bluuebunny/crossref_metadata_2025 huggingface dataset:
The Value of Informal Approaches to Mathematics Instruction and Remediation
Authors: Baroody Arthur J.
Published: October 2020

School mathematic is my terious, foreign, and threatening to many students. For too many children, school mathematics is something that happens to them rather than something that they make happen. Some children are so overwhelmed that they are intellectually and emotionally paralyzed by school mathematics. How can we make school mathematics sensible, familiar, and enjoyable to children—especially those with learning difficulties? Ginburg (1982) suggest that we should relate formal mathematical instruction to a child's informal knowledge and skills, which are often based on counting. This principle is applicable to children across the whole range of abilities, from kindergarten through eighth grade, and to a range of topics in school mathematics. Such an approach may help children feel more in control of their work and, as a result, feel better about themselves and school mathematics.
The Number Namer: An Aid to Understanding Place Value
Authors: Beattie Ian D.
Published: October 2020

Modern instructional methods are aimed at developing understanding. It is generally agreed that an optimal learning sequence of mathematical concepts moves from concrete to semiconcrete (iconic) to abstract learning experiences. Such a sequence involves objects and their manipulation, pictures or models of real objects, and, finally, the use of abstract symbols.
Research Report: The Process of Counting
Authors: Suydam Marilyn N.
Published: October 2020

Recently, a number of researchers have directed their attention to the process of counting, sometimes in connection with studies of problem solving or of addition and subtraction. The emphasis is on identifying how children learn to count or use counting; however, how the ideas children discover on their own might be applied in instruction is the ultimate goal. For instance, children might be led to discover new ideas by examining their own invented strategies.
Curricular Issues
Authors: Moser James M.
Published: October 2020

This article does not address the particular issue of whether manipulatives should be used in an elementary school mathematics program that includes arithmetic, geometry, statistics, elements of algebra, measurement, and the solution of routine and nonroutine problems in each of these areas. That issue has been addressed elsewhere in this special issue. Rather, the assumption is made that the reader is at least favorably inclined toward the use of manipulative materials. The issues that are considered here are posed as questions that teachers should resolve for themselves. Even though teachers may be convinced that manipulatives should be a part of instruction, some theoretical basis for their use should be developed. The writer is indebted to Richard Skemp (1981) for the quotation. “There is nothing quite so practical as a good theory.” Skemp does not take credit for it, claiming he heard it from someone else, who may even have read it somewhere. No matter, it is sound advice.
Research Report: Manipulative Materials and Achievement
Authors: Suydam Marilyn N.
Published: October 2020

From an analysis of sixty-four research studies at the elementary school level, Parham (1983) reported a decided difference in the achievement scores of students who had used manipulative materials and those who had not. Those who used manipulative materials scored at approximately the eighty-fifth percentile; those not using manipulative materials scored at the fiftieth percentile. This finding is clearly in agreement with an earlier review by Suydam and Higgins (1977), a major finding of which was that lessons using manipulative materials have a higher probability of producing greater mathematics achievement than do lessons in which such materials are not used.
Bridging the Gap between the Concrete and the Abstract
Authors: Heddens James W.
Published: October 2020

How can teacher help students make the transition from the use of manipulative material to abstract mathematical symbols? Many student have difficulty understanding mathematic because they are unable to make a connection between the physical world and the abstract world.
Developing Numeration Concepts and Skills
Authors: Harrison Marilyn , Harrison Bruce
Published: October 2020

A key Piagetian concept highlighted by Flavell (1963, 368) is that intellectual development is marked by a gradual transformation of overt actions into mental operations. The carrying out of actions by pupils with the decreasing support of external objects can be facilitated by having them use physical objects that exemplify given concepts, patterns, or operations; followed by pictorial or semisymbolic representations; and then imagined object and operations while keeping symbolic records. Subsequently the abstract concepts emerge in a form that can not only be used meaningfully but also be reinterpreted, as required, in terms of previous levels of representation, rather than exist purely at a symbolic level.
Special Learners
Authors: Thornton Carol A. , Wilmot Barbara
Published: October 2020

Children with special learning needs fall into two distinctly different group—the learning handicapped and the mathematically gifted. In this article consideration is given to the important role of manipulatives in mathematics instruction for each of these group of students.
Activities for Algebra
Authors: Williams David E.
Published: October 2020

More year ago than I care to remember. I and two other teachers team-taught three eighth-grade preal-gebra classes. One of the other teachers wanted to use manipulative materials, games, and activities, with these high-achieving students but was over-ruled by his two teaching partners during a planning session. I and the other disssenting teacher considered the plans to be frivolous, a waste of precious teaching time. Frustrated, the teacher abandoned the activities, acquiescing to the decision of the two senior members of the team.
Representational Models from the Learners' Perspective
Authors: Schultz Karen A.
Published: October 2020

