bluuebunny/crossref_metadata_2025 huggingface dataset:
How Fast Do Trees Grow? Using Tables and Graphs to Explore Slope
Authors: Joram Elana , Oleson Vicki
Published: December 2020

Slope is one of the most important mathematical concepts that students encounter in middle school and high school. The concept of slope “weaves its way through algebra, geometry, trigonometry, and calculus” (Anderson and Nelson 1994, p. 27), yet research with high school students shows that they can often calculate slope but are unable to interpret its meaning (Newburgh 2001; Stump 2001). This is a familiar problem in mathematics education; students learn a formula for solving a particular type of problem without conceptually understanding the relevant mathematical ideas. One way to enhance students' understanding of slope is to introduce it conceptually in middle school (Beckmann and Rozanski 1999), with the understanding that this conceptual foundation will provide a basis for the formal definition of slope to be introduced later on. In this article, we describe two related lessons for sixth-grade students that were designed to accomplish this goal.
“I Can't Write All the Way to 100”: Recognizing Students' Emerging Algebraic Strategies
Authors: Bannister Vanessa R. Pitts , Wilkins Jesse L. M.
Published: December 2020

In Principles and Standards for School Mathematics (NCTM 2000), understandings of patterns, relations, functions, mathematical models, and quantitative relationships are recognized as key facets of algebraic thinking. In essence, algebraic thinking “embodies the construction and representation of patterns of regularities, deliberate generalization, and most important, active exploration and conjecture” (Chambers 1994, p. 85). Algebraic thinking should function as a means of shifting from arithmetic concepts to algebraic concepts (Chappell 1997). This shift would have occurred if there exists reasoning about relationships between quantities, rather than the specific quantities themselves (Ferrini-Mundy, Lappan, and Phillips 1997; Yackel 1997). Research shows that this arithmetic to algebraic shift is difficult for students (Stacey and MacGregor 2000). Therefore, it is imperative to explore students' reasoning as they approach problems that elicit algebraic thinking. For this reason, we will present and discuss samples of student work regarding problems that promote algebraic thinking.
Integrating Content to Create Problem-Solving Opportunities
Authors: Beigie Darin
Published: December 2020

Problem solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned. (NCTM 2000, p. 182)
10 Practical Tips for Making Fractions Come Alive and Make Sense
Authors: Clarke Doug M. , Roche Anne , Mitchell Annie
Published: December 2020

Fractions are difficult to teach and to learn, but they should not be viewed as a lost cause. Important research in Australia and elsewhere can help us focus on the big ideas in ways that can help students in the middle years make sense of fractions. We share insights from our research, advice on what is important and less important, and practical classroom approaches and activities that our experience in a variety of professional development settings indicates can make fractions come alive for middle-grades teachers and students.
Building for the Future: The Mathematics of Architecture and Design
Authors: Reynolds Suzanne L. , Mahan Michael , Lee Barbara , DelRisco Gilda
Published: December 2020

The Connection Standard in Principles and Standards for School Mathematics (NCTM 2000) emphasizes the importance of interdisciplinary experiences in the teaching and development of mathematical ideas. This article describes a four-week summer enrichment program relating mathematics to real-life applications in architecture and design, produced by the faculty from Kean University (through an Improving Teacher Quality Partnership grant awarded by the state of New Jersey). The focus of this summer program was to provide both a professional development opportunity for teachers and preservice teachers from the university as well as an opportunity for accelerated learners to have authentic hands-on experience in the critical content areas of data analysis, measurement, and proportionality. The students selected were rising sixth and seventh graders from an urban New Jersey school district.
An Experiment in Using Portfolios in the Middle School Classroom
Authors: Maxwell Vicki L. , Lassak Marshall B.
Published: December 2020

Standard classroom tests tend to showcase what students know at that point in time and do not usually reflect real-life mathematics. They are not always accurate indicators of what students understand and how they understand it. I wanted to use an assessment tool that would allow students a better opportunity to exhibit mathematical growth in understanding and attitude over time. This tool should also give students the chance to show that they could communicate in a mathematical context, exhibit problem-solving techniques, and make mathematical connections to other subject areas. From these ideas and my review of the research on assessment, I decided to use portfolios as an assessment tool in one of my eighth-grade prealgebra classes.
Improving the Planning and Teaching of Mathematics by Reflecting on Research
Authors: Hoffman Lauren R. , Brahier Daniel J.
Published: December 2020

Much has been written in the past decade about the Third International Mathematics and Science Study (TIMSS), including the 1999 Video Study and the TIMSS 2003 test. The primary goals of the study were to evaluate mathematics achievement levels in several countries and to attempt to understand and explain the reasons for the differences (Stevenson 1998). The TIMSS case studies were compiled in Germany, Japan, and the United States and included videos of class sessions that were analyzed by an international team of researchers (TIMSS Video Mathematics Research Group 2003). The summaries of the research have been a basis for suggestions and changes in American classrooms. Specifically, the study clearly showed differences in the ways that teachers in Japan and in the United States conduct their classes that, in turn, may affect the academic achievement of their students. Several of the key findings will be discussed in this article. Then, based on the research, questions will be presented and recommendations for teachers will be suggested concerning how U.S. mathematics teachers might rethink their methods of planning and teaching.
Percents Can Make Sense
Authors: Zambo Ron
Published: December 2020

