bluuebunny/crossref_metadata_2025 huggingface dataset:
Teacher to Teacher: Cardboard Tubes Bring Geometry from Home
Authors: Hayes Nakonia
Published: December 2020

No matter what their ages, students who see mathematics used in the real world-a world they really do see and use every day-begin to appreciate the relationship between mathematics and their lives. Spending a lot of money to see that spark of connection in students' eyes is unnecessary. even with today's glitzy, worldly attractions. For example, the cardboard tube found inside a roll of tissue or paper towels can help students examine and extend their understanding of geometric properties of parallelograms.
Teacher to Teacher: The Mathematics in Your Note Paper
Authors: Camblin Bruce A.
Published: December 2020

A wonderful manipulattve for setting up geometry and fractions rests on almost everyone's desk. I am referring to those square pads of note paper and those sticky notes we all use. Doing some simple paper folding with these square slips of paper can greatly increase a student's understanding of fractions.
Assessing Problem-Solving Thought
Authors: Leitze Annette Ricks , Mau Sue Tinsley
Published: December 2020

Recent reform efforts that are based on the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) call for an increase in problem solving as part of the mathematics curriculum for students at all levels. Teachers can use problemsolving activities for multiple purposes, such as developing critical-thinking skills, data-organization skills, communication skills, and a risk-taking attitude, as well as making connections among mathematical topics. Regardless of the curriculum goal, teachers face many challenges in finding suitable activities and then assessing the work that students do on these activities. Neither task is easy.
Mathematical Connections from Biology: Killer Bees Come to Life in the Classroom
Authors: Rubink William L. , Taube Sylvia R.
Published: December 2020

The middle school mathematics curriculum emphasizes integrated curriculum projects in response to the need to give students opportunities to explore and broaden areas of investigations. These interdisciplinary experiences help students understand the challenges faced by professionals. Although many educators concur that data analysis and statistics taught in the mathematics classroom should use data from real-world situations (NCTM 1989), mathematics teachers often need additional resources, both human and material, and must search beyond their textbooks for exciting activities. One way to breathe more life into mathematics teaching is to bring in ideas from other fields of study, particularly career opportunities that involve collecting and analyzing data. We describe a field-tested interdisciplinary-unit activity that involves collecting data about honeybees. In this activity, middle school students gain hands-on experiences with collecting, transforming, and analyzing data by using actual techniques employed by entomologists, the scientists who study insects. Ultimately, students acquire a sense of the methodology that scientists use to obtain a quantitative “view” of the world, one in which they conceptualize objects as things that can be measured (Thompson 1995).
Reflecting on Students' Understanding of Data
Authors: McClain Kay
Published: December 2020

Ongoing discussions about students' experiences with data analysis include debates about the role of graphical representations in supporting students' understandings (Shaughnessy 1992; Lehrer and Romberg, in press; Hancock, Kaput, and Goldsmith 1992). More specifically, do students first need to know how to construct various types of graphs before they can engage in an analysis of data, or can they learn how to construct various types of graphs by engaging in data analysis? Further, can they engage in data analysis before they have acquired a conceptual understanding of the multiple forms of data representation? These discussions also include a debate over the role of representative values, such as the mean, mode, and median (Mokros and Russell 1995). What are students' understandings of representative values? How should representative values be introduced? What role does students' understanding of representative values play in their ability to analyze sets of data?
The Decorative Ornamental Ironwork of New Orleans: Connections to Geometry and Haiti
Authors: Germain-McCarthy Yvelyne
Published: December 2020

Think about new orleans. images of the wrought-iron balconies and doors of the French Quarter probably come to mind. Wrought iron was first brought to New Orleans from Spain in 1790. During the next twenty years, a number of free, mixed-race Haitians fled the Haitian slave revolts and entered the southern ports of Savannah, Charleston, and New Orleans. The Haitian refugees who came to Louisiana between 1791 and 1809 were better trained and better educated than were the inhabitants of the Louisiana territory, and “their influence insured that the state would have a Creole flair for years to come” (Hunt 1988, 58).
Assessment: So Math Isn't Just Answers
Authors: Brown-Herbst Kari
Published: December 2020

In 1994 the state of alaska began to develop alternative assessments, especially performance assessments. Twelve schools throughout the state were invited to participate in this pilot project. Unalaska City School was one of these schools, and I was one of the teachers. As part of the project, I studied performance assessments—what they were and how to administer and score them. I knew that my own chapter tests did not adequately measure students' mathematical knowledge. By the end of that year, I used performance assessments regularly and had put in place a portfoliolike system for tracking students' learning.
Homogeneous Groups Develop Thoughtful Mathematics
Authors: Bulgar Sylvia A. , Tarlow Lynn D.
Published: December 2020

