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What are the pros and cons of offline vs. online learning? In what scenarios are each useful?

Indian mythology have taught us that the children of kings are usually sent to GURUKULS or ashrams of intellectual gurus to attain knowledge and wisdom. Be it Lord Ram who was sent to guru Vashishta or be it Samrat Ashoka who was sent to guru Chanakya . The main motive of sending these princes to GURUKULS was to inculcate seeds of discipline, determination, sincerity, increase levels of concentration apart from learning.But now it seems that the trend has changed nine folds. Instead of going to coaching which is analogous to GURUKULS where experienced and knowledgeable teachers impart concepts, students wish to stay home and have an online session.It’s not what is right and what is wrong, it is about comfort - how comfortable is the student in online class or offline class and what are the merits and demerits of both the option. Both the options have their merits and de-merits, lets get to the key pointers one by one.One of the most important trait which a student must possess while preparing for an exam is having SINCERITY & CONTINUITY. Considering the offline mode of education, it guarantees continuity as there are regular classes which are conducted in stipulated time schedule. No matter what happens the class is bound to be conducted for say 3hrs or 4hrs what so ever the duration is allotted. However the scene is different when it comes to online mode, as here the student is “King of his Kingdom”, he can make things happen according to his whims and fancies. Due to flexibility in timings in online class the sense of continuity has to be attained by student which needs utmost dedication and self motivation from the student’s side towards the goal. And this continuity is usually NOT achieved or it is very difficult to achieve. A student can devote 3hrs of concentration in an offline class but devoting 3hrs in front of screen with earphones is very difficult. There are certain human body limitations like eyes in front of screen gets tired, brain gets tired, continuous usage of headphones irritates ears after sometime which is not the case in offline class. Learning is a continuous process and one cannot ignore the fact that -Long hours Continuity in studies is required to crack competitive exam and nothing can be better than offline class in this aspect.Next point is FAST FORWARD feature in online classes. Students should understand, we can watch a movie in fast forward mode but understanding concepts with increased speed of lecture can never happen. But because of long hours sitting in front of screen and listening lectures, usually there is a DECREASE IN PATIENCE LEVEL and student turns on the fast forward mode to finish the lecture and thinks the absorption of concepts will be at same level but this is not the case. Whereas, in Offline class there is no screen and considerable duration of breaks are given in between to maintain the concentration and patience level of students.We all have surely learnt one thing from the movie 3 idiots “ Agar dost fail ho jaye toh bura lagta hai , lekin agar dost first ajaye toh zyaada bura lagta hai “. So if the student has to taste the PEER MOTIVATION or COMPETITIVE SPIRIT then offline mode gives that. By simply watching the friend near you getting correct answers to the most of in-class worksheet questions, the student gets the zeal to work even more hard. However, in online mode student’s friend is he himself. In other words, to get the best output of online coaching student has to be self dependent and self motivated. Again doing all these by himself or herself is really a tough task, and through online mode of coaching that level of competitive spirit is very difficult to achieve. So, in this aspect also offline coaching gets an upper hand.The next thing which matters the most is COMMUTATION TIME: this is one of the deciding factors when the student have to choose between the two modes. The online mode saves a lot of travelling time in comparison to offline mode which can prove to be beneficial for serious aspirants as they can utilise this time in covering/ finishing some topics. But, staying in the nearest hostel to the coaching centre is a very good option to nullify this aspect also. As offline coaching definitely has upper edge over online so this factor of residing near the coaching institute will give all benefits of offline class which is otherwise difficult to achieve in online mode.Also, the online mode of education helps students to LEARN AT THEIR OWN PACE : This feature is specifically important for slow learners. Not only slow learners but every student can re-visit the session again in case a part is not understood at once. Whereas, in offline mode the student cannot retake the class - and when this is the case the students is ATTENTIVE in class and tries to grasp all concepts delivered at his/her best level. Though a repeat mode is there in online class but usually it is found due to this repeat mode student becomes less sincere and thinks there is another attempt to watch the same lecture. This decrease in level of seriousness and CONCENTRATION costs the final result and hence generally offline class takes thumbs up in this aspect also.Moreover, in the offline coaching class,the aspirants can have a DIRECT & PERSONAL DOUBT CLEARING SESSION with the concerned faculty for better understanding of concepts.It is a common scenario that best of the coaching institutes are available only in big cities or metropolitan cities. For such problems the online education mode is the saviour, as it provides a path for the students from small towns and cities to access the best knowledge, delivered by the best faculty right on their home study table. Thus, the problem of RE-LOCATING to a new city is solved through online course.The above listed points are just parameters which aspirants must acknowledge, if they face dilemma while choosing between the two. Along with this the students must also look after the reputation of the institute, teachers and the content delivery. The better and experienced the teachers are, the better reputation the institute has and the chances to get through the targeted exam increases.Both the modes of education have their own pros and cons, one must be very alert while choosing amongst the two. Online courses of small duration can be good but for basics and fundamentals long term, comprehensive classroom courses are definitely better option.In the end what matters is the student’s selection and student is the one who can make it happen.All the best and keep working hard to accomplish your goals.

