Smith, David Eugene
Of all the inventions of man, which one is the most internationally understood? It is not the language by which we express our thoughts, for millions and millions of educated people in various parts of the world cannot understand a single word that we may say. It is not our architecture, for the Zulu cannot or does not grasp the significance of a Burmese temple or of a modern skyscraper. It is not the motion picture, for although the nomad may be interested in what he sees upon the screen when he visits a town in an oasis in the desert, he neither knows nor cares anything about the machine nor can he generally understand the words that appear in print below the picture.
Numbers and Numerals tells how our knowledge of numbers developed in the past and what it means to us all today.
David Eugene Smith - 1935
This is the second of two lectures in the Julius and Rosa Sachs Foundation Lectures for 1934-1935, delivered before the faculty and students of Teachers College on February 11 and 18, 1935.
David Eugene Smith - 1935
The author has occasionally indulged in the easiest form of criticism, the destructive type. An effort has been made, however, to set up an ideal of the culture which a teacher should possess, an ideal not easily reached. It has been suggested that our normal schools possess a large element of abnormality and that the numerous teachers colleges which have come into being of late years do very little for the training of teachers.
The world in general and the educational world in particular is always trying to escape from tyranny, and it always does it in just one way, by the substitution of a new tyranny for an old one. In algebra we have had the dictation of constituted authorities of all sorts ranging from the state to committees of teachers and to individual superintendents and principals.
Well aware that I, to-day, break all precedents in the nature of a presidential address, it is proper that I offer due apology and confess my unconventionality. It is proper, too, that I should frankly say that I am aware that my message is wanting in the rigor of demonstration to which our science accustoms us;
The problem which has been assigned to me this afternoon is to set forth the features to be kept in mind in the teaching of mathematics for citizenship.
Among the games which have served to amuse people for generations past are several which have considerable value as arithmetical drill.
The position of the number rhyme in the history of education has already been discussed. In this chapter it is proposed to give a few translations from the famous Epigrams of Metrodorus, or Metrodoros, to take the Greek form of his name. These will serve to show the nature of the rhyming problem as it first appeared in the early Greek algebra, and may have some educational value in the present search for means to make the introduction to this science more interesting to a certain type of mind.
CHAPTER VII NUMBER GAMES BORDERING ON ARITHMETIC AND ALGEBRA Mathematical puzzles, tricks, and curious cases have always appealed to the interest of young and old.
The science of mathematics is taught in our schools for several reasons. For some the chief reason is that the subject is or may be practical. For others it is taught because it is a stepping stone to such sciences as astronomy and physics. Others see in it an exercise in terse, logical expression, such as is found in no other subject in the curriculum.
When the subject of number games shall be adequately treated, and the long and interesting story comes to be told of how the world has learned to handle the smaller numbers quite as much through play as through commerce, the climax will probably be found in the chapter relating to the Battle of Numbers, the Rith-momachia of the Middle Ages.
It is with some hesitancy that a chapter like the present one, tracing the history of a single number game, even though it be the most famous one, has place in an article on the number games of the school.
CHAPTER V THE ORIGIN AND DEVELOPMENT OF THE NUMBER RHYME The predilection for rhythm among primitive peoples is well known. The childhood of the race had the same delight in meter that the childhood of the individual has in every generation.
PREFACE In one of the graduate classes of Teachers College it is the custom to devote part of each year to a study of educational problems in the field of mathematics in other countries. As a rule the students in this class are teachers of some experience. All of them are college graduates and all are desirous of knowing the best that the world is doing in the teaching of their chosen subject.
Donald T. Page & David Eugene Smith - 1912
In early times there was little need for schools of navigation. The young sailors could learn the most important requirements by experience on board ship, and could learn from the older sailors what few scientific principles were necessary. The leisure time on a. voyage and during the winter season was employed to advantage in such studies.
