bluuebunny/crossref_metadata_2025 huggingface dataset:
ONE POINT OF VIEW: NCTM service and the “Arithmetic Teacher”
Authors: Price H. Vernon
Published: October 2020

The Golden Jubliee Year of the NCTM has just ended, and we are embarked on a new era of service. During the pa t fifty years our member hip has grown from zero to a current enrollment of nearly 50,000 individu al members and 30,000 additional subscribers. We are recognized as the largest professional organization in the world with a major interest in mathematics and/ or mathematics education. Such growth is achieved by providing services wanted and needed by large numbers of teachers.
Interviews to assess number knowledge
Authors: O'Brien Thomas C. , Richard June V.
Published: October 2020

Increasing attention has been given the great researcher Jean Piaget by American educators in recent years. It is the current fashion for mathematics educators to look to Piaget for answers to all sorts of questions ranging from grade-placement of topics to the relative merits of “discovery” teaching. It should be noted, however, that Piaget is not an educator. He is first and foremost an epistemologist—his research is concerned with the nature and development of knowledge, particularly in children—and the attention he has given to education is minute in comparison with the wealth of attention he has devoted to epistemological problems.
“Parallelograms”: a simple answer to drill motivation and individualized instruction
Authors: Tucker Benny F.
Published: October 2020

To drill or not to drill—that is the question that “bugs” many teachers of elementary mathematics. Some who would have us teach “modern” mathematics have said that drill and memorization have no place in contemporary mathematics programs. Experience and common sense tell us that these spokesmen of “modern math” are either wrong or misunder tood. Indeed, few have actually said that memorization and drill do not have their place in mathematics education. Most experts agree that both must be an integral part of our programs but that they must be kept in proper perspective.
Evaluation scale for a teaching aid in modern mathematics
Authors: Leeseberg Norbert H.
Published: October 2020

Currcnt literature in mathematics frequently refers to the use of teaching aids and the development of mathematics laboratories. This trend can be helpful, but professional personnel must be careful when selecting aids to learning. The questions most often brought up by administrator and boards of education center around the justification of the purchase and the use of the aids. The “Evaluation Scale for a Teaching Aid in Modern Mathematics” shown on the opposite page has been developed to assist teachers responsible for initiating purchases of such material.
Things you can try: Activities with easy-to-make triangle models
Authors: Backman Carl A. , Smith Seaton E.
Published: October 2020

Strips of colored tagboard and brass fasteners can be used to make a set of models for demonstrating a variety of properties of triangles. The models are inexpensive, durable, and easy to store, and they can be made by the classroom teacher for use in classroom demonstrations. They also are easily assembled by pupils for use in mathematics-laboratory activities.
IDEAS
Authors: Immerzeel George , Wiederanders Don
Published: October 2020

Ideas presents an easy way for you to try out a variety of ideas that have a high level of student interest. Each issue of the Arithmetic Teacher includes several worksheets on a key concept of mathematics. At Jeast one of these worksheets will be appropriate for your students.
Research on attitudes toward mathematics
Authors: Aiken Lewis R.
Published: October 2020

It is sometimes forgotten that in addition to learning principles, facts, and methods in school, children learn attitudes, values, and appreciations there and, it is hoped, develop a desire for further learning. Terms such as attitude, value, and appreciation refer to affective objectives of instruction, objectives that should constitute a part of learning every school subject.
Rainy-day games
Authors: Gessel Robert C. , Johnson Carolyn , Boren Marty , Smith Charles
Published: October 2020

There are many occasions when “rainyday games” have their special value and usefulness, and, of course, rainy-day games may even be played on sunny days too. The games described here give children a chance to ask questions that enable them to obtain information; the information is then used to reach a conclusion.
Things you can try: “Contig”: a game to practice and sharpen skills and facts in the four fundamental operations
Authors: Broadbent Frank W.
Published: October 2020

“Contig” is a game that intermediategrade children love to play and with which they can challenge their older brothers, sisters, or parents. Four things are required -three dice, a score pad, markers, and a Contig board. The dice, pad, and markers are readily available, and a supply of boards can be made by reproducing the sample provided at the end of this article on sheets of paper.
Discovering structure through patterns
Authors: Meconi L. J.
Published: October 2020

One of the emphases in our modern mathematics programs is to have children investigate and discover many of the myriad of patterns that exist in all phases of elementary school mathematics. It is in the area of investigating patterns that children frequently demonstrate an amazing degree of creativity.
A Comparison of Two Methods of Compound Subtraction: THE DECOMPOSITION METHOD AND THE EQUAL ADDITIONS METHOD
Authors: Rheins Gladys B. , Rheins Joel J.
Published: October 2020

The Last Decade has seen the movement for teaching arithmetic meaningfully become dominant in American schools. In consonance with it, a rationale has developed for teaching the processes of arithmetic in accordance with its basic concept.
Arithmetic for Today's Six- and Seven-Year-Olds
Authors: Gunderson Agnes G.
Published: October 2020

