@@finmat95 Think about it. If you were a natural-born talent, where do you think that talent came from? It came from your ancestors; now, how did your ancestors get that skill if they had no ancestors that had the skill? By practice, so create your own talent, create your own skill.
Love this. Always enjoyed math, but I struggled during my final year in high school (depression, family problems -> not doing my assignments). This killed my motivation. I thought I was incapable of doing math (my parents telling me I sucked at it didn't help either) even skipped Analysis and Linear Algebra lectures (I'm in CS) because I thought I couldn't understand. Years later, I stumbled upon some math videos on UA-cam, and I got hooked. Started studying Real Analysis and Linear Algebra on my own, and found out I wasn't as bad as I and the others thought I was. Math is for everyone.
gosh I resonate with you. I also had depression and insane love for math at the same time during high school. I hope you can keep your hopes high and do what you love!
I’m in my last year of high school but I’ve also been going through the worst time of my life mentally. I’ll still push through though because CS unis are too competitive over here(I’m in Canada) and anything below a 90 average is considered mediocre to them
This gap between math and the arithmetic we do the first like 12 years or so is pretty hard to overcome when it hits you out of nowhere. It helps a ton to have this kind of slap to the face in high school with something like a proof based trig class. I didn't have this rude awakening with physics because I DID enjoy reading my textbooks but a lot of people did. I think what a lot of people confuse for talent is just early exposure. If you grew up in a household where you got access to things before people would see them in school it made school much, much easier.
My fundamentals of linear algebra teacher used to tell us that some people gets the degree in 10 years after like 80% people failed the first exam. After failing the subject, I understood what he meant: just accept it and try it again until it works. Now I go much more relaxed knowing that I don't have to be a genius to do stuff
danish high school student here. i was talking with my maths teacher after class specifically about natural talent vs practice. my perspective has always been that hard work can beat talent when talent doesnt have hard work. i think being talented is something that anyone can achieve after some time. every tree grows differently, and same is your learning experience. in my opinion, being a natural born talent isn't nearly as valuable as someone who worked hard for it. eventually the person with natural born talent will hit a wall for the first time in their life and be unable to solve something, and it will all go down hill from there, whereas hard work will simply meet yet another unfamiliar obstacle and tackle it. Obviously it's not as black and white as this, but i think this is a good illustration. really great video and im proud of you for making it so far.
I found what you are talking about to be the issue when I got into my second year of university, programming bachelor. I've got asperger, and math and logic has often been quite easy for me to understand. The first year was quite easy, barely had to even go to the lectures to be able to pass. That all changed at the start of the second year. All of a sudden, I needed to actually work to be able to learn and understand what I was doing. This was something I had never had to do to this degree before. I like being challenged, but all of a sudden I was being challenged every single day nonstop instead of challenging myself every now and then, which I was used to. Suddenly I was up against a wall, and I wound up barely managing that year at all. I feel like I've learned a lot about myself though from that, and am currently in the middle of my third year. Gradually figuring out how to be able to work every day, as well as building up a work mode for myself to be able to handle all the stress that non stop difficult topics put on me.
As someone who converted from linguistics/literature to mathematics in high school and recently graduated as an honorary student with a bachelor's in math, I really support these claims with all my heart and experience. All my knowledge is built upon the hard work I put in, make it using different note taking techniques to completely change my way of learning depending on the subject. Making attempts to solve the problem in your learning is much more important and is an essential skill to develop. I think the main difference begins here. Just understanding how some functional analysis theorems work took me weeks to understand and visualize in my head, and will probably take much more in graduate studies. But in the end, I know I'll do it somehow, or at least will do everything in my power to achieve my desired understanding. So, learning how to learn and finding out which methods work best for you is vital. Then you add consistency and boom! The sooner you start developing that, the better.
I cannot tell you how comforting it is to hear from someone like you that you looked up the answers to problems. Quite often I will run into a situation where I think to myself "Either I will just not know how to solve this or I look it up and maybe understanding the solution will help me get better in the future" but I still feel bad about it every single time.
I have no talent in maths. I am just average. In year 10 I was consistently failing tests, getting as low as 30% sometimes (partly due to a bad teacher and also not doing the class work or homework problems). One day, my maths teacher told the entire class arrogantly that “if you are bad at maths, you will never improve”. It infuriated me so much, so I had to prove her wrong. I studied my absolute arse off for the final maths exam of the year. Doing page after page of the maths textbook, and redoing previous test questions I did. I even requested for her to give me practice tests from other schools. To my absolute amazement, I got 93% on the exam. I looked at her with a smug grin on my face. It just shows that spite is the greatest motivator there is (and not getting into trouble from my parents for failing.) Ok, on a more serious note, you can always do well in anything with practice. But you need to practice smart, not hard. Indeed, I practiced not just by doing questions, or writing notes passively, but actually deriving formulas myself by hand, and talking to myself about how concepts work. This is active learning, and I think it helped me significantly. I also just developed an interest in maths from the study process, and I did things like trying to derive Pythagoras and doing maths Olympiad problems in my spare time. I would have thought that was torture a mere few months before. At uni, I decided to do the “advanced” maths stream of my course (Data Science) because I was interested in it. I was terrified I couldn’t actually do it since I had not natural talent, but it turned out I did pretty alright in most of the courses (not the maths proof courses though). This was just due to smart practice. In short, even if you don’t have natural talent, it’s not a barrier at all. You just need a combination of an interest and smart practice.
This is very inspiring! It reminds me of my physics teacher who had something similar happen to him! He told us how he was an average at best student in math and once a teacher told him something along the same lines of "You'll never be good at maths" and he got so fed up with that he started studying very intensely and went on to go to Oxford and study Mathematics there! Truly impressive how sheer will and practice can take you truly far... One of my smartest and most dedicated teachers by far
I was very interested in math as a kid, but everyone around me was pretty much uninterested or struggled with it. Even the brightest individuals I knew barely finished high school. Moreover, the math teachers up to K-12 were often unprepared. Consequently, I didn't know how to learn math or what I should aim for. At the age of 11, I knew calculus was cool, but I stumbled in the first 10 pages of the book because I couldn't grasp the idea behind the proof of infinite prime numbers, which it presented as obvious. I think I only truly understood what math is about in my second year of college and felt like I had been missing out all my life until then. Also, I was so far behind that I had no chance of keeping up with anything more advanced than engineering math. After graduating, I returned to study mathematics (currently in my second year, but with a lot of credits from engineering, so I'm a bit ahead). Proof-based mathematics still intimidates me today. I'm slowly progressing through a Topology course but can't write the proofs without guidance at each step. It seems like everyone else can grasp new concepts easily, discuss them with the teacher, point out mistakes on the blackboard, and even suggest new ways to prove the theorems in Munkres' book (although more than half of the students in this class have dropped out by this point). Math is challenging and requires an incredible amount of time to understand its basic concepts.
@@skywalkerr580don’t try to make other people feel bad because your attention span is too short to read more than a sentence at a time, this dude had a very interesting story to talk about so it makes me sad to see you unable to see that
Don't be discouraged brother. Keep it up your hard work. I too struggle a lot. There are many like You and me . Just keep learning. With time , everything will make sense. All the best ❤
It is a natural feeling, but the grass is always greener. Those people that are more talented feel the same as you. You'd be surprised at how much practice can put you ahead of the curve where natural talent left you. Keep going.
@@johnwest6690humans are shallow and this is the opposite ironically. It’s intrinsic in everyone that they want to be the best and once you figure out you will never be truly the best in your specific hobby, you feel disdain for the world as a result.
Thank you so much for talking about how your Dad set you up with good habits and helped you with homework. My parents are great and I love them but they never helped me with homework besides getting frustrated when my teachers called home say I wasn't doing it. The way I see it is the only way to make up for all that is to never get discouraged, and remember what I want to accomplish and what it costs when I start giving myself to much of a break .
Honestly kinda my inspiration Currently taking trig in 12th grade Having only read calculus made easy and currently doing higher algebra by Hall and Knight doesn’t feel like enough preparation for what is to come Hopefully I can be like you someday though
Don't sweat too much about being "not prepared enough"; just know that math is all about getting a feel for the tools you learn in the class: practicing with purpose. If there's some integral: Integral(sec^3(x) dx) which you'd learn how to solve in calculus 2, it's all about understanding what tools work with this problem, and what tools don't. The only way to find out is using the tools that DONT work on it and see what that looks like, then you'll be better be able to appreciate the RIGHT tools. This is practicing with purpose, so you truly know your way around the problem; so on the exam if you forget what the right tool is, you can try using a random tool and realize "Oh yeah, ive seen this mess before, I have to try something else...hmmm what else did I learn to help me..." and then you'll land on the right tool and get the right answer! All math tools have a derivation behind them. Someone found out a way to get to the formula - so sometimes it can be useful to see how they got there. other times, not so useful. very little in math do you need to "Just memorize it lol". Don't be afraid to ask the professor these kinds of questions in their office hours, they will always be very happy to get a question like that and are always there to help you with any kind of specific problem you struggle with.
I have encountered a chad who I can really say that, he IS natural born for math. He's a classmate of mine who does not study at all during the semester(according to his roommates, 3 of them), speedruns real analysis in a night, and ends up scoring higher than me. The whole experience is quite frustrating since I studied my ass off for that exam, but I think that one of the functions of colleges, to have that one off experience and to use it to expand ones horizon.
