[Photo by luis de bethencourt.]
I love quotations, don’t you? Everything I might possibly want to say, someone else has already said it better than I ever could.
Now I’ve put together all of my blackboard quotes from the homeschool co-op classes, as well as a few longer quotations I used in past blog posts, and archived them in one convenient place. Browse and enjoy!
For me, it begins with pushing students into the verbal — that intersection of math and language. Tom Barrett’s “Fizz and Martina” series pose cartoon-like problems that students enjoy. But I’ve been able to harness the power of the series by stealing one question from them: Explain how you solved the problem. Do not use numbers in your explanation.
Once students can verbalize their thinking apart from the numbers, I can ask them to describe patterns they see — and give those names. Suddenly, the “names”, or equations are not abstract. They apply to something concrete that can be applied to the 100th term situation or 1000th term situation.
My school math was abstract because I first learned to manipulate equations and then learned to dread the application of equations to story problems. Start with stories. Find patterns. Give the patterns names (equations).
I must study politics and war that my sons may have liberty to study mathematics and philosophy.
Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
Arithmetic is neither fish nor beast; therefore it must be foul.
Now there’s three.
Mate again, now five.
And again, now there’s eight bunnies.
[Now thirteen bunnies, and counting. Mom will be pissed off.]
If inside a circle a line
Hits the center and goes spine to spine
And the line’s length is “d,”
The circumference will be
d times 3 point 1 4 1 5 9.
Mathematical Poetry site
Life without geometry is pointless.
Logic is a systematic method of coming to the wrong conclusion with confidence.
(similar to a comment by Morris Kline)
Natural numbers are better for your health.
Simple Simon met a pi man
Going to the fair.
Said Simple Simon to the pi man,
“You have unusual ware.
The pie’s I’ve seen before were round
But, gosh, your pi’s r2.”
Mathematical Poetry site
The human mind has never invented a labor-saving machine equal to algebra.
From The Quote Garden
In both math and writing, the core idea that you are trying to express exists somewhere in the aether. In both math and writing, you start out staring at the blank page, trying to figure out how to summon the idea, make it yours, and commit it to the page.
In both math and writing, you make false starts (unless you are very lucky) and work hard (unless you are very lucky) to express the idea with precision and clarity. In both math and writing, your familiarity with the idea that you are trying to express and your prior practice at expressing ideas can sometimes give you a head start in knowing in which direction to start.
Math is writing. Most of math is persuasive writing; math is an exquisitely structured argument.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
There’s a tendency for adults to label the math that they can do (such as identifying patterns, choosing between competing offers in a supermarket, and challenging statistics published by the government) as “common sense” and labeling everything they can’t do as “math” — so that being bad at math becomes a self-fulfilling prophecy.
Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.
One thing to keep in mind is that mathematics is a story and that teachers are a story tellers; the teaching and curriculum sequences are there to help you with the structure of the story. If you can bring the story of mathematics to life then you will have a much better chance of reaching all your students. What is easier: memorizing the story of the three little pigs, or learning to tell the three little pigs story on your own? Which is more satisfying?
Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.
In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.
In other words these children, by avoiding the early drill on combinations, tables, and that sort of thing, had been able, in one year, to attain the level of accomplishment which the traditionally taught children had reached after three and one-half years of arithmetical drill.
— L. P. Benezet
The Teaching of Arithmetic I: The Story of an experiment
There is no philosophy which is not founded upon knowledge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathematician.
I recognize the lion by his paw.
— Jacob Bernoulli
on reading an anonymous solution to a problem,
which he realized was Isaac Newton’s work
It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can would once have believed in witchcraft.
Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in the narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!
The solution “cost me study that robbed me of rest for an entire night.”
— Johann Bernoulli
of a problem that stumped mathematicians (including his brother Jacob) for years
But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found.
How dare we speak of the laws of chance? Is not chance the antithesis of all law?
Brain: an apparatus with which we think we think.
Teaching any subject has a funny way of educating the teacher at least as much as the student.
The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.
Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.
…a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.)
The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.
Logic is invincible, because in order to combat logic it is necessary to use logic.
— Pierre Boatroux
Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”
Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing.
