bluuebunny/crossref_metadata_2025 huggingface dataset:
An iconoclastic elementary school mathematics program
Authors: Earl Walter
Published: October 2020

There are many different approaches to the teaching of mathematics. They range from the formal, axiomatic type of presentation to the strictly utilitarian emphasis. The first is concerned with the logical structure of mathematics and seeks to transcend any particular relevance to ordinary experience. The second approach concentrates upon the development of a manipulative skill with some important mathematical operations. Most of the teaching at the junior high school and senior high school levels would today probably fall somewhere between these two extremes. In any of these methods, however, the teacher frequently finds that he is giving little attention to the development of mathematical intuition and what might be called a “freedom from the discipline.” It is the purpose of this paper to describe a program that attempts to emphasize and enhance the students' intuitive abilities.
Building “computers” for nondecimal number systems
Authors: Rabinowitz Frederick R.
Published: October 2020

A rash of comic-strip entries provided an unexpected form of motivation for a Philadelphia sixth-grade class. Mrs. Katharine McFadden's class at the John H. Webster School used them as a springboard for studying various systems of numeration. The class had been introduced to the binary system early in the year, had achieved very well, and had enjoyed the hours working with it. But when the cartoons on “new mathematics” appeared later in the year, and interest again became high, the pupils sought additional outlets for their interest in numeration systems.
Editorial comments: As we read
Authors: Inskeep James E.
Published: October 2020

A good teacher knows his children. A good teacher with experience knows intuitively how well his children learn. What do you as a teacher have to say about the education of your youngsters?
A rationale in working with signed numbers—revisited
Authors: Cohen Louis S.
Published: October 2020

In the November 1965 issue of The Arithmetic Teacher the article “A Rationale in Working with Signed Numbers” was presented. The article concerned a technique for teaching addition and subtraction of signed numbers. The present article will cover the multiplication and division of signed numbers.
A unit of measure
Authors: Mcclintic Joan
Published: October 2020

Ask a group of children for help in solving a problem, and little wheels begin turning.
A teaching aid for signed numbers
Authors: Pratt Edna M.
Published: October 2020

Each year the time comes when it is necessary to illustrate the multiplication of signed numbers. There are many approaches to helping the student develop insight and skill. Some of the more common are multiplying two negative numbers using the number line, 1 * an inductive approach,2 the modern mathematical proof approach,3 the statement approach,4 or the model approach,5 but some young mathematicians still have a hard time understanding these ideas. The following is a very effective device which may be used to help students discover and then reinforce the operation of multiplication of signed numbers.
The digit it is!
Authors: Mann Lee
Published: October 2020

I was one of a negative-minded in-service group recently foisted off on a positive young college professor who was to enlighten us on the new modern math for teachers. Light did flash, for he was very knowing, as commutative as anyone I ever met and definitely mod. In fact, every word he used was so new that he had to make up extras as he went along, just to keep abreast in his field.
A study of the interaction of socioeconomic groups and sex factors with the discrepancy between anticipated achievement and actual achievement in elementary school mathematics
Authors: Unkel Esther
Published: October 2020

America is proudly committed to the education of all the children in our nation. This results in challenging problems for the classroom teacher who wishes to give the best instruction to every child. Children of different socioeconomic backgrounds may be found in the same classroom, but there is little information as to the effect of socioeconomic status on achievement in elementary mathematics. Arithmetic tests reveal a child's achievement; but they do not indicate the degree to which he is measuring up to his potential with regard to his actual achievement in arithmetic, his chronological age, his grade level, and his mental ability. Boys and girls receive the same instruction, but there exists only scanty and often conflicting evidence as to whether one sex is able to attain higher achievement levels generally in mathematics than the other sex. Little information is available concerning the ability of children in varying socioeconomic groups to achieve in the two different kinds of learning in elementary mathematics: problem solving and computational work. Do socioeconomic status or sex differences have an effect on the ability of children to achieve in problem solving and computational work? The current concern with problems of education in culturally deprived areas increases the need for more information regarding this question.
The Witch's Best Game
Authors: Smith C. Winston
Published: October 2020

As students encounter three-digit subtraction, many who previously have “mastered” two-digit subtraction falter, not only because of the complex regrouping required in some problems, but also because of the wider selection of regroupings to consider. While two-digit subtraction requires the student to choose between two alternatives—no regrouping or regrouping from tens to ones—three-digit subtraction can call for (1) no regrouping, (2) regrouping from tens to ones, (3) regrouping from hundreds to tens, or (4) regrouping from hundreds to tens and from tens to ones. This range of choices, when linked with the understanding and use of basic subtraction facts and regrouping itself, creates for some a task that is very perplexing.
FRIO, or FRactions In Order
Authors: Drizigacker Rowena
Published: October 2020

