Disney explained it best in 'Donald's Classroom' 64 years ago
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Donald opens a door, numbers appear from everywhere, a cartoon is heard saying "P equals 3.141592653589747, etc. etc." Where am I, Donald wonders, only to be answered by the narrator's voice, “In the land of mathematics, the land of great adventure. This is a journey into the wonderland of mathematics."
So begins Walt Disney's excellent 1959 show Donald in Schoolland and theCommonSense; is republishing it as this year we celebrate its 100th anniversary.
The narrator takes Donald on a tour of ancient Greece, tells him about Pythagoras, the father of mathematics and music. He tells him that "You will find mathematics in the most unlikely places...". He explains to him that Pythagoras discovered the octave with a ratio of two to one and that from this harmony and numbers the present musical scale was developed. And that he was the one who discovered that the pentagram was full of mathematics. “Hidden within the pentagram is a secret for creating a golden rectangle, which the Greeks admired for its beautiful proportions and magical properties. (...) The rectangle can reproduce itself indefinitely", he characteristically tells him.
He tells him about the magical spirals that repeat the proportions of the golden section at infinity and explains that "for the Greeks, the golden rectangle represented a mathematical law of beauty. We find it in their classical architecture, the Parthenon. (…) In the centuries that followed, the Golden Rectangle dominated the idea of beauty and architecture throughout the Western world. Notre Dame Cathedral is a great example. Renaissance painters were well aware of this secret. Today, the Golden Rectangle is very much a part of our modern world. Modern painters have rediscovered the magic of these proportions. Indeed, this ideal analogy is found in life itself.'
Donald doesn't seem to care... but the narrator goes on to explain that “All of nature's works have a mathematical logic and her designs are limitless. The magical proportions of the golden ratio are often found in the spirals of nature's designs. The abundance of mathematical forms brings to mind the words of Pythagoras. Everything is arranged according to the number and the mathematical figure. Yes, there is mathematics in music, in art, in almost everything. And as the Greeks had guessed, the rules are always the same."
And he begins to explain to him that mathematics is found in games. Chess for example, a game played in squares, a mathematical contest between two minds, a game enjoyed for centuries by kings and emperors. “In fact, Lewis Carroll, a famous mathematician with a literary mind, used chess as a setting for his classic story. Through the looking glass, Alice came face to face with a not-so-friendly group of chess pieces. (…) Chess is a game of calculated strategy and since the board is geometric, the moves are mathematical. Checkmate and it's game over. Practically all games are played in geometric areas.'
He then gives him other examples of how mathematics exists in various games, baseball, football, basketball, billiards... how everything is based on geometric shapes and how without mathematics we would not even be able to keep score.
“And now,” he tells him, “you're ready for the most exciting game of all. And the playing field for this game is in the mind. Oh, look at your state of mind. Outdated ideas. Confusion. False concepts. Superstitions. Confusion. To think properly, we will have to clear the space…”
Perhaps this is the most unlikely point of the show.
Donald tries to "clean" and "arrange" his mind by formalizing what he encounters. "The shape of things is first discovered in the mind" the narrator urges him and begins to explain the meaning of shapes.
"The mind is the birthplace of all human scientific achievements. The mind knows no bounds when used properly. No pencil is sharp enough to draw as finely as you can imagine, and no paper large enough to hold your imagination. In fact, only in the mind can we conceive the Infinite. Mathematical thinking opened the doors to the exciting adventures of science. Each discovery leads to many others, an endless chain.
These are the doors of the future and the key is mathematics. The limitless treasures of science are locked behind these doors. In time, they will be opened by the curious and inquiring minds of future generations.
In the words of Galileo, mathematics is the alphabet with which God wrote the universe."
End.
Maybe "Donald in the Classroom" is still the best way to explain what math is to a 5-year-old, a 15-year-old and an adult at the same time...
*Cover photo: Paul Ehrenfest at the blackboard lectures at the University of Michigan, Ann Arbor / repository.aip.org