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BetterExplained: Math Lessons That Explain Concepts (betterexplained.com)
666 points by rfreytag on Dec 22, 2015 | hide | past | favorite | 126 comments

This site is a hidden gem on the internet. This guy deserves a lot more credit. I think he's doing something at the level of Khan Academy, but for seemingly simple concepts we all take for-granted.

An article I really enjoyed was his explanation of Quake's inverse square root method: https://web.archive.org/web/20150530232103/http://betterexpl...

What I also appreciate is that he admits when he doesn't understand something and doesn't pretend to give an incomplete, vague explanation and just straight up says he does't fully get it (like why the specific 'magic' number in the Quake algorithm). Thankfully he linked the original paper and I read up on it myself. But his initial explanation provided a solid background for the paper.

Hey, Kalid from BetterExplained here, thanks! My general philosophy is to be really, really honest with myself if I understood something. It's ok (really!) to admit when we haven't fully understood a concept.

Want a fun example? How about percentages. Yeah, that thing we mastered in 4th grade or whatever. That thing we use every day.

Well, did you know that

a% of b = b% of a?

Let's say you want 16% of 25. Ugh. Ok, let's multiply it out... divide by 100... no!

How about 25% of 16? Well, that's 4. Easy. But are they the same thing?

a% of b = a/100 * b

b% of a = b/100 * a

Either way, it's ab/100. Now every percentage problem has a 50% chance of being expressed more easily. How did we miss this? (Argh!!!)

Math is full of insights like that. Trig functions (sine, cosine, tan, etc.) are actually themselves percentages. A sine of .95 means you are at 95% of your maximum height (where the max is the hypotenuse). Sine and cosine are unitless numbers, and that's why the can be each other's derivatives (the percentage change of a percentage change...). So many things click! What else have we overlooked?

Anyway, really appreciate the note!

The percentages detail is very cool, never thought about it! I also think of sin/cos as percentages, but there's one small detail that can be easy to miss. While sine and cosine generate "percentages" between -1 and 1 they don't generate them uniformly, because their derivative is not a linear function! If we were to sample sine or cosine between 0 and PI at uniform intervals we would see that the percentages would "cluster" around the peaks and valleys because that's where the function "decelerates" (or the derivative decreases).

This bit me once!

Yes! That's a great point. Similar, if you ask people to pick a random point in a circle (random angle, random radius length) you don't get the random distribution you were expecting.


I have a question, but first a compliment :) I am enamored with this site. It validates a lot of my learning approaches and is helping me now when I was asking some of these 'how to learn' questions. Thank you!

The question: In my experience, 'generalized overview to specific' is too much required attention span for some listeners when trying to explain; Do you have a recommendation for being able to convey ideas to people that are tired/low attention in high stress situations? Example: office work and project work.

Referenced here: http://betterexplained.com/static/articles/adept-method/ "Start with a rough analogy and sharpen it until you’re covering the technical details."

Again, this site is amazing. If you don't get to the question, I am still happy to have found your site and hear how passionate you are about it. Thank you!

Hey! Thanks for the compliment, it really means a lot when the site resonates.

I'm not super experienced with lectures (mostly writing), but I think it works in office settings. Check out this talk from Simon Sinek:


People are usually motivated by the why, the mission, the story. Then you get into the what and how it's accomplished. With math, it can be similar: the "why" sets the stage. Humans in general prefer a narrative to a list of facts (Hacker News readers excepted :-).)

Thank you sir! That is revealing and informative.

Keep up the good work :)

Thanks, I appreciate it! :)

Euler's Identity is often explained in terms of mysterious language as well. Like, e^i*pi = -1 makes a lot more sense when you consider the exponential function as "proportional growth", i as "sideways", and pi as "halfway around a circle" - all Euler's Identity does is rephrase "going halfway around a circle" in terms of continuous and proportional sideways growth.

Exactly! I hate the Taylor Series explanation of Euler's formula. "Oh, just take the most analytic definitions of e^x, sin(x), cos(x), mix-n-match, and it works!" All symbols, no intuition.

If we see each concept individually (continuous growth + rotation) we can deduce that we get something like "continuous rotation" or a circular orbit.

And if we use a complex number (a + bi, not purely imaginary i) we get a spiral pattern. Euler's Formula becomes "obvious" dare I say ("obvious after the greatest mathematician figured it out for us".)


I've wasted so much of my life.

Me too. I was even a math geek but didn't understand most of it. I knew all the rules for equations plus heuristics for how to apply most of them. I can only imagine how much more effective I'd have been if there was a constant, parallel learning process focusing on intuitive understanding of all foundational concepts in various math branches.

I think I left a palm-print on my forehead when I saw that.

The trig part was intuitively obvious for me, but the % part was wow!

I love this approach to learning. It's something I've been really working to take on in my studies (and my own projects).

Awesome. I think it's a general principle (blurry-to-sharp) that works for many fields.

> a% of b = b% of a?

wow this is mind-blowing stuff! nice explanation too!

Paid you a great compliment in my main comment. This...

"a% of b = b% of a?"

Seems like a bit of a cheat as it's too obvious. I did 25% of a 100 to keep it simple as we use quarters and dollars a lot. Also can visualize it as a rectangle or stack of boxes that I take one chunk out of. Maybe use word "whole" or phrase "all of" for 100% to make 2nd part more obvious. I took a chunk out of all of that stack. All of this chunk equals the chunk I took. See? Too obvious. Stay on the harder shit like e, trig, etc.

