Mathematics education

"Because of white racism's ability to bludgeon us into believing that we are inferior beings and therefore incapable of learning math and the sciences, we must spend a significant amount of our learning and teaching time unlearning and unteaching. This is to say, for example, that when a brother or sister reaches the freshman college level, he or she has already been subjected to at least 17 years of conditioning that dictates: "You are too black and too ignorant to understand such lily-white and intelligent things as math, chemistry, physics, etc." Thus, a major part of the teacher's initial instructional time must be spent dealing with the psychological block against learning math—or any of the sciences."

"Histories make men wise; poets, witty; the mathematics, subtile; natural philosophy, deep; morals, grave; logic and rhetoric, able to contend."

"By the beginning of the seventeenth century we may say that the fundamental principles of arithmetic, algebra, theory of equations, and trigonometry had been laid down, and the outlines of the subjects as we know them had been traced. It must be, however, remembered that there were no good elementary text-books on these subjects; and a knowledge of them was therefore confined to those who could extract it from the ponderous treatises in which it lay buried. Though much of the modern algebraical and trigonometrical notation had been inroduced, it was not familiar to mathematicians, nor was it even universally accepted; and it was not until the end of the seventeenth century that the language of the subjects was definitely fixed. Considering the absence of good text-books, I am inclined... to admire the rapidity with which it came into universal use, than to cavil at the hesitation to trust to it alone which many writers showed."

"Those intending to continue in mathematics or science or technology... believe that a survey of the main directions along which living mathematics has developed would enable them to decide more intelligently in what particular field of mathematics, if any, they would find a lasting satisfaction. ...It is astonishing how few students entering serious work in mathematics or its applications have even the vaguest idea of the highways, the pitfalls, and the blind alleys ahead of them. Consequently, it is the easiest thing in the world for an enthusiastic teacher... to sell his misguided pupils a subject that has been dead for forty or a hundred years, under the sincere delusion that he is disciplining their minds. With only the briefest glimpse of what mathematics in this twentieth century—not in 2100 B.C.—is about, any student of normal intelligence should be able to distinguish between live teaching and dead mathematics. He will be less likely to drown in the ditch or perish in the wilderness."

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"The student can actually carry out the mathematical tasks in an authentically historical setting. He can do long division like the ancient Egyptians, solve quadratic equations like the Babylonians, and study geometry just as the student in Euclid's day. To get involved in the same processes and problems as the ancient mathematicians and to effect solutions in the face of the same difficulties they faced is the best way to gain appreciation of the intelligence and ingenuity of the scholars of early times."

"Students enjoy... and gain in their understanding of today's mathematics through analyzing older and alternative approaches."

"1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics."

"The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression."

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

"The author holds that our school curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled many a fine mind. It is the aim of this book to restore this cultural content and present the evolution of number as the profoundly human story which it is. ...the historical method has been freely used to bring out the rôle intuition has played in the evolution of mathematical concepts. And so the story of number is here unfolded as a historical pageant of ideas, linked with the men who created those ideas and with the epochs which produced the men."

"The systematic exposition of a textbook in mathematics is based on logical continuity and not on historical sequence; but the standard high school course in mathematics fails to mention this fact, and therefore leaves the student under the impression that the historical evolution of number proceeded in the order in which the chapters of the textbook were written. This impression is largely responsible for the widespread opinion that mathematics has no human element. For here, it seems, is a structure that was erected without a scaffold: it simply rose in its frozen majesty, layer by layer! Its structure is faultless because it is founded on pure reason, and its walls are impregnable because they were reared without blunder, error or even hesitancy, for here human intuition had no part! In short the structure of mathematics appears to the layman as erected not by the erring mind of man but by the infallible spirit of God. The history of mathematics reveals the fallacy of such a notion."

"Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise."

"A finished or even a competent reasoner is not the work of nature alone... education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one... in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not. ..Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:— 1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general. 3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion. 4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if... reason is not to be the instructor, but the pupil. 5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. ...These are the principal grounds on which... the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life."

