bluuebunny/crossref_metadata_2025 huggingface dataset:
Take Time for Action: Moving to Proportional Reasoning
Authors: Slovin Hannah
Published: December 2020

Ratio and Proportion are essential topics in the middle-grades mathematics curriculum. The ability to reason proportionally is important in students' transition from elementary school mathematics to high school mathematics and science. The study of ratio and proportion lays a foundation for first-year algebra, and subsequent mathematics and science study assumes that students can reason proportionally. Many textbooks, however, limit the problems involving proportions to those that emphasize procedural competence rather than conceptual understanding. These problems usually require students to use ratios and set up equations. Students may solve such problems “correctly” using the traditional algorithm of cross-multiplying and dividing, but their solutions do not necessarily give us much information about their understanding of the multiplicative relationships involved in proportions.
Reflections on Practice: Facilitating Meaningful Discussion of Mathematics
Authors: Sherin Miriam Gamoran
Published: December 2020

Prompting students to talk about mathematics is an important goal of mathematics education reform. Teachers are encouraged to provide opportunities for students to discuss their ideas about mathematics and to listen closely to what students say. Yet managing such discourse is no easy task and can present teachers with many new challenges (Ball 1991; NCTM 1991).
Teacher to Teacher: Credit Card Quiz
Authors: Marks Jeff
Published: December 2020

The students began the class period with a dreaded pop quiz. A few moans were uttered as the students, a nearly equal mix of sixth, seventh, and eighth graders, took out their pencils and paper. The quiz (see fig. 1) was revealed to the students, and the moans gave way to excited chatter and laughter. And so began the prealgebra class's journey into issues involving credit cards. The challenge for the teacher was to meld the content from the textbook with a real-life area of concern by using teaching strategies based on multiple intelligences.
Now and Then: From Shadows to Surveying, Thales of Miletus
Authors: Johnson Art
Published: December 2020

CYNTHIA BOSIVERT and her partner work in private, but in full view. In fact, you might have seen Cynthia or someone who has the same job that she does. Cynthia works with a graduated rod and a theodolite on a tripod—she is a surveyor. Her job begins long before she takes her instruments to a site. “Before a survey team even leaves the office, there is a lot to do,” says Cynthia. “The crew has to look up the original land deed at the local Registry of Deeds to see how the original deed describes the boundary lines. Older deeds might identify a tree or a large rock as an essential part of a property line, and when you get out to the site, it isn't there any more.” Regardless of how the original deed reads, the survey crew prepares a deed sketch to help guide them when they get to the actual site.
Enriching Students' Mathematical Intuitions with Probability Games and Tree Diagrams
Authors: Aspinwall Leslie , Shaw Kenneth L.
Published: December 2020

Middle school students often develop games to help make such decisions as who goes first or who gets the biggest piece of cake. Many students seem to have good intuition for determining which games are fair. Students feel comfortable with flipping a coin or playing “odd it out” to determine a winner.
Sumgo Here and Sumgo There
Authors: Meel David E.
Published: December 2020

The idea of “Sumgo” was suggested by the game of bingo and the need to illustrate the utility of educational games, help students practice skills, and introduce new concepts. This game was designed to investigate an interesting distribution while practicing a computational skill. As a result, the activity described in this article focuses on the concepts of sample spaces and exact probabilities while providing practice in addition. In designing “sumgo,” I envisioned a mathematics class actively engaged with the game while practicing addition and learning about data interpretation, experimental and theoretical probability, and the consequences of randomness.
Math at Work: Running by the Numbers
Authors: Milliken Paul
Published: December 2020

IN 490 B.C., THE MESSENGER PHEIDIPPEDES ran twenty-six miles to Athens carrying the news of Greek victory at the battle of Marathon. He delivered the news and dropped dead from the effort. Today, we celebrate that famous run with one of the most demanding events in human athletics, the marathon. Like Pheidippedes, the modern runner strives to complete the distance in as little time as possible. Unlike that early messenger, today's competitors undergo extensive training to ensure that they remain alive when they have finished the run. Kevin Smith uses mathematics to help runners prepare for marathons.
Bad Jokes Make Good Mathematics
Authors: Lipp Alan
Published: December 2020

The joke was never very funny, but it can be used to develop a mathematical method. In this article, this poor joke is used as a starting point for creating alternative approaches to solving a wide variety of traditional algebra word problems.
Using Attribute Blocks to Develop a Conceptual Understanding of Probability
Authors: Quinn Robert J.
Published: December 2020

Middle school students continue to rely on concrete experiences to construct knowledge but are starting to develop the ability to think abstractly (NCTM 1989, p. 68). Thus, the middle school curriculum should provide a “bridge between the concrete elementary school curriculum and the more formal curriculum of the high school” (NCTM 1989, p. 102). This article describes a series of activities using attribute blocks designed to help middle school students construct knowledge about, and develop conceptual understanding of, probability. Depending on the ability levels of the students, these activities can be completed in a single twohour time block or can be spread over three or four sessions of about one hour each. These introductory probability explorations are appropriate for seventh graders but can be adapted for students at other levels. Attribute blocks are frequently used in the primary grades, but this article shows that they can be quite useful in the middle grades, as well.
Using Models to Build an Understanding of Functions
Authors: Cramer Kathleen
Published: December 2020

