bluuebunny/crossref_metadata_2025 huggingface dataset:
Reliability and validity of learning style inventory for primary school students based on the Myers-Briggs learning style model (ISUMB2-O)
Authors: Ranđelović Dušan , Pavićević Miljana
Published: November 2020

The goal of this study was to examine reliability and validity of learning styles inventory ISUMB2-O for primary school children. This inventory is based on Myers-Briggs learning style model, and the model itself is based on Jung's personality types. According to this model, the individual's profile is determined based on four pairs of opposite preferences: extraversion/ introversion, sensing/intuition, thinking/feeling, and judgement/perception. Reliability data (in a sample of 100 students of the 4th grade of primary school in Nis) obtained, both for individual dimensions (expressed by the Cronbach's alpha coefficient) and based on item analysis, indicate that this is a stable instrument. Three out of four dimensions have the reliability over 0.93 expressed by the Cronbach's alpha. Validity was verified in two ways: a) Construct validity was checked by using factor analysis in order to check the factor structure of the instrument, and b) external validity was checked by using the degree of correlation of values obtained based on the dimensions of learning styles with the instrument ISUMB2-O and on the teachers' assessment of individual poles of the examined dimensions of learning styles in their students based on the descriptions they received. The four-factor solution is the optimal and best possible one for the given instrument and it can explain 57% of the total variance. There is a statistically significant positive correlation between the results for the dimensions of the inventory of learning styles and teachers' assessment of individual poles of the examined dimensions of learning styles in their students based on the descriptions they received. This correlation value is high in the case of dimensions E-I, S-N and J-P, and of medium intensity when it comes to the dimension F-T. No differences in learning styles were observed from gender standpoint, while the dimensions sensing-intuition and judgment-perception were found to be related to school achievement. The conclusion is that the instrument showed good psychometric characteristics, primarily from the aspect of extremely high reliability, high values for item-total correlation, satisfactory validity construct, and also due to adequate external validity, which indicate that the instrument measures the characteristics for which was originally intended. Limitations of the research were also highlighted, especially those related to the sample size.
Raleigh Schorling—Teacher
Authors: Edmonson James B.
Published: November 2020

While the major contributions of Professor Raleigh Schorling in teaching, research, and writing were largely in the field of mathematics, he made an enviable reputation in other educational work, He had in influential part in the organization of the Lincoln School of Teachers College, Columbia University, and served as the first principal of the school. At the University of Michigan he organized the University High School and served as its first principal. He also planned and directed the first program in student teaching offered by the School of Education of the University of Michigan. In addition to his University duties he served on numerous state and national committees and was an exceedingly valuable member of such committees because of his unusual resourcefulness, marked initiative, and fine professional spirit.
A Laboratory Course in Making Visual Aids for Mathematics Education
Authors: Schorling Raleigh , Jones Phillip S.
Published: November 2020

A Novel course is being given at the University of Michigan for prospective teachers of mathematics. It provides an opportunity to make aids that illustrate basic concepts and principles of mathematics. Although teachers of mathematics have long used such learning aids as blackboards, graphs, models, etc., the fact is that most mathematics classrooms are too barren of things- of aids that illustrate concepts and principles. The student teacher completing this course may be far too busy when he starts carrying a heavy load on his first job to make learning aids himself. However, with the info rmation gained he should be able to so manage that each semester a few ingenious high school pupils will make them for use by the class, and thus contribute effective learning aids to the permanent equipment of the mathematics classroom.
Functional Mathematics—Grades Seven Through Twelve
Authors: Gager William A.
Published: November 2020

Is November 1917 the Florida State Department of Education and the college of Arts and Sciences at the University of Florida sponsored a study under my direction to determine ways of improving certain parts of the present secondary mathematics curriculum. Thirty-six secondary mathematics teachers, representing all areas of subject matter, all types of schools, and all sections of the state were selected to make the study. Work on the project was begun at the University of Florida on June 14, 1948.
Devices for a Mathematics Laboratory
Authors: Berger Emil J.
Published: November 2020

This section is being published ns an avenue through which teachers of mathematics can share favorite learning aids. Readers are requested to send in descriptions and drawings of devices which they have found particularly helpful in their teaching experience. Send all communications concerning Devices for a Mathematics Laboratory to Emil J. Berger, Monroe High School, St. Paul, Minnesota.
The Unbroken Mystery of the Broken Number
Authors: Neureiter Paul R.
Published: November 2020

Whereas the measurement of understanding in arithmetic has received some attention in recent years,1 we do not possess comprehensive data which would allow us to draw something like a profile map of understanding as actually achieved under the present regimen of arithmetic instruction. It can be safely asserted, however, that there is a sharp drop in understanding as pupils, gingerly guided by their teachers, descend from the rather brightly lighted plateau of the natural numbers into the misty lowlands of the artificial numbers, which for elementary school pupils are confined to fractions.
A Mathematical Dr. I. Quiz-em Program
Authors: Diggins Julia
Published: November 2020

