bluuebunny/crossref_metadata_2025 huggingface dataset:
Teaching Mathematics Then and Now: Still Hazy after All These Years? A Reaction Paper to Algebra in the Elementary Schools
Authors: Bay-Williams Jennifer M.
Published: December 2020

In his song “Still Crazy after All These Years,” Paul Simon (1974) writes, “I seem to lean on old familiar ways.” Shipley's article “Algebra in the Elementary Schools,” written in 1912, offers the opportunity for us to reflect on what we have accomplished, what we have learned, and what remains hazy even 100 years later with respect to teaching and learning mathematics. Do we still lean on old familiar ways of teaching algebra? This article has many messages that are relevant to us today in thinking about algebra and about teaching and learning mathematics in general. In addition, it raises the following questions: How does Shipley's conception of algebra compare with our conception of it today? What broader issues does he address that are still prevalent today? How far have we come in our implementation of his ideas? What aspects of teaching and learning mathematics in elementary school remain hazy after all these years?
Fostering Mathematical Thinking and Problem Solving: The Teacher's Role
Authors: Rigelman Nicole R.
Published: December 2020

Describes the teacher's role in promoting mathematical thinking and problem solving in the classroom—identifying critical teacher actions and decisions; considering how beliefs influence the teacher's actions and decisions; and suggesting implications for teachers and students.
Problem Solvers: Sharing Cookies
Authors: Leitze Annette Ricks , Stump Sheryl
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.
From the Classroom: Reflections on Teaching an Unanswered Question: Finding Factors of Large Numbers
Authors: McMaken-Marsh Anna
Published: December 2020

Teaching Mathematics through Problem Solving: Prekindergarten–Grade 6 (Lester and Charles 2003), published by the National Council of Teachers of Mathematics, advocates mathematics learning that is challenging for students. The authors of the articles in this compilation insist that rather than telling students how to do mathematics, we need to set up, in wisely structured ways, problems that will engage students in discovering mathematical strategies and principles. These situations need not involve elaborate reallife situations—they may be simple mathematical equations. What makes them problematic is that we do not give students one set of steps for solving the problem.
The Mathematics Survey: A Tool for Assessing Attitudes and Dispositions: Reflect and Discuss
Authors: Whitin Phyllis E.
Published: December 2020

The need for positive attitudes and dispositions permeates the teaching and learning of mathematics. What students believe about mathematics influences what they are willing to say publicly, what questions they are likely to pose, what risks they are willing to take, and what connections they make to their lives outside the classroom (Borasi 1990; Whitin and Whitin 2000). Unless students have a realistic sense of mathematical applications in real-life contexts, they are unlikely to see themselves pursuing courses in advanced mathematics or choosing mathematicsrelated careers (Picker and Berry 2001).
Rulers of Different Colors: Inquiry into Measurement
Authors: Clarkson Lesa M. Covington , Robelia Beth , Chahine Iman , Fleming Michelle , Lawrenz Frances
Published: December 2020

Do your students have difficulty determining what all the lines on a ruler mean? Do they struggle with linear measurement? Do they find using nonstandard measures baffling? Are you looking for other strategies to demystify the ruler? The following lesson addresses students' difficulties with linear measurement while incorporating national mathematics and science standards.
From the Classroom: Memorable Menu Math: The Journey from Teaching Procedurally to Teaching Conceptually
Authors: Thrift Michelle L. , Ortiz Enrique
Published: December 2020

Menus may seem an unlikely topic for a mathematics unit. Over the years, however, restaurant menus have proved a great way to reinforce mathematical concepts in my third-grade classroom.
Engaging Preservice Teachers and Elementary-Age Children in Transformational Geometry: Tessellating T-Shirts
Authors: Shockey Tod L. , Snyder Karen
Published: December 2020

Elementary-age children and preservice teachers learn about and use transformational geometry and creating tessellations. Tessellations and transformational geometry are taught two both groups in different ways. The elementary children create a T-shirt made from tessellations, whereas, the preservice teacher makes real world connections
Making Cute Count
Authors: Dark Denise
Published: December 2020

A holiday quilt project in a kindergarten classroom becomes a focus for exploring patterns, shapes, measurement, spatial relationships, and number sense. Cooperative learning, small group work, problem solving, and communication of mathematical ideas enhance the completion of the project.
From the Classroom: I Know an Old Lady: Using Children's Literature to Explore Patterns
Authors: Mattone Lauren
Published: December 2020