From the time that students enter kindergarten and throughout their early elementary school years, they should have multiple experiences exploring patterns. The study of patterns for middle school students should shift to the study of functions (NCTM 1989). The question that this article addresses is how to plan and organize instruction for middle-grades students to help them develop an understanding of function.
Revisiting a Difference in Squares
Authors: Slavit David
Published: December 2020

While on a drive, three-year-old Rudi sees a cow and yells out, “Horsey! Horsey!” Although the word is wrong, this child has correctly abstracted his experiences with similar animals into one general idea and called it “horsey.” Rudi has made an important generalization that allows him to think about all such animals as a single group.
New Approaches to Algebra: Have We Missed the Point?
Authors: Thornton Stephen J.
Published: December 2020

Curriculum movements in the United States and Australia, characterized by such documents as Curriculum and Evaluation Standards for School Mathematics(NCTM 1989) and A National Statement on Mathematics for Australian Schools (AEC 1991), have challenged the conventional view of algebra as formal structure, arguing that algebra is fundamentally the study of patterns and relationships. Increased emphasis has been given to developing an understanding of variables, expressions, and equations and to presenting informal methods of solving equations. The emphasis on symbol manipulation and on drill and practice in solving equations has decreased (NCTM 1989).
The Tangram Conundrum
Authors: Thatcher Debra H.
Published: December 2020

The tangram is a versatile manipulative used to teach and learn such concepts as shapes, spatial relationships, fractions, percents, area, and perimeter. When I introduce it to undergraduate preservice teachers, they are drawn to the tangram because of the creative ways in which the shapes can form images; the book Grandfather Tang's Story (Tompert 1990) illustrates this quality well. The preservice teachers are surprised to find that this simple manipulative can challenge their understanding of some mathematical concepts taught in middle school. This article describes an investigation conducted by preservice teachers enrolled in a mathematics methods course.
Math at Work: Samantha Brown: Jill-of-All-Trades
Authors: Little Catherine
Published: December 2020

Empty rooms are full of possibilities, and those possibilities excite Samantha Brown, a self-professed Jill-of-all-trades. Samantha runs a small business that specializes in the finishing touches that turn a house into a home. She paints, sews, handles small carpentry jobs with ease, and has an eye for putting her work together with artistic flair. Estimating, measuring, budgeting, and all manner of problem solving fall into the realm of Samantha's everyday experience.
A Personal Journey: Integrating Mathematics and Service Learning
Authors: Donnell Barbara D.O
Published: December 2020

The NCTM's standards have affected me as a teacher in more ways than I can count. As a teacher of mathematics in the early 1990s, I realized the importance of professional development. My interest was so deep that the local educational service center trained me to teach other teachers about the Standards and their application.
Circles and the Number Pi
Authors: Tent Margaret W.
Published: December 2020

When i first started teaching middle school students about circles and the number π, my task as I understood it was straightforward: Teach them that π = 3.14 or 22/7, and teach them the formulas for finding the area and circumference of a circle. I poured in the information, but a lot of it spilled out and was lost. Here are two examples of my “wisdom” as it came out at the other end of the spout: (1) “π = 3.14, since nobody likes fractions anyway” and (2) “There are two formulas that you have to use, πr 2 and πd. Sometimes you use one formula and sometimes the other. If you are lucky, you will guess right.”
Mathematical RootsL: Magic Squares: Discovering Their History and Their Magic
Authors: Anderson Dawn
Published: December 2020

Magic squares have long been considered a mathematical recreation providing entertainment and an interesting outlet for creating mathematical knowledge. An nth-order magic square is a square array of n2 distinct integers in which the sum of the n numbers in each row, column, and diagonal is the same.
Challenge Problems: Love Them, or Hate Them, but Learn from Them
Authors: DeYoung Mary J.
Published: December 2020

The students had just handed in their final challenge problem of the first semester when Rachel asked to commemorate the moment. The more than twenty challenge problems that had been assigned to the students over the previous months had sometimes generated feelings of bewilderment, frustration, and anger as students struggled for solutions, but the problems had also produced feelings of elation and accomplishment when the light of understanding broke through the clouds of confusion.
The Possibility of Perfection
Authors: Masse Leonard N.
Published: December 2020

A lesson which draws on events in Major League Baseball to illustrate the relationship between experimental and theoretical probability. Activity sheet, with answers, included.
Innovation in Curriculum: Context in Mathematics Curricula
Authors: Meyer Margaret R. , Dekker Truus , Querelle Nanda
Published: December 2020