Why is Python so popular despite being so slow?

Yes, it can be up to 200x slower than C family. It's not just that its interpreted, since Lua can also be interpreted but is much faster. Mike Pall, the Lua genius, has said that it's because of some decisions about language design, and at least one of the PyPy guys agreed with him.Have tracing JIT compilers won? (read the whole thing).By the way, I should note that although web apps isn't my thing, latency does matter there too. Although even slow sites are relatively fast these days, I notice that we haven't yet hit the point where faster no longer matters. For some older comments from days where the numbers were bigger see here:Marissa Mayer at Web 2.0Google VP Marissa Mayer just spoke at the Web 2.0 Conference and offered tidbits on what Google has learned about speed, the user experience, and user satisfaction.Marissa started with a story about a user test they did. They asked a group of Google searchers how many search results they wanted to see. Users asked for more, more than the ten results Google normally shows. More is more, they said.So, Marissa ran an experiment where Google increased the number of search results to thirty. Traffic and revenue from Google searchers in the experimental group dropped by 20%.Ouch. Why? Why, when users had asked for this, did they seem to hate it?After a bit of looking, Marissa explained that they found an uncontrolled variable. The page with 10 results took .4 seconds to generate. The page with 30 results took .9 seconds.Half a second delay caused a 20% drop in traffic. Half a second delay killed user satisfaction.This conclusion may be surprising -- people notice a half second delay? -- but we had a similar experience at Online Shopping for Electronics, Apparel, Computers, Books, DVDs & more. In A/B tests, we tried delaying the page in increments of 100 milliseconds and found that even very small delays would result in substantial and costly drops in revenue.Being fast really matters. As Marissa said in her talk, "Users really respond to speed."A major problem for the future is datasets keep getting bigger and at a rate much faster than memory bandwidth and latency improves. I was speaking to a hedge fund tech guy at the D language meetup last night about this. His datasets are maybe 10x bigger than 10 years ago, and memory is maybe only 2x as fast. These relative trends show no sign of slowing - so Moore's Law isn't going to bail you out here.. He found that at data sizes for logs of 30 gig Python chokes. He also said that you can prise numpy from his cold dead hands as its very useful for quick prototyping. But, contra Guido, no Python isn't fast enough for many serious people, and this problem will only get worse. It has horrible cache locality, and when the CPU has to wait for memory access due to the data not being in the cache you may have to wait 500 cycles.Locality of referenceThat's maybe something rather valuable - you want productivity and abstraction, but not to have to pay for it. Andrei Alexandrescu may have one answer:http://bitbashing.io/2015/01/26/d-is-like-native-python.htmlThe Case for DProgramming in D for Python ProgrammersIn the past few decades, pursuing instant gratification has paid handsomely in many areas. In a world where the old rules no longer applied and things were changing quickly, you were much better off trying something and correcting course when it didn't work rather than being too thoughtful about it from the beginning.That seems to be beginning to change, and increased complexity is a big part of that. Rewriting performance-sensitive bits of your code in C sounds like you get the best of both worlds. For some applications that may be true. In other cases you may think that you had walked into a trap - so gratifying to have your prototype working quickly, but it may be that before you know it the project is bigger than you imagined, and at that stage it's not so easy to rewrite bits (and as you do that you now have to manage two code bases in different languages and the interface between them, and keep them in sync).Cython with memory views also seems like a great option, until you realize that you can't touch Python objects if you are writing a library (whether for your own use or for others) there is some possibility that you might want to use your code without engaging the GIL (global interpreter lock) - ie in multi-threaded mode. So in that situation you may end up depending on external C libraries for some purposes as python is off-limits. And that's fine, but yet more complexity and dependencies.On the other hand, here is how you can call Lua from D (and vice-versa is equally simple). So you get the benefits of native code with productivity and low-cost high-level abstraction but can still use a JITed scripting language if it suits your use case.JakobOvrum/LuaDimport luad.all;void main() {  auto lua = new LuaState;  lua.openLibs();  auto print = lua.get!LuaFunction("print");  print("hello, world!"); } Here’s how you write an Excel function in D that can be called directly as a worksheet function (I wrote the library with my colleagues helping):D Programming Language Discussion Forumimport xlld;  @Register(ArgumentText("Array to add"),  HelpTopic("Adds all cells in an array"),  FunctionHelp("Adds all cells in an array"),  ArgumentHelp(["The array to add"]))  double FuncAddEverything(double[][] args) nothrow @nogc {  import std.algorithm: fold;  import std.math: isNaN;   double ret = 0;  foreach(row; args)  ret += row.fold!