FRIO is a game that will help students understand the relative value of common fractions. It can be played and enjoyed by third graders as well as the uppergrade students and even by their parents. It can be made as simple or as difficult as desired.
Visual aid for multiplication and division of fractions
Authors: Olberg Robert
Published: October 2020

One of the most graphic ways of presenting the concept of multiplication and division of fractions is through the use of subdivided rectangular regions.
Now try this–in multiplication
Authors: Junge Charlotte W.
Published: October 2020

1. When the multiplier is a composite number: a) Separate the multiplier into 2 (or more) factors. b) Multiply the multiplicand by one of these factors, and that product by the other factor(s).
Report of the Nominating Committee
Authors: Allen Frank B. , Beckmann Milton , Hess Adrien L. , Ingraham Mary L. , Longshore Blanche , Lowry William C. , Lund Jens L. , Peak Philip , Schult Veryl
Published: October 2020

The Committee on Nominations and Elections presents its nominees for offices to be filled in the 1967 election. The president-elect is elected for one year, after which he serves two years as president and, finally, one year as immediate past president. The vice-presidents are elected for a term of two years, whereas the directors are elected for a term of three years.
Geometry alive in primary classrooms
Authors: Black Janet M.
Published: October 2020

Planning an experimental course in geometry for forty-two Grade 3 students was quite a challenge.
Some problems in geometry
Authors: Sullivan John J.
Published: October 2020

There are many interesting problems that usually begin with questions like “What is the shortest path …?” or “What is the greatest area …?”
Now try this—in multiplication
Authors: Junge Charlotte W.
Published: October 2020

A fter developing understanding and competence in multiplication, able students may find these “contractions” interesting and challenging. Some students may be content to learn to use these shortcuts, others may want to find explanations and develop a rationale. Both are given.
Extent of implementation of CUPM Level I recommendations
Authors: Fisher John J.
Published: October 2020

Approximately five years have passed since the publication of the recommendations of the Committee on the Undergraduate Program in Mathematics (CUPM), Panel on Teacher Training, of the Mathematical Association of America.1 This report recommended that the preservice preparation of elementary school teachers include, as minimum mathematical education, twelve semester hours (four courses). These recommended courses, known as the CUPM Level I, were to include (1) the structure of the real-number system and its subsystems (six semester hours), (2) algebra (three semester hours), and (3) geometry (three semester hours).
How effective are modern mathematics workshops?
Authors: Creswell John L.
Published: October 2020

Many changes have come about in most subject matter areas of the elementary school curriculum during the last ten years. One of the most profound of these has been in the area of elementary school arithmetic, referred to by such phrases as “modern mathematics,” “the new mathematics,” etc. These changes have created a tremendous problem in teacher training and retraining and have resulted in additional requirements in mathematics for elementary school teacher certification.
What is “readiness”?
Authors: Kaplan Jerome D.
Published: October 2020

One often hears the word “readiness” in the discussions or conversations of elementary or preschool teachers. A common remark is that some child is “not ready for number concepts.” The use of “ready” in this manner is vague: it does not indicate exactly what is meant by “being ready.”
The largest number that can be written in any base
Authors: Monsour Noah
Published: October 2020

Recently, in our seventh-grade accelerated mathematics class, we were discussing the largest number which could be written using only three digits. Many of the students thought the number was (999), until we discovered a three-digit number that was so large that, if we used one inch to write each digit, the number would measure over 12,000 miles; then later, we discovered a three-digit number even larger.
Instagrams, or instant histograms
Authors: Manheim Jerome H.
Published: October 2020

Conventionally, data collection and data display are separate operations. The former is often accomplished by a “tally sheet.” These “raw” data are then summarized for presentation in one form or another. Below is a table typical of data collection and summary that can be used to construct a histogram (or equivalently a bar graph).1
A model for arithmetic of signed numbers
Authors: Luth Lois M.
Published: October 2020

To enable elementary school children to perform the arithmetic of signed numbers easily and correctly, I use the story of “Hy in the Sky.” Hy is a man living in a house in the sky to which are attached an undetermined number of balloons and sandbags. A helicopter, from which Hy can buy more balloons and sandbags, comes each morning.
A fifth-grade student discovers zero
Authors: Sigurdson S. E. , Boychuk Halia
Published: October 2020

During a recent lesson, a classroom of Grade 5 students realized a basic concept in mathematics and science, a concept not often achieved by adults. The arbitrary nature of zero is a point not even mentioned in the traditional classroom; however, these pupils (not exceptional children) “discovered” the idea themselves during a session conducted in accordance with some in-service materials for elementary school teachers prepared by Dr. Robert Davis of the Madison Project.1
Offerings in mathematics of NET stations, 1965-67
Authors: Clary Robert C.
Published: October 2020

