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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## Other Posts In This Series

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