There are 3 kinds of mathematicians: those that can count ...



Quotes on Mathematics and Geometry

Is Teaching Mathematics Important? --- How do we respond?

"I think there's no way they should have to teach [math] now. We have computers. We no longer need to know why[pic]." (Rosie O'Donnell–Newsweek, April 9, 2001 arguing that there's no place for mathematics in the curriculum)

Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important... The knowledge doesn't make life any sweeter or happier, does it? (Helen Keller 1880 - 1968)

I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it. (Malcom X, 1925-1965)

Mathematicians boast of their exacting achievements, but in reality they are absorbed in mental acrobatics and contribute nothing to society. (Sorai Ogyu, 1666 - 1729)

Is How We Teach Mathematics Important?

[Asked whether he would like to see an experimental demonstration of conical refraction]

No. I have been teaching it all my life, and I do not want to have my ideas upset. (Isaac Todhunter 1820 - 1910)

Teach to the problems, not to the text. (Kim Nebeuts, Return to Mathematical Circles, 1988)

To state a theorem and then to show examples of it is literally to teach backwards. (Kim Nebeuts, Return to Mathematical Circles, 1988)

We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others. (Blaise Pascal, 1623-1662)

You know we all became mathematicians for the same reason: we were lazy. (Max Rosenlicht, 1949)

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, Teaching Teachers, Teaching Students, 1981)

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 Polyá, 1887-1985)

He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical. (Henry David Thoreau, 1817-1862)

Measurement

Measure what is measurable and make measurable what is not so. (Galileo Galilei, 1564-1642)

Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space. (W. B. Frankland, The Story of Euclid, 1901)

In the physical world, one cannot increase the size or quantity of anything without changing its quality. Similar figures exist only in pure geometry. (Paul Valéry, 1871-1945)

Geometry

The only right angle from which to approach any problem is the try angle. (author unknown)

Life without geometry is pointless. (author unknown)

He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side. (Plato, ca 429-347 BC Republic)

Poetry is as exact a science as geometry. (Gustave Flaubert, 1821-1880)

Inspiration is needed in geometry, just as much as in poetry. (Aleksandr Sergeyevich Pushkin, 1799 - 1837)

And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art... (Albrecht Dürer, 1471-1528)

Where there is matter, there is geometry. (Johannes Kepler, 1571-1630)

Geometry is a skill of the eyes and the hands as well as of the mind. (Jean Pedersen)

God is like a skillful Geometrician. (Sir Thomas Browne, 1605-1682)

The ludicrous state of solid geometry made me pass over this branch. (Plato, ca 429-347 BC Republic)

Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it is precisely this sort of mathematics which is of practical value. (Branko and Shephard Grunbaum, 1926- ?)

A work of morality, politics, and criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry. (Bernard Le Bovier Fontenelle, 1657-1757)

Logic, Proof, and Reasoning

“Obvious” is the most dangerous word in mathematics. (Eric Temple Bell, 1883-1960)

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. (Eric Temple Bell, 1883-1960)

Mathematics consists of proving the most obvious thing in the least obvious way. (George Polyá, 1887-1985)

Words differently arranged have a different meaning and meanings differently arranged have a different effect. (Blaise Pascal, 1623-1662)

The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality. (C. O. Oakley, American Mathematical Monthly, 1949)

Whoever ... proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured. (Albrecht Dürer, 1471-1528)

Proof is the idol before whom the pure mathematician tortures himself. (Sir Arthur Eddington, 1882-1944)

The truth of a theory is in your mind, not in your eyes. (Albert Einstein, 1879-1955)

Perhaps the greatest paradox of all is that there are paradoxes in mathematics. (Kasner, E. and Newman, J., Mathematics and the Imagination, 1940)

I have had my results for a long time, but I do not yet know how I am to arrive at them. (Karl Friedrich Gauss, 1777-1855)

There are no deep theorems-only theorems that we have not understood very well. (Nicholas P. Goodman, The Mathematical Intelligencer, 1983)

It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, `Yes, it is true, but you won't be able to understand the proof.' (Ronald Graham, Scientific American, 1993)

