Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in university. I studied EE not compsci. Things like algorithmic complexity I had to learn for myself and from Knuth. I'm trying to grok group theory right now to help with my understanding of crypto. Before this, it was never mentioned throughout my education, so I don't know what courses you would have had to take to learn that. The fact that I didn't even know group theory was important to crypto until after I had made the choice strikes me as a bad sign.
The common theme at every level is learning cherry-picked skills, before you're even told what the branches of mathematics even are. Everything seems disjointed because you're not taught to look past the trees for the forest. Most people infact, even technical folk, go through their entire lives without knowing the forest even exists. Any idiot can point to a random part of their anatomy and posit that there's a field of study dedicated to it. The same goes for mechanics or computer science. You just can't do that with mathematics as a student.
I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.
So many ideas seem closed to mortals because of the nature of mathematics.
Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.
This is SO TRUE.
The same thing happens to me regularly, and not just with "computer science" but with other technical fields, hard sciences, and mathematics. The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal.
For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions.
Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand when working and sharing "code" with some peers in the know but for stuff like wikipedia articles it makes things appear more complex and obscure than they should.
I don't mind if people call a local function variable "n" in some code, it's usually non ambiguous. But if you export a variable "n" in an external API you will be screamed at. Why is it ok for maths?
To give a quick example, the letter R in maths can mean the set of real numbers. It can also (with a different typeface!) mean Ramanujan summation. Oh wait, you're doing physics? Then R is the gas constant, silly.
Also, i is the imaginary unit. Except in physics, then it's j, because i is used for currents. Makes sense.
I've argued the same with a mathematician friend of mine. I hate academic papers because of their seemingly convoluted and backwards way of explaining things. His answer was that papers were not made to convey thoughts to laymen, they were made to communicate facts and proofs with as little ambiguity as possible, optimized for reading by other mathematicians. It's meant to be high bandwidth (hence the terse style and lack of intuitive explanation) and low ambiguity (hence the seemingly backwards order of explaining things and the pages of "a is blah, b is foo").
It's an interesting discussion because from the mathematician's perspective, they don't see why they should cater to anyone who doesn't bother absorbing the lingua franca and the method of delivery. Countering that is the philosophical argument that information should be as available as possible. Countering that is how practical and useful that is, and whether the cost / benefit would be worth it.
I still think we can have our cake and eat it too, but I'm not sure. I think if the purpose is merely to transmit proofs and axioms unambiguously, I think we can have a language that performs just that and nothing else. I think stuff like this exists, but I don't know why it isn't the standard to publish with it.
Then the explanation can reside alongside this unambiguous description, and can take whatever liberties it pleases.
I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.
It's like this all the time in math papers. It often seems in the end like the ideas themselves are fairly straightforward, and it shouldn't have taken that long to understand. I think, though, that if you actually sat down and tried to explain it in more intuitive terms, you'd find that you might not be able to. True, you could find a way to convey the general idea, but without the technical details, (1) while non-mathematicians may get a surface-level understanding faster, even mathematicians will not grasp the technical aspects, and (2) it will be very hard for anyone to extend your work or for anyone to apply it to another situation, so it will only be applied in the specific contexts that you explained it in.
Language isn't built to communicate math, so doing so effectively will be either difficult to understand or imprecise. Many people claim that it would be easy to explain deep math concepts with "a couple of block diagrams and a few short paragraphs"s, but I'd challenge them to write a textbook on abstract algebra, or topology, or something like that before they make that claim.
The reason is that math papers are usually tailored to another mathematician and thus abuses that to avoid a few communication pitfalls.
Firstly, it's broadly considered to be the case that mathematical ideas are not understood until you've gotten them "from a few different angles". Math builds upon itself so much that an idea may be almost useless on its own and produce true value in lying at the nexus between many convergent ideas. For instance, statistics as a field enjoys a very nice convergent point between logic, measure theory, and information theory (among others). Approaching it from all of these positions can lead to important mental breakthroughs and a paper or book author desires to appeal to these various "roads" to their topic. Without providing that context it might be said that the presentation is very sparse.
Secondly, almost conversely, each reader is likely to be more familiar with a subset of the possible roads to the author's topic of interest. By covering as many roads as possible as they approach their goal topic they provide more chances for the reader to pick an approach they find most comfortable and follow it (lightly ignoring the rest) to the goal. A simple block diagram might be the best way to present it to you, but only a private tutor could specialize their presentation so much.
---
There's an art to reading a math paper when you're an outsider to the primary topic. You want to breeze through the paper at a high level first, slowly collecting the exposition points which are most applicable to your own method of understanding. After that, iteratively deepen your reading while looking up topics which you feel you almost-but-perhaps-not-quite-enough understand. You can very easily read a paper and get enormous value while failing to connect to 60-70% of what's written.
> I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper...
I systematically ignore maths in CS papers. If the concept isn't described with code or pseudo-code at least, I will look at who cites the paper, and look for someone who has replicated enough in code. 90%+ of the time I can avoid dealing with the maths entirely, or get enough details that I can glean the rest without bothering trying to actually understand the maths properly.
Usually I find the maths tends to obscure vast amounts of missing details.
Of course, how well that works depends on the specific field.
Would anyone be interested in seeing a "Rap Genius for academic papers" to address the third paragraph of parent comment? To those who aren't familiar with Rap Genius, I basically just mean a place where academic papers can be annotated and explained in plain language by the community. Would you read/contribute?
> Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.
You have the benefit of hindsight. Everything is obvious in hindsight. All the research I've done is obvious and straight forward, if only I had known what I know now and would be able to draw a few simple diagrams.
That is, until you realise it's not. In those days and weeks spent deciphering mathematics you are actually learning a lot. I cannot count how many times I've been reading mathematics and struggled for weeks on a concept. Then one day it clicks and it all makes sense. Then I re-read the description again and the answer is clear as day. The answer was always there, I just hadn't learnt enough to appreciate it.
