bluuebunny/crossref_metadata_2025 huggingface dataset:
Focused Strategies for Middle-Grades Mathematics Vocabulary Development
Authors: Rubenstein Rheta N.
Published: December 2020

Principles and Standards for School Mathematics reminds us that communication is central to a broad range of goals in mathematics education (NCTM 2000). These goals include students' being able to (1) organize and consolidate mathematical thinking; (2) communicate coherently with teachers, peers, and others; (3) analyze and evaluate others' strategies; and (4) use language to express mathematics precisely. One part of communication is acquiring mathematical language and using it fluently. This article addresses learning vocabulary as one dimension of mathematics communication.
A Look at the Development of Data Representation and Analysis in Curriculum Focal Points: A Quest for Coherence
Authors: Schielack Jane F. , Seeley Cathy
Published: December 2020

One of the questions often raised with regard to NCTM's Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence is, “Where are data representation and analysis?” In this month's Focal Points feature, we take a look at an area of content—data representation and analysis—that is important enough to warrant a separate strand in Principles and Standards for School Mathematics (NCTM 2000) but that does not appear in the title of a content Focal Point until Grade 8. However, in the Appendix of Curriculum Focal Points, in comparison with Principles and Standards, it is clear that the content of the Data Analysis strand presented in each of the pre- K–2, 3–5, and 6–8 grade bands in the Standards is included in the specific grade-level descriptions in Curriculum Focal Points. Skills for data representation begin in prekindergarten and appear throughout later grades and are presented as connections to the concepts and skills involved in the Geometry and Measurement Focal Points. In Kindergarten through Grade 7, data representation and analysis in the Connections to the Focal Points sections also provide meaningful contexts as students apply their developing knowledge of number and operations. In Grade 8, the analysis of data through descriptive statistics becomes a Focal Point in itself.
See a Different Mathematics
Authors: Stallings L. Lynn
Published: December 2020

Sporting News, a popular sports magazine, runs certain stories under the heading of “See a different game.” These stories are in-depth analyses of strategy. The slogan means that if you know more about the strategy of the game, you will see the game differently. If students are pushed to think about mathematics more deeply and flexibly, they will, in effect, “see a different mathematics.” We can change our game plans by strategically posing mathematics problems so that students get that chance. As we add to our playbook of problem-posing strategies, our students will come to “understand a different mathematics.”
Solve It!: A Mix of 1, 2, 3, 4, 5, 6
Authors: Mooney Edward S.
Published: December 2020

Find the sum of all positive six-digit integers whose digits are a permutation of the digits in the number 123456 without listing them all.
Reflections on a Research-Inspired Lesson about the Fairness of Dice
Authors: Groth Randall E.
Published: December 2020

The relationship between educational research and teaching practice has typically been stormy. Researchers and teachers sometimes seem to be operating under different agendas. Researchers are often interested in theoretical and general questions, whereas teachers are generally interested in solving problems related to situations that arise in classes each day (Labaree 2003). The valued currency in the research community is work that develops and refines theory, whereas practical application is valued in the community of educational practice (Silver 2003). Given these contrasting objectives, the idea of linking research with practice can seem daunting. It has become imperative, however, for dialogue to occur between the two communities in light of legislation mandating research-based practices in schools (U.S. Department of Education 2002).
Dynamic Concrete Instruction in an Inclusive Classroom
Authors: Witzel Bradley S. , Allsopp David
Published: December 2020

NCTM advocates the use of multiple representations and manipulatives for building mathematical understanding (NCTM 2000). Engaging students in mathematics using manipulatives can have a powerful effect on learning. The use of manipulatives is especially effective for students with high-incidence disabilities, such as learning disabilities (LD), attention deficit hyperactivity disorder (ADHD), and mild to moderate mental disabilities (MD). Allowing students to use manipulatives gives them the opportunity to experience multisensory learning, which helps promote success for students with highincidence disabilities. Experiences with manipulative objects can help students develop conceptual understanding. If used appropriately, manipulatives can also help those with high-incidence disabilities connect conceptual understandings of mathematics to the process of doing mathematics (i.e., using effective problem-solving strategies) throughout their K–12 education.
Integrating Mathematics and Social Issues
Authors: Harrell Gregory K.
Published: December 2020

The Connections Standard in grades 6–8 in Principles and Standards for School Mathematics recommends that middle school students “recognize and apply mathematics in contexts outside of mathematics” (NCTM 2000, p. 274). This goal can be reached by providing students with rich problem contexts that involve connections to the real world (NCTM 2000). To find such contexts, mathematics teachers can look to the local community, because our culture influences the mathematics we do and influences the issues that are important to us. If teachers present students with interdisciplinary experiences within the context of local community issues, students will understand the usefulness of mathematics and it will help them develop the skills and knowledge necessary to become active participants in their communities (Zaslavsky 1996).
Solve It! The Three Bears Cookie Store
Authors: Mooney Edward S.
Published: December 2020

