Who are these signs written for? It's really *You, stop* or *You, yield* or *You, be alert for bears* (I'm not doing it).

After internalizing a language, we can take hints without explicit instructions. But, to put it politely, math isn't usually well-internalized.

Let's get clear about who the "math signs" are referring to.

Imaginary numbers are often defined as , and written this way they're utterly baffling.

A better restatement is:

which is still confusing. But what about this:

It's getting clearer. The instructions are: "Start with 1, multiply by i, multiply by i again, and (somehow) end up at -1".

What could do that? Realizing we *start at 1 and end up at -1* helps us visualize something like this:

Aha! is a change (visualized as a rotation) that moves us from positive to negative in two steps (More about imaginary numbers.)

Writing with a clear subject is confusing. (Don't get me started with $i = \sqrt{-1}$)

Missing the implied subject of "1" in caused me years of confusion. I wish this sign was hanging on the classroom wall:

Why is for any value of x? How do we ask for 0 of something and get 1 back out?

Again, let's break it down with a simple example. Here's a typical exponent:

But it's missing a subject. It's written better as:

We start at 1 (our default multiplicative scaling factor), scale by x, then scale by x again, ending up at . The size of the exponent (2) tells us the number of times to use our "times x" scaling machine.

Stepping back, multiplication is about scaling: 3 is really "1 * 3", or the unit quantity enlarged 3 times.

If we want t scale by x (one time), we write:

What if we don't plan on using our scaling machine at all?

The notation is a bit weird, but I'm using empty parenthesis to indicate a lack of action. See, the zero in is that of indifference -- taking no action -- and not obliteration. "Using" means we haven't scaled our original quantity at all.

Subtle, right?

We can take this "growth machine" idea further with the Expand-o-tron 3000.

Let's combine insights. What does a strange exponent like represent?

With our new "implicitly start at 1" perspective, it's really:

Start at 1 and then apply the growth engine. Here, growth is aimed sideways () with enough fuel to last for half a circle (article and video).

The essence of Euler's Identity is that we are starting at 1 and transforming it with a spin. We aren't creating a negative number out of seemingly positive exponents directly.

Calculus has numerous notational shortcuts. When we write:

it really means:

which really means:

Here's the tricky part. There isn't a single "dx", there's a whole chain of them along the number line. The sentence is something like:

*"Hey everyone on the number line! You're all spaced "dx" apart. Take your current position and square it. Then I'll come by and add you all up."*

The integral addresses not a single "you" like 1.0, but "them", the countless positions on the number line.

Find the implied subject in an equation, then work to shorten it (*Bears!*).

Happy math.

]]>"Argh, why couldn't they have explained it like this the first time?"

The difference between a semester of pain and instant understanding was one stupid, missing analogy. It still riles me up thinking about how close I came to missing the key concept (and disliking math).

In class there's the lesson about a specific formula, sure, but the meta-lesson is *how well the experience went*.

What worked? What didn't? How can we get more of the first and less of the second?

Over time, I realized individual topics were chances to explore what truly worked when learning. Not what a learning theorist or book said (*Flashcards! Mnemonics! Just study harder!*), but what *actually* worked for you.

A few of my scattered meta-lessons:

- Analogies, while imperfect, are a huge jump start. It's motivating to get the ball rolling and course correct along the way, vs. waiting to line things up perfectly.
- Nearly every explanation is improved with a visual or diagram.
- Humor and empathy put the reader at ease so they can tell you when they're
*actually*confused (vs. mindless head nodding). - Share the gotchas. The Wise Teacher hiding the 14 mistakes he made when learning the topic does students a disservice.

Every lesson is a chance to silently wonder "How well did that work?".

These days, I use ADEPT as a running checklist for how I get things to click:

This was pulled from actual frustrations (*Why can't they share a plain-English version first? A diagram?*) and I'm sure you'll have modifications of your own.

Don't just take a single lesson away from a lecture, article, or video. Think *How well did that work for me?* and build your learning approach around the best parts.

Happy math.

