
Significant Movements in Secondary School Mathematicsby Raleigh Schorling  1917 Your program committee has asked me to review the significant movements that are effecting the reorganization of subject matter and methods of instruction in secondary school mathematics.1 The recent vigorous criticisms by men in responsible positions make it appear unnecessary to state the need for the discussion of this topic. These widely published criticisms2 have evoked extensive discussion during the past year in educational meetings and literature. For example, in our city mathematics club we have followed with considerable interest the controversy between Professor David Snedden3 and the committee appointed by the New England Association of Teachers of Mathematics. Furthermore, in November the Central Association of Science and Mathematics Teachers took official action to cooperate with the Mathematical Association of America in the consideration of these criticisms by a national committee. This evidence of the importance of these criticisms makes it appear advisable to consider what possibilities these significant movements offer in overcoming profound discontent with the results of mathematics teaching. Your program committee has asked me to review the significant movements that are effecting the reorganization of subject matter and methods of instruction in secondary school mathematics.1 The recent vigorous criticisms by men in responsible positions make it appear unnecessary to state the need for the discussion of this topic. These widely published criticisms2 have evoked extensive discussion during the past year in educational meetings and literature. For example, in our city mathematics club we have followed with considerable interest the controversy between Professor David Snedden3 and the committee appointed by the New England Association of Teachers of Mathematics. Furthermore, in November the Central Association of Science and Mathematics Teachers took official action to cooperate with the Mathematical Association of America in the consideration of these criticisms by a national committee. This evidence of the importance of these criticisms makes it appear advisable to consider what possibilities these significant movements offer in overcoming profound discontent with the results of mathematics teaching. The first reaction to Professor Snedden's letter to the New England Committee seemed to be a resentment to statements that are easy to make but very difficult to prove. The "principle of exclusion" was interpreted as an "excess of enthusiasm for the new to discard what is good in the old." In advocating the principle of exclusion the critics of mathematics appeared to violate the first principle of discussion, in that they advocate something radically new without assuming the burden of proof. However, a more mature consideration of the questions raised by Professor Snedden has resulted in a discernment of the constructive phases of his program. This reaction is evident in the second statement4 by the New England Committee and the kindly reception of Professor Snedden's views in the 1916 meeting of the Central Association of Science and Mathematics Teachers at Chicago which culminated in the appointment of a "Committee on the Reorganization of the Science Courses." It is possible that secondary mathematics has come to the position of "Root, hog, or die." At any rate, the advice of the New England Committee that, "It behooves those who believe in high school mathematics to state their case, not trusting merely to the protection of educational inertia," seems well considered. The case of our critics, roughly, rests on the following indictments: (1) Mathematics as organized and taught is not a function of the present educational epoch. Originally it owed its prominent position in the school program to an alleged disciplinarian value. It has not been proved, so critics assert, that the methods of mathematics have farreaching intellectual significance. (2) It is implied that its utilitarian values are limited. (3) The traditional excessive formalism of algebra and the supposed rigid deductive logic of geometry have excluded the consideration of the vital needs of the students. In this respect the principles of the psychology of secondary school mathematics have been violated. (4) Since the development of geometry historically, other subject matter has also been developed which is more directly related to community needs. (5) As an efficient hurdle or barrier to the successful pursuit of professional courses in college, mathematics is too expensive in time and energy. (6) The objectives of mathematics instruction have not been determined by an inductive study of modern industrial and professional life, and hence, since no one knows what the utilitarian value of secondary mathematics is we need to apply the principle of exclusion. This principle would have us eliminate all material for which a definite use cannot be established. (7) The large number of failures5 in mathematics in the different high school courses, in college entrance examinations, and even in college freshmen courses indicates that the aims, other than selection, are not being accomplished. There are no data based on scientific investigation which enable us to answer these criticisms, and certainly this discussion should not be interpreted as a conscious attempt to defend existing conditions. Various attempts have been more or less entertaining but not particularly instructive. In this case, like others in which the factors involved are difficult to determine, the argument threatens to be vague and prolonged. DISCIPLINARY VALUE While this