Math Intuition Cheatsheet

by jgrodziski • 12 years ago

478 64 12 years ago

fenomas• 12 years ago

This is terrific. I'm mathy enough to know e to several places, and use it in programs occasionally, but if you'd asked me how it might be intuitively derived I'd have been at a loss. And that's just for something inside my comfort zone - the article on linear algebra is just as illuminating.

This looks like it will be my commute reading for the week. But more than that it just makes me happy that it exists.

kalid17• 12 years ago

Thanks, glad to hear you're enjoying it. I realized that until I could explain a concept intuitively, I didn't really understand it -- and it became a mission to find some analogy that really clicked for me. Hope it gives you some entertaining reading.

manualwise• 12 years ago

Same here. I always was super interested in mathematics and still am. To have this resource to learn about where it all "came from" and how it actually works is great. Will read it on my commute as well!

akrolsmir• 12 years ago

I'd read Yudkowsky's popular "An Intuitive Explanation of Bayes' Theorem" as well as a few others, but the article on this site made me feel like I actually understood it for the first time. The examples were clever, short, and simple, while still demonstrating how powerful Bayes' Theorem can be.

The concise explanations here seem more helpful than semester-long university courses I've taken. I definitely look forward to exploring his other articles on math and programming.

kalid17• 12 years ago

Glad it was helpful!

There's a strange tendency in technical explanations to expand, expand, expand. A heuristic I have: Imagine you're writing a letter to yourself, before you started the course. There's no reason to waste your own time, just cover the key difficulties and how to overcome them, in simple language you would have understood.

(Oh, this letter also costs $5 a word, so let's make them count!)

adamwong246• 12 years ago

Sure could have used this site a couple years ago when I was in school!

bithive123• 12 years ago

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 variable naming schemes, and arbitrary use of jargon where simpler terms would suffice.

A friend of mine who is a physicist once complained to me that every time he had to install some scientific software package on his computer, he had to deal with a litany of arbitrary things which seemed to have nothing to do with his task. I countered with my experience learning math and joked that at least programmers are willing to occasionally refresh our idioms and notations to better reflect our mutual understanding.

kalid17• 12 years ago

(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 candle in the darkness (http://betterexplained.com/calculus/lesson-1) is to start with intuitive notions (X-Raying a pattern into parts, Time-Lapsing parts into a whole) and then gradually introducing the terms. Eventually, if people are interested, we can get into the theory (which is like getting into the Peano axioms of arithmetic, if you care to go that deep.)

I think in math there's a tendency to express things in the lowest-level "machine code" we can. We need more comments and pseudocode outlines :).

JacobAldridge• 12 years ago

Kalid, I just read your article on Prime numbers (not-quite randomly selected from the homepage). I just wanted to say how well written I found it - even beyond the technical side of things, your writing style, tone, and humility ("if I didn't mess it up") are fabulous.

claudius2• 12 years ago

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 makes sense to start from an intuitive, rough idea, but in college/uni, I much preferred classes that started from first principles (axiomatic QM, sets → topology → metric spaces → calculus/algebra) etc. rather than the weird classes which define nothing properly and put in a “this is true, but I won’t tell you why :P” every other lecture. Which approach is preferable obviously also depends on your aims, but I found it hard to get a thorough understanding of a topic without a first-principles-based derivation at least at some point.

__Joker• 12 years ago

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.

vanderZwan• 12 years ago

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?)

nabla9• 12 years ago

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.

mathattack• 12 years ago

Kalid - First of all, your site kicks ass! Keep doing what you're doing. You probably can't even imagine the 2nd and 3rd order impacts of the benefits you provide. Someone gets one insight from your site, that makes something they learn in school more comprehensible, and then it explains it to someone else, who then can learn more quickly....

Bithive123 - I found that practicing Project Euler has helped my understanding a lot. Part of it was that it made math a joy again, and I tied it to learning new programming languages.

Pxtl• 12 years ago

To be fair, we drag around (and rigorously defend) many bad artifacts of past programming languages. Case sensitivity? Double-equals-sign for testing equality?

But still, programming seems far more willing to fix mistakes than hard sciences/math. We've been clanking around with Pi instead of Tau for how many centuries?

djrtwo• 12 years ago

When I took my first programming course, one of things I was most surprised by was case sensitivity. It's kind of funny thinking about it now my perspective is entirely different -- "D" and "d" aren't the same value so why would they mean the same thing?

Is case sensitivity a bad artifact? It doesn't seem like it is to me anymore, but my former self sure thought so.

chaoxu2• 12 years ago

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. Math below college are taught by math educators. They are no mathematicians and they do things differently.

Things in college are rigorous. Should college students learn calculus in a way that make engineers happy and mathematicians cringe? "Learning things not from first principles but 'intuition' of the real world? what is this? physics 101? ".

Different professor/department views it differently, depend on the view, you will get different education.

There are places offering business calculus. I personally believe to be much better way to teach calculus to people who don't really need to know how things work, but still need to know how to use it(at least for simple functions appearing in the real world).

programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque variable naming schemes, and arbitrary use of jargon where simpler terms would suffice.

Why should all mathematicians suddenly write things with the exact same convention just so people can learn it easier? Likely, most people who benefits from these won't advance mathematics in anyway.

Here is what I see when you wrote these things, but might sounds more interesting in a programming perspective.

"The barrier of not able to program comes from the number of programming languages, it be nice if we can all agree on the same programming language so I never have to learn another programming language again."

