bluuebunny/crossref_metadata_2025 huggingface dataset:
A mathematics laboratory—from dream to reality
Authors: Davidson Patricia S. , Fair Arlene W.
Published: October 2020

Just a year ago last September, the mathematics laboratory program at the Oak Hill Elementary School was still a dream. A few plans had been made during the previous spring for an initial purchase of materials, and the first job in the fall was to find “the place.” It was decided to convert a small storage room on the basement floor into a mathematics laboratory. By the end of the year, the dream had become a reality—the laboratory approach to learning mathematics had become a vital part of the mathematics program in the whole school.
Fostering enthusiasm through child-created games
Authors: Golden Sarah R.
Published: October 2020

How can the young child develop enthusiasm for mathematics, maintain needed skills, gain mathematical insight, and work at his individual level in a relaxed classroom climate of success?
A look at nets of cubes
Authors: Wardrop R. F.
Published: October 2020

The particular discovery process presented in this paper can best be initially motivated in the classroom by an informal group discussion.1 Ask the students to sketch with a pencil and ruler all of the possible patterns of six connected squares that can be drawn so that when the pattern is cut out and folded on the sides of each square the paper will form a cube. Squares are connected if and only if they have an edge in common. This initial discussion session is best started by sketching one example on the board and asking for volunteers to add new possibilities. This is also an appropriate opportunity for a discussion of the use of a systematized approach to any proposed problem.
Capsule History of the NCTM
Authors: Hlavaty Julius H.
Published: October 2020

The history of mathematics education in America is the story of a long and exciting adventure. It is the subject of a forthcoming NCTM Yearbook, A History of Mathematics Education in the United States and Canada. The following is a capsule account of the direct involvements and the tangential contacts of the National Council with that history during the past fifty years.
In the classroom: How to get subtraction into the game
Authors: Marion Charles F.
Published: October 2020

Games like Monopoly in which the number of spaces a player advances is determined by the throwing of two dice provide a pleasant way for children to practice thirty-six of the one hundred basic addition facts.
Mathematics and the low achiever
Authors: Chandler Arnold M.
Published: October 2020

As part of its Golden Jubilee activities, the National Council of Teachers of Mathematics will present a set of exciting new instructional materials designed to help low achievers learn mathematics. The characteristics of individuality, activity, success, meaning, and novelty are emphasized in these materials.
Another use for binary numerals
Authors: Dunkum William
Published: October 2020

One morning, while engaged in preparing problems in permutations and combinations, I found it necessary to list the subsets of the set S = {A, B, C, D, E}. As is well known, if there are n elements in a set S, then there are 2” subsets of the set, including the empty set Ø and the set S itself. However, tabulating all of the subsets is rather tedious and, even if done carefully, listing them often produces duplications or omissions which must be eliminated by cross-checking.
An investigation of the effect of an operationally defined word on conservation-of-number responses
Authors: Fletcher Robert F.
Published: October 2020

In recent years much interest has been exhibited in the problem of conservation of number. Accompanying this interest has been a sizable amount of controversy centering on the question of whether a child's apparent nonconservation of number is largely a language-caused phenomenon
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

The editor wishes to express appreciation to the many persons who have volunteered their services in the production of The Arithmetic Teacher. The list include our authors and referees as well as the editors of the journal. To members of the Washington office: Charle Hucka, production editor; Jame Gates, executive secretary; Carol McCamman, managing editor; Thomas Slaughter, advertising and circulation manager; and their asistant— a word of thanks is given at this time for their generous and devoted service. Special appreciation is expressed to the members of the editorial board. Each editor has assisted in unique ways and has given dedicated thought, time, and energy to the production of the journal.
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

It has been a privilege and a pleasure to serve as Editor of The Arithmetic Teacher. This May issue of the journal completes my four-year appointment as Editor, and I should like to take this opportunity to express my sincere appreciation to the many persons who have helped produce the journal. The members of our excellent Editorial Board were given recognition in the April issue, but a particular word of thanks goe to Jame Inskeep who has given thoughtful guidance and counsel and who helped with deci ion-making as well as writing “As We Read” for each issue. In addition, I should like to say a pecial thank-you to the members of the NCTM Board of Directors; to the members of the Publications Committee; to Charles Hucka, James Gates, and the staff of the Washington office, who have given dedicated assistance; to all persons who submitted manuscripts for consideration for publication: to our referee; and to all readers but especially to those who have written tellers to me regarding the journal. Your helpful suggestions, encouragement, a istancc. and confidence in our project have been deeply appreciated.
Mathematical heritage of Zambia
Authors: Careccio John
Published: October 2020