During the school year 1965/66 the Film and Television Committee of the National Council of Teachers of Mathematics became interested in preparing a brochure of offerings in mathematics at all levels via television.
Does base four bewilder you?
Authors: Schupbach Sister Joseph M.
Published: October 2020

Are you completely confused in your understanding or teaching of base four? For the low-average fourth-grade students of Our Lady Queen of Peace School, study of this base has been an enjoyable learning experience.
Mrs. Murphy's pies–an introduction to division by fractions
Authors: Freeman William W. K.
Published: October 2020

I'm going to tell you a story,” said the teacher. “Last night T went to a church fair.
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

This issue of The Arithmetic Teacher completes our publication schedule for the current school year. The next issue of your magazine should be delivered to you early in October.
The metric system in the elementary grades
Authors: Helgren Fred J.
Published: October 2020

We are all interested in a better education for our children, a better education in less time and with less expenditure of effort and of the teacher's time. Educators say that they are not emphasizing the study of the metric system until industry makes the change, and industry says that it cannot change because all the help is educated in the use of the English system—a vicious circle if there ever was one; for the metric system has been the legal system of measure for 100 years, is used extensively in this country, is destined to become the only system of measure in this country, and is the language of measure throughout the world.
Multiplication—repeated addition?
Authors: Bechtel Robert D. , Dixon Lyle J.
Published: October 2020

The elementary school teacher of today needs to have a comprehensive view of eleme ntary mathematics. The present emphasis on the “new” mathematics requires an understanding of number systems such as the natural number system, the whole number system, the rational number system, and the real number system, together with an understanding of some elementary concepts from geometry. This understanding of mathematics can no longer be limited to specific areas covered at one level in graded material, but should encompass the structures of different number systems. One must recognize the role of a specific topic in mathematics in relation to the overall structure of the systems under consideration. Failure to do this often leads to confusion for a student who must relearn or radically alter a previously learned mathematical principle. Such a situation should be avoided.
President's report
Authors: Johnson Donovan A.
Published: October 2020

The past year has been a year of much activity for the NCTM. To improve mathematics education, financial and human resources have been involved in publications, conferences, committee meetings, projects, and many other services for members and for Affiliated Groups. We have not solved all of our problems, and some things have been left undone, but many Council activities have been very successful.
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

We have sensed a need to help writers determine the type of mathematics research material that we should like to publish.
Multiplication of integers
Authors: Hollis Loye Y.
Published: October 2020

One of the major purposes in teaching mathematics is to develop understandings through an exploration of the “Whys?” of mathematics. Most teachers realize that presenting a rule to the learner without exploring its logic and/ or developing its mathematical justification does not usually lead to the development of basic understandings. This knowledge has motivated educators to develop new and improved teaching strategies. One of the most promising strategies involves the development of a situation that will enable the learner to discover the desired concept. Although it may be an unrealistic objective to have the learner discover all the mathematical rules, the more he is involved in this type of activity, the greater will be his depth of understanding of mathematics.
Measures–common and uncommon
Authors: Slaninka Ardys
Published: October 2020

Do you know what a ruple, a drachm, a hogshead, a hand, a span, or a cubit is? If you know, do your students know?
Don't sell short the distributive property
Authors: Mcdougall Ronald V.
Published: October 2020

Can you prove addition of fractions to your students? If your answer is yes, with the thought of dividing the proverbial pie into equal-sized sections and showing their sum, then you are not proving the concept but giving an example. Too frequently, examples are passed off as proofs for many of our mathematical concepts.
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

Many of our readers have noted that we try to emphasize a certain topic each month. This procedure innucnces our selection of articles, and it sometimes delays the publication of a ma nuscript. However, reports from our readers indicate that they a re pleased with the procedure, and we plan to continue it.
Quantitative aspects of at the Primary stage
Authors: Newbury N. F.
Published: October 2020

It is generally agreed that the learning of Science by children at the Primary stage can be profitable if their everyday experiences involving observation, discussion, and recording are fully utilised. Piaget and experienced teachers may, however, disagree on the age at which any precise understanding and appreciation of Science, logical argument, and scientific attitudes are possible. The long-term aim should certainly include the testing of hypotheses and the formation of general principles.
A laboratory plan for teaching measurement in Grades 1–8
Authors: Swart William L.
Published: October 2020

Give a layman the job of teaching an Australian aborigine how to weigh kangaroo steaks, and the layman's request for equipment will probably include a set of scales and various objects to weigh. When we, as professional teachers, are assigned a similar task with children, we request a book with pictures of scales and pictures of weights.
Functions
Authors: Esty Edward
Published: October 2020

Some of the recent elementary school arithmetic textbooks introduce functions, a topic formerly appearing no earlier than in high school. The University of Illinois Arithmetic Project has long used functions (called “jumping rules” by the Project) in classes for elementary school children.
Patterns and creative thinking
Authors: Perkins Ruth M.
Published: October 2020

