bluuebunny/crossref_metadata_2025 huggingface dataset:
Is Algebra in the Cards?
Authors: Bradley Elizabeth H.
Published: December 2020

When my colleague, pat, a special-education teacher, telephoned one afternoon, my answering machine explained, “We can't come to the phone because we're real busy building a great dodecahedron.” She left this message, “Well, we're not building any kind of 'hedron, but we've got a card trick that we'd like you to explain to us.” She went on to explain the “trick” in such long-distance detail that I had to listen to the message three times! Thus began a wonderful exploration into a simple card trick, which I shared with my eighthgrade students. The trick opened opportunities for my students to discover patterns, work with variables, and create tables. Through writing and discussion with others, students attempted to explain why the trick works and outlined rules for the patterns discovered. More advanced students might use this trick to pursue the concepts of functions, composition of functions, and using functions to solve a real-world problem.
May's Menu of Problems
Authors: Georgeson Joseph , Laughlin Connie
Published: December 2020

A light fare to whet your mathematical appetite.
Mathematics as Reasoning—Episodes from Japan
Authors: Sawada Daiyo
Published: December 2020

As recommended in the NSTM's curriculum and evaluation Standards for School Mathematics (1989, 29), the study of mathematics should emphasize reasoning so that students can—
Not Just a Fairy Tale
Authors: Johnson Rose M.
Published: December 2020

A crime committed many years ago has never been solved. As the judge listens carefully to the complaint presented by the Three Bears, she is distracted by Goldilocks's gumpopping during the court session. The preponderance of the evidence points in the Three Bears' favor, but how will the judge assess the damages committed by Goldilocks? What is fair punishment for Goldilocks's crime? Students meet in the judge's chamber to answer these questions.
Using House Plans to Teach Ratio, Proportion, and More!
Authors: Weidemann Wanda , Hunt Jane Braddock
Published: December 2020

Imagine a group of middle school students armed with stakes, string, tape measures, and a video camera walking to a large open area near their homes and talking excitedly about their homework. What could cause such enthusiasm?
Teacher to Teacher: Math Counts: It Really Works!
Authors: McCoy Sara , Reinke Kathryn S.
Published: December 2020

I teach a course, “topics of middle school Mathematics,” at Oklahoma State University. One of my students is involved with tutoring for MATHCOUNTS, and I encouraged her to write a short article about her experiences. This article is an excellent example of how students come to realize that what they learn in their teachereducation courses is valid.
Mathematics Notebooks in Middle School and Junior High School
Authors: Price J. J. , Canarecci Mary , Conrad Jim , Ehresnzan Doreen , Foster Carmie , Mark D. Harris , Martin Kathy , Mullendore Tammy , Rice Thomas K. , Wrighthouse Penny
Published: December 2020

Current educational philosophy places strong emphasis on student portfolios. Although some states require portfolios, many mathematics teachers are unsure about how to implement their use in classrooms. One way to develop a pool of items for potential inclusion in the mathematics portfolio is to have students keep a working portfolio in the form of a notebook.
Exploring and Changing Visions of Mathematics Teaching and Learning: What Do Students Think?
Authors: Jayne Fleener M. , Dupree Gloria Nan , Craven Lary D.
Published: December 2020

Change is hard. We can have little doubt that to survive in the twenty-first century, students need different skills from those touted by eighteenth-century educational wisdom. Over and over, in national reports, legislative mandates, edicts from school boards, concerns of parents, messages from principals, and discussions in the teachers' lounge. we address how schools are failing in so many ways to meet the needs of all our students. Mathematics and science, in particular, are the content areas most often cited as the sieves for separating and isolating students, where the haves experience meaningful mathematics and the have-nots experience boring, repetitive, and meaningless mathematics.
Encouraging Mathematical Thinking
Authors: Turner Julianne C. , Styers Karen Rossman , Daggs Debra G.
Published: December 2020

With these words, the NCTM (1989, 65) portrays a dilemma familiar to many middle-grades teachers. Although many teachers strive to involve their students in active and challenging problem-solving activities, students' past experiences may have instilled preconceptions that mathematics is mechanical, uninteresting, or unattainable. In addition, many teachers lack models and examples of how to design mathematics instruction so that it fosters students' engagement. Because the middle grades are crucial years for developing students' future interest in mathematics, middle-grades teachers must take seriously the challenge of presenting mathematics as an exciting discipline that is relevant and accessible to all students. For the past two year, we have been experimenting with approaches that will inte rest students in challenging mathematics while supporting them in constructing meaning.
Cartoon Corner
Authors: Fisher Julie
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 assigmnent. 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.
Honest Numbers: A Mathematics and Languages Connection
Authors: Bezuszka Stanley J. , Kenney Margaret J.
Published: December 2020

