bluuebunny/crossref_metadata_2025 huggingface dataset:
LCM—Let's Put It in Its Place
Authors: Ewbank William A.
Published: October 2020

By coincidence, the topic of least common multiple (LCM) had four separate mentions in the Arithmetic Teacher's December 1984 issue. Two letter refer to an article in the April 1984 issue (Lamb 1984); a Korean student introduces a “nover” method of finding LCMs that is the method I learned in England fifty year ago and that I have taught to U.S. college students for the last twenty years; and Doyle gives a computer program for generating LCMs (Doyle 1984). This discussion is very good, but when all is said and done, why is this rather cumber orne skill taught at all? I uggest it is a leftover from the skill-and-drill pencil-and-paper curriculum of our grandparents.
Teaching Mathematics with Technology: Multiples
Authors: Jensen Robert J.
Published: October 2020

Throughout children's study of mathematics, the concept of multiples and the skills associated with multiples play an important role. Initially, children's love of rhymes such as “two, four, six, eight, meet me at the garden gate” is extended to include skip counting with numbers other than 2. Skip counting forms the basis for the production of tables of basic multiplication facts. Children are later expected to acquire skill in accessing these multiples in random order (i.e., knowing that 7 times 2 is 14 without having to skip count by twos until 14). In intermediate grades the concept and skills related to multiples are extended to include comparisons between sets of multiples in search of common multiples or, even more restrictively, the least common multiple (LCM) of two or more numbers.
Up and Down Like a Hailstone in a Storm Cloud
Authors: Rubillo James M.
Published: October 2020

Number theory has fascinated mathematicians, numerologists, and puzzle solvers for generations. The great appeal of this branch of mathematics stems from the simple nature of the tools required to explore an interesting idea. These tools usually consist of persistence, ingenuity, and the four basic operations that are taught in the elementary grades. This article outlines an unsolved problem from number theory that can be explored by young students. The problem does not have a long or clear history. It was first mentioned during the 1930s and has periodically reappeared as a mathematical curiosity. The problem is known a the “3n + 1 problem” or the “hailstone problem.” Both names are obscure and do not lend an insight into the nature of the problem.
One Point of View: We Get What We Ask For
Authors: Higgins Jon L.
Published: October 2020

In his “One Point of View” editorial (October 1986), Robert B. Davis concludes his discussion of under-standing and learning as follows: “Should all children learn in terms of understanding instead of rote and ritual? I think so, but sometimes a few children force me to teach them a meaningles ritual.”
Conservation—a Meaningful Gauge for Assessment
Authors: McClintic Susan V.
Published: October 2020

Piaget's work on number conservation and one-to-one correspondence has had a valuable impact on teachers' understandings of the development of mathematics concepts in young children. However, a potential misapplication is the tendency to use Piaget's tasks of conservation of number and one-to-one correspondence as a sort of “entrance exam” into classroom number work. Whereas children who can conserve participate fully in classroom number activities, non-conservers may receive no mathematics instruction at all. Or, they may passively observe instruction that is beyond their abilities while teachers wait for conservation to occur. This article proposes that teachers should examine the elements of mathematical understandings that precede conservation and one-to-one correspondence and tailor classroom instruction to benefit both conserving and nonconserving students.
The Early Development of Number Relations
Authors: Walle John A. Van de
Published: October 2020

How many things can you think of that might be useful for a third-grade student or even an adult to know about the number seven? To know about seven certainly means something more than the ability to count seven objects and so name them as a set of seven. It involves the knowledge that seven is more than four, that it is one less than eight, two more than five, can be “made” from three and four as well as two and five, that seven and three more will make ten, and so on. If we consider the question “What is seven?” for a while, we may very well come to the conclusion that seven is not a simple concept that a student either has or does not have. Seven is a complex set of ideas.
Fractions in the Early Childhood Mathematics Curriculum
Authors: Bezuk Nadine S.
Published: October 2020

Fractions! This word evokes anxiety, discomfort, and even fear in many children (and even adults) and dismay in the teachers who have to teach them. However, experiences with fractions are part of children's everyday lives. Half a peanut butter sandwich, half an a pple, and a quarter of a dollar are fractions that children often encounter. The early childhood curriculum can capitalize on children's interest in their environment and their awareness of the existence of fractions in their world while laying the foundation for some important mathematics learning.
Research into Practice: Place Value for Tens and Ones
Authors: Payne Joseph N.
Published: October 2020

Two major ways to view the numbers to 100 are counting by one and grouping by tens and ones.
Teaching Mathematics with Technology: Concept Formation in Geometry: A Computer-aided, Student-centered Activity
Authors: Jensen Robert J.
Published: October 2020