Pretend that the name of thi journal is the Arithmetic Learner. I want to change your perspective. Follow me as we try to consider some things from the young learner's point of view.
One Point of View: Mental Computation and Computational Estimation—Their Time Has Come
Authors: Reys Barbara J. , Reys Robert E.
Published: October 2020

Changing curricula in mathematics is more difficult than moving an old graveyard in January. Nevertheless, cries for changing our mathematics programs are coming from many directions as ideas for a forward-looking, futuristic mathematics curriculum are offered. Although calls for specific curricular changes are varied, all seem to agree on one thing: mathematics programs must give significantly more attention to the development of skills in mental computation and estimation and much less attention to traditional written algorithms for computation.
Ideas
Authors: Duncan David R. , Litwiller Bonnie H.
Published: October 2020

To maintain students' computational skills, practice is needed. Students find practice activities more in teresting if they are provided in new and novel situations. It's a bonus if students discover number patterns as a result of the computations.
Research Report: Individualizing Mathematics Instruction
Authors: Suydam Marilyn N. , Schoen Harold L.
Published: October 2020

Mathematics teachers at all levels are faced with the problem of providing instruction for classes of (usually twenty to thirty-five) students who have differing backgrounds, abilities, interests, and learning styles. In response to this universal and perennial problem, educators have devised various ways to organize the classroom, the instructional approach, and the curriculum better to accommodate these individual differences. Willoughby (1976), for example, describes twenty approaches to individualizing mathematics instruction, some of which replace traditional teacher-directed instruction, whereas others only supplement it. Research provides some guidance concerning the effectiveness of many of these approaches.
Balancing a Checkbook: Children Using Mathematics Skills
Authors: Paull Sandra
Published: October 2020

My fourth and fifth grader needed to practice addition and subtraction skills in a practical way. Many realized that adults commonly use these kills to balance checkbooks. The use of checkbook balance registers in class seemed to be an opportunity for children both to experience a ritual they will perform in the future and to practice kills they needed to review. Little did I know how this simple activity would develop into a eries of le sons that included general computation, estimation, problem solving, and economics.
A Den of Thieves Investigates Division
Authors: Cheek Helen Neely , Olson Melfried
Published: October 2020

The algorithm for long division is probably the most complicated procedure we teach in elementary school arithmetic. Troutman and Lichtenberg (1982) classify division situations into two types, subtractive and distributive. They suggest that the subtractive interpretation is the easiest for children to grasp but claim that “to date, the subtractive procedure has not gained wide acceptance among elementary school teachers” (p. 116). Thus, division situations are usually interpreted as the distributive type.
The Super Brains
Authors: Powers Sandra M.
Published: October 2020

In September 1980. an idea germinated in our county for increasing the ability of elementary school student to solve problems. Since that time. student and teachers from twent-eight county school have become involved. This article describes the simple idea that has been received so enthusiastically.
A Seven-Part Invention: A Playlet in One Act with Overtures
Authors: Burton Grace M.
Published: October 2020

According to Vincent Glennon, mathematics education is like a three-legged milking stool: to be in balance it must be supported equally by a knowledge of mathematics, pedagogy, and psychology. This playlet focuses on the last of these supports.
Cards, a Good Deal to Offer
Authors: Clements Douglas H. , Callahan Leroy G.
Published: October 2020

Activitie with number card. can provide a wide variety of exploratory experiences in mathematics. In addition to being inexpensive and easy to make, the e materials offer young children the opportunity to explore numbers without the fru tration of writing them. Since the cards can be moved or rearranged, solutions can be changed easily. Children who arc not confident in mathematics may find this attribute particularly attractive. Their anxiety about having teacher or peers see mistake may be lessened.
Research Gives Calculators a Green Light
Authors: Hembree Ray
Published: October 2020

In December 1974, the National Council of Teachers of Mathe matic issued a far-reach ing statement that urged the use of hand-held calculators in schools (NCTM 1974, 3). The core of mathematic instruction in the elementary chool grade was computation, pleasantly coinciding with the calcula tor's first purpose, and a host of other intentions for its use were envisioned (Suydam 1976, 12): to aid algorithmic instruction, facilitate concept deve lopment, reduce the need for memorization, enlarge the scope of problem solving, motivate students, and encourage discovery, exploration, and creativity. With a cost that was rapidly shrinking, the device appeared too powerful and potentially omnipresent to ignore.
One Point of View: How Many Ways Can You “Understand”?
Authors: Davis Robert B.
Published: October 2020

Readers are encouraged to react to these editorials by writing to the author. with copies to the Arithmetic Teacher for consideration in “Readers' Dialogue.” Please double-space all leuers that are to be co11sideredfor publication. Editorials from ret1ders tzre welcomed.
Building Understanding with Blocks
Authors: Beattie Ian D.
Published: October 2020

The model of instruction that goes from the concrete to the abstract is widely accepted. Texts that describe me thods of teaching e lementary school mathematics use it, and instructors of mathematics methods classes for prospective elementary school teachers advocate its use. The intent of such texts and c lasses is obviously to prepare teachers who are familiar with such a model and can apply it in the classroom.
Ideas
Authors: Towsley Ann E.
Published: October 2020

