bluuebunny/crossref_metadata_2025 huggingface dataset:
Mathematical Lens: Iron Gate, Greenfield, Massachusetts
Authors: Turton Roger W.
Published: January 2021

The undulating curve at the top of an iron gate at a local cemetery in Greenfield, Massachusetts, brings to mind several different mathematical functions. Roger Turton took the photographs and submitted the accompanying questions and answers.
Introduction
Authors: Lott Johnny W.
Published: January 2021

In 1972, the Mathematics Teacher published a series of three articles in “The Forum,” a section of the journal devoted to diverging opinions with respect to the role of geometry and the best approach to it. The February issue addressed the question “What should become of the high school geometry course?” The articles in “The Forum” in February were written by Howard F. Fehr, Frank M. Eccles, and Bruce E. Meserve. I chose the article by Fehr as one that has had an effect on high school curricula and still poses some answers to the original question today. The question was the subject of a panel discussion at the August 2006 MathFest of the Mathematical Association of America sponsored by the NCTM/MAA Joint Committee on Mutual Concerns.
Maximizing the Fit of a Box Spring Mattress up a Stairwell
Authors: Buhl David A. , Morissette Michael A. , Wolff Jeffrey
Published: January 2021

A class investigation into the mathematics of determining the maximum size of a box spring mattress that could be carried up a stairwell. The Geometer's Sketchpad was used for the problem exploration in order to help avoid common mistakes when deriving the mathematical model. The numerical solution was described using graphical method.
How Far Up Am I? The Mathematics of Stadium Seating
Authors: McGraw Rebecca , Romero David , Krueger Robert
Published: January 2021

A lesson in which students develop understanding of linear functions investigating the height of the football stadium bleachers in relation to the given row. Students collect data related to their stadium seats and use this data to create tables, draw graphs, and derive equations. Whole class discussion concludes this activity.
From the 1910s: Our Critics and their Viewpoints
Authors: Minnick J. H.
Published: January 2021

The article originally printed in 1916 presents positions of several influential critics of theory of mental discipline and formal education. Although they disagree on reasons, they all wanted to reduce the role mathematics from a facility that helps students to develop their mental powers to a utility that suits a set of vocations.
From the 1930s: Gestalt Psychology and Mathematical Insight
Authors: Hartmann George W.
Published: January 2021

The article, originally printed in 1937, attempts to introduce Gestalt psychology - “geometry of the mind” to enhance teaching of mathematics. The author describes and analyzes three main propositions of the theory and points to a features applied to mathematical research. In conclusion he highlights that teachers of mathematics should act on recognition that content of mathematics has to be rediscovered and created de novo by every learner.
From the 1940s: The Progressive Nature of Learning in Mathematics
Authors: Brownell William A.
Published: January 2021

One editorial from February 1941 and the second from February 1942 editions. The first one is a reaction on another call to replace mathematics in schools with teaching students to think by “studying current controversial questions”. The second one constitutes response to school officials preaching the elimination of mathematics from the secondary schools. It discusses two letters from the Navy officials indicating critical shortage of qualified recruits due to the lack of mathematical skills of high school graduates.
From the 1950s: The Role of Insight in the Learning of Mathematics
Authors: Fehr Howard
Published: January 2021

The questions of problem-solving and the nature of trial and error, and the insight in arriving at meaningful solutions to thought problems, have been investigated by a rather large number of psychologists. These researchers have been interested in the scientific aspects of the process, in finding out what it is that goes on in the mind or the organism as they solve problems. They are busy with the logical structure and explanation of thought, analysis, insight, etc. They have not, however, been concerned with how to initiate or to direct mental processes that lead to successful solutions. This latter is the task of the educator and the teacher, namely, to apply to the teaching process the findings of psychologists so that they may develop a greater problem-solving ability on the part of their students.
From the 1960s: Are Students' Questions a Valid Criterion for Evaluating Creative Teaching?
Authors: Adkins Julia
Published: January 2021

The article describes the author's discovery made in a plane geometry class that type of questions asked by students may reflect creativity of teaching as opposed to traditional teacher's evaluation by quality of questions asked by the teacher. Describing her experience the author comes to defining of two levels of creative teaching: level one where the teacher asks thought-provoking questions and a higher level where students generate questions leading towards discovery of new relationships.
From the 1960s: On Learning Mathematics
Authors: Bruner Jerome
Published: January 2021