vagueness is more or less characteristic of the whole indictment, it is particularly true of the argument dealing with formal discipline. The situation is roughly as follows: For two decades we have been subjected to a partisan discussion with little sobriety of statement on either side. The real problem is that of degree of transfer of power. Measurement of the higher mental processes is not sufficiently refined for the determination of the degree of transfer. Professor Thorndike as the author of the doctrine of identical elements has furnished the necessary ammunition for the attack on disciplinarians. On the other hand, Professor Judd,6 as an advocate of generalized experience, has pointed out that the opponents of formal discipline have never really succeeded in finding anyone who is actually upholding the straw man that they are knocking down. The argument concerning the validity of the statistical manipulations of some of the experiments claiming to show little positive transfer has resulted in further confusion to the ordinary teacher of mathematics. Professor Hancock's study7 shows that the thinking public is as firmly convinced as ever of the high educational value of mathematics. The result is that the ordinary teacher of mathematics is confused as to the issues of formal discipline. It may appear later in the discussion that he could devote his energy more profitably to the scrutinizing of his materials and methods. At any rate, if there is any doubt in his mind that the argument of formal discipline pro and con has rested on vague and meager scientific data, he needs to read the excellent summary of statistical experiments by Rugg.8 UTILITARIAN VALUE When we come to the consideration of the utilitarian indictment we are struck with the fact that our critics appear to be reiterating ideals that have been periodically advocated in the past, and to be expressing an impatience with the inertia of a system that prevents us from realizing those ideals. The fact of the matter is that progress is being made in the movement to reconstruct mathematics, and the leaders are no other than mathematicians themselves. It is difficult to find a more severe critic of mathematics than John Perry,9 himself a professor of mathematics in the Royal College of Science, London. As chairman of the board of examiners of the board of education in the subjects of engineering, applied mechanics, practical mathematics, and more technical subjects, Professor Perry has charge of the education of some hundred thousand apprentices and has had therefore a very wide opportunity to observe the utilitarian values of mathematics. Professor J. W. A. Young has given us a clear statement of the Perry Movement in his "Teaching of Mathematics," Chapter VI. Moreover, I shall quote some of the most vigorous sentences used by Perry in his address to show his position on the subject of utilitarian value and to indicate the clear connection that his statements have with our more recent critics: "I belong to a great body of men who apply the principles of mathematics in physical science and engineering. We pay the teachers of mathematics to give us something that will be useful in our education and useful in life, useful to understand our position in the universe. Surely we have a right to ask the mathematicians to look at this matter from our point of view, and ask if it is not possible to help us without hurting themselves and their study." "In these days all men ought to study natural science. Such study is practically impossible without the study of higher mathematical methods than those of the mere housekeeper." "As examples of methods necessary even in the most elementary study of nature, I may mention—the use of logarithms in computation, knowledge of and power to manipulate formulae; the use of squared paper; the methods of calculus. Dexterity in all of these is easily learned by all young boys." Perry then proceeds to point out that the English polytechnic10 fails to give this needed practical training. In fact, he asserts that an English boy with the usual ignorance of mathematics and physical science when leaving school, "pitchforked" into a workshop where nobody thinks it his duty to give him instruction, ignorant of theory all his life, getting no scientific education whatsoever, cannot be a much worse engineer than the product of a polytechnic in which the authority insists on the learning of every detail of an elaborate life course of study, when not the smallest part of it is known to English engineers of practically the same actual value. "There can be no harm in saying that as there is no real study of natural science which is not quantitative, it must be through mathematics," and "My engineering friends think I have an exaggerated notion of the importance to all men of possessing a love for mathematics. But they have not had my experience. They have not seen its usefulness all through a man's life as I have seen it." I am not at all sure that these disconnected quotations make Perry's position clear, but I shall presently cite references in which it is shown that Perry, on the basis of his large practical experience, was profoundly convinced (1) that mathematics has extensive utilitarian values, and (2) that mathematics as taught in secondary schools failed to prepare the student to meet these utilitarian needs. It appears that quantitative thinking and the technique of quantitative expression play a far larger part in human experience than current opinion realizes. We note that current magazines and the daily papers include a very large proportion of mathematical terms and concepts, the meaning of which must be clear in