bluuebunny/crossref_metadata_2025 huggingface dataset:
Editorial comments: As we read
Authors: Hamilton E. W.
Published: October 2020

Diversity characterizes this issue. The breadth of interest which a teacher of arithmetic must have is amazing when you begin to list the topics. Herbart introduced the term “many-sided” into the educational literature over one hundred years ago. We now recognize even single fields of knowledge as many-sided. If beauty is in the eye of the beholder, it might also be said that a point of view is in the eye of the viewer. For instance, a topic new to many elementary teachers is developed in two articles. Salsburg uses the notion of a modulus to introduce several simple theorems about number from a rather ingenious point of view, and Hildebrand uses the same mathematical notion as a vehicle for an excellent illustration of induction and discovery.
An odd use for odd numbers
Authors: Salsburg David
Published: October 2020

Is the sum of two odd numbers always even? Is the product of two odd numbers always odd? These are questions which can open a wide new field of enrichment for the bright primary student who quickly learns his number facts and is impatient to try something new.
Experiment in enrichment—fourth grade
Authors: Hildebrand Francis H.
Published: October 2020

To determine what sophisticated mathematics could be utilized for enrichment at the elementary level, a pilot program was set up in a fourth grade at the John D. Pierce Laboratory School, Northern Michigan College. The writer conducted a series of demonstrations and discussions, held twice a week from February through May, 1962. A group of twelve fourth graders was scheduled for forty-minute periods but some extended beyond by five or ten minutes. Two interested fifth graders joined the ranks in March.
The “cancellation” bug-a-boo
Authors: Alan Riedesel C.
Published: October 2020

At a recent conference a number of intermediate- grade teachers were discussing problems that troubled them in their arithmctic teaching. One of the topics on which the discussion centered was “cancellation” as applied to the multiplication of fractions. Many teachers of a rithmetic and mathematics believe that part of the difficulty may very well be in the term “cancellation” and contend that the term “reduction” is much more in keeping with the mathematical principle involved.
Pupil understanding of the numeration system
Authors: Flournoy Frances , Brandt Dorothy , Mcgregor Johnnie
Published: October 2020

How well does the pupil understand basic principles underlying the decimal system of numeration? To develop, administer, and study the results on tests of understanding the numeration system was the purpose of this investigation.
In the classroom: Individualized patterns of thinking—results the same
Authors: Wert Fay
Published: October 2020

How many ways are there to get to a destination? Well, one could go by air, train, bus, car, or maybe by foot. It all depends upon where you start and upon the possible ways of getting from one place to another. Children do just that.
In the classroom: Concepts to enhance the study of multiplication
Authors: Haines Margaret
Published: October 2020

The use of a system called Napier's Rods will make multiplication an interesting study for both fast and slow learners. These were invented about 1617 by a Scottish mathematician named John Napier for performing multiplications.
Children learning mathematics
Authors: Burn Margaret H.
Published: October 2020

The teacher of mathematics, whether she wishes it or not, is in control of two vast considerations—the growing child and the subject, mathematics. Both are vitally involved and entwined one with the other. The child either achieves something toward whole growth, becoming richer in his growing personality and more balanced in his judgment because of his finding of weighed thinking, which is his own and not imposed by another; or he submits because he has to, and through no fault of his own, to the memorizing of ways and means, techniques, and formulas as decided for him, regardless, often, of his own thinking possibilities. This latter course is to be deplored, for it results in ceiling the able and frightening the weak, while with all, that which is the essence of life to mental growth and well-being—the child finding for himself, at his own level, his own thinking—is steadily stifled.
Greater flexibility in abstract thinking through frame arithmetic
Authors: King Elizabeth B.
Published: October 2020

Arithmetic can be much more than getting answers in problem solving and computation. It is not an end in itself, but can and should lead to discoveries and insights into mathematical laws and principles.
Goals for arithmetic teaching
Authors: Coleman Mary Elisabeth
Published: October 2020

What is good teaching of arithmetic? Certainly the basic characteristics of good arithmetic teaching are those of good teaching in any subject. As an overall generalization, we can say that good teaching is that teaching which develops an able and independent learner. After enjoying the better part of a month in observing fifty different teachers of arithmetic in elementary schools, I found certain characteristics present in the teaching which the building principals and I agreed were “good.”
First graders' number concepts
Authors: Holmes Emma E.
Published: October 2020

Prior to school entrance, children learn informally about groups, or sets, of objects and the elements comprising sets. They learn to distinguish one object from another. They begin to define sets by means of significant properties. For example, a child knows the set, his family, as a group of people related to him. He may know that his family has four mem hers. Furthermore, he may be aware that a set can be described in terms of its subsets. The family, for example, is made up of parents and children.
An experimental study of programmed versus traditional elementary school mathematics
Authors: Banghart Frank W. , Mclaulin John C. , Wesson James B. , Pikaart Leonard
Published: October 2020