>So many ideas seem closed to mortals because of the nature of mathematics.
I disagree with this statement 100%. No ideas are closed because of mathematics. The ideas are only closed if you are not willing to put in the time.
Group theory is a standard topic in a
college course, say, at the junior level,
in 'abstract algebra'. So, the other
topics typically are rings, fields,
vector spaces, maybe the construction of the
rationals and reals. The course might
also toss in a little on number theory.
Anyone with a ugrad pure math major is supposed
to know what a group is and the early, standard
theorems.
Group theory was used by E. Wigner for the
quantum mechanics of molecular spectroscopy so
that at times some chemistry students want to
know some of group theory and group representations.
My ugrad honors paper was on group theory.
I published a paper showing how a group of
measure preserving transformations could lead
to a statistical hypothesis test useful for
'zero-day' monitoring for anomalies in
server farms and networks.
Group theory pops up occasionally. Get
a good, standard text in abstract algebra
or two or three and spend a few evenings.
If you get to Sylow's theorem, you are likely
deep enough for starters. Group theory is
very clean, polished stuff and can be fun.
Go for it.
Exactly, many lessons are written to be defended from assault, not to be inviting. A castle, while romantic, is not as comfortable as a hotel.
(My personal mission is to find/share the aha! moments that actually make the details click. Why do we force everyone to laboriously discover them for themselves? Can't want talk about the underlying insights directly?)
Maybe not the corruption angle so much; but special interests. Now every book is designed to not offend the politically correct Californians, or the religious right in Texas, and to convince people that they are keeping up with the latest fads - http://www.edutopia.org/muddle-machine
> Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus.
I've been a secondary math teacher in the UK and I want to defend this point a little.
The job of a secondary math teacher at this level is to teach everyone math, particularly including a majority who don't have a strong interest and won't go on to study more mathematics. You probably underestimate how difficult a job this is.
With that in mind, this is already quite a long list of diverse topics. You neglected to mention an introduction to number, up to the real numbers, perhaps because you now think it is obvious. You were taught that.
I personally try to teach 'looking forward': explaining how these concepts link towards what direction you might take in the future.
However, it's very difficult to cover the whole scope of mathematics and mathematical subjects. For example, I personally knew only a little about what's relevant to EE (although I've learned over time). But, it's not that you could skip anything from the school curriculum anyway - my general advice is that potential EE students need to show interest enough to study independently outside of school.
Something like algorithmic complexity, you should be learning from Knuth. Well done for that: there are not many educational experiences better than learning independently from someone who has devoted his life to making his subject accessible.
Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+)
For the benefit of those not familiar with the UK school system:
- Compulsory education goes up to the age of 16 (roughly 11 'grades')
- Many people continue at school until around 18 (an additional 2 years, called the 'sixth form')
- These final 2 years of school can be taken at a secondary school (~= high school in the US) or at a college (which may specialise in these 2 years only, or may go beyond). This college is not the same as a university, although there are some areas where they may compete for the same students
- Most people who enter university as undergraduates do so after completing these additional 2 years, and the exams which partially determine university acceptance are taken when people are about 17 or 18
I went through the UK education as well, did you do just E math? You can take Additional Math along with it, that covers complex/imaginary numbers as well as Poisson sequencing/binomial theorem.
The thing is, I use very little abstract math in programming. Doing analytic philosophy actually did more than mathematics. The math stuff I do in programming is super basic stuff like trigonometry (sometimes trig with differentiation if I want to be fancy and use inflexion on curves) for GPS triangulation etc. I've never had to do anything more complex than that
I'm a big fan of A Levels (my experience in late 90s) because of the depth they can bring. Between the ages of 16 and 18 I was exposed to group theory, n dimensional linear algebra, (basic) proofs, curvature, complex numbers, various series and convergence. It was very eye opening. And the same is true of virtually all disciplines. The insight into a discipline A Levels gives, at a quite young age, cannot be achieved with a broad syllabus where depth is diluted.
> group theory right now to help with my understanding of crypto
Im also trying to understand crypto better, from a mathematical perspective. Apart from group theory, what else should I have an understanding of before going further? Do you have any resources that you would recommend?
The progression of math knowledge in the US is more or less the same, since our high school is the equivalent to your last two years of secondary school combined with college.
I currently teach math to at-risk students. I don't read all of these submissions about math education, but I skim the comments on most of them. The comments people make change the way I teach math.
I have always done a decent job of teaching math. I focus on helping students understand concepts, even when they are focusing on mechanics. I use words like "shortcut" and "more efficient method" rather than "trick" when showing students more efficient ways to solve problems. I have students do problems and projects that relate to their post-high-school goals.
But with the routines of school life, I get away from the fun of math from time to time. The comments on these submissions often remind me to go in and just tell stories about math:
- "Hey everyone, did you know that some infinities are bigger than other infinities?"
- "Hey everyone, do you have any idea how your passwords are actually stored on facebook/ twitter/ etc.?"
- "Have any of you heard the story about the elementary teacher who got mad at their class, and told everyone to add up all the numbers from 1 to 100? One kid did it in less than a minute, do you want to see how he did it?"
Thanks everyone, for sharing your perspective on your own math education, and about how you use math in your professional lives as well. Your stories help.
I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety strikes, and they freeze up.
Some people find eventually find their way around this first road block, and future discontinuities in understanding become less stressful, and eventually understood to be a completely normal part the process.
But the usual experience is that a person's math confidence is blown and as the math truck barrels on ahead, they never catch up. They understandably accept the identity of not being "good at math".
What's missing in math pedagogy at most schools is a systematic way to deal with the discontinuities when they strike, especially that first time. We can prepare students to deal with that panic. The tough part is that the math teacher probably has 90 students on roster, but the discontinuity could hit pretty much any given lesson, for some given student.