Problem Solving

Everything should be made as simple as possible, but not simpler. (Albert Einstein, 1879-1955)

Each problem that I solved became a rule which served afterwards to solve other problems. (Rene Descartes, 1596-1650)

An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity. (Howard Eves, In Mathematical Circles, 1969)

I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery. (Rene Descartes, 1596-1650)

An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them. (Werner Heisenberg, 1901-1976)

An engineer, a physicist, and a mathematician were all sleeping in their respective rooms when the house they were in caught fire in the middle of the night. The engineer woke up, saw the fire and quickly threw a bucket of water on the fire. The fire was put out, but the room was soaked, and much was ruined.

The physicist woke up, and saw the fire. He quickly calculated how much heat was in the fire, went to the sink and filled up a bucket with exactly the right amount of water. He threw the water on the fire, which was put out by the water without damaging anything else.

The mathematician woke up like the others and like the physicist calculated how much water would be needed to put out the fire. Then, he went back to sleep, satisfied that there was a solution… (Jon Meilstrup)

Mathematics in General

Do not worry about your difficulties in mathematics; I assure you that mine are greater. (Albert Einstein, 1879-1955)

Life is good for only two things, discovering mathematics and teaching mathematics. (Siméon Poisson, 1781-1840)

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. (Nikolai Lobatchevsky, 1793-1856)

Mathematics is a game played according to certain simple rules with meaningless marks on paper. (David Hilbert, 1862-1943)

A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence. (Howard Eves, In Mathematical Circles, 1969)

Perfect numbers like perfect men are very rare. (Rene Descartes, 1596-1650)

There are 3 kinds of mathematicians: those that can count and those that cannot. (author unknown)

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. (Paul Adrien Maurice Dirac, 1902- ?)

If you ask mathematicians what they do, you always get the same answer. They think. (M. Egrafov, Mathematics Magazine, 1992)

Mathematics is a language. (Josiah Willard Gibbs, 1839-1903)

It has been said that figures rule the world. Maybe. But I am sure that figures show us whether it is being ruled well or badly. (Johann Wolfgang von Goethe, 1749-1832)

Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes. (Richard W. Hamming, Mathematical Maxims and Minims, 1988)

I am interested in mathematics only as a creative art. (Godfrey H. Hardy, 1877-1947)

Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years.

(Hermann Weyl,1885 – 1955, Said in 1950 in The American Mathematical Monthly)

Technology

I cannot do it without comp[u]ters. (William Shakespeare, 1564 - 1616)

The real danger is not that computers will begin to think like men, but that men will begin to think like computers. (Sydney J. Harris, Return to Mathematical Circles, 1988)

I will use a teaspoon to till a flower pot,

a spade to till a flower bed,

a tiller to till a garden,

and a tractor and plow to till a field.

I will not drive a tractor and plow into my house to till a flower pot. (Timothy Peil, 1995)

Jokes for Mathematics and Geometry

Measurement

You are lost in the woods, what is the furtherest you can be from getting out? (Solution: half way.)

You come across a hole which measures five feet by three feet by two feet. How much dirt is in the hole? (Solution: the amount of dust that is in the air.)

Geometry

What did the acorn say when he finally grew up? (Solution: Geometry.)

A hunter left his camp walking five miles south and two miles east. He shot a bear and walked five miles north back to camp. What color was the bear? (Solution: white, since the hunter must be near the North Pole.)

Can you build a square cabin in such a way that all four corners point north? (Solution: yes, build the cabin at the South Pole.)

What is an insect that is not feeling well? (Solution: A secant.)

The towns of Equa, Later, and All gave a welcome home parade for a returning hero. The parade route traveled between the three cities which are located at the vertices of an equilateral triangle, and the three roads are flat, direct, and straight. The parade went at a constant rate. It took 80 minutes to go from Equa to Later and 80 minutes to go from Later to All, but it took an hour twenty to go from All to Equa. Explain the discrepancy. (Solution: There is no discrepancy, since 1 hour 20 minutes equals 80 minutes.)

What branch of mathematics is studied by small children? (Solution: ‘Top’ology.)