The past few years have seen the emergence of five mathematics curricula developed with support by the National Science Foundation for the middle grades: Mathematics in Context (MIC) (National Center for Research in Mathematical Sciences Education and Freudenthal Institute 2001), Math Thematics (Billstein and Williamson 1999), Connected Mathematics (CMP) (Lappan et al. 1998), MathScape, a View of the World from a Mathematical Perspective (Education Development Center 1998), and Pathways to Algebra and Geometry (Institute for Research on Learning 1997). One striking similarity that characterizes these middle school curricula, as well as their reform counterparts at the elementary and secondary school levels, is the pervasive use of context. Using a definition suggested by Borasi (1986), context is “the situation in which [a] problem is embedded” (p. 129). Context is usually supplied by the text of the problem, but it can also be contained in pictures, diagrams, or tables. A quick review of any of the curricula mentioned above shows that context is plentiful and varied. This abundance of context is in marked contrast with traditional textbooks, in which context appears only in brief introductions or end-of-section story problems.
Problems that Encourage Proportion Sense
Authors: Billings Esther M. H.
Published: December 2020

Several problems promoting proportion sense, reasoning about quantities, and the various relationships among quantities in proportions.
Miniature Toys Introduce Ratio and Proportion with a Real-World Connection
Authors: Drum Randell L. , Petty Wesley G.
Published: December 2020

IN THE REAL WORLD, A RATIO IS NEVER STATED without terms of reference for the quantities in the ratio. The objects mentioned above represent the application of the ratio and are just as important when teaching ratios as the numbers that form the ratios themselves.
Determining Probabilities by Examining Underlying Structure
Authors: Norton Robert M.
Published: December 2020

One way of grabbing students' interest in probability is to present the rules of a game of chance, ask them whether the game is fair, and have them use their intuition to give a ballpark estimate of the probability of winning. The issue of fairness can be counted on to arouse the interest of many students, and fairness of games is a topic that has found its way into textbooks (see, e.g., Chapin et al. [1994]). Having students commit to an answer helps to arouse and sustain their interest because they want to see whether their intuition is correct. Students also learn that giving an impulsive, intuitive, and incorrect answer to a probability question is easy, but a good approach to finding the correct answer is to examine how each possible outcome is produced under the rules of the game. Giving students experiences in looking at the underlying structures of a number of games offers them opportunities to enhance their probability intuition.
Take Time for Action: Mathematical Argumentation: Putting Umph into Classroom Discussions
Authors: Stein Mary Kay
Published: December 2020

Perhaps the most difficult recommendation of the NCTM's Standards to put into practice is that of orchestrating classroom discourse—moving from a teacher-centered classroom to one that is centered on student thinking and reasoning. Some researchers argue that traditional “chalk and talk” classrooms put all the intellectual authority in the hands of the teacher and little or no responsibility for thinking and reasoning on the shoulders of the students. Classroom discussions, in contrast, are viewed as encouraging students to construct and evaluate their own knowledge, as well as the ideas of their classmates. Few examples or guidelines exist, however, to help teachers orchestrate such discussions.
Exploring, Inventing, and Discovering Mathematics: A Pedagogical Response to the TIMSS
Authors: Martinez Joseph G. R.
Published: December 2020

A three-step process - exploring, inventing, discovering - that helps students derive concepts actively from problem solving.
By the Unit or Square Unit?
Authors: Ferrer Bellasanta B. , Hunter Bobbie , Irwin Kathryn C. , Sheldon Maureen J. , Thompson Charles S. , Vistro-Yu Catherine P.
Published: December 2020

These questions address the relationship between area and perimeter and challenge children to think beyond rules that they may have been taught in school.
A Simple Solution…or Is It?
Authors: Yolles Arlene
Published: December 2020

Article presents various students' solutions to a rate problem. Student work is included.
Multiplication from Lilavati to the Summa
Authors: Berg Rebecca
Published: December 2020

Many of the multiplication techniques used in fifteenth- and sixteenth-century Europe can be traced to India, specifically through a book called Lilavati (The Beautiful) by Bhaskara (1114–85). Although many of us find mathematics to be beautiful, the name is unusual for a mathematics book. According to a translation by the fifteenth-century Persian poet Fyzi, a story lies behind the book's name. Astrologers predicted that Bhaskara's daughter, Lilavati, would have an unfortunate marriage unless she married at a certain time on a certain day. Lilavati, in anticipation of her marriage at this special time, was watching a water clock. Unfortunately, when she bent over the water clock, a pearl dropped off her headdress and stopped the water from flowing out of the clock. Lilavati missed the time for the ceremony, and because she did not want to risk an unhappy marriage, she never wed. To offer Lilavati solace, Bhaskara named his book after her because it would be a book that “will last to the latest times” (Burton 1988, p. 236; Calinger 1999, p. 279).
Assessment in Action: Mrs. Grant's Measurement Unit
Authors: Steele Diana F.
Published: December 2020

