Analogy: Math and Cooking

If we had a recipe for pie, there's a few things we might expect. If we doubled the ingredients, we'd probably get double the pie. But if we took the ingredients and cooked them separately, we wouldn't expect to put them together and get our pie back. You can't bake the eggs, flour, and sugar in separate ovens and plop them together at the end.

In math, we can get misleading intuitions about what can (or can't) be rearranged.

After learning addition, we've memorized facts like 2 + 4 = 6. But this might stray into the idea that "whenever I see 2 and 4, I can simplify to 6".

Although 2 + 4 = 6, but "baked(2) + baked(4)" is not "baked(6)". Baking unmixed ingredients in the exponential oven we get:

$2^2 + 4^2 \neq 6^2$

We can only confidently say:

$(2 + 4)^2 = 6^2$

We combine the ingredients, then bake the result. Exponents, like baking an apple pie, modify the original ingredients so they can't be easily combined later. While we might recognize the original 2 and 4, they aren't directly available. Two baked pies can't be smashed together to consolidate the filling.

This confusion gummed me up in calculus, when learning derivatives (the bad boy of baking).

In algebra, we internalize rules like:

\displaystyle{x^2 \cdot x^4 = x^6}

But our intuition leads us astray when we get to the derivative.

\displaystyle{\frac{d}{dx}(x^2) \cdot \frac{d}{dx}(x^4) \neq \frac{d}{dx}x^6}

because

\displaystyle{2x \cdot 4x^3 \neq 6x^5 }

Raw polynomials can be multiplied, but the derivatives of multiplied polynomials can't be rearranged so easily. Multiplication makes functions interact in a way that makes taking the derivative more complex:

Working through the Product Rule we get:

\displaystyle{\frac{d}{dx}(x^2 \cdot x^4) = (\frac{d}{dx}x^2)x^4 + (\frac{d}{dx}x^4)x^2 = (2x)x^4 + (4x^3)x^2 = 6x^5}

When learning Calculus, I was confused how standard interactions (like multiplication) needed special handling. I thought I was done learning new rules for "arithmetic".

But no: functions, when multiplied, interact in funky ways. See how each side grows its own sliver of area (df * g and dg * f)? The functions being multiplied are "baked together" and the overall effect depends on them both, simultaneously. We can't examine them in isolation (e.g., df or dg by itself).

Now, there are setups when the inputs can be processed separately and combined later (linear algebra). The cooking equivalent might be a smoothie: An apple/banana smoothie mixed with a peach/mango smoothie is the same as blending all ingredients in the beginning.

A common assumption is that operations are usually linear, but $\sin(a + b) \not= \sin(a) + \sin(b)$ and $(a + b)^2 \not= a^2 + b^2$. Sorry, we have to carefully cook the ingredients if we want the math to taste right.

When our intuition for a math rule doesn't make sense, ask "Are we making a pie, or a smoothie?"

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Math and Analogies

Here's a talk on "Math And Analogies" I did for the Center on Contemporary Art (CoCA) in Seattle.

Transcript:

math-analogies-jpg.002

What's this? Well, it's a number. Probably written down a few thousand years ago. Great. And people, I suppose, could have read it back then if they spent enough time. And if we want, we can figure it out by spending time on it too (sigh).

math-analogies-jpg.003

So let's see. We have M, that's 1000. Then we have CM, that's 100 less than 1000, or 900. Then we have XC, which is 10 less than 100, or 90. Then we have VII, which is two more than 5, or 7. So the number is 1997.

math-analogies-jpg.004

OK, that wasn't fun. The question: why was that hard? Why was that difficult? Why was one representation so much easier? Well, there's a few answers.

math-analogies-jpg.005

One is that hey, we're just not smart enough.

Sorry guys, our brains just can't handle Roman numerals. I mean, computers can use them, why can't we? We're just not smart enough. And we're just bad at math.

That's one answer, and maybe it's possible that humans are incapable of understanding numbers.

math-analogies-jpg.006

But I think the better answer is that those numbers were poorly designed. Roman numerals aren't a great representation of numbers. We're trying to keep track of numbers in terms of two more than five, and one less than ten. We can't just say 9, we need to say "one less than ten", and so on. It makes things really complicated and there's not really a good reason for it. The result is that it's cumbersome, hard to use, and our brains are doing extra work they don't have to.

math-analogies-jpg.007

My goal for learning is to find the analogies that help things make sense. The analogy of numbers being in terms of "two more than five", "one less than ten" isn't a great system. Decimals are much better and that's what we use today.

The idea is that an analogy is like a raft you use to cross a river. The river is the problem, the concept, something that you want to understand, and the raft is how you approach it.

If you don't have an analogy at all, yes, maybe you're strong enough to swim to the other side by pure brute force, but usually you have a mental model that helps you with the problem. The problem is some mental models aren't as good as others. The idea of Roman numerals, while it "works", is not as good as decimals. We're taking a weaker raft to cross the river.

Let's take a look at a few mental models we've tried so far.

math-analogies-jpg.008

Rocks were the first mental model for counting. The word calculus (calculate) comes from "pebble", and the idea that numbers are basically rocks being counted is a powerful analogy. It sounds simple, but it's powerful.

The reason is we've taken out all the differences. There's a bunch of rocks in this photo. Different sizes, colors, weights, everything -- but we're counting them as rocks. And that's the big idea: we take things that are different in countless ways, but find the unifying principle behind them. And now we're counting that.

math-analogies-jpg.009

So the first concept (mental model) is that numbers are like rocks. Numbers are these physical things that unify disparate items and we can count them together and it works pretty well. One, two, three, four, five, six, seven, eight, nine, ten. With enough time we can count a herd of sheep. We can count money, it works pretty well.

math-analogies-jpg.010

But what's this?

Well, it's zero. Zero rocks, zero sheep, zero billion dollars are right there. Zero of the most beautiful painting in the world is right there.

Zero is a really weird concept. And when you've tied your concept of a number to physical items, concepts like zero become difficult.

Now we might work through it and say, "OK, let's allow for nothingness". So we can have "nothingness" as well as "somethingness".

math-analogies-jpg.011

OK, how about this? What's less than nothing? What's a negative rock?

Is it a rock that you owe? Well, if you're holding it, it's not negative, right? Is negative money something I owe you? Again, if I gave it to you, it might be negative to me, but it sure seems positive to you. There's a lot of things that get messed up when we're trying to use a very physical analogy. So the analogy, the raft is starting to break down. It's not helping us understand things.

math-analogies-jpg.012

In fact, even in 1759 people thought negative numbers were confusing. They "darken the very whole doctrines of the equations", right? They just didn't make sense. It was a weird concept and the problem was our analogy just couldn't handle it.

math-analogies-jpg.013

But of course, any modern third grader can tell you we have the concept of a number line. Moving our mental model to a number line let negatives and zero be part of the picture.

Instead of zero as a void, it became the neutral point, the center, the place that every other number moves away from. That's a really cool concept -- positive and negative numbers aren't that different. They're left and right directions away from the same starting point.

Something that was really, really difficult became easy with the right analogy. Let's keep going.

math-analogies-jpg.014

How about something like this? i squared equals -1. This is super confusing. Our raft is starting to shake.

Is there any number that when you square it becomes a negative? Zero squared is still zero. A positive number, when squared, is still positive. A negative number, when squared, becomes positive.

So, it seems like there's no solution. But again, this is like trying to find that negative rock. We're trying to apply the analogy in a way that limits our thinking. The analogy isn't the ultimate truth, right?

There might be a different way of thinking that makes it make sense, but we're stuck in this old system.

math-analogies-jpg.015

Here's an idea: If we have a number line going left and right, why not another number line going north and south?

We have East/West, why not North/South too? Let's allow this extra dimension where numbers can move up and down, left or right.

The imaginary dimension lets us get to negatives in two steps. Saying "i squared is negative one" is really saying: I'm starting at 1. We multiply by i, multiply by i again, and get to -1. So i represents a 90-degree rotation, and rotating twice points you backwards. That's what "i squared equals -1" means.

math-analogies-jpg.016

Finding the right analogy makes something that seems baffling click, all because we're using the right metaphor. We're thinking about it the right way. So, we've gone from rocks to lines to directions and that seems pretty good.

math-analogies-jpg.017

Let's put this into action a little bit. Here's a very famous equation. Euler's Identity is one of the most beautiful in math.

If you ask a mathematician or a physicist, this is usually considered the most beautiful identity in all of mathematics. It relates to so many different concepts.

math-analogies-jpg.018

The problem is it's really confusing and again, mathematicians thought it was just crazy when it came out. e to the i times pi equals negative one. We're taking these crazy constants: pi is irrational. i is this weird dimension, and e is another irrational number. All of these combined to equal -1 in such a clean way. It's wild.

And so people thought that OK, there are proofs, but it can't be understood. Again, that's coming from out existing analogies. If we have poor analogies for what numbers are, then yes, concepts like this are difficult, but with the right analogies they can make sense.

math-analogies-jpg.019

So let's take a crack at it. I like to mentally colorize equations. I want to understand what every single part of the equation is trying to say.

Normally we look at equations as a mix of symbols, but if we can identify what each part is doing, it becomes a sentence. It becomes a story of what's actually happening. So let's start.

math-analogies-jpg.020

e represents growth. It could be an entire separate video, but e is the concept of continuous growth, and i is the concept of going sideways (not the normal direction).

So the concept is that we have growth, but instead of growing like we want, we're growing sideways. And how long do we do this for?

math-analogies-jpg.021

Pi is halfway around a circle. A unit circle has two pi as its full circumference. So a single pi is just halfway around the circle.

We have the concept of growth going sideways that is lasting for half a circle. And if we plot it out, it looks like this:

math-analogies-jpg.022

We have growth, but it's going sideways, and we have enough fuel in the growth engine to point us backwards. So the equation is saying that this system of growth, pushing sideways, with enough fuel to last for half a circle, will point you backwards to negative one.

1 was the implied starting point. Start growing sideways with enough fuel to go half a circle and you'll end up backwards. So the key is we understood a concept that was baffling using a very simple diagram and analogy.

Now to really test it, let's try a few scenarios.

What if we don't put in any fuel?

What if we want to grow sideways, but we decided not to grow it all. We have the engine ready to go, but we don't fuel it up. If we put zero here instead, then we would stay at the starting point of 1.

Mathematically, we can say anything to the zeroth power is 1, which is true, but it's more helpful to think "I have this engine setup but didn't put any fuel into it, I never used it." Then you'll stay at your starting point.

If I put in twice as much fuel (2 * pi), I should circle all the way around. And again, we can mathematically say that if we square both sides, the result should be 1, but I like to think of it as using twice the fuel and going all the way around.

math-analogies-jpg.023

This is how to take an analogy and put it to work, helping you understand.

The main takeaway for me is that good analogies make math a joy. An idea that's baffling can become simple, even enjoyable, if we see it the right way. That's the heart of my philosophy. If I'm confused by something, I don't think my CPU isn't good enough. I just think I had the wrong analogy.

Happy math.


More reading:

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Colorized Math Equations

Years ago, while working on an explanation of the Fourier Transform, I found this diagram:

colorized fourier transform

(source)

Argh! Why aren't more math concepts introduced this way?

Most ideas aren't inherently confusing, but their technical description can be (e.g., reading sheet music vs. hearing the song.)

My learning strategy is to find what actually helps when learning a concept, and do more of it. Not the stodgy description in the textbook -- what made it click for you?

The checklist of what I need is ADEPT: Analogy, Diagram, Example, Plain-English Definition, and Technical Definition.

Here's a few reasons I like the colorized equations so much:

  • The plain-English description forces an analogy for the equation. Concepts like "energy", "path", "spin" aren't directly stated in the equation.
  • The colors, text, and equations are themselves a diagram. Our eyes bounce back and forth, reading the equation like a map (not a string of symbols).
  • The technical description -- our ultimate goal -- is not hidden. We're not choosing between intuition or technical, it's intuition for the technical.

Of course, we still need examples to check our understanding, but 4/5 ain't bad!