Don't know if it's more clear than you have explained, that d sigma is n-1 dimensional Hausdorff measure, which measures the n-1 -d volume of a n -d space, like lines in R2, areas in R3
I've come across these guys a lot. There can be other hidden factors like he may be thinking about math while walking. I knew a guy who did this and was thinking about math even when it seemed like he was just relaxing. This can actually be more useful or powerful than regular studying if you can manage to do it. Also you don't know what he went through in previous years so that may have set things up so this course would be easier for him. Better not to leap to huge assumptions about talent when you don't know his full story.
Don't be impressed. I do this for every one of my courses in university. Currently a junior in neuroscience taking physics 2 and calculus 2 along with some neuro courses. Hard work is more impressive. I do it because I'm lazy, and I know that I can get away with doing it and getting all A's. I know it won't get me far
Awesome video! It is almost as if being naturally gifted is a deception to get you to do further studies in maths, and then it turns out that your intuition is not worth so much anymore. Anyway if you are passionate in mathematics rather than being passionate about always being correct then it should be all the same to you even if it is a struggle to get to where you want to be.
I think many people in a PhD program feel this way. I wanted to know more and more about the hard maths and bit off more than I could chew. But I still wanted to know how it was done, so stuck with it.
Man, your channel is amazing, it really serves as a way to modulate my study environment and motivate me to study, thank you very much! What a find, don't stop making videos!
It’s like exercising with bad genetics, you can become much stronger and more aesthetic than you are and more than someone with good genetics who doesn’t workout as hard. But when hard work and genetics combine, you can’t compete with that. It’s similar with Maths, I believe. I’m a Maths teacher, who wasn’t the best at it, always had to work really hard and spend a lot longer than the top students to get the same results, but was capable of top results when I really pushed myself. I’m always improving even now and trying to impair the insights and intuition that took me years, to students in a shorter period of time. It’s probably the case for all things that hard work beats talent but only when talent doesn’t work.
the geneticspill is hard to swallow, im pissed off im only 130IQ, 10-20 years ago, yeah that would probably be good but now, man it's below average in the top level fields
yh exactly, theres a few metastudies on the role talent and hard work play in sports and at all levels talent is more important. obviously you improve at something the more you practice it but how quickly you improve/at what point you plateau depends on natural ability. rly it comes down to how you want to spend your time
I’m in my second year of my math masters and I feel the same way… trying to overcome the proof barrier but it is getting a bit easier. Taking a lower level real analysis but taking Abstract algebra which is quite tricky/fast paced. Hopefully I can get to a PhD 🤞🏼🤞🏼
Thank you for this perspectives. It’s scares me because I’m in the phase of still breezing through the math, but seeing the work that some of my friend are doing makes me realize that I haven’t gotten to the hard stuff yet.
I feel like this unintentionally proves that it is pretty much all about hard work. Ever since he was a kid he had an interest in science and math, worked at it with the help of his father which probably gave him a pretty good foundation that allowed him to excel in early classes that don't require too much then as he took more advanced classes that required more work and probably without the help of his father he had to start applying himself more.
It is true that I was just naturally interested in the subject so practicing did not seem like work for me, but I would say that I had to put far less effort in studying algebra and geometry than most of my peers in HS, even though the content was just as fresh for me as it was for them. But eventually everyone hits a wall and for me, it was proof writing.
@@PhDVlog777 that's interesting and it might come down to natural talent, but im not sure I am convinced. You said that your dad was very involved in your education and was a former teacher, did your peers have a similar situation? Even if they had tutors were they interested from childhood in STEM or at all? I get that I don't have the full picture, but I find it hard to believe that natural talent exists in any meaningful capacity when there are so many factors to consider.
My man, u can have 2 different kids, both with an equal interest in math, one born with an iq lets say 20% higher, both study the same amount of time. Who do you think (in the same amount of time) will learn more, will they learn equally? Of course not. Also before anyone mentions it high iq doesnt automatically give you math knowledge, you still have to learn it first, but people with "natural born talent" learn it way way faster. Anyone can learn abstract algebra, its just someone smarter might understand it in a year, someone in 2, but it will take your average human wayy more time and work to be put it.
Your videos are amazing man. I love them so much. Very realistic and practical for all people that want to advance in Math. Keep it up with the great uploads
This is such an inspiring story but also empathetic (and to be honest sympathetic) towards my current plight of getting a math degree. Thank you. Also, my bro graduated with an environmental science degree which is ironic since you got that for your bachelor's. Ironic! :)
I'm a math major and I'm in my second semester of real analysis, my first proof-based course, right now and it is kicking my butt. It's really reduced my love for the subject because I just don't grasp what's happening half the time and I've been thinking about dropping my major and just doing computer science, but you are really encouraging me to keep trying to do what I love.
I think real analysis can depend on what lecturer you get (like most I guess). Real analysis was the subject I came closest to failing because I couldn’t understand the lecturer. Did fine in complex analysis and measure theory later on with better teachers. Although I definitely did proof based courses before that like linear algebra and calculus.
"To use mathematics effectively in applications, you need not just knowledge but *skill* . Skill can only be obtained through practice.", Mary Boas in "To The Student" of her "Mathematical Methods in the Physical Sciences". This is true for both applied (computational-oriented) and pure (theorem-proof oriented) courses.
I was not even the best in my highschool class in math but still one of the best without putting in any effort into studying. In high school I only cared about videogames and playing guitar so I did not do that well in my final exams, but still got into a very good university. When it came time to choose what to do in uni, I naturally went to study maths and computer science because those had felt easy to me without actually putting in any effort. Well after first semester of Linear algebra and real analysis I basically fucked up both classes with very bad grades. I was at edge of going full on to programming side because I thought that I just had no natural talent to grasp these higher level proof based classes. Well after my freshman year, I spent entire summer studying hard to try to understand the stuff and picked up a book in topology to self study for the first. It was first really regirous math subject I took. After that summer I took a challenge and hopped into two graduate level class and I somehow managed to pass the classes with better grades than the elementary undergrad courses on my freshman year. Then going into stuff like abstract algebra felt like a breeze. Somehow I turned around in studies just by finding out that math is actually interesting theory building. Now im doing my master's thesis in geometric analysis and teaching and we shall see if I get a Phd spot/funding. Tldr: I really feel like natural talent in math doesn't carry you far.
Thinking of doing something similar to your channel but for my physics PhD. Love your videos and find them deeply relatable. Really enjoy the story time format.
I noticed that I had a sort of latent talent for math. I sucked at trig and Calculus (well, not as much as some, I just displayed no real inclination for it), but once I got to proof writing and discrete mathematics, it all changed. Propositional logic and meta-mathematical concepts all made perfect sense to me. I had a hard time remembering all of the rules for computation in Calculus and Linear Algebra, but writing proofs was a piece of cake for me. Even though I ended up getting a C in Calculus 2+3, I swapped my major to math and got a 98 in Abstract Algebra, then a 95 in Topology that same semester. It got trickier once I started doing Analysis and Representation theory, but I somehow went from someone who had a hard time in freshman courses to someone who got accepted into some extremely competitive math graduate programs.
School: 900 College: 1200 University: 1500 Phd: 1800 Natural born talented guy: 1100, he works to get an extra 700 and gets the PhD. Natural born less talented guy: 900, he works to get an extra 900 and gets the PhD. Average guy: 700, he works an extra 1100 to get the PhD. Cold hard truth here. But, being resilient and hard working is also a talent so it’s easier for a resilient average guy to get a PhD then, a lazy genius. Crazy thing is there are people who are more resilient than you and smarter 🤷♂️ But, don’t just guess if they are one or the other because you could really not know shit.
This aligns with my math experience. Everything through Trig was pretty easy and I was able to just rely on "natural ability" without much effort. Calc 1-2 was doable, but took a little work, Calc 3 was just tedious and basically turned me off from math. Linear Algebra was a return to easy mode for me. That's as far as I went with it for the most part while working on my CS degree.
Ive rediscovered my interest in math while solving some random financial questions online. I was genuinely good in math in my school years, not a genius, I just liked to study it, and I wanted to be an architect but I undergraduated in an area that has nothing to do with math at all. I have studied the subjects for 4 years, and completely forgot math existed. But while trying to solve those random problems I felt so happy, like I have this pure love for math or something.
I’m taking metric spaces and topology this semester. Just this afternoon I spent 4 hours working on a single problem. Talking to my classmates before class they made the solution sound like it took them no longer than 15 - 20 minutes . For me it wasn’t clear, and my internal dialogue was suggesting that I quit mathematics at several points during my problem set. My internal discussion said “You think you’re smart.You’re not smart. It took you four hours to produce this solution. Pathetic. Other kids can produce this in less than half an hour.” Once I had solved the problem, I felt such a great sense of peace. My internal discussion was now re-affirming thoughts of self love, tranquility and open mindedness when moving onto the next problem. Despite my difficulties in mathematics, and I personally have found a lot of inner peace studying mathematics. And my intuition keeps telling me that I need to continue. My curiosity to learn more mathematics like Functional Analysis, Symplectic Geometry, Irrational Pattern Functions, Poly Dimensional Topology.