NCTM's Professional standards for teaching Mathematics (1991) describes teaching as a complex series of decisions that includes selecting, creating, and adapting appropriate instructional tasks; establishing and sustaining a positive learning environment; and analyzing the learning results of chosen tasks. For teachers to perform all these activities, they must continually integrate assessment into their instruction. This continual assessment serves as the means for monitoring students' understanding and making instructional decisions. Chambers (1993) noted, “Every instructional activity is an assessment opportunity for the teacher as well as a learning opportunity for the student” (p. 17).
Food for Thought From Jay's Diner: Raking Leaves at School
Authors: Sherard Wade H.
Published: December 2020

The Editorial Panel is presenting the following problem to stimulate submissions to “The Thinking of Students” department. We encourage teachers to try this problem with students and analyze the different ways that they approach the problem.
Mathematical Roots: Euclidean Tools and the Creation of Euclidean Geometry
Authors: Kastberg Signe E.
Published: December 2020

Used by mathematicians over two thousand years ago, the compass and straightedge are still used by mathematics students today. The straightedge is similar to the ruler in that the user can draw a straight line with it, but the straightedge is not marked and cannot be used to measure things. Although many different types of compasses exist, your students probably use the modern compass. The characteristic that makes the modern compass special is tension, which allows the opening between the lead and the pointer to be fixed. By fixing the opening, the user can draw a circle of a given radius and copy a line segment.
Building Conceptual Bridges from Arithmetic to Algebra
Authors: Martinez Joseph G. R.
Published: December 2020

How can we help students make the transition from elementary school mathematics to middle school algebra? Although algebraic thinking and some important concepts are now routinely taught in the early grades, some students still approach algebra as an alien domain with an incomprehensible language.
A Rhythmic Approach to Geometry
Authors: Flores Alfinio
Published: December 2020

Several authors have used string designs to help students develop and understand mathematical ideas. Different aspects of this activity can be emphasized. First, students can focus on the beauty of the final string design, which in many instances, is related to mirror and rotational symmetries (Pohl 1986). Second, string designs can be used to highlight relations among such numbers as common factors, multiples, and prime numbers (Bennett 1981; Perl 1981). String designs can also give rise to interesting mathematical curves (Millington 1996), such as parabolas, pursuit curves, spirals, cardioids, and many more.
Making Observations Interactive
Authors: Albert Lillie R. , Mayotte Gail , Cutler Sohn Sheila
Published: December 2020

This article focuses on a yearlong collaborative project to study the role of interactive observational assessment in student learning of mathematical concepts. Interactive observational assessment is a pedagogical approach that invites engagement between teacher and student through written dialogue to help students develop their understanding of mathematics. Students are engaged in a mathematical problem while the teacher circulates among them to observe and make note of their work. Rather than nod and say, “Yes, you have it” or tell the students that they “have not found the solution yet,” the teacher responds to them by writing questions that challenge their thinking and mathematical reasoning. Knowing the type of question needed for this purpose and knowing how to ask it effectively are fundamental to this process.
X-tending the Fibonacci Sequence
Authors: Moran Glenn T.
Published: December 2020

Article outlines a lesson on the Fibonacci and Lucas sequences, giving opportunity for computation practice, mental mathematics, and proof; for algebra students, the article discusses an extension for solving simultaneous equations.
Divisibility Tests: So Right for Discoveries
Authors: Bennett Albert B. , Ted Nelson L.
Published: December 2020

Using base-ten pieces to illustrate divisibility tests with single-digit divisors. Several visual aids are provided.
Spotlight on the Principles/Standards: Building Mathematically Powerful Students through Connections
Authors: Thomas Christine , Santiago Carmelita
Published: December 2020

Connections in mathematics can be implemented in ways that create excitement in the classroom, develop in students a love for doing mathematics, and foster students' natural inclination for pursuing mathematical tasks. According to the Curriculum and Evaluation Standards for School Mathematics, “If students are to become mathematically powerful, they must be flexible enough to approach situations in a variety of ways and recognize the relationships among different points of view” (NCTM 1989, p. 84). Principles and Standards for School Mathematics (NCTM 2000) further asserts that students develop a deeper and more lasting understanding of mathematics when they are able to connect mathematical ideas. The 1989 and 2000 Standards clearly delineate the power and importance of connections in the mathematics curriculum. This article examines and compares curricular recommendations for connections in the two documents.
Families Ask: Why Not Just Use a Formula?
Authors: Gilliland Kay
Published: December 2020