I know so many people who have come back to intermediate math later in life and breezed through it, armed with intellectual confidence gained from other fields. They look back and wonder how they came to be so intimidated by math in their younger days. We've got to give younger people the tools and knowledge for overcoming this intimidation at a younger age. We've got to kill "I'm just not good at math".
Mathematicians are indeed lost and confused but in a very different way from beginning students. One must put in one's dues in what Terence Tao calls the "rigorous" phase before one can become productively confused in the "post-rigorous" phase.
http://terrytao.wordpress.com/career-advice/there%E2%80%99s-...
I completely agree about the power of math, and why programmers should learn it. There are two problems with math:
(1) Math is IMHO the worst taught of all academic subjects.
It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in this context. I have literally seen math texts that never explain the language itself, introducing reams of new math with no definitions for mathematical notation used.
I've looked for a good "dictionary of math" -- a book that explains every mathematical notation in existence and what it means conceptually -- and have never found such a thing. It's like some medieval guild craft that is passed down only by direct lineage among mathematicians.
Concepts are often never explained either. I remember struggling in calculus. The professor showed us how to do a derivative, so I mechanically followed but had no idea why I was doing what I was doing. I called up my father and he said one single sentence to me: "A derivative is a rate of change."
A derivative is a rate of change.
I completed his thought: so an integral is its inverse. Bingo. From then on I understood calculus. The professor never explained this, and the textbook did in such an unclear and oblique way that the concept was never adequately communicated. It's one g'damn sentence! The whole of calculus! Just f'ing say it! "A derivative is a rate of change!"
(2) The notation is horrible.
If math were a programming language it would be C++, maybe even Perl. There are many symbols to do the same thing. Every sub-discipline or application-area of mathematics seems to have its own quirky style of notation and sometimes these styles even conflict with each other.
Yet baroque languages like C++ and Perl at least document their syntax. If you read an intro to C++ book it begins its chapter on templates by explaining both what templates are for and the fact that type<int> means "type is templated on int."
Math doesn't do this. It doesn't explain its syntax. See point #1 above.
I agree wholeheartedly with how frustrating it is. I think part of the problem is that really great mathematicians are encouraged to stay as far away from teaching (and improving their teaching) as possible, and great teachers are often discouraged from pursuing more mathematics for a variety of reasons. And when I personally teach calculus I make sure to explain derivatives in the way you want in the very first day (before describing limits or anything else).
As to your second point, I think notation is a big problem, but it's a bit of a straw man. With very few exceptions that I doubt you would ever find yourself in, I have never met a professor or mathematician that would not explain notation if you asked (gladly stopping in the middle of a lecture or talk to clarify). There is still a lot of it, but every mathematician who is presenting the mathematics can explain the notation to any degree of precision you could ever want, and I have very few colleagues who have never stopped someone for this reason.
I think the bigger problem is trying to read mathematics by yourself, without the ability to ask questions. And even after understanding the notation, I feel programmers have bigger problems, which I've expanded more on in this post [1], the main difference between learning programming being there are simply more free and open resources for learning programming. This is probably because programmers invented the internet and filled it with their favorite content first.
But one point I make is that mathematical notation is inherently ad-hoc, and the only kinds of notation that stick around are the kinds that get used ad-hoc enough times to become standard. And even then people will make up their own notation for no other reason than that it's their favorite (Physicists are really good at this, and perhaps ironically it drives mathematicians crazy). Because of that (and because notation is introduced often to be rigorous, not to explain a concept) you're unlikely to ever find such a dictionary. Sorry :(
I highly recommend reading this. I didn't agree 100% with everything, and you probably won't either, but it's an excellent insight into what learning and doing math is about, and what it's like.
I'd love to read alternate viewpoints, but this is an excellent read.
The entire post was enjoyable but I found the last paragraph to have the most actionable advice:
What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion.
I really enjoyed this because it captures so much of the frustration that felt early in my programming career - especially in college when I had classmates several years my junior who were (as far as I could tell) mathematics and programming wunderkinds. I also think that this is the sort of rhetoric that should be used to begin teaching children basic mathematics and more advanced concepts as well, because I still recall many of my classmates in elementary and even highschool who simply felt like failures or that they weren't smart enough to understand things because they didn't "get" it the first, or fourth, or fiftyth time.
"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something."
This is true for all research.
And I don't mean just the physical sciences either. Historians and sociologists are also chronically "lost and confused." Otherwise it wouldn't be a topic worth of study.
This is why students who are "good at X", whether it be math, German, sports, or programming, may become frustrated when they find out that "good at researching X" is a very different matter.
I think it's a good point, but I still think the kind of lost and confused in mathematics is more embarrassingly extreme. Imagine a few hundred historians trying to discern when King George I died, and after 50 years of work they conclude, "All we know for sure is that it was between the day he was born and yesterday." A startlingly large part of mathematics feels like this.
And I think the reason is that "prevailing theories" mean nothing in mathematics.
I started off doing a combined maths and computer science degree.
With both computer science and maths you are chronically confused. The difference being with computer science it doesn't matter so much if you don't understand something, if you can get it to work you know you are on the right track. Maths is much more progressive, each proof builds on a previous one. So if you fail to understand one step you are screwed from that point on.
After the first year I realised I didn't actually enjoy being permanently confused and so I ditched the maths to focus on computers. I do regret this. It didn't take long at all before I forgot all that knowledge I had spent years sweating over.
I wish this post was around when I finished my undergraduate degree in Mathematics. I would have taken my adviser's advice to go to grad school. At the time, I remember telling him that I feel like a barely made it through the program. Apparently I wasn't alone. Amazing the difference 25 years and the internet makes.
This misses the dangerous part, which is mathematicians in groups can confuse each other into accepting ideas which are basically nonsensical, especially if the counter argument relies on some obvious but intuitive observation of reality but cannot be easily formalised within their chosen framework of the moment.
As a consequence of this it wouldn't surprise me if the overwhelming majority of maths was actually incoherent nonsense and that the people that understood this thought they were just very confused due to being shouted down all the time, when the really confused people are the ones oblivious to their own situation.