P.S. My buddy Nasos runs the excellent MetaLearn podcast and we have several chats about learning, a previous interview is below:

]]>Transcript:

What's this? Well, it's a number. Probably written down a few thousand years ago. Great. And people, I suppose, could have read it back then if they spent enough time. And if we want, we can figure it out by spending time on it too (sigh).

So let's see. We have M, that's 1000. Then we have CM, that's 100 less than 1000, or 900. Then we have XC, which is 10 less than 100, or 90. Then we have VII, which is two more than 5, or 7. So the number is 1997.

OK, that wasn't fun. The question: why was that hard? Why was that difficult? Why was one representation so much easier? Well, there's a few answers.

One is that hey, *we're just not smart enough*.

Sorry guys, our brains just can't handle Roman numerals. I mean, computers can use them, why can't we? We're just not smart enough. And we're just bad at math.

That's one answer, and maybe it's possible that humans are incapable of understanding numbers.

But I think the better answer is that those numbers were poorly designed. Roman numerals aren't a great representation of numbers. We're trying to keep track of numbers in terms of two more than five, and one less than ten. We can't just say 9, we need to say "one less than ten", and so on. It makes things really complicated and there's not really a good reason for it. The result is that it's cumbersome, hard to use, and our brains are doing extra work they don't have to.

My goal for learning is to find the analogies that help things make sense. The analogy of numbers being in terms of "two more than five", "one less than ten" isn't a great system. Decimals are much better and that's what we use today.

The idea is that an analogy is like a raft you use to cross a river. The river is the problem, the concept, something that you want to understand, and the raft is how you approach it.

If you don't have an analogy at all, yes, maybe you're strong enough to swim to the other side by pure brute force, but usually you have a mental model that helps you with the problem. The problem is some mental models aren't as good as others. The idea of Roman numerals, while it "works", is not as good as decimals. We're taking a weaker raft to cross the river.

Let's take a look at a few mental models we've tried so far.

Rocks were the first mental model for counting. The word calculus (calculate) comes from "pebble", and the idea that numbers are basically rocks being counted is a powerful analogy. It sounds simple, but it's powerful.

The reason is we've taken out all the differences. There's a bunch of rocks in this photo. Different sizes, colors, weights, everything -- but we're counting them as rocks. And that's the big idea: we take things that are different in countless ways, but find the unifying principle behind them. And now we're counting that.

So the first concept (mental model) is that numbers are like rocks. Numbers are these physical things that unify disparate items and we can count them together and it works pretty well. One, two, three, four, five, six, seven, eight, nine, ten. With enough time we can count a herd of sheep. We can count money, it works pretty well.

But what's this?

Well, it's zero. Zero rocks, zero sheep, zero billion dollars are right there. Zero of the most beautiful painting in the world is right there.

Zero is a really weird concept. And when you've tied your concept of a number to physical items, concepts like zero become difficult.

Now we might work through it and say, "OK, let's allow for nothingness". So we can have "nothingness" as well as "somethingness".

OK, how about this? What's less than nothing? What's a negative rock?

Is it a rock that you owe? Well, if you're holding it, it's not negative, right? Is negative money something I owe you? Again, if I gave it to you, it might be negative to me, but it sure seems positive to you. There's a lot of things that get messed up when we're trying to use a very physical analogy. So the analogy, the raft is starting to break down. It's not helping us understand things.

In fact, even in 1759 people thought negative numbers were confusing. They "darken the very whole doctrines of the equations", right? They just didn't make sense. It was a weird concept and the problem was our analogy just couldn't handle it.

But of course, any modern third grader can tell you we have the concept of a number line. Moving our mental model to a number line let negatives and zero be part of the picture.

Instead of zero as a void, it became the neutral point, the center, the place that every other number moves away from. That's a really cool concept -- positive and negative numbers aren't that different. They're left and right directions away from the same starting point.

Something that was really, really difficult became easy with the right analogy. Let's keep going.

How about something like this? i squared equals -1. This is super confusing. Our raft is starting to shake.