Information regarding the culture of Africa needs to be collected and saved before it is lost forever to the increasing tempo of modernization. This calls for preserving the stories, dances, music, and also other traditions such as measurement. People have always used some form of measurement for time, distance, weight, and volume. The author was able to collect measurement information by using university students who sought out the oldest people of their villages to find out how these things were done before the European types of measurement replaced the African methods.
Ancient systems of numeration—stimulating, illuminating
Authors: Cowle Irving M.
Published: October 2020

Elementary school children find the study of ancient systems of numeration a fascinating topic. Often an interest is kindled by this initial brush with these early civilizations that stimulates many a youngster to continue the study and investigate further, in depth, on his own. This child is richer for the experience and, properly guided, the entire group will benefit as well when the researcher reports his new discoveries to his classmates.
Some historical whats, hows, and whys in teaching arithmetic
Authors: Clason Robert G.
Published: October 2020

The purpose of this article is to examine some aspects of two changes in arithmetic teaching from 1800 to the “new” mathematics: changes related to psychological theory and changes related to the foundations of mathematics, particularly of the number concept.
Mathematics educators must help face the environmental pollution challenge
Authors: Henderson George L. , Van Beck Mary
Published: October 2020

If pollution of earth's air will cause heat from the sun to be trapped, melting the polar ice caps and raising the level of the oceans 60 feet, how many people will either die or be left homeless?
Tying things together with braids
Authors: Hennemann Willard W.
Published: October 2020

Since most of my teaching involves preservice and in-service courses in content and methods for teachers of elementary school mathematics, I have been concerned about providing my students with experiences that convey the objectives of a contemporary mathematics curriculum.
A SPACE TO LIVE IN
Authors: Daniells Roy
Published: October 2020

I am deeply honoured and more than a little embarrassed to appear before this august gathering—this set of integers, as I believe you term yourselves. We simple humanists have another name for you. As we refer to a flock of sheep, a band of thieves, a covey of partridges, or a pride of lions, we refer to a gathering like this as a magnanimity of mathematicians, signifying a greatness of spirit. You will readily understand that, as a mere humanist, in this company I feel as an avocado pear would feel in an assemblage of crystals–soft, tender, oleaginous, with a great stone inside hopeful of germination, in contrast to the shining consistency, the lucid and transparent candour of your own adamantine minds.
Using parts of ten
Authors: Sister Mary Gemma S.C.
Published: October 2020

We see below the parts of ten.
Do we need separate rules to compute in decimal notation?
Authors: Henry Boyd
Published: October 2020

It is perplexing indeed that we continue to teach decimal fractions as if they were enti tic quite apart from common fraction. Typically, we first teach student to add, subtract, multiply, and divide with common fractions. Then in due course, as if decimal fraction (commonly called decimal) were a totally different concept from common fraction, we teach a new set of seemingly unrelated rules and regulations for moving the decimal point about when computing with decimal fractions. It is the purpose of this article to observe that once the child has learned to work with common fractions with orne degree of success, he has very little additional to learn about decimal fractions. It is all a matter of notation.
Mary and John
Authors: Hollands Roy D.
Published: October 2020

The teacher had just marked and returned each child's number book. She reminded the children to count the number of check marks to sec if they got the same as the total he had written.
Things you can try: Drawing 3-D figures from 2-D templates
Authors: Ibe Milagros D.
Published: October 2020

One of the sad things about many current lessons on plane figures in elementary school geometry is the teaching of shapes only in terms of what they appear to be and not in terms of their potential or hidden possibilties for mathematical understanding and beauty.
ONE POINT OF VIEW: The morning after
Authors: Van Engen Henry
Published: October 2020

Events in the past ten years have made it necessary and desirable for colleges and universities to increase their course requirements in mathematics for elementary school teachers. There has been little or no opposition to this trend except as one finds it in individual colleges when there is a proposal to change course requirements. In mathematics the change has been in the direction of doubling or trebling the number of credits in mathematics required of prospective elementary teachers.
Blast-off mathematics
Authors: Beougher Elton E.
Published: October 2020

A more appropriate and informative title for this discussion might be “Astronomy, Space Travel, and Arithmetic.” An attempt will be made here to support the claim that the study of arithmetic and of topics from space science can be coordinated in the elementary school program. Further, it is claimed that this coordination can enhance both the mathematics and science areas and that each can promote learning in the other. The author hopes to provide some ideas that teachers might use to give variety to their lessons.
Mid-nineteenth century methods for the 1970s
Authors: Grossman Anne S.
Published: October 2020

In the present world of innovation and rapid communication, educators show much fervor for change and for implementing new programs and materials in their schools and classrooms. Because of this search for the “new,” many old and useful methods and materials used by previous generations for teaching mathematics are too often forgotten or exchanged for what we like to believe are more contemporary ones. This emphasis on what's new in the teaching of mathematics seems to obscure the main purpose of teaching mathematics. Old arithmetic is still being taught in the elementary school. It might also be wise to use some old strategies and effective methods from the past in the modern cl assroom to accomplish our goals in the 1970s.
The sieve of Eratosthenes
Authors: Dubisch Roy
Published: October 2020

