bluuebunny/crossref_metadata_2025 huggingface dataset:
Assessment: Create Assessments That Do It All
Authors: Cohen Ira S. , Fowler Joan V.
Published: December 2020

How would you like to hear such comments about your mathematics tests from students? In this article, ideas for an assessment system that reflects NCTM's Curriculum Standards (NCTM 1989) are presented. In this system, assessment tasks are created to support instruction that values a mastery of the “basics,” mental mathematics, mathematics vocabulaty, problem solving and writing in mathematics, calculator mathematics, and mathematics understanding. The motivation behind each task is the need to measure both a student's knowledge and his or her depth of understanding. In addition, as the students have said in their own words, these assessment are intended to be a learning experience. The assessment presented also adhere to the principle that “if it is important, assess it.”
What Is the Score on Scored Discussions?
Authors: McGrath Kathleen L.
Published: December 2020

I believe that humans do not exist in a vacuum and that one of my most important jobs as a teacher is to foster and support the progress of my students as lifelong learners who are productive, successful members of groups working on a common goal or problem. Cooperative learning is a wonderful tool for this purpose.
Standards 2000: Refining Our Efforts
Authors: Lappan Glenda
Published: December 2020

Since its inception in 1920, the National Council of Teachers of Mathematics I (NCTM) has assumed the responsibility of providing broad national leadership in mathematics education. To establish a focus for discussion on improving the mathematics education of every student in states and provinces across North America, NCTM published three Standards documents from 1989 to 1995, addressing curriculum, teaching, and assessment. The Standards present a focus and vision-not a single path or algorithmfor building excellent mathematics programs.
Reflections on Practice: Exploring How One Problem Contributes to Student Learning
Authors: Friel Susan N.
Published: December 2020

Today, problem solving is very much a focus of the mathematics curriculum. Increased emphasis has also been placed on listening to and using student-generated solution strategies to explore the process of problem solving and develop mathematics concepts. Most recently, the Third International Mathematics and Science Study (TIMSS) has highlighted the differences in mathematics teaching that occur among countries (TIMSS International Study Center 1997). These differences emphasize the role of problem solving and the benefits of engaging students in high-level tasks that provide opportunities for rich exploration and discussion.
International Perspectives: The Triple Jump: A Potential Area for Problem Solving and Investigation
Authors: Clarke Doug
Published: December 2020

Many teachers comment that interesting appucations of ratio are hard to find. Teachers also find that many students have difficulty with proportional reasoning in general, and ratio in particular. In this article, I talk through a process that led to the discovery of the role of ratios and percentages in discussions about triple jump performances, and to a range of tasks that provide important information about students' understanding of key concepts in mathematics.
Problem Posing and Critiquing: How It Can Happen in Your Classroom
Authors: English Lyn D. , Cudmore Donald , Tilley Dominic
Published: December 2020

Student-generated problems can offer a rich and motivating source of problem activities for middle school children. Traditionally, students have solved our problems, problems that we think will be of interest and relevance to them. The problems that we choose for students, however, are not always the ones that they would choose. Furthermore, the problems that we create are usually solvable, have “correct” expression, and normally contain appropriate and sufficient information. Real-life problems, however, do not always come so neatly packaged, and neither do student-generated problems. Being able to deal with nontextbook problems is becoming increasingly necessary in our society; we need to ensure that our students can meet this challenge.
Discourse, of Course: Encouraging Genuine Mathematical Conversations
Authors: Van Zoest Laura R. , Enyart Ann
Published: December 2020

Discourse is one area of the nctm's professional teaching standards for School Mathematics (1991) that causes many teachers particular difficulty. Mathematics teachers have a long history as lecturers. Although “initiation-reply-evaluation” (Richards 1991) sequences between the teacher and students are not uncommon, genuine mathematical conversations are rare in most classrooms (Weiss 1994). Discourse can be a problem area for teachers when they do not realize how important it is and have not seen or experienced dynamic classroom discourse. Once a teacher has seen students defending their mathematical ideas, questioning other students' ideas, and helping clarify the mathematics to one another, the importance of discourse becomes clear.
Teacher to Teacher: The Tables Turn: Students Evaluate Their Mathematics Class
Authors: Kaplan Cindy
Published: December 2020

Near the end of the school year, our graduating eighth-grade students are required to fill out a questionnaire to evaluate their mathematics class. They do so anonymously. The first eleven statements require the stude nts to “strongly disagree,” “disagree,” “agree.” or “strongly agree.” Note that they are given no middle ground; students must agree or disagree to some extent.
Now & Then: Dancing in Time and Space: A Step Back in Time
Authors: Greeley Nansee , Offerman Theresa Reardon
Published: December 2020

