bluuebunny/crossref_metadata_2025 huggingface dataset:
Spotlight on the Principles/Standards: Algebra in the Middle Grades
Authors: Lambdin Diana V. , Lynch R. Kathleen , McDaniel Heidi
Published: December 2020

When asked, “How long do you think it would take you to ride a fifty-mile bicycle race?” a class of Indiana sixth graders offered answers ranging from two to twelve hours. Some admitted that their answers were blatant guesses, whereas others offered interesting and thoughtful rationales for their estimates: a car can drive fifty miles in about one hour, and a cyclist might take twice as long for the same distance; because bicycling on hills is tiring, race times will depend on how hilly the course is; and an individual race takes longer than a relay race because team members can take turns bicycling and resting.
Teaching Toward the Big Ideas of Algebra
Authors: Woodbury Sonia
Published: December 2020

IN WHAT WAYS DO WE WANT MIDDLE-GRADES STUDENTS TO UNDERSTAND ALGEBRA? Hiebert and Carpenter (1992) describe the need for students to gain both procedural knowledge and broadly connected conceptual knowledge to understand mathematics. A knowledge of rules and procedures provides students with tools for efficient problem solving. However, in learning the procedures of algebraic manipulation, for example, students often develop what Skemp (1978) calls an “instrumental understanding” of algebra. He explains, “It is what I have in the past described as ‘rules without reasons,’ without realizing that for many pupils… the possession of such a rule, and the ability to use it, was what they meant by ‘understanding’ ” (p. 9). Skemp contrasts instrumental understanding with “relational understanding,” which “consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point” (p. 14).
Three Balloons for Two Dollars: Developing Proportional Reasoning
Authors: Langrall Cynthia W. , Swafford Jane
Published: December 2020

Ellen, Jim, and Steve bought three helium-filled balloons and paid $2 for all three. They decided to go back to the store and buy enough balloons for everyone in the class.
Unpacking the Nature of Discourse in Mathematics Classrooms
Authors: Knuth Eric , Peressini Dominic
Published: December 2020

The role of discourse, although always central in education and learning, is receiving increased attention in classrooms today as mathematics educators strive to better understand the factors that lead to increased learning.
Writing Tasks that Succeed
Authors: Liedtke Werner W. , Sales Judith
Published: December 2020

The items in Figure 1 were designed for you, the reader, to offer a hint about the issues addressed in this article. You can also compare your responses to these true-or-false statements with those presented by the authors.
Statistics Fever
Authors: Russo Linda M. , Passannante Marian R. C.
Published: December 2020

A Statistics units in which students learn to determine factors related to their health while employing community resources and spreadsheet technology. Activity sheets are included.
Take Time for Action: Assessing Geometric and Measurement Understanding Using Manipulatives
Authors: Strutchens Marilyn E. , Harris Kimberly A. , Martin W.GARY
Published: December 2020

Studying geometry benefits students in a number of ways. Geometry enables students to represent and make sense of the world, analyze and solve problems, and represent abstract symbols pictorially to facilitate understanding (NCTM 2000). Similarly, measurement establishes important connections between school mathematics and everyday life. However, students often have very little understanding of geometry and measurement concepts (Martin and Strutchens, in press). More often than not, students are asked to memorize geometric properties rather than to experience geometry through nature walks or worthwhile tasks that involve hands-on explorations. Further, students learn measurement through memorizing formulas rather than exploring the underlying concepts.
Mathematics Detective: Numbers Large and Small
Authors: Krusi Jean
Published: December 2020

Very large and very small numbers are often used in our world, but they can be hard to really understand. How big is a million? How small is a millionth? What can you think of that is extremely large? How can we measure such large things? What can you think of that is very tiny? How do we measure very small things? Here are two advertisements that use very small and very large numbers.
From Polygons to Poetry
Authors: Long Vena M.
Published: December 2020

MY ODYSSEY AS A TEACHER OF MATHEMATICS began in the late '60s in the rural midwest. Professional development in those days was a matter of personal development. My most treasured resource was the Mathematics Teacher. It was virtually my only opportunity to “converse” with others about mathematics and about teaching. Each month I devoured it from cover to cover on the evening of its arrival and took something from it into the classroom the next day.
Developing Number Sense on the Number Line
Authors: Bay Jennifer M.
Published: December 2020

One of the most important lessons that I have learned as a teacher is that seemingly boring problems on paper can come alive if I can find a way to lift them off the page. This transformation took place when the number line in my classroom became a brightly colored rope that stretched the length of the room, held by a student at each end. I first saw this idea as an approach to help young children order numbers from 1 to 10, then adapted it for middle school students. The scope of the activity eventually expanded to include explorations of large numbers, rational numbers, and algebra. As I saw improvement in students' conceptual understanding and their enjoyment of the life-sized number line, I used it more often in my classroom. I also found that the activities with the number line involved communication, reasoning, and justification— important processes in learning mathematics (NCTM 1989, 2000).
Riddles, Puzzles, and Paradoxes: Having Fun with Serious Mathematics
Authors: Evered Lisa J.
Published: December 2020