The article from the December of 1960 describes complex process of learning mathematics from a position of psychologist. The author thoroughly defines the subject of discussion and then focuses on a role of each of four components of learning: discovery, intuition, translation into language of mathematics and readiness to learn new material. He analyzes every topic providing insights into a learning process and recommendations valuable for educators.
From the 1980s: President's Annual Address: 58th Annual Meeting
Authors: Hill Shirley A.
Published: January 2021

The main focus of the president's address is communicating recommendations for school mathematics summarized in the document “An Agenda for Action”. It defines three major problems and presents analysis of a set of recommendations on how to improve the situation. Recommendations covers eight major categories, from “Problem solving” to “Public support for mathematics instruction”. It is noted that problem in mathematics education can only be solved by the cooperation of all sectors of our society.
The Magic of Balanced Groups: Educational Applications of Magic Squares
Authors: Bossé Michael J. , Nandakumar N. R. , Ore Melanie L.
Published: January 2021

Provides students with many interesting observations regarding the nature of magic squares, magic rectangles, and quasi-magic squares and provides tools for teachers to group students into balanced cooperative learning groups.
Proof for Everyone
Authors: Olmstead Eugene A.
Published: January 2021

How all students can apply the algebra skills they have learned to problem solving through investigation, along with different levels of proving the conjectures algebraically.
Encouraging Preservice Mathematics Teachers as Mathematicians
Authors: Burroughs Elizabeth A.
Published: January 2021

An assignment that asks preservice secondary mathematics teachers to make connections between the mathematics they know and the mathematics they will teach. It describes how one preservice teacher's project resulted in a physical representation of the statement and proof that the sum of cubes of the first n natural numbers is equal to the square of their sum.
Sharing Teaching Ideas: Graphing Families of Curves Using Transformations of Reference Graphs
Authors: Kukla David
Published: January 2021

For years I have tried to help students understand algebraic functions. I thought the prevalence of graphing calculators in the classroom would remove the veils from their eyes, but I found instead that students did not understand functions well enough to be able to use their calculators effectively.
Connecting Research to Teaching: Using a New Synthesis of Reading to Promote a Community of Mathematicians
Authors: Clair Janet St.
Published: January 2021

The literature documents the crucial need for students to be able to act like “mathematically literate persons.” A key feature of the idea of this community of mathematically literate persons is oral and written communication (Lindquist and Elliott 1996; NCTM 2000; Silver and Smith 1996).
Squaring Matrices: Connecting Mathematics and Science
Authors: Horton Robert M. , Wiegert Elaine M. , Marshall Jeff C.
Published: January 2021

We have all heard that mathematics is the language of science, yet many teachers have not had sufficient opportunities to learn how to use mathematics to provide insight into scientific ideas. Consequently, in many mathematics classes, students learn procedures yet have difficulty understanding why they are learning them. In working with both high school and college students, we have found that matrix multiplication is one of these procedures. In fact, when we introduced our students to problems similar to the ones presented here, we discovered that, although they had been taught how to multiply matrices, they did not understand the rationale behind the procedure. Further, they were unaware how useful this procedure could be for studying some important realworld phenomena. In this article, we present a technique for matrix multiplication that is appropriate for algebra 2 or late algebra 1 students and provides a means for investigating food chains, a rich source of study in both science and mathematics classes.
Mathematical Lens: Quilt Blocks
Authors: Zanetti Mary
Published: January 2021

Throughout the countryside around the town of Neversink, New York, one encounters artful displays of quilt blocks. Produced as part of a beautification project, these colorful squares currently adorn the sides of more than thirty barns and other buildings. Some are as small as 2 feet square, others as large as 8 feet square. They provide a springboard for many types of mathematical discussions.
Technology Tips: When Is a Good Fit Not a Good Fit? Dynamic Regression with the TI-Nspire Graphing Calculator
Authors: Edwards Michael , Meagher Michael , Özgün-Koca S. Asli
Published: January 2021