The Bear family decided to open a cookie store that featured bear-sized cookies. Mama and Baby Bear decorated the cookies, and Papa Bear sold the cookies. Mama Bear could completely decorate a cookie every 7 minutes. Baby Bear could completely decorate a cookie every 12 minutes. Unfortunately, Papa Bear was eating one of the cookies every 9 minutes. If they open the store and have no cookies decorated, how long will it take them to produce 500 decorated cookies that can be sold?
The Thinking of Students: Number Golf
Authors: Mooney Edward S.
Published: December 2020

The following “Solve It” task appeared in the October 2006 issue of this journal:
Quick Reads: Another Good Idea: A Yin and Yang Approach to Area and Irregular Shapes
Authors: Yopp David , Jinyan Zhou
Published: December 2020

The Measurement Standard for grades 6–8 in NCTM's Principles and Standards for School Mathematics (2000) emphasizes that students should develop formulas to find the areas of triangles, parallelograms, and trapezoids and develop strategies to find the areas of more complex shapes. “Developing formulas” often involves doubling or cutting and translating pieces of basic shapes, such as triangles, parallelograms, and trapezoids. Strategies for working with more complex shapes often include partitioning them into triangles, rectangles, or parallelograms. The yin and yang activity offers an irregular shape to investigate and extends and reinforces these strategies. Behind the name and the objects in the activity is the yin and yang story, which gives students the opportunity to reflect on different cultures and may help them remember the doubling strategy for finding area. This activity has been successfully implemented with students in China by Zhou Jinyan and with students in the United States by David Yopp.
Teaching Multiplication Algorithms from Other Cultures
Authors: Lin Cheng-Yao
Published: December 2020

Different multiplication algorithms are known from different cultures. For example, the Hindus used a vertical and crosswise method; the Egyptians and Russian peasants used doubling; the Japanese used an abacus; and the Chinese used counting rods. These methods and techniques have been used to help people around the world in their daily lives, for example, in building houses, counting agricultural products, or conducting commercial transactions in the marketplace.
Is Silence Golden? What Silent Participants Might Be Learning in Discourse-Rich Classrooms
Authors: Hillen Amy F. , Smith Margaret S.
Published: December 2020

Communication is an important part of classrooms in which students are engaged in challenging mathematics (NCTM 2000). Students communicate by “participating in social interaction, sharing thoughts with others, and listening to others share their ideas” (Hiebert et al. 1997). As students take part in sharing their mathematical thinking, they have opportunities to make and justify conjectures, consider the validity of others' ideas, engage in mathematical argumentation, and reflect on their current understandings (NCTM 2000). Thus, the mathematical discourse in classrooms provides important opportunities for students to learn mathematics (Hiebert et al. 1997; NCTM 2000).
Mathematical Explorations: A Treasure Hunt: Reflecting, Translating, and Rotating Points on a Coordinate Map
Authors: Bellomo Carryn
Published: December 2020

Middle school teachers in Clark County School District, Nevada, have found the following geometry activities effective with sixth-, seventh-, and eighth-grade students when they study translations, rotations, and reflections of points in the plane. Students use a local map, in this case a university campus map, to find particular points under a translation, rotation, reflection, or composite of these symmetries. Teachers might choose to customize the lesson for their students with a map from their community, preparing clues similar to those in this activity using familiar locations.
Teaching Algebra without Algebra
Authors: Kalman Richard S.
Published: December 2020

In 1953, I was recruited for my high school's math team. That event changed my life. No longer simply repeating techniques shown previously, I was taught to focus intensively on problem solving. As a result, I majored in mathematics in college.
Using Quilt Blocks to Construct Understanding
Authors: Westegaard Susanne K.
Published: December 2020

A quilting magazine's thirteen-part series titled “Taking the Mathematics Out” prompted the mathematics teacher and the quilter in me to think, “Say it isn't so.” Give up the chance to use a four-patch quilt block to work with writing and plotting coordinate pairs and writing equations for lines? Not use nine-patch quilt blocks to look for examples of reflections, translations, rotations, and changes in scale? Never use quilt blocks to classify triangles and quadrilaterals, find perimeter and area, look for angle relationships, or practice with similar figures?
Card Sorts, State Tests, and Meaningful Mathematics
Authors: Chauvot Jennifer B. , Benson Sharon L. D.
Published: December 2020

“Data-driven decision making” is a phrase that is now commonplace in educational conversations. It is used when school districts discuss strategies to help meet the needs of all students while closing the achievement gap between subgroups of students. Within mathematics education, there is a growing need to educate prospective and practicing mathematics teachers about No Child Left Behind accountability measures and corresponding data within a given state in a way that emphasizes meaningful mathematics instruction that is consistent with national recommendations (NCTM 2000). Given pressures that teachers face within high-stakes teaching environments, it is important to analyze assessment tools, such as state-mandated tests, in an efficient, enlightened manner. A goal of such an analysis is to acknowledge accountability through constructive discourse while continuing to emphasize good mathematics teaching. Such discourse about accountability can take on many forms (Chauvot 2006). This article shares card-sorting activities that use items from state-mandated tests with prospective and practicing mathematics teachers. These activities give teachers tools to further accountability measures while exploring reformminded mathematics instruction.
Developing Mathematical Understanding through Multiple Representations
Authors: Tripathi Preety N.
Published: December 2020