The results from the Third International Mathematics and Science Study (TIMSS), which tested a half million eighth-grade students in fortyone countries, have recently been publicized. Students in the United States ranked below average in mathematics, whereas students in Singapore earned top scores. Examining how students in Singapore study mathematics should provide useful information to mathematics educators on how to improve the performance of students in the United States. Problem solving is emphasized in Singapore, where students are expected to struggle with problems that have real-life implications.
Take Time for Action: Perimeter or Area? Which Measure Is It?
Authors: Chappell Michaele F. , Thompson Denisse R.
Published: December 2020

During the past twenty years, documents have recommended that the mathematics curriculum include measurement for all grades, K–12 (NCTM 1980, 1989). Indeed, students interact daily with measurement in their physical environment, for example, by finding the distance from home to school, their height and weight, and wall space for posters. Adolescents bring to the classroom varied conceptions of measurement, which may be in the form of basic applications or general formulas. All too often, a fundamental understanding of these ideas is sacrificed while students learn general formulas. This situation is particularly true for attributes of perimeter and area. To what extent do middle school students possess a conceptual understanding of these measurement concepts?
Posing Problems: Two Classroom Examples
Authors: Leung Shukkwan S. , Wu Rui-Xiang
Published: December 2020

IN ONE NCTM YEARBOOK TITLED PROBLEM Solving in School Mathematics, Butts (1980) reminded teachers of the importance of posing problems properly. What happens if we fail to do so and pose a problem that cannot be solved? This embarrassing moment can be made profitable if we ask students to help fix the problem. Here we share two lessons in which students help teachers pose problems. In the first lesson, fifth graders found a mistake in a proportion problem, and the teacher asked students to help her pose it properly. In the second, a geometry problem with insufficient information taken from an eighth-grade mathematics test was posed intentionally to future teachers in a teacher-training college as a problem-posing challenge. We first describe the two lessons and then close with suggestions on having learners help teachers pose problems properly.
Making Mathematics Real: The Boston Math Trail
Authors: Rosenthal Matthew M. , Ampadu Clement K.
Published: December 2020

What exactly is a math trail? Why were groups of students walking around Copley Square in Boston with calculators, notebooks, tape measures, and protractors? What were they measuring and calculating? Were they future engineers? All we can tell you for certain is that the students learned a great deal, they had fun, and they were able to answer for themselves the ageold question “When are we ever going to use this math?”
Socrates and the Three Little Pigs: Connecting Patterns, Counting Trees, and Probability
Authors: Thompson Denisse R. , Austin Richard A.
Published: December 2020

Explorations of concepts of chance should be a part of the middle school curriculum, as indicated in the mathematics curriculum frameworks developed by several states (Florida 1996; South Carolina 1993; New Jersey 1996). The challenge for teachers is to find contexts that interest middle school students and motivate them to explore these ideas.
Using Dragon Curves to Learn about Length and Area
Authors: Smith Lyle R.
Published: December 2020

Art and Mathematics have often been combined to spark the imaginations of middle school students. “Dragon curves,” which are made with three tiles (see fig. 1), can be used to create fascinating patterns and to help students understand the concepts of length and area. Dragon curves have long been used in mathematical recreations. For example, Gale (1995) and Knuth and Knuth (1973) have written about art, games, and other activities that dragon curves can inspire. Dragon curves are generated by algorithms that create fractal images. The three tiles shown in figure 1 can be used to approximate these images.
Take Time for Action: Proportional Reasoning
Authors: Miller Jane Lincoln , Fey James T.
Published: December 2020

Developing facility with proportional reasoning should be “one of the hallmarks of the middle grades mathematics program” (NCTM 1998, 213). Such reasoning has long been a problem for students, however, because of the complexity of thinking that it requires. Several standards-based curriculumreform projects have explored new approaches to developing students' proportional reasoning concepts and skills. Instead of offering direct instruction on standard algorithms for checking equivalence of ratios or solving proportion equations, these new approaches encourage students to build understanding and strategies for proportional reasoning through guided collaborative work on authentic problems.
Zooming In on Slope in Curved Graphs
Authors: Beigie Darin
Published: December 2020

Students in a middle school mathematics club used the zooming technology of Green Globs and Graphing Equations (Dugdale and Kibbey 1996) to study slope in curved graphs. Seventh and eighth graders investigated some elementary curved graphs by zooming in on evenly spaced points along a graph until the graph appeared linear and slope could be calculated. The slopes at the various points were then plotted on a separate grid and joined to make a graph of the slope itself and to discover the algebraic equation describing the new graph. The zooming technology gave the students a concrete, visual context in which to learn about the idea of slope in a curved graph and to study how slope varies along a curved graph.
From Tessellations to Polyhedra: Big Polyhedra
Authors: Peterson Blake E.
Published: December 2020