In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (NCTM) acknowledges the importance of exploring mathematical ideas from multiple points of view: “Different representations often illuminate different aspects of a complex concept or relationship…. The importance of using multiple representations should be emphasized throughout students' mathematical education” (2000, p. 68). In particular, NCTM notes that the introduction of technology in school mathematics classrooms provides new ways for teachers and their students to explore connections among representations: “Computers and calculators change what students can do with conventional representations and expand the set of representations with which they can work” (2000, p. 68). In this article, we discuss an interesting finding that our students made as they explored linear regression with a teacher-constructed TI-Nspire calculator document. The calculator's capability to link variables across two or more pages in the same document led students to findings that are important yet rarely discussed in school mathematics textbooks.
Technology Tips: Exploring the Cereal Box Problem with Fathom
Authors: Wanko Jeffrey J.
Published: January 2021

A number of experiments modeled in mathematics classes involve repeating an action until a specific condition is met. One might be interested in seeing how long it takes until a sum of 12 is rolled on two fair dice or until a specific name is drawn from a hat. These problems are easy to model by using physical materials and running the experiment a few times. But to see what happens over the long run, statistical modeling software is a useful alternative.
Activities for Students: Slicing a Cube
Authors: Gómez Emiliano
Published: January 2021

As mathematics educators, we know that mathematical maturity and good habits of mind need to be developed and encouraged continually. For students to form these habits, they need opportunities to think and explore mathematically. The activity “Slicing a Cube,” in which students are asked to decide whether or not certain figures may be obtained when a plane slices through a cube, provides such an opportunity.
Rethinking the Teaching of Systems of Equations
Authors: JÉrôme Proulx , Beisiegel Mary , Miranda Helena , Simmt Elaine
Published: January 2021

Highlighting multiple representations of systems of equations deepens student understanding.
Connecting Research to Teaching: Collaborative Teacher Inquiry through the Use of Rich Mathematics Tasks
Authors: Bornemann Greta , Haury Scott M. , Slavit David
Published: January 2021

As teachers trying to increase our students' excitement about mathematics, we have asked ourselves lots of questions about how to improve our practice. But what does improve mean? And how do we know when our practice has been improved? Every teacher aiming to refine his or her practice must wrestle with these fundamental questions.
Sound Off!: Two High School Mathematics Curricular Paths—Which One to Take?
Authors: Reys Robert , Reys Rustin
Published: January 2021

High schools are requiring students to complete more years of mathematics in order to graduate (Reys et al. 2007). This requirement raises several questions for schools, teachers, students, and parents. In particular, what mathematics should students study, and how should that mathematics be organized? High school mathematics programs today use two different mathematics course sequences. One sequence focuses each course on a specific subject (algebra, geometry, algebra, or precalculus), while the other integrates mathematical strands throughout each course. Choosing between subject-based and integrated course sequences stimulates discussions about-and often controversy over—which organizational choice is best and for whom.
Mathematical Lens: Annuals, Perennials, Periodicals
Authors: Clemens Meg
Published: January 2021

Photograph 1 shows a metallic garden border in Canton, New York. One of Meg Clemens's students used a similar photograph of this antique-looking edging for a project involving modeling digital images found in the neighborhood.
Who Was the Real William Shakespeare?
Authors: Edwards Michael Todd
Published: January 2021

Connecting language arts and mathematics, students use data analysis and readability measures to identify the Bard.
Empowering Students through Data
Authors: Barnes Paige A.
Published: January 2021

This activity helps students understand data analysis concepts and acquire skills to take control of their learning.
Activities for Students: Estimating African Elephant Populations (Part 2)
Authors: Fuentes Sarah Quebec
Published: January 2021

The March 2009 “Activities for Students” presented the first installment of a two-part activity on counting African elephant populations. Part 1 offered two scenarios and asked students to estimate populations by using data given in each one. In part 2, we present the final scenario, noting that the mathematics involved is somewhat more sophisticated than that required for last month's activities. For readers' ease, we re-present the history portion of last month's installment.
Designing a WebQuest
Authors: Salsovic Annette R.
Published: January 2021

WebQuests—inquiry-based lesson plans that harness the technology of the Internet—are easy to implement.
Activities for Students: Fitting Curves to Pottery
Authors: Harris Richard A. , Matthews Catherine E.
Published: January 2021

