bluuebunny/crossref_metadata_2025 huggingface dataset:
Using Tiered Lessons in Mathematics
Authors: Pierce Rebecca L. , Adams Cheryll M.
Published: December 2020

The movement toward inclusion has impacted classrooms by requiring teachers to respond to a broader range of academic needs. How can we possibly reach all the students in our classrooms when they are academically diverse, have special needs, and are ESL learners? One answer to this question is to differentiate instruction.
Let's Do Math: Wanna Play?
Authors: Lach Tisa M. , Sakshaug Lynae E.
Published: December 2020

Several articles have extolled the virtue of using games in mathematics class to explore various topics or skills. This article describes how an action research study was implemented to learn how students interacted with games in three important areas of mathematics.
Teacher to Teacher: Figural and Numerical Modes of Generalizing in Algebra
Authors: Rivera Ferdinand D. , Becker Joanne
Published: December 2020

Induction Plays A Central Role In Performing generalization and abstraction, two important processes that are necessary and highly valued in all areas of mathematics (Kaput 1999; Mason 1996; Romberg and Kaput 1999; Schoenfeld and Arcavi 1988). From 2000 to 2004, at least 30,000 eighth-grade students in northern California were tested on algebra tasks that asked them to construct linear patterns of the form y = mx + b. The students were expected to generalize using explicitly defined functions, including selecting, converting flexibly among, and employing various representations for, the patterns. Five years of data collection and analysis of students' work have shown that only three-fourths of the eighth graders tested could successfully deal with particular cases of linear patterns in visual and tabular form, and that less than one-fifth could use algebra to express correct relationships or to generalize to an explicit, closed formula (Becker and Rivera 2004).
The Thinking of Students: What's the Area?
Authors: Kersaint Gladis
Published: December 2020

The following problem appeared in the December 2004/January 2005 issue of this journal: Square ABCD has the centers of four circles of radius 5 cm as its vertices. Find the area of the shaded region. Ten teachers submitted the work of their students, illustrating their methods for solving this problem. Kathy Hawes, an eighth-grade mathematics teacher at Graham Middle School in Mountain View, California, wrote that she found this problem “particularly interesting, because it featured geometry and students often have little chance to work on geometry challenges in today's climate of algebra for all.”
How Does Your Doughnut Measure Up?
Authors: Maida Paula , Maida Michael
Published: December 2020

An activity in measurement and geometry in which sixth-grade students explore the volume and surface area of a donut. Through calculation, creative thinking, and discovery, the students enhanced their geometric vocabulary and reasoning. This articles includes student work, a lesson plan, and samples of activity sheets.
Solve It!: Cookies
Authors: Kersaint Gladis
Published: December 2020

The editorial panel is presenting the following problem to stimulate submissions to “The Thinking of Students” department. We encourage teachers to try this problem with students and analyze the different ways that they use to solve it. Feel free to photocopy this problem for use in your classroom.
Families Ask: Some Students Do Not Like Mathematics
Authors: Martinie Sherri
Published: December 2020

Children who claim not to like mathematics are often the same ones who demonstrate little persistence and motivation. They are sometimes unwilling to try new problems and will often respond to questions with an “I don't know.” Even one student who professes not to like mathematics is too many. Sometimes this attitude can be contagious. The feeling of inadequacy that accompanies this attitude can lead to disruptive behavior that interferes with the learning of all students. Something must be done to improve this attitude, but what?
Mathematics Detective: How to Box a Greeting
Authors: Plymate Lynda
Published: December 2020

Greeting cards are sent and received for a variety of occasions. They can also be recycled in many ways and for a mathematical use, as well. This activity is centered on turning a greeting card into a gift box. In addition to exploring some mathematics, you will also produce a nice box that can be used to store a gift or a personal treasure.
Focus of the Year: What Do We Know about Eighth-Grade Achievement?
Authors: Sowder Judith , Wearne Diana
Published: December 2020

The national assessment of Educational Progress (NAEP) is used by the federal government and by states to gauge achievement in several subject areas, including mathematics. The results of the NAEP tests in mathematics at the eighth grade are used here to help us explore students' mathematics achievement over the decade from 1990 to 2000. In particular, we use these data to counteract the media portrayal of students' achievement in mathematics as steadily declining.
The Thinking of Students: Ticket Sales
Authors: Kersaint Gladis
Published: December 2020

The following problem appeared in the october 2004 issue of this journal: A college sold tickets to a play at $4 per ticket. Anyone who attended and purchased a ticket at the door had to pay $5 a ticket. A total of 480 people attended the play, and the revenue from the ticket sales was $2100.
Discovering Euler Circuits and Paths through a Culturally Relevant Lesson
Authors: Robichaux Rebecca R. , Rodrigue Paulette R.
Published: December 2020