Now… Michael Anthony Began What Would Be His Career as a dancer when he was in elementary school. His parents had e nrolled him in both piano and dance classes. and he soon discovered his love of music. During junior high school, he succumbed to peer pressure and dropped dance to play sports. Michael's agility and conditioning from dance, however. proved he lpful on both the court and the field, and he soon became a valued member of his junior high school football and basketball team. By the time he started high school, he realized how much he missed dance. While continuing his sports, he returned to dance lessons and found the theater to be a wonderful place to incorporate dance with high school life. After high school, Michael attended the Boston Conservatory. majoring in musical theater and minoring in dance. He believes that his strong mathematic background significantly helped him in his music-theory courses and that it is an asset in his career as a director, composer, and choreographer.
Walking Around: Getting More from Informal Assessment
Authors: Cole Karen A.
Published: December 2020

AS A TEACHER, YOU ARE ALWAYS ASsessing your students. You walk around as they work on tasks or projects, observing groups, conversing with students, spot-teaching concepts and skills, and checking for understanding. This article describes similar work done through the Middle School Math through Applications Project (MMAP)—a comprehensive, project-based middle school mathematics curriculum project funded by the National Science Foundation. Its units are based on engaging scenarios in which students take the role of such mathematicsusing professionals as architects, biologists, and cartographers. We asked ourselves, “How can we organize this natural process to make better use of the precious information we get through informal contact with students?” We discuss some valuable techniques that MMAP teachers and researchers developed for organizing informal assessment so that it produces a coherent story of student progress; helps students make more progress with greater focus; and complements other types of embedded assessments, such as journal writing.
Mathematics Detective: Ads That Are BAA-AD!
Authors: Spangler David B.
Published: December 2020

Something is wrong with each of the advertisements to the right. Are the mistakes honest errors—or is someone trying to “pull the wool over our eyes”? Sometimes it is hard to tell for sure. But we can put on our detective hat to investigate what these advertisements are saying mathematically. The questions that follow will help us determine whether the bananas are “a-peeling,” whether the lamps are a “shady” deal, and whether the CDs should be “music to your ears.”
Take Time for Action: Using Performance Graphs to Assess Problem Solving
Authors: Evered Lisa , Uy Frederick L.
Published: December 2020

Assessing students' problem-solving competence is A perennial challenge for middle school teachers. “Textbook problems” that require only numerical responses do not give teachers adequate information about the student's understanding and misunderstandings. For a study conducted at the United Nations International School (UNIS) in New Yort City, we designed problem-solving test items that could be scored on two dimensions: the dimension of accuracy and under. standing and the dimension of information density, that is, the quantity of in. formation used by the student in formulating a response. This methodology led us to construct two-dimensional performance graphs for each student on each item. These graphs proved helpful in assessing student competencies and diagnosing student difficulties in the important area of problem solving.
Now and Then: Fiber Meets Fibonacci; The Shape of Things to Come
Authors: Johnson Art
Published: December 2020

AN OLD SAYING GOES SOMETHING LIKE THIS, “ARTISTS ARE born, not made.” For Billie Ruth Sudduth, this statement is not quite true. Billie Ruth, who lives in the North Carolina mountains, makes baskets that are prized by collectors from all across North America and have been displayed in the Renwick Gallery of the Smithsonian Institution in Washington, D.C. She is internationally known for her basket artistry and was the first woman to be designated a Living Treasure by the state of North Carolina. But she was not always a basket maker.
Innovation in Curriculum: Teaching Mathematics with MiC: An Opportunity for Change
Authors: Meyer Margaret R. , Ludwig Matthew A.
Published: December 2020

This “Innovations in Curriculum” article is a view of one implementation as seen through the eyes of a developer and a classroom teacher. It is a reminder that each experience with change creates challenges and has multiple perspectives. To help facilitate who is saying what, the developer's comments are in italics.
Dispelling Myths about Reform in School Mathematics
Authors: Curcio Frances R.
Published: December 2020

For decades, recommendations for improving the teaching and learning of Mathematics have been made (NCTM 1970). With the exception of the launching of Sputnik in 1957, no other single event seems to have sparked as much public controversy and debate in school mathematics as the advent of the “standards movement.”
Building on: Building Up: Proportional-Reasoning Activities for Future Teachers
Authors: Parker Melanie
Published: December 2020

The conceptual domain of ratio and proportion may be a bridge that permits access to higher-level thinking in mathematics (Lesh, Post, and Behr 1988). Future teachers who have developed a variety of rich representations and flexible ways of thinking about proportional relationships are well positioned to offer instruction that helps students move from using informal strategies to expressing proportional relationships in algebraic terms. I have found that exploring informal “building up” strategies with preservice teachers leads nicely to formalizing proportional relationships algebraically. In this article, I first share what I mean by “building up” strategies. A description and discussion of selected activities that have been used successfully with preservice teachers follow.
Teacher to Teacher: A Girls' Math Olympiad Team
Authors: Volpe Betty J.
Published: December 2020