Children are masters at discovering patterns in their environment. “Look, Ms. Mattone— we're sitting boy, girl, boy, girl.” I love to take advantage of these moments, to point out to my kindergarteners how they are using mathematics without even noticing. I also love showing how picture book authors and illustrators have discovered patterns and used them in exciting, creative ways. Picture books have a familiar currency to them, and children readily respond to problems posed by favorite characters and adventures. Each year, I create lessons for my kindergarten class based on popular picture books that explore pattern recognition and development. These lessons introduce the basic concepts of algebraic thinking and provide my students with the foundation they need to complete higher-order mathematics down the road.
Math by the Month: Read a Story, Discover the Math
Authors: Chick Linda , Holmes Andrea S. , McClymonds Nicole , Musick Steve , Reynolds Patti , Shultz Gilda
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 these as problems of the week. Because no solutions are suggested, students will look to themselves for mathematical justification, thereby developing the confidence to validate their work.
Investigations: Developing Algebra-'Rithmetic in the Elementary Grades
Authors: Suh Jennifer M.
Published: December 2020

The “Investigations” department features children's hands-on and minds-on explorations in mathematics and presents teachers with open-ended investigations to enhance mathematics 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 classroom settings.
Design of Activities on Numerical Representations Based on Cognitive Research
Authors: Kalifatidou Eleftheria R.
Published: December 2020

Unlike words that refer to concrete things in the real world, numbers refer to entities we must extract from our environment. First, we must be able to quantify sets of things and construct some mental representations of the determined quantities in order to relate them to the proper number words or Arabic numbers.
Supporting Teacher Learning: Teachers Learning about Data Display: Connecting Mathematics and Science Inquiry
Authors: McDuffie Amy Roth , Morrison Judith A.
Published: December 2020

Collecting, analyzing, and displaying data provide rich opportunities to connect mathematics and science concepts. However, mathematics and science teacher educators rarely work together to design tasks that connect mathematics and science. In this article, we describe collaboration between a mathematics teacher educator and a science teacher educator that included the design of an inquiry-based project for preservice elementary teachers to draw on the natural connections of these disciplines. We also discuss preservice teacher learning outcomes from the project and present recommendations for teacher educators.
Arithmetic for First Graders Lacking Number Concepts
Authors: Kamii Constance , Rummelsburg Judith
Published: December 2020

Among the first graders who came to our Title-I school one year, we found twenty-six children who had no understanding of number concepts. In the assessment at the beginning of the school year, these children could not conserve number with eight counters. They could count out four chips, but when we hid some of the chips and asked, “How many am I hiding?” the children gave random answers, such as, “Ten.” Our challenge was that we were required by law to teach an hour of arithmetic to these children every day despite the fact that they had not yet developed an understanding of number concepts.
From the Classroom: Taking Attendance: A Starting Point for Explorations in Number Sense
Authors: Mattone Lauren
Published: December 2020

Taking attendance is a wonderful way to encourage young children to explore important ideas, such as one-to-one correspondence, hierarchical inclusion (the idea that every number contains all numbers that precede it), and partwhole relationships.
Learning to Solve Problems in Primary Grades
Authors: Whitin Phyllis , Whitin David J.
Published: December 2020

Problem solving lies at the heart of mathematical learning. Children need opportunities to write, discuss, and solve problems on a regular basis (NCTM 2000). The problems must incorporate grade-appropriate content and be “accessible and engaging to the students, building on what they know and can do” (Lester and Charles 2003, p. xi). Teachers also play a key role in establishing a classroom environment that encourages children to solve problems in different ways and develop habits of mind that include persistence, risk taking, and a questioning attitude (Whitin and Whitin 2003). This article describes how a class of second- and third-grade students learned to create and solve their own problems.
Alice in Numberland: Through the Standards in Wonderland
Authors: Christy Donna , Lambe Karen , Payson Christine , Carnevale Patricia , Scarpelli Debra
Published: December 2020

What can Alice, the Queen of Hearts, and other characters from Alice's Adventures in Wonderland (Carroll 1865, 1960) teach children and their families about mathematics? The possibilities are endless. For example, Alice stretches and shrinks during her adventures: measurement! Measurement is one of five Content Standards with which all children need rich experiences as part of their mathematical education (NCTM 2000).
Instructional Strategies for Teaching Algebra in Elementary School: Findings from a Research-Practice Collaboration
Authors: Earnest Darrell , Balti Aadina A.
Published: December 2020

Incorporating algebra into the elementary grades has become a focus for teachers, principals, and administrators around the country. Algebra is commonly regarded as a gateway to future opportunity (e.g., Moses and Cobb 2001), and elementary mathematics standards at both the state and national levels now reflect this effort to provide students with opportunities to learn critical concepts before middle and high school (California Department of Education 1997; NCTM 2000). However, implementing algebra standards at the elementary level is challenging—how do mathematics educators effectively and meaningfully incorporate algebraic ideas into K–5 curriculum? When elementary teachers are unfamiliar with early algebra, lessons designed and labeled as algebraic may become arithmetic exercises; the algebra then remains hidden from both the teacher and students in the implementation. The result is that the algebra standard is only superficially addressed.
Sharing: Cat Games and Cookies Special Education Students Investigate Division
Authors: Taber Susan B. , Canonica Michele
Published: December 2020

