I cringe when hearing "Math teaches you to think".
It's a well-meaning but ineffective appeal that only satisfies existing fans (see: "Reading takes you anywhere!"). What activity, from crossword puzzles to memorizing song lyrics, doesn't help you think?
Math seems different, and here's why: it's a specific, powerful vocabulary for ideas.
I’ve studied probability and statistics without experiencing them. What’s the difference? What are they trying to do?
This analogy helped:
- Probability is starting with an animal, and figuring out what footprints it will make.
- Statistics is seeing a footprint, and guessing the animal.
What’s algebra about? When learning about variables (x, y, z), they seem to “hide” a number:
What number could be hiding inside of x? 2, in this case.
It seems that arithmetic still works, even when we don’t have the exact numbers up front.
I usually avoid current events, but recent skirmishes in the math world prompted me to chime in. To recap, there’ve been heated discussions about math education and the role of online resources like Khan Academy.
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Last time we tackled derivatives with a “machine” metaphor. Functions are a machine with an input (x) and output (y) lever. The derivative, dy/dx, is how much “output wiggle” we get when we wiggle the input:
Now, we can make a bigger machine from smaller ones (h = f + g, h = f * g, etc.).
The jumble of rules for taking derivatives never truly clicked for me. The addition rule, product rule, quotient rule — how do they fit together? What are we even trying to do?
Here’s my take on derivatives:
- We have a system to analyze, our function f
- The derivative f’ (aka df/dx) is the moment-by-moment behavior
- It turns out f is part of a bigger system (h = f + g)
- Using the behavior of the parts, can we figure out the behavior of the whole?
Why do analogies work so well? They’re building blocks for our thoughts, written in the associative language of our brains.
At first, I thought analogies had to be perfect models of the idea they explained. Nope.
“All models are wrong, but some are useful” – George Box
Analogies are handles to grasp a larger, more slippery idea.
I’ve just added a new feature to the site: an Aha / FAQ section for each article.
You can add an aha! moment or question, and vote / discuss them individually. This extends aha.betterexplained.com, making mini-posts for key ideas in an article.
How do you wish the derivative was explained to you? Here's my take.
Psst! The derivative is the heart of calculus, buried inside this definition:
But what does it mean?
Let's say I gave you a magic newspaper that listed the daily stock market changes for the next few years (+1% Monday, -2% Tuesday...).
I see the dot product as directional multiplication. But multiplication goes beyond repeated counting: it’s applying the essence of one item to another.
Normal multiplication combines growth rates: “3 x 4″ can mean “Take your 3x growth and make it 4x larger (i.e., 12x)”.
Logarithms are everywhere. Ever use any of the following phrases?
- 6 figures
- Double digits
- Order of magnitude
You’re describing numbers in terms of their powers of 10 — a logarithm. Ever mention an interest rate or rate of return? It’s the logarithm of your growth.