THE PRINCIPLES OF MATHEMATICS IN RE

THE PRINCIPLES OF MATHEMATICS IN RELATION TO ELEMENTARY TEACHING
BY A. N. WHITEHEAD.
We are concerned not with the advanced teaching of a few specialist mathematical students, but with the mathematical education of the majority of boys in our secondary schools. Again these boys must be divided into two sections ; one section consists of those who desire to restrict their mathematical education, the other section is made up of those who will require some mathematical training for their subsequent professional careers, either in the form of definite mathematical results or in the form of mathematically trained minds.
I shall call the latter of these two sections the mathematical section, and the former the non-mathematical section. But I must repeat that by the mathematical section is meant that large number of boys who desire to learn more than the minimum amount of mathematics. Furthermore most of my remarks about these sections of boys apply also to elementary classes of our university students.
The subject of this paper is the investigation of the place which should be occupied by modern investigations respecting Mathematical Principles in the education of both of these sections of schoolboys.
To find a foothold from which to start the enquiry, let us ask, why the nonmathematical section should be taught any mathematics at all beyond the barest elements of arithmetic. What are the qualities of mind which a mathematical training is designed to produce when it is employed as an element in a liberal education ?
My answer, which applies equally to both sections of students, is that there are two allied forms of mental discipline which should be acquired by a well designed mathematical course. These two forms though closely allied are perfectly distinct.
The first form of discipline is not in its essence logical at all. It is the power of clearly grasping abstract ideas, and of relating them to particular circumstances. In other words, the first use of mathematics is to strengthen the power of abstract thought. I repeat that in its essence this has nothing to do with logic, though as a matter of fact a logical discipline is the best method of producing the desired effect. It is not the philosophical theory of abstract ideas which is to be acquired, but the habit and the power of using them. There is one and only one way of acquiring the habit and the power of using anything, that is by the simple commonM. c. II. 29 450 A. N. WHITEHEAD place method of habitually using it. There is no other short cut. If in education we desire to produce a certain conformation of mind, we must day by day, and year by year, accustom the students' minds to grow into the desired structural shape. Thus to teach the power of grasping abstract ideas and the habit of using them, we must select a group of such ideas, which are important and are also easy to think about because they are clear and definite. The fundamental mathematical truths concerning geometry, ratio, quantity, and number, satisfy these conditions as do no others. Hence the fundamental universal position held by mathematics as an element of a liberal education. But what are the Fundamental Mathematical truths concerning Geometry, Quantity and Number ? At this point we come to the great question of the relation between the modern science of the Principles of Mathematics and a Mathematical Education. My answer to the question as to these Fundamental Mathematical truths is that, in any absolute sense, there are none. There is no unique small body of independent primitive unproved propositions which are the necessary starting-points of all mathematical reasoning on these subjects. In mathematical reasoning the only absolute necessary presuppositions are those which make logical deduction possible. Between these absolute logical truths and so-called Fundamental truths concerning Geometry, Quantity and Number, there is a whole new world of mathematical subjects concerning the logic of propositions, of classes, and of relations. But this subject is too abstract to form an elementary training ground in the difficult art of abstract thought. It is for this reason that we have to make a compromise and start from such obvious general ideas as naturally occur to all men when they perceive objects with their senses. In Geometry, the ideas elaborated by the Greeks and presented by Euclid are, roughly speaking, those adapted for our purpose, namely, ideas of volumes, surfaces, lines, of straightness and of curvature, of intersection and of congruence, of greater and less, of similarity, shape, and scale. In fact, we use in education those general ideas of spatial properties which must be habitually present in the mind of anyone who is to observe the world of phenomena with understanding. Thus we come back to Plato's opinion that for a liberal education, Geometry, as he knew