The purpose of this experiment was to compare programmed materials with nonprogrammed materials. The subject matter was fourth-grade arithmetic. Comparisons were made for total arithmetic scores, problem solving, comprehension, and boy-girl difference.
Editorial comments: As we read
Authors: Hamilton E. W.
Published: October 2020

Bonuses sometimes received from research are the unexpected outcomes that may overshadow the main effects being studied and the “straw in the wind” out-comes that someone may then pick up to study as a main effect. Both of these are present in this month's issue.
To invert or not to invert
Authors: Bray Claud J.
Published: October 2020

The teaching of the division of fractions has been and continucs to he a conccm of Leachcrs and wri ters in Lhc field of teaching arit hmctic. Should children merely be told to in vert nnd multiply, or should Lhey divide by the common-denominator method, or would some other method better promote learning?
An evaluation of topics in modern mathematics
Authors: Peck Hugh I.
Published: October 2020

If any one thing can be said concerning mathematics in schools today it is that mathematics is in tbe process of change. Mathematics teachers find themselves involved with “ncw” or “modern” mathematics. More and more modern mathematics textbooks are appearing as each month passes. As the modern mathematics becomes a reality in our school system, evaluation of such a program becomes important.
Let's modernize graph teaching
Authors: Meadows George C.
Published: October 2020

If you have ever corrected stacks of graphs drawn by junior high school students, you must have asked yourself, “Isn't there an easier way to teach these concepts?” We are not speaking of coordinate work in algebra, but just the simple circle, bar, and line graph representations used in daily life.
What has eighteen legs and catches flies?
Authors: Adkins Bryce E.
Published: October 2020

Teachers often use “Just for fun” worksheets as a SOUJ"ce of supplementary work for pupils who complete their arithmetic assignments in less than the allotted work period. One such successful worksheet idea, which often carries over to become unassigned homework, combines the appeal of riddles and codes. An example of such a worksheet follows. Its underlying idea, of course, can be adapted to fit any riddle, while the actual examples or problems should be devised to fit the maturity of the pupils.
Research on elementary school mathematics—1962
Authors: Weaver J. Fred
Published: October 2020

The references included in this 1962 listing embrace (1) normative and experimental studies which report specific data or findings on a problem associated with or closely related to mathematics instruction in the elementary school, and (2) bibliographies, summaries, and critical discussions which relate significantly to such normative and experimental studies.
Minutes of the Annual Business Meeting
Authors: Karnes Houston T.
Published: October 2020

The Annual Business Meeting was scheduled to follow immediately after the 8:30 A.M. general session during which Pres. Frank B. Allen presented a paper entitled, “The Council's Drive to Improve School Mathematics.” * Contained in this paper was a fine survey of the work of the Council during the past year and indications for important projects to be done in the future. Julius H. Hlavaty presided at this general session.
Modern mathematics and good pedagogy
Authors: Fehr Howard F.
Published: October 2020

Like a bolt out of the blue, education in America has suddenly become alive and exciting with new thinking concerning the intellectual purposes and subject matter of formal education. It is the intention of the writer to review some of the reasons for this activity, especially with regard to mathematical education; to briefly survey what we mean by the adjectives “new” or “modern”; and, finally, to present a picture of the content, purposes, and trends of emerging mathematical education.
The uneven progress of the revolution in elementary school mathematics
Authors: Marks John L.
Published: October 2020

Most of those in the mathematics community recognize the existence of a Revolution in Mathematics—that recent burst of interest which is accompanying the expanding discoveries and increased application of mathematics to science and industry—and with it, the striking changes being made in the mathematics curriculum from kindergarten through the university graduate level. There is strong approval and support of the need for change in our mathematics programs. Further, there is overwhelming acceptance of the general direction of change—toward a program that contains sound mathematics and is built on a psychology of teaming with understanding of mathematical ideas as a most important goal. It is generally agreed that if the rcvolution suddenly ceased today, the improvement in mathematics education would be tremendous, and would be a healthy influence on the curriculum for years to come.
Self-estimates of ability to teach arithmetic
Authors: Groff Patrick J.
Published: October 2020

Are pre-service teachers well prepared to teach elementary school arithmetic? It Seems that the evidence needed to answer this question reasonably should be taken from many sources. One of these sources would be a study of pre-service teachers' knowledge of arithmetic. Fulkerson has provided some evidence regarding this. He concluded from his study “ … that far too many of the 158 prospective elementary teachers studied herein have an insufficient. knowledge of arithmetic to teach the subject effectively.”1
We Quote
Authors: Graham J. D.
Published: October 2020

