bluuebunny/crossref_metadata_2025 huggingface dataset:
Questioning Zero and Negative Numbers
Authors: Wilcox Virginia B.
Published: December 2020

Last year, I was fortunate enough to take the Developing Mathematical Ideas (DMI) course on algebraic reasoning taught by Ruth Balf from the University of Washington, Seattle. DMI deepens participants' mathematics knowledge while also modeling how to listen to students and ask the right questions at the right time. As Willis and others note, “When students are learning mathematics, they use their existing knowledge and understandings to make sense of new ideas” (2007, p. 22). DMI allows teachers to uncover that existing knowledge. One of the primary messages is that children can and do construct their own meaning behind many complicated mathematical procedures and concepts. Taking a DMI course allows participants to return to their own classrooms and students with a fresh perspective.
Math by the Month: Presidential Parade
Authors: Gallego Christine , Saldamando Dora , Tapia-Beltràn Grace , Williams Karolyn , Hoopingarner Thomas C.
Published: December 2020

“Math by the Month” activities are designed to engage students to think as mathematicians do. Students may work on the activities individually or in small groups, or the whole class may use them as problems of the week. No solutions are suggested, so students look to themselves for mathematical justification and develop confidence to validate their work. The following activities are designed to help you and your students explore our presidential election and history through mathematical investigations involving number sense and operations, data collection, patterns and algebra, measurement, communication, and representation.
Enhancing Mathematical Learning in a Technology-Rich Environment
Authors: Suh Jennifer M. , Johnston Christopher J. , Douds Joshua
Published: December 2020

In Principles and Standards for School Mathematics (NCTM 2000), the Technology Principle asserts: “Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning” (p. 24). More specifically, a technology-rich environment for mathematical learning influences five critical features of the classroom (Hiebert et al. 1997): the nature of classroom tasks, the mathematical tool as learning support, the role of the teacher, the social culture of the classroom, and equity and accessibility. An essential question when working in a technology-rich mathematics environment is how technology can be used (appropriately) to enhance the teaching and learning of mathematics.
From the Classroom: Designing and Implementing Worthwhile Tasks
Authors: Breyfogle M. Lynn , Williams Lauren E.
Published: December 2020

Teachers often need to alter mathematical tasks that they find in their district-adopted set of curriculum materials or develop new ones if none is present on a particular topic. However, how to best go about this work is not always clear. How do you make effective decisions about alterations? What should you keep in mind as you consider developing tasks to help your students with a particular idea or misconception? These and other questions were central in our minds as we developed a task to help students learn about elapsed time.
Math by the Month: Probability
Authors: Gallego Christine , Saldamando Dora , Tapia-Beltrán Grace , Williams Karolyn , Hoopingarner Thomas C.
Published: December 2020

“Math by the Month” activities are designed to engage students to think as mathematicians do. Students may work on the activities individually, in small groups, or as problems of the week for the entire class. Because no solutions are suggested, students look to themselves for mathematical justification, thereby developing confidence to validate their work. The following activities are designed to help you and your students investigate probability through mathematical investigations involving number sense and operations, statistics, data collection, communication, and representation.
What Makes a Video Clip Interesting?
Authors: Linsenmeier Katherine A. , Sherin Miriam Gamoran
Published: December 2020

Classroom video excerpts are often used to help preservice and practicing teachers explore students' mathematical ideas. This article describes several types of video clips that the authors have found to be particularly productive for this purpose.
Zoos, Aquariums, and Expanding Students' Data Literacy
Authors: Mokros Jan , Wright Tracey
Published: December 2020

Field trips provide compelling venues for teaching students to work with observational data and make important connections between mathematics and science.
Math in the Milpa
Authors: Barta James , Sánchez Lucas , Barta Jacob
Published: December 2020

Travel with us to the rural Mayan village of Santa Avelina, Guatemala, to learn how to develop culturally relevant lessons for your own classroom.
Math Club Problem Posing
Authors: Simic-Muller Ksenija , Turner Erin E. , Varley Maura C.
Published: December 2020

An after-school mathematics program for Latino students focuses on field trips to explore the mathematical practices of the community's businesses.
Reasoning and Sense Making
Authors: Martin W. Gary , Kasmer Lisa
Published: December 2020

What does a new policy publ ication for high school mathematics have to do with you and with elementary school mathematics? Everything! Learn why.
From Calculating to Calculus
Authors: Steckroth Jeffrey J.
Published: December 2020

When his elementary task for preservice teachers became an excursion across grade levels, this former secondary school teacher gained appreciation for how closely connected elementary school multiplication is to the algebraic thinking of secondary school mathematics.
Links to Literature: The Polar Express to Early Algebraic Thinking
Authors: Johanning Debra , Weber William B. , Heidt Christine , Pearce Marian , Horner Karen
Published: December 2020