I think three objects will go into three boxes in nine ways, Miss Tucker,” said Jim.
The plight of a child
Authors: Dickson Judy Ticknor
Published: October 2020

Sharon is a third grader who is doing very poorly in mathematics. Her mother felt she needed a tutor, so I took the job. Sharon was quick and accurate with her basic facts of addition. Her troubles started when she became involved in adding numbers where regrouping was involved.
Designing research studies in elementary school mathematics education
Authors: Riedesel C. Alan , Sparks Jack N.
Published: October 2020

A recent study of journal-published research articles concerned with elementary school mathematics revealed that the majority of the studies published between 1900 and 1965 suffer from errors concerned with research design.1 Three facets of research design were studied: How well docs the design answer the research question? (50.7 percent of the studies were satisfactory); How adequately does the design control variables? (29. 7 percent of the studies were satisfactory); How properly is the sample selected for the design and purpose of the research? (27.7 percent of the studies were satisfactory).
Mathematics education in the elementary schools of the Soviet Union
Authors: Carlton Virginia
Published: October 2020

A group of mathematics educators, twenty from the United States and one from England,* visited mathematics education facilities in the Soviet Union (Moscow, Kiev, and Leningrad), from August 27 to September 9, 1966. The tour was sponsored by the NCTM under the supervision of the Committee on I nternational Mathematics Education, with some financial support from the National Science Foundation. The tour followed the conclusion of the International Congress of Mathematicians which was held in Moscow from August 16-26, 1966. The main purpose of the group was to visit persons and institutions which are involved in the education of the mathematically talented student. Two major topics on which members of the group focused during the tour were: (1) curriculum revision and research and (2) special schools and devices for stimulating interest in superior students.
Arithmetic is a joyous experience for elementary school children in Great Britain
Authors: Unkel Esther
Published: October 2020

What is being done in mathematics in England in the infant and primary schools (akin to our elementary schools, Grades 1 through 6)? I wanted to see for myself, so I went to England to find out.
Multiplication—logical or pedagogical?
Authors: Rappaport David
Published: October 2020

It appears that my article1 on multiplication has aroused considerable discussion, mostly critical of my position. Most of the criticism centers around the fact that repeated addition does not explain all multiplication, such as 1×3 or 0×3. In a lengthy article, Wcaver2 describes the various interpretations of multiplication. Although he presents a very strong case for rectangular arrays, he was gracious enough to point out that I was expressing my reluctance to see the repeated addition interpretation slighted or completely eliminated in the contemporary elementary school mathematics programs. Bechtel and Dixon3 were not so generous. They characterize my position as a reflection of poor mathematics and also poor pedagogy.
Editorial comment: As we read
Authors: Inskeep James E.
Published: October 2020

To invert or not to invert? Fractions, rationals, number lines, folded paper! This copy of The Arithmetic Teacher deals with these and related ideas. Perhaps no area of the traditional mathematics curriculum caused more consternation than that of common fractions. Nor are our modern curricula free from difficulties in this area. Terminology has become of increasing concern and is constantly subject to review. New and old models, fresh ideas, time-tested suggestions are desired fa re for the teacher of today who would improve the teaching of common fractions. This issue of the journal provides some excell ent contributions for improvement of teaching. There are also articles and departments of interest in other areas.
Addition of unlike fractions
Authors: Adachi Mitsuo
Published: October 2020

The teaching of addition of unlike fractions can be a very uninteresting process. Is there a more creative approach to the teaching, one that can challenge students to discovery of generalizations? Can we structure instances so that students themselves will develop the algorithm, “Change the fraction to equivalent fractions with common denominators and proceed as in the addition of like fractions”?
Fractions as operators
Authors: Fehr Howard F.
Published: October 2020

In the educational literature much information exists on the teaching of fractions and operations on fractions. This literature includes an abundance of suggestions for developing meanings that may be assigned to a fraction in elementary school instruction. However, except in mathematical treatises, there is not an abundance of information for teachers intended for challenging a deeper understanding of fractions. In this article there is presented one way of conceiving of fractions as mathematical entities with the purpose of broadening the teacher's knowledge of this aspect of arithmetic.
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

We wish to express special appreciation to our referees, who review manuscripts and make recommendations concerning publication. In many instances, our referees make suggestions that lead an author to revise his manuscript so that it is suitable for publication. Help such as this is invaluable. We are grateful for each reviewer.
Teach so your goals are showing!
Authors: Sandel Daniel H.
Published: October 2020

Each discipline needs to be examined from time to time. The revolution in school mathematics has caused us to evaluate our programs more critically, sharpen our thinking, reshape our goals, and examine our procedures in the light of what our students are capable of doing and what we believe are desirable outcomes.
Aesthetics in elementary mathematics
Authors: Rosenberg Edwin A.
Published: October 2020