Supermarkets often list sale prices in terms of how many items for how much. For example, recently I saw that Snickers bars were advertised at 8 bars for $3 and cans of SpaghettiOs were selling at 4 for $5. Are those good prices? How much is it if I buy only one bar or can? The price of one bar or one can is called the unit price. One way to think about unit price in the setting of these sale prices is to think about fraction-to-percent equivalents. This article describes instructional activities designed to help students develop both an understanding of, and mental math abilities for, the multiple relationships that exist among fractions and percents and unit prices.
Mathematical Explorations: Variables and Spreadsheets Connect with Real-World Problems
Authors: Niess Margaret L. , Sadri Pejmon , Lee Kwangho
Published: December 2020

Spreadsheet software is generally available in schools and is in wide-spread use in business. The use of spreadsheets can help students make mathematical connections with problems in the world around them. Many real-life quantitative problems require algebra for decision making. Examples include the impact of rising gas prices on family budgets; the amount of gasoline left in the tank of a car and the distance to the nearest gas station; and the level of monthly income versus money needed to pay for food, rent, utilities, and clothing. Although these issues do not require complex mathematics, they do require knowledge of basic algebra involving variables and equations.
Oranges, Posters, Ribbons, and Lemonade: Concrete Computational Strategies for Dividing Fractions
Authors: Kribs-Zaleta Christopher M.
Published: December 2020

Oranges, posters, ribbons, and lemonade in a mathematics classroom sound like recipe ingredients for a class party. Instead, they were used as storyproblem contexts and concrete models to help students develop computational strategies for dividing fractions.
Analyzing Students' Use of Graphic Representations: Determining Misconceptions and Error Patterns for Instruction
Authors: Scheuermann Amy , Garderen Delinda van
Published: December 2020

Central to the saying that “A picture is worth a thousand words” is the notion that it can harbor a wealth of information. A picture or, specifically, a graphic representation, such as a diagram, table, or chart, can contain much information about what a student understands and can do in mathematics. That said, how often do we carefully examine the graphic representations our students generate when doing mathematics?
The Role of Representations in Fraction Addition and Subtraction
Authors: Cramer Kathleen , Wyberg Terry , Leavitt Seth
Published: December 2020

Why do so many middle school students find fraction addition and subtraction difficult despite the fact that they have studied this topic since third or fourth grade? The Rational Number Project (RNP) (Cramer and Henry 2002; Cramer, Post, and delMas 2002) with support from the National Science Foundation is currently engaged in a teaching experiment with sixth graders from a large urban district in the Midwest to address this question.
Teaching and Learning Mathematics through Hurricane Tracking
Authors: Fernández Maria L. , Schoen Robert C.
Published: December 2020

During hurricane season, maps that track predicted storm paths are commonly seen on television and the Internet. The Weather Channel often receives number-one viewership ratings in regions encountering a major weather event, such as a hurricane or tornado (Kloer 2001). Mathematics teachers can tap into students' curiosity and interest about hurricanes to develop their understanding of mathematical ideas within a real-life context. In this article, we discuss observations and findings after implementing mathematics tasks based on data about hurricanes. Finding patterns and relationships, creating and interpreting graphs, and examining rates of change are just a few of the topics that can be studied. We developed these tasks as part of the Students' Transition Toward Algebra project and have used them with both middle school teachers and students.
The Importance of Equal Sign Understanding in the Middle Grades
Authors: Knuth Eric J. , Alibali Martha W. , Hattikudur Shanta , McNeil Nicole M. , Stephens Ana C.
Published: December 2020

The equal sign is perhaps the most prevalent symbol in school mathematics, and developing an understanding of it has typically been considered mathematically straightforward. In fact, after its initial introduction during students' early elementary school education, little, if any, instructional time is explicitly spent on the concept in the later grades. Yet research suggests that many students at all grade levels have not developed adequate understandings of the meaning of the equal sign (Baroody and Ginsburg 1983; Behr, Erlwanger, and Nichols 1980; Falkner, Levi, and Carpenter 1999; Kieran 1981; Knuth et al. 2006). Such findings are troubling with respect to students' preparation for algebra, especially given Carpenter, Franke, and Levi's (2003) contention that a “limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra. Virtually all manipulations on equations require understanding that the equal sign represents a relation” (p. 22).
Quick Reads: Another Good Idea: Chalking Up Success: An Exam-Review Strategy
Authors: Rogers Jason
Published: December 2020