Honestly, certain numbers are called honest. In fact, honest numbers exist in many of the world's languages and dialects. This brief article explores the meaning of honest numbers and their relation to arbitrarily chosen nonnegative integers. It also promotes the pursuit of honest numbers in various languages. Several projects and explorations for students to undertake are outlined. These open-ended projects not only connect different aspects of mathematics but also encourage students to investigate the cultures and languages present in today's diverse classrooms. The study of honest numbers yields interesting patterns among numbers and words. In this context, several opportunities for analyzing data and making conjectures arise.
Racing to Understand Probability
Authors: Van Zoest Laura R. , Walker Rebecca K.
Published: December 2020

One curriculum change proposed by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, 109) is an increased focus on probability in the middle grades.
Developing Conceptual Understanding of Relations and Functions with Attribute Blocks
Authors: Quinn Robert J.
Published: December 2020

Much attention has recently been focused on students in the middle grades. The unique difficulties and characteristics of this group have prompted the National Council of Teachers of Mathematics to refer to them as “children in transition” (NCTM 1989, 68). Middle school students continue to rely on concrete experiences to construct knowledge but are starting to develop the ability to think abtractly (NCTM 1989, 68). Thus, the middle school curriculum should build “a bridge between the concrete elementary school curriculum and the more formal curriculum of the high school” (NCTM 1989, 102). The series of lessons described in this article extends the concrete experiences that many elementary students have had with attribute blocks to provide middle school students with informal opportunities to explore the concepts and properties of relations and functions.
The Fractions of a Day
Authors: Edelman Leslie
Published: December 2020

In algebra class, students often have a difficult time understanding the notion that the fraction of the work that one person does plus the fraction of the work that the other person does will total the whole job. That is, it will equal 1. In my heterogeneous sixth-grade-mathematics group, we do a project that allows students to derive fractions with varying denominators, estimate that fractional representation in a pie chart as a piece of the whole. and add their various fractions to find the sum equal to one whole. The hope is that this process will ease the difficulty of their future reasoning in algebra.
Teacher to Teacher: A Big Gulp Activity
Authors: Kelly Bruce
Published: December 2020

The following instructions explain the Big Gulp activity and how to implement it in the classroom.
November's Menu of Problems
Authors: Georgeson Joseph , Laughlin Connie
Published: December 2020

A light fare to whet your mathematical appetite.
December's Menu of Problems
Authors: Georgeson Joseph , Laughlin Connie
Published: December 2020

A light fare to whet your mathematical appetite.
Math Is Functional! A Math Fair for Kids
Authors: Reys Barbara J. , Wasman Deanna G.
Published: December 2020

How do you give preservice teachers authentic opportunities to design and facilitate mathematics activities to motivate students? How do you involve parents in looking at mathematics from a nontraditional viewpoint? We found a way to address each of these important goals by sponsoring a Saturday-morning Math Fair for Kids using NCTM's promotional theme “Math Is FUNctional” (with the emphasis on FUN).
Using Bar Representations as a Model for Connecting Concepts of Rational Number
Authors: Middleton James A. , Heuvel-Panhuizen Marja van den , Shew Julia A.
Published: December 2020

Middle Grades Students Should be able to understand, represent, and use numbers in a variety of equivalent forms, including fractions, decimals, and percents. They should develop number sense for fractions and other representations of rational number. Students should also be able to represent such relationships in graphical form (NCTM 1989).
Estimate of the Week
Authors: Gulley Wendy
Published: December 2020

Imagine an extracurricular mathematical contest in which 60 percent of your school's student body voluntarily participates each week. Imagine the excitement on students' faces in a school assembly, where the results of this contest is the most anticipated announcement of the week. And imagine that this contest is easy for you to run, costs nothing, and gives students experience with important skills and concepts in the middle school curriculum.
On My Mind: Classroom Realities We Do Not Often Talk About
Authors: Leinwand Steven J.
Published: December 2020

After spending time in diverse schools, I often return to the Professional Standards for Teaching Mathematics (NCTM 1991) to help sort through what I have seen and heard. I use the Standards document to track the gap between our vision and the day-to-day realities I observe. A well-worn touchstone is Standard 6: Analysis of Teaching and Learning, which states the following (p. 63):
Mathematical Investigations—Powerful Learning Situations
Authors: Chapin Suzanne H.
Published: December 2020

What is a mathematical investigation? Why should I consider using instigations in my classroom? What types of investigations are appropriate for my students? We should ask these and related questions when we consider using alternative approaches to teach new content. Many teachers are now presenting in-depth investigations to students as one way of exploring and learning new mathematics.
Multiplication with Fractions: A Piagetian, Constructivist Approach
Authors: Warrington Mary Ann , Kamii Constance
Published: December 2020

