bluuebunny/crossref_metadata_2025 huggingface dataset:
Discovering properties of the natural numbers
Authors: Griffin Harriet
Published: October 2020

Have you ever known the joy of discovering a mathematical rule or relation? Surely you can recall the feeling of satisfaction that comes from finding a proof of a theorem, whether or not you formulated its statement. In this paper we propose to review some of the basic rules about t he natural numbers 1, 2, 3, etc., often called the whole numbers or arithmetic integers, to show that these numbers provide the superior student in the upper grades of the elementary school with many opportunities for discerning mathematical properties. Such considerations will not only sharpen the student's insight into like problems, but will also help him to develop a relish for such thoughtful activity. Investigations he can make independently, after some fundamental ideas have been explained, will also convince him of the need for distinguishing between the process of examining objects to draft a statement about a characteristic they possess and the task of constructing a proof to show whether or not the statement is correct.
The challenge of mathematics in the elementary school
Authors: Struthers Joseph A.
Published: October 2020

What mathematics is appropriate to the elementary school? Logical implication and contradiction? Plotting ordered pairs of real numbers as points on the coordinate plane? The slope-intercept method of graphing linear equations in two unknowns? “Nonsense,” says one, “those concepts are far too complicated for elementary children.” “Remember,” say another, “we teach children, not subject matter.” Let's see what is going on in some Colorado classrooms.
Developing arithmetical inquiry with enrichment aids
Authors: Crescimbeni Joseph
Published: October 2020

Instructional materials used in the elementary school classroom have undergone radical changes in the past few years, especially in the field of arithmetic. Yet, regardless of the types and kinds of materials that are developed for basic curriculum instruction, enrichment aids are necessary as supplemental and motivational learning devices for all children.
Parent education
Authors: Fred Weaver J.
Published: October 2020

Teacher education has been, and for some time will continue to be, a problem of major concern as we seek to implement effectively various contemporary programs of mathematics instruction. The Focal Point section and other portions of The Arithmetic Teacher have reflected this concern frequently through articles, reports, bibliographies, editorial comments, and so on. A variety of approaches and materials have been suggested for inservice education and also for pre-service education
From number lines to 2-D space concepts
Authors: Overholt Elbert D.
Published: October 2020

The number line has many uses in the elementary school classroom. It may help the child visualize the basic processes, clarify concepts of decimal fractious, and provide many other useful, semiconcrete experiences. This discussion considers some of the relationships that exist among number sentences, the basic processes, number line games, and the association of number pairs with positions in space.
Use of class time in eighth-grade arithmetic
Authors: Zahn Karl G.
Published: October 2020

In recent years, nationwide attention has been focused on mathematics teaching in the schools. The basic concepts of arithmet ic are taught in the elementary school. If a student is to succeed in a high school college preparatory program, understanding of these concepts must be achieved by the completion of the eighth grade. This achievement is measured by a standardized arithmetic achievement test. The score on this test is usually a determining factor as to whether or not a student will be admitted to a ninth-grade algebra class.
Pattern for discovery: prime and composite numbers
Authors: Hewitt Frances
Published: October 2020

A technique, based upon the manipulation of square regions, may be used to help children discover that the counting numbers may be classified according to the number of factors of each.
Games for the early grades
Authors: Deans Edwina
Published: October 2020

Games are one means of helping young children develop number concepts.
In-service education and the learning of conceptual mathematics
Authors: Whitman Nancy C.
Published: October 2020

One of the primary concerns of iu-service programs in mathematics is to improve the teacher's conceptual knowledge of mathematics. This is not to imply that the computational and applications aspects of mathematics are of les er importance, for they are of equal importance, but because the conceptual aspect of mathematics in the past has been badly neglected. This paper is concerned with the gain in conceptual knowledge, in particular of arithmetic, of teachers after haYing had inserYice education in mathematics.
A concerned teacher makes the difference
Authors: Paschal Billy J.
Published: October 2020

The chief cause of low achievement of disadvantaged children is that too many teachers and principals honestly believe that these children are educable only to an extremely limited extent. Disadvantaged children rarely exceed the expectation level of their teachers, and it is devastating for these children to have teachers who have a low expectation level for their learning. In fact, children with adequate intellectual endowment may do poorly in school because they perceive themselves as not able. The teacher's negative image of disadvantaged children lowers the children's self-perception or self-image and affects their academic achievement and classroom behavior.
More games for the early grades
Authors: Deans Edwina
Published: October 2020

Play a game of guessing and checking the weights of different children and of different objects in your room. Let several children lift an object in turn. Each will guess the weight and record his guess. Then the object is weighed to see which children came closest to the correct weight. Children are encouraged to use good judgment in making guesses. For example, if a child who has just been weighed stands beside the child who will be weighed next, there is some basis for comparison.
M is for MEANING in mathematics
Authors: Watanabe Ellen
Published: October 2020