Computers are well suited for generating geometric figures of random size and shape. Children are able to form conceptual generalizations when given various examples and counterexamples of a particular class of objects. This month's department suggests one possible way to combine this technological facility of the computer with students' natural abilities for concept formation into a powerful learning tandem. Small groups of students will be asked to create a “family album” for a specific shape. In the process, students will first use a computer to generate random samples of this shape and to obtain measures of specific attributes for each figure. Then they will use a ruler and protractor to reproduce these examples for their albums. Finally, they will develop a list of properties for the shape they are investigating. In the construction of their own descriptions for a particular geometric shape, students will creatively engage in the doing of mathematics. The principal tools for this activity will be a computer, the Geometric preSupposer software, a metric ruler, and a protractor.
Teaching Mathematics with Technology: Ratios
Authors: Jensen Robert J.
Published: October 2020

Learning about ratios empowers children with a language and procedure for comparing relative quantities in an efficient way. Understanding ratios is also a necessary precursor to meaningful work with proportional reasoning. The fact that many adults do not reach the Piagetian level described as “formal thought” is often attributed to weaknesses in their proportional reasoning skills. Aside from justifying the study of ratios as a critical component of formal thought, the varied real-life applications for ratios (cooking, model building, map reading, converting among and between measurement systems, calculating odds, scale drawing, unit pricing, etc.) make this area particularly meaningful to students. This month's department suggests a series of ratio activities for children that make use of a computer to produce graphically collections of objects in a grid with variable dimensions.
Research into Practice: Similarity in the Middle Grades
Authors: Lappan Glenda , Even Ruhama
Published: October 2020

Simila rity is an important topic in geometry, basic to understanding the geometry of indirect measurement, proportional reasoning, scale drawing and modeling, and the nature of growing. When United States’ teachers were asked to rate the importance of this topic for the Second International Study of Mathematics, they rated similarity of plane figures as being important for all students in grade 8.
One Point of View: Accomplishments of the 1980s
Authors: Willoughby Stephen S.
Published: October 2020

Typical of the 1980s is the fact that I was asked to write this evaluation of the 1980s two years before the decade ends. We have been quick to talk about our goals and accomplishments but slow to make meaningful change.
Geometric Extravaganza: Spicing Up Geometry
Authors: Larke Patricia J.
Published: October 2020

If we can have science fairs, why not geometry fairs? They are excellent ways for elementary teachers to add pizzazz to the teaching of geometry. A geometry fair or geometric extravaganza is a display or exhibit of geometry projects representing the students' culminating work in a geometry unit. The purposes of a geometry fair a re (I) to remind students of important geometric terms and concepts; (2) to enable students to explore the world of lines, angles, points, and geometric shapes; (3) to help students identify and construct geome tric shapes and designs; (4) to help students prepare projects using their knowledge of geometry and creativity; and (5) to help students share work with othe rs. thus building pride in their work.
Principles for Principals
Authors: Hynes Michael C.
Published: October 2020

This is the first installment of a column that will focus on issues related to elementary and middle school mathematics. Each month a principal will be highlighted and principles for administrative involvement in mathematics instruction will be preented.
A Cognitive Approach to Writing Instruction for Children Classified as Mentally Handicapped
Authors: Baroody Arthur J.
Published: October 2020

Many children with learning handicaps have difficulty learning to write numerals. To be effective, remedial instruction must redress the real problem. Otherwise, students (and teacher) may suffer unnecessary frustration. What, then, is the source of numeral-writing difficulties? And what can be done to teach or remedy this skill effectively?
Mathematics on the Home Front
Authors: Flexer Roberta J. , Topping Carolyn L.
Published: October 2020

Parents teach their children all the time. They teach, almost from the moment of birth, as they respond to, and interact with, their infants. They teach about the world that surrounds children, about sights and sounds, and about other stimulation. They teach language to their offspring almost without conscious effort. They are teaching language when they talk to their infants, when they smile at them as they make their first sounds, and when they encourage, ever so patiently, those first words. It is no accident that a child's first word often is “ma” or “da.”
MATHCOUNTS: The Ultimate in Problem Solving
Authors: Cox Sherry L.
Published: October 2020

Mathcounts is a mathematics competition for seventh and eighth graders sponsored in part by the National Council of Teachers of Mathematics, the CNA Insurance Compani es, the Nationa l Society of Professional Engineers, the National Aeronautics and Space Administration, the General Motors Foundation, and the Cray Research Foundation. It offers students an opportunity to expand the ir appreciation of mathematics and to improve their problemsolving abi lities.
Diagnosis in Mathematics: The Advantages of an Interview
Authors: Liedtke Werner
Published: October 2020