— R. C. Buck
Teaching Machines and Mathematics Programs
American Math. Monthly
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.
— James Caballero
Everybody a mathematician?, CAIP Quarterly 2 (Fall, 1989)
quoted by Donald McQuarrie, Mathematics for Physical Chemistry
In mathematics, the art of asking questions is more valuable than solving problems.
Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was younger, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”
“Can you do addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”
“I don’t know,” said Alice. “I lost count.”
— Lewis Carroll
Through the Looking Glass
“When I use a word,” Humpty Dumpty said, in a rather scornful tone, “it means just what I choose it to mean — neither more nor less.”
“The question is,” said Alice, “whether you can make words mean so many different things.”
“The question is,” said Humpty Dumpty, “which is to be master — that’s all.”
If a child is to keep alive his inborn sense of wonder, he needs the companionship of at least one adult who can share it, rediscovering with him the joy, excitement and mystery of the world we live in.
Education is the key to unlock the golden door of freedom.
You can only find truth with logic if you have already found truth without it.
I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.
— Agatha Christie
“I think you’re begging the question,” said Haydock, “and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that, you would go round the bend. Let me assure you of that!”
Free math (Available here Monday through Friday). But you must bring your own container, and you must fill it with much or little according to its capacity and the amount of work that you are willing to do. The learning assistant (sometimes euphemistically called a “teacher”) will provide expertise, advice, guidance, and will set an example. But in the final analysis it is you who must do the work needed for your learning… Here it is — this wonderful stuff called math. If you want it, come and get it. If you don’t want it, kindly step out of the way — as not to impede the progress of those who do. The choice is yours.
— L. M. Christolphe, Jr.
quoted in the Mathematical and Educational Quotation Server
We make a living by what we get, but we make a life by what we give.
I had my seniors write me an essay about their relationship with math. Not much instruction, just tell me what math was like growing up, good, bad, whatever. In maybe 10 cases they all pointed out that somewhere between elementary school and middle school math went from something they could see and understand to something they no longer got. Every one of them said the same thing, I loved math until middle school. What in the world changes in middle school?
I am in the habit of beginning each class by apologizing to my learners. I’ll teach the class better next time because of what I learn from my interactions with them and from their feedback. I remind them that they are free to take the class next year – when it is improved. No one takes me up on that, but it sets the tone that I expect to grow as an educator.
The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.
He who asks a question is a fool for five minutes;
he who does not ask remains a fool forever.
Learning without thought is labor lost;
thought without learning is perilous.
Too often, kids learn a distaste for the subject without ever having the chance to see what there is to love in mathematics. For too many, the understanding of math isn’t particularly enduring, while their dislike of the subject is.
I do my best to make my students think, but they still try to become good little algorithm followers.
Intuition is great. Inductive logic is great. But it just isn’t enough. Back it up. Verify it. Embrace the conflict that arises when what you thought was true turns out to be, well, not so much.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.
“I suppose you are two fathoms deep in mathematics, and if you are, then God help you. For so am I, only with this difference: I stick fast in the mud at the bottom, and there I shall remain.”
We use only 10% of our brains… Imagine how smart we would be if we used the other 60%!
I was x years old in the year x2.
— Augustus De Morgan
(when asked about his age)
It is not enough to have a good mind. The main thing is to use it well.
Each problem that I solved became a rule which served afterwards to solve other problems.
Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.
Mathematical thinking is not the same as doing mathematics — at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box — a valuable ability in today’s world.
— Keith Devlin
Introduction to Mathematical Thinking
These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music. A page of sheet music represents a piece of music: the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds. The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.
— Keith Devlin
Introduction to Mathematical Thinking
What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.
When a kid is feeling bad about being stuck with a problem, or just very anxious, I sometimes ask him to make as many mistakes as he can, and as outrageous as he can. Laughter happens (which is valuable by itself, and not only for the mood — deep breathing brings oxygen to the brain). Then the kid starts making mistakes. In the process, features of the problem become much clearer, and in many cases a way to a solution presents itself.