I think tutoring ("teaching assistant" as you guys say in America) for a course is really underrated as a way to get better. I have much better recall of the material from courses I tutored than courses I took but didn't tutor. Most of the maths and computer science courses that I took and never tutored for, I took them so long ago that I can barely remember much of what we covered. I'd pick it up if I had to go back through it, or if I needed to pick it up. But I can't recall anything but the very basics from the top of my head. But courses I tutored? I can remember most of what was covered in those courses many years later. You see every mistake and misconception anyone could possibly make and you have to get good at explaining things in 5 different ways, because different students need things explained differently to understand them.
teaching assistant in the US usually means you are actually hired by the teacher of the class to lead discussions or grade homeworks or tutor as well. Tutoring is a part of the teaching assistant job in the US, but it is not all of it.
Love your videos. I have a bachelor's in mechanical and have been graduated for 10 years but always had a interest and hunger for more math. I took calc 1 2 and 3, DifEq, LA, Mathematical Methods, and a few fluid mechanics course which is really just applied math mainly PDEs. That being said I wish I had the time or chance when I was in school to take more. But since being out I have done a fair amount of self study on topics interested in which is alot of fun. Following your PhD journey and learning about the process has been interesting and enjoyable
I cant speak for everybody but first year in undergrad was the hardest part. then halfway around year 2 when i got into algebra and topology the understanding of these topics was much more intuitive for me.
a natural talent combined with good parenting creates a passion which drives your practice. Your k-12 experience sounds exactly like mine except my parents didn't help me or care about anything I was interested in
Talent determines the upper limit, while effort determines the lower limit. So you aren’t going to reach even your lower limit without putting in effort.
i feel as if talent determines where you start, and how quickly you learn and build on that. but with effort you can improve both of those things. but it would take so much effort and time to get there that it may not be practically worth it. which is probably the reason why we often end up losing interest in the things that we're bad at. one more thing is, if you're just naturally better at something you have a very powerful positive feedback loop to keep you going, to keep you motivated. it makes that thing fun. if you're playing soccer, and play well and do cool stuff, you're going to enjoy that more than everybody else. or grinding video games for eg. while on the other hand, if you struggle all the time then your entire focus is on catching up. never being the first one to get somewhere is discouraging. you not only struggle with getting better, but also struggle with a whole bunch of things for that reason that bring down your productivity even more. you're much better of not worrying about any of it.
What you described with "borrowing now" from others is exacty what I had to do with advanced electromagnetics. I just couldn't fully solve many of the problems intially. Now, looking back I have a much better intuition for the subject.
As a university student on my 4th year, I find math is much easier if you try to understand what the numbers, symbols and processes represent. If you can find some good geometric interpretation of what the math is doing, it becomes so much easier to visualize the problem and understand how to solve it. Most of the time, I don't do a lot of practice problems, because so many of them don't help me get an intuitive undersatnding of how the processes work. If I can understand the how and why, I only need a small ammount of practice usually. The disadvantage is when teachers don't give any good explanations, and just tell us to solve lots of practice problems, I become completely lost. Fortunately, the internet is absolutely packed with math videos that give lots of different explanations, some of which are much better than what my professors give.
My experience is similar. Things were straightforward from K-12, years 1-3 of college math were a combination of enthusiasm/talent (ODE, PDE, algebra, analysis, linear, etc), but then year 4 math and beyond was purely hard work (topology, functional analysis, algebraic topology, differential geometry, etc).
There is really a strong parallel with learning music and mathematics. Especially in the "borrowing knowledge" topic. You learn other people's songs when trying to apply the concepts of how to "do" music, there isn't any shame in this; it is encouraged. Why we didn't apply this logic to math innately kinda baffles me.
I changed from applied math (which was very much not applied, mostly just proof stuff) to mechanical engineering for the exact same reason. I can say that it was absolutely the best choice I made. It brought back all of the calculus and differential equations and actually used them for useful things. Having a course like welding and aerodynamics is also a plus :)
Everything was very easy before university for me. Now in university I realize how my lazyness and lack of study skills have an impact on my performance.
I am quite talented at most common academic subjects and throughout my life I made this apart of my identity. I never really had to try for most if not all of my subjects causing a lazy attitude when it came to acadwmica. My high school years I can't really just fuck around for all my subjects so I start lacking in most. Math was one of the few that I didn't need to put much effort in and again at some point I started identifying with math.I also started to view it as a hobby and with these combined motivation I study math. As a junior in highschool I have taught myself math associated with the first two years of college and ill graduated highschool (hopefully) having taken classes for calc 3 and linear algebra. I try to teach myself proof based mathematics even going so far as to try easier sides such as number theory. Though in the end I feel most of the reason I do this is out of paranoia. Paranoia that one day I'll suddenly no longer be able to advance in math. Paranoia that I'll loose all interest in math. That was my main motivation for teaching myself math far above my level. I can really relate to the way you view reading and how you got back into math almost at an impulse as I see myself doing something simular.
Recently,I was struggling with linear algebra like matrices form and dot product in machine learning So sad … after Reading comments and watching videos I am inspired
When people say Asians are good are math, it has little to do with talent. The culture believes in hard work and that’s why successful asian students have been drilled in math since a young age. Unless you’re Terrance Tao
I think the constant hard working nature of those Asian countries being passed down throughout generations legit builds that naturally, innate, inborn talent they have to be academic monstrosities. I think all talent is derived from our ancestors and how hard they worked.
I would say to some extent that I got better at academics the further I went on. I remember also having troubles with Algebra I, but by the time I made it to calculus it all made sense and clicked really quickly. I feel like I glided through by electrical engineering degree, but I’m starting to have troubles in graduate school now.
I have a bit of a contrarian view of things. Everyone extols the virtues of hard work/practice over talent, but I feel it's smarter to choose a field you're already talented in. If you want to coast you can just get by without putting in much effort; if you want to be the best of course you've got to practice and have discipline; but in my opinion the best way is to learn skills that have synergy with your technical skillset that few others in your discipline can do, which will dramatically boost your odds of success. For example as a mathematician if you learn to get really good at public speaking, UA-cam, marketing, psychology (specifically the art of inspiring others, and also encouraging others who are struggling), and the science of learning you can become a famous maths educator like how Neil deGrasse Tyson is for science. It's infinitely easier than trying to become better than Terrence Tao at maths.
I’m in high school right now and I’ve been self studying a lot of math (right now I’m going through algebra chapter 0 by aluffi) including proof based stuff, I feel like I’m in a horror game right now, I’m terrified of when the bubble is going to burst and my natural talent is going to run out. Cause I thought it was going to be calf and then I did calc and it was fine and then I thought it was going to be proofs and I did that just fine, then I thought it was gonna be topology and I did that fine, I feel like I’m going to get humbled soon.
I think talent encompasses being good at everything in Mathematics, including having a good memory and good calculation skills, along with of course mathematical genius, and other things. What thing you were talking about is 'decent mathematical ability'.
I think one can distill down the “Natural Born Talent” vs “Practice” discussion down to “Intuition”. Specifically, the moment that new materials doesn’t become intuitive is the moment that one’s natural born talent becomes obsolete. Unfortunately, there are few human minds in history that materials above a bachelor’s degree are intuitive. Those few are legends in Academia.
Great video! as someone who doesn't have alot of natureal talent it's refreshing to see other people go thought the same struggles as i'm going through. Do you have any idea want you would like to do after your PhD, have you got any carrers in mind?
I'd say it's 60% talent and 40% hard work. I also think that the people that are drawn to math tend to be more naturally talented. Math certainly attracts those with analytical mind.
One of my boxing coaches told me that “talent will always beat hard work, but that talent will easily be hidden by your own habits”. It’s not the greatest quote, but for someone that used his genetics to pass through early stages of training/camps, I can humbly say that hard work can nullify your lack of talent, as one 3rd fight showed me that 😅 I still hate when people say “I can’t do something, because I’m not good at it”…like you barely even tried, no other task in like will be easier then the one you’ve been half ass-ing for a few years now. I think everyone has talents, and when you work hard enough you’ll start to see those talents much more clearly.
I know this isn't directed towards me but I'm going through that book rn and also taking a course on complex analysis and it's pretty good :) clarifies some of the more simple elements of complex numbers and such, and explains things well
Have you ever done any math competitions in highschool? (AMC, AIME, USAMO, IMO, etc.) Have you also heard of/done the Putnam contest? (I believe that is for uni students)
Amazing that you can now look up graduate course homework problems on the internet. If we were stuck, all we had were the professor or other students in class.
You can, but it’s not really helpful. You can write the solutions to get you by on the homework, but despite the wealth of resources online, I still end up needing to go to my professor to get explanations and build by understanding.
@@OmgitsNeens Oh, I don't disagree at all. There is no substitute for a mentor helping you to put everything into context. Still, the fact that all solutions of baby Rudin are available for free in 10 seconds online still blows my mind.
I feel like I had a similar undergrad experience. Except afterwards I never went back and I still regret it. It's not really financially viable at this point. I will still always find math interesting and continue to study it (slowly) on my own.
I gave up on math. It was kind of my dream but I couldn't put the required effort towards it. It doesn't help that I have ADHD and some other problems that affected me more than it should. I'm 31 and am finally finishing a computer science degree, but with 80% of math courses completed. I've changed majors 3 times from engineering to another engineering then math then compsci. Pathetic. Now I just feel like it is too late for everything.
ADHD is the most underestimated disorder add insult to injury also belittled and sold by those who have it as a super power only to realize much later how disadvantaged one is.
A “naturally talented” math person here. 100% attest that people who put lots of effort in are better than me. For perspective, I have done my bachelors in CompSci. I took up to Calc 3 and have a Data Science background. My number sense is about the same as someone with a PhD (particularly with statistics), but my ability to actually perform calculations pales in comparison to anyone who actually practices. Talent will make it easier to learn, and it gives you a good intuition, but that means very little in the real world if you don’t apply yourself to the discipline.
