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We can't expect an average man to be able to learn, comprehend and apply concepts at the rate of an intellectual.
What is an average man? There was a kid in school who was so dumb that he held the entire class back. He was always 4 answers behind everyone else and the teachers always had to help him catch up while we all waited. He was particularly bad at math.
A couple of years later I met him and he asked if I would accompany him to the track. He fancied himself a horseplayer. I went out of curiosity. I was amazed to find that he could calculate complicated odds and payoffs in his head much faster than I could. More than that he understood probability, he had taught himself. So you see, he was not dumb, he just needed to be motivated.
Maybe he wasn't dumb all along. Maybe people assumed he was dumb just because he was slow. maybe he just wrote slow, or spent more time thinking about the questions. Even if he was a bit slower than the other people, he had a interest in maths so developed it. So many people don't put in the work.
Richard Feynman had a rather decent IQ, yet he contributed to physics in a major way. Talent would help, but I think one can run as fast as Bolt if they train daily for multiple yours for several years.
I've met Kelvin here - he's a very smart engineer. But let me assure you he got there due to his creativity and hard work along with curiosity, much rather then IQ (which I'm sure he has a fairly high of).
That must have been an enriching experience to meet him. I would actually like to see a study done on this. They should take groups of people with no more than average IQ's or even low ones, and provide them with all the facilities and education they need, then see how much they can contribute.
I actually would be a test subject of my own volition. I have always been called creative a lot in the past and I do have lots of ideas. I am the kid who would stare into space and daydream in class. My intelligence is probably only a high average or slightly higher than that at best though. I want to see how far I can go with mathematics though. I might choose to adopt a science subject so I can experiment and try to innovate.
My problem has always been people thinking I'm stupid because of my bad grade and lack of qualifications from school. I went from an 11 year old education to doing 16 year old exams without being taught the material so it's natural i would get low grades. My chances of getting into a STEM career area is almost non existent. Unless I do something exceptional with proof. nethertheless I will do mathematics in my own time then see if I can get into STEM later.
I love to learn from experimentation but the books don't teach that way. I need to learn a lot more math and science before I can start to play with it. I'll start learning little bit of programming now though since you only need basic arithmetic and algebra to start.
I agree. In my opinion, a genius should be defined by his hard work, passion and contribution to their field. Nowadays, IQ > 150 is the main deciding factor.
I actually think it's a combination of genetics and hardwork and a variety of other factors. Take for example an olympic sprinter. They trained hard to be able to run that fast. However the average man could train that hard and couldn't expect to run as fast a Usain Bolt does. I don't think everyone has the capacity to make breakthroughs. Although probablyl more men than we allow have the ability to contribute to scientific fields. We can't expect an average man to be able to learn, comprehend and apply concepts at the rate of an intellectual. I just think traits like persistence, curiosity, creativity and an ability to think outside the box should be considered more. Because there are plenty of intelligent people who go to university don't do much with it, yet you have people creating things without that much formal education.
Google or Youtube this: 15-Yr-Old Kelvin Doe Wows M.I.T.
I understand perfectly. Sounds like a very practical sort of person. Probably not a genius though.
Oh I think you misunderstood a little detail. She never claimed to be a genius. She had middle class parents. She went to a decent university and got a decent job and I suppose you need decent intelligence to do that. In my personal opinion her reasoning is dismall but she's good at rote memorization.
She said to someone else "If you were a genius then you would figure a way out of your situation." She said that in the context of climbing the socio-economic ladder. Not that the person claimed to be a genius.
Hi;
Heck, we certainly should at least consider something that a guy like Newton says.
A girl was saying that if someone was a genius then they would figure a way to climb up the socio-economic. Since that's what she wants to do.
Has she had any luck?
She has a $50k = £30 job with a car and her own little apartment. The socio-economic success is relative to what you consider successful. Being a woman she loves her security and wont take the risk of trying to start her own business. She might pursue a PHD in hopes she gets increased pay
A girl was saying that if someone was a genius then they would figure a way to climb up the socio-economic. Since that's what she wants to do. There is a quote that went something like "Humans think they are smarter than doplhins because they have built civilizations and dolphins swim around having fun all day. Dolphins think they are smarter than humans for the exact same reason."
Sometimes I wonder what intelligence, intellect and genius even mean. I know their dictionary definitions but people will call an intelligent person stupid just because the person is ignorant. Or they call a person stupid for making a mistake they wouldn't. They will call a person smart because they're good at memorizing and regurgitation. Or they will say the person is smart because they know a lot of information.
The definition of intelligence is the ability to acquire and apply knowledge. Intelligence is said to be fixed but as you can probably realize it's easier to learn things that are similar to what you already know. Therefore having knowledge already in theory could increase intelligence.
