bluuebunny/crossref_metadata_2025 huggingface dataset:
The Thinking of Students: “Tri-Square” Numbers
Authors: Mahoney Devin
Published: December 2020

As part of our seventh-grade mathematics curriculum, we were assigned an independent mathematics project to investigate interesting number patterns. We were asked to write a paper on what we chose to investigate, to discuss the investigation, to predict outcomes, to verify predictions, and to propose areas for further study.
On My Mind: Cooperative Groups Can and Often Should Be Homogeneous
Authors: Hersberger Jim
Published: December 2020

To maximize the benefits of cooperative learning groups, the students within each group should be heterogeneous in ability and personal characteristics” (Artzt and Newman 1990, 449). “Although groups can be formed by random assignment, heterogeneous grouping ensures a mixture of mathematical achievement, gender, and race/ethnicity. An occasional use of sociometric choice by students is possible, but homogeneous grouping is usually not recommended” (Davidson 1990a, 57). “Groups may organize themselves on the basis of friendship or common interests or they may be carefully composed by the teacher to be heterogeneous with respect to abilities, sex, personality types, race, or other variables” (Dees 1990, 175).
Meeting the Challenges of Diversity and Relevance
Authors: Smith Margaret Schwan , Silver Edward A.
Published: December 2020

Current reform efforts have emphasized the need to change the way that mathematics is taught and learned so that all students have access to a mathematics education rich in opportunities for thinking, reasoning, and problem solving. Reaching all students may not be easy, however, since students in a mathematics classroom may be considerably diverse, not only with respect to prior mathematics achievement but also with respect to ethnicity, language, and life expelience. The challenge is even greater because teachers often do not share the ethnicity, primary language, or life experiences of the children they teach. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the Professional Standards for Teaching Mathematics (NCTM 1991) urge teachers to consider classroom diversity by-
And the winner Is…
Authors: Harlos Carol Ann
Published: December 2020

The more generallzed curriculum op the middle school years presents an excellent opportunity for students to begin to seek patterns and to make math ematical predictions. The following activity offers such an experience. The students' background necessary for this activity includes understanding multiples and powers, finding simple fonnulas, and evaluating fonnulas. Although the activity is written in student-ready fonn, teacher direction is usually necessaty to help most students complete the investigation.
Professional Development: Teachers' Communication and Collaboration Keys to Student Achievement
Authors: Sherman Helene J. , Jaeger Thomas
Published: December 2020

The curriculum and evaluation standards for School Mathematics (NCTM 1989) and the Professional Standards for Teaching Mathematics (NCTM 1991) have served as both stimuli for, and responses to, numerous formal and informal programs, conferences, and conversations calling for educational reform and improvement in mathematics teaching. After all the plans are drawn and all the objectives are written, however, reform is most likely to occur and make a lasting difference when teachers are aware of the need for improvement, have a voice in planning it, and derive a real sense of professional satisfaction from implementing the instructional changes.
Now & Then: From Königsberg to Columbus: Topology—from Bridges to Bagels
Authors: Barnes Sue , Michalowicz Karen Dee
Published: December 2020

Now … everyone has heard the expression. He must have his wires crossed—meaning that someone is mixed up, confused, or thinking along a different line from everyone else. One person who definitely does not have his wires crossed is Michael Wimberly, a programmer analyst for Ohio's Department of the Treasury.
Mathematics Investigator
Authors: Weber William B.
Published: December 2020

The Sears Tower in Chicago is the tallest office building in the world. About 1.3 million people visit the Sears Tower Skydeck each year to view the spectacular Chicago skyline. The Skydeck is located on the 103d floor, 1353 feet above the streets of Chicago. A visitor can reach the Skydeck by an express elevator that travels 1600 feet per minute or can choose to walk approximately 2100 steps. From the Skydeck, observers can see the major buildings and landmarks in downtown Chicago, including Soldier Field, Wrigley Field, Comiskey Park, the John Hancock Building, Shedd Aquarium, Adler Planetarium, and the Amoco Building. If visiting the Skydeck on a clear day, the view extends approximately fifty miles in all directions.
Teacher to Teacher: Mathematical Connections: Proportions and Modeling in the Solar System
Authors: Helme Brian D.
Published: December 2020

The connections between science and mathematics become quite clear to middlegrades teachers who teach both subjects. Each subject can reinforce the other. Modeling size and distance in the solar system is one science topic that connects directly to mathematics.
On My Mind: Proof and the Middle School Mathematics Student
Authors: Sconyers James M.
Published: December 2020

Is proof perceived as being rigid and formal? Something that students should first encounter in high school? Does a concern involve students' having difficulty when they finally confront the idea of proof, perhaps in their high school geometry class? One likely reason for this unease with proof is that it is so often left out of any work in mathematics until students reach high school. They are then overwhelmed, since it is so unfamiliar. This outcome is not inevitable. Middle school students are capable of grasping the basic logic of proof and should be given the opportunity to encounter it.
Rich Problemss—Big Payoffs
Authors: Jones Graham A. , Thornton Carol A. , McGehe Carol A. , Colba David
Published: December 2020