Learning mathematics has traditionally been thought of as a sequential progression. Children learn to count to ten, then to twenty, and then to one hundred. They learn to add without regrouping and then with regrouping. We teach addition before multiplication and the two-times table before the sixtimes table. We usually teach division as a separate unit after multiplication. Organizing mathematics textbooks as a sequential progression of skills promotes this perspective on teaching mathematics. When we, a special education teacher of fourthgrade students and a mathematics teacher educator at a university, joined forces to teach a unit on division to a class of ten fourth graders, we purposefully adopted a very different perspective in planning the learning activities. We found it useful to think of students' knowledge as individual networks of concepts, ideas, and procedures that were linked together in distinctive ways. Many students in the class shared some knowledge, but each student also had unique interests, talents, knowledge, and—especially—an idiosyncratic organization of that knowledge.
Using Your Inner Voice to Guide Instruction
Authors: Mohr Doris J. , Walcott Crystal Y. , Kastberg Signe E.
Published: December 2020

Planning mathematics instruction has become a challenging dilemma. Teachers are asked to address standards and use tools such as textbooks and manipulatives as well as understand and challenge student thinking. At the heart of this planning dilemma are seemingly conflicting tasks: Teachers must (1) design instructional opportunities that provide the potential for children to grow mathematically and (2) plan from the perspective of students' existing knowledge.
Disequilibrium and Questioning in the Primary Classroom: Establishing Routines that Help Students Learn
Authors: Carter Susan
Published: December 2020

Jean Piaget (1970) defined disequilibrium as a conflict between new ideas and current conceptions. Introducing this concept to my first graders improved formative assessment results and the overall classroom climate during mathematics class. Compared to my earlier students, my first graders made more progress with less difficulty. Engaging with a new concept allows students to feel successful as they struggle to integrate it into their current thinking. Although they have not yet mastered a particular mathematical concept, students can feel success if they can understand that the struggle is an expected, essential part of learning. By identifying their own disequilibrium, my students were able to continue to struggle with a concept or problem without “getting it right.”
The Workshop Way: Creating a Stimulating Intellectual Environment
Authors: Apanay Malinda F. , Vavoso Autumn C.
Published: December 2020

How can you show your thinking? The question resonated in our heads as we walked over and glanced at Cassi's blank paper (see fig. 1). Years ago, we might have felt we were missing the mark as her teachers. But because we had worked so hard to establish the intellectual environment in this class, we now had the foresight to interact with her during the work period. We stopped long enough to hear her plan to “act it out” as a way to represent her problem-solving strategy. Cassi began her dramatization. Besides capturing the audience's attention, she showed the class how she used the hundreds chart to help her solve the problem.
Investigations: Taking Time to Understand Telling Time
Authors: McMillen Sue , Hernandez Beth Oritz
Published: December 2020

This department features children's hands-on and minds-on explorations in mathematics and presents teachers with open-ended investigations to enhance mathematics 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 classroom settings.
A Venue for Peer Reviews
Authors: Groth Randall E. , Burgess Claudia R.
Published: December 2020

Online conversations help teachers engage in constructive criticism and attend more carefully to aligning lesson plans with problem solving.
Math by the Month: Medieval Times
Authors: Wilburne Jane M.
Published: December 2020

Journey back in time with us to the Medieval Period, also known as the Middle Ages, where this month we solve math problems set in contexts such as castles, jousting tournaments, royal families, and palatial feasts.
Back Talk: Ben's Thinking about Multiplication
Authors: Kline Kate
Published: December 2020

Ben's teacher gave him the following problem to solve and asked him to show his work on paper so that he would be able to explain his thinking to her.
Problem Solvers: October 2009
Authors: Hoekstra Ken
Published: December 2020

Recycling is one simple action that everyone on the planet can take to help reduce global warming. According to Ask.com, by “recycling half of your household waste, you can save 2400 pounds of carbon dioxide annually.” This installment of“problem solvers”challenges students to use logical thinking and apply their knowledge of mass to measure the recyclable materials gathered by the student council at Greendale Elementary School. Exploring this problem may even inspire your students to initiate their own recycling program.
Early Algebra to Reach the Range of Learners
Authors: Schifter Deborah , Russell Susan Jo , Bastable Virginia
Published: December 2020