order to be appreciated. The critics say that such mathematics should be taught to all. Current practice would have us believe that secondary mathematics functions as a means to prepare secondary school students to pursue intelligently the study of the physiological, biological, and sociological sciences. The critics say by all means teach the mathematics needed to the student who expects to specialize in any of these subjects, but leave the others (the great majority) alone. Here it is important to recognize that the field of subject matter involved is far more extensive and consequently affects a greater number of secondary school students than is commonly realized. It is a fact that mathematics is an essential prerequisite to courses in scientific agriculture, engineering, physics, chemistry, art (drawing, designing, modeling, life and still life drawing, handicraft), architecture, pharmacy, dentistry, navigation, astronomy, naval and military engineering, domestic science, insurance, forestry, commerce and administration (use of graph and formula), and railway administration. To this list we must add certain courses that are becoming more and more statistical, i.e., more exact, as for example political economy, sociology, hygiene, sanitation, and education. Visits to the various commercial clubs and a reading of the literature of trade journals impress us with the extensive use of the graph and the formula (the two most important topics of secondary mathematics) in the business world. Furthermore, we may add medicine to this list. Although some prominent educators assert that secondary mathematics is not essential to medicine, we have not yet discovered a reputable physician or surgeon who is willing to consider seriously such remarks with reference to his needs. It is difficult to see how a student with no training in mathematics could pursue the science courses in his preparatory years, much less read the technical literature while engaged in practice. In this connection, it is interesting to note that Dr. Alexis Carrel, with the help of the mathematician du Nouy, discovered a formula, by no means simple, whose graph predicts the course of the healing of a wound with uncanny accuracy. In fact it is possible to predict by the graph and the formula the exact day of recovery if the patient is treated by the Carrel irrigation method. We therefore fear that the physician untrained in mathematics would be confused in the reading of the literature describing the marvelous work of Dr. Carrel in the trench hospitals. Then too, we note that a considerable number of journalists have reasonably clear ideas of mathematical concepts, and perhaps we should insist that mathematics be requisite for future journalists, if they are to interpret properly human experience and discuss in scientific terms the wide field of human activities covered by the extensive list of sciences and "near sciences" which directly involve mathematics. Furthermore, in this complex mechanical age, the shopman's hopes of some day becoming more than a part of the machine he operates, appear to depend upon his ability to read the literature dealing with the process with which he is concerned, or his ability to interpret general instructions intelligently. These instructions are extremely formal, being usually expressed in complex formulas. The cry of the shop is for more useful mathematics. It is not yet clear what form shop mathematics will take, but shopmen like Perry insist that the problem is one of putting an advanced type of mathematics into a very simple form. It is just possible that a general course, emphasizing the use of the simple equation and formula along with the elementary principles of geometry and trigonometry, will prove the best preparation for vocational courses. At present, the shop instructors must assume the problem at this point and teach the mathematics from this point as an integral part of the shop work. This extensive list of subjects for which mathematics is a prerequisite is sufficient to show that a considerable portion of secondary school students is affected. Nor are we concerned only with the everincreasing number of secondary school students •who pursue these subjects in higher educational institutions. Universal education is the order of the day. These courses are being offered in community centers, evening schools, correspondence schools, extension departments, and in unorganized form by the public press. The successful citizen of the next generation may find it necessary to be able to interpret scientific literature dealing with his special field. In view of the extensive list of human activities which include mathematics as a prerequisite (assuming that those who determine these prerequisites know what constitutes adequate preparation in their own fields), it must be granted that mathematics meets a considerable social need on the basis of utility, independent of any disciplinary value. The immediate problem of the ordinary teacher of secondary school mathematics appears to be the scrutinizing of his materials and methods so as to meet efficiently these needs without being concerned about disciplinary values. However, our critics are undoubtedly correct when it is asserted that no one knows the extent to which these fields use the material found in the elementary courses, or the extent to which we fail to meet these utilitarian needs. Therefore we need to make the most careful inductive study of modern life to answer these questions. Having made such a comprehensive inductive study of modern conditions, we may venture to construct a course in mathematics. This course will not be designed to meet