Finding Unity in the Math Wars

I usually avoid current events, but recent skirmishes in the math world prompted me to chime in. To recap, there’ve been heated discussions about math education and the role of online resources like Khan Academy.

As fun as a good math showdown may appear, there’s a bigger threat: Apathy. And Justin Bieber.

Educators, online or not, don’t compete with each other. They struggle to be noticed in our math-phobic society, where we casually wonder “Should algebra be taught at all?” not “Can algebra be taught better?”.

Entertainment is great; I love Starcraft. But it’s alarming when a prominent learning initiative gets less attention than a throwaway pop song (Super Bass: 268M views in a year; Khan Academy: 175M views in 5 years). Online learning is a rounding error next to Justin Bieber — “Baby” has 700M views alone.

What do we need? The Math Avengers. Different heroes, different tactics, and not without differences… but everyone fighting on the same side. Against Bieber.

I could be walking into a knife fight with an ice cream cone, but I’d like to approach each side with empathy and offer specific suggestions to bridge the gap.

The Big Misunderstanding

Superheroes need a misunderstanding before working together. It’s inevitable, and here’s ours (as a math relationship, of course):

Bad Teacher < Online Learning < Good teacher

The problem is in considering each part separately.

  • Is Khan Academy (free, friendly, always available) better than a mean, uninformed, or absent teacher? Yes!

  • Is an engaging human experience better than learning from a computer? Yes!

But, really, the ultimate solution is Online learning + Good Teachers.

Tactics differ, but we can agree on the mission: give students great online resources, and give teachers tools to augment their classroom.

Why Do I Care?

I love learning. Here’s my brief background so you can root out my biases.

I was a good student. I was on the math team and hummed songs like “Life is a sine-wave, I want to de-rive it all night long…”. I drew comics about sine & cosine, the crimefighting duo. You might say I enjoyed math.

I entered college and was slapped in the face by my freshman year math class.

Professors at big universities must know everything, right? If I didn’t get a concept, something must be wrong with me, right?

I had a WWII-era, finish-half-a-proof-in-class, grouch of a teacher. I bombed the midterm and was distressed. Math… I loved math! I didn’t mind difficulties in Physics or Spanish. But math? What I used to sing and draw cartoons about?

Finals came. While cramming, I found notes online, far more helpful than my book and teacher. I sent an email to the class, gingerly suggesting BY EUCLID YOU NEED TO READ THESE WEBSITES THEY ARE SO MUCH BETTER THAN THE PROFESSOR. The websites turned up on an index card in the computer lab that evening. How many of us were struggling?

I was studying, staring at a blue book when an aha! moment struck. I could see the Matrix: equations were a description of twists, turns and rotations. Their meaning became “obvious” in the way a circle must be round. What else could it be?

I was elated and furious: “Why didn’t they explain it like that the first time?!”

Paranoid I’d forget, I put my notes online and they evolved into this site: insights that actually worked for me. Articles on e, imaginary numbers, and calculus became popular — I think we all crave deep understanding. Bad teaching was a burst of gamma rays: I’m normally mild mannered, but enter Hulk Mode when recalling how my passion nearly died.

My core beliefs:

  • A bad experience can undo years of good ones. Students need resources to sidestep bad teaching.

  • Hard-won insights, sometimes found after years of teaching, need to be shared

  • Learning “success” means having basic skills and the passion to learn more. A year, 5 years from now, do people seek out math? Or at least not hate it? (Compare #ihatemath to #ihategeography)

(Oh, I had great teachers too, like Prof. Kulkarni. The bad one just unlocked the Hulk.)

An Open letter to Khan Academy and Teachers

I recently heard a quote about constructive dialog: “Don’t argue the exact point a person made. Consider their position and respond to the best point they could have made.”

Here’s the concerns I see:

Packaging and presentation matters

Yes, other resources and tutorials exist, but there’s power in a giant, organized collection. We visit Wikipedia because we know what to expect, and it’s consistent.

Khan Academy provides consistent, non-judgmental tutorials. There are exercises and discussions for every topic. You don’t need to scour YouTube, digest hour-long calculus lectures, or open up PDF worksheets for practice.

So, let’s use the magic of friendly, exploratory, bite-sized learning of topics.

Community matters

Teachers and online tools don’t “compete” any more than Mr. Rogers and Sesame Street did. They’re both ways to help.

I do think the name “Khan Academy” presents a challenge to community building. Would you rather write for Wikipedia or the Jimmy-Wales-o-pedia?

Wikipedia really feels like a community effort, and though there are alternatives, in general it’s a well-loved resource.

I think teachers may hesitate to use Khan Academy, not out of jealousy, but concern that a single pedagogical approach could overpower all others. Let’s build an online resource that can take input from the math community.