Aritrmetic For Today's six and seven-year-olds is made up of activities as well as commercially prepared materials (workbooks and textbooks); such activities as playing store with real or toy money; constructing a grocery or postoffice; making number charts and arithmetic scrapbooks for the reading table; measuring objects in the schoolroom; measuring each other's heights and weights; activities centering around a unit on time-telling; a unit on money; a unit on measures; and many others. With such a varied content, arithmetic should always be interesting and challenging to our children from the first day of school.
Which Way Arithmetic?
Authors: Engen H. Van
Published: October 2020

Something Is Happening to the way arithmetic is being taught in the elementary school. This fact is so obvious it need not be documented. A few minutes spent comparing the books of today with books published just a few years ago will show this to be true. What is producing this change?
Eopdica
Authors: Hicks Carl D.
Published: October 2020

Eopdica is an acronym for Electrically Operated Panel for Displaying Illuminated Colored Arrays. It is the name I have given to a device that was created for the purpose of presenting a variety of mathematical concepts.
Inching our way towards the metric system
Authors: Vervoort Gerardus
Published: October 2020

What is heavier, a pound of gold or a pound of feathers?”
Geometric activities for later childhood education
Authors: Henderson George L. , Collier C. Patrick
Published: October 2020

Elementary and upper grade teachers should consider giving geometry a prominent place in their mathematics programs because of the wealth of problem-solving activities with geometric content. Furthermore, the omission of geometry from a mathematics program will limit a teacher's opportunity to develop a student's nonnumerical, mathematical abilities. In this article, some fundamental object ives of mathematics arc advanced and some means of accomplishing these objectives are suggested.
Inventing a numeration system
Authors: Smith Karl J.
Published: October 2020

Many of the current textbooks discuss and develop historical numeration systems, as well as numeration systems with bases other than ten. To allow the class to invent their own numeration system could be a worthwhile activity. This paper shows how this might be done.
Things you can try: Happiness is some intriguing numbers
Authors: Sparks Billie Earl
Published: October 2020

The story has been told of a little boy who refused to do the multiplication problems that bad been assigned. When the teacher asked him why he wasn't doing his work, be replied with an answer that is full of significance for all teachers: “How many of them do I have to do to convince you that I can multiply?” He had done a couple of problems and got them right. Why should he do anymore?
Sorting, Classifying, & Logic
Authors: Cruikshank Douglas E.
Published: October 2020

The central role of mathematics in the elementary school is well established. Increasing awareness of how children learn mathematics, however, has brought about changes in techniques of presentation and in content emphasis. Such is the natural evolution of a subject. The Piagetian influence, in particular, has focused interest on orga nization and classification skills, leading to a later grasp of logical concepts. These prelogical activities are discussed in the following paragraphs. The activities are intended for use with children whose ages range from five to nine. The specific age level at which any activity is introduced or extended is determined, of course, by the teacher in the context of a particular teaching situation.
Using research in teaching: Creative ability in mathematics
Authors: Balka Don S.
Published: October 2020

Most elementary teachers, at some point in time, have commented on the creative behavior of individual students: “Johnny is so creative in all the art projects he has done for me.” “Carolyn just amazes me with her original stories and poems.” Teachers have observed, usually in art or language arts, that some children have more “creative ability” than others; but what is creativity? Can a student's creative ability be found only in certain areas? Why should we identify students with creative ability?
Building reading skills in the mathematics class
Authors: Hater Mary Ann , Kane Robert B. , Byrne Mary Ann
Published: October 2020

Do children know how to read mathematics books? Ordinarily not. Thus, teachers are faced with the alternatives of conti nually reading and interpreting instructional materials for their pupils, using such materials only as sources of teaching ideas and exerci es, or teaching children to read mathematics.
Mathematics, questions, and “schools without failure”
Authors: Perl Teri
Published: October 2020

All education stems from questions … questions asked by others or those we ask ourselves.” In his book Schools Without Failure, William Glasser has explored the problems of fa ilure in schools and has described some solutions and approaches to solutions that he believes may help to eliminate failu res. The asking of good questions is an important technique in dealing with such failure. Teachers who wish to use this technique effectively must understand how different questions generate diffe rent levels of thinking and then they must learn how to usc questions to achieve their objectives. Glasser feels that one of the important reasons for failures in the educational process has been the tendency for teachers and educators to ask only one type of question. Until the “new math,” mathematics education in particular has been guilty of this limitation.
“Equation” means “equal”
Authors: Bernstein Barbara Elaine
Published: October 2020

In late elementary school, students arc introduced to translating sentences into simple algebraic equations. For example, students may be asked to express the following in mathematical symbols: Bill has th ree more model planes than Ken. Many students find it difficult to distinguish whether B + 3 = K or B = K + 3 is correct (where B equals the number of planes Bill has and K equals the number of planes Ken has). I have found a procedure that helps students over this common stumbling block.
Beyond computation
Authors: Moody William B.
Published: October 2020