For some time I have been increasingly annoyed by what I regard as a serious flaw in the presentation in many texts of the celebrated sieve of Eratosthenes for obtaining prime numbers. Both the efficient procedure as conceived by Eratosthenes (see, for example, Boyer [1968], Eves [1953], or Heath [1921]) and the inefficient procedure frequently presented begin by listing ail the natural numbers up to some fixed number n. (I am aware that we Jist the numerals for the numbers, but I don't want to be bothered later by having to say “the number named by the numeral”!)
Paper folding and equivalent fractions—bridging a gap
Authors: Bohan Harry
Published: October 2020

Much emphasis has long been placed on the value of using concrete models and manipulative devices in the early stages of the development of mathematical ideas. As Johnson suggests, however, it is entirely possible for a teacher to use instructional aids and get little benefit from them. “Some use them but fail to take students from the concrete representation to the concept behind it. Others use an inadequate or inappropriate aid” (Johnson 1967, p. 20). Brownell emphasizes that the processes children use at the concrete and semiconcrete levels must be closely related to the processes they will use later in dealing with abstract symbols (Brownell 1928, p. 24).
Seductive shibboleths
Authors: Weaver J. Fred
Published: October 2020

A portent of things to come? … “Yesterday in our math lab one group worked with tangrams, and they just loved it. Look at the cute pictures some of the children drew using the tangram pieces.”
What's Your Position On… the role of experience in the learning of mathematics?
Authors: Biggs Edith E. , Hartung Maurice L.
Published: October 2020

Learning mathematics, like learning language and the aesthetic subjects, is an active process and should call into play our imaginative and creative powers so that the subject is a delight to pupils of all ages. Investigations of every aspect of mathematics—number knowledge itself and written calculations, as well as measures, shapes, and representations of all kinds—are included. In the title of this article, the word experience could very well be changed to investigations. Investigations, to me, cover a wider range of activities than experiences, which arc often thought of as practical experiences.
Grouping of objects as a major idea at the primary level
Authors: Moser James M.
Published: October 2020

The Commission on Mathematics, in its well-known report written in 1959, called for the “judicious usc of unifying ideas” in mathematical instruction. While it is true that the report was written for the main purpose of improving secondary mathematics instruction, this particular recommendation has just as much importance for elementary education as it does for secondary.
Using research in teaching: Thinking about measurement
Authors: Steffe Leslie P.
Published: October 2020

It is essential that teachers of mathematics have a good grasp of the thinking basic to the mathematical content they present to their students. When confronted with the general area of thinking, mathematics educators (including teachers) have traditionally turned to psychology for help. Some mathematics educators are currently turning toward theories of cognitive development in their quest to understand thinking basic to mathematics. The search is rich in content and rewarding in terms of implication for mathematics teaching. The implications are not necessarily in terms of pedagogical polemics but lie in the insight that teachers can gain into the thinking of the young child and the origins of that thinking.
Elevator numbers
Authors: Becker Stanley
Published: October 2020

All teachers are constantly on the lookout for games and devices that will stimulate interest in mathematics. The following description of a device called “Elevator Numbers” readily fills that need. It enables children to practice adding numbers to produce sums from 10 to 20 or more.
Ideas
Authors: Immerzeel George , Wiederanders Don
Published: October 2020

Ideas presents an easy way for you to try out a variety of student activities that we have found to have a high level of student interest. In each issue of the Arithmetic Teacher several work heels that focus on a key concept of mathematics are presented. At least one of these worksheets will be appropriate for your students.
Using research in teaching: Some factors associated with pupils' performance levels on simple open addition and subtraction sentences
Authors: Weaver J. Fred
Published: October 2020

During the early school years, children's introduction to various aspects of addition and subtraction of whole numbers takes place within the context of “basic facts.”1
Minutes of the Annual Business Meeting
Authors: Gates James D.
Published: October 2020

President H. Vernon Price called the Annual Business Meeting to order at 9:00 A.M.
The mathematics laboratory for the elementary and middle school
Authors: Barson Alan
Published: October 2020

It is impossible to give an all-inclusive definition of a mathematics laboratory becausct of the various uses and styles of organization that exist. However, there are significant characteristics of good mathematics laboratories that can be described.
Individualized instruction: Developing broadened perspectives
Authors: Trafton Paul R.
Published: October 2020

The development of mathematics programs for elementary school pupils that are meaningful and productive, as well as related to the needs, interests, and abilities of children, is the concern of mathematics educators and teachers alike. However, a particular urgency surrounds tbe task today, for it is all too evident that the learning of mathematics is still not a successful experience for many boys and girls.
Individualized instruction: Speaking from reality
Authors: Galton Grace K.
Published: October 2020