Human interaction matters

It’s easy to misunderstand Khan Academy’s goal. I’ve seen many of their blog posts and videos, and believe Khan Academy wants to work with teachers to promote deep understanding.

But, some news coverage shows students working silently in front of computers in class, not watching at home to free up class time for personal discussions.

The teacher doesn’t appear to be involved or interacting, and that misuse of a learning tool is a nightmare for teachers who want a personal connection. Let’s have an online resource that directly contributes to offline interactions also.

Experience matters

I’ve seen that insights emerge hours (or years) after learning a subject. For example, we’ve “known” since 4th grade what a million and billion are: 1,000,000 and 1,000,000,000.

But do we feel it? How long is a million seconds, roughly? C’mon, guess. Ready? It’s 12 days.

Ok, now how long is a billion seconds? It’s… wait for it… 31 years. 31 years!

That’s the difference between knowing and feeling an idea. Passion comes from feeling.

Teachers draw on years of experience to get ideas to click — let’s feed this back into the online lessons.

Students matter

We teach for the same reason: to help students. Here’s a few specific situations to consider.

For many, Khan Academy is their only positive math experience: not teachers, or peers, or parents, but a video. Sure, it’s not the same as an in-person teacher, but it’s miles beyond an absent or hostile one. If an education experience gets someone excited to learn, and coming back to math, we should celebrate.

Remember, despite years of positive experiences and acing tests, a sufficiently bad class nearly drove me away from math. Resources like Khan Academy offer a lifeline: “Even with a bad teacher, I can still learn”.

When someone is interested, we need to feed their curiosity. I get a lot of traffic from Khan Academy comments — how can we help students dive deeper, without making them trudge randomly through the internet?

Lastly, we all learn differently. I generally prefer text to videos (faster to read, and I can “pause” with my eyes and think). Some like the homemade feel of Khan’s videos. Others might like the polished overviews in MinutePhysics. You might prefer 3-act math stories or modeling instruction.

Let’s offer several types of resources for students to enjoy.

Calling the Math Avengers

Still here? Fantastic. To all teachers, online and non:

  • What specific steps can we take to align our efforts?

One idea: Make a curated, collaborative, easy-to-explore teaching resource.

Khan Academy is well-organized: each topic has a video and sample problems. How about sections for complementary teaching styles, projects, and misconceptions?

Imagine a student could select their “Math hero” as Khan Academy or PatrickJMT or James Tanton and see lessons in the style they prefer (like Wikipedia, curate the list to “notable” resources).

Imagine teachers could explore the best in-class activities (“What projects work well for negative numbers?”).

Whatever the style, make it easy for other educators to contribute. Want project-based videos? Sure. Need step-by-step tutorials? Great. Prefer a conceptual overview? No problem.

Each teacher keeps their house style. Let Hulk smash, and Captain America handle the hostage negotiations. Use the hero that suits you.

(It’s a public google doc you can copy and edit)

Perfect? Nope. But it’s a starting point to think about how we can work together.

Let’s focus on the overlap and align our efforts: different heroes, different tactics, and on the same side.

Other Posts In This Series

  1. How to Develop a Mindset for Math
  2. Developing Your Intuition For Math
  3. Learn Difficult Concepts with the ADEPT Method
  4. Brevity Is Beautiful
  5. Learning To Learn: Embrace Analogies
  6. Learning To Learn: Pencil, Then Ink
  7. Intuition, Details and the Bow/Arrow Metaphor
  8. Finding Unity in the Math Wars
  9. Why Do We Learn Math?
  10. Math As Language: Understanding the Equals Sign
  11. Learning math? Think like a cartoonist.
  12. Learning To Learn: Intuition Isn't Optional
  13. Avoiding The Adjective Fallacy

Questions & Contributions


  1. One idea: Make a curated, collaborative, easy-to-explore teaching resource.

    As someone who follows KA daily, this is actually a goal of KA. Khan said in an interview that the ultimate goal is to have multiple teachers for a certain topic with games, simulations, and projects to customize curricula.

    Khan Academy also accepts requests to donate videos by emailing sample videos.

  2. @Michael: Awesome, thanks for the info! I follow KA casually and wonder how many other teachers know about this? I think the word needs to get out.

    It’d be great to have a collection of 2-3 different video styles for each topic [conceptual vs. tutorial vs. story-based].

  3. The problem with a lot of the ‘Anti-Khan’ movement, as I see it, is that these people are all physicists or mathematicians who teach at that level. Having worked in these fields for decades, they possess an extremely well-grounded high-level understanding of their material: They read and derive proofs, they work at building new physical models, etc. In Devlin’s words: “…all the other KA critics in the educational world are interested in facilitating something quite different: real learning among their students.”

