bluuebunny/crossref_metadata_2025 huggingface dataset:
On My Mind: Not All Manipulatives and Models Are Created Equal
Authors: Roberts Sally K.
Published: December 2020

The vision of the mathematics curriculum articulated in Principles and Standards for School Mathematics (NCTM 2000) calls for students to construct their own understanding of mathematical ideas by making, refining, and exploring conjectures based on evidence and use of a variety of reasoning and proof techniques (p. 3). For many of us who struggled to learn mathematics through a chalk-and-talk, do-it-my-way approach to mathematics instruction, the notion of using models and manipulatives to help the learner construct mathematics knowledge is both refreshing and exciting.
Issues of Language: Teacher Insights from Mathematics Lessons in Chinese
Authors: Anhalt Cynthia O. , Ondrus Matthew , Horak Virginia
Published: December 2020

A fundamental concern in mathematics education is to understand the connection between the mathematics and the students who are trying to learn the mathematics. Even under ordinary conditions, it may be difficult for teachers to completely understand the challenges that students face. This can be especially true for teachers of English language learners (ELL). Given the steadily increasing population of U.S. students who are classified as limited in English proficiency, it is imperative that teachers understand the perspective of an ELL student in an English-speaking classroom. Meaningful professional development can further this type of understanding. In this article, we describe a professional development experience in which twenty-two teachers from schools with large Latino student populations participated in two mathematics lessons taught in Chinese. The goal for the activity was to allow the teachers to experience challenges similar to those that many students face. Thus, in addition to describing the mathematics lessons, we examine some of the participating teachers' reflections and insights.
Mathematical Explorations: Batty Functions: Exploring Quadratic Functions through Children's Literature
Authors: Roy Jessica A. , Beckmann Charlene E.
Published: December 2020

Many middle school students do not realize that functions and mathematical relations are present in their everyday lives. For example, the number of calories a student burns is a function of how long the student runs; the amount of pizza available to be eaten is a function of the diameter of the pizza. These examples of functions found in our daily lives can be described in a variety of ways, including with tables, graphs, and possibly symbolic generalizations. Many students do not realize that functions and relations exist outside of the mathematics classroom and that they can be represented in numerous ways (Billings and Lakatos 2003).
Solve It! Pip Seek
Authors: Mooney Edward
Published: December 2020

A domino is a rectangular tile. The dots, called pips, represent the two numbers on the tile. The domino pictured below shows the pair 1–5. A Double 6 set of dominoes consists of every possible pairing of numbers from 0 to 6 (i.e., 0–0, 1–0, 1–1, …, 6–6). Sets are also available in Double 9, Double 12, Double 15, and Double 18. How many pips are on a Double 9 set of dominoes?
Understanding Students' Problem-Solving Knowledge through Their Writing
Authors: Steele Diana F.
Published: December 2020

The Curriculum and Evaluation Standards for School Mathematics, published in 1989 by NCTM, placed an emphasis on the importance of communication for learning and doing mathematics. Writing is one way to achieve this valuable communication in the classroom. In 2000, Principles and Standards for School Mathematics emphasized writing as being an important aspect of communication: “Writing in mathematics can also help students consolidate their thinking because it requires them to reflect on their work and clarify their thoughts about the ideas developed in the lesson' (NCTM 2000, p. 61). Several studies have also shown ways that writing can be an effective tool for improving students' learning of mathematics (Bell and Bell 1985; Pugalee 2001; Steele 2005). Pugalee (2001) found in his work with students that writing could potentially help students increase their problemsolving ability in mathematics.
Are We Golden? Investigations with the Golden Ratio
Authors: Reeder Stacy L.
Published: December 2020

What do our skeletons have in common mathematically with nature, Greek statues, the Parthenon, and Leonardo da Vinci's artwork? Fascinating Fibonaccis: Mystery and Magic in Numbers (Garland 1987) sheds light on the answer to this question and provides the opportunity for readers to discover other wonderful connections among mathematics, art, and nature. Making connections within mathematics and between mathematics and other content areas is one of the NCTM's goals for students. In my opinion, it is perhaps one of the most important goals. When students begin to make connections in mathematics and also between mathematics and other aspects of their world, they begin to see its beauty and its mystery. It becomes intriguing and meaningful to them, and their learning begins to have personal relevance.
Fermi Questions
Authors: Taggart Germaine L. , Adams Paul E. , Eltze Ervin , Heinrichs John , Hohman James , Hickman Karen
Published: December 2020

How middle school students view mathematics is a function of what they learn and how they learn it. Evidence from actual classrooms shows that a serious disconnection sometimes occurs between what students think mathematics can deliver and the real world (Burrill 1997). Students must have the opportunity to discover multiple ways to solve real-life problems through problem solving, using estimation and conjecture, and developing critical communication skills in the classroom.
Quick Reads: Promising Partnerships: Gearing Up for Mathematics
Authors: Simmons Charlotte , Byrne Jesse
Published: December 2020