& Now, while the content of school mathematics remained relatively static, there were changes in mathematics which have been described “as so extensive, so far reaching in their implications, and so profound that they can be described only as a revolution.” Because of these changes, during the last ten years there has developed a rising tide of opinion around the world that a far-reaching change in content and approach to school mathematic. was necessary to meet the needs of the second half of the twentieth century.
Frames, frames, and more frames
Authors: Folsom Mary
Published: October 2020

Few deny that the mathcmatirs curriculum of the elementary school needed revision. However, the rush to modernize this curriculum has given rise to some rather questionable practices. Experimental programs claim that the changes to be made should result in good, teachable mathematics. One could hardly argue with this objective.
Aligning traditional with new mathematics
Authors: Jude Lynch Sister Ann
Published: October 2020

The chief challenge in elementary mathematics today is the gradual adjusting to the new methods and ideas while still retaining much that is traditional. My problem has been how to continue to use a textbook that is a moderately traditional one while moving into that teaching area that involves the new concepts in mathematics. After much frustrated planning, I remembered a famous cliché used by one of our old-time Supervisors of Schools. “You do not teach a text, you teach a child.” The next step was to move to the question of how to teach these children some of the necessary concepts inherent in the new mathematics program while concomitantly using the traditional text.
In the Classroom: Making multiplication colorful
Authors: Bachrach Beatrice
Published: October 2020

The meaningful approach to the teaching of arithmetic has very definite advantages over the mechanical approach of years ago. The meaningful approach, with its emphasis on understanding, has done much to create interest, provide stimulation, and permit greater success in arithmetic for the students of average and superior mentality. But what of the slowlearner and retarded child?
In the classroom: Sixth graders compose space problems
Authors: Reuter Kathleen
Published: October 2020

The problems included in this report were a natural outgrowth of a somewhat detailed discussion of Gordon Cooper's orbital flight. Attention was called to both the scientific and mathematical impact of this historical endeavor. The teacher took advantage of the children's enthusiasm and interest to extend vocabulary and to further t he development of mathematical concepts.
The new mathematics
Authors: Corle Clyde G.
Published: October 2020

The elementary school tends to be a static institution. In spite of the fact that reorganization practices Over the last several decades have brought together many classrooms into larger attendance units, the autonomy of the teacher has not changed. She is an individual practitioner who retains the right to teach pretty much what she wishes, in whatever way she wants to teach it. Thus, changes come slowly, and curriculum revisions of any kind must await the disposition of the teacher.
In the classroom: An approach to subtraction using easy facts
Authors: Irwin Evelyn S.
Published: October 2020

Can the number of subtraction facts which must be learned in the primary grades be reduced? If so, must they be teamed later, or is it possible to perform subtraction without learning the facts with minuends greater than 10 that we presently consider essential?
New numerals for base-five arithmetic
Authors: Hilaire Paul , Westphal Walter
Published: October 2020

It may be of interest to teachers to discover that much can be learned (and perhaps taught) about our base-ten number system by considering a system having a base other than “ten”. Some of us have been teaching a long time and perhaps have forgotten how difficult the concepts of mathematics arc for the young child. To illustrate this difficulty, let the teachers be exposed to a set of number symbols they have never seen before and then try to work with them. The usual presentation of other number bases employs Hindu-Arabic numerals. This results in much difficulty because one must “unlearn” most of the number facts previously known. Children are usually uncomfortable in the presence of a statement like “4+4 = 10”, a true statement in baseeight arithmetic. The authors of this paper have developed a system of numerals which makes the transition from base-ten to another base less precipitous and perhaps more graciously understood. These numerals have had strong intuitive appeal to children. A child seems to come out of the experience with a better understanding of a number system.
Making statistical maps
Authors: Archbold John C.
Published: October 2020

How many elementary students become involved to some degree with statistical maps? Probably most of them do, since density and distribution concepts taught in the arithmetic program are supplemented with statistical maps found in social studies texts. A distribution or statistical map may convey information about resources, economic activities, or population data. In the past, excessive use of pictorial detail sometimes detracted from the effectiveness of these maps. How often the student bas been bewildered by a clutter of steers' heads here, a shock of corn there, and a general hodgepodge of symbols! Three standard cartographic techniques used in the construction of distribution maps will be illustrated and discussed here. There are many others—one-, two- and three-dimensional.
A reply to Mr. Rodman's critique
Authors: Brownell William A.
Published: October 2020

I am very glad that Mr. Rodman is publishing his critique of my research, and I am happy to have this opportunity to say so in print. I share his conviction that disagreements should be aired, as fully with respect to research as with respect to educational problems in general. To answer Mr. Rodman's objections will, to my regret, take considerable space in this journal, perhaps more space than can be justified.
Another reply to Mr. Rodman
Authors: Andrews E. E.
Published: October 2020

