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:
“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.
“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.
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.
@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.
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.
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
@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 !
@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!
@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 :).
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…
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…
@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