Most teachers have been in this situation: Only one week remains before final exams, and the weather outside is warm and sunny. The students are feeling the fatigue that comes with nine months of sustained learning, and teachers are focusing on getting through the next weeks of instruction and end-of-school paperwork. All this considered, the question remains: How do we get students actively engaged in the learning process while balancing fatigue and warmer weather? The wondrous world of sidewalk chalk might be part of the solution.
Contemporary Curriculum Issues: Statistics in the Middle Grades: Understanding Center and Spread
Authors: Kader Gary , Mamer Jim
Published: December 2020

The GAISE report emphasizes the importance of students having experience with statistical thinking throughout the pre-K-12 curriculum. Students' encounters with statistics in the middle grades should build on their foundational experiences from the elementary grades and provide a link to the inferential types of statistical thinking developed at the high school level. Middle-grades students should be actively involved in the statistical problem-solving process described in the GAISE report. That process involves (1) formulating a question that can be addressed with data, (2) collecting data to address the question, (3) analyzing the data, and (4) interpreting the results.
Mathematical Explorations: It Is a Piece of Cake: Algebraic Thinking in a Real-Life Situation
Authors: Lee Ji-Eun , Kim Kyoung-Tae
Published: December 2020

Picture a family gathering on a nice summer day; in addition to delicious food, fun activities, and much laughter and talking, what else could be included? Although it might seem odd, our family gathering resulted in an important mathematics teaching and learning lesson.
Developing Spatial Sense and Communication Skills Reflect and Discuss
Authors: Richardson Kerri , Stein Catherine
Published: December 2020

Do you find that your students sometimes struggle when communicating their mathematical ideas? Do you believe that developing good spatial sense is important in the mathematics classroom? To jump-start a mathematical conversation among you and your students, complete this exercise: Glance at figure 1a for no longer than three seconds, then draw what you saw. When you are done, look at the image and compare it with your drawing. What did you see? How did you draw it? Ask a colleague to do the same. Did he or she draw the diagram differently? How did you describe the image, and how did your colleague?
Harry Potter and the Coding of Secrets
Authors: Chua Boon Liang
Published: December 2020

When students learn mathematics, they tend to understand and remember the mathematical processes that they find practical and relevant. It is not surprising then that Principles and Standards for School Mathematics (NCTM 2000) encourages teachers to help students discover the relevance of mathematics in their daily lives. To help young people realize that mathematics is alive and constantly evolving, it is important for teachers to share information with students about new developments in mathematics. However, it is often difficult to find source material that illustrates appropriate applications of mathematical concepts. In addition, finding items that are interesting and comprehensible to students is even harder. The topic of functions is a case in point. Therefore, this article describes some teaching ideas based on cryptography, a practical application of functions and their inverses that students should be able to comprehend.
Differentiating Instruction in Mathematics for the English Language Learner
Authors: Murrey Deandrea L.
Published: December 2020

A commonly held notion is that students learning English will do well in a mathematics classroom because mathematics is not a subject dependent on language proficiency. However, a student who is an English language learner (ELL) may struggle to succeed in the mathematics classroom where only English is taught. In 2004, there were approximately 5 million children in grades K–12 who were considered ELLs (NCELA 2004). Students from various language backgrounds may be learning English at the same time that they are learning new concepts in mathematics taught through English. Further, students are also learning the academic language of mathematics. The NCTM states: “All students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding” (NCTM 2000, p. 5). There are strategies for teaching mathematics to students who are ELLs. However, in addition to using these strategies, mathematics teachers also need to provide explicit language instruction for those students learning English (Rothenberg and Fisher 2007). By differentiating instruction in mathematics for English language learners, teachers can plan and provide access to mathematics curriculum for all students, with the added goal of language instruction for students learning English.
Pen Pals: Practicing Problem Solving
Authors: Lampe Kristen A. , Uselmann Linda
Published: December 2020

What is a good mathematics problem? What is a good mathematics problem for students at a particular grade level? How can I make mathematics interesting to students and fun? Preservice teachers explored these questions as they participated in a pen pal project with fifth and sixth graders. The teachers presented the middle schoolers with custom-made, meaningful mathematics problems. The students honed their communication skills and received personalized feedback about their learning. Along the way, many became more motivated by mathematics than they had been, and some, stimulated by the interaction with college students, began thinking about the future and college.
Quick Reads: Another Good Idea: The Zero Box
Authors: Friedmann Paul
Published: December 2020

The concept of adding and subtracting positive and negative numbers is not easy for middle school students to grasp. An effective bridge from the use of manipulatives and pictures to the use of traditional symbolic representation (number sentences) and mathematical rules is needed so that students can internalize the process. The zero box is just such a bridge.
The Gifted Student
Authors: Chval Kathryn B. , Davis Jane A.
Published: December 2020

Teachers who differentiate learning with meaningful tasks for students at all levels can keep their entire class engaged.
Talking Generally or Generally Talking in an Algebra Classroom
Authors: Koellner Karen , Pittman Mary , Frykholm Jeffrey
Published: December 2020