There is a quote that went something like "Humans think they are smarter than doplhins because they have built civilizations and dolphins swim around having fun all day. Dolphins think they are smarter than humans for the exact same reason.
Genius is defined as "exceptional intellectual or creative power or other natural ability." Again people in prison are known for getting very creative because they have time to think. So I believe Newtons quotation has some merit.
I mean people like Einstein, Isaac Newton, Nikola Tesla, John Nash, etc go down in history for their contributions to the fields. Even people like Richard Feynman with non exceptional IQ's contributed aa significant amount to the field. Why do we see so many people that are held as geniuses never doing stuff but memorizing thousands of digits of pi.
Lets use Daniel Tammet as an example. He contributed more for society by being a test subject rather than help create anything. He had exceptional memory but is only known to have used it learning languages and taking part in memory contests.
If you do, it's false. Counterexample: n, x is 2, y is 1
2/(2-1) doesn't equal (2+1)/2
This is proof by what? So you only need one instance of a counterexample to prove it's false but you need to use variables to prove something is true. Well I found I found some numbers that made it true and I was going to use it to solve a problem but realized that it might not be true for all numbers and turns out it isn't.
Those prices on from abebooks.co.uk Serge Lang's boook on basic mathematics is going for twice the price of that on Amazon.co.uk. His Geometry book is around £18 more on other sites. Gelfrands boks are a bit more costly on Amazon too. So I might not be able to get anything for very cheap but at least these are cheaper than the aops books.
Do Serge Lang's books have any challenging questions or good tips like the aops videos do?
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x - y
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Actually I'm struggling to find those Sege Lang books for cheap. These are the prices I have found:
Basic Mathematics - Serge Lang £ 19.74
Geometry - Serge Lang £ 12.41
Algebra - Gefrand £ 15.51
Trigonometry - Gelfrand £20.06
Total = £69.82 = $116.10
I'm almost addicted to the AMC questions at this point. I just do them for fun. For me they're analogous to people that do word puzzles or sudoku on the train.
Patternman:
That's fantastic! You seem really enthusiastic about learning mathematics and I think you'll be very successful. I can go on and on about how I think the education system is nonsense. Their method: formula/trick, memorize, ace the exam. Method of the books I recommended: intuition, derivation, prove, remarks which works so much better. Take your time, but work at a steady pace. Spivak in one year is a reasonable and definitely doable goal.Note that you will need more than NCERT to move to Spivak (in my opinion). Here's a part of the list I used which is sufficient for Spivak (or Stewart, Courant, Apostol).
NCERT Books
Basic Mathematics by Serge Lang
Geometry by Kiselev (volume 1: planimetry) or by Lang (and Gene Murrow - highly recommended) or by Harold Jacobs but I recommend Lang's Geometry the most
Algebra by Gelfand
Trigonometry by GelfandGo through all of those and you'll be ready for Spivak. This also gives me the sense that you are serious for learning math in a rigorous way, which is exactly what these books are for. "For Dummies" and such books, including high school books except NCERT and Singapore, are written by crackpots. Best of luck!
Bobbym, not on the job?
I have done an AMC and Out of 24 questions I got 4 wrong and 4 questions I couldn't answer. Of the ones I didn't answer 1 was about the calender because I don't know how many days are in each month, and the other was something on magic triangles. I don't know what magic triangles are. These AMC papers can be fun but I feel strapped for time. A few of the problems can be done in 1 minute but the majority take me around 5 minutes to do. That's 2 hours to do 1 paper and then probably 45 minutes to review it adding their solution methods to my repertoire. So I may only do 2 papers a week or something from now on. I'm happy that a couple of my solutions weren't on their list also.
As for those books I don't know if I'll buy them. The art of problem solving videos give tips for methods of solving problems rather than just giving you the rules and facts. That is how the videos taught and the excerpts look the same. I'll go through the NCERT books 9, 10 and maybe 11 if I can find it anywhere. I'll do the occasional AMC problems and then probably buy the aops books bit by bit. If I bought them all it would cost me a small fortune of £284 if I can afford them. I'll definitely get the number theory book. Depending on how it goes and if I decide to go to university for a maths heavy degree then I might buy them and go through instead of the others you recommended.
Don't you recommend going through an easier calculus book before tackling Spivak's? I have heard of people getting A's for their A level mathematics in the UK but st with Struggling heavily with Spivak's. Does that book reveal your real mathematics aptitude because I spoke to someone who was going to do a maths degree but decided not to after reading that book.
