Dudley What is Mathematics For Underwood Dudley 1937

Dudley: What is Mathematics For?

Underwood Dudley (1937 ~ )

Moving Beyond Myths (National Academy of Sciences, 1970) Myth: Most jobs require little mathematics. Reality: The truth is just the opposite. 除非把「數學」等同於「算術」， 否則找不到充分的證據。

Everybody Counts (National Research Council, 1989) Over 75 percent of all jobs require proficiency in simple algebra and geometry, either as a prerequisite to a training program or as part of a licensure examination. 同樣找不到充分的證據。

代數教科書的廣告詞 “Career Application”—Includes explanations, examples, exercises, and answers for work in electronics; civil/chemical engineering; law enforcement; nursing; teaching; and more. Shows students the relationship of chapter concepts and job skills—with applications developed through interviews and market research in the workplace that ensure relevance.

教科書裡的題目都是生造的 The reason that problems drawn from everyday life do not appear in the texts is not that textbook authors lack energy and initiative; it is that they do not exist.

常有的錯覺 Supposing that problems once solved must be solved anew every time they are encountered. People seem to think that because something involves mathematics it is necessay to know mathematics to use it.

Dudley 的觀察結果 I am not saying that mathematics is never required in the workplace. Of course it is, and it has helped to make our technology what it is. However, it is needed very, very seldom, and we do not need to train millions of students in it to keep business going.

為什麼公眾仍然支持學校裡 要教很多數學？ The reason for this, I am convinced, is that the public knows, or senses, that mathematics develops the power to reason. It shows, better than any other subject, how reason can lead to truth.

為什麼公眾仍然支持學校裡 要教很多數學？ That is what mathematics education is for and what it has always been for: to teach reasoning, usually through the medium of silly problems.

為什麼公眾仍然支持學校裡 要教很多數學？ What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better.

Polya 論小學數學教育的目的 What is the aim of teaching mathematics in the primary school? It is better to consider the most general question: What is the aim of the schools? And the better question is: What do people generally think is the aim of the schools?

Polya 論小學數學教育的目的 We have therefore two kinds of aims in the schools. We have good and narrow aims. The schools should turn out employable adults — adults who can fill a job. But a higher aim is to develop all the resources of the growing child in order that he can fill in the job for which he is best fitted.

Polya 論小學數學教育的目的 So the higher aim, I express it so, is to develop all the inner resources of the child.

Polya 論小學數學教育的目的 Now what about mathematics teaching? Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. An adult who is completely illiterate is not employable in a modern society.

Polya 論小學數學教育的目的 However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking.

Polya 論小學數學教育的目的 The thinking that you can learn in mathematics is, for instance, to handle abstractions. …When we solve a practical problem, then from this practical problem we must first make an abstract problem. Mathematics applies directly to abstractions.

Polya 論小學數學教育的目的 But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems.

Polya 論小學數學教育的目的 I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems.

Dudley 與 Polya 論數學教育目的 Reasoning Thinking Problem solving

William Thurston (1946 ~ ) 1982 Fields獎得主

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 What is mathematics education good for? First, mathematics is a basic tool of everyday life. Second, mathematical reasoning is an important part of informed citizenship. Third, mathematics is a tool needed for many jobs in the infrastructure of our increasingly complex and technological society. Fourth, mathematics is intensively used (and sometimes abused) in most branches of science.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 To me, these utilitarian goals are important, but secondary. Mathematics has a remarkable beauty, power, and coherence, more than we could have ever expected. It is always changing, as we turn new corners and discover new delights and unexpected connections with old familiar grounds. The changes are rapid, because of the solidity of the kind of reasoning involved in

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 Mathematics is like a flight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process of sharpening our perception so that we discover patterns that are everywhere around.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 My experience as a mathematician has convinced me that the aesthetic goals and the utilitarian goals for mathematics turn out, in the end, to be quite close. Our aesthetic instincts draw us to mathematics of a certain depth and connectivity. The very depth and beauty of the patterns makes them likely to be manifested, in unexpected ways, in other parts of mathematics, science, and the world.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 To share in the delight and the intellectual experience of mathematics—to fly where before we walked—that is the goal of a mathematical education.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 數學教育的問題 A major source of problems, and a major barrier to solutions of the problems, is the stratification and compartmentalization of the immense mathematical education enterprise, from kindergarten through graduate school. In particular, there is very little communication between high school and college teachers of mathematics.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 Mathematics is a Tall Subject One feature of mathematics which requires special care in education is its ‘height, ’ that is, the extent to which concepts build on previous concepts. . This means it is possible to build conceptual structures which are at once very tall, very reliable, and extremely powerful.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 Mathematics is a Broad Subject Mathematics is also very broad. There are many subjects that are never discussed in the mainline curriculum which culminates in calculus. The subjects that are discussed have many interesting side-branches that are never explored.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 These two trends (lengthening and narrowing of school mathematics) have been hastened by the growing reliance on standardized tests. . 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.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 The long-range objectives of mathematics education would be better served if the tall shape of mathematics were de-emphasized, by moving away from a standard sequence to a more diversified curriculum with more topics that start closer to the ground.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through some process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. . The insight that goes with this compression is one of the real joys of mathematics.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 After mastering mathematical concepts, even after great effort, it becomes very hard to put oneself back in the frame of mind of someone to whom they are mysterious.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 The hard-earned and powerful tools which are available almost unconsciously to mathematicians, but not to students, make it hard for mathematicians to learn from their students. This puts a psychological barrier in the way of listening fully to students.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 It is better to keep interesting unanswered questions and unexplained examples in the air, whether or not the students, the teachers, or anybody is yet ready to answer them. The best psychological order for a subject in mathematics is often quite different from the most efficient logical order.

W. Thurston, Mathematical education, ar. Xiv: math/0503081 v 1 4 Mar 2005 We should present mathematics to our students in a way which is at once more interesting and more like the real situations where students will encounter it in their lives—with no guaranteed answer.

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數學教育所為何來？ 至少不該忽視： Algorithmic reasoning thinking problem solving