Experience with explicitly stating generalizations and finding examples, counterexamples, and proofs—algebraic reasoning—lets young students think more about the principles that underlie their work and can support those who struggle as well as those who excel.
From the Classroom: Wanted: your number sense!
Authors: Lambertus Amanda , Keene Karen , Coats Helen
Published: December 2020

Before technology became so prevalent, wanted posters were a means that law enforcement officers used to communicate information about alleged criminals. These posters depicted a sketch or a photograph of the wanted person; information regarding the person's physical attributes, such as height, eye color, and appearance; alternative names that the person had been know to use; and the crimes for which they were accused.
Preschoolers' number sense
Authors: Moomaw Sally , Carr Victoria , Boat Mary , Barnett David
Published: December 2020

Consider using a game-based assessment of number sense in young children, including those at risk because of socioeconomic level, disability, or the necessity of learning a second language.
Classroom characters coach students to success
Authors: Edwards Sharon A. , Maloy Robert W. , Anderson Gordon
Published: December 2020

Learning mathematical problem solving is as easy as 1, 2, 3 when teachers use flexible instructional strategies.
Early Childhood Corner: Sand and water table play
Authors: Wallace Ann H. , White Mary J. , Stone Ryan
Published: December 2020

Observing in Mary White's kindergarten classroom is like watching a beehive: hustle and bustle all around. Children work puzzles, create artwork, build with blocks, read books, and write their own stories.
Models for special needs students
Authors: Treahy Diana , Gurganus Susan
Published: December 2020

Teachers describe five proven, effective co-teaching strategies for collaborative partnerships with special education instructors and other professionals.
Communication speaks
Authors: Kinman Robin Lynn
Published: December 2020

“Communication is not a passing fad! [It] belongs in the very heart of every math class,” says this fourth-grade teacher from Missouri.
News&Views: Is math politically neutral?
Authors: Felton Mathew D.
Published: December 2020

The broader messages that we communicate through our choice of mathematical problems—how to do math, what math is for, and who it applies to—may privilege some students and exclude others. If “excellence in mathematics requires equity—high expectations and strong support for all students” (NCTM 2000, p. 12), one way to work toward equity within the classroom is to be careful about the messages we send to students through our problem choices.
Math by the Month: Economic insights through problem solving
Authors: Crawford Heather Glynn
Published: December 2020

The dramatic fluctuations of the U.S. and world economies during the past two years have seemed like a wild ride on an out-of-control roller coaster. Children are watching and listening with confusion. This set of problems is intended to help students better understand a few economic basics and put them on the path to being wise consumers.
A lesson-planning model
Authors: Linder Sandra M.
Published: December 2020

With attention to both content and context, this alternative model for instructional planning emphasizes the mathematical needs of students.
First grader's number knowledge
Authors: Thomas Jonathan N. , Tabor Pamela D. , Wright Robert J.
Published: December 2020

Whether working side by side with your colleagues—or miles apart—sharing observations of youngsters' understanding of Number and Operations has important implications for your instructional practices.
Building money skills, one cabin at a time
Authors: Macken Carol
Published: December 2020

You need not look far, maybe just within your own wallet, to see that we no longer use coins and paper currency as we used to. Debit, credit, and gift cards have become standard fare for retail transactions. Beyond altering how adults use money in the realm of daily transactions, these changes have a trickledown effect on our youngest learners. In years past, primary-grades students began the process of learning to count money having already handled coins, for instance, when purchasing small items, such as candy.
Implementing the Assessment Standards for School Mathematics: NCTM's Assessment Standards: A Document for All Educators
Authors: Joyner Jeane M.
Published: December 2020

I have been thinking about changing the way I assess students in my class room. Parents and students like my class, and I consider myself to be a good teacher. Now the changes I have made in how I am teaching mathematics are making me really queslion the ways in which I am assessing my stude I am just not clear on where and how to begin.”
Research into Practice: Using Sharing Situations to Help Children learn Fractions
Authors: Empson Susan B.
Published: December 2020

It is surprising to learn that first graders know a lot about fractions. That is what two first-grade teachers and the author discovered when they collaborated on a five-week fraction unit. This article describes the highlights of a case study of fractions in a first-grade class then presents some preliminary findings suggesting that third-, fourth-, and fifth-grade children can learn fractions in similar ways.
Promising Research, Programs, and Projects: Project Math: Good Beginnings
Authors: Thornton Carol A.
Published: December 2020

Broadening the mathematics content background and changing the instructional strategies of teachers in grades K-3 are the foci of Project MATH: Good Beginnings, housed at Illinois Slate University. The program is directed by Carol Thornton, professor of mathematics education at Illinois State University, and Judy Wells, a mathematics specialist from the Shaker Heights school district in Shaker Heights, Ohio.
Promising Research, Programs, and Projects: Collaboration Really Works
Authors: Corbin Gloria L.
Published: December 2020