"Trig functions (sine, cosine, tan, etc.) are actually themselves percentages. A sine of .95 means you are at 95% of your maximum height (where the max is the hypotenuse)."

Boom! Excellent example. I understood it almost entirely in equations back when I did it. Outside of some examples with trees and stuff we rarely got to sit on what the terms mean. So, let's see if I follow that.

So, a sine of 0.95 is like putting a protractor on a picture of a right triangle and marking a chunk of it that goes 95% to the top of that? If I did it visually, that is. Looking up the other two's definitions I found someone already did the visual thing I was attempting although not quite there yet in presentation (see pictures w/ angles):


SO, if we do it visually on those pics, does the percentage the sine represents start as a line coming from bottom-right to hypoteneuse? And where do the other two start? Or is my intuition screwing with me? ;)

Just saw your earlier comment, thank you!

1) Yep, the regular percentage formula is pretty basic. Mostly, I like it because we've overlooked something that's been under our noses for years or decades. What else have we been missing?

2) For trig, check out:


The traditional way of showing sine/cosine/tan (as on that page) leaves out the surrounding context where the percentage comes to life. Let me know if that link above clears thins up.

the trig function trick is pretty neat! but i wonder if i only think that because i know trig already, and would've gotten more confused with more analogies

Yeah, to be honest, I'm writing for someone who saw the textbook definition of trig but didn't have it click. I'm not sure how it would work as a sole treatment. (However, I suspect the vast majority of people reading the article are learning trig in school alongside an existing lesson, vs. randomly reading about it with no context. The people who google "trig" for fun having never heard of it... are strange beasts.)

This is fascinating! I wanted to know more about the history of that function at id, which lead me to this beyond3d post (which was the topic of the reddit post that BetterExplained listed as a source):


A slight correction: the technique existed long before the paper, so I wouldn't exactly say that its the "original" paper.

The paper was written by Chris Lomont in 2003, when the gamedev.net community first started investigating the code (I thought Beyond3D did their investigation earlier, but it looks like they did it in 2004). There is a different paper that published a similar idea in 1997 (Floating Point Tricks), but its not the one linked. The code first appeared in the wild on comp.graphics.algorithms around 2002, apparently.

I remember this because I followed the discussion on Gamedev.net when Chris Lomont wrote his paper. Some years after, I found the Beyond3D articles where they traced the origin back to Ardent Computer.




Here's the best explanation I've seen: http://h14s.p5r.org/2012/09/0x5f3759df.html

Why did you use the archive.org link?

I just love this guy's ADEPT method of exposition (Analogy, Diagram, Example, Plain English explanation, and just then Technical definition). [1]

That is exactly how I like to be introduced to any new concept, in special when I'm a complete newcomer to the field and can't relate it to previous ideas in it. Judging by the reactions to the BetterExplained site, other people agree with that.

[1] http://betterexplained.com/articles/adept-method/

I've found that I also need to understand the historical context/impetus... I need to be shown and eventually understand why something was developed when it was developed. I need chronology of thought and ideas.

For instance, learning about Gödel's theorems without having the Hilbert back-story explained. Learning about Leibniz's and Newton's calculus without learning about infinitesimals. And so on.

There's generally a reason _why_ concepts are born _when_ they are born. If you think about your maths classes, sometimes you're instructed to learn a method because it is useful and because it has real-world applications but it I don't think anybody is ever first taught algebraic geometry properly, if I may use that word. I don't think kids are taught the geometry is one thing and algebra is another and that different spaces can have different metrics. Am I making sense here? Do people see what I'm trying to get at?

Am I arguing for HADEPT? :) (Historical context, Analogy, Diagram, Example, Plain English explanation, and just then Technical definition)

Great point. I find myself looking at the history of the idea when writing up a post. Did you realize negative numbers were only accepted in the late 1700s? That the Fourier Transform was originally rejected as untrue when first presented, by world-famous mathematicians even?

(Yet we require students to internalize it without issue in a single lecture.)

Historical context is huge. I think I'm now stuck with this ADEPT name but maybe it fits into the Plain English portion :).

> Did you realize negative numbers were only accepted in the late 1700s?

I did not know that! This makes total sense. I'd like to know more about that. When you think about it, only whole positive numbers make sense from a quantitative perspective. One thing, two things, three things, and so on. What's half-a-thing? Right? a half-a-thing is still just one thing, if you know what I mean. And how can no thing (nothing) be a number? And how can negative numbers be "numbers". It has always struck me that imaginary numbers are really badly named. Zero and the negative numbers are just as 'imaginary', equally unintuitive from a certain perspective.

I applaud what you're doing. I think there is a metric-tonne of dogma and bad naming schemes in the standard maths curriculum. Remember in software engineering they say that naming things is one of the hardest parts of the task? I think the same applies to maths, perhaps more so.

Exactly! There's a quote from a famous mathematician at the time that the negatives "Darken the very whole doctrines of the equations". If positive is good, negative must be evil right? And how can "less than nothing" exist? I love the philosophical implications of it.

Ugh, tell me about the naming. "Imaginary numbers?" How about "rotated numbers". Nobody complains "Hey, when will I ever use the second dimension?". But "imaginary numbers" are setup to be eye-rolled.

I'd recommend "The Joy of x" by Steven Strogatz. It's full of that sort of story.