merely college entrance requirements, as in the past, but rather to meet the needs of a variety of institutions ranging from those connected with large industrial organizations to the university itself. It may be a general introduction to the field of mathematics. It may even be termed a culture course in mathematics. Moreover, it will probably be a course of mathematics, not a course about mathematics. On the content side, it will enable the student to use the simpler forms of the equation, to interpret statistical material graphically, to understand the uses of a formula, to interpret a wealth of illustrative geometrical material, and to generalize and make meaningful the arithmetic. Such a course may also include practical devices, e.g., trigonometric ratios, logarithms, the slide rule, etc., as tools for solving the practical type of problem. If the spirit of the discussions of the monthly meetings of our mathematics club (Chicago and vicinity) is a reliable index, then we need anticipate no difficulty in coming to an agreement with reference to the major topics of this outline. It is probable, too, that the educational world would endorse the requirement that pupils should be exposed for one year, either in junior or senior high school, to some such general introductory course in mathematics. Having determined these objectives it will not suffice merely to incorporate this most probable usable material; we need to give special consideration to the psychology of applications. Experience supports the conclusion of Professor Judd11 that a student is not performing the same mental process when he masters a principle as he does when he uses the principle. In this connection (utilitarian values), I call your attention to several opinions that have been frequently quoted by the critics of mathematics with serious results. First I refer to the following: "The actual amount of algebra needed by a foreman in a machine shop can be taught in about four lessons, and the geometry or mensuration that he needs can be taught in eight lessons at the most. The necessary trigonometry may take eight more, so that it is entirely feasible to unite these three subjects."12 As far as I have been able to discover, the interesting experiment here suggested has never been tried. One illustration which shows the use that is being made of this quotation is found in Mr. Parker's "Method of Teaching in High School."13 After quoting this opinion (along with two others) with evident satisfaction, Parker concludes— "These extended quotations from such a standard authority as Professor Smith may be accepted as stating authoritatively the relation of geometry to social needs as the latter have varied historically." There are four facts that have a clear bearing on the use of this quotation: (1) The opinion is unsupported by verifiable statistical data; (2) while Professor Smith is a recognized "standard authority," it happens that his great work for secondary mathematics does not consist of the determination of the utilitarian values of mathematics; (3) the critics, while accepting Professor Smith's opinion as a "standard authority," are entirely unwilling to follow him to his final conclusions. They would not entertain the thought of permitting Professor Smith to dictate the mathematics curriculum. Parker, having made use of this opinion to serve a desired end, then rejects Professor Smith's other opinions in the sentence immediately following the sentence quoted; (4) there is evidence that this unfortunate statement has either been misinterpreted or else Professor Smith has reversed himself in the following quotation: "Even aside from the value of the subject per se, which seems somewhat a matter of fashion rather than scientific decision at present, it does not stand on the same foundation as the classics, since the practical applications of mathematics are multifarious, and we have only to show the lines of these applications to secure a position that is not likely to yield to any such assaults as are now being made."14 Equally interesting is the following quotation which is appearing more and more frequently from Professor Millikan: "There is no mathematics needed in elementary physics, even as it is now taught, except the simplest algebraic equations with one unknown, and the single geometrical proposition of the proportionality of the sides of similar triangles."15 Professor Millikan then relates how his elevenyearold boy (in the eighth grade) had been taught how to handle similar triangles and solve algebraic equations with one unknown. He then concludes: "At the present time, then, all the rudimentary mathematics needed for high school physics is being taught in the grades. If it were not, we could teach it in connection with the physics in an extra half hour of time." Now I happen to know that the instructor in physics in the case of the boy cited spent many half hours teaching this same boy the mathematics involved in physics. But aside from this irrelevant fact since we are dealing with only one individual, we may examine Professor Millikan's own high school texts.16 In "A First Course in Physics," Professor Millikan has avoided the use of formulae which are a matter of common information; e.g., S=vt+½gt2, F=9/5C+32, etc. But we note that he nevertheless includes many quadratic equations which do not appear simple. When we turn to the laboratory text we find many of our old friends;e.g.,;and l_{2}l_{1}=b using this value of (l_{2}l_{1}) compute K; Efficiency=. All are found in the first seventeen experiments. We need not proceed to the more difficult for it already appears that Professor Millikan would be an exceedingly