Mr. Rodman's article and his reported results give every indication of an intelligent and imaginative arithmetic program in his school. There is, in fact, little evidence of “professional disagreement” between our article and his comments.
In the classroom: Pop goes an idea
Authors: Swenson Esther J.
Published: October 2020

Teachers and children in modern elementary schools are well acquainted with such manipulative aids as wooden blocks, tongue depressors, slips of paper or cards for pocket charts, and various other objects which may be bundled to make basic collections to fit base and place concepts. These work very well for smaller groups such as tens or sixes or eights. Most of them become cumbersome when the base is squared. A “base of bases” such as “ten tens” represents a rather large collection to hold or to pick up easily.
A lesson on absolute value
Authors: Cohen Donald
Published: October 2020

This lesson evolved as a result of the author's desire to develop the students' readiness for an algebraic approach to sequences and limits. It was taught to students in Grades 4-7, and to college freshmen.
Zero difficulties in multiplication
Authors: Harvey Lois F. , Kyte George C.
Published: October 2020

The purpose of this research was to determine the effects of an instructional program designed to isolate specific errors involving zero in multiplication of whole numbers and to deal with them by remedial teaching. The study was based on a sampling of 15 sixth-grade classes representative of the various community conditions in a city school system. Since all classes were heterogeneous in nature, the sample of more than 500 students was considered adequate for this study. There were 517 students included in the 15 classes.
Formal preparation for early childhood arithmetic
Authors: Bravo Anne R.
Published: October 2020

One fact that stands out clearly in early childhood education is that children will be coming to school at an earlier age. The logical reason for this is the emphasis being given to the total education for disadvantaged children. This program calls for school entrance at ages three, four, and five so that these young children can acquire an adequate background for the intensive educational programs which await them. It is also true that children of working parents are coming to school earlier and in larger numbers than ever before. Since this trend will surely continue, teachers must plan formally for these newcomers. Formal preparation does not entail a listing of number facts, but it does require an understanding of the number experiences that children are having and have had before coming to school.
In the classroom: A different way of subtracting
Authors: Dye David
Published: October 2020

Some procedures which are taught in the study of mathematics are refined for the student in subsequent years. For example, division is taught in modern programs as repeated subtraction, and after the students understand the division process, it is altered to shorten the computation. Again, in secondary mathematics, shortcuts and computational aids are introduced to lighten the load for the pupil.
The informal-intuitive versus the formal-deductive approach to learning
Authors: Clark John R.
Published: October 2020

In elementary school mathematics education one of the many problems to be resolved is that of the formal-deductive versus the informal-intuitive approach to learning.
Evaluation of an accelerated arithmetic program
Authors: Jacobs James N. , Beery Althea , Leinwohl Judith
Published: October 2020

In 1959–60 the Cincinnati Public Schools initiated a five-year experiment of arithmetic acceleration in selected schools in Grades 3 through 6. The plan was to provide a more challenging program of arithmetic instruction for capable pupils.
Rhymes—modern math style
Authors: Smith Frank
Published: October 2020

Elementary teachers throughout the nation are experiencing changes in arithmetic materials and methods. Through reading, workshops, and college courses they are learning more about systems of numeration.
… and now synthesis: a theoretical model for mathematics education
Authors: Glennon Vincent J.
Published: October 2020

The title of this paper is evidence of my long-standing concern for the third aspect of the Hegelian triad—thesis, antithesis, and synthesis—as it applies to mathematics education. Hegel, we recall, interpreted societal change as a reconciliation of opposites, a thesis and its antithesis, into a higher union or integration which he called synthesis. In a very real sense, we can use this conceptual model to interpret the changes that have taken place in school mathematics programs in recent years (and even over the past several thousand years) as man has attempted to select and transmit this segment of the culture.
Structure and the algorisms of arithmetic
Authors: Lowry William C.
Published: October 2020

We teach children the characteristics of our numeral system: that this system utilizes the idea of place value; that it has a base of ten; that it has ten symbols; and that with these characteristics, certain grouping and operation symbols, and the principles referred to below we can write numbers large and small in a great variety of ways. We teach children that certain principles govern the operations on these numbers: closure, commutativity, associativity, distributivity, under appropriate operations. We do not always use the commonly accepted mathematical names of these characteristics and principles the first time we refer to them; sometimes we use other names more appropriate to the maturity of the child.
Algebraic concepts in the elementary school
Authors: Laroe Rachael A.
Published: October 2020

The traditional subject matter of algebra has been described as generalized arithmetic; so, the basic concepts related to the rational number system are begun in elementary arithmetic and continued in the study of algebra. Algebra can really be considered a second step in abstraction rather than a different subject.
Statistics made simple
Authors: Grass Benjamin A.
Published: October 2020

