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!

*[Photo by luis de bethencourt.]*

### Browse by the Author’s Last Name

A – B – C – D – E – F – G – H – I/J – K – L – M – N – O – P/Q – R – S – T – U/V – W – X/Y/Z – Reference: Websites and Quote Books

## A

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).

—Janet Abercrombie

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

—John Adams

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.

—Woody Allen, quoted in H. Eves Return to Mathematical Circles

Unending digits …

Why not keep it simple, like

Twenty-two sevenths?

—Luke Anderson, via TeachPi.org

Arithmetic is neither fish nor beast; therefore it must be foul.

—Anonymous

Hare.

Hare.

They mate.

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.]

—Anonymous, from the comments on the Fib

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.

—Anonymous, Mathematical Poetry site

Life without geometry is pointless.

—Anonymous

Logic is a systematic method of coming to the wrong conclusion with confidence.

—Anonymous, (similar to a comment by Morris Kline)

Natural numbers are better for your health.

—Anonymous

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 r^{2}.”

—Anonymous, Mathematical Poetry site

The human mind has never invented a labor-saving machine equal to algebra.

—Anonymous, 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.

—Anonymous

You understand something if you have the ability to view it from different perspectives. Changing your perspective makes your mind more flexible, it makes you open to new things, and it makes you able to understand things.

—Roger Antonsen, Math is the hidden secret to understanding the world

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.

—Archimedes

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.

—Mike Askew, Rob Eastaway, Old Dogs, New Math: Homework Help for Puzzled Parents

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.

—Richard Askey, quoted in Elementary Mathematics for Teachers

## B

One thing to keep in mind is that mathematics is a story and that teachers are 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?

—Scott Baldridge, comment on the ProfoundUnderstanding Yahoo group

Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?

Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.

When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary. *But they are small matters compared to real arithmetic thinking.*

*If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more.* And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.

—Ruth Beechick, An Easy Start in Arithmetic (Grades K-3) (emphasis mine)

Everyone thinks it goes smoothly in everyone else’s house, and theirs is the only place that has problems. I’ll let you in on a secret about teaching: there is no place in the world where it rolls along smoothly without problems. Only in articles and books can that happen.

—Ruth Beechick, You Can Teach Your Child Successfully (Grades 4-8)

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.

—E. G. Begle

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.

—Daniel Bernoulli

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.

—Jacob Bernoulli, Ars Conjectandi

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 inhere.

What joy to discern the minute in infinity!

The vast to perceive in the small, what divinity!

—Jacob Bernoulli, Ars Conjectandi

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.

—Johann Bernoulli

How dare we speak of the laws of chance? Is not chance the antithesis of all law?

—Joseph Bertrand, Calcul des probabilités

Brain: an apparatus with which we think we think.

—Ambrose Bierce, The Devil’s Dictionary

Teaching any subject has a funny way of educating the teacher at least as much as the student.

—Chris Birk, How I Became a Better Writer Thanks to Distracted, Hungover College Kids

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.

—Ben Blum-Smith, Comment on Math Mama’s post What must be memorized?

It turns out that the people who do well in math are those who make connections and see math as a connected subject. The people who don’t do well are people who see math as a lot of isolated methods.

—Jo Boaler, Math Connections

…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.

—Ralph P. Boas [You will have to scroll down a bit to find Boas’ essay.], Lion Hunting and Other Mathematical Pursuits

Logic is invincible, because in order to combat logic it is necessary to use logic.

—Pierre Boatroux

Only dead mathematics can be taught where competition prevails: living mathematics must always be a communal possession.

—Mary Everest Boole

I rather like the idea that the Farey Sequences are named after someone who noticed a pattern and asked a question — and not even the first person to notice the pattern, ask the question, or provide the answer. As math teachers, we teach plenty of indifferent mathematicians who wake up when they experience the joy of discovering something that is new to them, not necessarily new to the whole world.

—Debra K. Borkovitz, Farey Fraction Visual Patterns

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.”