Thanks! I was really hesitant to force an acronym (it actually started as ADE) but then I realized I could work my way up to the full technical definition. Really glad to hear it's resonating. (Sometimes things happen out of order, i.e. you start with a plain English definition. But the idea is to have all 5 parts if you want to truly master a concept.)

Agreed, but it depends. The analogy can make things harder if you're not intimately familiar with the analogy. Take the below, an excerpt from the explanation on prime numbers for example, and consider you know 0 chemistry (not unlikely if you're reading an intro on prime numbers):

----- > I'm no chemistry expert, but I can see a relationship to the primes. Chemical elements have properties based on their location in the periodic table of the elements:

Atoms in group 8A (Neon, Argon) are the noble gases. They don't react and won't blow up in your face. Atoms in group 4A (Carbon, Silicon) bond well. They're great building blocks for other elements. Atoms in group 1 (Sodium, Potassium, etc.) are very reactive. Drop 'em in water and see them explode. And in organic chemistry there's an idea of a functional group: several atoms can determine the class of the entire molecule. For example:

Alcohols are a certain carbon-hydrogen chain with an OH group at the end. Methanol, ethanol, and other alcohols share similar properties because of this OH functional group. Those are the basics, if I didn't mess it up. Now let's see what happens when we treat numbers like chemicals.

First Example: Guessing Evenness In general, an organic chemical contains carbon (not quite, but it's a good starting point). No matter what elements you mix together, if you never add any carbon then you can't create an organic compound. -----

Anyway, a single example doesn't negate your point. I love analogies in learning, but one has to be careful to pick analogies from a level of understanding (way) below what you're trying to explain. I guess kids are introduced to primes and chemistry at roughly the same age, but I'd have picked a non-academic analogy to explain an academic concept. But even then, it's tricky. For example I've been confused by my fair share of 'sports analogies' in secondary school books, for sports I happened not to have ever tried or knew the rules for. But really, the analogy should be completely supplemental, and if possible marked off in a side box that people can, but not should, read for better understanding if it helps them. I find many school books do this really well, but I haven't seen it translated to web content as much somehow. For example, on Evenness he'll continue by explaining how if you have a factor of 2 in your number (e.g. 24 = 2^3 * 3), then no matter what, the number is even, likening it to an organic chemical which contains carbon no matter what (though, noting a caveat without going into it). I don't think that analogy is very strong, it's confusing if you don't know chemistry, and it's pretty redundant if you do. In fact I'd personally be better of without it, and understood Kalid's normal explanation without issue. Yet I had to read through something about Atoms in group 4A and their properties, unsure whether I could just skip it or whether it was important to grasp some larger point. Anyway I was already familiar with primes but my 12 year old self probably would've been confused with the chemistry analogy.

Thanks for the feedback! Agree analogies are context (and time) sensitive. As soon as you make a reference the clock starts ticking about how long it would remain relevant.

For this specific example, I was writing to a high-school version of myself who wanted to really get an intuition for primes. What can we deduce from a prime factorization, are there other ways to think about it? (Number theory is studied later, even though numbers are introduced early.)

For a younger child, I'd probably use Lego or Minecraft to show how numbers can have "building blocks". And if you didn't use any Redstone as a building block, there won't be any Redstone in the result. (I.e., a number which never had the "2" building block added, will never be even.)

Thanks for the comment!

Internet Archive has a copy: https://web.archive.org/web/20151221053341/http://betterexpl...

>>>>> Please donate to the Archive while you are at it! :-)

Google cache: http://webcache.googleusercontent.com/search?q=cache:http://...

I needed a refresher on trig last year for some robotics simulations I was working on. The trig lesson was extremely helpful: http://betterexplained.com/articles/intuitive-trigonometry/

And fwiw, I finally learned an intuitive understanding of radians by reading the Tau Manifesto (not on BetterExplained). It would be awesome if BetterExplained used tau instead of pi in the lesson on radians, but that's a minor nitpick for a very helpful set of lessons.

Out of curiosity do you get confused when people talk about 360 degrees, or 90 degree angles, or turning around 180 degrees, etc? Most people I know intuitively understand degrees and could easily understand radians the same.

Why should BetterExplained use tau? Tau is a fairly niche idea that will only confuse way more than explain. That's a bit like saying BetterExplained should be written in Esperanto.

Personally while I agree that historically pi would have been better off if defined as 6.28... instead of 3.14... (And even told my students this long before tau manifesto came out) I don't feel that a factor of 2 warrants using up a whole other Greek letter.

(I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).

The anecdata that tau is helpful to learners is, IMHO, pretty strong at this point. Facts beat theory. And since anecdata is all we have, well, it's all we can use to talk about.

I think it's way, way too easy for people on the terminal end of education to forget just how easy it is to get tripped up by very, very little things. Especially if they were themselves "good at math" and really aren't bothered by extra factors of 2 flying around. This is not the normal experience.

I just read the Tao Manifesto and - while I was skeptical at first - I have to agree that it is much easier.

Note that this is just my opinion, but I have kids and this gives me an alternative viewpoint when they have questions. I wonder what my more math oriented friends will think about Tau

Of course the right constant historically should have been 6.28... And not 3.14... Nobody's disputing that (or at least I'm not).

The question is whether you improve things by abandoning a universal standard with hundreds of years of support, momentum, and ginormous corpus of math/science literature all to save a 'simple' factor of two.