IN 1994, THE WORLD CUP SOCCER CHAMPIONships were held in various cities and in a variety of stadiums across the United States. Unlike American football, soccer is played almost exclusively on natural grass. The need for grass presented a problem for Detroit because its stadium, the Silverdome, is an indoor field with artificial turf. Growing grass in domed stadiums has not yet been successful, so the World Cup organizers turned to the soil scientists at Michigan State University. The scientists grew the grass outdoors on large pallets and then moved these pallets into the stadium in time for the games.
Christi Makes Sense of Sixth-Grade Mathematics
Authors: Ridlon Candice L.
Published: December 2020

IN FALL 1997, A LOCAL PUBLIC MIDDLE school granted me permission to try a new curriculum with twenty-six sixth graders for nine weeks. I wanted to study children in ordinary classrooms using commonly available materials, like an overhead projector and worksheets. These students were randomly selected from 153 low achievers in mathematics on the basis of scores from the Iowa Test of Basic Skills (ITBS)—scores below the fortieth percentile—and recommendation by a classroom teacher.
The Math Club for Girls and Other Problem Solvers
Authors: Karp Karen S. , Niemi Rhonda C.
Published: December 2020

You have just entered ‘My Plane,’” reported Captain Karen Lee, pilot at the United Parcel Service hub at the Louisville (Kentucky) International Airport. A group of girls from Meyzeek Middle School listened intently to Captain Lee's every word as they tried out the few passenger seats, sat in the pilot's cockpit, tried on her cap and blazer, and even tinkered with the dials and switches. Captain Lee works at one of the largest employers in our area, UPS, where the Math Club for Girls and Other Problem Solvers was touring the facilities of this package-delivery giant. Our interest in this senior pilot's career was the incentive for the group's visit, yet the club also observed the center's activities. At the most hectic point of the visit, fifty planes were being packed with cargo.
Now & Then: Community Planning through Data Analysis, John Graunt: The Founder of Statistics
Authors: Johnson Art
Published: December 2020

Now… A NEW HOUSING DEVELOPMENT IS BEING BUILT ON THE outskirts of town. Is the closest elementary school large enough to accommodate the expected influx of new students? Members of local and regional planning commissions consider such questions and help communities adapt to growth and change. Andrew Singelakis is the head of the Regional Planning Commission in south-central New Hampshire. He uses mathematics to help towns and cities in his part of the state plan and evaluate community services, from bus routes and new schools to traffic lights and tollbooths. To help communities plan for the future, Andrew uses mathematics to analyze data from the past and present. He also relies on a reasonable sense of both current trends and human nature.
Teacher to Teacher: Improving Students Mathematical Communication and Connections Using the Classic Game of “Telephone”
Authors: Bay Jennifer M. , Ragan Gay A.
Published: December 2020

Remember playing the game of “Telephone” as a child? Someone whispered a statement to the person next to him or her, who then whispered to the next person in line. As each person passed the message on, everyone eagerly waited to see how it was altered and who was guilty of changing it. Our activity, “equation telephone,” was inspired by this game. Each pair or group of students is given an equation and develops a story to fit the equation. The story is passed to the next group, which, in turn, must create an equation to fit the story—not solve it, just write it. That equation is passed to the next group, which has to write another story, and so on, until the equation returns to the original group. We found that this activity greatly improved our students' ability to interpret written information and recognize equivalent forms of an equation. The following sections share the details of this activity, along with examples of student responses.
Understanding Student Responses to Open-Ended Tasks
Authors: Moskal Barbara M.
Published: December 2020

Communicating mathematical knowledge is a challenge for students, and assisting students in clearly expressing their mathematical ideas is a challenge for teachers. Open-ended tasks give students opportunities to select their own approaches for both solving problems and expressing mathematical ideas (Billstein 1998; Conway 1999). Students' responses to these tasks give teachers evidence of their students' problem-solving and communication skills.
The Role of Definition
Authors: Keiser Jane M.
Published: December 2020

Much of formal geometry is based on unambiguous definitions that can be used to develop and prove theorems. Is it possible that such clear and precise definitions can hinder conceptual understanding? Just what role should definitions play in the mathematics classroom? Should students be told definitions? Or should they be asked to develop them? In this era of educational reform, we are emphasizing student involvement in discourse.
Improving Data Analysis through Discourse
Authors: McClain Kay , McGatha Maggie , Hodge Lynn L.
Published: December 2020

The National Council of Teachers of Mathematics has been advocating the importance of effective communication in classrooms since the release of its Standards documents (NCTM 1989, 1991). This emphasis is echoed in Richards's (1991) description of an inquiry classroom (see also, e.g., Ball [1993]; Cobb, Wood, and Yackel [1991]; Lampert [1990]). In this setting, the teacher's role is to guide the negotiation of classroom norms to enable the teacher and students together to engage in meaningful mathematical discussions, which include asking questions, solving problems, posing conjectures, and formulating and critiquing mathematical arguments. An increased emphasis on communication in the mathematics classroom allows students the opportunity to discuss and validate mathematical ideas and to make and evaluate conjectures and arguments.
Counting Attribute Blocks: Constructing Meaning for the Multiplication Principle
Authors: Bird Elliott
Published: December 2020