Any review of the arithmetic materials used in the schools reveals a mechanical approach to handling many of the principles of mathematics. Often the goal seems to center on attempting to enable the student to compute and arrive at a specific answer to a given problem. Little attention is given to problem-solving approaches and to tasks that raise additional questions beyond the obviously desired answer. Even the more recent publications lack original, open-ended situations that might promote student activities conducive to exploring interrelationships within mathematical topics.
Using research in teaching: Division involving zero: an experimental study and its implications
Authors: Grouws Douglas A. , Reys Robert E.
Published: October 2020

Division involving zero seems to cause more than ordinary difficulty for students. This appea rs to be true in all cases—whether the division example has zero as the dividend, the divisor, or the quotient. Most research related to the operation of division has concentrated on division algorithms; it has not directed attention to division examples that involve zero only in the divisor or dividend.
Let's Do It!: Making and using attribute materials
Authors: Bruni James V. , Silverman Helene
Published: October 2020

Among the most popular “math materials” found in elementary school classrooms are commercially available attribute materi als such as attribute blocks. Using these materials children of all ages can be actively involved in many enjoyable activities that help to develop classification skills, vocabulary, and logical reasoning ability. Any number of different activities arc possible for the entire class, small groups, or individual children.
A thinking model for deductive proof
Authors: Shoecraft Paul
Published: October 2020

The objectives of elementary school mathematics fall under one of two headings—knowing or knowing how. “Knowing” refers to that core of facts, concepts, and processes believed essential for all students to master (e.g., the addition facts, the concept of addition, the addition algorithm). “Knowing how” refers to those problemsolving skills by which the facts, concepts, and processes are put to use (e.g., solving verbal problems in addition, solving addition problems in daily living, proving theorems associated with addition). The teaching of these skills is the ultimate aim of elementary school mathematics, since what we know cannot be applied in daily life unless we know how to use it.
Results and implications of the NAEP mathematics assessment: elementary school
Authors: Carpenter Thomas P. , Coburn Terrence G. , Reys Robert E. , Wilson James W.
Published: October 2020

The National Assessment of Educational Progress (NAEP) has now reported its first mathematics assessment.* This article will examine the NAEP mathematics assessment for the two youngest age groups: 9-year-olds and 13-year-olds. The mathematics assessment was done in 1972-73 and will be repeated each five years, with about half of the exercises repeated from one assessment to the next. Previous articles in the Arithmetic Teacher (Foreman and Mehrens, 1968; Martin and Wilson, 1974) have provided information on the nature of the exercises, the procedures for assessment, the purposes of assessment, and general information on NAEP.
Your Council at Work: Aspirations, Actualities, Anticipations
Authors: Gibb E. Glenadine
Published: October 2020

You may recall that one of the societal forces upon us at the NCTM 52d Annual Meeting in Allantic City was the wave of attacks that had been made throughout that year on the mathematics programs in our schools. Among the questions we asked were, “Have we been responsible for a ‘lost generation in mathematics? Where have we failed? What should be our next steps?” There was cause for reflection, assessment, and renewed commitment.
Using research in teaching: Subtraction: what do students know?
Authors: Carpenter Thomas P. , Coburn Terrence G. , Reys Robert E. , Wilson James W.
Published: October 2020

How well do pupils subtract? One would expect 13-year-olds to do much better than 9-year-old, but would they make similar errors? How would the performance of 17-year-olds and young adults on subtraction problems compare with other age groups? Are pupils today subtracting better or worse than similar pupils were years ago? These and other questions will be considered in this article that reports national performance levels of various age groups on selected subtraction problems used in the 1972–73 mathematics assessment of the National Assessment of Educational Progress (NAEP).
Some comments on teaching the metric system
Authors: Dubisch Roy
Published: October 2020

Let me begin by saying that I heartily approve of the NCTM slogan “Think Metric.” The United States is going metric—to the benefit of all of us. And, in another twenty-five years or so, we will have a generation of children in our primary schools for whom (thankfully) our present system of measurement will have only historical interest.
Let's Do It!: From shadows to mathematics
Authors: Bruni James V. , Silverman Helene J.
Published: October 2020

Shadows are fascinating! They come in a wide variety of shapes and sizes. Sometimes you see them, sometimes you don't. When they do appear, you can't pick them up, nor can you weigh them. Shadows change shape and direction at different times of the day.
History of mathematics in the training program for elementary teachers
Authors: Francis Richard L.
Published: October 2020

The implications of an appreciation of mathematics on the part of prospective elementary teachers are farreaching, considering the teaching responsibilities they are about to assume. In light of these far-reaching implications, the college mathematics program for elementary teachers presents a definite challenge. One promising and meaningful provision for developing mathematical appreciation falls into the area of the history of mathematics.
Decimals
Authors: Prielipp Robert W.
Published: October 2020