Educational discussions often center on it, administrators all boast of its use in their programs, teachers insist they use it, parents demand its implementation, and children enjoy it. Indeed, hardly anyone will dispute that within the past decade the term individualized instruction has been a much talked-about subject in the field of education. What has proven to be more controversial is what it really means and how can it be achieved.
Teaching the division-by-subtraction process
Authors: Swart William L.
Published: October 2020

The division-by-su btraction process taught in most of today's elementary mathematics series includes three levels of performance. These are “functional” levels. That is, a child can function on level 1 without ever moving to level 2m—and leave elementary school on that level.
Properties of polygonal regions
Authors: Coltharp Forrest L.
Published: October 2020

Elementary school science activities often involve determining the properties of a collection of objects gathered by the children or suggested by the teacher. This article uses this chi ld-centered activity to discover various properties of some polygonal regions.
Polyhedra construction
Authors: Lulli Henry
Published: October 2020

Since my first article on paper folding appeared in the November 1969 issue of the Arithmetic Teacher, many requests for other polyhedra have been received. (One Jetter was from a former professor of mine who is now in Florida. Supervisors from Texas wanted to know the tetrahedron pattern. Even my colleagues at Clay Junior High have asked for a quickie class assignment as a filler or for use by a substitute teacher.) Here are some answers to these requests. This article presents my method for constructing models of the tetrahedron and the octahedron.
Teaching concepts incorrectly
Authors: Bompart Bill
Published: October 2020

A very brief but provocative article by Julia Adkins appeared in the Mathematics Teacher several years ago (1964). It was entitled “Leave the Door Open!” and emphasized the importance of “providing an atmosphere conducive to the development of creativity,” (p. 486) by keeping questions open for later extensions. For instance, to tell students that it is impossible to subtract seven from three is to deny them the opportunity to think beyond the set of whole numbers. It is to be hoped that all mathematics teachers endorse this philosophy and try to “leave the door open” at every opportunity.
Simple materials for teaching early number concepts to trainable-level mentally retarded pupils
Authors: Armstrong Jenny R. , Schmidt Harold
Published: October 2020

As a part of a project designed to teach early number concepts to trainable-level mentally retarded pupils, a step-by-step series of materials was developed. The materials were characterized by being (1) sequential, (2) easily utilized by pupils of low-level motoric, cognitive, and verbal skills, and (3) easily constructed with inexpensive materials that are readily available to most classroom teachers.
Some concerns about the application of Piaget's theory and research to mathematical learning and instruction
Authors: Weaver J. Fred
Published: October 2020

Based on remarks that I have made on more than one occasion, some persons may view me as being anti-Piaget. I am not.
Understanding multiplication and division of multidigit numbers
Authors: Cacha Frances B.
Published: October 2020

Anyone who has taught multiplication and division of large numbers knows that they are difficult processes for children to learn. Even though learning mathematics through understanding, rather than by rote, has been emphasized for decades, methods of teaching multiplication and division of multidigit numbers have been abstract. Because of the lack of unde rstanding, these concepts are not retained.
Fourteen different strategies for multiplication of integers or why (−1) (−1) = +1
Authors: Peterson John C.
Published: October 2020

Every time a teacher plans and teaches a mathematics Jesson, some type of strategy is involved. A strategy for teaching a given mathematical concept, according to Johnson and Rising (1967), is the procedure or algorithm used to deal with the concept. For each lesson the teacher selects the strategy that, in his professional judgment, is most appropriate for the topic, the class, the objectives, and the teacher. As he gains experience, the teacher increases the number of strategies he can use for each topic. He gains knowledge as to the effectiveness of using each strategy with different students. Eventually, he is able to select strategies that are most effective for presenting a topic; he knows which strategies are better for reinforcing the topic; and he knows which ones can be used to reteach topics when students have not gained the necessary knowledge after one strategy has been applied.
What's Your Position On Behavioral Objectives?: Are behavioral objectives the answer?
Authors: Nichols Eugene D.
Published: October 2020

At several meetings of mathematics teachers and of college students, I have expressed some concern about the use of behavioral objectives as a way of prescribing the entire mathematics curriculum—or for that matter a curriculum in any subject area. A number of individuals have asked that I make some of my remarks avail able in print. This article is a response to those requests.
Let them fold
Authors: Grant Nicholas , Tobin Alexander
Published: October 2020

Too frequently students are plunged into the formal study of geometry without first passing through the experiential stages necessary to enable them to be receptive to the basic assumptions of the subject. Paper folding is a convincing method of providing physical experiences with geometry and geometric concepts. The material that follows was developed with a sixthgrade SEED class.1 It has been simplified so that it can be used in earlier grades.
Starting the new school year in mathematics
Authors: Ballew Hunter
Published: October 2020