At the beginning of this semester, a mathematics educator in a methods class asked her students to fold a 6 × 8 index card in half to form two congruent rectangles, each with the maximum possible perimeter. She was not surprised when the students reported that the perimeter remains the same whether the card is folded lengthwise or widthwise. On further investigation, the students were completely surprised to discover that, although the area remains the same either way, the perimeter can change. When students have explored such problems as “If the area of a rectangle is 30 square units, what can the perimeter be?” using a variety of strategies and materials, they are more likely to understand the relationship between area and perimeter than if they had simply memorized formulas for area and perimeter.
Anchored Learning in Context
Authors: Meyer Margaret R. , Diopoulos Georgianna
Published: December 2020

Article describes how a teacher adapts a lesson for a classroom of learning disabled students. The adaptations and suggestions given can be applied to a variety of different lesson situations.
Blueprint for Writing Middle School Mathematics
Authors: Baxter Juliet A. , Woodward John , Olson Deborah , Robyns Janet
Published: December 2020

Article describes strategies for using writing to enrich students' understanding in middle school mathematics. Sample prompts are provided.
Teacher to Teacher: Rewrite to Improve
Authors: Perlin Michael H.
Published: December 2020

Many teachers that i know are often approached by students requesting extra-credit assignments or other ways to improve their grades. These students who seek to gain additional credit or to better their own work deserve a meaningful response. How many of us have empathized with students to the point at which we suggest extra-credit work outside the curriculum? Although certainly valuable in its own right, researching a famous mathematician or an unrelated mathematical topic for bonus points does not often address the reasons for students' declining grades. Furthermore, this type of research is not likely to offer students insight into why they have not mastered the given concepts. Some students have become so used to detailing the work of famous mathematicians from library books and encyclopedias, which are often written several grade levels above their understanding, that they cannot respond appropriately when asked to do the more important job of reflecting on their own tests or quizzes. I offer my students additional credit for reflecting on and mastering mathematical concepts, a second-chance program that I refer to as “rewrites.”
Strategies for Helping Students Who Have Learning Disabilities in Mathematics
Authors: Steele Marcee M.
Published: December 2020

Most middle school students who have learning disabilities are taught mathematics in inclusive classrooms (Mastropieri and Scruggs 2000). This situation enables teachers to use teaching strategies that are designed to help students who have learning disabilities and that will improve the classroom experience for everyone. Fortunately, teachers can implement several such strategies and modifications easily, without too much additional preparation or instructional time. These ideas can help make the mathematics experiences of students who have learning disabilities and their teachers more positive and successful.
Do You See What I See?
Authors: Chávez Óscar , Reys Robert E.
Published: December 2020

Using spatial visualization to make connections among Zeno's paradox, geometry, fractions, infinite series and limits.
What Happens to Geometry on a Sphere?
Authors: Sharp Janet M. , Heimer Corrine
Published: December 2020

WE HAVE TO SHARE THIS WITH OUR students! They will love it!” This statement was all we could think about after a professional development session dealing with geometry. Spherical geometry challenged our capabilities in geometry but greatly interested us. Before we could teach our students about spherical geometry, we needed to learn more about this strange new world ourselves. In this article, we describe our discoveries and some of the activities we developed for our sixth-grade students.
Teacher to Teacher: A New Approach to an Old Order
Authors: Rambhia Sanjay
Published: December 2020

One of the many important ideas that we teach in the mathematics and prealgebra curricula is the concept of order of operations. However, it is a concept that many students consistently forget from year to year. Students invariably solve problems from left to right, regardless of the hierarchy associated with the operations. This article outlines a new approach to teaching this important concept.
The Versatile Magic Square
Authors: Watson Gale A.
Published: December 2020

Article demonstrates the transformations that are possible to construct a variety of magic squares, including modifications to challenge students from elementary grades through algebra. An example of using magic squares with students who have special needs is also shared.
Problem Solving, Patterns, Probability, Pascal, and Palindromes
Authors: Hylton-Lindsay Althea Antoinette
Published: December 2020

Article presents a problem-solving activity, the Birth Order problem, and several solution-seeking strategies. It includes responses of current and prospective teachers and a comparison of the various strategies.
Explorations with a Functioning Flex-O-Gram
Authors: Hines Ellen , Bridges Jesse
Published: December 2020

Article describes a processes to expand thinking about the formula for the area of a parallelogram by observing patterns in the operation of a flexible parallelogram.
Maps and Algebra
Authors: Gratzer William
Published: December 2020

The learning principle from principles and Standards for School Mathematics (NCTM 2000) states, “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge” (p. 20). How to help students develop such understanding is a question that confronts every middle school mathematics teacher. The method described in this article helps students use their life experiences to develop mathematical understanding by illustrating that linear equations can be solved using ideas with which they are already familiar.
Calendar Capers: A Star is Born
Authors: Barger Rita H. , Bryant Cynthia G.
Published: December 2020