I'm going to be rather dismissive in my reply, and for that, I apologize, because I'm not quite sure how else to respond.
This is more or less a non-issue. Thanks to mathematicians building on Euclid for the last 2300 years, we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning. If you take a theorem that is accepted as proven, you can almost definitely follow an immense chain of logic back to the fundamentals. It will take you a ridiculous amount of time to do so, but it is possible.
If you're referring to specific debates in the math community (e.g. "I feel that the general math community accepting the axiom of choice was a bad idea") then that's worth being specific about in your post.
You can stop worrying now: what you're afraid of doesn't actually happen, though I sort of see how someone who hasn't spent their life studying mathematics might worry that it does. Since you have to back up your ideas with proofs, you can't in the long run hoodwink people into accepting false statements.
You also seem to worry about mathematicians accepting perfectly consistent sets of ideas even when those ideas contradict "inuititive observation of reality". To that I can only say that mathematics is not a subject where intuitive observation of reality plays any major role. What decides whether some piece of math is good or not is whether it is logically consistent, found interesting by people, and useful, either in other parts of mathematics or in applications to the real world. Notice in particular, that if the applications work no-one cares if part of the math leading to them contradicts any given person's intuition. For example, physics uses real numbers a lot and your intuition might tell you they don't make sense because there can't be uncountably many different things of any kind. But physics work extremely well and the math it uses is consistent, so we use it even if it doesn't sit well with a few people.
Since fidotron has been piled on, let me defend the point in his/her post. Good mathematics has come out of being worried that what other mathematicians have done isn't quite right, and I think the perspective of the article doesn't acknowledge that.
For example: Cantor's theorem is quite true, only cranks doubt it [1]. But many mathematicians take it to have the corollary that cardinalities greater than that of the natural numbers exist, which does not follow: it is perfectly coherent to say that constructions such as the power set of natural numbers do not exist as a definite whole, and so do not have a cardinality. These kinds of doubt have driven constructivism which has led to interesting work in topology, measure theory, and type theory, and led to such useful applications as calculators for exact real arithmetic.
Cantor's paradise seems to be coherent (likewise I would be deeply surprised if large parts of mathematics turned out to be misconstrued) but the assumptions of large cardinal set theory are grandiose and poorly justified, and yet for a long time those people who wondered if it was wise to embrace the whole edifice were marginalised. It seems that now there are many more mathematicians who are interested in revisiting this perspective [2].
To put Wiles' metaphor in perspective, it is good if some mathematicians step outside the mansion from time to time, to see if the superstructure is up to all the crashing about that happens in the dark rooms.
I'd be a lot more worried about the danger you mention if you could give even one example of that happening, ever. What ideas are mathematicians confusing each other into accepting that are basically nonsensical?
You're right that there is a big social aspect to mathematics that has bitten people in the butt many times throughout history.
But to say that an "observation of reality" should have any effect on existing mathematics is silly. Though mathematics might take inspiration from the physical world, it is removed from it by design. When some observation of reality disagrees with mathematics, that usually means the mathematical framework in question needs to be generalized or specialized, not altered.
This has happened over the years, for example, with measure theory, Fourier analysis, Lie theory, computational complexity theory, and many others.
Fair enough. I tried in my youth to solve every problem I came across. There were many I couldn't solve. It took a while before I developed the wisdom and discipline not to solve every problem no matter how long it took. By a while I mean decades. I sacrificed the possibility of family life, have stopped talking to my uncomprehending stepfather, and have kept my social interactions to an absolute minimum to pursue my consuming interest. (I mention this as a point of pride.) I find myself continually astonished by the ingenuity of solutions I probably could never have imagined after years of work. Perhaps, after a lifetime of effort that must be continually maintained, I have attained the level an entering freshman at Harvard. At this stage, I may be reduced at best to connoisseurship of some aspects of mathematics.
Now for some reflections on attitudes. Mathematicians sometimes act as if they believe that expertise in mathematics transfers to expertise in mathematics education. Suppose you are a sensitive student, lacking in confidence. You open Korner's beautiful book on Fourier Analysis, and the first thing you are greeted with is "This book is meant neither as a drill book for the successful student nor as a lifebelt for the unsuccessful student." Korner does not mention other references suitable for the successful and the unsuccessful student. You take this comment to mean that Korner would let the unsuccessful student drown. There is no implication, but this is the psychological import, the implicature. Why mention the unsuccessful student at all? Why not say who the book is for, without planting this gratuitous image in the reader's mind? It would take some time to return to this book, to get past the wonder at a mind capable of such an incidental, dismissive, off-handed acknowledgement of "the unsuccessful student."
You could say this is "overthinking." Such remarks, microagressions as they are termed today, "perpetrated against those due to gender, sexual orientation, and ability status", are sometimes revealed in the asides of mathematical authors [1].
And now if only mathematics educators would evaluate their students on the state of their confusion!
Reading good foundational text books carefully
is darned good advice. But for solving every
exercise before moving on, no, that's not a
good idea. Instead, be willing to be happy
solving some 90-99% of the exercises. For the
rest, guess, with some evidence, that they are
incorrectly stated, out of place, just too darned
hard, or some such. If insist on solving 100%,
then get on the Internet and look for solutions.
Next, if read some foundational text books, then
in each subject also
read several competing text books, perhaps just one mostly but also
look at least a little at the others for views
from 'a different angle' that can be a big help.
Why? Because likely no text book is perfect and,
instead, in some places is awkward, unclear,
misleading, clumsy, etc. So, views from a
'different angle' can make it much easier to
learn both better and faster.
His description of doing applications by just
getting what really need and forgetting the rest
can be done but is not so good. Instead, having
a good foundation helps a lot. And, commonly
for an application in an important field, there
really is some good material in that field that
should understand with the application. Else
risk doing the application significantly less
well than could have.