    This is all well and good, but…there are steps to real learning. One’s intellect does not simply pop into the arena of true theoretical understanding. It is an endeavor that takes years and years of work and practice! A huge part of this practice is, like it or not, solving repetitive and formulaic math and science problems.

    Take, for example, conservative fields from vector calculus. In order to fully understand conservative fields, there are a lot of ‘qualifications’ that must be kept in mind–smooth curves in regions that are both connected and simply connected. Most courses teach these qualifications right away. (Mine did). Mathematicians would encourage this practice, always striving to be perfectly correct, but I absolutely abhor it. It’s far more valuable to get students working on calculations as soon as possible–the component test, finding potential functions, etc. Later, when a numerical understanding of the concept starts to arise, it’s much more effective to introduce the limitations inherent in the techniques that have been taught.

    In other words, there’s a tendency in mathematics today to put the horse before the cart, and teach students theoretical concepts before they’re fully equipped to understand them. KA has been so successful exactly because it procedurally and formulaically rejects this process. No wonder students love it and the Math Royalty don’t.

  4. Er…in my very first sentence, I meant to write “these people are all physicists or mathematicians who teach at the UNIVERSITY level.” Whoops.

  5. @Joe: Great point. I believe concepts need “progressive refinement”, i.e., you learn the high-level concept, try some examples, then deepen your understanding, try more examples, and so on.

    One analogy I use is “explaining a cat”. First you show a cat, explain its basic features (furry, has a tail & claws) and observe it. Then you might explain that all cats (tigers, housecats, bobcats) descended from some common ancestor. Some cats are extinct today (sabre-toothed tigers).

    Then, you explain that all cats share some common DNA [ACATACAT :)] which gives them their “catness”. This is the expert-level understanding [I’m vastly oversimplifying the biology here, but that’s the idea].

    It’s very easy, especially in technical fields, to jump to the DNA-level description without first walking through the “here’s a picture of a cat” level.

  6. We need a common ground to do everything. We need to know where to learn anything and where to teach anything. We need to know how to figure out how to do anything. We need a better way to search instead of typing keywords into search engines.

  7. @Name: Yep, we need a common, curated area. Students and teachers shouldn’t have to do random google searches to find the best resources for well-known topics.

  8. “Select Your Math Hero”, I choose…. Justin Bieber. What’s wrong with getting Bieber to teach algebra 101? As long as someone else write the lesson plan, of course. Can’t we just work together?

  9. Hi there,

    Technically, what you’ve provided under “The Big Misunderstanding” is a big misunderstanding in and of itself–that is not an equation, but an inequality. Just a bit of pedantry :)

  10. Each of us learn differently. My endless search online has landed me in I find intuitive methods taught by Kalid more appealing but I can’t keep my attention with Khan’s videos. Should I conclude that the Khan’s videos are not effective and his pedagogy is wrong? No! My friend’s kid loves Khan’s videos; millions of people love them.

    Instead of judging the teaching methods and content of others the critics should spend the time creating and sharing learning materials. Let us lead people away from endless entertainment to life long learning and teaching! How about federated online content, curated, and mapped to age groups?

  11. @Shiv I agree. While KhanAcademy is good for complete novices and getting the gist of things, I’ve always found videos tedious and boring. “What, a one-hour video! I don’t have that much time…” I find myself saying. I often find myself scanning through YouTube’s ‘interactive transcriptions’.

    With text (and picture!) articles, I can go at my own pace. Stop, go for a walk to _really_ digest the stuff and then come back and pick off where I left. I can also scan ahead and skip to a section and decide if it’s worth reading at all (problem with YouTube).

    Lastly, I think we should highlight the forums a bit more. I find them to be an excellent resource and allows for interaction and specific questions to be answered…

  12. @Christopher: If Bieber started doing math videos that would be seriously awesome.

    @D: Ah, good point. I’m going to continue to say “equation” though as “inequality” is a little obscure I think. But noted.

    @Shiv, YatharthROCK: Great points. Yep, everyone has a different preference [I prefer text too, easier to scan]. But those are individual preferences, and whatever tools get someone interested in the topic are the ones we should use.

    Forums are an interesting angle as well. I like them, but unfortunately they can be time consuming to follow. But they’re great for specific, immediate questions.

  13. @D: Yes, an equation requires an “equal(s)” sign (=) – hence the name. Let me add that some equations may use “≤” (less than or equal to) and/or “≥” (greater than or equal to) rather than simply “=”. Perhaps Kalid’s “equation” could be called a “formula” to completely satisfy everyone (except perhaps chemists, for whom “mathematical formula” would be 100% correct!).

  14. @kaild What an ironic discussion!

    Here’s your brilliant article, which will benefit all classes of people (i.e., those in the loop, those not) and here are the pedants, the “I-don’t-car-if-they’re-new-learning everything-should-be-technically-correct” people you mentioned trying to undermine the meaning of the post (which, I doubt they get).