My primary gripe with Tau is that its symbolic representation is not "backwards compatible". Ie, Hartl proposed to use yet another Greek letter which is widely used elsewhere in all sorts of formulae and constants, instead of something that would cause far less ambiguity. For example, a new symbol, as physicists did by introducing hbar instead of h (Planck constant in radians instead of cycles).

>Facts beat theory.

Not necessarily. Facts (in the sense of "anecdata") are simple observations.

The only make any sense against or in favor of something when we combine them with a theory why it is so.

OK, students faring better on exams when they are taught with tau is an argument in favor of tau only if we theorize (assume) that it was the tau/pi thing that made the difference.

Whereas it could just be that the second group of students was just better naturally (e.g. a good school vs a mediocre one), or just a chance outcome, due to the small sample we picked, or the teacher going for tau was also better at explaining and would have fared just as good if he also was the one to teach the pi group.

It is one of the great errors of the "SCIENCE!" attitude that it convinces people that if we do not have peer-reviewed papers, we must necessarily claim full and total ignorance and berate anybody who attempts to do otherwise.

I'm not obligated to stop using my brain just because we have no SCIENCE! papers to work with. In fact, you can't. You have no choice. You must do your best with the data you have, because we do not have "science" for all the questions we encounter every day.

Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching.

> It is one of the great errors of the "SCIENCE!" attitude that it convinces people that if we do not have peer-reviewed papers, we must necessarily claim full and total ignorance and berate anybody who attempts to do otherwise.

In fact, I think quite the opposite. Its one of the great errors that people assume peer-reviewed papers are always correct, a lot of the time they're not.

We should be sceptical of people's claims when not supported by evidence. That doesn't mean we should ignore said claims, just that we shouldn't blindly accept them. This is the point where we should go and get some evidence, try to get a few teachers to use tau from the beginning and see if that helps.

Though I do agree that we shouldn't be scared of doing things just because we don't categorically know that it is right.

> Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching.

Or thinking about the same concept in different ways help you to understand the concept better. Its certainly something I believe has helped me a lot previously.

When someone says 90 degree angle, does that confuse you?

"90 degrees" is not confusing to me in the same way that measurements in feet and miles is not confusing. I was raised in a system that emphasized degrees, feet, and miles (USA). But in engineering, metric is the standard/default system of measurement. And in software libraries (e.g. JavaScript's Math built-in lib) or many calculators, the default is radians, not degrees. So even though degrees is more natural for me, it was worthwhile to get a deeper understanding of radians. The Tau Manifesto got me there. (90 degrees is 1/4 turn around a circle, which happens to be expressed as 1/4*tau -- and I didn't even have to look that up!)

As it happens, I've tutored people in the middle of high school geometry. Radians confuse them. Full stop. If there is something that can make it easier to understand, while at the same time sacrificing absolutely nothing of mathematical consequence, it's a good thing.

What confuses me is not the relevant issue. I've got $BOATLOADS of higher ed. I'm not the interesting case.

Ok, we're far away from my original point which was only on using tau in one website.

My actual only personal argument against Tao as Hartl proposed it is the use of a standard Greek letter. Really would have preferred he picked something that was more "backwards compatible", exactly as physicists did with hbar.

If you're for reducing confusion, do you think adding a new fundamental constant to the body of mathematics that is a Greek letter already used for countless other variables in history, will this cause confusion between old and new mathematical science texts and papers?

Asking students of science for any texts and papers they read whether tau is 2pi or some other variable, and keeping track across them, seems more confusing to me than just consistently using pi and extra factors of 2.

You have a valid complaint about reusing a letter that already had meaning in other contexts. He does address that in the manifesto, though.

You can't just baldly assert that radians are just as easy to understand as degrees - clearly they require more effort to understand, from an intuitive point of view. It seems obvious to me why degrees are easier to handle than radians: 180 is an easier number to understand than pi.

People don't instinctively understand why on earth they would use a measurement unit where '1' of the unit has no obvious purpose. What can you do with a 1 radian angle? It's a bit less than the angle in an equilateral triangle. It's too big for usefully measuring things - most angles you come across will be somewhere between 0 and 3 radians, which is just weird.

What's that, you don't use radians like that? You use 'fractions of pi' radians? But my calculator can only approximate pi to 9 decimal places, so how does that help me?

It's incredibly easy for people to intuit that a ninety degree angle is the same as a 30 degree angle plus a 60 degree angle. But it takes deeper intuition about fractions to figure out that a pi/2 radian angle is the same as a pi/6 radian angle plus a pi/3 radian angle. People are not used to units of measure that are typically denominated with rational fractions of irrational numbers.

Every argument on radians you presented applies equally well to Tau as Pi.

My original point is that IMHO, mainstream radian educational materials should NOT use the lesser-known and niche 'tau' in lieu of the globally understood 'pi' (other than a possible footnote or sidebar). Ie - it can cause more confusion to the student, especially when connecting it back to the greater scope of their studies.

I wasn't addressing your tau/pi issue, but rather your assertion that if you understand '90 degrees' that you should be able to understand radians. That seems like a case of 'well I find it simple, why doesn't everyone else?'

Because tau is more intuitive and this is all about intuition.

If you were baking a pie (pi?) and someone told you how much of an ingredient to add per half pie, that would be less intuitive.

And no, I do not get confused by concepts like 90 degrees, but this is about inspiring intuition, not the ability to do calculations without error.

