bluuebunny/crossref_metadata_2025 huggingface dataset:
A teaching program for experimentation with computer-assisted instruction
Authors: Fejfar James L.
Published: October 2020

Although many perspicacious mathematics teachers are interested in learning more about computer-assisted in struction (CAI), the high costs and relative inaccessibility1 of these sophisticated systems have afforded little opportunity for firsthand experience with this technological advance which will surely have far-reaching effects on education. Recently, however, a program has been developed which will “convert” a small general-purpose computer, available at many schools and campuses, to an electronic teaching machine capable of teaching skill s with facts of both multiplication and addition.2 Furthermore, the author wishes to announce th at this program is available to readers of The Arithmetic Teacher. It is hoped that the dissemination of this program will lead to improved pupil achievement, increa ed teacher and parental interest in innovation, and the enhancement of research opportunities in other places as it has at Indiana State University.
Computer-enriched instruction for the elementary teacher
Authors: Wahl M. Stoessel
Published: October 2020

Mathematics teaching through the ages has responded to the needs of the business community and the requirements of the existing culture. In this computer age, more and more high schools are adapting courses to computer instruction. Where does the elementary teacher fit into this picture? In a recent article in The Arithmetic Teacher,2 Lee R. Wolfe lists many misconceptions that seventh graders have about computers. He suggests, “The schools must provide experiences in the curriculum which tend to develop basic computer concepts.”
A plea for improvement of research in mathematics education
Authors: Lipsey Sally Irene
Published: October 2020

In the article that follows, Professor Lipsey proposes a national organization for research in mathematics education. Such a plan (NORME) would seem to have much appeal, for it could coordinate efforts toward meaningful research dealing in mathematics education.
Forum on Teacher Preparation
Authors: Mueller Francis J.
Published: October 2020

The fertility of the human mind has produced a knowledge explosion of unprecedented proportions. Hopefully that same fertility will produce the means of making the most of it. As it is, only the happiest of morons among us are not deeply and personally troubled by the skirmishes we lose daily in an effort to “keep up.” Time for reading and study become relatively shorter, something to be guarded ever more jealously and pent ever more prudently. We simply cannot afford to waste any of it in stumbling search.
On whole number computation
Authors: Brumfiel Charles , Vance Irvin
Published: October 2020

Some months ago the authors read a small booklet that describes the Trachtenberg system, a method for addition, multiplication, division, and square root extraction which depends primarily upon the memorization of a great many rules and secondarily upon the use of mental arithmetic. Trachtenberg's approach to computation suggests some interesting experimentation in the teaching of whole-number arithmetic. We would like to point out some of the directions experimentation might take, in the hope that readers of this article will be stimulated to try them.
Properties of operations: a meaningful study
Authors: Oosse William J.
Published: October 2020

One of the problems faced by middleschool-aged children in the study of mathematics finds its root in the properties of binary operations when confronting various mathematical systems. By the time these children begin the study of the properties of each of the four basic operations, they have already done extensive work with the operation itself. That is, children have learned to add, subtract, multiply, and divide long before the properties of these operations are introduced formally. By the time formal study of the operations and their properties is begun, the students are often no longer able to distinguish the property under consideration from the operation itself. Thus it becomes quite difficult to explain to some children exactly what is meant by the words “closure, associative, commutative, identity, and inverse.” Even though a teacher may succeed in fixing these concepts in the minds of his students, he must then face the problem of convincing each student, in a meaningful way, that every operation that is associative is not necessarily commutative, and vice versa. This is not an easy task, because the student is generally aware that addition and multiplication in the set of natural numbers have both properties, whereas subtraction and division in the same set have neither; in fact they do not even possess the closure property!
The meaning of meaningful
Authors: Olson Lynn
Published: October 2020

The Fiji Islanders, once a warrior people, developed elaborate war dances, the movements of which symbolized important aspects of their fighting operations. The dance movements were the meaningful symbols of the real battles. Today the wars have been abolished, but the dances have been preserved and are still performed in full costume and ritual. The movements of the dance are now meaningless with respect to the original purpose of battle. Whatever meaning the dances now possess must relate to something else, such as an interest in form for its own sake.
A game with shapes
Authors: Gogan Daisy
Published: October 2020

During their lunch period, Tom and his friend Greg invented a new game they called “Shapes.”
Of number lines and regions
Authors: Holder Lois D.
Published: October 2020

When introduced to the concept of fractional numbers as quotients, some children have difficulty in understanding why a number line can be assigned many different values. They frequently insist that number Unes of the same length must, of necessity, have the same value.
Climb the ladder
Authors: Mauthe Albert H.
Published: October 2020

Here is a new game. It is called “Climb the Ladder.” The rules of the game are very easy to learn, and you will have lots of fun playing Climb the Ladder.
Manipulative materials, geometric interpretation, and discovery
Authors: Garner R. C.
Published: October 2020