Listening to students while they problem solve can produce valuable results.
Take Time for Action: Exploring Equivalent Fractions with the Graphing Calculator
Authors: Olson Melfried
Published: December 2020

When students begin learning about fractions, it is important for them to understand the relationships involved. The study of fraction concepts begins in the primary grades, and work with relational understanding starts in grades 3 through 5. In its Number and Operations Standards for grades 3-5, the NCTM (2000) states that students should “recognize and generate equivalent forms of commonly used fractions, decimals, and percents” (p. 148) and “investigate the relationship between fractions and decimals, focusing on equivalence. Through a variety of activities, they should understand that a fraction such as 1/2 is equivalent to 5/10 and that it has a decimal representation (0.5)” (p. 150). This article describes classroom episodes within a three-day span that focused on using TI-73 calculators to explore equivalent fractions. Sample student insights and struggles are included.
Math Roots: Ratios and Proportions: They Are Not All Greek to Me
Authors: Snow Joanne E. , Porter Mary K.
Published: December 2020

Today, the concept of number includes the sets of whole numbers, integers, rational numbers, and real numbers. This was not always so. At the time of Euclid (circa 330-270 BC), the only numbers used were whole numbers. To express quantitative relationships among geometric objects, such as line segments, triangles, circles, and spheres, the Greeks used ratios and proportions but not real numbers (fractions or irrational numbers). Although today we have full use of the number system, we still find ratios and proportions useful and effective when comparing quantities. In this article, we examine the history of ratios and proportions and their value to people from the past through the present.
Quick Reads: Another Good Idea: Toward a Definition of Reflection
Authors: Rollick Mary Beth
Published: December 2020

“As students progress through the grades, the mathematics about which they communicate should become more complex and abstract. Students' repertoire of tools and ways of communicating, as well as the mathematical reasoning that supports their communication, should become increasingly sophisticated. Support for students is vital” (NCTM 2000, p. 60). Students often make comments that indicate partial understandings and misconceptions. With these comments in mind, a teacher can ask questions that scaffold the students' understanding. This article addresses the partial understandings and misconceptions that arise from using everyday language to describe the geometric concept of reflection.
Representations from the Real World
Authors: Santulli Tom
Published: December 2020

Providing a real-world context and requiring multiple representations will promote students' understanding of problems.
The Money Context
Authors: Tabach Michal , Friedlander Alex
Published: December 2020

Using the context of saving money, prealgebra students learn algebraic operations without formal procedures.
Mathematical Explorations: Reasoning with Algebraic Contexts
Authors: Johnson Gwendolyn , Thompson Denisse R.
Published: December 2020

According to Principles and Standards for School Mathematics, two standards that should be addressed in all grades are (1) Algebra and (2) Reasoning and Proof (NCTM 2000). Both involve generalization and are, therefore, naturally related (Lannin 2003). There is often much discussion about the need to engage students with tasks related to algebra; however, it is unclear if as much attention is given to engaging students with reasoning and proof, although this topic is also fundamental to mathematics. As Hanna (2000) noted, “Students cannot be said to have learned mathematics, or even about mathematics, unless they have learned what a proof is” (p. 24).
Ramping Up on Fractions
Authors: Stephens Ana C. , Bottge Brian A. , Rueda Enrique
Published: December 2020

Building a skateboard ramp motivates students to practice and apply fraction operations and measurement while developing problem-solving skills.
Introducing Volume 15: Welcome to the New Volume Year
Authors: Trafton Patricia
Published: December 2020

The beginning of a new school year is an exciting time for teachers and administrators, as well as for students. The Editorial Panel is equally excited. It means new students and the start of a new volume year for the journal.
The Thinking of Students: The Three Bears Cookie Store
Authors: Mooney Edward S.
Published: December 2020

The student work included in this article illustrates the various ways that students were able to cook up a solution to the Cookie Store problem.
Constructing Complexity for Differentiated Learning
Authors: Little Catherine A. , Hauser Sherryl , Corbishley Jeffrey
Published: December 2020

You can differentiate tasks to challenge learners at different levels of readiness. Develop alternate versions to increase or decrease task complexity to meet learners' needs.
Building Numbers from Primes
Authors: Burkhart Jerry
Published: December 2020

Use building blocks to create a visual model for prime factorizations. Students can explore many concepts of number theory, including the relationship between greatest common factors and least common multiples.
Solve It!: Product-ku Light
Authors: Mooney Edward S.
Published: December 2020

Many of you are familiar with Sudoku puzzles.
Algebraic Reasoning through Patterns
Authors: Rivera F. D. , Becker Joanne Rossi
Published: December 2020

Findings, insights, and issues drawn from a three-year study on patterns are intended to help teach prealgebra and algebra.
Mathematical Explorations: Hot Wheels
Authors: Winsor Matthew S. , Lesser Larry
Published: December 2020