—Alexandre Borovik, quoted by Pat Ballew in A Nice Presentation of the Harmonic Series

There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else — but persistent.

—Raoul Bott, via The MacTutor History of Mathematics archive

Our own school experiences can make it hard for us to teach without being tempted to “help them master” a concept that they may or may not be ready to master. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.

—Julie Brennan, homeschooler, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

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

## C

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.

—Georg Cantor

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.”

—Lewis Carroll, Alice in Wonderland

“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.”

—Lewis Carroll, Through the Looking Glass

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.

—Rachel Carson

Education is the key to unlock the golden door of freedom.

—George Washington Carver

You can only find truth with logic if you have already found truth without it.

—G. K. Chesterton, The Man Who Was Orthodox

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, An Autobiography

“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!”

—Agatha Christie, The Mirror Crack’d (Miss Marple Mysteries)

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.

—Winston Churchill

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?

—Jonathan Claydon

The ability to create, and maintain, and manipulate shapes mentally — that’s the goal. Just like kids who can put numbers together in their heads, kids who can rotate, flip, and think of how shapes fit together in their heads have a powerful tool to analyze not only simple shape puzzles, but dividing up an area that’s a more complex room shape … to look at a piece of artwork … or look at a building … For these kids, all the world around becomes a playground to use mathematical ideas.

—Doug Clements, Problem Solving Development: Composing Shapes

You know what? Children like mathematics. Children see the world mathematically … When we do a puzzle, when we count things, when we see who’s got more, or who’s taller … Play and mathematics are not on opposite sides of the stage.

—Doug Clements, Why Early Childhood is the Right Time to Start Learning Math

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.

—David Coffey, What’s in a Name?

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.

—Marshall Cohen

He who asks a question is a fool for five minutes;

he who does not ask remains a fool forever.

—Confucius

Learning without thought is labor lost;

thought without learning is perilous.

—Confucius

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.

—Katherine Cook, My Mathematical Autobiography

The way we taught students in the past simply does not prepare them for the higher demands of college and careers today and in the future. Your school and schools throughout the country are working to improve teaching and learning to ensure that all children will graduate high school with the skills they need to be successful.

In mathematics, this means three major changes. Teachers will concentrate on teaching a more focused set of major math concepts and skills. This will allow students time to master key math concepts and skills in a more organized way throughout the year and from one grade to the next. It will also call for teachers to use rich and challenging math content and to engage students in solving real-world problems in order to inspire greater interest in mathematics.

—Council of the Great City Schools, Parent Roadmaps to the Common Core Standards- Mathematics

I do my best to make my students think, but they still try to become good little algorithm followers.

—David Cox, The 2 Product Property

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.

—David Cox, ( )conceptions

## D

PTALSMP 1: Ask questions. Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

PTALSMP 2: Play. See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

PTALSMP 3: Argue. Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

PTALSMP 4: Connect. Ask how this thing is like other things. Try your ideas out on a new problem. Ask whether and how these ideas apply to other situations. Look for similarities and differences. Seek out the boundaries and limitations of your techniques.

—Christopher Danielson, an expanded version of the standards originally posted in Ginger ale (also abbreviated list of Standards for Mathematical Practice)

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.

—David van Dantzig

“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.”

—Charles Darwin, quoted in the Platonic Realms collection

We use only 10% of our brains… Imagine how smart we would be if we used the other 60%!

—Ellen DeGeneres

I was x years old in the year x^{2}.

—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.

—René Descartes

Each problem that I solved became a rule which served afterwards to solve other problems.

—René Descartes

Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.

—René Descartes

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.

—Keith Devlin, Should Children Learn Math by Starting with Counting?

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.

—Maria Droujkova, Natural Math discussion of math club activities

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.

—Maria Droujkova, Taxophilia

We had a discussion at the end of the club on how we are all confused now, but pleasantly so, and how important it is to rejoice in confusion and to be comfortable with it. Adults often strive very hard to get rid of any and all possible traces of confusion for kids, making things dreadfully boring.