This project began in a very simple manner. In our school system district-wide examinations are prepared and given twice each year in every subject area. One Thursday morning in January, 1963, every fourth-grade child took his arithmetic examination. Since I knew well in advance of this date that the children would have had their fill of hard-core mathematics, and since the Farmingdale public school system is departmentalized, I felt it unwise to give my afternoon classes a regular lesson in mathematics. I had, therefore, prepared to read to one of my sections a story from The Mathematical Magpie, compiled by Clifton Fadiman. The name of the story was “Star Bright” by Mark Clifton. The story deals with a very young child, three years old at the start, whose obvious mental prowess upsets her father. She is more than capable; she is more than a genius. Through the use of extrasensory perception she is able to read people's minds and to transport herself into other time periods. She does this with a series of interesting geometric shapes and topological ideas, such as the Moebius strip, Klein's bottle, and the tesseract. The vocabulary of this story was quite stimulating to this group, and we started to discuss the meaning of extrasensory perception. I told the children that basically it meant the ability to “see” without using your eyesight.
In answer to your questions—
Authors: Fehr Howard F.
Published: October 2020

“What is a set?” It is generally recognized that the first step in the teaching of arithmetic in the elementary school is to develop an understanding of the counting or whole numbers, that is, the numbers 0, 1, 2, 3, and soon. At the university level, there are two prevalent ways of developing the theory of these numbers. The one is to assume the numbers exist as a set, induce on them five axioms, referred to as Peano's axioms, and then by pure logic develop all the properties of and operations on these numbers. In one sense this corresponds to the rote method of teaching in the elementary school where the children are taught to recite the names of these numbers in a given order, and then taught how to use them, that is, to manipulate the numerals naming these numbers according to certain rules and procedures.
Adding by endings: some important considerations
Authors: Rivera Emilio
Published: October 2020

Pupils should perform accurate mental computations in an efficient mannrr. To develop these skills in our students, we teachers have been attracted by various approaches to mental arithmetic. We have recognized that very often we can, with little effort, help our children perform computations mentally which we ourselves find difficult. In some cases this might involve t he teaching of “tricks,” but very often the road to ease in computational skill is clear and simple. It is surprising, for example, to witness the amazing interest (and improvement) which children show when we analyze a prosaic operation such as column addition from the point of view of adding by endings. We select this particular operation because it is perhaps the most common mental operation performed in social situations. Accurate mental addition of columns is a valuable skill which children can readily appreciate.
An additive numeral system related to place value
Authors: Vitt Goldie E.
Published: October 2020

The concepts relating to place value in our decimal system involve what are probably some of the most difficult learning experiences that the primary child encounters. While many young children are able to tell “how many ones,” ”how many tens,” “how many hundreds” in a numeral, the teacher sometimes has reason to suspect that this skill is a superficial one, and that many conventional methods of presenting material intended to teach place value do not promote real internalization of the concepts involved or provide realistic mean for evaluation of learnings.
The mathematics of a five-year-old girl
Authors: Sueltz Ben A.
Published: October 2020

This report is based upon the personal observation of a girl by her father for a period of three weeks at the time of the child's fifth birthday. All of the experiences described originated with the child with no suggestion or deliberate creation of mathematical situations by the parent. It should be pointed out also that during this interval of three weeks the father was not at home during normal busines hours and that the child had not yet started kindergarten. Furthermore, there had been no attempt made to develop mathematical understanding in the child except as the child asked questions. This child led a normal life in a small town and had no older brothers and sisters. Her intelligence is believed to be at least average.
Dividing by a fraction
Authors: Koenker Robert H.
Published: October 2020

If we are to teach for understanding, we must do more than have children memorize rules. There are several meaningful approaches to dividing by a fraction. A teacher should use at least one of these approaches and possibly two or three, depending upon the maturity and background of her pupils.
Game to review basic properties and vocabulary
Authors: Condron Bernadine F.
Published: October 2020

I want two different numbers which are greater than 5 but less than 10. Use these given numbers to fill in the corners indicated.
Focal point: Project Idaho
Authors: Kay Richard , Woodby Lauren G.
Published: October 2020

Teachers of mathematics at all grade levels face the problem of upgrading their knowledge in mathematics because the content now being taught includes so many idea. 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.
Math—not new
Authors: Hardin Blanche C.
Published: October 2020

How do I begin to explain the thinking process of the Second-grade student who figured the example shown above? Perhaps I should briefly recount the mathematical background that our school curriculum prescribes, which, no doubt, is similar to that prescribed in most schools in the United States: to take any mystery out of numbers and to show the reasoning, the “sense,” behind the mechanical processes.
Pegboard geometry
Authors: Smith Lewis B.
Published: October 2020