A biological miracle occurs each year around Labor Day when a child's activities and bodily functions are in some mysterious way suddenly assumed to be connected to the clock. Each September we begin to tell the child when to come to school, when to go home, when to eat, when to skip rope, and when to go to the bathroom. He has a time to read, a time to spell, and a time to do arithmetic. We expect the child to mold himself into restriction and work as joyously as he flowed into freedom and play the previous June. We expect him to change his routines as easily as we switch from standard to daylight saving time.
Teacher-made materials for teaching number and counting
Authors: Silverman Helene
Published: October 2020

Most primary school teachers found themselves in an awkward position this past year. Federal cutbacks and a series of freezes on spending made it extremely difficult to obtain equipment. After years of receiving elaborate equipment through specially funded projects, the reality of no continuing supply of equipment left many teachers unprepared.
A two-dimensional abacus—the Papy Minicomputer
Authors: Arsdel Jean Van , Lasky Joanne
Published: October 2020

Whitehead has written, “It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we a re doing. The precise opposite is the case. Civil ization advances by extending the number of important operations which we can perform without thin king about them” (Whitehead 1920, p. 6 1).
Things you can try: Using functional bulletin boards in elementary mathematics
Authors: Schall William E.
Published: October 2020

Visual aids—films, still pictures, models, bulletin boards, and so on—are among the most useful tools in education, but they do not teach without intelligent planning and use (Glenn O. Blough and Albert J. Hugget, Elementary School Science and How to Teach It [New York: Dryden Press, 1957], pp. 33–34). Bulletin boards can play an important role in today's mathematics program. However, a bulletin board, if it is to be successful in achieving its purpose, must gain and be worthy of the class's attention.
Zero is an even number
Authors: Lichtenberg Betty Plunkett
Published: October 2020

Every once in a while you will come across a clever article concerning what elementary school students have to say about zero. Some of the intuitive comments that are reported are sound, as well as amusing. However, there are some common misconceptions that should be eliminated.
Why Johnny can't count
Authors: Bender Albert E.
Published: October 2020

It is unfortunate that we have been unable to agree upon a common alphabet, a single device that would enable children to read more easily. Such is not the case with counting, since we find a common counting system used throughout the world. It develops logically, using Arabic numerals and the base ten.
Ideas
Authors: Immerzeel George , Wiederanders Don
Published: October 2020

Each IDEAS presents activities that are appropriate for use with students at the various levels in the elementary school.
Team learning
Authors: Young Carolyn
Published: October 2020

Some instructors find it difficult to motivate students in mathematics. A change in method—something new and different—can be an exciting experience for both student and teacher. Perhaps part of the answer lies in the enthusiasm of the teacher and his ability to convey this to the students. This teacher, for one, is very enthusiastic and excited about team learning.
Using research in teaching: Payoff in increased instructional time and enrichment activities
Authors: Rea Robert E. , French James
Published: October 2020

A recent research project assessed the merits of oral instruction in mental computation versus pencil and paper enrichment activities for sixth-grade pupils. An outcome of particular interest to teachers of elementary mathematics is that both treatments, as measured by standardized achievement-test scor es, resulted in a significant growth over a small number of instructional sessions.
Flexibility in the Arithmetic Program: To Promote Maximum Pupil Growth
Authors: Coburn Maude
Published: October 2020

I HAVE HAD THE PRIVILEGE of observing arithmetic in many classrooms where arithmetic has real meaning for children. Would you like to visit with me in some of these modern classrooms where children really live their arithmetic?
They Love Arithmetic!
Authors: Stephens Harold W.
Published: October 2020

Are children more intelliget today than they were years ago? Perhaps not, but teaching methods have changed drastically and teachers today are able to recognize a pupil's ability, bring it to the fore, and help develop it in ways that make even a class in arithmetic fun. I say even arithmetic because this subject seems to frighten so many students in the elementary school, and this intrinsic fear is often carried into high school and college.
Who Counts?
Authors: Hickey William S.
Published: October 2020

Once Upon A Time there was a little, three-year-old, yellow-haired boy, whose mother and father were sad because the little boy didn't know how to count.
Can 2 + 2 = 11?
Authors: Buckland G. T.
Published: October 2020

For Many Millions Of People the Sum of 2+2 has always been 4. When a statement like this is made there arise many questions in the minds of even the least curious. Questions such as these: If two plus two isn't four what can it be? How can it be anything else? Can you show that it can be otherwise? Do you have to be a “mathemagician” to show that 2+2 can be other than 4? The answers rest in the solemn thought that 2+2 is 4 because of the viewpoint taken. Without any twisting of figures, or without any magic, it can very easily be shown that 2+2 can be 11 as well as 4. By the best rules 2+2 + 4 is only a matter of viewpoint. An analogy suggests itself. If we take a trip to an amusement park and cast a glance at a distortion mirror we don't see ourselves as we think ourselves to be. Instead our positions might be reversed and we might find ourselves tall and skinny or short and pudgy when all the time we have been thinking otherwise. In other words we have received just another viewpoint on an old subject.
An Approach to Per Cents
Authors: Hibbard Wilbur
Published: October 2020

