Why Johnny Can't Add - KeynesYouDigIt/Knowledge GitHub Wiki

The story

Let us look into a modern mathematics classroom. The teacher asks, "Why is 2 + 3 = 3 + 2?"

Unhesitatingly the students reply, "Because both equal 5"

No, reproves the teacher, the correct answer is because the commutative law of addition holds. Her next question is, Why is 9 + 2 = 11?

Again the students respond at once: "9 and 1 are 10 and 1 more is 11."

"Wrong," the teacher exclaims. "The correct answer is that by the definition of 2,

9 + 2 = 9 + (1 + 1).

But because the associative law of addition holds,

9+(1+1)=(9+1)+1.

Now 9 + 1 is 10 by the definition of 10 and 10 + 1 is 11 by the definition of 11."

Evidently the class is not doing too well and so the teacher tries a simpler question. "Is 7 a number?" Thà students, taken aback by the simplicity of the question, hardly deem it necessary to answer; but the sheer habit of obedìence causes them to reply affirmatively. The teacher is aghast. "If I asked you who you are, what would you say?"

The students are now wary of replying, but one more courageous youngster does do so: "I am Robert Sinith."

The teacher looks incredulous and says chidingly, "You mean that you are the name Robert Smith? Of course not. You are a person and your name is Robert Smith. Now let us get back to my original question: Is 7 a number? 0f course notl It is the name of a number. 5 + 2, 6 + 1, and 8 - 1 are names for the same number. The symbol 7 is a numeral for the number.

The teacher sees that the students do not appreciate the distinction and so she tries another tack. "Is the number 3 half of the number 8?" she asks. Then she answers her own question: "Of course not! But the numeral 3 is half of the numeral 8, the right half."

The students are now bursting to ask, "What then is a number?" However, they are so discouraged by the wrong answers they have given that they no longer have the heart to voice the question. This is extremely fortunate for the teacher, because to explain what a number really is would be beyond her capacity and certainly beyond the capacity of the students to understand it. And so thereafter the students are careful to say that 7 is a numeral, not a number. Just what a number is they never find out.

The teacher is not fazed by the pupils poor answers. She asks, "How can we express properly the whole numbers between 6 and 9?"

"Why," one pupil answers, "just 7 and 8."

"No", the teacher replies. "It is the set of numbers which is the intersection of the set of whole numbers larger than 6 and the set of whole numbers less than 9."

Thus are students taught the use of sets and, presumably, precision.

A teacher thoroughly convinced of the vaunted value of precise language, and wishing to ask her students whether a number of lollipops equals a number of girls, phrases the question thus: "Find out if the set of lollipops is in one-to-one correspondence with the set of girls." Needless to say, she gets no answer from the students.

Bent but not broken, the teacher asks one more question: "How much is 2 divided by 4?"

A bright student says unhesitatingly, "Minus 2."

"How did you get that result?" asks the teacher.

"Well," says the student, "you have taught us that division is repeated subtraction. I subtracted 4 from 2 and got minus 2."

It wouId seem that the poor children would deserve some relaxation after school, but parents anxious to know what progress their children are making a1so query them. One parent asked his eight-year-old child, "How inuch is 5 + 3?" The answer he received was that 5 + 3 = 3 + 5 by the commutative law. Flabbergasted, he re-phrased the question: "But how many apples are 5 apples and 3 apples?"

The child didn't quite understand that "and" means plus and so he asked, "Do you mean 5 apples plus 3 apples?"

The parent hastened to say yes and waited expectantly.

"Oh," said the child, "it doesn't matter whether you are talking about apples, pears or books; 5 + 3 = 3 + 5 in every case."

Another father, concerned about how his young son was getting a1ong in arithmetic, asked him how he was faring.

"Not so well," the boy replied. "The teacher keeps talking about associative, commutative and distributive laws. I just add and get the right answer, but she doesnt like that.

