bluuebunny/crossref_metadata_2025 huggingface dataset:
Mathematics Investigator: It's About Time (Or Is It?)
Authors: Davis Robert L. , Ferreira Linda M.
Published: December 2020

Two weeks from today, most Americans will enjoy the annual luxury of an extra hour's sleep, having turned their clocks back. “Why only once a year?” says John Brady, who believes he has found a formula to give us that extra hour of sleep every Sunday.
The Thinking of Students: Math Island: An Adventure
Authors: Catlin Catherine
Published: December 2020

This story was written to go along with the map of an island. We had been learning about coordinates, scale, and keys. We enlarged an island to scale and then developed a theme with illustrations and a story. My story uses different types of numbers and gives them some personality. I hope your students enjoy my story.
Now & Then: Mapping Out a Career Path: From Sargon to Satellites
Authors: Barnes Sue , Michalowicz Karen Dee
Published: December 2020

Now… John Snyder has always loved maps, beginning with his elementary school days in Indianapolis. Everyone who knew John identified him with the maps he loved. Oftentimes the greeting cards he received carried a map motif. When John reached high school, his interest in maps was further developed as he studied trigonometry, calculus, and algebra and could apply the concepts developed in those courses to map projection, the science of transforming the earth's spherical surface to a flat, two-dimensional piece of paper. After high school, John's avid interest in chemistry, coupled with World War II's push for more scientists and engineers, helped him choose a career path as a chemical engineer.
On My Mind: It Is Time to Question Fraction Teaching
Authors: Groff Patrick
Published: December 2020

As teachers well know, fractions instruction has long been a dominant part of the mathematics curriculum of the middle school. In this respect, the six mathematics textbooks currently adopted for use in California's fifth and sixth grades on the average devote 15 to 17 percent of their pages, respectively, to the study of fractions. One sixth-grade text (Hake and Saxon 1992) dedicates 24 percent of its content to fractions.
No Kidding, My Mom's Got Homework?
Authors: Bratina Tuiren A.
Published: December 2020

The american public hears repeatedly that parents and guardians need to take a more active role in their children's education. The Carnegie Task Force (1989) recommends that families become reengaged in the education of their young adolescent children. Mathematics educators must promote practices that encourage involvement of those at home.
Activities from African Calendar and Time Customs
Authors: Shirley Lawrence H.
Published: December 2020

The connections standard of the curriculum and Evaluation Standards for School Mathematics (NCTM 1989) encourages looking for mathematics in a much wider range of applications and cultures than in the past. Teachers interested in applying this standard through a multicultural curriculum have found examples from various cultures around the world, including those of Africa.
February's Menu of Problems
Authors: Greer Gerry , Wantuck Larry
Published: December 2020

A light fare to whet your mathematical appetite.
Cartoon Corner
Authors: Lawrence Ann
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 extra credit 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, double-spaced, for review by the editor of this section. The cartoons must include the date and the newspaper syndicate that holds the copyright.
Mathematics Investigator: Florida Panther has Uncertain Future
Authors: Davis Robert L.
Published: December 2020

Ours isn't the same cat as an African lion that stretches to 9 feet from nose to tail and weighs 400 pounds; a Florida panther male is only 7 feet and weighs from 100 to 150 pounds. All big cats sleep away most of their hours, waking up chiefly at dusk or dawn to go about the business of hunting, and explorers probably saw them only indistinctly as shadowy forms slipping at a distance through marshy hammocks and palmetto/pine woodlands.
By Way of Introduction
Authors: McClain Kay
Published: December 2020

Too often, middle-grades students view mathematics as a series of discrete topics. They are unable to understand the connected nature of the discipline because the presentation of material often focuses on isolated concepts to be mastered for the moment. Students need to experience mathematics in such a way that they begin to view it as an integrated whole, acknowledging its relevance and usefulness both in and out of school. By offering students classroom experiences that encompass topics embedded in real-world contexts, in connections with other disciplines, and in connections within mathematics, the vibrant discipline of mathematics actively engages students in their own learning.
The Pond: Doing Research Together
Authors: Kinneavy Kevin
Published: December 2020

Interdisciplinary approaches to teaching all subjects are central to the middle school concept. Such approaches can be especially useful for teaching mathematics, a subject that has traditionally been the bane of many students' existence. The NCTM recognizes the importance of interdisciplinary instruction in its Curriculum and Evaluation Standards for School Mathematics (1989, 84).
Mathematics Investigator: You Hit the Jackpot!
Authors: Reeves Andy
Published: December 2020

A $1 million jackpot is not a $1 million prize; you'd be a thousandaire-not a millionaire.
Lines by the Book
Authors: Mann Robert
Published: December 2020