Collaborative planning among teachers can be a positive experience—for students and teachers alike—with a minimum amount of time requirdd.
Exploring Measurement Through Literature
Authors: Lubinski Cheryl A. , Thiessen Diane
Published: December 2020

This article focuses on how the children's book How Big Is a Foot? (MyHer 1990) was used to prompt measurement experiences that reflect ideas embedded in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989).
Rethinking “Concrete” Manipulatives
Authors: Clements Douglos H. , McMillen Sue
Published: December 2020

Close your eyes and picture students doing mathematics. Like many educators, the mental pictures may include manipulative objects, such as cubes, geoboards, or colored rods.
A Teacher's Journal: Manipulatives for the Metal Chalkboard
Authors: Sharp Janet M.
Published: December 2020

Manipulates represent one possible teaching tool for building a child's conceptual foundations. Once mathematical ideas are experienced at the concrete level, they must eventually he matched to a symbolic expression. In this way, conjectures can be explored and verified.
Links to Literature: Revisiting the Powers of 2
Authors: Curcio Frances R. , Zarnowski Myra
Published: December 2020

Concept development is not an all-or-nothing affair. Instead, understanding builds slowly through repeated, similar, and refined experiences requiring observation, problem solving, and reflection.
Promising Research, Programs, and Projects: Learning to Think through Reading and Mathematics
Authors: Miller Elizabeth D.
Published: December 2020

During my many years of teaching primary students. I have come to realize that children do not do enough problem solving. Although they can perform mathematical operations. many do not really understand what they are doing. Too often we “train” our students in specific strategies involving particular problems.
Preservice Elementary Teachers: Building Portfolios Around Students' Writings
Authors: Pokay Patricia , Tayeh Carla
Published: December 2020

Traditionally, college mathematics courses have rarely asked students to write or to reflect on their learning, concentrating instead on textbook exercises and problem sets in a lecture format. The intent of this study was to model the use of portfolio assessment in a college mathematics course for preservice teachers while focusing on the contributions of the students' writing in the mathematics classroom. In the study, portfolios were the vehicle for organizing the students' writings.
Implementing the Assessment Standards For School Mathematics: Understanding Children's Reasoning
Authors: Bright George W.
Published: December 2020

Each day, teachers make many instructional decisions on the spur of the moment. These decisions typically are in response to what children are doing or to what teachers believe children are thinking. Observing children works well as a way of knowing what they are doing, but the most effective way I know to get information about children's thinking is to ask them to explain how they found their answers to problems.
Research into Practice: Direct Modeling and Invented Procedures: Building on Students' Informal Strategies
Authors: Chambers Donald L.
Published: December 2020

Young children commonly solve mathematics problems by directly modeling the action or relationship described in the problem. They do not need to be taught how to use direct-modeling strategies, nor do they need such often-assumed prerequisite knowledge as number facts or computational algorithms.
Math by the Month
Authors: Castle Marrietta , Lewis Cathie
Published: December 2020

These activities are designed to appeal directly to students, giving them open-ended questions to engage their intellect. Students may work on the activities individually, in pairs, or in small groups. No solutions are given for activities so that students will look to themselves as the mathematical authority.
Links to Literature: Children's Literature: Impetus for a Mathematical Adventure
Authors: Gibbons Estelle
Published: December 2020

The day started like any other. My fifth graders and I had just finished reading a chapter of The Indian in the Cupboard (Banks 1980) (fig. 1), a wonderful fantasy about a toy plastic Indian, Little Bear, who magically comes to life, and the adventures he encounters while trying to survive in a giant world.
First- and Second-Grade Students Communicate Mathematics
Authors: Spungin Rika
Published: December 2020

Most preschool children are natural communicators. They are eager to explain and show all they know. They love to draw pictures and to tell related stories. They are not afraid to be challenged or to take risks. Let us not discourage these academically wonderful traits when children enter school but rather enhance them.
Math by the Month
Authors: Olson Melfried , Easley Barbara
Published: December 2020

These activities are designed to give students problem-solving situatons to engage their intellect while also furnishing a conceptual framework for further mathematical ideas. Students are encouraged to work on the activities independently, in pairs, or in small groups.
Links to Literature: Literature and Algebraic Reasoning
Authors: Lubinski Cheryl A. , Otto Albert D.
Published: December 2020

The K-4 Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) encourages teachers to broaden and develop their students' mathematical understandings by providing students with opportunities to explore and discuss patterns and relationships.
Promising Research, Programs, and Projects: Algebraic Thinking: A Professional-Development Theme
Authors: Ruopp Faye Nisonoff , Cuoco Al , Rasala Sue M. , Kelemanik M. Grace
Published: December 2020