Traditional Curriculum

Sequence:

Grades 1-6 are arithmetic
Grades 7-8 are algebra and light geometry
Grade 9 is algebra
Grade 10 is deductive geometry
Grade 11 is intermediate algebra and trigonometry
Grade 12 is solid geometry and advanced algebra

Criticisms:

These are mechanical processes which rely on memorization rather than understanding
The processes are disconnected
Geometry seems like a big jump because its style of reasoning is so different. Because it's such a jump, students just end up memorizing the proof.
A connection is never made to proofs in geometry and proofs in algebra
The traditional curriculum respects tradition too much- triangles are not necessary when there aren't as many prospective surveyors
It's cold and abstract, so what motivation does anyone have to learn it? The rewards are too far in the future, "to get into college" makes college sound bad, and not everyone will be a professional mathmatician, scientist, or engineer.
Technical language, symbolism, and abstract concepts obscure the reasoning. You can use social or legal problems, like differentiating between "all good cars are expensive but not all expensive cars are good." That's easier to grasp than "All parallelograms are quadrilaterals, but not all quadrilaterals are parallelograms."
These don't capture the beauty of mathematics. Subjects are not selected for their beauty, and adding fractions isn't beautiful. Learning French grammar doesn't capture the beauty of French literature.
Some students are attracted by the intellectual challenge (or because they like things they just happen to do well), but this is rare, and cold comfort to people who just find it confusing and arbitrary
If the applications are just as boring and disconnected as the theory, in what sense are they applications?
This is like requiring students to read musical notation without allowing them to play music
Thousands of textbooks have been published on the traditional math curriculum, the vary only in order of topics. They include low-value topics that help them sell the books better.

If what you're really teaching is thinking and not memorization, what could a student steal from an open book exam?

The Origin of the Modern Mathematics Movement

In the early 50's, there was widespread agreement that math education had been unsuccessful- low grades, low adult retention, low student interest
Driven by the space race, some math researchers tried to make a new math curriculum
They assumed the problem was that math was too old, students knew this, and therefore deduced that it was irrelevant
The proposed solution was the to teach newer forms of mathematics- abstract algebra, topology, symbolic logic, set theory, and boolean algebra

Some criteria one could evaluate the new mathematics on:

Is it correct? Useless, doesn't address how students will take to it
Will it develop mathematicians? This wouldn't be appropriate given the disproportionate number of students that will become mathematicians
The contents should contribute to the goals of school and be accessible to young people

Reform of mathematics education was necessary, but it's not clear that curriculum was the most important thing to tackle first. That effort might have been better spent on making better teachers, who in turn may have contributed to making a better curriculum.

The deductive approach to mathematics

Familiarity breeds uncriticalness

Since most students were learning by rote memorization, the new math proposed teaching logically, one step at a time
Commutative and associative properties are taught as ways to understand arithmetic, positive and negative integers
Some compromises have to be made because some subjects are too academic (irrational numbers, roots)
Axioms are things that are held as self-evident theorems are built with them
Many of these things tooks hundreds of years to gain acceptance- irrational numbers, 0, negative numbers, complex numbers. A lot of the work that was done with them was intuited, bumbled through, or just wrong
The use of letters to represent a class of numbers (eg. .ax + b, where a and b represent any real number) leads to general expressions that are true for any value of those numbers
Many early cultures used words instead of variables
Most of the greatest advancements in math were intuited, not deduced. The logic always came later.
You need to go through the same challenges as your ancestors, and experience the same conclusions to think at the same level they did. An educator can accelerate this.
You can't get someone to think scientifically by pretending all of the bumbling didn't happen, giving them neat and clean results. This ignores history.
Civilization advances by extending the number of important operations which we can perform without thinking about them
"I have my result, but I don't know how to prove it yet"

Rigor

Modernists believe that students are disturbed by assumptions, which is demonstrably false. It's a matter of intuitive vs. not-intuitive assumptions.
Rigor often seems like proving the obvious, and doesn't endear the discipline to students, and distracts from learning new things. The number of minor features is so large that the major features of the subject fail to stand out.
The rigor doesn't enhance a student's understanding, it destroys it
Rigor is also a moving target- 1800's standards were not acceptable in 1900, which were not acceptable in 1950
Rigor was never intended as an aid to pedagogy- it's the domain of professional mathematics