His description from Wiles is more or less okay
for doing some research but, really, not for
learning. And for research, more of a 'strategic'
overview, i.e., with the 'lay of the land',
would be good, i.e., for publishing not
just one okay, likely isolated,
paper but a series of better papers
that yield a nice 'contribution'.
This is true with many, many things. Very often it is the connections between ideas that yields the deep understanding, not the ideas themselves. Focusing too intensely on a single idea or subject results in not making connections and, consequently, not really understanding.
I agree. I think most if not everything he said can be applied to learning almost any discipline. I studied physics at university and I definitely recognize the authors description.
"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something."
That's true of everything. It's fear and anxiety that prevents a lot of people from learning and trying new things. I keep trying to tell students or family members when they are learning to do stuff on the computer, just right click everything, just google anything you can think of, don't worry about it being perfect, don't worry about breaking anything. You have to hold back showing them the "answers" or else they become dependent.
I think this is a good read, although I don't agree with all of it - I'm of the mind that there is immense value in being able to figure out difficult proofs. The process develops your logical ability.
That's what I tell people around me. Studying math is hard because it makes you feel stupid. You always feel lost, you always feel like you missed so many things when you're starting to learn a new thing, you always feel like your questions are stupid (until you get that the rest of the class is pointless as well).
Especially with talented professors (Lyon 1, France, the professors there are not really good educators, but they are geniuses), they make you feel bad for not understanding things that seem so simple to them.
Studying math is depressive if you take it too seriously.
It's strange to hear mathematics described more as a search for art and structure than computation. Unfortunately most of my math education was on the computational/applied side. I'm only getting into number theory and the more esoteric math later in life for fun. As a parent I think we can't let the school system destroy our kids love of math through too much rote learning. We have to make it fun for them. (Same with music btw)
Reminds me of this great quotation, which Oksendal places before the preface to his stochastic differential equations book:
We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things.
Posted outside the mathematics reading room, Tromsø University
[+] [-] nly|12 years ago|reply
The common theme at every level is learning cherry-picked skills, before you're even told what the branches of mathematics even are. Everything seems disjointed because you're not taught to look past the trees for the forest. Most people infact, even technical folk, go through their entire lives without knowing the forest even exists. Any idiot can point to a random part of their anatomy and posit that there's a field of study dedicated to it. The same goes for mechanics or computer science. You just can't do that with mathematics as a student.
I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating.
So many ideas seem closed to mortals because of the nature of mathematics.
[+] [-] cs702|12 years ago|reply
This is SO TRUE.
The same thing happens to me regularly, and not just with "computer science" but with other technical fields, hard sciences, and mathematics. The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal.
[+] [-] simias|12 years ago|reply
Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand when working and sharing "code" with some peers in the know but for stuff like wikipedia articles it makes things appear more complex and obscure than they should.
I don't mind if people call a local function variable "n" in some code, it's usually non ambiguous. But if you export a variable "n" in an external API you will be screamed at. Why is it ok for maths?
To give a quick example, the letter R in maths can mean the set of real numbers. It can also (with a different typeface!) mean Ramanujan summation. Oh wait, you're doing physics? Then R is the gas constant, silly.
Also, i is the imaginary unit. Except in physics, then it's j, because i is used for currents. Makes sense.
[+] [-] makmanalp|12 years ago|reply
It's an interesting discussion because from the mathematician's perspective, they don't see why they should cater to anyone who doesn't bother absorbing the lingua franca and the method of delivery. Countering that is the philosophical argument that information should be as available as possible. Countering that is how practical and useful that is, and whether the cost / benefit would be worth it.
I still think we can have our cake and eat it too, but I'm not sure. I think if the purpose is merely to transmit proofs and axioms unambiguously, I think we can have a language that performs just that and nothing else. I think stuff like this exists, but I don't know why it isn't the standard to publish with it.
Then the explanation can reside alongside this unambiguous description, and can take whatever liberties it pleases.
[+] [-] pcmonk|12 years ago|reply
It's like this all the time in math papers. It often seems in the end like the ideas themselves are fairly straightforward, and it shouldn't have taken that long to understand. I think, though, that if you actually sat down and tried to explain it in more intuitive terms, you'd find that you might not be able to. True, you could find a way to convey the general idea, but without the technical details, (1) while non-mathematicians may get a surface-level understanding faster, even mathematicians will not grasp the technical aspects, and (2) it will be very hard for anyone to extend your work or for anyone to apply it to another situation, so it will only be applied in the specific contexts that you explained it in.
Language isn't built to communicate math, so doing so effectively will be either difficult to understand or imprecise. Many people claim that it would be easy to explain deep math concepts with "a couple of block diagrams and a few short paragraphs"s, but I'd challenge them to write a textbook on abstract algebra, or topology, or something like that before they make that claim.
[+] [-] tel|12 years ago|reply
Firstly, it's broadly considered to be the case that mathematical ideas are not understood until you've gotten them "from a few different angles". Math builds upon itself so much that an idea may be almost useless on its own and produce true value in lying at the nexus between many convergent ideas. For instance, statistics as a field enjoys a very nice convergent point between logic, measure theory, and information theory (among others). Approaching it from all of these positions can lead to important mental breakthroughs and a paper or book author desires to appeal to these various "roads" to their topic. Without providing that context it might be said that the presentation is very sparse.
Secondly, almost conversely, each reader is likely to be more familiar with a subset of the possible roads to the author's topic of interest. By covering as many roads as possible as they approach their goal topic they provide more chances for the reader to pick an approach they find most comfortable and follow it (lightly ignoring the rest) to the goal. A simple block diagram might be the best way to present it to you, but only a private tutor could specialize their presentation so much.
---
There's an art to reading a math paper when you're an outsider to the primary topic. You want to breeze through the paper at a high level first, slowly collecting the exposition points which are most applicable to your own method of understanding. After that, iteratively deepen your reading while looking up topics which you feel you almost-but-perhaps-not-quite-enough understand. You can very easily read a paper and get enormous value while failing to connect to 60-70% of what's written.