For many students, the intuitive response to the problem was “Divide 650 by 4, since a rectangle has four sides. That equals 162.5 feet, so the area is 26406.25 square feet.” Surprised, Coiba probed, “How do you know that that is the maximum area?”
Promoting Mathematics Connections with Concept Mapping
Authors: Bartels Bobbye Hoffman
Published: December 2020

Recent documents calling for reform in mathematics education stress promoting mathematical connections during instruction (NCTM 1989; National Research Council 1989, 1990). Mathematical connections are important because they link mathematical concepts to each other and to the real world. When teachers emphasize these connections during instruction, mathematics becomes less compartmentalized and more cohesive and has relevance to real life. In addition, stressing mathematical connections during instruction helps students develop better and deeper understandings of mathematics.
Cartoon Corner
Authors: Cain Barbara
Published: December 2020

“Cartoon Corner” is designed to include cartoons that can be used to highlight mathematics in an interesting way. The questions accompanying the cartoons can be used as a large- or small-group activity, as homework, or as an extracredit assignment. Students might want to create their own mathematics-related cartoons that demonstrate in a humorous way how mathematics is used and misused. Readers are encouraged to submit three copies of cartoons and classroom-tested questions, doublespaced, for review by the editor of this section. The cartoons must include the date and the newspaper syndicate that holds the copyright.
November's: Menu of Problems
Authors: Stout David
Published: December 2020

A light fare to whet your mathematical appetite.
Professional Development: Reflective Journals: Enhancing Mathematics Staff Development
Authors: Burk Jill , Littleton Pam
Published: December 2020

In 1993 tarleton state university sponsored the Pre-Algebra Experience, which was developed to improve mathematics instruction in the middle grades with funding from the Eisenhower Mathematics and Science Program. The major goal of this project was to stress to middle school mathematics teachers the content necessary for students to succeed in high school algebra.
Power On! The Case of Jewels Jones
Authors: Fragale Kendra C.
Published: December 2020

Since geometry is such a visual subject, it can easily become an area in which technology should play a significant role. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 159) states that “computer graphics software that allows students to create and manipulate shapes provides an exciting environment in which they can make conjectures and test their attempts at a twodimensional visualization.”
Teacher to Teacher: Fictional Literature in Mathematics
Authors: Sullivan Emilie P. , Nielsen William
Published: December 2020

We have noticed that a number of fictional books for middle schoolers reinforce language-arts skills but few have episodes involving mathematics. Children featured in fiction often read books for pleasure or information. They write journals and refer to characters from books. Books may even portray children trying to teach other children how to read. But how often do middlelevel fictional books mention mathematics in such contexts? Do literary characters use mathematics in everyday situations? Do they teach others how to solve mathematics problems? What attitudes toward mathematics are projected in fictional works?
Professional Development: Lesson Design and Reflection
Authors: Schroeder Merrie L.
Published: December 2020

After twenty-five years of teaching junior high and middle school students, I might expect designing lessons to be a mindless routine. Yet I believe that today is an exciting time to be creating lessons. The excitement comes from seeing a subtle and powerful shift from thinking about lesson design as what I will do to prepare students to meet district outcomes to thinking about what learners will do to understand mathematics. This shift from “I” and “outcomes” to “what learners will do” has caused me to change from writing my lesson plans as a linear list of teacher activities to writing them as a matrix of student activities based on two dimensions of communication: four modes-oral, written, visual, and kinesthetic-and four audiences-self, partner, small group, and whole class. This matrix is shown in figure 1. The type of audience is indicated by the number of dots. One dot represents communicating with self, two dots represent communicating with a partner, and so on.
Math Options: Exploring Career Possibilities
Authors: Vanleuvan Patricia , Smith Barbara , Dion Gloria S. , Simon Deborah S. , Kaplan Cheryl
Published: December 2020

The mathematics education of students in America's schools has been the subject of renewed debate and distress reminiscent of the rumblings of the post-Sputnik era. The current discussion is fueled by a realization that American competition in the global marketplace of the twenty-first century will require significantly more mathematicians, scientists, and engineers. Many question our ability to prepare students who demonstrate the high levels of expertise required in these fields. First, they point to intemational studies indicating that the mathematics skills of students in the United States are below those of students in other countries (Travers and McKnight 1985; Stevenson, Lee, and Stigler 1986). A second concern is related to demographic trends and the changing composition of our workforce. The majority of entrants to the job market in the new century are predicted to be women and minorities, who traditionally have not pursued careers that require extensive training in mathematics and science (Johnson and Packer 1987). Although the percent of women working in mathematics or science-related careers has increased, from 8.6 percent in 1975 to 13.4 percent in 1986, it is still comparatively low (Linn and Hyde 1989).
The Thinking of Students: How to Turn Oranges into Moolah
Authors: Webb Laurye M.
Published: December 2020