This section is designed for teachers who wish to give students new insights imo familiar topics in grades K-8. This material can be reproduced by classroom teachers for use in their own classes wit hom requesting permission from the National Council of Teachers of Mathematics. Manuscripts for this section are welcomed for review. Guidelines for authors are available on request.
AT Classic: The Revolution in Arithmetic
Authors: Brownell William A.
Published: October 2020

At the suggestion of Glenn Nelson of Cedar Fulls. Iowa. the Editorial Panel has authorized the occasional republication of an article of exceptional merit from the period 1954-1975 that would interest today's readers. Invitations to submit five nominations were sent to individuals familiar with the journal during this period. This article was one of the most highly rated. Introduction comments have been prepared by James E. Inskeep, Jr., who selected the article.
Research Report: What Are Calculators Good For?
Authors: Suydam Marilyn N.
Published: October 2020

Calculators are good for promoting achievement: their benefit has been shown in reviews by Suydam (1982) and, more recently, by Hembree (1985). Suydam summarized findings and reported that achievement scores were as high or higher when calculators were used for instruction as when they were not. Hembree used a different technique to analyze the findings of seventy-nine research reports. At only one grade level did calculator use appear questionable: at every other level, he reported that the use of “calculators in concert with traditional instruction … can improve the average student's basic skills with paper and pencil, both in basic operations and in problem solving.” Moreover, “students using calculators possess a better attitude toward mathematics and an especially better self-concept in mathematics than noncalculator students.”
Sampling Treats from a School of Fish
Authors: Vissa Jeanne
Published: October 2020

“New York State Plans Fish Census” read the title of the newspaper article. The tagging method of doing an animal or fish census provided an interesting idea for a lesson with my seventh and eighth graders on estimation, proportion, central tendencies, and structured sampling. It was also an occasion for using food in a mathematics lesson, which was always a welcome treat.
Number Facts—or Fantasy?
Authors: Hopkins Martha H.
Published: October 2020

Articles in the Arithmetic Teacher have stressed the use of the history of mathematics to enhance motivation in the elementary classroom (Cowie 1970; Jackson 1964; Kreitz and Flournoy 1960; Krulik 1980; Willerding 1954). The increased emphasis on other numeration systems in the “new math” produced several articles that encouraged teachers to include historical information about the development of the decimal system in their programs of study (Baker 1963; Delaney 1963; Fisher et al. 1963; Schaaf 1961; Young 1964). The rationale for including these topics was that they would help students appreciate the development of our number system and would stimulate interest, enthusiasm, and a better understanding of our civilization.
One Point of View: We Need Fundamental Reform
Authors: O'Brien Thomas C.
Published: October 2020

Yes! American education is in need of improvement. But to say that the remedy is more of what has already failed—more hours, more days, more homework, and more tests—is like saying that more aspirin, with more careful monitoring of patients' blood chemistry, will cure cancer. It won't! What is needed is fundamental qualitative reform, not quantitative extensions of what has clearly failed. We need reform in the curriculum, in our view of knowledge, in the role of the learner, and in the role of the teacher. This reform has several dimensions.
Fractions and Panes
Authors: Edge Douglas
Published: October 2020

Recently I saw an interesting photograph of classroom windows that had been painted with colorful flowers by the children (fig. 1). One child had commented, “Now it always looks like spring outside.”
Computers in Schools: A Plea for a Thoughtful and Educated Approach
Authors: Baron Norman
Published: October 2020

Standing in the background, I watched a teacher of computer science demonstrate her favorite piece of software to a group of uninitiated educators. On the left side of the monitor's screen was a line drawing of a small bucket. On the right was an identical bucket, and separating the two buckets was a pit. In each bucket was a group of small circles. The activity on the screen ran automatically. One at a time, circles from the first bucket popped up out of the bucket and fell into the pit, followed by another and another until all the circles were in the pit. Circles then began to pop up out of the second bucket until they also were in the pit. The object of this activity was to teach children to count and add. By counting as the first bucket emptied and then the second, it was hoped that the student could answer the questions that followed. The questions were, How many marbles were in the first bucket? How many marbles were in the second bucket? How many marbles were there all together?
Research Report: Metacognition and School Mathematics
Authors: Callahan Leroy G. , Garofalo Joe
Published: October 2020

Metacognition refers to the knowledge and control one has of one's cognitive functioning, that is, what one knows about one's cognitive performance and how one regulates one's cognitive actions during performance. In recent years, a growing number of psychologists and educators have been discussing and investigating the role and significance of metacognition on various aspects of academic performance (Flavell 1979). Most of the theorizing and research concerning metacognition has been in reference to performance on reading and memory tasks (Baker and Brown 1984; Schneider 1985), but lately mathematics educators have begun studying the role of metacognition in the performance of mathematical tasks. These mathematics educators are convinced that what one knows or believes about oneself as a learner and doer of mathematics and how one controls and regulates one's behaviors while working through mathematical tasks can have powerful effects on one's performance (Garofalo and Lester 1985; Kilpatrick 1984; Schoenfeld, in press; Silver 1985).
Ideas
Authors: Barson Alan , Barson Lois
Published: October 2020