—Maria Droujkova, after a math circle exploration of infinity, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

For our children, we dream that mathematics…

…makes sense

…is more than just arithmetic

…is joyous

…makes them strong

…is meaningful

…is creative

…is full of fascinating questions

…opens up many paths to solutions

…is friendly

…solves big problems and makes the world better

…is a powerful tool they can master

…is beautiful

…lets them learn in their own ways

…is connected to their lives

…asks “why” and not just “how”

…opens the world

—Maria Droujkova, James Tanton, and Yelena McManaman, Avoid Hard Work

As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary and dangerous than sailing the high seas!

But most parents and educators don’t present math this way. They just want the children to develop their mathematical skills rather than going for something more nebulous, like the mathematical state of mind.

Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive.

Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.

—Maria Droujkova and Yelena McManaman, Adventurous Math For the Playground Set (Scientific American online)

One’s work may be finished some day, but one’s education never.

—Alexandre Dumas, [or perhaps, Alexandre Dumas?]

I sat in class three days ago and thought to myself, “They need a class called ‘Math as a second language’ or MSL for short.”

It is easy to understand what a median is, or what attributes a kite has, or why is a rectangle a square but a square not a rectangle… for a minute or a day.

It is easy to temporarily memorize a fact. But without true understanding of the concept those “definitions” fade. If the foundation of truly understanding is not there to begin with then there is little hope for any true scaffolding and even less chance of any true learning.

—Duncan, comment on Christopher Danielson’s Geometry and language

Logic, like whiskey, loses its beneficial effect when taken in too large quantities.

—Lord Dunsany, in J. R. Newman (ed.), The World of Mathematics

## E

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.

—Rob Eastaway, Mike Askew, Old Dogs, New Math: Homework Help for Puzzled Parents

I need to teach my students about three things: failing gracefully, persevering, and succeeding graciously.

—Hollis Easter, via Twitter

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.

—Ecclesiastes 9:11

Opportunity is missed by most people because it is dressed in overalls and looks like work.

—Thomas Edison

There is no expedient to which a man will not go to avoid the labor of thinking.

—Thomas Edison

Any fool can know. The point is to understand.

—Albert Einstein

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.

—Albert Einstein

I have no particular talent, I am only inquisitive.

—Albert Einstein

Pure mathematics is, in its way, the poetry of logical ideas.

—Albert Einstein

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.

—Paul Erdös

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.

—Paul Erdos

There is no Royal Road to Geometry.

—Euclid

… 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 …

—Leonhard Euler

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.

—Leonhard Euler

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.

—Leonhard Euler, quoted in the Mathematical and Educational Quotation Server

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.

—Howard Eves, An Introduction to the History of Mathematics

An expert problem solver must be endowed with two incompatible qualities: a restless imagination and a patient pertinacity.

—Howard Eves, In Mathematical Circles

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

## F

I was born not knowing and have had only a little time to change that here and there.

—Richard Feynman

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!

—Dan Finkel, Good Mistakes, Constant Mistakes

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.

—Robert Frost, The Road Not Taken

I am not a teacher, but an awakener.

—Robert Frost [I can’t find the original source. Does anyone know where this comes from?]

## G

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.

—Galileo Galilei, quoted by Clifford Pickover, A Passion for Mathematics

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.

—Martin Gardner, quoted by G. Simmons, Calculus Gems

In no other branch of mathematics is it so easy for experts to blunder as in probability theory.

—Martin Gardner, quoted in Behind Monty Hall’s Doors: Puzzle, Debate and Answer?

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…

—Martin Gardner, foreword to Harold Jacobs’ Mathematics: A Human Endeavor

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.

—Ralph Gardner, Making Math Fun (Seriously)

Math is like ice cream, with more flavors than you can imagine — and if all your children ever do is textbook math, that’s like feeding them broccoli-flavored ice cream.

—Denise Gaskins, About Me

Learning to think a problem through can be hard work—and that is exactly what makes it fun.