As part of an economics unit, my students investigated the feasibility of having a sandwich bar in our school's cafeteria. To complete their study accurately and effectively, they needed to learn and understand the mathematical concepts of cost per unit, percents, taxes, surveys, graphing, and geometry.
Professional Development: Caring Teachers Can Realize the Vision of the Standards
Authors: Robicheaux Ray
Published: December 2020

What does it mean to “know” mathematics? This philosophical question was initially debated before the framing of the Currculum and Evaluation Standards for School Mathematics (NCTM 1989). For several years, “knowing” mathematics has meant knowing a set body of rules and procedures; more specifically, “knowing” mathematics has meant “knowing how” to perform an algorithm. Students in the United States fared poorly on problem solving and reasoning questions as measured by the National Assessment of Educational Progress (Carpenter 1988); their status has been even worse in international comparisons in this area (Steen 1987). These combined findings have caused people in business and indusby, the mathematics and science community, and the educational community to call for reform of mathematics education. The question has become one of either “fixing” what we have or forging an entirely new approach that focuses on mathematics as a complex, holistic thought process featuring heuristics and analyses. The latter interpretation of “knowing” mathematics has prevailed, bolstered by a broad-based, grass roots following. to become the vision of the Standards.
The Civil War and the Standards: Some Mathematical Activities
Authors: Doyle Trent M.
Published: December 2020

When I lived in virginia several years ago, I developed a keen interest in Civil War history. The abundance of battlefield parks, museums, and cemeteries located throughout the state generated and sustained my interest to learn more.
How Big Was the Roman Empire?
Authors: Baker Camille
Published: December 2020

This activity was designed as part of a sixth-grade interdisciplinary unit. “Seeing the World through the Eyes of Ancient Greeks and Romans.” In addition to learning about Greek and Roman geography, economics, government, and societies in social-studies class. students studied ancient scientists, physicians. and inventors in science class. They also explored Greek and Roman myths, religions, languages, and ideas in language-arts classes. In mathe matics classes, students experimented with the golden ratio and the pentagram. wrote an essay on how the Greeks used mathematics to understand their world, examined Greek and Roman architecture, and investigated the physical size of the Roman Empire. To culminate the unit, students worked in small groups on special projects, such as building a scale model of the Parthenon, measuring and creating a cale drawing comparing the soccer field with the Pantheon, creating and performing original myths or plays depicting life in ancient Greece and Rome, and constructing simple machines or demonstrations of the scientists' work in Greek and Roman times.
Data with Snap, Crackle, and Pop
Authors: Hofstetter Elaine B. , Sgroi Laura A.
Published: December 2020

Often when statistical concepts are introduced to intermediate-grade students, data sets are supplied by the teacher or the textbook to give students the oppottunity to find measures of central tendency. to create graphs, or to analyze information. Students, therefore. often find themselves obtaining such measures as the mean area of a collection of states in the United States or the range of population for various regions. Alternatively, sports data are used for many of the same purposes. However, students aged 11 through 14 are characterized by their need to connect to their lives information introduced in school. Many of these students are likely indifferent to populations and areas of states, and the appeal of sports data is not universal. In addition, these data sets are given to the students. Students are not afforded the opportunity to collect, create, or construct their own data from aspects of their own lives that are motivational and exciting.
Using Brainstorming Webs in the Mathematics Classroom
Authors: Hanselman Cheryl A.
Published: December 2020

Despite the national council of teachers of Mathematics's call to educators for a shift in emphasis in mathematics education, many children still have not changed their perception of what mathematics is all about Although many educators have taken on the challenge to make the Curriculum and Evaluation Standards (NCTM 1989) come alive by shifting the emphasis to making connections among concepts, many students remain in classrooms where mathematics concepts are taught in isolation and are viewed as a series of procedures. These students perceive that the most important aspect of solving a problem is arriving at some “answer,” regardless of whether it makes sense. Consider how quickly and emphatically a student can suggest using 1.2 as the decimal form of one-half. After all, 1.2 has the 1 and the 2, and the decimal! Students who have been taught only isolated procedures often do not know any other strategies to help them find a relationship between the fraction and its decimal form. One technique that helps students learn to connect concepts, not isolate them, is the use of a graphic organizer, the web.
On My Mind: New Tricks for Old Dogs
Authors: Meyer Margaret
Published: December 2020