Is there any number that when you square it becomes a negative? Zero squared is still zero. A positive number, when squared, is still positive. A number number, when squared, becomes positive.

So, it seems like there's no solution. But again, this is like trying to find that negative rock. We're trying to apply the analogy in a way that limits our thinking. The analogy isn't the ultimate truth, right?

There might be a different way of thinking that makes it make sense, but we're stuck in this old system.

Here's an idea: If we have a number line going left and right, why not another number line going north and south?

We have East/West, why not North/South too? Let's allow this extra dimension where numbers can move up and down, left or right.

The imaginary dimension lets us get to negatives in two steps. Saying "i squared is negative one" is really saying: I'm starting at 1. We multiply by i, multiply by i again, and get to -1. So i represents a 90-degree rotation, and rotating twice points you backwards. That's what "i squared equals -1" means.

Finding the right analogy makes something that seems baffling click, all because we're using the right metaphor. We're thinking about it the right way. So, we've gone from rocks to lines to directions and that seems pretty good.

Let's put this into action a little bit. Here's a very famous equation. Euler's Identity is one of the most beautiful in math.

If you ask a mathematician or a physicist, this is usually considered the most beautiful identity in all of mathematics. It relates to so many different concepts.

The problem is it's really confusing and again, mathematicians thought it was just crazy when it came out. e to the i times pi equals negative one. We're taking these crazy constants: pi is irrational. i is this weird dimension, and e is another irrational number. All of these combined to equal -1 in such a clean way. It's wild.

And so people thought that OK, there are proofs, but it can't be understood. Again, that's coming from out existing analogies. If we have poor analogies for what numbers are, then yes, concepts like this are difficult, but with the right analogies they can make sense.

So let's take a crack at it. I like to mentally colorize equations. I want to understand what every single part of the equation is trying to say.

Normally we look at equations as a mix of symbols, but if we can identify what each part is doing, it becomes a sentence. It becomes a story of what's actually happening. So let's start.

e represents growth. It could be an entire separate video, but e is the concept of continuous growth, and i is the concept of going sideways (not the normal direction).

So the concept is that we have growth, but instead of growing like we want, we're growing sideways. And how long dow we do this for?

Pi is halfway around a circle. A unit circle has two pi as its full circumference. So a single pi is just halfway around the circle.

We have the concept of growth going sideways that is lasting for half a circle. And if we plot it out, it looks like this:

We have growth, but it's going sideways, and we have enough fuel in the growth engine to point us backwards. So the equation is saying that this system of growth, pushing sideways, with enough fuel to last for half a circle, will point you backwards to negative one.

1 was the implied starting point. Start growing sideways with enough fuel to go half a circle and you'll end up backwards. So the key is we understood a concept that was baffling using a very simple diagram and analogy.

Now to really test it, let's try a few scenarios.

What it we don't put in any fuel?

What if we want to grow sideways, but we decided not to grow it all. We have the engine ready to go, but we don't fuel it up. If we put zero here instead, then we would stay at the starting point of 1.

Mathematically, we can say anything to the zeroth power is 1, which is true, but it's more helpful to think "I have this engine setup but didn't put any fuel into it, I never used it." Then you'll stay at your starting point.

If I put in twice as much fuel (2 * pi), I should circle all the way around. And again, we can mathematically say that if we square both sides, the result should be 1, but I like to think of it as using twice the fuel and going all the way around.

This is how to take an analogy and put it to work, helping you understand.

The main takeaway for me is that good analogies make math a joy. An idea that's baffling can become simple, even enjoyable, if we see it the right way. That's the heart of my philosophy. If I'm confused by something, I don't think my CPU isn't good enough. I just think I had the wrong analogy.

Happy math.

]]>Einstein’s Theory of Special Relativity has many consequences, including the famous equation E = MC² (Energy = Mass x Speed of Light²) and the notion that measurements of space and time are relative to the observer.

In this 30-minute episode we discuss:

- Einstein's backstory and how it influenced his work
- The mechanics of the equation in a way that you can understand
- The implications of the equation for our view of the Universe

It was a lot of fun -- hope you enjoy it.

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