Middle school teachers frequently use brainteasers to add enjoyment to their mathematics courses. This article suggests ways in which this enjoyment can lead to serious mathematical investigations.
How Big is Your Foot?
Authors: Weinberg Suzanne Levin
Published: December 2020

Concepts relating to fractions and measurement are difficult for students in the upper elementary and middle school grades to grasp (Bright and Heoffner 1993; Coburn and Shulte 1986; Levin 1998; Thompson 1994; Thompson and Van de Walle 1985; Witherspoon 1993). As a first-year teacher, I learned the value of relating difficult concepts, especially abstract concepts, to students' real-world experiences. The “How Big Is Your Foot?” project grew out of a question that I asked my eighth-grade students during my first year of teaching. We had just finished studying conversions in the metric system and had begun working with conversions in the customary system. As a warmup question, I asked my students to describe the distance from my desk to the door of the classroom. I wrote their responses on the chalkboard as they called out estimates: 1 meter, 60 meters, 25 feet, 300 inches, 300 centimeters.
Teacher to Teacher: Unexpected Riches from a Geoboard Quadrilateral Activity
Authors: Britton Barbara J. , Stump Sheryl L.
Published: December 2020

Exploring the simple quadrilaterals in a three-peg-by-three-peg section of a geoboard has proved to be a very stimulating activity in elementary mathematics content and methods classes. In the April 1998 issue of Mathematics Teacher, Kennedy and McDowell described a wonderful activity that focused on counting quadrilaterals on geoboards of various sizes. Our focus in this article is on the properties of quadrilaterals. The activity that we describe is often used as an introduction to a unit on geometry. Tom Lewis, who is with the Moline Public Schools in Illinois, also used these activities in his fifth-grade classroom and reports similar results.
Promoting Algebraic Reasoning Using Students' Thinking
Authors: Bishop Joyce W. , Otto Albert D. , Lubinski Cheryl A.
Published: December 2020

The changing applications of mathematics have contributed to a shift from the perception that mathematics is a fixed body of arbitrary rules to the realization that the discipline is “a vigorous active science of patterns” (National Research Council 1989, p. 13). NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) recommends using patterns to promote mathematical understanding and, in particular, algebraic reasoning. A number of other mathematics education reform documents make similar recommendations (e.g., AAAS [1989]; National Research Council [1990]; Steen [1990]; NCTM [2000]). Researchers have begun to identify different approaches that students use to reason about patterns (Bishop 1997; MacGregor and Stacey 1993; Orton and Orton 1996; Stacey 1989). Research also shows that using students' thinking about patterns can help them develop a better understanding of mathematical concepts and the representations that reflect those concepts (Carey 1992; Fennema, Carpenter, and Peterson 1989). This article illustrates how students' thinking about geometric patterns can be used to help them develop algebraic reasoning and to make sense of mathematical notation and symbols.
Spotlight on the Principles/Standards: Geometry in the Middle Grades: From Its Past to the Present
Authors: Chappell Michaele F.
Published: December 2020

Geometry, literally referring to the earth's measure, has fascinated individuals for centuries. To examine, understand, and effectively operate in our physical world, mathematicians in particular have formulated abstract models of geometric ideas, have investigated existing relationships among models, generated appropriate conjectures, and communicated arguments to validate related properties. These actions have contributed largely to the subject as it is currently known.
The Thinking of Students: Free Movie Passes
Authors: Sherard Hamp
Published: December 2020

Ann Sullivan, who teaches at St. Norbert School, Paoli, PA 19301, gave the problem to her sixth-grade Advanced Math class. The students solved the problem immediately, finding how many tickets of each color were under the seats of the theater. She then asked them to find different ways to solve the problem. Their various strategies showed evidence of algebraic thinking, whether they used arithmetic or algebra. In each situation the students reasoned that the tickets had to be distributed in sets of 48 tickets, (3 + 19 + 15 + 11), resulting in 432 ÷ 48, or 9, complete sets of tickets hidden under the seats of the theater.
Standards-Based Teaching: Alive and Well in Portugal
Authors: Johnson Iris DeLoach
Published: December 2020

NCTM'S Professional Standards for Teaching Mathematics (1991) emphasizes that teachers are “the key” to changing mathematics teaching and learning. Given that mathematics reform movements have never brought about “large-scale changes in teachers' behavior and teaching practices” (Hitch 1990, p. 2), Willis (1992) lamented that “whether the standards will actually produce sweeping changes in the way mathematics is taught and learned in U.S. classrooms remains to be seen” (p. 1). With Principles and Standards for School Mathematics (NCTM 2000), we continue to ask the vital question, How can we induce teachers to implement the Standards?
On My Mind: Which One, America?
Authors: Price Tom J.
Published: December 2020

