Mathematics Education
We must look to our own faculty for discerning those fine connective things—community of aim, interformal analogies, structural similitudes—that bind all the great forms of human activity and aspiration—natural science, theology, philosophy, jurisprudence, religion, art and mathematics—into one grand enterprise of the human spirit.
— Cassius Jackson Keyser (1912), American mathematician, educator, author, philosopher
Shameless ignorance in regard to such serious intellectual conquests as are embodied in the mathematical literature does not represent a normal condition on the part of those interested in the history of the human race.
On the contrary, such shamelessness is evidence of the lack of the proper aids to enter this literature.
— George Abram Miller (1916), American mathematician-historian, educator, author of Historical Introduction to Mathematical Literature
…It seems quite unrealistic to judge a curriculum by its general outline, or to judge a course by its syllabus. We can “cover” very impressive material, if we are willing to turn the student into a spectator.
But if you cast the student in a passive role, then saying that he has “studied” your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does.
— Edwin E. Moise (1965), American mathematician, educator, reformer, literary critic, author
Engineering is the customization of abstract scientific principles to create processes or products that human beings want in their social or everyday use… Likewise, in school mathematics, one should tailor all those principles to the needs of human beings.
In this case, the human beings are teenagers and young children; they have specific needs. Therefore, it’s our job to customize the mathematics we know into a form that they can use, that they can understand, without sacrificing the mathematical principles, of course.
— Hung-Hsi Wu (2010), contemporary mathematician, educator, author, reformer
One's conception of what mathematics is affects one's conception of how it should be presented. One's manner of presenting it is an indication of what one believes to be most essential in it...The issue, then, is not, What is the best way to teach? but, What is mathematics really all about?
Controversies about…teaching cannot be resolved without confronting problems about the nature of mathematics.
— Rueben Hersh (1979, 1986), American mathematician, educator, author, popularizer of math, philosopher
In regard to this question of words, there is also in the new mathematics books a great deal of talk about the value of precise language - such things as that one must be very careful to distinguish a number from a numeral and, in general, a symbol from the object that it represents.
The real problem in speech is not precise language. The problem is clear language. The desire is to have the idea clearly communicated to the other person.
It is of no real advantage to introduce new subjects to be taught in the old way.
— Richard Feynman (1965), influential American physicist, Nobel Laureate, educator, author
The desire to make sure that students see mathematics as a coherent whole. This is certainly how mathematicians see it, and to us it is one of the major attractions of the field: mathematics makes sense and helps us make sense of the world.
For me, perhaps the most discouraging aspect of working on K–12 educational issues has been confronting the fact that most Americans see mathematics as an arbitrary set of rules with no relation to one another or to other parts of life. Many teachers share this view. A teacher who is blind to the coherence of mathematics cannot help students see it.
— Roger Howe (1999), contemporary American mathematician, educator, author, reformer
In American elementary mathematics education, arithmetic is viewed as negligible, sometimes even with pity and disdain—like Cinderella in her stepmother’s house. Many people seem to believe that arithmetic is only composed of a multitude of “math facts” and a handful of algorithms. . . Who would expect that the intellectual demand for learning such a subject actually is challenging and exciting?
— Liping Ma (2001), contemporary mathematician, educator, reformer, author
No one, I believe, may contest the normal mathematician's right as a mathematical student or investigator to be quite indifferent as to the social value or the human worth of his activity. Such activity is to be prized just as we prize any other natural agency or force that, however undesignedly, yet contributes, sooner or later, directly or indirectly, to the wealth of mankind.
But when the mathematician passes from the role of student or investigator to the role of teacher, that right of indifference ceases, for he has passed to an office whose functions are social and whose obligations are human. It is not his privilege to chill and depress with the encasing fogs of the iceberg. It is his privilege and his duty, in so far as he may, to disclose its "million-lustered" splendors in all their power to quicken and illuminate, to charm and edify, the whole mind.
— Cassius Jackson Keyser (1912), American mathematician, educator, author, philosopher
For me it was a revelation to see an algorithm which, if you let it, would develop the theory for you. Ever since that time, it has been my aim to make theory and the numerical methods of solving problems as unified as possible.
I guess you can say in some sense it has always been my aim to unify and simplify. I believe that the simplification very often occurs through obtaining meaningful examples, examples, which, if you understand them, don’t need a lot of theory—the examples carry the story.
— Albert W. Tucker (1983), influential Canadian mathematician, educator, historian, author
One can invent mathematics without knowing much of its history. One can use mathematics without knowing much, if any, of its history. But one cannot have a mature appreciation of mathematics without a substantial knowledge of its history.
— attributed to Abe Shenitzer (1921 - 2022), Polish born mathematician, educator, author, translator, Holocaust survivor
Psychologically, the teaching of abstractions first is wrong. Indeed, a thorough understanding of the concrete must precede the abstract.
Abstract concepts are meaningless unless one has many and diverse concrete interpretations well in mind. Premature abstractions fall on deaf ears.
— Morris Kline (1973), American mathematician-historian, popularizer of mathematics, author of Why the Professor Can’t Teach
The task of the educator is to make the child’s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.
— Henri Poincare (1899), influential French mathematician, physicist, philosopher, educator, author of Science and Method
…mathematics is used and can only be learned and taught as an integral component of a larger sense-making resource system which also includes natural language and visual representation.
A semiotic perspective helps us understand how natural language, mathematics, and visual representations form a single unified system for meaning-making.
— Jay Lemke (2003), contemporary American physicist, semiotician, science educator, author
What does it profit a man to learn to manipulate the symbols of mathematics if he knows not their meaning?
— Robert Hayden (1981), American mathematician, educator, author
When we come to examine the question of the real reason for the study of mathematics to-day, we find that we seek a receding and an intangible something which quite baffles our attempts at capture.
— David Eugene Smith (1900), American mathematician-historian, educator, translator, author, one of the founding fathers of mathematics education as a separate discipline
The main contribution that an expository writer can make is to organize and arrange the material so as to minimize the resistance and maximize the insight of the reader and keep him on the track with no unintended distractions.
What, after all, are the advantages of a book over a stack of reprints? Answer: efficient and pleasant arrangement, emphasis where emphasis is needed, the indication of interconnections, and the description of the examples and counterexamples on which the theory is based; in one word, organization.
The discoverer of an idea, who may of course be the same as its expositor, stumbled on it helter-skelter, inefficiently, almost at random.
If there were no way to trim, to consolidate, and to rearrange the discovery, every student would have to recapitulate it, there would be no advantage to be gained from standing “on the shoulders of giants”, and there would never be time to learn something new that the previous generation did not know.
— Paul Halmos (1970), Hungarian-American mathematician, educator, author, popularizer of mathematics