busy man for one half hour if he attempted to teach similar triangles, laws of proportion, graphing of functions and solution of quadratic equations, not to emphasize the solution of the complicated formulae of his laboratory manual. Of course we do not care whether this material is called mathematics or physics. Personally, I should prefer to teach such mathematics in that practical and therefore more meaningful connection, but we need to object to the false implications that have been read into this quotation. THE MOVEMENT TO PSYCHOLOGIZE MATHEMATICS We may now return to the discussion of the third indictment listed against mathematics; namely, that mathematics as organized and taught is not based on sound psychological principles. The psychologizing of mathematics was one of the important points of Perry's program, as may be inferred from the following quotation: "As soon as we give up the idea of absolute correctness we see that a perfectly new departure may be made in the study of mathematics. I feel sure that our system of teaching boys elementary mathematics as if they were all going to be pure mathematicians must be altered. I say that the old Alexandrian method is bad. We teach all boys what is called mathematical philosophy that we may catch in our net the one demigod doing our best to ruin all the others. I think that men who teach demonstrative geometry and orthodox mathematics generally are not only destroying what power to think exists, but are producing a dislike, a hatred for all kinds of computation, and therefore for all scientific study of nature, and are doing incalculable harm."17 Perry then proceeds to argue skillfully for the acceptance as axiomatic of a larger number of basic principles and by the emphasis of the practical, i.e., through experimentation and through graphical methods generally, proceeds to give students the essential notions of trigonometry, analytic geometry, and the calculus. In short, he would have us proceed in secondary mathematics very much as we do in arithmetic when we give conviction as to the validity of numerous formulae but realize that it would be folly to give rigid logical proofs. He asserts that the orthodox logical sequence of principles in mathematics should be replaced by the psychological procedure implied in his program. In his insistence on a psychological versus the logical in the earlier years of the secondary school and in the need for special study and emphasis of the problem of application, he advocated two important principles of the psychology of our special subject as explicitly expressed in the recent psychology18 of our special subject to which we shall refer later. Incidentally, we may note that the movement for the cultivation of special psychological studies in each field along with a comprehensive study of general psychology seems to be amply justified by its constructive suggestions to secondary mathematics. THE MOVEMENT FOR COMPOSITE COURSES The importance of the Perry movement was called to the attention of American secondary teachers by Professor E. H. Moore in his presidential address, "On the Foundations of Mathematics," to the American Mathematical Society (1903) .19 This address has proved one of the most constructive papers in American secondary mathematics. The following quotations are most significant in the present discussion: "As a pure mathematician, I hold as the most important suggestion of the English movement the suggestions just cited, that by emphasizing steadily the practical sides of mathematics, that is, arithmetical computations, mechanical drawing, and graphical methods generally, in continuous relation with problems of physics and chemistry and engineering, it would be possible to give very young students a great body of the essential notions of trigonometry, analytic geometry, and the calculus." Concerning method, "Perry is quite right in insisting that it is scientifically legitimate in the pedagogy of elementary mathematics to take a large body of basal principles instead of a small body, and to build the edifice upon the larger body, for the earlier years, reserving for the later years the philosophic criticism of later years." In other parts of the address Professor Moore advocates the laboratory method of instruction and a specific practical and theoretical scientific training of secondary school teachers. This material has recently appeared in the form of an emphasis on supervised study and practice teaching. After fourteen years have passed it is indeed interesting for us to consider Professor Moore's following question: "Would it not be possible to organize the algebra, geometry, and physics of the secondary school into a thoroughly coherent fouryear course, comparable in length and closeness of structure with the four years' course in Latin?" The discussion of this question has been the direct inspiration for the movement of composite courses. The significance of the composite course is no doubt clear. The movement bears on our topic because it offers the most promising instrument for overthrowing the traditional watertight compartment system with a gain of mathematical power and no apparent loss in general training. The organization of mathematical material in this form leaves the door wide open for possible utilitarian values. As Professor Karpinski points out: "All good teachers have always correlated."20 The difficulty has been the lack of material in usable form. The beginning of the correlation movement was difficult for the following reasons: (1) the traditional training and routine experience of the early writers unfitted them for the task; (2) others were expected to correlate