What purpose is served by studying multiple representations of a mathematical idea? Let me begin by recounting a story. Once upon a time, five people who had never seen an elephant were blindfolded and led to an elephant and asked to describe what they thought the creature looked like. The people were allowed to touch the animal. They approached it from different directions. One felt the tail, the second found its leg, the third reached for the trunk, another felt its ear, and the fifth person touched the great stomach of the animal. Each one then described his or her version of the animal and each was, in turn, completely surprised at the contrasting view offered by the others in the group. Finally, the blindfolds were removed, and the five looked at the animal that each had described. Each person realized that not only was his or her perspective valid, albeit in a limited way, but also that this applied equally to every member of the group. Only after opening their eyes (literally and metaphorically) and seeing the animal from a variety of angles could they appreciate the complete picture.
Promoting Mathematics Accessibility through Multiple Representations Jigsaws
Authors: Cleaves Wendy Pelletier
Published: December 2020

There have been innumerable times when, attempting to solve a mathematical problem, I have been unsure of how to start, how to proceed, or how to interpret the problem correctly. Other times, when attempting to solve a mathematics problem with a group of colleagues, I would compare strategies with a partner only to suddenly “see” a simple, straightforward approach. I would tell myself, “If only I had thought to look at it that way” or “I cannot believe how easy this problem is now that I see how you approached it.” When working with students who have had limited success with mathematics, we want to provide the tools that allow them to look at problems from different perspectives. The ability to examine problems using varied approaches is one of the most important characteristics of good problem solvers. Other characteristics include independence, flexibility in thinking, and determination, and a willingness to take risks.
Teacher to Teacher: Literature in the Mathematics Classroom: Introducing The Inch Boy to Middle School Students
Authors: Foss Susan M.
Published: December 2020

I was sitting in a faculty meeting when my principal suggested that teachers of all disciplines should read to their students. I remember thinking to myself, These are middle school students. They should know how to read already. My principal continued by explaining that reading aloud to a class is an excellent and effective way to in-still the love of literature in students. I was doubtful, and thought, I'm a math teacher. We don't have time to read books in math. As if he were reading my mind, the principal gave examples of how teachers of all disciplines, including mathematics, could incorporate literature into their lessons. I was skeptical but decided to give it a try.
What Do Students Need to Learn about Division of Fractions?
Authors: Li Yeping
Published: December 2020

What should students know and be able to do after learning division of fractions? Many mathematics educators believe that it is not enough for students' knowledge of the division of fractions to be limited to the invert-and-multiply algorithm. In fact, many in the field have argued that students' learning of mathematics needs to go beyond rote memorization of procedures (Brownell 1938; NCTM 2000). It is important for teachers to carefully consider what students need to learn beyond the algorithmic procedure; developing students' conceptual understanding of division of fractions is not a trivial task.
Introducing Volume 14: Reflection Time
Authors: Wilkerson Trena
Published: December 2020

It is with excitement that we bring you volume 14 of Mathematics Teaching in the Middle School (MTMS). You will find articles that offer specific ideas on teaching a variety of topics in middle-grades mathematics, delve into critical issues about curriculum, and engage learners (and teachers) in stimulating mathematical experiences.
The Representational Value of Hats
Authors: Watson Jane M. , Fitzallen Noleine E. , Wilson Karen G. , Creed Julie F.
Published: December 2020

The literature that is available on the topic of representations in mathematics is vast. One commonly discussed item is graphical representations. From the history of mathematics to modern uses of technology, a variety of graphical forms are available for middle school students to use to represent mathematical ideas. The ideas range from algebraic relationships to summaries of data sets. Traditionally, textbooks delineate the rules to be followed in creating conventional graphical forms, and software offers alternatives for attractive presentations. Is there anything new to introduce in the way of graphical representations for middle school students?
NCTM's Advocacy for Mathematics Education
Authors: Krehbiel Ken
Published: December 2020

In 2003, the Board of Directors of the National Council of Teachers of Mathematics (NCTM) identified political advocacy as one of its key strategic priorities for the future. At that time, the Council embarked on a plan to lay the groundwork for more concentrated activities in advocacy. Since then, the Council can point to some significant successes, but it has also laid the foundation for future opportunities for the organization and its members.
Subtraction of Positive and Negative Numbers: The Difference and Completion Approaches with Chips
Authors: Flores Alfinio
Published: December 2020

Diverse contexts such as take away, comparison, and completion give rise to subtraction problems. The take-away interpretation of subtraction has been explored using two-colored chips to help students understand addition and subtraction of integers (Bennett and Musser 1976). This article will show how the difference and completion interpretations of subtraction may be illustrated with colored chips to help students develop a better understanding of the subtraction of negative and positive numbers. Black chips will represent positive numbers, and white chips will represent negative numbers. The basic principle is that chips of different colors will cancel each other when joined together; this pair of chips is called a zero-pair. Several zero-pairs together will still have a zero value. Each collection in figure 1 represents zero.
Take Time for Action: Students' Geometric Thinking about Rotations and Benchmark Angles
Authors: Olson Melfried , Zenigami Fay , Okazaki Claire
Published: December 2020