Geometry has now won for itself a place in the elementary school mathematics curriculum. Although emphasis varies from publisher to publisher, there is no question that geometry is included in each major series. The Cambridge Report urges a more distinguished place for it in the curriculum of the future.
Patterns in arithmetic: a three-year report
Authors: Fred Weaver J.
Published: October 2020

Many of our readers will recall thai the April, 1963, issue of The Arithmetic Teacher included a report of a visitation team to the Wisconsin School of the Air “Patterns in Arithmetic” [PIA] project, directed by Professor Henry Van Engen of the University of Wisconsin.1 Of current interest is a three-year report covering the work of this project in the Milwaukee public schools, Grades 4–6, for the 1960–61, 1961–62, and 1962–63 school years.2
Numeration—a fresh look
Authors: Peterson Wayne
Published: October 2020

Numeration is the topic most closely associated with the “new mathematics” in the minds of many. This is evidenced by the rash of articles which has appeared in the arithmetic teacher and elsewhere.
Modern mathematics in a toga
Authors: Johnson Paul B.
Published: October 2020

A world famous physicist spoke on mathematics and science in the modern age at a recent conference of mathematics teachers. At the close he called for questions from his more than one thousand listeners. “Ask any question you like.” A tense silence gripped the auditorium. After some moments a voice rose, “If no one else has a question, I've always wanted to know something. How do you multiply in Roman numerals?”
Geometry concepts in Grades K-3
Authors: Felder Virginia
Published: October 2020

If nothing else, teach children geometry to open their eyes—to help them sec patterns, to observe symmetry. Look at your hands. What do you see? I see two symmetric objects that are reflections of each other, one-to-one correspondence between parts, such as fingertips, finger joints, palms—maybe not the scars!
Moot mathematics
Authors: Muller Adelyn
Published: October 2020

A great discrepancy exists in the levels of mathematics taught in elementary schools across the nation. Surely, where a problem is acute, one would expect to find the greatest endeavor. Unfort unately, this is not true. However, one does find what seems to be a universal fear among elementary teachers of all modern mathematics programs. It was while trying to alleviate the continually widening breach between the necessary badeground in yesterday's and today's mathematics for elementary teachers that a rather unique operation developed, which we have tabbed Operation Stopgap.
Operation bottle caps
Authors: Bacon Marjorie
Published: October 2020

For many years I have used counting disk, popsicle sticks, tongue depressors, and various other items as counting devices. My first-grade children loved to write on the disks, chew them, and lose them on the floor. The sticks and depressors proved to be wonderful building objects, things to flip, and creative design material.
Minutes of the Annual Business Meeting
Authors: Karnes Houston T.
Published: October 2020

The Annual Business Meeting was scheduled to follow immediately after the 9:00 A.M. general session, during which President Bruce E. Meserve presented a paper entitled “The NCTM, Its Growth and Growing Pains.”* Contained in this paper was a fine survey of the work of the Council during the past year and indications for important projects to be done in the future. Donald Marshall, Vice-President, presided at this general session.
Federal funds for the improvement of mathematics education
Authors: Scandura Joseph H. , Johnson Donovan A. , Thomason Gladys M.
Published: October 2020

Congress has authorized a large sum of money for the improvement of education by passing the Elementary and Secondary Education Act of 1965. Since these funds will be available for projects in all subject areas, teachers and mathematics educators should make every effort to see that school planning makes provision for improved mathematics programs. This can be done by providing school and state leaders with the necessary information and statements of need which can be incorporated into their requests for funds.
Logic In the construction of magic squares
Authors: Keepes Bruce D.
Published: October 2020

Both in Adkins' article1 and in Hewitt's article2 on the construction of magic squares, the squares were constructed by mechanical means. The student only needed to memorize a set of rules to construct the magic squares. Another way of approaching the problem of constructing magic squares is through the application of logic and some principles of our numeration system. This method should contribute to the student's understanding of mathematics.
Nominations for the 1966 election
Authors: Peak Philip , Jones Philip S. , Hawthorne Frank , Lewis Eunice , Hess Adrien , Lowry William , Lund Jens L. , Allen Frank B. , Ingraham Mary L.
Published: October 2020

The following named persons have been nominated for the indicated offices in the National Council of Teachers of Mathematics by the Nominations and Elections Committee. The Committee's report was approved by the Board of Directors at its August, 1965, meeting in Vancouver, British Columbia. Biographical material and photographs of these nominees will be published in the January, 1966, issues of The Arithmetic Teacher and The Mathematics Teacher.
Editorial comments: As we read
Authors: Hamilton E. W.
Published: October 2020