[M]ore low-income families will be well-informed of postsecondary options and financial aid, and take the steps necessary for admission into an institute of higher education. With GEAR UP, more low-income students will go to college and succeed. (Silver, n.d.)
A Look at the Development of Algebraic Thinking in Curriculum Focal Points
Authors: Seeley Cathy , Schielack Jane F.
Published: December 2020

As more and more students are expected to study algebra in middle school or high school, much attention has been given to the development of algebraic reasoning. In Curriculum and Evaluation Standards for School Mathematics (1989), the NCTM advocated incorporating algebraic thinking from the early grades on. Principles and Standards for School Mathematics (2000) further strengthened this stand by highlighting a new Standard on Representation, a foundational tool for algebra. Whether high school students are expected to enroll in a course called agebra 1 or an integrated high school course that contains algebraic content, it is clear that the United States is placing more emphasis on preparing for the use of algebraic concepts and skills.
Take Time for Action: Investigating Students' Thinking about Functional Relationships
Authors: You Zhixia
Published: December 2020

How do your students perform when you ask them to compare linear and nonlinear relationships? Do they use process-oriented (procedural) approaches or object-oriented (structural) approaches? Can they integrate the two? Are those who connect process and object perspectives more successful in comparing functional relationships?
Division by a Fraction: Assessing Understanding through Problem Writing
Authors: Barlow Angela T. , Drake Jill Mizzell
Published: December 2020

As performance-based curricula and teacher accountability gain greater emphasis, teachers need avenues to ensure that their students are learning the mathematics content being delivered. According to the NCTM's Assessment Standards for School Mathematics (1995), assessment practices should enable teachers to assess students' performance in a manner that reflects what students know and can do. Unfortunately, the typical classroom assessments, such as chapter tests, homework assignments, and the like, rarely accurately reflect the depth of mathematical understanding expected to meet performancebased standards like those found in NCTM's Principles and Standards for School Mathematics (2000).
Quick Reads: Another Good Idea: Home Area and History
Authors: Scaptura Christopher N.
Published: December 2020

While reading the Washington Post one morning, I saw an article about the average size of American houses throughout history (Higgins 2007). According to the article, houses are currently about three to four times larger than they were in 1790. Not only that, the average size of a family during colonial times was much larger than today. Early American families averaged 7 people, whereas modern families average 2.6 people. These facts made me think of our current study of American history in social studies and our approaching unit on area in mathematics. I wanted to find a way to incorporate this housing information into a lesson for my students.
Student Representations at the Center: Promoting Classroom Equity
Authors: Imm Kara Louise , Stylianou Despina A. , Chae Nabin
Published: December 2020

The NCTM's Standards (2000) suggest that a representation is not only a product (a picture, a graph, a number, or a symbolic expression) but also a process, a vehicle for developing an understanding of a mathematical concept and communicating about mathematics. To serve as a vehicle in learning and communication, however, a representation must be personally relevant and meaningful to a student. Therefore, when choosing a representation to explore with a group of students or when reviewing student work, we ought to consider everything that students bring to the classroom. Even at a young age, students come to school with their own, often culturally influenced, valid representations (Lave 1998). Because those representations have been crafted, interpreted, and modified by the students themselves, they become vital to classroom instruction. To dismiss what students bring naturally to the classroom reduces mathematics to a one-way transaction between teacher as expert and student as novice, confirming the notion that a student's own thinking and all that he or she brings to mathematics is marginal at best. By relocating student-generated representations to the center of the instruction, the nature of how students experience mathematics changes dramatically. It reconsiders mathematics as a vibrant dialogue among different but equally valued thinkers. This deliberate approach to the teaching of mathematics, we believe, becomes vital if we are serious about creating greater equity for our students.
Developing Meaning for Algebraic Symbols: Possibilities and Pitfalls
Authors: Lannin John K. , Townsend Brian E. , Armer Nathan , Green Savanna , Schneider Jessica
Published: December 2020

An important goal of school mathematics involves helping students use the powerful forms of representation that have been developed over the centuries through the work of mathematicians throughout the world. However, challenges exist in encouraging students to develop meaning for the mathematical symbols used in formal algebra. Research has demonstrated that students often fail to develop a deep understanding of the meaning of symbolic representations of variables (e.g., Booth 1984; Clement 1982), so much so that Thompson (1994) found that a limited understanding of the meaning of variables negatively impacts students who later take college calculus. The question arises as to how we can develop meaning for formal algebraic symbols in the middle grades so that instruction can build on this meaning throughout students' high school and college experiences.
Sense-able Combinatorics: Students' Use of Personal Representations
Authors: Tarlow Lynn D.
Published: December 2020