Deep in the heart of southern Louisiana's Cajun country, middle school students enter a mathematics classroom. They hear Cajun music playing in the background and observe cattails and cypress knees “growing” along the bottom of the classroom walls. Foam crawfish lie on plastic bayous on the tables around the room. The room is filled with excitement for what is about to occur. These students begin to question each other, and ask: “Is this math?” “Perhaps we're going to skip math and do social studies?” “Maybe we are going to learn how to do a Cajun line dance?”
One, Some, or None: Finding Beauty in Ambiguity
Authors: Berkman Robert M.
Published: December 2020

This is a scene from a fifth-grade mathematics class: Students are in pairs, chatting eagerly; some are making sketches, others are bending straws and rearranging them on the desk, and still others are scribbling a variety of words. After two minutes, I hit the bell and say, “Okay contestants, time to vote. Hands behind your backs!” The room is silent as arms pull away from the table. “Okay, choose!” I call out, and hands stretch forward. Some students curve their fingers to make an “O”; others point a thumb upward.
March Math Madness: The Mathematics of the NCAA Basketball Tournament
Authors: Colgan Mark D.
Published: December 2020

Each year, I look forward to two exciting events during March: the NCAA basketball tournament and spring break. I find that students really enjoy exploring the mathematics involved in the NCAA basketball tournament, and these activities offer a fun way to either begin or review a unit on probability and statistics. Engaging students in solving problems related to their interests can also motivate them before or after spring break.
Algebraic Thinking and Geometry
Authors: Grandau Laura , Stephens Ana C.
Published: December 2020

Research on the learning and teaching of algebra has recently been identified as a priority by members of the mathematics education research community (e.g., Ball 2003; Carpenter and Levi 2000; Kaput 1998; Olive, Izsak, and Blanton 2002). Rather than view algebra as an isolated course of study to be completed in the eighth or ninth grade, these researchers advocate the reconceptualization of algebra as a strand that weaves throughout other areas of mathematics in the K–12 curriculum.
A Science Application of Area and Ratio Concepts
Authors: Horak Virginia M.
Published: December 2020

“CAN THERE BE MONSTER INSECTS?” “Are 'human' giants of fables and fiction possible?” “Why do I sink into fresh snow and a rabbit does not?” “Why are my footprints in wet sand deeper than those of a seagull?” Middle school students can readily investigate these questions when mathematics and science are integrated. This article will focus on the last two questions, which provide rich links among animal characteristics, area of footprints, and the use of ratios in proportional reasoning.
Lettuce Learn Math: Teaching Mathematics with Seeds and Centimeters
Authors: Rickard Laura N. , Wilson Colette
Published: December 2020

To the reader contemplating the relationship between lettuce cultivation and mathematical concepts, visualize marking off a rectangle that is 32 feet long and 12 feet wide. Imagine driving 14 pieces of rebar (metal concrete reinforcing bar) into the ground at even intervals on the long sides of the rectangle.
Body Data
Authors: Bombaugh Ruth , Jefferys Lynn
Published: December 2020

Perhaps one of the strongest connections between the national standards in mathematics and science education is their shared emphasis in all these areas: students collecting, organizing, and describing data; constructing appropriate charts and graphs to summarize the data; making predictions based on the data; and testing these predictions and conclusions (NCTM 1989, 2000; NRC 1996).
Integrating Curriculum through Themes
Authors: Horton Robert M. , Hedetniemi Traci , Wiegert Elaine , Wagner John R.
Published: December 2020

Integrating mathematics, science, language arts, and social studies within the middle school curriculum can be an important and worthwhile endeavor. With integration, students realize that, at least in the real world, disciplines do not exist in perfect isolation and that the separations so often seen in school are arbitrary and, at times, unnecessary. Although any one of these disciplines can be the center of the integration, mathematics may be the most natural choice, especially when we focus on mathematical models, descriptions of real-world phenomena through mathematics. The Connections strand of Principles and Standards for School Mathematics states that students across grade levels should be able to “recognize and apply mathematics in contexts outside of mathematics” (NCTM 2000, p. 64). Students can naturally make connections when the mathematics they are learning is presented through problems emanating from other disciplines, particularly in science. In turn, students may grasp underlying concepts of the other disciplines better when they view them through a mathematical lens.
Teacher to Teacher: How Do Students Think?
Authors: Cohen Robin
Published: December 2020

Mr. Adams and Mr. Brooks played a game of one-on-one basketball. Mr. Adams won by 15 points. In one quarter they scored the same number of points, in another quarter Mr. Brooks outscored Mr. Adams by 10 points, and in another quarter Mr. Adams outscored Mr. Brooks by 5 points. Fill in this chart to indicate how many points were scored by each player.
Science and Mathematics in Balance
Authors: House Peggy
Published: December 2020

Have you ever attached a pair OF cardboard “feet” to an inflated balloon and tossed the balloon into the air? The balloon always lands on its “feet.” Why? The answer involves the scientific concepts of gravity and center of gravity as well as several important geometric principles. Many children's toys, based on these principles, can be the vehicle for valuable investigations linking science and mathematics.
Implementing Reform Practices in a Middle School Classroom
Authors: Cady JoAnn
Published: December 2020