Ideas this month focuses on the attributes of numbers. The activities will highlight such attributes as one more than, equal to, greater than, less than, LCM, prime, and factor, Since teachers need to give students practice in understanding and using the vocabulary and characteristics of numbers, the following activities provide this practice in a motivational game-oriented situation using dice and game boards.
Assessing For Learning: Using Calculators in Assessing Mathematics Achievement
Authors: Williams David E. , McAloon Ann , Robinson G. Edith
Published: October 2020

In it “Position Statement on Calculators in the Mathematics Classroom” the National Council of Teachers of Mathematics recommends that calculators be integrated into all aspect of school mathematics, including class work, homework, and evaluation (NCTM 1986). This author cited the need for a comprehensive calculator proj ect encompassing all facet as of elementary mathematics education, a project that should include the development of a calculator-integrated curriculum. an extensive training program for teachers, the development of curriculum-support materials, change in textbook, workshops for parents and community group, and a change in evaluation of mathematics achievement (Williams 1987).
Teaching Mathematics with Technology: Introducing Factors with a Tiling Simulation
Authors: Jensen Robert J.
Published: October 2020

The concept of factor is often confusing for students when it is first introduced in the elementary curriculum. Motivating the development of new ideas through problem-solving episodes can be a fruitful approach before formal instruction begins. A tiling problem is posed here for your students to investigate using the microcomputer as a tool. Since factors, common factors, and the greatest common factor play a crucial role in the solution to problems of this type, this advance activity will give students a specific context upon which to build meaning for these concepts when they are fo rmally introduced.
One Point of View: Mathematics for the Young Child—Not Arithmetic
Authors: Wolfinger Donna M.
Published: October 2020

For young cnildren, those in pre-school through first grade, the world is a stage complete with props and scenery, a world to be manipulated and discovered. But because of the emphasis on standardized curriculum and testing, this discovery through manipulation has been seriously curtailed in many mathematics programs for young children. Instruction is focusing on correct answers to computational problems. First graders are frequently taught material once covered in the second or third grade, and they learn through paper-and-pencil exercises and memorization that are too abstract for them. Young children are being presented with a mathematics program in which the computations of arithmetic are excluding the conceptualization of mathematics.
Montessori: Right or Wrong about Number Concepts?
Authors: Bauch Jerold P. , Hsu Huei-hsin Joyce
Published: October 2020

When Maria Montessori “invented” her method for teaching the street urchins from the slums of Rome in the early 1900s, she showed remarkable insight into children's needs and unique ways to help them learn. Montessori was truly a scientific person, earning the first medical degree granted to a woman in Italy and studying such diverse fields as pedagogy, anthropology, and psychiatry. When she was asked to develop a preschool program for the street children of Rome, she made a quantum jump forward in both instructional theory and teaching materials.
One Point of View: Choosing a Calculator: Four-Function Foul-ups
Authors: Demana Franklin , Osborne Alan
Published: October 2020

The Agenda for Action champions taking full advantage of the power of calculators and computers at all grade levels. The evidence of numerous article and research studies supports the use of calculators in the classroom. Concerned mathematics educators should now spend their energy on issues affecting how learners can best use the calculator rather than on whether the calculator should be used.
One Point of View: Let's Talk about Talking Mathematics
Authors: Liedtke Werner
Published: October 2020

A elementary chool teachers we continually strive to make our mathematics teaching as effective as we possibly can. One way of reaching this goal involves changing the proportion of time allotted for development mental activities and individual seatwork for a lesson. Some evidence suggests that increasing the time spent on developmental tasks and student participation and thus decreasing the time students spend on practice can have a positive effect on mathematics achievement. This evidence implies that we, as teachers, should search for and collect strategies that are conducive to the implementation of developmental activities.
Learning and Fun with Geometry Games
Authors: Bright George W. , Harvey John G.
Published: October 2020

Much of the geometry taught in elementary school deal with measurement and with the properties of polygons and solids. Most of this content can be included in instructional games similar to those illustrated in this article.
Ideas
Authors: Barson Alan , Barson Lois
Published: October 2020

Ideas for this month is a set of activities based in geometry. Through geometry students can practice area measure, work with whole-number and fractional computation, and develop problem-solving skills in spatial relationships. The activities that follow will offer such experiences.
Discovering Perimeter and Area with Logo
Authors: Binswanger Richard
Published: October 2020