IN 1991 I BEGAN TO COACH THE SIXTHgrade Math Olympiad Team in Candlewood Middle School, a public middle school for grades 6 through 8. The Mathematical Olympiads for Elementary and Middle Schools (MOEMS) is a nonprofit public foundation that provides opportunities for children through grade 6 to experience creative problem solving in a nonthreatening competitive setting throughout the school year. The Math Olympiads holds five olympiad contests, which are given at monthly intervals beginning in the middle of November. Thus, each school has about two and one-half months to get ready for the olympiads. Each olympiad contest contains five verbal problems, each with a time limit. Each team may have a maximum of thirty-five participants. When the olympiads conclude in the middle of March, about two and one-half months remain to discuss and review the olympiad problems and to introduce new topics.
Reflections on Practice: Relating Fractions and Decimals: Listening to Students Talk
Authors: Oppenheimer Lauren , Hunting Robert P.
Published: December 2020

Despite the great amount of time that middle-grades teachers devote to teaching fractions and decimals, converting between these two representations continues to be a difficult task for students. According to the results of the sixth National Assessment of Educational Progress (NAEP) conducted in 1992, although 90 percent of eighth-grade students correctly paired a simple fraction with its pictorial representation, only 63 percent of students successfully shaded a fractional portion of a given rectangular region using equivalent fractions. Likewise, 92 percent correctly identified 14.9 seconds as being the decimal representation closest to 15 seconds, but when comparing common fractions with decimal notation, only 51 percent of eighth-grade students chose 1/2 as being the fraction closest to 0.52. Twenty-nine percent of eighth graders chose the fraction 1/50 as being closest in value to 0.52 (Kouba, Zawojewski, and Struchens 1997).
Innovation in Curriculum: Reflections on Developing Formal Mathematics and the Connected Mathematics Project
Authors: Zawojewski Judith S. , Robinson Marlene , Hoover Mark
Published: December 2020

Sound conceptual understanding is crucial to the development of students' formal mathematical knowledge. However, this conceptual understanding rarely develops from a single lesson, but rather from related experiences over time. For example, to use and apply formulas for area in a powerful way, students need to understand that the conventional measure of area is based on counting squares that cover a surface, and further, they need to understand the ways in which the concept of area as covering-with-square-units is related to the formulas for area.
Teacher to Teacher: Geometry Projects Linking Mathematics Literacy Art and Technology
Authors: Little Catherine
Published: December 2020

How do you spark your students' interest in geometry? Which assignments connect geometry with other disciplines? Which assignments call on students' initiative and creativity?—Editorial Panel
Putting Math in Motion with Calculator-Based Labs
Authors: Doerr Helen M. , Rieff Cathieann , Tabor Jason
Published: December 2020

Interpreting graphs is widely recognized as being an important goal of mathematics study. Yet students face many conceptual obstacles in learning to make sense of positionversus-time graphs. A calculator-based motion lab allows students to bring these graphs to life by turning their own motion into a graph that can be analyzed, investigated, and most important, interpreted in terms of how they actually moved. The investigation of motion can become a rich site for building students' intuitions about the concept of rate of change and for developing skills in creating and interpreting graphs. The kinesthetic activity of motion becomes a powerful means for students to understand position-time graphs. The study of changes in motion need not be reserved for students in precalculus or calculus classes.
Take Time for Action: Data Collection: Getting to Know Your Students' Attitudes
Authors: Strutchens Marilyn E.
Published: December 2020

DO YOUR STUDENTS LIKE MATHEmatics? Are they confident in their ability to do mathematics? All too often, we focus on ensuring that students develop mathematical content knowledge, paying little attention to such affective issues as students' attitudes toward mathematics and their beliefs about their abilities to do mathematics. Yet instruction affects students' attitudes toward, and beliefs about, mathematics; likewise students' attitudes influence mathematics instruction (McLeod 1992; Reyes 1980). Students should learn to value mathematics and become confident in their ability to do mathematics (NCTM 1989).
Now and Then: Displaying Data Is As Easy As Pie! Polar Area Graphs: Early Pie Charts
Authors: Johnson Art
Published: December 2020

Imagine having to cut your household budget by 10 percent. Where would you make the cuts? Would you eat less food? Change the climate control in your home? Buy fewer new clothes? At times, we all must live on a budget and cut back on expenses. For Mark Conrad, these tasks are a bit more challenging. Mark is in charge of a $60 million school budget for Nashua, New Hampshire. When Nashua's mayor or the city council asks the board of education to cut expenses, Mark is at the center of debate about whether the cuts are even possible and where they might be made.
Sticks to the Roof of Your Mouth?
Authors: Friel Susan N. , O'Connor William T.
Published: December 2020

IS IT SMOOTH, OR DOES IT HAVE SMALL BITS of peanuts in it? Does it taste sweet and nutty, or does it have a slightly burned taste? Is it overly sticky or dry? These are just some of the characteristics that are considered when peanut butters are rated on quality (Consumers Union 1990). This topic appears to interest many people; Consumer Reports has published two studies about peanut-butter quality since 1990 (Consumer's Union 1990, 1995). How does a consumer use these data to determine what may be a “best buy”? One eighth-grade class explored this question with dataanalysis activities (Lappan et al. 1998).
Capture and Recapture Your Students' Interest in Statistics
Authors: Morita June G.
Published: December 2020