Announcer: Mathematics Incorporated presents Dr. I. Quiz-em, the mental banker, master of wit and information. We hope you will enjoy his friendly smile and the cash, for your correct answer.
The Place of Mathematics in Human Affairs, and Related Curriculum Problems
Authors: Betz William
Published: November 2020

One does not have to he a professional student of mathematics to become convinced that the bearings of mathematics have world-embracing dimensions. In an almost unique sense, this huge area of learning touches the entire physical universe, and it permeates many aspects of the industrial, technical, and social fabric which human ingenuity has created on this revolving globe of ours. Obviously, then, we must greatly restrict the scope of our discussion, confining ourselves to the more modest question, how the school can or should react to this dramatic story in the education of American youth.
Mathematical Recreations
Authors: Bakst Aaron
Published: November 2020

The problem of teaching quadratic equations is always plagued by a dearth of motivating materials. Usually, in desperation, we are prone to resort to an admission that quadratic equations rarely offer an opportunity of presenting and utilizing “real” situations.
Pisa, Galileo, Rome
Authors: Kiely Edmond R.
Published: November 2020

In Professor W. R. Carnahan's article, “Mathematics an Essential of Culture,” which appeared in the November, 1950 issue of this magazine, there occur certain oft-repeated statements which, in the light of recent research, appear in need of revision. These statements concern Galileo's experiments with falling bodies from atop the Tower of Pisa and Galileo's clash with ecclesiastical authorities. As a matter of fact the reiterated statements concerning Galileo and Aristotle, and Galileo and the Catholic Church tend to obscure Galileo's real merits as a scientist. Even in Professor Carnahan's article we do not have as detailed a discussion of Galileo's contributions to the science of mechanics, on which his right to a place in the history of science properly rests, as we should have expected.
Evaluation of Determinants
Authors: Bakst Aaron
Published: November 2020

The proposed method of evaluation of determinants, however unorthodox it may seem, has, for some unknown reason, escaped the attention of authors and teachers of algebra, although references to it (and complete developments) have appeared in the literature on the subject.1 As a method for the evaluation of determinants it is general and it is, in some respects, more effective due to its simplicity. This proposed method is not offered with the view of supplanting and replacing the traditional method of evaluating determinants by means of direct expansion or by expansion by means of minors. However, it may appeal to some classroom teachers as a process which contains elements of simplicity.
Relationships for Radii of Circles Associated with The Triangle
Authors: Freitag Herta Taussig
Published: November 2020

In the April 1951 issue (p. 270), the department of Mathematical Recreations requested problems concerning the relationships of certain radii within a given triangle. The following have proved of interest.
Devices for A Mathematics Laboratory
Authors: Berger Emil J.
Published: November 2020

This section is being published as an avenue through which teachers of mathematics can share favorite learning aids. Readers are requested to send in descriptions and drawings of devices which they have found particularly helpful in their teaching experience. Send all communications concerning Devices for a Mathematics Laboratory to Emil J. Berger, Monroe High School, St. Paul, Minnesota.
The Small High School
Authors: Price H. Vernon
Published: November 2020

It is only natural for a teacher in the Midwest to be interested in the small high school and to recognize that a school of this type has special problems of its own. My personal interest in these problems was heightened by our experiences in re-writing the Iowa Course of Study in Secondary School Mathematics a few years ago. We soon discovered that one of our major tasks was to make provision for the needs of the small schools; indeed, it is essentially true that the course of study was written for these schools.
Mathematical Recreations
Authors: Bakst Aaron
Published: November 2020

Beginning with this issue of The Mathematics Teacher this department proposes to offer a series of problems and exercises which should lend to recreat ional activities in the classroom. Some of these problems may be found in books which are now out of print. Thus, their publication is, in the opinion of this department, justified. It is doubtful whether many of the classroom teachers have such sources handy. As far as possible, each issue will be devoted to a special topic. In this issue we will consider a series of problems which are associated with the number π.
Notes on The History of Mathematics
Authors: Sanford Vera
Published: November 2020

The social unrest in England that accompanied the shift from agriculture to grazing as the foreign market demanded more wool and paid better for it, was a serious situation in England in the sixteenth century. It is the basis for one of the few problems which the “Scholer” proposed to the Master in Recorde's Ground of Artes (c. 1542). Its presence in this volume makes one realize that problems of sociological and economic significance have been the concern of textbook writers even before the twentieth century.
Using Algebra in Teaching Geometry
Authors: Fehr Howard F.
Published: November 2020

It has been frequently stated that we should use algebra to a greater extent in teaching geometry. The purposes fordoing this have not always been made explicit.
What does “if” mean?
Authors: May Kenneth O.
Published: November 2020