Creating Colorized Equations

I colorized a few of my favorite math topics below. Making the colorizations was surprisingly fun. Like writing a haiku, there's a game to trimming down a concept to its essence.

Colorized: e (universal base of growth)

colorized e

The number e (2.718...) is the base of growth, generated from universal ideas. Take unit interest, with unit time and compound it perfectly. Read article.

Colorized: Euler's Formula

colorized euler's formula

Euler's Formula is one of the most important in math, linking exponents, imaginary numbers, and circles. The intuition: constant growth in a perpendicular direction traces a circle. Read article.

Colorized: Fourier Transform

colorized fourier transform

The Fourier Transform builds on Euler's Identity. Using your circular path, spin a signal at a certain speed to isolate the "recipe" at that speed (like separating a smoothie into its ingredients). Read article.

Colorized: Pythagorean Theorem

colorized pythaogrean theorem

The Pythagorean Theorem is usually thought to apply to triangles only. In fact, it applies to any shape, any type of 2d area. Triangles are just a convenient starting point. Read article.

Colorized: Bayes Theorem

colorized bayes theorem

Bayes Theorem has a simple intuition: evidence must be diluted by false positives. (Cry wolf and you won't be trusted.) Read article.

Going Forward

These colorized equations are an experiment in conveying the most intuition in the simplest package possible. We don't need VR/AR, holograms, or brain-computer interfaces to understand math -- have we exhausted the possibilities of crayon on a piece of paper?

My short-term goal is create colorized equations for the top 25 equations on the site. Then (not trying to look directly at the sun), gather colorizations for the top (100?) topics we're meant to learn in high school and college.

The idea is to find explanation styles that work, and do more of them.

Happy math.

Appendix: How to Create Your Own

I have a half-built visual tool to make these. For now, here is the LaTeX template I used:

https://www.overleaf.com/read/cvmtqywqgvvw

Appendix: Initial Feedback

The idea got a strong reception on Twitter (thanks Jan):

The top piece of feedback was having accessible versions for color blind readers; I plan to make options available here too.

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Learning Math (Mega Man vs. Tetris)

Mega Man was one of my favorite video games. You're a little cyborg, running through levels and fighting end bosses:

iceman

Here's the trick: every boss has a weakness. After you beat Fire-man with your regular gun, you earn a fire weapon. This makes your upcoming fight with Ice-man easier, which helps defeat the next boss, and so on.

mega man

fireman iceman

So why's this special?

In Mega Man, you look forward to fighting more bosses.

Every level is a chance to permanently upgrade your abilities, not a grind you're trying to survive.

The Tetris Mindset

Think about Tetris: would you look forward to a variety of new shapes appearing?

new tetris shapes

Heck no. Tetris can be fun in a "survive hordes of incoming zombies" sort of way, but in terms of learning, it's a frustrating, Sisyphean task. Every new piece is something to move beyond, not a learning opportunity. It's a test to find your breaking point.

In Mega Man, the game gets easier the more bosses you have. It's specifically designed for you to improve over time. Guess which game has 10+ sequels?

Conquering Euler's Formula

Math learning can follow the Mega Man pattern. If we want to beat "Dr. Euler", we need to beat his henchmen and master their weapons:

euler's formula mega man

  • Rad-man, once defeated, lets you think with radians, not just degrees
  • Power-man lets you understand the base and power of an exponent
  • I-man lets you unlock the rotation of imaginary numbers
  • Pi-man lets you think in cyclic patterns
  • Multi-man lets you understand how quantities transform each other
  • E-man helps you visualize continuous change
  • And finally, Dr. Euler lets you understand the role of imaginary exponents: $e^{ix} = \cos(x) + i\sin(x)$

After defeating the henchmen — truly understanding them and mimicking their abilities— Dr. Euler becomes defeatable. And after that, Boss Fourier. Then Captain Convolution.

There many more challenges on the horizon… and that's great! Every formula, once mastered, is a power to use.

When learning, I ask: "Did I internalize the concept so much I look forward to seeing it?". Learned ideas become allies, a decoder key to help unlock future equations.

The Mindset Shift

I constantly seek analogies for learning because my understanding has improved most from perspective shifts, not from studying specific concepts.

Despite years of math classes, I lacked intuitions on e, i, pi, radians, logs, exponents… (the list goes on). Math class involved grinding through Tetris levels, moving past the concepts and not absorbing them fully. Imaginary numbers were not friends I looked forward to seeing in an equation.

An Aha! moment helped me see learning as a set of additive skills we could internalize, and the process and challenges became something to look forward to. I hope the same shift happens for you. Wouldn't it be great to look forward to adding new abilities to your arsenal?

Happy math.

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Studying a Course (Machine Learning) with the ADEPT Method

What's the point of taking a class?

  • Build a lasting intuition for the key ideas.
  • During the course, understand it enough to solve problems.
  • After the course, enjoy it enough to revisit.

That's why I learn things. Non-goals are transcribing what a teacher says, or cramming only to forget everything. (Yeah, it's a game we play, but we're stepping off the treadmill and only cheating ourselves. Most subjects have useful insights buried somewhere.)

So, here's my strategy when studying:

  • If an idea clicks, write down the Aha! moment in language you'd use yourself.

  • If it doesn't, write down the Huh? moment. Move on and try again later (such as with the ADEPT method). adept method

Keep it simple, like the KonMari method of organizing: Look at everything in your house. Does it spark joy? Keep what does, thank and donate what doesn't.

A simple study plan: Go through the material. Did it click? Write down what helped, otherwise look for a better explanation.

Actual Example: Stanford Machine Learning Course (Coursera)

My current learning project is the Machine Learning Class on Cousera. I've read a smattering of blog posts, the subject is growing, and after my friend asked me to join the class, I had to sign up. (It's great.)

Here's where I'm keeping my notes, Aha, and Huh moments:

Machine Learning Notes on Google Docs

Studying a Course (Machine Learning) with the ADEPT Method

This is one of the best learning experiences I can remember. A few examples:

An Aha moment for each prerequisite

For the major concepts the course depends on, I keep a 5-second summary in mind. This underlying concept, why does it exist? In plain English, what does it mean?

  • Linear Algebra: spreadsheets for your equations. We "pour" data through various operations.
  • Natural log: time needed to grow. Helps normalize widely varying numbers.
  • e^x: models continuous growth, has a simple derivative.
  • Gradient: direction of greatest change, helps optimize.
  • Calculus -Art of breaking a system into steps. With the gradient, we can move in the best direction.

I reference these snippets as I encounter new formulas.

Huh moment: Need to clarify a Formula

clarify formula

There was a formula that I expected to be positive ("cost" should be positive), yet it had a negative sign out front. What gives?

It turns out I had forgotten a part of the derivation, where we expected the natural log to be negative. (This happens when we take the logarithm of numbers less than 1 — in other words, we are going "back in time" and shrinking.)

I would have preferred the equation written another way, and I made a note of this Huh? moment.

Huh moment: Why is it named that?

Early in the course, we define a "cost" function which tracks the difference between our predictions and the real value.

clarify formula again

Why not call this difference something normal, like error?

It turns out "cost" is used because later in the course, we have items to minimize (like the number of variables in our model) which are not directly related to the error. The "cost" captures things outside the model, like the complexity we have. (If two models make equally accurate predictions, prefer the simpler one.)

Ah, "cost" can include fuzzier concepts. (I'd still prefer that laid out up-front.)

annotate formula with aha moment

Aha moment: Summarize the Course in Plain English

As I go through the course, I have a plain-English definition in mind. What's it all about?

Machine Learning: Create models with Linear Algebra, then improve them with Calculus.

  • Linear Algebra lets us use many (tens, hundreds, thousands) of variables in a "math spreadsheet".
  • Calculus lets us improve our spreadsheet via feedback on how well it's working. Using functions like e^x, ln(x), x^2, etc. make it easy to take derivatives. Absolute value, if/then statements, etc. aren't easy to work with.

Now my thinking becomes: What types of predictive models can I make? If Linear Algebra can describe it, let's use it.

The Result: Notes You'll Actually Enjoy Reading

After the course is done, you're left with a set of notes that make sense to you: the Ahas, Huhs, and other gotchas. (This website is a running collection of mine.)

Future learning gets that much easier. Remember how you were confused about a topic a few years ago? Well, let's read the explanation you wrote to yourself on how to overcome it. Over time you build up a massive collection.

Other tips:

  • Embrace your confusion. The hesitation you feel when you see a formula is ok. Try to break down each part of the equation, ask what it means, make note of what is confusing and return over time. Every positive sign, every variable, why are they there?

  • It's ok to forget things - I do all the time. I just want a list of intuitions to load up when needed. Often a single phrase or diagram will bring it all back.

  • These notes are meant for you. Make them fast and quick. (My notes eventually become articles, but they stay informal and for my own use till then.)

  • The textbook already exists. Don't simply copy what the teacher/book said, add what you need to make it clear.

Show me your Aha! moments

This course is among the most fun I've had -- this is what learning should feel like, exploration with constant refinement. I'm curious to see if this approach helps you too.

For your next course, try keeping your notes in a single Google doc. Write down your Aha! and Huh? moments. Send me a link and I'll add them to this list:

I'm curious to see what works for you, feedback is always welcome.

Happy math.

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Empathy-Driven Mathematics

Charades is probably my favorite game. On the surface it's about communication, sharing an idea through a limited medium (gestures).

But it's really about empathy and reading your audience. What does the other person know? Are they getting lost? Are they having fun? (Am I having fun?)

What looks like a communication obstacle to an alien observer is an enjoyable experience for the human participants. Sure, there's an idea to convey, but maybe there's a clever, funny, or astoundingly simple way to convey it. Aha!

Math teaching should be the same: convey ideas with empathy for your audience.

Math Empathy Checklist

I use the ADEPT method to remind me of what helps me learn: an Analogy, Diagram, Example, Plain English, Technical Definition.

adept method

But when sharing a math idea, I have a different mental checklist. No convenient acronym, just a list of questions to ponder:

How long did the idea take to be discovered? Accepted?

If an idea was debated for centuries before being accepted, shouldn't that be taught?

Sure. Ok. How many of you know that negative numbers were called numeri absurdi? Only accepted (in the West) in the 1800s?

When we have struggles with new concepts (like imaginary numbers, also considered absurd), reference similar struggles in the past. Hey, you're confused? Good. So was everyone else, and here's how we resolved it.

How long did you take to internalize the idea?

Hey you (yes you, the teacher) -- what struggles did you have when learning?

Did imaginary numbers click instantly, without doubt? Did the Fourier Transform just snap into place on your first reading?

(You'd think so, given the unblinking, matter-of-fact treatment in most lessons. Argh!)

If you, the teacher, struggled with an idea, don't hide it: what tripped you up, how did you resolve it, and what issues do you still have?

I needed simulations before I understood the Fourier Transform: playing around with them made it click. Instead of writing down the definition, share the "behind-the-scenes" of what helped.

How comfortable are students asking questions?

Learning is a back-and-forth process. If students don't have questions, they either understood it perfectly, or they are scared/uninterested.

In charades, we can easily see if the other player is confused or having a good time.

Are we trying to be defensive, or helpful?

Academic writing is a bomb shelter, built to be defended from critics. Stable, rock-solid, but not welcoming.

math defense vs approachability

I'd prefer to make a beach bungalow you look forward to visiting. Yeah, the banana-leaf roof is leaky, and no, Dwight, it cannot withstand an aerial assault from AGM-114 Hellfire missiles. But we'll have a great time all the same.

Lessons barricaded with prefaces and caveats indicate you are protecting yourself, not trying to be helpful. (If students began Calculus without a month studying limits, they might (gasp) not have a rigorously defensible understanding on Day 1!)

At some point you reinforce the bungalow, don't start there.

Goal: Are we making students awesome?

Make your students awesome. I want readers to learn things in minutes that took me a decade to untangle. (Kathy Sierra has a great talk about making users awesome.)