The Measurement Standard states that understanding angles and angle measurement is important in the middle grades (NCTM 2000). To minimize misconceptions that the measure of an angle is determined by the length of its rays (Keiser 2000) or by its interior “space,” our curriculum research and development team decided to introduce angles and angle measurement using transformational geometry, specifically, the motion of rotation. Our class included ten fifth-grade students who had introductory experience with transformational geometry in the fourth grade. Our goals were for students to understand that 360 degrees measured a full rotation and to use this information to determine benchmark angles of 90 degrees, 180 degrees, and 270 degrees. To help achieve these goals, clockwise and counterclockwise movements were also emphasized. We hope the description of our students' developing understandings of angles and angle measurements will help you explore your students' thinking about these concepts.
Historical Research: How to Fit Minority and Women's Studies into Mathematics Class
Authors: Sáraco Margaret
Published: December 2020

Ask middle school students to name their favorite musicians, athletes, or actors, and they will tell you everything about them: statistics, hair color, who they are married to, where they live, their accomplishments, and more. Students are exposed to celebrities every day through television, movies, radio, and the Internet. Isn't it time we expose our students to some mathematical heroes?
Fraction Representation: The Not-So-Common Denominator among Textbooks
Authors: Hodges Thomas E. , Cady JoAnn , Collins R. Lee
Published: December 2020

Using visual representations, such as symbols, drawings, and graphs, helps middle school students reason about and understand mathematics. These representations support students' learning and help them communicate their mathematical ideas. Representations also help them organize their thinking, make connections among mathematical concepts, and model the mathematics that they see in the real world (NCTM 2000). The middle school mathematics curriculum seeks to move students in a logical progression from concrete models to drawings and pictures and finally to abstract symbols. Representations can assist students in making this transition.
Is 1 a Prime Number? Developing Teacher Knowledge through Concept Study
Authors: Davis Brent
Published: December 2020

At a meeting of eleven middle school mathematics teachers who gather monthly to discuss various topics, this question arose: “Is 1 a prime number?” The response proved to be a little surprising. The answer can be either yes or no, depending on the particular set of images and metaphors for multiplication that are discussed.
Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding
Authors: Webb David C. , Boswinkel Nina , Dekker Truus
Published: December 2020

A common challenge for middle-grades mathematics teachers is to find ways to promote student understanding of mathematics. When an algorithm for adding or subtracting fractions is explained as clearly as possible and students have opportunities to practice it, the reality is that many students will continue to confuse the procedures and forget how they work. The primary intervention used to address student confusion is to reteach common procedures and give additional practice in the hope that the students will understand over time.
1P + 4R = 5D: An Equation for Deepening Mathematical Understanding
Authors: Hackbarth Alan J. , Wilsman Margaret J.
Published: December 2020

No one will argue with the NCTM's assertion that students' implementation of and flexibility with varied representations are central to studying mathematics for deeper understanding (NCTM 2000). In our work with grades 6–8 mathematics teachers, however, we found reasons to be cautious when teaching activities that use multiple representations to develop a deeper understanding of procedures and concepts. In postactivity reflections, our teacher-learners consistently expressed a newfound appreciation for the variety of valid representations and strategies that can be used to solve a problem. This is a critical point at which to capture the learners' attention by asking this important question: “Why do the various representations and strategies lead to the same valid solution?”
Thinking through a Lesson: Successfully Implementing High-Level Tasks
Authors: Smith Margaret S. , Bill Victoria , Hughes Elizabeth K.
Published: December 2020

Mathematical tasks that give students the opportunity to use reasoning skills while thinking are the most difficult for teachers to implement well. Research by Stein and colleagues (Henningsen and Stein 1997; Stein and Lane 1996; Stein, Grover, and Henningsen 1996) makes the case resoundingly that cognitively challenging tasks that promote thinking, reasoning, and problem solving often decline during implementation as a result of various classroom factors. When this occurs, students must apply previously learned rules and procedures with no connection to meaning or understanding, and the opportunities for thinking and reasoning are lost. Why are such tasks so difficult to implement in ways that maintain the rigor of the activity? Stein and Kim (2006, p. 11) contend that lessons based on high-level (i.e., cognitively challenging) tasks “are less intellectually ‘controllable’ from the teacher's point of view.” They argue that since procedures for solving high-level tasks are often not specified in advance, students must draw on their relevant knowledge and experiences to find a solution path. Take, for example, the Bag of Marbles task shown in figure 1.
An Investigation of Solar Noon
Authors: Stewart H. Bruce , Reininger Maria E. , Smudzinksi Walter A.
Published: December 2020