—Denise Gaskins, Let’s Play Math: How Families Can Learn Math Together—And Enjoy It

Learning math is an adventure into the unknown. The ideas we adults take for granted are a wild, unexplored country to our children.

—Denise Gaskins, Let’s Play Math FAQs: Introduction

When you struggle with a math concept and conquer it, it will make you free. You don’t have to be afraid of it anymore.

—Denise Gaskins, FAQ: Lifelong Learning for Parents

Turn each equation into a little story. The mental picture helps your child reason out the relationships between the numbers and symbols.

—Denise Gaskins, FAQ: Trouble with Worksheets

Our main goal in mental math is that students recognize their options and build flexibility, not that they work each calculation as fast as possible.

—Denise Gaskins, FAQ: He Won’t Stop Finger-Counting

Explore big concepts. Math that captures a child’s imagination can make the more tedious work seem bearable.

—Denise Gaskins, FAQ: Trouble Finding the Right Math Program

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.

—gasstationwithoutpumps

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange: if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

—Karl Friedrich Gauss, Letter to Bolyai

Algebra is the beginning of a journey that gives you the skills to solve more complex problems. So, try not to think of Algebra as a boring list of rules and procedures to memorize. Consider algebra as a gateway to exploring the world around us all.

—Jason Gibson, Why Study Algebra?

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.

—John Golden, Elementary Homeschool

The most profound learning often takes place silently and invisibly, in between activities and away from prying eyes. It is here that all those pieces of information, having been shaved from actual experience, are pulled inward to jostle against one another in various combinations and arrangements until gradually, or sometimes suddenly, a new understanding emerges.

—Holly Graff, unschooler, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

The ultimate goal of mathematics is to eliminate all need for intelligent thought.

—Ronald L. Graham, quoted in Out of the Mouths of Mathematicians

by Rosemary Schmalz

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.

—Herb Gross, quoted by Jerome Dancis in Reading Instruction for Arithmetic Word Problems

The essence of mathematics is not to make simple things complicated but to make complicated things simple.

—Stanley Gudder

## H

“I hate fractions.”

“They probably hate you, too. The question is, which one of you will be master.”

—Jonathan Halabi, “I hate fractions” at jd2718

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.

—Jonathan Halabi, Outlook on teaching math

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.

—Monty Hall, quoted in Behind Monty Hall’s Doors: Puzzle, Debate and Answer?

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.

—Paul Halmos

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.

—Paul Halmos

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.

—Gordon Hamilton, Commercial Games 8

The experience of mathematics should be profound and beautiful. Too much of the regular K-12 mathematics experience is trite and true. Children deserve tough, beautiful puzzles.

—Gordon Hamilton

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.

—Godfrey H. Hardy, A Mathematician’s Apology

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.

—Godfrey H. Hardy, A Mathematician’s Apology

Get in the math car with a list of destinations and no map. Take whatever route you want, and marvel at the things you discover along the way.

—Nick Harris, @Mr_Harris_Math

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.

—David Hawkins, The Spirit of Play [pdf, 1.4MB], quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

MORE HASTE

(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.

—Piet Hein, Grooks

Problems worthy

of attack

prove their worth

by hitting back.

—Piet Hein, Grooks

One cat just leads to another.

—Ernest Hemingway

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.

—Herman Henkel, [Did he perhaps mean Hermann Hankel?] quoted by Noah benShea, Great Quotes to Inspire Great Teachers

If teachers would only encourage guessing. I remember so many of my math teachers telling me that if you guess, it shows that you don’t know. But in fact there is no way to really proceed in mathematics without guessing. *You have to guess!* You have to have intuitive judgment as to the way it might go. But then you must be willing to check your guess. You have to know that simply thinking it may be right doesn’t make it right.

One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.

—Leon Henkin, from “Round and Round at the Round Table” in Teaching Teachers, Teaching Students: Reflections on Mathematical Education

Mathematics is a game played according to certain simple rules with meaningless marks on paper.