Faculty members at Hawbridge School of the Environment for Grades 9–12, located in Saxapahaw, North Carolina, have begun to implement a number of interdisciplinary units across the curriculum. At Hawbridge, this approach means that each teacher in each class devotes a significant amount of time to teaching a themed unit during a specific time frame. This article will detail the integrative aspects of the pottery unit we taught in three different high school mathematics classes: algebra 1, algebra 2, and precalculus.
Modern Geometric Algebra: A (Very Incomplete!) Survey
Authors: Suzuki Jeff
Published: January 2021

Representing products as rectangles can be used to introduce, connect, and reinforce concepts across mathematics.
Listening to Geometry
Authors: Cooper Brett D. , Barger Rita
Published: January 2021

Studying geometric transformations in music—a real-world application of mathematics—deepens students' understanding of the underlying principles.
On Fences, Forms, and Mathematical Modeling
Authors: Legé Jerry
Published: January 2021

Modeling and then designing a picket fence allows students to use real-life mathematics to develop a mathematical habit of mind.
Ten Things to Consider When Teaching Proof
Authors: Cirillo Michelle
Published: January 2021

What I wish I had known about teaching proof before I taught geometry
Soft Drinks, Mind Reading, and Number Theory
Authors: Schultz Kyle T.
Published: January 2021

A mathematical proof inspired by a mind–reading trick found online can ready algebra students for more rigorous thinking.
The Back Page: Medians of a Triangle: The Search for Invariants
Authors: Garrison Lionel
Published: January 2021

When I begin teaching a class in geometry, one of my fundamental concerns is to broaden students' conceptions of the very nature of mathematics. Much of their previous work in arithmetic and elementary algebra has left students with the impression that doing mathematics consists primarily of memorizing and executing algorithms. In geometry, I want them to start exploring, with the aid of The Geometer's Sketchpad® (GSP), and making conjectures about the objects of study. Afterward we can attempt to prove their conjectures or find counterexamples to them.
Mathematics Problems from Ancient Egyptian Papyri
Authors: Howard Christopher A.
Published: January 2021

Age-old problems from the Rhind and Moscow Papyri can enliven the curriculum while making students aware of mathematics history.
Delving Deeper: Sizing Up Class Size: A Deeper Classroom Investigation of Central Tendency
Authors: Lesser Lawrence M.
Published: January 2021

Being able to select, use, and interpret measures of center is expected of all secondary students (NCTM 2000, 2006). Discussing average class size can be a motivational vehicle for exploring this topic because students (and teachers) at all grade levels notice when they have significantly bigger classes and high school juniors and seniors see average class size touted in the publicity brochures they receive from colleges. Also, educators, administrators, policy makers, and parents are concerned about the impact class size may have on student achievement and equity (e.g., Finn, Gerber, and Boyd-Zaharias 2005; Nye, Hedges, and Konstantopoulos 2004; Pong and Pallas 2001). Finally, using situations that readily yield results that students initially find counterintuitive can be motivating (Lesser 1998).
The Back Page: Discovering a Derivative
Authors: Goetz Albert
Published: January 2021

Teachers can often make up for a lack of creativity by keeping up with the literature. I never considered myself an imaginative or creative teacher, but I was well read; I attended lots of conferences and workshops, and I borrowed and copied whatever I could.
Proof-Related Reasoning in High School Textbooks
Authors: Johnson Gwendolyn J. , Thompson Denisse R. , Senk Sharon L.
Published: January 2021

The authors provide a framework for investigating proof-related reasoning in high school algebra and precalculus textbooks and suggest ways to increase students' opportunity to learn proof-related reasoning.
Investigating the Fundamental Theorem of Calculus
Authors: Johnson Heather L.
Published: January 2021

A student's response to a lesson on the fundamental theorem of calculus expands the lesson to include representing accumulated area as a function.
Delving Deeper: Growth Rates and the Marvelous Geometric Sequence
Authors: Burke Maurice J. , Hodgson Ted R.
Published: January 2021