This article describes how mathematics and science were integrated in a single lesson, in the context of an extended eighth-grade science unit on astronomy. The lesson involved applying principles of geometry and using measurement and calculations to interpret observations of the sun's apparent movement across the sky during the day. An upright pointer, which is the modern equivalent of an ancient device called the gnomon, produced shadows that were then tracked. The observations of the shadows were used to find the direction of geographic north.
Solve It!: Rule of Ten
Authors: Mooney Edward S.
Published: December 2020

Malik and Melinda each have an unmarked 4-inch wooden ruler. They realized that if they placed only two wellchosen marks on the rod, they could measure any inch unit length from 1 inch to 4 inches. Malik marked the 1-inch length and the 3-inch length. Melinda marked the 1-inch length and the 2-inch length.
Students as Performance Mathematicians
Authors: Gadanidis George , Hughes Janette , Borba Marcelo C.
Published: December 2020

What if we take the view, as an elementary school teacher once noted, that mathematics “can be discussed with your family and friends just like you would a favorite book or new movie”? (Gadanidis, forthcoming).
The Pizza Problem: A Solution with Sequences
Authors: Shafer Kathryn G. , Mast Caleb J.
Published: December 2020

As a mathematics educator working with preservice education majors, one of my primary goals is to provide students with problem-solving experiences. This is accomplished through the use of the “problem of the week,” or POW. Each problem is selected with a different strategy in mind. The students work independently on one problem each week; suggestions and class discussions then occur when necessary. About a month into the fall semester, I assign the Pizza problem.
Mathematics Circles: A Structured Approach to Problem Solving
Authors: Kridler Patricia G. , Moyer-Packenham Patricia S.
Published: December 2020

When I taught reading, I frequently used literature circles to help students structure their group discussions. This reading strategy involves assigning a different role to each student in a small group as he or she reads a section of the same book. Some of the roles include Illustrator, Discussion Director, Summarizer, and Travel Tracer. This strategy allowed each student to focus on a different aspect, such as vocabulary or connections, as he or she read an assigned section in a book. When the students finished their section in the book, they would meet as a group and put all the pieces together in their discussions. The intent of literature circles is “to allow students to practice and develop the skills and strategies of good readers” (DaLie 2001, p. 96). Literature circles are flexible and adaptable; what transpires during a group meeting depends on the book, the roles, and the students' interactions with the materials and with one another.
Using, Seeing, Feeling, and Doing Absolute Value for Deeper Understanding
Authors: Ponce Gregorio A.
Published: December 2020

Several activities that are based on the ancient method of false position, also called false assumption, are presented in this article as a way to motivate students to find the solution of literal equations in beginning algebra. With a historical connection, the activities described engage students in solving linear equations in a novel way. Extension exercise included.
Using the Ancient Method of False Position to Find Solutions
Authors: Edwards Thomas G.
Published: December 2020

During my twenty-four years as a middle school and high school teacher, I observed that students were often fascinated by vignettes from the history of mathematics. When the vignette had an ancient Egyptian setting, that background added a certain mystique.
Supporting Mathematics in Other Subject Areas
Authors: Stump Sheryl , Clark Susan E. , Mitchell Melissa , Roebuck Kay
Published: December 2020

Mathematics can be intertwined with other fundamental subjects, such as health, science, and social studies, to support learning in all these areas.
Solve It!: Survey Says
Authors: Mooney Edward S.
Published: December 2020

In a recent survey, men and women were asked where they primarily buy gifts. Do they shop in stores, shop online, or buy from catalogs?
The Thinking of Students: Christie's Cake
Authors: Mooney Edward
Published: December 2020

The following “Solve It” task appeared in the March 2007 issue of this journal: Christie made a cake in an 8 in. × 8 in. pan, which she wants to share with her 3 friends. She wants to divide the cake equally among all 4 people while giving each person a different shaped piece of cake.
Journey into a Standards-Based Mathematics Classroom
Authors: Brucker Elizabeth L.
Published: December 2020

Story problems—once feared— became the most powerful tool in this teacher's classroom.
Winning Big Money
Authors: McCue Camille
Published: December 2020

Students explore the concept of expected value by examining the television game show Deal or No Deal.
Decomposing Solids to Develop Spatial Sense
Authors: Obara Samuel
Published: December 2020

Students manipulate three-dimensional models, which allows them to improve their spatial sense.
It's a Fird! Can You Compute a Median of Categorical Data?
Authors: Leavy Aisling M. , Friel Susan N. , Mamer James D.
Published: December 2020

Students work with data of interest to them, which promotes understanding of categorical and numerical data as well as measures of central tendency.
Take Time for Action: From Kinesthetic Movement to Algebraic Functions
Authors: Goral Mary G.
Published: December 2020

Are students able to work through a series of geometric spatial activities, discover a pattern, and find an algebraic function? Can they move from using spatial intelligence to number sense to algebraic reasoning? Are they able to connect geometric thinking and algebra, physical models, and numeric relationships? Friel, Rachlin, and Doyle (2001) state, “Explorations that develop from problems that can be solved by using tables, graphs, verbal descriptions, concrete or pictorial representations or algebraic symbols offer opportunities for students to build their understandings of mathematical functions” (p. v). Further, combining physical spatial activities with algebraic reasoning can better engage beginning algebra students in the task at hand. According to Jensen (2001), the kinesthetic arts can provide a significant vehicle that can enhance content-area learning.
Giant Pencils: Developing Proportional Reasoning
Authors: Megan Che S.
Published: December 2020

