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.

## Other Posts In This Series

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

## Leave a Reply

27 Comments on "Intuition, Details and the Bow/Arrow Metaphor"

One way to REALLY improve intuition in math (and science) is to learn the history behind it. The history often provides the conceptual development of the subject that most textbooks skip. For example, ancient Babylonian tablets and Greek, Arabic, Hindu texts are a record of the “firsts” in math. Imagine developing a new topic by introducing these sources. It also makes the math more human – there’s the first person to develop it and there context to do so is such and such.

@Mark: I completely agree. e and natural log were discovered in the context of computing interest. Complex numbers weren’t fully utilized until the geometric interpretation came about. There are so many aspects where knowing the history makes you understand the thought process that went into the idea.

Per (following up) from my post in http://aha.betterexplained.com/posts/174/analogy-intuition–details-are-like-a-bow–arrow , here is Conrad Wolfram’s take on this issue.

http://www.youtube.com/watch?v=60OVlfAUPJg

I personally really like the idea (use the computational power of computers to help teach math nowadays). A lot more “high-level” thinking (real world – delving in IDEAS themselves) can be done by students; as opposed to drowning solely in the nitty-gritty “low-level” thinking (symbol manipulation, hand calculations, but LACKING the big picture of the IDEA), typical of traditional math education. *(high-level and low-level used from a computer science perspective – e.g. C++ vs Assembly)

Also, it’s a perfectly feasible real-world solution – the question is whether the educational institutions will slowly begin to adapt this style of teaching (knowing how stubborn and resistant to change they tend to be.) From an individual (self-learner) standpoint though, it is CERTAINLY feasible, and a VERY exciting idea! :D

A great status-quo breaker – thanks again for a marvelous article, Kalid! :D (My thought is that your site, combined with Lockhart’s Lament AND Wolfram’s idea – would be a very powerful movement indeed!)

Great piece!! Keep it up; the world needs to hear your message.

eh.. rigor can lead to intuition, but perhaps more vice versa.

i agree though, there should be a balance.

@Stan: Thanks for that pointer, I had forgotten about that TED talk! Yes, I really agree with his approach — we need to learn how to use and manipulate the tools we have, to really get an intuition for them. Memorizing giant tables of integrals isn’t doing anybody some good. See, the funny thing is after getting that intuition you become more interested in learning the nitty gritty also (since you’re interested in the subject in general).

And yep, I think think a joint movement would be fun… everybody is putting together the pieces :).

@intuit: Thanks!

@PL: Thanks for the comment — yep, a balance is needed. I err on the side of the minimum level of rigor needed to begin talking, then getting a deep intuition, then sprinkling in more rigor as needed.

This is amazing! Thank you…thank you!

@theObserver: You’re welcome! Glad you liked it :).

Kalid, Very well said.

I just want to add one thought. What you said about maths applies equally well to other subjects.

Of late, I have started reading metallurgy books and many times I have wondered how wonderful it would be if somebody wrote a book on metallurgy like Kalid does on maths.

I believe that any science or engg should be and can be explained in simple words.

I remember your words: Learn the bow and one arrow; more arrows will follow.

@Nandeesh: Thanks for the comment!

Yep, I agree, the bow/arrow concept is not just limited to math (math is a “fun” area to tackle because if we can make it understandable, or even enjoyable, think what’s possible for a subject which isn’t as generally reviled).

The more I think about it, the more I realize the value of truly understanding one bow and one arrow. Appreciate the note!

[…] Shared by Jeff A, Posted on BetterExplained […]

I honestly never loved maths in high school. Today I realize that what i learnt in high school was not maths at all. It was bihearted literature………just a pattern of alphabets and symbols memorized in a specific pattern(formulas).

Infact even students actually studying literature know more about their subject than what I or even the maths teachers knew about maths.

It was just recently that i actually started learning intutively.

And today I realize that physics…..which i always loved and maths….which i never loved were actually one and the same.

I feel i am the only person in my vicinity who truly understands maths.

There was a time when i used to truly respect people who used to bring full marks in maths. However….no longer.

With Respect

Binnoy

@Binnoy: Awesome, glad you’re starting to see the intuition behind things :).

I love your way of explaining things. I’ve read two articles so far, including this one, and they were both awesome. :) Love the metaphor and the whole philosophy. Thank you!!!