To be honest 2 days of watching the art of problem solving videos and doing 1 AMC has made me a significantly better at mathematics already, especially considering the time. I am better at using the rules I already know and understand them better. for example I have started using the commutative, associative, and distributive rules just to make the calculations easier. I now split the numbers into factors, convert them to fractions and cancell etc.
Watching the videos, doing the problems on AoPS + Brilliant & seeing the solutions makes me a lot better than class ever did. You can experiment and pick up little things like when you multiply by 5 you just half the number and put a 0 on the end.
This whole experience has made me unhappy with the education system here, even though I didn't finish it. The maths curriculum here is very watered down except maybe in the grammar schools and private schools. They just tell you a rule and a method then you got to use them on the most trivial problems. You go through the work book in class then eventually move on. The teachers make you think you're smart because you got a B or A, even though you don't fully understand. If kids have just learned indices you could ask them what is 2^3^3, or ask why n^0=1 and nearly all would be bewildered.
If I continued at this pace for 1 year I could probably be doing Spivak's exercises in approximately a years time. I'm unemployed at the moment so I have the free time but I might be working in a month or so. So the learning might not go that fast depending on what responsibilities are at the time. If I can find 3 hours of energy and time a day to the it then I'll be able to reach my goals.
Do you guys actually spend that long thinking about the math problems? The problems you're solving must be a lot harder than these then. These took longer than the average problems in textbooks that only take 5-30 seconds to do but none of the questions I answered from aops took more than a couple of minutes to solve. I had an idea of how to solve the ones I didn't answer but I wouldn't be able to answer them under exam considitions. It would be cheating to google some information then answer these questions right?
Also I like that they have multiple solutions to their problems. A few times my solution wasn't even there when I checked them. I can print of the questions from there and practice them anywhere. The NCERT books haven't arrived yet but I should be getting them on the weekend.
Why not AoPS questions? They are from contests like AMC. Anyways, NCERT has good problems. Print of problems from "Algebra" by Gelfand.
I was intimidated but I checked them out and can answer the lowest level ones. They have good videos on there too. I struggled on some of the problems and am ashamed because they're for kids.
mathepolis is nice but I would prefer to be able to print worksheets out and work on them m away from the computer. I get tired of staring at a screen. Brilliants questions can be a bit challenging for me and I don't wish to tackle them now until I get better. I will not even consider questions from aops right now.
Thank you for the greetings everyone. I really appreciate the help. I just ordered the NCERT books 9 and 10. Also oes anyone know anywhere I can find practice questions and answers online anywhere?
Thank you wise Yoda. These resources seem to go into a lot more depth than what I'm usually given. Those ncert books look good and cover more than the books we get here. Those problem solving books look great and are what I have been looking for. I am ready for that rigour but the price puts me off lol. I think I'll leave those till later. Thanks again ShivamS.
Welcome PatternMan!
It's sad that you couldn't complete your education, but better late then never. I think you have covered much more then early high school. Vectors and matrices are only scarcely covered in the 12th grade in high school in USA. However, you are missing a few topics. I suggest you fill your gaps in Algebra 2 and Elementary functions (precalculus) by going through a standard book, rather then learning topic by topic. As for books, try "Basic Mathematics" by Serge Lang, "Algebra" by IM Gelfand. A great set of books are NCERT books (www.ncert.nic.in/ncerts/textbook/textbook.htm). They are the primary set of books used by students in India, and covers man topics. You might want to start at grade 9 and move forward quickly. Another set of books, though expensive, is Art of Problem Solving. They are a bit expensive, but they give you great understanding. You can supplement all this with khanacademy.org or examsolutions.Good luck!
I forgot to put basic functions down too. I breifly covered that when they started replacing standard quadratics with functions. Thank you for the help. What do you find is the best method to learn mathemtics? I usually just follow the method, then try to understand it, then look at the proof or justification, then practice some problems.
Hello I'm looking to pursue mathematics as a hobby but might consider taking it up professionally depending on my progress and how much I like it. I was always good at math and seem to pick it up pretty quickly. I am here to discuss mathematics with people that like it because I there''s something special about hearing a person speak about a topic they're passionate about.
Since then I covered: prime numbers, fractions/ratio/proportion/percentages, probability, statistics/graphs, indices, surds, linear equations & graph, Quadratic equations/factoring/solving,completing square, pythagorus theorom, circle theoroms, vectors, matrices, truth tables, basic set theory, basic sequences and algebraic proof.
I think my knowledge is somewhere around GCSE or the beginning of A level or early high school in the USA. I want to understand all the fundamentals first so I can move on to calculus then the college/university level mathematics.
So if you please would help, I want to know what topics I need to cover and in how much depth? I would also like to know the best books around for teaching myself mathematics. I also want to know the best way you learn mathematics.