Take your class fishing without getting wet! Linking our mathematics lessons with the real work of real scientists provides lasting motivation for our students. In this hands-on activity, students gain the feeling of power and ownership of a statistical method that they themselves derive—and they have fun!
Juniper Green
Authors: Stallings L. Lynn , Bullock Patricia L.
Published: December 2020

Juniper green is the name of a school in England and of an enjoyable game invented by teacher Richard Porteous (Stewart 1997). By playing Juniper Green, students strengthen their number sense by using mental mathematics to find and check factors of numbers up to 100 and multiples up to 100. Figuring out an optimal strategy for playing introduces students to some interesting ideas about numbers. Students enjoy this game so much that they play it outside class. One needs to play Juniper Green to fully appreciate its worth; it does not take long to play, requires few materials, and is still challenging and enjoyable after being played several times.
Mathematics Detective: Mode Code
Authors: Spangler David B.
Published: December 2020

Cryptology is the Science of Secret Communication. The word derives from the Greek kryptos, meaning “hidden,” and logos, meaning “word.” People have wanted to communicate secretly for a long time. In fact, code writing is about as old as writing itself.
Assessing Open-Ended Problems
Authors: Conway Kathleen D.
Published: December 2020

AN IMPORTANT ASpect of the change that needs to be made to help our students survive in the twenty-first century involves enhancing their ability to think creatively and to solve problems. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) lists problem solving as one of the four process standards for students in kindergarten through twelfth grade and suggests using openended problems to address these standards. To assess students' solutions to open-ended problems, one must use techniques that are very different from those used in traditional assessment. The method proposed here defines and describes the use of the measures of fluency, flexibility, and originality to assess students' responses to open-ended problems.
Making Sure That Your Mathematics Curriculum Meets Standards
Authors: Kulm Gerald
Published: December 2020

Mathematics curriculum materials have come under heavy criticism in the wake of the results of the Third International Mathematics and Science Study (TIMSS). According to the TIMSS analysis, the mathematics curriculum has been characterized as splintered and unfocused, and United States mathematics textbooks have been described as “a mile wide and an inch deep” (Schmidt, McKnight, and Raizen 1996). Meanwhile, the National Science Foundation has funded the development of new mathematics materials that are designed to align with national standards (NCTM 1989) and to address the perception that current textbooks need improvement. These new materials are now becoming available through commercial publishers.
Sharing Reflections on Teaching
Authors: Lernhardt Melissa M. , Clement Lisa
Published: December 2020

WE HAVE RECENTLY STARTED TEACHING MATHEMATICS COURSES THAT WE HAD NOT taught previously. We wanted to be effective at helping our students, who are prospective elementary teachers, understand the mathematics that we were trying to teach, and we thought that reflecting together in a systematic way would help us meet that goal. We developed a process of reflecting that we believe a school or community of teachers can organize and adapt to suit their own needs. By reflecting, we mean giving careful thought and consideration to classroom events. In this article, we would like to share our reflecting process. We also want to share what we learned by participating in this process together.
Now We Are Nine…
Authors: Wood Eric
Published: December 2020

AN EXCITING YEAR IS IN STORE FOR Mathematics Teaching in the Middle School (MTMS): with this fifth volume, we reach our full complement of nine issues! And with the ninth issue comes an additional opportunity to serve you–our readers–by presenting more articles that address your needs as middle-grades-mathematics educators. For those who first conceived of this journal for the middle grades, for the first Editorial Panel that launched it, and for the subsequent panels that have continued to shape and nurture it, this volume is the culmination of much thought and effort. The years of development to a nine-issue volume have been richly rewarded by the involvement and enthusiasm of you–our readers.
Making Charts: Do Your Students Really Understand the Data?
Authors: Feicht Louis
Published: December 2020

Alook at any chart in the local or national newspapers reveals that they are much flashier than the charts is typically created in our mathematics or science classes. They tend to draw conclusions for the readers in their captions and titles. While looking for ways to make students' science-fair projects stand out to the public and the judges, I thought that this use of more descriptive titles might grab the judges' attention and give the students an edge over the competition. Using conclusions as chart titles and captions became one of the most important teaching and learning ideas I have used.
Innovation in Curriculum: Growth through Change
Authors: Van Boening Lois
Published: December 2020

AS A MATHEMATICS TEACHER, ONE OF MY primary goals has been to help all students attain success in mathematics. I believe that if students are to achieve in mathematics, they must be motivated to learn. To help provide this motivation, I let them know that I love mathematics and that mathematics is a fundamental part of their lives. Making a commitment to continue learning about how children learn mathematics best is an indirect way that I try to achieve this goal.
On My Mind: Reduce Skills Teaching in the Mathematics Class
Authors: Devlin Keith
Published: December 2020

