Mathematically, we can write:
And to a calculator, these are the same. Are they? There's a suspicion nothing (0) and complete cancellation (1 - 1) aren't quite identical.
In physics, there's the notion of a stable and unstable equilibrium. Take two pencils. Lay one on the table, balance the other on its tip.
They're both 'balanced'. There's zero motion. Yet one is a precarious position, carefully opposing the pull of gravity, while the other lays peacefully.
Lie on the floor for 10 minutes. Hold the plank pose for 10 minutes. From a physics perspective, no work was done (nothing moved), but your quivering arms tell a different story.
In algebra, we constantly factor equations to find roots.
Why? In short, we want to find the "neutral zones" where all forces balance.
Factoring
means "Is there a value where x^2, 2x, and 3 cancel each other out?". We arrange the scenario so the neutral zone is where we want to be (such as having no error, or having competing goals align).
There is often a "trivial solution", where we can plug in x=0 and all inputs disappear (lying the pencil on the table... or just taking it away!). However, we're more interested in finding a "neutral zone", where multiple, existing forces balance.
Programming languages distinguish "void/undefined/null" (a value is not set) and "having a value of emptiness".
var i; // i is undefined
i = 0; // i is now set to 4 bytes of "nothingness"
If we imagine data storage as a light switch, we have
By itself, var i
is just a name or pointer, but it's not yet referring to anything (not even nothingness). It's not that Gazasdasrb means "nonsense", it's that Gazasdasrb has no meaning at all.
Many math explanations say you "can't divide by zero". It's not that you can't, it's that it's undefined. What does division by zero mean? What does Gazasdasrb mean?
If we pick a specific value for the result of a division by zero (let's say 3/0 = 15) then we immediately have contradictions (this means 15 * 0 = 3).
We avoid this trouble by saying division by zero is "undefined", or "we haven't got around to picking a value, nyah". In some games, the only winning move is not to play.
(Sometimes we define a value for strange expressions (such as 0^{0} = 1), if it's useful and doesn't lead to contradictions.)
Calculus dances with the concept of zero. Beyond the study of limits and infinitesimals, we are curious about the meaning of "zero change".
When I say a function isn't changing ("the derivative is zero"), it's usually not enough information. Are we not changing because we're at a minimum, a maximum, or precariously balanced between a hill and ravine?
There are tricks, like the second-derivative test, to see what type of "zero change" we have.
Society sets many goals for itself. Here's one: reduce littering. Given our "multiple zero" interpretation, we could accomplish this with:
It's the same result -- clean streets -- but what strategy do we prefer?
In general, any negative influence (unemployment, crime, pollution, etc.) can be seen through the lens of prevention or cure, an absence vs. meticulous cancellation. The reading is 0 in both cases, and it's up to us to make the distinction. (Sir, unfrozen Caveman Og is asking about Wooly Mammoth attacks again. Should we sell him more repellent?)
In Eastern philosophy there's the notion of non-doing or Wu Wei. Our brains think of "non doing" as sitting lazily on the couch. But maybe it's another type of zero. (Again, hold a plank for 10 minutes and tell me nothing happened.)
This essay is quick armchair philosophy from an equation. The words "something comes from nothing" aren't convincing. But if I write 0 = 1 - 1, boom, an idea snaps into place. How did 5 symbols convince you in seconds? Isn't that amazing?
Calculations are nice, but not the end goal of math education. Intuition means you're comfortable thinking, daydreaming, and exploring a concept with math as a guidepost. Now imagine having this comfort with the notions of shape, change, and chance (geometry, calculus, statistics).
Let's learn to sing with math, baby.
]]>If you want a path that doesn't expect perfect motivation, shares insights in minutes (not weeks), and aims for lifelong insight, this guide is for you.
My learning strategy is to ask honest (sometimes uncomfortable) questions about what's really working.
No games, no kidding ourselves, just:
Here's my wishlist for a learning guide. Elon Musk talks about thinking from first principles, starting with fundamental truths and working forward from there^{1}. Who cares what's being done now, what's our goal?
Priority #1 for any class is: Do not create hate for the subject.
Imagine 99% of people in a skiing class never ski again. They cringe at the thought. We wouldn't console ourselves thinking "Oh, skiing teaches important physical skills that apply to other fields." We'd think "That skiing class is awful and needs to change."
Sure, not everyone will love skiing (or cooking, or math), but they shouldn't detest it. Temporary understanding is not worth permanent aversion.
So, what Calculus introduction made me excited to learn more?
For me, it was seeing how patterns can be cleverly split and re-assembled:
Most courses march you through weeks of theory "appreciate" these diagrams in week 11. Ugh. The big picture helps me appreciate the details, not the reverse.
A typical discussion:
"I want to learn Calculus. What should I do?"
"Here's a [full book/course/MOOC]. It's months of effort, I didn't do it myself, but here you go."
In other words, "go the library and read for 100 hours". The real question:
I'm interested in the subject. Is there a plan that worked for you?
Motivation is limited. Traditional classes "work" -- because students are under immense pressure to finish (tuition, peer pressure, fear of not graduating).
Online courses without this pressure have single-digit completion rates. We can pretend students "got something" from the experience, just like you "got something" from a movie you walked out of. We can't change the goalposts to "something is better than nothing" halfway through.
Realistic advice on what worked with my limited motivation (even as a math hobbyist!) is:
Get an Aha! moment in minutes that motivates me to keep going (a cool diagram, example, or simulation).
Take a progressive journey where even if I stop after an hour, I have some helpful insights (vs. an hour of stretching in the parking lot).
Maintain a desire to revisit the subject by having an approachable, gentle introduction. I'll then keep coming back to fill in gaps over time.