Words can be slippery customers in mathematics as well as in other fields. The only way to keep them from tripping the students (and the teachers) is to really understand their use. Here the use of “if” is clarified.
Historically Speaking,–: “A new ballad of Sir Patrick Spens”
Authors: Jones Phillip S.
Published: November 2020

Professor Norman Aiming first called the editor's attention to “A New Ballad of Sir Patrick Spens” appearing in Q Anthology, a selection from the prose and verse of Sir Arthur Quiller-Couch.1
Points and Viewpoints: A challenge to educators
Authors: Van Engen Henry
Published: November 2020

A column of unofficial comment
Some arithmetical fundamentals of value to junior high school teachers
Authors: Kingston J. Maurice
Published: November 2020

A discussion of some elementary points in mathematics so frequently misunderstood by teachers and of great import for the student's future success in the study of algebra
Tips For Beginners: Better bulletin boards
Authors: Jackson Humphrey C.
Published: November 2020

Many teachers of mathematics have found the bulletin board a helpful aid in teaching, and a challenge to their ingenuity and that of their pupils. From my experience in using bulletin boards, I have accumulated some ideas which I should like to share with other teachers in the paragraphs that follow.
Integrated mathematics and physics
Authors: Smith Malcolm
Published: November 2020

Those teachers interested in a correlated or functional approach to mathematics will enjoy reading about an effort to teach physics and mathematics in the same course at the Putney School.
The algebra of sets in the teaching of trigonometry
Authors: Schumaker John A.
Published: November 2020

While the use of the algebra of sets suggested in this article is not extensive, it does serve to make use of ideas that are presently being proposed by some groups for high school classes.
Mathematics in the Junior High School: As mathematics teachers we're foundation engineers
Authors: Eagle Edwin
Published: November 2020

Engineers are sometimes faced with the problem of a building foundation that has become incapable of carrying its load. In some cases, because of poor engineering in initial stages, an adequate foundation had never been built.
Using celotex for graphing
Authors: Reinhold Margaret
Published: November 2020

This past year, an intermediate algebra class at Wilmington High School had an interesting and valuable experience in using acoustical celotex boards for graphing.
The geometry of the fixed-compass
Authors: Hallerberg Arthur E.
Published: November 2020

“Collapsible-Compass” geometry was handed down to us by the Creeks. A rigid-compass geometry is a fixed item in our high school curriculum. Students are not reminded that there are geometries which presuppose the use of tools other than the rigid compass they use.
Tips for Beginners: Oral testing
Authors: Wolff Harry
Published: November 2020

One of the important aspects of a complete evaluation program in mathematics is that of systematic oral testing. Every time we ask a question of a class or direct one to a particular individual we are, or should be, trying to discover some particular facet of the educational or learning process. Since oral questioning is a form of testing, I suggest that we give it the same careful consideration which characterizes our attempts to construct written tests. If oral testing is given conscientious planning and if it is applied systematically, it will easily establish itself as a functional and essential part of our evaluation program.
Useful generalizations of the concept of function
Authors: Robinson George A.
Published: November 2020

A function is a special kind of set of ordered pairs of elements, but these elements can be numbers, ordered pairs of numbers, or statements.
Chains of reasoning in geometry
Authors: Wiseman John D.
Published: November 2020

The idea of proof as a chain of reasoning may be reinforced by having pupils make chain models of theorems they have proved.
Points and Viewpoints: Thanks a million
Authors: Pingry Robert E.
Published: November 2020

A column of unofficial comment
A simple example of a noncommutative algebra
Authors: Wendt Arnold
Published: November 2020

Students may not consider the commutative principle very important if they never see an example where multiplication is noncommutative.
Vectors – an aid to mathematical understanding
Authors: Smith Dan
Published: November 2020

Instruction in vectors offers many opportunities to teach fundamental mathematical ideas.
Parabolas and Pythagorean triples
Authors: Gruhn E. W.
Published: November 2020

At their intersection, a cone, a cylindrical surface, and planes have co-ordinates which may be Pythagorean triples.
Some heretical thoughts from an orthodox teacher
Authors: Manheimer Wallace
Published: November 2020

“The classroom teacher has the right to disagree, of course, with the currently fashionable prediction that the mathematics curriculum in the secondary schools will be altogether different in twenty years.”
The Weequahic configuration
Authors: Ranucci E. R.
Published: November 2020

An exercise in visualization in three-space.
Linear programming, an aid to decision making
Authors: Kane Robert B.
Published: November 2020

Some simple examples of linear programming, one of which is in three variables.
Providing for the basic student in the junior high school
Authors: Mehl William G.
Published: November 2020

Some helpful suggestions are presented for working with the special student considered in this article.
Experimental Programs: The objectives and activities of the School Mathematics Study Group
Authors: Wagner John
Published: November 2020

At the present time many schools are experimenting with a variety of curriculum plans, teaching methods, and written materials. Beginning with this issue, The Mathematics Teacher is planning to provide information on these many programs through this new department, “Experimental Programs.”
Mathematics and industry
Authors: Wilcox Marie S.
Published: November 2020