A tremendous Kapiolani School PTA hit was recorded a year ago (March, 1965) when the teachers of Kapiolani School in Hilo, Hawaii, presented an unusual and informative Math Fair for its PTA members. Planning and working under the leadership of Mrs. Samang L. Kong, principal, the faculty presented a Math Fair which portrayed a chronological and systematic development of basic number concepts starting in the kindergarten and Grade 1 and extending through Grade 6. Hundreds of number devices, colorful student charts, arithmetic aids, each with an accompanying explanation, were colorfully and vividly arrayed by grade levels throughout the huge school cafeteria.
Editorial comments: As we read
Authors: Hamilton E.W.
Published: October 2020

Several times in the last six years, I have pointed timidly to discrepancies or deficiencies in articles which we have published. Many times, articles have been held up pending the addition of supporting tables and elaboration before they were presented to our readers. Even then we have not always come as close to the standards set by Dr. Brownell as we should have liked to have done.
The structure of a fraction
Authors: Larson H. L.
Published: October 2020

Just as the linguist is endeavoring to find “basic thought units” that form the underpinnings of language structure, so the mathematician is searching for the simplest ideas which formulate the beginnings of conceptual thought. Educators are now aware that the teacher's intuition is often the block that prevents communication with the student. Or, to put it differently, the teacher often expects the student to understand the concept at much too high a level in the introductory lesson. All too often, the student must accept the “explanation” without real understanding, which means dull and obedient memorization. In time, the concept takes on some semblance of meaning as the student's own experiences relate to it. However, if the educator could develop the concept from the basic thought unit in the first place, the student would recognize this structure in many and more sophisticated environments without having memorized much of anything.
Graphing linear equations— discovery lesson
Authors: Gold Sheldon
Published: October 2020

A discovery lesson sometimes turns out to be a discovery for the teacher as well. This was my recent experience. In my three seventh-grade classes and one eighth-grade class, I devote one lesson each week to teaching modern mathematics using Madison Project materials.
Don't forget the parents!
Authors: Daniels John W.
Published: October 2020

In order to move forward in education, we must not overlook the parents of children who are involved in the study of modern mathematics. We must keep in mind that the concern of these parents, who have not been exposed to any modern mathematics, is genuine. Therefore, we should strive to help parents to achieve a better understanding of modern mathematics.
Mathematics teachers, on guard!
Authors: Meserve Bruce E.
Published: October 2020

It has become an NCTM tradition to have a presidential “State of the Council” address immediately preceding our annual business meeting. Last year I described in detail our very healthy, professionally active, financially sound organization. This year the title that I have selected is purposely ambiguous. I hope that each of you will find in my remarks several challenges, several alerts for your daily professional activities, and the recognition of several situations that you will want to encounter with great care: Mathematics teachers, on guard! Alert! Watch out!
All is not nonsense
Authors: Ferrer Mcfarland M.
Published: October 2020

It is interesting to see the sudden jumping on the bandwagon to criticize the “new math.” Among its critics are some who formerly advocated its characteristic topics but who are now trying to remove them from curriculum developments.
A game of fractions
Authors: Junge Charlotte W.
Published: October 2020

This game may be played by two, three, or four children.
A bee on a point, a line, and a plane
Authors: Sganga Francis T.
Published: October 2020

Most of the new elementary school mathematics programs place a great deal of emphasis upon such basic concepts as the number line, one-to-one correspondence, and number pairs. While there is general agreement that these concepts are important, a close study of how these concepts are developed indicates that it would be quite helpful to teachers and their students if greater emphasis were placed upon the significance of these concepts.
The number line in the junior high school
Authors: Hursh Ronald E.
Published: October 2020

There has never been a more interesting time to be teaching mathematics. The new programs are rich in subject matter and are a challenge to students and teachers a like. We are building mathematical concepts in addition to drilling on fundamentals in the same amount of time which previously had been used for drilling alone. This presents a substantial problem to which, I am sorry to say, I do not have the total solution. My concern is with being able to teach all of the mathematics desired in the time that is available. There is one device of instruction in the new materials that I believe has great potential in conveying mathematical ideas quickly and accurately. This device is the number line.
Children use signed numbers
Authors: Cochran Beryl S.
Published: October 2020

The arithmetic of signed numbers is one of the “big ideas” in mathematics that I find young children almost compelled to invent for themselves.
The Trachtenberg system as a motivational device
Authors: Yates William E.
Published: October 2020