@JZ: Thanks for the kind words! Glad you’re enjoying it :).

Thanks Kalid.

Very very god metaphor! Surely it will increase my teaching performance. I teach Operational Research for a business course in a local University and my great challenge is to make my students to get the intuition about the problem nature before teaching them the “arrows” of simplex and so on.

@Valter: Thanks for the comment, I’m happy if the analogy helps you convey how to develop intuition to your students!

First, thank you for this amazing blog and the insightful articles that make math so easy.. ok maybe not so easy but easier than usual :D.

I really like the idea of the bow and arrow metaphor, but wanted to know how could I implement it into every day learning (high school, college etc.).

@Sh: Thanks :). Applying the bow/arrow metaphor is tricky, it’s more of a general attitude to take: make sure you build up your bow skills (intuition), and not spend too much time on arrows (memorizing facts). I have an article on building your intuition which might be helpful: /articles/developing-your-intuition-for-math/

Thanks for this insite. I have a daughter in HS who is having difficulty in math. I hope this site will help her.

@J: You’re more than welcome — I hope she finds it useful too!

I hadn’t seen that video: “Conrad Wolfram: Teaching kids real math with computers”. He makes some good points. It reminds me of “A Mathematician’s Lament” by Paul Lockhart.

I agree that there is a HUGE gap between mathematics at school and mathematics at work. I also agree with him that there is too much emphasis on doing the same calculation for the hundredth time. I have experienced his idea of using a computer to make mathematics more interactive when I did a course on Operations Research. But I’m glad he acknowledged that computers could be used wrong.

The problem is, mathematics could be improved a lot(!)—even without computers. And we aren’t doing it! Instead we bombard students with theories, lemmas and so on. We teach them how a formula is derived; we might(!) explain what it physically means, but we rarely explain what it means mathematically. Which is what you do with this site. I have had very few teachers actually try explaining mathematics really well. I was fortunate to be good at it.

I like the fact that Conrad Wolfram takes an engineering approach: (1) take a real world question; (2) model it mathematically; (3) solve the mathematical problem; (4) check the solution against the real world. I also like that he only showed interactive programs and computer algebra systems.

It’s unfortunate that Conrad Wolfram didn’t talk about the history of mathematics. Because many of the new concepts in mathematics were introduced specifically to be able to solve real world problems.

@alex: Definitely, Wolfram’s talk has some good overlap with Lockhart’s lament.

I totally agree with you about the focus — I’ve seen dozens and and dozens of “proofs”, but seldom did they stick. I realized I had to (for myself) explain things in a way that actually worked, which meant focusing on intuition and how learning math changed the way I thought.

I really like Wolfram’s approach, and explicitly making the 1-4 steps known. Computation is such a small part of understanding, it’s the most mechanical and yet the most focused-on. I think the reason it’s emphasized is because many educators have never seen the inherent beauty, and focus on the part that can be measured (similar to someone who has never actually heard a song, assigning sheet music transcriptions and thinking that is the study of music).

The history of mathematics is actually where I go to look for insights :). For e, for example, it was discovered in the context of interest rates, so tracking where interest comes from, how it creates its own interest, etc. sheds a lot of light on the subject.

Thanks for this site… very inspiring!

Thanks Seun!

Hi Kalid, enjoy your site immensely, keep adding more!

when you say learning how to learn is not learning more and more arrows, but mastering a single bow and a single arrow, does that imply that specialists are better at learning new things, i.e. they are also better generalists? And on a sidenote, american liberal education that teaches multiple things in college seems to be giving students just the arrows, and not allowing students to master one bow and one arrow, thus fail? (or maybe it’s doing well, what do you think?) erm… this puzzle has bugged me for a long time, so I’d like your opinion.

Hi psoe, thanks for the note! I don’t like to make clearcut distinctions like specialist vs. generalist, but I think someone who is really, really good at something has figured out a) how to practice b) how to work through difficulties c) what it feels like to “know” something. The so-called “generalist” might actually be really good at A, and ok at B-Z, but would still have that mastery experience with A. In more abstract terms, if you can get into the “flow” state with one endeavor, you’ll know what it’s like to be in the zone with a new one (and everyone has hit that “in the zone” feeling at some point).

It’s really hard for me to comment on education systems as a whole. In general, I think flexibility of options good, and the focus on lots of arrows (or not) is a combination of the attitudes and efforts of the teacher and student.