On the basis of piaget's (1954, 1960) Constructivism, Kamii (1989, 1994) has demonstrated that children in the primary grades can invent their ownprocedures for solving multidigit problems with whole numbers. A significant finding of this research is that when children are not taught algorithms, such as those of “carrying” and “borrowing,” their number sense and knowledge of place value are far superior to those of students who have been taught these rules. Warrington (1997) extended this work to the fifth- and sixth-grade level and described an approach to “teaching” division with fractions without teaching the algorithm of “invert and multiply.” This article describes a constructivist approach to multiplication with fractions.
Now & Then: Words, Words, Words: Ancient Communication
Authors: Greeley Nansee , Offerman Theresa Reardon
Published: December 2020

Now… The insightful use of words is what most people think of when they consider the work of a journalist, especially that of a reporter. However, numbers play a large part, as well. Mike Fisher, sports journalist for the Dallas Cowboys; columnist for the Fort Worth Star-Telegram; and host of the radio show, Fish for Lunch, which airs daily on KLIF radio in Dallas, Texas, knows that a good understanding of numbers, particularly statistics, is essential to being a sports journalist.
Teacher to Teacher: Truss(t)ing Triangles
Authors: Bartels Bobbye Hoffman
Published: December 2020

Often middle school students see no connection between geometry and real life. The following activity was designed to help make this connection for seventh-grade students participating in a Saturday academy. The activity centers on an elementary investigation of the rigidity characteristic of triangles, a concept seldom mentioned in K-8 mathematic textbooks but essential to the construction of structures that have to absorb tremendous forces and not collapse. Although this activity was completed outside the traditional mathematics class, it can be adapted to a school schedule and completed over two or more class periods.
How Many Blocks?
Authors: Battista Michael T.
Published: December 2020

Fifth graders ben and Jessica were trying to determine the amount of space contained in their classroom. They chose as their unit of measure the 9-inch-by-9-inch-by-18-inch concrete block that had been used to construct the classroom walls. Because other students in the class had mentioned this idea, the teacher had previously brought an actual block into the classroom so that students could see all three of its dimensions. To find the number of blocks in the horizontal layer that covered the floor, they counted the blocks that occurred lengthwise in each of two adjacent walls and multiplied. They then multiplied that product by the number of blocks showing in the height of the room. For example, for a small room, such as in figure 1, they would have multiplied 3 times 3, for the floor; times 5, for the height; and claimed that 45 blocks fill the room.
Exchange: Geometry in the Middle School: An Exchange of Ideas and Experiences
Authors: Collier C. Patrick , Pateracki Tina
Published: December 2020

Geometry and measurement represent two of the nine content strands in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and thirteen of the forty-nine objectives for grades 5-8. In addition, several other content strands have explicit connections to geometry. For example, some of the “patterns and functions” deal with geometric patterns, and some of the probability is in geometric settings. The Standards document describes a vision for the role of geometry and measurement that is different from, and larger than, the traditional role. No longer is it to be a “frills” topic relegated to the back of commonly used textbooks nor restricted to identifying figures and reciting a litany of their properties. Instead, geometry should encompass a variety of experiences in which middle schoolers create their own understanding of the relationships in our multidimensional world. Geometric models should be used to analyze, interpret, and solve problems from probability, algebra, and other areas of mathematics. That is, geometry should take a place among the major topics of emphasis in the mathematics study of a middle school student.
March's Menu of Problems
Authors: Davidenko Susana
Published: December 2020

A light fare to whet your mathematical appetite.
Exploring Geometry with Technology
Authors: Manouchehri Azita , Enderson Mary C. , Pugnucco Lyle A.
Published: December 2020

The study of geometry in grades 5-8 should incorporate opportunities for students to engage in exploring and analyzing geometric shapes to conjecture about geometric relationships through data collection and model construction, according to the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). In this fashion, students will develop an intuitive understanding of geometric concepts and learn to reason formally and informally. Moreover, it is hoped that through such processes, students will formulate relevant definitions and theorems. The Standards document also encourages the use of computer technologies in middle school mathematics instruction. This suggestion was based on the assumption that interactive environments provided by appropriate geometry software have the potential to foster students' movement from concrete expetiences with mathematics to more formal levels of abstractions, nurture students' conjectuting spirit, and improve their mathematical thinking. Although the NCTM's visions for the geometry curriculum and for methods of teaching geometry in the middle levels are certainly attractive, many teachers are concerned about what software is useful for the middle school population, how such software can be used in instruction. what issues are associated with their use, and what the consequences are of learning and teaching mathematics within such environments.
“You Are Cleared to Land”
Authors: Billstein Rick
Published: December 2020