The poorly motivated arithmetic student reacts, not to the long-range value of a given quantity of arithmetical knowledge, but to the immediate use to which it can be put. In the final accounting, the value of this knowledge is determined by the student, not by the teacher. To help the student develop this sense of value we attempt to increase the meaningfulness of arithmetic by providing activities to demonstrate its practical application in everyday situations. Such an approach is usually successful in reaching the majority of these students. But not all will respond to this form of motivation. To some slow learners in the classroom these activities are only briefly amusing, and as a result the pupils are unable to become more than momentary participants in the learning situation.
Geometry for the disadvantaged
Authors: Paschal Billy J.
Published: October 2020

One day a critical Visitor came by the elementary school where an experimental research study in teaching modern mathematics to disadvantaged first graders was being conducted in order to see how the children were doing. She selected a little boy who was a repeater and asked him what they were studying. He proudly stood up and without hesitation said, “Circul ar regions, rectangular regions, and triangular regions!” The little boy, with his big eyes sparkling in anticipation, anxiously awaited another question that never came. The critic smiled in speechless amazement and left the room.
Arithmetical brain-teasers for the young
Authors: Winthrop Henry
Published: October 2020

An interest in numbers is sometimes stimulated in the elementary grades by arithmetical brain-teasers. Provided below are a number of arithmetical brain-teasers that range in difficulty from items that may be employed in the lower grades to items that can be used in those schools than have introduced a little elementary algebra in the seventh and eighth years. Without more ado let me present these in an order that reflects, in my judgment, their increasing difficulty.
Geometry, yes—but how?
Authors: Smith Lewis B.
Published: October 2020

Reading, writing, and arithmetic. No one has yet suggested that these traditional three R's be expanded to include geometry. But though it is not listed separately with the top three, geometry is in fact becoming an increasingly important subject in our school program.
Some mathematical ideas involved in the Mirror Cards
Authors: Walter Marion
Published: October 2020

Look at the pictures in Figure 1. You will probably notice that the pictures have something in common. What this common property is will be described differently by different people.
Making a counting abacus
Authors: Cunningham George C.
Published: October 2020

Use of the counting abacus (Fig. 1) may begin in the first grade and continue through high school level in general math courses. It is a very effective tool in showing the action of addition, subtraction, and regrouping. By changing the number of beads used, a teacher may show these same actions in other base systems.
The quest for an improved curriculum
Authors: MacLean J. R.
Published: October 2020

For the past several years teachers have been almost overwhelmed by successive waves of propaganda that has generally been critical of our “traditional” mathematics programs. Unless set notation and symbolism, non-decimal numeration, modular arithmetic and strict attention to correct terminology (such as distinction between number and numeral) were included in our courses of study, we were not up-todate. Our students would be ill-prepared for their responsibilities in the Space Age and we were effectively sabotaging the future of our country.
Next steps in school mathematics
Authors: Johnson Donovan A.
Published: October 2020

Our society is a dynamic, changing society in which knowledge increases rapidly and unpredictably. Technological advances make available devices which have wide application for new types of instruction in our schools. Federal support for education provides funds to develop new programs for learners of varied ability. Experimental groups continue to prepare revised curricula for the elementary school. Textbook companies are continuing to publish new texts. In terms of these factors, what can we conjecture about the resultant changes in school mathematics? Certainly the adoption of an up-to-date text will not keep us in step very long.
Comparative achievement with departmentalized and self-contained classroom organization
Authors: Price Edette B. , Prescott Arthur L. , Hopkins Kenneth D.
Published: October 2020

The most advantageous organizational structure for elementary classrooms has been the subject of much discussion in educational literature. Departmentalization was common practice through the twenties and thirties, declined during the forties, and then returned in popularity. 1, 2 The departmentalized classroom organization provides a “teacher-specialist” for each of the subjects taught. In the self-contained classroom, one teacher instructs a class of children of a given grade in all the curricular areas. The advocates of departmentalization have emphasized the superiority in having a specialist teaching each subject, the implication being that greater learning would accrue.
Now what?
Authors: Froelich Effie
Published: October 2020

I work the hardest on mathematics! I get my lowest marks in math! But I learn the most and just love to come to class."
Unusual problem solving
Authors: Johnson David C.
Published: October 2020

The primary concern of this article is to exhibit a situation for developing the problem-solving ability of the reader. However, the sample problem is also very appropriate for your own students. The problem is of a different nature from those of the preceding article, by Louis Cohen. This problem does not lend itself to the immediate writing of, and subsequent solution of, a mathematical sentence. Rather, the basic objective in attempting to solve the problem is that the individual should discover and generalize from certain given information. This is one of the most important objectives of mathematics education.
Extending the impact of research on mathematics education
Authors: Fred Weaver J.
Published: October 2020