Many, if not most, students have collected or played with Hot Wheels® cars. Therefore, I decided that middle-grades students could explore the mathematical concept of scale factor involving these miniature cars. This activity has been successfully used with students of all ability levels, including special education students.
The Thinking of Students: Creative “Solve It!” Solutions: Rule of Ten
Authors: Mooney Edward S.
Published: December 2020

The student work in this article illustrates their understanding of how inches measure up.
Getting a Bead on It
Authors: Ferrucci Beverly J. , McDougall Jennifer , Carter Jack
Published: December 2020

For a convenient, concrete way of showing linear equations and their key components, ask your students to string beads.
Stories about Never-Ending Sums
Authors: Friedlander Alex
Published: December 2020

Infinity and infinitely small numbers pique the curiosity of middle school students. Examples such as the story of Achilles and the Tortoise promote questions about domain, representations, and infinite sums–all of which may not get answered until students reach high school.
Mathematical Explorations: Exploring the Volume of Mayan and Egyptian Pyramids
Authors: Alejandra Sorto M.
Published: December 2020

Middle school teachers in Central Texas and rural Guatemala have some mathematics in common.
Math for Real: A Three-Girl Family
Authors: Tillema Erik S.
Published: December 2020

Mr. and Mrs. Ortenzi want to have three children. They have picked out three names for a boy and three names for a girl. Since Mrs. Ortenzi wants to have three girls, she would like to learn about the chance that this might occur. To help her determine her chance of having three girls, solve the problems below.
Quick Reads: Another Good Idea: Podcasting for Understanding
Authors: Pannell F. Andy , Hutchison Linda
Published: December 2020

The sixth graders were in my face. “How'd I do?” “What did I get?” “Do you even have them graded yet?” I had graded their unit tests on operations with fractions, and the results were catastrophic. The class of twenty-four students had a range of 0 points up to a perfect 32 out of 32, with a mean score below 50 percent. How could their grades be so low after we had spent six weeks on this material?
Tech-Knowledgy and Diverse Learners
Authors: Suh Jennifer M.
Published: December 2020

When teachers know how to effectively use the unique features of computer applications, they can address the varying cognitive strengths and needs of different students.
Constructing Spatial Understanding
Authors: Obara Samuel
Published: December 2020

Dynamic geometry software can help preservice middle school teachers and their students draw connections between two-dimensional nets and threedimensional solids.
Mathematical Exploration: Probability Experiments with Shared Spreadsheets
Authors: Beigie Darin
Published: December 2020

Probability Experiments with Shared Spreadsheets Probability experiments illustrate how apparently random events are ultimately governed by the laws of probability. A large number of trials are usually necessary for experimental data to converge to a theoretical prediction. Having students electronically combine their data into a shared spreadsheet provides an efficient and powerful way to collectively analyze a large amount of information. In this activity, seventh-grade students use a shared spreadsheet to collaboratively investigate what happens when they roll a pair of dice. They compare the case of two 6-sided dice with the case of a 4-sided die rolled with an 8-sided die.
Deductive Puzzling
Authors: Wanko Jeffrey J.
Published: December 2020

Students can improve their deductive reasoning and communication skills by working on number puzzles.
The Whole Is the Sum of Its Parts
Authors: Bieda Kristen N.
Published: December 2020

Scenarios of students using manipulatives to explore and justify mathematical ideas show how they are poised to generate effective proofs.
Informing Practice: They “Get” Fractions as Pies; Now What?
Authors: Petit Marjorie M. , Laird Robert , Marsden Edwin
Published: December 2020

Picture a fifth-grade classroom where students are learning about fractions. Fraction circles are being used to introduce the topic and build on the basic concepts. Near the end of the fraction unit, the teacher asks his students to compare the relative sizes of 3/7 and 7/8. All but three of his students reach for their fraction circles instead of reasoning about the relationships between the two fractions. The teacher had expected many students to compare the given fractions to the benchmark measurement of 1/2, recognizing that 3/7 is less than 1/2 and 7/8 is greater than 1/2.
Wake Up, Because Math Matters
Authors: Gilpin Jeanny
Published: December 2020

Read one teacher's story about building a Math Matters club, and why her students arrived at school before first bell to participate enthusiastically.
Quick Reads: Another Good Idea: Capture the Flags
Authors: Howe John B. , Badillo Joy A.
Published: December 2020

At the middle school level, nothing is more important than a hands-on application of mathematics. It offers students a way to progress from a concrete level of understanding to a more abstract form of thinking. Capture the Flags is one in a series of outdoor activities that gives our students a concrete application to solidify their understanding of a pencil-and-paper classroom activity. It also helps students improve their spatial-reasoning ability.
Think outside the Polygon
Authors: Graf Andrea B.
Published: December 2020

After constructing irregular polygons, students use geometry software to explore rotations and reflections and build spatial-visualization skills.
Quick Reads: Another Good Idea: Kickin' It with “Integer Football”
Authors: Blackford Kelli Lane
Published: December 2020

