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.

Table of Contents

## 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

- Developing Your Intuition For Math
- Why Do We Learn Math?
- How to Develop a Mindset for Math
- Learning math? Think like a cartoonist.
- Math As Language: Understanding the Equals Sign
- Avoiding The Adjective Fallacy
- Finding Unity in the Math Wars
- Brevity Is Beautiful
- Learn Difficult Concepts with the ADEPT Method
- Intuition, Details and the Bow/Arrow Metaphor
- Learning To Learn: Intuition Isn't Optional
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Learning to Learn: Math Abstraction
- Learning Tip: Fix the Limiting Factor
- Honest and Realistic Guides for Learning
- Empathy-Driven Mathematics
- Studying a Course (Machine Learning) with the ADEPT Method

## Leave a Reply

54 Comments on "Finding Unity in the Math Wars"

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.

@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].

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.

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

@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.

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.

@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.

“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?

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 :)

Each of us learn differently. My endless search online has landed me in betterexplained.com. 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?

@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…

@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.

@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!).

@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 BTWP.P.S: Sorry for my apparent lack in eloquence that others seem to possess in quantities that make mine insignificant@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!

well said Kalid.

@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

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

@kalid Markdown is still not implemented :(

@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].

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.

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.

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.

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.

[…] Finding Unity in the Math Wars — 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.” I like this! (and the point that math teachers fighting with each other is missing an opportunity to fight for the existence of math education) (ps, “unity … math”, I see what you did there) […]