Algebra in the early grades (pre-K-2) is a relatively new focus area in the mathematics education community. As students, many of today's elementary school teachers did not study algebra until they were in high school.
Problem Solvers: Problem: North Dakota's centennial quilt
Authors: Small Marian
Published: December 2020

Quilt investigations, such as the Barn quilt problem in the December 2008/January 2009 issue of Teaching Children Mathematics and its solutions in last month's issue, can spark interdisciplinary pursuits for teachers and exciting connections for the full range of elementary school students. This month, North Dakota's centennial quilt problem blends the mathematical strands of measurement, data and probability, geometry, and number.
Math by the Month: The math in names
Authors: Columba Lynn , Hammond Thomas , Waddell Lanette
Published: December 2020

What is in a name? Actually, quite a lot of math! Join us as “math by the month” challenges students to apply their knowledge of data analysis, geometry, and algebraic thinking to solve this collection of math problems.
Paint bucket polygons
Authors: Edwards Michael Todd , Harper Suzanne R.
Published: December 2020

Use photo editing software as a teaching tool to bring inaccessible polygon definitions within reach of your students' understanding.
Supporting language learners
Authors: Cady Jo Anny , Hodges Thomas E. , Brown Clara
Published: December 2020

Incorporating these instructional practices for two mathematical tasks into lessons to support English learners gives them access to an excellent, equitable program.
Math by the Month: On the map with Google
Authors: Hammond Thomas , Waddell Lanette R.
Published: December 2020

These activities develop pattern-seeking, prediction, and explanation. All the problems use Google™ maps (http://maps.google.com/) and some use little-known features of the tool. Before sharing these problems with your students, explore the site to become familiar with what the tool can do. Be careful, though–you may find yourself spending more time playing with its features than you had planned!
Star Students Make Connections
Authors: Marshall Anne Marie , Superfine Alison Castro , Canty Reality S.
Published: December 2020

Discover strategies to engage young math students in competently using multiple representations.
Using number sense to compare fractions
Authors: Bray Wendy S. , Abreu-Sanchez Laura
Published: December 2020

Third-grade teachers found that giving particular attention to the use of realworld contexts, mental imagery, and manipulatives brought success to problem solving as students moved from using models to reasoning.
What math knowledge does teaching require?
Authors: Thames Mark Hoover , Ball Deborah Loewenberg
Published: December 2020

Effectively helping others learn is demanding work that necessitates sensibility as well as specialized knowledge and skill.
A Teacher's Journal: A Fourth-Grade Family Math Night
Authors: Hall Jane B. , Acri Rita P.
Published: December 2020

(September entry) Our fourth graders arrived this year full of excitement about mathematics. We hope that we can “dust ourselves off” and once again keep the subject alive and interesting for them.
Early Childhood Corner: Take the Magic Out of Your Classroom!
Authors: Andrews Angela Giglio
Published: December 2020

From ancient times, people have attributed phenomena that they do not fully comprehend to the supernatural and to magic, which is “the production of results through mysterious influences or unexplained powers” according to the 1984 Random House College Dictionary.
Communicating About Fractions with Pattern Blocks
Authors: Caldwell Janet H.
Published: December 2020

One of the central concerns of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) is helping students develop in-depth understanding of the central ideas of mathematics.
Using Concept Diagrams to Promote Understanding in Geometry
Authors: Shaw Jean M. , Thomas Conn , Hoffman Ann , Bulgren Janis
Published: December 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and the van Hiele model for geometric thought (Crowley 1987) advocate increasing students' understanding of geometric properties and relationships as they enter the intermediate anil middle grades.
Perimeter Patterns
Authors: Austin Richard A. , Biafore Patricia
Published: December 2020

The following geometry activity featuring the concept of perimeter was carried out over one week in a fifth-grade class. As the students worked in small groups, they explored mathematics in ways aligned with the Curriculum and Evaluation Standards for School Mathematics for grades 5-8 (NCTM 1989).
Investigations: Count on Mathematics for Number Sense
Authors: Speer William R. , Brahier Daniel J.
Published: December 2020

The “Investigations” section recognizes the importance of children's exploring hands-on and minds-on mathematics and, therefore, presents teachers with open-ended explorations to enhance mathematical instruction. These meaningful tasks are designed to evolve as they are conducted.
Research into Practice: What Do Children Understand about Average?
Authors: Russell Susan Jo , Mokros Jan
Published: December 2020

The statistical idea we come across most frequently is the idea of average. Children in fourth grade and beyond fairly easily learn to apply the algorithm for finding the mean, but what do they understand about the mean as a statistical idea?
Reviewing and Viewing
Authors: Whitin David J.
Published: December 2020