In Most Schools we teach fractions, decimals and percentage in this order. It is a logical sequence leading from the general case to a more restrictive case.
Hindu-Arabic Numerals
Authors: Sanford Vera
Published: October 2020

The Numerals We Use seem SO simple and practical that we take them for granted. We tend to belittle them because they are familiar. On the other hand, we are painfully aware of the difficulties children have in learning to use them. So it is appropriate to survey the development of Hindu-Arabic numerals, not because we will at once relay the information to our pupils, but because we do our best teaching when the subject the children are investigating is one that commands our respect and admiration.
Another look at fractions
Authors: Ness Harald M.
Published: October 2020

Students at all levels—in elementary and high school classes, and in lower-level college courses—have difficulty with fractions. There is considerable confusion in the minds of students when they have to operate with fractions.
Adding fractions using the definition of addition of rational numbers and the Euclidean algorithm
Authors: Sherzer Laurence
Published: October 2020

Adding fractions by first finding the lowest common denominator has for a long time been assumed to be the best approach to this operation. Is this really the most efficient approach? Is it pedagogically the most practical?
Things you can try: Cleo's clock
Authors: Kaprocki Cleo
Published: October 2020

In my teaching I have developed and successfully used the idea of associating the number line and the clock. Since the children had had experience with the number line, my idea was to relate this experience to the telling of time.
No time on their hands
Authors: Bachrach Beatrice
Published: October 2020

The telling of time is basic to the activities of daily living. The clock is constantly dictating what we must do. The very youngest of elementary school students find clock reading useful and meaningful in conducting their lives. It is a skill vital to their functioning independently—it is relevant.
Ideas
Authors: Immerzeel George , Wiederanders Don
Published: October 2020

Each ideas presents activities that are appropriate for use with students at various levels in grades K-8. When you have chosen the activities that are, most appropriate for your students, remove tire sheets and reproduce the copies that you need. After a sheet has been used, add your own comments and file the materials for future use.
Exemplify what you explicate
Authors: Sowell Evelyn
Published: October 2020

That professor! What he says is great—but that idea just won't work in the classroom!” These statements may be common among some teacher education students. Such comments are now heard much less frequently, however, around the University of Houston. The mathematics education faculty is experimenting with a competency- based program, as part of a collegewide endeavor, that requires prospective teachers to actually use in their classrooms what they hear and read about teaching. Initial experiences with this program suggest several advantages both for teachers in training and for teacher educators. This article describes some features and benefits of one type of approach to competency-based mathematics education.
A modulo line
Authors: Neuner Albert R.
Published: October 2020

Modulo arithmetic has been described as a “clock” arithmetic. A modulo system, being a finite system, provides an excellent setting for the study of arithmetical concepts. Most students readily grasp the concepts of addition and subtraction through the use of tables. Some of the slower students have difficulty with the multiplication and division concepts. The tables can be constructed through the use of “clocks” on which the elements have been represented. It is also possible to illustrate the concepts of modulo, or clock, arithmetic on the more familiar number line.
Metrication in Britain
Authors: Williams Elizabeth
Published: October 2020

In 1965 the British Government announced its support for the plan put forward by the Confederation of British Industry (CBI) for the gradual changeover of trade and industry to metric units. When in 1968 the CBI gave 1975 as the date by which it was asking individual industries to have completed their plans for the transfer, educationists and administrators realised that they must immediately consider the effects of these decisions on schools. It was evident that the majority of children then at school would start work in a world that had adopted metric units for general use. They would never have to use British units for their adult calculations connected with shopping, travelling, farming, or construction work.
Ideas
Authors: Immerzeel George , Wiederanders Don
Published: October 2020

Each IDEAS presents activities that are appropriate for use with students at tlte various levels in tlte elementary school. After you have chosen the activities that are most appropriate for your students, remove the activily sheets and reproduce the copies you need. After a sheet has been used, add your own comments and file the malerials for future use.
A slide rule for elementary school
Authors: Holz Alan W.
Published: October 2020

Physical devices that illustrate mathematical concepts can enhance the the mathematical understanding of elementary school children. Many of the concepts that children learn in elementary school involve the basic operations of addition and multiplication and their inverses. The simple slide rules described here could be used in several ways to help children understand these basic operations and perhaps make learning about them more fun.
One point of view: Four in one
Authors: Rising Gerald R.
Published: October 2020