Students are often dazzled by the prodigious growth rate of the geometric sequence gn = 2n and the geometric series whose partial sums are Sn = 1 + 2 + 4 + 8 + … + 2n−1 = 2n − 1. Teachers sometimes note that the geometric sequence is the discrete “form” of an exponential function, which is characterized by very rapid growth. In particular, exponential functions grow faster than polynomial functions. A rigorous explanation of this claim is left to the calculus class in which students examine the relative growth rates of functions by using L'Hopital's rule. However, even by using tools developed in algebra and precalculus, teachers can explain to their students (or lead them to a justification) that the geometric sequence gn = 2n grows faster than any polynomial sequence. These tools are the binomial theorem, Pascal's triangle, mathematical induction, and an understanding of the end behavior of exponential and polynomial functions (i.e., what happens to the graphs as x approaches infinity).
Learning from Student Approaches to Algebraic Proofs
Authors: D'Ambrosio Beatriz S. , Kastberg Signe E. , dos Santos João Ricardo Viola
Published: January 2021

Generalizing from number patterns provides insights into the proof process and a basis for a hypothetical teaching trajectory.
Engaging with the Art and Science of Statistics
Authors: Peters Susan A.
Published: January 2021

Statistics uses scientific tools but also requires the art of flexible and creative reasoning.
Connecting Research to Teaching: Slope, Rate of Change, and Steepness: Do Students Understand these Concepts?
Authors: Teuscher Dawn , Reys Robert E.
Published: January 2021

How do we mathematics teachers introduce the concepts of slope, rate of change, and steepness in our classrooms? Do students understand these concepts as interchangeable or regard them as three different ideas? Here we report the results of a study of high school Advanced Placement (AP) Calculus students who displayed misunderstandings about the meaning of these three concepts. We then share an example of a typical rate-of-change problem commonly found in mathematics textbooks that may contribute to the development of these misconceptions. Finally, we provide a classroom task that supports the discussion of the differences among these three concepts, and we ask readers to reflect on ways to help students differentiate among them.
Activities for Students: Investigating Star Polygons
Authors: Starling Tina T. , Hollebrands Karen F.
Published: January 2021

With the geometry curriculum already packed with content, who has time to introduce anything new? Many students already have difficulty with regular polygons to begin with—wouldn't an additional topic for polygons be adding fuel to the fire? Perhaps. However, if activities are carefully chosen, students can actively review prerequisite skills as well as benefit from being asked to think critically in a new way.
Translations toward Connected Mathematics
Authors: Applebaum Mark , Leikin Roza
Published: January 2021

The translation principle allows students to solve problems in different branches of mathematics and thus to develop connectedness in their mathematical knowledge.
ProofBlocks: A Visual Approach to Proof
Authors: Dirksen Jennifer , Dirksen Nathaniel , Cheng Ivan
Published: January 2021

An inherently visual technique, ProofBlocks lends itself to the use of manipulatives, thus making proof more accessible to more students.
Connecting Research to Teaching: Comparing Pays Off
Authors: Star Jon R. , Kenyon Martina , Joiner Rebecca M. , Rittle-Johnson Bethany
Published: January 2021

Consider the following, perhaps familiar, scenario. A mathematics teacher is circulating around the classroom, looking over the shoulders of students who are busy solving linear equations such as 3x + 2 = 5x + 8. The teacher notices that one student, Paul, persists in using his own somewhat idiosyncratic and quite inefficient strategy (see fig. 1). Although Paul's strategy is not fundamentally incorrect, the extra steps required can lead to more calculation errors and wasted time.
The Back Page: My Favorite Lesson: Paper Pool
Authors: Vennebush G. Patrick
Published: January 2021

The Paper Pool lesson on Illuminations (http://illuminations.nctm.org) asks students to look for patterns as they investigate pool tables of various sizes. Students appreciate the challenge of determining the number of hits that occur on an m × n table, and I enjoy the amount of mathematics that this lesson allows them to explore.
Media Clips: Gasoline Prices / Cigarette Butts
Authors: Gibbs Crystal
Published: January 2021

… The cost of gas dropped across North America in recent weeks, partially because reserves have increased. Analysts have linked it to people driving less.… It is predicted that Canadians will drive 800 kilometres less this year versus a year ago.
Mathematical Lens: Reflections on Vacation Spots
Authors: Çağlayan Günhan
Published: January 2021