Manipulation of materials, geometric interpretation, discovery—no one of these concepts is a panacea in the teaching of mathematics in the elementary school. Each has its place, however, and all are useful in helping the child to gain an understanding of polygonal areas and how to compute them.
A pictured approach to an idea for division
Authors: Schloff Charles E.
Published: October 2020

The following is an illustration of the development of a grouping concept in division which has been used to introduce children to the abstract concept of the division of whole numbers.
Informal geometry through symmetry
Authors: Dennis J. Richard
Published: October 2020

The geometry component of the elementary school program is the basis of much I discussion today. Many of the efforts in this area of mathematics have approached elementary school geometry from the point of view of Euclid's postulates. Another approach to geometric topics is based on the idea of symmetry.* A unit using this approach would involve five to six weeks of class time and would do much to augment the elementary school geometry program.
Start small and grow
Authors: Smith Wilson E.
Published: October 2020

The question arose as to the number of different license plates that the state of Missouri could issue. The following solution was developed by two fifth-grade boys.
In the classroom: Bang, buzz, buzz-bang, and prime
Authors: Allen Ernest E.
Published: October 2020

Games often provide an excellent technique to stimulate motivation and maintain interest. This article discusses four versions of a simple counting game. Though the game is easily played by primary youngsters, it is also applicable to older children and adults.
Editorial feedback
Authors: Thomas Richard L.
Published: October 2020

It may be interesting to note that Petronella Hiehle's enrichment ideas in her article in the November 1968 issue of The Arithmetic Teacher may be extended to use at the eighth- or ninth-grade level.
President's report: The state of the Council—Golden Jubilee Year
Authors: Hlavaty Julius H.
Published: October 2020

For the National Council of Teachers of Mathematics, this is a year that looks in two directions-back, with fond nostalgia, on our fifty years of activity in improving mathematics instruction, and forward to many more years of continued contribution in this area. We are also ending a first year of “austerity,” an austerity dictated by our desire to gird our loins for the tasks ahead.
Minutes of the annual business meeting
Authors: John Lenore S.
Published: October 2020

President Julius H. Hlavaty called the Annual Business Meeting to order at 9:35A.M.
As we read
Authors: Inskeep James E.
Published: October 2020

Children with problems? Problems with children? This issue of The Arithmetic Teacher has many suggestions in answer to these two important questions. Disadvantaged child ren and children with “learned” handicaps need our particular attention if we are to truly promote mathematical literacy for all. The im portance of interesting children in mathematics and the need to remedy difficulties that they encounter are aptly illustrated in the articles offered this month. The teacher will find many practical suggestions. Supervisors of teachers and those who have responsibility for their education and in-service training will also find challenging material. This month's issue deals with disadvantaged children and remedial mathematics.
Low achiever lesson in primes
Authors: Davies Robert A.
Published: October 2020

Having students recognize primes is usually a long process with the low achiever. The following activities give learners an enjoyable opportunity to discover primes for themselves.
National Council of Teachers of Mathematics: Golden Jubilee Year activities
Authors: Brant Vincent , Brown Kenneth E. , Gates James D. , Hlavaty Julius H. , Karnes Houston T. , Schult Veryl
Published: October 2020

Reaching a fiftieth birthday or anniversary is always cause for a celebration-but what kind of a celebration? Like a rooster, shall NCTM stand on a fence and crow about what it has done? Like Lot's wife, shall NCTM look back to contemplate the past and run the risk of turning into a pillar of salt? Like a prophet, shall NCTM look into the future and make great predictions?
Prime numbers from the multiplication table
Authors: Stenger Donald J.
Published: October 2020

Topics from number theory are being studied in the intermediate grades of the elementary school and in the junior high school. Their inclusion in the curriculum can be supported by many reasons. However, the purpose of this article is not to defend the pedagogical value of teaching number theory; but to offer a suggestion as to how it can be taught in these grades. This paper will be limited to the development of prime and composite numbers to demonstrate a technique of teaching them through the use of the multiplication table. The technique is one in which the class explores the multiplication table to discover and analyze certain patterns as they relate to prime and composite numbers.
Capacity comparisons by children
Authors: McClintic Joan
Published: October 2020

Unexpectedly a long roll of carpet was delivered to our kindergarten room. Unrolled, it produced a “magic carpet” of green, which initiated an understanding of capacity of containers by the children. Standing on the carpet, there seemed ample room for all of us, as one child threaded quickly in and out among his classmates. Sitting down, where our feet had been, produced a slightly more crowded appearance to our “magic carpet,” and our helper had to slow his pace as he journeyed among the children.
What sets are not
Authors: Vaughan Herbert E.
Published: October 2020