As we move forward in the twenty-first century, information and its communication have become at least as important as the production of material goods, and the nonmaterial world of information processing requires the use of discrete mathematics (NCTM 1989). Combinatorics, the mathematics of counting, plays a significant role in discrete mathematics. It is usually described as having three parts: counting (how many things meet our description), optimization (which is the best), and existence (are there any at all). The NCTM is explicit about the importance of students learning discrete mathematics: “As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum” (NCTM 2000, p. 31).
Exploring Segment Lengths on the Geoboard
Authors: Ellis Mark W. , Pagni David
Published: December 2020

Given a 5-peg × 5-peg geoboard, how many different lengths can be made by stretching a rubber band to form an oblique (nonhorizontal, nonvertical) segment between any two pegs? This question launches an investigation that requires students to make connections to the Pythagorean theorem, congruence, and combinations. It also leads to the question, “Can the results be generalized to an n-peg × n-peg geoboard?” (Note that we use “n-peg × n-peg” to refer to the dimensions of the geoboard to prevent confusion with the length of the side of the geoboard, which would be n − 1.) With its use of visual representation and potential mathematical ideas that can be explored from simple to complex, this task makes meaningful mathematics accessible to learners with varied levels of prior knowledge. In addition, mathematics vocabulary such as oblique and congruent are introduced in a meaningful context, making the lesson ideal for developing academic language.
Investigating Mathematical Thinking and Discourse with Ratio Triplets
Authors: Canada Dan , Gilbert Mike , Adolphson Keith
Published: December 2020

As teachers, we know that real mathematical understanding includes the higher-level skills of comparing, contrasting, and making connections among different computational approaches. What mathematical connections can students make when asked to explore a task that is posed in three different ways? We wanted to investigate the range of mathematical approaches that students might use, what connections they might make between parallel tasks, and how their conceptual understanding might be affected. The activity, called Ratio Triplets, referred to a set of three versions of one particular mathematics problem. The essential question of the activity asks students to determine which of two packages of ice cream was the better buy: A 64-ounce container selling for $6.79 or a 48-ounce container selling for $4.69.
Promoting Decimal Number Sense and Representational Fluency
Authors: Suh Jennifer M. , Johnston Chris , Jamieson Spencer , Mills Michelle
Published: December 2020

The abstract nature of mathematics requires the communication of mathematical ideas through multiple representations, such as words, symbols, pictures, objects, or actions. Building representational fluency involves using mathematical representations flexibly and being able to interpret and translate among these different models and mathematical concepts. This article shares a collaborative lesson study experience in planning and teaching a unit on decimals. Participants included fifth- and sixth-grade teachers and lesson study facilitators, including a university mathematics educator, a doctoral student, and a school mathematics specialist. The lesson was taught in a fifth-grade class with a high population of English language learners (ELL) and special needs students. The overarching goal of the lesson study was to develop students' representational fluency and mathematical proficiency with decimals. While working with teachers, the lesson study facilitators shared related research on representations and the importance of selecting and evaluating effective mathematical models to give the diverse population access to decimal concepts. The lesson study facilitators' goal was to heighten teachers' awareness of the importance of multiple representations and introduce a planning process that allows teachers to select models in a thoughtful and critical way that would facilitate the teaching and learning of a mathematics concept.
Solve It!: Window Panes
Authors: Mooney Edward S.
Published: December 2020

Diana designs stained-glass windows. She has a large supply of square glass window panes. Some panes are yellow, and some panes are red. She wants to place them in windows like the one below in such a way that no two windows are the same, even when the design is rotated or reflected. How many different window designs can she create using any combination of red and yellow panes?
Quick Reads: Promising Partnerships: Learning 24-7: Changing Attitudes about Mathematics
Authors: Shannon-Smith Elaine , Fawcett Gay
Published: December 2020

We introduced a mathematics workshop for teachers by asking participants to give a simile for mathematics: “Math is like a _______.” The responses provided insight into the perceptions of the people we would be working with over the next three days.
Mathematical Explorations: Getting a Bee in Math Class
Authors: Sharp Brian
Published: December 2020

In 2007, DreamWorks Animation released Bee Movie, an animated film cowritten and produced by comedian Jerry Seinfeld. In the film, honeybee Barry B. Benson is on a quest to change his predetermined career path, namely, a life of making honey. The filmmakers used various characteristics of honeybees to create an entertaining movie. Teachers can also use facts about honeybees to create interesting mathematics activities. This article describes two activities that combine the characteristics of honeybees with important mathematical concepts.
The Pythagorean Theorem with Jelly Beans
Authors: Yun Jeong Oak , Flores Alfinio
Published: December 2020

principles and standards for school mathematics advocates an experimentation approach to middle-grades geometry study (nctm 2000). Students are asked to explore and examine a variety of geometric shapes and discover their characteristics and properties using hands-on materials. They also create inductive arguments about the pythagorean relationship. This empirical approach to the pythagorean theorem, for example, will lay the foundation for analytical proofs.
Elastic, Cottage Cheese, and Gasoline: Visualizing Division of Fractions
Authors: Peck Sallie , Wood Japheth
Published: December 2020

