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I wouldn't be surprised if this held true for particular subjects that are generally thought of as "hard" as well. My favorite is mathematics. I used to have a really tough time with maths and used to hate it. Then, the smartest friend I have (who coincidentally did his PhD in financial mathematics with Freddy Delbaen back then -- for those who don't know this area, that means he was really good) once made an offhand remark along the lines of "Most people think of mathematicians as 'geniuses' -- if only. You have no idea how much you simply have to learn by heart and how much work it can be." That sort of pushed me over the edge and I started to re-learn mathematics from the ground up, starting with, yes, high-school material, working my way up with MIT OCW courses (http://ocw.mit.edu) and eventually being able to take PhD classes in statistics in the space of about 2-3 years (incl. periods of part- and full-time work). Not bad for a guy who couldn't tell the difference between cos and sine functions when he started. And all because I shed the notion of "math is for talented people" and started to actually work my ass off. It was surprisingly "easy" once I put in the work and I also noticed how shortcuts taken in one part would hurt me down the road later so it really was all about remaining disciplined.



I wouldn't be surprised if this held true for particular subjects that are generally thought of as "hard" as well. My favorite is mathematics.

A new favorite quotation of mine about mathematics learning:

"Mathematics must be written into the mind, not read into it. 'No head for mathematics' nearly always means 'Will not use a pencil.'" Arthur Latham Baker, Elements of Solid Geometry (1894), page ix.

Hat tip to the Bay Area Math Circle for the quotation reference.


As I have commented before here, mathematics is a skill and needs practice. If you don't work examples and problems you won't learn math, at best you will learn about math. I taught myself first-year calculus twenty years ago, but didn't keep up with it. I have recently started reviewing it, and because I sat down and worked things through, it is actually coming back pretty easily (more so than I expected at least).


I agree that the path to insight goes through a lot of work, including memorization and repetition, but I know people who have taken that path and ended up with little to no insight. For example, my sister had "no head for math" but got As in high school calculus through sheer hard work and repetition. She worked and reworked homework problems so much that she could solve almost any test problem by remembering the homework problem most similar to it and using the homework solution to guide her. To this day she insists that she never understood the first thing about it even though she was one of the better students in an honors calculus class. She went on to do the same thing in college calculus (and got As once again) but she felt lost the whole time and said that if any test she took had come back marked "F" instead of "A," she would not have been surprised in the least.


I just finished a course on integral calculus...one of my classmates said he put in 20-30 hrs/week and he ended up dropping the class - he just couldn't understand the problems and he was a terrible test taker. I put in a lot of work in the first month or so then the rest of the course sort of just sailed by. I guess my only point is, YMMV...


If he struggled that much with integral calc the cause is likely a poor foundation in High-school level mathematics.


I agree, the hardest parts of calculus are the algebraic manipulations required, the basic ideas are relatively straight-forward.


Integral calculus is easy - it's all about calculation. Where math got hard for me was when the focus shifted from calculation to proof (abstract algebra, number theory, real analysis). I thought I could coast by with the same sort of intuitive understanding I had with calculus, but I was wrong and I didn't have the self discipline that was required to do well.


With proof you have to do a whole host of them first. I remember, filling a book with proof. Eventually, a pattern will emerge and they would be easy. At first, I used to hate them but practise enough times and you would want them to show on the exams.


For me it was quite the opposite. I didn't begin to show any particular interest in mathematics until proof based courses came into the picture. I would certainly not have earned a degree in mathematics were it not for my interests being piqued by a capricious used bookstore purchase of E. Kampke's Set Theory when just out of high school. It was the axiomatic proof based method that appealed to me. Calculation, on the other hand, seemed to my naive 17-year-old self to be terribly banal.

To put things in perspective, and to possibly invalidate the general application of my insight on the matter, Calculus was in fact the only course I passed my final semester of high school. I received a D shortly before I dropped out altogether. The D score was earned only after being the only student to ace the final, a task which was itself only possible after I proved the first fundamental theorem to myself (thanks to an especially verbose description of it in one of the exam questions) during the course of the test.

To this day I find that the actual solving of equations to be tedious and can only be interested in problems tenable to axiomatic and algorithmic approaches. Thats where all the fun is imho. Who cares about actually determining a number (or equation)? [the answer: all the smartest people do.]


How much time did you spent on it? I mean, daily, or weekly? I'm starting to work on the same direction, but I work a lot during the day as a developer, so I'm rather exhaust when I'm over with it, and usually don't have more than 1 hour a day or little more to spend on learning maths. I guess it's too little?

I'm serious though, I'm learning with textbooks, etc.. and I'm a dropout so I have some of the basics already.


What I tried to do when I was working full-time was: Get up early in the morning, review some definitions, then read perhaps only one theorem, try to prove it yourself, then read the proof in the book, then try to find a problem in the exercises that would use that theorem or refers to it, and take those problems with me to work. At work, I could then think about it while doing less demanding tasks or just during breaks. Then, after work, I was exhausted so I only read a little bit and looked at new definitions or so. Lots of time, I also just goofed off (yeah, it took discipline and energy to get something done when working full days). On weekends, I could naturally do more. Still, this was my slowest time, for obvious reasons, and I could never have gotten through the material if I had been working full-time all the time.

In my first year of a doctoral program, I could study from after lunch until next morning and I concentrated on getting the fundamentals right. The greatest book, the one where things began to click, was Rudin's "principles of mathematical analysis". Before I came to this book, I had wasted a lot of time finding my style of learning or, if you will, discovering my preferences. But if you already have the basics and are not going to waste as much time as I was trying to figure out what is what, you might pull it off much quicker.

Good luck! To me it was a cool time and I finally got rid of my math inferiority complex ;)


Thank you very much! I thought you were going to say it couldn't be done while working full time. You give me hope.


When I first studied calculus I was lucky to have a job that gave me lots of free time. I spent about four hours a day, almost every day for about 4 months to cover the first two semesters of calculus (I started getting tired of it and only read the book for the third semester's worth, which is why I don't count myself as having learned it).


I have the same experience, my major is filled with statistics and math and coming out of high school I didn't have the right prerequisites. Nonetheless, a decent amount of interest and some hard work and I got up to speed in no time. Calculus was a breeze after that. It goes for a lot of things really, when you're interested in the subject and willing to work for it, it becomes so much easier. I now just have a "How hard can it be?" attitude towards stuff.




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