Few persons, if any, would deny that “the Golden Age of educational research is now with us” [8, p. 367].1 This is true for educational research in general, and for research on mathematics education in particular.
Conservation of length and the teaching of linear measurement: a methodological critique
Authors: Sawada Daiyo , Nelson L. Doyal
Published: October 2020

Important insights into the young child's conception of basic mathematical and scientific concepts have arisen out of the experimental work conducted by Jean Piaget [5, 6, 7]. * In assessing the degree to which a child has acquired a concept, Piaget often uses the criterion of conservation. He has shown that children of four to six do not in general conserve certain basic properties of objects or sets of objects. For example, a young child may think that the number of blocks in a row is increased if the row is elongated, or that a wooden rod becomes shorter when it is rotated through 90 degrees. Such a child is said to be a “nonconserver” of number in the first case, and a “nonconserver” of length in the second case.
Measurement in the elementary school
Authors: Paige Donald D. , Jennings Margaret
Published: October 2020

This paper represents a survey of thirtynine complete or partial series for elementary school as concerns their presentations of various areas of measurement. Some of the methods of teaching measurement will be discussed and evaluated. In addition, some suggestions for the improvement of the teaching of measurement will be suggested.
Teaching time concepts to culturally disadvantaged primary-age children
Authors: Dutton Wilbur H.
Published: October 2020

Children's time concepts have their beginning in daily experiences and at a very early age. The sequence of events such as eating, sleeping, playing, or going to school is the beginning of an understanding of time and the ordering and sequencing of time.
The importance of definitions in mathematics: zero
Authors: Newman Claire M.
Published: October 2020

The elementary school mathematics teacher has an important job—that of laying the foundation for his pupils' future work in mathematics. However, in his efforts to teach mathematics so that children understand the concepts behind their work, there may be a tendency to overlook the fact that all ideas in mathematics cannot be justified. The very nature of mathematical thought is such that there are some basic ideas we must agree to accept.
Applications of binary notation
Authors: Perisho Clarence R.
Published: October 2020

Binary notation is always fascinating because of its simplicity. There are only two digits, 0 and 1. The arithmetic is simple: there are only four possible basic facts for addition and only four for multiplication. However, many applications do not involve addition or multiplication but are concerned with situations where the notation itself is interesting or useful.
Colors, geometric forms, art, and mathematics
Authors: Hillman Thomas P.
Published: October 2020

With the rising popularity of “pop” art, “op” art. and stunning geometric designs in femi nine fashions, the author is reminded how easily art and mathematics can be closely related, especially in the upper elementa ry school or junior high school classrooms. For students who have been convinced that mathematics is always associated with calculations or manipulations (or even hallucinations!), the following suggestions can arouse new interest in mathematics—and perhaps reestablish vigorous efforts when regular mathematical topics are pursued again. These lessons arc introductory art lessons that have ma thematical overtones, and it is im portant that the student not become too burdened with the mathematical ideas at the expense of losing interest in art. The concepts in both subject areas are simple and basic. and each student is given an opportunity to discover relationships for himself—in art as well as in mathematics.
Tinkertoy geometry
Authors: Richards Pauline L.
Published: October 2020

Finding suitable instructional aids for my fifth-grade geometry class was a problem until I discovered the versatility of Tinkertoys.
One
Authors: Jordan Arthur E.
Published: October 2020

Did you ever have the opportunity to “brainstorm” a mathematical idea? Near the close of a course in “Problems in Teaching Arithmetic in the Elementary School” this technique was employed to review some of the structural concepts of contemporary elementary mathematics. How important is one? Does one have any unique qualities, functions. or properties? The statements which follow were selected from the responses of the class members.
An investigation leading to the Pythagorean property
Authors: Junge Charlotte W. , Froelich Effie
Published: October 2020

It becomes very frustrating for a high school pupil to try to understand a long, involved, and rigorous proof, when he has no fundamental grasp of what he is trying to prove.
Suggested content for research articles published in “The Arithmetic Teacher”
Authors: Riedesel C. Alan
Published: October 2020

Over the years since its advent in 1954 The Arithmetic Teacher has developed into the largest publisher of research articles concerned with elementary school mathematics. In fact, out of 799 research studies identified by Suydam1 (published in fifty journals between 1900 and 1965), 158 appeared in The Arithmetic Teacher. This number represents more research reports in elementary school mathematics than were published by any other journal. This publication record is very appropriate since The Arithmetic Teacher reaches a wide audience of persons interested in mathematics on the elementary school level. It is read by classroom teachers, supervisors, principals, mathematics educators, and educational researchers interested in elementary school mathematics.2 With this audience, which varies greatly in research background, it is essential that reports be accurate, informative, concise, and readable. It is our contention that a research article can be so clearly written that it is informative to both the technical and nontechnical reader.
A comparison of three stratagems for teaching mathematical concepts and generalizations by guided discovery
Authors: Henderson Kenneth B. , Rollins James H.
Published: October 2020