>(I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).

I suggest using pi, but with the number 2 before it. /s

Hey, thanks for the feedback! Glad to hear the trig article helped (one of my favorite aha moments -- all the trig functions, including stuff like the exotic cosecant, can be drawn together.)

I'm a fan of Tau as an interesting simplifying concept but don't think mentions of pi should necessary be removed. My intuition is that radians are "distance traveled" as you go along a circle. Pi is the "neutral to max to neutral" distance and Tau is the full cycle (neutral, max, neutral, min, neutral). Ideally we just think about "distance" and context determines whether we want the journey to be "there and back" or just "there".

i (mostly) agree. i think thinking about a fraction of a full circle is the easiest way to work with trig. having said that, the mental overload on your average high school student flipping between pi and tau is too much. my personal compromise is to never cancel out the '2' and always work with 2pi. it makes some equations look a bit odd from the textbook but whacks me in the forehead to remember to think about circles...

No they should not use Tau, because the point of education is to give you knowledge that you can build on later. Nothing else uses Tau, so teaching you Tau does not prepare you for further education.

I'm not even gonna touch the Tau vs. Pi debate because it's like a sports team rivalry, and not a rational discussion.

> Nothing else uses Tau, so teaching you Tau does not prepare you for further education.

Nitpicking, but that is a self-defeating argument. If everybody used Tau, there would be no problem in teaching it.

Not that I disagree that doing the bootstrapping its usage and reaching there would be a problem, but all by itself is not a reason to not try changing it.

I've been using BetterExplained to review concepts I thought I mastered (based on great test scores and grades) but years later realized I had zero intuition on. It has been, by far, the most useful single source of math content I've found on the internet. The world needs more people like you.

Two Questions:

1) What are your thoughts on interactive content like ExplainedVisually? I've been thinking about doing something similar for data structures / algorithm topics. How much of learning math concepts is exploratory vs learn-by-doing?

2) Are you running BetterExplained as a side business, or full-time? And if you're willing to share techniques and numbers for the entrepreneurial HN community, what are some things you've done to market it, monetize, etc and what were the results?


Really glad to hear the site helps with new insights (especially for someone who has aced the tests/grades). I was in a similar boat, qualified on paper, but not in my heart of hearts.

1) In general I like any efforts to explain things in new ways, and ExplainedVisually is great. My philosophy is that teaching is like humor. You want to make something funny, present it well, but not overexplain it. Let's people enjoy the joke. If you do too much handholding, you ruin the surprise and it's not fun to be told "Ok, the punchline is coming up...". It's not an exact science, but you get a nose for when something is truly illuminating vs. trying to chew your food for you.

2) BetterExplained is a side business. Happy to share numbers, etc. I have some earlier posts about ebook sales and techniques:


I don't think I should be taken as a model of learning though (I write very infrequently) but thankfully math is evergreen. Many of my most popular articles are 5-7 years old.

I do want to dedicate more time to it. I realize I was afraid of ruining my love of learning by turning it into a profession, but I'm slowly coming to grips with it. That's one of the hardest parts for me actually, feeling I'll kill the golden goose by squeezing too hard.

I'll probably do a blog post / postmortem on marketing, numbers, etc. so keep an eye out :).

1) "Not over explain it" -- that's really good advice. You don't want to take the "aha" moment away from someone.

2) Thanks for the link! Definitely looking forward to your blog post/postmortem (don't let BetterExplained die!). Have you found learning to be less enjoyable by writing about it, or do you end up discovering a dozen other tidbits of math magic you want to share with everyone?

Two more questions if you don't mind :)

3) What are your thoughts on word problems?

For example, in most linear algebra textbooks, you are given matrix and are asked you to process it. Rarely are you given a word problem and are asked to think through the entire process (data and operations matrices setup --> processing --> meaningful end result).

4) For inspiration, what are you experiences explaining concepts in a cross-disciplinary manner?

When I was a student, I never understood why a concept is important. Homework problems were abstracted out of all real-world context to train for mechanical problem solving. Only now, after exploring data and writing algorithms in health, journalism and finance, have I finally been able to answer the question I always had as a kid: "why is this stuff useful?"

(Looking back at comments, replying to this a bit later.)

1) Exactly. It's like spoiling a movie.

2) Hah, the postmortem is more about the Reddit interaction (write up about went well / things I'd change). I'm planning on working on the site as long as I can. It's a life mission at this point.

Learning has stayed enjoyable, I tend to write insights that really strike me and get excited to share. (Which leads to me studying it more and figuring out new insights.)

When learning is drudgery (this happens often), I tend to let the topic sit a bit, and I don't write publicly about it. The articles on the blog are what genuinely get me excited about the topic. I do think there's usually a way to see a topic that makes it come alive.

As an example, I'm working on quaternions. I have a large list of notes here: http://aha.betterexplained.com/t/quaternion/267 and I'm slowly getting an intuition that I'll then work into an article.

3) I like word problems because they force us to ask the uncomfortable question of whether we can think with the material (vs. follow the steps). That said, this check of whether you're thinking or following steps can be accomplished with other types of questions too. For me the method isn't as important as the outcome.

4) Good question. So far, my audience is typically people who are self-motivated (i.e., they have a test, are curious, need homework help, etc.) vs. giving a talk to a potentially uninterested audience. (Not intentionally uninterested, but a volunteer audience.)

