Math Education Quotations
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- "The second most important job in the world, second only to being a good parent, is being a good teacher." (S.G. Ellis)
- "… math is just logic with numbers attached." (Marilyn vos Savant born August 11, 1946 is an American magazine columnist, author, lecturer and playwright who rose to fame through her listing in the Guinness Book of World Records under "Highest IQ".
- ”Less is more.” (Robert Browning; English poet and playwright; 1812–1889; in a 1855 poem titled Andrea del Sarto.
Introduction
The mathematics and the teaching of mathematics have a very long history. A little bit of this history can be captured in quotations from math researchers, math educators, math users, and others.
Notice the "Less is more." quotation given above. A short quotation (such as the one-liners that we frequently see) can be thought of as a very short article or story. An important idea is captured in just a few words. Here is a version of this idea that comes from a famous mathematician, Blaise Pascal:
- "I have made this letter longer than usual, only because I have not had the time to make it shorter." (Blaise Pascal, French mathematician and scientist; 1523–1662.)
Mathematics is a language that helps to communicate precisely and in relatively short length. However, it takes careful thought as well as a lot of time and effort to write mathematics well.
The quotations given here have been divided into three categories—mathematics, math education, and general education. Sometimes a quotation could just as well fit into one category as the other.
Suggestions to Secondary School Math Teachers
- Each week, post a different math-related quote on your bulletin board.
- Once in a while, have a contest among your students to find the "best math-related quote."
Quotations About "What is Mathematics?
This section contains a few often quoted statements that help to provide insight into "What is mathematics?"
"No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof." (Leonardo da Vinci; Italian scientist, mathematician, engineer, inventor, anatomist, painter, sculptor, architect, botanist, musician and writer; 1452–1591; quoted in Concepts of Mathematical Modeling by Walter J. Meyer.)
"Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency." (René Descartes, French philosopher, mathematician, scientist, and writer; 1596–1650.)
"Each problem that I solved became a rule which served afterwards to solve other problems." (René Descartes, French philosopher, mathematician, scientist, and writer; 1596–1650.)
"The most powerful single idea in mathematics is the notion of a variable." (Alexander Keewatin Dewdney, 1941–, Canadian, computer scientist, mathematician, and philosopher.)
"One of the most important concepts in all of mathematics is that of function." (T.P. Dick and C.M. Patton.)
"How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality?" (Albert Einstein; German physicist; 1879–1995.)
"Nature's great book is written in mathematics." (Galileo Galilei, 1564–1642, Italian physicist, mathematician, astronomer, and philosopher.
“Mathematics is the queen of the sciences.” (Carl Friedrich Gauss, German mathematician, physicist, and prodigy; 1777–1855.)
"Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing his difficulties." Evarist Galois, 1811-1832)
"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." (Hermann Hankel; German mathematician; 1839–1873.)
"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." (G. H. Hardy; English mathematician; 1877–1947.)
"The mathematician's patterns, like the painter's or the poet's, must be beautiful." (G. H. Hardy; English mathematician; 1877–1947.)
Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey Harold Hardy; English mathematician; 1877–1947.)
“Statistics: the mathematical theory of ignorance.” (Morris Kline; American mathematician and a writer on the history, philosophy, and teaching of mathematics; 1908–1992.)
"It is impossible to be a mathematician without being a poet in soul." (Sophia Kovalevskaya; the first major Russian female mathematician, and also the first woman who was appointed to a full professorship in Europe in 1889; 1850–1891.)
"A mathematical formula should never be "owned" by anybody! Mathematics belong to God." (Donald Knuth; American computer scientist; 1938–.)
“God created the natural numbers. All the rest is the work of man.” (Leopold Kronecker; German mathematician and logician; 1823–1891.)
"If I have seen further it is by standing on the shoulders of giants." (Isaac Newton; English mathematician & physicist; 1642 - 1727; Letter to Robert Hooke, February 5, 1675.)
"To speak freely of mathematics ... I call it the most beautiful profession in the world ..." (Blaise Pascal; 1523–1662.)
“Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems.” (George Polya; math researcher, researcher and educator; 1877–1985.)
"Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform." (Bertrand Russell; English philosopher, historian, logician, and mathematician; 1872–1970.)
"Mathematics, rightly viewed, possesses not only truth but supreme beauty—a beauty cold and austere, like that of sculpture." (Bertrand Russell; English philosopher, historian, logician, and mathematician; 1872–1970.)
"Mathematics, in the common lay view, is a static discipline based on formulas...But outside the public view, mathematics continues to grow at a rapid rate...the guide to this growth is not calculation and formulas, but an open ended search for pattern." (Lynn A. Steen; American mathematician and math educator.)
"… algebra is the language of mathematics, which itself is the language of the information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995). It is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education." (Lynn A. Steen, American mathematician and math educator.)
“One has not even a slight idea of what mathematics is, one does not suspect its extraordinary scope, the nature of the problems that it proposes and solves, until one knows what a function is, how a given function is studied, how its variations are followed, how it is represented by a curve, how algebra and geometry aid each other mutually, how number and space illustrate one another, how tangents, areas, volumes are determined, how we are led to create new functions, new curves, and to study their properties. Precisely these notions and methods are needed to read technical books in which mathematics is applied. They are indispensable to whoever wishes to understand the rapid scientific movement, the manifold scientific applications of our times which day by day tend to modify more profoundly our fashion of thinking and of living.” (Jules Tannery; French mathematician; 1848–1910.)
"Numbers constitute the only universal language." (Nathanael West; American novelist; 1903–1940.)
Quotations About Math Education
"What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?" (Benjamin Franklin.)
"A great teacher makes hard things easy." (Ralph Waldo Emerson; American essayist, philosopher, poet, and leader; 1803–1882.)
"Each problem that I solved became a rule which served afterwards to solve other problems." (René Descartes, French philosopher, mathematician.)
"Nature's great book is written in mathematics." (Galileo; 1564 – 1642.)
"The book of nature is written in the language of mathematics. (Galileo; 1564 – 1642.)
"The greatest calamity in the history of science was the failure of Archimedes to invent positional notation." (Carl Friedrich Gauss.)
Murray Gell-Mann. 16 TED minute March 2007 video: Beauty and truth in physics. Gell-Mann is a Nobel Prize winning physicist. Quoting from the video:
- A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have.
- As we go to higher and higher energies (smaller and smaller distances), the next onion skin (manifestation of the basic law) resembles the previous one to some extent. The result is that newly encountered phenomena are described rather simply, and therefore elegantly, in terms of mathematics close to what was already developed for phenomena studied earlier.
"The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method, ..." (Paul Halmos, 1916—2006.) Quoting from the Wikipedia:
- In the American Scientist 56(4): 375-389, Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways.
- Halmos's 1985 "automathography" I Want to Be a Mathematician is an outstanding account of what it was like to be an academic mathematician in 20th century America. He called the book “automathography” rather than “autobiography”, because its focus is almost entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means, and is a favourite of many teachers of mathematics:
- "Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?”
"In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generations adds a new story to the old structure." (Hermann Hankel.)
"Science is what we understand well enough to explain to a computer. Art is everything else we do." (Donald Knuth; American computer scientist; 1938–.)
"Logic is the art of going wrong with confidence." (Morris Kline; American mathematician and a writer on the history, philosophy, and teaching of mathematics; 1908–1992.)
“Statistics: the mathematical theory of ignorance.” (Morris Kline; American mathematician and a writer on the history, philosophy, and teaching of mathematics; 1908–1992.)
“A proof tells us where to concentrate our doubts.” (Morris Kline; American mathematician and a writer on the history, philosophy, and teaching of mathematics; 1908–1992.)
"It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could be safely relegated to anyone else if machines were used." (Gottfried Leibniz; German philosopher and mathematician; 1646–1716
"The greatest unsolved theorem in mathematics is why some people are better at it than others." (Adrian Mathesis; mathematician.)
"Learning to solve problems is the principal reason for studying mathematics." (National Council of Teachers of Mathematics.)
"Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state." (Plato; Classical Greek philosopher, mathematician, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the western world; 428/427 BC– 348/347 BC.)
"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 Poincare; French physicists and mathematician; 1854–1912.)
"I am too good for philosophy and not good enough for physics. Mathematics is in between." (George Polya.)
A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise. (George Polya.)
"The point is to make math intrinsically interesting to children. We should not have to sell mathematics by pointing to its usefulness in other subject areas, which, of course, is real. Love for math will not come about by trying to convince a child that it happens to be a handy tool for life; it grows when a good teacher can draw out a child's curiosity about how numbers and mathematical principles work. The very high percentage of adults who are unashamed to say that they are bad with math is a good indication of how maligned the subject is and how very little we were taught in school about the enchantment of numbers."Alfred Posamentier, Professor of Mathematics Education at the City College of New York, 2002 New York Times article.
"At the age of eleven, I began Euclid, with my brother as my 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; 1872–1970, English philosopher, historian, logician, and mathematician.)
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." (Bertrand Russell; English philosopher, historian, logician, and mathematician; 1872–1970.)
"We cannot hope that many children will learn mathematics unless we find a way to share our enjoyment and show them its beauty as well as its utility." (Mary Beth Ruskai; Tufts University.)
“Civilization advances by extending the number of important operations which we can perform without thinking of them.” (Alfred North Whitehead; British Mathematician and Philosopher; 1861-1947.)
“The art of progress is to preserve order amid change and to preserve change amid order.” (Alfred North Whitehead; British Mathematician and Philosopher, 1861-1947.)
"In formal logic, a contradiction is the signal of defeat, but in the evolution of real knowledge it marks the first step in progress toward a victory." (Alfred North Whitehead; British Mathematician and Philosopher, 1861-1947.)
Quotations About Education in General
"They know enough who know how to learn." (Henry B. Adams; American writer; 1883–1918.)
"A teacher affects eternity; he can never tell where his influence stops. (Henry B. Adams; American writer; 1883–1918.)
"He who dares to teach must never cease to learn. (Anonymous.)
“Education is a human right with immense power to transform. On its foundation rest the cornerstones of freedom, democracy and sustainable human development.” (Kofi Annan, Ghanaian diplomat, seventh secretary-general of the United Nations, winner of 2001 Nobel Peace Prize.)
"Where the needs of the world and your talents cross, there lies your vocation." (Aristotle.)
"In short, learning is the process by which novices become experts. " (John T. Bruer. Schools for Thought, 1999, page 13.)
"Your work is to discover your work and then with all your heart to give yourself to it." (Buddha
"Education is the key to unlock the golden door of freedom." (George Washington Carver.)
"Learning without thinking is labor lost; thinking without learning is dangerous." (Chinese Proverb.)
"Give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime." (Chinese proverb.)
“Any teacher that can be replaced by technology—should be.” (Arthur C. Clarke; British science fiction author; 1917–2008.)
"Genius is one per cent inspiration, ninety-nine per cent perspiration." (Thomas Alva Edison.)
“I never teach my pupils; I only attempt to provide the conditions in which they can learn.” (Albert Einstein.)
"The value of an education ... is not the learning of many facts but the training of the mind to think something that cannot be learned from textbooks." (Albert Einstein.)
“Education is what remains after one has forgotten what one has learned in school. “ (Albert Einstein)
“Wisdom is not a product of schooling, but of the life-long attempt to acquire it.” (Albert Einstein)
“The aim (of education) must be the training of independently acting and thinking individuals who, however, can see in the service to the community their highest life achievement.” (Albert Einstein)
"The second most important job in the world, second only to being a good parent, is being a good teacher." (S.G. Ellis)
“You must be the change you wish to see in the world." (Mahatma Gandhi, Indian political leader.)
"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, The Disciplined Mind: What All Students Should Understand, Simon & Schuster, 1999.)
"... pedagogy is what our species does best. We are teachers, and we want to teach while sitting around the campfire rather than being continually present during our offspring's trial-and-error experiences." (Michael S. Gazzaniga, 1998, p 8)
"In a completely rational society, the best of us would be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and the highest responsibility anyone could have." (Lee Iacocca, American industrialist; born in 1924.)