Textbooks are a wonderful resource for the mathematics teacher, but at times they can also be limiting. Many teachers, for example, feel compelled to follow the curriculum in the text and, as a result, often lack time to incorporate other rich, engaging mathematical activities. It is often easier to present a topic “by the book” rather than develop alternative lessons—especially if these alternative lessons involve finding, building, or purchasing enough manipulatives or related materials for the entire class to use. Consequently, it is important that teachers have access to activities that can be easily integrated into any class or any curriculum—regardless of text or available materials. I have found that this Lines by the Book investigation is such an activity.
Turning Origami into the Language of Mathematics
Authors: Cipoletti Beth , Wilson Nancy
Published: December 2020

NCTM (1989) proposes using everyday objects, such as paper, to enable students to explore geometric relationships and vocabulary. Paperfolding and other types of hands-on activities have been found to increase students' ability to communicate mathematically and foster their understanding of mathematical concepts. These tasks help students move from concrete to abstract reasoning. Origami projects use everyday objects, incorporate geometric relationships, create opportunities for communication, and produce aesthetically pleasing objects to share with others while providing an opportunity to acquire knowledge that bridges diverse cultures. Using origami activities in the classroom allows opportunities for teacher-to-student, student-to-student, and school-to-community communication using geometric language.
Express Yourself with Algebra? This Game Delivers!
Authors: Moseley Bryan
Published: December 2020

Students often have trouble understanding the mathematical meaning of algebraic expressions. The game discussed in this article is a fun activity I developed for students to communicate about algebraic expressions in a fantasy setting that makes the content meaningful. The game has been tested successfully in three different prealgebra classrooms as part of a thematic unit that encouraged students to consider the functional relationships involved in running a pizzeria (Brenner, Mayer, Moseley, Brar, Duran, Smith-Reed, and Webb 1997). However, the game's flexibility allows it to be adapted to a standalone activity that meets the needs of a particular classroom.
The Art of Posing Problems and Guiding Investigations
Authors: Dugdale Sharon , Matthews James , Guerrero Shannon
Published: December 2020

Peek into his classroom any day of the week and you are likely to find Mr. Hill (a pseudonym) either engaged in fast-paced dialogue with his students or circulating around the room while small groups delve into a problem that the class has formulated. The atmosphere is lively and intense, and there is a sense that something big is in the process of being discovered. With decades of teaching experience at a range of grade levels, Mr. Hill is also a popular leader at teacher workshops where he immerses participants in mathematical investigations appropriate for use with their students.
The Thinking of Students: Overlaps
Authors: Weeden Katherine
Published: December 2020

I PRESENTED THE OVERLAPS PROBLEM THAT appeared in the February 2004 issue of the journal to my students as a “Snow Day Problem.”
Using Ancient Egyptian Fractions to Review Fraction Concepts
Authors: Edwards Thomas G.
Published: December 2020

Much of What We Know Today About the mathematics of ancient Egypt is contained in a papyrus scroll that was copied from an earlier scroll by the scribe Ahmes in about 1650 BME (before the modern era) (Boyer 1968). A fascinating feature of ancient Egyptian mathematics is its treatment of common fractions. In most cases, the Egyptians used only unit fractions, that is, fractions with numerators of 1. The one common exception is 2/3, and they would occasionally use fractions of the form n/(n + 1). However, both forms are complements of unit fractions.
Collaborating to Introduce Algebra
Authors: Van Dyke Frances , Tomback Jessica
Published: December 2020

During Spring 2003, a Teacher and A university professor worked together to introduce algebra using a functional approach in a sixth-grade classroom in a Washington, D.C. public school. The university professor, Franny Van Dyke, was on leave from teaching for the spring semester, had a daughter in the class, and had a strong interest in introducing functions using graphs. The teacher, Jessica Tomback, felt that her students were ready for algebra and was intrigued by the idea of working with graphs. Van Dyke came into the classroom for one hour each week for three months. The following is an account of the project from each of their perspectives.
Take Time for Action: Studying Student's Reasoning in Writing Generalizations
Authors: Krebs Angela S.
Published: December 2020

How do Your Students do When you Ask them to generalize? Do they reason numerically or geometrically? Can they make connections among representations? Algebraic reasoning and representations are foundational for middle-grades students as they prepare to enter high school and pursue future studies. Many policymakers suggest an “algebra for all” approach (Silver 1997). Although there is no consensus on what this algebra might look like, many agree that understanding patterns, relations, and functions and representing and analyzing mathematical situations with variables are important components of algebra (NCTM 2000). Studying patterns in middle school builds a foundation for more formal investigations with functions in high school.
Rethinking Fair Games
Authors: Coffey David , Richardson Mary
Published: December 2020

Take a Moment to Think About how you would answer this question: What does it mean for a game to be fair? When asked this question, many people focus on playing fair or the fact that neither player cheats.
Spatial Thinking Tasks Can Change Students' Attitudes
Authors: Ameis Jerry A.
Published: December 2020