Every year, I teach the basics of the metric system to my seventh graders. Every year, several of them ask me the same haunting question: “Mr. Price, why doesn't the United States just switch to the metric system?” In my opinion, after talking to many adults, the answer seems mainly to come down to sheer American pride. We Americans have the attitude that the world revolves around us and that all the rest of the human population on this planet should do everything that we do.
Creating Connections: Promoting Algebraic Thinking with Concrete Models
Authors: Chappell Michaele F. , Strutchens Marilyn E.
Published: December 2020

The recent “Algebra for all” era has meant “the best of times and the worst of times” in many middle schools. At one extreme, many adolescents delight in the opportunity to study algebra or algebraic thinking and perform well in this course of study. At the other extreme, too many adolescents encounter serious challenges as they delve into fundamental ideas that make up this essential mathematical subject. Instead of viewing algebra as a natural extension of their arithmetic experiences, significant numbers of adolescents do not connect algebraic concepts with previously learned ideas. For instance, data from the Third International Mathematics and Science Study (TIMSS) showed that at the international level, only 47 percent of the seventh graders and only 58 percent of the eighth graders were able to recognize that m + m + m + m was equivalent to 4m (Beaton et al. 1996).
Power over Trash: Integrating Mathematics, Science, and Children's Literature
Authors: Ronau Robert N. , Karp Karen S.
Published: December 2020

WE HEARD THAT REMARK OVER AND over when we visited a class of sixth graders for a week in a nearby middle school. Through an integrated approach that incorporated literature to define a topic—in this instance, garbage—we linked concepts and activities in mathematics and science. This article shares a strategy for teaching organization, analysis, and representation of data using manipulatives and graphing calculators.
Making Connections with Prime Numbers
Authors: Yolles Arlene
Published: December 2020

Few topics in mathematics present more possibilities for enrichment and lesson extensions than prime numbers. Each year in my fifth-grade class, I discover more and more ideas to add to this already rich subject. The study of prime numbers offers opportunities for enriching students' mathematical language, making connections with related topics, discussing the historical development of mathematics, exploring the Internet, and gaining further understanding of mathematics topics already covered. As a teacher, I approach prime numbers enthusiastically, and my students come away from the study with a wealth of knowledge and understanding and, more important, a desire for further exploration.
Spotlight on the Principles/Standards: Data in the Middle Grades: A Probability WebQuest
Authors: Arbaugh Fran , Scholten Carolyn , Essex N. Kathryn
Published: December 2020

“Spotlight on the Standards” focuses on the grades 6–8 content and process standards found in NCTM's Principles and Standards for School Mathematics (2000). The articles compare NCTM's Curriculum and Evaluation Standards for School Mathematics, published in 1989, with the Principles and Standards relating to the middle grades and suggest ways that teachers might incorporate Standards-based practices into their instruction.
Math at Work: Lily Law: Urban Planner
Authors: Little Catherine
Published: December 2020

Have you ever felt as if you were in a concrete jungle? Why do some communities have a cozy feeling, whereas others feel cold? Would you prefer to live in a new subdivision or an older one? These questions may all depend on personal preference, but they interest developers who want to sell you the homes that they build. Recently, builders have noted a renewed interest in developing land for a city feel. New developments with curb-side parking and front porches are springing up all over the place. At one time, homes in a subdivision were built to look exactly the same, but now they exhibit more variety, such as detached and semidetached homes in the same area and houses of different widths.
Developing an Area Formula for a Circle with Goldilock and the Three Bears
Authors: Ameis Jerry A.
Published: December 2020

Developing formulas by identifying patterns is an element of algebraic thinking (NCTM 2000). This article describes a way to develop a formula for the area of a circle that involves active learning with patterning and estimating. This approach was tried with Erika, a fifth-grade student, and offers a meaningful learning experience for students at the lower end of the middle grades.
Subtracting Integers: An Affective Lesson
Authors: Petrella Gerald
Published: December 2020

Developing meaning for the procedure for subtracting integers was a troublesome issue for me for many of the twenty years that I taught middle school mathematics. Early in my career, I taught one or another of the various conventional methods for prompting and justifying the integer-subtraction procedure, with mixed results. These efforts typically included such approaches as the charged-particle model or the number- line model. Student reactions to these lessons characteristically conveyed a sense of confusion, consternation, and frustration. Was this lack of success the result of a good model being badly implemented, a flawed model, indifferent students, or a combination of all or some of the above? The fact was that these conventional approaches, as implemented in my classes, did not enhance the learning environment. On the contrary, the model itself was a distraction.
Spotlight on the Principles/Standards: Measurement in the Middle Grades
Authors: Berry Robert , Wiggins Joyce
Published: December 2020