Teachers can help students develop proportional reasoning and explore measurement with an open-ended activity.
Talking and Writing in the Classroom
Authors: Fello Susan E. , Paquette Kelli R.
Published: December 2020

Students' problem-solving skills are enhanced when they communicate their mathematical ideas.
Patterns with Checkers
Authors: Lewis Kathy
Published: December 2020

While playing a popular game with checkers, students experience patterns that can be expressed using symbols.
Mathematical Roots: The Chinese Remainder Theorem
Authors: Burroughs Elizabeth A. , Evans Tyler J.
Published: December 2020

This problem can be solved by a system of linear congruence. This linear-congruence topic was studied by ancient Chinese mathematicians and examined further by Euler in the eighteenth century and Gauss at the turn of the nineteenth century (Yan and Shiran 1987, p. 166). This category of problems is accessible to middle school mathematicians and addresses the topics of factors, multiples, and relatively prime numbers. This gem of an historical mathematics problem is at an appropriate level for middle-grades students and provides a rich problem-solving opportunity.
Traveling from Arithmetic to Algebra
Authors: Darley Joy W.
Published: December 2020

A number line can be a useful visual tool to help students progress from arithmetic to algebra in their quest to understand variables.
Families Ask: Algebra in the Middle Schools
Authors: Coates Grace Dávila
Published: December 2020

Our middle school is encouraging all students to take algebra in the eighth grade. I took it in ninth grade. Isn't algebra a high school course? My child has done OK in math so far but may not want to take algebra next year. What's the rush?
Problem Solving and the English Language Learner
Authors: Brown Clara Lee , Cady Jo Ann , Mark Taylor P.
Published: December 2020

Various techniques can enhance mathematics instruction for English language learners.
Promoting Efficient Strategy Use
Authors: Townsend Brian E. , Lannin John K. , Barker David D.
Published: December 2020

Helping students determine different strategies to explore algebraic reasoning tasks can help them draw accurate conclusions.
Orchestrating Discussions
Authors: Smith Margaret S. , Hughes Elizabeth K. , Engle Randi A. , Stein Mary Kay
Published: December 2020

Five practices constitute a model for effectively using student responses in whole-class discussions that can potentially make teaching with high-level tasks more manageable for teachers.
Mathematical Explorations: Exploring Unknown Probabilities with Miniature Toy Pigs
Authors: Nelson Cheryl Q. , Williams Nicole L.
Published: December 2020

The typical probability experience for middle-grades students involves flipping coins and rolling dice. In both of these situations, probability can be discussed from experimental and theoretical viewpoints. It is not likely that your students have ever experienced a situation in which the theoretical probability is unknown. The following activities investigate the tossing of a toy pig (see fig. 1), with an unknown probability of each landing position (Whyte 2006). In other words, the theoretical probability of each landing position cannot be determined. Hence, students contend only with finding an experimental probability for each possible outcome.
What Are You Worth?
Authors: Stephan Michelle L.
Published: December 2020

Use concepts from finance, specifically, assets and debts, to give students a real-world understanding of integer concepts and operations.
A Framework for Analyzing Geometric Pattern Tasks
Authors: Friel Susan N. , Markworth Kimberly A.
Published: December 2020

Teachers can use geometric patterns to promote students' understanding of functional relationships.
Using Steffe's Advanced Fraction Schemes
Authors: McCloskey Andrea V. , Norton Anderson H.
Published: December 2020

Recognizing schemes, which are different from strategies, can help teachers understand their students' thinking about fractions.
The Thinking of Students: Window Panes
Authors: Mooney Edward S.
Published: December 2020

The solution methods that students employed to solve this problem involved both symmetry and guess and check. The methods also opened up windows of understanding for them.
So Many Graphs, So Little Time
Authors: Wall Jennifer J. , Benson Christine C.
Published: December 2020

Most graphs fall into five distinct categories. Show students these five categories and help them understand the graphs that they are sure to see in real life.
Provoking Intellectual Need
Authors: Lim Kien H.
Published: December 2020

Tasks that generate a need for mathematical intellect generate motivation and promote success in learning mathematics. Some specific tasks and ways to generate new ones are offered.
Correlating Mathematics and Science
Authors: Mireles Selina Vásquez
Published: December 2020

Learn to connect similar concepts in math and science that have different vocabulary and different meanings to provide a deeper understanding of each subject.
Convincing and Justifying through Reasoning
Authors: Mueller Mary F. , Maher Carolyn A.
Published: December 2020

While building models to address questions about fractions, we explore the types of reasoning that naturally emerge and the environment that motivates students to engage in proof.
Math in the Box
Authors: DeYoung Mary J.
Published: December 2020