Appreciation of mathematics is repeatedly and properly touted as one objective of instruction in mathematics. Formally. appreciati on may be interpreted as sensitive awa reness, especially of aesthetic qualities. Those who pursue mathematical studies often obtain aesthetic satisfactions in perceiving relationships, structures. and the “elegant” nature of ce rtain solutions.
The road to inverse and multiply
Authors: Bates Thomas
Published: October 2020

Whether we are pupils or teachers, in most mathematics lessons we are constantly shifting our thought back and forth between what we generally regard as the “real” world of physical objects and the “nonreal” world of mathematical abstractions. The shift is often quite unconscious because the dichotomy between the two worlds is not always clear-cut, and this seems particularly to be the case in arithmetic lessons. In arithmetic the two worlds seem to be separated (and joined) by an ill-defined and nebulous region which I like to think of as a “patterns of action” zone, a zone into which from one side we project mental images of physical things and mental images of sensorymotor actions, and from the other side mental images of mathematical structure. In this zone there is often a merging of physical world language and mathematical language, a sort of lingua franca, as when we say—and presumably also imagine-“How many twos are there in six?” I think it is in this zone, rather than in the world of physical things or in the world of abstract mathematical structure that most arithmetic becomes understood by children.
“Interest getters”
Authors: Zahn Karl G.
Published: October 2020

It is generally recognized that it is important for a teacher to maintain pupil interest in arithmetic. The use of a variety of learning activities is an excellent technique for retaining pupil interest. One technique is to use one class period per week for presenting “interest getters.” Friday is usually a good day to present these, for the teacher can review the past week's activities and then present some stimulating items to create and maintain pupil interest.
A message to teachers of elementary mathematics
Authors: Hlavaty Julius H.
Published: October 2020

High school teacher of mathematicsand I include myself among them—have long been accustomed to criticisms from colleagues at the college level about the teaching of mathematics in the high schools. Occasionally we have reacted to these criticisms by criticizing our colleagues in the elementary school. Yet yea rs of teaching have convinced me that we all, by and large, do the best we can at all levels. Indeed, I have a profound admiration for the elementary school teachers who, facing very broad responsibi lities, achieve great success in all areas, including mathematics.
Tick-tack-four
Authors: Jordan Diana
Published: October 2020

Children are intrigued by numbers and their relationships to one another. If we, as teachers, use this curiosity, it is possible to develop many games and activities in arithmetic which can serve as replacement for traditional drill.
Multiplication unlimited!
Authors: Hullihan William F.
Published: October 2020

Several years ago, while I was teaching an eighth-grade room of over-age boys, the students and I discovered some new ways of using the standard multiplication “chart.”
Odd, isn't it?
Authors: Smith Frank
Published: October 2020

Most references define an odd number as being any counting number which, when divided by two, produces a remainder of one. This means that all other counting numbers are even. Some references state that all counting numbers that are multiples of two, including two, are even. All remaining numbers in the set are odd.
Research on mathematics education, Grades K-8, for 1967
Authors: Riedesel C. Alan , Suydam Marilyn N. , Pikaart Len
Published: October 2020

This is the eleventh of a series of annual listings of research concerned with elementary school mathematics. During the very important period of change in elementary mathematics education from 1957 to 1966 the summaries were compiled by Dr. J. Fred Weaver of the University of Wisconsin.1 We hope that this listing will prove to be as valuable as the previous ones.2
Patterns of intersection
Authors: Allen Bruce A.
Published: October 2020

An interesting pattern of intersections was discovered by an eighth-grade student after an introduction to a geometry lesson on how lines in a plane may be associated.
President's report: The state of the Council, 1967/68
Authors: Johnson Donovan A.
Published: October 2020

Now that my term as president is completed, I can look back and assess the activities of the NCTM. The past two years have been a time of much activity for me−a time to meet many new friends, a time to learn about exciting innovations in mathematics education, a time to extend the activities of the NCTM, a time to consult with leaders in many fields. Of course, I should add that it has been a time to worry about NCTM finances, to be concerned about the critics, to fly away from home in fair weather or foul.
Through clouds of failure into orbit
Authors: Travers Kenneth J.
Published: October 2020

If we are not to live in embarrassed dread in the century ahead, we will have to understand the mathematical ideas that lie behind the scientific advances that are shaping the modern world.— Jerome S. Bruner.
The development of teaching materials for low-achieving pupils in seventh- and eighth-grade mathematics
Authors: Keiffer Mildred C.
Published: October 2020

Cincinnati teachers of seventh- and eighthgrade mathematics have long been concerned about the problem of suitable text materials for low-achieving pupils in their classes. As far back as 1956 they began to try to write materials themselves to fill the void they saw. They collected samples prepared by individual teachers and, relying upon experience and judgment, worked at refining these samples into generally useful lessons. These were collected in the Cincinnati Curriculum Bulletin (No. 113), which was published in 1961, and were available on order from the central office. The materials were in the form of studyguide sheets and were designed to be used as the basis for discussion and written work in class.
Elementary school metric geometry
Authors: Krause Eugene F.
Published: October 2020