Math happens when we notice similarities and differences. This is math proper. You purposefully create differences, keeping similarities, and observe what happens. There are layers and layers of noticing to be had. We need to return to activities again and again to reach more layers. That’s why geeks are often told, “You have too much time on your hands!” when an outsider realizes how much time is spent with a single activity. There are riches to be had ONLY if you spend the time.
One’s work may be finished some day, but one’s education never.
Logic, like whiskey, loses its beneficial effect when taken in too large quantities.
There’s a tendency for adults to label the math that they can do (such as identifying patterns, choosing between competing offers in a supermarket, and challenging statistics published by the government) as “common sense” and labeling everything they can’t do as “math” — so that being bad at math becomes a self-fulfilling prophecy.
I have seen something else under the sun: The race is not to the swift or the battle to the strong, nor does food come to the wise or wealth to the brilliant or favor to the learned; but time and chance happen to them all.
Opportunity is missed by most people because it is dressed in overalls and looks like work.
There is no expedient to which a man will not go to avoid the labor of thinking.
Any fool can know. The point is to understand.
Gravitation cannot be held responsible for people falling in love. How on earth can you explain in terms of chemistry and physics so important a biological phenomenon as first love? Put your hand on a stove for a minute and it seems like an hour. Sit with that special girl for an hour and it seems like a minute. That’s relativity.
I have no particular talent, I am only inquisitive.
Pure mathematics is, in its way, the poetry of logical ideas.
In a cat’s eye, all things belong to cats.
— English proverb
When I was a child, the Earth was said to be two billion years old. Now scientists say it’s four and a half billion. So that makes me two and a half billion.
Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
There is no Royal Road to Geometry.
… I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. … True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand …
Now I will have less distraction.
— Leonhard Euler (1707-1783)
[upon losing the use of his right eye]
Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – … whose sum was given by Leibniz as 1/2, although others disagree. … Understanding of this question is to be sought in the word “sum”; this idea, if thus conceived — namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken — has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.
The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some “dreaming” time.
An expert problem solver must be endowed with two incompatible qualities: a restless imagination and a patient pertinacity.
There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every new discovery in mathematics results from an attempt to solve some problem.
— Howard Eves
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians
I was born not knowing and have had only a little time to change that here and there.
It’s amazing that this vision of math as “getting to the right answer on your first try” even exists. I have to make, unmake, remake so many mistakes to get where I’m going. I think all mathematicians work that way. Somehow, a big part of the experience of math is trouble. Frustration is the status quo. But when you get something—the thrill!
A math student’s best friend is BOB (the Back Of the Book), but remember that BOB doesn’t come to school on test days.
— Josh Folb
(I couldn’t find a link about the man himself, although many people like his quote enough to include it on their webpages.)
Imagine that you wanted your children to learn the names of all their cousins, aunts and uncles. But you never actually let them meet or play with them. You just showed them pictures of them, and told them to memorize their names. Each day you’d have them recite the names, over and over again. You’d say, “OK, this is a picture of your great-aunt Beatrice. Her husband was your great-uncle Earnie. They had three children, your uncles Harpo, Zeppo, and Gummo. Harpo married your aunt Leonie … yadda, yadda, yadda.
— Brian Foley
Times Tables – The Worst Way to Teach Multiplication
Two roads diverged in a wood, and I —
I took the one less traveled by,
And that has made all the difference.
That vast book which stands forever open before our eyes, the universe, cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals — the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.
In no other branch of mathematics is it so easy for experts to blunder as in probability theory.
The continuing hullabaloo about the “new math” has given many a parent a false impression. What was formerly a dull way of teaching mathematics by rote, so goes the myth, has suddenly been replaced by a marvelous new technique that is achieving miraculous results throughout the nation’s public schools.
I wish it were true — even if only to the extent implied by entertainer (and math teacher) Tom Lehrer in his delightfully whimsical recording on “The New Math”: “In the new approach, as you know, the important thing is to understand what you’re doing, rather than to get the right answer.”
