Quotations XXII: Six Months in the Dark

[Photo by frozenchipmunk.]

School is in session, which means I am once again searching out pithy, inspirational quotations for my chalkboard. Some recent tidbits…

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

Gary Player
Quoted in Precalculus Mathematics in a Nutshell

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Quotes XXI: How Is Logic Like Whiskey?

Photo by Brian – Progressive Spin.

Logic is the science of making valid deductions and proofs — and it is also a fruitful topic for blackboard quotes. Here are a few of my favorites:

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

G. K. Chesterton
The Man Who Was Orthodox

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Quotations XX: How Old Was Erdös?

Math blackboard
Photo by foundphotoslj.

Here are a few mathematical gems from my co-op class blackboard:

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

John Adams

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

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Quotations XIX: How Do We Learn Math?

He doesn’t learn algebra
in the algebra course;
he learns it in calculus.

I have been catching up on my Bloglines reading [procrastinating blogger at work — I should be going over the MathCounts lesson for Friday’s homeschool co-op class], and found the following quotation at Mathematics under the Microscope [old blog posts are no longer archived].

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Quotations XVIII: The Art of Asking Questions

Priorities, motivation, and a little bit of math from my blackboard…

We make a living by what we get, but we make a life by what we give.

Winston Churchill

In mathematics, the art of asking questions is more valuable than solving problems.

Georg Cantor

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Quotations XVII: If People Don’t Believe That Mathematics Is Simple…

Quotes from my blackboard during October:

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

Vernon Law

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

David Eugene Smith

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Quotations XVI: Back to the Blackboard

Classes are back in session at our homeschool co-op, so I am again collecting short quotes for the blackboard. Here are the ones I used in September:

Any fool can know. The point is to understand.

Albert Einstein

Life without geometry is pointless.


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

Marvin Minsky

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Quotations XV: More Joy of Mathematics

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

Dirk J. Struik
A Concise History of Mathematics

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

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Cat Quotations

Princess Kitten

I found this worksheet on the KISS Grammar website, and I loved the quotations so much I just had to share them:

In a cat’s eye, all things belong to cats.

— English proverb

One cat just leads to another.

— Ernest Hemingway

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Quotations XIV: The Joy of Mathematics

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

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

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The Procrastinating Blogger Award

Procrastination is the art of keeping up with yesterday.

Don Marquis

Joyful Days kindly nominated me for the Thinking Blogger Award back in the days of the dinosaurs. Well, she isn’t that old, really — it was only last April. I am grateful to her for thinking of me, and ever since then I have been thinking deeply about whom to nominate in my turn. Or, to be more precise, I printed out the nomination post as a reminder, and then it got lost in a pile of “to sort/read/file” papers on a shelf under my desk…

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Quotations XIII: Mathematics Education Is Much More Complicated than You Expected

Registrations have been rolling in for our homeschool co-op, and the most popular classes are full already. Math doesn’t seem to be a “most popular” class. I can’t imagine why! Still, many of my students from last year are coming back for another go, and I am getting spill-over from the science class waiting list.

Anyway, I have started planning in earnest for our fall session. As usual, I look to those wiser than myself for inspiration…

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

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Math Jokes

Blame it on MathNotations and his Corny Math Jokes (which actually included one I hadn’t heard before) — or maybe I have been reading too many of Chickenfoot’s strange tales — but anyway, I’m in a mood for humor.

So here are a couple of old favorites:

Eric W. Weisstein
from MathWorld–A Wolfram Web Resource

Hat tip: These had gotten lost in the dustbunnies of my memory until I saw the Frivolous Theorem mentioned recently at Art of Problem Solving.

Edited to add: Scott at Grey Matters recently updated his Mathematical Humor post, which may be where I had originally read these. He links to several more great MathWorld jokes, including the ever-tasty Pizza Theorem.

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Quotations XII: Mathematicians at Play

This week’s quotes for teachers:

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

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

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See You Later! (Quotations XI)

We will be heading out soon on vacation, so I will not be blogging for awhile. The rest of this week is devoted to packing. (I hate packing!) But before I leave, here is a longish quote on teaching math from the book I am reading this week: Ian Stewart’s Letters to a Young Mathematician.