Decimals— in particular, the decimal representation of rational numbers—can be one of the most interesting topics of arithmetic.
Property games
Authors: Peterson John C. , Dolson Gene
Published: October 2020

Property blocks are often used in kin dergarten and first grade to help teach children to distinguish shape, color, size, and thickness.
Mathematics as a creative language
Authors: Aiken Lewis R.
Published: October 2020

Whether one subscribes to a product or process definition of creativity, the abstract science of mathematics is one of the most original creations of the human mind. As A. N. Whitehead (1958) noted, with the possible exception of music it is difficult to conceive of a more purely creative endeavor than modern mathematics, dealing as it does with the abstractions of things divorced from any specific physical sensations or feelings.
The Pythagorean theorem: a useful geometric tool for approximating √2
Authors: Williams Gail Atneosen
Published: October 2020

My class of prospective elementary school teachers discovered a geometric method for approximating √2 that uses the Pythagorean theorem. What follows is an account of one very exciting class period.
New math or old math?—the wrong question
Authors: Kaufman Burt A. , Haag Vincent H.
Published: October 2020

The recent wave of criticism of the “new math” is aimed at excesses in its so-called new content, just as the proponents of the reform movement of the sixties decried the deficiencies of the traditional curriculum.
IDEAS
Authors: Bright George W.
Published: October 2020

Ideas for this month was inspired by two articles that have appeared in previous issues of the Arithrnetic Teacher. The IDEAS activities show one way to translate the content of these articles into classroom instruction. Other kinds of activities also could be developed.
Ideas About Problem Solving: A Look At Some Psychological Research
Authors: Lester Frank K.
Published: October 2020

Those of us who have studied or taught mathematics at any level know that problem solving is at the heart of doing mathematics. In fact, many teachers and mathematicians would agree that doing mathematics is essentially solving problems. Assuming that problem solving is what mathematics is all about, there is no need for me to make a case in support of the increased emphasis that is being placed on problem solving in the elementary school mathematics curriculum.
You Can Teach Problem Solving
Authors: Leblanc John F.
Published: October 2020

Teaching problem solving is a problem, but like most other problems it can be solved. And like other teaching situations, there is no formula that can be automatically applied to guarantee success in teaching problem solving. There are, however, some guidelines and hints that can go a long way in aiding a teacher in problem-solving instruction.
Research on Problem Solving: Implications for Elementary School Classrooms
Authors: Suydam Marilyn N. , Weaver J. Fred
Published: October 2020

Teachers and pupils both know that verbal problems cause problems. Because of this, and because it is considered to be an ultimate goal of mathematics instruction, researchers have devoted much attention to problem solving over the years.
Bring a Bike to Class
Authors: Ames Pamela
Published: October 2020

Bicycles can provide a series of problem situations that can be provocative, productive, and extiting. The situations can be very “hands on” in the initial stages and yet lead to some important conclusions involving mathematical decision-making. Even though students have a surprising lack of basic understanding about these vehicles, they have wide experience that can provide a rich learning environment. The problems associated with bicycles reach across many areas of mathematics and make the student call on many different mathematical skills.
Quantifying Chance
Authors: Souviney Randall J.
Published: October 2020

“Of all the people who died of lung cancer in 1977, 70 percent had smoked cigarettes sometime during their lives.” “Four out of five dentists interviewed recommend sugarless gum for their patients who chew gum.” To understand these statements, an intuitive knowledge of certain probability concepts is required on the part of the reader.
Let's Do It: The Surprising Circle!
Authors: Dana Marcia , Lindquist Mary Montgomery
Published: October 2020

What do you know about a circle? Often we consider only its measurements and fail to look at its other properties. What happens when you put several circles together or look at parts of a circle? Are looks deceiving? A circle looks so perfect and round, but can it fool you? Can it lead you on extensive searches? Can it produce unusual designs? What can you do with circular shapes and parts of circular shapes?
A Measurement Fantasy
Authors: Rees Jocelyn Marie
Published: October 2020

“Look at that green spaceship go! Who's flying it?”
Methods and Materials for Learning Subtraction
Authors: Bradford John W.
Published: October 2020

Reading the article, “Diagnosing and Remediating Systematic Errors in Addition and Subtraction Computations,” by L. S. Cox in the Using Research in Teaching section of the February 1975 issue of the Arithmetic Teacher has prompted me to write this paper describing some of my experiences in a combination third and fourth grade classroom.
Making Inferences from Marbles and Coffee Cans
Authors: Webb Leland F. , McKay Jamie D.
Published: October 2020

More and more today we are realizing the probabilistic nature of our world and, accordingly, trying to pro-vide certainty where reason to believe is the most we can hope for. Daily, public-interest polls report our preferences, circle graphs detail how our tax dollars are spent, and percentages indicate how we are going to vote in a future election. Politicians travel and speak according to statistics that have been collected.
A Manipulative Aid for Adding and Subtracting Integers
Authors: Tim Grady M. B.
Published: October 2020