People use measurement ideas everyday because the ideas arise out of the need to compare attributes of the real world and because humans are naturally inclined to make comparisons. Comparison is the basis for measurement. We make simple comparisons, such as that “the width of the desk is about an arms length,” and complex comparisons expressed in terms of precise numerical measurements. Although measurement encompasses such topics as time, temperature, length, perimeter, and volume, this article specifically focuses on how middle school students learn about angle measurement.
Take Time for Action: One Fraction Problem, Many Solution Paths
Authors: Yang Der-Ching , Reys Robert E.
Published: December 2020

Developing number sense is an important goal of mathematics programs. For example, the Number and Operations Standard in NCTM's Principles and Standards for School Mathematics states, “Central to this Standard is the development of number sense”(NCTM 2000, p. 32). Number sense is a complex process involving many different components of number and operations on numbers (McIntosh, Reys, and Reys 1992). Furthermore, number sense is of international interest and concern. Research documenting that students who are skilled in written computation but who do not necessarily show good number sense has helped stimulate increased attention to developing number sense in Taiwan (Reys and Yang 1998).
Innovation in Curriculum: My Involvement in Change
Authors: Stevens Bridgette Almond
Published: December 2020

MY FOURTH YEAR OF TEACHING WAS like a new beginning. Why? Because I felt as if I were a first-year teacher all over again at a middle school in Iowa. That year was my first experience using Mathematics in Context (MiC 1998), an NCTM Standards– based middle school mathematics program that encompassed not only a different textbook but different ways of teaching and helping children learn mathematics.
Your Students' Images of Mathematicians and Mathematics
Authors: Picker Susan H. , Berry John S.
Published: December 2020

What images do your students have of mathematicians and mathematics? Finding out more about your students' images of mathematics can help you understand their attitudes toward, misconceptions about, and opinions of the subject. One way to discover these attitudes is to ask your students to create a drawing of a mathematician. You may be surprised at the results!
Innovation in Curriculum: Fostering Student Discourse: Don't Ask Me! I'm Just the Teacher!
Authors: Frykholm Jeffrey A. , Pittman Mary E.
Published: December 2020

Throughout the past several years, middle-grades mathematics curricula have undergone a significant shift. Recently developed curriculum programs based on both recommendations of the NCTM and contemporary learning theories now emphasize problem solving, critical thinking, mathematical connections, and mathematical communication in ways that they did not before. As these powerful curriculum programs continue to find a stronghold in our middle schools, new implications and roles for both teachers and students are becoming clear.
Transition Toward Algebra
Authors: Fernandez Maria L. , Anhalt Cynthia O.
Published: December 2020

IN PREPARING STUDENTS FOR ALGEBRA, EDucators must consider the range of mathematics and the continued development of mathematics as a science of pattern and order. Informal explorations involving physical models, data, graphs, and symbols are vital to a solid foundation for students' transition into algebra. Also important is developing a deep understanding of such concepts as decimals, fractions, ratio and proportion, measurement, integers, functional relationships, and variables. As students in elementary school and middle school work with number and operation and other mathematics strands, the focus should be on informal discussion and investigations that lead students to build, describe, represent patterns, develop and apply relationships, make and verify rules or generalizations, and explore mathematical properties (NCTM 1989, 2000).
On My Mind: Why Don't Teachers Know All the Ways?
Authors: Perkins Isabel , Flores Alfinio
Published: December 2020

TO HELP ALL STUDENTS BECOME CONFIDENT in mathematics, teachers need to know and accept a variety of ways to tackle a problem. This point was made clear to us by a student named Steven. Because of an unstable family situation, he had attended a different school nearly every year, and almost every time he moved, he was told that the way he learned mathematics at his previous school was “wrong.” The “right” way was the way presented by his current teacher. Steven's attempts to keep straight all the various algorithms he had been taught had resulted in a confused and very frustrated boy. He would often hide behind a façade of “superior knowledge,” and because he was older than the other boys, he acted superior or arrogant.
Exploring Functions with Dynamic Physical Models
Authors: Hines Ellen
Published: December 2020

Functions are an important component in the study of mathematics (NCTM 1989, 2000). Learning about the concept of functions can be a natural way for students to “mathematize” the real-world relationships that they observe. Everyday life abounds with opportunities for students to observe and describe dynamic relationships that can be classified as functions.
Fibonacci: Beautiful Patterns, Beautiful Mathematics
Authors: Miller Catherine B. , Veenstra Tamara B.
Published: December 2020

Using examples from nature to introduce the Fibonacci sequence and then describing some fascinating mathematical patterns, using multiples and GCFs, that exist among these numbers. Three activity sheets, with solutions, are included.
Mathematical Roots: The Other Life of Florence Nightingale
Authors: Franklin Christine Annette
Published: December 2020