As a Middle School Instructor, I categorized myself as a teacher who followed the NCTM's recommendations for teaching mathematics in the middle school (NCTM 2000). However, after entering a doctoral program in mathematics education, I realized that there was more to implementing the Standards in my classroom than using manipulatives and asking “why” questions. My research focused on preservice teachers' implementation of practices recommended by NCTM during their clinical experiences and as first-year teachers. These novice teachers were overwhelmed when placed in their own classrooms and had difficulty implementing Standards-based practices during their first year of teaching. Limited planning time, limited class time to engage in discussion, traditional textbooks, and assessment pressures were cited as constraints that prevented them from implementing Standards-based practices (Cady 2002).
Introduction to Volume 12
Authors: Lawrence Ann
Published: December 2020

We are excited to introduce volume 12 of our journal in this, the 100th anniversary of NCTM's first journal, the Mathematics Teacher. As we celebrate, our goal, as always, is to enhance the dialogue among the community of mathematics educators interested in the middle grades. We would like to challenge our readers to support your journal this school year through the following opportunities.
What's on Your Radar Screen? Distance-Rate-Time Problems from NASA
Authors: Condon Gregory W. , Landesman Miriam F. , Calasanz-Kaiser Agnes
Published: December 2020

Imagine that you are an air traffic controller on a busy evening, responsible for directing 100 planes per hour heading to a major metropolitan area. (see fig. 1.) your job is to keep all these planes safely separated and on time to their destinations. To do this, you must quickly and accurately solve a multitude of distancerate- time problems. This air traffic control scenario provides an exciting, challenging, and meaningful context for prealgebra and elementary algebra students to “model and solve contextualized problems using various representations” (nCTm 2000, p. 222) that address diverse learning styles and preferences as well as different levels of mathematics preparedness.
Understanding the Properties of Arithmetic: A Prerequisite for Success in Algebra
Authors: Tent Margaret W.
Published: December 2020

As early as fourth grade, we begin introducing students to the properties of arithmetic. The commutative property of addition and the commutative property of multiplication allow us to change the order of the numbers without changing the value of the expression. The associative property of addition and the associative property of multiplication state that regrouping numbers when adding and/or multiplying does not change the solution. The distributive property of multiplication over addition allows us either to find the addend of the numbers, then multiply by the factor, or to first multiply the factor by each addend, then find the sum. Finally, the identity properties of addition and multiplication state that the value of any real number remains unchanged if zero is added to it or if it is multiplied by one, respectively.
Let the 100th Anniversary Begin
Authors: Tunis Harry
Published: December 2020

This academic year marks the 100th volume of the Mathematics Teacher (MT), the NCTM's publication for secondary teachers and teacher educators. We will celebrate this centennial anniversary in myriad ways, including retrospective features in each school journal. On this special occasion, let's begin by looking back to see how Mathematics Teaching in the Middle School (MTMS) evolved from this first school journal.
The Revolution in Arithmetic
Authors: Brownell William A.
Published: December 2020

During the last half-century major changes have occurred in our conception of arithmetic as a school subject. These changes have resulted from both the study of arithmetic itself and from influences from movements and developments outside the subject matter; and they have affected both the content of arithmetic and the methodology of presenting that content to children.
Math Roots: Understanding Aztec and Mayan Numeration Systems
Authors: Walmsley Angela L.E.
Published: December 2020

Numbers have been recorded in a variety of ways throughout time. For example, the Babylonians used marks pressed in clay; the Egyptians used papyrus and ink brushes to create tally marks; and the Maya introduced a symbol for zero (Billstein, Libeskind, and Lott 2001). All these ancient peoples used numerals, or written symbols, to express what they meant mathematically. They created their own numeration system, which is a collection of uniform symbols and properties to express numbers systematically. The Hindu- Arabic system is one such numeration method; however, understanding others can reveal to students that our current system finds its roots in what has come before.
On My Mind: A National Mathematics Curriculum for the United States: Two Perspectives
Authors: Borst Robert W. , Rorvig Vickie
Published: December 2020

Should a national curriculum be developed and mandated for K–12 schools in the United States? That is, should a set of learning expectations outlining what mathematics students at each grade level must study and learn be articulated for every school in the nation?
A Worthwhile Mathematical Task for Students and Their Teachers
Authors: Chamberlin Michelle T. , Zawojewski Judith
Published: December 2020

Worthwhile mathematical tas ks not only prompt students to learn mathematics, they also prompt teachers to learn and improve their teaching in their own mathematics classrooms. When teachers use worthwhile tasks, they have to learn “what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge” (NCTM 2000, p. 19).
Relational Understanding and Instrumental Understanding
Authors: Skemp Richard R.
Published: December 2020