The word “geometry” comes from the Greek words gê, meaning “earth,” and metron, méaning “to measure.” Thus, etymologically, the term “metric geometry” is redundant and the term “nonmetric geometry” is ridiculous. In modern usage, however, nonmetric geometry refers to the parts of geometry one can study with-out using the concept of measure, while metric geometry refers to the geometry one can do using the measure concept. The purpose of this article is to explain the concept of measure and to illustrate some of its applications.
Discovery in number operations through geometric constructions
Authors: Allen Neufeld K.
Published: October 2020

A sharp pencil, straightedge, and reams of scrap paper are the only ingredients necessary to set a group of curious students to exploring the number system and discovering ideas that seem so obvious to them. The obvious is often uninteresting, but approaching the obvious by a new path often turns monotonous practice into exciting rediscovery.
An application of modular number systems
Authors: Adkins Julia
Published: October 2020

Some mathematics books for Grades 4—6 include a section on modular number systems (also called modular arithmetic or clock arithmetic). A simple application of this type of material is described below
Some activities with operation tables
Authors: Ginther John L.
Published: October 2020

The following ideas regarding operations have stimulated much useful discussion in the author's classes. The subject matter involved should be well within the grasp of pupils in the upper elementary grades. The fact that the operations presented are unfamiliar to the students has an advantage: they lack previous experience to fall back on“ they must do some thoughtful analysis of the various properties, patterns, and questions that arise.
Take a mathematical holiday
Authors: McCalla Elaine
Published: October 2020

The day before vacation usually brings an air of carefree excitement into the classroom as the children come, bubbling over with thoughts of “No school tomorrow!” To the teacher. educationally speaking, I'm afraid it often means a wasted day. I have found a way to put some of this restlessness to work, and so I invite you to join me as we “take a mathematical holiday”—the day before Christmas vacation.
Number line: Versatility
Authors: Milne Esther
Published: October 2020

How can you show a number line at innumerable positions? How can you, at the same time, show that the positive integers may be on either side of the origin? Is there a simple device you can construct to illustrate all possibilities?
Promoting problem–solving skills through nonverbal problems
Authors: Trueblood Cecil R.
Published: October 2020

Solving verbal problems is considered to be a very important part of elementary school mathematics programs. This is based on the assumption that there will be transfer to solving problems faced by pupils in a variety of physical world situations. With textbook problems, the poor reader often cannot abstract the essential elements of a problem situation because of his low level of reading ability. The teacher therefore needs verbal problems that present less interference to the development of problem-solving skills.
Focus on research: Lectures versus manuals in the education of elementary teachers
Authors: Pigge Fred , Brune Irvin H.
Published: October 2020

Teachers of methods in elementary mathematical education usually seek somehow to create, maintain, and extend students' mathematical understandings. Often their students come to them with three to six semester hours of credit in mathematics for futu re elementary teachers. Not similarly countable, however, are the students' stores of usable mathematical concepts. Nearly always their instructors recognize that these students need a new look, if not a thorough review, into some key ideas. So the question arises, How best can we handle such a sharpening of mathematical understandings?
Goals for mathematical education of elementary school teachers
Authors: Morley Arthur
Published: October 2020

The Cambridge Conference Teacher Training Report1 is about matters of vital concern to most countries, and so perhaps some comment from England is allowable.
A natural way to teach division of rational numbers
Authors: Vance Irvin E.
Published: October 2020

In working with students in Grades 5–9, preservice teachers, and in-service elementary teachers, we ask the question: “Can you solve the problem 14/9, ÷ 2/3 by dividing 14 by 2 and 9 by 3?”
A different look at decimal fractions
Authors: Wilson Patricia , Mundt Delbert , Porter Fred
Published: October 2020

In this article we are attempting to look at decimal fractions in a different way than the ordinary. A new way is used to lead to the representation of nondecimal fractions as decimals. The discovery of a way to tell whether a fraction will be a repeating decimal is the final objective.
Division of fractional numbers
Authors: Heddens James W. , Hynes Michael
Published: October 2020

The following axioms for the teaching of mathematics are offered in the Twentyfourth Yearbook of the National Council of Teachers of Mathematics.1
Computers and art
Authors: Mott-Smith John
Published: October 2020

The digital computer is a research tool in my work. In the past several years I have made hundreds of computer-derived photographs to illustrate scientific studies related to computer processing of visual information.
Sample “computer”
Authors: Mcdermott John J.
Published: October 2020