Probability and Statistics are important parts of the school mathematics curriculum. Many students believe that these areas are recent additions to the field of mathematics. Probability and statistics, however, have been actively studied for more than three hundred years. James Bernoulli (1654–1705), a Swiss mathematician, developed important probability concepts. Johann Carl Friedrich Gauss (1777–1855) was a German mathematician who studied the distribution that takes on the famous bell-shaped curve. Another statistician from England, R. A. Fisher (1890–1962), argued for the importance of randomness when designing an experiment. All these men are well-known statisticians who made important contributions to the field of statistics. An individual who is not commonly discussed as an early contributor to the study of statistics is the famous English nurse Florence Nightingale (1820–1910).
Food for Thought from Jay's Diner: Erin Cleans Her Room
Authors: Sherard Wade H.
Published: December 2020

The Editorial Panel is presenting the following problem to stimulate submissions to “The Thinking of Sudents” department. We encourage teachers to try this problem with students and analyze the different ways that they approach the problem.
Mathematical Notations and Procedures of Recent Immigrant Students
Authors: Perkins Isabel , Flores Alfinio
Published: December 2020

Mathematics is often referred to as a universal language. Compared with the differences in language and culture faced by students who are recent immigrants, the differences in mathematical notation and procedures seem to be minor. Nevertheless, immigrant students confront noticeable differences between the way that mathematical ideas are represented in their countries of origin and the manner that they are represented in the United States. If not addressed, the differences in notation and procedures can add to the difficulties that immigrants face during their first years in a new country.
On My Mind: Let's Take Another Look at Pi Day
Authors: Trotter Terrel
Published: December 2020

IN OUR EARNEST DESIRE TO EDUCATE OUR students about the great importance and significance of the number p by celebrating with special mathematical activities on March 14, we just may be teaching a concept that we had not intended or that we are overlooking altogether. I am referring here to the fundamental concept of the place-value structure of our decimal number system.
Math at Work: Hockey Stats: Data Collection on Ice
Authors: Harris Joanne
Published: December 2020

With three seconds left on the clock, Kevin gets the puck on a breakaway. He shoots. Paul makes the save.” The buzzer sounds, and the game is over. “Paul now holds the league record for shutouts.”
Building Explicit and Recursive Forms of Patterns with the Function Game
Authors: Rubenstein Rheta N.
Published: December 2020

The function game is a powerful and motivating tool for engaging middlegrades students in mental mathematics, problem solving, communication, and inductive reasoning (Rubenstein 1996). The game can also be used to help students achieve the goals of NCTM's Algebra Standard for grades 6–8; that is, to “represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules” (NCTM 2000, p. 222). (For a simple electronic version of the game, use the applet on the CD-ROM in Cuevas and Yeatts [2001].) This article will show how the function game format serves as a launchpad to help students build, distinguish, and translate between two basic forms of patterns.
Food for Thought From Jay's Diner: The Trojan Horse Derby
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.
Helping Students Make Sense of Algebraic Expressions: The Candy Shop
Authors: Gregg Diana Underwood , Yackel Erna
Published: December 2020

Increasing emphasis on “Algebra for all” (NCTM 1997a, 1997b) compels educators to identify and address fundamental ideas that build the foundations for algebraic thinking and reasoning. Identifying these foundational concepts and developing appropriate instructional approaches are the focuses of our work. One area in which students often experience difficulty is adding and subtracting algebraic expressions. Although students may be able to memorize a procedure, such as “distribute the negative” when subtracting algebraic expressions, they are often unable to make sense of this procedure. Our work suggests that part of students' difficulty in this area is that they do not conceptualize an algebraic expression as a composite unit. In the paragraphs below, we explain what is meant by composite units and how this construct helped frame our development of an instructional sequence to help students make sense of, and find meaning in, algebraic expressions and operations on algebraic expressions.
Integrated Instruction: Is it Science? Is it Mathematics?
Authors: Frykholm Jeffrey A. , Meyer Margaretr
Published: December 2020

Although much has been said in recent years about the value and importance of integrating the curriculum, many questions remain. Is integration always desirable? Can mathematics and science be taught as equals? What are the implications for student learning? Can integrated instruction meet state and national standards for each discipline? Is true integration even possible? In this article, we look closely at these issues: first, from the perspective of mathematics teachers, then from the perspective of science teachers. We also clarify both the potential of integration and the various problems that teachers must recognize and address if integrated instruction is to be successful. When appropriate, we have included content examples that have rich potential for integration but nevertheless present their own unique challenges and issues.
Teacher to Teacher: Making Minutes More Meaningful: The Ninety-Minute Mathematics Class
Authors: Broman Sarah
Published: December 2020

Students asked, “Can we take a break?” After hearing this question and others like it day after day, I realized teaching eighth-grade mathematics in a block schedule involves more than cramming two lessons into one class period. My students were not engaged in the tried-and-true lessons that I had been using for the previous four years in forty-five-minute class periods. They were bored listening to lectures that lasted nearly seventy minutes and felt overwhelmed with the lengthy assignments. I must admit that I was not crazy about block scheduling either. I, too, was bored and overwhelmed by the task of keeping interested while I talked all period. Obviously, I needed to improve my instructional strategies so that everyone, including me, would be more engaged in what was happening in mathematics class.
Diet, Ratios, Proportions: A Healthy Mix
Authors: Telese James A. , Abete Jr. Jesse
Published: December 2020