It is convenient to begin with a story that was told to me by a mathematics supervisor. One of the teachers whom this supervisor was observing began a lesson by saying “Children, open your desks and take out your elements.” Each child took out some colored counters. Next, “Now take out your sets.” Each child took out some loops of differently colored yarns. “Now, put your green elements in your yellow set.” And so on. This teacher was being very modern by using the word “set” in class—of course, she was using it very incorrectly. A set is not a loop of yarn. And, it has no color.
Early research contributions to elementary school mathematics
Authors: Burns Paul C. , Davis Arnold R.
Published: October 2020

In the first month of the Golden Jubilee Year, it seems appropriate to reflect upon selected pioneer researches on elementary school mathematics. This article presents thirty-four examples of research that touch the following eight topics: beginning instruction; content selection; role of drill and practice; basic operations; problem solving; readability and vocabulary; disability; history and summaries of research. Since most of the researches presented here are well known, the methods and results will be described only briefly; more important, they illustrate the tremendouszeal and insight with which elementary school mathematics, and the methods of teaching it, have been studied from the time of the initial professionalization of the subject matter for the elementary school (1915-20) to about the conclusion of World War II in 1945. This period corresponds roughly to the first twenty-five years of the National Council of Teachers of Mathematics.
In the classroom: Four-by-four magic square for the new year
Authors: McCombs Wayne E.
Published: October 2020

There is an easy way to construct fourby-four (4 × 4) magic squares that will total any desired number. Before we look at this method, the term “4 × 4 magic square” should be explained. A 4 × 4 magic square is a square divided equally into sixteen smaller squares, arranged to form four rows and four columns.
In the classroom: The greatest—a game
Authors: Ruderman Harry D.
Published: October 2020

This game requires the selection of numbers randomly. One simple way to do this is to use ten ordinary 3” × 5” blank cards marked on one side with the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9—one numeral on each card. The cards arc shuffled face down and a card is selected, say the top card. The number shown on this card is the number to be used for the game. After a selection of a card is made and its number is used, the card is returned to the deck so that all ten cards will be used for the next random selection of a number.
Symmetry
Authors: Forseth Sonia D. , Adams Patricia A.
Published: October 2020

Combining art projects with math and science is an ideal way of promoting creativity in children. They become aware of how to use math and science; they learn to make and solve creative problems; and they have fun while doing it!
The danger/value of leaping to conclusions
Authors: Picard Anthony J.
Published: October 2020

Every teacher who has faced the problem of motivating slow learners knows there is not enough time to achieve higher level objectives, such as formulating generalizations and testing these generalizations. The following experiences with a group of slow seventh graders may provide some help with both of these important activities.
The child with a learning disability in arithmetic
Authors: Ruth Homan Doris
Published: October 2020

children with learning disabilities in arithmetic are often identified as being unable to learn arithmetic even when tests have indicated there is no reason they cannot do so. Teachers are perplexed regarding the discrepancy between the child's learning capacity and his achievement.
Fibonacci numbers: fun and fundamentals for the slow learner
Authors: Loftus Sonja
Published: October 2020

The problem of making mathematics more a live and more interesting for the slow learner is a challenge for teachers. Too often the slow learner is asked to repeat and repeat the same material year after year. If we as teachers can find material to stimulate students, it can be fun for us too. This paper describes my experiences with teaching slow learners Fibonacci number a topic new to me.
Teaching a unit on the computer to academically talented elementary school children
Authors: Horn Lister W. , Gleason Gary M.
Published: October 2020

The last half of the twentieth century is likely to be known as the age of the computer revolution. The computer has become an integral part of many areas of life and is likely to assume greater importance in the future. This tool, like any other, must be understood and its use mastered to provide the greatest possible benefits. The present generation of elementary school children will be intimately involved with the computer as a tool and with the social implications of its use. This unit was designed to provide insights into the nature of the computer and the type of thinking that must be practiced to utilize the machine. As the computer becomes more and more important in our society. it will be increasingly necessary to provide all students with a background in this area.
In the classroom: Fourth graders develop their own subtraction algorithm
Authors: Mae Silvey Ida
Published: October 2020

Children in the schools of today are being encouraged to experiment, to estimate, to conjecture, and to verify ideas. The development of a spirit of adventure and excitement through such activities has been noted by many teachers. However, it is easier to talk about such a spirit than to create it. Certainly it is more easily attained if guidelines for solutions to problems are kept loose and many possible avenues to these solutions are explored.
In the classroom: Multiplication football
Authors: Fennell Francis M.
Published: October 2020

Football in the classroom? Why not! As any experienced teacher knows, review of the basic multiplication facts can be quite boring to intermediate-grade children; football is another matter.
Sparking interest in the mathematics classroom
Authors: Bradfield Donald L.
Published: October 2020