Before a teacher can offer meaningful mathematics remediation, data about a student's strength and weaknesses need to be collected. Although written responses on an activity sheet or on a test may indicate areas of concern, many times more information is required before accurate guesses about appropriate remediating task can be made. We have found that a guess based solely on examining written responses may in fact be incorrect.
Word Problems: Do I Add or Subtract?
Authors: Mahlios Jan
Published: October 2020

Studies of word problems and children's solutions to them have produced distinct and detailed classifications of such problems. Classifications of problems are often based on the active or static nature of the problem, on a set-subset relationship, or on comparisons of distinct sets. Within each problem class, the placement of the unknown factor creates more categories. An in structional technique is presented here to help student create a symbolic representation that may facilitate the decision-making process of choosing an appropriate operation to solve the problem.
Mathematics + Social Studies = Learning Connections
Authors: Spence Carolyn , Martin Carol S.
Published: October 2020

All too often the school day of a six-year-old is fractured into thirty-minute segments, of compartmentalized learning. When each subject is dealt with in isolation, children are often robbed of the opportunity to make connections from one part of the curriculum to another. Since establishing such relationships lies at the heart of the educational process, early childhood teachers face the daily challenge of building bridges from one subject area to another.
Results of the Fourth NAEP Assessment of Mathematics: Trends and Conclusions
Authors: Carpenter Thomas P. , Lindquist Mary M. , Brown Catherine A. , Kouba Vicky L. , Silver Edward A. , Swafford Jane O.
Published: October 2020

This article is the third one to appear in the Arithmetic Teacher reporting the results of the fourth mathematics as essment of the National Asse sment of Educational Progress (NAEP). The first two article reported achievement of third-and seventh-grade students in pecific content area (Kouba et al. 1988a. 1988b). This article discusse general achievement trend and change in performance over time. A parallel article on trend in performance of older students appears in this month's Mathematics Teacher (Silver et al. 1988).
Research into Practice: Using Money to Teach about the Decimal System
Authors: Schliemann Analúcia Dias
Published: October 2020

Children learn much mathematics in everyday life. When they buy things, for example, they have to count their money and calculate their change. Counting money is a special type of counting. On the one hand, when children count objects, they establish a one-to-one correspondence bet ween objects and number words; only absolute value, as opposed to relative value, is involved. On the other hand, when counting money children must bear in mind both types of value; while they deal with the coins one by one (absolute value), they must also take into account the relative value of the coins. Counting money thus helps children understand the decomposition of numbers; a large amount (for example, sixty-eight) is made up of smaller amounts—tens and ones—that can be repeated (six tens and eight one). All these ideas—relative and absolute value, decomposition, and repetition of equal values—are basic both to the understanding of the decimal system and to the undertanding of important properties of arithmetic operations.
Principles for Principals: Tips for Management of Manipulatives
Authors: Kamp Donald A. , McGilvray Midge
Published: October 2020

Fourth-grade teacher Susan O. Strauss registered an SOS with her principal, “I went to the mathematics in-service training and learned how to teach fractions using pattern blocks. I know we have block, but I can't find them or the accompanying booklet. I asked three teachers, and no one seems to know where they are. I don't know if it is worth the hassle. I can't go through this every time!”
Integrating Arithmetic and Geometry with Numbered Points on a Circle
Authors: Newman Claire M. , Turkel Susan B.
Published: October 2020

Over the years recommendations have been made to treat mathematics as a unified discipline by showing how the various branches are interrelated. Learning theorists indicate that ideas presented in a unified form are far more intelligible than those that seem to be nothing more than an unorganized mass of rules to be memorized. It is helpful for the learner to see how various mathematical topics fit together.
Teaching Number Sense
Authors: Howden Hilde
Published: October 2020

Of all the Standards in the K–4 component of NCTM's working draft of Curriculum and Evaluation Standards for School Mathematics, the one concerning number en e raised the most questions from teachers, parents, and administrators in session held to acquaint our educational community with the Standards. These four questions were most frequently asked: What is number sense? Why is number sen e important? How is number cnse taught? How is number sense measured?
Applying Number Sense to Problem Solving
Authors: Dougherty Barbara J. , Crites Terry
Published: October 2020

NCTM's Commission on Standards for School Mathematics (1987) has identified problem solving and number sense as important components of an effective mathematics program. This emphasis is generating attempts to understand the problem-solving process better and to incorporate the results into classroom practice. In keeping with the thrust, this article discusses the interrelationships between problem solving and number sense in light of difficulties experienced by students participating in the problem-solving process.
No Answer Please
Authors: Glatzer David J. , Glatzer Joyce
Published: October 2020

Students often give answer without regard to number sense.
Principles for Principals: Facilitating Change
Authors: Leiva Miriam A.
Published: October 2020