One of my favorite far side cartoons features Rex the Wonder Dog. Rex is shown balancing an elaborate array of objects while traversing a tightrope on a unicycle. The caption reads, “High above the hushed crowd, Rex tried to remain focused. Still, he couldn't shake one nagging thought: He was an old dog and this was a new trick.” Maybe that cartoon speaks to you the way it does to me. As one of the developers of the middle-grades curriculum Mathematics in Context (MiC), one of the Standardsbased middle school curriculum projects funded by the National Science Foundation, I have used that cartoon many times to describe to teachers, young and old, how it might feel to be a teacher who is about to implement a mathematics curriculum such as MiC. I can usually tell from the nervous laughter that although they might not be old, they recognize that the new Standards-based curricula will require them as teachers to learn “new tricks.”
You Are Cleared to Land
Authors: Billstein Rick
Published: December 2020

ON A NOT-SO-ROUTINE FLIGHT TO AN NCTM meeting, my colleague and seatmate, Jim Williamson, and I had ample time to chat and ponder our surroundings. As happens many times in the West in the winter, heavy snow caused a long delay. While we waited on the runway for the airplane to be de-iced, we noticed that runway numbers were posted in pairs, as in Runway 2-20. We also noticed that some numbers had an L or an R attached to them, as in Runway 2L. This observation was the only setting we needed to start a real-world investigation on how airport runways are named. Because it is hard to generalize from the small sample of numbers that we could see from the airplane, we realized that we needed more data. We talked to a pilot, who furnished us with several old maps of airport runways. He did give us a hint that the runway numbers had something to do with the compass headings of the runway from magnetic north. We were now in business: we had an interesting problem, we had data, and we had time to talk about it and think about it. This atmosphere is the type for which we strive in a middle school mathematics class. The following activity is based on our investigation and can be used for middle school mathematics.
Cracking the Code
Authors: Marshall Paul , Varro Laszlo
Published: December 2020

Since the earliest days of recorded history, cryptography—the science of encrypting and deciphering messages—has played an important and quite frequently decisive role in influencing world events. The defeat of the Persian forces of Xerxes in the Battle of Salamis near Athens in 480 B.C. was primarily due to a secret message successfully sent from Persepolis to the Greeks five years earlier, warning them about the Persian preparation for war. Two millennia later, in 1587, the interception and deciphering of encrypted messages between the imprisoned Mary Queen of Scots and her rebellious followers led to her execution and the crushing of hopes to restore Catholicism in England. The Allied Forces' access to the highly sophisticated Enigma codes and code machines in World War II was a serious blow to Nazi naval warfare and significantly contributed to the collapse of the Third Reich. Currently, with millions of financial transactions conducted over the Internet every day, the importance of encrypted messages has become even more obvious.
Building Percent Dolls: Connecting Linear Measurement to Learning Ratio and Proportion
Authors: Moss Joan , Caswell Beverly
Published: December 2020

There was an air of excitement and anticipation in the grade 5/6 class as the students consulted with one another and put the final touches on their percent measurement dolls. The doll-making unit, a favorite with the students, was a culminating activity in an ongoing research project for learning rational number and proportion. The students, who attend a laboratory school associated with the Ontario Institute for Studies in Education, and their teacher, Beverly Caswell, had just spent the last five mathematics classes working on the measurement, design, and building of these dolls. They had been invited to present their creations to a group of preservice teachers and to explain the mathematics that had been involved.
Moving Forward and Backward with Palindromes
Authors: Schiller Diane , Charles Mary
Published: December 2020

What is a palindrome? It is a number, word, or phrase that reads the same backward or forward. The Greek word palindromos means “running back again.” The Greek poet Sotades is generally credited with creating the first word palindrome around 300 BC. The earliest references to a numeric palindrome, 12345654321, are found in Indian Sanskrit mathematical literature written around AD 850.
What Does It Mean to Do Math in Today's Classrooms?
Authors: Martinie Sherrie , Marcoux Cheryl
Published: December 2020

According to Marilyn Burns, students will define mathematics based on how they do math. If the majority of their time is spent reciting a list of properties and following a sequence of steps to solve equations, students will think that algebra involves only memorizing a list of rules. If they spend most of their time in geometry practicing vocabulary, they will look at it as simply being a subject in which they spout back definitions. If they spend the majority of their time on paper-and-pencil computation, they will think that mathematics simply involves performing isolated operations on numbers (Burns 1995). On the other hand, students will have a very different definition of the subject if the focus in the mathematics class is on understanding concepts and developing meaningful procedures. Students can learn all of the activities above in a way that will allow them to apply these skills to new situations or know when to use a particular skill or piece of factual knowledge.
Food for thought from Jay's Diner: Marble Tournament
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 it.
Spotlight on the Principles: The Equity Principle through the Voices of African American Male Students
Authors: Berry Robert
Published: December 2020

Calvin is a sixth-grade African American male student in a school district in the southeastern United States. As an elementary school student, Calvin earned the highest level of achievement on his state's standardized mathematics test in grades 3, 4, and 5. In addition, he scored in the 98th percentile on the Iowa Test of Basic Skills in mathematics. On all objective measures in mathematics, Calvin performed well and, in most cases, excelled. In addition, he earned A's and B's consistently in mathematics. Calvin stated that mathematics is his favorite subject and that mathematics comes naturally to him and is easy. He loves challenging mathematics problems and mathematics puzzles.
Proportional Reasoning: Lessons from Research in Data and Chance
Authors: Watson Jane , Shaughnessy J.
Published: December 2020

PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics (NCTM 2000) places proportionality among the major concepts connecting different topics in the mathematics curriculum at the middle school level (p. 217). What concerns us about many of the problems presented to students, however, is that they are often posed purely as a ratio or proportion from the start. Often the statement of a problem is a giveaway that a proportion is involved. For example, the question “If 15 students out of 20 get a problem correct, how many students in a class of 28 would we expect to get the problem correct?” does not tap the depth of proportional reasoning that is required for meaningful problem solving.
A Chilling Project Integrating Mathematics, Science, and Technology
Authors: Schooler Susan Rodgers
Published: December 2020

Principles and Standards for School Mathematics states that in the middle grades “measurement concepts and skills can be developed and used throughout the school year rather than treated exclusively as a separate unit of study” (NCTM 2000, p. 241). One part of my attempt to do that with my seventh-grade students involved a collaborative activity carried out with the school's technology education teacher, Todd Kelley. We decided to combine our classes for one period of the day for two weeks. We both taught seventh graders during that one period: my students were in an Advanced Math class; his students were in an elective Technology Design class. My students had just completed a unit on calculating surface areas and volumes; they were expected to be able to use formulas to find the surface area and volume of basic three-dimensional shapes, including prisms, cylinders, spheres, cones, and pyramids. The technology students had been working on the IDEATE design process, which gives six specific steps that students should follow in a design project: Identify and define the problem; Develop the design brief; Explore possible alternatives; Accumulate and assess the alternatives; Try out the best solution; and Evaluate the results (Todd 1996, p. 17). We wanted to give our students a chance to practice the skills learned from both classes by applying them to a realistic situation and in a setting that gave them maximum room for creativity, problem solving, and divergent thinking. With the help of our media specialist, Kris McGlaun, we were able to create a unit that accomplished these goals.
Students' Strategies for Fair Shares
Authors: Driskell Shannon O. S.
Published: December 2020

Children often begin to construct an informal understanding of fractions before entering school as they learn to share their crayons or snacks fairly with friends. NCTM (2000) recommends that teachers recognize and build on each child's informal knowledge of fractions during grades K–2. In grades 3–5, children should be actively engaged in constructing conceptual knowledge about fraction concepts, with an emphasis on computational fluency as they progress into grades 6–8. The NCTM (2000) further suggests that “The study of rational numbers in the middle grades should build on students' prior knowledge of whole-number concepts and skills and their encounter with fractions, decimals, and percents in lower grades and in everyday life” (p. 215).
Changing Views: Fearless Families Conquering Technology Together
Authors: Hillman Susan , Malotka Cathy M.
Published: December 2020

“THAT'S CHEATING!” SAID A STUDENT, when I announced that calculators would be used to solve the problems we would be working on in class. When I asked him why, he said, “My teacher told me that using calculators was cheating in math.”
The Value of Multiple Solutions
Authors: Kalman Richard
Published: December 2020

All too often, a simple exercise IS dressed up with a few words, designed to look like it might interest a young student, and called “a problem.” These affectations do not make it a problem. A problem should not just force a student to decode words but must “unleash the thinking part of my brain,” one elementary school student said when describing the questions asked on competitions created by the Mathematical Olympiads for Elementary and Middle Schools. A problem is a question for which the reader has no clear-cut algorithm or procedure already in place.
Celebrating 100 Years of Flight: Testing Wing Designs in Aircraft
Authors: Pugalee David K. , Nusinov Chuck , Giersch Chris , Royster David , Pinelli Thomas E.
Published: December 2020

December 17, 1903, Marked a Mile-stone in history when the Wright brothers launched the first powered flight. This initial flight lasted only 12 seconds and traveled only 120 feet. However, one of the later flights that day produced better results, traveling 852 feet in 59 seconds, and was met with great enthusiasm for the possibility of sustained controlled travel. Since that momentous day, flight has changed dramatically, with jets cruising at speeds around 600 miles per hour (mph) at altitudes around 40,000 feet.
Computer-Generated Fractal Art
Authors: Beigie Darin
Published: December 2020

Computer Software Can be a Powerful tool in both the study and creation of fractals. The Snowflake Fractal Generator (Shodor Education Foundation, www.shodor.org) is an appealing example of software that allows students to make fractals similar to the classic Koch snowflake (see fig. 1).
Paving a Way to Algebraic Word problems Using a Nonalgebraic Route
Authors: Amit Miriam , Klass-Tsirulnikov Bella
Published: December 2020

Middle School Teachers Working with students who are making their first steps in solving algebraic word problems are familiar with this phenomenon: young students can successfully cope with intricate real-life mathematics, such as spending pocket money or planning an excursion under constraints, but they become helpless when faced with an ordinary textbook problem.
Mathematics Detective: Mathematics Sleuth for a Day
Authors: Spangler David
Published: December 2020