Previous articles sponsored by this committee have reported assistance which industries have been giving to students and teachers of mathematics. Herein we should like to consider a way in which schools may assist industry and at the same time assist their own students—by giving additional consideration to the training of technicians.
How to make mathematics more attractive
Authors: Dadourian H. M.
Published: November 2020

Some suggestions for the teacher who is attempting to help his students find mathematics as interesting as he does himself.
Historically Speaking,—: The evolution of extended decimal approximations to π
Authors: Wrench J. W.
Published: November 2020

In his historical survey of the classic problem of “squaring the circle,” Professor E. W. Hobson [1]* distinguished three distinct periods, characterized by fundamental differences in method, immediate aims, and available mathematical tools.
Historically Speaking,—: Augustus De Morgan and modern mathematics
Authors: Constantia Mary
Published: November 2020

The last decade, 1950-60, has aptly been called a period of “Revolution in Mathematics.” The decade ushered in in 1960 has already begun to reap the benefits of this revolution in an improved mathematics program from the kindergarten up to and including the graduate school.
Enrichment: a geometry laboratory
Authors: Dunn-Rankin Peter , Sweet Raymond
Published: November 2020

The Class waited expectantly as the student's wax-paper folding was projected upon the screen. The opaque lines enveloping the hyperbola. stood out clearly and effectively. Next, a parabola's envelope was shown and a comparison made. Now I he tudents gra pcd significantly the distinction between these curves, a concept that, heretofore had been confusing because of the similariLy in their equations.
The linear equation and absolute values
Authors: Forges Arthur
Published: November 2020

Equation containing absolute value notation
Concave polygons
Authors: Berman Roslyn M. , Berman Martin
Published: November 2020

Finding the sum of the interior and exterior angles of a concave polygon
Experimental Programs: An experiment in programmed instruction in junior high school
Authors: Randolph Paul H.
Published: November 2020

Only a brief glance through the last few years of THe Mathematics Teacher will reveal that there are at least two topics which have created considerable discussion; namely, the role of “modern mathematics” in the high school curriculum, and the applicability of programmed learning to mathematics. Thus, when the author had a chance to work with a group of junior high school students, this seemed like an ideal opportunity to test the value of these ideas.
Points and Viewpoints: On numbers and numerals
Authors: McLean Robert C.
Published: November 2020

A column of unofficial comment
The Limit Concept in the SMSG Revised Sample Textbooks
Authors: Hight Donald W.
Published: November 2020

How much instruction should high school teachers give toward understanding the limit concept?
Some Axioms for Teaching Real Exponents
Authors: Falbo C. E.
Published: November 2020

A mathematically productive way to enrich the mathematics curriculum is to axiomatize the essential topics.
Snowflake Curves
Authors: King Bruce W.
Published: November 2020

A problem that involves concepts and methods important in the mathematical development of students
Pythagorean Mathematics in the Modern Classroom
Authors: Shulte Albert P.
Published: November 2020

Mathematics mixed with mysticism
Reflections of a Retiring Teacher of Mathematics
Authors: Fawcett Harold P.
Published: November 2020

A veteran in NCTM urges that this is no time for little thoughts.
Intersection and union word puzzle
Authors: Nichols Thomas D.
Published: November 2020

Some fascinating drill
International Mathematical Education: The new mathematics program in the Danish gymnasium
Authors: Rindung Ole
Published: November 2020

In 1958, the structure of the Danish school system was changed, and in connection with this the curricula for each subject at all school levels were revised. This provided an opportunity to create a mathematics curriculum which reflects the trends reported in recent international discussions on the problem of modernization of mathematics teaching. This newly created curriculum is now being used in grades 8 to 11, and next year will go into operation in grade 12, the last secondary school year.
A Boolean switchboard
Authors: Horton George W.
Published: November 2020

A practical look at the algebra of switching
Project Idaho
Authors: Kay Richard
Published: November 2020

Teachers of mathematics at all grade levels face the problem of upgrading their knowledge in mathematics, because the content now being taught includes many ideas that are unfamiliar to many teachers. This upgrading problem exists for experienced as well as inexperienced teachers at all levels, but it is especially acute for experienced elementary school teachers who are using a contemporary mathematics book.
A recursion formula for solving n linear equations in n variables
Authors: Kattsoff Louis O. , Simone Albert J.
Published: November 2020

A machine-age look at a familiar problem
An examination of the differences resulting from iteration
Authors: Walton Howard L.
Published: November 2020

Iteration methods can enrich the regular curriculum.
A reachable research area
Authors: Sharpe Benjamin
Published: November 2020

How to provide a feeling of original discovery that may even be valid
More on Pascal's Triangle and powers of 11
Authors: Mueller Francis J.
Published: November 2020

Another look at divisibility
Divisibility test for 7
Authors: Feiner Henri
Published: November 2020