—David Hilbert, quoted by Nicholas Rose, Mathematical Maxims and Minims

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.

—I.N. Herstein

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.

—Rudbeckia Hirta, Learning Curves blog

Audrey seemed, for once, at a loss for words. She was thinking about the question.

I try to stay focused on being silent after I ask young children questions, even semi-serious accidental ones. Unlike most adults, they actually take time to think about their answers and that often means waiting for a response, at least if you want an honest answer.

If you’re only looking for the “right” answer, it’s fairly easy to gently badger a child into it, but I’m not interested in doing that.

—Thomas Hobson, Thank You For Teaching Me

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.

—Hilda Phoebe Hudson

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.

—E. B. Huey, quoted in Elementary Mathematics for Teachers

## I/J

When my students complain that I’m smarter than them, I counter that I’m just at a higher level of misunderstanding.

—Bruce James

A great many people think they are thinking when they are really rearranging their prejudices.

—William James

The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.

—William James, via the Furman University Mathematical Quotations Server

…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.

—Thomas Jefferson, quoted by J. Robert Oppenheimer in the Mathematical Quotations Server

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.

—Jennifer in MamaLand, Spotted (myself!) around the Web…

Nothing produced such odd results as trying to get even.

—Franklin P. Jones

## K

Games put children in exactly the right frame of mind for learning difficult things.

Children relax when they play — and they concentrate. They don’t mind repeating certain facts or procedures over and over, if repetition is part of the game.

Children throw themselves into playing games the way they never throw themselves into filling out workbook pages.

The games solidify the achievements of children who are already good at math, and they shore up children who need shoring up. They teach or reinforce many of the skills that a formal curriculum teaches, plus one skill that formal teaching sometimes leaves out — the skill of having fun with math, of thinking hard and enjoying it.

If you play these games and your child learns only that hard mental effort can be fun, you will have taught something invaluable.

—Peggy Kaye, Games for Math

All it takes to do mathematics is opportunity, a frustrating problem, and a bit of stubbornness.

—Ellen Kaplan, math circle leader, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

A good student is one who will teach you something.

—Irving Kaplansky

The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.

—John Kemeny

The greater our knowledge increases, the greater our ignorance unfolds.

—John F. Kennedy

Where there is matter, there is geometry.

—Johannes Kepler

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.

—James Key

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.

—John Maynard Keynes

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.

—Morris Kline, in N. Rose, Mathematical Maxims and Minims

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.

—Leopold Kronecker

## L

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.

—Pierre-Simon de Laplace, quoted in H. Eves Return to Mathematical Circles

I am nearly driven wild with the Dorcas accounts, and by Mrs. Wakefield’s orders they are to be done now.

I do hate sums. There is no greater mistake than to call arithmetic an *exact *science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Number which it requires a mind like mine to perceive.

For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different.

Again if you multiply a number by another number before you have had your tea, and then again after, the product will be different. It is also remarkable that the Post-tea product is more likely to agree with other people’s calculations than the Pre-tea result.

Try the experiment, and if you do not find it as I say, you are a mere sciolist*, a poor mechanical thinker, and not gifted as I am, with subtle perceptions.

Of course I find myself not appreciated as an accountant. Mrs. Wakefield made me give up the book to [my daughter] Rose and her governess (who are here), and was quite satisfied with the work of those inferior intellects.

—Maria Price La Touche, The Letters of A Noble Woman, London: George Allen & Sons, 1908

**sciolist:* (archaic) A person who pretends to be knowledgeable and well informed. From late Latin *sciolus *(diminutive of Latin *scius *‘knowing’, from *scire *‘know’) + *-ist.*

Experience is the hardest kind of teacher. It gives you the test first, and the lesson afterward.

—Vernon Law

The only teaching that a professor can give, in my opinion, is that of thinking in front of his students.

—Henri Léon Lebesgue

But in the new (math) approach, the important thing is to understand what you’re doing, rather than to get the right answer.