when they were unsympathetic with the movement; and (3) in one or two instances the authors seem to be writing the material for the sake of correlation instead of using the correlation principle where it naturally makes the topics taught more meaningful. The result was that the early material was not teachable by the ordinary teachers. In spite of these early handicaps, the movement has gained in momentum. I have not been able to find any material which weakens the program outlined in Professor Moore's vision. The composite course movement has profited by certain explicit principles furnished us by the special psychology of secondary mathematics. This appears in (1) the avoidance of excessive formalism, (2) realization of the psychological difficulties involved in the organization of space relations, (3) recognition of space as a vivid form of experience capable of analysis and comparison and "the most available instrument for the training of students in application of mathematics,"21 (4) acceptance of the principle that students cannot in general apply mathematical information without special consideration of the psychology of application, and finally, (5) arrangement of subject matter to consist of a bringing together of the simpler principles of algebra, geometry, analytics, trigonometry, and calculus, and binding up inductively by experiment to the complex problems in these separate fields. THE JUNIOR HIGH SCHOOL MOVEMENT Another movement involved in this discussion is the junior high school movement. We are told that the junior high school will close the breach between the elementary schools and the high schools; that the earlier years of a child's life will be enriched by the introduction of science, civics, art and knowledge of human life; that the movement means a clear gain in time of one year, possibly more; that it will stimulate the child's interest and enthusiasm and will increase the probability of his pursuit of higher education. To mathematics teachers the movement is one of vital importance because it promises to be a valuable instrument for breaking up the inertia of our traditional subject matter, organization, and methods of instruction. The eighth grade and first year high school cannot be liberalized by shoving the conventional material down a year or so. The reorganization must be more fundamental. Even the more conservative writers of junior high school texts are reducing the excessive formalism of algebra and introducing geometry by a general approach to the deductive form. In some junior high schools eighth and ninth grades are studying the simple principles of trigonometry in the attempt to make material available which is interesting and more probably usable. We may expect teachers who have had to readjust their methods and material to this new situation to be more willing to leave the beaten paths. If the junior high school mathematics is to consist of the simpler and more practical principles of secondary mathematics, combined with arithmetic and taught inductively by experiment, then it is a most welcome wedge to break up the present system. THE MOVEMENT FOR IMPROVED TECHNIQUE It is important to point out that a considerable part of the recent criticisms of mathematics should have been entered in the accounting as the results of poor teaching. We read in methods texts that mathematics is one of the easiest subjects to teach because of its superior organization. You are familiar with the practice of school executives who "farm out" a class or two to almost any teacher who happens to have a vacant period regardless of whether this teacher has had special training in the subject matter or special technique. Statistical studies show that contrary to current opinion it is extraordinarily difficult even for good teachers to teach mathematics in the high school at a high level of success. The realization of the need of special technique is of firstrate importance. Recently this technique has been improved by the incorporation of supervised study and the application of the principle of individual differences. The available literature shows the possibilities of transforming the older form of recitation in which a large portion of the time is spent in having students recite in turn on work previously assigned, to a more profitable form of recitation in which (1) the home work is rapidly disposed of; (2) the major portion of the recitation offers opportunity for reflective thinking in the inductive approach to new principles; (3) the presentation of new material is followed by periods of supervised study at the points of psychological needs. Experience with practice students shows the serious need of special training in technique and the psychology of the special subject for the development of such a recitation. Practice teachers when beginning their work frequently go through a "showing" process in attempting to teach: they have the logical arrangement of subject matter, not the psychological. In my own experience with practice teachers it has been difficult to convince them that the learner approaches the subject in a different way. Again and again a practice teacher will say: "I don't see why I could not drive this thing home when it was so 'dead easy' and in such definitely organized form." The trouble lay in the fact that the teacher had forgotten the inductive propaedeutic development that had led to the easily comprehended mathematical concept in her own mind. She sees the end in view as she drives forward with the class in the attempt to hurdle all intervening steps. After a few fatal experiences, the practice teacher begins to see what the instructor of her special