W. F. Enteman & David Eugene Smith - 1912
In this review of a report on the “Commercial Problems in the Mathematical Instruction of the Higher Schools,”i will be found, as in the original publication, (a) some discussion of the field of commercial problems, (b) a brief review of text-books, both old and new, (c) a discussion of the important aids in teaching the subject, and (d) some general conclusions. One of the aims of the report is to call attention to the cause for the existence of commercial problems in the mathematical instruction in Germany.
A. T. French & David Eugene Smith - 1912
This article contains a brief digest of a much more extended discussion under the same titlei in one of the reports of the International Mathematics Commission.
Cilda Langfitt, Katherine Simpson & David Eugene Smith - 1912
The report here reviewedi gives the important regulations that were passed in regard to the state examinations in Prussia and the United North German States from 1810 to 1898, and also the regulations concerning the examinations in Braunschweig and Mecklenberg.
The material for this report was collected by Dr. Zühlke from three sources,—the official regulations on the subject of line-drawing and descriptive geometry, the literature in the department of mathematics, and the results of a tour of investigation undertaken during the previous year.
What we most desire is that our teachers of secondary mathematics shall be thoroughly familiar with their field far beyond the demands of the curriculum and that they be masters of it on its historical, its practical, and its theoretical side.
CHAPTER II EVOLUTION OF THE REFORM IN GERMANY1 The report by Dr. Schimmack, referred to in the footnote, is divided into two sections; the first dealing with the reform and its progress from 1840 to 1907, and the second dealing with the reform and its progress from 1907 to the present day.
Miriam E. West & David Eugene Smith - 1912
CHAPTER III SECONDARY SCHOOLS OF HESSE AND BADEN The two reports that are reviewed in this chapter1 treat of the organization, the curriculum, and the methods of mathematical instruction in the higher schools of Hesse and Baden.
Katharine S. Arnold, Ruth Fitch Cole & David Eugene Smith - 1912
CHAPTER IV THE SECONDARY SCHOOLS OF THE HANSEATIC STATES The report on mathematical instruction in the Gymnasien and Realschulen in the Hanseatic states, Mecklenburg and Oldenburg,1 was prepared from the results of investigations carried on by sending questionnaires to the various educational centers.
CHAPTER V THE SECONDARY SCHOOLS OF WÜRTTEMBERG1 This report is divided into seven chapters. The first chapter treats of the various schools in Württemberg. The second and third chapters give an account of the mathematics course offered and of the various reforms which have been proposed in the Gymnasium and Realgymnasium respectively.
The object of this report by Professor Wieleitner is to give a survey of the evolution of mathematics in the Bavarian high schools, together with additional information concerning the training and extension work of teachers in Bavarian Gymnasien, and in the humanistic schools in particular.
The well-known American authority, Professor J. W. A. Young, who is quoted by Dr. Lietzmann in this report, explains the superiority of the German schools over the American schools in this manner: “The causes of the excellence of the Prussian work in mathematics may be classed under three heads: (1) The central legislation and supervision.
Maurice Levine & David Eugene Smith - 1912
In the first chapter Dr. Wirz describes the present organization of the higher schools in Elsass-Lothringen (Alsace-Lorraine). This is followed by a historical and critical survey of the development of instruction in mathematics, especially with regard to the curriculum, from the time of the French control in 1870 to the present day.
This article is an attempt to state briefly the main points brought out in a report recently published by Dr. Jahnke of Berlin on the mathematics of the higher industrial schools.1 The report deals with the special schools of mining, military science, forestry, agriculture, and commerce.
This report concerning mathematics instruction in the German middle technical schools of the machine industry consists of six chapters which treat of the development of these schools, their organization, the mathematics instruction given in them, the text-books, the method of treatment of the different subjects of instruction, and the preparation of the teacher of mathematics.
The evolution of the teaching of primary arithmetic extends over a period of about two hundred years, although numerous sporadic efforts at teaching the science of number to young children had been made long before the founding of the Francke Institute at Halle.