An interesting test for you to unravel
Continued square roots
Authors: Rotando Louis M.
Published: November 2020

An interesting sequence involving recursive definition
Buffon's needle: stochastic determination of π
Authors: Thumm Walter
Published: November 2020

A discussion of the Buffon problem, including the presentation of several solutions which may be of interest and value for classroom use in secondary schools and junior colleges
Relations and functions
Authors: Stiel Edsel F.
Published: November 2020

“We shall explore the relation concept both from a naive and from a mathematical viewpoint”
Commuting linear functions and fixed points
Authors: Seguin Charles P.
Published: November 2020

An invitation to research, accompanied by at least one elegant proof
The use and abuse of programed instruction
Authors: Smith Leander W.
Published: November 2020

The first of several reports on programing
A General Divisibility Test
Authors: Pruitt Robert
Published: November 2020

If you scan the recent literature for teachers of arithmetic, you will notice much activity in the area of divisibility tests for whole numbers. This is unquestionably the part of elementary number theory that requires little mathematical sophistication. Accordingly, it has much appeal for elementary school and junior high students.
Congruency of Triangles by Aas
Authors: Ryoti Don
Published: November 2020

After a discussion of the SAS, ASA, and SSS statements about congruency of triangles, the natural question arises whether any other combinations of three pairs of parts of two triangles are sufficient for congruency. The other possible combinations are AAS (two angles and a noil- included side), SSA (two sides and a nonincluded angle), and AAA. It is to be understood here that the nonincluded sides are corresponding. A similar statement applies with respect to the nonincluded angles.
Variations on a Theme y = 2x − 3
Authors: Wernick William
Published: November 2020

It Happens frequently in the arts that the work of one gifted creator is the inspiration for a later one, who may extend or embellish the earlier work, or simply use it as a basis for further artistic effect.
On An Easy Construction of a Computing Machine
Authors: Hausner Melvin
Published: November 2020

In Common with most universities and colleges in the country, New York University offers a survey course in mathematics to liberal arts students who will not major in mathematics or any of the exact sciences. At NYU it is a four-credit course called “Introduction to Mathematics,” although it is known to our computing machine by the more endearing name “A63.0001.” Because of the profound effect that computing machines have on our lives, and because of the mathematical nature of their theory and logic, I decided, on an experimental basis, to devote a few lectures to this topic. This note is devoted to what happened. It is my belief that a similar technique would work very well in the high schools. It should be remembered that the students in the class, as nonscience majors taking a minimal requirement course, were on the average less mathematically talented and trained than a group of scientifically minded high school juniors and seniors.
International Mathematical Education: The Teaching of Vectors in the German Gymnasium, II
Authors: Athen Herman
Published: November 2020

The first part of this article appeared in the immediately preceding issue of this journal (April 1966).
Mathematics and the Teaching of Science
Authors: Bowen John J.
Published: November 2020

The relationship between science and mathematics has existed at least since Archimedes used geometry to establish the science of statics. Even the formerly descriptive biological sciences now use mathematics extensively, and some of this has filtered down to the high schools where students in the “new” biology are using the concepts of statistical sampling and logarithmic growth curves.
The Public Image of “New Mathematics”
Authors: Mueller Francis J.
Published: November 2020

Except, perhaps, for the controversy of several years ago that swirled around “Why Johnny Can't Read,” nothing else in curriculum history seems to have aroused as much public discussion as “new math.”1 As a consequence, the subject is now a perfectly acceptable conversation piece at just about any kind of social gathering, from fashionable cocktail parties to boondoggling sessions around the office water cooler.
Using Geometry to Prove Algebraic Inequalities
Authors: Garfunkel J. , Plotkin B.
Published: November 2020

In Line with the increased emphasis on the study of inequalities and the greater stress on integrating the various branches of mathematics taught in the high schools, this article treats algebraic inequalities geometrically. We further believe that geometry, which is taught early in the student's career, is usually too soon forgotten, and using geometry with his later work in algebra will help him to correct this. To this end we offer a geometric approach for illustrating and proving some basic algebraic inequalities.
Experimental Programs: Classroom Management
Authors: Kaplan Jerome D.
Published: November 2020

The recent excitement in the educational community about technology has masked one of the crucial underlying problems created by new devices and ideas— the management of the classroom. Major transformations are taking place in the organization of schools and classrooms because of the impact of technology.
A Time for Testing
Authors: Gurau Peter K.
Published: November 2020

I like the teaching profession. I enjoy my students, my subject, and the general atmosphere. My work gives me a sense of accomplishment.
Counterpoint as an Equivalence Relation
Authors: Delman Morton
Published: November 2020

Most textbooks are unable to cite any familiar examples of an equivalence relation other than “equality” and geometric “similarity.” It has occurred to me that musical counterpoint is a perfect example of an equivalence relation, because it illustrates the three required properties: reflexiveness, symmetry, and transitiveness.
The Shape and Size of the Earth
Authors: Fischer Irene
Published: November 2020