"I have tried to say to students of mathematics that they should read the classics and beware of secondary sources. This is a point which Eric Temple Bell makes repeatedly... in '... that the men of whom he writes learned their mathematics not by studying in school or by reading textbooks, but by going straight to the sources and reading the best works of the masters who preceded them. It is a point which in most fields of scholarship at most times in history would have gone without saying. ...The purpose of a secondary source is to make the primary sources accessible to you."

"Should I teach them from the point of view of the history of science, from the applications? My theory is that the best way to teach is to have no philosophy, [it] is to be chaotic and [to] confuse it in the sense that you use every possible way of doing it. ...so as to catch this guy or that guy on different hooks as you go along, [so] that during the time when the fellow who's interested in history's being bored by the abstract mathematics... the fellow who likes the abstractions is being bored another time by the history—if you can do it so you don't bore them all, all the time, perhaps you're better off. ...I don't know how to answer the question of different kinds of minds with different interests... after many, many years of trying to teach and trying all kinds of different methods, I really don't know how to do it."

"More often than we are aware, it is the jargon which is the hurdle that a student cannot overcome rather than the mathematical concepts being introduced."

"Some of the ancient methods of calculation are particularly suited to mental arithmetic. ...Multiplication by a power of two is easily performed by successive doubling—a method fundamental to Egyptian multiplication (and division). Many 'tricks'... have been known for centuries. ...think of multiplication by a number close to a power of 10. How many children have been asked laboriously to multiply by 97 instead of multiplying by 100, [multiplying the original number again] by 3, and subtracting? 'Trick' methods... very often... can introduce principles...(100 - 3)n = 100n - 3n...is to make use of the distributive property... though... we do not have to put this in such technical language in the classroom."

"One of the great mistakes in the teaching of algebra is to present it as if it were a subject unrelated to arithmetic. ...Many schoolchildren... arrive at the conclusion that algebra and arithmetic are different subjects. ...the history of mathematics teaches us that even cubic and quartic equations were successfully solved without the benefit of modern notation. ...The use of words instead of symbols can illustrate how close the formulations are to the original... problems."

"The contemporary decline in interest in geometry and its gradual disappearance from school curricula... should be deplored... Geometry is the most visual of the mathematical disciplines. It is not in principle divorced from numbers, and hence neither is it divorced from algebra. Many a pupil's understanding of algebraic proofs would be considerably reinforced by... visual geometrical proofs which were the hallmark of Greek mathematics and to some extent of Arab mathematics also. ...where a geometrical proof is clear and immediate, as... with... many algebraic identities such as (a \pm b)^2 = a^2 \pm 2ab + b^2\!, the geometry should not be forgotten. The Greeks were some of the greatest teachers of all time... [and] geometric algebra was in many ways [their] greatest achievement ..."

"Reflect on how many real problems of life require solutions which are meaningless unless they are whole numbers. The absence of Diophantine analysis from school curricula is difficult to justify. The methods... are not beyond the capabilities of schoolchildren."

"We cannot expect to present at pre-university level the more abstract approaches to the definition[s]... of the late nineteenth and early twentieth century. We... accept... that the natural numbers are 'given'... we do not need to define them. Our pupils will discover ways in which numbers relate to the real world for themselves—all we have to do is to provide the environment in which this can happen easily and effectively."

"The games which can be built up from the simple idea of dots and lines... can be a productive source of teaching material. After all, s provided the Pythagoreans and neo-Pythagoreans with important theorems... It is well worth while... looking at the games played by the undeveloped peoples of the world. ...These ...are much closer to reality than... sophisticated and expensive forms of entertainment... which... choke the natural initiative... Simple mathematical games ...stimulate initiative and suggest other games which children can invent for themselves... For children, life is naturally simple. Let us not complicate it for them sooner than is necessary, least of all in our teaching of mathematics... which already has enough complications of its own."

"It is a student's understanding of mathematics, above all other subjects, which suffers most from unenlightened teaching methods. ...the troubles may well stem mainly from the first year or two of the child's encounter with numbers... if children come to fear them or to be bored with them, they will eventually join the ranks of the present majority... If, on the other hand, numbers are made a... source of adventure and exploration from the beginning, there is a good chance that the level of numeracy in society can be raised... There is a real role here for the history of mathematics—and the history of number in particular—for history emphasizes the diversity of approaches and methods which are possible and frees us from the straitjacket of contemporary fashions in mathematics education. It is, at the same time, interesting and stimulating in its own right."

"Geometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it."

"The history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. ...If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys."

"Using the history of algebra, teachers of the subject, either at the school or at the college level, can increase students' overall understanding of the material. The "logical" development so prevalent in our textbooks is often sterile because it explains neither why people were interested in a particular algebraic topic in the first place nor why our students should be interested in that topic today. History, on the other hand, often demonstrates the reasons for both. With the understanding of the historical development of algebra, moreover, teachers can better impart to their students an appreciation that algebra is not arbitrary, that it is not created "full-blown" by fiat. Rather, it develops at the hands of people who need to solve vital problems, problems the solutions of which merit understanding. Algebra has been and is being created in many areas of the world, with the same solution often appearing in disparate times and places. ...professors can stimulate their students to master often complex notions by motivating the material through the historical questions that prompted its development. In absorbing the idea, moreover, that people struggled with many important mathematical ideas before finding their solutions, that they frequently could not solve problems entirely, and that they consciously left them for their successors to explore, students can better appreciate the mathematical endeavor and its shared purpose."

"Algebra is a machine, or more accurately, a collection of machines. There is a machinery to factor expressions, to decompose complicated fractions into simpler ones, and so on. The object of the conversion in every case is to obtain a form more useful for the problem in hand. ...Elementary algebra as a whole is a huge machine to mechanize thinking. ...The mechanization of processes that have to be used repeatedly is... a great gain, since one does not have to think about them. They become habitual like washing and dressing. ...in itself elementary algebra is of no great interest. ...generally speaking, a machine is valuable because it turns out a useful product. ... The individual techniques of algebra are like single notes selected at random from large and magnificent musical compositions. ...these notes ...employed in the investigation of more significant undertakings, help to form beautiful patterns of reasoning. ...Unfortunately, the usefulness of the techniques of algebra has caused many people to mistake the means for the end and to emphasize these menial techniques to the exclusion of the larger ideas and goals of mathematics. The students who are bored by the process of algebra are more perceptive than those who have mistakenly identified algebraic processes with mathematics."

"The usual courses present segments of mathematics that seem to have little relationship to each other. The history may give perspective on the entire subject and relate the subject matter of the courses not only to each other but also to the main body of mathematical thought."

"The usual courses in mathematics are... deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master almost any difficulty, and that subjects are completely thrashed out and settled. The succession of theorems overwhelms the student... The history, by contrast, teaches us the development of a subject is made bit by bit with results coming from various directions. We learn, too, that often decades and even hundreds of years of effort were required before significant steps could be made. In place of the impression that subjects are completely thrashed out one finds that what is attained is often but a start, that many gaps have to be filled, or that the really important extensions remain to be created."

"There is a danger to the humanities in the present educational crash programs designed to produce a large number of mathematicians, physical scientists, engineers, and technical workers. ...Part of the... objective of the present work is to supply material which can serve as a cultural background or supplement for all those who are receiving rapid, concentrated exposure to recent advanced mathematical concepts, without any opportunity to examine the origins or gradual historical development of such ideas. Hence, although designed for the layman, this book would be helpful in courses in the history, philosophy, or fundamental concepts of mathematics."

"I am well aware of the negative attitude of so many students toward the subject. There are many reasons for this, one of them no doubt being the esoteric, dry way in which we teach the subject. We tend to overwhelm our students with formulas, definitions, theorems, and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority. The history of mathematics is a good way to correct these impressions."

"The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality."

"Zoologists maintain that the embryonic development of an animal recapitulates in brief the whole history of its ancestors throughout geologic time. It seems it is the same in the development of minds. The teacher should make the child go over the path his fathers trod; more rapidly, but without skipping stations. For this reason the history of science should be our first guide."

"Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument."

"Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery."

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information."

"Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing."

"I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation. ...I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life."

""Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result. ...we may wish a better characterization ..."successive trials" or "successive corrections" or "successive approximations." ...You use successive approximations when ...looking for a word in the dictionary ...A mathematician may apply the term ...to a highly sophisticated procedure ...to treat some very advanced problem ...that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation ...may appear as successive approximations to the truth. Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for ...many ... situations, straightforward algebra is more efficient than successive approximations."