Of interest to a teacher who wishes to teach by guided discovery would be evidence about the relative efficacy of patterns or stratagems he could use. The present study sought to obtain evidence about three such stratagems.
Frequencies of unwritten algorisms
Authors: Pigge Fred L.
Published: October 2020

The material presented in this article is an extension of a research study designed to determine the relative effectiveness of three methods of teaching addition and subtraction of fractions. The results of the original study were published in the January 1965 issue of The Arithmetic Teacher1.
Editorial comment: As we read
Authors: Inskeep James E.
Published: October 2020

Creative teachers seldom miss the relationship between science and mathematics. Considerable appeal has been made to the apparent similarities found in the mathematics and science curricula. The man in the street almost always puts the scientist and the mathematician in the same category. However, some mathematics teachers and mathematicians react unfavorably to the identification of mathematics with science. The purist will make a distinction. The practitioner will laugh and teach what his children need to know. We do not teach mathematics or science exclusively in the elementary school; we teach mathematics and science. A child leaving our elementary and junior high schools will need to know some of the deductive, axiomatic nature of mathematics. He will also need to know the thrill of experimentation and the excitement of inferential reasoning from empirical data. But, more than this, he will need to know how to mix the two in studying mathematics and science and in applying them to the world about him. This issue of THE ARITHMETIC TEACHER develops the theme of mathematics and science.
Mathematics in elementary science
Authors: Blanc Sam S.
Published: October 2020

Mathematics and science seem to be terms that are joined together in our common language almost as firmly as day and night, salt and pepper, and love and marriage! Everywhere, that is, but in our typical elementary school program. Here, for some reason, a dichotomy of emphasis occurs. As every experi enced teacher knows, mathematics is commonly taught quite separately from science. Children may not be led to discover the relationships that exist between these two major investigative activities of the human mind.
Calculators in the classroom
Authors: Van Atta Frank
Published: October 2020

At the risk of starting anew the arguments about computers versus computation, it might be worthwhile to examine the contention that many problems that cannot be done by the pupil alone can be handled by a “pupil + calculator” combination.
Horizontal enrichment with graphs
Authors: Schell Leo M.
Published: October 2020

Many line graphs used in elementary school mathematics describe linear functions of some kind, e.g., time-earnings relationship. But in reality, most such correlations are not neat point-to-point relationships but rather are an estimate based on a line drawn through a scatter plot. The following material, designed to supplement a unit on graphs or on functions without “stealing” content from succeeding grades, dealt with both rectilinear and curvilinear functions where the plotted points do not lie on a straight or a smooth line.
Mathematical spelunking
Authors: Dodge Winston E.
Published: October 2020

It seems as though a few of the bright students do all of the discovering, and the advantages of the method are denied to the bulk of the class.”
The status of research on elementary school mathematics
Authors: Suydam Marilyn N.
Published: October 2020

During the past few years, an increasing amount of emphasis has been given to research. The need for direction in the teaching-learning process in all areas of the curriculum has been intensified, and the development of curriculum reform movements such as that of “modem mathematics” has accentuated that need. Research should be a source of knowledge about curriculum content and methods and a foundation for decisions about curriculum innovations.
Geometric representation of binomial by binomial—laboratory style
Authors: Peskin Anne S.
Published: October 2020

Although the flannel board with felt pieces is usually used in the elementary grades, I have a lso found it to be very valuable as a visual aid in the upper grades. The pieces a re movable and removable, and the pupils can experiment with the arrangements of the pieces. One topic that can be made exciting by its use is the multi plication of (a ± b) (a ± cb) where c is an integer and a > cb by choice.
From the editor's desk
Authors: Brydegaard Marguerite
Published: October 2020

We appreciate letters and manuscripts from our readers. As noted in the January issue, we receive relatively few quality articles concerning learning in the primary grades. We extend an invitation to you to submit for consideration for publication manuscripts that develop fresh and inspiring ideas.
Humpty-Dumpty's lesson in arithmetic
Authors: Adkins Bryce E.
Published: October 2020

Many elementary school teachers are aware that Charles L. Dodgson, writing under the pseudonym of Lewis Carroll, was the author of Alice in Wonderland and Through the Looking Glass. They may also know that he was a mathematician and logician; his writing includes at least ten books pertaining entirely to mathematics and logic.1 Yet these same teachers, who read of Alice's adventures to their pupils, often fail to grasp the significance of Humpty-Dumpty's statement, “When I use a word, … it means just what I choose it to mean—neither more or less.”2 Yet herein lies the key to one of the difficulties that plague elementary school teachers when they try to increase their knowledge of mathematics—that is, the use of different terms by different authors to name the same ideas. Even more confusing to these teachers is the assignment of different meanings to the same terms.
Methods, math, and mothers, or what can a poor parent do?
Authors: Snedeker Noreen D.
Published: October 2020