Representation is one of the five Process Standards from Principles and Standards for School Mathematics. This Standard proposes that prekindergarten through grade 12 instructional programs should enable students to “select, apply, and translate among mathematical representations to solve problems” (NCTM 2000, p. 67). To implement a program that meets this Standard, teachers themselves must have attained this level of proficiency.
Mathematical Roots: The Mystery of the Disappearing Calculators
Authors: Lin Cheng-Yao , Lemons Joshua K. , Moser Morgan E. , Smith Melissa A.
Published: December 2020

Cryptology is a branch of science dealing with secret communications (Krystek 2000). The word cryptology is derived from the Greek word kryptos, meaning “hidden,” and logos, meaning “word.” Communicating through hidden messages has been in existence almost as long as written language.
How Many Piano Tuners Are in Our City?
Authors: Kenyon Glenn C.
Published: December 2020

Posing Fermi questions to students allows them to use their innate number sense and encourages mental calculation.
Mathematical Explorations: A Fair Game? The Case of Rock, Paper, Scissors
Authors: Nelson Cheryl , Williams Nicole
Published: December 2020

Rock, paper, scissors is a popular game. Middle-grades students often use it to choose the person who will go first in some activity or larger game. This game also provides a rich context to explore the relationship between experimental and theoretical probability through the concept of a fair game. Research finds that students learn best from classroom activities that connect to real-world problems (NCTM 2000; Sutton and Krueger 2002).
Exploring Results of the NAEP: 1980s to the Present
Authors: Kloosterman Peter , Rutledge Zachary , Kenney Patricia Ann
Published: December 2020

Results of the long-term trend assessment (LTT) for middle-grades students show positive advancement.
Trusting Students
Authors: Kastberg Signe E. , Norton Anderson , Klerlein Jacob T.
Published: December 2020

Trusting students' mathematical goals can result in new understandings that enable teachers to modify their instruction.
Patterns, Quantities, and Linear Functions
Authors: Ellis Amy B.
Published: December 2020

Students explore linear functions through patterns and the measurement of real-world quantities.
Burning the Candle at Just One End
Authors: Lim Kien H.
Published: December 2020

Using nonproportional examples helps students determine when proportional strategies apply.
What Can We Do with a Common Numerator?
Authors: Danielson Christopher
Published: December 2020

Comparing fractions with a common numerator may not be news, but exploring the use of common numerators for adding fractions was an adventure into unchartered territory for aspiring teachers.
On My Mind: Teacher as Musician versus Teacher as Composer
Authors: Meyer Margaret R.
Published: December 2020

You have probably heard this analogy: Learning math skills without concept development and problem solving is like practicing musical scales and never getting to play music.
The Thinking of Students: Creative Solutions to “Solve It!”: Animal Farm
Authors: Mooney Edward S.
Published: December 2020

The student work included in this article illustrates the various ways that students were able to think outside the box, or pen, as they solved this meaty problem.
A Worthwhile Task to Teach Slope
Authors: Wagener Lauren L.
Published: December 2020

Consider the possibilities when teaching slope. It is a fundamental algebraic concept that has connections to advanced mathematics and to the world outside the classroom.
Proof: Examples and Beyond
Authors: Knuth Eric J. , Choppin Jeffrey M. , Bieda Kristen N.
Published: December 2020

Asking middle school students to verify the math they do requires them to think about proof. By doing so, students construct arguments in the middle school and are more ready for proof in high school.
Bursting with Geometry
Authors: Cipoletti Beth , Wilson Nancy
Published: December 2020

By building a Lithuanian sunburst star with straws, students explore a number of geometric relationships and reinforce their mathematical vocabulary.
Rubrics as play
Authors: McGatha Maggie B. , Darcy Peg
Published: December 2020

Consider the role of rubrics in how students present their ideas. Viewed as more than an assessment, rubrics can also provide feedback to teachers, further student understanding, and encourage articulation of ideas.
Solve It!: Cat and Mouse
Authors: Mooney Edward S.
Published: December 2020

In the video game Cat and Mouse, you receive 4 points for each cat caught and 11 points for each mouse caught. What scores are impossible to get in this game?
Bridging the Math Gap
Authors: Switzer J. Matt
Published: December 2020

Knowing both the content and manner in which students learn math in elementary school can help teachers bridge the gap between elementary school and middle school (and then high school) math.
Mathematical Exploration: Trapezoids to Triangles
Authors: Young Elaine
Published: December 2020