There are so many serious problems that face us today, so many unopposed entrepreneurs pushing us in dangerous directions, so few things being said to combat the deterioration of school mathematics and school mathematics teaching, so many of us focusing on the easy (but wrong) answers to the hard questions, that writing an editorial is too frustrating. For that reason I have chosen to write a series of miniviews, encapsulating in editorial precis my opinions-too briefly of course to carry the argument, but at least, I hope, raising the issues. If they encourage response, fine. If they earn me the right to expand on any one, all the better.
Activities in mathematics for preschool children
Authors: Liedtke W. W. , Nelson L. D.
Published: October 2020

Many worthwhile mathematics activities that involve the manipulation of concrete materials and provide a background for significant mathematical processes and ideas can be designed for preschool chtldren. Challenging and interesting situations, which use material familiar to the child, can be devised. The child's responses to such situations can be used to identify his level of understanding and to establish appropriate patterns of communication.
On beyond the mathematics laboratory
Authors: Wilderman Ann Mcpherson , Krulik Stephen
Published: October 2020

The December 1971 issue of the arithmetic teacher was entirely devoted to mathematics laboratories. While the material presented in that issue was quite exciting, thought-provoking, and informative, it did raise a very basic question which should be considered; What can a school or individual teacher do if a centralized mathematics laboratory is not possible?
Function generating problem: The row chip switch
Authors: Greenes Carole Ellen
Published: October 2020

Teachers are always searching for motivating mathematical problems to enhance current mathematics programs. Such function generating problems as the tower puzzle, switch 'em (peg game), lucky seven, squares on a checkerboard, and regions in a rectangle help to enliven the curriculum.
Using research in teaching: Teaching for generalization: An array approach to equivalent fractions
Authors: Beardslee Edward C. , Gau Gerald E. , Heimer Ralph T.
Published: October 2020

One of the difficulties encountered when teaching the concept of equivalent fractions is that a student may be able to demonstrate equivalence with particular pairs of fractions, but all too often is unable to generalize the concept to other pairs of fractions. For example, a student may be able to establish the equivalence of 1/3 and 3/9 but encounter difficulty when confronted with the pair of fractions 5j 15 and 7/21. With this situation in mind, the authors conducted a series of investigations on how to effectively teach the concept and insure its generalization (Beardslee 1972, Gau 1972).
Perceptual burdens in learning mathematics
Authors: Osborne Alan R.
Published: October 2020

Teachers use a number of instructional strategies that depend upon physical and pictorial devices. The accumulated experiences of generations of teachers and research evidence concerning children's learning indicate the wisdom of using physical embodiments of mathematical ideas as a base for teaching.
Teacher-directed evaluation of mathematics laboratories
Authors: Wilkinson Jack D.
Published: October 2020

The evaluation of mathematics laboratories has been a thorn in the side of mathematics educators and teachers for the past decade. When is an activity called a mathematics laboratory lesson? Should a teacher emphasize process or cognition in a mathematics laboratory? Are student attitudes really improved when students work in a mathematics laboratory? Is the mathematics laboratory most succesful with lower-ability students? Should our school start a mathematics laboratory next year? Clear-cut answers to these and other attending questions have not surfaced. There is an obvious need for research dealing with mathematics laboratories.
What can a teacher learn about a pupil's thinking through oral interviews?
Authors: Lankford Francis G.
Published: October 2020

In addition to the written record a pupil makes as he uses pencil and paper to compute, it is often helpful to know the thinking associated with that record. Recent experience in a research study indicates that knowledge of a pupil's thinking, as he computes, can be determined by carefully conducted individual interviews. The study has been reported under the title Some Computational Strategies of Seventh Grade Pupils (Lankford 1972). It was based on interviews with 176 seventh-grade pupils enrolled in schools in Richmond, Virginia; Northumberland County, Virginia; Washington, D. C.; Atlanta, Georgia; Detroit, Michigan; and Denver, Colorado. In each school system, all pupils in one intact class in a single school were used.
The status of national assessment in mathematics
Authors: Martin Wayne H. , Wilson James W.
Published: October 2020

In this issue, which focuses on evaluation in mathematics education, it is both timely and appropriate to consider the status of the first assessment of mathematics conducted by the National Assessment of Educational Progress (NAEP). During the 1972–73 academic year, a national sample of 9-year-olds, 13-yearolds, 17-year-olds (including high school dropouts and early graduates), and young adults between the ages of 26 and 35 were assessed to determine their levels of attainment in certain mathematical knowledges and skills. National Assessment is now in the process of analyzing the mathematics data and has begun work on a cooperative project with the National Council of Teachers of Mathematics (NCTM) to consider the interpretations and curricular implications of the mathematics results. The purpose of this article is to explain how NAEP's mathematics assessment differs from the usual normreferenced testing programs and to preview what types of information will result from the assessment data.
National Council of Teachers of Mathematics: Nominees for the 1975 Election
Authors: Greenholz Sarah , Begnaud Lurnice , Brydegaard Marguerite , Edwards E. L. , Ferguson W. Eugene , Hlavaty Julius H. , Howitz Thomas , McNabb William K. , Schaaf Oscar
Published: October 2020