—Tom Lehrer

Someone asked me if I was ever sorry I had chosen mathematics. I said, “I didn’t choose! Mathematics is an addiction with me!”

—Marguerite Lehr

My long-term goal is for my kids to be independent learners, but the best way for that to happen is for me to be by their side now.

—Lucinda Leo at *Navigating by Joy*

As soon as you understand 2 x 4, you can’t believe there was a time when you didn’t understand it.

—Cynthia Copeland Lewis

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.

—Abraham Lincoln, quoted by William Dunham, The Mathematical Universe

If you take any 4-sided shape at all — make it as awkward and as ridiculous as you want — if you take the middles of the sides and connect them, it always makes a parallelogram. Always! No matter what crazy, kooky thing you started with.

That’s scary to me. That’s a conspiracy.

That’s amazing!

That’s completely unexpected. I would have expected: You make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t — it’s always a slanted box, beautifully parallel.

WHY is it that?!

—Paul Lockhart, promotional video for Measurement

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, 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.

—Tony Lucchese

## M

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?

—Liping Ma, Arithmetic in American Mathematics Education: An Abandoned Arena?

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.

—David Marain, Odds and Evens Week of 12-1-10

Procrastination is the art of keeping up with yesterday.

—Don Marquis

Mathematics depend upon the teacher rather than upon the textbook and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas, what Coleridge calls the ‘Captain’ ideas, which should quicken imagination.

—Charlotte Mason, Toward A Philosophy of Education

The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child.

—Charlotte Mason, Home Education

We take strong ground when we appeal to the beauty and truth of Mathematics; that two and two make four and cannot conceivably make five, is an inevitable law.

It is a great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence — that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter.

—Charlotte Mason, Toward A Philosophy of Education

In a word our point is that Mathematics are to be studied for their own sake and not as they make for general intelligence and grasp of mind. But then how profoundly worthy are these subjects of study for their own sake, to say nothing of other great branches of knowledge to which they are ancillary!

—Charlotte Mason, Toward A Philosophy of Education

The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving, though some people find it too rare for their breathing. It differs from most mountainous countries in this, that you cannot lose your way, and that every step taken is on firm ground. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth.

—Charlotte Mason, Ourselves

A child’s intercourse must always be with good books, the best that we can find… We must put into their hands the sources which we must needs use for ourselves, the best books of the best writers.

For the mind is capable of dealing with only one kind of food; it lives, grows and is nourished upon ideas only; mere information is to it as a meal of sawdust to the body.

—Charlotte Mason, Toward A Philosophy of Education

I do a mean T. Rex impression and the class was convulsed in giggles — the perfect way to enter a “hard” math lesson. I chucked the planned lesson for the day, and we went with the dinosaurs, and eventually various other creatures with different numbers of digits. I asked the class how the T. Rex would count. After all, it has only three fingers. I’ll admit to a lot of roaring and stomping as I, the T. Rex, became more and more frustrated trying to write a note to my mother in which I wanted to tell her that I had eaten those four velociraptors.

—Michelle Martin, elementary teacher, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

It is a great nuisance that knowledge can only be acquired by hard work.

—W. Somerset Maugham

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.

—H.L. Mencken

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.

—Dan Meyer

You don’t understand anything until you learn it more than one way.

—Marvin Minsky

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*.

—Edwin E. Moise

Arithmetic is being able to count up to twenty without taking off your shoes.

—Mickey Mouse

## N

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.

—Kate Nowak, Virtual Conference on Core Values: Mistakes are Made

## O

If this feels hard, that doesn’t mean you’re a failure. It means you’re doing the right thing to get better.

—Ben Orlin, Learning to rock-climb is changing how I’ll teach math

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.

—William F. Osgood, quoted in Out of the Mouths of Mathematicians by Rosemary Schmalz

## P/Q

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.

—Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics

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.

—Rózsa Péter, quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

Being in very nature God, He did not consider equality with God something to be grasped, but made Himself nothing…

—Philippians

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.

—Plato, Phaedo

The more I work and practice, the luckier I seem to get.