Changes are taking place in the teaching and learning of mathematics—students are actively involved in doing mathematics with manipulatives and models; they are discussing problems among themelves with their teacher's guidance; they are exploring alternate ways to solve problems; and they are posing questions and examining solu tions (Dossey et al. 1988). This student-oriented, problem-solving approach is advocated by NCTM's Curriculum and Evaluw ion Standards for School Mathematics (Commission on Standards for School Mathematics of the NCTM 1987) and supported by the finding of the 1986 National As essment of Educa tional Progress. Principals, the instructional leaders in the schools. can be facilitator of change, and their support of innovative programs and activitie is crucial to reforming school mathematics.
One Point of View: Textbooks and Curriculum—Whose Dilemma?
Authors: Keedy M. L.
Published: October 2020

Are textbook content and curriculum necessarily the same? We in the teaching profession insist that they are not, but to our chagrin we must admit that they almost always are. Logically, it follows that to revise the curriculum we must make changes in texts. The recommendations in NCTM's Curriculum and Evaluation Standards (Commission on Standards for School Mathematics of the NCTM 1987), for example, will never be widely implemented unless they are incorporated into commercially published texts.
Research into Practice: Arithmetic as Problem Solving
Authors: Lampert Magdalene
Published: October 2020

Teachers often feel torn, especially in upper elementary school mathematics classes, between spending time on problem-solving work that will get students to understand mathematics and spending time on developing computational skills. The conflict is especially strong because the computational skills in the curriculum at this level are complex: “long” multiplication and division, relating fractions to decimals and percents, and operations on fractions. The procedures involved in doing these computations involve many steps, and students often have difficulty remembering what to do and in what order.
“Look Ahead” Activities Spark Success in Addition and Subtraction Number-Fact Learning
Authors: Thornton Carol A.
Published: October 2020

The ability to count with skill and understanding is a useful and important problem-solving skill in mathematic. Counting is a child's natural approach to number work in kinder-garten and first grade and is the basis for finding solutions in addition and subtraction, both within and apart from problem-solving contexts. Place yourself in a first-grade classroom in early February and “listen” to these children.
Developing a Problem-Solving Lesson
Authors: Cemen Pamala Byrd
Published: October 2020

Problem solving is one of the most important mathematical abilities that teachers can foster in students, as evidenced by its prominent role in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and An Agenda for Action (1980). Because of the increased availability of calculators and computers, knowledge of which computations are necessary is more important than proficiency in carrying out those computations. As a result, elementary school teachers are being encouraged and entreated to teach problem solving. However, teaching such a topic at higher cognitive levels is far from easy; the key is to be able to ask good questions. Because excellent problem-solving questions are seldom created “on the spot,” teachers will benefit from writing lesson plans that include questions they can ask at crucial moments, keeping in mind they may not need the questions at all.
IDEAS
Authors: Silverman Helene
Published: October 2020

The IDEAS section for this month focuses on the reading, recording, and analyzing of schedules. Students will be encouraged to discover patterns and to reorganize the data in various forms.
Six Hard Pieces
Authors: Jamski William D.
Published: October 2020

During a recent class with inservice elementary school teachers, my students and I worked with a standard set of area-interrelated tangram pieces that contained a square (S), a parallelogram (P), and five (right isosceles) triangles: two little (L); one medium (M); and two big (B). Figure 1 includes them with dotted segments to show relative areas clearly.
Suzuki Meets Pólya: Teaching Mathematics To Young Pupils
Authors: Hazlewood Donald G. , Stouffer Sandy , Warshauer Max
Published: October 2020

The Suzuki method (Suzuki 1969, 1981, 1984) of teaching young pupils to play the violin has been widely used with remarkable results. Suzuki credits his approach to a simple observation, namely that all Japanese children learn to speak their native language and, thus, must have high ability for learning. Suzuki reasoned that this ability, if properly developed. would enable children to learn to play the violin. Since his basic premise was that all children are capable of high achievement, the problem for any child who could not learn music must be in the way music was taught.
Implementing The Standards: Patterns, Relationships, and Functions
Authors: Rowan Thomas E. , Howden Hilde
Published: October 2020

In the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). function is viewed as both a concept and a process. As a concept, it is the study of regularity and quantification of phenomena, which is the essence of mathematics. As a process, students apply the concept to analyze relationships throughout the curriculum by using tables, graphs, verbal and mathematical rules, and models to represent them.
Polygons On a Number Lattice: Sums, Products, and Differences
Authors: Litwiller Bonnie H. , Duncan David R.
Published: October 2020

Every teacher wants to find meaningful situations in which his or her students can practice computational skills, either with paper and pencil or using the calculator. One way in which this practice can be meaningful is to couple it with the discovery of numerical patterns. We shall present several problems that combine practice and discovery. These problems wi ll all involve a number lattice in the form of a triangle, as shown in figure 1.
Implementing The Standards: Assessing Students' Learning To Inform Teaching: The Message In Nctm's Evaluation Standards
Authors: Thompson Alba G. , Briars Diane J.
Published: October 2020