it, is the queen of sciences. In addition to Geometry, there remain the ideas of quantity, ratio, and of number. This in practice means, Elementary Algebra. Here the prominent ideas are those of ' any number,' in other words, the use of the familiar x, y, z, and of the dependence of variables on each other, or otherwise, the idea of functionality. All this is to be gradually acquired by the continual use of the very simplest functions which we can devise, of linear functions, graphically represented by straight lines, of quadratic functions graphically represented by parabolas, and of those simple implicit functions graphically represented by conic sections. Thence, with good fortune and a willing class, we can advance to the ideas of rates of increase, still confining ourselves to the simplest possible cases. THE PRINCIPLES OF MATHEMATICS 45 1 I wish here emphatically to remind you that both in Geometry and in Algebra a clear grasp of these general ideas is not what the pupil starts from, it is the goal at which he is to arrive. The method of progression is continual practice in the consideration of the simplest particular cases, and the goal is not philosophical analysis but the power of use. But how is he to practise himself in their use ? He cannot simply sit down and think of the relation y = x + 1, he must employ it in some simple obvious way. This brings us to the second power of mind which is to be produced by a mathematical training, namely, the power of logical reasoning. Here again, it is not the knowledge of the philosophy of logic which it is essential to teach, but the habit of thinking logically. By logic, I mean deductive logic. Deductive logic is the science of certain relations, such as implication, etc., between general ideas. When logic begins, definite particular individual things have been banished. I cannot relate logically this thing to that thing, for example this pen to that pen, except by the indirect way of relating some general idea which applies to this pen to some general idea which applies to that pen. And the individualities of the two pens are quite irrelevant to the logical process. This process is entirely concerned with the two general ideas. Thus the practice of logic is a certain way of employing the mind in the consideration of such ideas; and an elementary mathematical training is in fact nothing else but the logical use of the general ideas respecting Geometry and Algebra which we have enumerated above. It has therefore, as I started this paper by stating, a double advantage. It makes the mind capable of abstract thought, and it achieves this object by training the mind in the most important kind of abstract thought, namely, deductive logic. I may remind you that other choices of a type of abstract thought might be made. We might train the children to contemplate directly the beauty of abstract moral ideas, in the hope of making them religious mystics. The general practice of education has decided in favour of logic, as exemplified in elementary mathematics. We have now to answer the further question, what is the rôle of Logical Precision in the Teaching of Mathematics ? Our general answer to the implied question is obvious, Logical Precision is one of the two objects of the teaching of mathematics, and it is the only weapon by which the teaching of mathematics can achieve its other object. To teach mathematics is to teach logical precision. A mathematical teacher who has not taught that, has taught nothing. But having stated this thesis in this unqualified way, its meaning must be carefully explained; for otherwise its real bearing on the problem of education will be entirely misunderstood. Logical precision is the faculty to be acquired. It is the quality of mind which it is the object of the teaching to impart. Thus the habit of reading great literature is the goal at which a literary education aims. But we do not expect a child to start its first lesson by reading for itself Shakespeare. We recognize that reading is impossible till the pupil has learnt its alphabet and can spell, and then we start it with books of one syllable. In the same way, a mathematical education should grow in logical precision. It 29—2 452 A. N. WHITEHEAD is folly to expect the same careful logical analysis at the commencement of the training as would be appropriate at the end. It is an entire misconception of my thesis to construe it as meaning that a mathematical training should assume in the pupil a power of concentrated logical thought. My thesis is in fact the exact opposite, namely that this power cannot be assumed, and has got to be acquired, and that a mathematical training is nothing else than the process of acquiring it. My whole groundwork of assumption is that this power does not initially exist in a fully developed state. Of course like every other power which is acquired, it must be developed gradually. The various stages of development must be guided by the judgment and the genius of the teacher. But what is essential is, that the teacher should keep clearly in his mind that