Find out how to make an origami paper box and explore the algebra, geometry, and other mathematics that unfolds.
Cultivating an Area Model
Authors: Tillema Erik S.
Published: December 2020

A rich understanding of multiplying binomials can be developed by working through a series of concrete problems.
Capitalizing on the Unexpected
Authors: Taber Susan B.
Published: December 2020

When students use manipulatives, teachers can take advantage of their surprising solutions to engage the class in rich and productive mathematical discussions.
Math Roots: Can a Sixth Grader Do Trig
Authors: Vajiac Bogdan
Published: December 2020

Can sixth graders learn trigonometry? Although this material is taught at the high school and college levels, most sixth graders are smart enough to learn it. In fact, South Korean, Romanian, and American schools abroad introduce basic trigonometry in middle school. U.S. schools can do it, too. With the right preparation, any student can learn a bit of trigonometry; a simple curiosity is enough to get started.
Take Time for Action: How Students Interpret Graphs
Authors: You Zhixia
Published: December 2020

How do your students perform when you ask them to interpret graphs that are set in a real-life context? Do they interpret a graph as a literal picture (an iconic interpretation) of a situation? Or do they use a symbolic interpretation?
Laying Siege to Parabolic Understanding
Authors: Rick John W.
Published: December 2020

An attack on a storm a wall of domino castle using a trebuchet is one way to help students understanding while learning about medieval mathematics.
The Power of Incorrect Answers
Authors: Hoffman Brittany L. , Lynn Breyfogle M. , Dressler Jason A.
Published: December 2020

To bolster students' ability to prove as well as develop mathematical argumentation skills, create an environment in which students must regularly explain and justify their thinking.
Take Time for Action: Connecting Multiple Intelligences with Popcorn
Authors: Leinenbach Marylin T. , Raymond Anne M.
Published: December 2020

Students often have to practice mathematics skills that may seem disconnected from one another and from the students' own lives. How do you help students form meaningful connections?
Quick Reads: Another Good Idea: Active Involvement for Clear Assessment
Authors: Wasserman Mitchell
Published: December 2020

Math teachers want to know if students get it when a new mathematics concept has been taught. Because we are not our students' mind readers, it is difficult to know who does and does not understand. Students often will not admit that they do not understand and are embarrassed to raise their hands when we ask, “Are there any questions?”
Solve It!: Macedonian Flag
Authors: Mooney Edward S.
Published: December 2020

The Republic of Macedonia is a European country bordered by Serbia, Albania, Bulgaria, and Greece. Its flag consists of a yellow sun projecting eight yellow rays of light on a red background. The length of the flag is twice as long as the width. Approximately what portion of the flag is yellow? (A black-and-white working version of the flag is provided below.)
The Area Model of Multiplication of Fractions
Authors: Tsankova Jenny K. , Pjanic Karmen
Published: December 2020

The algorithm for multiplying proper fractions is often taught by asking students to notice patterns when finding part of a fractional part. An area-model approach will extend students' understanding.
From Inductive Reasoning to Proof
Authors: Yopp David A.
Published: December 2020

When students learn strategies for identifying key ideas in inductive arguments, these ideas can be extended to provide more formal proofs.
The Thinking of Students: Creative “Solve It!” Solutions: Survey Says
Authors: Mooney Edward S.
Published: December 2020

Students bought into the mathematics they found in this problem that kept on giving.
Take Time for Action: Step Up to Nonlinear Functions
Authors: Morge Shelby P. , Pusey Eleanor L.
Published: December 2020

Do your students think that all functions are linear? Do they have experience working with nonlinear functions? If so, how do they think about those functions?
100 Students
Authors: Riskowski Jody L. , Olbricht Gayla , Wilson Jennifer
Published: December 2020

After constructing and conducting a survey, students explored concepts of statistics, considered distribution of cultural groups, and produced a multimedia summary of their work.
Dividing Fractions and Problem Solving
Authors: Cramer Kathleen , Monson Debra , Whitney Stephanie , Leavitt Seth , Wyberg Terry
Published: December 2020

See how a class of sixth graders used concrete and pictorial models to build meaning for arithmetic operations with fractions.
Calculus in the Middle School?
Authors: Barger Rita H. , McCoy Ann C.
Published: December 2020

The path of a thrown ball is one of many examples that can connect middle school math to calculus. Showing this calculus connection early may motivate students to undertake increasingly advanced high school and college math.
Contemporary Curriculum Issues: Transitions from Middle School to High School: Crossing the Bridge
Authors: Brown Lisa C. , Seeley Cathy L.
Published: December 2020

As students transition from middle school to high school, the academic landscape changes. This transition is complicated by a number of challenges—some general and some specific to mathematics—faced by students, educators, and families.
Mathematical Exploration: Tennis Balls, Lines, and Geometric Transformations
Authors: Taylor Sharon E. , Mittag Kathleen Cage
Published: December 2020

Activities that use linear data as a model will help students understand linear functions. In this activity, students collect data and make connections to concepts used in geometry and measurement. Rather than teaching introductory algebra and geometry, this activity is intended to serve as a culminating task to reinforce material already learned.
Math for Real: A Speck of Dust
Authors: Tillema Erik S.
Published: December 2020

