Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Math Intuition Cheatsheet

betterexplained.com

Post reply on HN
Print view View original

31–40 of 68 posts

Re: Math Intuition Cheatsheet

#31

Post by claudius »

pja wrote: post #28
bjeanes wrote: post #17

Earlier quoted context omitted.

Salesforce's Apex language is case insensitive. It leads to a a lot of confusion, frustration, and distress when different programmers have their personal preferred style of referring to a class or var or when the lazy ones just write everything in lowercase. Never again.

Would the best of all possible worlds be a case sensitive language where once you defined a name all other names that differed only in the case of the characters would be impossible to define? That way names would have to be referred to as written, but you would avoid the potential confusion of having multiple names that differed only in the case of their characters. Lots of programmers would probably regard this as…

I like there to be a difference between a variable “p” and a variable “P” (the former, for example, might be a vector, whereas the latter could conceivably be a projector onto this vector). Especially when doing numerical work, I found it to be extremely helpful to stick to established notation in the field also in the code intended to implement the algorithms written in this notation. In this sense, I find it sometimes sad that C++ doesn’t allow boldface/curly variable names :)

Re: Math Intuition Cheatsheet

#32

Post by jgrodziski »

For me the best one from Kalid is the explanation of the fourier transform: http://betterexplained.com/articles/an-interactive-guide-to-...

This article is simply amazing, moreover Kalid is an humble person and references his inspirations from several people over the web. Thank you very much Kalid for the time you dedicated to share all this wonderful resources.

Re: Math Intuition Cheatsheet

#33

Post by arianvanp »

I started my 1st year in college doing both math and computer science majors. I really love math but I just hate how (especially calculus and linear algebra courses) they just don't work on growing any intuition at all. in contrast, in my CS major we had a linear algebra class (ugh overlapping classes, huh) as well and we used computer graphics as a case study to learn about the basic concepts which was really fun and enlightening.

Eventually I just ragequit math because I didn't have the patience and time to search for the intuition myself, which is a pitty. I'm really happy I followed the classes about mathematical proofs, groups etc because I really enjoyed those but others were just painful.

Anyhow recently I discovered this blog too up my math concerning computer science is this: http://jeremykun.com/. It's a tad more advanced than this but I'd really recommend it. His blog is freaking amazing and always a joy to read. Math as a sidehobby it is! :)

Re: Math Intuition Cheatsheet

#35

Post by richardjdare »

chaoxu wrote: post #29
bithive123 wrote: post #5

I am often saddened by how much easier it has been for me to acquire computer science versus mathematical literacy. Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus, the biggest barriers have turned out to be things programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque…

> Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus Why would you need to prove central limit theorem in order to use calculus? What kind of course teach it that way? It would sound like something to prove for a honors calculus class that prepares students to become mathematicians. Anyways, about math education. Mat…

>>Why would you need to prove central limit theorem in order to use calculus? What kind of course teach it that way? It would sound like something to prove for a honors calculus class that prepares students to become mathematicians.

Yes, but not knowing calculus, how would he know that? Or any of those things that you have pointed out? They are certainly not taught at school (where we are unhelpfully taught an entirely different "mathematics" to that which you get at university, and will often require for advanced software work) and none of this stuff is ever pointed out in maths books.

Instead, the reader of a maths textbook is unsettled by the constant unspoken assumptions and unexplained concepts that he is already assumed to know. There is in university level maths books what looks like an almost wilful disregard for how people are actually taught mathematics at high school. How do students going from school to university cope? is there some secret occult ritual where all this knowledge is transmitted?

That's how it looks for those of us who are trying to teach ourselves, and it is in marked contrast to the experience of learning almost anything else, particularly programming.

Re: Math Intuition Cheatsheet

#36

Post by __Joker »

kalid wrote: post #8
bithive123 wrote: post #5

I am often saddened by how much easier it has been for me to acquire computer science versus mathematical literacy. Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus, the biggest barriers have turned out to be things programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque…

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

Unfortunately, most of the mathematical text miss the historical context which they are developed in. I personally enjoy a text in any subject, where the author develops the content with historical perspective.

Any learning course should include one textbook which develops the content from an historical perspective.

Re: Math Intuition Cheatsheet

#37

Post by vanderZwan »

kalid wrote: post #8
bithive123 wrote: post #5

I am often saddened by how much easier it has been for me to acquire computer science versus mathematical literacy. Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus, the biggest barriers have turned out to be things programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque…

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

Same thing with logarithms, which were invented to simplify multiplying two huge numbers. Combined with log tables (think paper LUT for humans) that became a simple matter of looking up two log conversions, adding them, then looking up the inverse of the answer.

Is that what we learn in high school? Nope. So everyone is left wondering what the hell they are good for (or were, before we had calculators) the first time around.

(btw, as far as I can tell you don't have that bit of history on your site yet either - might be an interesting addition?)

Re: Math Intuition Cheatsheet

#38

Post by hibikir »

kalid wrote: post #25
claudius wrote: post #24

Earlier quoted context omitted.

> Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." It depends. In high school, it certainly mak…

I definitely agree -- context matters. For a potential math major [i.e. people for whom metric space refers to a measure and not a flat in Europe :)], you definitely want the ground-up understanding. In CS it's similar, where you learn about transistors, logic gates, ALUs, CPUs, machine code, compilers, along with high-level languages. But, some people just need the HTML "Hello World" to make their webpage. (In the m…

That's how they taught CS at your school?

Where I studied, we did have classes on all the low level stuff, but we didn't start there: I took a class taught using a high level language every semester. CS-100 was intro to how computers worked, but right along with it, you had 101 teaching C.

If you are going to end up teaching software engineering material, compiler design and such, you just can't have people that have barely done any programming for a year or two, or all they are going to learn is boxes next to each other, instead of being able to actually write a simple compiler in the compiler class.

Re: Math Intuition Cheatsheet

#39

Post by marcosdumay »

vanderZwan wrote: post #37
kalid wrote: post #8

Earlier quoted context omitted.

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

Same thing with logarithms, which were invented to simplify multiplying two huge numbers. Combined with log tables (think paper LUT for humans) that became a simple matter of looking up two log conversions, adding them, then looking up the inverse of the answer. Is that what we learn in high school? Nope. So everyone is left wondering what the hell they are good for (or were, before we had calculators) the first time…

Well, I learned that at school. It still didn't answer what the hell logarithms are good for. Everyone was left wondering why we were learning something that can be replaced by calculators.

Seeing some modern application of lagarithms would be great. Even just making a log-log graph at some point would answer every question. But those are not at the official curriculum.

Re: Math Intuition Cheatsheet

#40

Post by nabla9 »

kalid wrote: post #8
bithive123 wrote: post #5

I am often saddened by how much easier it has been for me to acquire computer science versus mathematical literacy. Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus, the biggest barriers have turned out to be things programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque…

(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little ca…

I find your emphasis of building intuition very important.

Personal anecdote related to Pythagorean theorem/euclidean distance.

When in ninth grade, I asked my math teacher to explain why why the the standard deviation is like it is and why the square root and not something else. He could not explain that to me satisfactorily.

I spend few hours looking at the formula myself and plotting and drawing different datasets until I made the following association: if dataset is like N dimensional vector and its values x1, x2, .. xN are coordinates, then I'm looking at euclidean distance from mean plus adjusting for the number of dimensions. I think this single moment of cryptic formula making sense changed my attitude towards statistics and math. Dataset ~ vector and data-point ~ vector-coordinate association helps to understand statistics using geometrical intuition.

Post reply on HN