When I was doing my Master's (physics) there was a PhD student who was brutal. While his peers PhD students studied for hours, days, endless nights for the qualifying exam, dude just did his homework and had a side startup (unrelated to physics)...dude was amazing. Ended getting an honors PhD. So yes, being smart does help, like it or not.
don't you think he might have had a more efficient process? he must have been really confident and didn't struggle with any sorts of anxiety, especially while juggling with a startup at the same time? Personally the less anxious I become in my life the clearer, more focused and fluidly I seem to think.
@@matviyk3066 One night (must have been like 7pm or so), I was heading out of Campus with my GF (at that time) and we saw him an offered him a ride since he was heading in the same direction. We got to talk for about an hour or so (lots of traffic in a big city!) and it was very insightful. I talked with some friends who had gone to the same University as he did for his Undergrad and the stories were out of this world....I think he is the smartest person I have ever met, along another friend I had while at my Master's....they were both amazing, but the first guy was just out of this world...
Are you a UofT student? Those used stickers that you have on your books seem strikingly similar to what we have over at University of Toronto when our library/bookstore sells books.
I naively dropped mathematics in school but then in college fell in love with economics so now I'm trying my best to get into a quantitative economics masters program, starting my real analysis course very soon 😺
Its mostly talent to be honest. You can practice as much as you want when it comes to math and science if you dont have the talent for it you're not gonna go very far
Talent is undoubtedly way more valuable than hard work Anyone who studies the most can invent stuff, however that's not the case in real life, we remember people like Nikola Tesla, Dmitri Mendeleev, James Clerk Maxwell & Michael Faraday not because they worked hard but because they thought out of the box Someone who works hard can solve 50 application (level 3) questions in 30 minutes maybe Someone who has talent can solve evaluation (level 5) questions and invent or discover new concepts that open up whole new fields, meaning they prevail at the creation level (level 6) Whether we like it or not that's how people's brains are differentiated in terms of practical ability
such a great motivation... i'm in basicly the same shoe (currently physics undergrad) and it felt real good to see i'm might not hopeless... :) i'll tattoo this on me "Eventually practice becomes talent".. xddd (sry for bad english, still learning :))
I personally believe that the difference between high school-level math and more advanced math lies in the type of the information required by the brain to processes. High school-level math is mostly about using relationships between different symbols, which is all visual information. Advanced level math in college or grad school is more about reasoning, which necessarily requires processing information through words (i.e., verbal information). I personally believe this is the main difference between those who have to do more hard work while doing advanced math (or any type of subject that requires an advanced level of reasoning) and those who are deemed "naturally talented". I believe the "naturally talented" people process verbal information in a different way. I'm still not sure how their brains process verbal information though (it's a work in progress).
Processing verbal information is also a skill. I would like to use interpreters as an example. It has been scientifically established that an interpreter is not a person of natural born talent, but instead a skilled professional. Interpreting is much more than translating words and spitting them out, it is a process that includes receiving information, decoding it and then building it back up again in the target language as it would be expressed in the most natural way for the given context of the situation. I would encourage anyone to try interpreting from one language to another. It will train your mind to hold more information at any given time and be able to remember and reproduce more and more accurately.
I have a question as someone that nearly failed undergraduate mathematics, but really excelled at high school maths, and is now in a master's that needs more applied maths. How do you go about identifying where your gaps in knowledge are, and how deep to go?
It is difficult to say, but I would look at the classes needed to get to a BS in applied math, and at the very least take notes of text books used in those subjects. For example, if you need probability theory, then I would read some of Chung or Billingsley to get familiar with the topic. Problem solving is best but more time consuming. At the very least, you are looking at a few months of playing catchup, and you are never really certain where to start and stop. But this was my approach to filling in the gaps with measure theory and complex analysis and it seemed to work. I hope this helps!
@@PhDVlog777 Thank you! Tbh, it feels really overwhelming to start. Unsure if you have experience with it, but did you mainly fill all of these gaps yourself? Did you find tutors? Or a mix of both?
I think talent helps you to a certain degree but at some point you have to learn a lot, even if your talented. But many talented people learn to learn when they are young, so they struggle in college.
I am a 3rd year engineering student and I experienced the same. k12- no work needed. Y 1&2 - easy topics but tedious calculations. Y 3- Why am by being taught by Eminem.
What are your thoughts on going against your talents or doing something you are not talented in? I prefer fiction over non-fiction for example and enjoy things outside of my field but I chose the field, cs, in order to force my brain to think a certain way. It was more like an admiration for people's logic and creativity in this field and even though I held almost no talent in it, I am still trying to get a degree in it, hoping by raw effort and practice I can become very comfortable and strong in it.
It may be a little extra work but I do not think there is anything wrong with picking a degree outside your natural skill set. A lot of people attending university do not have a particular set of skills (*Liam Neeson intensifies*) and they just pick a major and go with it.
Something you said there spooked me a little. You mentioned that you almost only read class notes and barely read the textbook which didn’t work out for you and I’m worried I’m ending up in a similar situation. I am taking a measure theory class now and the prof. is a bit on the eccentric side. His lectures deviate from the textbook quite a bit, to the point were notation is almost completely different, a bunch of definitions and especially theorems/lemmas/propositions and their proofs are done in an almost completely different way, and the way he progresses through certain chapters in the book, and what he decides to talk about and what not to talk about have been a little bit different from how the book actually does it. So what has ended up happening is that I’m almost entirely basing my work around notes from his lectures and only using the book to double check/look up some stuff as well as for a few exercises that he has picked out. Based on your own experience, am I shooting myself in the foot, or does the situation warrant not using the textbook too much? It’s getting to the point where I’m struggling to follow the proofs and theorems in the book due the different notation and so on.
Use the textbook because it's new to you. For your professor, he knows it so he doesn't look at the textbook. That's why he deviates. I think if you build a strong knowledge base from the textbook, going to class will be fun because you get the extra stuff that's not in the book so it will only add to you. What I'm saying is, reading the tetxtbook will only add to you and not reading it will only take away from you.
It depends on what your professor expects from you. If the prof asks easy questions, and knowing thy lecture notes is sufficient, then you will probably be OK, but I would have to sit in the class myself. I know this: my notes from analysis were unreadable and I had to read other textbooks to keep up. Problem solving was also far more beneficial than studying lecture notes and reading ever were. My advice is to (if you are not already) speak with your instructor and learn what he expects from you. Speak with others in your department that have had his class before and learn from their stories. This is what I did and it worked for me. Good luck!
During my 1-10 years I used to suck at math . I used to fail maths classes and when my wud question the teachers they wud blame it on my previous maths teachers. Only when I started working on maths problems on my own during 11th and 12th is when I actually started to see the beauty behind it .
@15:05 "Eventually practice becomes talent"
What a great mindset to keep in mind during our studies. Thank you.
unfortunately it's false.
@@finmat95 its not that it is, just that it seems like it from an outside perspective. practice does help though, a lot.
@@finmat95shut up.
continue making excuses because you're afraid to do hard work@@finmat95
@@finmat95 Think about it. If you were a natural-born talent, where do you think that talent came from? It came from your ancestors; now, how did your ancestors get that skill if they had no ancestors that had the skill? By practice, so create your own talent, create your own skill.
Love this. Always enjoyed math, but I struggled during my final year in high school (depression, family problems -> not doing my assignments). This killed my motivation. I thought I was incapable of doing math (my parents telling me I sucked at it didn't help either) even skipped Analysis and Linear Algebra lectures (I'm in CS) because I thought I couldn't understand. Years later, I stumbled upon some math videos on UA-cam, and I got hooked. Started studying Real Analysis and Linear Algebra on my own, and found out I wasn't as bad as I and the others thought I was. Math is for everyone.
gosh I resonate with you. I also had depression and insane love for math at the same time during high school. I hope you can keep your hopes high and do what you love!
I’m in my last year of high school but I’ve also been going through the worst time of my life mentally. I’ll still push through though because CS unis are too competitive over here(I’m in Canada) and anything below a 90 average is considered mediocre to them
What videos did you stumble upon?
@@ZKrow3823b1b, Numberphile, Dr Trefor Bazett
Maybe you couldn't understand it earlier? As your brain matured, your ability to understand did, too.
The capacity to learn is a gift; The ability to learn is a skill; The willingness to learn is a choice.” ― Brian Herbert
This gap between math and the arithmetic we do the first like 12 years or so is pretty hard to overcome when it hits you out of nowhere. It helps a ton to have this kind of slap to the face in high school with something like a proof based trig class. I didn't have this rude awakening with physics because I DID enjoy reading my textbooks but a lot of people did. I think what a lot of people confuse for talent is just early exposure. If you grew up in a household where you got access to things before people would see them in school it made school much, much easier.
absolutely. all the geniuses in this world had access to resources. ill slap myself in the face with this stuff, made me lol, as im in HS.
bro you are so right ! thank you for your insightful take
My fundamentals of linear algebra teacher used to tell us that some people gets the degree in 10 years after like 80% people failed the first exam. After failing the subject, I understood what he meant: just accept it and try it again until it works. Now I go much more relaxed knowing that I don't have to be a genius to do stuff
this is why i like self-learning. no pressure.
danish high school student here. i was talking with my maths teacher after class specifically about natural talent vs practice. my perspective has always been that hard work can beat talent when talent doesnt have hard work. i think being talented is something that anyone can achieve after some time. every tree grows differently, and same is your learning experience. in my opinion, being a natural born talent isn't nearly as valuable as someone who worked hard for it. eventually the person with natural born talent will hit a wall for the first time in their life and be unable to solve something, and it will all go down hill from there, whereas hard work will simply meet yet another unfamiliar obstacle and tackle it. Obviously it's not as black and white as this, but i think this is a good illustration. really great video and im proud of you for making it so far.