There I was making peanut butter and jelly sandwiches for the lunch box, when my son said,
Cognitive outcomes of the SMSG mathematics program in Grades 4, 5, and 6
Authors: Grafft William D. , Ruddell Arden K.
Published: October 2020

Evaluation has not been a strong feature of the new elementary school mathematics projects. Objective statistical investigations are rare. The paucity of evaluative data has been defended on the ground that an overwhelming job of curriculum reform had to be accomplished quickly. Some say time and resources have not permitted broad-scale appraisal. [1] While these arguments may have merit, there is a need for research which will provide data on the performance of students in these programs.
“Arithmecode” puzzle
Authors: Winick David F.
Published: October 2020

The dashes below, labeled with Roman numerals, represent the words of a quotation or phrase. Each dash stands for a single letter. For example, five dashes represent a five-letter word. To determine each word, the reader must examine the clues by the corresponding Roman numerals and identify the number described. Once the number is identified it must be decoded by selecting the appropriate code letter for each digit of the number. Each digit corresponds to exactly one letter of the word in the same order, but there may be two or three code letters for any given digit.
Fractions in the new elementary curricula
Authors: Botts Truman
Published: October 2020

The title of this article may be something of a misnomer, for I am going to be primarily conce rned with fractions in the newer curricula for elementary school teachers, rather than for the children themselves.
The pegboard as a fraction maker
Authors: Cunningham George S. , Raskin David
Published: October 2020

A modern approach to fractional numbers emphasizes structure as well as the interrelationship of muJtiplication, division, addition, and subtraction. The danger of too much abstraction can be averted by the use of physical manipulative materials as models for the concepts to be understood. It is helpful if a device is readily available as a model. The present article will show the use of the pegboard in developing an understanding of fractional numbers.
The Concept of Area
Authors: Bourne H. N.
Published: October 2020

Can Children of primary school age be helped to form concepts of area by means of practical experiences? The formation of a true concept of area is not easy for children before the age of 9 or ro years. The assessment of size when two dimensions are involved is obviously a more difficult mental operation than when one only has to be considered. Piaget in many of the experiments he conducted with young children often came across this difficulty they have when making a judgment involving two different factors simultaneously as for instance when considering the numbers of objects differently spaced.
Sets—natural, necessary, (k)nowable?
Authors: Geddes Dorothy , Lipsey Sally I.
Published: October 2020

Is the idea of set appropriate for the elementary school curriculum? Elementary school teachers and others have not welcomed the inclusion of this new topic wholeheartedly. Some say, “Too much rigor” Others maintain that the language is too technical. They sec confusion instead of clarity. They recognize that the notions of counting and numeration grew out of man's need to keep track of his possessions but ask, “What historical rationale can be given for the notion of set?” Since the habit of thinking in terms of sets has become routine for mathematicians and other scientists,1 it is important for us to seek answers to the questions and concerns relating to sets raised by the elementary school teacher.
Attitudes of prospective elementary school teachers towards arithmetic
Authors: Reys Robert E. , Delon Floyd G.
Published: October 2020

What are the attitudes toward arithmetic of preservice elementary education majors? At what educational level were these attitudes developed? Are the attitudes of preservice elementary education majors altered by courses in their mathematics preparatory program? Questions such as these are of paramount importance to those engaged in preparation of elementary school teachers.
Mathematics laboratories and teachers‘ centres—the mathematics revolution in Britain
Authors: Biggs Edith E.
Published: October 2020

Britain one of the educators quoted most often is Professor A. N. Whitehead. “Every child should experience the joy of discovery” has been our slogan for more than twenty years. In this respect, America has been at least partly responsible for the revolution which is taking place in British education. Although the changes did not start with mathematics, it is in mathematics that the greatest and perhaps most exciting changes are taking place today.
A mathematics program for disadvantaged Mexican-American first-grade children
Authors: Castaneda Alberta M.
Published: October 2020

This paper presents the rationale for and the content of a mathematics program written for and taught to a group of disadvantaged Mexican-American first-grade children. The point of view basic to the writing and teaching of the program was that mathematics instruction is essentially a task of fostering concept formation and language acquisition, and it was felt that this point of view would be particularly appropriate to mathematics instruction for disadvantaged Mexican-American children whose instruction would be in their second language and whose experiential background had, in all probability, been limited by the poverty of their families and the restr ictions of their ethnic group membership.
Interrelationships between mathematics and art for the kindergarten
Authors: Swartz Evelyn
Published: October 2020