Acentral theme of NCTM's evaluation standards (National Council of Teachers of Mathematics. Commission on Standards for School Mathematics 1989) is that assessment is an integral part of teaching. One of the many reasons to assess students' learning is to find out where we teacher are in relation to our goals. Another reason is to be able to inform students, parents, or administrators about students' progress. Regard less of the relative merits of these and other reasons for assessing students' learning, the main purpose of assessment should be to inform teaching.
Selected Children's Books For Geometry
Authors: Thiessen Diane , Matthias Margaret
Published: October 2020

What experiences have your students had with geometry both outside and within the classroom? Do your students view the study of geometry as interesting and functional? Do they appreciate the beauty of geometric design? How is geometry presented in your textbook? Are the lessons on exploring shapes limited to having the students name such figures as triangles and squares or spheres and cubes? Are the lessons on area limited to substituting measures in formulas? Are the lessons on constructions limited to step-by-step instructions? Has geometry been reduced to the study of definitions? Or are the lessons designed with activities in which students develop geometric concepts and relationships through explorations?
Acting Out Numbers
Authors: Strand Susan
Published: October 2020

The numeral 123 is written on the chalkboard. “Oh. that's one hundred thirty-two.” says six-year-old Wendy. “No. it's twelve hundred three.” asserts her partner.
Number-Lattice Polygons and Patterns: Sums and Products
Authors: Duncan David R. , Litwiller Bonnie H.
Published: October 2020

Elementary school teachers are always interested in furnishing computational review for their students. Students are more motivated if the review takes place in a nonroutine setting. It is also a bonus if number patterns are discovered in the process.
Teaching Mathematics with Technology: Data Base Programs in the Mathematics Classroom
Authors: Wiebe James H.
Published: October 2020

A data base is a set of information typically stored on a computer-related medium like a floppy diskette. A data base manager (DBM) is a computer program that is used to create and manipulate data bases. A data base is the computerized equivalent of a file cabinet, with “records” (analogous to file folders) containing information about individuals. A single piece of information, like a telephone number or address, is contained in a field, or category.
Developing Spatial Skills With Three-Dimensional Puzzles
Authors: Izard John
Published: October 2020

Spatial skills are an important component of the mathematics curriculum. Both teachers and students have to make use of their spatial skills when using two-dimensional diagrams to represent three-dimensional objects and when interpreting photographs or television images (see, e.g., Pohl [1986]; Greenfield [1987]: Smith [1987]; Bright and Harvey 11988]: Carroll [1988]).
Discovering Pi — Two Approaches
Authors: Gerver Robert
Published: October 2020

It is unreasonable to expect students to become adept in the measurement of the area and circumference of circles without a thorough introduction to pi. Why is it that the phrase “the area is 49π square inches” means nothing to so many elementary and secondary school students? Although pi's role in mathematic is seldom clownplayed. it is rarely developed in the context of the formula A = πr2. Perhaps every student needs to take a “discovery” look at pi.
Teaching Money with Grids
Authors: Stevenson Cathy L.
Published: October 2020

Counting money and making change are often difficult concepts for students in the primary grades to grasp. Boys and girls often do not comprehend the relative values of the coins and how to combine these values to determine amounts of money. This problem often seems to exist even when students are able to give the name of the coins and their individual value by rote memory.
One Point of View: “How Can I Get Others to Implement the Standards? I'm Just a Teacher!”
Authors: Hitch Chris
Published: October 2020

After a great deal of anticipation, the NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989) has arrived, and NCTM members have been inundated with information about this new and exciting document.
Implementing The Standards: The Vision of Problem Solving in the Standards
Authors: Campbell Patricia F. , Bamberger Honi J.
Published: October 2020

Problem solving has been espoused as a goal in mathematics education since the late 1970s, with focused attention ansmg from NCTM's An Agenda for Action (1980). But problem solving should be more than a slogan offered for its appeal and widespread acceptance. It should be a cornerstone of mathematics curriculum and instruction, fostering the development of mathematical knowledge and a chance to apply and connect previously constructed mathematical understandings. This perception of problem solving is presented in the Curriculum and Evaluation Standards for School Mathematics (Standards) (NCTM 1989, 23, 75). See table 1. Indeed, as noted in the Standards, “students need to work on problems that may take hours, days, and even weeks to solve. Although some may be relatively simple exercises to be accompplished independently, others should involve small groups or an entire class working cooperatively” (NCTM 1989, 6).
One Point of View: Using Manipulatives Successfully
Authors: Joyner Jeane M.
Published: October 2020