Mathematics teachers are frequently confronted by students who enter their classrooms asking, “Are we going to do anything fun today?” or “Do we have to do work?” These questions seem to imply that mathematics classes are dull and routine. Motivating students to want to participate actively in a mathematics lesson can be a daunting task at times. One way to increase through a problem-based activity that focuses on the importance of health issues and establishes a connection between health and mathematics (Telese 1998).
Supporting Reform through Performance Assessment
Authors: Kitchen Richard , Cherrington April , Gates Joanne , Hitchings Judith , Majka Maria , Merk Michael , Trubow George
Published: December 2020

According to reform documents, results from performance assessment tasks give teachers immediate feedback about students' mathematical strengths and weaknesses (NCTM 1995). However, little research has been done to demonstrate just how teachers take advantage of this feedback. Furthermore, less is known about how teachers benefit from collaboratively writing, revising, implementing, and scoring performance tasks.
Take Time For Action: Reasoning about Linear Equations
Authors: Gay A.Susan , Keith Charlotte J.
Published: December 2020

Have you heard of or used semantic feature analysis? It is a literacy strategy that can help students determine relationships among related vocabulary terms (Cooter and Flynt 1996). In mathematics classrooms, the use of semantic feature analysis is relatively new.
Build Your Own Fraction Computer!
Authors: Glidden Peter L.
Published: December 2020

The NCTM's Curriculum and Evaluation Standards (1989) called for increased emphasis on promoting students' conceptual understanding of fractions and fraction operations; this call was reaffirmed in Principles and Standards for School Mathematics (NCTM 2000). Currently, many manipulatives, including pattern blocks, fraction circles, fraction squares, geodot paper, and fraction strips, are available to help teachers promote this understanding. This article describes another manipulative, the fraction computer, that I have found helpful for teaching fraction addition and subtraction.
Three-Dimensional Geometry and Crystallography
Authors: Haws LaDawn
Published: December 2020

An activity involving multilink blocks that helps students develop three-dimensional spatial skills. The activity is motivated by the work of Nobel Prize winner Dorothy Crowfoot Hodgkin.
Ratios of Linear, Area, and Volume Measures in Similar Solids
Authors: Hartzler Stanley J.
Published: December 2020

PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics states that students need to understand that the areas of similar shapes “… are proportional to the squares of the corresponding side lengths. Similarly, through investigation they should recognize … that their volumes are proportional to the cubes of those lengths” (NCTM 2000, p. 245). The purpose of this article is to describe one activity that fulfills this recommendation and may have ancillary benefits, as well.
Teacher to Teacher: Order of Op Hop
Authors: Nurnberger-Haag Julie
Published: December 2020

One fun, multisensory method of teaching the order of operations is called Order of Op Hop. It uses a map that resembles a hopscotch board to teach the order both visually and kinesthetically (fig. 1). After taking notes on grouping symbols and the order of operations, students and I go outside with sidewalk chalk to draw the order-of-operations map and hop through the sequence. When using the map with my high school prealgebra and algebra students, I explain that we are going to explore the order of operations using visual and kinesthetic learning styles.
Food for Thought from Jay's Diner: The Hot Dog–Eating Contest
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 approach the problem.
Interdisciplinary Projects Enhance Teaching and Learning
Authors: Olson Jo Clay
Published: December 2020

When i began teaching twenty-five years ago, the standards (NCTM 1989, 2000) did not exist and interdisciplinary connections were seldom made. My first opportunity to explore subject-area connections occurred when a science teacher commented on the inability of students to calculate percentages or the speed of a small car rolling down a ramp. We organized curricular topics that allowed me to teach skills, such as finding percentages, before students applied them in the science lab. Not surprisingly, students performed better on the unit tests and seemed to remember the skills for longer periods of time. From this experience, I began to search for ways to help students understand that mathematics was more than following procedures.
On My Mind: Your Manuscript: From Beginning to End
Authors: Thompson Denisse
Published: December 2020

Now that the school year is well under way, it is a good time to think about sharing your favorite mathematics teaching idea with your colleagues. One way to do so is to write an article and submit it to Mathematics Teaching in the Middle School. Your idea would then be available to more than 30,000 other educators who receive the journal as part of their membership in NCTM or have access to the journal at their school libraries.
Why Students with Special Needs Have Difficulty Learning Mathematics and What Teachers Can Do to Help
Authors: Allsopp David , Lovin Louann , Green Gerald , Savage-Davis Emma
Published: December 2020