The Nominations Committee needs your help in selecti ng nominees for the 1975 election. At that time a president for NCTM and four new members for the Board of Directors will be elected.
Area approximations by use of random numbers
Authors: Hall Harvey N.
Published: October 2020

Stimulating and meaningful procedures for developing mathematical concepts are always needed in the teaching of mathematics. The methods that are presented here will add considerably to the building of concepts related to area and randomness. Very little is required in the way of materials or special skills, and the procedures have the advantage of being open-ended enough to encourage adaptations by the individual teacher or student.
Number sentence trios
Authors: Bidwell James K.
Published: October 2020

When students are studying number sentences it is fun and important to examine the solution set for the three related sentences that use >, =, and <
Magic squares: extensions into mathematics
Authors: Sawada Daiyo
Published: October 2020

Many a potentially dull practice session on addition and subtraction has been transformed into a lively encounter with mathematics through the introduction of magic squares. Figure 1 shows three of the more familiar ways in which magic squares are used. Normally work with magic squares in the elementary school does not go beyond the types of examples shown there. Further exploration of magic squares is usually limited to finding simple procedures for constructing 3 × 3 squares, 4 × 4 squares, and n × n squares where n is odd. The procedures are frequently presented in a rather rote fashion since the emphasis is on practice for addition, not on the properties of magic squares per se.
An algorithm for reducing fractions
Authors: Bell Kenneth M. , Rucker Donald D.
Published: October 2020

In the teaching of elementary mathematics, one of the areas that seems to give the most trouble is that of the reduction of simple fractions. For example, let us consider a proper fraction of the form a/b. We can reduce a/b to lowest terms by finding the greatest common divisor of a and b, denoted by GCD (a, b), and factoring it out of both numerator and denominator. This method, of course, calls for an understanding of factoring. Factoring is not beyond the realm of understanding of the students, but the method is so time-consuming and laborious that most students, especially the slower ones, are disenchanted with it and do not use it. Students then resort to the old standby method of trial-and-error, accompanied by a few rules of divisibility, which is, of course, also very time-consuming and laborious on many fractions. Too many students become discouraged with the whole process and tend to give up, thereby retarding their learning of how to properly reduce fractions.
Junior high school students as teaching aides
Authors: Herzig Shoshana
Published: October 2020

This article was intended originally to describe the planning and development of the mathematics laboratory at the Kalaheo Intermediate School in Kailua, on the Hawaiian island of Oahu. It soon became apparent, however, that it was the story of the student-aide program that should be told because the student aides were one of the most important factors in making the mathematics laboratory a success. Like many other innovations, necessity brought the student-aide program into being.
An overlooked aspect of modern mathematics
Authors: Mc Nerney Charles R.
Published: October 2020

When one speaks of the so-called modern mathematics, quite frequently the first, and sometimes the last, thought that comes to mind is new subject matter. Uneasy minds are sometimes put to rest when modern mathematics is described by such glib phrases as “the best of the old with the thoroughness of the new” and “a balance between understanding and skills.” Occasionally, the term “new emphasis” creeps into the conversation, but much more frequently the term “structure” is mentioned and the pulse rate is again sent leaping.
Pictures, graphs, and transformations a distorted view of plane figures for middle grades
Authors: Swadener Marc
Published: October 2020

With the advent of the “new” mathematics movement about a decade ago, several new topics and concepts have been incorporated in school mathematics. The common example of this is the presentation of the idea of set at all levels of school mathematics. The introduction of new concepts has been on-going; a concept that has recently gained recognition is transformations on the plane.
Reinforce division by learning ratios
Authors: Arnold William R.
Published: October 2020

At the sixth-grade level, pupils are expected to learn several concepts that are related to ratios in one way or another; for example, rational number, decimal, percent, and constant. Learning and relating these concepts is affected by factors which, to many teachers, seem difficult to reconcile. The process of abstracting the concepts, learning symbolic representations of the concepts, and making a shift from concrete or visual to formal or abstract ways of thinking about these concepts are among the factors that must be taken into consideration. With all of this, teachers consistently express concern for how to employ interesting, meaningful ways to teach the concepts without detracting from a need to teach basic skills.
Mathematics and chess
Authors: Robitaille David F.
Published: October 2020

There are a number of mathematical problems associated with the game of chess and with chessboards. The first of three such problems discussed in this article involves magic squares. It is not my purpose to describe methods of generating magic squares here, so only a brief discussion is included. (There are many references available on the subject of constructing magic squares, including an article by Ralph Munger in the February 1973 issue of the MATHEMATICS TEACHER.)
IDEAS
Authors: Immerzeel George , Wiederanders Don
Published: October 2020

Activities that contribute to the student's personal understanding of key concepts in mathematics