The students at the girls’ school in which I work seemed not to be as adroit in internalizing their mathematics, as they were in merely recalling it. They were attempting to please their teacher, afraid to take risks, and too often parroted answers. The girls seldom searched for what might lie beyond a concept but were satisfied just to understand it. We believed that some portion of this. reluctance was due to the way we taught mathematics as a didactic subject. with students getting their ideas almost exclusively by listening.
President's Report: Your NCTM—Who Are We? What Do We Believe? What Are We Doing? Where Are We Going?
Authors: Dossey John A.
Published: October 2020

The questions posed by the title of this article are ones that all professional organizations should take time to consider on a regular basis. If one does not know the intent of the membership. nor their beliefs. then the orga nization may not be positioning itself to serve their interests a best it could. Further, without knowledge of the members’ beliefs and desires, it is difficult to move ahead with vigor and have the co nviction that the necessary support will materialize to strengthen the gains made.
A Case for a Logo-based Elementary School Geometry Curriculum
Authors: Battista Michael T. , Clements Douglas H.
Published: October 2020

Geometry is an essential part of mathematics. Unfortunately, according to evaluations of mathematics learning, such as the National Assessment of Educational Progress (NAEP), students are failing to understand basic geometric concepts and are failing to develop adequate geometric problem-solving skills (Carpenter et al. 1980; Fey et al. 1984; Kouba et al. 1988). This poor performance is due, in part, to the current elementary school geometry curriculum, which focuses on recognizing and naming geometric shapes and learning to write the proper symbolism for simple geometric concepts (cf. Carpenter et al. [1980]; Flanders [1987]). In contrast, we believe that elementary geometry should be the study of objects, motions, and relationships in a spatial environment (Clements and Battista [1986]; cf. Trafton and LeBlanc [1973]). First experiences with geometry should emphasize informal study of physical shapes and their properties and have as their primary goal the development of students' intuition and knowledge about their spatial environment. Subsequent experiences should involve analyzing and abstracting geometric concepts and relationships in increasingly formal settings.
Jar Lids—an Unusual Math Manipulative
Authors: Langbort Carol R.
Published: October 2020

For the past fifteen years, many educators have made an effort to incorporate the use of concrete materials into the elementary school mathematics curriculum. These manipulatives are, in fact. useful with students at all levels; however, some teachers still reject using them. The authors of both the 1985 Math Framework for California Public Schools and the 1987 Mathematics Model Curriculum Guide, K–8 recommend the use of concrete materials for all learners, including high school students and adults. Certainly, manipulative material should be used without question in the elementary grades.
AT Classic: A Philosophy of Arithmetic Instruction
Authors: Fehr Howard
Published: October 2020

Fehr covers a wide range of topics in this article by necessity, because of his broadly conceived view of arithmetic instruction. He reminds us that arithmetic instruction must be determined by a set of several interrelated concerns: societal and children's needs, the structure of the discipline, and a knowledge of how children learn. Perhaps more than anything, though, I consider this article a classic because of Fehr's discussion of problem solving. To Fehr, problem solving is the ultimate goal of arithmetic instruction—a position that has a very familiar, contemporary ring. However, Fehr's views about how to develop problem-solving ability. which were an outgrowth of his commitment to field theories of learning, are less familiar and largely ignored. Since some recent studies tend to support Fehr's views, they warrant the attention of classroom teachers and researchers alike. By reprinting this article from April 1955. it is hoped that another perspective will be injected into the debate on how to teach problem solving. (The reader is referred to another classic by Fehr, “Theories of Learning Related to the Field of Mathematics,” in The Leaming of Mathematics: Its Theory and Practice, the Twenty-first Yearbook of the NCTM.)
The 12.7 cm Hot Dog Corporation
Authors: Martin Hope
Published: October 2020

A corporation was formed in the United States fourteen years ago that does not appear on the New York Stock Exchange or any other. It is alive and well and prospering at Northwood Junior High School in Highland Park, Illinois. The “12.7 cm Hot Dog Corporation” is owned and operated by a group of about forty-five eighth graders, who make all the executive decisions concerning the sale of hot dogs, chips, soda pop, and popcorn at home boys' and girls' basketball games and wrestling meets. I started the corporation to bring “real life” into the classroom and encourage students to use their mathematics skills to make the decisions necessary to run a successful business.
Number Tablets for Primary Grades
Authors: Ewbank William A.
Published: October 2020

Number tablets are simple and versatile teaching devices. These small square cards numbered 1–50 (or 1–20 for beginners) can be turned facedown, shuffled, and moved around in game and puzzle situations. The number tablets (referred to as NT hereafter) can be any suitable size, but I find that 1-inch (2.5-cm) squares are convenient. Over 600 NTs of this size can be cut from one standard sheet of posterboard, and pupils can write the numerals on the blank cards (fig. 1).
Two Technological Fables
Authors: Forish-Ferguson Laura
Published: October 2020

