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Paul Lockhart first "became interested in mathematics when he was 1...

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I teach maths in secondary school. Every time a student makes a cla...

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A Mathematician’s Lament

by Paul Lockhart

musician wakes from a terrible nightmare. In his dream he finds himself in a society where

music education has been made mandatory. “We are helping our students become more

competitive in an increasingly sound-filled world.” Educators, school systems, and the state are

put in charge of this vital project. Studies are commissioned, committees are formed, and

decisions are made— all without the advice or participation of a single working musician or

composer.

Since musicians are known to set down their ideas in the form of sheet music, these curious

black dots and lines must constitute the “language of music.” It is imperative that students

become fluent in this language if they are to attain any degree of musical competence; indeed, 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.

As for the primary and secondary schools, their mission is to train students to use this

language— to jiggle symbols around according to a fixed set of rules: “Music class is where we

take out our staff paper, our teacher puts some notes on the board, and we copy them or

transpose them into a different key. We have to make sure to get the clefs and key signatures

right, and our teacher is very picky about making sure we fill in our quarter-notes completely.

One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit

because I had the stems pointing the wrong way.”

In their wisdom, educators soon realize that even very young children can be given this kind

of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t

completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply

won’t apply himself to his music homework. He says it’s boring. He just sits there staring out

the window, humming tunes to himself and making up silly songs.”

In the higher grades the pressure is really on. After all, the students must be prepared for the

standardized tests and college admissions exams. Students must take courses in Scales and

Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college

when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high

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. Nevertheless, it is important that every

member of society be able to recognize a modulation or a fugal passage, regardless of the fact

that they will never hear one. “To tell you the truth, 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. I had this one kid, though, man was she sensational! Her sheets were

impeccable— every note in the right place, perfect calligraphy, sharps, flats, just beautiful.

She’s going to make one hell of a musician someday.”

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Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy

dream. “Of course!” he reassures himself, “No society would ever reduce such a beautiful and

meaningful art form to something so mindless and trivial; no culture could be so cruel to its

children as to deprive them of such a natural, satisfying means of human expression. How

absurd!”

Meanwhile, on the other side of town, a painter has just awakened from a similar

nightmare…

I was surprised to find myself in a regular school classroom— no easels, no tubes of paint.

“Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh

grade we mostly study colors and applicators.” They showed me a worksheet. On one side were

swatches of color with blank spaces next to them. They were told to write in the names. “I like

painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!”

After class I spoke with the teacher. “So your students don’t actually do any painting?” I

asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main

Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and

apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that.

Of course we track our students by ability. The really excellent painters— the ones who know

their colors and brushes backwards and forwards— they get to the actual painting a little sooner,

and some of them even take the Advanced Placement classes for college credit. But mostly

we’re just trying to give these 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.”

“Um, these high school classes you mentioned…”

“You mean Paint-by-Numbers? We’re seeing much higher enrollments lately. I think it’s

mostly coming from parents wanting to make sure their kid gets into a good college. Nothing

looks better than Advanced Paint-by-Numbers on a high school transcript.”

“Why do colleges care if you can fill in numbered regions with the corresponding color?”

“Oh, well, you know, it shows clear-headed logical thinking. And of course if a student is

planning to major in one of the visual sciences, like fashion or interior decorating, then it’s really

a good idea to get your painting requirements out of the way in high school.”

“I see. And when do students get to paint freely, on a blank canvas?”

“You sound like one of my professors! They were always going on about expressing

yourself and your feelings and things like that—really way-out-there abstract stuff. I’ve got a

degree in Painting myself, but I’ve never really worked much with blank canvasses. I just use

the Paint-by-Numbers kits supplied by the school board.”

***

Sadly, our present system of mathematics education is precisely this kind of nightmare. In

fact, if I had to design a mechanism for the express purpose of destroying a child’s natural

curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being

done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-

crushing ideas that constitute contemporary mathematics education.

Everyone knows that something is wrong. The politicians say, “we need higher standards.”

The schools say, “we need more money and equipment.” Educators say one thing, and teachers

3

say another. They are all wrong. The only people who understand what is going on are the ones

most often blamed and least often heard: the students. They say, “math class is stupid and

boring,” and they are right.

Mathematics and Culture

he first thing to understand is that mathematics is an art. The difference between math and

the other arts, such as music and painting, is that our culture does not recognize it as such.

Everyone understands that poets, painters, and musicians create works of art, and are expressing

themselves in word, image, and sound. In fact, our society is rather generous when it comes to

creative expression; architects, chefs, and even television directors are considered to be working

artists. So why not mathematicians?

Part of the problem is that nobody has the faintest idea what it is that mathematicians do.

The common perception seems to be that mathematicians are somehow connected with

science— perhaps they help the scientists with their formulas, or feed big numbers into

computers for some reason or other. There is no question that if the world had to be divided into

the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the

latter category.

Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical,

subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or

physics (mathematicians conceived of black holes long before astronomers actually found any),

and allows more freedom of expression than poetry, art, or music (which depend heavily on

properties of the physical universe). Mathematics is the purest of the arts, as well as the most

misunderstood.

So let me try to explain what mathematics is, and what mathematicians do. I can hardly do

better than to begin with G.H. Hardy’s excellent description:

A mathematician, like a painter or poet, is a maker

of patterns. If his patterns are more permanent than

theirs, it is because they are made with ideas.

So mathematicians sit around making patterns of ideas. What sort of patterns? What sort of

ideas? Ideas about the rhinoceros? No, those we leave to the biologists. Ideas about language

and culture? No, not usually. These things are all far too complicated for most mathematicians’

taste. If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is

beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest

possible things are imaginary.

For example, if I’m in the mood to think about shapes— and I often am— I might imagine a

triangle inside a rectangular box:

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I wonder how much of the box the triangle takes up? Two-thirds maybe? The important

thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I

talking about some metal triangle forming part of a girder system for a bridge. There’s no

ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing,

amusing yourself with your imagination. For one thing, the question of how much of the box the

triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully

made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes

its size from one minute to the next. That is, unless you want to talk about some sort of

approximate measurements. Well, that’s where the aesthetic comes in. That’s just not simple,

and consequently it is an ugly question which depends on all sorts of real-world details. Let’s

leave that to the scientists. The mathematical question is about an imaginary triangle inside an

imaginary box. The edges are perfect because I want them to be— that is the sort of object I

prefer to think about. This is a major theme in mathematics: things are what you want them to

be. You have endless choices; there is no reality to get in your way.

On the other hand, once you have made your choices (for example I might choose to make

my triangle symmetrical, or not) then your new creations do what they do, whether you like it or

not. This is the amazing thing about making imaginary patterns: they talk back! The triangle

takes up a certain amount of its box, and I don’t have any control over what that amount is.

There is a number out there, maybe it’s two-thirds, maybe it isn’t, but I don’t get to say what it

is. I have to find out what it is.