Think of any dusty surface. How did the dust get there? What is in dust? How many specks of dust can be found in one swipe of a dust cloth?
Take Time for Action: The Percent Predicament
Authors: Olson Melfried , Zenigami Fay , Slovin Hannah
Published: December 2020

Have you ever thought that a student grasped the ideas you were hoping they would learn only to find out that you were mistaken?
Quick Reads: Another Good Idea: Poematics: Exploring Math through Poetry
Authors: Ayebo Abraham , Wiest Lynda R.
Published: December 2020

Abraham Ayebo and Lynda R. Wiest One important key to a student's success in math is being able to communicate mathematically. To do so, students need to develop a language for expressing mathematical ideas (NCTM 2000).
Shrinking Your Class
Authors: Herron-Thorpe Farren L. , Olson Jo Clay , Davis Denny
Published: December 2020

Engineering principles and fun materials provide a context to help students develop their understanding of fractions, measurement, and scale factors.
The Value of Guess and Check
Authors: Guerrero Shannon M.
Published: December 2020

Providing students a structure for employing the guess-and-check method for solving word problems leads to an understanding of quantitative relationships and builds a bridge to algebra.
Assessing Understanding through Reasoning Books
Authors: Roberts Sally K. , Tayeh Carla
Published: December 2020

My Mathematical Reasoning Book can help students communicate their thought processes to a larger audience. To work effectively, a plan of action requires focused and deliberate instruction.
Map Scale, Proportion, and Google Earth
Authors: Roberge Martin C. , Cooper Linda L.
Published: December 2020

Using images from GoogleTM Earth, students cross both mathematical and geographic boundaries while learning about map scale and proportional reasoning.
Making Technology Work
Authors: Soucie Tanja , Radović Nikol , Svedrec Renata
Published: December 2020

Although technology can open the door to better teaching and learning, sometimes it is not the most effective tool.
Graphing Calculators as Tools
Authors: Browning Christine A. , Garza-Kling Gina
Published: December 2020

The many uses of handheld technology for learning mathematics inherently involve specialized knowledge that is framed as TPACK: technology, pedagogy, and content knowledge.
Quick Reads: Another Good Idea: From Geometric Shapes to Algebraic Symbols
Authors: Vinogradova Natalya
Published: December 2020

Connections between algebra and geometry are so explicit that it is difficult to firmly separate these two branches of mathematics. Geometric representations are often used to help students visualize algebraic expressions.
Take Time for Action: Comparing the Volumes of Rectangular Prisms
Authors: Assuah Charles K. , Wiest Lynda R.
Published: December 2020

Can middle-grades students determine which of two rectangular prisms has a larger volume? Can they do so without using a formula? Geometry, and particularly the concept of volume, is important in many subjects, such as physics and chemistry. Carpenter and Romberg (2004) note that students greatly enhance their mathematics knowledge when they make generalizations and construct arguments to justify their methods through problem solving. To realize this goal, teachers should encourage and accept students' self-initiative, allow students' responses to help shape lessons, and inquire about students' understanding of concepts before sharing their own (Brooks and Brooks 1993). We describe a task that requires students to use their own nonformulaic methods and reasoning to investigate volume.
Contemporary Curriculum Issues: Developing Mathematical Habits of Mind
Authors: Mark June , Cuoco Al , Goldenberg E. Paul , Sword Sarah
Published: December 2020

Mathematical habits of mind include reasoning by continuity, looking at extreme cases, performing thought experiments, and using abstraction that mathematicians use in their work (Cuoco, Goldenberg, and Mark 1996; Goldenberg 1996). Current recommendations emphasize the critical nature of developing these habits of mind: “Once this kind of thinking is established, students can apply it in the context of geometry, trigonometry, calculus, data and statistics, or other advanced courses” (Achieve 2008, p. 4).
The Intersection of Language and Mathematics
Authors: Swanson Patricia E.
Published: December 2020

Use both context and language to teach integer concepts in academically and linguistically diverse classrooms.
Using Tables to Bridge Arithmetic and Algebra
Authors: Brown Susan A. , Mehilos Megan
Published: December 2020

The TABLE feature on a graphing calculator is a valuable tool that students can use as they move from specific numbers and calculations to a focus on variables, patterns, and relationships.
Mathematical Exploration: Mean, Median, or Mode: Which One Is My Pencil?
Authors: Mittag Kathleen Cage , Taylor Sharon E. , Fies David
Published: December 2020

Methods for finding measures of central tendency are usually taught in middle school. Determination of mean, median, and mode are often presented as rote processes. In addition, graphical representations of data, such as stem-and-leaf and box-and-whisker plots, are also introduced. However, in many instances, no conceptual meanings are associated with the algorithms or graphical methods.
Fractions: A New Slant on Slope
Authors: Cheng Ivan
Published: December 2020

By using lines to represent fractions, algebra students are able to gain a new perspective on slope and find a new way to perform operations on fractions.