Giving impressively rigorous definitions on day 1 doesn't make students awesome. Ignoring historical and personal confusion doesn't make students awesome. Organized chapters of theorem/proof/exercise doesn't make students awesome.

Share what actually worked, in a way you would have liked to see it.

Happy math.

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Learning to Learn: Math Abstraction

We simplify complex ideas to understand them. When working well, math makes things simpler. (Occasionally the medicine is worse than the disease.)

Let's change the generic "Math teaches you to think" to a more specific "Math helps us simplify ideas". We hide detail after detail to reveal an essential truth.

Is this style of thinking necessary? Required for survival? Usually not. But it's often interesting.

What's the simplest drawing you can recognize as a face? What's the simplest joke that's still funny? The simplest exercise that grows a muscle? Would knowing that improve your art, humor, or fitness?

Abstracting Counting

What's in this picture?

lions

A computer says "millions of pixels" and you say "three lions". In seconds you threw away countless details to reveal a deeper insight.

What happened? We abstracted the scenario into something simpler.

math abstraction examples from image to number

  • Remove background from foreground
  • Remove differences between each animal
  • Remove "animal-ness" (treat lions as generic "lines")
  • Remove need to count objects with literal lines
  • Remove need to specify a fixed number ("n happens to be 3 today")

We tend to call the steps we're explicitly aware of "math". Once it becomes natural, it's just "effortless seeing". (Wouldn't it be nice to move more concepts into the "effortless" category?)

Abstracting Learning

Let's try the "math simplification" on a bigger idea: learning.

What does learning involve? At its bare essence, what do we need?

Learning = Insight + Enthusiasm

Insight (for me) comes from Analogies, Diagrams, Examples, Plain English descriptions, and Technical definitions. (Read more about the ADEPT method)

Enthusiasm comes from humor, warmth, empathy, and being treated like a human (not math robot).

A good lesson has both. But wait: is enthusiasm enough by itself? Hrm. Maybe it's better written:

Learning = Insight * Enthusiasm

  • If you have 0 for either, you aren't really learning.
  • "Negative insight" is learning something false.
  • "Negative enthusiasm" is hating something, even to the point of discouraging others.
  • "Negative insight with negative enthusiasm" could be discouraging others from learning something false (which is good, right?).

This is just playing with words and pseudo-equations. Sure. But seeing how enthusiasm impacts education reveals a truth: an educational experience can become negative when enthusiasm points the wrong way.

The equations above don't have to be "right". They're helping us work through an idea. The math approach is to isolate the key factors and figure out how they're related.

Abstracting Technology

For something like a car, the key elements seem to be:

Car = Propulsion * Control

Traditionally, the details of propulsion involve a gas engine, and the details of control require a human driver. But we're interested in abstraction: are these details we can hide?

Maybe propulsion can be electric. Maybe control can be from a computer. A self-driving electric car satisfies the essence of the equation with different details. (Just like 1 apple + 2 apples = 3 apples works as well as 1 lion + 2 lions = 3 lions.)

Asking the right question is difficult, and critical. For this problem, what are the essential variables? What counts, and what can be thrown away?

Abstracting Programming

A lot of people argue that "math helps your programming". Yes, but not in the way you think.

Most programmers don't use anything beyond algebra and basic statistics. (Yes, yes, if you're working on a video game physics engine you can sit down.)

The key lesson from math is how it abstracted the vast complexity of the world. Here are a few fundamental types of "quantity":

  • integers (whole numbers)
  • floats (decimal numbers)
  • hexadecimal numbers (whole numbers with a simpler way to use powers of 2)
  • null (an unset number, different from zero)

Programmers don't need math skills so they can crank through arithmetic. They need math to see examples of the world getting simpler.

Any piece of data (text, images, video, etc.) can be expressed as a giant list of numbers, a combination of the above elements. That's pretty simple.

What other metaphors from math (functions, structure, change, chance) can help us simplify our code?

The Ladder of Abstraction

Bret Victor has a wonderful essay on the Ladder Of Abstraction.

If a new concept is difficult for me, I wonder if I'm at the right level of detail. There's no all-purpose answer like "less detail is better". Sometimes you're staring at your feet and need to zoom out, sometimes you're in the clouds and need to zoom in.

Analogies, Diagrams, Examples, Plain English and Technical definitions, throw them at the wall and trial-and-error a way to better understanding. Like getting an eye exam, move closer or further from an idea until it snaps into focus.

Happy math.

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Learning Tip: Fix the Limiting Factor

What's a tough concept you finally figured out? (For me, it was the imaginary numbers. I'll never forget that Aha! moment.)

Ok. For that difficult concept, what finally made it click? It's usually:

  • An analogy
  • A diagram
  • An example
  • A friendly, plain-English description

Rarely is it because we're lacking:

  • A technical definition
  • A new technology
  • A gamified incentive
  • "more time" (I lack motivation, not time)

The limiting factor -- the thing holding me back -- is how I approach a concept.

The Roman Numeral Problem

Imagine you're teleported to a Roman classroom. Kids -- heck, the adults -- are struggling with multiplication. (IV times VII is really hard!)

What do you fix: Flip the classroom? Gamify things? Invent a printing press to distribute more worksheets?

Helpful, in time. But the first fix should be a simple discussion:

Hey, I'm from the future. Yes, it's pretty nice. But first, we need to fix your concept of a number. Individual lines for digits is cumbersome. Instead, think in groups of ones, tens, hundreds, and so on. Now multiplication is built into your numbers, and arithmetic gets a lot easier. Let me show you...

Boom. The "Roman Numeral Problem" is not fixed with better tech. Just a better understanding.

Ok. Imagine a time traveler (you, 6 months from now) is going to tutor you today. What would they suggest?

  • Imaginary numbers: Don't try to conceptualize "the square root of a negative number". That's really clunky. Think about rotations instead.

imaginary number

  • Trigonometry: Don't try to memorize SOH-CAH-TOA and a mess of equations. Visualize a single diagram and the connections jump out.

trig overall diagram

trig identities

  • Calculus: Don't force yourself through epsilon-delta proofs. Practice breaking things apart, putting them together, and get a feel for the patterns.

calculus circle sphere

  • Fourier Transform: Don't simply memorize the formula. Internalize the notion of a cycle recipe and practice going from a pattern, to the recipe, and back.

fourier constructive interference

Focus on the problems a time traveler would fix first.

While drafting this post, a comment came in:

So, I just started learning about imaginary numbers in math class, and I was so confused. I understood the idea, but not the practical application or really what i was. I am a person who needs to understand a concept fully, I have trouble accepting that i=the square root of -1. I was googling it and I only got more confused. Then, I found your article on imaginary numbers, and all of a sudden, I got it. I could visualize it, even though I have no specific examples of their importance, I can understand why and how they could be important. It clicked. It doesn't make me want to go do my worksheet on adding and subtracting them, but in math tomorrow, I will be a much happier camper. -Abby

It drives me crazy to see endless tutorials on imaginary numbers that don't address the fundamental confusion of how a negative number can have a square root. You can give me all the videos and interactive quizzes you want, I'm not truly learning until you explain the notion of a rotation.

This misprioritization shows up everywhere:

  • Fix the plot, then worry about special effects.

  • Fix the recipe, then worry about decor.

  • Fix the melody, then worry about the instruments.

  • Fix the analogy, then worry about the presentation format.

Identify what's held you back and fix that first.

Happy math.

Appendix: Technology As The Education Fix

Technology helps with certain limitations (access, distribution, cost). But the quality of the source material is still up to us. I'd prefer handwritten letters with Socrates to a HD video conference with Carrot Top.

Veritasium has a great video on these lines ("This Will Revolutionize Education"):

If we think the limiting factor in education is still distribution, we'll focus on technical solutions.

But you know what? We've had Shakespeare online for a few decades now. Modern kids must be poetry experts because of free access to quality literature, right?

It's not an access problem any more. It's a motivation, interest, enthusiasm, understanding-what's-actually-going-on problem. Let's fix that first.

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Honest and Realistic Guides for Learning

Summary: Check out the Calculus Learning Guide. I'm sharing an honest, realistic learning plan for what helped me enjoy the subject.

calculus-learning-guide-screenshot

If you want a path that doesn't expect perfect motivation, shares insights in minutes (not weeks), and aims for lifelong insight, this guide is for you.

First Principles of Education

My learning strategy is to ask honest (sometimes uncomfortable) questions about what's really working.

No games, no kidding ourselves, just:

  • Did the concept I'm learning click?
  • If not, can I find a better explanation? (And let's share it.)

Here's my wishlist for a learning guide. Elon Musk talks about thinking from first principles, starting with fundamental truths and working forward from there1. Who cares what's being done now, what's our goal?

Principle: Avoid teaching hatred of a subject

Priority #1 for any class is: Do not create hate for the subject.

Imagine 99% of people in a skiing class never ski again. They cringe at the thought. We wouldn't console ourselves thinking "Oh, skiing teaches important physical skills that apply to other fields." We'd think "That skiing class is awful and needs to change."

Sure, not everyone will love skiing (or cooking, or math), but they shouldn't detest it. Temporary understanding is not worth permanent aversion.

So, what Calculus introduction made me excited to learn more?

For me, it was seeing how patterns can be cleverly split and re-assembled:

Most courses march you through weeks of theory "appreciate" these diagrams in week 11. Ugh. The big picture helps me appreciate the details, not the reverse.

Principle: Give Realistic Advice

A typical discussion:

"I want to learn Calculus. What should I do?"

"Here's a [full book/course/MOOC]. It's months of effort, I didn't do it myself, but here you go."

In other words, "go the library and read for 100 hours". The real question:

I'm interested in the subject. Is there a plan that worked for you?

Motivation is limited. Traditional classes "work" -- because students are under immense pressure to finish (tuition, peer pressure, fear of not graduating).

Online courses without this pressure have single-digit completion rates. We can pretend students "got something" from the experience, just like you "got something" from a movie you walked out of. We can't change the goalposts to "something is better than nothing" halfway through.

Realistic advice on what worked with my limited motivation (even as a math hobbyist!) is:

  • Get an Aha! moment in minutes that motivates me to keep going (a cool diagram, example, or simulation).

  • Take a progressive journey where even if I stop after an hour, I have some helpful insights (vs. an hour of stretching in the parking lot).

  • Maintain a desire to revisit the subject by having an approachable, gentle introduction. I'll then keep coming back to fill in gaps over time.

Principle: Don't Ignore Difficulties

For fun, find a lesson on imaginary numbers.

  • Does it acknowledge negative numbers were also distrusted?

  • Is the name "imaginary" described as an insult, given by people who didn't understand the concept?

  • Does the teacher mention their own confusion? (Or did imaginary numbers just click?)

  • Is there a real-world application? (If not, is this because it truly doesn't exist, we haven't tried to look, or it isn't important for learning?)

This type of lesson is a giant pet peeve. The flow is "Here's a confusing concept. I was confused myself, but I won't tell you that. Memorize the definition, apply it in these practice problems, and we'll call it a day."

Argh, this drives me nuts. It reinforces the stereotype that math class is a game of moving symbols around. (This symbol multiplied by this other symbol makes -1. Tada!)

It's ok to lack an intuition; I lack it for most things. But hiding our initial confusion implies the subject isn't confusing.

Principle: Expect to get it wrong sometimes

There's a common trope of the smart-aleck student trying to "outsmart" the teacher. Do basketball players try to "outsmart" their coach?

The flawed assumption is teachers must be some omniscient authority giving you access to precious knowledge. The knowledge is out there, it's not like the teacher invented the math herself. Instead, imagine a coach who is trying to improve your understanding.

Coaches can be wrong, sure. But they've seen many struggle with the same issues you're facing, and are trying to help. It's ok if Lebron James can dunk better than his coach.

The math may be perfect and unchanging, but the way it's taught is not. Let's make it easy to improve lessons and not expect perfection the first time.

Principle: We have Different Goals

Most courses assume you want mastery of the subject. That's fine, but is it necessary?

There are several levels of music understanding:

  • Intuitive Appreciation: Just enjoying the music.