I found what you are talking about to be the issue when I got into my second year of university, programming bachelor. I've got asperger, and math and logic has often been quite easy for me to understand. The first year was quite easy, barely had to even go to the lectures to be able to pass. That all changed at the start of the second year. All of a sudden, I needed to actually work to be able to learn and understand what I was doing. This was something I had never had to do to this degree before. I like being challenged, but all of a sudden I was being challenged every single day nonstop instead of challenging myself every now and then, which I was used to. Suddenly I was up against a wall, and I wound up barely managing that year at all. I feel like I've learned a lot about myself though from that, and am currently in the middle of my third year. Gradually figuring out how to be able to work every day, as well as building up a work mode for myself to be able to handle all the stress that non stop difficult topics put on me.
As someone who converted from linguistics/literature to mathematics in high school and recently graduated as an honorary student with a bachelor's in math, I really support these claims with all my heart and experience. All my knowledge is built upon the hard work I put in, make it using different note taking techniques to completely change my way of learning depending on the subject. Making attempts to solve the problem in your learning is much more important and is an essential skill to develop. I think the main difference begins here.
Just understanding how some functional analysis theorems work took me weeks to understand and visualize in my head, and will probably take much more in graduate studies. But in the end, I know I'll do it somehow, or at least will do everything in my power to achieve my desired understanding.
So, learning how to learn and finding out which methods work best for you is vital. Then you add consistency and boom!
The sooner you start developing that, the better.
You will never be Terrence Tao or Gauss.
go beat Von Neumann with hard work
I disagree. Genuine hard work talent eventually becomes exceptional and revolutionises the field they're in
I cannot tell you how comforting it is to hear from someone like you that you looked up the answers to problems. Quite often I will run into a situation where I think to myself "Either I will just not know how to solve this or I look it up and maybe understanding the solution will help me get better in the future" but I still feel bad about it every single time.
I have no talent in maths. I am just average. In year 10 I was consistently failing tests, getting as low as 30% sometimes (partly due to a bad teacher and also not doing the class work or homework problems).
One day, my maths teacher told the entire class arrogantly that “if you are bad at maths, you will never improve”. It infuriated me so much, so I had to prove her wrong. I studied my absolute arse off for the final maths exam of the year. Doing page after page of the maths textbook, and redoing previous test questions I did. I even requested for her to give me practice tests from other schools.
To my absolute amazement, I got 93% on the exam. I looked at her with a smug grin on my face. It just shows that spite is the greatest motivator there is (and not getting into trouble from my parents for failing.)
Ok, on a more serious note, you can always do well in anything with practice. But you need to practice smart, not hard. Indeed, I practiced not just by doing questions, or writing notes passively, but actually deriving formulas myself by hand, and talking to myself about how concepts work. This is active learning, and I think it helped me significantly. I also just developed an interest in maths from the study process, and I did things like trying to derive Pythagoras and doing maths Olympiad problems in my spare time. I would have thought that was torture a mere few months before.
At uni, I decided to do the “advanced” maths stream of my course (Data Science) because I was interested in it. I was terrified I couldn’t actually do it since I had not natural talent, but it turned out I did pretty alright in most of the courses (not the maths proof courses though). This was just due to smart practice.
In short, even if you don’t have natural talent, it’s not a barrier at all. You just need a combination of an interest and smart practice.
This is very inspiring! It reminds me of my physics teacher who had something similar happen to him! He told us how he was an average at best student in math and once a teacher told him something along the same lines of "You'll never be good at maths" and he got so fed up with that he started studying very intensely and went on to go to Oxford and study Mathematics there! Truly impressive how sheer will and practice can take you truly far... One of my smartest and most dedicated teachers by far
Powerful story. This applies for everything in life.
looks like she was a good teacher afterall
@@ownagesniper1 💀fuuuck no
What is smart practice according to you??
I was very interested in math as a kid, but everyone around me was pretty much uninterested or struggled with it. Even the brightest individuals I knew barely finished high school. Moreover, the math teachers up to K-12 were often unprepared. Consequently, I didn't know how to learn math or what I should aim for. At the age of 11, I knew calculus was cool, but I stumbled in the first 10 pages of the book because I couldn't grasp the idea behind the proof of infinite prime numbers, which it presented as obvious. I think I only truly understood what math is about in my second year of college and felt like I had been missing out all my life until then. Also, I was so far behind that I had no chance of keeping up with anything more advanced than engineering math. After graduating, I returned to study mathematics (currently in my second year, but with a lot of credits from engineering, so I'm a bit ahead). Proof-based mathematics still intimidates me today. I'm slowly progressing through a Topology course but can't write the proofs without guidance at each step. It seems like everyone else can grasp new concepts easily, discuss them with the teacher, point out mistakes on the blackboard, and even suggest new ways to prove the theorems in Munkres' book (although more than half of the students in this class have dropped out by this point). Math is challenging and requires an incredible amount of time to understand its basic concepts.
If I didn't know any better, I would think that you were me lol
Holy yap
@@skywalkerr580don’t try to make other people feel bad because your attention span is too short to read more than a sentence at a time, this dude had a very interesting story to talk about so it makes me sad to see you unable to see that
Don't be discouraged brother. Keep it up your hard work. I too struggle a lot. There are many like You and me . Just keep learning. With time , everything will make sense. All the best ❤
@@skywalkerr580 You are literally commenting under a 30 minute long video story lmao. Are you always this dense?
I'm sometimes tempted to give up on everything I like because I can't stand the people who are just naturally more talented than me
That's pretty shallow
"There's no miracle people"
I feel this so hard lol
It is a natural feeling, but the grass is always greener. Those people that are more talented feel the same as you. You'd be surprised at how much practice can put you ahead of the curve where natural talent left you. Keep going.
@@johnwest6690humans are shallow and this is the opposite ironically. It’s intrinsic in everyone that they want to be the best and once you figure out you will never be truly the best in your specific hobby, you feel disdain for the world as a result.
Thank you so much for talking about how your Dad set you up with good habits and helped you with homework. My parents are great and I love them but they never helped me with homework besides getting frustrated when my teachers called home say I wasn't doing it.
The way I see it is the only way to make up for all that is to never get discouraged, and remember what I want to accomplish and what it costs when I start giving myself to much of a break .
Honestly kinda my inspiration
Currently taking trig in 12th grade
Having only read calculus made easy and currently doing higher algebra by Hall and Knight doesn’t feel like enough preparation for what is to come
Hopefully I can be like you someday though
Keep at it 👍
Don't sweat too much about being "not prepared enough"; just know that math is all about getting a feel for the tools you learn in the class: practicing with purpose.
If there's some integral:
Integral(sec^3(x) dx) which you'd learn how to solve in calculus 2, it's all about understanding what tools work with this problem, and what tools don't.
The only way to find out is using the tools that DONT work on it and see what that looks like, then you'll be better be able to appreciate the RIGHT tools. This is practicing with purpose, so you truly know your way around the problem; so on the exam if you forget what the right tool is, you can try using a random tool and realize "Oh yeah, ive seen this mess before, I have to try something else...hmmm what else did I learn to help me..." and then you'll land on the right tool and get the right answer!
All math tools have a derivation behind them. Someone found out a way to get to the formula - so sometimes it can be useful to see how they got there. other times, not so useful. very little in math do you need to "Just memorize it lol". Don't be afraid to ask the professor these kinds of questions in their office hours, they will always be very happy to get a question like that and are always there to help you with any kind of specific problem you struggle with.
I have encountered a chad who I can really say that, he IS natural born for math. He's a classmate of mine who does not study at all during the semester(according to his roommates, 3 of them), speedruns real analysis in a night, and ends up scoring higher than me. The whole experience is quite frustrating since I studied my ass off for that exam, but I think that one of the functions of colleges, to have that one off experience and to use it to expand ones horizon.
Don't know if it's more clear than you have explained, that d sigma is n-1 dimensional Hausdorff measure, which measures the n-1 -d volume of a n -d space, like lines in R2, areas in R3
Hard work always impressed me more than talent anyway.
I've come across these guys a lot. There can be other hidden factors like he may be thinking about math while walking. I knew a guy who did this and was thinking about math even when it seemed like he was just relaxing. This can actually be more useful or powerful than regular studying if you can manage to do it.
Also you don't know what he went through in previous years so that may have set things up so this course would be easier for him. Better not to leap to huge assumptions about talent when you don't know his full story.
Skill issue
Don't be impressed. I do this for every one of my courses in university. Currently a junior in neuroscience taking physics 2 and calculus 2 along with some neuro courses.
Hard work is more impressive. I do it because I'm lazy, and I know that I can get away with doing it and getting all A's. I know it won't get me far
Awesome video! It is almost as if being naturally gifted is a deception to get you to do further studies in maths, and then it turns out that your intuition is not worth so much anymore. Anyway if you are passionate in mathematics rather than being passionate about always being correct then it should be all the same to you even if it is a struggle to get to where you want to be.