ผลลัพธ์ (ไทย) 1:

หลักการของคณิตศาสตร์เกี่ยวกับสอนระดับประถมศึกษาโดย A. N. WHITEHEADเรามีความกังวลไม่ มีสอนขั้นสูงของนักคณิตศาสตร์ผู้เชี่ยวชาญบาง แต่ มีการศึกษาทางคณิตศาสตร์ส่วนใหญ่ของเด็กผู้ชายในโรงเรียนมัธยมของเรา อีกครั้ง เด็กผู้ชายเหล่านี้ต้องแบ่งออกเป็น 2 ส่วน ส่วนหนึ่งประกอบด้วยผู้ที่ปรารถนาที่จะจำกัดการศึกษาคณิตศาสตร์ อื่น ๆ ส่วนประกอบที่จะต้องฝึกอบรมคณิตศาสตร์อาชีพมืออาชีพที่พวกเขาตามมา ในรูปแบบของผลลัพธ์ทางคณิตศาสตร์แน่นอน หรือ ในรูปแบบของจิตใจฝึก mathematically ฉันจะเรียกของเหล่านี้สองส่วนหลังส่วนคณิตศาสตร์ และอดีตส่วนคณิตศาสตร์ไม่ แต่ต้องซ้ำว่า โดยส่วนคณิตศาสตร์ตั้งใจ ที่จำนวนมากของเด็กผู้ชายที่ต้องการเรียนมากกว่ายอดเงินต่ำสุดของคณิตศาสตร์ นอกจากนี้ หมายเหตุของฉันเกี่ยวกับส่วนเหล่านี้ของเด็กผู้ชายส่วนใหญ่กับยังเรียนประถมศึกษาของนักศึกษามหาวิทยาลัยของเรา เรื่องของกระดาษนี้เป็นการตรวจสอบสถานที่ควรครอบครอง โดยเคารพหลักการทางคณิตศาสตร์ในการศึกษาของทั้งสองส่วนนี้ของ schoolboys สืบสวนสมัยใหม่ ในการค้นหา foothold ที่จะเริ่มต้นที่คำถาม ให้เราถาม ทำไมส่วน nonmathematical ควรได้รับการสอนคณิตศาสตร์ใด ๆ ที่นอกเหนือจากองค์ประกอบ barest ของเลขคณิต คุณภาพของจิตใจที่ฝึกฝนทางคณิตศาสตร์ถูกออกแบบมาเพื่อผลิตเมื่อเป็นลูกจ้างเป็นองค์ประกอบในการศึกษาแบบเสรี คืออะไร คำตอบของฉัน ซึ่งใช้ทั้งส่วนของนักเรียนเท่า ๆ กัน คือ ว่า มีสองรูปแบบพันธมิตรวินัยทางจิตใจซึ่งควรได้รับตามหลักสูตรคณิตศาสตร์ออกแบบมาดี แบบฟอร์มเหล่านี้สองอย่างใกล้ชิดว่าพันธมิตรได้แตกต่างกันอย่าง The first form of discipline is not in its essence logical at all. It is the power of clearly grasping abstract ideas, and of relating them to particular circumstances. In other words, the first use of mathematics is to strengthen the power of abstract thought. I repeat that in its essence this has nothing to do with logic, though as a matter of fact a logical discipline is the best method of producing the desired effect. It is not the philosophical theory of abstract ideas which is to be acquired, but the habit and the power of using them. There is one and only one way of acquiring the habit and the power of using anything, that is by the simple commonM. c. II. 29 450 A. N. WHITEHEAD place method of habitually using it. There is no other short cut. If in education we desire to produce a certain conformation of mind, we must day by day, and year by year, accustom the students' minds to grow into the desired structural shape. Thus to teach the power of grasping abstract ideas and the habit of using them, we must select a group of such ideas, which are important and are also easy to think about because they are clear and definite. The fundamental mathematical truths concerning geometry, ratio, quantity, and number, satisfy these conditions as do no others. Hence the fundamental universal position held by mathematics as an element of a liberal education. But what are the Fundamental Mathematical truths concerning Geometry, Quantity and Number ? At this point we come to the great question of the relation between the modern science of the Principles of Mathematics and a Mathematical Education. My answer to the question as to these Fundamental Mathematical truths is that, in any absolute sense, there are none. There is no unique small body of independent primitive unproved propositions which are the necessary starting-points of all mathematical reasoning on these subjects. In mathematical reasoning the only absolute necessary presuppositions are those which make logical deduction possible. Between these absolute logical truths and so-called Fundamental truths concerning Geometry, Quantity and Number, there is a whole new world of mathematical subjects concerning the logic of propositions, of classes, and of relations. But this subject is too abstract to form an elementary training ground in the difficult art of abstract thought. It is for this reason that we have to make a compromise and start from such obvious general ideas as naturally occur to all men when they perceive objects with their senses. In Geometry, the ideas elaborated by the Greeks and presented by Euclid are, roughly speaking, those adapted for our purpose, namely, ideas of volumes, surfaces, lines, of straightness and of curvature, of intersection and of congruence, of greater and less, of similarity, shape, and scale. In fact, we use in education those general ideas of spatial properties which must be habitually present in the mind of anyone who is to observe the world of phenomena with understanding. Thus we come back to Plato's opinion that for a liberal education, Geometry, as he knew it, is the queen of sciences. In addition to Geometry, there remain the ideas of quantity, ratio, and of number. This in practice means, Elementary Algebra. Here the prominent ideas are those of ' any number,' in other words, the use of the familiar x, y, z, and of the