In teaching basic operations with integers, the number line is sometimes as confusing as it is helpful. There is, however, a manipulative technique that students can understand. They can experience success with an abstract idea and have confidence in their results that will help as they later learn the rules for adding and subtracting integers.
My Favorite lesson: Addition through Palindromes
Authors: Nichol Margaret
Published: October 2020

One lesson that I especially look forward to teaching each school year is reviewing addition of whole numbers through the use of palindromes. I have found this method of review to be highly motivating for older students who, although they need a review in basic operations, think that addition has already been covered and that addition drill i omething done only by the lower grades. Drill, however, is still needed by many older students. But drill alone isn't the answer. The lesson must include an element of challenge along with an interesting goal to be reached. The use of palindromes meets these requirements so well that the lesson is not only effective but fun to teach as well.
The Future of Fractions
Authors: Usiskin Zalman P.
Published: October 2020

Is it true, as many assert, that common fractions will become obsolete?
The Multiple Triangle: Experiences in Discovery
Authors: Ouellette Hugh , Gannon Gerald
Published: October 2020

Imagine an arm balance with a pan at each end. On one pan we have the enemy of the young—boredom; on the other pan, the guardian angel of the young—discovery. We want the balance to be tipped in favor of the guardian angel, but as every teacher knows, this is not an automatic result. It takes a lot of planning and hard work. To counter boredom the teacher must create a learning environment for the student that is exciting, useful for maintaining basic skills, and fun to explore. In order to excite and involve students we must present a well-balanced program that con tains some novel, challenging, and surprise-filled problems. In the process of solving the problems, the students learn the basic skills through their experiences in gathering data.
A Case for Probability
Authors: Jones Graham
Published: October 2020

During the past twenty years, many educators have supported the inclusion of probability in the mathematics curriculum at all levels. Although the efforts of these educators have had some success at the upper secondary level, they have not resulted in universal acceptance of a systematic study of probability in elementary and lower secondary mathematics.
A Case and Techniques for Estimation: Estimation Experiences in Elementary School Mathematics—Essential, Not Extra!
Authors: O'Daffer Phares
Published: October 2020

A few years ago, the estimation exercise shown in figure I appeared in some text materials for the middle grades. The source went on to say that the Oxford Dictionary defines the word “ floccipaucinihilipilification” as “the action of estimating is worthless.” Although the majority of elementary school mathematics teachers would probably disagree with the implications of this word, many children leave the elementary school with little, if any, skill in estimation. A student teacher recently raised the inevitab le question when she said, “I can't find any source where it tells how to do it. What is it and what good is it?”
Ant, Aardvark, and Fudge Brownies
Authors: Green Dianne M.
Published: October 2020

A mathematics lesson that really turns my first-graders on captures their attention by the use of cartoon characters and a plate of fudge brownies that I bake for the children. I have cutouts of Orville Ant, Aardvark, and fudgebrownie bushes that the children can manipulate while they hear the story “Aardvark and Orville.”
One point of view: The problem of solving story problems
Authors: Zweng Marilyn J.
Published: October 2020

Why can't kids solve story problems? The two overly simplistic answers typically given to this question are “they can't read” and “they can't compute.” True, if children can't read or if they can't compute they're not going to be successful in solving verbal problems, but many students can do both and nevertheless perform poorly. Furthermore, there is good evidence that lack of reading ability and computational skill are not the major factors contributing to poor problem-solving performance. Knifong and Holtan (1977) found that the majority of students who solved problems incorrectly could read the problems orally and could retell the story of the problem in their own words; that is, they could comprehend the problem. The authors conclude, “Even assuming a biased interpretation, it is difficult to imagine that poor reading accounts for more than 1O% of these erred problems.” With respect to computation, the results of the 1972 NAEP Mathematics Assessment provide a convincing argument against the “they can't compute” explanation for problem-solving failures.
Magic Triangles—More Discoveries!
Authors: Zalewski Donald L.
Published: October 2020

After reading Yates's (1976) enriching experiences with magic triangles I immediately was “hooked” on trying to figure out why his pattern worked and if any solutions for side sums of 18 and 22 existed. My investigations verified, modified, and contradicted Yates's observations.
Teacher-Made Tests: Development and Use
Authors: Mckillip William D.
Published: October 2020

Teacher-made tests can be the most useful tests for you and your students because they are your tests: They are made for your students, your program, and your objectives; the results are obtained immediately; and you know the testing situation and you interpret the results. Teacher-made tests are the most personal and personalized part of a testing program.
Diagnostic Teaching for Elementary School Mathematics
Authors: Fowler Mary Anne
Published: October 2020