On a not-so-routine flight to an NCTM meeting, my colleague and seat-mate, Jim Williamson, and I had ample time to chat and ponder our surroundings. As happens many times in the West in the winter, heavy snow caused a long delay. While we waited on the runway for the airplane to be de-iced, we noticed that runway numbers were posted in pairs, as in Runway 2-20. We also noticed that some numbers had an L or an R attached to them, as in Runway 2L This observation was the only setting we needed to start a real-world investigation on how airport runways are named. Because it is hard to generalize from the small sample of numbers that we could see from the airplane, we realized that we needed more data. We talked to pilot, who furnished us with several old maps of airport runways. He did give us a hint that the runway numbers had something to do with the compass headings of the runway from magnetic north. We were now in business: we had an interesting problem, we had data, and we had Time to talk about it and think about it. This atmosphere is the type for which we strive in a middle school mathematics class. The following activity is based on our investigation and can be used for middle school mathematics.
The Thinking of Students: Marcy's Dot Pattern
Authors: Kenney Patricia Ann , Zawojewski Judith S. , Silver Edward A.
Published: December 2020

Suppose that you gave the problem in figure 1 to your students to solve and asked them to explain how Marcy could determine the number of dots in the twentieth figure without drawing all twenty pictures. What kinds of strategies do you think that your students would suggest to Marcy? Will some strategies be more efficient than others? Will some students disregard the directive and draw all or some of the twenty pictures to solve the problem? Also, will their explanations be complete and clear?
May's Menu of Problems
Authors: Quinn Anne Larson , Barb Cynthia A. , Lasher Pameia R.
Published: December 2020

A light fare to whet your mathematical appetite.
Mathematics Detective: Calendar Search: Every Day Is Special!
Authors: Campbell Larry N.
Published: December 2020

At the beginning of the school year we often look at our calendar to circle dates that are important to us. Have you ever noticed that some dates seem to be mathematically special? What about today's date? Suppose that it is September 15. We could define the date 9/15 as a common-factor date because the month and day numbers have the common factor 3. Some other common-factor dates are 9/3, 9/18, and 9/21. What are the remaining common-factor dates in September?
Polygon Capture: A Geometry Game
Authors: Carroll William M.
Published: December 2020

The curriculum and evaluation standards for School Mathematics (NCTM 1989) calls for an increased role for geometry in the primary and middle school curricula. An important mathematical strand in its own right, geometry also provides opportunities to promote and assess mathematical communication, reasoning, and problem-solving skills. Unfortunately, many students lack the vocabulary and the conceptual understanding needed to desctibe geometric relationships. This atiicle describes a game, Capture the Polygons, that I have designed to help middle school students think about geometric properties and the relationships among them. A version of the game has been tested in firth- and sixth-grade classes as part of the field test of Fifth Grade Everyday Mathematics (Bell et al. 1995). Observations of classes playing the game, as well as feedback from their teachers, indicate that students find the game challenging but fun. Depending on the background of the students, it can be played at different levels of difficulty.
Cartoon Corner
Authors: Fisher Julie A.
Published: December 2020

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

Incorporating the rubrjc scorjng results of assessments into a grading system has posed problems for many teachers. Using a rubric to grade problem solving, for example, is generally understood. But how do you assign points, percents, or grades for the purpose of reporting progress to parents or administrators? This article describes a way to assign points to the problem-solving process.
Innovations in Curriculum: Implementing Change: Rewards and Challenges
Authors: Tetley Linda
Published: December 2020

Welcome to a new department called “lnnovations in Curriculum.” It was prompted by some exciting new curricular materials that are currently available for middle school mathematics classes.
Cartoon Corner
Authors: Fisher Julie A.
Published: December 2020

“Cartoon Corner” is designed to include cartoons that can be used to highlight mathematics in an interesting way. The questions accompanying the cartoons can be used as a large- or small-group activity, as homework, or as an extracredit assignment. Students might want to create their that demonstrate in a humorous way how mathematics is used and misused. Readers are encouraged to submit three copies of cartoons and class room-tested questions, doublespaced, for review by the editor of this section. The cartoons must include the date and the newspaper syndicate that holds the copyright.
Promoting Mathematical Discourse: Learning from Classroom Examples
Authors: Manouchehri Azita , Enderson Mary C.
Published: December 2020

The NCTM's Professional Standards for Teaching Mathematics (1991) has directed attention to “discourse” in the mathematics classroom. This document recommends that mathematics instruction should promote students' discourse by orchestrating situations in which each individual's thinking is challenged and by asking students to clarify and justify ideas. “Discourse,” as described by the Standards document, highlights the way in which knowledge is constructed and exchanged in the classroom (Ball 1992). Teaching mathematics from the perspective of developing mathematical discourse requires building a new vision for mathematics classrooms and poses a major challenge for mathematics teachers at all levels. This challenge was recognized by D'Ambrosio (1995). She identified the need to build environments in which students construct a “personal relationship” with mathematics as one of the most important requirements for promoting and sustaining the type of discourse envisioned by the reform movement. In such environments, students engage in authentic mathematical inquiry; act like mathematicians as they explore ideas and concepts; and negotiate the meanings of, and the connections among, those ideas with others in class (D'Ambrosio 1995).
Teachers' Action in Community Problem Solving
Authors: Pugalee David , Malloy Carol
Published: December 2020