  • Natural Description: Humming a tune you heard or made up.

  • Symbolic Description: Reading and writing the sheet music.

  • Theory: Explaining how harmonies work, why minor scales are somber, etc.

  • Performance: Playing the official instruments.

In language learning, there is an ILR scale from no profiency to native fluency. Not everyone studying Calculus needs to become Isaac Newton. Can we have a path that goes as far as we need?

An Honest and Realistic Learning Plan

Combining these insights, I've made a Calculus Learning Guide.

calculus-guide-detail

The principles, as I tried to apply them:

  • It's honest. It's the explanation that actually inspired me, not the theoretical explanation that requires weeks of discipline for some future payoff.

  • It acknowledges limited motivation. How far can you get in 1 minute? 10 minutes? An hour? Pretty far, I think. And getting a win in 10 minutes means you'll come back for more.

  • It's updatable. With lessons based primarily on text, we can easily update, re-arrange, add, edit, fix. Other formats are essentially a bet we got it right the first time.

  • It acknowledges levels of understanding. Most people just want an appreciation for Calculus. Technical performance is a goal we can separate, organize, and build a path to.

  • I eat the veggies myself. This guide has "gut checks" like "Can I describe an integral in everyday terms?" and "Can I derive the product rule on my own?". This is how I actually refresh my Calculus understanding.

In my ideal world, every Wikipedia topic would have a guide that took you from the 1-minute version to a full technical understanding. Go as far as you wish, make meaningful progress at each step, and have fun along the way.

Happy math.


  1. Musk mentions not "reasoning by analogy", or assuming a conclusion is true based on what happened in another scenairo. This is different from "understand by analogy", getting the gist of an idea and then working to the technical version. The analogy is a raft to cross the river, to be left behind once you're on land. 

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Avoiding The Adjective Fallacy

You’re reading this, so I’ll assume your English is pretty good. What’s wrong with these phrases?

  • Old little lady
  • Red big dog
  • Vietnamese spicy food

Do you have a logical reason for why they sound strange? Or are they just off?

You probably didn’t think, “In 3rd grade I mastered the Royal Order of Adjectives:

  1. Determiner
  2. Observation
  3. Size
  4. Shape
  5. Age
  6. Color
  7. Origin
  8. Material
  9. Qualifier

… and upon applying them, noticed several errors. Old little lady is incorrect because rules #3 and #5 are swapped — a childish mistake, really. The next…”

Ugh. Describing Gran Gran isn’t a logic puzzle. But guess what students learning English are taught?

royal order of adjectives

Even as a native speaker, could you construct this chart? Is this how you’d teach someone English?

The Adjective Fallacy is trying to learn by mastering the formal rules. Just because a concept can be rigorously defined doesn’t mean we should study it that way.

We didn’t become good at English by studying a chart: we developed an ear for the language and know how it should sound. And “old little lady” sounds off.

Similarly, getting good at math doesn’t mean marching through a gauntlet of rules on every problem. It’s having a native speaker’s feeling about what works or doesn’t.

“303 x 13 = 5074” looks strange, but not because we computed the left-hand side. It’s weird because odd numbers can’t multiply to become even (intuition). The last digit of the result should be 3×3= 9. 5074 is too large, since 300 x 10 (similar numbers nearby) is only 3000. Our Spidey Sense is blaring that the computation looks wrong.

My learning goal is knowing enough to make rough predictions on my own. I want a horse sense for algebra, calculus, trig, and even imaginary exponents, without scurrying off to apply an equation.

Rules aren’t inherently bad: they summarize, resolve ambiguous cases, and help us practice our weak spots. The question is how much to use them when starting off.

Learn enough rules to get started – don’t attempt to master them from the outset. See examples in a larger context and let the pattern-matching machinery of your brain get to work.

Learning Math

Math is a language too. Here’s a gut check: Would my current math study technique have helped me learn English?

If an English class spent a month on the adjective chart we’d have a talk with the teacher. But a Calculus class that spends weeks on the formal theory of limits is typical. Can we admit that studying this much detail, this early, doesn’t build fluency?

Pondering that question made me realize I had large gaps in trigonometry and calculus. I could only describe concepts using the adjective chart I’d memorized with a furrowed brow. (I’ll describe my grandma, just give me a minute!)

Enough was enough: embrace approaches that actually help you, like seeing the big picture first. In Calculus, that might mean seeing an integral in the first lesson:

calculus disc rings

calculus unroll rings

That’s what Calculus does: break a shape into pieces (the derivative), and glue it together in various ways (the integral). If you like this style of teaching, check out the full Calculus series.

A typical calculus syllabus covers integrals in week 12, after months of “building a foundation”. Better not use a complete sentence until we’ve studied adjectives, nouns and verbs separately, right? (My hand wringing could solve the energy crisis.)

The path to understanding isn’t always the most structured.

Happy math.

Update: This concept is called tacit knowledge, or “we know more than we can tell” (Michael Polanyi). Tacit knowledge is acquired through experience, and complements the explicit knowledge written as rules.

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Learning To Learn: Intuition Isn’t Optional

My learning progress skyrocketed after adopting a new standard: Intuition Isn't Optional.

Imagine a chef who follows a new recipe to the letter. No matter how it looks, no matter the reviews the recipe has, if the dish doesn't taste good we know something is wrong. A sense of taste is the ultimate cooking tool.

When learning, we defer to external indicators (tests, teachers) to inform us we've learned something. External standards are made to be objective and easily-verified (Did you pick the correct answer?), but the important, subjective question is how well a concept sits in your mind. Did you actually experience it?

My checklist of truly learning a topic means it is:

  • Understandable: Did I have an aha! moment? Can I explain the concept in simple language? Does it connect to other topics I know?

  • Memorable: Do I have an analogy, diagram, or example that will stick with me for months or years?

  • Enjoyable: Do I want to revisit or use this knowledge? Don't study literature in a way that makes you hate reading.

That's my current definition of "intuitive understanding", and for subjects I care about, I keep digging until I have all three aspects.

It's ok to take your time (calculus took years to become enjoyable) and it's ok to not care about everything equally (biology isn't particularly compelling for me). I firmly believe any subject can become intuitive if I put in the effort to find analogies, diagrams, examples, plain-english descriptions, and technical details (the ADEPT method).

So, how do you set your own learning standard?

Step 1: Study Famous Learners

Let's not recreate the wheel: famous learners have already described their thinking process, which we can adopt. It's not about memorizing Einstein's Theory of Relativity, it's about internalizing the mindset that could lead to that idea.

Here's a few viewpoints that resonated for me:

"Education is what remains after one has forgotten what one has learned in school." —Albert Einstein

“The only real valuable thing is intuition.” —Albert Einstein

  • True learning goes beyond memorized facts. While I can forget the equation of a circle, I can't forget that it's round. And knowing it's perfectly round quickly leads me back to the equation.

"The noblest pleasure is the joy of understanding." —Da Vinci

  • True understanding implies joy. And practically, you'll only continue studying what you like.

"To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.” —George Póyla

“Education is the kindling of a flame, not the filling of a vessel.” —Socrates

  • We aren't robots, and we should embrace the subjective aspects of learning. A teacher's goal goes beyond knowledge-transfer to enjoyment-transfer.

The Humane Representation of Thought from Bret Victor

  • There are deeper, richer levels of understanding than what's traditionally used. Explore a higher standard.

"I think most people can learn a lot more than they think they can. They sell themselves short without trying. One bit of advice: it is important to view knowledge as sort of a semantic tree — make sure you understand the fundamental principles, ie the trunk and big branches, before you get into the leaves/details or there is nothing for them to hang on to." —Elon Musk

  • Your own standards greatly influence your understanding. External tests won't check if facts are comfortably connected.

I have a larger collection of quotes that help align my thinking.

Step 2: Ask Questions That Check Your Standards

After rummaging through quotes that resonate, build a set of questions that capture your standard. For me, it became:

  • Do I have a visceral, ingrained analogy? Can it help solve problems?
  • Can I explain the concept to others? Do they want to explain it to their friends afterwards?
  • Will I remember the essential idea after a few months or years?
  • Can I find something to enjoy in the topic? Will I return after I inevitably forget 95% of it?

Questions seem to prompt more interest than a statement: "Do I have an analogy?" vs. "I must have an analogy".

With this approach, strange corners of math I didn't previously enjoy (like Euler's Formula) became mysteries to solve: what is the insight here? Can I express it in a plain-English sentence? (Here's a shot: Continuous rotation means you're moving in a circle.)

Setting new standards helps take control of your education and overcome longstanding demons.

When people say "I hate math" I doubt they actually hate numbers (arithmetic), patterns & relationships (algebra), or shapes (geometry). They hate lessons that don't contain insight, enjoyment, and basic human empathy. It's fine to be disinterested in Ancient Egyptian Civilization, but hate comes from getting lost on a tour and spending the night near a sarcophagus.

These are the questions that helped me: what are your standards for learning?

(Thanks to Scott Young, Uri Bram, and Tom Miller for brainstorming ideas.)

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Learn Difficult Concepts with the ADEPT Method

After a decade of writing explanations, I’ve simplified the strategy I use to get new concepts to click.

Make explanations ADEPT: Use an Analogy, Diagram, Example, Plain-English description, and then a Technical description.

ADEPT method of learning

Here’s how to teach yourself a difficult idea, or explain one to others.

Analogy: What Else Is It Like?

Most new concepts are variations, extensions, or combinations of what we already know. So start there!

In our decades of life, we’ve encountered thousands of objects and experiences. Surely one of them is vaguely similar to this new topic and can be the starting point.

Here’s an example: Imaginary numbers. Most lessons introduce them in a void, simply saying “negative numbers can have square roots too.”

Argh. How about this:

  • Negative numbers were distrusted until the 1700s: How could you have less than nothing?
  • We overcame this by realizing numbers could exist on a number line, allowing us to move forward or backward from zero.
  • Imaginary numbers express the idea that we can move upwards and downwards, or rotate around the number line.

Instead of just going East/West, we can go North/South too – or even spin around in a circle. Neat!

Analogies are fuzzy, not 100% accurate, and yet astoundingly useful. They’re a raft to get across the river, and leave behind once you’ve crossed.

Diagram: Engage That Half Of Your Brain

We often think diagrams are a crutch if you aren’t macho enough to directly interpret the symbols. Guess what? Academic progress on imaginary numbers took off only after the diagrams were made!

Favor the easiest-to-absorb explanation, whether that comes from text, diagram, or interpretative dance. From there, we can work to untangle the symbols.

So, here’s a visualization:

imaginary numbers

Imaginary numbers let us rotate around the number line, not just move side-to-side.

Starting to get a visceral sense for what they can do, right?

Half our brain is dedicated to vision processing, so let’s use it. (And hey, maybe for this topic, twirling around in an interpretative dance would help.)

Example: Let Me Experience The Idea

Oh, now’s our chance to hit the student with the fancy terminology, right?

Nope. Don’t tell someone the way things are: let them experience it. (How fun is hearing about the great dinner I had last night? The movie you didn’t get to see?)

But that’s what we do for math. “Someone smarter than you thought this through, found out all the cool connections, and labeled the pieces. Memorize what they discovered.”

That’s no fun: let people make progress themselves. Using the rotation analogy, what happens after 4 turns?

imaginary number rotation

How about 2 turns? 4 turns clockwise?

Plain-English Description: Use Your Own Words

If you genuinely experienced an idea, you should be excited to describe it:

  • Imaginary numbers seem to point North, and we can get to them with a single clockwise turn.
  • Oh! I guess they can point South too, by turning the other way.
  • 4 turns gets us pointing in the positive direction again
  • It seems like two turns points us backwards

These are all correct conclusions, just not yet written in the language of math. But you can still reason in plain English!

Technical Description: Learn The Formalities

The final step is to convert our personal understanding to the formal notation. It’s like sharing a song you’ve made up: you can hum it to yourself, but need sheet music for other people to use.