So we get to play and imagine whatever we want and make patterns and ask questions about

them. But how do we answer these questions? It’s not at all like science. There’s no

experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a

figment of my imagination. The only way to get at the truth about our imaginations is to use our

imaginations, and that is hard work.

In the case of the triangle in its box, I do see something simple and pretty:

If I chop 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.

That means that the triangle must take up exactly half the box!

This is what a piece of mathematics looks and feels like. That little narrative is an example

of the mathematician’s art: asking simple and elegant questions about our imaginary creations,

and crafting satisfying and beautiful explanations. There is really nothing else quite like this

realm of pure idea; it’s fascinating, it’s fun, and it’s free!

Now where did this idea of mine come from? How did I know to draw that line? How does

a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck.

That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason.

There is something so wonderfully transformational about this art form. The relationship

between the triangle and the rectangle was a mystery, and then that one little line made it

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obvious. I couldn’t see, and then all of a sudden I could. Somehow, I was able to create a

profound simple beauty out of nothing, and change myself in the process. Isn’t that what art is

all about?

This is why it is so heartbreaking to see what is being done to mathematics in school. This

rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be

memorized and procedures to be followed. In place of a simple and natural question about

shapes, and a creative and rewarding process of invention and discovery, students are treated to

this:

Triangle Area Formula:

A = 1/2 b h h

b

“The area of a triangle is equal to one-half its base times its height.” Students are asked to

memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill,

the joy, even the pain and frustration of the creative act. There is not even a problem anymore.

The question has been asked and answered at the same time— there is nothing left for the

student to do.

Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing

interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it

helps you to create richer, more nuanced works of art. But it’s not the fact that triangles take up

half their box that matters. What matters is the beautiful idea of chopping it with the line, and

how that might inspire other beautiful ideas and lead to creative breakthroughs in other

problems— something a mere statement of fact can never give you.

By removing the creative process and leaving only the results of that process, you virtually

guarantee that no one will have any real engagement with the subject. It is like saying that

Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be

inspired by that? (And of course it’s actually much worse than this— at least it’s understood that

there is an art of sculpture that I am being prevented from appreciating).

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.

The art is not in the “truth” but in the explanation, the argument. It is the argument itself which

gives the truth its context, and determines what is really being said and meant. Mathematics is

the art of explanation. 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 proofs— you

deny them mathematics itself. So no, I’m not complaining about the presence of facts and

formulas in our mathematics classes, I’m complaining about the lack of mathematics in our

mathematics classes.

f your art teacher were to tell you that painting is all about filling in numbered regions, you

would know that something was wrong. The culture informs you— there are museums and

galleries, as well as the art in your own home. Painting is well understood by society as a

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medium of human expression. Likewise, if your science teacher tried to convince you that

astronomy is about predicting a person’s future based on their date of birth, you would know she

was crazy— science has seeped into the culture to such an extent that almost everyone knows

about atoms and galaxies and laws of nature. But if your math teacher gives you the impression,

either expressly or by default, that mathematics is about formulas and definitions and

memorizing algorithms, who will set you straight?

The cultural problem is a self-perpetuating monster: students learn about math from their

teachers, and teachers learn about it from their teachers, so this lack of understanding and

appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation

of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of

symbols, creates its own culture and its own set of values. Those who have become adept at it

derive a great deal of self-esteem from their 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. Math is not

about following directions, it’s about making new directions.

And I haven’t even mentioned the lack of mathematical criticism in school. At no time are

students let in on the secret that mathematics, like any literature, is created by human beings for

their own amusement; that works of mathematics are subject to critical appraisal; that one can

have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if

it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and

elegant? Does it get me closer to the heart of the matter? Of course there’s no criticism going on

in school— there’s no art being done to criticize!

Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them,

that we think it’s too hard? We seem to feel that they are capable of making arguments and

coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply

that we as a culture don’t know what mathematics is. The impression we are given is of

something very cold and highly technical, that no one could possibly understand— a self-

fulfilling prophesy if there ever was one.

It would be bad enough if the culture were merely ignorant of mathematics, but what is far

worse is that people actually think they do know what math is about— and are apparently under

the gross misconception that mathematics is somehow useful to society! This is already a huge

difference between mathematics and the other arts. Mathematics is viewed by the culture as

some sort of tool for science and technology. Everyone knows that poetry and music are for pure

enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from

the public school curriculum) but no, math is important.

SIMPLICIO: Are you really trying to claim that mathematics offers no useful or

practical applications to society?

SALVIATI: Of course not. I’m merely suggesting that just because something

happens to have practical consequences, doesn’t mean that’s what it is

about. Music can lead armies into battle, but that’s not why people

write symphonies. Michelangelo decorated a ceiling, but I’m sure he

had loftier things on his mind.

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SIMPLICIO: But don’t we need people to learn those useful consequences of math?

Don’t we need accountants and carpenters and such?

SALVIATI: How many people actually use any of this “practical math” they

supposedly learn in school? Do you think carpenters are out there

using trigonometry? How many adults remember how to divide

fractions, or solve a quadratic equation? Obviously the current

practical training program isn’t working, and for good reason: it is

excruciatingly boring, and nobody ever uses it anyway. So why do

people think it’s so important? I don’t see how it’s doing society any

good to have its members walking around with vague memories of

algebraic formulas and geometric diagrams, and clear memories of

hating them. It might do some good, though, to show them

something beautiful and give them an opportunity to enjoy being

creative, flexible, open-minded thinkers— the kind of thing a real

mathematical education might provide.

SIMPLICIO: But people need to be able to balance their checkbooks, don’t they?

SALVIATI: I’m sure most people use a calculator for everyday arithmetic. And

why not? It’s certainly easier and more reliable. But my point is not

just that the current system is so terribly bad, it’s that what it’s missing

is so wonderfully good! Mathematics should be taught as art for art’s

sake. These mundane “useful” aspects would follow naturally as a

trivial by-product. Beethoven could easily write an advertising jingle,

but his motivation for learning music was to create something

beautiful.

SIMPLICIO: But not everyone is cut out to be an artist. What about the kids who

aren’t “math people?” How would they fit into your scheme?

SALVIATI: If everyone were exposed to mathematics in its natural state, with all

the challenging fun and surprises that that entails, I think we would

see a dramatic change both in the attitude of students toward

mathematics, and in our conception of what it means to be “good at

math.” We are losing so many potentially gifted mathematicians—

creative, intelligent people who rightly reject what appears to be a

meaningless and sterile subject. They are simply too smart to waste

their time on such piffle.

SIMPLICIO: But don’t you think that if math class were made more like art class

that a lot of kids just wouldn’t learn anything?

SALVIATI: They’re not learning anything now! Better to not have math classes at

all than to do what is currently being done. At least some people

might have a chance to discover something beautiful on their own.

SIMPLICIO: So you would remove mathematics from the school curriculum?

SALVIATI: The mathematics has already been removed! The only question is

what to do with the vapid, hollow shell that remains. Of course I

would prefer to replace it with an active and joyful engagement with

mathematical ideas.