A curriculum with goals for students of valuing mathematics, being confident in their abilities, making mathematical connections, becoming mathematical problem solvers, and learning to reason and communicate mathematically is a call for classrooms in which students are actively involved in learning. It is a call for teachers to establish environments that encourage the use of manipulatives to assist students in attaining these goals proposed by the NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989). A major difficulty, however, is how to manage the materials efficiently.
Implementing The Standards: Meanings of Operations
Authors: Trafton Paul R. , Zawojewski Judith S.
Published: October 2020

“Carlos finally understands subtraction! He got nine of the ten exercises correct, and every one of them involved renaming.” Statements like this confuse students' computational proficiency with their understanding of an operation. Ironically, this student's understanding of the operation of subtraction may be very shallow. We often give too little attention to building and assessing concepts of operations, yet this is one of three important components of teaching and learning about number that are included in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). The first of the three components is the standard on number sense and numeration. Concepts of operations (see table 1) is the second and builds on number sense. The third of the three number standards, whole-number computation, must build on the other two areas. A deep understanding of the concepts associated with an operation results from careful attention to many important real-world and mathematical ideas and relationships.
Teaching Mathematics With Technology: Calculators and Division
Authors: Usnick Virginia E. , Lamphere Patricia M.
Published: October 2020

Calculators help teachers teach in ways thot encourage students to become actively involved in their learning. This personal involvement is certainly consistent with the goals set forth in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). Calculators also allow the exploration of exercises that, although beyond the computational expertise of the students, may furnish opportunities for development of higher-order thinking skills
The Number Line And Mental Arithmetic
Authors: Kurland Theodore E.
Published: October 2020

Some students have easily recognized difficulties with addition and subtraction. Some have no trouble adding or subtracting single-digit numbers when the sums are less than ten (7 + 2, 5 + 4, etc.) but have to resort to their fingers for sums greater than ten (7 + 5, 8 + 6, etc.). Other students have no difficulty adding numbers whose sums are greater than ten, such as 7 + 5, but have difficulty determining their differences, like 12 − 7. Finally, some students have no difficulty adding 7 + 5 or 8 + 4 but cannot mentall y add 17 + 5 or 18 + 4 or recognize the connection between the sum of singledigit numbers and numbers inc reased by orders of ten. Any one or a combination of these difficulties may appear when students compute; moreover, these problems will continue to plague them, curtailing their confidence and development in mathematics and mental arithmetic.
Creating Magic Squares
Authors: Lyon Betty Clayton
Published: October 2020

For centuries, magic squares have held a fascination for young and old. Those who are challenged by magic squares are always searching for new ways to generate them. Recall that a magic square is an n × n array of numbers in which the sum of the numbers in each row, each column, and each diagonal is the same. It can be shown that a given square is “ magic” by adding the proper components but it is more exciting to create a new square and show that it is magic.
Transition Boards: A Good Idea Made Better
Authors: Sutton John T. , Urbatsch Tonya D.
Published: October 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) recognizes that addition and subtraction computations remain an important part of the school mathematics curriculum and recommends that the emphasis be shifted to the understanding of concepts. Transition boards are simple devices to aid students' conceptual understanding.
Two-Digit: Addition and Subtraction: What Works?
Authors: Thompson Frances
Published: October 2020

Students are introduced to two digit addition and subtraction during the second grade. This is their first encounter with the idea of regrouping in computation. Previous computation has been with singledigit numerals in the basic addition and subtraction facts. Much groundwork is necessary in numeration before students are introduced to twodigit computation. Students must have an understanding of how tens and ones are related in our base-ten notation. They need many varied experiences involving regrouping 10 ones for 1 ten or changing 1 ten for 10 ones before an algorithm is even introduced. But how to introduce the addition and subtraction algorithms effectively is the real problem.
Ideas
Authors: Barnett Carne S. , Young Sharon L.
Published: October 2020

The ideas section for this month focuses on collecting and analyzing data. with a focus on consumer awareness and judgment. Students have opportunities to sort, to compare the means for a set of data, to design their own investigations, to argue and defend points of view, and to make graphs. This month's reproducible sheet for the IDEAS section can be used by multiple grade levels. They include an activity sheet for parents to use with their children, three class-activity sheets, and a data sheet. You may find that several of the activities will be appropriate for your students.
American Indians, Mathematical Attitudes, And The Standards
Authors: Taylor Lyn , Stevens Ellen , Peregoy John J. , Bath Barbara
Published: October 2020

Readers may wonder what American Indians. mathematical attitudes, and the Standards (Curriculum and Evaluation Standards for School Mathematics [NCTM 1989]) have in common. This article presents practical activities relevant to teaching mathematics in the lndian culture. These activities also serve as a means for incorporating a cross-cultural component in the non Indian classroom. The activities exemplify the spirit of the NCTM's Standards and have been used successfully with American Indian middle school students in summer mathematics camps. We believe that they are appropriate for use in elementary and secondary school mathematics classes. This belief is based on research done with other populations and on our own in-class experiences.
Think, Tell, Share—Success for Students
Authors: Thornton Carol A.
Published: October 2020