The fundamental counting principle (fcp) is a “big idea” in mathematics. It says that if an event E can occur in a ways and event F can occur in b ways, then event E followed by event F can occur in a × b ways. It is easy to understand and easy to apply. For instance, if Ng has 3 sweatshirts and 2 pairs of jeans, he has the possibility of 3 × 2, or 6, different outfits. Yet many learners whose view of mathematics is strictly procedural have little conceptual understanding of the fcp. This deficiency becomes evident when they try to remember formulas or procedures, such as those for counting various types of arrangements or subcollections. The activity described in this article offers a meaningful introduction to the principle.
Writing a PEMDAS Story
Authors: Golembo Vadim
Published: December 2020

IN MY SEARCH TO FIND A MEANINGFUL WAY to teach a “because that's the way it is” concept, such as the order of operations, I found that giving students a creative context in which to explore this concept is ultimately more interesting and more worthwhile to them. This project asks students to relate mathematical operations to realworld situations, communicate mathematical concepts in writing, share their ideas with their peers, and most important, exercise their creativity.
The Thinking of Students: Make Mine Lemon-Roasted
Authors: Sherard Hamp
Published: December 2020

The following “Food for Thought” problem appeared in the January 2000 issue of the journal: The sales manager for Chicken Shack Restaurants is studying the sales figures of two new varieties of chicken sandwiches recently introduced at eight of its restaurants. For four weeks, three restaurants have been selling lemon-roasted-chicken sandwiches, and for three weeks, the other five restaurants have been selling barbecued- chicken sandwiches. During this testing period, 6000 lemon-roasted-chicken sandwiches were sold, and 7200 barbecued-chicken sandwiches were sold. Which of the two new chicken sandwiches should the sales manager report as having the better sales?
Gaining Insight into Students' Thinking through Assessment Tasks
Authors: Stylianou Despina A. , Kenney Patricia Ann , Silver Edward A. , Alacaci Cengiz
Published: December 2020

The task in Figure 1A, which was given to middle school students, asked them to find the number of dots in the twentieth step of the pattern. As shown in figure 1b, the answers that students gave ranged from 20 dots to well over 60 million dots. If your students gave these answers without providing work or explanations, would you be able to tell how they obtained them? Probably not. Without looking closely at the students' work or explanations or talking to them about their solution strategies, it is difficult to understand how they were thinking about the pattern task and how their thinking could have produced such a wide range of answers.
An Assessment Model for the Mathematics Classroom
Authors: Moskal Barbara M.
Published: December 2020

This article describes a composite of the experiences of two teachers to illustrate how an assessment event might occur in a classroom. The composite is a result of multiple observations made during the 1994–95 school year of two sixth-grade mathematics classrooms (Moskal 1997). The scenario that begins this article was developed from a task created by the assessment team of a project called Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) (Lane 1993).
A Raisin in the Sun: Fostering Cultural Connections with a Classic Movie
Authors: Chappell Michaele F. , Thompson Denisse R.
Published: December 2020

WE LIVE IN AN ERA IN which students are more media savvy than ever. Throughout their school years, students encounter numerous media, such as videotapes, CD-ROMs, video games, and the World Wide Web. When planning mathematics instruction, it seems reasonable to use media resources, such as classic movies, that have rich potential for engaging students in the learning process.
Reader Reflections: Triumph through Adversity: Supporting High-Level Thinking
Authors: Henningsen Marjorie A.
Published: December 2020

Worthwhile mathematical tasks are central to students' learning because “tasks convey messages about what mathematics is and what doing mathematics entails” (NCTM 1991, p. 24). Such tasks engage students in looking for patterns, explaining ideas, justifying conclusions, testing conjectures, framing and posing problems, making decisions, comparing and contrasting problem structures and multiple solution strategies, and reasoning across multiple representations. Frequent experience with such tasks increases students' opportunities to do, think, and reason about mathematics and, therefore, increases their opportunities to learn.
Spotlight on the Standards: Using Communication to Develop Students' Mathematical Literacy
Authors: Pugalee David K.
Published: December 2020

When students are given the opportunity to communicate about mathematics, they engage thinking skills and processes that are crucial in developing mathematical literacy. The importance of communication is evidenced through NCTM's recognition of this skill as one of the five process standards in mathematics, in both the 1989 and 2000 Standards documents (NCTM 1989, 2000).
Zero: A Special Case
Authors: Wilson Patricia S.
Published: December 2020

Zero is an important number in mathematics today, but it has not always been so! People used numbers for thousands of years before they used zero. In fact, people did not even have a symbol for zero until after the common era (C.E.) began. Today, we use zero to help represent large numbers, such as one million, 1 000 000, or small numbers, such as onethousandth, 0.001. We use zero in unique ways when we add, subtract, multiply, and divide. Although zero is an integer, it is neither positive nor negative. For these reasons, mathematicians often test their theories and ideas using zero as a special case.
Algebraic Thinking through Origami
Authors: Higginson William , Colgan Lynda
Published: December 2020

