part of finishing up their degree. Having said that, I know people in say computer science who struggled through Calculus not really caring for the subject. I can't imagine how things were for people in less technical disciplines. While Calculus is certainly important for CS majors (well probably about as much as say randomized algorithms), I can't help but ask why these classes are not replaced by a class in first order logic, especially if the end goal is to train people to think mathematically.
 I agree that formal logic is probably a more relevant math-y course than calculus for the average CS student. But how to design these requirements is a longstanding problem, partly due to disagreements over what should be in a "technical core", and partly due to staffing and issues of uniformity.Calculus is probably more relevant than symbolic logic for many kinds of engineering, for example, so the course is already being offered. It's also seen by (most) mathematicians as a more central subject, so it's what goes in the core if the mathematics department is given responsibility for designing the core. Additional complications arise over what to do with things like statistics (in some places it's in the math department, while in others it's a separate department).
 Oh, I am not disagreeing with the importance of Calculus for engineering. However, calculus as a core tool for say biology majors? While, I know that there are a ton of biological models (say in population dynamics) that require knowledge of calculus, I feel like for most people, this becomes one of those hate it and get through it classes and we really need to rethink its importance.
 No disagreement there, and especially the way it's normally taught now, which can feel more like a course on special-case methods for performing symbolic integration. There is some understanding you gain by knowing how to work symbolic integrals (the chain rule, integration by parts, etc., etc.), but I wouldn't put it near the top of the list of things all students must know. The actual integration can be done fine by Mathematica or Maple; what's important is what you do with that.
 Spot on. All I know is the situation at Berkeley, but I imagine it's similar at other universities. The lower-division math curriculum (particularly intro calculus and linear algebra) is under pressure from a bazillion different departments to cover such a wide range of topics that 90% of the material is useless to 90% of the students.For example, a friend of mine was a forestry major, and the only reason she needed calculus was for population modeling. She knows exponentials and logistics better than I do! But she had a hell of a time passing calculus because she had trouble with the other 90% of the course. Keep in mind that this is a discipline that until maybe a decade ago required zero math.Maybe at least at the lower-division level, applied math needs to be a bit more targeted...? But that would require us letting go of the notion that math is intrinsically edifying, and that people "ought to", "must", or "should" know it.
 I think the problem itself comes from Math departments. Typical undergrad math curricula involves throwing a mix of math at students in the hope of training them to get mathematical literacy. From personal experience, I suspect it works because of the pressure and the massive amounts of math you are exposed to over a short period of time. This clearly is not true for people in other disciplines which is why you have these haphazard courses like Linear Algebra and Calculus thrown at them in the hope that they "get math" and also "learn something useful". It is an ugly situation where a lot of people just end up hating math (which considering that their sample size is small is clearly a bad thing to do but then again, they are probably never going to learn statistics to figure that out :).
 Calculus is certainly not important for CS majors. Compared to, say, linear algebra.I always recommend http://steve-yegge.blogspot.com/2006/03/math-for-programmers... on math for programmers.
 >> Calculus is certainly not important for CS majors. Compared to, say, linear algebra.I dunno. I am trying hard to refrain from saying X is not important for C.S. because the nature of this site is that you are going to find a ton of C.S. people who point out that they would be crippled with the lack of X :-)Having said that, I read Steve's article many many years ago and upon reflection, I have realized that Steve is way better in Math than he lets on. Some of the techniques there are designed not for you to "get" Math but to get some sort of cultural awareness of Math (which of course is a fun pursuit on its own). The reason why casual foraging of a ton of math has never worked for me when I started learning math is because one of the reasons Math is incredibly hard is that it is inherently a personal journey. Getting math sometimes requires working at it patiently for weeks before you finally understand how to wrap your brain (which really means making connections with how you think about something that you previously learned and this new math) around it. This requires hard work mostly in the form of rather mind numbing problems which at first glance can be boring and feel like you are being told to "wash on wash off" (to use a Karate Kid reference). However, the subconscious is a powerful tool and helps by developing weird connections. As I got more and more trained in math, I realized these connections became much much easier. I suspect this is why math researchers are able to look at paper and quickly grasp the intuition behind it.
 Re: to get some sort of cultural awareness of Math, aka Liberal Arts degree in mathematicsThe Princeton Companion to Mathematics is the best tool I know for this job. This is a bedtime reading, no problem solving needed. A bit like reading, say, Python Modules of the Week. You'd need to actually use that module in your code to "get" it, but just knowing that some module exists is very valuable. Same goes for math.