In this article, the author defines relational and instrumental understanding. He then explains the impact he feels these two disparate goals have on the attitudes and understanding of students. We believe the reader will find his ideas about the teaching and learning of mathematics remarkably contemporary and thought-provoking.
Families Ask: Geometry: More than Just Shapes
Authors: Allen David
Published: December 2020

Think back to the geometry you experienced as an elementary school student. Now recall a problem from high school geometry. Often, geometry tasks at the younger grades are limited to identifying shapes or labeling properties; in high school, students are expected to use abstract reasoning to prove a complex relationship. Instruction in geometry has traditionally been overlooked during middle school, which causes a gap between elementary school experiences and the thought processes required in high school.
Teacher to Teacher: Mathematics and Science: The Beginning of a Beautiful Relationship
Authors: Davis Jon D. , Kent Sally
Published: December 2020

Engaging in meaningful problems outside of school rarely involves only mathematics knowledge. Oftentimes, multiple subject areas are involved when solving problems that professionals routinely encounter outside of school. unfortunately, middle and secondary students often experience subject areas as isolated islands within the ocean of their experiences. Efforts to bring the closely related subject areas of mathematics and science together have been promoted in documents by NCTM (1989, 2000) and the American Association for the Advancement of Science (1989).
Using Engaging Contexts to Introduce Concepts
Authors: Scanlon Regina M.
Published: December 2020

Do you have a favorite way to introduce a concept, lesson, or unit? I have several favorites that I use almost every year. I like them because they grab the students' interest, help them grasp some basic concepts, and can be used as reference points during later instruction or practice.
Measuring Tremendous Trees: Discovery in Action
Authors: Maxwell Sheryl A.
Published: December 2020

Twenty-four middle school students gathered around their teacher, curiously anticipating the upcoming activity. They were enjoying the weather and being outside—a different place to hold their mathematics class. The day before, they experienced a minidiscovery lesson about isosceles right triangles. Today, they were to link this concept to a tree-measuring activity that was designed by a teacher educator at a nearby university.
Mathematics in the History of Civilization
Authors: ALTSHILLER COURT NATHAN
Published: December 2020

Ever wonder if there was a connection between societal events and the making of mathematics? Read on to explore the impact that society has had on mathematics and that mathematics has had on society and then ponder what the future might hold.
How to Buy a Car 101
Authors: Flores Charity A.
Published: December 2020

What seventh-grade student is not interested in learning mathematics without the textbook? Better yet, what student does not want to research ways to spend money? Problem-Based Learning (PBL) is a motivational, effective approach to teaching curriculum standards using technology and a new perspective to teaching.
Locusts for Lunch: Connecting Mathematics, Science, and Literature
Authors: Austin Richard A. , Thompson Denisse R. , Beckmann Charlene E.
Published: December 2020

Locusts for lunch? if you watch reality television, lunching on locusts is tame compared with some of the edibles that participants in Fear Factor, the Amazing Race, Survivor, and other programs are required to eat. Locusts are considered delicacies in many parts of the world.
A Model for Understanding Understanding in Mathematics
Authors: Davis Edward J.
Published: December 2020

Although this article was written almost thirty years ago, its content is relevant to today's conversations about the roles of conceptual understanding and procedural fluency in the teaching of mathematics.
Counting Dots and Measuring Area: Rich Problems from Japan
Authors: Peterson Blake E.
Published: December 2020

In fall 2003, I had the opportunity to conduct some research on the student-teaching process in Japan. During my seven weeks of research at the junior high school affiliated with Ehime University in Matsuyama, Japan, I observed mathematics lessons taught by student teachers as well as many more lessons taught by experienced teachers. The basis for most of these lessons was wonderfully rich mathematics problems. In these lessons, a problem was posed to the students, time was given for them to explore the problem, and then solutions were discussed. Similar problem-based lessons can be found in The Teaching Gap (Stigler and Hiebert 1999) and The Open-Ended Approach: A New Proposal for Teaching Mathematics (Becker and Shimada 1997).
Using Error Analysis to Teach Equation Solving
Authors: Hawes Kathy
Published: December 2020

In many mathematics classrooms, students are responsible for correcting their own homework. Placing a red X next to an incorrect answer is often the extent of homework checking because students often lack the ability to find and correct mistakes in their own work. Without this skill, students lose a valuable opportunity to discuss homework in a meaningful way.
Not Just for Computation: Basic Calculators Can Advance the Process Standards
Authors: Moss Laura J. , Grover Barbara W.
Published: December 2020

The National Council of Teachers of Mathematics (NCTM), the International Society for Technology in education (ISTe), and the conference board of the mathematical Sciences (cbmS) are among the many organizations that have advocated the use of various forms of technology (e.g., computer programs, graphing/scientific/ four-function calculators) to educate students. a major reason for this advocacy is captured well in the National Technology Standards comments on using technology in mathematics learning activities. “Technology can play a role in enhancing mathematical thinking, student and teacher discourse, and higherorder thinking by providing the tools for exploring and discovering mathematics” (ISTE 2000, p. 96).
Cell Phone Coverage Area: Helping Students Achieve in Mathematics
Authors: Zbiek Rose Mary , Reed Shari Ann , Boone Tracy
Published: December 2020