Since the publication of the original Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), we have come to embrace the importance of the “big ideas” of mathematics—the complex web of ideas that goes beyond content and includes reasoning, problem solving, and communication using multi-ple representations. Big ideas in classrooms give students opportunities to see mathematics as an integrated whole and to make thoughtful connections among related concepts and applications to other domains. This article illustrates how mathematical insights can arise naturally from a rich learning task when the teacher facilitates a stimulating and responsive dialogue.
Reflections on Practice: Manipulatives: One Piece of the Puzzle
Authors: Stein Mary Kay , Bovalino Jane W.
Published: December 2020

Getting students to think about mathematics in ways that go beyond using procedures to solve routine problems is an important goal of mathematics reform. Manipulatives can be important tools in helping students to think and reason in more meaningful ways. By giving students concrete ways to compare and operate on quantities, such manipulatives as pattern blocks, tiles, and cubes can contribute to the development of well-grounded, interconnected understandings of mathematical ideas.
Families Ask: Understanding Mathematics and Basic Skills
Authors: Brahier Daniel J.
Published: December 2020

Since the publication of Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and Principles and Standards for School Mathematics (NCTM 2000), many school districts and teachers have implemented new curriculum materials to achieve the vision of the Standards. In addition, many educators have adopted a constructivist viewpoint in their teaching practices, resulting in hands-on lessons for children and the use of real-life problems, visual and hands-on approaches, and invented strategies for solving problems. In the context of reform, however, parents and community members are prone to question whether their children will actually “learn the basics” when engaged in hands-on, real-life investigations.
The Value of Mistakes
Authors: Eggleton Patrick J. , Moldavan Carla C.
Published: December 2020

The importance of building from mistakes in mathematics instruction to develop better reasoning and problem-solving abilities.
Burgers, Graphs, and Combinations
Authors: Hardy Michael D.
Published: December 2020

One warm friday night, A friend and I decided to pop into a Steak 'n Shake for a burger. We noticed an interesting claim on the place mat. According to the restaurant chain, a customer could order from the sandwich platter menu for a year without ordering the same meal twice. Wondering aloud whether the claim was true, my friend and I decided to have some fun and determine the answer. We arrived at a solution, quickly, but I also thought that this question would give middle school students an interesting opportunity to build connections between mathematical concepts and realistic situations.
Using a Lifeline to Give Rational Numbers a Personal Touch
Authors: Weidemann Wanda , Mikovch Alice K. , Hunt Jane Braddock
Published: December 2020

When was the last time that parents told you they enjoyed their children's homework assignment because it gave them a chance to communicate with their children? That is exactly what happened in one middle-grades classroom. Jane Hunt, a seventh-grade mathematics teacher, used a number-line activity based on students' individual timelines to help her students understand the concepts of integers and rational numbers. These personal lifelines connected mathematics with language arts and other subjects. The project allowed for concept development through real-life experiences, with the added bonus that many parents became enthusiastic about opportunities to share family history with their children.
From Description to Proof
Authors: Groot Cornelis de
Published: December 2020

These quotes came from sixth-grade students during several geometry lessons. Because of my interest in how students stimulate one another's reasoning in mathematics, I spent some time in classrooms with teachers who were willing to step back and let their students express their thinking, and whose careful modeling would resonate in the students' reasoning at a later time. The two vignettes that follow illustrate the fact that student-to-student discourse and careful teacher modeling support a transition path to more formal mathematical reasoning.
Probability on a Budget
Authors: Ewbank William A. , Ginther John L.
Published: December 2020

Common dice numbered 1 to 6 are useful for simple mathematical situations including probability. But if faces like 1, 1, 3, 5, 5, 6 were substituted, the probability with two different dice changes. Includes a cut out for creating Paper Cubes.
Eleven: The First Uninteresting Number?
Authors: Maor Eli
Published: December 2020

Afamous parody tells of a mathematician who tries to prove that all counting numbers are interesting. Assume that they are not. The number 1 is obviously interesting, as is 2—the first even number, the only even prime, and the base of the binary system. Next comes 3, which is the sum of 1 and 2 and hence is interesting. Certainly 4 is interesting, if for no other reason than the fact that 4 = 2 + 2 = 2 × 2 = 2 2. And so the story goes, until we reach the first presumed uninteresting counting number. But of course, the fact that it is the first such number makes it interesting! Thus, all numbers are interesting.
Mathematics in the South Korean Flag
Authors: Kim Ok-Kyeong , Simms Ken
Published: December 2020