In the play and movie The King and I, the British teacher Anna tells the royal children of Siam that the earth is a sphere. And the crown prince says to his father, the king, in utter disbelief, “Does she not know that the earth is flat, and that the whole world is carried by four elephants standing on the back of the Big Turtle?”
Mathematics and Art
Authors: May Kenneth O.
Published: November 2020

“But mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius. This must be known.”—Marston Morse, Yale Review, Summer 1951.
Providing Motivation for the Chain Rule
Authors: Holden Lyman S.
Published: November 2020

The purpose of this article is to describe how a certain graphing technique can be used to increase the heuristic appeal of a general formula from the differential calculus. The formula to which reference is made is the formula for the derivative of the composition of two functions (the chain rule). We shall interpret this rule in relation to the graph of the function that is the composition of two functions. Using the geometric interpretation of the derivative as the slope of a curve, it will be possible to provide improved motivation for the rule.
The Effects of a Unit in Probability and Statistics on Students and Teachers of Ninth-Grade General Mathematics
Authors: Shulte Albert P.
Published: November 2020

The major emphasis of the “revolution in mathematics” at the secondary level has been on the improvement of courses in the college preparatory sequence. From the inception of the “revolution” until quite recently, little attention has been given to the ninth-grade general mathematics course. In the past few years there has been a substantial in crease in research and experimentation with ninth-grade general mathematics. Aside from these experimental situations, however, little actual change in the course has taken place. This does not mean that teachers or students have been generally satisfied with the course. It is the writer's experience, from observation of classes and talking with teachers and administra tors, that the general mathematics course is dislikod by most teachers and many students.
The Special Case May Be the Hardest Part
Authors: Thompson Richard B.
Published: November 2020

The use of mathematical induction seems to be a difficult topic to teach effectively because most applications of induction are either trivial or occur in complicated settings that involve too many ex traneous concepts. Recently, while teaching an elementary course, I gave an assign ment which, to everyone's surprise (my self included), led us to discover one of the basic properties of induction in a par ticularly simple context. We were discussing Pascal's triangle and the binomial co-efficients when I suggested that my stu dents give an inductive proof of the following formula.
Have You Read …?
Authors: Peak Philip
Published: November 2020

It is always good to look ahead and see what we might expect. It is also well to see what others are experting, This is such au article which was a paper presented to the American Statistical Society by the author, and this is what he sees. Expectations are limitless, but resources are not; therefore, sometimes there must be a crunch between the two. We must make rational choices, based on the best data we can gather. We must also recognize the probabilities at tached to these choices. We must face reason able social control, and, through balance between units of government, there must be social planning but individual and institutional de grees of autonomy. We must plan into the future, not stumble into it. Man and institution must aggressively renew, redesign, and revitalize our thinking and our processes.
International Mathematical Education: Note on the First International Congress on Mathematical Education
Authors: Fehr Howard F. , Becker Jerry P.
Published: November 2020

The First International Congress on Mathematical Education is now history. It was held in Lyon, France, August 24-30, 1969. Approximately 700 mathematics educators from 37 countries participated in this event. The United States was well represented, having approximately 100 participants. Only France, the host country, had a larger representation.
Reviews of films
Authors: Coxford Arthur F.
Published: November 2020

The film makes effective use of points, color, a metronome, and different-sized arrows to demonstrate the physical principles at work as forces act upon bodies in space. The points are the bodies, the arrows are forces, and the metronome marks equal time intervals. Several scenes in the film reminded this viewer of a billiard game where the points were the balls and the arrows were cue sticks.
GJY Insert, Never Underestimate the Inner-City Child
Authors: Greenholz Sarah , Keiffer Mildred
Published: November 2020

Teaching in slum schools is hard work. Children who have learned to distrust adults, to expect failure in school, and to settle most of their problems with their fists are not easy to instruct in the classroom. Such children are those who are growing up in the crowded ghetto areas in the great cities. Housing is sub-standard, minority groups are well represented and the welfare worker is a fami liar person.
Truth-Table Models of Mendelian Trait Segregation
Authors: MacDonald Theodore H.
Published: November 2020

With the increasing emphasis in the classroom being placed on “;learning by discovery,” high school students are encouraged to see interrelationships within subject areas embraced by their courses. This has led, not unnaturally, to interdisciplinary analyses of certain key themes in science and in life generally at the high school level. One of the fundamental tasks of any science teacher is to exhort his students to find connections between physics, chemistry, biology, and mathematics as a means of unifying an approach to the physical sciences.
Students Become Math-Minded through league Influence
Authors: McCormick Martha Ann
Published: November 2020

Can a college stimulate interest in mathematics in the high schools of its area? Can it, encourage more effective teaching at the high school level? Can it create rapport between the high school mathematics teachers and the college mathematics staff? We at Missouri Southern College believe the answer is YES ! We feel the MSC Math League has started us well on the road to achievement of these goals.
Computer Arithmetic
Authors: Franta W. R.
Published: November 2020