The joys of working with middle school students in a mathematics classroom may be tempered by the challenge of meeting external expectations. However, it is possible to carefully design and implement a lesson that both embodies the spirit of Principles and Standards for School Mathematics (NCTM 2000) and meets the needs set forth by our state assessments. Such a lesson can address a big mathematical idea through related activities that develop skills and concepts, involve reasoning and writing, and help students use mathematics in the world around them.
Using Sorting Networks for Skill Building and Reasoning
Authors: Andre Robert , Wiest Lynda R.
Published: December 2020

“When are we going to do that tarp thing again?” Andre's middle school students often ask. “When you can tell me what it is properly called,” is his response. “We should practice our math on the sorting network,” the students recall. This is a regular request that author Robert Andre often honors because he has found that his students truly enjoy the opportunity to walk about the classroom practicing skills and mental math rather than sit in their chairs working with pencil and paper.
Alternative Uses for Junk Mail: How Environmental Print Supports Mathematical Literacy
Authors: Koellner Karen , Wallace Faith
Published: December 2020

With the recent call for teachers to foster reading across the content areas (Bean 2000; NCTM 2000), mathematics teachers are looking for sources that both support complex mathematical content and are rich in literacy. These sources are often difficult to find, especially for middle-grades teachers. One rich source is found everyday in our mailboxes, stuffed in our front doors, and thrown on our driveways. Environmental print (Harris and Hodges 1995) or realtime text includes junk mail, spam, advertisements, and magazines. These items are in great supply and rich with mathematics content (Wallace, Clark, and Cherry 2005). for example, advertisements often require the calculations of costs, tax, and delivery charges as well as comparisons between buying in bulk and buying individual items. Credit card solicitations require comparisons of variables. In all these sources and in life generally, knowledge of statistics and probability is essential in understanding leading and misleading information.
Families Ask: Showing Your Work: Beyond Following Steps
Authors: Bay-Williams Jennifer M.
Published: December 2020

While talking with parents and other noneducators, I try to be careful when using education jargon; however, that does not always mean that we are communicating about the same thing.
On My Mind: Mathematics through the Eye of the Camera
Authors: Iuzzolino Carol A.
Published: December 2020

Many movies include a scene or two containing mathematics content to achieve various effects, such as to show the importance of education, to emphasize a point the filmmaker wants to clarify, and to add humor. many of us have been dazzled by the complexity of mathematics in A Beautiful Mind and Good Will Hunting. I know the level of mathematics is beyond the middle school years, but the passion for the subject matter might inspire some students. In Rain Man, Dustin hoffman plays the autistic brother of Tom cruise and proves that he can calculate square roots and do complex decimal computations in his head. These scenes hint at the importance of mental-math skills, a concept that we need to emphasize in the middle grades. It also shows that calculations can be done accurately and rapidly, even when the person doing the calculation has no understanding of what the numbers mean. middle school students are used to a world of television and video games and learn a lot from these visual aids. So why not use films to help us teach concepts or make students smile about mathematics?
The Algebra of the Arches
Authors: Buerman Margaret
Published: December 2020

Several years ago, i read an item in Mathematics Teaching in the Middle School from an author who used information from her summer vacation to design lessons for a middle school mathematics class (fisher 1999). on one summer vacation while hiking through arches national park in moab, utah, i thought that surely there was mathematics in the science and geology of the sandstone rock formations around me. The park office and souvenir store sold books containing information that would help me discover some of the mathematics of the arches (Baars 1993; Johnson 1985).While reading these books and having taught middle school mathematics for many years, it occurred to me that lessons could be written about large numbers. The sandstone cliffs, from which the arches were formed, date back 140 million years; in addition, they are on the colorado plateau, which dates back 200 million years. The erosion that formed currentday arches began 2 million years ago. a lesson on geometry and measurement could also be written, since the area was part of the paradox Basin, a 10,000 squaremile depression. however, could some relevant algebra be found in the data about the history and geology of arches national park?
The Future of Fractions
Authors: Usiskin Zalman P.
Published: December 2020

In the 1970s, the movement to the metric system (which has still not completely occurred in the United States) and the advent of hand-held calculators led some to speculate that decimal representation of numbers would render fractions obsolete. This provocative proposition stimulated Zalman Usiskin to write “The Future of Fractions” in 1979. He challenged this opinion by illustrating how it was based on flawed assumptions and erroneous reasoning. Some twenty-eight years later, Usiskin follows up on his fraction remarks in “Some Thoughts about Fractions.” In this work, he bolsters his previous arguments concerning the welfare of fractions and asserts that their study will remain an important part of school mathematics.
Short Takes: Another Good Idea: More Than Meets the Eye: An Interdisciplinary Exploration
Authors: Wallace Edward C.
Published: December 2020