((a, b) => b.isNaN ? 0.0 : a + b)(0.0);  return ret; } My point is that it’s a false dichotomy. It’s not fast and painfully unproductive vs slow and productive. You can have both if you have a bit of imagination and are prepared and able to make decisions based on the relevant factors rather than social proof.What Knuth actually said is a little more nuanced than the soundbite that his words have become (often used in conversation to terminate thought on a topic, when a little time pondering one's particular use case would bear dividends). He was saying don't waste time worrying about little low-level hacks to save a few percent unless you know it's important.; he wasn't talking about big choices like which language (and implementation) you use.There is no doubt that the grail of efficiency leads to abuse. Programmers waste enormous amounts of time thinking about, or worrying about, the speed of noncritical parts of their programs, and these attempts at efficiency actually have a strong negative impact when debugging and maintenance are considered. We should forget about small efficiencies, say about 97% of the time: premature optimization is the root of all evil. Yet we should not pass up our opportunities in that critical 3%. A good programmer will not be lulled into complacency by such reasoning, he will be wise to look carefully at the critical code; but only after that code has been identified. It is often a mistake to make a priori judgments about what parts of a program are really critical, since the universal experience of programmers who have been using measurement tools has been that their intuitive guesses fail.Page on archive.orgConsidering performance as one factor when you pick which framework you will be wedded to isn't premature optimization. It's prudent forethought about the implications because it's easier to take the time to make the right decision today than to change it later.By the way, he also said in the same article (we tend to hear what we want to hear and ignore the rest):In our previous discussion we concluded that premature emphasis on efficiency is a big mistake which may well be the source of most programming complexity and grief. We should ordinarily keep efficiency considerations in the background when we formulate our programs. We need to be subconsciously aware of the data processing tools available to us, but we should strive most of all for a program that is easy to understand and almost sure to work. (Most programs are probably only run once; and I suppose in such cases we needn't be too fussy about even the structure, much less the efficiency, as long as we are happy with the answers.)Python that isn't too clever may be easier to understand than old-school C/C++, but I am not sure that this is always the case when heavy metaprogramming gets involved (and nobody forces you to write old-school C/C++ today). Static typing does bring benefits in terms of correctness and readability, and some very smart people have spoken about this in explaining their choices of less popular frameworks:Caml Trading talk at CMUThere simply isn't an answer that applies to everyone - it depends on your circumstances, and what you are trying to achieve. But I would encourage you in the medium term to consider the possibility that python isn't the only high-level productive language that may be used to solve general purpose sorts of problems. And some of these other languages don't involve this kind of performance penalty, are better on the correctness front, and can interface to any libraries you might wish to use.I've alluded to one already, but there are others. Lua may be too simplistic for some, but it is fast. Facebook use it in their torch machine learning library, and you can run this from an ipython notebook. It's a big world out there - popular solutions get chosen for a reason, but when things change the currently popular solution isn't always the best option for the future.Addendum: a self-proclaimed 'Python fanboy' complained that I did not answer the question in this response. I think I did, although it's true that I would score nul points on the modern A-level style box-ticking approach to scoring exams. Whether that is a negative thing depends on your perspective!Those who are watching closely will also notice that the question I answered is different from the one into which it has been merged. So if you object about that part, take it up with the guy that merged it.Obviously Python is popular because it's gratifying to get quick