dependence of variables on each other, or otherwise, the idea of functionality. All this is to be gradually acquired by the continual use of the very simplest functions which we can devise, of linear functions, graphically represented by straight lines, of quadratic functions graphically represented by parabolas, and of those simple implicit functions graphically represented by conic sections. Thence, with good fortune and a willing class, we can advance to the ideas of rates of increase, still confining ourselves to the simplest possible cases. THE PRINCIPLES OF MATHEMATICS 45 1 I wish here emphatically to remind you that both in Geometry and in Algebra a clear grasp of these general ideas is not what the pupil starts from, it is the goal at which he is to arrive. The method of progression is continual practice in the consideration of the simplest particular cases, and the goal is not philosophical analysis but the power of use. But how is he to practise himself in their use ? He cannot simply sit down and think of the relation y = x + 1, he must employ it in some simple obvious way. This brings us to the second power of mind which is to be produced by a mathematical training, namely, the power of logical reasoning. Here again, it is not the knowledge of the philosophy of logic which it is essential to teach, but the habit of thinking logically. By logic, I mean deductive logic. Deductive logic is the science of certain relations, such as implication, etc., between general ideas. When logic begins, definite particular individual things have been banished. I cannot relate logically this thing to that thing, for example this pen to that pen, except by the indirect way of relating some general idea which applies to this pen to some general idea which applies to that pen. And the individualities of the two pens are quite irrelevant to the logical process. This process is entirely concerned with the two general ideas. Thus the practice of logic is a certain way of employing the mind in the consideration of such ideas; and an elementary mathematical training is in fact nothing else but the logical use of the general ideas respecting Geometry and Algebra which we have enumerated above. It has therefore, as I started this paper by stating, a double advantage. It makes the mind capable of abstract thought, and it achieves this object by training the mind in the most important kind of abstract thought, namely, deductive logic. I may remind you that other choices of a type of abstract thought might be made. We might train the children to contemplate directly the beauty of abstract moral ideas, in the hope of making them religious mystics. The general practice of education has decided in favour of logic, as exemplified in elementary mathematics. We have now to answer the further question, what is the rôle of Logical Precision in the Teaching of Mathematics ? Our general answer to the implied question is obvious, Logical Precision is one of the two objects of the teaching of mathematics, and it is the only weapon by which the teaching of mathematics can achieve its other object. To teach mathematics is to teach logical precision. A mathematical teacher who has not taught that, has taught nothing. But having stated this thesis in this unqualified way, its meaning must be carefully explained; for otherwise its real bearing on the problem of education will be entirely misunderstood. Logical precision is the faculty to be acquired. It is the quality of mind which it is the object of the teaching to impart. Thus the habit of reading great literature is the goal at which a literary education aims. But we do not expect a child to start its first lesson by reading for itself Shakespeare. We recognize that reading is impossible till the pupil has learnt its alphabet and can spell, and then we start it with books of one syllable. In the same way, a mathematical education should grow in logical precision. It 29—2 452 A. N. WHITEHEAD is folly to expect the same careful logical analysis at the commencement of the training as would be appropriate at the end. It is an entire misconception of my thesis to construe it as meaning that a mathematical training should assume in the pupil a power of concentrated logical thought. My thesis is in fact the exact opposite, namely that this power cannot be assumed, and has got to be acquired, and that a mathematical training is nothing else than the process of acquiring it. My whole groundwork of assumption is that this power does not initially exist in a fully developed state. Of course like every other power which is acquired, it must be developed gradually. The various stages of development must be guided by the judgment and the genius of the teacher. But what is essential is, that the teacher should keep clearly in his mind that

ผลลัพธ์ (ไทย) 3:

หลักการของคณิตศาสตร์ในความสัมพันธ์กับการสอนระดับประถมศึกษา โดย เอ็นเฮด
.