The national council of teachers of Mathematics advocates a balanced approach of teaching both procedural and conceptual knowledge (NCTM 2000). In practice, however, students with special needs often receive a great deal of algorithmic instruction because mastering algorithms is what we “see” them struggle with the most. Even with a heavy dose of algorithmic instruction, many of these students still have difficulty performing algorithms efficiently. Furthermore, without developing conceptual understanding while learning algorithms, these students will never understand foundational mathematical concepts.
Take Time for Action: Investigating Mathematical Thinking Using Open Writing Prompts
Authors: Aspinwall Leslie , Aspinwall Julie S.
Published: December 2020

Students' responses to writing prompts focused on mathematics concepts and terminology can provide teachers with information about students' development in both affective and cognitive domains. By the time they enter fifth grade, children have established beliefs about mathematics. They may also deepen their mathematical understanding as they attempt to express their thoughts and ideas in their own developing mathematical language.
Food for Thought from Jay's Diner: Candy Store Mathematics
Authors: Kersaint Gladis
Published: December 2020

The editorial panel is pre-senting 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.
Improving Middle School Teachers' Reasoning about Proportional Reasoning
Authors: Thompson Charles S. , Bush William S.
Published: December 2020

Article describes a professional development project to increase teachers' understanding of proportional reasoning, the thinking patterns associated with proportional reasoning, and the applications of proportional reasoning across the middle-grades curriculum.
Food for Thought from Jay's Diner: Married People
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.
The Giants Project
Authors: Weinberg Suzanne Levin , Hammrich Penny L. , Bruce Matthew H.
Published: December 2020

What is proportional reasoning? many word problems emphasize realworld situations, with the focus on a single variable. Word problems involving proportional reasoning situations are more complex. Solving these types of problems requires students to identify the two variables involved, then identify the relationship between these two variables. Helping students develop a true understanding of proportional reasoning cannot be accomplished with one isolated topic or situation. For students to use proportional reasoning skills, especially in situations involving rational numbers or unfamiliar story settings, several prerequisite topics must be mastered, including equivalence, fraction, ratio, and scale (Levin 1999; Weinberg 2002). Until these foundational topics are seen as interconnected concepts rather than as isolated skills, students may learn how to solve a single proportion item but fail to recognize the many proportional instances found in and out of the classroom.
Crossing the Bridge to Formal Proportional Reasoning
Authors: Chapin Suzanne H. , Anderson Nancy Canavan
Published: December 2020

Mathematics educators agree that ratio and proportion are important middle school mathematics topics. In fact, researchers have stated that proportional reasoning involves “watershed concepts” that are at the “cornerstone of higher mathematics” (Lamon 1994; Lesh, Post, and Behr 1988). Yet assisting students in developing robust understanding of the many concepts and procedures that are related to using ratios, rates, and proportions is not straightforward. For example, being able to reason proportionally and being able to represent that reasoning symbolically do not always go hand-inhand. As with many complex topics, students' understanding grows with time and experience.
Using a Pattern Table to Solve Contextualized Proportion Problems
Authors: Sharp Janet M. , Adams Barbara
Published: December 2020

Students in a sixth-grade classroom we visited were celebrating a classmate's birthday and enjoying fun-sized bags of Peanut M&M's candies. We overheard them discussing their curiosity at the small number of blue M&M's each of them had received in their small bags. Because the students were occupied in an informal, party atmosphere, we were pleasantly surprised to hear one student, Rickea, comment on a related mathematical issue. She speculated that the teacher's class-sized bag would have relatively few blue M&M's, as well. What a wonderful teaching opportunity for ratios and proportions Rickea's casual comment posed! In this article, we describe (1) how we built a week-long, problem-based unit around Rickea's original proportion question and (2) the effectiveness of using problem solving to help Rickea and her classmates construct knowledge about ratio and proportional thinking.
The Legend of Paul Bunyan
Authors: Buhl David , Oursland Mark , Finco Kristin
Published: December 2020

One of the greatest forms of entertainment that is passed down through generations is the art of storytelling. Stories told from generation to generation have entertained children around the world for centuries. Many favorites for children have been myths and folktales that entail measurements that pique our curiosity. Some examples include Johnny Appleseed; Nessie the Loch Ness Monster; and Paul Bunyan and Babe, his blue ox. These are just a few of the better-known characters of such tales. When used in the classroom, storytelling often captivates the imagination of students, thus enhancing the learning process.
Lisa's Lemonade Stand: Exploring Algebraic Ideas
Authors: Billings Esther M. H. , Lakatos Tracy
Published: December 2020