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SIMPLICIO: But how many math teachers know enough about their subject to

teach it that way?

SALVIATI: Very few. And that’s just the tip of the iceberg…

Mathematics in School

here is surely no more reliable way to kill enthusiasm and interest in a subject than to make

it a mandatory part of the school curriculum. Include it as a major component of

standardized testing and you virtually guarantee that the education establishment will suck the

life out of it. School boards do not understand what math is, neither do educators, textbook

authors, publishing companies, and sadly, neither do most of our math teachers. The scope of

the problem is so enormous, I hardly know where to begin.

Let’s start with the “math reform” debacle. For many years there has been a growing

awareness that something is rotten in the state of mathematics education. Studies have been

commissioned, conferences assembled, and countless committees of teachers, textbook

publishers, and educators (whatever they are) have been formed to “fix the problem.” Quite

apart from the self-serving interest paid to reform by the textbook industry (which profits from

any minute political fluctuation by offering up “new” editions of their unreadable monstrosities),

the entire reform movement has always missed the point. The mathematics curriculum doesn’t

need to be reformed, it needs to be scrapped.

All this fussing and primping about which “topics” should be taught in what order, or the use

of this notation instead of that notation, or which make and model of calculator to use, for god’s

sake— it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason.

To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration;

to be in a state of confusion— not because it makes no sense to you, but because you gave it

sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to

be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive,

damn it. Remove this from mathematics and you can have all the conferences you like; it won’t

matter. Operate all you want, doctors: your patient is already dead.

The saddest part of all this “reform” are the attempts to “make math interesting” and

“relevant to kids’ lives.” You don’t need to make math interesting— it’s already more

interesting than we can handle! And the glory of it is its complete irrelevance to our lives.

That’s why it’s so fun!

Attempts to present mathematics as relevant to daily life inevitably appear forced and

contrived: “You see kids, if you know algebra then you can figure out how old Maria is if we

know that she is two years older than twice her age seven years ago!” (As if anyone would ever

have access to that ridiculous kind of information, and not her age.) Algebra is not about daily

life, it’s about numbers and symmetry— and this is a valid pursuit in and of itself:

Suppose I am given the sum and difference of two numbers. How

can I figure out what the numbers are themselves?

Here is a simple and elegant question, and it requires no effort to be made appealing. The

ancient Babylonians enjoyed working on such problems, and so do our students. (And I hope

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you will enjoy thinking about it too!) We don’t need to bend over backwards to give

mathematics relevance. It has relevance in the same way that any art does: that of being a

meaningful human experience.

In any case, do you really think kids even 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, an

anodyne to the practical workaday world.

A similar problem occurs when teachers or textbooks succumb to “cutesyness.” This is

where, in an attempt to combat so-called “math anxiety” (one of the panoply of diseases which

are actually caused by school), math is made to seem “friendly.” To help your students

memorize formulas for the area and circumference of a circle, for example, you might invent this

whole story about “Mr. C,” who drives around “Mrs. A” and tells her how nice his “two pies

are” (C = 2πr) and how her “pies are square” (A = πr

2

) or some such nonsense. But what about

the real story? The one about mankind’s struggle with the problem of measuring curves; about

Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which

is more interesting— measuring the rough dimensions of a circular piece of graph paper, using a

formula that someone handed you without explanation (and made you memorize and practice

over and over) or hearing the story of one of the most beautiful, fascinating problems, and one of

the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles

for god’s sake!

Why aren’t we giving our students a chance to even hear about these things, let alone giving

them an opportunity to actually do some mathematics, and to come up with their own ideas,

opinions, and reactions? What other subject is routinely taught without any mention of its

history, philosophy, thematic development, aesthetic criteria, and current status? What other

subject shuns its primary sources— beautiful works of art by some of the most creative minds in

history— in favor of third-rate textbook bastardizations?

The main problem with school mathematics is that there are no problems. Oh, I know what

passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here

is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad

way to learn mathematics: to be a trained chimpanzee.

But a problem, a genuine honest-to-goodness natural human question— that’s another thing.

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 the mindless regurgitation of

formulas and algorithms (together with contrived exercises designed to make use of them).

A good problem is something you don’t know how to solve. That’s what makes it a good

puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as

a springboard to other interesting questions. A triangle takes up half its box. What about a

pyramid inside its three-dimensional box? Can we handle this problem in a similar way?

I can understand the idea of training students to master certain techniques— I do that too.

But not as an end in itself. Technique in mathematics, as in any art, should be learned in context.

The great problems, their history, the creative process— that is the proper setting. Give your

students a good problem, let them struggle and get frustrated. See what they come up with.

Wait until they are dying for an idea, then give them some technique. But not too much.

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So put away your lesson plans and your overhead projectors, your full-color textbook

abominations, your CD-ROMs and the whole rest of the traveling circus freak show of

contemporary education, and simply do mathematics with your students! Art teachers don’t

waste their time with textbooks and rote training in specific techniques. They do what is natural

to their subject— they get the kids painting. They go around from easel to easel, making

suggestions and offering guidance:

“I was thinking about our triangle problem, and I noticed something. If the triangle is really

slanted then it doesn’t take up half it’s box! See, look:

“Excellent observation! Our chopping argument assumes that the tip of the triangle lies

directly over the base. Now we need a new idea.”

“Should I try chopping it a different way?”

“Absolutely. Try all sorts of ideas. Let me know what you come up with!”

o how do we teach our students to do mathematics? By choosing engaging and natural

problems suitable to their tastes, personalities, and level of experience. By giving them time

to make discoveries and formulate conjectures. By helping them to refine their arguments and

creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and

open to sudden changes in direction to which their curiosity may lead. In short, by having an

honest intellectual relationship with our students and our subject.

Of course what I’m suggesting is impossible for a number of reasons. Even putting aside the

fact that statewide curricula and standardized tests virtually eliminate teacher autonomy, I doubt

that most teachers even want to have such an intense relationship with their students. It requires

too much vulnerability and too much responsibility— in short, it’s too much work!

It is far easier to be a passive conduit of some publisher’s “materials” and to follow the

shampoo-bottle instruction “lecture, test, repeat” than to think deeply and thoughtfully about the

meaning of one’s subject and how best to convey that meaning directly and honestly to one’s

students. We are encouraged to forego the difficult task of making decisions based on our

individual wisdom and conscience, and to “get with the program.” It is simply the path of least

resistance:

TEXTBOOK PUBLISHERS : TEACHERS ::

A) pharmaceutical companies : doctors

B) record companies : disk jockeys

C) corporations : congressmen

D) all of the above

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The trouble is that math, like painting or poetry, is hard creative work. That makes it very

difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work

of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules

than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an

actual book with a point of view.