Both Johnny and J anie missed six out of ten addition problems on their mathematics papers yesterday. As aresult of this poor performance, they are assigned ten more addition exercises for today under the assumption that attempting to do more examples of a similar nature will help them comprehend the concept of addition and improve their skill in solving addition problems. How often is this futile procedure repeated in the school day of Johnny, Janie, and their peers? What are the alternatives?
Ideas
Authors: Tabler M. Bernadine , Hall Jacobson Marilyn
Published: October 2020

The IDEAS this month asks students to make estimates and to be alert to the reasonableness of results. Estimating answers and getting approximate resuits are important skills for consumers when they are doing quick calculations. Alertness to the reasonableness of a result is a valuable skill to accompany calculator usage.
Children's Surprising Knowledge of Arithmetic
Authors: Ginsburg Herbert P.
Published: October 2020

Citing the work of Piaget, developmental psychologists often tell us that the young child, from age three to about five or six, is incapable of basic mathematical concepts. There is some truth to this, but new research offers some surprises: it shows that the young child is in some ways a skilled mathematician.
Let's Do It: Transition Boards: Moving from Materials to Symbols in Addition
Authors: Thompson Charles S. , Walle John Van de
Published: October 2020

Frequently children who can solve multidigit addition or subtraction problems by using place-value pieces such as beansticks and loose beans have subsequent difficulty in solving the same problems using only paper and pencil. Many other children are very successful in solving multidigit addition or subtraction problems on paper but cannot solve them by using manipulative materials. The latter quite likely indicates a lack of understanding of the pencil-and-paper method. Both groups of children need assistance in bridging the gap between the use of manipulative materials and the use of numerals only.
Homer's Theorem
Authors: Richards W. S.
Published: October 2020

Far, far away, through this galaxy and midway through the receding galaxy of Plebus, is a magical, mystical planet know as Circleland. Whimsical little folk known as Loopers live there.
Let's Do It: Transition Boards: Moving from Materials to Symbols in Subtraction
Authors: Thompson Charles S. , Walle John Van de
Published: October 2020

One of the common errors made by child ren when working subtraction problems like 293 – 165 is that of subtracting the smaller amount in each column from the larger amount in that column. Children defend this procedure to us and say, “You said that we couldn't subtract a larger number from a smaller one.” Though these children have learned something about subtraction, they clearly have not learned that the amount named by the bottom number must be subtracted from the amount named by the top number.
Let's Do It: A Triangle Treasury
Authors: Walle John Van de , Thompson Charles S.
Published: October 2020

In this article, fifteen triangle shapes cut from poster board or construction paper provide the springboard for discussions of a variety of geometric concepts at all levels of sophistication, from grades 3 to 8, with many activities suitable for K to 2. Some time and care must be taken in making the patterns for the fifteen triangles, but once this is done, duplicates of each shape can easily be traced and cut.
Using Dice: From Place Value to Probability
Authors: Pereira-Mendoza Lionel
Published: October 2020

Teachers are constantly searching for ideas for practice activities. Over a period of time the author has managed to accumulate a multitude of different activities with dice involving topics such as place value, addition and subtraction of whole numbers, probability, and so on—hence the title. Some of these activities may be familiar. All have been field-tested.
Doing Mathematics Is Problem Solving
Authors: DeVault M. Vere
Published: October 2020

Problem solving in mathematics, like writing in the language arts, requires competence with certain basic skills or tools. Clearly, one cannot solve many mathematical problems without some notion of the basic facts, without competence in computation, understanding of operations, or the ability to sequence tasks in logical order. In today's elementary school mathematics curriculum, however, so much time is devoted to the development and perfection of tools that little time is left for using these tools in problem solving. Skill in the use of these tools in problem solving is learned behavior and the mathematics curriculum must provide ample and appropriate experience and practice in helping children learn to use these tools.
Fractions: Results and Implications from National Assessment
Authors: Post Thomas R.
Published: October 2020

Despite the recent introduction of the calculator and emphasis on the metric system, fractions continue to occupy an important place in school mathematics programs. Persuasive arguments for continued emphasis can be made not only because of their social utility but also because of their place in the development and structuring of mathematical ideas to follow. Accordingly, the National Assessment of Educational Processes (NAEP) allocated a substantial portion of the second mathematics assessment to fractions.
Calculator—Taking the First Step
Authors: Bestgen Barbara J.
Published: October 2020

How should I describe my students' reactions as they first begin to use calculators? Motivated? Interested? Intrigued? Excited? It is hard to convey the initial enthusiasm students display as they begin to experiment with the typical four-function calculator in a classroom setting.
Computer Game Playing—“Turn-On” to Mathematics
Authors: Winner Alice-Ann , McClung Margo D.
Published: October 2020

At the upper elementary level, teachers of mathematics are faced with two problems. They must reteach skills that the children have been exposed to, but have not mastered. At the same time they should awaken children to the joys and excitement of mathematics.
Speed of the Earth
Authors: Vest Floyd
Published: October 2020