    You are KA here and the pedants, the mathematicians. Can’t we all get along?

    P.S: No offence to anyone here BTW
    P.P.S: Sorry for my apparent lack in eloquence that others seem to possess in quantities that make mine insignificant

  15. @YatharthROCK: “Can’t we all get along?” So why are you falsely accusing others of “undermining” and “not getting” the meaning of Kalid’s excellent post? Please don’t be so intemperate, but share in the learning!

  16. @Ralph Yeah, sorry. I need to be a little more discrete when I rant, ad when I do, try not to overtone things too much and let a healthy discussion continue. #letsgetalong

  17. @YatharthROCK: Sure thing! Still, I hope you don’t tone down your enthusiasm and passion! Best wishes to all (*]*).

  18. @Ralph, YatharthROCK: Thanks for the discussion, and for being civil. It’s very easy to misinterpret statements online or take them out of context, so I appreciate it.

    The points about the language are well-taken though — a few people have been confused by it [esp. since equation implies an equal sign somewhere]. I’ve rephrased it to “math relationship”.

    @Cherae: Thanks!

    @YatharthROCK: I turned on markdown but realize a lot of older comments were written without it in mind (and things like (*]*) don’t show up properly). However, you can use “i” tags if you like. I may turn on markdown eventually after auditing some previous comments [shuddering at the thought, it’s a lot].

  19. I can’t speak for Khan Academy, having only learned of it from an educational specialist a few days ago. However, having seen the light come on for a child around similar concepts, it is crystal clear to me that there are alternative ways to teach and to learn. Because everyone learns differently, our society would benefit by embracing solutions that recognize this reality and allow our educational system to address the need.

  20. Hi Lisa, thanks for the comment. I totally agree — there’s no reason we need to limit ourselves to one teaching style, any more than we limit our reading to one author.

  21. Hi Kalid – I’ve enjoyed your articles here and this one really forced me to pause and consider what goes into the success of people like George Polya or Martin Gardner in explaining complicated ideas to “the uninitiated”, so to speak. To me it seems like their writing always shows a talent for approaching topics as stories to be told – faithfully, accurately, but also in a way that engages the reader’s sense of “setting”, “characters”, “plot development” and “resolution”.

    And it makes sense that nudging a student/reader’s brain into listening-to-stories mode would help so much with comprehension and retention. Storytelling is universal: every known culture on the planet passes on stories. Stories engage both the language center and parts of the brain involved in predicting other people’s actions. For thousands of years they’ve been a focus of social activity; people bond over anecdotes. They capture a sense of raw possibility (leading to what-ifs, branching story arcs, alternate endings, prologues, etc.) by building language structures that map to events so that the listener can come to grips with different possibilities by shuffling those structures around, composing the pieces together in a potentially huge number of different ways.

    Well, we do an extremely similar thing in mathematics, don’t we? Choose a setting for the story: draw assumptions from observations of something we want to model, or from results whose stories we’ve already told. Cast and develop the characters: which lemmas, principles, intuitions, conjectures and solid, long-standing theorems will get involved? Carry the plot forward: describe how each step follows from the last, how the interplay between characters unfolds, letting the reader fill in routine details or ones that are more fun to imagine independently of the writer. Conclude: lead to the consequence(s) of what was assumed at the start.

    I think it would be very worthwhile, at least in some areas of teaching, to approach mathematics as a giant, still-unfolding story to be told.

    For instance, there are so many ways to tell the story of calculus and I wish one of them in particular could be known to high school students. To roughly outline what I mean: At first we had the set {0,1} where multiplying any two of its elements results in something that belongs to the set (it’s closed under multiplication). And that’s nice because x*y = 1 if and only if x = 1 and y = 1, so treating 1 as “true” and 0 as “false” we can capture the notion of logical “and” as multiplication. There’s another operation, addition, that allows us to count one thing. We can say x + y = 1 if either x or y is 1… but 1 + 1 doesn’t belong to this set so the idea comes up that we’d like to give names to expressions like 1 + 1 + … + 1 and write calculations in terms of those names. Closing {0,1} under the operation of addition we get the set ℕ = {0,1,2,3,…} of natural numbers where 2 is the name of 1 + 1, 3 is the name of 1 + 1 + 1 and so on.