methods course has been talking about; she realizes that the logical arrangement of subject matter in her mind must be set aside and be displaced by a new arrangement built up in the light of a practical psychology, which teaches the approach to a concept from the point of view of one who learns versus the method of one who knows. Besides the emphasis on practice teaching courses, which fortunately is now gaining momentum, we need large supervision in order to insure the training of teachers in service. However, it is difficult to be optimistic about supervision. The large high school departments without organization or departmental meetings, familiar to all of us, make it appear that adequate supervision is the exception rather than the rule. Adequate supervision demands that every school system have at least one efficiency expert in the teaching of secondary mathematics, in order to promote the training of teachers in service. The functions of this efficiency expert might include the following: (1) the teaching of a section regularly in one of the high schools (shifting at the end of each term); (2) presiding at a mathematics faculty meeting held regularly in each of the high schools; (3) supervision of mathematics in all high schools of the city, the major part of the time being given to the particular building in which teaching is done during a given semester; (4) serving as a clearinghouse for the methods and special technique used in the mechanics of the schoolroom. But the detailed discussion of this suggestion falls outside the limits of this discussion. THE STANDARD TESTS MOVEMENT Finally, we come to the consideration of the standard test movement which has recently invaded the secondary field. It has a bearing on our topic because these tests promise to improve teaching technique. This is particularly true of the RuggClark Tests.22 These tests state emphatically that, "Their more important function, however, is that of diagnosis: they reveal difficulties in learning and point out needed changes in teaching emphasis."23 This phase of the testing movement is the most promising phase in algebra, and is of course involved in the study of individual differences. Such tests furnish us a standard collection of facts in the experience of teachers and bring them to bear in the analysis of individual difficulties. No teacher should fail to make himself familiar with the detailed report of common errors presented by the RuggClark Tests. The relative difficulty of the various processes are there revealed. Though teaching methods will undoubtedly profit by the wide application of these tests along the line of diagnosis, we need to examine critically the implications of the second function of the tests stated by the authors as follows: "They provide a yard stick for measuring the results of instruction and for determining the relative efficiency of teaching and the progress that teachers and pupils are making."24 This statement is subject to loose implications which may be the cause of a considerable amount of apathy toward the algebra tests. I have been trying to find out what is going on in the minds of mathematics teachers who fail to become enthusiastic over such algebra tests as are available. Roughly, the case against tests appears to be the following: (1) A fundamental disagreement exists concerning relative values and aims of a first year course in mathematics. Aims and values that are determined either by the "arm chair method" or by current practice as revealed in textbooks disregard the real needs of the students. The indefensible emphasis on factoring bears on the issue. There appears to be no internal evidence in these tests indicating a large familiarity with a movement to reorganize and vitalize secondary mathematics. The question is: Are the values of mathematics so sufficiently known that the error of teaching or learning can be determined? The early tests in every subject must need confine themselves to the formal aspects, to those mental traits which lie on the surface and hence are easily measured. It does not follow that there are not other values which as yet defy measurement which are more valuable than these formal external traits. The difficulty of measuring the higher mental processes is revealed in that these tests do not establish a scale for verbal problems but merely determine the relative difficulty of a list of such problems. (2) The authors assert that "Teachers and administrators who have been loath to adopt 'measuring methods' in this school practice and who have hesitated to introduce practice exercises do so on the ground that such a procedure will 'mechanize' teaching."25 Of course this objection is really implied in (1) above. The teacher anxious to make a good showing and knowing that these tests are being used as a yardstick to determine her teaching efficiency, and conscious of the fact that these tests measure ability only in the easily measured formal processes, will put the emphasis at that point. The question is, does any one know the state of diminishing returns in these formal processes? Indeed, efficiency in this might easily be a measure of the teacher's inefficiency. In following so closely the Courtis idea, the authors seem to have assumed that the same thing which happens in arithmetic is true in algebra. In view of the fact that tests tend to dominate both subject matter and method, it is the part of wisdom to question the objectives set forth. (3) The phrase "yardstick for measuring" implies a knowledge of maximum error. Promising as the testing movement is, have the processes of measurement been sufficiently refined to justify the term "yardstick" or