Reform efforts in mathematics have as A goal the development of learners who are mathematically literate. As a result of this emphasis, teachers are being asked to think about mathematics instruction as the merger of pedagogical knowledge, content knowledge, and knowledge of how to integrate both into school mathematics (Mathematical Sciences Education Board 1996). But as they modify their practices for this merger, they struggle with how to structure learning situations for optimum learning; how and when to use grouping instead of didactic instruction; and when to use inquiry-based activities that allow students to investigate, conjecture, and test hypotheses in learning mathematics instead of “teaching by telling” the mathematics (Futrell 1996; Smith 1996).
By Way of Introduction: Data and Chance
Authors: Pateracki Christina
Published: December 2020

Asolid foundation in mathematics and science is increasingly necessary to navigate this changing technological and information age” (U.S. Department of Education 1998). This statement is particularly true of probability and statistics. We need only to pick up the newspaper to find numerous examples of how information has been quantified and presented to us, the public, in a variety of graphical and narrative forms. To be an astute consumer of this information, we must have a solid understanding of data and chance.
Exploring Probability through an Evens-Odds Dice Game
Authors: Wiest Lynda R. , Quinn Robert J.
Published: December 2020

Middle-Grades students should have opportunities to experiment actively with situations that model probability, including “making hypotheses, testing conjectures, and refining their theories on the basis of new information” (NCTM 1989, 111). These experiences should include a discussion of theoretical probabilities, where appropriate, and the use of correct mathematical terminology and expressions.
Means and MADs
Authors: Kader Gary D.
Published: December 2020

Included activity builds a framework for developing students' understanding of the notion of variation from the mean. The activity is a modification and an extension of one described by Friel, Mokros, and Russell (1992).
International Perspectives: Cat and Mouse
Authors: Lannen Brian
Published: December 2020

Here is a story about “That Favourite Teaching Resource,” which, despite its simplicity, I believe helps address some serious issues in mathematics education. I obtained it from AAMT [Australian Association of Mathematics Teachers], and I am pleased to offer it back to our readers with some feedback on my experience with it. The resource is a game, widely applicable and easy to implement. You could be using it today!
Teacher to Teacher: The Time-Line Project
Authors: Tayeh Carla , Pokay Patricia A.
Published: December 2020

How do you weave the history of mathematics into your classroom activities? How can you promote greater awareness among your students about the richness that different cultures have contributed to the development of mathematics?
Reflections on Practice: Modifying Our Questions to Assess Students' Thinking
Authors: Chappell Michaele F. , Thompson Denisse R.
Published: December 2020

Among assessment's varied purposes is monitoring students' progress toward learning and achievement (NCTM 1995; Webb 1992). Written assessment items in the form of openended questions enlighten teachers about the extent to which students understand underlying mathematical concepts. Cai, Lane, and Jakabcsin (1996) assert that these types of items are rich and informative, bringing “mathematical communication into assessment” (p. 138). Such communication helps in assessing not only the mathematics that students know but also their understanding of the mathematics (NCTM 1989, 1995).
Much of What I Needed to Know I Learned by Teaching Mathematics
Authors: Collier C. Patrick
Published: December 2020

What would you say if asked to deliver a commencement address and could not refuse? I found myself trying to answer that question. The only parameters were to keep it to ten minutes or less and make it appropriate to the occasion. I knew that it had to reflect my concept of education as a lifelong endeavor, that it had to reflect my experience as a mathematics educator, and that it should contain some of the important themes from my work with problem solving for teachers. As I composed the speech, I recalled how the elements of problem solving extend far beyond the mathematics classrooms. I have gotten very positive responses from people who are familiar with the current role of problem solving in the mathematics curriculum and from those who respond to the more general themes. I gave this address on 16 May 1998 at the University of Wisconsin—Oshkosh. It has also appeared in the June 1998 issue of Intersection, a newsletter published monthly by the NCTM and the Exxon Education Foundation.
From The Giver to The Twenty-One Balloons: Explorations with Probability
Authors: Lawrence Ann
Published: December 2020

Basing an activity in the mathematics classroom on an event or a story from children's literature offers a vivid, engaging context for mathematical investigation.
Reflections on Practice: Multiple Strategies = Multiple Challenges
Authors: Meyer Margaret R.
Published: December 2020