From the glamorous stars of the entertainment world come the clues for sparking interest in the mathematics classroom. Basically, these stars make use of three kinds of entertainment—comedy. mystery, and drama. Through proper use of these methods of entertainment, the mathematics classroom can come alive with diversions and excitement. The intent of this article is to illustrate how to spark interest in the mathematics classroom by utilizing types of recreational mathematics that are categorized under three entertainment media headings. Suggestions will be made as to how to assimilate each type into a regular program of mathematical instruction.
Editorial comment: As we read
Authors: Inskeep James E.
Published: October 2020

The teaching of measurement is usually included in elementary school mathematics. Uses for the ideas of measurement are applicable to all segments of the curriculum. Practical measurement forms a significant part of needed skills for child and adult alike. A cursory glance at the need and the application of measurement will give ample evidence to its importance. Teaching the ideas of measurement is not an easy task. Examples of measurement must come from the real world to effectively illustrate this important subject. Measuring lends itself to activity-oriented experiences, and yet we still find teachers listing equivalents and expecting children to memorize them. This issue of The Arithmetic Teacher is devoted to the teaching of measurement in the elementary schools and represents the position that measurement should be experienced.
The angle mirror outdoors
Authors: Woodby Lauren G.
Published: October 2020

Measurement activities outside the classroom have a special appeal to children if they use some optical device. A simple and useful optical instrument can be made by fastening two small mirrors so that their faces form a dihedral angle of 45†. The instrument is sometimes called an “optical square” because it can be used to establish perpendicular lines.
“We made it and it works!“ the classroom construction of sundials
Authors: Stoessel Wahl M.
Published: October 2020

In the early days, each household had its own sundial where time was read from the slowly moving shadow traced on the dial face by the movement of the sun across the skies. “Dialling”1 was a subject included in the school curriculum. Later, the standardization of time and the general use of accurate clocks rendered sundials obsolete. Although there has been renewed interest in these time-telling garden ornaments, many of today's children have never seen them function.
Functions
Authors: Johnson David C. , Cohen Louis S.
Published: October 2020

One of the most interesting and important unifying concepts in school mathematics is the function concept.1 One often hears such expressions as “the function of the brain is to enable a person to make rational decisions” or “the function of a lamp is to give light.” However, the mathematical use of the word “function” is linked to the historical use of the word by mathematicians and scientists. Historically, mathematicians and scientists have used the word “function” to illustrate how one condition or “state” affects another. For example, they use such phrases as “distance is a function of time,” “water pressure is a function of depth,” and “the area of a circle is a function of its radius.” You will note that these expressions suggest pairs—distance and time, pressure and depth, area and radius. In the following discussion we will first review the mathematical definition of function, and then present some practical uses of functions. Part 2 of this article give examples of functions that can be used to describe some real life situations.
Through the years: individualizing instruction in mathematics
Authors: Glenadine Gibb E.
Published: October 2020

Annivcrsaries provide a time for celebration, a time for reflection, and a time for speculation. The Golden Jubilee Year (1970) of the National Council of Teachers of Mathematics is no exception as one identifies issues of long standing that confront teachers of mathematics.
The geometric continuum
Authors: Fawcett Harold P.
Published: October 2020

The first geometric transformation that had any real impact on my intellectual life occurred in the early fall of 1908 when I was a student in a small Canadian school. I disliked school and was in rebellion against those local forces designed to have me behave as though I did like it. Nothing seemed to improve until one day a new principal was assigned to our little school of twenty-three children; I have been forever grateful for her strong and sympathetic leadership. She was friendly, fair and firm, commanding our respect and stimulating our finest potentialities. One day she spoke of the famous tides of the Tantramar River which we could see in the distance, of their pressure on the sturdy dikes, built to keep them under control, and ot those rare occasions when the dikes yielded and the surrounding marshes were flooded with the restless waters from the bay. From this familiar scene she led us to the valley of the Nile where, it is recorded, the tides of an unruly river were indirectly responsible for the development of those elementary Euclidean insights so essential in establishing the boundaries of flooded property. She spoke our language. She stirred our imagination, and that drab little room which had been the source of so much personal agony actually became an inviting highway to learning. It was this “geometric transformation” that led me into the teaching of mathematics; and I cite this experience because it “ transformed” my life and continues to exercise a strong influence on the position I hold with respect to the role of geometry in the school program.
Forum on teacher preparation: Development of elementary school mathematics teaching in the United States
Authors: Mueller Francis J. , Burns Paul C.
Published: October 2020

The methods component of mathematics education in the United States has seldom been static. Particularly interesting is the cyclic nature of recurring issues and their varying proposed soltllions.
Division with common and decimal fractional numbers
Authors: Dilley Clyde A. , Rucker Walter E.
Published: October 2020