We have been told that the speed of the earth is about 67 000 miles per hour. How do we know this, since the sun is so far away and the circle of the path of the earth around the sun is so very large? People are so small in relation to these distances, how can such a result be determined?
Problem Solving in the Intermediate Grades: Helping Your Students Learn to Solve Problems
Authors: Worth Joan
Published: October 2020

This article is written for the intermediate grade classroom teacher because your students are at a crucial stage in the development of mathematical skills and attitudes. Intermediate graders are both ready and able to learn to solve problems and to become proficient problem solvers.
Patterns in Multiples
Authors: Robold Alice I.
Published: October 2020

Patterns in color, design, sound, and rhythm are fascinating and enjoyable to children. All of these can be incorporated in learning experiences that help children see mathematical relationships in multiples of numbers.
Whither Research on Compound Subtraction: (A Communication to the Editor)
Authors: Weaver J. Fred
Published: October 2020

YOUR NOTE FOLLOWING a recently published research article on compound subtraction by Rheins and Rheins in THE ARITHMETIC TEACHER1 was a signi ficant one in several respects. Most importantly, it directed attention to t he complexity of factors which must be considered in any comprehensive evaluation of the relative merits of methods of compound subtraction.
A Meaningful Approach to Cancellation
Authors: Peoples John A.
Published: October 2020

Much progress has been made by arithmetic teachers and textbook writers in developing arithmetic along meaningful lines, but there still are phases of the subject for which a meaningful approach has not been developed. A very useful concept, for which a complete rationalization has not been incorporated in current textbooks, is the one of “cancellation,” —or in more meaningful terms, “reducing before multiplying.”
The Abacus and Multiplication
Authors: Devault M. Vere
Published: October 2020

IN THE NOVEMBER, 1955, issue of this JOURNAL, Mr. Flewelling presented an excellent article on The Abacus as an Arithmetic Teaching Device. As pointed out in the Editor's Note following the article, “the greatest value in using an abacus comes from gaining insight into the number system and our modes of calculating.”
Factors Determining Attitudes Toward Arithmetic and Mathematics
Authors: Poffenberger Thomas , Norton Donald A.
Published: October 2020

In recent months, many scientists, educators and statesmen have referred to the alarming shortage of graduates in engineering, the physical sciences and mathematics. The shortage of persons trained in these fields is being felt in industry, the government and the military services and it is critical in education.
Meaning and Skill—Maintaining the Balance
Authors: Brownell William A.
Published: October 2020

The subject given me poses a question. It is no academic question arising out of purely theoretical considerations. It is assumed that both meaning and computational competence are proper ends of instruction in arithmetic. It is implied that somehow or other both ends are not always achieved and that there is evidence that this is so.
Arithmetic via Television: A Course for Teachers and Parents
Authors: Brune Irvin
Published: October 2020

Teaching via the medium of television means hard work. Apparently, though, the results compensate for the efforts. The following comments represent the views of one teacher who taught one telecourse.
The Controversy Regarding the Teaching of Higher-Decade Addition
Authors: Flournoy Frances
Published: October 2020

There is considerable disagreement regarding the teaching of higherdecade addition. It seems to be a question of both “when and how.” A study of published research at this time does not reveal any studies which might help in solving the problem of when and how to teach higher-decade addition.
Non-Pencil-and-Paper Solution of Problems: An Experimental Study
Authors: Petty Olan
Published: October 2020

Frequently one hears the argument that an excessive amount of activity with pencil and paper occurs when children attempt to solve arithmetical verbal problems. One further hears the contention that extreme emphasis on the pencil-and-paper type of arithmetic has created the following undesirable practices and situations in the arithmetic program: complete lack of, or too little use of, mental arithmetic; careless habits, involving division of attention, resulting from continuous presentation of the printed statement of the problem while solving it with pencil and paper; failure to use methods of handling arithmetical situations that approximate more nearly methods of handling similar quantitative situations encountered in other areas of learning and in the everyday affairs of life; evaluation of a child's ability in arithmetic almost entirely upon the basis of pencil-andpaper work; limited use of different sensory modes of presentation of work in arithmetic; and complete lack of, or less efficient use of, “short-cuts.”
Why “Indent” in Multiplication?
Authors: Hickerson J. Allen
Published: October 2020

WE MAY BE DOING MORE HARM than good when we teach children deductively to use many of the traditional techniques of computation. “Carrying” in addition; “borrowing” in subtraction; “indenting” in multiplication; “multiplying-subtracting-bringing down” in division; “canceling” in multiplication of fractions; “inverting and multiplying” in division of fractions; “counting decimal places” in multiplication of decimals; and “moving decimal points” in division of decimals are cases in point. To illustrate why this may be so, let us consider “indenting” in multiplication, although any one of the other techniques mentioned rould serve equally well as an example of deductive teaching.
Graphs Can Jazz Up the Mathematics Curriculum
Authors: Christopher Leonora
Published: October 2020