A CENTRAL FOCUS OF MANY DATA ANALYsis units at the middle school level is the use of graphical displays to discover facts that are not obvious from the raw data alone. Trends and relationships that are hidden in the numerical displays are often quite obvious if the appropriate graph is constructed.
Probability Games from Diverse Cultures
Authors: McCoy Leah P. , Buckner Stefanie , Munley Jessica
Published: December 2020

To make mathematics relevant and meaningful for all students, it is important that we embrace a wide variety of real-world applications. Diverse cultures provide rich and interesting contexts in which students can experience and explore mathematics. One of the five Process Standards is Connections (NCTM 2000). The probability activities discussed here help students make connections across mathematics concepts through games from diverse cultures.
Teacher to Teacher: Solving Quadratic Equations by Completing Squares
Authors: Vinogradova Natalya
Published: December 2020

Should we teach quadratic equations in the middle school? Many teachers might regard this as a difficult task. However, I have found that using a hands-on visual model can make quadratic equations accessible to middle school students. The question is not when we can teach a topic but how to teach a topic to students with a particular level of mathematics knowledge.
By Way of Introduction: Mathematics and the Arts
Authors: Lawrence Ann , Yamagata Candace
Published: December 2020

We have dedicated this focus issue of MTMS TO sharing ways that middle school teachers of mathematics are integrating the arts in their classes. In the broadest sense, this integration can help students build a fundamental understanding of and appreciation for both mathematics and the arts.
Teaching Mathematics through the Art of Kolam
Authors: Chenulu Syamala
Published: December 2020

One goal of the NCTM's connections Standard is that mathematics instruction pre- K–12 should “enable all students to recognize and apply mathematics in contexts outside of mathematics” (NCTM 2000, p. 64). Art of all kinds provides opportunities to address this goal. Moreover, many mathematics educators, including myself, believe that it is important and beneficial to provide a multicultural perspective in our classrooms. “Knowledge of the ideas of others can enlarge our view of what is mathematical and, in particular, add a more humanistic and global perspective to the history of mathematics. This enlarged view, in which mathematical ideas are seen to play a vital role in diverse human endeavors, provides us with a richer and fuller picture of mathematics” (Ascher 2002, p. 200).
Weaving Plaids Based on (a ± b)2
Authors: Bolte Linda A.
Published: December 2020

The mathematical basis of weaving designs can be explored by middle-grades students from a variety of traditional perspectives—symmetry, reflections, number patterns, and ratios. However, one unique interaction between weaving pleasing designs and mathematics that was explored in the 1940s and 1950s by Ada K. Dietz offers students a different perspective on the interactions between mathematics and the art of weaving.
Just Five Does It: Using Five Numbers to Make Patterned Squares
Authors: Kenney Margaret J. , Bezuszka Stanley J.
Published: December 2020

All students can succeed in creating mathematical art. This article offers some ideas about creating mathematical art based on number patterns. It includes a process that integrates number and geometry and results in attractive and useful patterned squares. Students may perform the activity with paper and pencil or use computer software like The Geometer's Sketchpad. The diagrams in this article were completed using Sketchpad. The article also features the use of conjecturing as an important element in mathematics instruction.
Menu of Problems: April 2007
Authors: Rock David
Published: December 2020

A man was born on February 29, 1928. He died on January 29, 2004. How many actual birthdays did he get to celebrate during his life (not counting the day he was born as a birthday)? Hint: Don't forget about leap years.
Connecting the Mobiles of Alexander Calder to Linear Equations
Authors: Hedin David S.
Published: December 2020

“That thing is tight!” students did not know the word for it, but those learning English were attracted by the mobile that I carried through the halls of our southern California middle school. They asked me, “You teach math, don't you?” The mobile's arms carved the air as the moving sculpture jiggled under my hand, captivating my students.
Short Takes: Promising Partnerships: The Mathematics of Radial Designs
Authors: Wilkes Gretchen
Published: December 2020

Mathematics and the arts are often viewed as having little to do with each other, when in fact the two are considered universal languages and share many of the same structures and patterns. Students' interest can often be stimulated when they are able to perceive this interconnection, and they begin to view art and mathematics through a different lens.
Master of Tessellations: M. C. Escher, 1898–1972
Authors: Ranucci Ernest R.
Published: December 2020

This article gives a brief account of the contributions to geometry of a well-known artist.
On My Mind: University Faculty—Making Classroom Connections
Authors: Thompson Denisse R. , Kersaint Gladis
Published: December 2020

University teacher educators typically wear many hats. Their many roles may include providing professional development (i.e., workshops) for practicing teachers on various topics (e.g., use of technology, teaching English Language Learners); teaching mathematics or methods courses to teachers or teacher candidates enrolled in undergraduate or graduate teacher education programs; and working with various groups or entities related to policy issues in mathematics education (e.g., teacher licensure, high-stakes assessment). Although mathematics teacher educators contribute to the field in various ways, a perception exists that university faculty are in an “ivory tower,” having few or ancient connections to schools. In some cases, their credibility may be questioned because of the time that has elapsed since they were in a classroom full time.
Measurement and Fair-Sharing Models for Dividing Fractions
Authors: Gregg Jeff , Gregg Diana Underwood
Published: December 2020