The unit described in this article connects many topics of mathematics. It was designed to help students become mathematically literate. To truly learn, as opposed to just learning for a test, the mathematics skills that students will need throughout their lives, they must be given opportunities to make conjectures, investigate, explore, discover, and solve problems. This unit encompasses all these skills. It also enables students to use reasoning and logic; to systematically collect, organize, and describe data; and to construct, read, and interpret displays of data while developing an understanding of mean, median, mode, and range. Finally, the unit also integrates science, history, reading, and art with mathematics in a captivating way that allows students at all levels to both learn and enjoy.
Classify & Capture: Using Venn Diagrams and Tangrams to Develop Abilities in Mathematical Reasoning and Proof
Authors: Moyer Patricia S. , Bolyard Johnna J.
Published: December 2020

Article describes a game of classifying objects according to their attributes using a three-set Venn diagram, then analyzing the classification of the objects to determine their accuracy. Students attempt to capture opponents' game pieces (tangrams) by making mathematical arguments that identify misplaced pieces.
Developing Algebraic Reasoning Through Generalization
Authors: Lannin John K.
Published: December 2020

NCTM's (2000) recommendations for algebra in the middle grades strive to assist students' transition to formal algebra by developing meaning for the algebraic symbols that students use. Further, students are expected to have opportunities to develop understanding of patterns and functions, represent and analyze mathematical situations, develop mathematical models, and analyze change. By helping students move from specific numeric situations to develop general rules that model all situations of that type, teachers in fact begin to address the NCTM's recommendations for algebra. Generalizing numeric situations can create strong connections between the mathematical content strands of number and operation and algebra (as well as with other content strands). In addition, these generalizing activities build on what students already know about number and operation and can help students develop a deeper understanding of formal algebraic symbols.
Teacher to Teacher: The Pentomino Square Problem
Authors: Jamski William D.
Published: December 2020

During part of a presentation to some middle school mathematics students, right before a vacation from school, we looked at pentominoes. (Pentominoes are shapes made from five congruent squares attached to each other along full sides.) See figure 1's illustration of the definition; two examples are pentominoes, one is not.
How Many Times Does a Radius Square Fit into the Circle?
Authors: Flores Alfinio , Regis Troy P.
Published: December 2020

This article presents alternative and interesting approaches to exploring. The first part of the article describes an empirical method that allows students to estimate the area of a circle with surprising accuracy. This method can be used as an extension of the approach of using a grid to approximate the area of a circle. The use of a grid is conceptually illuminating; it emphasizes the idea that the area is measured in square units and that areas of shapes that do not have straight sides can nevertheless be measured with square units. An example of such an approach is described in “Covering a Circle” (Lappan et al. 1998); however, when students actually count squares, the values they find may not be exact. For example, in a sixth-grade classroom, when students counted the squares in a grid containing a circle with radius 7, their answers ranged from 142 to 178 square units for the approximate area of the circle. Dividing such numbers by the square of the radius, that is, 49, we obtain values ranging from 2.9 to 3.6. The area approximation was not exact enough to determine the value of π with two significant digits. In the first activity we describe, the goal is to help students understand that the area of the circle is about 3.1 times the area of the radius square (that is, a square with a side length that is equal to the radius).
Encouraging Students to Think about How They Think!
Authors: Reeves Charles A. (Andy) , Reeves Rosemarie
Published: December 2020

Article suggests teachers encourage metacognition in their students by becoming knowledgeable about the effects of Einstellung and using that knowledge in their teaching practices. Appendix includes directions for student activities.
Proportionality: A Unifying Theme for the Middle Grades
Authors: Lanius Cynthia S. , Williams Susan E.
Published: December 2020

The mathematics studied in high school and beyond is organized around large themes, such as algebra, geometry, trigonometry, calculus, statistics, and so on. Even though the boundaries between these topics blur at times, the themes give connections and structure to the mathematics studied. For example, a huge body of knowledge is recognized as algebra. Students know that they are doing algebra when they formulate and solve equations, even when they perform these tasks in geometry or calculus class. In contrast, the study of mathematics in the middle grades, with the exception of prealgebra, lacks an overarching theme. Most middle-grades' courses, generally known merely as sixth-, seventh-, or eighth-grade mathematics, lack big ideas and tend to be a mass of discrete topics with little connection or continuity. The notion that prealgebra is a course to be done right before algebra, as opposed to a long-term theme to be developed over several years, is a prime example of this situation.
Patchwork Quilts: Connections with Geometry, Technology and Culture
Authors: McCoy Leah P. , Shaw Jean M.
Published: December 2020

Patchwork quilts are an important part of American culture and history. The patchwork designs are geometric, and early American women used mathematics and artistry as they sewed warm covers for their families. The history of quilts can be traced through several cultures, including that of Native Americans, western pioneers, slaves escaping through the Underground Railroad, and immigrants from Europe and Asia. Often, students have seen quilts in their homes and are interested in exploring the patterns. Quilt making is related to family heritage, and the study of quilts may connect young students with their grandparents or family histories.
Revisiting the Sum of Odd Natural Numbers
Authors: Kalman Richard
Published: December 2020