Prices on software, books, and materials are subject to change. Consult the suppliers for the current prices. The comments reflect the reviewers' opinions and do not imply endorsement by the Notional Council of Teachers of Mathematics.
Connecting Decimals and Other Mathematical Content
Authors: Thompson Charles S. , Walker Vicki
Published: December 2020

Experiences with many children in the middle grades indicate that they have poor decimal concepts and lack fundamental skills in working with decimal values. For example, when asked to identify which of 0.36 or 0.339 is greater, children frequently choose 0.339 because 339 is greater than 36.
Promising Research, Programs, and Projects: The Effect of Personalized Word Problems
Authors: Hart Janis M.
Published: December 2020

Many students have difficulty converting a word problem into the necessary mathematical form needed to solve the problem. They seem unable to create a mental representation that links the text of the word problem to appropriate mathematical expressions.
Making Cents
Authors: Larsen Nancy J.
Published: December 2020

My classroom needed some new playground equipment, hut we had no money with which to purchase it. That situation was the motivation behind a mathematics unit that brought a practical, hands-on experience to my second-grade class in the form of collecting pennies.
A Teacher's Journal: The Nature and Experience of Teaching
Authors: Battan Gayle B.
Published: December 2020

I love teaching! It is as simple as that. Teaching for me is a combination of a profession and a hobby—not a hobby that is frivolous and decorative but one that is rewarding and fulfilling.
Investigations: Reflections of Mathematics
Authors: Speer William R. , Dixon Juli
Published: December 2020

The “Investigations” department recognizes the importance of children's exploring hands-on and minds-on mathematics and, therefore, presents teachers with open-ended explorations to enhance mathematical instruction. These meaningful tasks are designed to evolve as they are conducted.
The Inside-Out Box: An Analysis of Structures and Space
Authors: Fowler Marilyn L.
Published: December 2020

The NCTM's Standard address the need for children to develop spatial sense and explore the effects of transforming, combining, sub-dividing, and changing geometric figures (1989). Yet our students show weakness in these skills when compared with children of other nations (Stigler, Lee, and Stevenson 1990, quoted in Grouws 1992). The positive correlation between spatial abilitie and mathematics achievement at all levels (Fennema and Sherman 1977; Fennema and Shetman 1978; and Guay and McDaniel 1977; all cited in Grouws 1992) shows that teachers need to present activities that can develop children's spatial abilities.
Math by the Month
Authors: LeBeau Sue
Published: December 2020

The Math by the Month activities are designed to appeal directly to students. Students may work on the activities individually, in pairs, or in small groups.
Investigations
Authors: Speer William R.
Published: December 2020

The “Investigations” department recognizes the importance of children's exploring hands-on and minds-on mathematics and, therefore, presents teachers with open-ended explorations to enhance mathematical instruction.
Experiences with Patterning
Authors: Ferrini-Mundy Joan , Lappan Glenda , Phillips Elizabeth
Published: December 2020

Over the past decade we have learned that children are capable of mathematical insights and mathematical invention that exceed our expectations. We have also learned that we, as teachers, contribute to—or suppress—this insight and inventiveness in our students by the choices we make.
A Teacher's Journal: Liz's Pattern
Authors: Land Jill E. , Becher Paul G.
Published: December 2020

A recent mathematical experience with Liz Leatherberry, a student in sixth grade, emphasized to us the importance of early learning, which can form a solid base for algebraic knowledge.
Functions from Kindergerten Through Sixth Grade
Authors: Willoughby Stephen S.
Published: December 2020

One of the most important and pervasive concepts in mathematics is that of a function. High school and college students often have trouble with the function concept because of the abrupt and abstract way in which it is introduced.
Mathematics of Change: Ins and Outs or Ups and Downs
Authors: Tierney Coroclia C. , Namirovsky Ricardo
Published: December 2020

We describe here an activity with variations useful throughout the elementary school grades that involve children using small numbers to do a sequence of additions and subtractions that we call “changes.”
Math by the Month: Classification and Logical Reasoning
Authors: Olson Melfried , Olson Judith
Published: December 2020

These activities are designed to appeal directly to students. Students may work on the activities individually, in pairs, or in small groups. No solutions are suggested or given for the activities so that students will look to themselves as the mathematical authorities, thereby developing the confidence and critical-thinking necessary to validate their work.
National Council of Teachers of Mathematics: Choices and Challenges
Authors: Burrill Gail
Published: December 2020

Life has changed enormously since I grew up in Fisher, Minnesota. Over the years, we faced many challenges and made choices that helped us meet those challenges as we kept up with changes in agriculture, manufacturing, transportation, entertainment, and education.
By Way of Introduction: Beyond the Classroom
Authors: Edge Douglas
Published: December 2020