Except for long division, it is probably true that the most difficult topics for elementary school students to learn are division with common and decimal fractional numbers. In both cases, the algorithms are usually taught by referring to the principle for multiplying by I.
A critical review of the Hindu-Arabic numeration system
Authors: Hess Adrien L.
Published: October 2020

During World War II a group of clerks in the Army Transport Service found it economical to use base twelve to package objects that were to be loaded into planes or ships. They found that a group of twelve objects could be packaged as 1 × 12, 2 × 6, or 3 × 4 but that a group of ten objects could be packaged only as 1 × 10 and 2 × 5. The greater number of ways of packaging in twelves made possible better loading of a plane or a ship. [I]
Editorial comment: As we read
Authors: Inskeep James E.
Published: October 2020

Lip service to the importance of mathematics in today's world comes easily. Nearly every teacher of mathematics can give numerous examples of the way mathematics is related to the man in the street. The housewife and the plumber will readily assent to the need for mathematics in daily life. There is widespread belief that mathematics is vital and practical yet many reachers continue to teach it as merely another academic subject.
Never Underestimate the Inner-City Child
Authors: Keiffer Mildred , Greenholz Sarah
Published: October 2020

Teaching in slum schools is hard work. Children who have learned to distrust adults, to expect failure in school, and to settle most of their problems with their fists are not easy to instruct in the classroom. Such children are those who are growing up In the crowded ghetto areas in the great cities. Housing is sub-standard, minority groups are well represented and the welfare worker is a familiar person.
In the classroom: Double and double again
Authors: Schloff Charles E.
Published: October 2020

In multiplying two-digit numbers, it is oftentimes easier to mentally double one of the multiplicative factors and add rather than multiply in the more conventional fashion, to arrive at a correct solution quickly.
In the classroom: The number line and division
Authors: Jarosh Sharon C.
Published: October 2020

The number line can be very effective in illustrating addition, subtraction, and multiplication problems. However, when introducing division, it should be made clear to the student that there are two types of division problems and although the number line will give the correct answer to both, it does not give an accurate picture of both.
A graphic representation of prime and composite numbers
Authors: Schafer Dorothy R.
Published: October 2020

An introduction to prime and composite numbers in the middle grades often includes the construction and use of the Sieve of Eratosthenes. The technique of crossing out all multiples of 2 that are greater than 2, multiples of 3 greater than 3, and so on, results in crossing out certain numbers repeatedly. For instance, 30 is eliminated as a multiple of 2, 3, and 5.
Models for fractional numbers–a quiz for teachers
Authors: Sowder Larry
Published: October 2020

In this article, the tables are turned and we give a quiz for teachers. It consists of three items, with comments on the answers, and a concluding summary that includes a consideration of the implications involved.
Using research in teaching: Isolation of factors that influence the ability of young children to associate a solid with a representation of that solid
Authors: Brumbaugh Douglas K.
Published: October 2020

The purpose of the study was to investigate the ability of young children to comprehend the relationships th at exist between a solid and its representations expressed in the form of sketches or photographs.
One Point of View: The New Formalism
Authors: Engen Henry Van
Published: October 2020

Once upon a time arithmetic as taught to elementary school children was really bad. It was highly formal. There was no discovery of ideas, and it consisted almost entirely of rigid rules for computation.
A Discovery Lesson in Elementary Mathematics
Authors: Smith Lewis B.
Published: October 2020

Guided discovery lessons may lead to unexpected discoveries as we11 as to those that were intended by their teachers. The following account is based upon the experiences of a small group of advanced college students in a course designed to fulfill the spirit of the Nuffield Foundation film, I Do and I Understand. The first part of this article describes a lesson brought by a student with a specific goal—to enable the learner to predict the height of an unknown right triangle when certain other information iS provided. Because of a prevailing spirit of inquiry additional available discoveries were made. The reader will realize that the experiences described here involve not only ratios and proportions but also similarity, primes, composites, and common factors.
Ideas
Authors: Immerzeel George , Wiederanders Donald
Published: October 2020

Ideas is presented to provide you with an easy way to try out some ideas that we have found interesting in our classroom. In each issue we plan to present several worksheets. One of these worksheets will be appropriate for your students.
Number Patterns from Digit Sums
Authors: Trotter Terrel , Klaver Irene
Published: October 2020

Discovery can be an important process in elementary mathematics education. The unit described here encourages self-discovery and allows the student to occasionally make a discovery that is new even to the teacher. All that the unit requires is a pencil, some paper, a creative imagination, and a “Let's see what would happen if …” attitude toward the problem. It is based on the concept of the digit sums of numbers, specifically of sets of numbers that have a pattern linking them together.
Using research in teaching: Research suggestions: Use of time in teaching elementary school mathematics
Authors: Riedesel C. Alan
Published: October 2020