The primary motivations to learn are probably:

* practicality

* curiosity

* beauty / awe

* sense of accomplishment

Depending on your audience you'd have to tailor it. But I think beauty/awe is more powerful than we think. Even for a technical talk, I'd get people see the aha! moment. It's the sugar that helps the medicine go down.

Interesting set of articles. "Intuition isn't Optional"[1] could explain why pursuing simple solutions and good naming is important in software.


Nice point. Yes, naming can prime people to understand or be confused by a topic. Renaming the "imaginary numbers" to the "rotated numbers" would make them orders of magnitude easier to learn. "What's so strange about the second dimension?" vs. "How do I understand the square root of -1?".

This site is absolutely excellent. There are a lot of things in math that I sort of had a handle of the mechanics of, but less so the intuitions for, which this site fills in well:

I really liked the explanation of sine as something that makes things 'circly'.


Does the site take external contributions?

By popular demand: http://patreon.com/betterexplained


Sorry, I was asking if you accept article contributions.

as a reader, I think that that should be nice. Also, it would be nice to have a forum where ideas get exchanged. Then Kaled can decide if any of the discussions should make it into the site as an article ?

Thanks for the feedback! I have a community in progress at http://aha.betterexplained.com -- I have some collaboration ideas I'll be announcing early next year :).

Ah! Yes, I'm planing on starting up a community area for contributions, to make collaborative guides to topics. Hope to make an announcement early next year :).

I've been considering a patreon page, but the best way is to help share the site with people learning math (students, family friends, etc.). A genuine recommendation is one of the best things I can ask for. Thank you!

The article on math intuition plus the e article were incredible. I always knew how to work with them and sort of what they meant but never really intuitively. The articles were first time in a while that happened. Reminds me when I first learned the often-hated word problems were the best part of math that explained how to actually map it to real world. Equations took less thought for me so I avoided lots of word problem practice. Was glad I shifted back a bit before calculus or I'd never be able to explain what it was good for.

This guy's site should get more attention and probably an award. I'll be experimenting with less-math-inclined people to see how effective it is.

Yeah, the e article is awesome. Cleared up a lot of cobwebs in my brain..

Hey nick! Thanks for kind words, it's really touching to know when things are clicking. I'd love to see how things go with people that aren't math experts. My philosophy is I found one math up the mountain, it worked for me, but there's other ways too. It's awesome when people take the diagrams/analogies and adapt them for their own style. I saw your other comment too, will reply up there.

After reading several articles I came to the conclusion that every Wikipedia article on math concepts which are covered on betterexplained should start with the link to betterexplained. I know it's not going to happen, but it would benefit hundreds of thousands of people.

I feel like it's often easier to learn concepts from equations. The betterexplained articles are so wordy and contain so many analogies (some of which are leaky abstractions) that there's all this noise around the core concepts. Equations cut away all of that noise. And it's not like I'm some huge maths nerd; I've always found language and philosophy far easier to learn than mathematics.

Oddly enough, I find betterexplained much more useful when I have already grasped the core concepts, because it does a great job of connecting to other concepts.

Thanks for the feedback! I see the articles as a supplement to the formal description people usually have in hand when they are googling for help :). For the succinct formal definition I definitely recommend Wikipedia or Mathworld.

Wow, glad to hear the articles had a positive impression :). I'm bouncing ideas for intuitive guides for any Wikipedia topic. After the 10 min intro, you can read the Wikipedia article for more details.

Is there anything similar to this for physics? I'm taking an introductory mechanics class and I can't help but think there must be a much more intuitive, logical way to solve even complicated multi-step problems than just falling into algorithmic pattern-based resolution.

I haven't done much physics, but Feynman is my teaching inspiration. He has his famous lectures on Physics:


(Put online by Michael Hartl, of TauDay fame among others.)

I haven't gone through them extensively but Feynman was a master of making the complex simple.

Tangent: with author permission, these lessons are loaded into the recommendation system for Knewton's free learning service: https://www.knewton.com/

Awesome! (Kalid from BetterExplained here.) This goes for everyone, all the articles are under a CC license.

Knewton has done a good job of curating!

I bought the book a while back, and I still struggled with it. I have very low math skills. My pre algebra teacher was a joke, and everything building on that flew right over my head. I basically talked my counselor into letting me graduate even though I didn't pass basic math graduation requirements in college. So I'm a pretty clean slate, and this was quite difficult for me. I'm a pretty good developer, but telling the computer to do math for you based on looking up what it needs to do is much different from understanding the concepts myself which has been a real difficult thing for me. Kahn academy is the worst, like bringing back old terrible memories of repeating hellish math problems that I don't really get, but still want to move on out of boredom. Such it is.

Thanks for the feedback! The book is more of a "top concepts I wish I knew" but not really a ground-up tutorial.

For starting from a clean slate, you might like:


Once you can visualize the basic operations (add, subtract, multiply, divide), every new math operation becomes a lot easier (complex numbers = rotations, exponents = growth, combining them = orbiting a circle...).

When you say the opposite can be the multiplication of -2, did you actually mean the multiplication of -1? It's a bit unclear in the example whether the "loss of two" means 1 * -1 = -1 and therefore a relative loss of two, or 1 * -2 = -2 or a relative loss of 3. I always figured opposite meant inverting either the fraction so 2/1 becomes 1/2 or it meant multiplying by -1, effectively toggling the negative sign.