What has been said of mental arithmetic naturally leads to some question as to the nature of the written work. What shall this be? If the difference in longitude between two ships (since standard time by one system or another is now coming to be universal on land) is 33° 45', how shall a pupil find the difference in time? Here are a few possibilities: images
Before considering the curriculum in arithmetic it is well to devote a little attention to certain great principles that teachers have as a whole agreed upon, in theory if not in practice. Some of these have already been discussed in this article; others will strike the reader as rather trite, which simply means that they are generally accepted; and others will not appeal to all. They will, however, be found to be suggestive of the thought of leaders at the present time, and they may, though stated dogmatically, form the basis for profitable discussions by teachers.
The Leading Mathematical Features: There should in this year be a thorough review of the fundamental operations with integers. This should be followed by the same operations with the common fractions and denominate numbers of business. Percentage may be begun, although in some places it is better to postpone this until the following year. Review: There is usually a new text-book begun in this grade, and this, if properly arranged, offers plenty of material for the review above mentioned, with numbers that are appropriately larger. Teachers should undertake this review in the spirit and for the reason suggested in section XIII.
As already stated, the two most noteworthy changes in arithmetic in recent years have related to the nature of the problems and the arrangement of material. The latter has been the result of a more or less serious study of child psychology, namely of the powers of the individual in the various school years.
Of all the terms used in educational circles "Method" is perhaps the most loosely defined. Efforts have been made to limit its meaning, to divide its responsibilities with such terms as "Mode" and "Manner," but it still stands and is likely to stand as a convenient name for all sorts of ideas and theories and devices.
The objection to the expression "Mental Arithmetic" is fully a generation old. It is argued that written arithmetic is quite as mental as any other kind, and that the opposite to written is oral. As to this there can be no argument, but the word "mental" has so long been used to apply to that phase of arithmetic that is not dependent upon written help that, like a person's proper name, it need not be held strictly to account for what it literally signifies.
There is only one test for a question involving a single operation. Either the answer is right or it is wrong. If the problems require some interpretation, a teacher may properly mark both for operations and for method; that is, a pupil may perform his operations correctly, but may have misinterpreted the meaning of the problem.
The questions of mental and written arithmetic lead naturally to that of the analyses to be expected on the part of children. What is their object, what should be their nature? How extensively should they be required?
There has of late years been a tendency throughout the country to make arithmetic, as other subjects, more interesting to children. What the real motive was it is hard to say, since it was probably somewhat subconscious. Such statistical information as we have shows arithmetic always to have been looked upon by children as one of the most interesting subjects of the course, so that the reason was not that it was relatively a dull study. Possibly the desire was that the work of the teacher should become easier through increased interest on the part of the pupils. But whatever the reason it cannot be questioned that, other things being always kept equal, there is a great gain in increasing the interest in any kind of work.
Nothing new goes into arithmetic without a protest, and so for what goes out. Nevertheless there has been an evolution here as everywhere else, and this evolution has made for the betterment of the subject. To take a concrete illustration, the first printed arithmetic had no symbols of operation.
Without taking up the important but rather axiomatic question of reviews from time to time, it seems proper to mention one phase of the matter that is too frequently forgotten. Any one who has ever had much to do with the supervision of the grade work in arithmetic is struck by the general complaint that children are never prepared to enter any particular grade.
It is well, before leaving this general discussion, to consider a few of the subjects for legitimate experiment in the teaching of arithmetic that might occupy the attention of schools like those connected with Teachers College.
Besides these subjects that may be designated as more or less general, there are many details that demand investigation. These are partly arithmetical and partly psychological in nature, and belong quite as much in one field as another.
The first question that naturally arises in connection with the arithmetic of the first grade is as to whether or not the subject has any place there at all.
Whether or not arithmetic has a definite time allotment in the first grade, it usually has one in the second, although some teachers oppose it even here. The argument already advanced holds the more strongly here, especially as in many schools, the child is quite prepared to use a text-book by the middle of this year. The Leading Mathematical Features: In schools of average advancement, where the question of language is not as serious as in some cities in the East, children in this grade may be expected to complete the addition tables and to learn the multiplication tables to 10 X 5.