Each time a new topic is introduced in elementary school mathematics, the teacher is faced with a number of questions concerning the approach, sequence, and materials that should be used. Suggestions from resea rch are often available for aid in answering these specific questions. Also a number of qu estions concerning use of time, organization, and approach are pertinent to improving the teaching of the majority of topics in elementary school mathematics. The material th at follows poses several questions concerned with use of teaching time for elementary school mathematics instruction and gives uggestions from research for each question.
Fun with pegs and pegboards
Authors: Brong Tedi
Published: October 2020

Pegboards and pegs seem to be available in most kindergarten classrooms. Too often they sit on a shelf because there are no directions available for their use. Many kindergarten teachers allow the children to devise their own ways of using them, which is, of course, desirable. However, pegboards and sticks a re suited to introducing and developing several arithmetical concepts. The activities described here have helped me as I have worked with kindergarten and first-grade children. My experience indicates that children enjoy these activities and learn from them.
Sue's secret mathematics: One child's view of finite differences
Authors: Kessle Bernard M.
Published: October 2020

This is a true story. It happened one day as Sue—my daughter, who was 12 at the time—and I played a mathematics game.
“Remainders” that shouldn't remain
Authors: Connelly Ralph , Heddens James
Published: October 2020

Children have been required for too long to do page after page of examples of “uneven” division—division of a whole number by a whole number which results in a nonwhole-number quotient (Spitzer 1967, chap. 7). The “extra” number in the quotient—the “amount left over”—has been considered the “remainder” in a division example.
There always are more questions
Authors: Cruikshank Douglas E. , Flandre Charles De
Published: October 2020

The difference between the elementary mathematics curriculum of two decades ago and that of the last decade is in what we chant. Previously we chanted the “times tables”nd played fast-moving games of “flash card.” Today, after nearly fifteen years of revolution in elementary school mathematics, we chant the associative property and ask, “What plus what equals what plus what?” And we still cling tenaciously to the tables and flash cards. The chant curriculum has caused a continuing emphasis on lower-level cognitive skills, particularly rote memorization. We are not much better off today than two decades ago.
Teaching—for what?
Authors: Maertens Norbert , Schminke Clarence
Published: October 2020

As classroom teachers and, concomitantly, students of the teaching of elementary school mathematics, we readily accept the idea that learning is accomplished on a variety of levels. Further, our teaching methods generally reflect this. When we want quick, automatic response to number facts, we use flash cards, timed tests, and the like; but when we want to develop understanding and the abiUty to solve problems, we use quite different procedures. In the latter instance we are likely to pace instruction more evenly, provide many concrete materials, encourage pupil inquiry through group discussion and give appropriate attention to the mathematics of our environment. Experience has taught us, however, that when we attempt to evaluate such instructional efforts, we seem much less effective than we assumed ourselves to be during the original teaching process. Our instruments are often haphazard and ill-conceived, and our interpretations of test results can be equally vague and sterile. The material that follows presents a hi erarchical framework for evaluation—a framework that can help us systematically examine different levels of learning and subsequently determine appropriate evaluation activities more closely associated with tbe original purposes of the instruction.
Judging mathematical statements in the classroom
Authors: Jansson Lars C.
Published: October 2020

In a third grade classroom the teacher asks the pupils to measure the line segment shown in figure 1 to the nearest half-inch. Bob places his ruler as shown, and says that the segment is 3½ inches long because 3½ is the nearest half-inch mark on the ruler.
Mathematics laboratories
Authors: Matthews Geoffrey , Comber Julia
Published: October 2020

The “math lab” has been described as an English invention, and there may be some truth in this. Certainly, the English have taken up the idea th at young children learn by doing, a central idea of Piaget. The message is simple: Young children learn through becoming involved through acting and through experiencing mathematics rather than just through “lea rning” it from a book.
Using research in teaching: Laboratory settings in mathematics: What does research say to the teacher?
Authors: Vance James H. , Kieren Thomas E.
Published: October 2020

In a recent issue of the Arithmetic Teacher, J. Fred Weaver (1971) warns that haphazard or faddish use of the notions of “experience” and “activity” in mathematics may be undermining the achievement of the essential cognitive goals of the mathematics program. Thus, any conscientious teacher should ask, “What can I expect as o utcomes of a mathematics laboratory?” In trying to answer this question, one might turn to reported research and evaluation concerning mathematics laboratories and the activity type of programs. In doing so, one must realize that there are real limitations on the answers one will find. The limitations are both quantitative and qualitative in nature. While there seem to be ever-increasing amounts of manipulative and mathematics-laboratory materials available, there are a relatively small number of good research or evaluation efforts. Furthermore, because these studies took place in real classrooms with real children, pecial limit ation come with any conclu ions d rawn from them.
More than just a name
Authors: McMahon William E.
Published: October 2020