Mathematics is an art, and art should be taught by working artists, or if not, at least by people

who appreciate the art form and can recognize it when they see it. It is not necessary that you

learn music from a professional composer, but would you want yourself or your child to be

taught by someone who doesn’t even play an instrument, and has never listened to a piece of

music in their lives? Would you accept as an art teacher someone who has never picked up a

pencil or stepped foot in a museum? Why is it that we accept math teachers who have never

produced an original piece of mathematics, know nothing of the history and philosophy of the

subject, nothing about recent developments, nothing in fact beyond what they are expected to

present to their unfortunate students? What kind of a teacher is that? How can someone teach

something that they themselves don’t do? I can’t dance, and consequently I would never

presume to think that I could teach a dance class (I could try, but it wouldn’t be pretty). The

difference is I know I can’t dance. I don’t have anyone telling me I’m good at dancing just

because I know a bunch of dance words.

Now I’m not saying that math teachers need to be professional mathematicians— far from it.

But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it?

If teaching is reduced to mere data transmission, if there is no sharing of excitement and

wonder, if teachers themselves are passive recipients of information and not creators of new

ideas, what hope is there for their students? If adding fractions is to the teacher an arbitrary set

of rules, and not the outcome of a creative process and the result of aesthetic choices and desires,

then of course it will feel that way to the poor students.

Teaching is not about information. It’s about having an honest intellectual relationship with

your students. It requires no method, no tools, and no training. Just the ability to be real. And if

you can’t be real, then you have no right to inflict yourself upon innocent children.

In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you

can take classes in early childhood development and whatnot, and you can be trained to use a

blackboard “effectively” and to prepare an organized “lesson plan” (which, by the way, insures

that your lesson will be planned, and therefore false), but you will never be a real teacher if you

are unwilling to be a real person. Teaching means openness and honesty, an ability to share

excitement, and a love of learning. Without these, all the education degrees in the world won’t

help you, and with them they are completely unnecessary.

It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are

naturally curious like anyone else. Just talk to them! And more importantly, listen to them!

SIMPLICIO: All right, I understand that there is an art to mathematics and that we

are not doing a good job of exposing people to it. But isn’t this a

rather esoteric, highbrow sort of thing to expect from our school

system? We’re not trying to create philosophers here, we just want

people to have a reasonable command of basic arithmetic so they can

function in society.

12

SALVIATI: But that’s not true! School mathematics concerns itself with many

things that have nothing to do with the ability to get along in society—

algebra and trigonometry, for instance. These studies are utterly

irrelevant to daily life. I’m simply suggesting that if we are going to

include such things as part of most students’ basic education, that we

do it in an organic and natural way. Also, as I said before, just because

a subject happens to have some mundane practical use does not mean

that we have to make that use the focus of our teaching and learning.

It may be true that you have to be able to read in order to fill out

forms at the DMV, but that’s not why we teach children to read. We

teach them to read for the higher purpose of allowing them access to

beautiful and meaningful ideas. Not only would it be cruel to teach

reading in such a way— to force third graders to fill out purchase

orders and tax forms— it wouldn’t work! We learn things because

they interest us now, not because they might be useful later. But this

is exactly what we are asking children to do with math.

SIMPLICIO: But don’t we need third graders to be able to do arithmetic?

SALVIATI: Why? You want to train them to calculate 427 plus 389? It’s just not a

question that very many eight-year-olds are asking. For that matter,

most adults don’t fully understand decimal place-value arithmetic, and

you expect third graders to have a clear conception? Or do you not

care if they understand it? It is simply too early for that kind of

technical training. Of course it can be done, but I think it ultimately

does more harm than good. Much better to wait until their own

natural curiosity about numbers kicks in.

SIMPLICIO: Then what should we do with young children in math class?

SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon,

Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose

them to situations where deductive reasoning is necessary. Don’t

worry about notation and technique, help them to become active and

creative mathematical thinkers.

SIMPLICIO: It seems like we’d be taking an awful risk. What if we de-emphasize

arithmetic so much that our students end up not being able to add and

subtract?

SALVIATI: I think the far greater risk is that of creating schools devoid of creative

expression of any kind, where the function of the students is to

memorize dates, formulas, and vocabulary lists, and then regurgitate

them on standardized tests—“Preparing tomorrow’s workforce today!”

SIMPLICIO: But surely there is some body of mathematical facts of which an

educated person should be cognizant.

SALVIATI: Yes, the most important of which is that mathematics is an art form

done by human beings for pleasure! Alright, yes, it would be nice if

people knew a few basic things about numbers and shapes, for

instance. But this will never come from rote memorization, drills,

lectures, and exercises. You learn things by doing them and you

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remember what matters to you. We have millions of adults wandering

around with “negative b plus or minus the square root of b squared

minus 4ac all over 2a” in their heads, and absolutely no idea whatsoever

what it means. And the reason is that they were never given the

chance to discover or invent such things for themselves. They never

had an engaging problem to think about, to be frustrated by, and to

create in them the desire for technique or method. They were never

told the history of mankind’s relationship with numbers— no ancient

Babylonian problem tablets, no Rhind Papyrus, no Liber Abaci, no Ars

Magna. More importantly, no chance for them to even get curious

about a question; it was answered before they could ask it.

SIMPLICIO: But we don’t have time for every student to invent mathematics for

themselves! It took centuries for people to discover the Pythagorean

Theorem. How can you expect the average child to do it?

SALVIATI: I don’t. Let’s be clear about this. I’m complaining about the complete

absence of art and invention, history and philosophy, context and

perspective from the mathematics curriculum. That doesn’t mean that

notation, technique, and the development of a knowledge base have no

place. Of course they do. We should have both. If I object to a

pendulum being too far to one side, it doesn’t mean I want it to be all

the way on the other side. But the fact is, people learn better when

the product comes out of the process. A real appreciation for poetry

does not come from memorizing a bunch of poems, it comes from

writing your own.

SIMPLICIO: Yes, but before you can write your own poems you need to learn the

alphabet. The process has to begin somewhere. You have to walk

before you can run.

SALVIATI: No, you have to have something you want to run toward. Children can

write poems and stories as they learn to read and write. A piece of

writing by a six-year-old is a wonderful thing, and the spelling and

punctuation errors don’t make it less so. Even very young children can

invent songs, and they haven’t a clue what key it is in or what type of

meter they are using.

SIMPLICIO: But isn’t math different? Isn’ t math a language of its own, with all

sorts of symbols that have to be learned before you can use it?

SALVIATI: Not at all. Mathematics is not a language, it’s an adventure. Do

musicians “speak another language” simply because they choose to

abbreviate their ideas with little black dots? If so, it’s no obstacle to

the toddler and her song. Yes, a certain amount of mathematical

shorthand has evolved over the centuries, but it is in no way essential.

Most mathematics is done with a friend over a cup of coffee, with a

diagram scribbled on a napkin. Mathematics is and always has been

about ideas, and a valuable idea transcends the symbols with which you

choose to represent it. As Gauss once remarked, “What we need are

notions, not notations.”