IT IS MY BELIEF THAT WE NEED TO REDUCE significantly the time we spend teaching basic skills in grades 7–12 mathematics classes.
Graphs in Real Time
Authors: Beckmann Charlene E. , Rozanski Kara
Published: December 2020

Slope is introduced in various ways in current middle and secondary school mathematics textbooks. In one approach, students graph a line and determine the vertical and horizontal distances from the axes to a point on the line, determining the coordinates of the point. Students then calculate the ratio of vertical distances to horizontal distances for several pairs of points on the line and discover that the ratio is always the same. At this point, this constant ratio is defined as the slope of a line. Students are then asked to determine to what aspect of the graph the slope corresponds and how it relates to the equation of the line (Phillips 1991, 55–57).
International Perspectives: Active Learning of Mathematics
Authors: Smith Jim
Published: December 2020

Since the cockcroft report in 1982, there has been an increasing emphasis on the use of active learning in school mathematics, with a typical and influential view being propounded in Better Mathematics: “Mathematics can be effectively learned only by involving pupils in experimenting, questioning, reflecting, discovering, inventing and discussing. Mathematics should be a kind of learning which requires a minimum of factual knowledge and a great deal of experience in dealing with situations using particular kinds of thinking skills” (Ahmed 1987, 24).
National Council of Teachers of Mathematics: Revitalizing and Refocusing Our Efforts
Authors: Lappan Glenda
Published: December 2020

WE ARE CURrently standing at one of the most important crossroads we have faced as a Council and as a profession in the past twenty years. We have more than a decade of direction setting and progress under our belt. And we have learned a lot. We have every right to celebrate our successes even as we look to the future and the continuing need for improvement.
The Thinking of Students: Krystal's Method
Authors: Johnson Art
Published: December 2020

Our students are continually constructing their own understanding of mathematics as they discover and then learn it in our classes. Sometimes their personal constructs involve new approaches to traditional problems, which is exactly what happened with Krystal Demmons, a student in my geometry class.
Take Time for Action: Dominoes as Fractions: Misconceptions and Understandings
Authors: Ward Robin A.
Published: December 2020

Research by peck and jencks (1981) and van de Walle (1990) indicates that the concept of fractions is often difficult for students in early grades to master. Additionally, the results of the fourth NAEP assessment test suggest that children have a limited understanding of rational numbers (Kouba et al. 1988). Introducing abstract symbols, terminology, and forms of representation without first developing meaning on the basis of students' experiences and readiness may result in some of the difficulties they exhibit (NCTM 1994). The following activity invloving dominoes fosters insights into the strategies that my students used and the misconceptions that they exhibited while studying fractions and considering their relative magnitude in relation to other fractions.
Evaluating Instructional Materials
Authors: Bernhard Jamal , Lernhardt Melissa Mellissinos , Miranda-Decker Rose
Published: December 2020

IN 1996, WE WERE ASKED TO EVALUATE A new set of mathematics instructional materials for the middle school classroom. The instructional unit included a wide assortment of materials, so we decided to begin with some explicit criteria on which to base our evaluation. We turned to several sources, including the National Council of Teachers of Mathematics (NCTM) and local school districts, seeking evaluation guidelines that were in the spirit of the NCTM's Standards documents (1989, 1991, 1995).
Mathematically Rich, Equitable Game Software
Authors: Murray Megan , Mokros Jan , Rubin Andee
Published: December 2020

IT IS NO SECRET THAT MIDDLE SCHOOL STUdents spend significant amounts of free time playing software games, many of which fall into the “edutainment” category and many of which engage children in practicing mathematical skills. Middle school children and their parents spend a great deal of money on such software, often in the hopes of raising levels of mathematical achievement. The reasoning is as follows: If mathematical mastery can be accomplished in an entertaining and engaging manner, it is a win-win situation. Drill-and-practice games are especially popular, and in some instances they lead to modest increases in computational speed and accuracy (Becker 1990). But is it possible for children to go beyond drill and practice while using computer games? What are the ingredients of mathematical games that help students deepen their mathematical thinking? And to what extent are effective mathematical games also accessible to a range of children?
Genetics as a Context for the Study of Probability
Authors: Brahier Daniel J.
Published: December 2020

Two teachers recently interviewed a number of middle school students to assess their understanding of probability. Students were shown a bag containing ten red marbles and ten white marbles and were asked to pull two marbles at random from the bag. Before looking at the marbles, they were asked, “Do you think that the two marbles you pull out are more likely to be both red, both white, or one of each, or are the chances of any of these combinations equal?” The students almost invariably said that since three possibilities were involved—red-red, white-white, and red-white—any of the combinations had a one in three chance, so no particular result was any more likely than another. Of course, the teachers watched with wonder as these students completely ignored the notion that redwhite and white-red are different combinations, so the chances of pulling one of each is twice that of pulling out two reds or two whites.
Teacher to Teacher: Oh, Good, It's Tuesday!
Authors: Georgakis Pauline
Published: December 2020