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….

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Quotations X: The Royal Road

More quotations especially for teachers:

There is no Royal Road to Geometry.


Teaching is the royal road to learning.

Jessamyn West

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

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Historical Tidbits: Alexandria Jones

[Read the story of the pharaoh’s treasure: Part 1, Part 2, and Part 3.]

Alexandria JonesHere are a few more tidbits from math history, along with links to relevant Internet sites or books, and three more math puzzles for you to try. I hope you find them interesting.

Next time, a new adventure (sort of)…

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How Can We Teach Problem Solving?

We continue to plan our co-op courses for next fall. Some of the classes I had hoped for will not happen, and my children are going to have to make some tough choices between the remaining topics. Unfortunately, they have not yet mastered the ability to be in two classrooms at once.

I have three math courses to plan, and I think I will focus as much as I can on teaching math through problems, even at the elementary level. These are once-a-week enrichment classes for homeschooled students, so I assume they have a “normal” math program at home. I want to introduce a few topics they might not otherwise see, to deepen their understanding of the topics they have studied, and to give them a taste of that “Aha!” feeling that comes from conquering a challenging problem. Has anybody done something like this, and can you recommend some good resources?

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Quotations IX: A Good Student Is…

Aaaargh! My Internet service is on the brink again. But since I had to run into my husband’s office to check email, I’ll take a few minutes to post a quick note. Here are a couple of quotes especially for teachers:

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

Henri Léon Lebesgue

A good student is one who will teach you something.

Irving Kaplansky

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Quotations VIII: The Essence of Mathematics

Our homeschool co-op classes are done for the semester, so this will be my last compilation of blackboard quotes for awhile. I love collecting quotations, however, so I will be treating you to some of my favorites “just for teachers” over the summer. Stay tuned!

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

George Polya

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

George Washington Carver

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Happy Pi Day I

photo by jorel314 (cc by 2.0)

photo by jorel314 (cc by 2.0)

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

I don’t remember anyone ever mentioning Pi Day when I was in school, but any excuse to celebrate math sounds like fun. March 14 at 1:59 (a.m. or p.m.) is about as close as the calendar can get to 3.14159…

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In Honor of the Standardized Testing Season…

[Feature photo above by Alberto G. (CC-BY-SA-2.0) via flickr.]

The school experience makes a tremendous difference in a child’s learning. Which of the following students would you rather be?

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


“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.”

“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.”

“Nine from eight I can’t, you know,” Alice replied very readily: “but—”

“She can’t do Subtraction,” said the White Queen. “Can you do Division? Divide a loaf by a knife — what’s the answer to that?”

“I suppose—” Alice was beginning, but the Red Queen answered for her. “Bread-and-butter, of course.”

“She can’t do sums a bit!” the Queens said together, with great emphasis.

— Lewis Carroll
Through the Looking Glass

…in other words…

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

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Two Fathoms Deep and Stuck in Muck

“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

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Common Sense and Calculus

One more quote from W. W. Sawyer’s Mathematician’s Delight before I have to return the book to the library:

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.

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Mathematics and Imagination

Comments by W. W. Sawyer, in his wonderful, little book, Mathematician’s Delight:

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.

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Harmonic Series Quotation

If you’d like to start your week with a laugh, here’s a great quote:

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.”

—Rudbeckia Hirta
Learning Curves Blog: The Harmonic Series
quoting Alexandre Borovik

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Why Study Mathematics?

by d3 Dan via flickr

[Rescued from my old blog.]

What teacher hasn’t heard a student complain, “When am I ever going to have to use this?” Didn’t most of us ask it ourselves, once upon a time? And unless we choose a math-intensive career like engineering, the truth is that after we leave school, most of us will never again use most of the math we learned. But if math beyond arithmetic isn’t all that useful, then what’s the point?

If you or your student is singing the Higher Math Blues, here are some quotations that may cheer you up — or at least give you the strength of vision to keep on slogging.

We study mathematics…

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Math Quotes V: A Man Is Like a Fraction…

[Rescued from my old blog.]

The blackboard quotes for my math class have been a bit more philosophical the last few weeks:

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

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