The Language of Mathematics

Aimed at precision, not understanding
This produces extreme volumes of abstract terminology
No addition and subtraction: "Binary operations"
Language cuts corners if it communicates more effectively- avoid excess verbiage if there's no danger of error (eg. Robert Smith the name and Robert Smith the boy)
Formal definitions do not aid instruction: Students know what a triangle is and do not have to be taught that it consists of the union of three noncollinear points and the line segments joining them. That's much harder than just saying "triangle"
Introducing opaque, abstract terms puts a huge burden on your memory
You can teach a lot of terms without actually teaching any math. Terminology, especially pretentious terminlogy is no subtitute for substance.
Symbols are dense, and often language is much more clear. Symbols don't clarify words, it's the other way around.
Musical notation is not music, and you wouldn't prioritize notation over being able to play
Overly precise language can introduce doubt about what is being said just because it so precise

Math for Math's Sake

New math concepts grow out of a practical need, not because the theory is so extensible
Math isn't an abstract thing that's applied to the world; it's a practical thing that can viewed abstractly. Why do we teach the abstraction?
This often happened because the curriculum was written by pure mathematicians, who taught it to teachers, and none of them know how to actually apply it
Math is not an isolated, self-sufficient body of knowledge. It exists primarily to help man understand and master the physical, economic, and social worlds. Otherwise, it wouldn't have any place in school.
Isolating math robs it of its meaning
Math is not a natural human interest, because it's so abstract. Law, economics, biology, and a lot of other more vital fields are natural interests that provide challenging problems.
You can't appreciate the varieties of structures unless you've seen enough things that might make you want to investigate them

The New Contents of the New Mathematics

What we had before, plus set theory, number bases, congruences, inequalities, matrices, symbolic logic
The majority of it is still the old stuff, so not new
"New" fields (linear programming, ops research, game theory, quality control) all rely on traditional mathematics
Mathematics is cumulative, old stuff is foundational
Set theory operations are not a good use of limited time. It was added to make the New Mathematics sound sophisticated.
Interest in music theory does not produce musicians. Knowing color theory doesn't make you a painter. While set theory is at the foundation of sophisticated and rigorous math, it's no use in learning how to do elementary math.
Set theory for elementary math is a hollow formalism that encumbers ideas that are far more easily understood intuitively
Most of the traditional stuff is there, it just got pushed earlier, when it's less applicable.
Symbols don't control or direct thinking, they only communicate something that was thought. You still think in language.
Knowing a general theorem is not particularly useful or impressive when you only half-know one application of it
Highly abstract concepts can't be taught at an elementary level
The more general an abstract concept is, the emptier it is
It's not efficient to teach an abstract concept early because it covers several concrete cases at once. A good understanding of the concretes is an understanding of the abstractions, not the other way around. A thorough understanding of the concrete must precede the abstract- an abstract concept is meaningless until you have a lot of diverse concretions in mind.
Abstractions must grow on people, you can't hand them out. They need to be right in front of them, and then you can point them out. Students can mouth the abstractions, but that's not the same as understanding those abstractions.

The Testimony of Tests

Most tests require students to answer a fair number of questions in a limited amount of time. If it were a real question that needed a real result, it would take too long.
Most tests are just tests of memory. If they were tests of thinking, why wouldn't they all be open book?
Accurate assessment of the programs is hard, because the teachers are often self-selecting, the students are usually unusually good, and many of the new math approaches only superficially adopted the program (to satisfy interested administrators).
Telling students they're in a special program makes them try extra hard (the Hawthorne Effect)
The pressure on young people to secure a college education has brought many more students to college who are not as well prepared in all subjects and who are less motivated.
Mathematics is being sped up when slowing down would be wiser
You should teach things in the order they were discovered, not the order of their foundations. Retrace the steps that led to it.