Students often struggle with adding and subtracting integers. To help alleviate this struggle, I found a way to relate integers to a topic that is practically universal: football. I developed the game of “integer football” as a way for students to practice the skills of adding and subtracting integers and to show them how these concepts work outside the classroom.
Master Geometry while Coteaching
Authors: Hunt Jessica H.
Published: December 2020

All students benefit from lessons adapted to a team-teaching model.
The Story of the Trojan Octagon
Authors: Derado Josip , Garner Mary , Edwards Belinda P. , Garrett Violette L.
Published: December 2020

Using Flatland: The Movie to set the stage, students employ imagination and geometric constructions to reinforce their understanding of polygons.
Math for Real: How Old Is That Cactus?
Authors: Edenfield Kelly W. , Araujo Zandra de
Published: December 2020

Large cacti called saguaros (pronounced so-WAH-ros) have astounding growth rates and long life spans. Kelly took a picture of Zandra standing next to a saguaro. They then wondered about the height and the age of the cactus in the photograph. Use the following information to help them satisfy their curiosity.
Exploring Our Complex Math Identities
Authors: Leatham Keith R. , Hill Diane S.
Published: December 2020

Three reflective tasks for students at all levels help them and their teachers become more aware of their beliefs about mathematics.
Mathematical Exploration: Similar Triangles
Authors: Johnson Gwen
Published: December 2020

Within school mathematics, similarity is a topic that has many connections to real life. Similarity is related to scale drawings, such as those used in architecture, and models, including models of trains, cars, and boats. It is also used to solve problems that involve similar figures, which are common in middle-grades textbooks.
Using Bloom's Taxonomy with English Language Learners
Authors: Gómez Conrado L. , Kurz Terri L.
Published: December 2020

English language learners (ELLs) at the preproduction phase, or initial phase, of language proficiency have limited oral English language skills; however, they do not lack cognitive abilities (de Jong and Harper 2005). On the contrary, most ELLs possess previous language and academic experiences. They can understand and articulate through nonverbal means (such as by drawing pictures) much more than they can demonstrate through speaking. Because many teachers may not fully understand the relationship between cognitive ability and language proficiency, they fail to challenge ELLs with higher-level activities (de Jong and Derrick-Mescua 2003).
Making Sense of Decimal Multiplication
Authors: Rathouz Margaret M.
Published: December 2020

To build student confidence, to whole-number operations, geometry, and measurement.
Reasoning with Real-World Data
Authors: Wiest Lynda R. , Amankonah Frank O. , Thomas Troy A.
Published: December 2020

Entertain the use of popular songs and movies as data-analysis lessons to promote quantitative literacy.
Mathematical Explorations: Gauss's Idea: Take a Notion
Authors: Sinn Robb , Hix Sherry L.
Published: December 2020

A grandfather clock in Jodie's home chimes once at one o'clock, twice at two o'clock, and so on. If there are no chimes except at the top of the hour, how many times does the clock chime between one o'clock and ten o'clock?
Lesson Study on the Farm
Authors: Klein Robert M. , Hendrickson Katie A.
Published: December 2020

Stacking bales of hay in a barn provides the context for a team of teachers to develop a rich lesson involving geometric solids.
Proportional Reasoning with a Pyramid
Authors: Mamolo Ami , Sinclair Margaret , Whiteley Walter J.
Published: December 2020

A three-dimensional model and geometry software can help develop students' spatial reasoning and visualization skills.
Tailoring Tasks to Meet Students' Needs
Authors: McDuffie Amy Roth , Wohlhuter Kay A. , Breyfogle M. Lynn
Published: December 2020

Thread small changes seamlessly into high-level reasoning tasks to reach all students.
On My Mind: The Standards—Some Second Thoughts
Authors: Willoughby Stephen S.
Published: December 2020

Members of the National Council of Teachers of Mathematics can be proud of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). Not only has the Standards document set the course for improving mathematics education, but it has been imitated by virtually every other content-oriented teachers' organization. Thus, as well as recasting mathematics education, the NCTM has led the way toward recasting education generally. However, professional educators have an obligation to reconsider and reflect on the Standards themselves and any other recommendations made by responsible educators and organizations.
Now & Then: Ann Wagner, Mechanical Engineer: A Brief History of Mechanical Engineering
Authors: Bennett Betsy K.
Published: December 2020

Now … Ann Wagner is a Mechanical Engineer in the Facility Management Division of the Facilities Engineering and Technical Services at the 1121-acre Goddard Space Flight Center in Greenbelt, Maryland. She designs, improves, repairs, and replaces systems that deliver utilities-heat, light, cooling, water, and power-to the fifty buildings there. Goddard was established as the National Aeronautics and Space Administration's first space-flight center in 1959. Its mission, “to expand knowledge of the Earth and its environment, the solar system and the universe through observations from space,” tells the significance of the work here, which is to research and to support communication of all kinds throughout the world. The Hubble Space Telescope Control Center; the Flight Dynamics Facility, which supports space-shuttle operations; and vast communication networks are among the many resources at Goddard that simply cannot be shut down for some period of time for repairs to a utility cable, pipe, or pump.
A Teacher's Views on Classroom Assessment
Authors: Tonack De A.
Published: December 2020