The Leading Mathematical Features: In this year rapid written work is an important feature. The oral has predominated until now, but in Grade III the operations involve larger numbers than before, and the child begins to acquire the habit of writing his computations. Multiplication extends to two-figure multipliers and long division is begun. The most useful tables of denominate numbers are completed.
The leading Mathematical Features. In this the last year of the primary grades it is well to feel that the essentials of arithmetic have all been touched upon. It is, therefore, desirable to review the four fundamental operations, extending the multiplication and division work to include three-figure multipliers and divisors. The common business fractions should also be included, with simple operations as far as multiplication.
The Leading Mathematical Features: The leading features of this year should be percentage and its applications, particularly to discount, profit and loss, commission, and interest. Ratio and simple proportion may also be included.
The Leading Mathematical Features: As in the preceding grade, it is well to begin by a general review of the fundamental processes from a higher standpoint than before. Ratio and proportion are usually completed in this year, whether introduced here for the first time or not, and the applications naturally cover a broader field. Percentage is the leading topic of the year.
III. WHAT ARITHMETIC SHOULD INCLUDE If we taught arithmetic only for its utilitarian value, to fit a person for the computations that the average man needs to perform or know about in daily life, the range of subject matter would not be great. Addition, particularly of money, but not involving very large or numerous amounts, is probably the most important topic. Perhaps, for it is difficult to say with certainty, the simple fractions ½ and ¼ are next in line of relative importance, including ½ of a sum of money, ¼ of a length or weight, and so on. Very likely the making of change, one of the forms of subtraction, is next in frequency of use.
The edition of the TEACHERS COLLEGE RECORD for March, 1903, containing the article on Mathematics in the Elementary School, by my colleague Professor McMurry and myself, having long since been exhausted, and repeated requests having been made for a reprint of this article or for a new treatment of the question, it has been thought best to set forth again the principles guiding the department of mathematics of Teachers College in its work in the teaching of arithmetic. And since educational views change more or less from time to time, with the world, the country, the school, and the individual, it has seemed wiser to prepare a new article for the RECORD than to attempt a revision of the one prepared some six years ago.
In no way has arithmetic changed as much of late years as in the nature of the problems and the arrangement of the material. The former has come about from two causes, (1) the needs of society, and (2) the study of child psychology. The latter, the arrangement of the material, has been determined almost entirely by psychological considerations. In this chapter it is proposed to speak briefly of the former, the nature of the problems.
The following outline of theory and of subject-matter is proposed rather as a basis for discussion with students in professional courses than as a fixed body of thought for use in the elementary school. Yet the course of work here outlined is followed to a very large extent in Teachers College in its Horace Mann School of observation and its Speyer School of practice, although the arrangement of topics is necessarily different in the two schools.
OUTLINE FOR THE FIRST FIVE GRADES Grade I I. General Suggestions (a) In the spirit of the foregoing article, the purpose of mathematics for six-year-old children is to meet from their point of view, their daily need of number as it arises in their school studies and in their relations outside of school....
GENERAL DISCUSSION OF THE WORK OF THE LAST THREE GRADES (A) MODERN SUBJECT-MATTER The work of the last three grades is chiefly devoted to the application of arithmetic to the affairs of life.
IV. OUTLINE FOR THE LAST THREE GRADES Grade VI In this grade the reduction of common fractions to decimals, and vice versa, is the only topic in pure arithmetic demanding attention; except as the others enter into reviews. The year is given to applications, chiefly in percentage and denominate numbers.
The professional work in the Department of Mathematics during the first semester has been confined to the course in Education 23, on the Theory and Practice of Teaching Mathematics in Secondary Schools, and to the course in History of Mathematics. The course in Education 23 has included two lectures each week, with thirty hours of practical work in the Horace Mann School.