results (libraries help too), and until recently performance didn't matter much since processor speed and even the modest advances in memory latency and bandwidth leapfrogged for a while our ability to make use of them. So why not 'waste' a few cycles to make the programmer's life easier since you aren't going to be doing much else with them.One can't stop there though, because the future may be different. Sandisk will have a 16 Tb 2.5" SSD next year. It will cost a few thousand bucks, and isn't going to be in the next laptop I buy for sure. But you can see which way the wind is blowing, because when people have large amounts of storage available they will find a way to use it, and memory simply shows no sign of keeping up. They are talking about doubling capacity every year or two. So in 10 odd years thats 7 doublings, which is 128x bigger than present capacity. Yet memory might be only 2x faster. Looks like I'll be able to get Gigabit internet in my area soon enough (whether I'll move house to take advantage of it is yet to be decided). It's a matter of time before that's commonplace, I should think.On top of that, modern SSDs are pretty fast. You can get 2.1 Gb/sec sequential read throughput using an M2 1/2 TB drive that costs less than 300 quid. (That's raw data - possibly even higher throughput if the data is compressed and you can handle the decompression fast enough). Yet it seems like the fastest JSON parser in the world takes 0.3 seconds to parse maybe 200 Meg of data (so 600 Meg/sec). Parsing JSON isn't exactly the most expensive text-processing operation one might want to do. So it doesn't seem like one is limited by IO in this case, necessarily! And that's today. and trends are only going one way.What is the best language to use in those circumstances? How long do you expect your software to last?Addendum 29th October 2016.A paper published in January 2016 by the ACM observes the following. It may not be true for everyone, and may not be true for many for a while yet. But my experience has been that as storage gets bigger, faster, and cheaper, people find a way to use it and the size of useful datasets increase, and I think it truly is a case of William Gibson’s “The future is already here - just unevenly distributed”.Non-volatile StorageFor the entire careers of most practicing computer scientists, a fundamental observation has consistently held true: CPUs are significantly more performant and more expensive than I/O devices. The fact that CPUs can process data at extremely high rates, while simultaneously servicing multiple I/O devices, has had a sweeping impact on the design of both hardware and software for systems of all sizes, for pretty much as long as we've been building them.This assumption, however, is in the process of being completely invalidated.The arrival of high-speed, non-volatile storage devices, typically referred to as Storage Class Memories (SCM), is likely the most significant architectural change that datacenter and software designers will face in the foreseeable future. SCMs are increasingly part of server systems, and they constitute a massive change: the cost of an SCM, at $3-5k, easily exceeds that of a many-core CPU ($1-2k), and the performance of an SCM (hundreds of thousands of I/O operations per second) is such that one or more entire many-core CPUs are required to saturate it.This change has profound effects:1. The age-old assumption that I/O is slow and computation is fast is no longer true:this invalidates decades of design decisions that are deeply embedded in today's systems.2. The relative performance of layers in systems has changed by a factor of a thousand times over a very short time: this requires rapid adaptation throughout the systems software stack.3. Piles of existing enterprise datacenter infrastructure—hardware and software—are about to become useless (or, at least, very inefficient): SCMs require rethinking the compute/storage balance and architecture from the ground up.Addendum: March 2017Intel 3D XPoint drives are now available, although they aren’t cheap. Their I/O performance means it’s increasingly difficult to say that you’re necessarily I/O bound. Emerging storage today has 1,000 times better latency than NAND flash (SSD drives), and is only 10 times worse latency than DRAM. Overnight that means the bottleneck moved away from storage to processors, the bus, the kernel and so on - the entire architecture, but that includes applications and server processes. Guido’s claim that Python is fast enough may still be true for many applications. But not if you are handling decent amounts of data.These new storage technologies won’t change everything overnight. But they’ll get cheaper and more widespread quickly enough. And that will have implications for the future when it comes to making the right decisions about language choices. Because it’s empirically true that what’s possible as regards language implementations depends on awful lot on language design - they are intimately coupled. If you want to