The Deeper Reasons for the New Mathematics

Math has become divorced from the sciences, when it used to be integral to it
The great classical mathematicians were put their work to practical use
Math is too narrowly specialized and focused inward
Math is now split between "pure" and "applied"
It may come to pass that engineers and scientists come up with the new advancements in math, and mathematicians won't know about them because they are too isolated
Applications of math to science came from ideas that were inspired by science- not the other way around.
Most new math reformers were college professors, with hyper-narrow specializations and no experience outside of math. They acted as though pedagogy were only an implementation detail. Pedagogy is harder than mathematics.
Professional mathematicians are already motivated to pursue math, and don't take into account that other people don't see the point.
They are neither professional writers nor teachers
Mathematicians have a lot of hubris, and assume they're good at everything since they were smart enough to learn math
Mathematics is a simple subject compared to economics, psychology, or physics, and doesn't require a very broad background, and doesn't require the complexities of people. It's a refuge for people who shy away from people and the world.
Teachers have been over-awed by college professors, as have school administrators.

The Proper Direction for Reform

Math should be taught as a liberal art, where you learn about a subject and the role it plays in our culture and society
Math is our key to understanding the physical world, painting realisitcally, sound recording and reproduction, technology, music, biology, and medicine. Philosophical quests for truth are aided mathematically, and literature often concerns itself with the math of its time.
Knowledge is a whole, and math is part of that whole- it didn't develop apart from other activities and interests
We should teach the relationship between math and other human interests
You don't rob a future mathematician: You're showing more people how they might make a career using these skills
Students are not naturally motivated to learn an abstract. Their natural motivation is the real, largely physical problems.
You can teach a concept first, but you need to apply it immediately
Use of real problems not only motivates mathematics, but gives meaning to it
If you derive the math you teach from the real world, you don't need to figure out how to translate it back to the real world
Rather than trying to deduce, the students should be constructing mathematics with the aid of a teacher. This enables a student to discover, but requires the right questions to ask them to lead them there. The questions have to be reasonably answerable.
Teachers get anxious to "cover ground" that they hand the final statements down, and since the students don't have time to learn them, they memorize them.
Genetic principle: The historical order is usually the right order
Proof as a criterion for acceptance of a result and proof as pedagogy are unrelated
The Hindus, Arabs, and Europeans all reasoned about radicals through analogy. This lasted until the 19th century.
Math lab: Teaching 0 velocity with parabolas by throwing a ball in the air, shapes with geoboards, number lines with rods, waves with oscilliscopes, pendulums, springs & weights, tuning forks
Test mathematical ideas heuristically, which helps them stick and increases intuition
Intuition can lead to error, but fear of error can't be a deterrent
Logical presentation is at best subordinate and supplementary aid to learning, and at worst an obstacle. So present things as intuitively as possible, not as rigorously as possible.
The deductive proof is the final step. It only needs to convince the student, not a mathematician. The capacity to appreciate rigor is relative to the mathematical age of the student, not the age of the math
Little lies: teaching a student a proof is rigorous when it isn't is pedagogy. Our own appreciation for rigor changes over time, why can't theirs? Their is no perfect proof.
It doesn't matter if a student can define a polygon if they can recognize and work with them
Keep vocab down, and try to use words they already know. Verbalization comes after understanding. Same with symbols, which are intimidating.
Concentrating on curriculum is escaping realilty- we need better educated teachers with better liberal arts backgrounds.
The best choices for mathematics education professors would be broadly educated mathematicians with a genuine interest in education, but such people do not stand out in the mathematical world and would be harder to locate.

Quotes

With or without proof, the traditional method of teaching results in far too much of only one kind of learning - memorization. The claim that such a presentation teaches thinging is grossly exaggerated. By way of evidence, if evidence is needed, I have challenged hundreds of high school and college teachers to give open book examinations. This suggestion shocks them. But if we are really teaching thinking and not memorization, what could the students take from the books?

A few students are attracted to mathematics by the intellectual challenge or because they like what they happen to do well. The rare student who experiences this challenge may indeed be intrigued - as some mathematicians are - by the fact that there are only five regular polyhedra. However, as far as most students are concerned, the world would be just as well off if there were an inifinite number of them. As a matter of fact, there is an infinite number of regular polygons and no one seems depressed by this fact.