Indeed, there is something to be said for the old math when taught by a poorly trained teacher. He can, at least, get across the fundamental rules of calculation without too much confusion. The same teacher trying to teach new math is apt to get across nothing at all…
I can recall the deep satisfaction I felt on the all-too-rare occasions at school when the concepts or formulas fell into place. It seemed an entirely different discipline from writing, where something arises from a blank page through a combination of hard work and patience, with a sliver of creativity. With math, the experience is more like discovering something that’s always existed and finally decided to stop playing hard-to-get.
Doing math or computer programming at a professional level is a lot like writing. Sometimes it flows naturally and fluently, and sometimes it is blocked and it is like struggling to lift a huge boulder to get anything done.
The toughest thing for a homeschooler is the same as for a school teacher – shifting from a weak tea vision of math being grinding calculations to a rich frothy mug of math as an active way of thinking.
The ultimate goal of mathematics is to eliminate all need for intelligent thought.
As important as mathematics is, it is a distant second to the need for good reading comprehension. We teachers so often hear students summarize a course by saying, ‘I could do everything except the word problems.’ Sadly, in the textbook of life, there are only word problems.
The essence of mathematics is not to make simple things complicated but to make complicated things simple.
“I hate fractions.”
“They probably hate you, too. The question is, which one of you will be master.”
A child learns to count spoonfuls, learns to count people, learns to count fingers, learns to just plain count, and in the process acquires the abstract concept of, for example, “two.” The child takes ownership of this concept, and can reapply it freely. As adults we may take “two” for granted, but we have never met it, never touched it, never tasted it. It is one of the first completely abstract concepts that we ever owned.
If the host is required to open a door all the time and offer you a switch, then you should take the switch. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood. My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home.
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
Many teachers are concerned about the amount of material they must cover in a course. One cynic suggested a formula: since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick.
Board games are a celebration of problem solving, and problem solving is at the heart of a quality mathematics education… The mathematics might be hidden, but I guarantee you that it will be there.
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. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
There are many things you can do with problems besides solving them. First you must define them, pose them. But then, of course, you can also refine them, depose them, or expose them, even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found new goal they do lead to. It’s called play.
Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings.
(Inscription for a monument at the crossroads.)
Here lies, extinguished in his prime,
a victim of modernity:
but yesterday he hadn’t time—
and now he has eternity.
prove their worth
by hitting back.
One cat just leads to another.
In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.
Stu came to my office looking for a new major. Stu is bad at math and can’t handle the math sequence required of business majors. So Stu was wondering what majors require the lowest level math sequence that counts towards graduation.
I listed a few.
Stu was disappointed. Stu pointed out that you don’t usually think about people in those fields as making a lot of money. Stu lamented that everything that is in demand requires math.
To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.
Repetition progressively frees the mind from attention to details, makes facile the total act, shortens the time, and reduces the extent to which consciousness must concern itself with the processes.
When my students complain that I’m smarter than them, I counter that I’m just at a higher level of misunderstanding.
A great many people think they are thinking when they are really rearranging their prejudices.
…the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not be in indulged in by one who is to have a profession to follow for his subsistence.
I don’t love math nearly as much as I pretend I do when I’m teaching it or blogging about it or trying to enthuse my kids. I just believe — ever since an eye-opening university-level Mathematics in Perspective course — that math is taught VERY badly, bumbled and fumbled and as a result we have this societal fear of what is, essentially, a great big GAME.
Nothing produced such odd results as trying to get even.
A good student is one who will teach you something.
The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.
The greater our knowledge increases, the greater our ignorance unfolds.
Where there is matter, there is geometry.
Two contrasting attitudes:
non-math person: “Math is so abstract.” i.e. “hard to understand”
math person: We abstract *in order to understand.*
Part of our job is to teach people this latter mentality.
No knowledge of probabilities helps us to know what conclusions are true. There is no direct relation between the truth of a proposition and its probability.
Polya has become the Marx and Lenin of mathematical problem solving; a few words of obeisance need to be offered in his name before an author can get down to the topic at hand.
— Jeremy Kilpatrick [pdf 97KB]
Logic is the art of going wrong with confidence.
If you could lead through testing, the U.S. would lead the world in all education categories. When are people going to understand you don’t fatten your lambs by weighing them?