[+] [-] vidarh|12 years ago|reply
I systematically ignore maths in CS papers. If the concept isn't described with code or pseudo-code at least, I will look at who cites the paper, and look for someone who has replicated enough in code. 90%+ of the time I can avoid dealing with the maths entirely, or get enough details that I can glean the rest without bothering trying to actually understand the maths properly.
Usually I find the maths tends to obscure vast amounts of missing details.
Of course, how well that works depends on the specific field.
[+] [-] vargas84|12 years ago|reply
[+] [-] cabinpark|12 years ago|reply
You have the benefit of hindsight. Everything is obvious in hindsight. All the research I've done is obvious and straight forward, if only I had known what I know now and would be able to draw a few simple diagrams.
That is, until you realise it's not. In those days and weeks spent deciphering mathematics you are actually learning a lot. I cannot count how many times I've been reading mathematics and struggled for weeks on a concept. Then one day it clicks and it all makes sense. Then I re-read the description again and the answer is clear as day. The answer was always there, I just hadn't learnt enough to appreciate it.
>So many ideas seem closed to mortals because of the nature of mathematics.
I disagree with this statement 100%. No ideas are closed because of mathematics. The ideas are only closed if you are not willing to put in the time.
[+] [-] graycat|12 years ago|reply
Anyone with a ugrad pure math major is supposed to know what a group is and the early, standard theorems.
Group theory was used by E. Wigner for the quantum mechanics of molecular spectroscopy so that at times some chemistry students want to know some of group theory and group representations.
My ugrad honors paper was on group theory.
I published a paper showing how a group of measure preserving transformations could lead to a statistical hypothesis test useful for 'zero-day' monitoring for anomalies in server farms and networks.
Group theory pops up occasionally. Get a good, standard text in abstract algebra or two or three and spend a few evenings. If you get to Sylow's theorem, you are likely deep enough for starters. Group theory is very clean, polished stuff and can be fun. Go for it.
[+] [-] kalid|12 years ago|reply
(My personal mission is to find/share the aha! moments that actually make the details click. Why do we force everyone to laboriously discover them for themselves? Can't want talk about the underlying insights directly?)
[+] [-] wisty|12 years ago|reply
Maybe not the corruption angle so much; but special interests. Now every book is designed to not offend the politically correct Californians, or the religious right in Texas, and to convince people that they are keeping up with the latest fads - http://www.edutopia.org/muddle-machine
[+] [-] tokenrove|12 years ago|reply
Which is why everyone, but especially programmers, should learn a lot more math.
[+] [-] ronaldx|12 years ago|reply
I've been a secondary math teacher in the UK and I want to defend this point a little.
The job of a secondary math teacher at this level is to teach everyone math, particularly including a majority who don't have a strong interest and won't go on to study more mathematics. You probably underestimate how difficult a job this is.
With that in mind, this is already quite a long list of diverse topics. You neglected to mention an introduction to number, up to the real numbers, perhaps because you now think it is obvious. You were taught that.
I personally try to teach 'looking forward': explaining how these concepts link towards what direction you might take in the future.
However, it's very difficult to cover the whole scope of mathematics and mathematical subjects. For example, I personally knew only a little about what's relevant to EE (although I've learned over time). But, it's not that you could skip anything from the school curriculum anyway - my general advice is that potential EE students need to show interest enough to study independently outside of school.
Something like algorithmic complexity, you should be learning from Knuth. Well done for that: there are not many educational experiences better than learning independently from someone who has devoted his life to making his subject accessible.
[+] [-] rahimnathwani|12 years ago|reply
For the benefit of those not familiar with the UK school system:
- Compulsory education goes up to the age of 16 (roughly 11 'grades')
- Many people continue at school until around 18 (an additional 2 years, called the 'sixth form')
- These final 2 years of school can be taken at a secondary school (~= high school in the US) or at a college (which may specialise in these 2 years only, or may go beyond). This college is not the same as a university, although there are some areas where they may compete for the same students
- Most people who enter university as undergraduates do so after completing these additional 2 years, and the exams which partially determine university acceptance are taken when people are about 17 or 18
[+] [-] rachellaw|12 years ago|reply
The thing is, I use very little abstract math in programming. Doing analytic philosophy actually did more than mathematics. The math stuff I do in programming is super basic stuff like trigonometry (sometimes trig with differentiation if I want to be fancy and use inflexion on curves) for GPS triangulation etc. I've never had to do anything more complex than that
[+] [-] zhte415|12 years ago|reply
I'm a big fan of A Levels (my experience in late 90s) because of the depth they can bring. Between the ages of 16 and 18 I was exposed to group theory, n dimensional linear algebra, (basic) proofs, curvature, complex numbers, various series and convergence. It was very eye opening. And the same is true of virtually all disciplines. The insight into a discipline A Levels gives, at a quite young age, cannot be achieved with a broad syllabus where depth is diluted.
[+] [-] broolstoryco|12 years ago|reply
Im also trying to understand crypto better, from a mathematical perspective. Apart from group theory, what else should I have an understanding of before going further? Do you have any resources that you would recommend?
[+] [-] unknown|12 years ago|reply
[deleted]
[+] [-] Figs|12 years ago|reply
"If I had more time, I would have written a shorter letter." -- attributed to Pascal.
[+] [-] acjohnson55|12 years ago|reply
[+] [-] japhyr|12 years ago|reply
I have always done a decent job of teaching math. I focus on helping students understand concepts, even when they are focusing on mechanics. I use words like "shortcut" and "more efficient method" rather than "trick" when showing students more efficient ways to solve problems. I have students do problems and projects that relate to their post-high-school goals.
But with the routines of school life, I get away from the fun of math from time to time. The comments on these submissions often remind me to go in and just tell stories about math:
- "Hey everyone, did you know that some infinities are bigger than other infinities?"