Over a thirty-six month period beginning September 1992, teams of two elementary, two middle, and two high school teachers from each of seven school districts attended a series of biweekly seminars at Education Development Center (EDC). The seminars, called Teachers, Time and Transformations (TTT), were held during the school day as part of a professional-development program for K–12 mathematics teachers.
Tech Time: Mining Mathematics through the Internet!
Authors: Lynes Kristine
Published: December 2020

Mathematics is truly a universal language, and electronic communication is allowing people to communicate quickly and easily both locally and around the world.
Geometry and Language—A Natural Connection
Authors: Pereira-Mendoza Lionel
Published: December 2020

A quick reading of the literature indicates that the use of language, both written and oral, in mathematics has received a great deal of attention. The importance of language, and of writing in particular, is reflected in such documents as the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The question facing teachers is not whether oral and written language should be part of the mathematics curriculum but how language can be used effectively.
Links to Literature: Problem Solving with Franklin the Turtle
Authors: Harris Jacqueline
Published: December 2020

As a classroom teacher, I have struggled to find a solid approach to problem solving in mathematics. Most commercial teacher's manuals and books for teaching problem solving suggest practices that I feel are contrived and do not reflect real-life situations; moreover. I have found little connection between the proposed lessons in these books and students' success with problem solving.
A Teacher's Journal: Hands-On Addition and Subtraction with the Three Pigs
Authors: Bartek Mary Marron
Published: December 2020

Addition and subtraction with regrouping and across zeros can be difficult concepts for second and third graders. Although the traditional algorithms are not the only successful processes that students may use, they are commonly used in mathematics textbooks and, especially when zeros are involved, can be difficult processes to teach. I have found that several days spent with the Three Pigs can go a long way to increase students' understanding of the traditional algorithms.
Our Heritage: Learning Data-Management Skills Meaningfully
Authors: Pothier Yvonne M. , Nickerson Christine M.
Published: December 2020

During January, the local Heritage Association was planning and promoting various projects to celebrate Heritage Day in February. Mrs. Nickerson's fifth-grade classroom was beginning a unit on data management, to be assisted by Ms. Pothier, a university mathematics educator, for three mathematics classes a week.
Investigations: Timely Pursuits
Authors: Hopkins Martha H.
Published: December 2020

This department presents teachers with open-ended explorations to enhance mathematical instruction. Meaningful tasks evolve as they are conducted. They invoke problem solving and reasoning, require communication skills, and encourage connections among mathematical concepts. The ideas presented in this column have been tested in various classroom settings.
Tech Time's News from the Net: Houghton Mifflin Mathematics Center
Authors: Lazerick Beth
Published: December 2020

Early childhood and elementary teachers and supervisors, preservice students, and parents will want to check out the Houghton Mifflin Mathematics Center to access information on technology plans, Internet resources, and educational games online. It also has many noncommercial sections that are tilled with information and ideas.
Third-Grade Students Engage in a Playground Measuring Activity
Authors: Reynolds Anne , Wheatley Grayson H.
Published: December 2020

The K-4 section of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) emphasizes measurement concepts and processes, with attention to “actual measuring,” the “estimation of measurements.” and the “use of measurement … ideas throughout the curriculum” (p. 20). The development of concepts of measure is linked with the development of geometric ideas.
Promoting a Problem-Posing Classroom
Authors: English Lyn D.
Published: December 2020

Children have traditionally solved our problems—problems that we think will be of interest and relevance to them. We need to shift some of this responsibility to students and let them pose problems that they consider to be worthwhile pursuing.
Interpreting the Standards: Translating Principles into Practice
Authors: Schifter Deborah E. , O'Brien Deborah Carey
Published: December 2020

Since the publication of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the Professional Standards for Teaching Mathematics (NCTM 1991). such phrases as “mathematics should be taught for understanding.” “teachers should facilitate the construction of mathematical concepts,” and “classrooms should be student centered” have become identified with a reformed mathematics pedagogy.
A Teacher's Journal: Automobile License Plates
Authors: Durand Elaine S.
Published: December 2020

Students develop mathematical power when they are able to acknowledge, use, and assess the mathematics they encounter in their daily lives. An awareness of mathematics in the world in which one lives provides opportunities for building confidence, enhancing self-esteem, and stimulating creativity.
Problem Solvers: When Will We Reach One-Half?
Authors: Olson Melfried
Published: December 2020

The following problem is suggested for students in grades 1-6, although it may be appropriate for students in other grades as well.
Links to Literature: A Better Way to Share
Authors: Bergman Jill Duea
Published: December 2020