Mathematics teachers are always looking for innovative exercises or projects that can be used in the classroom to enhance students' opportunities to learn. One exercise that allows for mathematical thinking is exploring flags that have geometric features embedded in them. Having students search for patterns in these geometric features can develop algebraic thinking (for more information, see Selkirk [1992] and Hellman and Long [1993]). In addition, many flags can be used as contexts to develop conceptual knowledge of fractions and proportions.
Food for Thought From Jay's Diner: Selling Candy
Authors: Sherard Wade H.
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 approach the problem.
Rice + Tecnhology = an Exponential Experience!
Authors: Crocker Deborah A. , Long Betty B.
Published: December 2020

Using Technology in Mathematics and integrating mathematics and literature have become more common in recent years than in the past. Integrating mathematics, technology, and literature is even more powerful than combining mathematics with technology or literature alone. Using technology enables students to explore problems and mathematical ideas beyond those that might typically be found at a given grade level. Additionally, combining mathematics and literature gives the mathematics a context and enhances the literature. Many children's books lend themselves to integrating mathematics and literature. One of our favorites is One Grain of Rice (Demi 1997). We used this book in one of Phyllis Wisniewski's eighthgrade mathematics classes at Kings Creek Elementary School in Lenoir, North Carolina.
On My Mind: Numbers in the Garden and Geometry in the Jungle
Authors: Devlin Keith
Published: December 2020

Many children (and not a few adults, for that matter) find numbers either boring or intimidating or both. But practically everyone is fascinated by the living world, and the canny teacher can take advantage of this fact. A summertime mathematics lesson can be enlivened by a brief excursion into a garden to count petals and leaf patterns on flowers and plants, leading to the surprising discovery that even Mother Nature knows her arithmetic.
Looking Silly to Make Sense: A Number Patterning Activity
Authors: Castellan Catherine M.
Published: December 2020

What do you do with a class of eighth graders and fifteen minutes of free time? I remember asking myself that very question a number of years ago when I was a junior high school mathematics teacher with an eighth-grade homeroom. In those few moments, I came up with an activity that has served me well over many years. Since that time, I have expanded and modified the original activity as I adapted it to second and third grades, conference presentations, PTA meetings, faculty inservice workshops, undergraduate and graduate mathematics methods courses, and the occasional cocktail party.
Investigating Limits in Number Patterns
Authors: Beigie Darin
Published: December 2020

Technology gives students the ability to stretch beyond their immediate environment, to explore the power and magic of numbers by transforming the abstract into the concrete. The idea of a limit is one of the most fundamental concepts in mathematics, yet exposure to this concept often awaits introductory calculus, where the topic can seem abstract and forbidding. Spreadsheets and programmable calculators are powerful tools that enable middle school students to visualize and explore limiting behavior, allowing them to experiment and grapple with the notion of a limit in concrete settings. Such experiences can help plant seeds of understanding in an important, yet perhaps underexplored, topic in number sense.
Understanding Multiplication as One Operation
Authors: Azim Diane
Published: December 2020

Some of the wonder of numbers becomes apparent when we use numbers to perform calculations. Understanding multiplication with positive rational numbers is not a simple process. It requires reconceptualizing the meanings of multiplication with whole numbers to include numbers that are less than 1 or are mixed numbers. (Numbers in this article are all non-negative numbers.) To learn how multiplication works with fractions is to experience a whole new meaning for multiplication.
Food for Thought From Jay's Diner: The Rossi's New Kitchen Floor
Authors: Sherard Wade H.
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 approach the problem.
Writing Samples to Understand Mathematical Thinking
Authors: Goldsby Dianne S. , Cozza Barbara
Published: December 2020

NCTM's Principles and Standards for School Mathematics emphasizes the need for all students to organize and consolidate their mathematical thinking through communication and to communicate their mathematical thinking coherently to others (NCTM 2000). Writing helps students focus on their own understandings of mathematics: “Students gain insights into their thinking when they present their methods for solving problems, when they justify their reasoning to a classmate or teacher, or when they formulate a question about something that is puzzling them” (NCTM 2000, pp. 60–61).
Challenges in Implementing Strategies for Gender-Aware Teaching
Authors: Gilbert Melissa C. , Gilbert Lucia Albino
Published: December 2020

Several challenges middle school mathematics teachers may face as they apply the NCTM Equity Principle to their classrooms.
Families Ask: Homework: Practice for Students or a Snack for the Dog?
Authors: Gilliland Kay
Published: December 2020

Homework constitutes one of the most immediate and frequent contacts between families and the school. It is essential to establish good lines of communication between caregivers and schools. Homework generally has its most significant impact when it is commented on and graded by the teacher.
Using Active Learning Strategies to Motivate Students
Authors: Boyer Kimberly R.
Published: December 2020