Inviting and working with parents in our mathematics classrooms is not new. Two changes seem to have occurred: an increased demand for parental participation in the schooling of their children as well as interaction between parents and schools. The two quotations are but samples of these many calls: the first was made by Richard Riley, current United States secretary of education; the second was made by Jack Price, president of NCTM from 1994 to 1996. This particular focus issue reflects that demand.
In Our Opinion: School-Family Partnerships: Parents, Children, and Teachers Benefit!
Authors: Ford Marilyn Sue , Follmer Robin , Litz Kathleen K.
Published: December 2020

The advent of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) dramatically altered mathematics education. Like teachers, parents are coming to terms with shifting expectations and methods of instruction. Communicating with parents and involving them in their children's learning of mathematics help parents understand why and how mathematics education has changed.
Parents, Portfolios, and Personal Mathematics
Authors: Ensign Jacque
Published: December 2020

From the standpoint of the child, the great waste in the school comes from his inability to utilize the experiences he gels outside the school in any complete and free way within the school itself; while, on the other hand, he is unable to apply in daily life what he is learning in school. That is the isolation of the school—its isolation from life.
Investigations: Picture This
Authors: Hopkins Martha H.
Published: December 2020

The “Investigations” department recognizes the importance of children's exploring hands-on and minds-on mathematics and, therefore, presents teachers with open-ended explorations to enhance mathematical instruction. These meaningful tasks are designed to evolve as they are conducted. They invoke problem solving and reasoning, require communication skills, and encourage the identification of connections among various mathematical concepts and principles. To ensure usability, the ideas presented in this column have been tested in various classroom settings.
Links to Literature: Books + Manipulatives + Families = A Mathematics Lending Library
Authors: McCarty Diane
Published: December 2020

What do Alexander, Who Used to Be Rich Last Sunday (Viorst 1978), Theodoric's Rainbow (Kramer 1995), and Grandfather Tang's Story (Tompert 1990) have in common?
In My Opinion: Maintaining Our Integrity amidst Controversy and Attacks
Authors: Weissglass Julian
Published: December 2020

An old Chinese proverb states, “The tree may prefer calm, but the wind will not subside,” which may strike a responsive chord with educators who thought it possible to pursue the improvement of mathematics education free from controversy and political attack. It has turned out not to be so.
Using Spreadsheets to Promote Algebraic Thinking
Authors: Battista Michael T. , Borrow Caroline Van Auken
Published: December 2020

For students to find algebra conceptually meaningful, as well as useful in modeling and analyzing real-world problems, they must be able to reflect on, make sense of, and communicate about general numerical procedures (Kieran 1992). Such procedures consist of set sequences of arithmetic operations performed on numbers. Examples include computing an average and performing the standard division algorithm. Thinking about numerical procedures starts in the elementary grades and continues in successive grades until students can eventually express and reflect on the procedures using algebraic symbolism. This article outlines how such thinking can progress to algebraic reasoning and illustrates how computers used to promote this progression.
News from the Net: Visit These Sites
Authors: Lazerick Beth
Published: December 2020

Tech Time's News from the Net” began three years ago as an experiment. In 1995, Teaching Children Mathematics, its Editorial Panel, and the editors of “Tech Time” thought that teachers would like to read about exemplary Web sites. Those accessing the Internet knew how frustrating it was to try to find exceptional material that could be used to improve the teaching and learning of mathematics.
Early Childhood Corner: Kindergarten Is More Than Counting
Authors: Kline Kate
Published: December 2020

Teachers of primary-grade children realize the importance of helping their students develop an understanding of number relationships. It is important to encourage the kind of thinking that allows children to readily decompose numbers into parts and know how to put parts together to make a whole. This thinking sets the foundation for working with larger numbers, using reasoning to approach computation, and developing sophisticated mental strategies. Parker (1998) describes the importance of building what she calls “fluency with small numbers.” She defines fluency as being able to take apart and put back together numbers without even thinking, or with automaticity, and believes that before the third grade, children should be fluent with numbers to 10.
Kids + Conjecture = Mathematics Power
Authors: Cantlon Danise
Published: December 2020

My conjecture: Any number divided by twice itself equals 1/2, except 0.” Imagine this statement being read by a student in a third- and fourth-grade classroom. Then imagine a classroom discussion with students searching for examples and extensions. Students in such a situation are not being told about mathematics; rather, they are developing their mathematical power.
Investigations
Authors: Enright Brian E.
Published: December 2020

The “Investigations” department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents teachers with open-ended explorations to enhance mathematical instruction. These tasks invoke problem solving and reasoning, require communication skills, and connect various mathematical concepts and principles. The ideas presented here have been tested in various classroom settings.
Early Childhood Corner: What Comes Next? The Mathematics of Pattern in Kindergarten
Authors: Economopoulos Karen
Published: December 2020

