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As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure.
The lock-step approach of algebra, geometry, and then more algebra (but rarely any statistics) is still dominant in U. S. schools, but hardly anywhere else. This fragmented approach yields effective mathematics education not for the many but for the few primarily those who are independently motivated and who will learn under any conditions.
What humans do with the language of mathematics is to describe patterns... To grow mathematically children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.
Mathematics, in the common lay view, is a static discipline based on formulas...But outside the public view, mathematics continues to grow at a rapid rate...the guid to this growth is not calculation and formulas, but an open ended search for pattern.
In mathematics, if a pattern occurs, we can go on to ask, Why does it occur? What does it signify? And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears.
Intellectually, perspective [drawing] is a breakthrough, because here, for the first time, the physical space we live in is being depicted as ifit were an abstract, mathematical space. A less obvious innovation due to perspective is that here, for the first time, people are actually drawing pictures of infinities.
Mathematics is the abstract key which turns the lock of the physical universe.
For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.
The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events.
The infinite in mathematics is always unruly unless it is properly treated.
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
Biographical history, as taught in our public schools, is still largely a history of boneheads; ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant general the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.
Mathematics is no more computation than typing is literature.
I love mathematics...principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game the encompassing of the infinite.
...it is the greatest achievement of a teacher to enable his students to surpass him.
If present trends continue, our country may soon find itself far behind many other nations in both science and technology nations where, if you inform strangers that you are a mathematician, they respond with admiration and not by telling you how much they hated math in school, and how they sure could use you to balance their checkbooks.
There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little if any, original thought... No exercise, then, can always be done with reasonbable dispatch and with a miniumum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require though on the part of the student.
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem.
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.
The profound study of nature is the most fertile source of mathematical discovery.
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