เรามีความกังวลกับการสอนขั้นสูง ไม่กี่ผู้เชี่ยวชาญทางคณิตศาสตร์นักเรียน แต่ด้วยการศึกษาทางคณิตศาสตร์ของส่วนใหญ่ของเด็กผู้ชายในโรงเรียนมัธยมของเรา อีกครั้งเด็กเหล่านี้จะต้องแบ่งออกเป็นสองส่วน ; ส่วนหนึ่งประกอบด้วยผู้ที่ประสงค์ที่จะ จำกัด การศึกษาทางคณิตศาสตร์ของพวกเขาส่วนอื่น ๆขึ้นของผู้ที่จะต้องฝึกคณิตศาสตร์สำหรับการประกอบอาชีพที่ตามมามืออาชีพทั้งในรูปแบบของผลลัพธ์ทางคณิตศาสตร์ที่แน่นอน หรือในรูปแบบของความคิดทางคณิตศาสตร์ฝึก
จะเรียกหลังของทั้งสองส่วน ส่วนคณิตศาสตร์ อดีตไม่ใช่คณิตศาสตร์ ส่วนแต่ผมต้องขอย้ำว่าโดยส่วนทางคณิตศาสตร์จะหมายความว่าจำนวนมากของเด็กที่อยากเรียนมากกว่าจำนวนเงินขั้นต่ำของคณิตศาสตร์ นอกจากนี้ส่วนใหญ่ของข้อสังเกตของฉันเกี่ยวกับส่วนเหล่านี้ของเด็กผู้ชายที่ใช้ยังเรียนประถมของนักศึกษาของเรา
หัวข้อของบทความนี้คือการตรวจสอบสถานที่ที่ควรครอบครองโดยการเคารพในหลักการทางคณิตศาสตร์ที่ทันสมัย ในการศึกษาของทั้งสองส่วนของเด็กนักเรียนชาย .
หาตั้งหลักที่จะเริ่มถาม เราถามว่า ทำไมมาตรา nonmathematical ควรจะสอนคณิตศาสตร์ที่นอกเหนือจากองค์ประกอบของอาจารย์ของคณิตศาสตร์อะไรคือคุณสมบัติของจิตที่ฝึกคณิตศาสตร์ถูกออกแบบมาเพื่อผลิตเมื่อมันถูกนำมาใช้เป็นองค์ประกอบในการศึกษาเสรีนิยม ?
คำตอบ ซึ่งใช้อย่างเท่าเทียมกันทั้งในส่วนของนักศึกษา คือ มีสองรูปแบบของพันธมิตรจิตวินัย ซึ่งควรได้รับการออกแบบด้วยคณิตศาสตร์หลักสูตร ทั้งสองรูปแบบแม้ว่าอย่างใกล้ชิดพันธมิตรจะสมบูรณ์ที่แตกต่างกัน
รูปแบบแรกของวินัยไม่ได้อยู่ในสาระสําคัญของตรรกะทั้งหมด มันคือพลังของความคิดนามธรรมอย่างชัดเจนโลภ และเกี่ยวข้องกับสถานการณ์ที่เฉพาะเจาะจง ในคำอื่น ๆที่ใช้ครั้งแรกของคณิตศาสตร์คือการเสริมสร้างพลังของความคิดนามธรรม ผมขอย้ำว่าในสาระสําคัญของไม่เกี่ยวกับตรรกะ