"At each stage of its development the human race has had a certain climate of opinion, a way of looking, conceptually, at the world. The next glimmer of fresh understanding had to grow out of what was already understood. The next move forward, halting shuffle, faltering step, or stride with some confidence, was developed upon how well the [human] race could then walk. As for the human race, so for the human child. But this is not to say that to teach science we must repeat the thousand and one errors of the past, each ill-directed shuffle. It is to say that the sequence in which the major strides forward were made is a good sequence in which to teach them. The genetic method is a guide to, not a substitute for, judgement."

"Why should the typical student be interested in those wretched triangles? ...Without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science. We may question the propriety of allowing a learner, at the commencement of his Geometrical studies, to exhibit Geometrical demonstrations in Algebraical symbols. Surely it is not too much to apprehend that such a practice may occasion serious confusion of thought."

"Since formal mathematics is no more than a culturally-dependent system of aesthetics, while it may continue to be taught like Western music, there is no need to impose its consequences on K-12 children. What we need to teach children is practical mathematics... In particular, it may be worth re-examining whether one might want to teach as entirely separate subjects, from the outset, the two mathematical streams: practical mathematics and formal mathematics, with their distinct notions of number and proof."

"The education of the child must accord both in mode and arrangement with the education of mankind, considered historically. In other words, the genesis of knowledge in the individual, must follow the same course as the genesis of knowledge in the race. In strictness, this principle may be considered as already expressed by implication; since both being processes of evolution, must conform to those same general laws of evolution... and must therefore agree with each other. Nevertheless this particular parallelism is of value for the specific guidance it affords. To M. Comte we believe society owes the enunciation of it; and we may accept this item of his philosophy without at all committing ourselves to the rest."

"Nowhere in all ancient mathematics do we find any attempt at what we call demonstration. No argumentation was presented, but only the prescription of certain rules: "Do such and so." We are ignorant of the way the theorems were found... To those who have been educated in Euclid's strict argumentation, the entire Oriental way of reasoning seems at first strange and highly unsatisfactory. But this strangeness wears off when we realize that most of the mathematics we teach our present-day engineers and technicians is still of the "do such, do so" type, without much attempt at rigorous demonstration. Algebra is still being taught in many high schools as a set of rules rather than as a science of deduction. Oriental mathematics never seems to have been emancipated from the millenial influence of the problems of technology and administration, for the use of which it had been invented."

"One makes sense of , whether fictional or factual, by a mental construction that is sometimes called the world of story ...[T]he imaginative effort is a standard way of understanding what people say... In order to understand connected speech about concrete things, one imagines them. ...The capacity to do this... encourages empathy, but it also allows one to do mathematics. ...This is often fun, and it is a form of playing with ideas. ...This ludic aspect of mathematics is emphasized by in his semiotic analysis of mathematics and acknowledged by David Wells in his comparison of mathematics and games. ...The ludic aspect is something that undergraduates, many of whom have decided that mathematics is either a guessing game... or the execution of rigidly defined procedures, need to be encouraged to do when they are learning new ideas. They need to fool around with them to become familiar... Changing the s and seeing what a function looks like... to learn how the function behaves. ...Mathematical research involves a good deal of fooling around, which is part of why it is a pleasurable activity... This is not competitive..."

"A liberal Education ought to include both Permanent Studies which connect men with the culture of past generations, and Progressive Studies which make them feel their community with the present generation, its businesses, interests and prospects. ...When the Student is once well disciplined in geometrical mathematics, he may pursue analysis safely and surely to any extent. But though modern mathematics may thus be very fitly studied as a sequel to the older forms of mathematical science... these modern methods cannot supply the place of the ancient subjects as the Permanent Studies in our Educational course."

"The use of mathematical study, with which we have to do, is not to produce a school of eminent mathematicians, but to contribute to a Liberal Education of the highest kind."

"So far as civilisation is connected with the advance and diffusion of human knowledge, civilisation flourishes when the prevalent education is mathematical, and fades when philosophy is the subject most preferred."