Math is the sheet music we’ve agreed upon to share ideas. So, here’s the technical terminology:

  • We say i (lowercase) is 1.0 in the imaginary dimension
  • Multiplying by i is a 90-degree counter-clockwise turn, to face “up” (here’s why). Multiplying by -i points us South
  • It’s true that starting at 1.0 and taking 4 turns puts us at our starting point:

\displaystyle{1 * i * i * i * i = 1 }

And two turns points us negative:

\displaystyle{1 * i * i = -1 }

which simplifies to:

\displaystyle{i^2 = -1}

so

\displaystyle{i = \sqrt{-1}}

In other words, i is “halfway” to -1. (Square roots find the halfway point when using multiplication.)

Starting to get a feel for it? Just spitting out “i is the square root of -1” isn’t helpful. It’s not explaining, it’s telling. Nothing was experienced, nothing was internalized.

Give people the chance to make an idea their own.

The Mental Checklist

I used to be satisfied with a technical description and practice problem. Not anymore.

ADEPT is a checklist of what I need to feel comfortable with an idea. I don’t think I’ve actually learned a topic unless I have a metaphor that ties everything together. Here’s a few places to look:

Unfortunately, there aren’t many resources focused on analogies, especially for math, so you have to make your own. (This site exists to share mine.)

Modifying the Learning Order

It seems logical to assume we can present facts in order, like transmitting data to a computer. But who actually learns like that?

I prefer the blurry-to-sharp approach to teaching:

baseline vs progressive learning

Start with a rough analogy and sharpen it until you’re covering the technical details.

Sometimes, you need to untangle a technical description on your own, so must work backwards to the analogy.

Starting with the technical details:

  • Can you explain them in your own words?
  • Can you solve an example problem, describing the steps in your own words?
  • Can you create a diagram that represents how the concept fits together for you?
  • Can you relate the concept to what you already know?

With this initial analogy, layer in new details and examples, and see if it holds up. (It doesn’t need to be perfect, but iterate.)

If we’re honest, we’ll admit that we forget 95% of what we learn in a class. What sticks? A scattered analogy or diagram. So, make them for yourself, to bootstrap the rest of the understanding as needed.

In a year, you probably won’t remember much about imaginary numbers. But the quick analogy of “rotation” or “spinning” might trigger a flurry of recognition.

The Goal: Explanations That Actually Work

I’m wary of making a contrived acronym, but ADEPT does capture what I need to internalize a new concept. Let’s stop being shy about thinking out loud: does a fact-only presentation really work for you? What other components do you need? I have a soft, squishy brain that needs the connecting glue, not just data.

Scott Young uses the Feynman Technique to explain concepts in everyday words and work backwards to an analogy and diagram. (Richard Feynman was a world-class expositor and physicist, and one of my teaching heroes.)

Tom Roth wrote a nice summary for ADEPT, Feynman Technique, and others.

Beyond any technique, raise your standards to find (or create) explanations that truly work for you. It’s the only way to have concepts stick.

Happy math.

Bonus: BE ADEPT

“BE” is a nice prefix for the style to use when teaching:

  • Brevity is beautiful.

  • Empathy makes us human. Use your natural style, relate to common experience, and anticipate questions in your explanation.

I’ve yet to complain that a lesson respected my time too much, or related too well to how I thought.

Appendix: ADEPT Summaries

ADEPT is like a nutrition label for an explanation: what are the key ingredients?

Concept Euler’s Formula
Analogy Imaginary numbers spin exponential growth into a circle.
Diagram Learn Difficult Concepts with the ADEPT Method
Example Let’s figure out the value of 3^i. (It’s on the unit circle.)
Plain-English Raising an exponent to an imaginary power spins you on the unit circle. The same destination can be written with polar (distance and angle) or rectangular coordinates (real part and imaginary part).
Technical \displaystyle{e^{ix} = \cos(x) + i\sin(x)}

Concept Fourier Transform
Analogy Like filtering a smoothie into ingredients, the Fourier Transform extracts the circular paths within a pattern.
Diagram Smoothie being filtered: Learn Difficult Concepts with the ADEPT Method
Example Split the sequence (4 0 0 0) into circular components: Learn Difficult Concepts with the ADEPT Method
Plain-English / Technical Learn Difficult Concepts with the ADEPT Method

Learn Difficult Concepts with the ADEPT Method


Concept Distributed Version Control
Analogy Distributed Version Control is like sharing changes to a group shopping list with your friends.
Diagram / Example Learn Difficult Concepts with the ADEPT Method
Plain-English We check out, check in, branch, and share differences (“diffs”).
Technical git checkout -b branchname
git diff branchname

Combine ingredients with your own style. Steps might merge, but shouldn’t be skipped without a good reason (“Zombies coming, no time for biochem, use this serum for the cure.”). The site cheatsheet has a large collection of analogies.

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Learning math? Think like a cartoonist.

What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understand Ed O’Neill:

ed oneill original

A literal-minded artist might portray him like this:

realistic image

While the technical skill is impressive, does it really capture the essence of the man? Look at his eyes in particular.

A cartoonist might draw this:

cartoon image

Wow! The cartoonist recognizes:

  • The unique shape of his head. Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect.

  • The wide-eyed bewilderment. The whites of his eyes, the raised brows, the pursed lips – the cartoonist saw and amplified the emotion inside.

So, who really “gets it”? It seems the technical artist worries more about the shading of his eyes than the message they contain.

Numbers Began With Cartoons

Think about the first numbers, the tally system:

I, II, III, IIII …

Those are… drawings! Cartoons! Caricatures of an idea!

They capture the essence of “existing” or “having something” without the specifics of what it represents.

Og the Cavemen Accountant might have tried drawing individual stick figures, buffalos, trees, and so on. Eventually he might realize a shortcut: draw a line and call it a buffalo. This captures the essence of “something is there” and our imaginations do the rest.

Math is an ongoing process of simplifying ideas to their cartoon essence. Even the beloved equals sign (=) started as a drawing of two identical lines, and now we can write “3 + 5 = 8” instead of “three plus five is equal to eight”. Much better, right?

So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in after we’ve seen the essence.

Technically Correct: The Worst Kind Of Correct

We agree that multiplication makes things bigger, right?

Ok. Pick your favorite number. Now, multiply it by a random number. What happens?

  • If that random number is negative, your number goes negative
  • If that random number is between 0 and 1, your number is destroyed or gets smaller
  • If that random number is greater than 1, your number will get larger

Hrm. It seems multiplication is more likely to reduce a number. Maybe we should teach kids “Multiplication generally reduces the original number.” It’ll save them from making mistakes later.

No! It’s a technically correct and real-life-ily horrible way to teach, and will confuse them more. If the technically correct behavior of multiplication is misleading, can you imagine what happens when we study the formal definitions of more advanced math?

There’s a fear that without every detail up front, people get the wrong impression. I’d argue people get the wrong impression because you provide every detail up front.

As George Box wrote, “All models are wrong, but some are useful.”

A knowingly-limited understanding (“Multiplication makes things bigger”) is the foothold to reach a more nuanced understanding. (“People generally multiply positive numbers greater than 1, so multiplication makes things larger. Let’s practice. Later, we’ll explore what happens if numbers are negative, or less than one.”)

Takeaways

I wrap my head around math concepts by reducing them to their simplified essence:

  • Imaginary numbers let us rotate numbers. Don’t start by defining i as the square root of -1. Show how if negative numbers represent a 180-degree rotation, imaginary numbers represent a 90-degree one.

  • The number e is a little machine that grows as fast as it can. Don’t start with some arcane technical definition based on limits. Show what happens when we compound interest with increasing frequency.

  • The Pythagorean Theorem explains how all shapes behave (not just triangles). Don’t whip out a geometric proof specific to triangles. See what circles, squares, and triangles have in common, and show that the idea works for any shape.

  • Euler’s Formula makes a circular path. Don’t start by analyzing sine and cosine. See how exponents and imaginary numbers create “continuous rotation”, i.e. a circle.

Avoid the trap of the guilty expert, pushed to describe every detail with photorealism. Be the cartoonist who seeks the exaggerated, oversimplified, and yet accurate truth of the idea.

Happy math.

PS. Here’s my cheatsheet full of “cartoonified” descriptions of math ideas.

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Math As Language: Understanding the Equals Sign

It’s easy to forget math is a language for communicating ideas. As words, “two and three is equal to five” is cumbersome. Replacing numbers and operations with symbols helps: “2 + 3 is equal to 5”.

But we can do better. In 1557, Robert Recorde invented the equals sign, written with two parallel lines (=), because “noe 2 thynges, can be moare equalle”.

“2 + 3 = 5” is much easier to read. Unfortuantely, the meaning of “equals” changes with the context — just ask programmers who have to distinguish =, == and ===.

A “equals” B is a generic conclusion: what specific relationship are we trying to convey?

Simplification

I see “2 + 3 = 5” as “2 + 3 can be simplified to 5”. The equals sign transitions a complex form on the left to an equivalent, simpler form on the right.

Temporary Assignment

Statements like “speed = 50” mean “the speed is 50, for this scenario”. It’s only good for the problem at hand, and there’s no need to remember this “fact”.

Fundamental Connection

Consider a mathematical truth like $a^2 + b^2 = c^2$, where a, b, and c are the sides of a right triangle.

I read this equals sign as “must always be equal to” or “can be seen as” because it states a permanent relationship, not a coincidence. The arithmetic of $3^2 + 4^2 = 5^2$ is a simplification; the geometry of $a^2 + b^2 = c^2$ is a deep mathematical truth.

The formula to add 1 to n is:

\displaystyle{\frac{n(n+1)}{2}}

which can be seen as a type of geometric rearrangement, combinatorics, averaging, or even list-making.

Factual Definition

Statements like

\displaystyle{e = \lim_{n\to\infty} \left( 1 + \frac{100\%}{n} \right)^n}

are definitions of our choosing; the left hand side is a shortcut for the right hand side. It’s similar to temporary assignment, but reserved for “facts” that won’t change between scenarios (e always has the same value in every equation, but “speed” can change).

Constraints

Here’s a tricky one. We might write

x + y = 5

x – y = 3

which indicates conditions we want to be true. I read this as “x + y should be 5, if possible” and “x – y should be 3, if possible”. If we satisfy the constraints (x=4, y=1), great!

If we can’t meet both goals (x + y = 5; 2x + 2y = 9) then the equations could be true individually but not together.

Example: Demystifying Euler’s Formula

Untangling the equals sign helped me decode Euler’s formula:

\displaystyle{e^{i \cdot \pi} = -1}

A strange beast, indeed. What type of “equals” is it?

A pedant might say it’s just a simplification and break out the calulus to show it. This isn’t enlightening: there’s a fundamental relationship to discover.

e^i*pi refers to the same destination as -1. Two fingers pointing at the same moon.

They are both ways to describe “the other side of the unit circle, 180 degrees away”. -1 walks there, trodding straight through the grass, while e^i*pi takes the scenic route and rotates through the imaginary dimension. This works for any point on the circle: rotate there, or move in straight lines.

euler's formula

Two paths with the same destination: that’s what their equality means. Move beyond a generic equals and find the deeper, specific connection (“simplifies to”, “has been chosen to be”, “refers to the same concept as”).

Happy math.

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Why Do We Learn Math?

I cringe when hearing "Math teaches you to think".

It's a well-meaning but ineffective appeal that only satisfies existing fans (see: "Reading takes you anywhere!"). What activity, from crossword puzzles to memorizing song lyrics, doesn't help you think?

Math seems different, and here's why: it's a specific, powerful vocabulary for ideas.

Imagine a cook who only knows the terms "yummy" and "yucky". He makes a bad meal. What's wrong? Hrm. There's no way to describe it! Too mild? Salty? Sweet? Sour? Cold? These specific critiques become hazy variations of the "yucky" bucket. He probably wouldn't think "Needs more umami".

Words are handholds that latch onto thoughts. You (yes, you!) think with extreme mathematical sophistication. Your common-sense understanding of quantity includes concepts refined over millennia: base-10 notation, zero, decimals, negatives.