Mathematics Instruction is assuming a new direction that includes a new perspective on assessment. Instruction and assessment cannot be separated, since what is assessed conveys to students what is important, and how that learning is assessed must be congruent to the method and emphasis of instruction. Much of formal mathematics evaluation and grading relies on assessing a student's product, such as a paper-and-pencil test or homework. Informally, teachers assess students' processes just as much as, or more than, their products. When the process and product aspects are combined, mathematics evaluation might be described as perfonnance assessment, in which the teacher observes a student carrying out a process and evaluates the student's performance as well as the product. If the majority of mathematics evaluation is asswned to be a type of performance assessment, the challenge becomes to establish performance criteria including the specific behaviors, products, and qualities of a student's work.
Now & Then: Brokering in Money and Mathematics
Authors: Hilgart Faye
Published: December 2020

Now… Although loy gotham's officlal job title at Painewebber is that of an investment executive, she says, “I think of myself as a problem solver.” She listens to her client's needs and asks questions. How much income does the client want the investment to produce per month? How long can the money be invested? What is the purpose of the investment—for college tuition or a retirement fund? What kind of return on the investment does the client anticipate—a steady return with a low-risk stock or income that may vary widely from week to week with a high-risk stock? With this information in mind, Loy creates an investment plan that reflects the client's needs and preferences.
The Thinking of Students: Playing Mathematical Chairs
Authors: Greenwood Jay
Published: December 2020

The editorial panel presented this “Food for Thought” problem in the March-April 1996 issue of the journal:
November's: Menu of Problems
Authors: Greer Gerry , Wantuck Larry
Published: December 2020

A light fare to whet your mathematical appetite.
Teacher to Teacher: A Holiday Project
Authors: Kofod Maudrey Taranto
Published: December 2020

Any teacher with more than two weeks' experience with students knows that some things that are supposed to work well in the classroom somehow just do not make it. This one did.
Teaching the Mean Meaningfully
Authors: Uccellini John C.
Published: December 2020

Ask a group of middle school students what the average (mean) of 2, 8, 4, 6, 3, and 7 is; they will probably give the answer 5. Ask these same students what the number 5 represents in relationship to the six numbers given and the response usually heard is an explanation of the algorithm, “Add them up and divide by the number of them that you have.” The response is no different if the problem is given in a real-life context. For example, the foregoing six numbers could represent the number of pencils that six students have in their desks. In either situation, the almost universal response of students when questioned about the meaning of 5 from the “add and divide” algorithm demonstrates that students have not gained a conceptual understanding of this basic statistic. This same phenomenon exists throughout mathematics and is demonstrated whenever students try to explain subtraction by describing the vertical algorithm or the Pythagorean theorem by stating that c2 = a2 + b2. Through the use of simple manipulative activities and graphing, however, middle school students can be taught the mean meaningfully.
Division with Fractions
Authors: Nowlin Douald
Published: December 2020

Although operations with fractions have been in the elementary and middle school curriculum for many years, finding examples of practical problems that illustrate the usefulness of division with fractions and mixed numbers is not easy. Most real-world applications of rational numbers involve decimal numerals (Usiskin and Bell 1984), but examples of division with fractions and mixed numbers are often obviously contrived. In addition, many teachers and prospective teachers have difficulty constructing examples and concrete models for the operation of division with fractions (Borko 1992; Ball 1988).
Teacher to Teacher: Homework Partners
Authors: Kaplan Cindy
Published: December 2020

While working on difficult or complicated word problems in an algebra class, students have a tendency to become discouraged and give up. This frustration can often cause the students to lose their self-confidence. One solution to this frustration is a program that I have used successfully for several years—the assignment of “homework partners.”
On My Mind: “Algebra for All”—Increasing Students' Access to Algebraic Ideas, Not Just Algebra Courses
Authors: Silver Edward A.
Published: December 2020

Algebra has long been viewed as being a crucial component of American students' mathematics education. In fact, it has been called a “gatekeeper” because the successful completion of an algebra course is a prerequisite not only to further study in mathematics and other school subjects but also to many jobs and later opportunities. Because of the perceived importance of algebra, most American students, who complete high school, study algebra for two years. Nevertheless, the poor performance of twelfthgrade students on many algebra related tasks on the National Assessment of Educational Progress (NAEP) suggests that enrollment in these courses does not ensure that students acquire mastery of fundamental algebraic ideas (Mullis et al. 1991). Moreover, a lack of algebraic competence among even fairly successful high school graduates is evinced by the large number of remedial mathematics courses offered by the nation's colleges.
Activities to Help in Learning about Functions
Authors: Willoughby Stephen S.
Published: December 2020