Textbook writers also seem to take inordinate pride in brevity, which can often be interpreted as incomprehensibility. Reasons for steps are either not given or given so briefly as to make the presentation almost useless to the student. Many authors seem to be saying, "I have learned this material and now I defy you to learn it." Brevity in mathematical exposition is the soul of witlessness and obscurity.

Reform of mathematics education was called for, but there is a serious question as to whether curriculum was the weakest component and should have been tackled first. It would, I believe, be generally conceded that the policy of universal education pursued in the United States is highly commendable, but our country was not and still is not prepared to carry on such a .program. Certainly we do not have enough qualified teachers; therefore the education in many parts of this country is woefully weak. Were more good teachers available they wou1d have been able long ago, by acting in concert, to remedy the defects of the traditional curriculum. Since the teacher is at least as important as the curriculum, the money, time and energy devoted to curriculum reform might well have been devoted to the improvement of teachers.

If it took mathematicians a thousand years from the time first-class mathematics appeared to arrive at the concept of negative numbers - and it did - and if it took another thousand years for mathematicians to accept negative numbers - as it did - we may be sure that students will have difficulties with negative numbers. Moreover, the students will have to master these difficulties in about the same way that the mathematicians did, by gradually accustoming themselves to the new concepts, by working with them and by taking advantage of all the intuitive support that the teacher can muster.

One could of course argue that the growth of mathematics may indeed have proceeded as just described, but now that we have the proper logical structures for the number system, the calculus and other branches, we need not ask the students to repeat the fumblings of the masters. We can give students the correct approaches and they will understand them. This argument can be countered by the fact that the greatest mathematicians did try to build the logical foundations for the various subjects buf failed for centuries to do so. Their failure should serve as some evidence that the logical approaches are not easy to grasp. One can compress history and avoid many of the wasted efforts and pitfalls, but one caunot eliminate it. Of course, our students may be superior to the best mathematicians of the past.

Most proofs presented to students are also artificial for additiona1 reasons. When a mathematician seeks to prove a theorem he suspects is correct, he uses any means, however clumsy, indirect or devious, but perhaps more natural to the creative process, to make the proof. Once the theorem is proven he or his successors, now able to see how the essential difficulty was overcome, can usually devise a smoother or more direct proof. Some theorems have been reproven several times, each successive proof remodeling and simplifying the previous one and often including generalizations or stronger results. Thus, the final theorem and proof are far from the original natural thoughts. One should expect, then, that the student facing such a reworked, polished, and possibly more complicated resu1t would not be able to grasp it.

The insistence on a logical approach also deceives the student in another way. He is led to believe that mathematics is created by geniuses who start with axioms and reason directly from the axioms to the theorems. The student, unable to function in this manner, feels humbled and baffled but the obliging teacher is fully prepared to demonstrate genius in action. It is intellectually dishonest to teach the deductive approach as though the results were acquired by logic.

Asking the students to cite axioms in the elementary operations with numbers is like asking an adult to justify each action he takes after getting up in the morning. Why should he bathe? Why should he brush his teeth? Why wear clothes? And so on. If a man should consciously consider and answer these questions he would never get to work. Most of what he does in the morning must be habitual. There is a story that a centipede was walking along leisurely when it met a toad. The toad remarked to the centipede, "Isn't it wonderful? You have one hundred feet and yet you know when to use each one". Thereupon the centipede began to think about which foot to use next and was unable to move.

The mathematician says A, writes B, means C, but D is what it should be. And D is in fact a splendid idea which emerges from tidying up the mess.

The deductive structures which many mathematicians like to call mathematics are tbe dried-up stalks of the living plants. They are empty formalisms as opposed to real contents; they are shells of palaces.

Another argument advanced by the advocates of the new mathematics is that their emphasis on logical structure teaches students to think deductively. This is probably correct. But even if they do teach deductive logic, why is this so important? It is not the kind of thinking that is useful in everyday life. The big problems and even the little ones that human beings are called upon to solve in life cannot be solved deductively. There are no self-evident axioms from which one can deduce what career to follow, whom to marry, or even whether to go to the movies. The real decisions call for judgment, and this is entirely difierent from deductive reasoning which leaves no room for judgment. The legal mind, the business mind, and the political mind are much more relevant.