Students new to computer programming often write computer programs that fail to perform as expected. Seeking to correct the resulting deficiencies, the student often looks in vain for an error in logic, when the true cause of the trouble may be a failure to consider the nature of computer arithmetic.
Taxicab Geometry- a Non-Euclidean Geometry of Lattice Points
Authors: Byrkit Donald R.
Published: November 2020

During the past decade, much im-portance has been placed on the structure of mathematical systems. Structure has been the byword in the construction of texts and courses. For this task the framework of geometry has been found to be quite acceptable at the secondary level. The axiom system of Euclidean geometry is examined in depth, but usually very little attention is given to other axiom systems. Occasionally axiom systems are discussed per se, but examples are usually too difficult to develop (such as non-Euclidean systems), too limited in de-velopment (such as a finite geometry), or too trivial to create any interest.
Triangles And Trigonometry
Authors: Ailles Douglas S.
Published: November 2020

Many students rely too much on a set of trigonometric tables and spend a great deal of extra time, that could be used to better advantage, on these tables. I have found that by exploring the facets of the diagram shown in figure 1, one can discover many basic relationships.
A Non-Euclidean Distance
Authors: Clemens Stanley R.
Published: November 2020

There are basically two approaches to classical Euclidean plane geometry—the synthetic approach and the metric approach. The older of the two is the synthetic approach followed by Eucliding later by Hilbert. In the Eucliding treatment, one begin by assuming as undefined the relations of betweenness, congruence of segments, and congruence of angles. The metric treatments, initiated by G. D. Birkhoff in the 1930s, assumes the existence of the real numbers (or a set of postulates that guarantees the existence of the real numbers) and the existence of a distance function d and an angle-measure function m.
The Forum: The Utility of Behavioral Objectives: PRO A Valuable Aid to Teaching
Authors: Allendoerfer Carl B.
Published: November 2020

I AM writing as a comparative novice in the field of behavioral objectives, for I first heard of them only a few years ago. My first impression was that they were an overly formalized procedure for doing what any good teacher did anyhow and that they should remain the private property of the educational psychologists. Later, however, I reluctantly agreed to use them in the evaluation of the preliminary edition of a textbook that I was writing and to my great amazement discovered that they were of tremendous utility to me. So, like anyone who has just adopted a new religion, I am an enthusiast; and in this article I shall tell you why I am. Enthusiasm, however, should be tempered with judgment; and so I do not wish to claim that behavioral objectives will solve all our problems—they have an important place in mathematical education, but they should be used with appropriate restraint.
Another Look at the Roses
Authors: Grinstein Louise S.
Published: November 2020

IN TEACHING polar-coordinate curve-tracing, frequent examples cited are the so-called rose curves:
A Geometric Application of the “Shepherd's Principle”
Authors: Murrow Gene
Published: November 2020

Most readers are probably familiar with the standard problem “How many squares of all size are contained in an n × n checkerboard pattern of n2 1 × 1 squares?”
Residue Designs
Authors: Locke Phil
Published: November 2020

A decade ago, while teaching secondary mathematics in Franklin, Ohio, I discovered in the high school library an obscure mathematical tract having the above title.1 Intrigued, I decided to share this find with my algebra class.
Mathematical-Musical Relationships: A Bibliography
Authors: O'Keeffe Vincent
Published: November 2020

Since the time Pythagoras reputedly discovered a relationship between the length of a vibrating string and the musical harmonic series, scholars have known a connection between mathe-matics and music. Through the course of history, the alliance has been recognized in varying degrees as musicians either looked toward a more scientific view of art or zealously guarded their domain as a field based on intuition and inspiration.
The Election of a President
Authors: Sullivan John J.
Published: November 2020

THE election of a president in the United States has many mathematical aspects. It is certainly appropriate for teachers to exploit student interest in presidential elections to advance the aims of mathematics classes. The electoral college, apportionment of representatives, population data, and many other features of a presidential election can provide interesting and profitable mathematical exercises. Without difficulty one can devise good problems in arithmetic, geometry, algebra, and statistics.
An Instructive Approximation for the Sine Function
Authors: Rand Roger
Published: November 2020

BEGINNING trigonometry students Frequently inquire how the trigonometric tables can bo constructed. In particular, how can the tablc of sines he constructed?
Changing Bases by Direct Computation
Authors: Lee John W.
Published: November 2020

The usual method for changing from one base to another is to change one base to base ten and then to change to the other base by performing a sequence of divisions in base ten. The purpose of this article is to look at the change of base directly, without involving any base ten computations.
The Appreciation of Radian Measure in Elementary Calculus
Authors: Lipsey Sally Irene , Snow Wolfe
Published: November 2020