Basic to all teaching is the need to make it possible for children to learn to think, to make decisions, to investigate, and to solve problems. Motivating children to perform these tasks is the challenge of every teacher. One way that I have found to get children interested in mathematics is through the use of graphs. There are at least five reasons why graph work is effective.
Testing for Teaching
Authors: Beattie John , Algozzine Bob
Published: October 2020

The first week of classes was finally over. Ms. Jones could now take the tests that she was obligated to give and record the scores, as she had done for the past ten years. She noted mentally that Tommie Smith's mathematics score indicated that he was two years below his actual grade level. “I'll have to do something about that,” she thinks as she puts Tommie's cumulative folder in the file—where it belongs.
What's Going On
Authors: Hanel Ann Marie
Published: October 2020

Task force on computer education. The Montana State Superintendent of Schools has organized a sixteen-member task force to address the rapidly expanding use of computers in the schools. The task force consists of teachers from all levels of education, school administrators, and state department curriculum specialists. A handbook outlining suggestions and guidelines for districts that are developing computer education programs has been written by the task force. The handbook also includes software and hardware evaluation and selection guides, curriculum ideas, administrative applications, staff development plans, telecommunications usages, col lege preservice suggestions, and an extensive bibliography. A videotape presentation and a statewide staff development program will also be developed. For more information contact Dan Dolan, Math and Computer Education Specialist, State Capitol, Helena, MT 59620.
Powers and Patterns: Problem Solving with Calculators
Authors: Lappan Glenda , Phillips Elizabeth , Winter M. J.
Published: October 2020

With the publication of An Agenda for Action: Recommendations for School Mathematics of the 1980s, the NCTM has emphasized its support for helping students to develop and use problem-solving skills. The challenge for the teacher is to provide opportunities for the development of the e skill while teaching mathematical concept that comprise the basic curriculum. With the wide-spread availability of calculators, teachers have a tool that can be used to expand the study of many basic mathematical idea to include the development of problem-solving strategies. Calculations that would be so time consuming as to be impractical if they were done with paper and pencil, can be quickly done with a calculator.
Young Children Are Good Problem Solvers
Authors: Moser James M. , Carpenter Thomas P.
Published: October 2020

Do you think many first-grade children could solve this comparison problem? Joe won 6 prizes at the fair. His sister Connie won 9 prizes. How many more prizes did Connie win than Joe? Most curricular programs apparently assume that word problems are difficult for children of all ages, and that children must master symbolic addition and subtraction operations before they will be able to solve even simple word problems. Recent National Assessment of Educational Progress (NAEP) results (Carpenter et al. 1980) also give some credence to the belief that children are poor at problem solving. We believe. however, that young children are good problem solvers.
Let's Do It: Teaching Time with Slit Clocks
Authors: Horak Virginia M. , Horak Willis J.
Published: October 2020

The teaching of time in the primary grade should consist of activitie of two distinct types. The first focus on developing an understanding of the concept of time. This deals with the passage of time. the chronological equencing of events, and the duration of time periods. The second type focuses on telling time. This includes reading a clock, recording time, and predicting times after determined inter vals.
Let's Do It: “Button Bag” Mathematics
Authors: Horak Virginia M. , Horak Willis J.
Published: October 2020

Children in the primary grades need to be introduced to many activities that allow them to observe and organize their environment. Although these activities may not directly involve numbers, they form the basis for the understanding of more advanced number concepts. Recognizing patterns, forming one-to-one corres po nde nces, classifying, and ordering are a few skills that can be developed at an early level.
A Complete Model for Operations on Integers
Authors: Battista Michael T.
Published: October 2020

Teaching student the four basic operations on the set of integers in a meaningful way is a difficult task. The task could be made easier, however, if there was a single physical model for the integers in which all four basic operations could be represented. The number-line model, though useful, has serious shortcomings and is incomplete. As an alternative, some author have suggested the “positive-negative charge” model (Frand and Granville 1978, Grady 1978), but they used this model only for representing addition and subtraction. In this article I will describe the “charge” model and demonstrate how it can be extended to all four operations on the set of integers.
Microcomputers in a Sixth-Grade Classroom
Authors: Morris Janet P.
Published: October 2020

How should computer be used in the classroom? An Agenda for Action state that “computers should be integrated into the core mathematics curriculum,” that they “should be used in imaginative ways for exploring, discovering, and developing mathematical concepts,” and that the computer activities should “fit the goals or objectives of the program” (NCTM 1980, p. 9).
Curricular Issues: Teaching Rational Numbers
Authors: Payne Joseph N.
Published: October 2020

Vigorous debate is taking place on the relative emphasis to be given fractions and decimals. Pro and con arguments use the facts of poor performance on all state and national mathematics tests, the certain but slow introduction of the metric system in the United States, the widespread use of calculators, and the real-life uses of fractions and decimals