Geometry Provides Rich Opportunities for encouraging mathematical thinking, especially when making and validating conjectures and imagining and visualizing what happens when shapes are manipulated and altered (NCTM 2000). In particular, spatial thinking tasks can stimulate student interest in mathematics, especially for those who have been turned off the subject.
Mailing a Publication: Exploring Linear and Step Functions in a Real-World Context
Authors: Billings Esther M. H. , McClure Melanie Schultz
Published: December 2020

How the context of mailing a package was used with seventh-grade students to explore and connect the different representations of step and linear functions. Students will learn geometry concepts and algebra concepts through this hands-on, real world application.
Harry Potter and the Magic of Mathematics
Authors: McShea Betsy , Vogel Judith , Yarnevich Maureen
Published: December 2020

The Middle School Years Represent an important time in the formation of an individual's lifelong attitudes toward mathematics. Middle school teachers are faced with the challenge of convincing their students that mathematics is an exciting, useful, and creative field of study. Interdisciplinary approaches to mathematics have been useful in accomplishing this goal. In particular, connecting mathematics to literature is an inventive way to capture students' interests, since examples from literature can be used to teach important mathematical concepts in an exciting and innovative manner. Many classic literary texts are rich in mathematical content, including Alice in Wonderland by Lewis Carroll and Flatland by Edwin A. Abbott. However, to stimulate students' interest, it is important to find interconnections between mathematics and current popular children's literature.
Introduction to Volume 11: A Second Decade of Mathematics
Authors: Zilliox Joe
Published: December 2020

Volume 11 marks the beginning of a second decade for Mathematics Teaching in the Middle School (MTMS). The journal has continued to grow since NCTM initially responded to readers' concerns and developed a publication addressing the specific interests and needs of middle school teachers and students. This focus on teachers and students that guided editorial decisions throughout the first ten years will continue to influence the contents and quality of the journal into the future.
You Made It through the Test; What about the Aftermath?
Authors: Brown Scott A.
Published: December 2020

A writing activity that is used after graded exams are handed back to students. Article discusses the benefits of students reflecting on their performance by writing about their strengths and weaknesses on an exam. Includes examples of student writing, scoring rubrics, and suggestions for implementation.
Teacher to Teacher: Indian Lightning Multiplication
Authors: Lewis Jacob M.
Published: December 2020

In addition to teaching content area information, we as teachers also want to help our students develop a conceptual framework for mathematics. Without this conceptual framework, students often have trouble internalizing new information and mastering topics. For example, a student who does not really understand whole-number multiplication will more than likely have difficulty multiplying binomials. For this reason, in an introductory algebra course, a teacher needs a way to reinforce multiplication skills in an age-appropriate way while laying the groundwork for algebra topics involving multiplication.
Using Clock Arithmetic to Teach Algebra Concepts
Authors: Brown Elizabeth M. , Jones Elizabeth
Published: December 2020

Learning algebra concepts can be difficult for middle school students. One reason may be because we work in only one number system, the set of real numbers. Students have only one frame of reference to provide examples of abstract concepts, such as the additive and multiplicative identities, additive and multiplicative inverses, and connections among the operations. These concepts are essential in solving equations. For example, we can think of an equation like 3x + 4 = 7 in the following way: Begin with a number, multiply it by 3, and add 4. If the answer is 7, what number did we start with? To solve this type of equation algebraically, we just undo what was done to the original number, that is, we add -4, the additive inverse of 4, and multiply the result by 1/3, the multiplicative inverse of 3. We can also think of this as subtracting 4 and dividing by 3, because addition and subtraction are inverse operations as are multiplication and division. The ideas of inverse operations and inverse elements are, therefore, central to algebra. Many students understand these ideas well enough to do simple problems like 3x + 4 = 7 but get confused with more difficult problems, such as 3(x + 6) + 2 = 5x + 5, where it is not as easy to see the order in which to undo these operations. We use finite systems to help students understand these key concepts in algebra, including additive and multiplicative identities, additive and multiplicative inverses, closure, and the relationships between addition and subtraction and multiplication and division.
Giving Exponential Functions a Fair Shake
Authors: Wanko Jeffrey J.
Published: December 2020

In a traditional mathematics curriculum, middle school students' experiences with functions in algebra are usually limited to linear relationships. However, a number of realworld experiences (e.g., compound interest, population growth, and the amount of a drug remaining in a subject's bloodstream) involve exponential functions. In this article, I describe two problems involving exponential functions and some basic probability concepts that were used in an eighthgrade class, which led to greater student understanding about the general form of an exponential function. I also describe how students used graphing calculators to help them make sense of the values that are used in the general form and how they relate to the context of the problem.
Teacher to Teacher: What Is the Name of This Game?
Authors: Mahoney John F.
Published: December 2020

Students of all ages enjoy playing games. Many games illustrate or reinforce mathematical principles. The following game does both.
Mathematics of Alice's Adventures in Wonderland
Authors: Taber Susan B.
Published: December 2020

