Mathematics education

"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."

"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 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."

"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."

"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."

"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."

"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."

"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."

"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."

"Because of white racisms 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 teachers initial instructional time must be spent dealing with the psychological block against learning math—or any of the sciences."

"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."

"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 whos interested in historys 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 dont bore them all, all the time, perhaps youre better off. ...I dont 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 dont know how to do it."