make the most of emerging storage technologies, it’s unlikely in my view that Python will in general be the right tool for the job, even if it was a decade back.Some people here say some things that appear to make sense but are simply not right. Python is slow not because it is interpreted, or because the global interpreter lock (GIL) gets in the way of python threads - those things only make it worse. Python is slow because language features that have been there by design make it incredibly difficult to make it fast. You can make a restricted subset fast - there’s no controversy about that. But what I say is also what Mike Pall, the LuaJIT genius has said, and the authors of PyPy agreed with him.Here is what the author of Pyston - the Dropbox attempt to JIT Python (they gave up because it was just too difficult) - has to say about why Python is slow.Why is Python slowThere's been some discussion over on Hacker News, and the discussion turned to a commonly mentioned question: if LuaJIT can have a fast interpreter, why can't we use their ideas and make Python fast? This is related to a number of other questions, such as "why can't Python be as fast as JavaScript or Lua", or "why don't you just run Python on a preexisting VM such as the JVM or the CLR". Since these questions are pretty common I thought I'd try to write a blog post about it.The fundamental issue is:Python spends almost all of its time in the C runtimeThis means that it doesn't really matter how quickly you execute the "Python" part of Python. Another way of saying this is that Python opcodes are very complex, and the cost of executing them dwarfs the cost of dispatching them. Another analogy I give is that executing Python is more similar to rendering HTML than it is to executing JS -- it's more of a description of what the runtime should do rather than an explicit step-by-step account of how to do it.Pyston's performance improvements come from speeding up the C code, not the Python code. When people say "why doesn't Pyston use [insert favorite JIT technique here]", my question is whether that technique would help speed up C code. I think this is the most fundamental misconception about Python performance: we spend our energy trying to JIT C code, not Python code. This is also why I am not very interested in running Python on pre-existing VMs, since that will only exacerbate the problem in order to fix something that isn't really broken.I think another thing to consider is that a lot of people have invested a lot of time into reducing Python interpretation overhead. If it really was as simple as "just porting LuaJIT to Python", we would have done that by now.I gave a talk on this recently, and you can find the slides here and a LWN writeup here (no video, unfortunately). In the talk I gave some evidence for my argument that interpretation overhead is quite small, and some motivating examples of C-runtime slowness (such as a slow for loop that doesn't involve any Python bytecodes).One of the questions from the audience was "are there actually any people that think that Python performance is about interpreter overhead?". They seem to not read HN :)Update: why is the Python C runtime slow?Here's the example I gave in my talk illustrating the slowness of the C runtime. This is a for loop written in Python, but that doesn't execute any Python bytecodes:import itertoolssum(itertools.repeat(1.0, 100000000)) The amazing thing about this is that if you write the equivalent loop in native JS, V8 can run it 6x faster than CPython. In the talk I mistakenly attributed this to boxing overhead, but Raymond Hettinger kindly pointed out that CPython's sum() has an optimization to avoid boxing when the summands are all floats (or ints). So it's not boxing overhead, and it's not dispatching on tp_as_number->tp_add to figure out how to add the arguments together.My current best explanation is that it's not so much that the C runtime is slow at any given thing it does, but it just has to do a lot. In this itertools example, about 50% of the time is dedicated to catching floating point exceptions. The other 50% is spent figuring out how to iterate the itertools.repeat object, and checking whether the return value is a float or not. All of these checks are fast and well optimized, but they are done every loop iteration so they add up. A back-of-the-envelope calculation says that CPython takes about 30 CPU cycles per iteration of the loop, which is not very many, but is proportionally much more than V8's 5.I thought I'd try to respond to a couple other points that were brought up on HN (always a risky proposition):If JS/Lua can be fast why don't the Python folks get their act together and be fast?Python is a much, much more dynamic language that even JS. Fully talking about that probably would take another blog post, but I would say that the increase in dynamicism from JS->Python is larger than the increase going from Java->JS. I don't know enough about Lua to compare but it sounds closer to JS than to Java or Python.”