What we call "Math" are just the ideas we haven't yet internalized.

Let's explore our idea of quantity. It's a funny notion, and some languages only have words for one, two and many. They never thought to subdivide "many", and you never thought to refer to your East and West hands.

Here's how we've refined quantity over the years:

  • We have "number words" for each type of quantity ("one, two, three... five hundred seventy nine")
  • The "number words" can be written with symbols, not regular letters, like lines in the sand. The unary (tally) system has a line for each object.
  • Shortcuts exist for large counts (Roman numerals: V = five, X = ten, C = hundred)
  • We even have a shortcut to represent emptiness: 0
  • The position of a symbol is a shortcut for other numbers. 123 means 100 + 20 + 3.
  • Numbers can have incredibly small, fractional differences: 1.1, 1.01, 1.001...
  • Numbers can be negative, less than nothing (Wha?). This represents "opposite" or "reverse", e.g., negative height is underground, negative savings is debt.
  • Numbers can be 2-dimensional (or more). This isn't yet commonplace, so it's called "Math" (scary M).
  • Numbers can be undetectably small, yet still not zero. This is also called "Math".

Our concept of numbers shapes our world. Why do ancient years go from BC to AD? We needed separate labels for "before" and "after", which weren't on a single scale.

Why did the stock market set prices in increments of 1/8 until 2000 AD? We were based on centuries-old systems. Ask a modern trader if they'd rather go back.

Why is the decimal system useful for categorization? You can always find room for a decimal between two other ones, and progressively classify an item (1, 1.3, 1.38, 1.386).

Why do we accept the idea of a vacuum, empty space? Because you understand the notion of zero. (Maybe true vacuums don't exist, but you get the theory.)

Why is anti-matter or anti-gravity palatable? Because you accept that positives could have negatives that act in opposite ways.

How could the universe come from nothing? Well, how can 0 be split into 1 and -1?

Our math vocabulary shapes what we're capable of thinking about. Multiplication and division, which eluded geniuses a few thousand years ago, are now homework for grade schoolers. All because we have better ways to think about numbers.

We have decent knowledge of one noun: quantity. Imagine improving our vocabulary for structure, shape, change, and chance. (Oh, I mean, the important-sounding Algebra, Geometry, Calculus and Statistics.)

Caveman Chef Og doesn't think he needs more than yummy/yucky. But you know it'd blow his mind, and his cooking, to understand sweet/sour/salty/spicy/tangy.

We're still cavemen when thinking about new ideas, and that's why we study math.

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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.

Finding Unity in the Math Wars

Finding Unity in the Math Wars

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”.

Finding Unity in the Math Wars

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.

Finding Unity in the Math Wars

(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.

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Learning To Learn: Embrace Analogies

Why do analogies work so well? They’re building blocks for our thoughts, written in the associative language of our brains.

At first, I thought analogies had to be perfect models of the idea they explained. Nope.

“All models are wrong, but some are useful” – George Box

Analogies are handles to grasp a larger, more slippery idea. They’re a raft to cross a river, and can be abandoned once on the other side. Unempathetic experts may think the raft is useless, since they no longer use it, or perhaps they were such marvelous swimmers it was never needed!

Analogies are perfectly fine. But why do they work so well?

Our brains are association machines. Connections, relationships, patterns — we need meaning! Yet we present topics as if we could be programmed with raw information.

Consider the typical language class:

  • Here’s the grammar
  • Here’s the vocabulary
  • Put the vocab in the grammar and go!

We know how well that works. The mistake is thinking direct study of the grammar and vocabulary will build fluency — it’s a tough slog. I suspect a class of 80% speaking, listening, making idioms, building pronunciation and 20% vocabulary/grammar does much better than the reverse.

Start with simple analogies you deeply understand, then attach extra details.

Here’s an example: I can casually describe i (the imaginary number) as the square root of -1 and you can blindly accept it.

But you won’t really believe me until I start down the path of “Hey, numbers can be 2 dimensional, and i is a rotation into the 2nd dimension”. The word “rotation” stretches our brain about what a number could be — the number line may not be the final step. We’re having a real discussion and can start learning!

See, you’re extremely fluent with the idea of a line, and the idea of a second dimension, and we can work “i is a rotation” into that framework. In computer terms: we are programming with the native language of the machine. Our brain thinks with connections, so explain new data in terms of existing connections!

Although a subject can be distilled into rules and facts, drinking this concentrated math isn’t the best way to enjoy it. It’s not how our brains work, and presenting raw data suffers from a painful translation step.

I don’t think of algebra, trig and other math as a table of equations. It’s a web of connections and insights. But why show facts and hope you recreate the mental model in my head, instead of describing it directly?

No, no — let’s have a brain-to-brain. Here’s the analogies in my head, I want you to have them too.

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Intuition, Details and the Bow/Arrow Metaphor

My favorite analogies explain a thought and help you explore deeper truths. Here’s a metaphor that captures my stance on learning:

  • Rote details are arrows, intuition is the bow.

Our goal is to hunt down problems. You can use arrows alone, sure, but intuition is the framework that makes details astoundingly useful. Here’s a few insights I explored over a chat and in email (thanks Jay, Stan, Luke and David!).

Balancing Rigor and Intuition

I hate the false choice between rigor and intuition. We can have both! Each has a role to play:

  • Details (arrows) do the actual work, but are cheap & plentiful
  • Intuition (the bow) is the framework that makes the details effective (in theory, it’s optional; in practice, it’s not)
  • Hunting (effective problem solving) is the ultimate goal: how can the entire system help us?

Yes, you can have details without intuition: it’s chasing a buffalo, waving an arrow overhead. Victories are possible, though exhausting, and the process hardly encourages you to learn more. It’s like memorizing a math proof you don’t understand (honestly, I’d prefer the buffalo).

Intuition without rigor is bad, too: having a tool but never firing it in the real world. We need both: bows AND arrows, not bows OR arrows. (But between the two, give me a single arrow and let me practice my bowmanship).

The Bow in Your Head

Imagine a cavemen who sees me launch an arrow from a cave. The bow is hidden: because he’s only seen spears, he thinks I threw the arrow by hand. Probably with the help of shaman magic.

Our mental bows are similarly hidden, and the way some people use math seems like dark magic. We don’t see their intuition, just the problem being obliterated.

Let’s unlock those insights. My goal is to share the “bows in our heads”, not to show a flying arrow and having you invent the bow that launched it.

Educational Approaches

When teaching, it’s easy to focus on how many arrows you can name, classify, and stick into motionless targets. We pretend it’s preparation for the real world and “learning how to learn”.

I disagree. “Learning how to learn” means mastering a single arrow in a single bow. Truly mastering it. New techniques follow.

I’d rather train an Aboriginal hunter to use a modern rifle than a random person on the street. We laymen (myself included!) have no idea about aiming, tracking, breathing, timing, precision and the intangibles you discover when mastering a tool. I’ve heard about guns my entire life and played dozens of video games: that hunter could still outshoot me within 15 minutes of seeing his first rifle. It’s not the arrows, it’s the bow. That is “learning how to learn”.

In modern times: what’s the point of force-feeding students until they hate learning? Until they never read another classic book? Until they have a life-long aversion to math?

I’d prefer to have “basic” high schoolers that love middle-school algebra vs. “advanced” ones that hate calculus. Because in 5 years the ones who loved algebra will love calculus too.

(Psst… if students truly enjoy math in middle school, they’ll probably enjoy it in high-school, and end up graduating with calculus anyway. But if you start to expect that, you’re back to square one!)

The Dynamite Arrow

A thought: what about a dynamite arrow, the super-useful detail that levels the playing field? Couldn’t that equate the amateur and expert?

Perhaps. But even so, you need a decent shot to make the system effective. It’s no good shooting the dynamite arrow 10 feet, or in a backwards direction. And again: if an amateur was able to get that dynamite arrow, how quickly can the expert get it?

I’m not saying “never get more arrows”. I’m saying it’s pointless to collect more arrows until you can shoot the ones you have.

Finding the Essentials

Quick: You have 2 minutes to explain archery to caveman. Go!

Do you focus on arrow-building technique, chopping tress, finding rocks, etc.? Or do you say “Arrows are made of wood and have feathers to stabilize their flight. That’s enough there. Here’s how to make an awesome bow (longbow, crossbow, compound bow…)”.

Most systems have a “core operating principle”. The rest are details about arrows. For example: Cellular phones talk to a collection of “cells” which hand off the signal between towers as you move. Done. Cells = cellular telephone. GSM, CDMA, & friends are the language each tower understands, and not important to core understanding.

When explaining, look for the bows.

A Sign of Learning

How do you know someone truly enjoyed learning? They start asking archery questions, not arrow questions.

After you get that imaginary numbers are numbers in another dimension, it’s about 10 minutes until you have genuine interest in “Hey… could there be 3d or 4d numbers too?”.

Good luck getting that after teaching “The square root of -1 is i. No, I won’t tell you why, nobody told me why — just memorize it for the test!”.

Understanding is measured by the questions we ask, not the tests we answer.

(Since you’re curious: 4d numbers are called “quaternions” and used to model spinny things in video games. But beyond that, shouldn’t we be using a list or something? What’s that? You want to learn about linear algebra?)

Cheap Entertainment

I’ve always been bothered by “educational games” which are thin veneers on rote memorization. Memorizing the name of every US President doesn’t tell you much about history. Being a spelling bee champion doesn’t make you a good writer. Knowing pi to 1000 places doesn’t mean you understand the concept.

The cheap way to make math “fun” is to make a game of picking up arrows. The real way to make math fun is experience the joy of shooting an arrow on your own.

That said, skill-building games like Math Blaster can be awesome. It’s a matter of picking up the right arrows and not stopping there. A mathematician doesn’t enjoy crunching numbers any more than a writer enjoys conjugating verbs –it’s a necessary step to enjoy the art.

Upgrading Your Bow

We start with the same details, but have different ways of using them. Similar to how rifling (grooves in a barrel to spin a bullet) increases range, what tricks do we have?

I’ve gained immense value from upgrading the bow that holds the Pythagorean Theorem. That “arrow” ($a^2 + b^2 = c^2$) can be launched in so many ways — each year I find a new personal discovery (it’s not about distance; it can apply to any shape; it explains the gradient).

How can you upgrade your understanding to a longbow, a crossbow, a compound bow, an automatic machine-gun bow?

Onwards and Upwards

The “bow and arrow” metaphor keeps giving: I’ve found over a half-dozen interpretations I love, and I’m sure there’s more (share ’em if you’ve got ’em).

In a sentence: The joy and value of learning is in archery, not arrow-finding.

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Brevity Is Beautiful

Brevity is my favorite aspect of effective communication. We’re limited creatures, only able to handle a few thoughts at once — make them count!

Concise writing helps us share ideas, but we hamstring ourselves by trying to appear “substantial”. Let’s figure out how to avoid this trap.

Benefits of Brevity

Concise, efficient writing has non-obvious benefits:

We maximize information density.

We can hold about 7 digits in memory. Given limited room, a few powerful thoughts are better than a single dilute one.

What’s better: “x is the sum of two times y and three times z” or “x = 2y + 3z”?

Concise thoughts are more understandable. (By the way, math used to be written in English, as above. Egads.)

We respect the reader.

Long-winded diatribes are about the author: listen to me and look at what I know. Effective communication is about the reader: I’ve distilled hundreds of pages to these essential insights.

Information is everywhere, and I can eventually understand a topic by reading dozens of mediocre books. But time is limited — give me the source that communicates the most understanding in the least time.

We communicate raw thought.

Writing isn’t about words, it’s about recreating ideas:

  • Idea in my head → words are written → words are read → idea in your head

With good writing we hear the author’s voice, not our own thoughts deciphering their message. The ideal of communicating raw ideas appears in programming, design, art and even humor (“Brevity is the soul of wit”).

Obstacles to Brevity

If brevity is so desirable, why don’t we do it?