 How the heck is linear algebra important for CS majors? Where in CS is linear algebra useful where calculus isn't?
 Also graphics.That said, it's absurd to make a dichotomy between the two. Both are essential. Maybe not at first glance at a simple CRUD app, but both get surprisingly handy once you want a bit more out of it.
 Machine Learning, Graphics etcOn the other hand calculus is also used in machine learning...Plus optimisation via calculus is sometimes a better choice then other cs specific methods.
 And calculus is used in graphics.
 Machine learning, for example.
 Machine learning algorithms, at least the ones that I used, needs convex optimization to fully understand them which needs real analysis which is just calculus with more mathematical rigor.
 Your are confusing Math versus Applied Math. We take what we want to use in the study of engineering, physics, economics, finance and political science - pure Math develops those tools and they're highly useful.Is Math of a college level necessary for everyone? Certainly not. Should algebra be necessary for everyone - I think so (I think the same way about calculus).Nothing is necessary to perform your day-to-day duties - not even arithmetic - people have been getting by and been leading fruitful lives for thousands of years without that sort of stuff. So when people talk about learning Math as a life-skill they're dead wrong.But if you want to anything more that just get by - if you want to ask questions and support your arguments, analyze problems and come up with answers you can trust you gotta, oughta, shoulda learn Math.
 Well, it's not an example of Betteridge's law ;-)The political science professor in question might consider the fact that while many college students struggle with basic math, they struggle just as much with basic literacy (most undergraduates can't write grammatical sentences, use punctuation, don't know how to write an essay, and have a hazy grasp of the idea of citation). Extending his argument we might consider removing History and English from the curriculum.
 Go read "How should mathematics be taught to non-mathematicians?" for a mathematician's take on the topic.
 Is this problem unique to mathematics? Consider natural sciences, history, geography.It's easy to make up real-world examples of when each could be an advantage, but it's hard to say anything is necessary. It is what you make it.
 The article is not really talking about mathematics. It is talking about numeracy, which is on par with literacy in importance in my opinion.
 So...why shouldnt we phase out Poetry too?Is there a definitive study showing that training in poetry makes for more competent assembly line workers and lower unemployment levels?
 I think we should commission such a study. Those of us with English degrees will be perfectly suited to conducting this survey and delivering the results in epic poem format.
 If there is anywhere where some poetry class is a requirement for graduation for all majors, that requirement probably should be phased out.
 We probably should.
 I would say that the basic college level math courses have helped me tremendously in non-obvious ways. Companies run on spreadsheets and formulas, and to better predict and analyze situations, it is easier if all parties involved have a common understanding.Our economies are huge, number driven in almost every aspect. To ignore that or turn out graduates are less well rounded would not be more productive as a whole.
 The thing about mathematical literacy and financial contracts (credit agreements etc) is that if those contracts are constructed in such a way as to deliberately confuse the average customer/borrower, then increasing the average level of education wont actually help. The contracts will just get even more confusing to compensate.
 > Yes, young people should learn to read and write and do long division, whether they want to or not.Ironic, since in our modern computerized world algebra is important and long division is useless. The writer himself probably carries a powerful computer in his pocket.
 A little coherence goes a long way.
 ...and James Joyce showed you can get lit profs to take what you write seriously without any coherence
 ...only in Finnegans Wake. And let's be fair to English professors, many of them don't take that book too seriously. :)
 Yes.
 No.
 '...people brag all the time about not knowing how to do math'. This is true, I think, mostly of Anglo-Saxon countries. It is quite a disgusting attitude. In what way is someone's life-chances or overall happiness degraded by a better understanding or skill in the use of numbers and higher mathematical concepts? I consider this anti-maths meme one of the most pernicious aspects of our culture.
 I think this is misunderstanding of cause and effect, and correlation does not imply causation. High math skill does correlate with nerdiness, and ceteris paribus, nerdiness does degrade your life chances or overall happiness. It does not mean high math skill alone, if you are not nerdy, will degrade your life chances, but well, they do not have high math skill to understand that. :)
 I wasn't so much thinking about innate mathematical ability, more the mathematical education most kids should receive. Disparaging attitudes towards maths skills, which is a general rule, is pretty harmful to society, I think.

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