14

SIMPLICIO: But isn’t one of the purposes of mathematics education to help

students think in a more precise and logical way, and to develop their

“quantitative reasoning skills?” Don’t all of these definitions and

formulas sharpen the minds of our students?

SALVIATI: No they don’t. If anything, the current system has the opposite effect

of dulling the mind. Mental acuity of any kind comes from solving

problems yourself, not from being told how to solve them.

SIMPLICIO: Fair enough. But what about those students who are interested in

pursuing a career in science or engineering? Don’t they need the

training that the traditional curriculum provides? Isn’t that why we

teach mathematics in school?

SALVIATI: How many students taking literature classes will one day be writers?

That is not why we teach literature, nor why students take it. We

teach to enlighten everyone, not to train only the future professionals.

In any case, the most valuable skill for a scientist or engineer is being

able to think creatively and independently. The last thing anyone

needs is to be trained.

The Mathematics Curriculum

he truly painful thing about the way mathematics is taught in school is not what is missing—

the fact that there is no actual mathematics being done in our mathematics classes— but

what is there in its place: the confused heap of destructive disinformation known as “the

mathematics curriculum.” It is time now to take a closer look at exactly what our students are up

against— what they are being exposed to in the name of mathematics, and how they are being

harmed in the process.

The most striking thing about this so-called mathematics curriculum is its rigidity. This is

especially true in the later grades. From school to school, city to city, and state to state, the same

exact things are being said and done in the same exact way and in the same exact order. Far

from being disturbed and upset by this Orwellian state of affairs, most people have simply

accepted this “standard model” math curriculum as being synonymous with math itself.

This is intimately connected to what I call the “ladder myth”— the idea that mathematics can

be arranged as a sequence of “subjects” each being in some way more advanced, or “higher”

than the previous. The effect is to make school mathematics into a race— some students are

“ahead” of others, and parents worry that their child is “falling behind.” And where exactly does

this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve

been cheated out of a mathematical education, and you don’t even know it.

Real mathematics doesn’t come in a can— there is no such thing as an Algebra II idea.

Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of

the subject, and a teacher’s own path through the standard curriculum reinforces this myth and

prevents him or her from seeing mathematics as an organic whole. As a result, we have a math

curriculum with no historical perspective or thematic coherence, a fragmented collection of

assorted topics and techniques, united only by the ease in which they can be reduced to step-by-

step procedures.

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In place of discovery and exploration, we have rules and regulations. We never hear a student

saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I

found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have

teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no

mention of the aesthetics behind this choice, or even that it is a choice.

In place of meaningful problems, which might lead to a synthesis of diverse ideas, to

uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in

mathematics, we have instead joyless and redundant exercises, specific to the technique under

discussion, and so disconnected from each other and from mathematics as a whole that neither

the students nor their teacher have the foggiest idea how or why such a thing might have come

up in the first place.

In place of a natural problem context in which students can make decisions about what they

want their words to mean, and what notions they wish to codify, they are instead subjected to an

endless sequence of unmotivated and a priori “definitions.” The curriculum is obsessed with

jargon and nomenclature, seemingly for no other purpose than to provide teachers with

something to test the students on. No mathematician in the world would bother making these

senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re

equal for crying out loud. They are the same exact numbers, and have the same exact properties.

Who uses such words outside of fourth grade?

Of course it is far easier to test someone’s knowledge of a pointless definition than to inspire

them to create something beautiful and to find their own meaning. Even if we agree that a basic

common vocabulary for mathematics is valuable, this isn’t it. How sad that fifth-graders are

taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use

words like “conjecture,” and “counterexample.” High school students must learn to use the

secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’

(a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” ) That

this particular shorthand, a holdover from fifteenth century nautical tables, is still with us

(whereas others, such as the “versine” have died out) is mere historical accident, and is of utterly

no value in an era when rapid and precise shipboard computation is no longer an issue. Thus we

clutter our math classes with pointless nomenclature for its own sake.

In practice, the curriculum is not even so much a sequence of topics, or ideas, as it is a

sequence of notations. Apparently mathematics consists of a secret list of mystical symbols and

rules for their manipulation. Young children are given ‘+’ and ‘÷.’ Only later can they be

entrusted with ‘√¯,’ and then ‘x’ and ‘y’ and the alchemy of parentheses. Finally, they are

indoctrinated in the use of ‘sin,’ ‘log,’ ‘f(x),’ and if they are deemed worthy, ‘d’ and ‘∫.’ All

without having had a single meaningful mathematical experience.

This program is so firmly fixed in place that teachers and textbook authors can reliably

predict, years in advance, exactly what students will be doing, down to the very page of

exercises. It is not at all uncommon to find second-year algebra students being asked to calculate

[ f(x + h) – f(x) ] / h for various functions f, so that they will have “seen” this when they take

calculus a few years later. Naturally no motivation is given (nor expected) for why such a

seemingly random combination of operations would be of interest, although I’m sure there are

many teachers who try to explain what such a thing might mean, and think they are doing their

students a favor, when in fact to them it is just one more boring math problem to be gotten over

with. “What do they want me to do? Oh, just plug it in? OK.”

16

Another example is the training of students to express information in an unnecessarily

complicated form, merely because at some distant future period it will have meaning. Does any

middle school algebra teacher have the slightest clue why he is asking his students to rephrase

“the number x lies between three and seven” as |x - 5| < 2 ? Do these hopelessly inept textbook

authors really believe they are helping students by preparing them for a possible day, years

hence, when they might be operating within the context of a higher-dimensional geometry or an

abstract metric space? I doubt it. I expect they are simply copying each other decade after

decade, maybe changing the fonts or the highlight colors, and beaming with pride when an

school system adopts their book, and becomes their unwitting accomplice.

Mathematics is about problems, and problems must be made the focus of a students

mathematical life. Painful and creatively frustrating as it may be, students and their teachers

should at all times be engaged in the process— having ideas, not having ideas, discovering

patterns, making conjectures, constructing examples and counterexamples, devising arguments,

and critiquing each other’s work. Specific techniques and methods will arise naturally out of this

process, as they did historically: not isolated from, but organically connected to, and as an

outgrowth of, their problem-background.

English teachers know that spelling and pronunciation are best learned in a context of reading

and writing. History teachers know that names and dates are uninteresting when removed from

the unfolding backstory of events. Why does mathematics education remain stuck in the

nineteenth century? Compare your own experience of learning algebra with Bertrand Russell’s

recollection:

“I was made to learn by heart: ‘The square of the sum of two

numbers is equal to the sum of their squares increased by twice

their product.’ I had not the vaguest idea what this meant and

when I could not remember the words, my tutor threw the book at

my head, which did not stimulate my intellect in any way.”

Are things really any different today?

SIMPLICIO: I don’t think that’s very fair. Surely teaching methods have improved

since then.

SALVIATI: You mean training methods. Teaching is a messy human relationship;

it does not require a method. Or rather I should say, if you need a

method you’re probably not a very good teacher. If you don’t have

enough of a feeling for your subject to be able to talk about it in your

own voice, in a natural and spontaneous way, how well could you

understand it? And speaking of being stuck in the nineteenth century,

isn’t it shocking how the curriculum itself is stuck in the seventeenth?