Just four years ago we published an article by Dr. John Clark which reviewed a decade of experiment in arithmetic and pointed out some of the major aspects of t he emerging program. His long view of the educational process is always worth considering. This time he lists the changes in philosophy and psychology which over a span of six decades lead to the adoption of extreme positions and sometimes extreme reforms. There is in this review hope for the younger generation, some comfort for the older generation, and much guidance for t hose in their prime as they make the decisions of the day. And as usual with Dr. Clark, there is a clever story included.
Structure—key word of the sixties
Authors: Neufeld K. Allen
Published: October 2020

“ … The curriculum of a subject should be determined by the most fundamental understanding that can be achieved of the underlying principles that give structure to that subject.”*
A substitute teacher in mathematics
Authors: O'Neill Connie
Published: October 2020

Substitute teachers are in a class by themselves, in more ways than one. True, they are well paid, but their task is not an easy one—that of keeping learning running as smoothly as possible while the regular teacher is away. They work under the real handicap of one- or two-day stands, and unless they remain for several years as substitutes in the same system, they have little status in the classrooms. Status is something that quite naturally belongs to the regular teacher. To most pupils, the sight of a substitute promises a holiday, and the idea of work takes flight.
They are not “magic” number cards
Authors: O'Donnell J.R.
Published: October 2020

Today “magic” number cards can be found in a child's bubble-gum package or an executive's free gift package from an airline as he boards a plane. These cards are quite fascinating to the bubble-gum set as well as the busy executive.
Student acceptance of the new mathematics programs
Authors: Feldhake Herbert J.
Published: October 2020

Mention of the advantages and disadvantages of the new mathematics programs can arouse heated discussions in many segments of our society. Because of their unfamiliarity With the contents of these programs, many parents have been distrustful of these changes. Many reports about the success of the new mathematics programs have been in terms of measures of student achievement. Rarely have investigations been designed to measure the feelings and attitudes of the students toward these new programs. Rather, only a few teachers' subjective opinions of students' likes or dislikes have been mentioned about this aspect of the new mathematics.1 The discrepancy in these opinions confuses many parents about the usefulness of the new mathematics and makes the parents suspicious of its value.
Ten questions
Authors: Corle Clyde G.
Published: October 2020

The following questions were submitted by the elementary teachers of Clinton County, Pennsylvania, after a series of workshops in elementary school mathematics. The teachers asked for a written discussion of ten of the questions which appeared to be of unusual interest to the group.
A suggested plan for interclass grouping in arithmetic
Authors: Adams Anne H.
Published: October 2020

The use of grouping has achieved widespread use in the area of reading in the elementary school. This concept has also been advocated for all subject matter areas; however, because of large classes, limited time, and the number of content areas to cover, grouping has not always been possible in every subject
Fraught with naught
Authors: Muller Adelyn
Published: October 2020

Recently zero has become a protagonist in the area of mathematics in elementary schools. Teachers view with alarm any attributes of zero besides its placeholder function. Granted, if this were the periphery of zero's activity it would still warrant meritorious recognition.
Greatest common divisor and least common multiple
Authors: Zink Mary Hart
Published: October 2020

A simple and effective way of finding the greatest common divisor and the least common multiple of two or more numbers makes use of and strengthens the ideas of union and intersection.
The continuing work of the Cambridge Conference on School Mathematics (CCSM)
Authors: Hilton Peter
Published: October 2020

A group of 25 people with a strong interest in mathematics gathered together in Cambridge in the summer of 1963. The group consisted of university mathematicians representing various special fields in pure and applied mathematics, mathematical statisticians, experts in curricular problems in mathematics, industrial mathematicians, physicists, and chemists.
Factors relating to achievement with selected topics in geometry and topology
Authors: D'Augustine Charles H.
Published: October 2020

Groups responsible for the development, analysis, evaluation, and restructuring of topics in the elementary school mathematics curriculum have proposed the inclusion of many topics in geometry and topology heretofore not an integral part of elementary school mathematics curriculum. Two of these groups are the School Mathematics Study Group (Elementary) and the Greater Cleveland Mathematics Project. Both of these groups have produced materials which include many topics in intuitive geometry and topology.
Letter to the editor
Authors: Fitch Sally H.
Published: October 2020

We have been working on measurement in our sixth-grade class, comparing the measures in feet and in meters, in Fahrenheit and centigrade, and in grams and ounces. The class decided that we would use only decimal-baaed measurements in the classroom, regardless of what we had to do in the outside world.
Children's attitudes toward arithmetic
Authors: Abrego Mildred Brown
Published: October 2020

In the educational system of the United States the education of the whole child has been important. Home environment, school environment, heredity, and good health-physical and mental-have played their part in the maturation of the student. All of these influences can aid learning, but without the right attitude the child's full potential of growth in knowledge cannot be realized.
Number line multiplication for negative numbers
Authors: Coon Lewis H.
Published: October 2020