As part of a continuing effort in data analysis, Mr. Denis presented a ride-to-the-park graph (fig. 1) to his middle school students and checked their understanding of the way the data were entered. “What do the numbers across the bottom of the graph tell us about Alba's ride? What about those in the left-hand column? [Pointing to (9, 6)] This point is not above any number. What does it represent?”
IDEAS
Authors: Westley Joan E.
Published: October 2020

The Ideas section for this month focuses on gathering. displaying, and interpreting data on balloons in activities that integrate mathematics and science. Students are involved in sampling, taking polls and surveys, doing experiments, classifying, estimating, and measuring. This month's reproducible sheets for the IDEAS section are designed to be used by students in the early elementary grades (1-4). They include three classactivity sheets and one activity sheet for parents to use with their children. Consequently, a teacher may want to reproduce and use several sheets.
“Stocking Up” on Mathematics Skills
Authors: Bitner Joe , Partridge M. Elizabeth
Published: October 2020

These comments are some of those heard in a sixth-grade classroom as the students peruse the financial section of the newspaper. Why are these students so excited about stock quotations? They are playing the “stock-market game,” an activity that teaches mathematics skills and economic concepts through a meaningful, real-life experience.
A Historical Reconstruction of Our Number System
Authors: Krusen Kim
Published: October 2020

Imagine your class as a “primitive society” just on the brink of civilization. Your society has been using tally sticks to represent numerical quantities. But now that your society is becoming more involved in commerce with other societies, you need an easier way to represent large numbers and some structure so that numbers can be manipulated. You need an organized number system. Creating a number system from scratch was the recent task of my sixth-grade class. My objective was to offer a more humanistic approach for my students further to understand and appreciate the structure of our number system. As the teacher. I was armed with a general knowledge of the number systems of the great ancient civilizations, and my students were armed with an enthusiasm to be cave dwellers for the day instead of mathematics students. With these resources, we began our project.
Teaching Mathematics With Technology: Using Calculators in Mathematics Changes Testing
Authors: Harvey John G.
Published: October 2020

No matter the level at which we teach mathematics, we are being asked to incorporate calculators into our instruction, to teach students both calculator facility and effective ways of using calculators, and to encourage and expect those students to use calculators appropriately. As early as 1975, just three years after the introduction of Texas Instruments's Data Math calculator, the National Advisory Committee on Mathematical Education (NACOME) urged that calculators be used in mathematics instruction (NACOME 1975, 40–43). Five years later the National Council of Teachers of Mathematics recommended that “mathematics programs [should] take full advantage of calculators … at all grade levels” (NCTM 1980, 1).
Table-Top Mathematics—A Home-Study Program For Early Childhood
Authors: Ford Marilyn Sue , Crew Caroline Gibson
Published: October 2020

When parents come into the kindergarten or first-grade classroom for their first conference, the teacher often hears one of two concerns: “How can I help my child at home?” and “I can't work with my child at home; it's too frustrating.” Coming from parents who have little or no experience in the teaching of young children. these statements are valid and understandable. First, parents often do not know what to do to assist their child in extending the classroom learning at home. Their attempt is often directed back to the teacher by asking for extra worksheets to be completed after school. However, the young child who has worked hard at school should not be expected to return home, it down, and do worksheets. Children should be allowed a time to relax, just as adults require after a day at work. Homework should be in the form of an activity similar to the developmental, hands-on learning experiences used in the classroom.
A Mathematical Conversation With Fourth Graders
Authors: Goldenberg E. Paul
Published: October 2020

This article reports a mathematical conversation with some fourth graders. As part of an NSF-funded curriculum-development project conducted at Education Development Center, we wanted to see how students would think about decimals if their first encounter with them was as “more numbers between the familiar numbers.” Two groups of eight fourth graders were selected in a school serving a predominantly blue-collar semiurban community near Boston. One group was identified as “better than average but not the brightest,” whereas the other was decribed as having “real problems” and “little mathematical ability.” None of the sixteen students had apparently learned anything about decimals before.
Implementing The Standards: Teaching Informal Algebra
Authors: Schultz James E.
Published: October 2020

Algebra is described in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, 150) as “the language through which most of mathematics is communicated.” For many years algebra has often referred to a single course or two featuring manipulative skills and punctuated by other courses called prealgebra or geometry. In the spirit of the Curriculum and Evaluation Standards, it is more appropriate to think of algebra as a cohesive body of concepts, closely connected to other branches of mathematics, in which manipulative skills play a supporting rather than star role. In this context, the distinction between prealgebra and algebra is less apparent. Rather, a gradual building from informal to formal concepts takes place over most of the K–12 curriculum. In the continuing series on implementing the Standards, this article presents suggestions for developing algebraic concepts beginning in the early grades.
The Tangram— It's More Than an Ancient puzlze
Authors: Kriegler Shelley
Published: October 2020