MY SIXTH-GRADE STUDENTS LOOK forward to Tuesday every week. In fact, they cannot wait for Tuesday to come again, because every Tuesday they get together in small groups to solve problems. When they enter the classroom, the students ask for the problems, get in their groups, and begin. Soon the room is filled with the sound of problem solvers at work: “I've got an idea.” “Let's try this.” “How did you get that?” “I'm telling you, this will work.” “What if we …?” “Why did you do that?”
Reflections on Practice: Multiple Interpretations = More Challenges
Authors: Kouba Vicky L.
Published: December 2020

AS THE EMPHASIS IN CURRICULA AND ASSESSMENT shifts toward activities that are embedded in contexts, challenges and opportunities for thoughtful reflection on practice arise. Meyer (1999) presented multiple teaching and assessment challenges that teachers face when students use quite different ways to solve a problem. In addition to finding multiple solutions to a single, clear interpretation of a problem, groups of students may also have differing interpretations of contexts.
Teacher to Teacher: Finding Common Ground
Authors: Bradley Elizabeth H.
Published: December 2020

IF YOUR STUDENTS CONFUSE GREATEST common factor (GCF) with least common multiple (LCM), try this approach, which employs a diagram as a visual aid and can be used to reduce fractions.
How Do Students Think about Statistical Sampling before Instruction?
Authors: Jacobs Victoria R.
Published: December 2020

How would your students respond to the Raffle Scenario in figure 1? What information about your students' knowledge would help you plan instruction for statistical issues related to the Raffle Scenario? This article highlights students' thinking and instructional implications from two studies that examined upper-elementary students' written and oral explanations as they responded to survey situations.
Problem-Centered Mathematics Teaching
Authors: Lubienski Sarah Theule
Published: December 2020

Through my interactions with preservice and in-service teachers in California, Michigan, and New York, I have heard a variety of perspectives on instruction in problem-centered mathematics instruction. I have watched educators struggle to find a role for problem solving in their classes. What constitutes teaching through problem solving? Exploring answers to this question can raise awareness of one's own perspectives and lead to deeper knowledge of problem-centered mathematics teaching.
Teacher to Teacher: Inequalities: Part of Every Child's Life
Authors: Tent Margaret W.
Published: December 2020

Bring me numbers!” roared the queen. “Many numbers. Special numbers. Yes, bring me numbers that are less than 10. No, no! They must be greater than 5. Do you have a number? Do you have a number that will satisfy me? Ah! You do! A magnificent number, 7. Yes! A 7 will do nicely.
Mathematics Detective: Why 2K?
Authors: Krusi Jean B.
Published: December 2020

A lot of discussion has occurred in newspapers and magazines about whether the new millennium begins this year or next, and we have all seen headlines about potential Y2K problems with computers. Here are some questions and problems to celebrate the year 2000. Be a detective, and use the clues in these problems to solve them.
Free Pizza? Slim Chance!
Authors: Florence Hope
Published: December 2020

The following “Food for Thought” problem appeared in the March 1999 issue of the journal.
Concentration: Connecting Fractions, Decimals, & Percents
Authors: Sweeney Elizabeth S. , Quinn Robert J.
Published: December 2020

Fractions, Decimals, and Percents are often included in discussions about middle school mathematics. Unfortunately, these discussions also include groans of dissatisfaction, stemming from the lack of success that teachers often have in teaching these concepts. Many students fail to see the relationship among fractions, decimals, and percents. As one student put it, a decimal is “a thing that makes numbers even more confusing,” whereas another characterized a percent as “the way teachers give you points.” The fact that these topics are typically taught in isolation is the main source of dissatisfaction. Often, the only connection mentioned by textbooks is a cursory discussion of conversions. This article describes one innovative method that can help middle-grades students become more flexible in their ability to represent fractions, decimals, and percents, an outcome recommended by the NCTM's Standards.
Should the U.S. Emulate Singapore's Education System to Achieve Singapore's Success in the TIMSS?
Authors: Menon Ramakrishnan
Published: December 2020

Since the release of the timss results, a call has been issued to emulate the curriculum, if not the educational practices, of Asian countries that have done well on the TIMSS. California has even stated that its “new” mathematics curriculum will be fashioned after the “rigorous” Singapore mathematics curriculum. Before endorsing such views, it would be instructive to look at some possible reasons that Singapore's eighth graders performed so well in the TIMSS and some possible implications. I list five possible reasons for Singapore students' success in the eighth-grade TIMSS and ask the reader to reflect on the implications to mathematics teaching and learning in the United States under “Questions to ponder.”
Teacher to Teacher: Travel the World: An Addition Game
Authors: Owens Katherine D. , Sanders Richard L.
Published: December 2020