How did Yasha Berchenko-Kogan become so good at math, and at what age did he start doing rigorous proofs?

Boredom. As a kid, I was often bored. As any bored kid knows, making up arbitrary rules and following them is a great way to pass the time. Consider the card game war. Kids play that, and if there's just one kid, you might find them playing against themselves. Playing war against yourself is pretty much the same as doing basic arithmetic problems. You might object that one is a fun easy game and the other is hard schoolwork, but that distinction is a lie. In either case, all you're doing is blindly following rules for no particular reason. Fortunately, nobody made that distinction for me, and so doing arithmetic problems took its place among playing outside with friends, reading fiction, and playing with legos as a way to keep myself occupied.Resources. The biggest resource is the fact that nobody sabotaged my natural childish curiosity to explore math. To be good at math, you have to do lots of math. To do lots of math without getting distracted, you have to have fun doing it. To have fun doing it, you can't have people around you telling you that math is work, because, eventually, you'll believe them.After that, there's fun books. Those arithmetic problems, they came from Russian elementary school math books that my mom gave me. Those books were designed with kids in mind, so they were fun to read and kept your attention. I just took a look at them to refresh my memory: They've got dialogue, characters, pictures, plenty of examples, and tons of problems, each one unique. You'd call them "word problems," but there they're the rule rather than the exception. The other kind of problem is lame, not fun, and also not particularly educational. What's the point of knowing how to add if you don't know when to add? I'm quite happy that folks at Art of Problem Solving have written the books in the Beast Academy Bookstore. It's good that there are now textbooks in English that take into account that for children to read a book, you need to make the book fun for children to read.While I'm on book recommendations, I'll also recommend the recently translated Math from Three to Seven, which my mom used when I was about that age.If you've got good schools, which my parents put a lot of effort into making sure I had, then that's all you need to get the ball rolling. Tons of practice with arithmetic in elementary school is enough for people to label you as "good at math," so they put you in touch with more resources, which make you better at math, which makes more people put you in touch with even more resources, and so forth.It's a snowball effect. My middle school had a math club, where I picked up some tricks. Those few tricks, along with an understanding of arithmetic, are enough to do well at the Mathcounts competition. When I qualified for nationals, our math coach was kind enough to work with me one-on-one until the competition. Those skills were enough to get me accepted at Canada/USA Mathcamp, where I learned more problem solving skills and got a head start on the math I'd learn in college. The problem solving skills got me into MOP, where I learned more problem solving skills, which made me do well in contests, which got me into good colleges, where I learned lots of math well thanks in part to the head start I got at Mathcamp. Doing well in college classes got me recommendation letters for research opportunities, which taught me mathematical writing and speaking skills and got me into good grad schools, where I'm learning even more math.I really admire folks that are top-notch mathematicians who only decided to do math when they were already in college. I feel I've had it easy. Thanks to the opportunities I've had, I've been doing lots of math my whole life. It'd be weird if nothing came of it.Self-Concept. When I was 13, a friend pointed out a guy and said, "You know the AIME contest? Well, that guy qualified for the next level, and, what's more, he won it!" I was pretty amazed. I thought that he must be some sort of math genius with superhuman math powers.At that time, all of my successes had come without much effort. Sure, I'd take opportunities that were thrown in front of me, but I wouldn't set goals and work towards them. After all, what was the point? I'd be the kid that's "good at math" and get praise whether or not I put in effort. On the other hand, winning national contests was for people with superhuman math powers, not for people like me. The thought that training would get me there didn't occur to me.That changed with MOP, the U.S. summer program that trains people for the International Math Olympiad. When