I think many people in a PhD program feel this way. I wanted to know more and more about the hard maths and bit off more than I could chew. But I still wanted to know how it was done, so stuck with it.
Man, your channel is amazing, it really serves as a way to modulate my study environment and motivate me to study, thank you very much! What a find, don't stop making videos!
It’s like exercising with bad genetics, you can become much stronger and more aesthetic than you are and more than someone with good genetics who doesn’t workout as hard. But when hard work and genetics combine, you can’t compete with that. It’s similar with Maths, I believe. I’m a Maths teacher, who wasn’t the best at it, always had to work really hard and spend a lot longer than the top students to get the same results, but was capable of top results when I really pushed myself. I’m always improving even now and trying to impair the insights and intuition that took me years, to students in a shorter period of time. It’s probably the case for all things that hard work beats talent but only when talent doesn’t work.
Impart*
the geneticspill is hard to swallow, im pissed off im only 130IQ, 10-20 years ago, yeah that would probably be good but now, man it's below average in the top level fields
yh exactly, theres a few metastudies on the role talent and hard work play in sports and at all levels talent is more important. obviously you improve at something the more you practice it but how quickly you improve/at what point you plateau depends on natural ability. rly it comes down to how you want to spend your time
If what you are saying is true there is no point to living. If you can not win then why even play
this ^@@TwoTransmaxxingGamers
I’m in my second year of my math masters and I feel the same way… trying to overcome the proof barrier but it is getting a bit easier. Taking a lower level real analysis but taking Abstract algebra which is quite tricky/fast paced. Hopefully I can get to a PhD 🤞🏼🤞🏼
Man I don’t understand the maths in this channel but I keep coming back 😅
1:15
"Growing up, my dad-"
*instantly pauses* shhhheeeeeeeiiiit.
Thank you for this perspectives. It’s scares me because I’m in the phase of still breezing through the math, but seeing the work that some of my friend are doing makes me realize that I haven’t gotten to the hard stuff yet.
I feel like this unintentionally proves that it is pretty much all about hard work. Ever since he was a kid he had an interest in science and math, worked at it with the help of his father which probably gave him a pretty good foundation that allowed him to excel in early classes that don't require too much then as he took more advanced classes that required more work and probably without the help of his father he had to start applying himself more.
So it may not be a born talent , right?
It is true that I was just naturally interested in the subject so practicing did not seem like work for me, but I would say that I had to put far less effort in studying algebra and geometry than most of my peers in HS, even though the content was just as fresh for me as it was for them. But eventually everyone hits a wall and for me, it was proof writing.
@@PhDVlog777 that's interesting and it might come down to natural talent, but im not sure I am convinced. You said that your dad was very involved in your education and was a former teacher, did your peers have a similar situation? Even if they had tutors were they interested from childhood in STEM or at all? I get that I don't have the full picture, but I find it hard to believe that natural talent exists in any meaningful capacity when there are so many factors to consider.
@shadowshadow2724 no one can just excel without studying and practice
My man, u can have 2 different kids, both with an equal interest in math, one born with an iq lets say 20% higher, both study the same amount of time. Who do you think (in the same amount of time) will learn more, will they learn equally? Of course not. Also before anyone mentions it high iq doesnt automatically give you math knowledge, you still have to learn it first, but people with "natural born talent" learn it way way faster. Anyone can learn abstract algebra, its just someone smarter might understand it in a year, someone in 2, but it will take your average human wayy more time and work to be put it.
Your videos are amazing man. I love them so much. Very realistic and practical for all people that want to advance in Math. Keep it up with the great uploads
This is such an inspiring story but also empathetic (and to be honest sympathetic) towards my current plight of getting a math degree. Thank you.
Also, my bro graduated with an environmental science degree which is ironic since you got that for your bachelor's. Ironic! :)
I'm a math major and I'm in my second semester of real analysis, my first proof-based course, right now and it is kicking my butt. It's really reduced my love for the subject because I just don't grasp what's happening half the time and I've been thinking about dropping my major and just doing computer science, but you are really encouraging me to keep trying to do what I love.
What career in the future are you going to pursue from math major?
Not wise. I was a math major who went the cs/data route.
Interesting. I am a CS major about to switch to math.
I think real analysis can depend on what lecturer you get (like most I guess). Real analysis was the subject I came closest to failing because I couldn’t understand the lecturer. Did fine in complex analysis and measure theory later on with better teachers.
Although I definitely did proof based courses before that like linear algebra and calculus.
"To use mathematics effectively in applications, you need not just knowledge but *skill* . Skill can only be obtained through practice.", Mary Boas in "To The Student" of her "Mathematical Methods in the Physical Sciences".
This is true for both applied (computational-oriented) and pure (theorem-proof oriented) courses.
I was not even the best in my highschool class in math but still one of the best without putting in any effort into studying. In high school I only cared about videogames and playing guitar so I did not do that well in my final exams, but still got into a very good university.
When it came time to choose what to do in uni, I naturally went to study maths and computer science because those had felt easy to me without actually putting in any effort. Well after first semester of Linear algebra and real analysis I basically fucked up both classes with very bad grades. I was at edge of going full on to programming side because I thought that I just had no natural talent to grasp these higher level proof based classes.
Well after my freshman year, I spent entire summer studying hard to try to understand the stuff and picked up a book in topology to self study for the first. It was first really regirous math subject I took. After that summer I took a challenge and hopped into two graduate level class and I somehow managed to pass the classes with better grades than the elementary undergrad courses on my freshman year. Then going into stuff like abstract algebra felt like a breeze. Somehow I turned around in studies just by finding out that math is actually interesting theory building. Now im doing my master's thesis in geometric analysis and teaching and we shall see if I get a Phd spot/funding.
Tldr: I really feel like natural talent in math doesn't carry you far.
Thinking of doing something similar to your channel but for my physics PhD. Love your videos and find them deeply relatable. Really enjoy the story time format.
Hard work beats talent if talent doesn't work hard
Your videos are very soothing and nice to listen to in the background. Thank you
I noticed that I had a sort of latent talent for math. I sucked at trig and Calculus (well, not as much as some, I just displayed no real inclination for it), but once I got to proof writing and discrete mathematics, it all changed. Propositional logic and meta-mathematical concepts all made perfect sense to me. I had a hard time remembering all of the rules for computation in Calculus and Linear Algebra, but writing proofs was a piece of cake for me. Even though I ended up getting a C in Calculus 2+3, I swapped my major to math and got a 98 in Abstract Algebra, then a 95 in Topology that same semester. It got trickier once I started doing Analysis and Representation theory, but I somehow went from someone who had a hard time in freshman courses to someone who got accepted into some extremely competitive math graduate programs.
School: 900
College: 1200
University: 1500
Phd: 1800
Natural born talented guy: 1100, he works to get an extra 700 and gets the PhD.
Natural born less talented guy: 900, he works to get an extra 900 and gets the PhD.
Average guy: 700, he works an extra 1100 to get the PhD.
Cold hard truth here. But, being resilient and hard working is also a talent so it’s easier for a resilient average guy to get a PhD then, a lazy genius. Crazy thing is there are people who are more resilient than you and smarter 🤷♂️
But, don’t just guess if they are one or the other because you could really not know shit.
This aligns with my math experience. Everything through Trig was pretty easy and I was able to just rely on "natural ability" without much effort. Calc 1-2 was doable, but took a little work, Calc 3 was just tedious and basically turned me off from math. Linear Algebra was a return to easy mode for me. That's as far as I went with it for the most part while working on my CS degree.
ive been in kind of a slump lately and this video is what i needed. thank you
Wow I’ve never seen someone with exactly the same handwriting as me before. This feels like running into my twin on the street
Dude I love your videos. I resonate with your story a lot, keep it up and cheers to you!
Ive rediscovered my interest in math while solving some random financial questions online. I was genuinely good in math in my school years, not a genius, I just liked to study it, and I wanted to be an architect but
I undergraduated in an area that has nothing to do with math at all.
I have studied the subjects for 4 years, and completely forgot math existed.
But while trying to solve those random problems I felt so happy, like I have this pure love for math or something.
I’m taking metric spaces and topology this semester. Just this afternoon I spent 4 hours working on a single problem. Talking to my classmates before class they made the solution sound like it took them no longer than 15 - 20 minutes . For me it wasn’t clear, and my internal dialogue was suggesting that I quit mathematics at several points during my problem set. My internal discussion said “You think you’re smart.You’re not smart. It took you four hours to produce this solution. Pathetic. Other kids can produce this in less than half an hour.”
Once I had solved the problem, I felt such a great sense of peace. My internal discussion was now re-affirming thoughts of self love, tranquility and open mindedness when moving onto the next problem.
Despite my difficulties in mathematics, and I personally have found a lot of inner peace studying mathematics. And my intuition keeps telling me that I need to continue. My curiosity to learn more mathematics like Functional Analysis, Symplectic Geometry, Irrational Pattern Functions, Poly Dimensional Topology.
That's great to hear
I would argue that the ability to do hard work and not burn out is the natural born skill that's operative here.