Icouldn't believe my ears when i heard a colleague make this statement during my first year of teaching: “Eighth grade is the year that the students' brains are turned off.” Was it true? Were eighth-grade students really that hopeless? In my early years of teaching, I was on a mission to be the best mathematics teacher I could be by incorporating problem solving, reasoning, communication, and mathematical connections into each lesson. I wanted to take time to involve my students personally so that they could see how mathematics directly affects their lives. However, I quickly learned that there is “no one way to be a topnotch teacher” (Harmin 1998, p. 2).
Investigating Prime Numbers and the Great Internet Mersenne Prime Search
Authors: Wanko Jeffrey J. , Venable Christine Hartley
Published: December 2020

Middle school students learn about patterns, formulas, and large numbers motivated by a search for the largest prime number. Activities included.
Teaching Geometry and Measurement through Literature
Authors: Moore Sara Delano , Bintz William P.
Published: December 2020

AS TEACHER EDUCATORS, OUR WORK involves helping novice and experienced middle school teachers integrate literature in the mathematics classroom.
Middle-Level Students Learn Mathematics Using the U.S. Map
Authors: Koirala Hari P. , Goodwin Phillip M.
Published: December 2020

Middle school educators have widely accepted the importance of interdisciplinary curricula (Cook and Martinello 1994), and Principles and Standards for School Mathematics argues that mathematics should be linked with other subject areas at all levels (NCTM 2000). According to Principles and Standards, “thinking mathematically involves looking for connections, and making connections builds mathematical understanding” (p. 274). Therefore, connecting mathematics to other disciplines has become particularly important.
Using Student Contributions and Multiple Representations to Develop Mathematical Language
Authors: Herbel-Eisenmann Beth A.
Published: December 2020

A way to introduce and use mathematical language in mathematics classrooms that draws on multiple representations and student language.
Take Time For Action: The Number of Trials Does Matter
Authors: Aspinwall Leslie , Tarr James
Published: December 2020

PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics (NCTM 2000) calls for middle school students to develop and evaluate inferences that are based on data. Combining or “pooling” data from simulations reduces the likelihood of drawing incorrect inferences. But are your students aware that more accurate inferences can be drawn from larger sets of data and that smaller samples are more likely to produce unusual results?
Bullish on Mathematics: Using Stock Market Simulations to Enhance Learning
Authors: Alsup John K. , Altmyer Donald J.
Published: December 2020

Middle school students are rapidly developing a keen interest in the stock market and investments (Turner 2000; Kelly 2000). They are participating in online stock market simulations in record numbers, have started investment clubs at their schools, and are educating their parents about finding up-to-the-minute financial information on the Internet. For these students, a typical family dinner conversation might sound something like this: “Dump GM, Mom, and buy EMC Software. I went online and looked at some graphs. Its growth rate is awesome, and there is a huge demand for computer storage with the digital revolution. It's cheap at $25.”
Math at Work: Counting on a Good Flight
Authors: Telfer Cory
Published: December 2020

For thousands of years, people have been fascinated by flight. For Chuck Blain, a love of flying began with stories of an uncle who had flown a British Spitfire in World War II. Joining the Air Force out of high school gave Chuck his first opportunity to take flight. Later, while building up enough flight hours to become a commercial pilot, Chuck turned to teaching, but he never completely left flying behind.
Families Ask: Calculators in the Classroom
Authors: Gilliland Kay
Published: December 2020

Middle school teachers often hear similar questions about calculators: “Should I allow my child to have a calculator?” “Are calculators just a crutch?” “Is using a calculator for homework acceptable?”
Math at Work: Off to the Races with Melanie Paterson
Authors: Harris Joanne
Published: December 2020

MEET MELANIE PATERSON, race car driver. Melanie currently races in the Canadian Formula Ford Championship series. This form of racing is an entry-level professional open-wheel series. The cars have 1600cc/1.6L engines, race with no aerodynamic aids, and use street radial tires. From Formula Ford 1600, a racer can proceed into Formula 2000, Formula Atlantic, Indy Lights, and the CART (Championship Auto Racing Teams) racing series. With each progression in race series, more powerful engines and higher technology come into play. Melanie's goal is to eventually race in the CART series as well.
Food for Thought from Jay's Diner: Tina's Vacation
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 strategies that they use to solve the problem.
Conceptualizing Ratios with Look-Alike Polygons
Authors: Dwyer Nancy K. , Causey-Lee Betty J. , Irby Nekeya M.
Published: December 2020

IN THE MIDDLE GRADES, STUDENTS BUILD ON, and extend their understanding of, fractions. They begin to use fractions as ratios and expressions of part-whole and other relationships. Using fractions as ratios requires interpreting fractions differently than when they represent part of a region or set of objects. For example, the first term of a ratio may tell the number of objects in one set, and the second term may tell the number of objects in a completely different set. Figure 1 shows examples of several different kinds of ratios (Kennedy and Tipps 1997).
Using Literature to Teach Factorials
Authors: Blintz William P. , Moore Sara Delano
Published: December 2020