I was 14, the organizers of the summer program, in addition to inviting the top 30 high school students, wisely decided to also invite the top 30 freshmen, something that they have done every year since.I did pretty terribly at MOP. There were 4-hour tests with 4 problems, and I'd bash my head against the problems and solve one or none of them. I'd run out of ideas well ahead of the time limit, and spend the rest of the test listening to music or napping to pass the time. However, I came away from it thinking that, since the older students are all going to graduate, all I needed to do to win was to get near the top of this group of 30 freshmen over the next couple years. The task was very concrete, and seemed doable, as long as I put in the effort.I signed up for classes on Art of Problem Solving (AoPS). I started a math club at my school so that I had a team to go to contests with and a group to do problems with. Gradually, my contest math improved.With academic math, I ran into the same issue in college: I became complacent in my abilities, while at the same time discounting more capable people as super-geniuses, rather than paying attention to the fact that they went to seminars and carried around a math book that they read when the conversation in the lounge wound down. Midway through graduate school, things are better on this front. I feel that, if I work hard, then I will get a PhD with decent results, and if I don't work hard, then I won't.I think it's important across the board, not just in math, to be aware of what you can realistically accomplish, but only if you put in the effort. To go far, you need to put in a lot of effort. In order to put in a lot of effort, you need to believe that the outcome will be different if you put in the effort compared to if you don't.Interest. One of my earlier memories is my mom teaching me to count to twenty. I also remember my dad teaching me the English words for numbers up to a hundred. I remember subtracting, needing to borrow from zero, and asking my dad what to do. He told me to figure it out on my own, but I didn't know where to start. Another time, I remember my answers to arithmetic problems not matching up with the answers in the back of the book, and my grandma telling me about how you do multiplication and division first, and then addition and subtraction.I remember my mom asking me to define a square. Or rather, she asked me to pretend I was telling someone what a square was over the phone. I said that it's a shape with four sides. She drew a random quadrilateral. I said, "No, the four sides have to be the same." She drew a rhombus. I said, "No, they have to meet at right angles!" She finally drew a square.I remember these things because they were important to me, because I found them interesting. Everybody learns how to count to twenty. Everybody also learns how to draw a house, how to tighten a screw, and how to swing on a swing. The ones that stick in your memory, though, are the ones that you cared more about at the time.No matter the opportunities and resources, a square peg isn't going to very happy or successful in a round hole, and I think it's important to take some time to pursue things without worrying about what you'll get out of it, and see where that takes you. Maybe it will look more square-ish. I think that most successful people didn't get there by trying to emulate other successful people. The might have had some role models, but mainly they were pursuing their interests, which let them work twice as hard as everybody else, which, along with some luck, made them come out ahead. You're not going to be better at being Steve Jobs than Steve Jobs was, but, with effort, you might become the best at being you.If I had been someone else, instead of doing math while bored, I would have taken that pencil and paper and drawn the things around me. If I had been yet another person, I would have taken apart any electronics (or goldfish) that I could get my hands on to see how they worked. In yet another world, I would have folded that paper up and built things out of origami units. As it was, I put some dots around a circle and connected them to make stars, before Vi Hart made it cool:That's the end of my mathematical life story, but I want to address the other parts of the question in a postscript.Rigorous proofs. Asking me when I started doing rigorous proofs is like asking a writer when they started doing five-paragraph essays: It's missing the point of the endeavor. I recommend reading the blog post, There’s more to mathematics than rigour and proofs, by Terry Tao. Terry Tao talks about pre-rigorous, rigorous, and post-rigorous phases in mathematical