Maybe learning calculus itself is more akin to learning one programming language, but rarely calculus is taught in a way to end all learning(again, depend on professor/department). When mathematicians teach calculus, they would want to teach the principles in order for one to advance even further.

and now to each point(as someone who had pure math education)

1. "encouraging varied notations" This is not true. People might have different preference because which school/field they come from, I have yet to see anyone enjoying people come up with new notations when there are some established notation(s). In general, language use in the same field are mostly consistent (for a few decades, at least). No competent mathematician would have inconsistent notation in one single article.

2. "Terse notations" A few symbols captures a whole lot of ideas. It's not software engineering, the number of different variables/symbols in a paper is not the same order of magnitude as our programs. There are books with table of symbols of around 2 pages, but usually such a book would take years to master.

3. "opaque variable naming schemes", it's a convention can be learned by doing, and you don't even need to follow someone else's scheme. In fact, not knowing the convention doesn't mean anything. Most variables are used a few times and never used again. Just a few days ago I pick up a paper by the Hungarian combinatorial optimization authors, their notation is completely different from what I usually read. Once I read the introduction, I already internalized the "table of variables/notations" so I don't even need to refer back often. To programmers, this is like complaining different open source programs use different naming scheme for their API.

4. "Arbitrary use of jargon where simpler terms would suffice". I do not believe this happens often in mathematics. Mathematicians likes things to be elegance, simple and precise. There must be reasons for us to use a word, otherwise we won't use it.

5. "at least programmers are willing to occasionally refresh our idioms and notations to better reflect our mutual understanding." Mathematicians do this all the time, not just "occasionally". It doesn't happen in mature and well established topics where all the problems are dry. There are nonstandard analysis, complementary to the standard analysis. There are homotopy type theory to have another foundation for mathematics. Many things we seem to see it and think it's set into stone it's because there really isn't anything to be done.

richardjdare3• 12 years ago

>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.

j2kun2• 12 years ago

Your friend must have been trying to link a fast C or Fortran library to some higher level language, yeah? That's always a nightmare.

Sharma• 12 years ago

Thanks for posting this! I just read a small portion of Integration and trust me, it all make sense now!

Mistral• 12 years ago

Super, but no induction!

NoPiece• 12 years ago

I really like this cheatsheet, it is very inspiring actually. You should consider doing a version written for elementary school aged kids. You need to get buy in while they are still young!

kalid17• 12 years ago

Thanks! I'd love to do more elementary school topics. I still encounter revelations about things like basic counting (fencepost problem) and arithmetic.

My meta goal is to raise people's standards for what it means to understand something. ("If it didn't click, I should keep looking for a better explanation." vs. "It didn't click, I'm no good at this.")

pawannitj• 12 years ago

Are there other resources like this on Math?

omnibrain• 12 years ago

Those explanations remind me of Feynman's explanation of elliptical orbits.

curiousDog• 12 years ago

This is gold. Please do point to more resources like this.

ithinkso• 12 years ago

I like the idea but I personally really enjoy and prefer 'math english'. At first it may seems odd but after a while you get used to it and it's actually easier, you just scan through epsilons, deltas knowing exactly what they mean. Intuition has edge-case scenarios, double-meanings and almost-truths. I don't know why, but I think intuition (pseudocode math as someone had named it) gives a little bit of a 'illusion' of knowledge. You think you got it but are you really?

kalid17• 12 years ago

Great point. I use the word "intuition" but feel it might be overloaded, things with intuitive knowledge, psychics, going with your gut despite the facts, etc.

The sense I mean is closer to "grokking" or "clicking", when symbols and definitions actually mean something. You start thinking with the symbols, not despite them. You actually look forward to checking your understanding with real problems, because when it clicks the problems become easy :).

RobinL• 12 years ago

This September, I'm starting tutoring A level maths to underprivileged kids for a charity after helping them with GCSEs last year. This looks like it will be really helpful. Thanks!

kalid17• 12 years ago

That's awesome! I hope you can make use of the site. One goal for the site is to provide "ammo" to teachers who are looking for other ways to present difficult concepts.

gajomi• 12 years ago

What an excellent collection of articles (well, I only ready a few, but I would like to think my experience extrapolates to the rest)! But I don't think "cheatsheet" is really the right word. Really this is a collection of short expository writings on particular mathematical concepts. And very well done, I think, to the point that calling it a "cheatsheet" (which I associate with a haphazardly conjoined sequence of facts and tricks out of context) is to do it a disservice. Or maybe the name choice has some marketing value?

kalid17• 12 years ago

Glad you enjoyed them so far :).

"Cheatsheet" might not be the best term for the summary page, perhaps "quick reference". I picked cheatsheet because it's a little more approachable than "reference", which implies something formal (and perhaps offputting).

jgrodziski• 12 years ago

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.

kalid17• 12 years ago

Thanks so much, and for originally submitting the cheatsheet to HN. (I hope it helps people untangle concepts that confused me for literally a decade.)

I really appreciate the encouragement and hope to keep cranking as long as I can.

arianvanp• 12 years ago

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! :)

j2kun2• 12 years ago

A lot of mathematics people feel this way about programming, too. Both sides expect one to devote the time to build intuition to grok their subject. I think it's easier to learn programming coming from the mathematics side, because you're going from nobody holding your hand to a compiler giving you error messages. You can't interactively test whether your intuition in mathematics is sound (you can, but you have to play both the role of the compiler and the tester).

factsfinder• 12 years ago

Seems like there are a lot of things i need to learn in maths

agumonkey• 12 years ago

I love the content, just wishing for some of their views on explaining electronic circuit design.

FlyingLawnmower• 12 years ago

Great work. Please keep adding examples!

Math Intuition Cheatsheet

Related Stories