Alice's adventures in wonderland, the captivating book first published in 1865, began as a story told by a young mathematics lecturer to the three Liddell sisters, ages 8 to 13, during an afternoon of rowing on the river. The story might have evaporated into the summer's air but for Alice Liddell, then aged 10, who asked Charles Lutwidge Dodgson to write it down for her. Although Dodgson himself illustrated the first copy, which he gave to Alice as a Christmas gift in 1864, he later expanded the story, had it illustrated by the famous artist John Tenniel, and published it under the name of Lewis Carroll (Cohen 1995).
An “Arithmetic” Thinker Tackles Algebra
Authors: Armstrong Alayne C.
Published: December 2020

In her work concerning algebraic thinking, Kieran notes that students learning algebra tend to fall into two groups—“algebraic” thinkers who use undoing as a way to solve equations, and “arithmetic” thinkers who use trial-and-error substitution to solve equations. “Algebraic” thinkers rely on inverse operations; for example, this group would solve 5 + a = 12 by saying 12 – 5 = 7, ignoring the variable itself. When these students move on to more complex equations, such as 3a + 3 + 4a = 24, they tend to overgeneralize and get stuck (“24 divided by 4, minus 3, minus, um, no, divided by 3”). They are unable to balance the equation because they have not assigned enough significance to the role of the equal sign within the equation- solving process (Kieran 1988, p. 94). When arithmetic learners speak of their solutions, however, because they are using trial-and-error substitution, Kieran finds that they discuss the balance required between the two sides of the equation. She further states that of these two, “arithmetic” thinkers are using a method that “may provide a more intuitive basis for the more structural solving methods” (1992, p. 401). I was curious to see if an eighth-grade student whose thinking could be characterized as “arithmetic” would indeed find this type of thinking a help or a hindrance to her further development of algebraic concepts.
Using Creative Writing and Literature in Mathematics Classes
Authors: Halpern Cara M. , Halpern Pamela A.
Published: December 2020

Using literature in mathematics education is not a novel idea. Many books and articles have been written on how and why to introduce children's literature in elementary mathematics education over the past ten years, but little has been done at the middle school and high school levels.
Math Roots: Serving Up Sierpinski!
Authors: Adams Thomasenia Lott , Aslan-Tutak Fatma
Published: December 2020

The sierpinski triangle, created in 1916, has some very interesting characteristics. It is an impressive and valuable topic for mathematical exploration, since it combines Euclidean geometry (triangles and measurement) with fractal geometry. The Sierpinski triangle, also known as the Sierpinski gasket, is a fractal formed from an equilateral triangle. It is one of the most popular fractals to construct and analyze in middle school mathematics lessons. Since the 1960s, it has been possible to design fractals using a computer program, especially the complex examples that are often difficult to construct by hand. However, students can easily duplicate the Sierpinski triangle.
International Perspectives: Using Dialogue
Authors: New Rachel
Published: December 2020

In preparing to teach speed to a Y10 intermediate class in a multicultural secondary modern, I was aware of the difficulty in getting students not only to follow a complex logical argument, but also to be able to reproduce it in a variety of different problem situations. Normally one would set out the working out for the problem below as a sequence of calculations, with some verbal but fairly formal explanation in a whole class discussion. Some questions and answers would feature, and students would generally be able to follow the logic of each step.
Solve It!: Generosity
Authors: Kersaint Gladis
Published: December 2020

The editorial panel is presenting the following problem to stimulate submissions to “The Thinking of Students” department. We encourage teachers to try this problem with students and analyze the different ways that they use to solve it. Feel free to photocopy this problem for use in your classroom.
Teacher to Teacher: Playing around with Mono-pi-ly
Authors: Kroon Cindy
Published: December 2020

According to the geometry standard in Principles and Standards for School Mathematics, “In grades 6–8, all students should precisely describe, classify, and understand relationships among types of two- and three- dimensional objects” (NCTM 2000, p. 232). The Measurement Standard goes on to state, “In grades 6–8, all students should develop and use formulas to determine the circumference of circles” (NCTM 2000, p. 240). In addition, South Dakota's Measurement Standard for Grade 7 delineates what mathematics students should know, such as “Given the formulas, find the circumference, perimeter, and area of circles” (South Dakota Department of Education 2004).
By Way of Introduction: Connecting Mathematics and Science in the Middle School
Authors: Zilliox Joe , Schultz James
Published: December 2020

Although mathematics has been called the “Queen of the Sciences,” are your students missing the rich connections that are inherent between mathematics and science? Can they use their understanding of science and mathematics to answer these questions: How do we measure earthquakes? How much does it cost to grow your own lettuce? How can we map the seafloor? The articles in this focus issue show how classroom teachers engage their students in a science context to help them make that connection.
Using Engineering to Understand Reciprocal Functions
Authors: Sharp Janet , Zachary Loren , Luttenegger Greg
Published: December 2020