When some of the current mathematics programs and proposed art programs for the kindergarten are compared, many components common to both subjects are found. An understanding of the interrelationships which exist between the two makes it possible for the kindergarten teacher to capitalize on them and to reinforce and extend children's learning in both areas. Both can contribute to the conceptual development and the enhancement of visual perception of children so that they can learn to live comfortably in a world of technology and a world of art.
The years ahead
Authors: Gibb E. Glenadine
Published: October 2020

In many facets of our society—industry, commerce, government, and education—planning staffs must be at work, planning staffs committed to predicting the future. Plans are projected. Budgets are prepared. Tax rates are fixed. Funds are allotted. Production schedules are set. Research programs are initiated. To some extent, a crystal ball is used in forecasting because assumptions are made that some variable factors will be valid.
A cooperative university-district in-service program
Authors: Catmull A. Earl
Published: October 2020

A new series of textbooks came into the mathematics programs of the forty-four elementary schools of Granite School District in the 1965/66 academic year. During the previous school year, the district bad sponsored an in-service television program one afternoon each week. Each elementary principal and teacher was expected to take advantage of this opportunity in order to prepare for the coming new programs. It was a step in the right direction, but it was quite insufficient.
A method for changing numerals in certain nondecimal bases to numerals in other certain nondecimal bases, directly
Authors: Barrett M. J.
Published: October 2020

In “Numeration Systems for the Whole Numbers,” Booklet Number 3 of the NCTM's Topics in Mathematics for Elementary School Teachers (1964), on pages 22 through 25, a method is given for converting numerals in the binary system into numerals in the octal system of numeration by dividing the binary numeral into groups of three digits, beginning at the right and counting toward the left. Each group of three digits in the binary system is then made to correspond to a digit in the octal system. For example, the binary numeral 1110101 is used and divided into groups of three digits, thusly: 1-110-101. The 1 in the first left-band group equals one in the octal system. The 110 in the second group equals six in the octal system, and the 101 in the last group equals five in the octal system. A conversion table will more clearly show why this is so. (See Table 1.) Zeroes are prefixed to binary numerals 0, 1, 10, and 11 so that if these occur in other than the first left-hand group, one will know how to read them in the conversion process.
What can you do with an egg carton?
Authors: Baumgartner Margery
Published: October 2020

I love egg cartons. Doesn't everybody? Whole books have celebrated their potentialities for art projects, and now expensive objets d'art based on the lowly egg carton are appearing on the market. But egg cartons also have value for the beginning study of mathematics.
Just for Fun
Authors: Caldwell J. D.
Published: October 2020

Some of our readers might like to try the following diversion with their classes. As many times as I have seen it used, it never fails to stimulate pupil interest and further exploration. We might also note that this particular lesson is a good example of the inductive-deductive approach.
Learning laboratories in elementary schools in Winnetka
Authors: May Lola J.
Published: October 2020

The main purpose of the learning laboratories in the Winnetka Public Schools is to help children become independent learners. We believe children have great capacity for self-Iearning if they are given materials developed so they can proceed on their own, provided they have guides to help them when they have questions or make mistakes.
Educating mathematics specialists to teach children from disadvantaged areas
Authors: Rosskopf Myron F. , Kaplan Jerome D.
Published: October 2020

One of the major problems besetting mathematical education today is the educating of competent, imaginative teachers at all levels who are knowledgeable of mathematics, methodology, and psychology. The so-called reform movement in mathematics curricula during the past ten years has underscored the need for adequate educating of mathematics teachers. Although qualified teachers are required at all levels, a serious shortage exists at the elementary school level.
Interrelationships among mental abilities, reading, language arts, and arithmetic with the mentally handicapped
Authors: Cawley John F. , Goodman John O.
Published: October 2020

Academic achievement is the foremost component of education programs for mentally handicapped children. Specific to the area of arithmetic, the tendency among educators is to stress concrete dimensions2 without adequate emphasis upon verbal problem-solving, concept attainment, and numerical reasoning.3 The justification for this type of programming seems to emerge from the assumptions that have been drawn from measures of mental abilities.
Survey research: guidelines for status studies
Authors: Romberg Thomas A.
Published: October 2020

In the article that follows, Dr. Romberg deals with surveys that can be identified as status studies. Such studies are of great interest to teachers, school administrators, and the general public.
Editorial comment: As we read
Authors: Inskeep James E.
Published: October 2020

The new mathematics! Even the term has aged! Jokes about modern mathematics have acquired the graying and wrinkling stage that comes with age. Along with sets and other-than-base-ten numeration systems, geometry has often been classed as “new” mathematics. Interestingly, geometry is an ancient mathematical discipline, and yet the teachers presently working in the elementary schools received very little of it in their formal education.
An intuitive introduction to the Euclidean concept of betweenness
Authors: Denmark Tom
Published: October 2020