When teachers give a problem like “Find 3/4 of 60,” they can anticipate the ways in which students will respond. It is easy to tell whether the student knows how to do the problem and whether she or he got the right answer. The mistakes that students might make on this problem are also predictable. Other problems found in traditional middle school curricula might produce more variety in solution strategies, but usually few surprises greet the experienced teacher. Few judgments need to be made, because the strategy is either appropriate or not; and barring computational errors, the answer is either right or wrong. This familiarity with the mathematics, the problems, and potential student responses produces a desirable level of comfort for teachers, even for those who are sometimes uncertain about their own mathematical skill and understanding. It leaves them free to focus on preparing students for the next skill to be learned and assisting those students who need extra help to become proficient. What happens when the curriculum, some of the mathematics, the problems, and the solution strategies all seem new and unpredictable? What does this situation mean for the teacher and for the students?
Take Time for Action: Establishing Fraction Benchmarks
Authors: Reys Barbara J. , Kim Ok-Kyeong , Bay Jennifer M.
Published: December 2020

The use of benchmarks, such as 0, 1/2, and 1, for comparing the size of fractions is not emphasized very often in instruction. Comparing fractions with 0, 1/2, or 1 by comparing the numerator with the denominator or by mentally modeling the fraction is a powerful tool for gauging the size of fractions, making quick estimates, and judging the reasonableness of computed results (Bezuk and Bieck 1992). If students are asked to compare 5/8 with 4/9, the traditional technique is to find the common denominator, convert both fractions to equivalent forms using this common denominator, then compare numerators. However, this problem can be solved more efficiently by comparing each fraction with 1/2: 5/8 is larger than 1/2, since it is larger than 4/8; and 4/9 is less than 1/2, since 4/9 is less than 4.5/9, or 1/2.
Teacher to Teacher: Counting Grass
Authors: Little Catherine
Published: December 2020

I could see the look of shocked disbelief spreading from face to face. Mrs. Little had “really lost it this time.” She wanted the class to go out and count the number of blades of grass in the soccer field!
Mathematics Detective: McGwire, Sosa, and the Home-Run Champions
Authors: Spangler David S.
Published: December 2020

The 1998 major league baseball season was filled with excitement as Mark McGwire of the Saint Louis Cardinals and Sammy Sosa of the Chicago Cubs battled to break Roger Maris's Major League single-season home-run mark. Maris, while playing for the New York Yankees, hit 61 home runs in 1961. Once McGwire and then Sosa broke Maris's record, the two continued also to score a big hit with fans, as many people tried to predict just how many home runs each would finally hit for the season.
Powerful Connections: An Opportunity for Professional Growth
Authors: Panasuk Regina M. , Sullivan Mary M.
Published: December 2020

Professional development may take many forms, but long-lasting qualitative change can be achieved when teachers themselves select the methods by which to investigate more effective approaches for teaching and for supporting students' learning of mathematics (NCTM 1991). To ensure progress, teachers must construct their own professional knowledge and not simply experience an information transmission from in-service specialists.
Card Logic
Authors: Pagni David L. , Shultz Harris S.
Published: December 2020

IN MATHEMATICS WE LOOK FOR PATTERNS that can lead to an algorithm for solving a problem in the general case. An interesting problem that is accessible to almost any level from fifth grade to adult is described in this article. The object of the problem is to devise an algorithm that will allow the problem to be solved with different numbers of cards.
Exploring Factor Sets with a Graphing Calculator
Authors: Eisen Arron Paul
Published: December 2020

One component of the middle school curriculum is the concept of the factors of a number.
Reflections on Practice: Perimeter and Area through the van Hiele Model
Authors: Malloy Carol
Published: December 2020

IN MANY CLASSROOMS, MATHEMATICS IS TAUGHT by using examples to show students how to solve problems, then having the students complete large numbers of similar problems (Battista 1999). This process, called “parrot math” by O'Brien (1999), overlooks research showing that students (1) develop knowledge through interaction between the student and the knowledge, (2) do not think like adults, and (3) learn well through social interaction. Disregarding current research on how students learn mathematics and continuing the use of “parrot math” can be harmful to students' broader understanding of mathematical relationships (O'Brien 1999; Battista 1999).
Mathematics Detective: The Price Isn't Right
Authors: Spangler David
Published: December 2020

The stories that you are about to read are true; only the names have been changed. In each scenario, employees made mathematical errors and charged customers incorrect amounts. So “come on down,” and let us investigate what happened. Let us also do some mathematics to make sure that the price is right.
Teacher to Teacher: Ordered Pair Relations: a Performance Assessment
Authors: Allen Robert M.
Published: December 2020

Assessment goals may include having students demonstrate factual knowledge, apply skills, integrate previously learned skills with new concepts, and synthesize concepts from various disciplines. The assessment strategy described here seeks to help students integrate new concepts with previously learned topics from different strands and eventually apply the results of the problems to future situations.
Reinventing Scrabble with Middle School Students
Authors: Quinn Robert J. , Wiest Lynda R.
Published: December 2020