The north shore (new york) school District asks me periodically to teach a class of selected fourth- and fifth-grade students. On those occasions, I choose problems that employ important mathematics, use more than one concept, contain subtleties, or allow for a variety of solutions. In this article, I share students' thinking as they attempted to solve a nonroutine problem from a past contest created by the Mathematical Olympiads for Elementary and Middle Schools. My two major mathematical goals were to elicit creativity in devising solutions and to emphasize the idea that any problem might have several equally valid solutions.
Building on Students' Intuitive Strategies to Make Sense of Cross Multiplication
Authors: Boston Melissa D. , Smith Margaret S. , Hillen Amy F.
Published: December 2020

Middle-grades students' understanding of proportional relationships should be fostered through problem solving and reasoning (NCTM 2000). Toward this end, instruction in proportionality should expose students to a variety of strategies and allow students to gain experience modeling proportional situations (Langrall and Swafford 2000). Students should be given ample opportunities to develop intuitive strategies based on factor- of-change (“how many times as many”) relationships (Cramer and Post 1993). Research has shown that middle-grades students are more successful at method is appropriate to use” (NCTM 2000, p. 221). We begin our discussion by focusing on the events that unfold in Marie Hanson's sixth-grade classroom during a lesson on understanding ratios and proportions (Smith et al. forthcoming), and use this lesson as a context for considering how factor-of-change relationships might be used to assist students in understanding why cross multiplication works.
Assessing Proportional Thinking
Authors: Bright George W. , Joyner Jeane M. , Wallis Charles
Published: December 2020

Proportional thinking is an important part of mathematics in the middle grades and “connects many of the mathematics topics studied in grades 6–8” (NCTM 2000, p. 217). Partly in response to this need, NCTM's 2002 Yearbook titled Making Sense of Fractions, Ratios, and Proportions (Bright and Litwiller 2002; Litwiller and Bright 2002) addressed proportional reasoning across the grades but with special emphasis on the teaching and learning of this important area in the middle grades.
The Influence of Ancient Egypt on Greek and Other Numeration Systems
Authors: Zaslavsky Claudia
Published: December 2020

You may have learned how the ancient Egyptians wrote numbers. For example, for the number 600, you would write a symbol for a scroll six times. Actually, ancient Egypt had two main systems of writing: hieroglyphic and hieratic. Hieroglyphics, dating back over 5,000 years, were used mainly for inscriptions on stone walls and monuments. Hieratic writing was a cursive script suitable for writing on papyrus, the Egyptian form of paper. Much of our knowledge of ancient Egyptian mathematics comes from a papyrus written by the scribe Ahmose around 1650 B.C.E. Although he wrote in hieratic script, recent historians transcribed this document and others into hieroglyphics, giving readers the impression that all Egyptian writing was in hieroglyphics, the system that you may have learned.
Surprise! Turn Routine Problems into Worthwhile Tasks
Authors: Caulfield Richard , Harkness Shelly Sheats , Riley Robert
Published: December 2020

Textbook problems are often mundane, uninteresting, and allow for very little mathematical thinking or discourse. Because students in our township scored lowest in the category of ratio and proportional reasoning on our statewide assessment, we–three eighth-grade teachers–decided to work together to address this area of weakness. We modified a typical textbook problem related to this mathematical concept in an attempt to foster students' thinking, mathematical understanding, and discourse. In this article, we describe what happened in the classroom when we introduced our “modified” textbook problem.
Improper Applications of Proportional Reasoning
Authors: Dooren, Wim Van , Bock Dirk De , Verschaffel Lieven , Janssens Dirk
Published: December 2020

One of the major goals of elementary and middle-grades mathematics education is for students to obtain a deep understanding of the proportional model in a variety of forms and applications. However, the reinforcement of proportionality at numerous occasions in school mathematics, along with the teaching of some standardized methods for solving proportionality problems, appear to lead to a resistant tendency in some students and adults to see and apply proportions everywhere. This same application occurs in situations where another method of solution is appropriate. Along with mastering the proportional scheme, its misuse seems to appear, as well. This overgeneralization of proportion has many faces: It has been found at different age levels and in a variety of mathematical domains, such as elementary arithmetic (Cramer, Post, and Currier 1993), algebra (Matz 1982), geometry (De Bock, Verschaffel, and Janssens 1998, 2002) and probability (Van Dooren et al. 2002).
Using Origami to Promote Geometric Communication
Authors: Robichaux Rebecca R. , Rodrigue Paulette R.
Published: December 2020

Rigami has been used frequently in teaching geometry to promote the development of spatial sense; to make multicultural connections with mathematical ideas; and to provide students with a visual representation of such geometric concepts as shape, properties of shapes, congruence, similarity, and symmetry. Such activities meet the Geometry Standard (NCTM 2000), which states that students should be engaged in activities that allow them to “analyze characteristics and properties of twoand three-dimensional geometric shapes and develop mathematical arguments about geometric relationships” and to “use visualization, spatial reasoning, and geometric modeling to solve problems” (p. 41). This article begins with an explanation of the importance of communication in the mathematics classroom and then describes a middle school mathematics lesson that uses origami to meet both the Geometry Standard as well as the Communication Standard.
The Mathematics of Native American Star Quilts
Authors: Neumann Maureen D.
Published: December 2020