AT ONE TIME, NOT SO LONG AGO, we thought that we were a very odd pair. Although we were faculty members in middle school education at the same university, one of us was a reading educator and the other, a mathematics educator. What could we possibly have had in common? What could we talk about? And how could we talk about it to each other? We did not have the same professional backgrounds, and we surely did not speak the same language! Many of our colleagues just shook their heads and said that our collaboration would never work.
A Modeling Approach for Enhancing Problem Solving in the Middle Grades
Authors: Ferrucci Beverly J. , Yeap Ban-Har , Carter Jack A.
Published: December 2020

Modeling is a flexible, powerful, and engaging tool for middle school students to use to enhance their problem-solving skills. In this article, we describe one teacher's first efforts to develop her students' modeling expertise in a variety of arithmetic word problems. This modeling approach emanates, in part, from our observations of Singapore schools and curricula during the year 2001. We initially describe some of the basis for modeling that is grounded in current research and standards documents.
A Design for Ratio and Proportion Instruction
Authors: Abrahamson Dor , Cigan Christian
Published: December 2020

This article describes a method for teaching ratio and proportion in a fifthgrade classroom. Our unit design creates and follows a learning path that begins with revisiting multiplication as repeated addition. It then explores patterns in the multiplication table and moves to the concept of rate as a single column out of the multiplication table. Next, students discuss ratio as two linked rate columns, or a ratio table, then look at two rows out of a ratio table, or a proportion quartet. As a fifth-grade introduction to ratio and proportion, this unit focuses on integer cases, that is, situations in which the question mark in a:b = c:? is an integer. This emphasis fosters multiplicative reasoning in a developmentally appropriate numerical environment.
On My Mind: Getting Back to Class
Authors: Brahier Daniel J.
Published: December 2020

These are just a few of the expressions that I have heard tossed about by teenagers in my classroom this past year. I have even secretly pulled students aside after class and occasionally asked what a slang phrase meant. I have been serving as an administrator, consultant, and university professor for over a decade, and it had been thirteen years since I last taught mathematics on a full-time basis. I decided that it was time to immerse myself in the life of a teacher and took a sabbatical from my college-level work to step back into a classroom and teach full time for a year. The experience was as much of an education for me as it was for my students. I learned many important lessons during that year, but probably the most significant is to keep up the efforts to spread the word about reforms advocated by the NCTM's Standards. An inquiry approach to teaching and learning really does work.
Developing a Meaningful Understanding of the Mean
Authors: Bremigan Elizabeth George
Published: December 2020

One of the first statistical measures that students encounter in their study of mathematics is the arithmetic mean. The procedure for determining the arithmetic mean of a given set of numbers is relatively simple, because it requires only two computational skills: addition and division. Thus, students are often introduced to the arithmetic mean in grades 4 or 5. At this level, computation of the arithmetic mean is frequently presented as an application of division rather than as a statistical concept. Initially, the arithmetic mean is often called the average; in this article, the arithmetic mean will be referred to simply as the mean.
Representation as a Vehicle for Solving and Communication
Authors: Preston Ronald V. , Garner Amanda S.
Published: December 2020

Students in MS. Simpson's seventh-grade prealgebra class were challenged to use data to decide which of several class party plans was best. Because Takisha focused on price, she preferred using a table to justify her decision. Samantha used a written explanation to determine for herself which plan was best. Brandon's group used a table that indicated the price per person. However, when it came to convincing others of the best plan, some students chose other representations. For example, Brandon decided to use his graphing calculator to display linear graphs of the three plans. Melissa decided to use a triple bar graph. Tables, written explanations, rules, equations, and graphs were all important representations used to solve and communicate the results of the problem.
Heptades and Heptagons: The Historical Roots of 7
Authors: Funkhouser Charles P.
Published: December 2020

Rolling a 7 with a pair of dice, breaking a mirror and getting cautioned about 7 years of bad luck, drinking 7-Up, flying in a Boeing 747, working at something 24/7, staying open 7 days a week: What is it about the number 7 that has made it such a common part of our vocabulary and daily lives? Whether discussing good or bad fortune, measuring time, or exploring a numerical framework within religions of the world, the number 7 has a special place in the world of numbers, numerals, and culture. How 7 took on such a key role in world cultures and mathematics is a rich and diverse story spanning all seven continents.
Learning Geometric Concepts through Ceramic Tile Design
Authors: Bell Clare V.
Published: December 2020

Symmetry and geometric patterns are commonly used in the creation of designs that symbolize and contribute to the definition of culture. Native American weaving and pottery designs, Mexican tiles, and Islamic religious art are forms of cultural representation that rely heavily on a repetition of geometric figures and symmetry. These items are used as examples of geometric art for the lessons in this article (see fig. 1).
Turning Traditional Textbook Problems into Open-Ended Problems
Authors: Kabiri Mary S. , Smith Nancy L.
Published: December 2020

Article demonstrates how to take traditional textbook problems and rewrite them to be more open-ended to better meet the needs of diverse learners in a classroom.