Graphic representations of numerical data can illustrate relationships within the data that students might not otherwise notice. When students investigate common numerical data patterns like 2, 4, 6, 8 or 1, 3, 5, 7, they can easily recognize that differences between terms are constant.
Modeling the Seafloor
Authors: Clary Kim M.
Published: December 2020

Using geometry to do geology, “Modeling the Seafloor” is a handson activity in which students construct mathematical and physical models to represent a portion of the ocean's floor. While making a three-dimensional model of the seabed, students use fractions, decimals, conversions of decimals to fractions, scale drawings, formulas, visualization, and other skills. They see mathematics placed in a real-world context, and they use their mathematics skills as part of a process to achieve a meaningful end result.
Mathematics Detective: A Hidden Solution to the Water Jar Problem
Authors: Martinez Joseph G. R. , Martinez Nancy C.
Published: December 2020

The Water Jar problem, which is discussed here, has been around since the early 1800s. The problem involves finding a specific amount of water by pouring water in and out of two different containers. In fact, one movie highlighted elements of the Water Jar problem in its storyline. The leading characters, one a police sergeant and the other a businessman, become involved in solving puzzles and problems related to a crime. The crime, of course, involves the classic Water Jar problem. Two containers, an unlimited amount of water but a very limited amount of time, and a pressure-sensitive mechanism that will trigger a catastrophe add drama to the movie and highlight the need for clear-headed logic to avoid a potentially disastrous dilemma.
Developing Algebraic Thinking through Pattern Exploration
Authors: Lee Lesley , Freiman Viktor
Published: December 2020

Pattern exploration is A pivotal activity in all mathematics, indeed in all the scientific disciplines. Children who are attempting to express perceived patterns mathematically are in an excellent position to learn algebraic language and engage in algebraic activity. Principles and Standards for School Mathematics (NCTM 2000) acknowledges the relationship of pattern exploration and algebraic thinking by placing pattern work within the Algebra strand. Yet one can undertake considerable pattern exploration without engaging students in any algebraic thinking whatsoever and teachers may, themselves, be unclear about how patterns can be used to further algebraic thinking. Work with repeating patterns in the early grades, or teaching patterns as a “topic” in the middle grades, may not foster the development of algebraic thinking in students. In this article, we will address this question: How can teachers exploit pattern work to further algebraic thinking and introduce the formal study of algebra in middle school?
Why, Why Should I Justify?
Authors: Lannin John , Barker David , Townsend Brian
Published: December 2020

Suppose that in your classroom, as part of an algebraic reasoning unit, students were asked to find a general rule for the Beam Design problem in figure 1. Examine the rules and explanations below to determine those that you would view as acceptable and those that you believe would be unacceptable. Also consider how you would respond to students who provided such explanations.
Where is the Moon Tonight?
Authors: Mathews Susann M. , Cornell Kevin F. , Basista Beth A.
Published: December 2020

When students are asked to observe the Moon at the same time every night for a month, they remark on one of their first surprises: “The Moon wasn't there last night,” even though “it was there the night before.” During the course of a six-week investigation, seventy-five fifth-grade students discovered the reason for not always seeing the Moon at the same time in the same place.
Using Thought Bubble Pictures to Assess Students' Feelings about Mathematics
Authors: Zambo Debby , Zambo Ron
Published: December 2020

We live in a Mathematical World (NCTM 2000), and students differ greatly in their pleasure in doing mathematics. Take two students as examples. Student A is an adolescent in prealgebra who feels good about mathematics. Student A enjoys working on challenging problems and will often request more time to work on mathematics projects in class. This student loves working cooperatively with others on problems and works intensely and deeply when numbers are involved. Student A is excited about taking algebra next year and envisions mathematics as an integral part of both life and career when discussions turn to life beyond the classroom. Adolescents like student A enjoy the time they spend doing mathematics, and they succeed at it. Their success causes them to have positive feelings about mathematics, to feel happy and relaxed when they do mathematics, and to be motivated to accept the challenges that mathematics provides.
Do You Understand Your Algorithms?
Authors: Pickreign Jamar , Rogers Robert
Published: December 2020

Because an increasing number OF school districts requires the successful completion of an algebra course to graduate from high school, many middle school teachers are beginning to focus more attention on introducing algebraic thinking to their students (NCTM 2003). Consequently, it becomes important to consider ways to ensure that these experiences are meaningful and connected to arithmetical experiences from the earlier grades. We believe that presenting middle school students with activities that involve exploring computational algorithms—how and why they work—can contribute to the development and promotion of algebraic thinking.
Families Ask: Data Analysis and Statistics in the Middle School
Authors: Martinie Sherri
Published: December 2020