Once upon a time there was a super smart horse that could add, subtract, multiply, and divide. Someone suggested that the horse should try a bit of analytic geometry. The horse immediately died.

Moral of the story: You cannot put Descartes before the horse.

The Möbius strip is a thing

Which somewhat resembles a ring

But given the strength

To travel its length

You still haven’t done anything.

Problem Solving

A Student's Guide to Problem Solving

Joe Dobson, Mathematics Supervisor, Winston-Salem, N. C.

Rule 1: If at all possible avoid reading the problem. Reading the problem only consumes time and causes confusion.

Rule 2: Extract the numbers from the problem in the order in which they appear. Be on the watch for numbers written in words.

Rule 3: If rule 2 yields three or more numbers, the best bet for getting the answer is adding them together.

Rule 4: If there are only two numbers which are approximately the same size, then subtraction should give the best results.

Rule 5: If there are only two numbers in the problem and one is much smaller than the other, then divide if it goes evenly—otherwise, multiply.

Rule 6: If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.

Rule 7: If the rules 1–6 don't seem to work, make one last desperate attempt. Take the set of numbers found by rule 2 and perform about two pages of random operations using these numbers. You should circle about five or six answers on each page just in case one of them happens to be the answer. You might get some partial credit for trying hard.

Rule 8: Never, never spend too much time solving problems. This set of rules will get you through even the longest assignments in no more than ten minutes with very little thinking.

Mathematics in General

A mathematics teacher who talks at length affects both ends of students. One end is made to feel numb and the other end dumb.

Classic Paradoxes and Puzzle Problems

1. Assume the earth is a perfect sphere, 25000 miles in circumference at the equator. Suppose it has a single non-stretchable wire circling the earth at the equator. Then suppose 50 feet of wire is added to the original wire and this new wire forms a concentric circle around the earth. Could a person put a razor blade between the earth and this new wire? crawl under the wire? or walk under the wire?

2. A fly crawls the shortest possible distance from the lower corner of a room to the opposite upper corner. The room is 20 feet long, 16 feet wide and 10 feet high. What was the distance the fly crawled?

3. Materials: strip of paper and a scissors.

(a) Is it possible for a fly to cross both sides of a strip of paper without crossing an edge?

(b) Now if we split the strip lengthwise down the middle will we get two strips?

4. Prove: The circumference of a small circle equals the circumference of a larger circle.

Let a wheel with a circular hub roll along a horizontal plane until it has made one revolution, then the line[pic] is equal to the circumference of the wheel (large circle). The hub (small circle) has also made one revolution, since it is drawn on the side of the wheel, concentric with the larger circle. Hence, the circumference of the small circle rolls out the line[pic]. Therefore, the two circumferences are equal. (What is wrong with the proof?)

5. Prove: An obtuse angle is congruent to a right angle.

Let [pic] be an obtuse angle and [pic] be a right angle. Make[pic], join [pic], bisect it at [pic] and draw l perpendicular to [pic]at E. Also, bisect [pic] in [pic] and draw[pic]at F. The two perpendiculars meet in [pic]. Draw [pic]then [pic]and[pic]. Since[pic], the sides of the triangle [pic] are congruent, each to the sides of[pic]. Hence, these triangles are in every way congruent and the angle opposite the side [pic] is congruent to the angle opposite[pic]. Likewise, [pic]. [pic]

or [pic] So an obtuse angle is congruent to a right angle. (What is wrong with the proof?)

Solutions to Classic Paradox and Puzzle Problems

1. A person could easily walk under it. Since[pic], the added 50 feet in circumference would

add about 8 feet to the radius.

2. Approximately, 37.36 feet. Shortest distance between two points is a straight line. Layout the two walls as if they were one flat wall.

3. (a) Yes, construct a Möbius strip.

(b) No, just one longer strip.

4. The fallacy is due to the fact that there is slippage involved in the turn of the small wheel.

5. The fallacy is in the construction. If constructed correctly, [pic] will be on the opposite side of line

[pic] as C and [pic] will be on the same side of the line [pic] as B.

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