The function concept is perhaps as important as any concept in mathematics. It permeates all of mathematics, from first-year algebra through calculus and beyond, as well as most applications of mathematics. This article describes several activities and games that have been used by me and many other teachers to provide a dynamic and enjoyable early introduction to the function concept.
Building Foundations for Algebra
Authors: Stacey Kaye , MacGregor Mollie
Published: December 2020

Learning algebra is an important milestone in a student's mathematical development. It opens the door to organized abstract thinking and supplies a tool for logical reasoning. It gives a student the satisfaction of finding simplicity in what appears to be complex and finding generality in a collection of particulars. For example, phenomena as diverse as the growth of algae in a pond and the accumulation of bank debt manifest the same mathematical behavior; in each situation, a small rate of change accumulates over time to produce substantial changes in the totals. Even a tiny change in percentage growth can produce unexpected and extreme effects. This behavior is captured algebraically in the exponential function, but it is hard to describe clearly in words, as our foregoing attempt shows. For this reason, the language of algebra is the standard medium for precise communication about numbers and functions and is essential for higher mathematics. It is important that students learn to master this new language.
The Missing Link? Writing in Mathematics Class!
Authors: Ciochine John G. , Polivka Grace
Published: December 2020

The last thing I ever thought I would be stressing in my mathematics classes is the use of writing as a tool for teaching mathematics. On entering the teaching field as a middle school mathematics teacher, I thought, “Give me a book and I will be ready to educate all my students.” Wow! Was I wrong! After realizing that I needed something more than a book, I began to read professional articles and to talk with my colleagues. One colleague suggested that I do some writing with my students. My first reaction was that I teach mathematics, not English. After her constant urging, I decided I would try it, although I was sure I would not like it. What I have found is an interesting way to approach middle school mathematics while helping students develop their communication and reasoning abilities.
Mental Computation in the Middle Grades: The Importance of Thinking Strategies
Authors: Mcintosh Alistair , Reys Robert E. , Reys Barbara J.
Published: December 2020

At the primary-grades level, the benefits of developing and using mental strategies for computing have been well articulated (see, e.g., Beberman (1959); Brownell [1972); Cobb and Merkel [1989]: Kamii [1989]; Reys and Barger [1994): Shuard [1987); Trafton [1978)), and many primary-grades teachers are now encouraging students to invent and use thinking strategies as a way to facilitate their development of number sense. They are also dealing with the practical implications of implementing this approach to computation, which is very different from the traditional. rule-oriented, procedural approach to computation. At the middle-grades level, however, comparatively Little discussion related to the same issue has occurred. At this level, should students be encouraged to invent mental strategies for computing? Should standard written algorithms for computing continue to be taught? How does an emphasis on thinking strategies relate to the current emphasis on using the calculator as an efficient tool for computation?
Mathematics Detective: Exploring Graphs: WYSIWYG
Authors: Johnson Millie
Published: December 2020

Many kinds of graphs and charts confront us wherever we turn. We must be able to pull important information from them when needed, such as the length of time to store milk safely or how much air to put in certain shock absorbers. The following graphs have been taken from various media sources. The questions will guide you in interpreting these interesting graphs.
Now & Then: Measuring Angles in Physical Therapy: A New Angle on the History of Movement
Authors: Greeley Nansee , Offerman Theresa Reardon
Published: December 2020

Now… What do geometry and physical therapy have in common? “Angles!” according to Robin Nicholson. Robin is a physical therapist, often called a PT, in Saint Paul, Minnesota. She works at the HealthEast OPTimum Rehabilitation Center-Maplewood and sees about twelve patients a day for half-hour sessions. Patients come to her after they have recovered from surgery or from an injury to a muscle, ligament, or joint. She has worked with patients as young as five and as old as ninety. “In general,” she said, “the younger the patient, the quicker the recovery process.” Robin believes that the most important facet of recovery is the attitude of the patient.
500 + Writing Formats
Authors: McIntosh Margaret E.
Published: December 2020

As mathematics teachers, we realize that we need to have our students write as one of the ways of “learning to communicate mathematically,” as recommended by NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). The Standards document expands on this idea (p. 6) by stating,
On My Mind: From Doogie Howser to Dweebs— or How We Went in Search of Bobby Fischer and Found That We Are Dumb and Dumber
Authors: Sheffield Linda jensen
Published: December 2020

As my son and his friends moved from elementry school to middle school, I became acutely aware that many students who had always done well in school did not want to be identified as “smart kids” in middle school. Some of them were afraid that they would be picked on by the other students if it were known that they were bright. Others just wanted to fit in. The school had a very difficult time recruiting enough students to be on the MATHCOUNTS team; no girls would even by it. It simply was not cool to be smart-and at a suburban middle school that boasts of some of the highest test scores in the state.
Allium to Zircon: Mathematics & Nature
Authors: Harrell Marvin E. , Fosnaugh Linda S.
Published: December 2020

Nature reveals its splendor in the seasons, its grandeur in geophysical configurations, and its order in each organism's obedience to its laws. What can the study of nature bring to the mathematics classroom?