To think of all the amazing discoveries and profound revolutions in

mathematical thought that have occurred in the last three centuries!

There is no more mention of these than if they had never happened.

SIMPLICIO: But aren’t you asking an awful lot from our math teachers? You

expect them to provide individual attention to dozens of students,

guiding them on their own paths toward discovery and enlightenment,

and to be up on recent mathematical history as well?

17

SALVIATI: Do you expect your art teacher to be able to give you individualized,

knowledgeable advice about your painting? Do you expect her to

know anything about the last three hundred years of art history? But

seriously, I don’t expect anything of the kind, I only wish it were so.

SIMPLICIO: So you blame the math teachers?

SALVIATI: No, I blame the culture that produces them. The poor devils are

trying their best, and are only doing what they’ve been trained to do.

I’m sure most of them love their students and hate what they are being

forced to put them through. They know in their hearts that it is

meaningless and degrading. They can sense that they have been made

cogs in a great soul-crushing machine, but they lack the perspective

needed to understand it, or to fight against it. They only know they

have to get the students “ready for next year.”

SIMPLICIO: Do you really think that most students are capable of operating on

such a high level as to create their own mathematics?

SALVIATI: If we honestly believe that creative reasoning is too “high” for our

students, and that they can’t handle it, why do we allow them to write

history papers or essays about Shakespeare? The problem is not that

the students can’t handle it, it’s that none of the teachers can. They’ve

never proved anything themselves, so how could they possibly advise a

student? In any case, there would obviously be a range of student

interest and ability, as there is in any subject, but at least students

would like or dislike mathematics for what it really is, and not for this

perverse mockery of it.

SIMPLICIO: But surely we want all of our students to learn a basic set of facts and

skills. That’s what a curriculum is for, and that’s why it is so

uniform— there are certain timeless, cold hard facts we need our

students to know: one plus one is two, and the angles of a triangle add

up to 180 degrees. These are not opinions, or mushy artistic feelings.

SALVIATI: On the contrary. Mathematical structures, useful or not, are invented

and developed within a problem context, and derive their meaning

from that context. Sometimes we want one plus one to equal zero (as

in so-called ‘mod 2’ arithmetic) and on the surface of a sphere the

angles of a triangle add up to more than 180 degrees. There are no

“facts” per se; everything is relative and relational. It is the story that

matters, not just the ending.

SIMPLICIO: I’m getting tired of all your mystical mumbo-jumbo! Basic arithmetic,

all right? Do you or do you not agree that students should learn it?

SALVIATI: That depends on what you mean by “it.” If you mean having an

appreciation for the problems of counting and arranging, the

advantages of grouping and naming, the distinction between a

representation and the thing itself, and some idea of the historical

development of number systems, then yes, I do think our students

should be exposed to such things. If you mean the rote memorization

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of arithmetic facts without any underlying conceptual framework, then

no. If you mean exploring the not at all obvious fact that five groups

of seven is the same as seven groups of five, then yes. If you mean

making a rule that 5 x 7 = 7 x 5, then no. Doing mathematics should

always mean discovering patterns and crafting beautiful and

meaningful explanations.

SIMPLICIO: What about geometry? Don’t students prove things there? Isn’t High

School Geometry a perfect example of what you want math classes to

be?

High School Geometry: Instrument of the Devil

here is nothing quite so vexing to the author of a scathing indictment as having the primary

target of his venom offered up in his support. And never was a wolf in sheep’s clothing as

insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it

is school’s attempt to introduce students to the art of argument that makes it so very dangerous.

Posing as the arena in which students will finally get to engage in true mathematical

reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative

rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject,

and permanently disabling them from thinking about math in a natural and intuitive way.

The mechanism behind this is subtle and devious. The student-victim is first stunned and

paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly

and painstakingly weaned away from any natural curiosity or intuition about shapes and their

patterns by a systematic indoctrination into the stilted language and artificial format of so-called

“formal geometric proof.”

All metaphor aside, geometry class is by far the most mentally and emotionally destructive

component of the entire K-12 mathematics curriculum. Other math courses may hide the

beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured.

(Apparently I am incapable of putting all metaphor aside.)

What is happening is the systematic undermining of the student’s intuition. A proof, that is,

a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof

should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted

argument should feel like a splash of cool water, and be a beacon of light— it should refresh the

spirit and illuminate the mind. And it should be charming.

There is nothing charming about what passes for proof in geometry class. Students are

presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a

format as unnecessary and inappropriate as insisting that children who wish to plant a garden

refer to their flowers by genus and species.

Let’s look at some specific instances of this insanity. We’ll begin with the example of two

crossed lines:

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Now the first thing that usually happens is the unnecessary muddying of the waters with

excessive notation. Apparently, one cannot simply speak of two crossed lines; one must give

elaborate names to them. And not simple names like ‘line 1’ and ‘line 2,’ or even ‘a’ and ‘b.’

We must (according to High School Geometry) select random and irrelevant points on these

lines, and then refer to the lines using the special “line notation.”

You see, now we get to call them AB and CD. And God forbid you should omit the little bars

on top— ‘AB’ refers to the length of the line AB (at least I think that’s how it works). Never

mind how pointlessly complicated it is, this is the way one must learn to do it. Now comes the

actual statement, usually referred to by some absurd name like

PROPOSITION 2.1.1.

Let

A

B and

C

D intersect at P. Then ∠APC ≅ ∠BPD.

In other words, the angles on both sides are the same. Well, duh! The configuration of two

crossed lines is symmetrical for crissake. And as if this wasn’t bad enough, this patently obvious

statement about lines and angles must then be “proved.”

Proof:

Statement

Reason

1. m∠APC + m∠APD = 180

1. Angle Addition Postulate

m∠BPD + m∠APD = 180

2. m∠APC + m∠APD = m∠BPD + m∠APD

2. Substitution Property

3. m∠APD = m∠APD

3. Reflexive Property of Equality

4. m∠APC = m∠BPD

4. Subtraction Property of Equality

5. ∠APC ≅ ∠BPD

5. Angle Measurement Postulate

Instead of a witty and enjoyable argument written by an actual human being, and conducted

in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic form-

letter of a proof. And what a mountain being made of a molehill! Do we really want to suggest

that a straightforward observation like this requires such an extensive preamble? Be honest: did

you actually even read it? Of course not. Who would want to?

The effect of such a production being made over something so simple is to make people

doubt their own intuition. Calling into question the obvious, by insisting that it be “rigorously

B

C

D

A

B

C

D

A

P

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proved” (as if the above even constitutes a legitimate formal proof) is to say to a student, “Your

feelings and ideas are suspect. You need to think and speak our way.”