- "Hey everyone, do you have any idea how your passwords are actually stored on facebook/ twitter/ etc.?"
- "Have any of you heard the story about the elementary teacher who got mad at their class, and told everyone to add up all the numbers from 1 to 100? One kid did it in less than a minute, do you want to see how he did it?"
Thanks everyone, for sharing your perspective on your own math education, and about how you use math in your professional lives as well. Your stories help.
[+] [-] acjohnson55|12 years ago|reply
Some people find eventually find their way around this first road block, and future discontinuities in understanding become less stressful, and eventually understood to be a completely normal part the process.
But the usual experience is that a person's math confidence is blown and as the math truck barrels on ahead, they never catch up. They understandably accept the identity of not being "good at math".
What's missing in math pedagogy at most schools is a systematic way to deal with the discontinuities when they strike, especially that first time. We can prepare students to deal with that panic. The tough part is that the math teacher probably has 90 students on roster, but the discontinuity could hit pretty much any given lesson, for some given student.
I know so many people who have come back to intermediate math later in life and breezed through it, armed with intellectual confidence gained from other fields. They look back and wonder how they came to be so intimidated by math in their younger days. We've got to give younger people the tools and knowledge for overcoming this intimidation at a younger age. We've got to kill "I'm just not good at math".
[+] [-] yomritoyj|12 years ago|reply
[+] [-] api|12 years ago|reply
(1) Math is IMHO the worst taught of all academic subjects.
It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in this context. I have literally seen math texts that never explain the language itself, introducing reams of new math with no definitions for mathematical notation used.
I've looked for a good "dictionary of math" -- a book that explains every mathematical notation in existence and what it means conceptually -- and have never found such a thing. It's like some medieval guild craft that is passed down only by direct lineage among mathematicians.
Concepts are often never explained either. I remember struggling in calculus. The professor showed us how to do a derivative, so I mechanically followed but had no idea why I was doing what I was doing. I called up my father and he said one single sentence to me: "A derivative is a rate of change."
A derivative is a rate of change.
I completed his thought: so an integral is its inverse. Bingo. From then on I understood calculus. The professor never explained this, and the textbook did in such an unclear and oblique way that the concept was never adequately communicated. It's one g'damn sentence! The whole of calculus! Just f'ing say it! "A derivative is a rate of change!"
(2) The notation is horrible.
If math were a programming language it would be C++, maybe even Perl. There are many symbols to do the same thing. Every sub-discipline or application-area of mathematics seems to have its own quirky style of notation and sometimes these styles even conflict with each other.
Yet baroque languages like C++ and Perl at least document their syntax. If you read an intro to C++ book it begins its chapter on templates by explaining both what templates are for and the fact that type<int> means "type is templated on int."
Math doesn't do this. It doesn't explain its syntax. See point #1 above.
[+] [-] j2kun|12 years ago|reply
As to your second point, I think notation is a big problem, but it's a bit of a straw man. With very few exceptions that I doubt you would ever find yourself in, I have never met a professor or mathematician that would not explain notation if you asked (gladly stopping in the middle of a lecture or talk to clarify). There is still a lot of it, but every mathematician who is presenting the mathematics can explain the notation to any degree of precision you could ever want, and I have very few colleagues who have never stopped someone for this reason.
I think the bigger problem is trying to read mathematics by yourself, without the ability to ask questions. And even after understanding the notation, I feel programmers have bigger problems, which I've expanded more on in this post [1], the main difference between learning programming being there are simply more free and open resources for learning programming. This is probably because programmers invented the internet and filled it with their favorite content first.
But one point I make is that mathematical notation is inherently ad-hoc, and the only kinds of notation that stick around are the kinds that get used ad-hoc enough times to become standard. And even then people will make up their own notation for no other reason than that it's their favorite (Physicists are really good at this, and perhaps ironically it drives mathematicians crazy). Because of that (and because notation is introduced often to be rigorous, not to explain a concept) you're unlikely to ever find such a dictionary. Sorry :(
[1]: http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...
[+] [-] ColinWright|12 years ago|reply
I'd love to read alternate viewpoints, but this is an excellent read.
[+] [-] chwolfe|12 years ago|reply
What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion.
[+] [-] zacinbusiness|12 years ago|reply
[+] [-] dalke|12 years ago|reply
This is true for all research.
And I don't mean just the physical sciences either. Historians and sociologists are also chronically "lost and confused." Otherwise it wouldn't be a topic worth of study.
This is why students who are "good at X", whether it be math, German, sports, or programming, may become frustrated when they find out that "good at researching X" is a very different matter.
[+] [-] j2kun|12 years ago|reply
And I think the reason is that "prevailing theories" mean nothing in mathematics.
[+] [-] weavie|12 years ago|reply
With both computer science and maths you are chronically confused. The difference being with computer science it doesn't matter so much if you don't understand something, if you can get it to work you know you are on the right track. Maths is much more progressive, each proof builds on a previous one. So if you fail to understand one step you are screwed from that point on.
After the first year I realised I didn't actually enjoy being permanently confused and so I ditched the maths to focus on computers. I do regret this. It didn't take long at all before I forgot all that knowledge I had spent years sweating over.
[+] [-] bpyne|12 years ago|reply
[+] [-] fidotron|12 years ago|reply
As a consequence of this it wouldn't surprise me if the overwhelming majority of maths was actually incoherent nonsense and that the people that understood this thought they were just very confused due to being shouted down all the time, when the really confused people are the ones oblivious to their own situation.
[+] [-] pflats|12 years ago|reply
This is more or less a non-issue. Thanks to mathematicians building on Euclid for the last 2300 years, we have a system of mathematics built on a few basic principles (that you would not disagree with) and deductive reasoning. If you take a theorem that is accepted as proven, you can almost definitely follow an immense chain of logic back to the fundamentals. It will take you a ridiculous amount of time to do so, but it is possible.