That is also developed
copium u cant IMPROOV IQ@@hernanvallejo2929
Really needed this, been struggling with discrete math. Also you sound exactly like one of my discrete math TAs
I am not looking forward to discrete math
@@user-qy6tu9ip9v It's ok, you get through it, you survive, you get stronger
I think tutoring ("teaching assistant" as you guys say in America) for a course is really underrated as a way to get better. I have much better recall of the material from courses I tutored than courses I took but didn't tutor. Most of the maths and computer science courses that I took and never tutored for, I took them so long ago that I can barely remember much of what we covered. I'd pick it up if I had to go back through it, or if I needed to pick it up. But I can't recall anything but the very basics from the top of my head. But courses I tutored? I can remember most of what was covered in those courses many years later. You see every mistake and misconception anyone could possibly make and you have to get good at explaining things in 5 different ways, because different students need things explained differently to understand them.
teaching assistant in the US usually means you are actually hired by the teacher of the class to lead discussions or grade homeworks or tutor as well. Tutoring is a part of the teaching assistant job in the US, but it is not all of it.
Love your videos. I have a bachelor's in mechanical and have been graduated for 10 years but always had a interest and hunger for more math. I took calc 1 2 and 3, DifEq, LA, Mathematical Methods, and a few fluid mechanics course which is really just applied math mainly PDEs. That being said I wish I had the time or chance when I was in school to take more. But since being out I have done a fair amount of self study on topics interested in which is alot of fun. Following your PhD journey and learning about the process has been interesting and enjoyable
I cant speak for everybody but first year in undergrad was the hardest part. then halfway around year 2 when i got into algebra and topology the understanding of these topics was much more intuitive for me.
a natural talent combined with good parenting creates a passion which drives your practice. Your k-12 experience sounds exactly like mine except my parents didn't help me or care about anything I was interested in
fight until you know nothing but success.
Talent determines the upper limit, while effort determines the lower limit. So you aren’t going to reach even your lower limit without putting in effort.
i feel as if talent determines where you start, and how quickly you learn and build on that. but with effort you can improve both of those things. but it would take so much effort and time to get there that it may not be practically worth it. which is probably the reason why we often end up losing interest in the things that we're bad at.
one more thing is, if you're just naturally better at something you have a very powerful positive feedback loop to keep you going, to keep you motivated. it makes that thing fun. if you're playing soccer, and play well and do cool stuff, you're going to enjoy that more than everybody else. or grinding video games for eg. while on the other hand, if you struggle all the time then your entire focus is on catching up. never being the first one to get somewhere is discouraging. you not only struggle with getting better, but also struggle with a whole bunch of things for that reason that bring down your productivity even more. you're much better of not worrying about any of it.
Agreed. A lot of these comments are cope. You can work as hard as you want and never be Terrence Tao or Gauss.
What you described with "borrowing now" from others is exacty what I had to do with advanced electromagnetics. I just couldn't fully solve many of the problems intially. Now, looking back I have a much better intuition for the subject.
Success = Talent * Effort^Focus + Random Events.
Maybe talent*effort*focust + random events as being twice as focused makes you learn twice as much per unit of time studies.
As a university student on my 4th year, I find math is much easier if you try to understand what the numbers, symbols and processes represent.
If you can find some good geometric interpretation of what the math is doing, it becomes so much easier to visualize the problem and understand how to solve it. Most of the time, I don't do a lot of practice problems, because so many of them don't help me get an intuitive undersatnding of how the processes work. If I can understand the how and why, I only need a small ammount of practice usually.
The disadvantage is when teachers don't give any good explanations, and just tell us to solve lots of practice problems, I become completely lost.
Fortunately, the internet is absolutely packed with math videos that give lots of different explanations, some of which are much better than what my professors give.
I didn't "get" Math until my Sophomore Geometry class in high school.
My experience is similar. Things were straightforward from K-12, years 1-3 of college math were a combination of enthusiasm/talent (ODE, PDE, algebra, analysis, linear, etc), but then year 4 math and beyond was purely hard work (topology, functional analysis, algebraic topology, differential geometry, etc).
There is really a strong parallel with learning music and mathematics. Especially in the "borrowing knowledge" topic. You learn other people's songs when trying to apply the concepts of how to "do" music, there isn't any shame in this; it is encouraged. Why we didn't apply this logic to math innately kinda baffles me.
I changed from applied math (which was very much not applied, mostly just proof stuff) to mechanical engineering for the exact same reason. I can say that it was absolutely the best choice I made. It brought back all of the calculus and differential equations and actually used them for useful things. Having a course like welding and aerodynamics is also a plus :)
The best quote about talent vs practice is from Hajime no ippo:"Not everyone that works hard is rewarded, but all those who succeed have worked hard."
You can get decently far with only one of the two, but if you want to become the best at something you really need both
Everything was very easy before university for me. Now in university I realize how my lazyness and lack of study skills have an impact on my performance.
I am quite talented at most common academic subjects and throughout my life I made this apart of my identity. I never really had to try for most if not all of my subjects causing a lazy attitude when it came to acadwmica. My high school years I can't really just fuck around for all my subjects so I start lacking in most. Math was one of the few that I didn't need to put much effort in and again at some point I started identifying with math.I also started to view it as a hobby and with these combined motivation I study math. As a junior in highschool I have taught myself math associated with the first two years of college and ill graduated highschool (hopefully) having taken classes for calc 3 and linear algebra. I try to teach myself proof based mathematics even going so far as to try easier sides such as number theory. Though in the end I feel most of the reason I do this is out of paranoia. Paranoia that one day I'll suddenly no longer be able to advance in math. Paranoia that I'll loose all interest in math. That was my main motivation for teaching myself math far above my level. I can really relate to the way you view reading and how you got back into math almost at an impulse as I see myself doing something simular.
Recently,I was struggling with linear algebra like matrices form and dot product in machine learning So sad … after Reading comments and watching videos I am inspired
When people say Asians are good are math, it has little to do with talent. The culture believes in hard work and that’s why successful asian students have been drilled in math since a young age. Unless you’re Terrance Tao
I think the constant hard working nature of those Asian countries being passed down throughout generations legit builds that naturally, innate, inborn talent they have to be academic monstrosities. I think all talent is derived from our ancestors and how hard they worked.
I would say to some extent that I got better at academics the further I went on. I remember also having troubles with Algebra I, but by the time I made it to calculus it all made sense and clicked really quickly. I feel like I glided through by electrical engineering degree, but I’m starting to have troubles in graduate school now.
I have a bit of a contrarian view of things. Everyone extols the virtues of hard work/practice over talent, but I feel it's smarter to choose a field you're already talented in. If you want to coast you can just get by without putting in much effort; if you want to be the best of course you've got to practice and have discipline; but in my opinion the best way is to learn skills that have synergy with your technical skillset that few others in your discipline can do, which will dramatically boost your odds of success. For example as a mathematician if you learn to get really good at public speaking, UA-cam, marketing, psychology (specifically the art of inspiring others, and also encouraging others who are struggling), and the science of learning you can become a famous maths educator like how Neil deGrasse Tyson is for science. It's infinitely easier than trying to become better than Terrence Tao at maths.
I’m in high school right now and I’ve been self studying a lot of math (right now I’m going through algebra chapter 0 by aluffi) including proof based stuff, I feel like I’m in a horror game right now, I’m terrified of when the bubble is going to burst and my natural talent is going to run out. Cause I thought it was going to be calf and then I did calc and it was fine and then I thought it was going to be proofs and I did that just fine, then I thought it was gonna be topology and I did that fine, I feel like I’m going to get humbled soon.
Everyone hits the wall, but the comforting thing is, walls are always of finite height :) Keep going!
School - Happy Work-Sad😔
I think talent encompasses being good at everything in Mathematics, including having a good memory and good calculation skills, along with of course mathematical genius, and other things. What thing you were talking about is 'decent mathematical ability'.
Even in the level of master's degree in maths, it doesn't need to discuss about talent. Most of people can graduate if they study hard.
I think one can distill down the “Natural Born Talent” vs “Practice” discussion down to “Intuition”. Specifically, the moment that new materials doesn’t become intuitive is the moment that one’s natural born talent becomes obsolete. Unfortunately, there are few human minds in history that materials above a bachelor’s degree are intuitive. Those few are legends in Academia.
talent multiplied by practice, both can take it to 0, both are effective
Great video! as someone who doesn't have alot of natureal talent it's refreshing to see other people go thought the same struggles as i'm going through. Do you have any idea want you would like to do after your PhD, have you got any carrers in mind?
I like being a math teacher, but I also like day trading.
I'd say it's 60% talent and 40% hard work. I also think that the people that are drawn to math tend to be more naturally talented. Math certainly attracts those with analytical mind.
One of my boxing coaches told me that “talent will always beat hard work, but that talent will easily be hidden by your own habits”.
It’s not the greatest quote, but for someone that used his genetics to pass through early stages of training/camps, I can humbly say that hard work can nullify your lack of talent, as one 3rd fight showed me that 😅
I still hate when people say “I can’t do something, because I’m not good at it”…like you barely even tried, no other task in like will be easier then the one you’ve been half ass-ing for a few years now. I think everyone has talents, and when you work hard enough you’ll start to see those talents much more clearly.
“No one will know you’re talented if you don’t work hard”
Do you like Lang's Complex Analysis? And why?
I know this isn't directed towards me but I'm going through that book rn and also taking a course on complex analysis and it's pretty good :) clarifies some of the more simple elements of complex numbers and such, and explains things well
I like it, although I haven't picked it up in awhile. My favorite complex analysis book right now is Churchill-Brown-Verhey.