We are all aware of the fact that children learn when they enjoy and see a purpose in what they are doing. However, when children are required to master computational skills, teachers sometimes find it very difficult to create such a learning atmosphere.
A workshop on computer-assisted instruction in elementary mathematics
Authors: Jerman Max , Suppes Patrick
Published: October 2020

During the 1967/68 school year the Stanford project in computer-assisted instruction in elementa ry mathematics was expanded to include schools in Iowa, Kentucky, and Mississippi, in addition to schools in California. As many as 78 students were able to take arithmetic lessons simultaneously on instructional terminals operated by phone line from the computer of Stanford's Institute for Mathematical Studies in the Social Sciences. The instructional terminals (teletype machines with modified keyboards) were located one to a classroom in some schools and grouped in a single room in other schools. Before describing the workshop held at Stanford for Mississippi teachers, a brief description of the drill-and-practice program in arithmetic skills and concepts will be given.
Elementary graphing experiences
Authors: Pierson Robert C.
Published: October 2020

In this modern day and age people are confronted with various pictorial representations of data and, in the future, children will be faced with an ever-increasing mass of pictorial information. lt is urgent that students be able to understand these pictorial representations and from this understanding make intelligent interpretations.
Discovering patterns in addition
Authors: Edmonds George F.
Published: October 2020

I read Ruth Perkins' article in The Arithmetic Teacher for December 1967 with a great deal of interest. In it she describes a discovery made by a primary class about the number of addition names one can find for a whole number using two addends. This is exactly the kind of teaching we are trying to develop among the primary teachers in Kenya. My job consists of developing a modern mathematics curriculum for the primary schools of Kenya and assisting the teachers' colleges to prepare teachers who will be able to teach the new materials. We work with twenty-six primary schools within commuting distance of the Curriculum Development and Research Centre. The Centre is located in Nairobi. We also have model schools, which are under the direct supervision of the teachers' colleges. The materials we develop are tried our in these schools before they are published and made available to all the primary schools. We also prepare textual materials on methods for use in the colleges.
Learning structures for arithmetic
Authors: Bidwell James K.
Published: October 2020

Frequently, as teachers, we concentrate too much on the details of the week or of the unit. It is too easy to lose contact with the larger context of the arithmetic we teach. For this reason perhaps, lea rning theories stress the need of awareness of the larger plan: What does the student know? What will he eventually learn?
The sorry state of mathematics teacher education
Authors: Rising Gerald R.
Published: October 2020

The concern I express in this polemic is a simple but important one: Education programs for mathematics teachers are not only below any reasonably acceptable standard, but they are getting steadily worse!
Counting: new road to multiplication
Authors: Stern Jane L.
Published: October 2020

Children can be taught to count by threes, fours, sixes, sevens, eights, and nines in the same way they are taught to count by ones, twos, and fives. If children were taught how to count by the former numbers, the task of teaching multiplication would be as easy as teaching addition. Children seldom have trouble with the two-and five-times tables, since they have had previous counting experiences with these numbers.
Discovering exponents
Authors: Kamins Martin P.
Published: October 2020

The following chart should be put on the board prior to any instruction or discussion. It is more effective if the chart is present when the children a rrive in class, if possible.
Magic square patterns
Authors: Gorts Jeannie
Published: October 2020

A magic square, as we all know, is one in which the sum of the numbers—horizontally, vertically, or diagonally—is the same number. Suppose you want to make a magic square of 9 sections, whose sum is 15. If you divide your sum by the square root of the number of sections your magic square consists of, your result is the center number of your magic square. In this case, 15 divided by 3 equals 5. (Note: When you have an even number of sections, you will have not one center number, but four.)
Equivalence and equality
Authors: Sanders Walter J.
Published: October 2020

N ancy was setting the table for the evening meal. When it was time to put the Silverware on the table her little sister Valerie asked to help. Nancy said, “Get six teaspoons, six forks, and six knives from the silverware drawer and place them around the table.”
Towards the Golden Jubilee Year—1970
Authors: Hlavaty Julius H.
Published: October 2020

Thirty years ago in Atlantic City, I attended my first meeting of the National Council of Teachers of Mathematics. Everyone considered that meeting a great success, for there were 343 people present! Thirty years ago, the total membership of the Council was smaller than the number of those attending this present convention.
In Tennessee: Making use of the NCTM film series
Authors: Cannon James R. , Oakes James G.
Published: October 2020

The National Council of Teachers of Mathematics released a ten-film series in 1966, Mathematics for Elementary School Teachers. Designed for use in in-service and preservice programs, the color films treat the whole numbers at both the con: cept and classroom levels. A textbook has also been prepared to accompany the films.1
Sugar-cube mathematics
Authors: Higgins Jon L.
Published: October 2020

