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BetterExplained Books for Kindle and Print

Concrete math lessons that slice through the jargon.
Guides , , Popular Developing Your Intuition For Math

Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?

Suppose we want to define a “cat”:

Caveman definition: Evolutionary definition: Modern definition:

The modern definition is precise, sure. But is it the best ? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.

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seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is .

Let’s approach ideas from a different angle. I imagine a circle: the center is the idea you’re studying, and along the outside are the facts describing it. We start in one corner, with one fact or insight, and work our way around to develop our understanding. Cats have common physical traits leads to Cats have a common ancestor leads to A species can be identified by certain portions of DNA . Aha! I can see how the modern definition evolved from the caveman one.

But not all starting points are equal. The right perspective makes math click — and the mathematical “cavemen” who first found an idea often had an enlightening viewpoint. Let’s learn how to build our intuition.

Time for a math example: How do you define a circle?

There are seemingly countless definitions. Here’s a few:

The list goes on, but here’s the key: the facts all describe the same idea! It’s like saying 1, one, uno, eins, “the solution to 2x + 3 = 5″ or “the number of noses on your face” — just different names for the idea of “unity”.

But these initial descriptions are important — they shape our intuition. Because we see circles in the real world before the classroom, we understand their “roundness”. No matter what fancy equation we see (x 2 + y 2 = r 2 ), we know deep inside that a circle is “round”. If we graphed that equation and it appeared square, or lopsided, we’d know there was a mistake.

As children, we learn the “caveman” definition of a circle (a really round thing), which gives us a comfortable intuition. We can see that every point on our “round thing” is the same distance from the center. x 2 + y 2 = r 2 is the analytic way of expressing that fact, using the Pythagorean theorem for distance. We started in one corner, with our intuition, and worked our way around to the formal definition.

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