Have you ever done any math competitions in highschool? (AMC, AIME, USAMO, IMO, etc.) Have you also heard of/done the Putnam contest? (I believe that is for uni students)
iirc the Putnam contest is only for those who have not completed a bachelor's degree (Or masters, possibly)
No, my high school was small and we didn't have those opportunities. I didn't know what the IMO was until my second year of my masters degree.
Intelligence is 80% heritable
Amazing that you can now look up graduate course homework problems on the internet. If we were stuck, all we had were the professor or other students in class.
You can, but it’s not really helpful. You can write the solutions to get you by on the homework, but despite the wealth of resources online, I still end up needing to go to my professor to get explanations and build by understanding.
@@OmgitsNeens Oh, I don't disagree at all. There is no substitute for a mentor helping you to put everything into context. Still, the fact that all solutions of baby Rudin are available for free in 10 seconds online still blows my mind.
I feel like I had a similar undergrad experience. Except afterwards I never went back and I still regret it. It's not really financially viable at this point. I will still always find math interesting and continue to study it (slowly) on my own.
7:19 Wait couldn't you just prove it by truth tables?
I gave up on math. It was kind of my dream but I couldn't put the required effort towards it. It doesn't help that I have ADHD and some other problems that affected me more than it should. I'm 31 and am finally finishing a computer science degree, but with 80% of math courses completed. I've changed majors 3 times from engineering to another engineering then math then compsci. Pathetic. Now I just feel like it is too late for everything.
ADHD is the most underestimated disorder add insult to injury also belittled and sold by those who have it as a super power only to realize much later how disadvantaged one is.
A “naturally talented” math person here. 100% attest that people who put lots of effort in are better than me.
For perspective, I have done my bachelors in CompSci. I took up to Calc 3 and have a Data Science background.
My number sense is about the same as someone with a PhD (particularly with statistics), but my ability to actually perform calculations pales in comparison to anyone who actually practices.
Talent will make it easier to learn, and it gives you a good intuition, but that means very little in the real world if you don’t apply yourself to the discipline.
When I was doing my Master's (physics) there was a PhD student who was brutal. While his peers PhD students studied for hours, days, endless nights for the qualifying exam, dude just did his homework and had a side startup (unrelated to physics)...dude was amazing. Ended getting an honors PhD. So yes, being smart does help, like it or not.
don't you think he might have had a more efficient process? he must have been really confident and didn't struggle with any sorts of anxiety, especially while juggling with a startup at the same time? Personally the less anxious I become in my life the clearer, more focused and fluidly I seem to think.
@@matviyk3066 One night (must have been like 7pm or so), I was heading out of Campus with my GF (at that time) and we saw him an offered him a ride since he was heading in the same direction. We got to talk for about an hour or so (lots of traffic in a big city!) and it was very insightful. I talked with some friends who had gone to the same University as he did for his Undergrad and the stories were out of this world....I think he is the smartest person I have ever met, along another friend I had while at my Master's....they were both amazing, but the first guy was just out of this world...
is getting a math phd degree contingent on doing something 'thesis worthy/substantial' (like solving an unsolved problem or finding some new insight)?
Yes. More or less.
You need to have a successful thesis, which is deemed worthy by your thesis committee.
Are you a UofT student? Those used stickers that you have on your books seem strikingly similar to what we have over at University of Toronto when our library/bookstore sells books.
No he’s not . He studies in the US and is not from Canada
Ah, am in my third year of college, needed to hear this thank you
i always wonderd how the hell do people come up or find even an idea to do a phd in
I naively dropped mathematics in school but then in college fell in love with economics so now I'm trying my best to get into a quantitative economics masters program, starting my real analysis course very soon 😺
Good luck! :)
Its mostly talent to be honest. You can practice as much as you want when it comes to math and science if you dont have the talent for it you're not gonna go very far
Dont forget that even geniuses practiced problems
2:44 whaaaaaat! I'm still in high school but our story is quite similar
3:26 thanks for leaving in the voice crack
Talent is undoubtedly way more valuable than hard work
Anyone who studies the most can invent stuff, however that's not the case in real life, we remember people like Nikola Tesla, Dmitri Mendeleev, James Clerk Maxwell & Michael Faraday not because they worked hard but because they thought out of the box
Someone who works hard can solve 50 application (level 3) questions in 30 minutes maybe
Someone who has talent can solve evaluation (level 5) questions and invent or discover new concepts that open up whole new fields, meaning they prevail at the creation level (level 6)
Whether we like it or not that's how people's brains are differentiated in terms of practical ability
such a great motivation... i'm in basicly the same shoe (currently physics undergrad) and it felt real good to see i'm might not hopeless... :)
i'll tattoo this on me "Eventually practice becomes talent".. xddd
(sry for bad english, still learning :))
What do you think about the following statement: a talent that does not lead to practice is not a talent at all?
I personally believe that the difference between high school-level math and more advanced math lies in the type of the information required by the brain to processes. High school-level math is mostly about using relationships between different symbols, which is all visual information. Advanced level math in college or grad school is more about reasoning, which necessarily requires processing information through words (i.e., verbal information). I personally believe this is the main difference between those who have to do more hard work while doing advanced math (or any type of subject that requires an advanced level of reasoning) and those who are deemed "naturally talented". I believe the "naturally talented" people process verbal information in a different way. I'm still not sure how their brains process verbal information though (it's a work in progress).
Processing verbal information is also a skill. I would like to use interpreters as an example. It has been scientifically established that an interpreter is not a person of natural born talent, but instead a skilled professional. Interpreting is much more than translating words and spitting them out, it is a process that includes receiving information, decoding it and then building it back up again in the target language as it would be expressed in the most natural way for the given context of the situation. I would encourage anyone to try interpreting from one language to another. It will train your mind to hold more information at any given time and be able to remember and reproduce more and more accurately.
I have a question as someone that nearly failed undergraduate mathematics, but really excelled at high school maths, and is now in a master's that needs more applied maths. How do you go about identifying where your gaps in knowledge are, and how deep to go?
It is difficult to say, but I would look at the classes needed to get to a BS in applied math, and at the very least take notes of text books used in those subjects. For example, if you need probability theory, then I would read some of Chung or Billingsley to get familiar with the topic. Problem solving is best but more time consuming.
At the very least, you are looking at a few months of playing catchup, and you are never really certain where to start and stop. But this was my approach to filling in the gaps with measure theory and complex analysis and it seemed to work. I hope this helps!
@@PhDVlog777 Thank you! Tbh, it feels really overwhelming to start. Unsure if you have experience with it, but did you mainly fill all of these gaps yourself? Did you find tutors? Or a mix of both?
I think talent helps you to a certain degree but at some point you have to learn a lot, even if your talented. But many talented people learn to learn when they are young, so they struggle in college.
I am a 3rd year engineering student and I experienced the same.
k12- no work needed.
Y 1&2 - easy topics but tedious calculations.
Y 3- Why am by being taught by Eminem.
What are your thoughts on going against your talents or doing something you are not talented in? I prefer fiction over non-fiction for example and enjoy things outside of my field but I chose the field, cs, in order to force my brain to think a certain way. It was more like an admiration for people's logic and creativity in this field and even though I held almost no talent in it, I am still trying to get a degree in it, hoping by raw effort and practice I can become very comfortable and strong in it.
It may be a little extra work but I do not think there is anything wrong with picking a degree outside your natural skill set. A lot of people attending university do not have a particular set of skills (*Liam Neeson intensifies*) and they just pick a major and go with it.
I wish I have more math fundamentals during K-12 i feel like my teachers did me and the class a dis-service
This is going To be the most viewed Vid for the next six months
What is the name of the proposition at the end of the video?
Something you said there spooked me a little. You mentioned that you almost only read class notes and barely read the textbook which didn’t work out for you and I’m worried I’m ending up in a similar situation.
I am taking a measure theory class now and the prof. is a bit on the eccentric side. His lectures deviate from the textbook quite a bit, to the point were notation is almost completely different, a bunch of definitions and especially theorems/lemmas/propositions and their proofs are done in an almost completely different way, and the way he progresses through certain chapters in the book, and what he decides to talk about and what not to talk about have been a little bit different from how the book actually does it. So what has ended up happening is that I’m almost entirely basing my work around notes from his lectures and only using the book to double check/look up some stuff as well as for a few exercises that he has picked out.
Based on your own experience, am I shooting myself in the foot, or does the situation warrant not using the textbook too much? It’s getting to the point where I’m struggling to follow the proofs and theorems in the book due the different notation and so on.
Use the textbook because it's new to you. For your professor, he knows it so he doesn't look at the textbook. That's why he deviates. I think if you build a strong knowledge base from the textbook, going to class will be fun because you get the extra stuff that's not in the book so it will only add to you. What I'm saying is, reading the tetxtbook will only add to you and not reading it will only take away from you.
It depends on what your professor expects from you. If the prof asks easy questions, and knowing thy lecture notes is sufficient, then you will probably be OK, but I would have to sit in the class myself.
I know this: my notes from analysis were unreadable and I had to read other textbooks to keep up. Problem solving was also far more beneficial than studying lecture notes and reading ever were.
My advice is to (if you are not already) speak with your instructor and learn what he expects from you. Speak with others in your department that have had his class before and learn from their stories. This is what I did and it worked for me. Good luck!
During my 1-10 years I used to suck at math . I used to fail maths classes and when my wud question the teachers they wud blame it on my previous maths teachers. Only when I started working on maths problems on my own during 11th and 12th is when I actually started to see the beauty behind it .