There's a lot of mathematics in a twentyfive cent box of sugar cubes. They can be used to illustrate different numeration systems, the measurement of volume, multiplication of fractions, and even techniques of estimation. A twenty-five cent box of cocktail-sized sugar cubes contains 210 cubes of surprising uniformity. Suppose we begin by having students verify the number of cubes in the box. (You may wish to cover the numbers printed on the outside of the box. It is also advisable to set one box aside for eating purposes only!) The most direct way to determine the number of cubes is to count them one by one as they are removed from the box and piled on the table. Of course, if you're interrupted in this process and forget where you were, you have little choice but to start all over again. With a moderate number of distractions a counting job like th is could last all morning for many students!
An intuitive approach to square root
Authors: Boomer Lyman W.
Published: October 2020

While teaching exponential notation to junior high school students, I have found it useful to introduce the concept of square root. The students see that the skills that they have learned are tools for solving “theoretical” problems. Since they have difficulty understanding the meaning of square root, it is well to begin with what they do understand, namely, the whole numbers and their squares.
Focus on research: Research on mathematics education, grades K–8, for 1968
Authors: Riedesel C. Alan , Suydam Marilyn N.
Published: October 2020

This is the twelfth of a series of annual listings of research concerned with elementary school mathematics.1 The annotated format used for the 1967 listing has been retained.
Forum on teacher preparation: Preservice textbooks: an analysis
Authors: Cruikshank Douglas E.
Published: October 2020

Teachers need to be well prepared in order to understand contemporary mathematics and to provide worthwhile learning experiences for their elementary school youngsters. As a result of changes that have occurred in mathematics and elementary school mathematics instruction, efforts are being made to improve the preservice and in-service education of elementary teachers. Two examples are The Arithmetic Teacher's “Forum on Teacher Preparation,” and the 1967 summer conference at East Lansing, cosponsored by the Science and Mathematics Teaching Center of Michigan State University and the National Science Foundation.1
Finding averages with bar graphs
Authors: Ball Joyce
Published: October 2020

Our school still uses the numerical marking system for reporting to students and parents. I have found over a period of several years that pupils in the fourth grade are apt to look at the single mark on papers that are returned daily or weekly, and have little sense of any cumulative effect over the ten weeks' marking period. This is true both of students doing superior work and those doing below-average work. While the process of averaging marks is not a complicated one, it presents two kinds of problems for the typical fourthgrader: (1) It may involve more complicated arithmetic than he is able to handle at the early stages of the fourth grade; and (2) The possibility of error both in addition and division is so large that the end result may be far from accurate.
Golden fruit salad—a little fishy!
Authors: Cramer Ward
Published: October 2020

Make fruit salad with goldfish?” “Of course,” says five-year-old Chris, a member of our kindergarten class.
Do your first graders measure up? (A report of a unit with disadvantaged learners)
Authors: Bachrach Beatrice
Published: October 2020

Give them an inch”—and they'll be on the road to measurement. Although measurement is rarely included as a topic in the first grade curriculum, we found it of considerable value—academic as well as psychological—to our disadvantaged first graders. Measurement is a topic for which children have a great deal of enthusiasm. They associate measurement with an advanced level of mathematics and find the study challenging and rewarding.
Development of a pool of mathematics test items for grades 7 and 8
Authors: Horn Dorothy M.
Published: October 2020

In 1964 the Ontario Department of Education introduced curricula for grades 7 and 8 based on the new approach to mathematics. As the curricula were gradually introduced into the schools, the need was felt for aid in evaluating the work of students in the new mathematics. It was decided that a pool of test items would prove a helpful resource for classroom teachers by providing materials for a flexible curriculum. These multiple-choice items were not expected to replace teacher-made tests of the traditional type.
Square root+
Authors: Frederiksen John G.
Published: October 2020

Most modern mathematics textbooks favor the approximation method for computing square roots. The so-called long division method is not often mentioned. The approximation method is sound in principle, and it finds favor with many students because of the logic of its continuing process. Less capable students, however, complain that the approximation method has “too much figuring” and “too many chances for error.”
Presenting multiplication of counting numbers on an array matrix
Authors: Schrage Merry
Published: October 2020

Multiplication tends to imply an avenue of excitement and adventure for children. During my ten years as a grade three teacher, I have yet to find a year when some eager child hasn't asked the question, “When are we going to learn multiplication?” Prior to third grade the children are exposed to readiness activities in preparation for multiplication. However, the real drive to master facts and algorisms is in the third-grade mathematics sequence in Dade County, Florida.
A prime quotient study for junior high school students
Authors: Hildebrand Francis H.
Published: October 2020

Student motivation is always a central concern of the junior high school teacher, particularly in mathematics. In the present article we offer a sample unit of mathematics instruction suitable for junior high school students, with two purposes in mind: (1) to illustrate a current theory of learning and motivation, and (2) to offer a model for the construction of other such units. The unit is purposely built around division, since that topic is commonly disliked by both teachers and students.