—Gary Player, quoted in Precalculus Mathematics in a Nutshell

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.

—Henri Poincaré, quoted by Theoni Pappas, More Joy of Mathematics

No other statistical puzzle comes so close to fooling all the people all the time.

—Massimo Piattelli-Palmarini, on the Monty Hall problem, quoted in The Power of Logical Thinking

A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem.

—George Pólya, How to Solve It

Beauty in mathematics is seeing the truth without effort.

—George Pólya

It is better to solve one problem five different ways, than to solve five problems one way.

—George Pólya

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.

—George Pólya, How to Solve It

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.

—George Pólya, How To Solve It

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.

—George Pólya, Mathematical Discovery, Volume II

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.”

—George Pólya, How To Solve It

Circles to square and cubes to double would give a man excessive trouble.

—Matthew Prior

## R

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!

—Constance Reid

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

Just like the games we play, the fun in learning mathematics is in the challenge.

—Erlina Ronda, The fun in learning mathematics is in the challenge

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.

—Andy Rooney

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.

—Malke Rosenfeld, The Elephant in the Room

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.

—Hugo Rossi, Mathematics Is an Edifice, Not a Toolbox

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.

—Gian-Carlo Rota

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.

—Rudy Rucker

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.

—Damon Runyon

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.

—Bertrand Russell, The Autobiography of Bertrand Russell

## S

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.

—W. W. Sawyer, Vision in Elementary Mathematics

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.

—W. W. Sawyer, Mathematician’s Delight

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.

—W. W. Sawyer, Mathematician’s Delight

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.

—W. W. Sawyer, Prelude to Mathematics, [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.]

I used to think my job was to teach students to see what I see. I no longer believe this. My job is to teach students to see; and to recognize that no matter what the problem is, we don’t all see things the same way. But when we examine our different ways of seeing, and look for the relationships involved, everyone sees more clearly; everyone understands more deeply.”

—Joe Schwartz, Then and Now

If you would make a man happy, do not add to his possessions but subtract from the sum of his desires.

—Seneca

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.

—Nick Senger, Quotes about Reading for Students to Paraphrase

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.

—Anne Shirley (fictional), Anne of Avonlea by Lucy Maude Montgomery

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.

—David Eugene Smith

What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?

—David Eugene Smith

During off-times, at a long stoplight or in grocery store line, when the kids are restless and ready to argue for the sake of argument, I invite them to play the numbers game.

“Can you tell me how to get to twelve?”

My five year old begins, “You could take two fives and add a two.”

“Take sixty and divide it into five parts,” my nearly-seven year old says.

“You could do two tens and then take away a five and a three,” my younger son adds.

Eventually we run out of options and they begin naming numbers. It’s a simple game that builds up computational fluency, flexible thinking and number sense. I never say, “Can you tell me the transitive properties of numbers?” However, they are understanding that they can play with numbers.

I didn’t learn the rules of baseball by filling out a packet on baseball facts. Nobody held out a flash card where, in isolation, I recited someone else’s definition of the Infield Fly Rule. I didn’t memorize the rules of balls, strikes, and how to get someone out through a catechism of recitation.

Instead, I played baseball.

—John Spencer, Memorizing Math Facts

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.

—Ian Stewart, The Magical Maze

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.

—Ian Stewart, Letters to a Young Mathematician

Mathematics is a vast adventure; its history reflects some of the noblest thoughts of countless generations.

—Dirk J. Struik, A Concise History of Mathematics

## T

I have studied many philosophers and many cats. The wisdom of cats is infinitely superior.

—Hippolyte Taine

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!

—James Tanton, Thinking Mathematics

…And I am going to panic, because we got 16, but the problem doesn’t want 16. It wants 15. So I have two choices right now: Give up, and cry, and just go home.

Or use my common sense. What would I like this to be?

