Descartes’ PB&J (or, the Clever Idea at the Heart of High School Math)

In cooking, the biggest breakthroughs come from surprising but natural combinations. Shrimp… and walnuts. Chicken… with peanut sauce. Cookie dough… in ice cream. Mouthwatering as the separate ingredients may be, the real magic happens when you throw them together. If you’re striving for complexity, excitement, and overriding deliciousness, then the unexpected harmony of disparate flavors is your holy grail.

And what’s true for the kitchen goes double for the math classroom. The best math explores deep connections between seemingly unrelated objects.

Take the coordinate plane. Since prehistory, humans have explored geometry—the relationships among shapes. And for over a millennium, mathematicians have investigated algebra—abstract patterns among numbers. But it took the philosopher Renee Descartes to unite the two, to realize that shapes and equations share the same DNA.

Descartes’ breakthrough was to take a pair of numbers—say, x and y—and represent it as a point on a piece of paper. Every pair of numbers yields a point, and every point signifies a pair. Though not entirely new—Descartes built on similar ideas from other mathematicians—the Cartesian plane was a striking step forward. What made it so radically inventive, so uniquely powerful?

It allowed algebraic questions to be reinterpreted as geometric ones—and vice versa.

Take this problem: If x + y = 10, and 2x + 3y = 24, then what could x and y be? We can solve this problem using the standard algebraic techniques; or we can turn it into a geometry problem:

Or take this challenge: Prove that the diagonals of a rectangle are congruent. We could respond in the Euclidean geometric style, using congruent triangles; or we could convert it into an algebra problem:

Descartes had achieved a peanut-butter-and-jelly-caliber breakthrough. The two seemingly disparate fields of geometry and algebra were no mere strangers. They were identical twins—dressed in different clothes, but spawned from the same logical genome.

The hallmark of powerful mathematics is its ability to translate smoothly between different types of problems. Performing those acts of translation requires a theoretical breakthrough like Descartes’, a realization that two seemingly separate universes are actually one.

Virtually our entire high school math curriculum now consists of playing out the consequences of Descartes’ innovation. Flip through any textbook, and you’ll see Descartes’ sticky, peanut-butter-and-jelly-coated fingerprints on almost every page.

19 thoughts on “Descartes’ PB&J (or, the Clever Idea at the Heart of High School Math)

  1. This is why I love math! Surprising and deep connections between seemingly disparate facets of mathematics, such as this connection between equations and shapes, give us a vague glimpse of the true nature of mathematics.

    I didn’t enjoy mathematics until I realized that there was something profound, indescribable, and seemingly unreachable, hidden deep beneath the ideas of numbers and equations, which somehow gives birth to those numbers and equations. Math began to excite me when I realized, in the words of Leonhard Adleman, that mathematics “is less related to accounting than it is to philosophy.”

    Please write more about surprising and deep connections like this!

    1. Thanks for reading! That’s a nice quote about accounting vs. philosophy. Accountants may use some math, but I’d say mathematicians use lots of philosophy.

    1. It’s a shame your class didn’t state the connection explicitly! I think a lot of math teachers feel too busy to zoom out and talk about the big picture for a bit, but I find such time is almost always well-spent.

  2. Always funny and interesting. I don’t know how you do to have always more interesting things to say every week. It’s truly impressive.

    1. Yeah, WordPress made it harder to do them and then I got lazy. But yours is the third complaint I’ve gotten, so I should probably step my game back up.

  3. Rachel Lee
    I always knew that math was built on top of each other, and that what you learn in one chapter of the textbook you will use that strategy for the next. Although I personally don’t like math, I like how that you can build your math skills on one another and be able to solve harder questions. I think this makes math a little easier than it would have been if there were no connections at all

  4. Victor van Niekerk
    I always get the feeling that people who don’t like math are actually just missing the point of math. It’s not about throwing random numbers around, nor is it about wasting time finding useless letters and values. It’s about learning to see how things connect. There are few things in the world as effective to growing brain capacity and exploring critical thinking as doing math.

  5. Daniel lee
    Just like how the result of Columbian Exchange brought the bitter tasting cacao bean and the sweet sugarcane to make a delicious bar of chocolate, we can relate to this on how Descartes found a connection between geometry and algebra. This connection brings math into a whole new level of understanding because this makes many ways to solve for an equation. I believe that some combinations, such as this example, of subject/topic will become beneficial to us for the future.

  6. I’m not really a fan of math, but through the experiences i’ve had in math classes i’ve figured out why its so important. When you go outside and see some building, you wonder how they built it so perfectly, they did it with math. But math isn’t all about that one thing. There are a million ways that math helps us in the world. It helps us to determine connections to stuff. I think that is what makes math so cool, is to see what connections you can discover through it.

  7. Julia You
    I do know that math is related to each other in many ways, and that what we are doing now do covers all stuff that we did when we were younger. Also, if you do not remember what we have been learning, you might have hard time learning new things, as everything is related to each other. I actually thought that Algebra and Geometry is unrelated, but then I found that it is related in many ways. I like learning something new, and it is always amazing to learn new stuff in math, and learning how those formula work on numbers.

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