If geometry is to be an integral part of the mathematics curriculum for Grades K–6, then careful consideration must be given to the provision of a sound basis for the development of geometric concepts. One fundamental notion which is often overlooked or neglected is the concept of a point B being between two points A and C. Since the Euclidean definition of a point lying between two other points is essential to the formulation of definitions for line segments, rays, and lines, this concept should be introduced early in the children's study of geometry
Tantalizing ternary
Authors: Ranucci Ernest R.
Published: October 2020

Arithmetic computation in bases other than ten is a fertile field to explore. Especially interesting are certain problems that arise in the base three (ternary) system.
Teachers, researchers: Two proposals
Authors: Alan Riedesel C.
Published: October 2020

The purpose of this month's “Focus on Research” is to make two suggestions that are designed to increase participation in this column. They are also designed to increase dialogue between classroom teachers and researchers concerned with elementary school mathematics.
Improving problem-solving skills
Authors: Sims Jacqueline
Published: October 2020

In the last few decades, a number of procedures have been developed for improving problem-solving ability. This article presents some examples of interesting approaches to the problem.
Mathematicalosterms
Authors: Mathison Sally
Published: October 2020

If you had trouble deciphering the title of this article, perhaps you should read no further. Robert E. Reys' “Mathematical Word Search” that appeared in the April 1967 issue of THE MATHEMATICS TEACHER gave me the idea for this puzzle. I was sure that my junior high students would enjoy a puzzle of this kind, but Mr. Reys made use of words such as ellipse, congruence, calculus, polynomial, contrapositive, etc. Junior high students would find these words as nonsensical as the whole array of letters. So I decided to create my own puzzle using mathematically related words that a junior high student would recognize. It is a challenge to hide as many of these words as possible in a 20-by-20 square. Try it and see.
1969 Elections Report of the Committee on Nominations
Authors: Beckmann Milton W. , Gibb E. Glenadine , Longshore Blanche , Meserve Bruce E. , Rickey Agnes Y. , Schult Veryl , Steffens Herbert R. , Trimble Harold C. , Wooton William
Published: October 2020

The Committee on Nominations presents its nominees for offices to be filled in the 1969 elections. The president-elect is elected for one year, after which he serves two years as president and, finally, one year as immed iate past president. The vicepresidents are elected for a term of two years, whereas the directors are elected for a term of three years.
We all teach mathematics
Authors: Wilson Malcolm G.
Published: October 2020

Many people think of a subject in the school curriculum as one to be taught by the person employed by the school board for that purpose. Although they do not consider themselves teachers, most people teach at one time or another.
Introduction to ratio and proportion
Authors: Brousseau Roland L. , Brown Thomas A. , Johnson Peter J.
Published: October 2020

The objective of this lesson is to introduce fifth- and sixth-grade students to ratio and proportion through a discovery situation. This approach would also be applicable for junior and senior high general mathematics students. This technique should allow students to find patterns that will enable them to verbalize algorithms for the solution of proportions. The following is from an actual demonstration with fifth graders.
A proposal for the improvement of the mathematics training of elementary school teachers
Authors: Spitzer Herbert F.
Published: October 2020

Leaders in mathematics education have long recognized that the elementary school teacher's Jack of mathematical background was a major deterrent to improvement of instruction in this curricular area. While a number of recommendations for improving the mathematics part of teacher-education programs were made before the new mathematics era,1 very little change in the mathematics preparation of elementary school teachers took place until after new mathematics began to be widely used in the elementary schools. Then, when teachers began to meet elements of strange new content, the need for additional mathematical knowledge became readily apparent.
In the classroom: Pegboard multiplication of a fraction by a fraction
Authors: Nelson Diane , Nelson Marvin N.
Published: October 2020

When a fraction is divided by a proper fraction, the answer is always larger than the dividend. In comparison, when a proper fraction is multiplied by a proper fraction, the answer is always smaller than either fraction involved. This is rather confusing to students who have observed that the product of two whole numbers is larger than either factor, except when one or both factors are 0 or 1. They have also observed that the answer when dividing whole numbers, excluding 1 as a divisor, is smaller than the dividend.
Editorial comment: As we read
Authors: Inskeep James E.
Published: October 2020

Fascinating to some and frightening to others, the computer has emerged from its infancy to become a gangling adolescent with a voracious appetite for new challenges! And yet, the computer offers wonderful possibilities to free men from routine and release them to genuinely creative behavior. For years a primary objective of elementary school mathematics instruction was to make “computers” of persons. The application of the computer should now permit us to make “persons” out of persons in our curriculum. This month's issue of The Arithmetic Teacher features articles dealing with the computer, its use in mathematics education, and its significance for the classroom.