I was Very Much Intrigued by your “On My Mind: New Tricks for Old Dogs” in the August 2004 issue of Mathematics Teaching in the Middle School.
Examining Mathematics Tasks from the National Assessment of Educational Progress
Authors: Lambdin Diana V. , Lynch Kathleen
Published: December 2020

How Would Your Middle-Grades students approach this algebraic thinking task?
Geometric Probability and the Areas of Leaves
Authors: Hoiberg Karen Bush , Sharp Janet , Hodgson Ted , Colbert Jim
Published: December 2020

Why Do Some Plants Have Larger leaves than other plants? Why do the overall shapes of different kinds of leaves vary? How does one determine the area of a peculiarly shaped leaf? Plant biologists are extremely interested in these questions. A biologist might wonder, for instance, which of two plant species carries out the most photosynthesis. Since the amount of light a plant can absorb for use in photosynthesis is related to its area, the biologist might investigate the photosynthetic capacity of two species by comparing the amount of chlorophyll in leaf pieces of the same area.
Sorting Symbol Strings
Authors: Novak Jodie D. , Jacobs Judith E.
Published: December 2020

Atheme Throughout All Grade Bands of the Algebra Standard of the NCTM's Principles and Standards for School Mathematics is the ability of students to “represent and analyze mathematical situations and structures using algebraic symbols” (p. 222). In the band for grades 6–8, this theme is further articulated as asking students to “develop an initial conceptual understanding of different uses of variables” (p. 222). Although variables sometimes occur alone, more often they occur in expressions, equations, and inequalities. We will refer to letters and numbers combined in equations, inequalities, and expressions as “symbol strings” as do Chazan and Yerushalmy (2003). The role of the variable is often determined by the symbol string in which it occurs; therefore, if students understand the different kinds of symbol strings, they will understand the roles that variables play. We have developed activities that ask students to identify, describe, compare, and classify symbol strings—in other words, to develop a feel for symbol strings. Chazan and Yerushalmy (2003) discuss it in this way: “Skilled performance [in school algebra] involves developing a feel for symbol strings … that indicates what sorts of creatures they are and what should be done with them.”
Families Ask: Why Not Just Tell Students How to Solve the Problem?
Authors: Bay-Williams Jennifer M. , Meyer Margaret R.
Published: December 2020

If you have been attempting to implement the NCTM's Process Standards (Problem Solving, Reasoning and Proof, Communication, Representation, and Connections), then you likely have been posing rich problems for students to solve. As students encounter difficulties, you try to ask questions to guide their thinking, without taking away the challenge of the task. This major shift away from telling students how to solve the problem and having them practice a standard way, however, can result in students reporting to their families that you will not help them in mathematics class.
The Sum of Cubes: An Activity for Review and Conjecture
Authors: Siegel Murray H.
Published: December 2020

Teachers share various student solution approaches from a “Food for Thought from Jay's Diner” problem.
A Problem-Solving Alternative to Using Key Words
Authors: Clement Lisa L. , Bernhard Jamal Z.
Published: December 2020

The pitfalls of using key words to support students when problem solving, and an alternative way (quantitative analysis) to support students' sense-making. Research from this article will show teachers how to use quantitative analysis to help guide problem solving in the classroom.
Every Story Tells a Picture
Authors: Maus John W.
Published: December 2020

One of the Most Wonderful Things about mathematics is that no matter how long you have been teaching, it continues to reveal itself in surprising ways and in unexpected places. One particular revelation occurred to me while reading a book by one of my favorite authors, Kurt Vonnegut, who connected mathematics to literature in a way that I had not thought of before.
John Henry: The Steel Driving Man
Authors: Murphy David E. , Gulley Laura L.
Published: December 2020

Children's Literature Can Stimulate creative ways to introduce new mathematics concepts or reinforce previously learned concepts to students. Primary teachers realize that children's literature is a powerful tool to entice and motivate students to actively explore mathematics. We believe that middlegrades teachers can also connect mathematics and literature to provide meaningful contexts in which students can construct understanding of mathematics ideas. We decided to explore the classic tall tale of John Henry to show how box plots (box-andwhisker plots) can be used to visually display and compare data sets. We took this idea into a sixthgrade classroom for a “John Henry Day.”
Poetry in Motion: Using Shel Silverstein's Works to Engage Students in Mathematics
Authors: Bay-Williams Jennifer M.
Published: December 2020