— Jonathan Kozol
at Westfield State College’s 157th Commencement
The Good Lord made all the integers; the rest is man’s doing.
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
Experience is the hardest kind of teacher. It gives you the test first, and the lesson afterward.
The only teaching that a professor can give, in my opinion, is that of thinking in front of his students.
But in the new (math) approach, the important thing is to understand what you’re doing, rather than to get the right answer.
Someone asked me if I was ever sorry I had chosen mathematics. I said, “I didn’t choose! Mathematics is an addiction with me!”
As soon as you understand 2 x 4, you can’t believe there was a time when you didn’t understand it.
I told myself, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means.” So I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what “demonstrate” means, and went back to my law studies.
The mathematical question is “Why?” It’s always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.
— Paul Lockhart
author of Measurement
Math class is not terribly different from shop class. Both are all about tools and toolboxes. Math class is no more about mathematics than wood shop is about craftsmanship and design.
In American elementary mathematics education, arithmetic is viewed as negligible, sometimes even with pity and disdain—like Cinderella in her stepmother’s house. Many people seem to believe that arithmetic is only composed of a multitude of “math facts” and a handful of algorithms. . . Who would expect that the intellectual demand for learning such a subject actually is challenging and exciting?
Recognize that every math program, whether more traditionally skill-based or reform-oriented (more problem-solving, projects, less drill) has its merits and its weaknesses. Whether you believe there is too much emphasis on basic facts (less likely!), or not enough, you can supplement with the myriad of resources on the web.
Procrastination is the art of keeping up with yesterday.
It is a great nuisance that knowledge can only be acquired by hard work.
The best teacher is not the one who knows most, but the one who is most capable of reducing knowledge to that simple compound of the obvious and wonderful.
The physical five oranges goes up the ladder to the picture of the five oranges which goes up to the representation of the five oranges as a numeral. This points in the direction of a definition of abstraction: when we abstract we voluntarily ignore details of a context, so that we can accomplish a goal.
You don’t understand anything until you learn it more than one way.
It seems quite unrealistic to judge a curriculum by its general outline, or to judge a course by its syllabus. We can “cover” very impressive material, if we are willing to turn the student into a spectator. But if you cast the student in a passive role, then saying that he has “studied” your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does.
Arithmetic is being able to count up to twenty without taking off your shoes.
Whatever you do, for goodness sake, don’t tell them the right answer. (Like, ever. Let them come to consensus. Learn how to ask helpful questions without giving away the store.) Unless for some reason you want to completely shut down discussion. And thinking.
The clearer the teacher makes it, the worse it is for you. You must work things out for yourself and make the ideas your own.
Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren’t ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive.
I love mathematics … principally because it is beautiful, because man has breathed his spirit of play into it, and because it has given him his greatest game — the encompassing of the infinite.
Being in very nature God, He did not consider equality with God something to be grasped, but made Himself nothing…
I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive.
The more I work and practice, the luckier I seem to get.
The mathematician does not study pure mathematics because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful.
No other statistical puzzle comes so close to fooling all the people all the time.
A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem.
Beauty in mathematics is seeing the truth without effort.
It is better to solve one problem five different ways, than to solve five problems one way.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.
The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving…The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the class. He writes a, he says b, he means c; but it should be d. Some of his sayings are handed down from generation to generation.
- “In order to solve this differential equation you look at it till a solution occurs to you.”
- “This principle is so perfectly general that no particular application of it is possible.”
- “Geometry is the science of correct reasoning on incorrect figures.”
- “My method to overcome a difficulty is to go round it.”
- “What is the difference between method and device? A method is a device which you used twice.”
Circles to square and cubes to double would give a man excessive trouble.
Mathematics is a world created by the mind of men, and mathematicians are people who devote their lives to what seems to me a wonderful kind of play!
You have to make them laugh. You must never underestimate the power of laughing in a maths classroom.
— Carol Roberts
quoted in Laughing lesson Adult learning
In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.
A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking, and frustration. And proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.
The 50-50-90 rule: Anytime you have a 50-50 chance of getting something right, there’s a 90% probability you’ll get it wrong.
