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):
 (amazon) |
 (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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Have a question? Know an explanation that caused your own a-ha moment? Write about it here.
Thanks for the good writing and ideas!
“Ink” helps propagate the Curse of Knowledge that blinds experts from seeing plight of a novice.
Pencil gives hope that goals are achievable. When great writers pull back the curtain and show piles of scratched up drafts and when you see great presenters like Steve Jobs needing hundreds of hours of practice before a show to make it look “effortless,” it gives hope that a large part of talent is realizing the need for hard work and a willingness to stick through valleys of the pencil days to reach the destination of ink.
It’s as if ink hides most of the story, but that’s what seems what the world desires because the pencil days can be so arduous and common.
Jeff Moser — January 14, 2010 @ 7:27 am
“Myth of the Perfect Formula”–yes. I have this problem with woodworking (and similar) projects. I procrastinate not because I don’t want to do it but because I don’t want to redo it. Meanwhile I could built and destroyed 17 drafts in that amount of time and have the perfect final version by now.
David — January 14, 2010 @ 8:00 am
Great analogy – thanks for the insight!
Cl — January 14, 2010 @ 9:05 am
love these thoughts.
I was recently listening to some controversy over a school in the Bronx with a sub 50% graduation rate that was on the verge of getting shut down. Teachers and supporters of the school said the kids were improving in ways that weren’t quantifiable by traditional measures. I would guess the emphasis on the rigor portion probably has something time concerns, with schools having to make cases for themselves every academic year.
Rigor doesn’t last very long does it? I only remember a few math formulas, and they’re usually the ones I took the time necessary to derive. But the learning the history behind the pencil lines takes a lot of time, and after a while everyone (including yourself) starts asking you what the hell you’re drawing.
ash — January 14, 2010 @ 11:18 am
I’ve seen this mentioned elsewhere as well — different perspectives of the same core issues.
One source was a blog entry:
http://www.lifebeyondcode.com/2009/12/26/why-some-smart-people-are-reluctant-to-share/
The other a presentation given by a physics professor:
http://www.youtube.com/watch?v=WwslBPj8GgI
paul — January 14, 2010 @ 12:55 pm
@Jeff: Thanks for the note! I really like that point about how much practice is required for something to look “effortless”. Heck, we forget that it took a few months for us to learn to stand upright and walk, and it’s something that still incredibly hard for a robot to do.
@David: Funny — I procrastinate / delay the same when writing these posts! There’s an interesting story about a pottery professor: for some students, he graded them on the *best* pot they made during the semester.
For another set, he judged them on the *quantity* of pots they made that semester.
What happened? The students who had to make the most pots ended up making the best ones, since they had so much practice. The others, who concentrated on making the one perfect pot, didn’t end up making as high quality ones
. I can read these stories but it’s still so hard to get it to settle in my head.
@Cl: You’re welcome!
@ash: Thanks. That’s a great point — unfortunately, “rigor” is very easily measurable, but the underlying understanding is not (so we end up testing for ability to reproduce material, not rigor). But as you say, memorized facts fade away, and what’s left after years is whatever intuitive understanding you managed to pick up.
@paul: Thanks for the links! I enjoyed that blog post, it does capture the problem of internalizing difficult-to-learn knowledge and later thinking it’s “obvious”. I’m looking forward to checking out that video too.
Kalid — January 15, 2010 @ 4:31 pm
http://en.wikipedia.org/wiki/Dreyfus_model_of_skill_acquisition
You should read about the Dreyfus model of skill acquisition, it describes more about how an expert isn’t the best person to be teaching a novice.
Jason Axelson — January 18, 2010 @ 1:23 am
Excellent!!Thanks again for another great article!!
Karan Goel — January 19, 2010 @ 8:42 am
I think the other important element to the pencil is that it allows us to correct our mistakes during the process before we are ready to publish our final results. The pencil is the tool that connects understanding to results.
online colleges — January 21, 2010 @ 12:53 pm
Hi Khalid, funny thing is that I actually made this same connection after reading one of your articles on the discovery of pi. After each iteration, you get closer to the answer. I guess its up to you when its good enough to ink in
There is an analogy for you: ink as limits!