The number line, a traditional tool in arithmetic classes, continues to find greater acceptance in some of the new mathematics programs. The logic behind some of the applications of “cricket- Jumps,” vectors, or hops1 to and fro on the line seems to give an elementary school child a model suited to his learning level. Thus the jumps of the earlier grades are replaced by moves in the intermediate grades, which are replaced by vector representations in the upper grades. Figure 1 shows the addition of 3 to 2, written 2+3.
Inductive teaching vs. deductive teaching
Authors: Rupkey Brother John Bosco
Published: October 2020

Most arithmetic teachers throughout the country today are involved in modern mathematics programs. They can hardly avoid it, since modern mathematics texts have been widely adopted in their school systems. However, the mere possession of a modern text by the teacher and students does not mean that the objectives that underlie the modern mathematics movement in the schools arc being realized.
A proclamation of ideas
Authors: Kavett Phyllis F.
Published: October 2020

Students at Newark State College in the course entitled, “Mathematics in the Elementary School,” made a display of mathematics teaching aids in a well-traveled area of their college. This display was an activity which developed from one of the most popular assignments in the course.
A mathematics program for slow learners at the junior high level
Authors: Lerch Harold H. , Kelly Francis J.
Published: October 2020

A pproximately one-third of the students in Grades 7 to 12 arc unsuccessful in meeting the requirements of our educational system. There is evidence that these students discontinue schooling before graduation and repeat this failure experience in the vocational and civic community. Two important factors contributing to this lack of adjustment between the school and a sizable percentage of its student population are the inappropriateness of the students' educational experiences and a lack of school climate suitable to their personal development. A significant percentage of these failure-prone students may be characterized as “slow learners.” The verbal learning capacity of these students is above that of students selected for special programs for the mentally ret arded, but below the average. They lack the basic educational skills and are handicapped in the development of healthy personal and vocational characteristics. In many school situations, slow learners demonstrate little change in achievement from year to year, and their lack of success is frequently associated with early school leaving.
African mathematics program (continued)
Authors: Weaver J. Fred
Published: October 2020

Focal Point for the October, 1965, issue of the arithihetic teacher was devoted to information about the African Mathematics Program (XII, 472–480), the first part of the African Education Program to begin its work.
Mathematics of measurement
Authors: Smart James R. , Marks John L.
Published: October 2020

Measurement is a major topic in elementary school mathematics, appearing in every grade from one through eight. The importance of measurement is indicated not only by its wide application in everyday life, but also because its basic ideas are studied in mathematics classes through the graduate level of the university. Despite the wide applicability of measmcment and its continuous appearance in the curriculum, there is a great gulf between how the practical man measures, reports his measurements, and judges their preciseness, and the mathematician's convention for dealing with measurements.
Concept determines process–a look at the fraction symbol
Authors: Hannon Herbert
Published: October 2020

Most of our everyday life experiences include some aspects of quantity. Such quantitative relationships must be identified, measured, and expressed in suitable number language if a satisfactory interpretation of their interrelation with the other aspects of the total situation is to be made.
A survey of arithmetic intraclass grouping practices
Authors: Brewer Emery
Published: October 2020

Intraclass grouping for reading instruction in the elementary school is common practice.1 Individual pupil differences in arithmetic learning ability are in all probability as great as such differences in ability to learn to read. So a question arises about the extent of intraclass grouping for arithmetic instruction. If such grouping is not common practice, what are the reasons?
Research on mathematics education, Grades K-8, for 1965
Authors: Fred Weaver J.
Published: October 2020

This is The Arithmetic Teacher's ninth annual research listing, and the tenth in a series that covers the 15-year period beginning with the calendar year of 1951.1
An example of informal geometry: Mirror Cards
Authors: Walter Marion
Published: October 2020

The need for informal geometry, especially in the earlier grades, is being recognized by educators, psychologists, and mathematicians. The Mirror Cards were created by the author to provide a means of obtaining, on an informal level, some geometric experience that combines the possibility of genuine spatial insight with a strong element of play.
Manipulative devices
Authors: Hamilton E. W.
Published: October 2020

Most mathematics teachers realize that there has been a proliferation of devices for use in the teaching of arithmetic. Manipulative devices have shared in this proliferation; and the commercial catalogs list all sorts of things, and many models of some of the more common devices. It therefore seems hopeless to try to list them all and to comment upon their merit or lack of merit. Rather, it seems appropriate to define manipulative aids rather narrowly and then deal with representative materials. This method should help the reader to develop discrimination in choice and use and to become alert to the advantages of a good aid, and to the pitfalls that can trap the user—both in the limitation inherent in the devices and in thoughtless use of them.