Although a substantial number of results in the calculus of trigonometric functions are contingent on expressing the measure of an angle in radians, many students persist in the use of degree measure and seem to feel more comfortable with it. They convert from radians to degrees, at times erroneously, for their computations and diagrams.
Spaces, Functions, Polygons, and Pascal's Triangle
Authors: Jansson Lars C.
Published: November 2020

Two recent trends in the teaching of mathematics can often go hand in hand. These two are the spiral approach to content development and the discovery-teaching/learning method. This paper will present some topics that illustrate how these approaches to content and method may complement each other. Space will not permit the full explication as it might occur in a classroom, but it is hoped that both the discovery approach to mathematical conjecture as well as the spiral nature of the topics will be evident. The specific topics discussed here range from those for the upper elementary school pupil to those for the high school senior—the exact placement depends not only on the topic itself, but on the language and notation as well.
A Generalized Base for Integers
Authors: Woo Norman
Published: November 2020

One of the most important notations in mathematics, the positional system of no-tation of numbers, seems to have been invented in India by the Hindus. It was certainly transmitted to the Western world by the Arabs in about the eighth century A.D. At first the Hindu-Arabic system of notation was of interest only in the marketplace, but eventually it came to be of interest to mathematicians, and it is still the object of considerable study.
Mathematics in Sports: Examples for General Mathematics
Authors: Duncan David R. , Litwiller Bonnie H.
Published: November 2020

Teaching general mathematics can be a rewarding experience. One challenge is to find topics that are interesting to the students. Statistics, when introduced in relation to sports, provides such topics.
Mathematics Education in Basel
Authors: Hughes Barnabas B.
Published: November 2020

Basel is located at the juncture of Switzerland, Germany, and France. In this paper, we use the word to refer to Basel City, politically independent from Basel Land, which completes the territory known as the canton of Basel, one of the twenty-two autonomous cantons in the Helvetic Confederation (official name of Switzerland).
Lower Columbia College Mathematics Learning Center
Authors: Spangler Richard
Published: November 2020

Ideally, teaching mathematics involves identifying the skill level of the individual pupil and then designing a program to carry him as far as he desires to advance. Unfortunately this arrangement is not economically feasible under the present traditional quarter or semester system in college, for this involves a teaching load of no more than ten students per instructor at any one time.
Problem Solving—Solution or Technique
Authors: Whirl Robert J.
Published: November 2020

As a mathematics teacher, I try to emphasize that the answer to a problem is not as important as the technique used in finding the answer. A good problem-solving technique is a major objective in mathematics.
Recreation: Hexiamonds
Authors: Spaulding Raymond E.
Published: November 2020

Hexiamond shapes can be used to generate a fascinating collection of open-ended puzzles. Solution of the puzzles and construction of new hexiamond shapes will challenge young geometers in junior or senior high school while developing fundamental concepts of symmetry and rigid motion.
The Secondary School Mathematics Curriculum Improvement Study: A Unified Mathematics Program
Authors: Fehr Howard F.
Published: November 2020

IN THE early 1950s the mathematicians at the university level began justifiably to criticize the mathematical preparation of high school students for university study. Partly because of this criticism, a number of reform movements arose, of which those of the University of Illinois Committee on School Mathematics, the Commission on Mathematics of the College Entrance Examination Board, and the School Mathematics Study Group were most notable. The activities of these reform groups brought about a great improvement of the mathematics curriculum, though maintaining the traditional separation of mathematics into yearly units of study, especially for algebra and geometry. Included in these programs were quite a few topics not found in the pre-1950 era, such as the structure of the several number systems, the early introduction of the function concept, and a more rigorous presentation of Euclidean synthetic geometry. Likewise, the notions of sets and set operations were used to the advantage of better understanding.
Activities: Discovery with Cubes
Authors: Reys Robert E.
Published: November 2020

Objectives: Students will visualize three-dimensional figures, construct a table, discover patterns in the table, and use patterns to make predictions.
More on the Mathematics of Musical Scales
Authors: Botts Truman
Published: November 2020

AN INTERESTING article by Paul S. Malcom entitled “The Mathematics of Musical Scales” appeared in the November 1972 issue of the Mathematics Teacher. The present article may be viewed as a sequel to that one, although it may also be read independently. We propose to explore in a bit more detail the just-intonation scale natural to stringed instrument players and singers and the relation to it of the equal-temperament scale used in tuning pianos. We shall make constant use of fructions, and along the way we shall encounter an analogue of the idea of compound interest and a simple exercise in exponents and logarithms. At the end, we shall comment very briefly on an advanced branch of mathematics, called harmonic analysis and involving trigonometric sine and cosine functions, some basic ideas of which are illustrated by the analysis of a musical tone into its harmonic components or overtones.
How to Break a Window without Calculus
Authors: Hughes John H.
Published: November 2020

IT IS always educational to find that widely differing methods may be applied to the solution of a problem, especially if the student learns that thinking about method can save a lot of work.