Whoops, might not have been clear enough. In more mathy terms:

Multiplication by 2 means "1 (starting point) times 2 (scaling)"

If we "do the opposite" we can take the inverse of the starting point or the scaling:

-1 (additive inverse of starting point) times 2 = -2

or reverse the operation

1 (same starting point) * 1/2 (scaling inverse) = 1/2

Of course, we assume the scaling term is what's being inverted, but it's important to think about the meaning. There's a hidden parameter for these operations and sometimes making it explicit can be helpful. (I.e., euler's formula, e^ix, is better seen as 1.0 * e^ix. That is, you are starting at 1.0 then doing a rotation.)

I would love to hear your thoughts about something I'm working on. Care to shoot me an email? jkun2@uic.edu

I wonder why more secondary teachers don't use these sorts of examples? Any teachers out there that can give insight?

The central bureaucracy has so overdetermined education that there is no longer any room for deviation. If it's not in the curriculum, there's no time for it, because there isn't hardly even any time to cover what is in the overdetermined curriculum.

If you support innovation in education anymore, your first goal now is to get the central bureaucracy out of it. Until that happens, nothing else can change.

Removing the central bureaucracy is a double edged sword. For every classroom you free to innovate, you also free a teacher to pan science, attack evolution, teach bible and religion in science class, etc.

Many of us grew up in school districts prior to the Bush and Obama centralization regimes and quite literally were taught that Evolution is a Satanic lie by public (non-private non-religious) schools.

Innovation is a double-edged sword in general. Doesn't mean we should stop.

Besides, are we not discussing exactly a situation in which the central bureaucracy has overspecified an inferior education? Nothing about "panning science, attacking evolution, teaching the bible and religion" has anything to do with whether or not there is a central organization; one stray election and you'll get all those things coming out of the central organization too.

The very topic of this discussion is that an inferior education technique is being mandated right now. The casual default presumption that the centrally-mandated education is perfect is already falsified by the evidence.

So are you saying we could fix the mandate and that would be good, or that specifying a minimum basic to which all teaching should be done is wrong in general? Or some third thing?

The trouble is that—as my teacher wife likes to say—once the door closes you can teach whatever you want, however you want. She says that in a positive way to dismiss some of the ridiculous expectations of the bureaucracy but I've seen plenty of teachers do troublesome things because of that.

My first reaction is to agree wit you, but then I thought about it and how do we know they don't? Sure the common wisdom is "teachers suck" and "our education system is out of date" but I haven't been in an elementary school math class in 25 years, let alone a majority of them to really have an informed opinion about this.

If it really is a problem, an obvious solution is to just supply prepared materials that are based on this to the schools, and that will do a lot to make it available, if indeed it does work as well or better than the current materials.

Good point. I wasn't suggesting teachers suck. In my experience, though, this approach is more accessible, but it isn't widely used. I'm not a trained educator, so I'm wondering if there we don't see more of this sort of teaching in the classroom.

Good question. I see the site as a complement to existing lesson plans. Some teachers have taken the material and presented it in their own style, which I absolutely love:



Video a teacher made:


I don't know anything about educational policy, classroom management, etc. I want to provide the best ingredients I can and let teachers make the best meal they can.

Even if they did, students don't care for most things, intelligently explained or not. That's a first problem to overcome.

According to...?

A lot of people love learning new things. It's the mode of learning them that is offputting to so many.

Have you spent any time teaching unmotivated students?

Yes, ideally, every student would be motivated to learn for its own sake, but for an individual say, 8th grade, teacher to get a class of kids to that point is an enormous task. And yes, perhaps the entire education system should be revamped so that kids never lose motivation, but how to do that is hardly a solved problem.

Which means "Here's the deeper principles that motivate this problem" is going to have a huge uphill battle against "just tell us how to do the problems that are going to be on the test", or worse, "this has no relevancy to my life, so I'm going to tune out this entire class".

I have taught kids ranging from 6 years old to 2nd year University undergrads. The really young kids were in the setting of a coding bootcamp while the 2nd year undergrads were in a Tutorial. I'm just prefacing my comment by saying that I've experienced a large range of ages and abilities.

I always see this excuse as a failure of the teacher, rather than the students. I believe students want to learn, and the non-motivation is usually a result of something that isn't so hard that the teacher can't get around it. I used to feel that way before and only gear my lessons towards the motivated ones (why should I waste my time on kids who dont want to learn?), but I realized that it was I who was not motivated enough to get through to those kids. The movie "Stand and Deliver" portrays what I'm trying to say in a really fun and useful way.

I think teachers need to be held to a higher standard and blaming their lack of success on students should be the last resort, after everything has been tried. Sure you'll get some really pathological cases where the student is absolutely unreachable, but I think that's so rare that it's not worth talking about.

That seems like an a-priory assumption.

I think it depends on overall context (socio-economic status of parents, what they see everyday, what other teachers do, what the policy of the school is in general, what their society at large perceives as success, etc.).

Empirical observation however -- and I've taught 2 different secondary schools myself although just for a couple of years -- tells us that some students are motivated and others are not. The teacher can try and nudge them towards the subject, but it wont do that much with most of the unmotivated students. My experience has been in what in the US you'd call "inner city" schools btw.

I'm not saying that this is an absolute rule, so individual counter-examples don't really negate this, unless they do indicate a reverse GENERAL trend. Sure, you could get a greatly motivated student even in a crappy school with crappy teachers, the question is how often would that happen.