Would you like to integrate geometric relationship, scale drawings. graphing, patterns, and logical thinking into a unit that requires problem solving and cooperation among students? Would you like a three-to-five-day lesson plan that requires a minimum of preparation on your part and maximum involvement by students? Would you like activities that seem to be enjoyed by upper elementary. middle, and high school students regardless of mathematical ability? Then lessons on tangrams are for you!
IDEAS
Authors: Bankard Dianne , Fennell Francis (Skip)
Published: October 2020

The ideas section for this month focuses on uses of numbers. The activities explore how students use numbers in school and home settings as a way for students and the teacher to get to know each other at the beginning of the school year. The activities involve data collection and interpretation and probability and statistics and emphasize a growth-and-development theme. The IDEAS activity sheets are designed to be used by multiple grade levels. Most of the four class-activity sheets can be used by either individual students or small cooperative-learning groups. Also included is an activity sheet to be used as a school-home connection. Encourage the students to complete this activity sheet as a parent-child project.
Understanding Partitive Division of Fractions
Authors: Ott Jack M. , Snook Daniel L. , Gibson Diana L.
Published: October 2020

When first-grade students start addition of whole numbers, they are given sets of objects to combine and count to determine the sum. Most educators agree that beginning students need experience combining group or objects, such as combining two pencils and three pencils to get a group of five pencils. Such activities are needed to give students an understanding of what addition means before moving on to faster — but less meaningful — ways to sum numbers, such as the use of an algorithm or calculator. This approach is based on the learning principle that meaning comes from knowing the things signified by the symbols. Thus, familiar, concrete experience — actual or recalled — should be a first step in the development of new abstract concepts and their symbolization. Though this principle of moving from the concrete to the abstract is extensively used in teaching the addition of whole numbers, it is used much less in teaching the division of fractions. The inevitable result in that a student may be able to use an algorithm by rote to divide fractions, but the skill is useless because it is devoid of meaning.
Catering to All Abilities through “Good” Questions
Authors: Sullivan Peter , Clarke David
Published: October 2020

“Good” questions are a way of stimulating better quality communication between teachers and students. “Good” questions have three features: the students are required to do more than simply remember a strategy to answer them; the students can learn in the process of answering the question; and the questions have several acceptable answers. This article considers how such questions can be used as a way of catering to the range of abilities found in most classrooms. It first dicusses the idea of mixed-ability teaching and then explains how good questions can be used in mixedability classes.
The NCTMs “Standards”: New Dimensions in Leadership
Authors: Carl Iris , Frye Shirley M.
Published: October 2020

Together, we have been doing something wonderful, something remarkable, and something that should be an inspirational example to everyone in the world of education. As Paul O'Neill, chair of the President's Advisory Council on Education, said at our press conference in Washington, D.C., through the NCTM's Standards documents our professional discipline has ensured its self-renewal.
There Is a Mathematics Club in My School!
Authors: Rogish Jacqueline M.
Published: October 2020

Clubs have always held an attraction for adults as well as children. They give everyone the sense of belonging to a group, build confidence in a specific area, and add to self-esteem. In that spirit, mathematics clubs have proved successful from elementary to high school. With the reorganization of the middle school, clubs will be an integral part of this restructuring process.
Calendar Mathematics
Authors: Chancellor Dinah
Published: October 2020

The calendar is designed to speak directly to students, giving them open-ended questions intended to engage their iniellectthrough their interests. Students are encouraged to work on the activities individually, in pairs, or in small groups. No answers are given for the calendar questions so that students are encouraged to look to themselves as the mathematical authority, thereby developing the confidence and critical-thinking skills necessary to validate their thinking.
Ed's “Strategy” for Solving Division Problems
Authors: Harel Guershon , Behr Merlyn
Published: October 2020

Prior to formal instruction, children possess considerable understanding of arithmetic and the strategies for solving basic arithmetic problems. Carpenter and Moser (1982), for example, have found that many children can successfully solve word problems that involve addition and subtraction before they receive any instruction in such problems.
IDEAS
Authors: Irons Calvin , Irons Rosemary , Fennell Francis (Skip)
Published: October 2020

The “IDEAS” section for this month focuses on the role of number and language in real-world situations. Students are asked to discuss, describe, read, and write about numbers they find in familiar real-world situations. The emphasis on using components of language helps students build a broader vocabulary of numbers than the traditional symbolic representation of numbers. The activities also help develop good number sense.