Now there is a place for formal proof in mathematics, no question. But that place is not a

student’s first introduction to mathematical argument. At least let people get familiar with some

mathematical objects, and learn what to expect from them, before you start formalizing

everything. Rigorous formal proof only becomes important when there is a crisis— when you

discover that your imaginary objects behave in a counterintuitive way; when there is a paradox

of some kind. But such excessive preventative hygiene is completely unnecessary here—

nobody’s gotten sick yet! Of course if a logical crisis should arise at some point, then obviously

it should be investigated, and the argument made more clear, but that process can be carried out

intuitively and informally as well. In fact it is the soul of mathematics to carry out such a

dialogue with one’s own proof.

So not only are most kids utterly confused by this pedantry— nothing is more mystifying

than a proof of the obvious— but even those few whose intuition remains intact must then

retranslate their excellent, beautiful ideas back into this absurd hieroglyphic framework in order

for their teacher to call it “correct.” The teacher then flatters himself that he is somehow

sharpening his students’ minds.

As a more serious example, let’s take the case of a triangle inside a semicircle:

Now the beautiful truth about this pattern is that no matter where on the circle you place the

tip of the triangle, it always forms a nice right angle. (I have no objection to a term like “right

angle” if it is relevant to the problem and makes it easier to discuss. It’s not terminology itself

that I object to, it’s pointless unnecessary terminology. In any case, I would be happy to use

“corner” or even “pigpen” if a student preferred.)

Here is a case where our intuition is somewhat in doubt. 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? What we have

here is a fantastic math problem! Is it true? If so, why is it true? What a great project! What a

terrific opportunity to exercise one’s ingenuity and imagination! Of course no such opportunity

is given to the students, whose curiosity and interest is immediately deflated by:

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THEOREM 9.5. Let ∆ABC be inscribed in a semicircle with diameter

A

C.

Then ∠ABC is a right angle.

Proof:

Statement

Reason

1. Draw radius OB. Then OB = OC = OA

1. Given

2. m∠OBC = m∠BCA

m∠OBA = m∠BAC

2. Isosceles Triangle Theorem

3. m∠ABC = m∠OBA + m∠OBC

3. Angle Sum Postulate

4. m∠ABC + m∠BCA + m∠BAC = 180

4. The sum of the angles of a triangle is 180

5. m∠ABC + m∠OBC + m∠OBA = 180

5. Substitution (line 2)

6. 2 m∠ABC = 180

6. Substitution (line 3)

7. m∠ABC = 90

7. Division Property of Equality

8. ∠ABC is a right angle

8. Definition of Right Angle

Could anything be more unattractive and inelegant? Could any argument be more

obfuscatory and unreadable? This isn’t mathematics! A proof should be an epiphany from the

Gods, not a coded message from the Pentagon. This is what comes from a misplaced sense of

logical rigor: ugliness. The spirit of the argument has been buried under a heap of confusing

formalism.

No mathematician works this way. No mathematician has ever worked this way. This is a

complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about

erecting barriers between ourselves and our intuition, and making simple things complicated.

Mathematics is about removing obstacles to our intuition, and keeping simple things simple.

Compare this unappetizing mess of a proof with the following argument devised by one of

my 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’re equal, which means it must be an actual rectangle.

That’s why the corner is always a right angle.”

Isn’t that just delightful? And the point isn’t whether this argument is any better than the

other one as an idea, the point is that the idea comes across. (As a matter of fact, the idea of the

first proof is quite pretty, albeit seen as through a glass, darkly.)

B

A

C

O

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More importantly, the idea was the student’s own. The class had a nice problem to work on,

conjectures were made, proofs were attempted, and this is what one student came up with. Of

course it took several days, and was the end result of a long sequence of failures.

To be fair, I did paraphrase the proof considerably. The original was quite a bit more

convoluted, and contained a lot of unnecessary verbiage (as well as spelling and grammatical

errors). But I think I got the feeling of it across. And these defects were all to the good; they

gave me something to do as a teacher. I was able to point out several stylistic and logical

problems, and the student was then able to improve the argument. For instance, I wasn’t

completely happy with the bit about both diagonals being diameters— I didn’t think that was

entirely obvious— but that only meant there was more to think about and more understanding to

be gained from the situation. And in fact the student was able to fill in this gap quite nicely:

“Since the triangle got rotated halfway around the circle, the tip

must end up exactly opposite from where it started. That’s why

the diagonal of the box is a diameter.”

So a great project and a beautiful piece of mathematics. I’m not sure who was more proud,

the student or myself. This is exactly the kind of experience I want my students to have.

The problem with the standard geometry curriculum is that the private, personal experience

of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a

rigid step-by step pattern of uninspired formal deductions. The textbook presents a set of

definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students

copy them into their notebooks. They are then asked to mimic them in the exercises. Those that

catch on to the pattern quickly are the “good” students.

The result is that the student becomes a passive participant in the creative act. Students are

making statements to fit a preexisting proof-pattern, not because they mean them. They are

being trained to ape arguments, not to intend them. So not only do they have no idea what their

teacher is saying, they have no idea what they themselves are saying.

Even the traditional way in which definitions are presented is a lie. In an effort to create an

illusion of “clarity” before embarking on the typical cascade of propositions and theorems, a set

of definitions are provided so that statements and their proofs can be made as succinct as

possible. On the surface this seems fairly innocuous; why not make some abbreviations so that

things can be said more economically? The problem is that definitions matter. They come from

aesthetic decisions about what distinctions you as an artist consider important. And they are

problem-generated. To make a definition is to highlight and call attention to a feature or

structural property. Historically this comes out of working on a problem, not as a prelude to it.

The point is you don’t start with definitions, you start with problems. Nobody ever had an

idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a

square and discovered that it could not be represented as a fraction. Definitions make sense

when a point is reached in your argument which makes the distinction necessary. To make

definitions without motivation is more likely to cause confusion.

This is yet another example of the way that students are shielded and excluded from the

mathematical process. Students need to be able to make their own definitions as the need

arises— to frame the debate themselves. I don’t want students saying, “the definition, the

theorem, the proof,” I want them saying, “my definition, my theorem, my proof.”

23

All of these complaints aside, the real problem with this kind of presentation is that it is

boring. Efficiency and economy simply do not make good pedagogy. I have a hard time

believing that Euclid would approve of this; I know Archimedes wouldn’t.

SIMPLICIO: Now hold on a minute. I don’t know about you, but I actually enjoyed

my high school geometry class. I liked the structure, and I enjoyed

working within the rigid proof format.

SALVIATI: I’m sure you did. You probably even got to work on some nice

problems occasionally. Lot’s of people enjoy geometry class (although

lots more hate it). But this is not a point in favor of the current

regime. Rather, it is powerful testimony to the allure of mathematics

itself. It’s hard to completely ruin something so beautiful; even this

faint shadow of mathematics can still be engaging and satisfying.

Many people enjoy paint-by-numbers as well; it is a relaxing and

colorful manual activity. That doesn’t make it the real thing, though.