Pattern blocks are often used in math class during geometry lessons to support concepts of shape, angle, tessellations, and symmetry. They are also used with fraction study (e.g., Lanius 2007; NCTM 2009a). Using pattern blocks to construct other larger geometric shapes can help students understand the relationships among various shapes. Algebraic relationships and functions emerge when students count blocks and measure figures. The activities in this article support the development of such geometric vocabulary as perimeter, area, sequence, and nonstandard measurement.
Math for Real: Food Labels
Authors: Hunt Margie J. Raub , Hunt Bill
Published: December 2020

Food labels provide important information that needs to be understood, not just read, particularly since two pressing health issues in our society are the overconsumption of fat and sugar.
A Historical Perspective from a to c
Authors: Edwards Thomas G. , Özgün-Koca S. Aslı
Published: December 2020

Using quadratic functions as an example, see how the evolution of technology has altered the expectations of students' understanding and critical thinking.
Digesting Student-Authored Story Problems
Authors: Alexander Cathleen M. , Ambrose Rebecca C.
Published: December 2020

When students are asked to write original story problems about fractional amounts, it can illustrate their misunderstandings about fractions. Think about the situations your students would describe to model 1/2 + 2/3.
Problems with nth-Term Problems: Reflect and Discuss
Authors: Matteson Shirley M.
Published: December 2020

Examine sixth-grade students' work to gain insight into the early stages of their algebraic understanding.
Informing Practice: A Model Approach to Problem Solving
Authors: DiMatteo Rachel Wing
Published: December 2020

Students who pursue careers in business, medicine, engineering, or architecture need robust mathematical backgrounds, with a particularly strong emphasis on problem solving. For example, when civil engineers are asked to determine the best placement for a cell phone tower, some of their thinking is related to schoolbased mathematical procedures and concepts. However, for the most part, they problem solve, using computer programs to construct mathematical models that relate the three-dimensional topography of the earth and the availability and cost of property for tower placement.
The Sweetest Chocolate Milk
Authors: Newton Kristie J.
Published: December 2020

A nonroutine fraction problem provides many solution paths.
Shifting Our Computational Focus
Authors: Keiser Jane M.
Published: December 2020

In the primary grades, as well as in the middle grades, students may spend more time on conceptual understanding.
See the Math Behind the Medicine
Authors: Saunders Marnie M.
Published: December 2020

To promote math and science, a researcher designed an activity to show students that biomedical fields are within their reach.
The Leap from Patterns to Formulas
Authors: Beigie Darin
Published: December 2020

Geometric problems help students progress from predicting numerical patterns to expressing algebraic generalizations.
Fold in Origami and Unfold Math
Authors: Georgeson Joseph
Published: December 2020

Students fold exciting shapes to explore how changes in one dimension affect measurements in another.
Dine on Rich Functional Examples
Authors: Robichaux Rebecca R. , Rodrigue Paulette R.
Published: December 2020

Starting with the Dinner Table problem and extending to other geometric examples, students become familiar with patterns and functions as well as graphing concepts.
Target Geometry and Probability Using a Dartboard
Authors: Williams Nicole L. , Bruels Carolyn
Published: December 2020

As middle-grades students explore area problems in geometry class, they typically calculate areas of circles and squares using formulas. Sometimes they explore problems using special polygons to compare a shaded area to an unshaded area. Students are often asked to simulate probability through the use of coins and dice. However, these situations contain little motivation for finding a solution. The following dartboard activity investigates how students can use simulations to determine experimental probability and build on intuition to make connections between calculating area and finding the theoretical probability of random events.
What If We Lived in Flatland?
Authors: Anderson Rick , Princko Jamie A.
Published: December 2020

Students traverse new mathematical territory as they imagine life in a two-dimensional universe.
All These Rays! What's the Point?
Authors: Roberts Sally K. , Tayeh Carla
Published: December 2020

Teacher candidates move beyond “Just give me a formula” to a deeper understanding by exploring the algebra found in geometric models.
The Truth about PEDMAS
Authors: Ameis Jerry A.
Published: December 2020

A hierarchy-of-operators triangle shapes students' conceptual understanding of the order of operations.
Integrate Technology with Student Success
Authors: Wolf Drew , Lindeman Patrick , Wolf Trent , Dunnerstick Robert
Published: December 2020

As teaching methods and teaching tools evolve, so too should the way we use technology in the classroom.
On My Mind Let's Be Realistic
Authors: Masingila Joanna O.
Published: December 2020

It is essential that teachers provide classroom activities that connect in-school and out-of-school mathematics to enrich each student's set of experiences and offer some common experiences for all students in the class. These activities need not be elaborate; motivating, rich activities can be developed by transforming textbook exercises into problems that require students to use their mathematical knowledge in ways that reflect how that knowledge is used in everyday situations.
Areas and Brownies
Authors: Fan C. Kenneth
Published: December 2020