This article introduces the “whole-number machine” and describes its use as an odometer in an addition game. This device was created for use with middle school students and can be used as a hands-on display as students review place-value concepts, practice computation of whole numbers, and explore large-number values and number names. The addition game described here has been tested with fifth-grade students whose motivation to play the game was influenced by the use of the “whole-number machine” as a calculating and recording device.
Let's Talk about the Weather: Lessons Learned in Facilitating Mathematical Discourse
Authors: Leonard Jacqueline
Published: December 2020

One idea that has gained considerable attention in mathematics education is that teachers should become facilitators of mathematical discourse (Bush and Kincer 1993; Hiebert and Wearne 1993; NCTM 1991). Mathematical discourse should encourage student interpretations of a mathematical task and student-tostudent interactions (Lo and Wheatley 1994). In the majority of American classrooms, few interactions occur, because teachers dominate the discourse by asking all the questions (Cazden 1988); and if students do not know, the teachers also give the answers. As a result, “teaching by telling” is the most common pedagogy. One alternative to this practice is constructivist teaching.
Now and Then: A Career That Counts, Herman Hollerith: Punching Holes in the Process
Authors: Johnson Art
Published: December 2020

Now… AMY SMITH IS INVOLVED IN THE ANALYSIS OF CENSUS DATA that come from mailed forms or from door-to-door surveys. She works at the Administrative Records and Methodology Research Branch (ARMR) of the Population Division of the Census Bureau, located outside of Washington, D.C. The Census Bureau is part of the Department of Commerce. According to Amy, “We have to develop and test the estimate procedures; then collect, tabulate and check the appropriate incoming data; and then analyze the population estimates. In checking the data, we look for data that do not align with previous data, allowing for reasonable changes since the last data were received. In addition, we often do not receive all the data that are needed to prepare the annual estimates. Then we must develop additional techniques to fill in the missing data.”
Using Concept Maps to Link Mathematical Ideas
Authors: Baroody Arthur J. , Bartels Bobbye H.
Published: December 2020

The National Council of Teachers of Mathematics's standards documents (1989, 1991, 1995) emphasize making and assessing connections. Why are connections so important? Research suggests that understanding can be viewed as a connection between two pieces of information (Ginsburg 1977), and an understanding of elementary concepts is essential for mathematical power, for example, applying school mathematics to everyday tasks, inventing mathematical procedures, understanding and solving genuine problems, and comprehending more advanced mathematical ideas. The degree of a student's understanding is determined by the number, accuracy, and strength of connections (Hiebert and Carpenter 1992; Resnick and Ford 1981). A concept is well understood if it has many links to other aspects of knowledge that are accurate and strong.
Some Big Ideas of Algebra in the Middle Grades
Authors: Edwards Thomas G.
Published: December 2020

Without question, mathematics in general, and algebra in particular, have served as “gatekeepers” to the study of other academic fields, such as engineering, the physical sciences, computer science, and medicine, as well as to increased vocational opportunities in technical support fields. As a result, middle school teachers have felt increased pressure both to teach algebraic concepts directly and to develop mathematical concepts in ways that will support students' formal study of algebra in the future. A recent call for manuscripts in Mathematics Teaching in the Middle School noted that “the rate of students' success with this subject has been linked to the careful, planned development of algebra as a way of thinking about and modeling various phenomena at every grade level” (NCTM 1999). Such a careful, planned development requires clearly identifying the “big ideas” of algebra that are appropriate to middle school.
Mathematics Detective: We All Scream for Ice Cream!
Authors: Krusi Jean B.
Published: December 2020

Even as the hot weather of summer fades into autumnal chill, we continue to eat the dessert of all seasons: ice cream. Americans eat more ice cream than people in any other country. Over two billion gallons of ice cream are produced in the United States each year.
Mathematical Roots: Looking at Probability through a Historical Lens
Authors: Jardine Dick
Published: December 2020

The mathematical theory of probability has a long history, mostly because of people's interest in gambling and games of chance. In completing this activity, students will not only develop an understanding of the mathematics of probability but also learn about some of the people responsible for creating that branch of mathematics.
Innovation in Curriculum: Connecting Integers to Meaningful Contexts
Authors: Kent Laura Brinker
Published: December 2020

Eaningful contexts can be used to enhance learning of significant mathematics and promote connections between students' experience and new mathematical content (NCTM 1998). In addition, contextualized situations, whether real or imaginary, help students make visual and sometimes physical links between their informal knowledge and formal mathematical ideas. For example, linear measurements of such objects as ladders or poles that are partially above and partially below water level can be used to introduce positive and negative numbers. Moreover, such contexts as board games and sports build on students' activities and interests outside the classroom.
Clinching First Place: Calculating the Magic Number
Authors: Quinn Robert J.
Published: December 2020