Through Brilliant, Clever Poems and prose, Shel Silverstein has worked his way into many classrooms and into the hearts of students and teachers. He wrote poems with an obvious knowledge that a child's mind is active and curious, and middle school students love reading and listening to his words. Imbedded in many of his poems and prose are opportunities to do mathematics in ways that will get students' minds “flickerin'.” Using a poem, picture book, or portions of a novel can raise the curiosity of middle school students and can increase their desire to solve mathematics problems. As students engage in solving literature-based mathematics lessons, they are applying mathematics in different contexts and making connections among mathematical ideas, which are expectations outlined in the Connections Standard in Principles and Standards for School Mathematics (NCTM 2000). In the following sections, four delightful Silverstein works prompted engaging mathematics explorations. Each selection focused on a different mathematical strand, although there is much integration of other concepts in the problems.
Making a Million Meaningful
Authors: Ellett Kim
Published: December 2020

Just How Much is 1 Million? IN a time when the world population has been over 1 billion for a century and our country's national debt is over 7 trillion dollars, 1 million of anything does not impress us much. After all, does anyone rush out to buy a lottery ticket when the jackpot is only $1 million? Never mind that it would take the average teacher over twenty years to earn that $1 million. For a number that has ceased to amaze us, 1 million is actually a very interesting number!
Angles, Time, and Proportion
Authors: Pagni David
Published: December 2020

When we use an analog clock to teach children about different angles formed by the hour hand and the minute hand, we have the added benefit of connecting a specific time with a specific angle measure. For example, what is the measure of the angle formed by the hour hand and minute hand at 3:00? See figure 1. Reasoning that the hour hand and minute hand must be perpendicular (a horizontal line intersecting a vertical line), we can conclude that the angle measure is 90°.
“Weigh” to Go! Exploring Mathematical Language
Authors: Adams Thomasenia , Thangata Fiona , King Cindy
Published: December 2020

Sometimes what we say or what we intend to say is not understood by those with whom we are speaking. In classrooms where a specialized vocabulary and structure of communication are necessary for academic success, teachers must be aware of any factors that might impede students' success. In mathematics, for example, language is the primary medium for constructing, sharing, and expanding ideas. Being both proficient with and understanding mathematical language in the form of vocabulary, symbols, expressions, definitions, and so on, are important. Therefore, it is critical for mathematics teachers to consider problems that students might be having with the language of mathematics.
Learning Measurement through Practice
Authors: Pumala Valerie A. , Klabunde Daric A.
Published: December 2020

Inherent in every middle school curriculum is the need for students to measure—the mass of iron filings for a chemical reaction, the appropriate volume when mixing paint, or the length of a board for a shelf. The list is never ending. As endless as the need to measure is the list of questions often heard in the classroom: “What side of the ruler are centimeters on?” “Is this in ounces or milliliters?” “This thermometer—is it in Celsius or Fahrenheit?” The underlying question that we as teachers ask is this: “Why are our students having so much difficulty measuring?”
Teacher to Teacher: Phenomenally Exciting Joint Mathematics-English Vocabulary Project
Authors: Seidel Cynthia , McNamee Julia
Published: December 2020

At the end of a recent school year, the English teacher on my team and I were looking for some fun learning activities that we could do with our eighth graders. At that point, the students were quite comfortable with us and our quirky senses of humor, and we trusted them with an opportunity to be creative and a bit silly while also producing a learning outcome. After brainstorming some ideas, we decided to create a vocabulary project. Since so many words that are commonly used in mathematics have a different meaning in the real world, it was fairly easy to create a list of mathematics-English vocabulary words (see worksheet 1).
Menu of Problems: May 2005
Authors: Burns Sue
Published: December 2020

A light fare to whet your mathematical appetite.
Spotlight on the Principles: The Pursuit of Mathematics for All!
Authors: Rubel Laurie , Meyer Margaret R.
Published: December 2020

“EXCELLENCE IN MATHEMATICS education requires equity—high expectations and strong support for all students.” Thus begins the Equity Principle found in Principles and Standards for School Mathematics (NCTM 2000, p. 12). It goes on to detail some of the dimensions of equity: high expectations, strong support, reasonable accommodations, worthwhile opportunities, and access to technology. People are unlikely to disagree with the stance implied by the Equity Principle. “Mathematics for all” is a rallying cry akin to “liberty and justice for all.” To disagree with the Principle would be like speaking out against motherhood or apple pie. Yet the pathway to mathematics for all is not always clear. What does equity in mathematics look like in our classrooms and schools? How would individual teachers proceed if the Equity Principle was guiding their teaching?
7 Things I Learned about Teaching (and Assessing) Mathematics from My Sensei (Karate Instructor)
Authors: Brahier Daniel J.
Published: December 2020

Three years ago, two of my sons (then eight and ten years old) asked me if they could take karate lessons at a local dojo (karate school) that taught a traditional Okinawan style. My wife and I agreed to allow them to start lessons under one condition—that I could take lessons along with them. While in high school, I had studied a similar style of karate for almost four years but had not returned to the discipline in twenty-five years. My sons were excited to have me aboard, and we started taking lessons at the lowest rank of white belt.
Walking through Space: A New Approach for Teaching Functions
Authors: Kalchman Mindy S.
Published: December 2020

