Some of Mr. Hansenís Favorite Quotations

(rev. 4/20/08, 8/2/09, 11/6/09, 3/27/010, 7/11/010, 8/21/13, 6/21/15)



The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low, and achieving our mark.



EERac (an anonymous poster on Scott Aaronsonís Shtetl-Optimized blog)

The potentially valuable aspect of studying high school math isnít learning to solve certain random types of problems. Itís learning to think clearly about deceptively-simple concepts. In doing this you learn to attack problems, generalize techniques, define concepts, make convincing arguments, and find counterexamples. If you need to know the quadratic formula for some weird reason, then all you need to do is look it up. The skills listed above arenít just useful to scientists, but to lawyers, writers, doctors, etc.

Nigel Goldenfeld

The only meaningful distinction between styles of mathematics is Applied Mathematics, and Mathematics that has yet to be applied.

G. H. Hardy

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

Sofia Kovalevskaya

It is impossible to be a mathematician without being a poet in soul.

Edna St. Vincent Millay

Euclid alone has looked on Beauty bare.

Isaac Newton

If I have seen farther than others, it is because I have stood on the shoulders of giants.

Heinz Pagels
[in 1988, when complexity research was in its infancy]

The great unexplored frontier is complexity ... I am convinced that the nations and people that master the new science of Complexity will become the economic, cultural, and political superpowers of the next century.

Horacio Porta, a professor of mine at UIUC

This is such a great theorem that it remains valid despite a change of notation in the middle of the proof.

J. J. Uhl, another professor whom I knew at UIUC

[Click here for Dr. Uhlís compilation of quotes from many sources.]

Unknown (oft-repeated quote, perhaps coined by someone in the test-prep industry)

A calculator is a tool that gives you the wrong answer faster.


Metaknowledge and Cognition

George Bernard Shaw

Beware of false knowledge; it is more dangerous than ignorance.

(Einstein supposedly said something similar: ďThe only thing more dangerous than ignorance is arrogance.Ē See another Einstein quote below.)

Alfred North Whitehead

It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

[For a plethora of other excellent quotes from Whitehead, click here.]



Benoit Mandelbrot*

Think not of what you see, but what it took to produce what you see.

* (as quoted by Keith Devlin in the NOVA program ďFractals: Hunting the Hidden DimensionĒ)

Henri Poincarť

Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.


Teaching and Learning


I hear and I forget. I see and I remember. I do and I understand.

Albert Einstein

Education is what remains after one has forgotten everything he learned in school.

Paul Lockhart

Teaching is not about information. Itís about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you canít be real, then you have no right to inflict yourself upon innocent children.

John Stuart Mill

A pupil from whom nothing is ever demanded that he cannot do, never does all he can.

William Butler Yeats

Education is not filling a bucket, but lighting a fire.



Frederick Brooks

Adding manpower to a late software project makes it later.

Bill Gates

Software is more important than hardware.


Mr. Hansenís Probably Original Quotations on Math, Education, Math Ed, Life, etc.

early 2000s

Math: the worldís only 100% baloney-free zone.

[Bragging time: If you Google baloney-free zone without quotes, Iím the #4 hit. If you Google ďbaloney-free zoneĒ with quotes around the words, Iím #2 and #3. And if you Google the words must pass quiz in any order, Iím the #1 hit in the world!]

ca. 2002

If math is to be worth anything, it has to be more about communication than computation. Computers do computation faster and more accurately than the best of us. You want answers? Computers can crank them out by the billions. Answers arenít worth all that much. Good questions are more interesting, and explanations and understanding are worth a lot. Focus on the things that are uniquely human. If all you can do is execute an algorithm or plug something into a formula, you havenít added any value.


I am well aware that in 20 years, the vast majority of my students will remember almost nothing about meóbasically, they will recall whether I was kind to them or mean to them, and thatís about it. But a few, perhaps 5%, will hopefully remember me as one of the handful of teachers who influenced their life in a meaningful, positive way. That tiny, tiny yield is what motivates me as a classroom teacher, and I know from talking to other teachers that this is a common sentiment. We went into the field precisely because it does not conform well to bottom-line thinking. If we had been more conventional in our thinking, we would have chosen a different profession.

. . . [W]e face a crisis in education. The crisis is real, and it is present at this very moment. The crisis is that we have misidentified the crisis.

In a misguided attempt to make education more accountable and achievement-oriented, we (as a nation) run the risk of removing all the education from the system. What will be left behind is training, 13 miserable years of it, with no break for art or music, and our nation will be all the poorer. The students know this. Letís listen to them for a change.


Studentsí time has a dollar value, too. Itís not only the labor hours of teachers that cost money. A school that organizes time and schedules for the convenience of adults and treats kids as if they should sit around, bored, most of the time . . . is a school that has no business surviving in the 21st century.


We are the miracles others are praying for.


Whoever thinks failure is a tragedy must think success on the first try is commonplace.


The fact that fireflies have a short season does not make them any less miraculous.


Do not be envious of people with lots of money or lots of things. Maybe thatís all they have.