A good piece of advice: *If you want something in life to work out the way you want it to work, just make it happen.*

I want that to be 16. How can I make that happen? Just add one. Bingo! It becomes 16. However, if you make changes in your life, you’ve got to deal with the consequences…

—James Tanton, Quadratics 2: The Algebra of Quadratics

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.

—Silvanus P. Thompson, Calculus Made Easy

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?

—Archbishop Tillotson

“[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.

—Edward Titchmarsh

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.

—Leo Tolstoy

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.

—Andrei Toom, Word problems: Applications vs. Mental Manipulatives

*New Year’s Day*

Now is the accepted time to make your regular annual good resolutions. Next week you can begin paving hell with them as usual.

Yesterday, everybody smoked his last cigar, took his last drink, and swore his last oath. Today, we are a pious and exemplary community. Thirty days from now, we shall have cast our reformation to the winds and gone to cutting our ancient shortcomings considerably shorter than ever. We shall also reflect pleasantly upon how we did the same old thing last year about this time.

However, go in, community. New Year’s is a harmless annual institution, of no particular use to anybody save as a scapegoat for promiscuous drunks, and friendly calls, and humbug resolutions, and we wish you to enjoy it with a looseness suited to the greatness of the occasion.

—Mark Twain, Letter to Virginia City Territorial Enterprise, Jan. 1863

## U/V

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.

—Unidentified student in Doug’s class, from the comments on Kate’s post Formulas? What Formulas?

Cats aren’t clean, they’re just covered with cat spit.

—Unknown

Dogs believe they are human.

Cats believe they are God.

—Unknown

I got rid of my husband. The cat was allergic.

—Unknown

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.

—Sue VanHattum, Math Mama Writes, Richmond Math Salon: A Sweet Sampling

What do mathematicians do? We play with math. What are little kids doing when they’re thinking about numbers, shapes, and patterns? They’re playing with math. You may not believe it yet, but you can have fun playing with math, too.

—Sue VanHattum, editor, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

Remember that joy and passion lead to more learning than duty ever did.

—Sue VanHattum, editor, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

Can you hear it? That’s the sound of the awesomeness approaching.

—Patrick Vennebush, Math Jokes 4 Mathy Folks

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

—John von Neumann

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.

—Johann Wolfgang von Goethe

The length of your education is less important than its breadth, and the length of your life is less important than its depth.

—Marilyn vos Savant

## W

The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.

—George Washington, quoted by William Dunham, The Mathematical Universe

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.

—Thomas J. Watson

Teaching is the royal road to learning.

—Jessamyn West

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

—Hermann Weyl, The American Mathematical Monthly, November, 1992.

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.

—Alfred North Whitehead, quoted in The Viking Book of Aphorisms

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.

—Alfred North Whitehead, An Introduction to Mathematics

From the very beginning of his education, the child should experience the joy of discovery.

—Alfred North Whitehead

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.

—Andrew Wiles, who finally solved Fermat’s Enigma

Mathematics: a wonderful science, but it hasn’t yet come up with a way to divide one tricycle among three little boys.

—Earl Wilson

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.

—Frederick Winsor

Black holes are where God divided by zero.

—Steven Wright

Five out of four people have trouble with fractions.

—Steven Wright

## X/Y/Z

The excitement of learning separates youth from old age. As long as you’re learning, you’re not old.

—Rosalyn S. Yallow, quoted in Rosalie Maggio (ed.), The Beacon Book of Quotations by Women

It is the process of sharing — of not only creative and insightful problem-solving approaches, but also memorable moments filled with camaraderie, generosity, and incomparable joy. That is why I love math.

—Luyi Zhang, math major, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

## Reference: Websites and Quote Books

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

Your quote about remembering 40% reminds me of another conclusion I once heard drawn from statistics: The speaker said that he had just read that a large majority of automobile accidents occur within 20 miles of a person’s home and at speed less than 40 miles per hour. So clearly, he said, if you want to avoid an accident, never drive at less than 40 miles per hour when you are within 20 miles of your home.

That’s a good one, Jay! I would love to use it as a blackboard quote. It’s a bit long, but with judicious editing…