    Now ℕ is a nice set because it’s closed under both multiplication and addition – so that gives plenty of useful ways to think about whole number quantities of things. Of course, annoyingly enough, 1 – 2 doesn’t belong to ℕ and so we end up wanting to close ℕ under the subtraction operation to get ℤ = {…,-2,-1,0,1,2,…}, the set of all integers. (Where -1 is a name for {0 – 1, 1 – 2, 2 – 3, …}, -2 is a name for {0 – 2, 1 – 3, 2 – 4 …}, 1 is now considered a name for {1 – 0, 2 – 1, 3 – 2, …}, etc.) The integers serve us well, until whole numbers aren’t enough for our purposes and we want to chop up quantities very finely so that we can use arithmetic to say things about small pieces adding up to whole pieces. So we make ℚ, the set of rational numbers, out of ℤ in a way similar to how we made ℤ out of ℕ: say two ‘pairs’ of integers are considered to be the same element of ℚ if they represent the same fraction, so for example 1/2 is a name for {1/2, 2/4, 3/6, …}. The rational numbers serve us even better than the integers because they give us a lot of control over how large or small our quantities are.

    Is it enough to close ℤ under division? For many purposes, sure, but now that we’re dealing with such fine-grained numbers we’ve become interested in challenges like calculating the slope of a curve at a point. But it’s not hard to draw a curve which has slope √2 at one of its points, and √2 can’t be written as a fraction! We can get as many rational numbers as we like, each one closer to the “instantaneous slope” than the number coming before it, by drawing better and better secant lines – expressing the slope as the limit operation applied to a sequence of rational numbers. Then to calculate with numbers like √2, we need the real numbers ℝ to be closed under the operation of taking a limit.

    There could be so much to gain from collecting good learning references and having them (along with comments, questions, notes and such) laid out in a format where they naturally “branch off” from the appropriate conceptual story arcs. I wish something like that already existed (seriously, the internet has already been around for how long now?) and would love to contribute what I could.

  22. You have made an excellent point here. It is not that I agree with the methods of Khan Academy, but honestly, their videos have helped a lot of students around the world, regardless of the soundness or unsoundness of their pedagogy.

  23. While we are sharing teaching resources I hope they are paired with some exercises/problems to allow students to proof what they know. What everyone misses about Khan Academy is John Resig’s (and others contribution) in terms of creating both a problem generating engine OPEN SOURCE as well as the new programming interface. This stuff in open source and folks are welcome to create and share what they do under a creative commons license. The big fight coming is between the publishers and the Math Avengers. I’m betting on the Hulk.

  24. One thing I keep noticing about people who take issue with what they call the “anti-Khan” movement – they …

    …. they make false assumptions and state them as fact.

    I teach very remedial students who really struggle with math. Not University level.

    The Khan Academy promotes itself as being amazing and revolutionary. The actual content for the basic level learnign is *fine* for people who need a brush up on procedure, but the man tells students that two plus itself is the same as two times one… no, that’s not the only basic, fundamental error he makes. My students already feel like failures — so when this “best thing in the world” doesn’t work for them, they are absolutely sure that it’s their failure. I don’t care about his nice tone and that it helps some people. I see the damage it does to others… and all he’d have to do is care enough to have videos that had good content instead of his one-off “hey, I don’t even know what I’m going to say half the time” stuff.

    My students deserve better.

  25. I found KA through your website and used it as a reference for an adult education class I teach. All my students though it was fantastic. PatricJMT came from a very quiet chap at the back of the class, and is just great for another view on things.
    Why teach maths (sorry, I’m a Brit and we need our ‘s’)- because we can and because it opens the mind to so many other things. Like thinking
    I’m with you, Khalid.

  26. @Kazuo: Thanks for the awesome comment. I completely agree about stories — our brains are wired to understand analogies, metaphors and the like. Long before writing was common, we communicated information via oral histories. Why not use our brains, which handle this so well, to understand math?

    Partly, I think this is why I need analogies / metaphors before really feeling comfortable with a topic. Just seeing a series of logical steps does not satisfy that part of my brain that demands a story-level understanding of what’s happening.

    In terms of contribution, I’d like to think more about what is possible here to help organize our efforts. Really appreciate the comment!

    @Guillermo: Exactly. Khan’s videos aren’t exactly how I would have done them, and that’s ok. They help a lot of people — that’s what counts.

    @mark: Great insight about the open-source / community nature of the contributions. Right now we’ve been silo’d for so long, with proprietary “textbook” knowledge locked behind publishers. We need Wikipedia-style, community contributions.

    I remember seeing that Donald Duck video a long time ago! Thanks for the reminder. I’d love to get such quality resources like this in one place.

    @Sue: Thanks for the comment. I agree, it’s important not to take any current incarnation of a particular video (or resource) as a final one. We need a process to continually refine our content, similar to Wikipedia articles [many started out quite meager, with a single sentence, and evolved from there]. The hugely important human element in teaching is providing the encouragement to avoid having students feeling like failures.

    @Steve: Thanks for the note. Exactly, students should use what works for them. There’s no reason we have to all listen to the same musician (or the same online resource). We prefer different styles.

