Mathematics Education
Mathematicians have a naive idea of pedagogy. They believe that if they state a series of concepts, theorems, and proofs correctly and clearly, and with plenty of symbols, they must necessarily be understood.
This is like an American speaking English loudly to a Russian who does not know English, in the belief that his increased volume will ensure understanding.
— Morris Kline (1977), American mathematician-historian, popularizer of mathematics, author of Why Johnny Can’t Add
Knowing something for oneself or for communication to an expert colleague is not the same as knowing it for explanation to a student…
Pedagogy is not something to be added after the fact to content. Pedagogy and content are inextricably interwoven in effective teaching. Pedagogy, like language itself, can either liberate or imprison ideas, inspire or suffocate constructive thinking.
— Hyman Bass (1997), contemporary American mathematician, educator, author, reformer
We don’t diagnose pneumonia with only a thermometer, and we don’t attempt to cure it by putting ice in a patient’s mouth. We should take a similarly enlightened attitude toward testing in mathematics education.
— William P. Thurston (1990), influential American mathematician, educator, Fields medalist
This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception:
either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud;
moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which the idea appears non-obvious.
— Mikhail Leonidovich Gromov (1992), contemporary Russian-French mathematician, educator, Abel Prize laureate, author
Hence: We desire emphatically, an enlivening of instruction in mathematics by means of its applications, but we desire also that the pendulum which in earlier decades perhaps swung too far in the abstract direction, should not now swing to the other extreme, but we wish to remain in the just mean [proper middle course]. To preserve the just mean is the problem and the art of the teacher.
— Felix Klein (1902), influential German mathematician, inaugural president of the International Commission on Mathematical Instruction (ICMI)
…each year of math is an over-stuffed grab bag of disconnected topics. The sheer volume of topics is so great that teachers must move at a pace far too fast to allow most students to develop any deep understandings. It is flatly impossible to “cover” all the required topics and also devote the time students need to understand them, so the goal of understanding is abandoned.
Once understanding is displaced as the central goal in math education, the sky's the limit, and any quantity of topics can be covered. The faster the race through topics, the greater the momentum, and it is momentum—not reasoned thought, or educational efficacy, or outcomes—that drives the entire enterprise of math education.
— Matthew Brenner (2011), contemporary American computer scientist, educator, author of The Four Pillars Upon Which the Failure of Math Education Rests (and what to do about them)
When clever people pride themselves on their own isolation, we may well wonder whether they are very clever after all. Our studies in mathematics are going to show us that whenever the culture of a people lose contact with the common life of mankind and becomes exclusively the plaything of a leisure class, it is becoming a priestcraft.
It is destined to end, as does all priestcraft, in superstition. To be proud of intellectual isolation from the common life of mankind and to be disdainful of the great social task of education is as stupid as it is wicked. It is the end of progress in knowledge.
No society, least of all so intricate and mechanized a society as ours is safe in the hands of a few clever people. The mathematician and the plain man each need the other.
— Lancelot Hogben (1936), English zoologist, medical statistician, popularizer of math and science, author of Mathematics for the Million
As the battles have raged in the history of Western thought, mathematics has been on the front lines…I want, then, to conclude by advocating that we teach mathematics not just to teach quantitative reasoning, not just as the language of science—though these are very important—
but that we teach mathematics to let people know that one cannot fully understand the humanities, the sciences, the world of work, and the world of man without understanding mathematics in its central role in the history of Western thought.
— Judith V. Grabiner (1988), contemporary American mathematician-historian, educator, author
To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.
— George Pólya (1977), Hungarian-Swiss-American mathematician, educator, popularizer of mathematics, author of How to Solve It
... in teaching on an elementary level one must tell the truth, nothing but the truth, but not the whole truth.
— attributed to Wallie A. Hurwitz (1886 - 1958), American mathematician, educator, philanthropist
As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art.
This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste.
But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities.
…In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.
— John von Neumann (1947), influential Hungarian-American mathematician, polymath, a founding father of digital computer science and game theory
Mathematicians depend ultimately on society for their livelihood and the privilege of working on something that passionately interests them. In return we must, in various ways, repay this debt and encourage our fellow citizens to take a friendly and tolerant view of our strange profession.
— Michael Atiyah (2009), influential British-Lebanese mathematician, Fields Medalist, Abel Prize laureate