If you're referring to specific debates in the math community (e.g. "I feel that the general math community accepting the axiom of choice was a bad idea") then that's worth being specific about in your post.
[+] [-] omaranto|12 years ago|reply
You also seem to worry about mathematicians accepting perfectly consistent sets of ideas even when those ideas contradict "inuititive observation of reality". To that I can only say that mathematics is not a subject where intuitive observation of reality plays any major role. What decides whether some piece of math is good or not is whether it is logically consistent, found interesting by people, and useful, either in other parts of mathematics or in applications to the real world. Notice in particular, that if the applications work no-one cares if part of the math leading to them contradicts any given person's intuition. For example, physics uses real numbers a lot and your intuition might tell you they don't make sense because there can't be uncountably many different things of any kind. But physics work extremely well and the math it uses is consistent, so we use it even if it doesn't sit well with a few people.
[+] [-] chalst|12 years ago|reply
For example: Cantor's theorem is quite true, only cranks doubt it [1]. But many mathematicians take it to have the corollary that cardinalities greater than that of the natural numbers exist, which does not follow: it is perfectly coherent to say that constructions such as the power set of natural numbers do not exist as a definite whole, and so do not have a cardinality. These kinds of doubt have driven constructivism which has led to interesting work in topology, measure theory, and type theory, and led to such useful applications as calculators for exact real arithmetic.
Cantor's paradise seems to be coherent (likewise I would be deeply surprised if large parts of mathematics turned out to be misconstrued) but the assumptions of large cardinal set theory are grandiose and poorly justified, and yet for a long time those people who wondered if it was wise to embrace the whole edifice were marginalised. It seems that now there are many more mathematicians who are interested in revisiting this perspective [2].
To put Wiles' metaphor in perspective, it is good if some mathematicians step outside the mansion from time to time, to see if the superstructure is up to all the crashing about that happens in the dark rooms.
[1]: https://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps (Postscript file)
[2]: http://homotopytypetheory.org/book/ has been very successful
[+] [-] tokenadult|12 years ago|reply
[+] [-] j2kun|12 years ago|reply
But to say that an "observation of reality" should have any effect on existing mathematics is silly. Though mathematics might take inspiration from the physical world, it is removed from it by design. When some observation of reality disagrees with mathematics, that usually means the mathematical framework in question needs to be generalized or specialized, not altered.
This has happened over the years, for example, with measure theory, Fourier analysis, Lie theory, computational complexity theory, and many others.
[+] [-] ChristianMarks|12 years ago|reply
Now for some reflections on attitudes. Mathematicians sometimes act as if they believe that expertise in mathematics transfers to expertise in mathematics education. Suppose you are a sensitive student, lacking in confidence. You open Korner's beautiful book on Fourier Analysis, and the first thing you are greeted with is "This book is meant neither as a drill book for the successful student nor as a lifebelt for the unsuccessful student." Korner does not mention other references suitable for the successful and the unsuccessful student. You take this comment to mean that Korner would let the unsuccessful student drown. There is no implication, but this is the psychological import, the implicature. Why mention the unsuccessful student at all? Why not say who the book is for, without planting this gratuitous image in the reader's mind? It would take some time to return to this book, to get past the wonder at a mind capable of such an incidental, dismissive, off-handed acknowledgement of "the unsuccessful student."
You could say this is "overthinking." Such remarks, microagressions as they are termed today, "perpetrated against those due to gender, sexual orientation, and ability status", are sometimes revealed in the asides of mathematical authors [1].
And now if only mathematics educators would evaluate their students on the state of their confusion!
[1] http://en.wikipedia.org/wiki/Microaggression
[+] [-] graycat|12 years ago|reply
Reading good foundational text books carefully is darned good advice. But for solving every exercise before moving on, no, that's not a good idea. Instead, be willing to be happy solving some 90-99% of the exercises. For the rest, guess, with some evidence, that they are incorrectly stated, out of place, just too darned hard, or some such. If insist on solving 100%, then get on the Internet and look for solutions.
Next, if read some foundational text books, then in each subject also read several competing text books, perhaps just one mostly but also look at least a little at the others for views from 'a different angle' that can be a big help. Why? Because likely no text book is perfect and, instead, in some places is awkward, unclear, misleading, clumsy, etc. So, views from a 'different angle' can make it much easier to learn both better and faster.
His description of doing applications by just getting what really need and forgetting the rest can be done but is not so good. Instead, having a good foundation helps a lot. And, commonly for an application in an important field, there really is some good material in that field that should understand with the application. Else risk doing the application significantly less well than could have.
His description from Wiles is more or less okay for doing some research but, really, not for learning. And for research, more of a 'strategic' overview, i.e., with the 'lay of the land', would be good, i.e., for publishing not just one okay, likely isolated, paper but a series of better papers that yield a nice 'contribution'.
[+] [-] nbouscal|12 years ago|reply
[+] [-] napowitzu|12 years ago|reply
[+] [-] j2kun|12 years ago|reply
[+] [-] helicon|12 years ago|reply
[+] [-] edtechdev|12 years ago|reply
That's true of everything. It's fear and anxiety that prevents a lot of people from learning and trying new things. I keep trying to tell students or family members when they are learning to do stuff on the computer, just right click everything, just google anything you can think of, don't worry about it being perfect, don't worry about breaking anything. You have to hold back showing them the "answers" or else they become dependent.
[+] [-] Bahamut|12 years ago|reply
[+] [-] baby|12 years ago|reply
Especially with talented professors (Lyon 1, France, the professors there are not really good educators, but they are geniuses), they make you feel bad for not understanding things that seem so simple to them.
Studying math is depressive if you take it too seriously.
[+] [-] mathattack|12 years ago|reply
[+] [-] GIFtheory|12 years ago|reply
We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things. Posted outside the mathematics reading room, Tromsø University