Patterning activities are a staple of the kindergarten curriculum. If you ask any kindergarten teacher, she or he is likely to consider the study of patterns to be an essential part of the mathematics program.
Problem Solvers
Authors: Olson Melfried
Published: December 2020

The goal of the “Problem Solvers” department is to foster communication among teachers by posing one problem each month for K-6 teachers to try with their students. Every teacher can become an author: pose the problem, reflect on your students’ work, analyze the classroom dialogue, and submit the resulting insights to this department. Remember that even student misconceptions are interesting.
Teaching the Values of Coins
Authors: Drum Randell L. , Petty Wesley G.
Published: December 2020

Teaching about coins can be a frustrating experience for many primary teachers. Teaching the names of the coins is not the problem. Instead, teaching about the values of coins is difficult.
Investigations: This Little Piggy
Authors: Brahier Daniel J. , Kelly Monica , Swihart Jennifer
Published: December 2020

This department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents teachers with open-ended explorations to enhance mathematics instruction. These tasks are designed to evolve as they are conducted. They invoke problem solving and reasoning, require communication skills, and connect various mathematical concepts and principles. The ideas presented here have been tested in various classroom settings.
Educating Hannah: It's a What?
Authors: Cockcroft Sir Wilfred H. , Marshall John
Published: December 2020

It's a what?” asked Hannah in a tone of disbelief. “It's a thick circle,” repeated her older sister, Sarah.
Math by the Month: So You Have a Problem?
Authors: Holly Karen
Published: December 2020

The “Math by the Month” activities are designed to appeal directly to students. Students may work on the activities individually or in small groups. No solutions are suggested so that students will look to themselves as the mathematical authority, thereby developing the confidence to validate their work.
Links to Literature: Interweaving Language and Mathematics Literacy through a Story
Authors: Harris Jacqueline
Published: December 2020

Programs that promote literacy development in the early grades, kindergarten through grade 2, are of prime interest to educators, particularly when they are also effective for ESL students. With interest, then, I talked with Silvana Bozzato, a kindergarten teacher at Our Lady of Victory School, set in a multicultural, downtown Toronto neighborhood. I wanted to know if she used in her program any books with a focus on mathematics, wondering how that practice might help students understand important mathematical concepts. Bozzato used many such books, and we agreed that she would model one lesson in which mathematics and language were interwoven, choosing a beautifully illustrated, lyrical story by Pat Brisson, Benny's Pennies (1993).
Some Like It Hot: Promoting Measurement and Graphical Thinking by Using Temperature
Authors: Moore Deborah A.
Published: December 2020

Looking for a hot topic of discussion? Try temperature. Although many applied-measurement activities are used in classrooms today, most focus on distance or position. The fundamental concepts of temperature and the change in temperature over time are often overlooked.
Early Childhood Corner: Collecting Data Outdoors: Making Connections to the Real World
Authors: Basile Carol G.
Published: December 2020

The most enjoyable experiences that I have had with young children are those that occur outdoors. Taking children on walks in the woods, at our local park, or simply around the school yard can prompt many discoveries about the natural world. As we walk, children gain knowledge and skills by using their senses to collect information about the world around them. Traditionally, we think of providing these experiences as part of children's scientific learning. However, direct observation is also an important piece of mathematical learning that is essential for identifying patterns, promoting problem solving, and developing spatial sense and reasoning.
Beyond Helping with Homework: Parents and Children Doing Mathematics at Home
Authors: Kliman Marlene
Published: December 2020

Ideas on helping parents do mathematics with their children as part of everyday family life.
The Importance of Spatial Structuring in Geometric Reasoning
Authors: Battista Michael T.
Published: December 2020

Second-grader Katy was shown that a plastic inch square was the same size as one of the indicated squares on the seven-by-three-inch rectangle displayed in figure 1a. She was then asked to predict how many plastic squares she would need to cover the rectangle completely.
Questioning Your Way to the Standards
Authors: Mewborn Denise S. , Huberty Patricia D.
Published: December 2020

The Professional Standards for Teaching Mathematics (NCTM 1991, 3) calls for five changes in the environment of the mathematics classroom.
Investigations: What Is Your Millennial Age?
Authors: Welchman Rosamond
Published: December 2020

This department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents teachers with open-ended explorations to enhance mathematics instruction. These tasks involve problem solving and reasoning, require communication skills, and connect various mathematical concepts and principles. The ideas presented here have been tested in various classroom settings.
A Teacher's Journal: 100 Activities for the 100th Day
Authors: Thomas Cynthia S.
Published: December 2020