Students' informal algebraic encounters in grades K–8 lay the foundation for more formalized study of algebra in the future (NCTM 2000; Yackel 1997). The study of functions and relationships is essential in algebra. Functions and relations involve variables and describe how the change in one variable causes a change in a second variable. Words, tables, graphs, and formulas can be used to describe these relationships, which can also be found in many aspects of real life. Consider the following two examples: (1) the darkness of a piece of toast is a function of the setting on the toaster (Ritchhart 1997) and (2) the length of time required to ride to school on the bus can also be a function of the distance between the bus stop and the school. These relationships were described in words, but they can also be described by using a graph, a table, and depending on the exact nature of the relationship, even a formula. Many children do not realize that their everyday lives are filled with such functions and relations (Ritchhart 1997), and they have limited opportunities to interpret and analyze change graphically. Because students experience difficulty in interpreting graphs (Janvier 1981), they must be given opportunities to both interpret and create them.
Sweet-Tooth Geometry
Authors: Scanlon Regina M.
Published: December 2020

Surface area and volume of prisms, pyramids, and cylinders are challenging topics for middle school students. For the past five or six years, at the end of a unit covering these topics, I have reinforced this material by doing a project that involves the construction of a “creature” by each student. The creature had to be made out of a pyramid and a prism. Later, I added a cylinder as another choice to use in making the project. I provide the students with materials (see fig. 1), a scoring rubric, and class time to work on the project.
Using Number Relationships for Estimation and Mental Computation
Authors: Menon Ramakrishnan
Published: December 2020

Article describes how students can compute and estimate mentally, and rapidly, by using appropriate number relationships.
Math Roots: Eureka! Or Don't Throw out the Crown with the Bathwater
Authors: Kaplan Gail
Published: December 2020

The life of archimedes was almost entirely devoted to mathematical research. Historians claim that Archimedes would forget about eating while working on a mathematical problem. Anecdotes that have been passed down through the centuries reveal what is known about the life and personality of this mathematician. One of the most famous stories about Archimedes is the story of the golden crown.
Lollipop Statistics
Authors: Goldsby Dianne S.
Published: December 2020

AS NCTM'S Principles and Standards for School Mathematics (2000) points out, students should work directly with data to understand the fundamentals of statistical ideas. Teachers should also introduce statistics in a way that will capture the attention of students of varying abilities and interests. The constructivist approach to teaching emphasizes the idea that students work better when presented with tasks that are meaningful and relevant; in other words, they expend energy on topics that interest them (Brahier 2000). One way to harness that energy in the classroom is to teach with music, an area of interest for most middle school and high school students. This article describes the use of the 1950s hit “Lollipop” (Ross and Dixon 1986), heard in the movie Stand by Me, as a launching point to introduce ideas of counting, working with frequency tables, and graphing data.
Using School Lunches to Study Proportion
Authors: Attia Tamar Lisa
Published: December 2020

Four—I can't eat eight” is the answer that baseball's Yogi Berra is supposed to have given when asked into how many pieces he wanted his pizza cut. Although Yogi Berra must have learned during his elementary education that the fractions 4/4 and 8/8 are equivalent, his famous pizza comment could illustrate an inability to apply a mathematical concept to a real-world situation. It could also represent a failure to meet the NCTM's Connections Standard for grades 6–8 that students be able to “recognize and apply mathematics in contexts outside of mathematics” (p. 274). A project I embarked on with my sixth-grade mathematics class required applying mathematics to analyze and write about a practical and relevant issue for them: the nutritional value of their school lunches.
Operation Central: An Original Play Teaching Mathematical Order of Operations
Authors: Blackwell Sarah B.
Published: December 2020

For too long, our thinking has been that certain subjects belong together and that others are unrelated. We often pass this idea on to students by characterizing them as either mathematics and science or language and history learners. As a teacher, I dread the comment “I'm no good in mathematics. I'm more of an arts person.” I am frustrated when I see students pigeonhole mathematics as an independent subject when it offers so much room for interdisciplinary study.
Portfolios and a Backward Approach to Assessment
Authors: Britton Kristine L. , Johannes Jennifer L.
Published: December 2020

What does a letter grade have to do with what has been learned? Has a student mastered a concept just because he or she has earned an A? When parents look at their child's report card, do they know what was taught or learned or what difficulties their child may be having? As students get older, it appears that they share less and less with their parents, which creates a gap between school and home. Results of a survey conducted in 2001 at Rice Lake Middle School in Rice Lake, Wisconsin, involving 154 students and 34 parents showed that only 22 percent of students bring schoolwork home to share with their parents on a regular basis. In the same survey, 82 percent of parents and students agreed that learning is more important than grades. Often, when parents do see something from school it is because the child needs help, rather than to celebrate the successes of that child.
Fishy Fun under the Sun
Authors: Moore Deborah A. , Schwarz Maria C.
Published: December 2020

MY FAVORITE SEQUENCE OF LESSONS involves fun and creative activities that are both interesting and accessible to middle school students. The activities described in this article give a teacher one week's worth of lessons that allow geometry and measurement as well as ratio and proportion to be addressed in a cooperative learning environment. These lessons integrate science and mathematics with hands-on exploration using manipulatives.