Students are more motivated to learn mathematics when they recognize that it has value. Collecting and analyzing data that is meaningful and interesting to students can emphasize valuable mathematics.
Learning from Voices in Classrooms
Authors: Kastberg Signe E. , Otoupal-Hylton Wendy
Published: December 2020

Wendy and i met and began working together in hopes of finding some consistency in what we each saw as our chaotic lives. Wendy began her second career in 2002 as a middle-grades teacher. She was responsible for teaching six middle school mathematics classes. Each class of twenty-two to twentyeight students consisted of a mixture of grades 6, 7, and 8 students of varying abilities. Students entering grade 6 had Wendy as their mathematics teacher for three years. She quickly found traditional texts unsatisfying for her students and sought alternative curricula.
If I Only Had One Question: Partner Quizzes in Middle School Mathematics
Authors: Danielson Christopher , Luke Michele
Published: December 2020

Imagine that you are visiting a seventhgrade classroom. You arrive shortly after class has begun to find students working in pairs and having discussions. You see the teacher monitoring the students' work, reading over their shoulders, and responding to an occasional question. It may be surprising to learn that today is quiz day. Students are taking a partner quiz.
Take Time for Action: Similar Shapes and Ratios
Authors: Frost Janet Hart , Dornoo Michael D.
Published: December 2020

A problem-solving activity on any mathematics topic offers opportunity for rich insight into students' thinking processes and levels of understanding. Problem solving is a reasoning-oriented activity rather than a rote application of memorized procedures. It can serve as an especially fruitful approach to the study of geometry. Since students often do not have enough experience exploring geometric relationships through inquiry and conjecturing (Mistretta 2000), they often have difficulty understanding and applying concepts of geometry (Battista 1999). When students are given the opportunity to explore figures, such as similar polygons, they are likely to encounter bumps along the path toward developing new geometric knowledge, and their teacher has the opportunity to observe their progress.
It's the Thought That Counts: Reflecting on Problem Solving
Authors: Roberts Sally , Tayeh Carla
Published: December 2020

We Still Talk About The Time We were at the airport waiting to fly to a conference, both of us carrying formidable stacks of our students' metacognitive journals to read on the plane. as we looked at each other's papers and thought about the work ahead, we could not help but ask ourselves why we gave this assignment. as we began to read and share excerpts from students' journals, we did not need to ask why; we knew.
Using Alice in Wonderland to Teach Multiplication of Fractions
Authors: Taber Susan B.
Published: December 2020

Alice's adventures in wonderland by Lewis Carroll is a captivating story that appeals to students in the middle grades. It also provides an opportunity to help students develop an understanding of some complex mathematical content, particularly the multiplication of fractions, which is introduced in the middle grades.
Engaging Contexts for the Game of Nim
Authors: (andy) Reeves Charles A. , Gleichowski Rosemarie Reeves
Published: December 2020

Middle school teachers realize the value of students playing games in mathematics classes if those games emphasize problem-solving strategies, algebraic reasoning, or spatial sense.
Mathematics Detective: Talk Isn't Cheap
Authors: Walton Candide , Hanebrink Tamela
Published: December 2020

The Richardson family is shopping. Lacy has found the perfect dress for a dance, and Kyle has located a video game. However, they have a problem. Neither of them has any money, nor do they know exactly which store their parents are in. The solution? Cell phones for everyone. Currently, only the parents have cell phones. If every member of the Richardson family, a family of four, had cell phones, then the children could contact their parents. Later, the family decides that, in general, cell phones for all will be beneficial for parents and children.
Psychology of Learning in the Junior High School
Authors: Fehr Howard F.
Published: December 2020

Although we now often use the term “middle school,” the educational tenets described in this article remain viable today. Students from the ages of 10 to 15 are prime for meaningful learning. The notion of a general mathematics curriculum containing substantial review of earlier concepts is not acceptable; significant mathematics and problem-solving techniques need to be taught. Just as it was fifty years ago, the role of the classroom teacher is significant in providing motivation and approval.
Instructional Games with Calculators
Authors: Judd Wallace
Published: December 2020

The powerful idea of this article illustrates how to use readily available technology to enhance the learning of concepts in ways that are fun for students. It can easily be adapted to all levels, from kindergarten through calculus.
Short Takes: Another Good Idea: Computing Quartiles: It's All in the Cards!
Authors: Barlow Angela
Published: December 2020

The procedure for computing quartiles is relatively simple; or is it? Step 1: Find the median of your data set. Step 2: Find the median of the lower half of the data set. Call this number Q1. Step 3: Find the median of the upper half of the data set. Call this number Q3. You are finished.
Solve It! Multiples of Four
Authors: Mooney Edward S.
Published: December 2020

The Editorial Panel is present ing 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 students solve it.
Magic with Mayan Math
Authors: Overbay Shannon R. , Brod Mary Jean
Published: December 2020