SIMPLICIO: But I’m telling you, I liked it.

SALVIATI: And if you had had a more natural mathematical experience you would

have liked it even more.

SIMPLICIO: So we’re supposed to just set off on some free-form mathematical

excursion, and the students will learn whatever they happen to learn?

SALVIATI: Precisely. Problems will lead to other problems, technique will be

developed as it becomes necessary, and new topics will arise naturally.

And if some issue never happens to come up in thirteen years of

schooling, how interesting or important could it be?

SIMPLICIO: You’ve gone completely mad.

SALVIATI: Perhaps I have. But even working within the conventional framework

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 real problem is that the bureaucracy does not allow

an individual teacher to do that. With a set curriculum to follow, a

teacher cannot lead. There should be no standards, and no curriculum.

Just individuals doing what they think best for their students.

SIMPLICIO: But then how can schools guarantee that their students will all have

the same basic knowledge? How will we accurately measure their

relative worth?

SALVIATI: They can’t, and we won’t. Just like in real life. Ultimately you have to

face the fact that people are all different, and that’s just fine. In any

case, there’s no urgency. So a person graduates from high school not

knowing the half-angle formulas (as if they do now!) So what? At least

that person would come away with some sort of an idea of what the

subject is really about, and would get to see something beautiful.

In Conclusion…

24

o put the finishing touches on my critique of the standard curriculum, and as a service to the

community, I now present the first ever completely honest course catalog for K-12

mathematics:

The Standard School Mathematics Curriculum

LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not

something you do, but something that is done to you. Emphasis is placed on sitting still, filling

out worksheets, and following directions. Children are expected to master a complex set of

algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part,

and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables

are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures,

akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are

handed out, and the students learn to address the church elders as “they” (as in “What do they

want here? Do they want me to divide?”) Contrived and artificial “word problems” will be

introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.

Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’

and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent

preparation for Algebra I.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this

course instead focuses on symbols and rules for their manipulation. The smooth narrative thread

that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance

algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no

characters, plot, or theme. The insistence that all numbers and expressions be put into various

standard forms will provide additional confusion as to the meaning of identity and equality.

Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of

students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy

and distracting notation will be introduced, and no pains will be spared to make the simple seem

complicated. This goal of this course is to eradicate any last remaining vestiges of natural

mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate

geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic

simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety

of standard formats for no reason whatsoever. Exponential and logarithmic functions are also

introduced in Algebra II, despite not being algebraic objects, simply because they have to be

stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder

mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

T

25

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory

definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of

a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and

obsolete notational conventions, in order to prevent students from forming any clear idea as to

what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All

Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and

symmetry. The measurement of triangles will be discussed without mention of the

transcendental nature of the trigonometric functions, or the consequent linguistic and

philosophical problems inherent in making such measurements. Calculator required, so as to

further blur these issues.

PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked

attempt to introduce late nineteenth-century analytic methods into settings where they are neither

necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to

obscure the intuitively clear notion of smooth change. As the name suggests, this course

prepares the student for Calculus, where the final phase in the systematic obfuscation of any

natural ideas related to shape and motion will be completed.

CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it

under a mountain of unnecessary formalism. Despite being an introduction to both the

differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be

discarded in favor of the more sophisticated function-based approach developed as a response to

various analytic crises which do not really apply in this setting, and which will of course not be

mentioned. To be taken again in college, verbatim.

***

And there you have it. A complete prescription for permanently disabling young minds— a

proven cure for curiosity. What have they done to mathematics!

There is such breathtaking depth and heartbreaking beauty in this ancient art form. How

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.

And it is school that has done this! What a sad endless cycle of innocent teachers inflicting

damage upon innocent students. We could all be having so much more fun.

SIMPLICIO: Alright, I’m thoroughly depressed. What now?

SALVIATI: Well, I think I have an idea about a pyramid inside a cube…

I teach maths in secondary school. Every time a student makes a claim that maths that they learn is not going to be of any use to them, I agree.
> by filling the circle with a polygon of a greater area and greater number of sides.
I think this could be expressed more clearly.
Is there a place on the internet where I can find problems and make my own intuitive solutions?
That’s a very interesting quote. It brings into question what Euclid thought of as a “line”, as opposed to a “straight line”.
Does P = NP?
Still a lot of unsolved math problems out there...https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
"A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre."
Quote by: Euclid, Elements, Book I
More on parallelograms from Paul Lockhart: https://www.youtube.com/watch?v=V1gT2f3Fe44
GO was invented in China more than 2,500 years ago and is believed to be the oldest board game continuously played to this day. Paul Lockhart teaches GO in many of classes, including calculus. His son Will made a fantastic GO documentary on Netflix: https://www.surroundinggamemovie.com/, and another son Ben is one of the best GO players in the world.
Gauss quotes don't fit a gaussian distribution, none are below average... https://en.wikiquote.org/wiki/Carl_Friedrich_Gauss
Standardized testing is a political issue, with every town, city, state and country comparing itself to every other. "Preliminary research by the Council of the Great City Schools, which represents large urban districts, found that students take an average of 113 standardized tests between pre-K and 12th grade. It said testing time for 11th graders was as high as 27 days, or 15 percent of the school year."
For more: https://www.pbs.org/newshour/education/congress-decide-testing-schools
Archimedes (287 - 212 BC) used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides.
For more: https://en.wikipedia.org/wiki/Method_of_exhaustion
<3
This 25-page essay was written in 2002, and was originally circulated in typewritten manuscript copies, and subsequently on the Internet. Paul Lockhart later turned this into a 140 page book. A Mathematician's Lament, often referred to informally as Lockhart's Lament, is a short book on the pedagogics and philosophy of mathematics. Characterized as a strongly worded opinion piece arguing for an aesthetic, intuitive and heuristic approach to teaching and the importance of mathematics teaching reforms, the book frames learning mathematics as an artistic and imaginative pursuit which is not reflected at all in the way the subject is taught in the American educational system.
Source: https://en.wikipedia.org/wiki/A_Mathematician's_Lament
Paul Lockhart first "became interested in mathematics when he was 14 (outside the classroom, he points out). He dropped out of college after one semester to devote himself exclusively to math. Based on his own research he was admitted to Columbia, received a PhD, and has taught at major universities. Since 2000 he has dedicated himself to "subversively" teaching grade-school math." Courtesy of Goodrerads: https://www.goodreads.com/author/show/6421135.Paul_Lockhart
This is what we still do. I am an engineering student at a reputed university in the States. It is pathetic. But sometimes I think that the teachers are doing this because it's easier to test us with predictable problems.
G.H. Hardy (1877 – 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of population genetics. In addition to his research, he is remembered for his 1940 essay on the aesthetics of mathematics, titled A Mathematician's Apology. One other notable quote from A Mathematician's Apology: "The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."
![Imgur](https://i.imgur.com/0GB3MbK.png)
Source: https://en.wikipedia.org/wiki/G._H._Hardy