A musician wakes from a terrible nightmare.
In her dream, she finds herself in a society where music education has been made mandatory.
Since musicians are known to set their ideas down in the form of sheet music, it is imperative
that students become fluent in this language.
It would be ludicrous to expect a child to sing a song or play an instrument without
having a thorough grounding in music notation and theory, playing and listening to music,
let alone composing an original piece are considered very advanced topics, and are generally
put off until college, and more often graduate school.
Of course, not many students actually go on to concentrate in music, so only a few will
ever get to hear the sounds that the black dots represent.
Most students just aren't very good at music.
They are bored in class.
Their skills are terrible, and their homework is barely legible.
Most of them couldn't care less about how important music is in today's world.
They just want to take the minimum number of music courses and be done with it.
I guess there are just music people and non music people.
Meanwhile, on the other side of town, a painter has just awakened from a similar nightmare.
She was in a regular classroom, no easels or tubes of paint.
Since the students don't apply paint until high school, they first need to study colours
and applicators.
In a few years, they'll be ready to take pre-paint by numbers, which prepares them for the main
paint by numbers sequence in high school.
The teacher said that the really excellent painters, the ones who know their colours
and brushes off by heart, get to the actual painting a little sooner.
And some of them even take the advanced placement classes for college credit.
But mostly, teachers are just trying to give the kids a good foundation in what painting
is all about.
So when they get out there in the real world, and paint their kitchen, they don't make a
total mess of it.
This setup is from the introduction to an essay called Lockhart's Lament, where the
author Paul Lockhart argues that our current math education is exactly this kind of nightmare.
The essay has the ability to blow your mind and change the way you think about learning
math.
So I'd like to share some of the ideas from it with you.
There's a link to the full essay in the description.
Lockhart was a successful research mathematician, before he decided to devote his time to teaching
school kids, and now teaches at a school in New York.
He claims that when students say that math class is stupid and boring, that they are
right.
He says that the first thing to understand is that mathematics is an art.
And the difference between math and other arts, like music or painting is that our culture
doesn't recognise it as such.
Mathematicians enjoy thinking about the simplest possible things.
And the simplest possible things are imaginary.
It gives an example of a triangle inside a rectangular box.
But he isn't talking about a drawing of the box or of any real physical object, just the
imaginary concept.
In math things of the way you want them to be.
You have endless choices, and there is no reality to get in your way.
You can then play with your imagination, and wonder how much of the box does the triangle
take up?
You might find something simple and pretty.
Such as if I chopped the rectangle into two pieces like this, I can see that each piece
is cut diagonally in half by the sides of the triangle.
So there is just as much space inside the triangle as outside.
And that means the triangle must take up exactly half the box.
Now, where did this idea come from?
How does one know to draw that line?
Well how does a painter know where to put his brush?
Inspiration, experience, trial and error and luck, that's the art of it, creating these
beautiful little poems of thought, these sonnets of pure reason.
In school, instead of a natural question about shapes and a creative and rewarding process
of invention and discovery, students are treated to: area is equal to half base times height,
a fact to be memorised.
The question has been asked and answered at the same time and there is nothing left for
the student to do.
Lockhart says that if you deny students the opportunity to engage in this activity, to
pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively
frustrated, to have an inspiration and to cobble together their own explanations and
proof.
Well, you deny them mathematics itself.
Those who have become adept at it derive a great deal of self esteem from this success.
The last thing they want to hear is that math is really about raw creativity and aesthetic
sensitivity.
Many a graduate student has come to grief, when they discover, after a decade of being
told they were good at math, that, in fact, they have no real mathematical talent, and
are just very good at following directions.
I felt a bit called out by this paragraph.
And here even after completing a university major in math, I was left wondering if I have
even really encountered real math at all.
And I wondered if I was good at math just because I was quite good at memorising things.
Lockhart says that this cultural problem is a self perpetuating monster.
Students learn about math from their teachers, and teachers learn about it from their teachers.
And people actually think they do know what math is about, and are apparently under the
gross misconception that mathematics is somehow useful to society.
I do quite relate to this next part where he says that kids don't want something that
is relevant to their daily lives.
You think something practical, like compound interest is going to get them excited?
People enjoy fantasy.
And that is just what mathematics can provide, a relief from daily life.
Maths and doesn't need to be made interesting by word problems.
It's already full of some great human stories.
What other subject shuns its primary sources, beautiful works of art, by some of the most
creative minds in history.
Math class lacks problems.
Genuine honest to goodness, natural human questions.
How long is the diagonal of a cube?
Do prime numbers keep going on forever?
Is infinity a number?
How many ways can I symmetrically tile a surface?
The history of mathematics is the history of mankind's engagement with questions like
these, not a mindless regurgitation of formulas and algorithms.
A good question does not just sit there in isolation, but serves as a springboard to
other interesting questions.
If a triangle takes up half its box, then what about a pyramid inside of a three dimensional
box?
There is a place for teaching technique when the student gets to a spot where they need
it to overcome a frustration in their own problem.
The trouble is that math, like painting or poetry is hard creative work.
And that makes it difficult to teach.
As a more serious example, let's take the case of a triangle inside a semicircle.
Now, the beautiful truth about this problem is that no matter where on the circle, you
place the tip of the triangle.
It always forms a nice right angle.
It's not at all clear that this should be true.
It even seems unlikely.
Shouldn't the angle change if I move the tip?
And what we have here is a fantastic math problem.
Is it true?
And if so, why is it true?
But students would be treated to this theorem.
Let the triangle ABC be inscribed in a semicircle with diameter AC, then the angle ABC is a
right angle.
You can compare this unappetizing proof with one devised by one of Paul's seventh graders.
Take the triangle and rotate it around, so it makes a four sided box inside the circle.
Since the triangle got turned completely around, the sides of the box must be parallel.
So it makes a parallelogram.
But it can't be a slanted box, because both of its diagonals are diameters of the circle.
So they are equal, which means it must be an actual rectangle.
And that's why the corner is always a right angle.
A good teacher can guide the discussion and the flow of
problems so as to allow the students to discover and invent mathematics for themselves.
The essay finishes by saying that it's ironic that people dismiss mathematics as the antithesis
of creativity.
They are missing out on an art form older than any book, more profound than any poem,
and more abstract than any abstract.
Now, this essay is a lament not really a guide on how to fix things.
It's one person's opinion on what's wrong with math education, and it was written in
2002.
So maybe some things have changed, but maybe a lot of things are still the same today.
Some of his ideas made me feel inadequate to ever call myself a teacher.
But I hope that keeping these ideas in mind would help all of us to become better teachers
and students.
This video was brought to you by my Patreon supporters, so a big thank you to them.
And a special shout out to today's Patron cat of the day Peach.
Thanks so much for watching to the end of the video, and I hope to see you next time.