I also don't agree that the teacher should be "held to a higher standard" (at least when meant to an extreme). Sure, there are indeed crappy teachers.

But students should come into school willing to learn and respecting the environment, something that's not always the case. It shouldn't be up to the teacher to do some special stunts to get the students basic attention -- instead of, say, playing mobile games on their smartphones, talking to each other loudly, even listening to music on headphones.

I think its the teachers job to earn the students' respect, by not only being a role model but also demonstrating genuine interest in the topic they are teaching. I think that when I show how deeply I love the topic I'm teaching, it rubs off on the students and they go along with it.

Also, you seem to be arguing from the perspective of what is rather than what ought to be. I'm arguing for a shift in perspective where teacher competence and enthusiasm and high expectations of students isn't something special or extra, but rather the norm.

>I think its the teachers job to earn the students' respect

Students should have a respect for school (and the role of the teacher) before any other kind of respect can be earned by the teacher as an individual.

Or, to put it another way, earning the students respect as a teacher is OK.

But having to earn the attention, and having to fight against students making noise, playing, ignoring the lesson etc, should not be the case.

>I think that when I show how deeply I love the topic I'm teaching, it rubs off on the students and they go along with it.

As I said, assuming the teacher is capable and passionate, it still depends on the students. Depending on the school/area/class/etc some students wouldn't care even if Alan Kay taught them programming and Richard Feynman did physics.

The idea that students will be captivated by a passionate and eloquent teacher doesn't always pan out in reality. A lot of times it's more like: https://www.youtube.com/watch?v=Bdf_XdDwc-o

>Also, you seem to be arguing from the perspective of what is rather than what ought to be.

Well, to get things to where it "ought to be" you should first tackle and work with "what is".

I'm really curious why it is that students seem to just inherently care about test scores. Surely they were born with such passion for seeing "A" over "B" over "C," and so on.

If they were taught to care about deeper principles from day one, I guarantee they would care about deeper principles by year 8.

That would take a society that also cares about deeper principles, starting with parents etc., and not just "getting into university" and "getting a good job" (and that's when it's not just "make shitloads of money").

right. Most ppl have no idea why they are being educated, what it means to be educated.

I love this site. This explained logarithms and exponential growth really well.

Glad to hear it helped, thanks.

Oh! Dude, you are amazing! Seriously, I don't normally say that to people - I can't thank you enough because you honestly got me over a few mathematics hurdles.

If you ever have a paid subscription, I'm buying :-)

Thanks Chris, really appreciate it. Hearing when the site clicks with someone is invigorating.

No real paid subscriptions yet but it's a great idea =).

I'm so glad this exists going into my next term involving a lot of maths that I just don't fundamentally understand; sure I can remember a set of rules after boring revision of sequences of instructions but I hate that.

I asked what a "dot product" _meant_ once and was just told how it could be used to determine these other values. I later learnt what it actually was by reading some renderer library code.

I had a similar "aha" moment when I read how you described sine/cosine/etc as percentage values in relation to positions around the circle. I hate how I was taught maths since secondary school (when I was so confused at trig/pythag lessons but learnt the concepts very quickly when I was coding around with positions and angles in a game engine mod to get a position x units in front of a character instead of doing homework)

I really think this site will help me in the coming year, thank you!

From: http://betterexplained.com/static/articles/rethinking-arithm...

1 * x^2 = 9

"What transformation (“times x”), when applied twice, will turn 1 into 9?"

That's such a great way of thinking about algebra. The "understood" 1 is equivalent to, e.g., an understood "YOU" in the English language.

Beautiful way to put it! We aren't always explicit about the subject of the transformation. Similarly, when working with exponents, I think:


is really

1 * e^ix

That is, we're starting with 1.0 and doing a transformation (rotation) on it.

This is great, wish someone would do this for more advanced math like topology, or other more abstract/pure math

Seems like the site has gone down with traffic and CloudFlare is serving up a generic 522 error page.

Yep, bummer I was looking forward to checking this out.

Check back later when traffic has died down. The site really is worth a read.

Thanks! Working on getting a fully static version of the site up. Argh, Wordpress with even caching plugins, still folds. (It's on reddit and getting 100x normal traffic). Eventually a postmortem will be in order.

Static links to some favorites:

• Imaginary numbers: http://webcache.googleusercontent.com/search?q=cache:http://...

• Understanding e: http://webcache.googleusercontent.com/search?q=cache:http://...

• Intuitive Trig: http://webcache.googleusercontent.com/search?q=cache:http://...

• Calculus intro: http://webcache.googleusercontent.com/search?q=cache:http://...

• Sine waves: http://webcache.googleusercontent.com/search?q=cache:http://...

• Euler's Formula: http://webcache.googleusercontent.com/search?q=cache:http://...

• Linear Algebra: http://webcache.googleusercontent.com/search?q=cache:http://...

Update, I've made a static version of the site (http://betterexplained.com/static/) which you can browse. I'm redirecting requests there.

Saw a lot of that yesterday, too

>Secondly, if he is not involved in criminal activity, why does Mr Claus use unbreakable encryption?

How do they know he encrypts everything?

shoutout to kalid - love your work.

Thank you!

Error 522

Aw, and here I was hoping for clips from SquareOne TV...

Is it a data mining site? I'm just entering captcha's for days..

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