I used to think that math was some kind of inaccessible, abstract magic trick, a sort of in-joke that excluded us common folk, but now I realize that math is completely not that at all. The reality of math as most of us know it is like that story where three men are standing in a dark room touching different parts of an elephant. None of them has the full picture because they’re only perceiving individual elements of the whole animal. The reality, I’m discovering, is that math is just like that elephant: a large, expansive, three-dimensional, intelligent, sensitive, expressive creature. The problem is that most of us have been standing around in that dark room since about kindergarten, grasping its tail, thinking “this is what math is and, personally, I don’t think it’s for me.” We’ve been blind to the larger, incredibly beautiful picture that would emerge if only we would turn on the lights and open our eyes.
The study of infinity is much more than a dry academic game. The intellectual pursuit of the absolute infinity is, as Georg Cantor realized, a form of the soul’s quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment.
It may be that the race is not always to the swift, nor the battle to the strong – but that is the way to bet.
At age eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.
Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.
People often make mistakes when they reason about things they have never seen. Imagination does not always give us the correct answer. We can only argue correctly about things of which we have experience or which are reasonably like the things we know well. If our reasoning leads us to an untrue conclusion, we must revise the picture in our minds, and learn to imagine things as they are.
When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination. Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.
If you cannot see what the exact speed is, begin to ask questions. Silly ones are the best to begin with. Is the speed a million miles an hour? Or one inch a century? Somewhere between these limits. Good. We now know something about the speed. Begin to bring the limits in, and see how close together they can be brought. Study your own methods of thought. How do you know that the speed is less than a million miles an hour? What method, in fact, are you unconsciously using to estimate speed? Can this method be applied to get closer estimates?
You know what speed is. You would not believe a man who claimed to walk at 5 miles an hour, but took 3 hours to walk 6 miles. You have only to apply the same common sense to stones rolling down hillsides, and the calculus is at your command.
The desire to explore thus marks out the mathematician. This is one of the forces making for the growth of mathematics. The mathematician enjoys what he already knows; he is eager for more knowledge.
And here is a bonus—an article by Sawyer about Hardy: A Mathematician’s Apology Revisited
Go down deep enough into anything and you will find mathematics.
— Dean Schlicter
[Does anyone know who he is? Plenty of people quote him, but I can’t find a single link about the man himself.]
If you would make a man happy, do not add to his possessions but subtract from the sum of his desires.
Paraphrasing is one of the most important skills we can teach junior high and high school students. Often they want to rush into interpreting and reacting to a text even before they know what it means. We teachers sometimes suffer from the delusion that since a student can read the words on the page, he or she understands what’s been read. But that’s not always true.
I had the most beautiful set of theories you ever knew when I started out as a schoolma’am, but every one of them has failed me at some pinch or another.
We’ve all heard that a million monkeys banging on a million typewriters will eventually reproduce the entire works of Shakespeare. Now, thanks to the Internet, we know this is not true.
— Robert Silensky
If we are to teach mathematics at all, real success is not possible unless we know that the subject is beautiful as well as useful. Mere utility of the moment without any feeling of beauty becomes a hopeless bit of drudgery, a condition which leads to stagnation.
What would mathematics have amounted to without the imagination of its devotees—its giants and their followers? There never was a discovery made without the urge of imagination—of imagination which broke the roadway through the forest in order that cold logic might follow.
What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?
I don’t want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that — by some mysterious agency — capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.
A second reason why few students ever realize that there is mathematics outside the textbook is that no one ever tells them that. I don’t blame the teachers. If your students are having problems remembering how to solve quadratic equations, the wise teacher will stay well clear of cubic equations, which are even more difficult. A process of self-censorship sets in. In order to avoid damaging the students’ confidence, the texts do not ask questions that the methods being taught cannot answer. So insidiously, we absorb the lesson that every mathematical question has an answer.
It’s not true.
Our teaching of mathematics revolves around a fundamental conflict. Rightly or wrongly, students are required to master a series of mathematical concepts and techniques, and anything that might divert them from doing so is deemed unnecessary. Putting mathematics into its cultural context, explaining what is has done for humanity, telling the story of its historical development, or pointing out the wealth of unsolved problems or even the existence of topics that do not make it into school textbooks leaves less time to prepare for the exam. So most of these things aren’t discussed.