Curricula in the middle school grades often overlook the fact that functions have spatial qualities that students can easily recognize and interpret. This oversight is critical; the visual images associated with the spatial aspects of mathematics may be the gateway to success for many visual learners who are at a loss when faced with the traditionally numerical focus of middle school mathematics. For example, a table of values is a numerical representation that is used primarily for recording and storing coordinate pairs after they have been calculated by plugging values into an equation. However, a table of values is also rich with spatial cues that speak to the shape (straight or curved), direction (increasing or decreasing), and steepness of a function's line or curve. Recognizing, interpreting, and understanding these cues allow students to develop important reasoning skills that are embedded in many of the NCTM's Standards for communication and problem solving.
The Thinking of Students: Have You Lost Your Marbles? Three Creative Problem-Solving Approaches
Authors: Edwards Brent
Published: December 2020

As an eighth-grade mathematics teacher, I have come to value the process of communicating about mathematics. I have learned that it is vital for students to explain and describe their work, not just simply submit their solutions. I use several methods to encourage my students to share their problem-solving processes, such as asking them to show their work on the board for the class, write essays, and answer a simple question like “How did you arrive at your answer?” Of these tasks, writing takes the most time and effort, but the results can be extraordinary. The formal essay requires students to organize their thoughts, prepare a logical sequence, and explicitly communicate the details of their work. This “thinking about thinking” translates to an improved problem-solving ability and greater conceptual understanding.
Incorporating Spatial Ability Instruction in Teacher Preparation
Authors: Sundberg Sue E. , Goodman Terry A.
Published: December 2020

A learning environment for the preservice mathematics teacher. Article discusses how development of spatial abilities can be incorporated into courses for these students. Includes lesson plans and activity sheets with answers.
Addressing Diversity in the Mathematics Classroom with Cultural Artifacts
Authors: Neel Kanwal Singh
Published: December 2020

To paraphrase john dewey, diversity is one word, but many things. It could describe students from diverse backgrounds and refer to ethnicity, culture, gender, language, learning style, socioeconomic level, intellectual ability, and physical capability. The question arises as to how one should teach mathematics to all learners, regardless of their diversity. The experiences that students bring to the classroom; the ways they interpret their learning experiences; and the knowledge, skills, and attitudes with which they leave the classroom will vary.
Embedding Algebraic Thinking throughout the Mathematics Curriculum
Authors: Vennebush G. Patrick , Marquez Elizabeth , Larsen Joseph
Published: December 2020

Like love, algebra is where you find it. You can locate it almost anywhere in the middle school curriculum if you know where to look and what to look for. But with so many demands on our time, we often forget to look. We take problems at face value, and we assume that a geometry problem is just a geometry problem or that a data analysis activity is only about data analysis. If we scratch below the surface, however, we can find rich opportunities for algebraic thinking lurking in number explorations, measurement tasks, and geometry investigations.
Families Ask: Games in the Middle School
Authors: Martinie Sherri
Published: December 2020

The purpose of the “Families Ask” department is to help classroom teachers respond to questions commonly asked by caregivers of their students. A commonly asked question will be posed; a rationale for the response will be presented for teachers; and a reproducible page will be offered for duplication and distribution to parents, other caregivers, administrators, or community members—anyone involved in the mathematical education of middle school children.
Developing Students' Understandings of Variable
Authors: Stephens Ana C.
Published: December 2020

Algebra's “Gatekeeper” status has prompted several in the mathematics education research community (e.g., Davis 1985; Kaput 1998; Olive, Izsak, and Blanton 2002) to urge educators to view algebra not as an isolated course but as a continuous K–12 strand that is present throughout the entire mathematics curriculum. Central to the transition from arithmetic to algebraic reasoning is the concept of variable (Schoenfeld and Arcavi 1988). Schoenfeld and Arcavi argue that despite its importance, most mathematics curricula offer little to assist students in developing ideas about this concept. They assert that instead of providing students opportunities to practice manipulating terms and solving for unknowns, teachers should encourage students to view variables as shorthand tools for expressing already-understood ideas about varying quantities. This article describes a mathematical problem that can encourage students to view variables in this way while confronting a common misconception.
Monkey Paws, English Pounds, and Leagues: Using Literature in the Middle School
Authors: Martinie Sherri , Marcoux Cheryl , Stramel Janet
Published: December 2020

In our district, all teachers are responsible for teaching reading and problem solving to all students. To meet this challenge, teachers were trained in reading instruction and in teaching problem-solving skills and strategies. These expectations encouraged all teachers to integrate reading and problem solving into their classroom instruction. The basis of the reading training stemmed from the book I Read It, But I Don't Get It (Tovani 2000). Tovani describes what good readers do to recognize and clear up confusion and suggests several “Fix Up” strategies. Our emphasis was on enabling students to “make a connection between the text and: Your life, your knowledge of the world, or another text” (p. 51).