Van De Walle (2007) describes dividing one fraction by another in this way: “Invert the divisor and multiply is probably one of the most mysterious rules in elementary mathematics” (p. 326). Tirosh (2000) concurs and cites research suggesting that “division of fractions is often considered the most mechanical and least understood topic in elementary school” (p. 6) and that students' performance on tasks involving division of fractions is typically very poor. These claims are reflected in the difficulties that college students experience in courses for mathematics for elementary teachers when they try to explain why the invert-andmultiply algorithm works. See the following problem.
Short Takes: Another Good Idea: Pasta Symmetry
Authors: Windle Tara
Published: December 2020

Students enjoy the chance to be creative, especially those in the middle grades. Teachers can channel that creative energy into an authentic assessment tool that students will love. Principles and Standards for School Mathematics states that students in middle school are expected to “apply transformations and use symmetry to analyze mathematical situations” (p. 232). Our students have also been challenged to “recognize and apply mathematics in contexts outside of mathematics” (p. 274) and to “create and use representations to organize, record, and communicate mathematical ideas” (p. 280). Using card-stock paper, glue, gold spray paint (optional), and as many varieties of pasta as I could find, I gave my sixthgrade middle school students the opportunity to convince me that they understood the concepts of reflectional and/or rotational symmetry while creating a unique piece of art.
Be Resolute about Absolute Value
Authors: Kidd Margaret L.
Published: December 2020

Conceptualization of absolute value. Article discusses reasons for absolute value being a distance from 0. Includes number line representations of absolute value.
Introducing Volume 13: A New Look
Authors: Wilkerson Trena
Published: December 2020

Mathematics Teaching in the Middle School has a new look! What an exciting way to start the new year! Whether you are a student, a teacher, or an administrator, we all get excited about beginning a new school year. Students get a new book bag, fun school supplies, and perhaps new clothes and new hairstyles. Teachers and administrators have worked all summer preparing new lessons, securing and organizing resources, and planning for the new school year. Everyone at NCTM working with MTMS has been busy, as well. You will find updated fonts, new layout styles, additional focused departments, and more ways to address the teaching and learning of mathematics in the classroom.
Masterpieces to Mathematics: Using Art to Teach Fraction, Decimal, and Percent Equivalents
Authors: Scaptura Christopher , Suh Jennifer , Mahaffey Greg
Published: December 2020

A way to introduce fraction, decimal, and percent equivalents by linking student-created Op Art with examples of twentieth-century abstract painting. Article discusses differentiating instruction according to student ability level. Includes examples of student work and worksheets.
Solve It! A Trillion Dollar Classroom
Authors: Mooney Edward S.
Published: December 2020

According to a September 2006 report by Forbes magazine, the combined wealth of America's 400 richest people is $1.25 trillion. According to the Bureau of Engraving and Printing, a dollar bill is 2.61 inches wide and 6.14 inches long. It is also 0.0043 inches thick.
A Context for Integer Computation
Authors: Gregg Jeff , Gregg Diana Underwood
Published: December 2020

The National Research Council (2001) notes that a variety of metaphors have been used to introduce negative numbers in school, “including elevators, thermometers, debts and assets, losses and gains, hot air balloons, postman stories, pebbles in a bag, and directed arrows on a number line” (p. 245). In addition, standard textbooks have typically employed either a chip model, where positive integers are represented by black chips and negative integers by red chips; a charged-field model, showing +/−; or a number-line model. Some contexts or models are not really metaphors because, for example, negative numbers do occur in temperatures, in accounting, and in some sports scoring (e.g., scores under par in golf). However, these contexts become metaphorical when they are used to model arithmetical operations with integers. Each model or context must construct a scenario that makes it plausible to explain, for instance, why subtracting an integer is the same as adding its opposite or why a negative times a negative is a positive.
How Does Your Mathematical Garden Grow?
Authors: Beck Shari A. , Huse Vanessa E. , Reed Brenda R.
Published: December 2020

Imagine a middle school mathematics classroom where students are actively engaged in a real-life application problem incorporating multiple Process and Content Standards as outlined by NCTM (2000). Sounds of mathematical communication arise as students use multiple representations to help connect mathematical ideas throughout the problem-solving process. Students apply various types of reasoning and explore alternate methods of proof while working attentively on applications that incorporate Number and Operations, Algebra, Geometry, and Measurement.
Implementation of the NCTM Curriculum Focal Points: Concept versus Content
Authors: Schielack Jane F. , Seeley Cathy
Published: December 2020