For the last class, I bake brownies. However, there's a catch! No one gets a brownie until someone can figure out how to cut the single layer into equally sized pieces for all. The result has always been a productive hour of creativity as the students work hard to earn a bite of the scrumptious, chocolate-filled treat.
Using a Database for Student Research
Authors: Healy Lulu , Hoyles Celia
Published: December 2020

Computer databases, widely available in schools, have the potential to offer new ways of approaching aspects of data handling within mathematics education. It is not always easy, however, to integrate the use of databases into everyday classroom life. Although databases can be used in ways that encourage students to test hypotheses, generate conjectures, and interpret trends in data, these results do not happen unless students are involved in meaningful inquiry in which they research an item of personal interest. The project described in this article is one of a series developed in collaboration with teachers to exploit the use of computer databases in the search for information. By orienting data handling around students' own research endeavor, we aimed to stimulate students' interest not only in the data themselves but also in the mathematical strategies underpinning the research process.
Building Bridges to Algebraic Thinking
Authors: Day Roger , Jones Graham A.
Published: December 2020

In spite of the fact that middle school teachers have been urged to develop algebraic thinking in their students (Lawson 1990; NCTM 1989; Phillips et al. 1991), the question of how to initiate that process may still be problematic for many teachers. Although it is important to suggest that explorations into algebraic thinking should emphasize “physical models, data, graphs and other mathematical relationships” (NCTM 1989, 102) and to recognize that “numerical and graphical problem-solving techniques become accessible strategies … through the use of technology” (Demana and Waits 1990, 53), teachers need help in building a bridge between their current instructional goals and new goals that emphasize an earlier introduction to algebraic thinking. In this article, we describe and illustrate an approach to algebraic thinking that is based on an extension of the problem-solving tasks typically investigated by elementary and middle school students.
Algebra for Everyone: Start Early
Authors: Fouche Katheryn K.
Published: December 2020

As more school systems adopt algebra as a standard for all students, the debate escalates about what content to include and which techniques are most effective. While researchers and practitioners sort out the details, some practices should now be incorporated into all programs designed to encourage algebraic ways of thinking.
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 assignment. Students might want to create their own mathematics-related cartoons that demonstrate in a humorous way how mathematics is used and misused. Readers are encouraged to Submit three copies of cartoons and classroom-tested questions, double-spaced, for review by the editor of this section. The cartoons must include the date and the newspaper syndicate that holds the copyright.
Exploring Patterns in Nonroutine Problems
Authors: Curcio Frances R. , Nimerofsky Barbara , Perez Rossana , Yaloz Shirel
Published: December 2020

For the past several years, we have been examining ways to introduce notions of algebra to our heterogeneously grouped middle school students. Through our experiences of designing instructional activities that integrate nonroutine, nontraditional problems. we found a meaningful way to involve students in exploring and formalizing patterns, conjecturing about the patterns they identify, verbalizing relationships between and among elements in patterns, and eventually generalizing and symbolizing the relationships—all essential components of algebraic thinking (Silver, in press).
Exploring Algebraic Patterns through Literature
Authors: Austin Richard A. , Thompson Denisse R.
Published: December 2020

Activities that engage middle school students in investigating and extending patterns are essential in developing students' algebraic thinking. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) for grades 5-8 and the position statement on “Algebra for Everyone” (NCTM 1994) describe the importance of using patterns and relationships to develop algebraic thinking.
Mathematics Projects Promote Students' Algebraic Thinking
Authors: Malloy Carol E.
Published: December 2020

Teachers in all curricular areas, especially science, have encouraged students to develop projects to extend their knowledge of specific phenomena. As middle school mathematics teachers grapple with methods to help students enhance algebraic thinking, they should consider the power of the mathematics project. My experiences with students have demonstrated that mathematics projects must be structured as investigations in which students work cooperatively and where the mathematics in the proj-ects is substantial. Projects can provide middle school students with the opportunity to investigate, conjecture, and reach mathematical conclusions that require algebraic thinking, as recommended in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). This article describes how various projects helped middle school students harness their experiences and construct mathematical conclusions through algebraic thinking Students employed algebraic thinking as they used number patterns and verbal rules to “explore the interrelationships of these representations” (NCTM 1989, 102) and reach conclusions in their projects.
Graphing in the Information Age: Using Data from the World Wide Web
Authors: Dixon Juli K. , Falba Christy J.
Published: December 2020