  27. Michael Arnel said that Khan’s “ultimate goal is to have multiple teachers for a certain topic…[and that] Khan Academy also accepts requests to donate videos by emailing sample videos.”

    I guess I would believe that if I saw even ONE video (coughPatrickJMTcough) on the Khan Academy site other than Sal’s. I know what he says is his “ultimate” goal. What about an actual interim goal? I don’t buy it.

  28. The underlying question seems to be who could be a (math) teacher? Is the teacher an expert in the area and knows all the answers and applies best methods to teach? If we assume this to be true then we will be disappointed. Instead, if we assume that the process of teaching is a learning activity where the teacher learns actively from the students. A good teacher actively seeks feedback is willing to change his/her world view so that everyone benefits and learns. In my view, the models for math are very precisely defined unlike the mental models that we have about the world that is partial but useful. The debate about math teaching by “non experts” will continue… Let us salute the people who have the courage to teach and change our (math) world-view for the better.

  29. Mathematics is best understood and taught as “how to get things done” – using practical examples. Pythagoras believed that everything could be explained by numbers. I’ve long thought that basic mathematics, physics and chemistry can be taught to children from the time they are able to count, by teaching them survival skills such as cooking, where concepts of volume, weight, size (incl. surface area), time, temperature and interaction and thermal conversion of ingredients are used to produce things they can actually eat! Then inform them they are in fact being research scientists as well as chefs!

  30. Well, I’ve visited this website some weeks ago and I liked what I saw (and see now). This time I’m not going to join the discussion about math teaching – I’m not even sure if this is the right place to put this – but just to mention some initiatives that I found useful and kind of similar (saving distances) to what you intend with Better Explained, and what Khan Academy intends, too. Maybe you already know some of them, but anyway, just wanted to point them out – though I haven’t tried them all. They are:
    * ‘Head First’ books series.
    * Coursera; an online learning platform.
    * Udacity; another online learning platform.
    * CodeSchool; – I haven’t tried it yet, but seems promising.

  31. @Roberto Great links! Love all those resources (except CodeSchool which I haven’t tried but have heard of before)

  32. @Reginalid: Yes, it may not be as easily discoverable, but there are some other contributors, like Vi Hart ( and Brit Cruise ( However, there’s not alternative explanations for the basic topics. One difficulty might be getting enough videos from another contributor to help provide a consistent experience.

    @Shiv: Yes, good teaching is maybe 30% content knowledge and 70% motivation, inspiration, empathy, etc. If you want pure content knowledge you can just read Wikipedia.

    @Ralph: Cooking is such an awesome example of “real world” math, chemistry, physics, etc. I’m surprised it’s not used more. Practical examples help solidify ideas, theory gives you new ones, then you get practical examples of those new ideas, and so on. I see it as a spiral.

    @Roberto: Awesome, thanks for the links. I read one of the “Head First” books on programming and enjoyed in. In general, I think we need a multitude of “teaching artists” to provide their expression. We don’t limit our appreciation of music to one band or composed, and similar to math. (The biggest hurdle is realizing math is something that can be appreciated like music or another art).

  33. In 7th grade I hated math. Boring teacher who was very uninspiring. In 8th grade I loved math. The teacher made it fun. Ever since then I have loved pursuing math and science. Got to college and learned calculus and sort of learned network analysis and math behind electronics. Then picked up a book called “Calculus Made Easy” which helped me create a PID loop in a microcontroller. It helped me understand calculus better than I ever learned from a teacher. The teacher was good too.

    How did I get here? I am trying to learn noise generation techniques for generating realistic terrains in a game engine. Somehow I ended up here. The point is that it was a good math teacher in 8th grade that helped me love the subject. I have had insights on my own (I need to write them down if I remember them) and completely agree on capturing what we can.

    It will be a battle to unseat the elitism inherent in higher learning and science. As a science lover I have watched the painful politics in science. One good example was Pons and Fleischman. The top physicists did not even review the theory or findings when they denounced anything had been discovered. It was all politics and protecting research grants. That is what online learning resources are fighting. There is a “provider” for higher learning and they don’t want competition.

  34. Hey K–

    As I’m sure you know, David Foster Wallace wrote a book on infinity called “Everything and More.” I just started it so I can’t comment yet on its overall quality, but in the preface he says something that reminded me on you. He says that he disliked and did poorly in every math class he ever took, save one, which was taught by “one of those rare specialists who can make the abstract alive and urgent.”

    Amen, brother.

  35. Hi Tim, really appreciate the note. I totally agree: we need concepts alive in our minds, or else we’re fooling ourselves about what we’ve really learned. Thanks for the support :).

  36. “But, really, the ultimate solution is Online learning + Good Teachers”

    You meant the ultimate solution is “Good Teachers + Online Learning” right?

