• 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” (a² + b² = c²) 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.

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; either is fine.

The key is delivering gems, not dirt. When writing, you know what you’re providing .

I still struggle with accepting that it’s ok, nay good, to share a single, concise thought if you think it’s a gem. There’s no need to pad to make it seem “substantial”.

Does anyone think the 278-word Gettysburg address isn’t meaty enough? Should Lincoln have stretched it out a bit?

Expert’s Guilt (The sky is not blue)

Brevity’s enemy is an armada of “helpful” information. Consider this: 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 say “The sky is blue” without a 3-page disclaimer, lest an expert 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.

I couldn’t help but compulsively photograph seemingly mundane things, awestruck by their design. I did get some quizzical looks when blabbering on about a straw or ladder. But there’s a few lessons:

• Unimaginative people ask “why?” when confronted with a new idea and sit there, lifeless. Instead, ask “why not?” and do something new.
• Not every invention pans out, and that’s fine. At least you are trying to improve things.
• Several minor, 2% improvements add up over time (see the rule of 72). Everyday efficiencies are great — you don’t always need a breakthrough to make a difference.

Browse the photos below and read my comments at flickr to see what the fuss was about. In upcoming posts I’ll expand on why these examples of innovation made me shriek with delight and what we can learn from them.

More generally, the Pareto Principle is the observation (not law) that most things in life are not distributed evenly. It can mean all of the following things:

• 20% of the input creates 80% of the result
• 20% of the workers produce 80% of the result
• 20% of the customers create 80% of the revenue
• 20% of the bugs cause 80% of the crashes
• 20% of the features cause 80% of the usage
• And on and on…

But be careful when using this idea! First, there’s a common misconception that the numbers 20 and 80 must add to 100 — they don’t!

20% of the workers could create 10% of the result. Or 50%. Or 80%. Or 99%, or even 100%. Think about it — in a group of 100 workers, 20 could do all the work while the other 80 goof off. In that case, 20% of the workers did 100% of the work. Remember that the 80/20 rule is a rough guide about typical distributions.

Also recognize that the numbers don’t have to be “20%” and “80%” exactly. The key point is that most things in life (effort, reward, output) are not distributed evenly – some contribute more than others.

Life Isn’t Fair

What does it mean when we say “things aren’t distributed evenly”? The key point is that each unit of work (or time) doesn’t contribute the same amount.

In a perfect world, every employee would contribute the same amount, every bug would be equally important, every feature would be equally loved by users. Planning would be so easy.

But that isn’t always the case:

The 80/20 rule observes that most things have an unequal distribution. Out of 5 things, perhaps 1 will be “cool”. That cool thing/idea/person will result in majority of the impact of the group (the green line). We’d like life to be like the red line, where every piece contributes equally, but that doesn’t always happen.

Of course, this ratio can change. It could be 80/20, 90/10, or 90/20 (remember, the numbers don’t have to add to 100!).

The key point is that most things are not 1/1, where each unit of “input” (effort, time, labor) contributes exactly the same amount of output.

So Why Is This Useful?

The Pareto Principle helps you realize that the majority of results come from a minority of inputs. Knowing this, if…

20% of workers contribute 80% of results: Focus on rewarding these employees.
20% of bugs contribute 80% of crashes: Focus on fixing these bugs first.
20% of customers contribute 80% of revenue: Focus on satisfying these customers.

The examples go on. The point is to realize that you can often focus your effort on the 20% that makes a difference, instead of the 80% that doesn’t add much.

In economics terms, there is diminishing marginal benefit. This is related to the law of diminishing returns: each additional hour of effort, each extra worker is adding less “oomph” to the final result. By the end, you are spending lots of time on the minor details.

A Fun, Non-Math Example, Please

Everything is nice and rosy in the abstract. I want to give you a real example. Take a look at this awesome video of an artist drawing a car in Microsoft Paint. It’s pretty phenomenal what can be accomplished with such a basic tool:

Now let’s deconstruct this video. It’s about 5 minutes long, so each minute is about 20% of the way to completion (of course the video is sped up, but we are only interested in relative times anyway). Take a look at how the car evolved over time:

1:06 (Level 1) – Wireframe
2:00 (Level 2) – Basic coloring
3:05 (Level 3) – Beginning details: rims, windshield
4:04 (Level 4) – Advanced details: shading, reflections
5:05 (Level 5) – Finishing touches: headlights, background

Now, let’s say the artist was creating potential designs for a client. Given 5 minutes of time, he could present:

• A single car at top quality (Level 5)
• A reasonably detailed car (Level 3) and a colorized wireframe (Level 2)
• 5 cars at a wireframe level (5 Level 1s)

“But #5 is way better than #1!!!” someone will inevitably shout.

The point isn’t that #5 is better than #1 — it clearly is. The question is whether #5 is better than five #1s, or some other combination.

Let’s say your customer doesn’t know whether they want a car, a truck, or a boat, let alone the color. Spending the time to create a Level 5 drawing wouldn’t make sense — show some concepts, get a general direction, and then work out the details.

The point is to put in the amount of effort needed to get the most bang for your buck — it’s usually in the first 20% (or 10%, or 30% — the exact amount can vary). In the planning stage, it may be better to get 5 fast prototypes rather than 1 polished product.

In this example, after 1 minute (20% of the time) we have a great understanding of what the final outcome will be. Most of the “work” is done up front, in the sense of deciding the type of vehicle, body style, and perspective. The rest is “filling in details” like colors and shading.

This isn’t to say the details are easy — they’re not — but each detail does not add as much to the picture as the broad strokes in the beginning. The difference between #4 and #5 is not as great as #1 and #2, or better yet, a blank drawing and #1 (the time from 0:00 to 1:06). You really have to look to see the differences on the car between #4 and #5, while the contribution #1 makes is quite obvious.

Concluding Thoughts

This may not be the best strategy in every case. The point of the Pareto principle is to recognize that most things in life are not distributed evenly. Make decisions on allocating time, resources and effort based on this:

• Instead of 1 hour on a rough draft for an article you may write, spend 10 minutes on 6 outlines for a paper / blog article and pick the best topic.
• Instead of investing 3 hours on a website, spend 30 minutes and create 6 different template layouts.
• Rather than spending 3 hours to read 3 articles in detail (which may not be relevant to you), spend 5 minutes glancing through 12 articles (1 hour) and then spend an hour each on the two best ones (2 hours).

These techniques may or may not make sense – the point is to realize you have the option to focus on the important 20%.

Lastly, don’t think the Pareto Principle means only do 80% of the work needed. It may be true that 80% of a bridge is built in the first 20% of the time, but you still need the rest of the bridge in order for it to work. It may be true that 80% of the Mona Lisa was painted in the first 20% of the time, but it wouldn’t be the masterpiece it is without all the details. The Pareto Principle is an observation, not a law of nature.

When you are seeking top quality, you need all 100%. When you are trying to optimize your bang for the buck, focusing on the critical 20% is a time-saver. See what activities generate the most results and give them your appropriate attention.