Mathematics is a vast adventure; its history reflects some of the noblest thoughts of countless generations.
I have studied many philosophers and many cats. The wisdom of cats is infinitely superior.
Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful!
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks… Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
How often might a man, after he had jumbled a set of letters in a bag, fling them out upon the ground before they would fall into an exact poem, yea, or so much as make a good discourse in prose? And may not a little book be as easily made by chance as this great volume of the world?
“[M]athematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary schools learning to detest mathematics… They return to the elementary school to teach a new generation to detest it.”
— Time magazine, June 18, 1956
quoted by George Polya in How to Solve It
It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.
A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction.
For example, coins, nuts and buttons are clearly distinct and countable and for this reason are convenient to represent relations between whole numbers. The youngest children need some real, tangible tokens, while older ones can imagine them, which is a further step of intellectual development. That is why coin problems are so appropriate in elementary school. Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into the transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds and because of this are appropriate as reifications of the idea of uniform movement, which, in its turn, can serve as a reification of linear function. Thus, we can move children further and further on the way of de-reification, that is development of abstract thinking.
I finally get it — you don’t have to worry about memorizing a bunch of formulas if you just understand where they come from. You can always figure them out again.
Cats aren’t clean, they’re just covered with cat spit.
Dogs believe they are human.
Cats believe they are God.
I got rid of my husband. The cat was allergic.
People have this notion that math is about getting a right answer, and the testing really emphasizes that notion. And that’s such a bad way to approach math because it makes it scary. When you look at little kids, they pose their own questions. They say, “Ooooh, what’s bigger than a million?” And they think about things their own way. At school, the teacher poses the questions, and the students answer their questions. Schooling is not a natural environment for learning.
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.
The length of your education is less important than its breadth, and the length of your life is less important than its depth.
The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.
Follow the path of the unsafe, independent thinker. Expose your ideas to the dangers of controversy. Speak your mind and fear less the label of ‘crackpot’ than the stigma of conformity. And on issues that seem important to you, stand up and be counted at any cost.
Teaching is the royal road to learning.
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
I will not go so far as to say that constructing a history of thought without profound study of mathematical ideas is like omitting Hamlet from the play named after him. But it is certainly analogous to cutting out the part of Ophelia. For Ophelia is quite essential to the play, she is very charming. . . and a little mad.
The study of mathematics is apt to commence in disappointment… We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this greatest science eludes the efforts of our mental weapons to grasp it.
From the very beginning of his education, the child should experience the joy of discovery.
Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark, unexplored mansion. You enter the first room of the mansion, and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch. You turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark.
Mathematics: a wonderful science, but it hasn’t yet come up with a way to divide one tricycle among three little boys.
Probable-Possible, my black hen,
She lays eggs in the Relative When.
She doesn’t lay eggs in the Positive Now
Because she’s unable to postulate how.
Black holes are where God divided by zero.
Five out of four people have trouble with fractions.
The excitement of learning separates youth from old age. As long as you’re learning, you’re not old.
Several of the quotes below have been pulled from these wonderful math and education websites:
- Furman University Mathematical Quotation Server
- MAA Digital Library Mathematical Quotations
- Mathematical and Educational Quotation Server at Westfield State College
- Mathematical Quotations
- Platonic Realms Interactive Database of Math Quotes
And if you enjoy these quotes, you may want to check your library for…
- A Dictionary of Quotations in Mathematics
- Dictionary of Mathematical Quotations
- Great Quotes to Inspire Great Teachers
- In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes
- Mathematically Speaking: A Dictionary of Quotations
- Memorabilia Mathematica: The Philomath’s Quotation Book
- Statistically Speaking: A Dictionary of Quotations
- Teacher’s Treasury of Quotations
- The Music of Reason: Experience the Beauty of Mathematics Through Quotations
- The Teacher’s Quotation Book: Little Lessons on Learning
- Out of the Mouths of Mathematicians: A Quotation Book for Philomaths