Middle school students should be offered opportunities to analyze interesting situations and should be encouraged to draw logical conclusions in the context of these situations. This article describes a lesson in which students are asked to apply their mathematical intuition to answer questions, determine possibilities based on the win-loss records of teams in a sports league, and develop a formula for calculating the magic number for clinching first place. The magic number is a term familiar to most sports fans. It refers to the sum of the number of wins by the first-place team and losses by another team (frequently the second-place team) in their remaining games that will ensure that the firstplace team finishes first at the end of the season.
Linking Algebraic Concepts and Contexts: Every Picture Tells a Story
Authors: Nickerson Susan D. , Nydam Cherie , Bowers Janet S.
Published: December 2020

Current trends in middle school curricula highlight the importance of integrating algebraic reasoning throughout the middle grades. The use of innovative technologies offers one exciting way to engage students in algebraic problem-solving activities. In this article, we report on our efforts to design for seventh graders a computer-based instructional unit that would not only address state standards but, we hoped, also lay the groundwork for the students' subsequent formal study of algebra. The software we chose, SimCalc MathWorlds, is available free of charge via the World Wide Web at www.simcalc. umassd.edu. Look on the first page of this site. With this program, students can animate characters on the screen by creating graphs of position and velocity.
Spotlight on the Principles/Standards: Problem Solving in the Middle Grades
Authors: Malloy Carol E. , Guild D. Bruce
Published: December 2020

IN WHAT WAYS WOULD YOU LIKE YOUR middle-grades students to experience problem solving in the mathematics curriculum? Do you want the curriculum to capture the excitement of geometry and measurement, algebra, statistics, and number relationships? Do you want it to help students understand and build new mathematical knowledge and explore new mathematical relationships? Do you want the curriculum to be filled with opportunities for students to ponder, create, and critique arguments about mathematics? If this is your vision for your students, then you should be pleased with, and excited by, the Problem Solving Standard in Principles and Standards for School Mathematics (NCTM 2000).
Math at Work: Basing a Career on Base Two
Authors: Tepper Jonathan R.
Published: December 2020

Have you ever wondered, “When will I use this in the real world?” Kris Kasias, a computer programming consultant, uses the mathematics that he learned in school on the job. Kris works with computers and deals with numbers and patterns every day.
Deciding When to Use Calculators
Authors: Thompson Anthony D. , Sproule Stephen L.
Published: December 2020

The influence of technology, particularly the calculator, in the middle school classroom has become a compelling issue for both practicing and prospective teachers. The National Council of Teachers of Mathematics (1989) encourages the use of calculators in the middle grades, but teachers face a number of difficulties when they introduce calculators in their classrooms. In our work with both prospective and practicing teachers, we frequently hear the same concerns, particularly from middle school teachers, about incorporating calculators into the curriculum. These teachers ask, “When should I use calculators?” and “What should students know before I allow them to use calculators?” In particular, teachers want to be able to justify their answers to these questions to other teachers and parents who might be concerned about including calculator use in the middle school curriculum. The larger question that teachers often ask is “On what basis do I make the decision to use calculators with my students?”
Teacher to Teacher: Mission Possible! Can You Walk Your Talk?
Authors: Johnson Iris DeLoach
Published: December 2020

The Mission Impossible television series is a thing of the past, yet Mission Impossible movies keep the mystery and intrigue alive. I take advantage of this intrigue to make one particular mathematics lesson more interesting for my students. This lesson uses a problemsolving approach to introduce an exploration of graphing-related concepts, such as intercepts, slope, and rates of change.
Listening to Middle School Students' Algebraic Thinking
Authors: Smith John P. , Phillips Elizabeth A.
Published: December 2020

NO PART OF THE K–12 MATHEMATICS curriculum is more fluid and controversial than introductory algebra. Content and assessment issues lie at the core of this debate: What algebra skills and understandings are important? What kind of evidence suggests that students possess these skills? Neither question can be answered in simple terms; in fact, no single “right” answer may exist for either one.
The Inka Quipu: Positional Notation on a Knotted Cord
Authors: Zaslavsky Claudia , Crespo Bianka
Published: December 2020

WE INHERITED A MARVELOUS SYSTEM of numeration from India by way of the Arab world. The Indo-Arabic (also called Hindu-Arabic, or, simply, Arabic) system of number notation is considered one of the most important inventions in the history of humankind. The amazingly efficient system encompasses the following features: (1) distinct symbols for the numbers from 1 to 9, (2) a symbol for 0, (3) place value for successive powers of base ten, and (4) the ability to write any numeral with just ten symbols.
Students' Building on One Another's Mathematical Ideas
Authors: Sherin Miriam Gamoran , Louis David , Mendez Edith Prentice
Published: December 2020

Three years ago, the authors of this article began to work together to develop a middle school classroom in which students talk about mathematics. We found that getting students to talk about mathematics was easy—in fact, students were eager to share their ideas. More challenging was finding ways to encourage students to interpret and comment on the ideas of their classmates. These kinds of exchanges were especially important to us because in the discourse community that we envisioned, the teacher is not alone in the position of clarifying and questioning student methods. Rather, students take an active role by commenting on and critiquing one another's mathematical ideas (Ball 1993; Brown and Campione 1996; Sherin, Mendez, and Louis 2000).