Schoolchild Guilt (aka the 10-page paper)

School assignments ask for pages of text, not ideas. The teacher really wants an essay with 3 meaningful insights, but that’s tough to specify. So instead he asks for a 10-pager, hoping some ideas are buried inside.

The assignment is easily gamed: take a few scattered thoughts, bump up the font and margins, and tada, we have 10 pages. We know this isn’t what the teacher wants, but it satisfies the letter of the law.

An analogy: A king secretly wants treasure. He asks his subjects to bring him a ton of dirt each, hoping for gems inside. They do, and on average there’s a single gem in each pile — but the king spends hours clawing through the dirt.

One day a peasant sees a lone gem on the beach. But because the king asked for dirt (he’ll be punished if he only brings a handful of “stuff”), he buries the gem in an enormous pile and delivers that to the king, who spends hours trying to find the jewel.

Is that what the king wanted? We writers are the peasants that bring material for you to sift through!

Getting Our Money’s Worth

Thought experiment: you see two reference books, one at 100 pages and the other at 200. Do you wonder if the smaller book could be concise and well-written, or do you immediately assume “bigger is better” and reach for the tome?

And that’s why publishers pad their books — we reward those with the most words, not the best ones. It’s akin to judging a portrait by how much paint was used, or a song by its length.

Brevity and Substance

My “brevity” means economy of words, saying what’s necessary and no more. “Necessary” could be a paragraph or 50 pages: the key is delivering gems, not dirt. While writing, you'll have a hunch :).

I still struggle to accept it’s ok, nay good, to share a single, concise thought if you think it’s a gem. There's no need to appear substantial.

Does anyone think the 278-word Gettysburg address isn’t meaty enough?

Expert’s Guilt (The sky is not blue)

Brevity’s enemy is an armada of helpful caveats. Quick question: is the sky blue?

Well, it’s black at night. And orange at sunset/sunrise. And grey when cloudy. In fact, it’s more likely to be non-blue than blue!

My goodness, I could never declare “The sky is blue” without a 3-page disclaimer, lest a meteorologist have my head.

No. Writing riddled with caveats is like the “Are you sure? Really sure?” dialogs we hate in software: yes, yes, we get it!

Models are simplifications, we all know this: assume an intelligent reader and don’t encumber your writing to satisfy every critic. Corner cases are exactly that, and should live away from the main text.

Examples of Brevity

I learn by reflecting on great examples — what makes them tick?

Computers and Programming

Ruby has wonderful shortcuts for everyday tasks.

value = parameter || getValue() || "default"

Which means “try to use parameter, then try getValue(), and if all else fails assign a default”. Ruby was the first language I felt I was reading without notational cruft getting in the way.

Kernighan and Ritchie’s The C Programming Language is the gold standard of technical manuals. Concise and useful, it has no desire to satisfy some publisher’s pagecount requirement: “C is not a big language, and it is not well served by a big book.”

Don’t Make Me Think! is an excellent usability guide. The title is the summary: keep things brainlessly easy. The book expands with examples, yet remains brief.

The unix command line (“cat foo.txt | sort | uniq -c | sort -rn”) is wonderfully concise and powerful: it’s hard to express the above more simply (output a file, sort the lines, count the unique ones, and sort again by that count in descending order).

Mathematics

As we saw with English vs. arithmetic, expressive notation helps us focus on the idea being conveyed.

Consider the difference between decimal and Roman numerals: how can you use math when it takes 5 minutes to decode MCMXCVII times XLII? Decimal notation is one of our greatest discoveries.

Quotes

Why do we love quotes? They are distilled thoughts! Great quotes help us experience an idea without getting lost in verbiage.

Some favorites:

  • “I have made this letter longer than usual because I lack the time to make it shorter.” –Blaise Pascal (It’s easier to plop down dirt than to dig through and pull out the gems)

  • “Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts. This requires not that the writer make all his sentences short, or that he avoid all detail and treat his subjects only in outline, but that every word tell.” –William Strunk Jr. (Efficiency is universally appreciated)

Economy of Motion

Great athletes and musicians are efficient. They move less and waste less than the rest of us, and do more with the same amount of time. Concise thoughts require less mental energy to understand.

Headlines

Top 10 lists grab our attention. Why? They imply someone has found the gems: we sifted through dozens of items and are bringing you the best. Unfortunately, these headlines have been abused to mean “Here are 10 random things”.

Cheatsheets

Cheatsheets are pure gems, going from A to B without distraction. The key is knowing the background of your audience. A physics cheatsheet is great for reference, not learning.

Final Thoughts

Reflection helps develop a learning philosophy. I discovered that my fear of not having enough substance was based on measuring dirt. Brainstorming, writing down ideas, and leaving the essentials is more than ok — it’s my ideal.

Remember: is our goal to satisfy a length requirement, impress with our vocabulary, or communicate effectively? Do readers a favor and give ‘em your best gems.

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Learning To Learn: Pencil, Then Ink

I loved drawing as a kid. A recent Aha! was realizing how similar the process of good drawing is to good learning -- they depend on recognizing and mastering underlying structures. My philosophy in 3 words:

Pencil, then ink.

It's simple, perhaps cliched, but powerful. Lee Ames had a great series of books on drawing (Draw 50 animals):

Learning To Learn: Pencil, Then Ink(amazon)Learning To Learn: Pencil, Then Ink(source)

The cover reveals it all. How do you draw an elephant?

  • Pencil the structure using ovals, rectangles, and so on
  • Ink the final result, taking the lines you want
  • Erase the underlying pencil structure, revealing the elephant

Why's this special? The key to learning is understanding the pencil structure, the scaffolding that's not always present in that final, finished elephant. Let's see how this analogy relates to learning.

Tracing Math

Is tracing different from drawing? You bet. Tracing is mimicry -- we don't know why a line is there. We just start in one corner and work our way around. Sure, we might make a pretty elephant -- but can we draw one with a different trunk? Standing on two hind legs? Probably not.

Math is similar: we "teach" by tracing a student through the steps of a proof. But there's an underlying pencil structure that was in the mind's eye of the proof's author that we're not seeing. We're walking the student along the drawing ("Here is the head, here is the trunk, here is the leg") without show the mindset that created the proof ("The head is an oval, connected to a larger oval for the body; the legs are cylinders, which we smooth out.").

If we're lucky, the student generalizes the steps and creates their own pencil structure.

But why? Why do we leave the most interesting part of understanding to private contemplation? I love discovering these "aha" moments that put the result into place -- what's the mental map that made the facts snap together?

When we share insights we can stop "tracing math" and begin drawing on our own. It's way more satisfying.

Creation Vs. Understanding

What's the point of education: the results or understanding?

If the goal are results, then art class should be about using stencils and tracing to make perfect representations. If the goal is understanding, then we should take out the pencils, make our lumpy apples and lopsided bananas and try our hand at still lifes.

It's seldom either-or: we want results and understanding. Unfortunately, we focus on results because they're easier to test (Can you plug X into these formulas and get the right answer?). I'm here to remind us that we need to understand what's happening too.

Rigor and Intuition

I've struggled how to reconcile rigor and intuition -- both have their role, but how do they fit? The drawing analogy captures my feelings:

  • Rigor (permanent inked lines) helps cement ideas after the intuitive pencil structure has been put into place

Focusing on rigor prematurely creates fear and trepidation -- What if I'm wrong? -- and encourages people to trace the inked results instead of learning how to experiment on their own. It makes you think math (or any subject) is something you get right or not at all. Which isn't the case -- many (most?) results have been developed intuitively and cemented later.

Rigor/ink is emphasized because it's the only thing visible; I want to champion the (now-invisible) pencil lines which laid the original groundwork.

The Myth Of The Perfect Formula

Before seeing the Ames book, I thought you drew by starting in one corner and filling in the figure. Some experts may do that (more later), but the "normal way" to draw is by starting with a penciled foundation.

We know writers need drafts. But do we allow drafts in learning? Are we so concerned with reproducing inked results that we discourage or ignore the pencil?

Math developed through wayward paths and missed connections, not always by the smooth progression we see in our classroom syllabus. Showing only the final results makes it appear like it's supposed to progress linearly and unwaveringly. Maybe discussing how zero, negatives, and imaginary numbers were initially distrusted (and embraced) would help us empathize with students embracing the idea.

I chuckle that we "matter of factly" introduce imaginary numbers when the experts of the time had objections. They're difficult, non-intuitive concepts (at first) -- it's ok to admit that we had some rough drafts crumpled in the corner.

Wayward paths can help us better understand the correct ones.

Learning: Seeing the Structure

After my frustrations with learning new concepts, I've taken the philosophy that some structure must exist. When I see a new concept (an inked bird, for example) I really think there must be some collection of shapes that make it make sense.

If I'm having trouble, I blame my approach -- I'm just not seeing the idea in the same way as its inventor. Maybe someone else has written about it, or there's an analogy to another idea.

But what if there's no underlying structure, just a perfect, inked elephant without eraser marks? It's possible.

After you internalize an idea, you start thinking directly in ink. We don't "draw" the letter A in pencil -- we just write A because we're so familiar with it. Practice moves ideas into the "ink-only" stage, which let us work on bigger ideas. For example, you need need to commit arithmetic to "muscle memory" before you can understand algebra. If you can write arithmetic, you can learn to draw algebra. Once you can write algebra, you can draw calculus. And so on -- if you don't get arithmetic, Calculus (with its pencil-lines in algebra) will still look like a jumble.

Revealing Structures

We often look back and add the original pencil lines to finished works:

  • Design Patterns in programming -- abstract ways to find similarities between programming solutions
  • Grid Layout in graphic design: A general structure to organize content
  • Monomyth in storytelling -- a common pattern popular stories take

Sometimes we create "nice-looking elephants" through trial and error. Later on, we realize there's a common structure that can simplify future efforts. True learning is about discovering and exploring these structures, not simply generating the pretty elephants.

Do Experts Teach Best?

Who should teach? The person who just "sees" the elephant from day 1, or the one who learned to break it down and construct it? Imagine taking an art class from Stephen Wiltshire (this panorama of Rome was drawn from memory):

Drawing a city after a helicopter ride is an amazing gift -- but I doubt it's transferrable. He goes far beyond the underlying pencil structure that "regular" artists would need.

Beginners need the pencil marks -- experts who've internalized them sometimes forget that. True learning happens when people can recreate that structure in their minds. When the experts can remember what it was like to not "see it all at once", then real learning can happen.

I want to share the pencil sketches that evolved into the elephant, instead of erasing them and pretending that I, too, can just draw from memory.

Final Thoughts

I'm sure there's more analogies hidden in there somewhere. The process of drawing -- pencil structure, inked result -- captures thoughts about learning that have been rattling in my head.

Don't learn by tracing: find (or invent!) those pencil structures. Seeing the pencil lines makes the idea become your own: you can modify it, combine it with others, or just appreciate it at a deeper level. And that's the joy of learning.

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Developing Your Intuition For Math

Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?

Suppose we want to define a “cat”:

  • Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry…
  • Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics…
  • Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT…

cat analogy

(Illustration courtesy Common Craft)

The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.

Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.

Let’s approach ideas from a different angle. I imagine a circle: the center is the idea you’re studying, and along the outside are the facts describing it. We start in one corner, with one fact or insight, and work our way around to develop our understanding. Cats have common physical traits leads to Cats have a common ancestor leads to A species can be identified by certain portions of DNA. Aha! I can see how the modern definition evolved from the caveman one.

But not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition.

What is a Circle?

Time for a math example: How do you define a circle?

Definitions of a circle

There are seemingly countless definitions. Here’s a few:

  • The most symmetric 2-d shape possible
  • The shape that gets the most area for the least perimeter (see the isoperimeter property)
  • All points in a plane the same distance from a given point (drawn with a compass, or a pencil on a string)
  • The points (x,y) in the equation x2 + y2 = r2 (analytic version of the geometric definition above)
  • The points in the equation r * cos(t), r * sin(t), for all t (really analytic version)
  • The shape whose tangent line is always perpendicular to the position vector (physical interpretation)

The list goes on, but here’s the key: the facts all describe the same idea! It’s like saying 1, one, uno, eins, “the solution to 2x + 3 = 5″ or “the number of noses on your face” — just different names for the idea of “unity”.