The author examines the mathematics involved in making star quilts and the cultural significance these quilts hold for the Lakota Sioux.
The Chess and Mathematics Connection: More Than Just a Game
Authors: Berkman Robert M.
Published: December 2020

Mathematics educators have known for a long time that competitive games can play a dual purpose in advancing our work. One type of game can act as a vehicle for mastering a certain skill, like “multiplication war,” in which each player lays down two cards and compares the products to determine who wins. A second type is useful for developing thinking strategies that have applications to mathematics. A simple game like “poison apple,” in which each child takes turns removing one or two apples from a tree until one person is stuck with the last apple, helps develops planning, strategizing, and pattern-recognition skills.
Coordinate Plane Set Detective
Authors: Beigie Darin
Published: December 2020

Coordinate plane graphing is usually introduced to students by asking them to plot specified ordered pairs and identify points on the coordinate plane. A popular example involves plotting a collection of ordered pairs and connecting the resulting points with line segments to form geometric shapes or pictures (e.g., see Mystery Media 1990). This article describes an extension beyond plotting specified points that involves plotting collections of ordered pairs according to prescribed set conditions, such as the one shown in figure 1. These coordinate plane sets can form points, line segments, rays, lines, curves, and regions. I give these problems in an informal manner to my prealgebra students to foster a more robust appreciation of ordered pairs and the coordinate plane before progressing to table-of-values calculations. The problems ask students to investigate the coordinate plane and make decisions about appropriate ordered pairs, to see patterns, to construct pairs, and to make comparisons. The problems encourage thinking and questioning that help with algebraic topics discussed later on, such as solving for the intercepts of an equation, graphing an equation like x = 5, invoking the zero product property, and graphing an inequality (e.g., see Larson, Boswell, Kanold, and Stiff 2001 for a discussion of these algebraic topics). Furthermore, these set problems generally stretch students' minds, often requiring open-ended search and discovery as well as careful investigation and attention to computational details involving absolute values, inequalities, integer operations, arithmetic expressions, and number theory.
Reasoning and Working Proportionally with Percent
Authors: Parker Melanie
Published: December 2020

Percent, an important topic in the middle school and junior high school curriculum, is commonly found in the everyday lives of students. This topic presents a wonderful opportunity for students to reason and work proportionally with quantities. Teachers have tried different instructional approaches with varying degrees of success, but percent remains a difficult topic for students to understand. Various representations have been used in instruction, such as fractions, decimals, pie charts, and number lines. Although the use of visual models is a step in the right direction, these visual representations are often abandoned as students progress into the topic. Emphasis is then placed on conversions, equations, and computations. However, these representations may not stress the very nature of the concept. Instruction in this important topic should stress what percent really is instead of concentrating on surface features of the numeral and the problems in which it appears.
Mathematical Roots: Benjamin Banneker and the Method of Single Position
Authors: Mahoney John F.
Published: December 2020

BENJAMIN BANNEKER was a farmer, scholar, mathematical wizard, and a selftaught astronomer and surveyor. An African American, he was born a free man in Maryland in 1731. At the age of twenty-two, using only a pocket watch as a guide, he built a wooden striking clock that kept accurate time and continued to strike until it burned in a fire shortly after his death. He wrote an almanac and ephemeris from 1791 through 1802—some were published and widely distributed in Pennsylvania, Delaware, Maryland, and Virginia. In 1791, Banneker received an appointment to assist in surveying the lines of the Federal Territory—a tenmile square now known as the District of Columbia. Banneker died in 1806 on his Maryland farm (Benjamin Banneker Association 2003).
Mathematical Modeling: Convoying Merchant Ships
Authors: Mathews Susann M.
Published: December 2020

Few events of the twentieth century have had as much impact as who won World Wars I and II. In both wars, Great Britain reduced the sinkings of merchant ships by German submarines through sailing their ships in groups (convoying). Before instituting convoys, Great Britain suffered severe losses to attack by German submarines. In World War II, Japan allowed merchant ships to sail individually. Japan's losses to U.S. submarines were a critical element in Japan's defeat (Roscoe 1949). In a convoy, many merchant ships sail in a large group under the escort of naval warships to protect the poorly armed merchant ships. In World War I, the British admiralty opposed sending merchant ships grouped together in convoys for several reasons that proved to be false. I proposed the problem of whether or not the British should convoy their merchant ships to my preservice teachers in a course in mathematical modeling for middle school teachers. While working on this problem, the students analyze and rebut each of the admiralty's arguments against convoying. Mathematical models are used to support the rebuttals.