Sebastian Marquez — January 26, 2010 @ 1:08 pm
@Kalid: Thanks so much! I’d like to mention something personal here. When I was younger, and learning the basics of maths, I’d never get it right. Many times, I ended up crying in the class (without being noticed) about not understanding something which EVERYONE seemed to be getting perfectly. Even now, when I’m studying maths in High school, those basics haunt me, and trip me up when I try to solve problems. It’s disheartening, but excellent blogs like yours make maths seem fun, and it is encouraging to know that its not completely my fault and I can move on without feeling like a failure. Thank you so much for sharing your “ah-ha!” moments (I’ve only had a few of them) and this excellent analogy which really highlights the problems in our education system and gives fresh heart to those like me. Please keep writing!
@Everyone who commented before me: Excellent comments! They help draw out the analogy further. Thanks to you too !
Charu — January 30, 2010 @ 5:41 am
@Jason: Thanks for that reference! I like this line: “The progression is thus viewed as a gradual transition from rigid adherence to rules to an intuitive mode of reasoning…”
Currently, the model seems to posit that we learn first by tracing the ink [rigid adherence to rules] and then start to see the underlying pencil structure [intuitive reasoning]. I wonder whether it’s necessary to struggle in the dark with the first stage, or whether glimpses of the underlying reasoning can shine through to help motivate. There’s often a rote memorization part, but if it can be put in the context of building “muscle memory” (vs. the ultimate goal) then I think education would be that much more enjoyable (i.e., learn the alphabet not because it’s intrinsically fun, but gives you the mental ‘muscle memory’ to write and read without issue). Great link!
@Karan: You’re welcome!
@online: I agree — the pencil is needed to make sense of the final results. Argh, I get too many people leaving useful but ad-text comments here, there’s rel=nofollow on all this stuff guys
. I might start rewriting people’s names if they’re too spammy.
@Sebastian: Oh, I hadn’t thought of that in terms of pi/limits! That’s very interesting, because the end result is the same, but if you don’t see the process of getting there the final meaning can be obscured. Interesting
.
@Charu: Wow, thank you for that personal insight — I’m really happy you stuck with it and are now overcoming those demons. I think we all have things like that in our past — for me, I’ve always been self-conscious / nervous about singing, it’s just something I don’t think I do well so feel embarrassed about it. But a lot of these things are in our heads, that’s probably one of the unstated roadblocks in learning — overcoming our own biases and burdens. Part of this blog is cathartic, just writing about the stuff that was difficult for me to grasp, and the joy when I finally did. I think most things are like that — there’s always another explanation out there that can help
.
Kalid — January 30, 2010 @ 2:05 pm
Thanks for the reply ! And you are absolutely right.
“probably one of the unstated roadblocks in learning — overcoming our own biases and burdens.” So true! You’ve written it very well. Thanks again
Charu — January 30, 2010 @ 6:41 pm
@Charu: You’re more than welcome, always happy for the conversation!
Kalid — February 1, 2010 @ 1:02 pm
Awesome post! must say your site is great! I’m a student from Pakistan, and I hate it when my teachers want me to memorize stuff. Instead of cursing them, I come here to learn fun stuff! Thanks for giving me that opportunity…
BadRobot14 — February 6, 2010 @ 2:06 am
check this one also – about 7-8 hours
http://www.dueysdrawings.com/colored_pencil_tutorial.html
Ala'a — February 12, 2010 @ 10:24 am
This reminds me of the connection between the gamma function and the factorial. I could recite that they were connected, but for the longest time, I hadn’t seen how anyone would be able to connect the two. Then, I learned Laplace transforms…
Nobody — February 13, 2010 @ 8:57 am
@Kalid – thanks so much for all these wonderful insights. I have been thinking alot recently about how learning is meant to be all about sharing those aha moments. In fact you have inspired me to start something similar with Physics – one of my favorite subject (but in which I have not found much sharing culture).
And it is so true that we forget so much if it is only rigor. to me for example. I have done some topics in Physics – twice, but still, I can’t really remember things. It is as though I am not the owner of my knowledge
Again thanks Kalid, and to all those who comment and further enrich the aha moment discussions
uche9012 — March 20, 2010 @ 8:50 am
Loved reading your insights… I really agree with it too!
NewReader — March 30, 2010 @ 7:45 pm
@BadRobot14: Awesome, glad to provide an outlet
.
@Ala’a: Thanks for the pointer, I’ll check it out.
@Nobody: Nice, I need to learn about Laplace transforms some day.
@uche9012: Awesome, really glad it’s helping! Good luck with your site — yes, physics is another one of those subjects where we tend to memorize things with rigor instead of understanding at an intuitive level. And the great irony is that physics is about the real world we live in!
@NewReader: Thanks!
Kalid — April 6, 2010 @ 2:25 pm