For fun, find a lesson on imaginary numbers.
Does it acknowledge negative numbers were also distrusted?
Is the name "imaginary" described as an insult, given by people who didn't understand the concept?
Does the teacher mention their own confusion? (Or did imaginary numbers just click?)
Is there a real-world application? (If not, is this because it truly doesn't exist, we haven't tried to look, or it isn't important for learning?)
This type of lesson is a giant pet peeve. The flow is "Here's a confusing concept. I was confused myself, but I won't tell you that. Memorize the definition, apply it in these practice problems, and we'll call it a day."
Argh, this drives me nuts. It reinforces the stereotype that math class is a game of moving symbols around. (This symbol multiplied by this other symbol makes -1. Tada!)
It's ok to lack an intuition; I lack it for most things. But hiding our initial confusion implies the subject isn't confusing.
There's a common trope of the smart-aleck student trying to "outsmart" the teacher. Do basketball players try to "outsmart" their coach?
The flawed assumption is teachers must be some omniscient authority giving you access to precious knowledge. The knowledge is out there, it's not like the teacher invented the math herself. Instead, imagine a coach who is trying to improve your understanding.
Coaches can be wrong, sure. But they've seen many struggle with the same issues you're facing, and are trying to help. It's ok if Lebron James can dunk better than his coach.
The math may be perfect and unchanging, but the way it's taught is not. Let's make it easy to improve lessons and not expect perfection the first time.
Most courses assume you want mastery of the subject. That's fine, but is it necessary?
There are several levels of music understanding:
Intuitive Appreciation: Just enjoying the music.
Natural Description: Humming a tune you heard or made up.
Symbolic Description: Reading and writing the sheet music.
Theory: Explaining how harmonies work, why minor scales are somber, etc.
Performance: Playing the official instruments.
In language learning, there is an ILR scale from no profiency to native fluency. Not everyone studying Calculus needs to become Isaac Newton. Can we have a path that goes as far as we need?
Combining these insights, I've made a Calculus Learning Guide.
The principles, as I tried to apply them:
It's honest. It's the explanation that actually inspired me, not the theoretical explanation that requires weeks of discipline for some future payoff.
It acknowledges limited motivation. How far can you get in 1 minute? 10 minutes? An hour? Pretty far, I think. And getting a win in 10 minutes means you'll come back for more.
It's updatable. With lessons based primarily on text, we can easily update, re-arrange, add, edit, fix. Other formats are essentially a bet we got it right the first time.
It acknowledges levels of understanding. Most people just want an appreciation for Calculus. Technical performance is a goal we can separate, organize, and build a path to.
I eat the veggies myself. This guide has "gut checks" like "Can I describe an integral in everyday terms?" and "Can I derive the product rule on my own?". This is how I actually refresh my Calculus understanding.
In my ideal world, every Wikipedia topic would have a guide that took you from the 1-minute version to a full technical understanding. Go as far as you wish, make meaningful progress at each step, and have fun along the way.
Happy math.
Musk mentions not "reasoning by analogy", or assuming a conclusion is true based on what happened in another scenairo. This is different from "understand by analogy", getting the gist of an idea and then working to the technical version. The analogy is a raft to cross the river, to be left behind once you're on land. ↩
Developer Tea Interview (Part 1, 30 mins)
Developer Tea Interview (Part 2, 60 mins)
Topics include:
The conversation was a blast, lots of great Aha! moments. I was really impressed by how thoughtful, friendly, and insightful Jonathan Cutrell makes the topics. You should definitely subscribe to the show.
]]>I'm using a theme that makes new layouts a snap, vs. coding everything myself. Now we can finally browse insights visually:
In a follow-up I'll share the "holy grail" web setup I've settled on, after years of struggle.
In my head, I thought of this site as a collection of "articles". Unfortunately, an article takes weeks or months to write (Fourier Transform, looking at you) even though a key insight might be a single phrase.
So, to untangle my brain, I realize there's a few types of updates:
Insights: A quick diagram, analogy, or example that might help. A 1-5 min read. (Which could be a tweet, if they weren't so temporary. It hurts to write something and have it disappear in the void.)
Articles: An end-to-end lesson that helps explain a topic. Usually 1000-2000 words, 15-30 min read.
Guides: A learning path through a collection of articles, with a rough timeline, skills to check, etc. Going through a guide could take an hour or week. The first guide is coming next week.
Along with math, I'd like to share lessons on the learning process, the marketing/positioning of the site, and the content creation hurdles I'm overcoming.
If I spent years confused about a concept, and had an Aha! moment, I want you to benefit. If I get tripped up on an idea (like thinking every post must be a full "article"), I want you to avoid it.
I've been experimenting with a community at http://aha.betterexplained.com. I like having a dedicated area for discussions, but it reduced the amount of quick feedback (typos, etc.) that improved articles. I've added the regular comment forms back in.
Here's the idea:
I'm starting to embrace the story of the potter: create more, and you'll improve. It doesn't help to sit on the perfect jewel. (And what you think is a perfect jewel now, won't be in a few months anyhow.)
Originally, I was worried about over-sharing (as if my problem has been writing too much), so I'm going to course-correct the other way: be less shy with what I write. Share the little things.
On that note, I recently did an interview on the Developer Tea Podcast on learning, the ADEPT method, and the site itself. I think you'll enjoy it:
Developer Tea Interview (Part 1, 30 mins)
Developer Tea Interview (Part 2, 60 mins)
Wow! That felt great to write. A quick update, ready in minutes, not the weeks a typical post takes. I'm enjoying this already.
Happy math.
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