In the information age, middle school students must be intelligent consumers of information. to instill critical thinking with respect to statistical data, the interpritation and creation of graphs are essential. Although vast amounts of information can be gleaned from traditional text sources, the World Wide Web (WWW) offers information that is updated far more frequently than most printed materials. Because of the motivational aspects and expedient nature of using data from the Web, this article focuses on its use; however, each of the activities can be adapted for use with traditional, text-based media.
How Children Think about Division with Fractions
Authors: Warrington Mary Ann
Published: December 2020

When children are allowed to create and invent, their fertile minds enable them to solve problems in a variety of original and logical ways. When their minds have not been shackled by rules and conventions, children are free to invent procedures that reflect their natural thought processes.
Three's the Charm
Authors: Muri Mari , Zawojewski Judi
Published: December 2020

The editorial panel is pleased to (welcome readers to volume 3 of Mathematics Teaching in the Middle School. Good things come in threes, and the journal is no exception. Three noteworthy events will characterize volume 3. The first is a special March-April 1998 issue with a focus on geometry. We chose this topic because it has traditionally been an area of poor performance by middle school students, as indicated by the results of the National Assessment of Educational Progress and the Third International Mathematics and Science Study. Yet geometry has the potential for creating exciting opportunities for “doing” mathematics in the middle school classroom. Further, with the increased availability of technology-based geometry tools, students' experiences with the topic can be enhanced through dynamic aud sophisticated investigations.
National Council of Teachers of Mathematics: Choice and Challenges
Authors: Burrill Gail
Published: December 2020

Life has changed enormously scince I grew up in Fisher, Minnesota. Over the years, we faced many challenges and made choices that helped us meet those challenges as we kept up with changes in agriculture, manufacturing, transportation, enter-tainment, and education.
Strategy Games: Treasures from Ancient Times
Authors: Gorman Jacqueline
Published: December 2020

The fifth-grade classroom was filled with the sounds of tension and concentration. I heard the quiet clicking of dry beans; hushed, intense discussions; and sighs of both frustration and satisfaction. My students were scattered around the room in small groups, leaning over game board and attempting to work out winning strategies for the game of kalah.
Stop Using Foul Language in the Mathematics Classroom
Authors: Hanselman Cheryl A.
Published: December 2020

Some words have no place in the mathematics classroom. Most teachers do not allow children to use foul language. Words that inappropriately refer to deities, bodily functions, or intimate relationships are banned because they are hurtful and because they create confusion between their meaning and perceived connotations. Teachers, however, unintentionally use other words that are confusing and hurtful to young mathematical minds. In my many years as a teacher of remedial-level mathematics, I have found that such terms as reduce, cancel, and invert and multiply have created deep-seated confusion and misconceptions in the minds of my students. I believe that they should be treated like foul language and banned from premature use in the mathematics classroom.
Many Faces Have I
Authors: Zilliox Joseph T. , Lowrey Shannon G.
Published: December 2020

We have always attempted to involve as many people as possible-teachers, students, and classroom visitors-in making decisions about the content and process of learning mathematics in our sixth-grade class. In so doing, we have all learned together. While investigating properties of polygon and polyhedra, a challenge was posed to the class. The challenge appeared in Focus, a publication of the Mathematical Association of America (MM) (1994). A class of gifted and talented student in Maryland had constructed a large polyhedron with faces that measured thirteen inche to a side. The new letter contained a challenge to reader to better this record.
Young Women Experience Mathematics at Work in the Health Professions
Authors: VanLeuvan Patricia
Published: December 2020

Middle school students often need assistance in connecting the mathematics they are learning in school to their future responsibilities as adults and to the skill requirements of the varied career options that will be open to them. Experiencing mathematics as it is actually used in the workplace may motivate students to continue mathematics coursework. Young women, in particular, are more apt to enroll in mathematics courses throughout high school when they see mathematics as useful for future study or for a career (Linn and Hyde 1989; Pedersen, Bleyer, and Elmore 1985; Tobin and Fox 1980).
Teacher to Teacher: The Human Scatterplot
Authors: O'Keefe James
Published: December 2020

Far too often, skills I graphing are isolated into two main categories-those used in algebra and those in statistics. Often lost in this dichotomy is the sense that all graphs represent the relationship between variables. Those variables can be x and y, but they can also be categories and frequencies, as in bar graphs, or measurable characteristics that can be correlated, as in scatterplots. The “human scatterplot” is an activity that is intended to bridge the gap between algebraic and statistical graphs. Students will experience a scatterplot as being a visual and kinesthetic tool for discovering patterns among variables and making predictions based on those patterns.
Mathematical Tasks as a Framework for Reflection: From Research to Practice
Authors: Stein Mary Kay , Smith Margaret Schwan
Published: December 2020