As February approached, I heard several primary-grade teachers talking about activities for the 100th day of school. This sounded like a worthwhile celebration for our whole school, which has about 220 students in grades pre-K—6. At a faculty meeting, I suggested that everyone join in the festivities. Of course, we had to come up with a variety of activities that would be appropriate for all grade levels and subject areas. When the 100th day arrived, we were ready! Everyone in the entire school participated. The secretary made a badge for each student who had been present for 100 days, the cafeteria workers marked each 100th tray to receive a special treat, the principal rang the school bell on the 100th minute, and the custodian just smiled as we put sticky-note estimates everywhere in the school.
The Mathematician as a Child and Children as Mathematicians
Authors: Banchoff Thomas F.
Published: December 2020

A mathematician reflects on his experiences as a young mathematician in grade school and describes his experiences working with K–6 teachers and students.
Investigating Mathematics as a Community of Learners
Authors: Drier Hollylynne Stohl
Published: December 2020

How third graders worked together to investigate relationships between fractions and percents using a spreadsheet tool. Highlights a student's creative approach to a proportion task.
Problem Solvers: What Can You Measure?
Authors: Olson Melfried
Published: December 2020

The goal of the “Problem Solvers” department is to foster improved communication among teachers by posing one problem each month for K–6 teachers to try with their students. Every teacher can become an author; pose the problem, reflect on your students' work, analyze the classroom dialogue, and submit the resulting insights to this department. Remember that even student misconceptions are interesting.
Teacher-Clinicians Encourage Children to Think as Mathematicians
Authors: Kaplan Rochelle G. , King Bonnie , Dickens Nancy , Stanley Viola
Published: December 2020

How teachers can use clinical interviewing techniques to bring out the mathematician in at-risk students.
In My Opinion: What Can We Expect from Research?
Authors: Hiebert James
Published: December 2020

As reform recommendations collide with traditional practices, the questions on the minds of many interested teachers and parents are these: Should we really adopt this new textbook series? Should we change the way we teach? What mathematics should students really be learning? It would be nice if research could give clear and simple answers to these kinds of “should” questions, but it does not and it cannot.
Enthusiastic Voices from Young Mathematicians
Authors: Steele Diana F.
Published: December 2020

Students should value mathematics in their lives and become confident in their abilities to solve problems, learn to reason, and communicate their mathematical ideas (NCTM 1989). Using these mathematical skills gives students school experiences that come close to the experiences of mathematicians. Such experiences help students learn what it means to know mathematics as a discipline as they actively construct their own knowledge and develop mathematical concepts.
News from the Net: Montessori Mathematics
Authors: Lazerick Beth
Published: December 2020

For Montessori teachers and for those of you who may just be interested in learning about the Montessori method of teaching mathematics, Shu-Chen Jenny Yen's Online Montessori Albums is a wonderful site to explore.
Exploring Children's Thinking about Mathematicians and Their Work
Authors: Rock David , Shaw Jean M.
Published: December 2020

Describes a survey of 215 children's impressions about mathematicians and discusses the children's responses.
Research, Reflection, Practice: Implications of Research on Children's Understanding of Geometry
Authors: Fox Thomas
Published: December 2020

In my mathematics methods course for preservice teachers, I ask my students to assess a group of elementary school students concerning their level of geometric reasoning. To do so, they use tasks that focus on assessing and extending students' geometric understandings. These open-ended tasks, along with a framework developed from research findings involving children's geometric reasoning, are described in this article. An important aspect of these tasks is that they focus on how students communicate their reasoning so that my preservice teachers can make more informed instructional decisions when planning a follow-up geometry lesson. Research on geometric reasoning has shown that a match between students' reasoning level and instructional tasks is crucial if meaningful learning is to occur (Crowley 1987).
Principles and Standards for School Mathematics: What's in It for You?
Authors: Joyner Jeane , Reys Barbara
Published: December 2020

Have you seen it? Have you read it? Have you begun talking with colleagues about the ideas that it presents? Have you reflected on how it will influence your instruction? Have you examined your curriculum materials in light of the expectations that it outlines?
Using Language and Visualization to Teach Place Value
Authors: Cotter Joan A.
Published: December 2020

For years, a slow weight gain for premature infants was accepted as normal. Recently, however, better understanding of the special nutritional needs of these infants has resulted in a substantial increase in weight gains. In the same way, mathematics education research is giving us insights into helping young children learn substantially more mathematics.
The Shape of Fairness
Authors: Penner Elizabeth , Lehrer Richard
Published: December 2020

Shape and form are often used as mathematical models of situations. For example, teachers explain that light travels in a line or that the shadow cast by a person is related by similar triangles to that cast by a flagpole. Yet despite the common use of mathematical models in the sciences and in design professions, children rarely have the opportunity to participate in this form of mathematical thinking. In this article, we describe how first and second graders modeled a “fair” playing space in a game of tag called “Mother, may I?” The children modeled the playing space by using a succession of different forms, such as lines and squares, to represent a fair game, discovering along the way the properties of each of the forms that made them less-than-ideal models of fairness. Participation in the game gave the children many opportunities to think about important concepts in measuring length and the idea of using form to model a situation.
Early Childhood Corner: Achieving Numerical Understanding for All Young Children
Authors: Fuson Karen C. , Grandau Laura , Sugiyama Patricia A.
Published: December 2020