Children and mathematics are natural enemies, or so the reults sometime seem to indicate. The chief cause of this conclusion seems to be that mathematics students are too often asked to do things they don't know how to do. One solution for this difficulty is ability grouping, but the next step spell trouble: what names to give the groups.
Project SOSO (Save Our Slow Ones)
Authors: Howell Daisy
Published: October 2020

During the spring semester of 1970, three members of the Mathematics Department of Delta State College were asked by the Greenwood, Mississippi, City School System to come and work with a group of elementary teachers who were teaching low-achieving students at the sixth-grade level. As a result of this teaching experience, a request was made to Delta State College for a small research grant from state funds allocated to colleges for such purposes. College officials were enthusiastic about the project and granted the request. Thus work began on Project SOSO (Save Our Slow Ones) in the summer of 1970 and was completed in midsummer of 1971. Basically, this project was designed to help elementary teachers improve methods and techniques for teaching lowachieving students in mathematics.
Things you can try: The concentration game
Authors: Hampton Homer F.
Published: October 2020

The use of drill has fallen into disfavor in recent years. In spite of the shortcomings of drill in and of itself, it does seem to make a worthy contribution to a student's learning experience. Perhaps we can improve the setting in which drill experiences are conducted and benefit from drill without incurring its faults. The current image of drill is vague, but its theme could well be “reoccurring but varied contacts with a concept, skill, or procedure.”
Using research in teaching: Seeking little Eulers
Authors: Bush Mary Thomas
Published: October 2020

Can teaching sixth and seventh graders be fun and creative? An enrichment unit that the author has found is both centers on problems of the type “Trace out a figure without lifting your pencil from the paper.” In particular, students were asked to trace out paths on certain figures called networks. The goal of this path study unit was to discover conditions under which it is possible or impossible to trace a path.
Problems associated with the reading of arithmetic
Authors: Barney Leroy
Published: October 2020

If a child is not successful in arithmetic, many teachers mistakenly assume the entire problem to be a “natural” dislike for the subject. The child's lack of success may be characterized by incomplete homework, inaccurate work, unwillingness to explore by himself the field of mathematics beyond those assignments made by the teacher in the standard text, inattentiveness, and a verbally pronounced discontent with the subject. Admittedly, each of these symptoms could be associated with a dislike of mathematics, but each of them could also be attributed to reading problems associated with mathematics and not to mathematics per se.
Divisibility rules for numbers expressed in different bases
Authors: Wardrop R. F.
Published: October 2020

The divisibility rules for numbers expressed in base-ten numerals are not usually taught to prospective elementary teachers as a separate unit, but in most curricula are integrated with the elementary arithmetic topics. As this author has noticed in teaching numerous off-campus courses for in-service elementary teachers, the majority of these teachers are not usually aware of divisibility rules other than those for two, five, and nine.
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 worksheets that focus on a key concept of mathematics are presented. At least one of these worksheets will be appropriate for your students
Piaget and pedagogy: fundamental relationships
Authors: Sawada Daiyo
Published: October 2020

Over the past ten years, and with increasing frequency, the name of Jean Piaget has become a familiar sight to the eyes of mathematics teachers, especially at the elementary school level. Numerous books have appeared (Almy 1966; Athey and Rubadeau 1970; Beard 1969; Copeland 1970; Furth 1970), along with a large number of journal publications, th at address themselves to the topic of Piagetian implications for teaching.
Giving meaning to the addition algorithm
Authors: King Irv
Published: October 2020

In the February 1971 issue of the Arithmetic Teacher, Henry Van Engen stated his belief that the formalism of the new mathematics programs is inappropriate for elementary school children. My experiences over the past few years lend strong support to Van Engen's statement, and in this article I examine one specific use of such formalism—the attempt to give meaning to the algorithms of arithmetic. After the attempt is analyzed, an alternative approach will be presented. Although my comments apply to all four basic operations, the discussion will be restricted to the addition algorithm.
“Pica-Centro”—a game of logic
Authors: Aichele Douglas B.
Published: October 2020

On my recent trip to Venezuela I came across an interesting and amusing game that I find is popular with many different groups of students. The game was explained to me, and in a short time I found that I could play “Pica-Centro” with some degree of competency. Since returning to the United States, I have shared this game with several teachers and colleagues who heartily endorse its use on an enrichment basis.
Using research in teaching: Text books, transitions, and transplants
Authors: Callahan Leroy G. , Passi Sneh Lata
Published: October 2020

There is little doubt that the textbook still plays an essential role in determining elementary school mathematics programs in the schools of this country. Although there appears to be a ground swell of interest in alternatives to the dominant role of the textbook, established methods of operating in elementary school classrooms are not quickly or easily changed. It is quite obvious on visits to typical classrooms that the textbook is a part of the “establishment.”
Mathematical concepts and the postage stamp
Authors: Niman John
Published: October 2020