The ancient Mayan system of numeration is quite sophisticated, yet the basics are beautifully simple. It is an appealing system to teach to students of almost any age. Mayan numerals are composed of only three symbols, but with the power of place value and a symbol for zero, large numbers can be concisely written. With their interest in astronomy and the passage of time, the Maya (located in what is now Central America and Southern Mexico) developed accurate calendars using this numeration system. Mayan merchants often used cacao beans in making computations to keep track of their business transactions (Closs 1986).
A Mathematical Private Eye
Authors: Lee Ji-Eun , Kim Kyoung-Tae
Published: December 2020

An instructional idea where students determine an individual's secret personal information using generalized algebraic thinking. Article discusses going from specific numerical calculation to generalized algebraic patterns of solutions. Includes sample problems with solutions.
Irrational Numbers Can In-Spiral You
Authors: Lewis Leslie D.
Published: December 2020

Introducing students to the pythagorean theorem presents a natural context for investigating what irrational numbers are and how they differ from rational numbers. This artistic project allows students to visualize, discuss, and create a product that displays irrational and rational numbers.
Guitars, Violins, and Geometric Sequences
Authors: Barger Rita , Haehl Martha
Published: December 2020

Middle schoolers are not the easiest students to motivate. There are so many things going on in their lives that often school is valued more for its social opportunities than for its educational activities. Imagine the reactions of your students as they come into your room, hearing guitar music coming from the CD player on your desk. Better yet, imagine that they walk in to see you sitting on a tall stool at the front of the room, strumming a guitar. Either way, you have set the stage for an engaging lesson on ratio, proportion, and proportional reasoning.
Math Roots: Mathematics and Visual Arts: Exploring the Golden Ratio
Authors: Jarvis Daniel
Published: December 2020

Mathematics and visual arts have long shared aspects of both form and function. One such ancient connection is found in the golden ratio. This article introduces the reader to this unique mathematical phenomenon in three separate contexts: (1) historical, (2) mathematical, and (3) pedagogical. A brief history of this fascinating number, various teaching strategies, and a project for middle school mathematics students will be explored. A list of resources is given for further study.
Escher in the Classroom
Authors: Britton Jill
Published: December 2020

I first read the article “Master of Tessellations: M. C. Escher 1898–1972” in 1974. I was teaching a class titled Recreational Mathematics at a junior college in Montreal at the time, and the article served as a basis of the Escherrelated material that I added to the course content. As the years passed, my knowledge of Escher and involvement with his geometric artwork grew. I spoke about Escher art at NCTM conferences; met Escher's eldest son, George, at a regional NCTM conference in Halifax; developed Escher materials for Dale Seymour publications; and gave a speech titled “Escher in the Classroom” at the International Escher Congress held in Rome, Italy, in 1998 (the centennial of Escher's birth). All of this began for me with a Mathematics Teacher article.
Anticipating Student Responses to Improve Problem Solving
Authors: Wallace Ann H.
Published: December 2020

I have been a Mathematics Teacher and Educator throughout all four publications of the Standards documents (NCTM 1991, 1995, 1989, and 2000). Over the years, while concentrating on improving various aspects of my teaching, specifically, improving my students' ability to problem solve, I have been perplexed to see students pick numbers out of a problem and perform an operation with no regard for the context. To address this issue by teaching problem-solving lessons made me realize that I did not know the difference between students solving a problem and actual problem solving. A lesson beginning with a problem or task does not make it a problem-solving lesson, especially when students would inevitably solve it the way I had intended. Instead of problem solving, my students were trying to figure out what I was thinking. To prevent the temptation of leading students in this way of thinking required careful planning of problem-solving lessons.
Emotion and Thought
Authors: Davis Robert B.
Published: December 2020

That emotions can both facilitate learning and block learning is an accepted psychological fact. The author considers those factors that he feels develop an optimal learning situation in the classroom.
It's All in the Cards: Adding and Subtracting Integers
Authors: Ponce Gregorio A.
Published: December 2020

The addition and subtraction of integers is one of the first major avenues, and roadblocks, to student success in the learning of algebra. Unfortunately, many students cannot make, or have a very difficult time making, the transition from working with whole numbers to working with integers. Students who struggle and do not succeed in making this transition undoubtedly find themselves at a disadvantage when trying to make sense of subsequent mathematical concepts that rely on their proficiency with and understanding of the addition, subtraction, multiplication, and division of integers. Moreover, since “algebra is a prerequisite for virtually all other mathematics” (Usiskin 2005, p. 10), one is compelled to find ways to help students make progress with challenging material. The following activity, which has been used in the classroom, gives teachers some techniques and an effective bridge to help students overcome this initial roadblock to algebra.
Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms
Authors: Smith Margaret S. , Hillen Amy F. , Catania Christy L.
Published: December 2020