Tops and Bottoms by Janet Stevens (1995) presents many opportunities for students to think mathematically within an entertaining, fictional context (fig. 1). After losing a bet in his race with Tortoise.
Tech Time: Symmetry in Season
Authors: Seidel Judith Day
Published: December 2020

As large, white snowflakes fall outside and stick to the window of the computer room, the fifth-grade mathematics class does not jump up to watch the storm. The reflections and rotations that enliven their computer screens are just as fascinating. During the winter term at the Brearley School, fifth graders use The Geometer's Sketchpad (Key Curriculum Press 1995) and a growing understanding of symmetry and angles to construct their own snowflakes.
In the Comfort of Their Own Homes: Engaging Families in Mathematics
Authors: Ehnebuske Jean M.
Published: December 2020

Is an effective way available to engage parents in mathematics activities with their children, help them reinforce concepts introduced in class, give them an opportunity to communicate regularly with teachers, and make them aware of the need for reform in mathematics education? You bet! A project that we first piloted in our district in 1995. and have since expanded, helped us accomplish those goals with great success.
Implementing the NCTM's Standards through Cognitive Coaching
Authors: Ray Theresa M.
Published: December 2020

Improving or expanding teaching skills requires hard work in which professional dialogue can he an indispensable tool (Joyce 1988). Educators are presented with a challenge in implementing the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). One way to implement the Standards in curriculum development and instruction is through cognitive coaching, a process that requires professional dialogue and can be a vehicle for change (Garmston. Linder, and Whitaker 1993; Sparks 1990). This article explains a project that used the process of cognitive coaching to implement the Standards and describes my own experience as a participant in the project. It is my hope that readers will make connections with their colleagues and be encouraged to put theory into practice in their own classrooms.
The “Measure Me” Doll
Authors: Ruggles JoLean , Slenger Barbara Sweeney
Published: December 2020

At the McKinley Thatcher School, the teacher of a kindergarten-first-grade class usually starts the school year with a “Me” unit to get to know the students. Their ages range from five to seven years, and the teacher keeps the students for both years. The “Me” unit, then, helps students to get acquainted and enables the teacher to note the growth changes from the previous year.
A Problem Worth Revisiting
Authors: Schulman Linda , Eston Rebeka
Published: December 2020

To build mathematical power in young learners, teachers need to pose problems of emerging relevance. To pose such problems is a guiding principle of a constructivist pedagogy that encourages children to investigate widely and deeply as they form their own understandings of important concepts (Brooks and Brooks 1993). Concepts do not develop quickly; broad generalizations take time to form. By revisiting the same questions throughout the school year, children are encouraged to amplify their mathematical thinking. Through these “revisits,” teachers can more easily note developmental growth and patterns among learners and focus their instruction on big ideas.
The Quarter Quandary: An Illustration of NCTM's Professional Teaching Standards
Authors: Mewborn Denise S.
Published: December 2020

The Professional Standards for Teaching Mathematics (NCTM 1991) identifies four elements of teachers’ work that have a significant impact on the mathematics learning that takes place in a classroom: tasks, discourse, environment, and analysis.
Math by the Month: Patterns All Around
Authors: Filliman Paula
Published: December 2020

These 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 authorities, thereby developing the confidence to validate their work.
Listening to Students: The Power of Mathematical Conversations
Authors: Atkins Sandra L.
Published: December 2020

The power of mathematical conversations was epitomized during a series of lessons in which fourth-grade students were encouraged to make conjectures and defend their conjectures to their classmates. The opening excerpt illustrates that the students had had limited experiences with angles in various orientations. Consequently, they had internalized a very narrow definition for a right angle. However, this limited definition did not become clear until we asked them to elaborate on their thinking.
Image Maker: Developing Spatial Sense
Authors: Wheatley Grayson H. , Reynolds Anne M.
Published: December 2020

Just as it is useful to have a mental map of streets in a city, it is also useful to have mental images of mathematical patterns and relationships. A student who has constructed a network of image-based mathematical meanings can more easily devise solutions to problems. If a student knows only one way and if the task is not straightforward, he or she may not be able to solve the problem. The ability to construct and transform mental images leads to flexibility and power. In doing mathematics, it is advantageous to know more than one way to solve a problem or complete a routine task. Mental images play a crucial role in all aspects of mathematics.
What Is in the Daily News? Problem-Solving Opportunities!
Authors: Silbey Robyn
Published: December 2020