To ask students to recognize the need for these missing axioms and theorem is to ask for a critical attitude and maturity of mind that is entirely beyond young people. If the best mathematicians did not recognize the need for these axioms and theorems for over two thousand years how can we expect young people to see the need for them?

In a rigorous development of geometry and algebra many intuitively obvious theorems must be proved. The students' conclusion will be that mathematics is largely concerned with proving the obvious.

The modernists apparently also want to keep their subject pure. They don't wish to sully it; they desire to remove the dross of earth from which mathematics has risen. But as they wash the ore they keep the iron and lose the gold. A perfect command of the English language is useless if a man has nothing to say, and pure mathematics has little to say to young students.

Descartes:

I found that, as for Logic, its syllogisms and the majority of its other precepts are useful rather in the communication of what we already know or... in speaking without judgment about things of which one is ignorant, than in the investigation of the unknown

W.S. Gilbert:

If youre anxious for to shine in the high aesthetic line as a man of culture rare,
You must get up all the germs of the transcendental terms, and plant them everywhere.
You must lie upon the daisies and discourse in novel phrases of your complicated state of mind, The meaning doesnt matter if its only idle chatter of a transcendental kind.

And every one wilI say
As you walk your mystic way,

"If this young man expresses himself in terms too deep for me, Why, what a very singularly deep young man this deep young man must be!"

The very point of an abstract formulation is that it unifies and reveals common properties in concrete and familiar branches of mathematics. Therefore, abstraction is not tbe first stage but the last stage in a mathematica1 development. It may give insight, but only into concrete structures already well learned. It unifies, but only what one already knows. Without much previous knowledge of concrete cases the abstract concepts remain empty, arbitrary children of mathematical fantasy. To confront youngsters with abstractions that lie above their level of maturity is to create bewilderment and revulsion rather than increased knowledge. In brief, the highly abstract concepts cannot be exploited at an elementary level.

In a real sense one cannot teach an abstraction. The difficulty posed to the student is analogous to giving him a correct biological definition of dogs and then showing him a poodle and a collie as examples. When presented with a bull terrier and asked if that is a dog, the student may still be baffled. The biological definition contains so many broad and technical terms that he may not really understand it; if so he cannot apply it.

The new mathematics advocates, countering the charge of too much abstraction, have cited tbe Harvard psychologist Jerome S. Bruner who said, "Any subject can be taught in- some intellectublly honest form to any child at any stage of development." The saving feature of this doctrine is its vagueness.

In mathematics, knowledge of any value is never possession of information, but "know-how". To know mathematics means to be able to do mathematics:to use mathematical language with some fluency, to do problems, to criticize arguments, to find proofs and, what may be the most important activity, to recognize a mathematical concept in, or to extract it from, a given concrete situation.-Therefore, to introduce new concepts without a sufficient background of concrete facts, to introduce. unifying concepts where there is no experience to unify, or to harp on the introduced concepts without concrete applications which would challenge the students, is worse than useless: premature formalization may lead to sterility; premature introduction of abstractions meets resistance especially from critica1 minds who, before accepting an abstraction, wish to know why it is relevant and how it cou1d be used.

The object of mathematical rigor is to sanction and legimize the conquests of intuition, and there never was any other object for it.

This genetic principle may safeguard us from a common confusion: If A is logically prior to B in a certain system, B may still justifiably precede A in teaching, especially if B has preceded Á in history. On the whole, we may expect greater success by following suggestions from the genetic principle than frorn the purely forrnal approach tomathematics.

Nor did these men hesitate to put to practical use the scientific knowledge that they and others had gathered. Newton studied the motion of the moon to help sailors determine their longitude at sea. Euler studied the design of ships and of sails, made maps, and wrote a masterful text on artillery. Descartes designed lenses to improve the telescope and microscope. Gauss not only made a survey of the electorate of Hannover, but worked on the improvement of the electric telegraph and the measurement of magnetism. These few examples could be multiplied a hundredfold. Almost all of these men not only saw the potential in the scientific knowledge they were helping to amass but were keenly concerned that the knowledge be utilized.