An experienced teacher has either a philosophical position or a gut feeling that reveals a bias toward the “how to” or the “what” of any subject to be taught. These two positions are often mutually exclusive; and thus the process-content debate has a long history in education. A good twenty years have elapsed since the hand-held calculator added its challenge to the debate. In the recent past, the applications of the personal computer in the school curriculum have come under discussion.
Teaching Mathematics with Technology: Mathematics and Spreadsheets
Authors: Bright George W.
Published: October 2020

Calculators are a useful tool for students working on problems involving a few calculations, but spreadsheets offer the assistance needed for problems requiring lots of calculations. Calculators typically display numbers one at a time, but spreadsheets can keep track of and display many numbers simultaneously. The power of spreadsheets, then, lets teachers expand both the content that is taught and the ways in which it is taught. Students ask “what if” questions by manipulating the data in spreadsheets and can learn useful skills for managing information.
Principles for Principals: Planning for 1989–90
Authors: Best Betty A. , Lilly Marsha W.
Published: October 2020

Every end is a new beginning. To begin the 1989-90 school year effectively, what commitments will you make to ensure a quality mathematics program?
Counting Skills–a Foundation for Early Mathematics
Authors: Frank Alan R.
Published: October 2020

The teaching of counting skills is an essential part of mathematics instruction programs for children, particularly young handicapped children (Baroody 1986). Counting skills are important in themselves. They are also important because they serve as prerequisites for many other mathematics skills (Silbert, Carnine, and Stein 1981). The purpose of this article is to discuss the vital role that counting plays in learning other mathematics skills and to suggest ways in which counting skills can be taught to young children experiencing learning difficulties in mathematics.
One Point of View: Manipulatives Don't Come with Guarantees
Authors: Baroody Arthur J.
Published: October 2020

Recent research indicates that to promote meaningful learning, mathematical instruction should begin with experience that are real to pupil and then proceed to the symbolic level (e.g., Baroody 1987; Davis 1984; Ginburg 1982). Clearly, manipulative can be an important tool in effort to make mathematics more real to pupil. However, to discourage their uncritical use, perhaps manipulative hould carry the following warning label: The Secretary of Education [or other appropriate authority] has not determined that using manipulatives is either a sufficient or a necessary condition for meaningful learning.
Calculating First Graders
Authors: Starkey Mary Ann
Published: October 2020

Calculators in first grade. Oh no! The students won't even know their basic facts.
Teaching Mathematics with Technology: Order of Operations
Authors: Wiebe James H.
Published: October 2020

The NCTM's recently released Curriculum and Evaluation Standards for School Mathematics (1989) recommends that calculators and computers be freely available to elementary school students for solving mathematical problems, exploring potterns and concepts, and investigating realistic applications. Most students, however, need some help in learning to use these tools, especially if they are using them with problems involving more than one operation, as many realistic applications do. They need to know that different calculators or computer software tools use different internal algorithms for finding answers and, thus, may give different answers to the same problems. They need to be taught how to enter multistep problems and evaluate the displayed result, that is, to do parallel mental computations. This article focuses on teaching elementary school students the order in which calculators and computer languages solve mathematical expressions.
PURR-R-R-FECTLY: Wild About Mathematics
Authors: Jacobs Jean W.
Published: October 2020

Can your school become “purr-r-r-fectly wild about mathematics”? Let me explain how our school did. Planning, organizing, and implementing a lively week of mathematics activities was one of the most challenging, yet rewarding, endeavors that our faculty, parents, and students have undertaken. The planning began early in December and culminated during the third week in April in a celebration of North Carolina's Math Month.
Research Into Practice: Parental Involvement in a Time of Changing Demographics
Authors: Secada Walter G.
Published: October 2020

The involvement of parents in their children's education can take a number of forms. In their synthesis of the research literature. Tangri and Moles (1987) outlined three dimen-sions of parental involvement. First, it can refer to service in schools (e.g., participating in school governance activities, working in classrooms as paid aides or volunteers). Second, it can refer to home-school relationships (e.g., written and phone communications. home visits by teachers, parent- teacher conferences at school, parent education and training sponsored by the school). Finally, parental involvement can refer to support of learning activities at home (e.g., assisting with homework, tutoring, providing educational enrichment activities). It should be noted that in this article, “parent” refers to any adult caregiver in the home.
How Does A Computer Subtract?
Authors: Evered Lisa
Published: October 2020

Students frequently believe computers perform computations almost magically when, in fact, most computer operations are quite simple. Your students will enjoy learning to subtract as a computer does. At the same time, they will discover that computer computations are not too complicated for them to understand after all.
Low Tech, 198, And The Geometry Of The Calculator Keys
Authors: Zollman Alan
Published: October 2020