thinking. Here's how it looks for writing:Pre-rigorous:I had cereal and orange juice for breakfast and for lunch I had a mushroom stew for dinner I had baked chicken and mashed potatoes and also a strawberry rhubarb pie which was the best ever, oh and I forgot also for lunch I also had fried rice and lemonade.Rigorous:I had three meals today. First, for breakfast, I ate cereal with skim milk, and I drank orange juice with pulp. Then, for lunch, I ate a mushroom stew with carrots, potatoes, and onions, I ate fried rice with mixed vegetables, and I drank lemonade. Finally, for dinner, I ate baked chicken with thyme and paprika, I ate garlic mashed potatoes, I drank water, and for dessert I ate a strawberry rhubarb pie. In conclusion, I ate a lot of good food today.Post-rigorous:I ate some awesome food today. I had cereal for breakfast and an excellent mushroom stew for lunch. My dinner was fantastic: Baked chicken spiced with thyme and paprika, and mashed potatoes on the side. The strawberry rhubarb pie for dessert was absolutely delicious!In the pre-rigorous phase, you spew ideas all over the place. In the rigorous phase, you focus on grammar and structure. In the post-rigorous phase, you focus on getting your ideas across to the reader. You emphasize what's important, and you vary your language and structure to keep the reader engaged.The analogy with writing is quite apt here: Mathematical writing has a lot more in common with the essays you write in high school humanities classes than with the work you show in high school math classes.Grammar and structure are necessary to good writing, but it's not the thing that made Harry Potter a bestseller or Shakespeare timeless. It's not even the ideas: There were tons of books about wizarding schools (and research institutes) before Harry Potter came along. Like with startups, ideas are a dime a dozen, and execution is key. Rowling was successful because she developed her world in impressive detail and then skillfully immersed the reader into it.It's the same with good mathematical writing. The author has to privately develop the argument in a lot of detail, but the purpose of the writing itself is to communicate ideas to the reader. Rowling doesn't need to tell us about Harry getting out of bed every day. If the day begins with Harry in class, we can imagine the preceding uneventful morning on our own, if we really want to. On the other hand, Rowling can't just tell us that Harry escaped and then proceed to the next plot point, even if we could imagine the escape on our own. We want to know all the details about how he escaped, and we want it to be exciting to read.Moreover, you could imagine a writer going from "pre-rigorous" writing to "post-rigorous" writing without ever reading a book on grammar or sitting down and learning about topic, supporting, and concluding sentences. The best authors were at one time kids writing run-on sentences with poor spelling. However, I can see them becoming who they are today without doing worksheets on where to put commas and without being told where every sentence needs to go in a five-paragraph essay. I suspect many of them learned grammar rules as they went along and picked up on what makes for good structure from the books they read and from writing feedback from teachers.Likewise, I and many others never got sat down and taught how to write rigorous proofs. A proof is just an explanation for why something is true. We started by writing crappy explanations that were unorganized, unclear, and filled with logical flaws that we didn't notice because it was hard enough for us to figure out what we meant, let alone for anybody else to. Over time, with practice and the help of coaches and teachers, we learned about ways to clearly structure proofs, ways to tell the difference between key steps and irrelevant details, ways to double check and debug proofs, and ways to engage the reader by emphasizing the purpose or consequences of our work.Calculus. I finished Calculus BC when I was 15. However, like "rigorous proofs," the age when you do calculus doesn't say all that much about your mathematical ability. I'd say it says more about your parents' ability to bully the school into letting you take math classes at a younger age.Although a math graduate student must, among other things, know calculus, there's tons of fundamental mathematics that doesn't require calculus at all, for example in combinatorics, number theory, and abstract algebra. Someone who has a solid foundation in these areas is much more ready to study advanced mathematics than someone who has taken high school calculus.

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