Young children aged 3 to 7 can learn a great deal about numbers. In a home or daycare environment, this learning can occur as children experience daily routines. Young children will learn to count, match, see, and compare numbers if caregivers or older children count, show objects, and point out small numbers of things. Such informal teaching can be done while children play, eat, get dressed, go up and down stairs, jump, and otherwise move through the day. These activities are engaging and fun but need to be encouraged and modeled by adults or more advanced children in the group. In larger day-care or school settings, numerical understanding results from similar informal learning opportunities combined with more structured experiences that enable all children to engage in supported learning activities with adult and peer modeling and help.
Are Rules Interfering with Children's Mathematical Understanding?
Authors: Behrend Jean L.
Published: December 2020

The small group of second- and third-grade students in the learning disabilities resource room chanted, “The ones are first. The ones are first. The ones are first!” “Remember,” their teacher said, “when you add numbers together, you start with the numbers in the ones place. The ones are first!” The teacher hoped that this chant would reduce computational errors. When students exhibit difficulties in mathematics, teachers often try a quick fix, such as giving a rule to follow. The rule may appear to solve the immediate problem but could actually interfere with students' development of mathematical understanding.
Problem Solvers: Responses to the Nets and Polyhedra Problem
Authors: Sakshaug Lynae
Published: December 2020

The problem from the September 2000 issue
Links to Literature: Using Representations to Explore Perimeter and Area
Authors: Moyer Patricia S.
Published: December 2020

In an elementary school classroom, as in real life, the lines between the content areas should be blurred, particularly between mathematical problem solving and mathematical situations contextualized in good literature. For that reason, I always look for interesting books about mathematical situations. Why use children's literature to teach mathematics? A good story often places mathematical problems in the context of familiar situations and is similar to, yet a much more elaborate version of, mathematical word problems. Assertions that children's inability to solve word problems results from their inability to read or to compute effectively simply are not true. The problem is that children do not know how to choose the correct operation or sequence of operations to solve the problem. To solve a problem situation presented in words, children need to be able to connect computational processes with appropriate calculations. Their difficulties lie in the fact that children simply do not understand the mathematics well enough conceptually to make the connection with the problem- solving situation. Using books with authentic problem situations may help children see that learning computation serves a real-life purpose.
Connecting Mathematics Instruction with the Families of Young Children
Authors: Kyle Diane W. , McIntyre Ellen , Moore Gayle H.
Published: December 2020

During the past four years, we have worked closely with our students' families, making efforts to get to know them better to improve classroom instruction in a multiage primary school classroom of six- to eight-year-olds.
Learning with Games of Strategy from Mongolia
Authors: Ascher Marcia
Published: December 2020

Students can derive many benefits by learning about games of strategy from diverse cultures. First, playing games of strategy and solving mathematics problems share several important features. Both activities have clearly defined goals, as well as a clearly defined set of permissible actions to reach the goals. The challenge is to select an action, but in doing so, the player or problem solver must consider the logical implications of a chain of steps. The implications include the subsequent options that are opened or closed by the solver's action and how these options help the solver make progress toward the goal. Thus, by playing games of strategy, students can begin to develop, and teachers can reinforce, valuable modes of thought. When the games are from a culture that is different from the mainstream culture of the class, the teacher has the opportunity to introduce the less familiar culture and describe the context in which the game was originally played. Students learn that people who may be widely dispersed in time and space have enjoyed the challenge of logical thought. They also learn that games are, and have been, integrated into many different aspects of life and that they are appreciated by both adults and children. Above all, students learn that people can create their own challenges and entertainments.
Patterns and Symmetry: Reflections of Culture
Authors: Moyer Patricia S.
Published: December 2020

Many contributions of diverse cultures foster a rich understanding of mathematics. Knowing how one's culture has contributed to mathematics and how these contributions enhance our cultural environment supports the acquisition of mathematical power. However, discussing culture in mathematics classrooms for a one-week celebration of women in mathematics or a one-month recognition of the contributions of African Americans is not enough. Cultural learning that recognizes race, ethnicity, gender, and social class should be woven into the fabric of mathematics lessons throughout the year. Yet many teachers have limited backgrounds in promoting culturally relevant mathematics in meaningful ways.
Problem Solvers: Have a Heart
Authors: Mann Robert
Published: December 2020

