## Lesson 3: Turning Knowledge Into Insight

Last time, we saw that counting could be tricky: are we tracking the fence posts, or the fence pieces?

I'm not ashamed to admit I needed to revisit the basics. Today, we'll mine our knowledge of arithmetic for fresh insights.

## Addition

We have a pretty good handle on addition, right? Ok. Let's say I hand you two quarters:

* What's the total value? 50 cents, clearly.
* What's the total weight? The combined weight, obviously (call it 5 grams + 5 grams = 10 grams).
* What's the total temperature? Um... what?

Sure, both coins have a temperature (room temperature), but you don't *add* them and say it's the equivalent of a single piping-hot coin. Age is similar: we don't think the two coins are the equivalent of a single antique.

There's an unstated intuition that addition can:

- Merge "combinable" properties like weight and value
- Group (but not merge) per-item properties like temperature, age, etc.

We've known this difference instinctively, but we wait until high school or college to label it:

- In math terms, we track non-combinable properties along different dimensions (x-axis, y-axis)
- Getting fancy, we learn about vectors (a list of properties that don't mix) and scalars (a single value that can be combined with other scalars)
- In programming, we learn that addition combines numbers (3 + 5 = 8), but just groups text ("cow" + "boy" = "cowboy", not "cboowy"). Addition works differently depending on the data type.

The key intuition is to ask "How does it combine?" when adding. In math class, we might formally ask "Are these quantities in the same dimension?".

## Multiplication

We first learn multiplication as repeated counting: 2 x 3 means 2 + 2 + 2.

But when editing an image and you see a "2x" button, do you expect it to duplicate the image or zoom in?

Like addition, multiplication seems to have a few interpretations:

- Repeated counting: 2x means changing the quantity
- Scaling up: 2x zoom means taking the original item and enlarging it
- Combining properties: length * width = area, where single dimensions are merged into 2d area

Picking the right interpretation can remove the pain behind confusing concepts. For example, how would you explain why a negative number times a negative number is positive?

Hrm. We can get technical and show how it's the only interpretation that keeps the laws of arithmetic intact. Not convincing.

A negative number is the opposite of a positive one. If savings are positive, debt is negative. If height is positive, going underground is negative.

So, what does it mean to multiply a negative by a negative? With the "multiplication combines properties" interpretation, we're taking the opposite of an opposite... which is the original!

-1 x -1 = 1 means "The opposite of the opposite of 1 is 1". If we have something like -3 x -3 = 9, the logic is similar, except now we're scaling while we take the opposite.

With the traditional "multiplication is counting" metaphor, we go crazy trying to imagine what negative counting means. (Quick, show me -3 cows, and then count them -3 times.)

## Start Labeling Your Intuition

A lot of my math difficulties come from applying the wrong metaphor to a problem, like trying to pull out a screw with pliers. Progress is possible, but laborious.

Intuition is my favorite learning strategy, but it needs to be articulated. (Quick, think about "Democracy" in abstract, without the term.)

After we've described our hunch ("Multiplication could be repetition, zooming, or transferring properties") we can run new problems through the checklist. Hopefully one approach clicks, and the problem unravels: doesn't "the opposite of an opposite" click nicely?

If nothing works, we rummage around until we find a new interpretation, which helps with later problems. It's like the Mega Man video game, where defeating a boss means you can use his power against other villains.

In a way, you *almost* look forward to being confused, because it means there'll eventually be a new intuition in your arsenal (though to be honest, I'd rather things just click the first time).

The long-term goal is to clearly describe our intuitions, and share them so we can benefit from each other's "Aha! That's how it works!" moments.

In the next lesson, we'll look at a few interpretations of exponents.

Happy math,

-Kalid