The National Council of Teachers of Mathematics recommends that … mathematics programs take full advantage of the power of calculators and computers at all grade levels” (NCTM 1980, 8). Somehow, without meaning to, the calculator has taken a subsidiary role to the computer in the implementation of NCTM's An Anemia for Action recommendation. High technology is getting the majority of education's emphasis, while low technology, namely the calculator, is not having the impact that it could in the elementary school. This situation prevails despite the fact that 98 percent of this country's population uses calculators in everyday mathematics applications (Saunders 1980). It is time to reaffirm the viable role of the calculator in mathematics education (NCTM 1987). This article presents an interesting aspect of the geometrical array of the keys on a calculator that can be turned into a problem-solving, problem-posing situation for the upper elementary or middle school classroom. Read this article with a calculator in hand.
Implementing The Standards: The Geometry Standards in K—8 Mathematics
Authors: Rowan Thomas E.
Published: October 2020

A group of fifth graders was observed doing a textbook exercise in which they were to identify which of a set of five geometric shapes were rectangle (fig. 1). Their efforts were interesting and illustrated the concerns that some mathematics educator have about our current teaching of geometry. It was obvious that most of the students thought they knew what a rectangle was. But even those who exhibited this confidence had difficulty identifying the proper figures and explaining to their classmates why a trapezoid is not a rectangle or that the sides of a rectangle do not have to be horizontal or vertical. as in shape C in figure 1. They seemed to have an incomplete knowledge of this geometric shape, which is usually introduced in kindergarten or first grade.
Making And Exploring Tangrams
Authors: Dunkels Andrejs
Published: October 2020

Tangrams have been around for considerable time. Many of us have used them for activities to promote our students' sense of shapes. The most popular approach seem to be to use commercially produced cardboard or plastic tangram set or alternatively a template from which the pupils cut out the pieces. Sometime the pieces are numbered from I through 7 for reference. This article repor ts my experience of having pupils create their own tangrams. This activity offers pupils a rich experience in geometry and spatial sense.
One Point Of View: Calculators and Common Sense
Authors: Higgins Jon L.
Published: October 2020

When NCTM released its Curriculum and Evaluation Standards for School Mathematics in March 1989, I looked for the reaction of the press to this very important document. In almost every newspaper the headline or the Sthru t of the article related to the Standards was that NCTM now recommended that calculator be used in mathematics classrooms. How discouraging! NCTM members know that in March 1989 that recommendation was very old news. Even worse was the fact that the more substantial recommendations of the Standards were generally ignored.
Station Break: A Mathematics Game Using Cooperative Learning and Role Playing
Authors: Rees Rebecca D.
Published: October 2020

“Station break” is a teachermade game that incorporates role playing and cooperative learning as a way to help learners solve practical mathematical problems. It is intended for elementary and junior high school students, especially those who have limited interest or ability in mathematics. By writing problems that integrate current topics of study with learners' interests and ability levels, teachers can make the subject more interesting to students. Students learn to develop both social and collaborative skills while they improve their skills in mathematics. Students and the teacher work cooperatively to make preparation for the game easy.
Implementing the Standards: Implications of NCTM's Standards for Teaching Fractions and Decimals
Authors: Payne Joseph N. , Towsley Ann E.
Published: October 2020

A second grader calls all parts “half” no matter how large they are. A fourth grader writes 3.10 as the decimal for 3/10. A sixth grader can compute 1/2 × 1/2 but cannot reason that one-half of one-half of a pizza is one-fourth of the whole. Most eighth graders cannot estimate the sum 7/8 + 12/13 from choices of 1, 2, 19, and 21. Classroom teachers easily recognize similar deficiencies and ask, How do I teach for greater success?
Research into Practice: Teaching Mathematics and Thinking
Authors: Silver Edward A. , Smith Margaret S.
Published: October 2020

One afternoon. eleven-year-old Michael stopped by the home of an adult friend and found her nearly buried in decorating books, charts. and samples. Michael's sudden appearance at the door was a welcome sight, and he was asked to assist his friend in the process of decorating her home office. Although quite sure he knew little about interior decorating, Michael agreed to lend a hand. The friends began to measure the room and calculate the areas of the ceiling, walls, and floor. They looked at wallpaper swatch books, paint-color charts, and rug samples from various manufacturers and discussed the differences in price, the expected coverage per gallon of paint and roll of wallpaper, and the relative quality and ea e of upkeep for different products. One tyle of wallpaper they liked had a horizontal stripe and required matching. After approximating how many rolls of striped wallpaper they would need to buy, they performed a similar approximation for a design that did not require matching. They looked at carpet amples, discussed the relative merits of light and dark colors and variou styles, and considered the issue of cost versus quality with respect to durability and stain resistance. Finally, they selected a set of materials to comple te the decorating project based on cost, product quality. maintenance required, and personal preference. After helping his friend with this task for several hours, Michael departed for his home and his dinner.
Using a Polar Coordinate System in the Classroom
Authors: Carroll William M.
Published: October 2020