Which letters of the alphabet occur most often in the English language? Which occur least often? The answers to these questions are essential to code breakers as they attempt to crack complex ciphers (Rosen 1988). This information can also help students understand the structure of language and form strategies when playing word games. The word game described in this article offers a context for integrating mathematics and linguistics and performing statistical analyses. Middle school students enjoy games that allow them to improve their mathematics skills in nonthreatening settings (Braxton et al. 1995). Further, games provide an opportunity for students to work cooperatively and use problemsolving skills (Leonard and Tracy 1993).
The Thinking of Students: The 100th Letter
Authors: Trotter Terrel
Published: December 2020

Solving nonroutine problems with multiple approaches is one of the goals promoted by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). I present two problems that seem to fit the bill, since my seventh-grade students worked on them in various ways.
International Perspectives: Universal Games from A to Z
Authors: Morris Helen
Published: December 2020

Games have been played for thousands of years, for power, for money, for amusement, all over the world. They began very simply, using the resources available at the time—sand, pebbles, coloured sticks, shells. Most of the games that we play today have descended from these simple ancient pastimes, although we now have access to superior resources, including computers programmed to beat us, and commercially manufactured packages. We cannot always afford to purchase the more sophisticated offerings, but we can always find items around that will still be able to provide children with experiences and excitement enjoyed around the world for the past 3000 years.
Why Is the Year 2000 a Leap Year?
Authors: Wilkins Jesse L. M.
Published: December 2020

Keeping track of time has intrigued people throughout history. The constant urge to harness time has resulted in many attempts to perfect the calendar. The study of calendars offers students many opportunities to investigate measurement issues associated with time, the revolution of the earth around the sun, and the historical development of the calendar as civilization became more dependent on keeping accurate time.
Reflections on Practice: Redefining Success in Mathematics Teaching and Learning
Authors: Smith Margaret Schwan
Published: December 2020

Amajor goal of current reform efforts is to help students learn mathematics with understanding. “Good” mathematical tasks are an important starting point for developing mathematical understanding, but selecting and setting up good tasks does not guarantee a high level of student engagement (Smith and Stein 1998). Using such tasks can, and often does, present challenges for teachers and students.
The Thinking of Students: The Chicken Problem
Authors: Reeves Charles A. (Andy)
Published: December 2020

The problem shown in Figure 1 (Burrill 1998) struck me immediately as one to try with my sixth graders. Before reading further, try to solve the Chicken Problem yourself.
On My Mind: What Can We Expect from Research?
Authors: Hiebert James
Published: December 2020

AS REFORM RECOMMENDATIONS COLLIDE with traditional practices, the questions on the minds of many interested teachers and parents are these: Should we really adopt this new textbook series? Should we change the way we teach? What mathematics should students really be learning? It would be nice if research could give clear and simple answers to these kinds of “should” questions, but it does not and it cannot.
Take Time for Action: Mean and Median: Are They Really So Easy?
Authors: Zawojewski Judith S. , Shaughnessy J. Michael
Published: December 2020

When are median and mean taught to students in your curriculum? Our first introduction to these terms as students was in college-level statistics courses in the mathematics department. Of course, we had learned to find arithmetic averages during elementary school as an application of long division and again in high school algebra when learning to use variables to represent relationships in equations. One of the authors taught middle school in the 1970s, and the other taught finite mathematics to college students in the 1970s. We were surprised to find these measures of central tendency in the books for middle school and equally surprised that college students had not previously seen median and mode. The procedure for finding the median is much easier than the one for finding the mean, so why not include it in the middle school curriculum? To teach the mean, all we had to say to students was that it was the same as the average that they had already learned in fifthand sixth-grade mathematics.
Innovation in Curriculum: Algebraic Concepts: What's Really New in New Curricula?
Authors: Star Jon R. , Herbel-Eisenmann Beth A. , Smith John P.
Published: December 2020

New mathematics curricula serve middle grades students well when they provide students with richer and more accessible introductions to a wide range of mathematical content. New curricula also serve teachers well when they lead us to examine and reflect on what and how we teach. When these curricula enter our working lives and conversations, we are often forced to question exactly what is “new” about them and how this “newness” may affect our students' learning. To address this issue and, we hope, to support further reflection and discussion, we take a closer and more careful look at what is new in one middle school curriculum's approach to algebra. The curriculum we examine is the Connected Mathematics Project (CMP) (Lappan et al. 1998), particularly the eighth-grade units, but the issue of what is new in algebra is relevant to many other innovative middle school curricula, as well.
Never Say Anything a Kid Can Say!
Authors: Reinhart Steven C.
Published: December 2020