According to the professional Standards for Teaching Mathematics (NCTM 1991), a primary factor in teachers' professional growth is the extent to which they “reflect on learning and teaching individually and with colleagues” (p. 168). Reflecting on their classroom experiences is a way to make teachers aware of how they teach (Hart et al. 1992) and how their students are thriving within the learning environment that has been provided. Although all teachers think informally about their classroom experiences, cultivating a habit of systematic and deliberate reflection may hold the key to improving one's teaching as well as to sustaining lifelong professional development.
Above and beyond AAA: The Similarity and Congruence of Polygons
Authors: Slavit David
Published: December 2020

The development of intuitive understandings of geometiy is highly important at the middle school level. One way to develop intuitive geometric understandings is to create a learning environment in which students are encouraged to explore relationships among various geometric shapes. The environment should give students opportunities to make conjectures; suitable materials and teacher guidance should be readily available to support attempts to verify these conjectures (Clements and Battista 1992). For this reason, the NCTM (1989) has encouraged teachers to take a perceptual-to-conceptual approach to teaching geometry, which involves the development of general geometric concepts through tangible and familiar experiences.
Middle School Students' Reasoning about Geometric Situations
Authors: Carroll William M.
Published: December 2020

Along with an increased emphasis on reasoning, communication, and problem solving, the National Council of Teachers of Mathematics has called for a change in assessment techniques (1989, 1991, 1995). In contrast to short-answer questions, assessments that elicit writing, diagrams, and other representations offer better windows into students' understandings and misconceptions about mathematics. This article describes some short geometry tasks that go beyond the simple recognition of figures and properties. Because they have been field-tested with students using various mathematics curricula, we have collected hundreds of student responses to these questions that seem to represent a good range of students' geometric thinking and development. From these responses, along with ideas about the development of geometric thinking (Fuys. Geddes, and Tischler 1988), we have developed scoring rubrics to go along with many of these questions. The rubrics and questions might be useful to middle school teachers who are developing short openended questions that encourage and assess students' thinking.
April's Menu of Problems
Authors: Davidenko Susana
Published: December 2020

A light fare to whet your mathematical appetite.
Cartoon Corner
Authors: Fisher Julie
Published: December 2020

“Cartoon Corner” is designed to include cartoons that can be used to highlight amthematics in an interesting way. The questions accompanying the cartoons can be used as a large- or small-group actitlity, 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 or cartoons and classroom-tested questions, double-spaced, for review by the editor or this section. The cartoons must include the date and the newspaper syndicate that holds the copyright.
Mathematics Fitness
Authors: Trezise Kathleen A.
Published: December 2020

Students who are involved in mathematics instruction that is relevant to life will understand that mathematics is a part of the real world and not just another school subject. Teachers who connect mathematics with other content areas and the real world afford students the opportunity to see the daily importance of mathematics. The activity that follows is consistent with the following NCTM curriculum and evaluation standards. In grades 5-8, the mathematics curriculum should include—
Problem Solving: Is More Than Solving Problems
Authors: Mikusa Michael G.
Published: December 2020

The curriculum and evaluation Standards for School Mathematics (NCTM 1989) states that one of its five general goals is for all students to become mathematical problem solvers. It recommends that “to develop such abilities, students need to work on problems that may take hours, days, and even weeks to solve” (p. 6). Clearly the authors have not taught my students! When my students first encountered a mathematical problem, they believed that it could be solved simply because it was given to them in our mathematics class. They also “knew” that the technique or process for finding the solution to many problems was to apply a skill or procedure that had been recently taught in class. The goal for most of my students was simply to get an answer. If they ended up with the correct answer, great; if not, they knew that it was “my job” to show them the “proper” way to go about solving the problem.
Roll the Dice —an Introduction to Probability
Authors: Freda Andrew
Published: December 2020

An understanding of probability and the related area of statistics is essential to being an informed citizen.” So begins Standard 11 of the NCfM's curriculum standards for grades 5-8 (NCTM 1989), which also states that “[s]tudents in the middle grades have an intense interest in the notions of fairness and the chances of winning games.” I have found a dice game that enables me to use this “intense interest” by giving students an opportunity to explore the idea of various outcomes and the likelihood of the occurrence of one of those outcomes. This dice game also helps to explain the benefits of using a large sample rather than a small one. The game is constructed to eem fair-it appears that each student has an equally likely chance of winning-but it is decidedly unfair. Part of the power of this activity comes from students' intrigue with this simple dice game that turns out to be much more involved than it appear to be at first glance.
Take Time for Action: Computation versus Number Sense
Authors: Reys Robert E.
Published: December 2020

How well can your students compute? What is their level of number sense? As teachers, we struggle with developing procedures and skills in our mathematics classes. We want students to be able to compute, and we also want them to understand what they are doing. Maintaining a proper balance between conceptual development and skills, whatever they are, presents a continuing challenge. You are invited to “Take Time for Action” and see how your students react to two very similar, but perhaps very different, situations.