The capacity to reason algebraically is critical in shaping students' future opportunities and, as such, is a central theme of K–12 education (NCTM 2000). One component of algebraic reasoning is “the capacity to recognize patterns and organize data to represent situations in which input is related to output by well-defined functional rules” (Driscoll 1999, p. 2). Geometric pattern tasks can be a useful tool for helping students develop algebraic reasoning, because the tasks provide students with opportunities to build patterns with materials such as toothpicks or pattern blocks. These materials help students “focus on the physical changes and how the pattern is being developed” (Friel, Rachlin, and Doyle 2001, p. 10). Such work might help bridge students' earlier mathematical experiences and lay the foundation for more formal work in algebra (English and Warren 1998; Ferrini-Mundy, Lappan, and Phillips 1997; NCTM 2000). Finally, the relationships between the quantities in pattern tasks can be expressed using symbols, tables, and graphs, as well as words. Thus, pattern tasks can also give students opportunities to make connections among representations—a key component in developing an understanding of function (Knuth 2000).
Quick Reads: Another Good Idea: Integers Made Easy: Just Walk It Off
Authors: Nurnberger-Haag Julie
Published: December 2020

Walk It Off is a multisensory method that I developed to teach students how to multiply and divide as well as add and subtract integers. In my experience, this method makes these processes much more effective, efficient, and entertaining than other approaches. Students have the opportunity to use the Process Standards while exploring a topic that is often taught strictly as algorithms without understanding. In addition to having visual, oral, and aural characteristics, Walk It Off is kinesthetic, because students literally walk off problems on number lines. The multiple learning modalities that this method uses are compatible with brain-based learning. Students become actively involved in doing calculations as they physically act out problems using the underlying concepts of integers. Although many books and lessons already use number lines to demonstrate addition and subtraction in various ways, the Walk It Off method makes it possible to use a number line for multiplication and division of integers as well. Using integers becomes easy, because students learn only two slightly different processes for the two basic groups of operations: one process for addition and subtraction and a similar process for multiplication and division.
Mathematical Roots: Get Primed to the Basic Building Blocks of Numbers
Authors: Robbins Christiana , Adams Thomasenia Lott
Published: December 2020

In his book Elements, Euclid established that certain numbers are the building blocks of our natural number system. He revealed that these natural numbers could be “decomposed” into their smallest units as products of specific numbers. Numbers that can only be factored by themselves and 1 are called “prime numbers” and comprise part of the basic building blocks of numbers.
A Look at the Development of Ratios, Rates, and Proportionality
Authors: Seeley Cathy , Schielack Jane F.
Published: December 2020

One of the most significant features of Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence is that the document highlights the coherent development across the grades of important ideas and the connections among these ideas. In this month's Focal Points article, we will take a look at one of the most important ideas in middle school mathematics: proportionality. The ability to recognize, understand, and work with ratios, rates, and proportional relationships is crucial to future success in algebra. When students are able to describe situations involving constant rates of change and discuss how two quantities are related proportionally, they will be prepared to understand linear functions. This understanding is a critical building block to the study of topics found in secondary mathematics.
Solve It! Fund Run
Authors: Mooney Edward S.
Published: December 2020

Aaron, Beth, Candace, Darvel, and Emil ran around a track as part of a fundraiser for their school. Aaron ran first, and then when he was tired, Beth ran. When Beth got tired, Candace ran, and so on. By the end of the fundraiser, they had run a total of 30 miles. Aaron and Beth combined ran 40 percent of the total distance. Beth and Candace combined ran 34 percent of the total distance. Candace and Darvel combined ran 40 percent of the total distance. And Darvel and Emil combined ran 41 percent of the total distance. How far did each person run?
Metric Madness
Authors: Kroon Cindy D.
Published: December 2020

What student could resist a rousing game of “metric madness”? Not only is it entertaining, it is also educational. While students are playing, they are also reviewing metric system vocabulary, measurements, and calculations.
On My Mind: Turning the Mathematics Classroom into an Intellectual Playground through Poetry
Authors: Hammett John E.
Published: December 2020

As Mrs. Kim balances her coffee cup on her manila folder filled with student papers and pushes her shoulder-strapped purse out of the way, she searches her overstuffed briefcase for her lanyard of classroom keys. Her mind, however, is already drifting back to the previous day's events when her middle school class had been energized by an exhilarating visit from a dynamic college professor who writes poetry. In response to the visit, her students enthusiastically composed the poetry contained within the folder she is holding.
Mathematical Explorations: What If We Were Built Like the Dinosaurs?
Authors: Fuentes Sarah , Garruto Patricia , Lockard Fran
Published: December 2020

Can you imagine having a three-footlong neck? What if triangular plates covered your back? How would you like to have arms so short that you could not feed yourself? In this twopart activity, middle school students are given the chance to use mathematics and science to address these questions with an artistic flair. Using ratios, students design new “humans” with features that are proportional to those of the dinosaurs. They then explore the implications for humans if these new features became real body parts.