    You have taken all of math education and reduced it to an argument about online learning and teachers. As if this is the heart of the problem.

    I find it amusing how easily we continue to push this meme of “math-phobia” and thus need to reorient all of mathematics. It’s acceptable to say I hate math or it’s not my subject or I’m not very good at it socially but to say I can’t read or write very well would cause embarrassment.

    Successive generations want life to be completely catered to them, everything has to come to them and society let this idea run amok.

    So while you’re discussing a math-phobia society and online learning versus classroom, teacher learning, tell us have you looked at why other societies in the world don’t have these problems or are far more successful in terms of math education? Do you think these societies are creating math plays and movies and juggling acts to ‘sell’ math to their populations?

    The private market wants to see public education destroyed. Here we give lip service to the idea that education is important. That died long ago and has been drowned out in a sea of consumerism.

    The simple fact is that other nations place a higher focus on learning and spend more on education than we are willing to do in America.

  37. This is a perfect critique of KA from the article you linked. In the internet age, hype, fanboyism and ‘branding’ are very regular phenomenons.

    Again, the private market is pushing procedural/standardized testing, turning education into a product. Around this procedural/standardized testing you grow entire ‘prep’ industries and tutoring facilities. All of whom have an interest in pushing more of these test which diminish learning and often stunt real learning.
    OF course this also creates an even more inequitable society.

    The Decline of the “Great Equalizer”

    Teachers are being asked to do more and more with less and less, it’s a prefect setup to see them fail. When they fail, the privateers are ready to swoop in and make money off the government. Socialism for the wealthy and the private is all good of course. The private market is interested in profits foremost, not human beings.

    “But the real problem is not the stuff on the KA site. Flawed as it is, it is, as I noted earlier, a lot better than many people have, or ever had, access to. The fact that many of Khan’s fans describe him as “the best teacher ever” speaks volumes about the poor quality of the mathematics education that many receive. I’ve visited many math classrooms both in this country and around the world, and I’ve seen great math teaching. You won’t find it on KA. Instead, you will find something else, something unique and of value.

    Sure, KA has lots of weaknesses and could be improved. That goes for any product. The real problem is that the US (and other nations) identify mathematics learning with instruction and passing procedural tests. In that world, KA meets a clear market need for instruction to help people pass procedural math tests. “

  38. “Can’t we all get along?”

    “But the real problem is not the stuff on the KA site. Flawed as it is, it is, as I noted earlier, a lot better than many people have, or ever had, access to. The fact that many of Khan’s fans describe him as “the best teacher ever” speaks volumes about the poor quality of the mathematics education that many receive. I’ve visited many math classrooms both in this country and around the world, and I’ve seen great math teaching. You won’t find it on KA. Instead, you will find something else, something unique and of value.

    Sure, KA has lots of weaknesses and could be improved. That goes for any product. The real problem is that the US (and other nations) identify mathematics learning with instruction and passing procedural tests. In that world, KA meets a clear market need for instruction to help people pass procedural math tests.

    In contrast, Ani, Noschese, Golden, Coffey, Meyer, Allain, and all the other KA critics in the educational world are interested in facilitating something quite different: real learning among their students.

    Sal Khan says he is trying to move into the real, conceptual learning space as well, but so far I have not seen much that would qualify, and as I noted earlier, my own interest in trying out the MOOC format notwithstanding, I have yet to be convinced that it is possible over the Web.

  39. @NotSoFast

    The problem I have with that is that… different people learn differently. For some people, Khan Academy does work. If it doesn’t, they can use something else.

    This is what some people (not necessarily yourself) don’t seem to understand. They speak of “real learning” and say that KA isn’t helpful. That depends entirely on the person. There is no one-size-fits-all solution, and in fact, that’s why I sort of disagree with the part of this article where he said that ‘good’ teachers are better than using resources (such as a computer) to self-educate. Depends on the person.

  40. Too many comments to read through in order to check if the following point has been made. Sure, music videos have a lot more views than online maths tutorials, but one doesn’t spend 2 hours in school every day watching music videos.

  41. I use both KA and betterexplained. Stumbled upon betterexplained first before a friend told me about KA. I just wanna say that even though betterexplained doesn’t come close to the amount of output KA has, I understand Kalid far more than I do on KA.

    This is in spite of KA using videos, which is supposed to be a better medium. I just feel like Kalid makes that extra effort with the intuition. Just look at the length of each article and the effort he puts in (with diagrams and all).

    Still I like both and a very noble effort by both you guys. Much respect to both of you. Not gonna diss Justin Bieber or Nicki Minaj or whatever. Entertainers are there to entertain, teachers are there to teach. If people like them then so be it, nothing wrong with that.