There are increasing demands on the mathematics teacher to integrate pupils' individual experiences and interests with the learning of mathematical concepts. Some of these demands can be met by using the many activities that are available in the mathematics laboratory. Other ways, not yet fully explored, also exist. For example, one can investigate mathematical relationships through art, dance, music, poetry, newspapers, comic strips, and hobbies.
Finding the missing addend, or checkbook subtraction
Authors: Johnson Paul B.
Published: October 2020

A subtraction algorithm that simplifies the balancing of one's checkbook as well as illustrates subtraction principles is a variant of the so-called Austrian algorithm for subtraction. It uses the “find-the-missingaddend” approach to subtraction and focuses attention on combinations of singledigit addends that give a sum with an indicated digit. It shortens the answering of problems like 126 − (28 + 15 + 46)
A discovery approach with ancient numeration systems
Authors: Keller Robert W.
Published: October 2020

Included in many recent elementary and junior high school mathematics textbooks is an introduction to one or more numeration systems. In addition to Roman numerals, which have been a part of the curriculum for many years, students now might learn about the Egyptian, the Mayan, the Babylonian, and other systems of numeration.
Things you can try: A student-constructed game for drill with integers
Authors: Cantlon Merle Mae , Homan Doris , Stone Barbara
Published: October 2020

In teaching a unit on integers, teachers are invariably faced with the age-old problem of how to provide interesting and meaningful drill. Popsicle sticks can be used to provide a motivating, “hands on” laboratory exercise in which the students build their own game. The anticipation of using what they are studying in a fun situation seems to be an added motivational factor for students. Tbe exercise shown in figure 1 is a suggestion for the first activity, in which the children make the materials and experiment with the “Popsicle-Stick Game.”
Using research in teaching: Open sentences: some instructional considerations from research
Authors: Grouws Douglas A.
Published: October 2020

Open sentences such as 21 + N = 59, 9 + N = 17, 18 − N = 9, and so on, occur regularly in elementary school mathematics textbooks. Children learn to solve sentences such as 9 + 6 = N and 38 − 19 = N, where the placeholder is by itself on one side of the equality sign, much earlier than open sentences where the placeholder is in a different position such as 14 − N = 8 or N + 17 = 56. Children's performance is not easily predicted when open sentences of the latter forms are involved. For example, it is not clear which, if any, of the open sentences 24 + N = 58, N + 24 = 58, 58 − N = 24, or N − 24 = 34 children have the most difficulty solving. Which would you predict?
Models and mathematics
Authors: Fennema Elizabeth H.
Published: October 2020

Although there is reasonable agreement about which mathematical ideas should be taught in the elementary school (Begle 1966) and that these ideas should be taught meaningfully (Dawson and Rud-dell [a] 1955), there is little agreement on how learning environments should be structured to facilitate this learning. One reason for this lack of agreement is an inadequate recognition of the role that concrete and symbolic models can and should play in facilitating the learning of mathematical ideas.
Elementary school mathematics: a word of caution and a question
Authors: Fletcher Robert F.
Published: October 2020

One of the most valuable features of the modern mathematics movement has been the relatively successful elimination of patently incorrect mathematics from modern elementary school mathematics programs. Perhaps it is now time that we ask ourselves, “Are we regressing?” A look at the treatment of the associative property found in some materials designed for elementary school use may point to the desirability of “a word of caution.”
National Council of Teachers of Mathematics: Auditor's Report
Authors: McGivern Raymond H.
Published: October 2020

We have examined the accompanying Balance Sheet of Funds of the National Council of Teachers of Mathematics as of 31 May, 1972, the related Statements of General Fund Revenues and Expenses, and the Statements of Fund Balances. Our examination was made in accordance with generally accepted auditing standards and accordingly included such tests of the accountitig records and such other auditing procedures as we considered necessary in the circumstances.
Codes for Boys and Girls
Authors: Willerding Margaret F.
Published: October 2020

In the november 1952 issue of The Mathematics Teacher Andree1 suggested a unit on cryptography as a motivating factor in the teaching of General Mathematics. As a direct result of this article an in-service class in cryptography was taught by the author during the fall semester of 1953–1954. The class consisted of elementary school teachers, high school mathematics teachers and an elementary school principal.
Development of Ability to Estimate and to Compute Mentally
Authors: Sauble Irene
Published: October 2020

At the present time in all arithmetic programs increased emphasis is being placed upon development of ability to compute mentally and to obtain reasonably close approximate answers for computations in all four processes. Attention to this aspect of arithmetic teaching is essential because it contributes to both of the commonly accepted large general aims in the teaching of arithmetic: the social aim and the mathematical aim.