All the applications of mathematics to science came from mathematica1 ideas which were inspired by science. No mathematician ever cooked up ideas useful to science by sitting in an ivory tower. It is true that ideas inspired by science later found unexpected application, but the ideas were sound to start with because they derived from genuine physical problems.

It is also easy to see why the texts are so poorly written. Professional mathematical writing has a style of its own. It is succinct, monotonous, symbolic and sparse. The chief concern is to be correct. On the other hand, good texts must have a lively style, arouse interest, tell students where they are going and why. Writing is an art and mathematicians do not cultivate it.

It is rational to present mathematics logically, but it is not wise. Consider the teaching of calculus. One knows that the calculus is built on the theory of limits and so may conclude that the way to teach calculus is to start with the theory of limits. The wise man would also consider whether young people can learn the theory of limits from scratch and whether they will want to learn it without motivation and prior insight.

Further, teaching 2 + 2 = 5 and then having to correct it is one thing but teaching subtraction as "taking away" and then introducing the notion that -2 is the additive inverse to 2 is another. For a youngster the latter is verbiage.

Hence the education for all these students should be broad rather than deep. It should be a truly liberal arts education wherein students not only get to know what a subject is about but a1so what role it plays in our culture and our society. Put negatively, there should be no attempt to train professionals in mathematics and little concern for what future study in mathematics may require.

If we are compelled for practical reasons to separate learning into mathematics, science, history and other subjects, let us at least recognize that this separation is artificial and false.

Alfred Whitehead:

In scientific training, the flrst thing to do with an idea is to prove it. But allow me for one moment to extend the meaning of "prove"; I mean - to prove its worth. . .

The solution which I am urging, is to eradicate the fatal disconnection of subjects which kills the vitality of our modern curriculum. There is only one subject-matter for education, and that is Life in all its manifestations. Instead of this single unity, we offer children Algebra, from which nothing follows; Geometry, from which nothing follows. . .

Let us now return to quadratic equations. . . . Why should children be taught their solution? . . .

Quadratic equations are part of algebra and algebra is the intellectual instrument for rendering clear the quantitative aspects of the world.

Alfred Whitehead:

Elementary mathematics must be purged of every element which can only be justified by reference to a more prolonged course of study. There can be nothing more destructive of true education than to spend long hours in the acquirement of ideas and methods which lead nowhere. There is a widely-spread sense of boredom with the very idea of learning. I attribute this to the fact that they [the students] have been taught too many things merely in the air, things which have no coherence with any train of thought such as would naturally occur to anyone, however intellectual, who has his being in this modern world. The whole apparatus of learning appears to them as nonsense

To present mathematics as a liberal arts subject requires a radical shift in point of view. The traditional and modern approaches treat mathematics as a continuing cumulative logical development. Algebra precedes geometry because some algebra is used in geometry. Trigonometry follows geometry because a modicum of geometry is used in the former subject. The new approath would present what is interesting, enlightening, and culturally significant - restricted only by a slight need to include earlier concepts and techniques that will be used later. In other words, we should be objective-oriented rather than subject-oriented.

Mathematics proper does not and perhaps should not appeal to ninety-eight per cent of the students. It is an esoteric study, entirely intellectual in its appeal and lacking the emotional appeal of, say, music and painting. The creative mathematician may derive some emotiona1 values such as satisfaction of the ego, pride in achievement, and glory - values none too noble, in any case.

There is a widely known story that a mathematician lecturing before a class got stuck in the middle of a proof. He went over to the side of the board, drew a few pictures, erased them and was then able to continue his lecture. What this story implies about pedagogy is seriously disturbing but it does speak for the uses of pictures.

The training of good teachers is far more important than the curriculum. Such teachers can do wonders with any curriculum. Witness the number of good mathematicians we have trained under the traditional curriculum, which is decidedly unsatisfactory. A poor teacher and a good curriculum will teach poorly whereas a good teacher will overcome the deficiencies of any curriculum.