  42. Hey, I’m a senior in high-school and I just discovered this site. I have to say, I believe that the biggest bottleneck in education is communication! The intuition/insight you try to get at in your posts is “the” idea; when people understand it so well that they almost feel it. The way new content is introduced in school is horrible; every new idea should start with something extremely basic. I see many of my peers struggle with concepts only because the concepts have been communicated in this technical, contrived manner. With the worst teachers, you can see a disconnect between what they are teaching and how they are talking to the class. It makes everyone disinterested. Teachers are trying to relate it to the “real world” (“Let me give you a real world example,”) Math, science, education in general shouldn’t be something you have to “get through”. I believe, if you’re not interested in it – don’t learn it! The thing is, everyone should be interested in math, English, and learning in general. We tend to bundle things up into “subjects” and then assign people to certain subjects, such as the “math person” or the “artistic person”, when in reality math and english and science are all as close to the “real world” as you can get. The stuff taught in school is not some magical, quirky thing reserved for those who love little puzzles (“nerds”), or have some deep need to express themselves artistically. Everyone should enjoy learning all of the subjects on some level, because they are life itself; they don’t exist on a separate plane!
    By the way, I realize I just babbled, but I couldn’t help but vent my views on education! Every thing else in the world is charging forward while education lags behind…

  43. Hi Alex, thanks for the note! I agree, communication is key: can we convey the heart of the idea? (And not all the technical gibberish that surrounds it, just get to the idea itself?). Once we have the intuition, we can “firm it up” with all the rules.

    I really don’t like the “math person” or “history person” distinction. We don’t say someone is a “reading person” or a “listening person”. Nope. These are skills which are within our grasp. Thanks for sharing your thoughts!

  44. I don’t know if this has been said above but KA is perfect people who aren’t majoring in anything math related and just want to get through their math classes which they are struggling with.

  45. Me too! (I entered college and was slapped in the face by my freshman year math class.) Personnaly, as I overheard, “not the sharpest knife in the drawer, but certainly not the scissors’. I had the head of the math department for algebra (might even have been a remedial class). This guy would, with a flair, start with his left hand and finish with his right and fill the two front boards and the three side boards per class. I dumbed it!
    Summer repeat I had a stubby cigar smoking personality who quipped QED, quite easily done and thumped his chalk out of the second floor window (last piece of chalk in the house and had to go find more). Made an A. The teacher can make all the difference. Had I had your site n-years ago I might have made the engineering cut.

  46. Hi, love the philosophy behind your site, I have a 7 year old, and been part of his math development. At that age (and younger) if you can’t explain at the intuitive level, they are not going to get it! So, a really good discipline for me! I have no choice but to go about it from the intuition/story-telling angle. I’d love to see you tackle elementary math, too. Maybe when you have your own kids, hey. 😉

    I hated math at school (an all too common experience, sadly) but they way I’m doing it with my kid, I’ve learned to like it, a lot. Math is his favourite thing, and it has become my favourite subject to do with him as a result (I am an English literature major). I do live in the fear though, of what you say, one bad math experience can undo years of good. :( I am wondering if it is too much to ask that an elementary school teacher, who is a generalist, can manage to teach math in an engaging way? I know your site usually attracts people who are at a more advanced level of math, but it nags me that at the elementary level, kids are not getting the sensitive handling required to cultivate the right attitude toward math – a curious, exploratory, experimental experience rather than rote learning. If we “lose” these kids even before they hit third grade, what then of the future of math in our country?

  47. I don’t know if anyone else mentioned this, but I believe TENURE is a very large contributor to bad teachers! When I was in law school I actually heard a professor boast to us (students) “go ahead and complain, I’m tunured.” These overpaid vermin have no place in an educational system, they are there simply for a check and the relative autonomy and prestige that comes with being a “college professor.” This is, imho, a primary reason why this country (US) has such serious issues with STEM education; and lack of a workforce with the necessary skills for modern business. This has been going on for a long long time, this country’s tenure system needs a serious overhaul; otherwise we continue down the same sad road.

  48. I am a 69 year old who struggled with calculus at school. I was utterly mystified, and was set to drawing conic sections because I liked drawing! Ever since then, I have had a hankering to understand calculus. Your website has encouraged me to set out on this exploration. Like you I have a poor memory, but once I understand something I know I can always retrieve that insight. Along the way, I like many of the things you have said- in particular:” Understanding is shown by the questions we ask, not the tests we pass.” Long may you continue to each!

  49. Don’t get it why there’s a war b/w KA and BE.
    Articles here are too good… but people who want to learn all the way from basics can surf KA.
    There can’t be a comparison b/w 2 sites whose approach are completely different.
    For comparing, comparison can be done with Where a lot of higher math is going on (with articles) and community posted questions

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