On September 12, 2006, the NCTM released a new lens through which to view the content presented in Principles and Standards for School Mathematics (NCTM 2000). In Principles and Standards, the Curriculum Principle calls for a mathematics curriculum to be coherent, focused on important mathematics, and well articulated across the grades. This means that an effective curriculum “gives teachers guidance regarding important ideas or major themes, which receive special attention at different points in time. It also gives guidance about the depth of study warranted at particular times and when closure is expected for particular skills or concepts” (NCTM 2000, p. 16). With the publication of Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, NCTM has taken a major step toward defining the mathematics that every student should know and be able to do. Where the Principles and Standards presented content across multiple grade bands, this new publication applies the Curriculum Principle to create a specific example of a focused and coherent curriculum, prekindergarten through grade 8.
C2 = Creative Coordinates
Authors: McHugh Shelley R.
Published: December 2020

When students are allowed to use art to show understanding of a mathematics concept, they become excited about mathematics because they have often found more success in art class. “C2 = Creative Coordinates” is the title of a project that combines both art and mathematics. The project lets students illustrate or find an image, then plot points to map their design on a grid.
Lattice Multiplication in a Preservice Classroom
Authors: Nugent Patricia M.
Published: December 2020

Maurer defines an algorithm as being “a precise, systematic method for solving a class of problems” (1998, p. 21). This article discusses lattice multiplication, which is an algorithm for multiplication, and my preservice students' conceptual understanding of it. The conceptualization of the algorithm from the multiplication of whole numbers to the multiplication of polynomials is also explored.
Take Time for Action: Student Approaches to Unitizing in Fair-Share Problems
Authors: Lamberg Teruni de Silva
Published: December 2020

How do your students unitize when solving problems that involve fair sharing? Do they visualize the unit, partition it when needed, and distribute the pieces equally? Can your students explain their thinking and keep track of what they are doing? Unitizing involves identifying a reference unit, such as a box of crayons or a candy bar, based on the problem context; breaking the unit into smaller chunks; and regrouping pieces to make the problem easier to solve (Lamon 1996). It is a way of conceptualizing an amount before, during, and after the sharing process (Lamon 1996).
On My Mind: The Finger and the Moon
Authors: Flores Alfinio
Published: December 2020

A Tanzanian proverb reads, “I pointed out to you the moon, and all you saw was the tip of my finger.” This proverb came to mind as I was looking at cases of students working with decimals and the difficulties they were having (Barnett, Goldenstein, and Jackson 1994). It dawned on me that part of the difficulty is that we use the decimal point as a pointer to indicate the place value of the digits around it and that many students focus on the finger (the decimal point) and not the moon (the place value).
Integrating Literature and Mathematics: A Mysterious Connection
Authors: Wilburne Jane Murphy , Napoli Mary
Published: December 2020

The stage was set. A preservice elementary teacher entered Mrs. Burnell's seventh-grade classroom, wearing a raincoat and holding a flashlight, and stole a poster placed on the front board. The middle school students were stunned. What just happened? Several minutes later, four preservice elementary teachers entered the classroom and began acting out a dramatization based on a chapter of Balliett's novel Chasing Vermeer (2004). They selected a scene in which characters in the book use pentominoes to try to solve a mystery, which was a perfect segue into the mathematics lesson that follows.
Connecting Measurement and Architecture: Building an Inflatable
Authors: Gray Elizabeth D. , Tullier-Holly Denise
Published: December 2020

Middle school students need to see connections between mathematics and the real world. However, they often learn mathematics as a set of distinct topics or separate strands, because a majority of the available textbooks tends to present it that way, and teachers tend to follow the textbooks. According to Principles and Standards for School Mathematics (NCTM 2000), our students should be made aware of mathematical connections explicitly so that the manner in which topics are connected is obvious. McClain (1996) suggests that if teachers offer classroom experiences in which students can see connections, then “the vibrant discipline of mathematics actively engages students in their own learning” (p. 682).
The Coat Check Problem: A S(t)imulating Lesson
Authors: Pagni David L.
Published: December 2020

California State University, Fullerton, partnered with local schools to offer a special program to introduce middle school students to a college campus and encourage future enrollment in college preparatory classes. As part of that program, I was asked to present a one-hour mathematics lesson to various groups of students. I decided to teach a lesson on probability because it offers hands-on experience for students and because middle school students should be able to “understand and apply basic concepts of probability” (NCTM 2000, p. 248). Based on previous teaching experience as well as work with over 300 students (10 groups of 30) who visited our campus, I found that middle school students have a good sense of the concept of the probability of an event. Many students have had experience simulating experiments by repeatedly tossing a coin, rolling a number cube, or drawing a chip from a bag. Furthermore, they can write probabilities for events, such as “get a tail”; “roll a 3”; or “draw a green chip” from a bag containing three green chips and two white chips, respectively. With this in mind, I introduced the Coat Check problem to build on that knowledge and introduce middle school students to a more complex form of simulation to estimate probabilities. In this article, I will share my methods and results so that middle school mathematics teachers can replicate the lesson.