If we had a recipe for pie, there's a few things we might expect. If we doubled the ingredients, we'd probably get double the pie. But if we took the ingredients and cooked them separately, we wouldn't expect to put them together and get our pie back. You can't bake the eggs, flour, and sugar in separate ovens and plop them together at the end.

In math, we can get misleading intuitions about what can (or can't) be rearranged.

After learning addition, we've memorized facts like 2 + 4 = 6. But this might stray into the idea that "whenever I see 2 and 4, I can simplify to 6".

Although 2 + 4 = 6, but "baked(2) + baked(4)" is not "baked(6)". Baking unmixed ingredients in the exponential oven we get:

$2^2 + 4^2 \neq 6^2$

We can only confidently say:

$(2 + 4)^2 = 6^2$

We combine the ingredients, then bake the result. Exponents, like baking an apple pie, modify the original ingredients so they can't be easily combined later. While we might recognize the original 2 and 4, they aren't directly available. Two baked pies can't be smashed together to consolidate the filling.

This confusion gummed me up in calculus, when learning derivatives (the bad boy of baking).

In algebra, we internalize rules like:

\displaystyle{x^2 \cdot x^4 = x^6}

But our intuition leads us astray when we get to the derivative.

\displaystyle{\frac{d}{dx}(x^2) \cdot \frac{d}{dx}(x^4) \neq \frac{d}{dx}x^6}

because

\displaystyle{2x \cdot 4x^3 \neq 6x^5 }

Raw polynomials can be multiplied, but the derivatives of multiplied polynomials can't be rearranged so easily. Multiplication makes functions interact in a way that makes taking the derivative more complex:

Working through the Product Rule we get:

\displaystyle{\frac{d}{dx}(x^2 \cdot x^4) = (\frac{d}{dx}x^2)x^4 + (\frac{d}{dx}x^4)x^2 = (2x)x^4 + (4x^3)x^2 = 6x^5}

When learning Calculus, I was confused how standard interactions (like multiplication) needed special handling. I thought I was done learning new rules for "arithmetic".

But no: functions, when multiplied, interact in funky ways. See how each side grows its own sliver of area (df * g and dg * f)? The functions being multiplied are "baked together" and the overall effect depends on them both, simultaneously. We can't examine them in isolation (e.g., df or dg by itself).

Now, there are setups when the inputs can be processed separately and combined later (linear algebra). The cooking equivalent might be a smoothie: An apple/banana smoothie mixed with a peach/mango smoothie is the same as blending all ingredients in the beginning.

A common assumption is that operations are usually linear, but $\sin(a + b) \not= \sin(a) + \sin(b)$ and $(a + b)^2 \not= a^2 + b^2$. Sorry, we have to carefully cook the ingredients if we want the math to taste right.

When our intuition for a math rule doesn't make sense, ask "Are we making a pie, or a smoothie?"

Other Posts In This Series

  1. Developing Your Intuition For Math
  2. Why Do We Learn Math?
  3. How to Develop a Mindset for Math
  4. Learning math? Think like a cartoonist.
  5. Math As Language: Understanding the Equals Sign
  6. Avoiding The Adjective Fallacy
  7. Finding Unity in the Math Wars
  8. Brevity Is Beautiful
  9. Learn Difficult Concepts with the ADEPT Method
  10. Intuition, Details and the Bow/Arrow Metaphor
  11. Learning To Learn: Intuition Isn't Optional
  12. Learning To Learn: Embrace Analogies
  13. Learning To Learn: Pencil, Then Ink
  14. Learning to Learn: Math Abstraction
  15. Learning Tip: Fix the Limiting Factor
  16. Honest and Realistic Guides for Learning
  17. Empathy-Driven Mathematics
  18. Studying a Course (Machine Learning) with the ADEPT Method
  19. Math and Analogies
  20. Colorized Math Equations
  21. Analogy: Math and Cooking
  22. Learning Math (Mega Man vs. Tetris)