Most elementary school students have experiences with a rectangular graphing system that helps to prepare them for ideas in algebra and analytic geometry. Mathematics books at the elementary school level contain a section on graphing, though it is often one of the later chapters, where it may be treated as supplementary material. In science and in social studies, students may find various data graphed or may have a project in which they have to graph daily temperatures, rainfall, or test scores. Biorhythms, record sales charts, or, for the more ambitious, the daily stock market give additional exposure. Designing graphics on the computer monitor often requires some placement by a rectangular coordinate system. By the time students meet the Cartesian coordinate system with its x- and y-axes and use it to graph equations, the idea and techniques should be somewhat familiar.
Pentominoes Revisited
Authors: Onslow Barry
Published: October 2020

Pentominoes, a name coined by Solomon W. Golomb in 1954, lend an opportunity for students from grade 3 on to examine many interesting patterns and relationships while solving problems. This article uses pentominoes to examine the concepts of congruence, similarity, transformations (flips, turns, and slides), tessellations (tiling), networks, perimeter, area, and volume in a problem-solving, cooperative learning environment.
Some Observations of Mathematics Teaching in Japanese Elementary and Junior High Schools
Authors: Becker Jerry P. , Silver Edward A. , Kantowski Mary Grace , Travers Kenneth J. , Wilson James W.
Published: October 2020

A U.S.–Japan Seminar on Mathematical Problem Solving was held at the East-West Center in Honolulu 14– 18 July 1986 (Becker and Miwa 1987). Among the seminar's proposals was that cross-cultural research on American and Japanese students' problem-solving behaviors be organized and carried out. The author were in Japan in the fall of 1988 to meet with their Japanese counterparts, plan research, and make visits to mathematics classrooms preliminary to conducting a two-year program of research. In addition to planning the research, we were on a fact-finding visit to classrooms to better acquaint ourselves with mathematics teaching and learning in Japan.
Mathematical Connections: Free Rides For Kids
Authors: Bohan Harry
Published: October 2020

If one of your goals is to teach students to think mathematically for themselves (Curriculum and Evaluation Standards for School Mathematics, 1989; An Agenda for Action 1980), you may want to take them for some “free rides.” During a fourthgrade unit on fractions I incidentally used the term “free ride” for the first time in the introduction to a lesson on multiplication of mixed numbers. The class was told they were about to be taken on what could be called a free ride. A free ride was described as a situation in which they were studying a topic that was new, but one that could be attained by using what they know.
Implementing The Standards: Probability
Authors: Fennell Francis (Skip) , Rowan Thomas E.
Published: October 2020

These questions are asked by teachers exploring probability with their students. Informal exploration of chance is central to the development of beginning concepts related to probability. Chance is something that most students have experienced in playing games, watching television game shows, and participating in sports. Probability is fun! It should be an important component of any K–8 mathematics program.
Using a Calculator to Develop Number Sense
Authors: Bobis Janette F.
Published: October 2020

The aim of mathematic education today include the development of students' “abilities to recognie mathematical patterns and relation hips” (Mathematics K-12, Statement of Principles 1989). Turkel and Newman (1988) claim that people who can “easily perceive number relationships” have a well-developed number sense.
Learning to Write About Mathematics
Authors: Wilde Sandra
Published: October 2020

Carey, a first-grade student, had just finished learning through the exploration of cardboard “pizzas” that fractions with I in the numerator get smaller as the denominators increase.
Evaluating Expressions: A Problem-Solving Approach
Authors: Sanfiorenzo Norberto R.
Published: October 2020

Grouping symbols are used to indicate the order of performing operations in evaluating expressions. The most commonly used grouping symbols are parentheses and brackets.
Research Into Practice: Helping to Make the Transition to Algebra
Authors: Kieran Carolyn
Published: October 2020

Knowledge is not an entity which can be simply transferred from those who have to those who don't.… Knowledge is something which each individual learner must construct for and by himself. This view of knowledge as an individual construction … is usually referred to as constructivism. (Lochhead. cited in Blais 1988, 624)
Implementing The Standards: Geometry through the Standards
Authors: Morrow Lorna J. , Rowan Thomas E.
Published: October 2020

An underlying view of mathematics education expressed in the Curriculum and Evaluation Standards (NCTM 1989) is that a student should be actively involved both mentally and physically in constructing his or her own mathematical knowledge: “The K-4 curriculum should actively involve children in doing mathematics. … [They should] explore, justify, represent, solve, construct, discuss, use, investigate, describe, develop, and predict” (NCTM 1989, 17).
The Power Of Five: An Alternative Model
Authors: Erp Jos W. M. van
Published: October 2020

A method of teaching basic addition and subtrctction that facilitates learning without relying on either counting or rote memorization was presented by Flexer (1986). It employs concrete models of numbers that not only emphasize the baseten system but also show single-digit numbers in terms of five. Her 5-frame is a modification of Wirtz's 10-frame (Wirtz 1980): the application of it is based on Thompson and Van de Walle's (1984) use of this 10-frame and on the method of mental regroping proposed by the Association of Mathematical In truction (AMI) in Japan (Hatano 1980: Ginbayashi 1984).