In An Agenda for Action, the NCTM asserted that problem solving must be at the heart of school mathematics (1980). Almost ten years later, the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) stated that the development of each student's ability to solve problems is essential if he or she is to be a productive citizen. The Standards assumed that the mathematics curriculum would emphasize applications of mathematics. If mathematics is to be viewed as a practical, useful subject, students must understand that it can be applied to various real-world problems, since most mathematical ideas arise from the everyday world. Furthermore, the mathematics curriculum should include a broad range of content and an interrelation of that content.
Investigations: Are You Puzzled?
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 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.
Math by the Month: Playground Mathematics: Playing with Space
Authors: Ham Jan
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: Measuring Up with The Principal's New Clothes
Authors: Wickett Maryann S.
Published: December 2020

The Principal's New Clothes by Stephanie Calmenson (1989) provides a delightful, whimsical context for investigations in measurement. This story is a modern-day adaptation of Hans Christian Andersen's The Emperor's New Clothes. Mr. Bundy, the principal of P.S. 88, is a very classy dresser. One day, Moe and Ivy come to town, offering to make Mr. Bundy a new suit from magic cloth. Children find the hilarious outcome very entertaining.
Using Alternative Algorithms with Preservice Teachers
Authors: Simonsen Linda M. , Teppo Anne R.
Published: December 2020

During their undergraduate program, preservice elementary teachers are expected not only to become generalists across a wide range of school subjects but also to develop pedagogical knowledge of the developmental and social needs of children. We have developed a twosemester freshman-level mathematics-content course that attempts to address multiple needs of preservice teachers. The goals of this course are to help preservice teachers (a) deepen their understanding of mathematical concepts; (b) restructure their attitudes toward, and beliefs about, the nature of mathematics and how it is learned; (c) investigate pedagogical issues; and (d) experience mathematics learning within a reformbased environment. These goals are interrelated and reflect the complexity of the nature of the knowledge and disposition, both mathematical and pedagogical, that preservice teachers are expected to develop.
Getting Students Actively Involved in Geometry
Authors: Robertson Stuart P.
Published: December 2020

For the past three years, I have begun my school year by having students write a “mathematical autobiography.” We talk about what an autobiography is and what a mathematical autobiography might be like. The students write about their interactions with mathematics, how they feel about it, and what they have done in mathematics. Their writings often reveal that students view mathematics as computation. They write about addition, subtraction, multiplication, and division; which operations they like to do and which ones they do not like. One activity that I use to address their one-sided view of mathematics is a geometry unit, which gives another view of mathematics and highlights how geometry surrounds students every day.
Investigations: What Do You Like to Read?
Authors: Urso Josephine
Published: December 2020

This department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents openended explorations to enhance mathematics instruction. These explorations 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.
Improving Problem Solving through Drawings
Authors: Kelly Janet A.
Published: December 2020

While working with third-, fourth-, and fifth-grade teachers in a National Science Foundation–sponsored project designed to enhance the mathematics and science teaching of in-service elementary teachers, we recognized that teaching mathematics problem solving was one of their greatest challenges. Discussions with the teachers revealed that most were using an algorithmic approach to problem solving with an emphasis on facts, rules, and procedures. Their students were being taught to solve word problems in a systematic, single-mode manner. We found that the teachers were most comfortable with the algorithmic approach because that is how they were taught mathematics when they were in school. As one teacher commented, “I was stunned to find out that not everyone worked math problems the same way.”
Dumpling Soup: Exploring Kitchens, Cultures, and Mathematics
Authors: Smith Nancy L. , Babione Carolyn , Vick Beverly
Published: December 2020

Many teachers understand the importance of discussing diversity and celebrating cultural heritage. One way that these two goals can be accomplished is by looking at the lives of others through multicultural literature. Teachers often use such literature in language arts and social studies but may be unsure how to use it in mathematics.
How Huge Is a Hundred?
Authors: Auriemma Susan Hampton
Published: December 2020

How much is a hundred?” How would your students respond to such a question? This article shares the experiences of my first graders as they participated in activities that develop number sense to answer this question. Teaching number sense to students in grades K–4 is an important goal of the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). The following activities are based on the Standards in many ways. They require problem solving, reasoning, communicating, and connecting mathematics to everyday situations that interest children. They provide opportunities to develop measurement and place-value concepts and to integrate reading, writing, drawing, and mathematics in ways that contribute to a cooperative learning environment.
Journals and Portfolios: Alternative Assessment for Preservice Teachers
Authors: Liebars Cathy
Published: December 2020

Traditionally, assessment in mathematics courses has consisted of tests, quizzes, and textbook exercises.
Reading, Writing, Arithmetic-Making Connections
Authors: Taylor Gwen
Published: December 2020

As teachers review the Standards for the English Language Arts (1996) and the Curriculum and Evaluation Standards for School Mathematics (1989), they can observe many similarities. Both publications call for changes in how instruction should take place in classrooms.
Problem Solvers: Marbles
Authors: Olson Judith
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.