But these initial descriptions are important — they shape our intuition. Because we see circles in the real world before the classroom, we understand their “roundness”. No matter what fancy equation we see (x2 + y2 = r2), we know deep inside that a circle is “round”. If we graphed that equation and it appeared square, or lopsided, we’d know there was a mistake.

As children, we learn the “caveman” definition of a circle (a really round thing), which gives us a comfortable intuition. We can see that every point on our “round thing” is the same distance from the center. x2 + y2 = r2 is the analytic way of expressing that fact, using the Pythagorean theorem for distance. We started in one corner, with our intuition, and worked our way around to the formal definition.

Other ideas aren’t so lucky. Do we instinctively see the growth of e, or is it an abstract definition? Do we realize the rotation of i, or is it an artificial, useless idea?

A Strategy For Developing Insight

I still have to remind myself about the deeper meaning of e and i — which seems as absurd as “remembering” that a circle is round or what a cat looks like! It should be the natural insight we start with.

Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them. Once the central concept is clear, the equations snap into place. Here’s a strategy that has helped me:

  • Step 1: Find the central theme of a math concept. This can be difficult, but try starting with its history. Where was the idea first used? What was the discoverer doing? This use may be different from our modern interpretation and application.
  • Step 2: Explain a property/fact using the theme. Use the theme to make an analogy to the formal definition. If you’re lucky, you can translate the math equation (x2 + y2 = r2) into a plain-english statement (“All points the same distance from the center”).
  • Step 3: Explore related properties using the same theme. Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.

Let’s try it out.

A Real Example: Understanding e

Understanding the number e has been a major battle. e appears all of science, and has numerous definitions, yet rarely clicks in a natural way. Let’s build some insight around this idea. The following section will have several equations, which are simply ways to describe ideas. Even if the equation is gibberish, there’s a plain-english idea behind it.

Here’s a few popular definitions of e:

Definitions of e

The first step is to find a theme. Looking at e’s history, it seems it has something to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.

Let’s look at the first definition, in the upper left. The key jump, for me, was to realize how much this looked like the formula for compound interest. In fact, it is the interest formula when you compound 100% interest for 1 unit of time, compounding as fast as possible.

  • Definition 1: Define e as 100% compound growth at the smallest increment possible.

The article on e describes this interpretation.

Let’s look at the second definition: an infinite series of terms, getting smaller and smaller. What could this be?

\displaystyle{e = {1 \over 0!} + {1 \over 1!} + {1 \over 2!} + {1 \over 3!} + \cdots}

After noodling this over using the theme of “interest” we see this definitions shows the components of compound interest. Now, insights don’t come instantly — this insight might strike after brainstorming “What could 1 + 1 + 1/2 + 1/6 + …” represent when talking about growth?”

Well, the first term (1 = 1/0!, remembering that 0! is 1) is your principal, the original amount. The next term (1 = 1/1!) is the “direct” interest you earned — 100% of 1. The next term (0.5 = 1/2!) is the amount of money your interest made (“2nd level interest”). The following term (.1666 = 1/3!) is your “3rd-level interest” — how much money your interest’s interest earned!

Money earns money, which earns money, which earns money, and so on — the sequence separates out these contributions (read the article on e to see how Mr. Blue, Mr. Green & Mr. Red grow independently). There’s much more to say, but that’s the “growth-focused” understanding of that idea.

  • Definition 2: Define e by the contributions each piece of interest makes

Neato.

Now to the 3rd, and shortest definition. What does it mean? Instead of thinking “derivative” (which turns your brain into equation-crunching mode), think about what it means. The feeling of the equation. Make it your friend.

\displaystyle{\frac{d}{dx}Blah = Blah}

It’s the calculus way of saying “Your rate of growth is equal to your current amount”. Well, growing at your current amount would be a 100% interest rate, right? And by always growing it means you are always calculating interest — it’s another way of describing continuously compound interest!

  • Definition 3: Define e as a function that always grows at 100% of your current value

Nice — e is the number where you’re always growing by exactly your current amount (100%), not 1% or 200%.

Time for the last definition — it’s a tricky one. Here’s my interpretation: Instead of describing how much you grew, why not say how long it took?

If you’re at 1 and growing at 100%, it takes 1 unit of time to get from 1 to 2. But once you’re at 2, and growing 100%, it means you’re growing at 2 units per unit time! So it only takes 1/2 unit of time to go from 2 to 3. Going from 3 to 4 only takes 1/3 unit of time, and so on.

The time needed to grow from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4… and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation.

ln(a) is simply the time to grow from 1 to a. We then say that “e” is the number that takes exactly 1 unit of time to grow to. Said another way, e is is the amount of growth after waiting exactly 1 unit of time!

  • Definition 4: Define the time needed to grow continuously from 1 to a as ln(a). e is the amount of growth you have after 1 unit of time.

Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!

What’s the Moral?

In math class, we often start with the last, most complex idea. It’s no wonder we’re confused — we’re showing DNA and expecting students to see the cat.

I’ve learned a few lessons from this approach, and it underlies how I understand and explain math:

  • Search for insights and apply them. That first intuitive insight can help everything else snap into place. Start with a definition that makes sense and “walk around the circle” to find others.
  • Develop mental toughness. Banging your head against an idea is no fun. If it doesn’t click, come at it from different angles. There’s another book, another article, another person who explains it in a way that makes sense to you.
  • It’s ok to be visual. We think of math as rigid and analytic — but visual interpretations are ok! Do what develops your understanding. Imaginary numbers were puzzling until their geometric interpretation came to light, decades after their initial discovery. Looking at equations all day didn’t help mathematicians “get” what they were about.

Math becomes difficult when we emphasize definitions over understanding. Remember that the modern definition is the most advanced step of thought, not necessarily the starting point. Don’t be afraid to approach a concept from a funny angle — figure out the plain-English sentence behind the equation. Happy math.

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How to Develop a Mindset for Math

Math uses made-up rules to create models and relationships. When learning, I ask:

  • What relationship does this model represent?
  • What real-world items share this relationship?
  • Does that relationship make sense to me?

They're simple questions, but they help me understand new topics. If you liked my math posts, this article covers my approach to this oft-maligned subject. Many people have left insightful comments about their struggles with math and resources that helped them.

Math Education

Textbooks rarely focus on understanding; it's mostly solving problems with "plug and chug" formulas. It saddens me that beautiful ideas get such a rote treatment:

  • The Pythagorean Theorem is not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more.
  • e is not just a number. It is about the fundamental relationships between all growth rates.
  • The natural log is not just an inverse function. It is about the amount of time things need to grow.

Elegant, "a ha!" insights should be our focus, but we leave that for students to randomly stumble upon themselves. I hit an "a ha" moment after a hellish cram session in college; since then, I've wanted to find and share those epiphanies to spare others the same pain.

But it works both ways -- I want you to share insights with me, too. There's more understanding, less pain, and everyone wins.

Math Evolves Over Time

I consider math as a way of thinking, and it's important to see how that thinking developed rather than only showing the result. Let's try an example.

Imagine you're a caveman doing math. One of the first problems will be how to count things. Several systems have developed over time:

number system table

No system is right, and each has advantages:

  • Unary system: Draw lines in the sand -- as simple as it gets. Great for keeping score in games; you can add to a number without erasing and rewriting.
  • Roman Numerals: More advanced unary, with shortcuts for large numbers.
  • Decimals: Huge realization that numbers can use a "positional" system with place and zero.
  • Binary: Simplest positional system (two digits, on vs off) so it's great for mechanical devices.
  • Scientific Notation: Extremely compact, can easily gauge a number's size and precision (1E3 vs 1.000E3).

Think we're done? No way. In 1000 years we'll have a system that makes decimal numbers look as quaint as Roman Numerals ("By George, how did they manage with such clumsy tools?").

Negative Numbers Aren't That Real

Let's think about numbers a bit more. The example above shows our number system is one of many ways to solve the "counting" problem.

The Romans would consider zero and fractions strange, but it doesn't mean "nothingness" and "part to whole" aren't useful concepts. But see how each system incorporated new ideas.

Fractions (1/3), decimals (.234), and complex numbers (3 + 4i) are ways to express new relationships. They may not make sense right now, just like zero didn't "make sense" to the Romans. We need new real-world relationships (like debt) for them to click.

Even then, negative numbers may not exist in the way we think, as you convince me here:

You: Negative numbers are a great idea, but don't inherently exist. It's a label we apply to a concept.

Me: Sure they do.

You: Ok, show me -3 cows.

Me: Well, um... assume you're a farmer, and you lost 3 cows.

You: Ok, you have zero cows.

Me: No, I mean, you gave 3 cows to a friend.

You: Ok, he has 3 cows and you have zero.

Me: No, I mean, he's going to give them back someday. He owes you.

You: Ah. So the actual number I have (-3 or 0) depends on whether I think he'll pay me back. I didn't realize my opinion changed how counting worked. In my world, I had zero the whole time.

Me: Sigh. It's not like that. When he gives you the cows back, you go from -3 to 3.

You: Ok, so he returns 3 cows and we jump 6, from -3 to 3? Any other new arithmetic I should be aware of? What does sqrt(-17) cows look like?

Me: Get out.

Negative numbers can express a relationship:

  • Positive numbers represent a surplus of cows
  • Zero represents no cows
  • Negative numbers represent a deficit of cows that are assumed to be paid back

But the negative number "isn't really there" -- there's only the relationship they represent (a surplus/deficit of cows). We've created a "negative number" model to help with bookkeeping, even though you can't hold -3 cows in your hand. (I purposefully used a different interpretation of what "negative" means: it's a different counting system, just like Roman numerals and decimals are different counting systems.)

By the way, negative numbers weren't accepted by many people, including Western mathematicians, until the 1700s. The idea of a negative was considered "absurd". Negative numbers do seem strange unless you can see how they represent complex real-world relationships, like debt.

Why All the Philosophy?

I realized that my **mindset is key to learning. **It helped me arrive at deep insights, specifically:

  • Factual knowledge is not understanding. Knowing "hammers drive nails" is not the same as the insight that any hard object (a rock, a wrench) can drive a nail.
  • Keep an open mind. Develop your intuition by allowing yourself to be a beginner again.

A university professor went to visit a famous Zen master. While the master quietly served tea, the professor talked about Zen. The master poured the visitor's cup to the brim, and then kept pouring. The professor watched the overflowing cup until he could no longer restrain himself. "It's overfull! No more will go in!" the professor blurted. "You are like this cup," the master replied, "How can I show you Zen unless you first empty your cup."

  • Be creative. Look for strange relationships. Use diagrams. Use humor. Use analogies. Use mnemonics. Use anything that makes the ideas more vivid. Analogies aren't perfect but help when struggling with the general idea.
  • Realize you can learn. We expect kids to learn algebra, trigonometry and calculus that would astound the ancient Greeks. And we should: we're capable of learning so much, if explained correctly. Don't stop until it makes sense, or that mathematical gap will haunt you. Mental toughness is critical -- we often give up too easily.

So What's the Point?

I want to share what I've discovered, hoping it helps you learn math:

  • Math creates models that have certain relationships
  • We try to find real-world phenomena that have the same relationship
  • Our models are always improving. A new model may come along that better explains that relationship (roman numerals to decimal system).

Sure, some models appear to have no use: "What good are imaginary numbers?", many students ask. It's a valid question, with an intuitive answer.

The use of imaginary numbers is limited by our imagination and understanding -- just like negative numbers are "useless" unless you have the idea of debt, imaginary numbers can be confusing because we don't truly understand the relationship they represent.

Math provides models; understand their relationships and apply them to real-world objects.

Developing intuition makes learning fun -- even accounting isn't bad when you understand the problems it solves. I want to cover complex numbers, calculus and other elusive topics by focusing on relationships, not proofs and mechanics.

But this is my experience -- how do you learn best? A few friends have written up their experience:

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