is this a loose implication? In presenting the case for these tests the authors deny the validity of the Monroe Tests because the Monroe Tests do not employ the cycle principle. It is asserted that since some problems in the Monroe Tests are several times as difficult as others, the efficiency of a pupil solving a given number of problems on any test cannot be validly compared with the efficiency of pupils solving half as many, a third as many, twice as many, etc. By this argument, designed to refine measurement and permit the comparison of individuals on the basis "times" as many, the authors appear to rule out their own tests as "a yardstick for measuring the individual." This may be illustrated by Test VII, which is arranged in seven cycles with three problems per cycle. "The following table reveals the relative difficulty of the problems of this test. The corresponding problems of successive cycles appear in any one column. To determine relative difficulty of successive problems in one cycle, read horizontally across the table." TEST VII, QUADRATIC EQUATIONS Number of Cycle First Problem in each Cycle Second Problem Third Problem 1 2 3 4 5 6 7 19.7 22.0 17.6 20.6 15.6 13.9 10.7 34.3 29.0 28.5 29.1 29.9 (39.1) 30.7 23.7 24.8 40.4 43.8 35.7 33.0 25.8 The crests of difficulty are the second problem in cycle 1, the second in cycle 2, the third in cycle 3, etc. Let us take a hypothetical case and suppose that student A has been able to do successfully nine problems, whereas student B is in the midst of the fifth. Shall we say that A is twice as efficient as B? A has successfully passed three crests whereas B may or may not be failing on the second. The point is illustrated even if we assume B to be finishing the eighth problem. It would be false to assume that A is 1 1/8 times as efficient. We are dealing here with the increment of ability which is exceedingly difficult if not impossible to evaluate. We may illustrate the point concretely as follows: It is reported that Charlie Chaplin has signed a contract calling for a million dollars per year salary, whereas North who imitates him gets but $1,000 per year. Shall we say that Chaplin is 1,000 times as efficient as North? The person who does not attend the "movies" regularly probably is unable to distinguish the two characters. Chaplin is just a little better than his imitator, but it is this small increment which is of greatest value because, as Thorndike points out, "... they occur seldom, become famous and are given large financial returns."26 Thorndike shows that though we may judge differences in intellectual products near the middle of the human range by their intrinsic quantity and quality, we shift our basis of judgment as the limit is approached. Thorndike is skeptical about the logic of the "times" judgment in mental traits and careful in the use of the phrase "times as efficient." The point is more easily illustrated in the discussion of improvement, as it is easier to see that an improvement from just being able to solve four problems per minute to just being able to solve five per minute is an altogether different improvement than an improvement from ten to eleven. In view of these facts, it is just possible that the authors of the RuggClark Tests in questioning the validity of the Monroe Tests have ruled out their own tests as a "yardstick" for measuring the relative efficiency of teachers and pupils. Incidentally, it would be easy to point out an additional fundamental objection to the Monroe Tests in their violation of a commonly accepted principle of mathematical pedagogy. The Monroe Tests served a useful purpose in making a substantial beginning in the secondary field testing movement, but a wide usage does not appear justifiable. We shall no doubt be bombarded presently with numerous tests which I fear will not be designed as carefully as the RuggClark Tests. For this reason, in our attempt to remove the profound discontent that is being so freely expressed, we need to guard jealously against any loose implications which might tend to swing mathematics teaching back into the very practices which would retard those movements designed to ameliorate the situation and to hasten a constructive program. 1 This material was presented at the University of Chicago Conference with Affiliated Secondary Schools (April, 1917). 2School Review, March, 1916; January, 1917; The Modern School, pamphlet published by General Education Board; Parker, Methods of Teaching in High School, pp. 8185; and Judd, Psychology of High School Subjects, pp. 123132. 3Mathematics Teacher, December, 1916. 4Mathematics Teacher, December, 1916. 5Judd, The Psychology of High School Subjects, p. 184. 6Ibid., p. 395. 7School and Society, June 19, 1915. 8The Experimental Determination of Mental Discipline in School Students. 9Address to the Section of Educational Science, Educational Review, 1902, pp. 158181. 10Educational Review, 1902, p. 167. 11Psychology of High School Subjects, p. 23. 12David Eugene Smith, The Teaching of Geometry, p. 90. 13p. 59. 14Mathematics Teacher, December, 1916, p. 79 15School and Society, January, 1916. 16Millikan and Gale, A First Course in Physics. Millikan, Gale, and Bishop, Laboratory Physics. 17Educational Review, 1902, p. 158. 18Judd, Psychology of High School Students. 19School Review, 1903, pp. 521538. 20Karpinski, School and Society, January, 1917. 21Judd, Psychology of High School Subjects, pp. 1132; in particular pp. 122123. 22School Review, February and March, 1917. 23Ibid. 24School Review, loc. cit. 25Report of Illinois High School Conference, SchoolResiew, March, 1917,p.200. 26Thorndike, Educational Psychology, Vol. II, p. 185.