The goal of the “Problem Solvers” department is to foster improved communication among teachers by posing one problem each month for teachers of grades K-6 to try with their students. Every teacher can participate: pose the problem, reflect on your students' work, analyze the classroom dialogue, and submit the resulting insights to this department. Every teacher can help us all better understand children's capabilities and thinking about mathematics by contributing to the journal. Remember that even student misconceptions are interesting.
Representations in Teaching and Learning Fractions
Authors: Watanabe Tad
Published: December 2020

Fraction concepts continue to be one of the most challenging topics for elementary and middle school children (see, e.g., Kouba, Zawojewski, and Strutchens [1997]). One important factor in teaching and learning fractions is the use of representations. This article addresses four issues surrounding this topic: (1) tools for representing fractions, (2) methods of representing fractions, (3) fraction notations, and (4) fraction language.
The Lunchroom Project: A Long-Term Investigative Study
Authors: Manchester Patricia
Published: December 2020

From the time they first enter school, children learn to collect information and present it graphically. In most early childhood classrooms, at some point during the year, children are polled on their favorite ice-cream flavors, pets, or colors; and in many schools, graphs of “how our class gets to school” show neat rows of buses, cars, and walking feet. As they progress through the grades, children are often asked to conduct their polls individually and to make up questions that their classmates can answer using their bar graphs or pictographs. Thoughtful teachers often ask their students to make predictions before they gather information and to write a paragraph or two summarizing their results.
Understanding and Supporting Children's Mathematical Vocabulary Development
Authors: Rubenstein Rheta N. , Thompson Denisse R.
Published: December 2020

Imagine a teacher running her hands across her desk as she tells her students, “A plane is a perfectly flat surface.” The students listen quietly, but one of them is thinking, “I thought a plane was something that flies.”
Learning Spanish while Practicing Mathematics Concepts and Skills: A Winning Combination
Authors: Kelly Gwendolyn N. , Kennedy Teresa J. , Eberhardt Megan B. , Austin Laurie K.
Published: December 2020

The minute you walked into the classroom, you could tell that something was different. The fifthgrade students were enthusiastically holding up laminated sheets of 8 1/2-by-11-inch cardboard, called response boards, with numbers on them. As the teacher asked “¿Qué es esto?” they replied with big smiles on their faces, “Seis más cinco son once” (“Six plus five is eleven”).
Promoting Communication in the Mathematics Classroom
Authors: Whitin Phyllis , Whitin David J.
Published: December 2020

Our two sons, who majored in statistics and civil engineering in college, were each required to take a technical writing course. The purpose of this course was to help students communicate mathematical and scientific ideas in a coherent and understandable way. The abilities to apply formulas or classify sets of data were not sufficient. Instead, these colleges wanted their graduates to be able to explain findings to different audiences in a logical and thoughtful manner and to represent these findings in different forms.
Early Childhood Corner: The Big Math for Little Kids
Authors: Balfanz Robert , Ginsburg Herbert P. , Greenes Carole
Published: December 2020

How many four-year-olds do you know who like to stomp, wiggle, and shout as they count to one hundred; go on a shape hunt to find examples of spheres, cubes, and rectangular prisms; help finish a story about a fantastic pasta maker by completing complex patterns; create and compare two towers of connecting cubes to figure out whose is taller and by how much; and use a map to locate objects in a room? Our answer is just about every four- and five-year-old we have observed over the past four years while developing the Big Math for Little Kids prekindergarten and kindergarten mathematics program. We began this process in the late 1990s because we were dissatisfied with the current state of early childhood mathematics. We observed teachers exposing young children to little or no interesting mathematics or instructing them to learn skills and concepts that they already knew (Greenes 1999). This observation contrasted with the wide range of mathematical ideas and skills that the children explored and employed during free play (Ginsburg 1999). We examined the historical record and learned that over the past two centuries, early childhood mathematics has gone through brief periods of richness followed by longer periods during which the mathematical interests and abilities of young children were seriously underestimated (Balfanz 1999). Aided by funding from the National Science Foundation, we decided to try to create an early childhood mathematics program that would build on the diverse mathematical interests and rich, implicit understandings of mathematics that young children hold.
Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician's Perspective
Authors: Bass Hyman
Published: December 2020

In recent years, few aspects of mathematics education have been as much discussed and debated as the notions of computational fluency and algorithms. A National Research Council report, Adding It Up: Helping Children Learn Mathematics (Kilpatrick, Swafford, and Findell 2001), offers an image of what it means to have skill with mathematics, or mathematical proficiency. This concept is helpful for moving beyond these debates. Mathematical proficiency includes five components: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (Kilpatrick, Swafford, and Findell 2001, p. 116). That these components are not separate but fundamentally intertwined is important to note. This article illustrates some of the ways in which the goal of computational fluency and an appreciation of mathematical algorithms are related to this larger concept of mathematical proficiency.