After extensive planning, i presented what should have been a masterpiece lesson. I worked several examples on the overhead projector, answered every student's question in great detail, and explained the concept so clearly that surely my students understood. The next day, however, it became obvious that the students were totally confused. In my early years of teaching, this situation happened all too often. Even though observations by my principal clearly pointed out that I was very good at explaining mathematics to my students, knew my subject matter well, and really seemed to be a dedicated and caring teacher, something was wrong. My students were capable of learning much more than they displayed.
Daily Journals Connect Mathematics to Real Life
Authors: Albert Lillie R. , Antos Jennifer
Published: December 2020

Why are we learning this? Why do we have to know how to do this? When are we ever going to use this outside of class?” Do these questions sound familiar to you? Students in mathematics classes commonly ask these questions when they are unable to make connections between what they are learning in the classroom and their daily lives. This article discusses the importance of relating mathematics to students' everyday lives. When children make connections between the real world and mathematical concepts, mathematics becomes relevant to them. As mathematics becomes relevant, students become more motivated to learn and more interested in the learning process. This article describes a journal-writing project developed in a fifth-grade total-inclusion classroom and specifies the major features of the writing project, including the framework used to assess student learning.
Fostering Mathematical Thinking through Multiple Solutions
Authors: Cai Jinfa , Kenney Patricia Ann
Published: December 2020

The reform movement in school mathematics advocates communication as a necessary component for learning, doing, and understanding mathematics (Elliott and Kenney 1996). Communication in mathematics means that one is able not only to use its vocabulary, notation, and structure to express ideas and relationships but also to think and reason mathematically. In fact, communication is considered the means by which teachers and students can share the processes of learning, doing, and understanding mathematics. Students should express their thinking and problem-solving processes in both written and oral formats. The clarity and completeness of students' communication can indicate how well they understand the related mathematical concepts.
Using Counterintuitive Problems to Promote Student Discussion
Authors: Maylone Nelson J.
Published: December 2020

The NCTM's Curriculum and Evaluation Standards document (1989) has sparked a revolution in curriculum and textbook design and, to some extent, the evaluation of students. The Standards encourage problem solving in context; increasing opportunities for students to discuss mathematics; and actively involving students in exploring, conjecturing, analyzing, and applying mathematics. Recommended instructional techniques include promoting group work, making presentations, conducting experiments, and working with manipulatives.
Teaching Algebra in the Middle Grades Using Mathmagic
Authors: Koirala Hari P. , Goodwin Phillip M.
Published: December 2020

A LARGE NUMBER OF MATHEMATICS EDUCATORS and teachers argue for including algebra in the middle school mathematics curriculum (Fouche 1997; Silver 1997). Recommended algebraic concepts to be taught in the middle grades include variable, expression, and equation (NCTM 1989), and middle-grade students should be able to “apply algebraic methods to solve a variety of real-world and mathematical problems” (NCTM 1989, 102). In spite of this emphasis on teaching algebra, a large number of middle school students, especially at the fifth- and sixthgrade levels, are never taught algebraic concepts.
Welcome to Volume 6!
Authors: Thompson Denisse
Published: December 2020

This volume sees the old millennium out, with gratitude for the five exciting years that have witnessed the establishment of the journal as a valuable resource for middle-grades mathematics educators of all descriptions. With the January 2001 issue, volume 6 takes the journal into the new millennium, with its continuing promise of service to all mathematics educators involved in middle-grades mathematics teaching and learning.
Choosing Problems with Entry Points for All Students
Authors: Stern Frances
Published: December 2020

Teachers frequently ask, “How can I teach everyone in my classroom when the mathematical knowledge and ability levels are so varied? Is it possible to teach meaningful mathematics without dividing the students into separate ability-level groups?” To solve this dilemma, I look for problems that can be explored on several levels.
Making Connections: A Thematic Mathematics Project for Grade 7
Authors: Smith David B.
Published: December 2020

Here is the teacher'S Dilemma: Students are motivated by new experiences in mathematics, but many lack the basic skills and understanding of concepts that they need to be successful when the pace of instruction quickens. For example, when introducing the multiplication property of equality to solve ax = b, I find that many seventh graders have forgotten how to multiply fractions and decimals. They are interested in the prospect of learning something new, but their enthusiasm is dampened because they cannot perform simple computations. Even after years of drill and practice, familiar algorithms have not reached long-term memory.
2 Is Not the Same As 2.0!
Authors: Drum Randell L. , Petty Wesley G.
Published: December 2020

The packaging on a set of allen hex key wrenches identifies their sizes as 2.0 mm, 3.0 mm, and 4.0 mm. A sector in a circle graph reports a company's percent of income from domestic sales as 42.30 percent. An illustration of a plate of steel available from an industrial supplier gives the diameter of a hole that has been bored in the plate as 8.2500 in.