"Apply yourself. Get all the education you can, but then, by God, do something. Don't just stand there, make it happen." (Lee Iacocca, American industrialist; born in 1924.)
“At every of these schools shall be taught reading, writing, and common arithmetick, and the books which shall be used therein for instructing the children to read shall be such as will at the same time make them acquainted with Graecian, Roman, English, and American history. At these schools all the free children, male and female, resident within the respective hundred, shall be intitled to receive tuition gratis, for the term of three years, and as much longer, at their private expence, as their parents, guardians or friends, shall think proper.” This is quoted from a bill brought before the Virginia Legislature in 1778 by Thomas Jefferson. The legislation was titled “A Bill for the More General Diffusion of Knowledge (1778)." It was not passed.
“Children are the message we send to the future.” (Abraham Lincoln) Also attributed to others, such as Marshall McLuhan.
“...we discovered that education is not something which the teacher does, but that it is a natural process which develops spontaneously in the human being. It is not acquired by listening to words, but in virtue of experiences in which the child acts on his environment. The teacher's task is not to talk, but to prepare and arrange a series of motives for cultural activity in a special environment made for the child.” (Maria Montessori; Italian physician and educator, a noted humanitarian best known for the philosophy of education which bears her name; August 31, 1870–May 6, 1952.)
"Once you have learned how to ask relevant and appropriate questions, you have learned how to learn and no one can keep you from learning whatever you want or need to know." (Neil Postman and Charles Weingartner. Teaching as a Subversive Activity.)
"The wisest mind has something yet to learn." (George Santayana.)
"In the book of life, the answers aren't in the back." (Charles Schulz; American cartoonist best known worldwide for his Peanuts comic strip; the quoted statement is from the comic strip character Charlie Brown; 1922–2000.)
"When you're average, you're just as close to the bottom as you are the top." Alfred North Whitehead; British mathematician and philosopher; 1861–1947.)
"From the very beginning of his education, the child should experience the joy of discovery." (Alfred North Whitehead; British mathematician and philosopher; 1861–1947.)
Miscellaneous Uncategorized and Unattributed
Copied from http://www.onlinemathlearning.com/math-quotes.htmlOnlineMathLearning.com:
- "Decimals have a point." http://www.onlinemathlearning.com/math-quotes.html
- "Natural numbers are better for your health."
- "Geometry is just plane fun."
- "Polar coordinates aren't just arctic fashions."
- - - - - - - - - -
"Calculus has its limits." (Quoted in many places.)
"Trigonometry is a sine of the times." (Author Unknown.)
""Small minds discuss persons. Average minds discuss events. Great minds discuss ideas. Really great minds discuss mathematics." (Anonymous.)
"How many times can you subtract 7 from 83, and what is left after wards? You can subtract it as many times as you want, and it leaves 76 every time." (Author Unknown.)
"The lottery is a tax on people who are bad at math." Quoting from http://www.ithadtobeyou.net/carpe/archives/001012.html:
- In fact, I go with the famous line from "Almost Live!" (an old late night Seattle show that used to come on before Saturday Night Live) that "the lottery is a tax on people who are bad at math".
Biologists defer only to chemists, chemists defer only to physicists, physicists defer only to mathematicians, and mathematicians defer only to God, although you'll be hard pressed to find one so humble. Anonymous quote found a http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?t=5890.
Some Quotation Sources
Crewton Raymone's House of Math. Quoting from the site:
- Here are some of my favorite Math quotes. You will find these quotes scattered about on every page of this site. Here they are, all in one place in no particular order. This is by no means a complete list. I'll be adding Math quotes here as well as other quotes on education and learning and then more general quotes that are semi-on topic. I collect quotes
Donald Knuth, computer scientist. Knuth is a world leader in computer science. Quoting Knuth: " I have been a happy man ever since January 1, 1990, when I no longer had an email address. I'd used email since about 1975, and it seems to me that 15 years of email is plenty for one lifetime."
101 Great Computer Programming Quotes.
Wiki Quotes:
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