While viewing photographs posted online at flickr.com, Günhan Çağlayan came across one titled “Vacation in Norway” (see photograph 1), taken by Mirjam Pouw of the Netherlands. Çağlayan reflected on how the beautiful reflections in the image could be modeled with graphs of various types of functions. Ron Lancaster, a co-editor of this department, followed up on Çağlayan's ideas and offers readers the following questions for further reflection.
Docfish: A Card Game with Factoring
Authors: Fishman Daniel M.
Published: January 2021

In this variation of the well-known card game Go Fish, players seek to pair prime factors contained within the cards' face values.
Delving Deeper: Enumeration of Rectangles in a Tableau Shape
Authors: Mingus Tabitha T. Y. , Grassl Richard M. , Diaz Ricardo , Andrew Lane , Parker Frieda
Published: January 2021

The Mathematics Teacher article “Keep Counting Those Boxes—There's More” (Grassl and Mingus 1998) illustrated the ways a problem can be extended to create further problems to explore. In analyzing the question “How many squares of all sizes does a checkerboard contain?” the authors tackled the fairly natural extension “How many rectangles does a checkerboard contain?” and then, finally, the question “How many rectangles does a subdivided rectangular m-by-n grid contain?” The authors were able to show that the answer is given by.
Media Clips: Tall Guys / Fibs
Authors: Lancaster Ron
Published: January 2021

Why do tall guys find it easier to get dates than us normal-sized guys?
The Mathematics of Garlic
Authors: Moore Nathan T. , Deming John C.
Published: January 2021

More than vegetables—proportional reasoning skills and an understanding of sophisticated statistical concepts—can grow from a backyard plot.
Funny Face Contest: A Formative Assessment
Authors: Colen Yong S.
Published: January 2021

Algebra students create their own funny faces onscreen after studying several parent functions and their transformations.
Activities for Students: How Long Does It Take for a Person to Sober Up? Some Mathematics and Science of DUI
Authors: Winston Bente B. , Zunker Matthew T.
Published: January 2021

The number of annual accidents and fatalities related to driving under the influence (DUI) across the United States is staggering. In 2008 alone, drivers under the influence caused 11,773 deaths, representing 32% of all highway fatalities that year. That is an average of one DUI-related death every 45 minutes, a number greater than three times the number of people killed on 9/11 and equivalent to the number of people killed if a large passenger airplane were to crash every week.
Chocolate Key Cryptography
Authors: Bachman Dale J. , Brown Ezra A. , Norton Anderson H.
Published: January 2021

This colorful illustration of a primary component of modern cryptography—the Diffie-Hellman key exchange—draws students into the secret world of message encoding and decoding.
Connecting Research to Teaching: Is 7/10 Always Equivalent to 700/1000?
Authors: Rubel Laurie H.
Published: January 2021

Consider the following problem: Compare the probabilities of the following two events: (1) 7 tails out of 10 tosses of a fair coin and (2) 700 tails out of 1000 tosses of a fair coin. Which of the events is more likely, or are they equally likely? (adapted from Fischbein and Schnarch 1997, p. 99)
Media Clips: Highly Paid Tweens / Brown-Bagging It
Authors: Slate Molly , Hammond Sarah E.
Published: January 2021

… Hollywood's 10 top-earning tweens collectively pulled down $107 million between June 1, 2007 and June 1, 2008. …
Exploring the Mathematics of Bouncing Balls
Authors: Vinogradova Natalya , Blaine Larry G.
Published: January 2021

Simple equipment—stopwatches, tape measures, and golf balls—allows students to explore geometric sums and quadratic equations and analyze experimental data.
Bouncing Balls and Graphing Derivatives
Authors: Cory Beth
Published: January 2021

Using motion detectors, calculus, and physics, students create height-, velocity-, and acceleration-versus-time graphs of a bouncing ball.
Technology Tips: Exploring Algebra and Geometry Concepts with GeoGebra
Authors: Garber Kim , Picking David
Published: January 2021

Throughout their K–12 education, students will spend a considerable amount of time developing mathematics competency. By the time they reach middle school and high school, the focus will be on becoming proficient in algebra and geometry. As a means to motivate and help students explore these topics, we recommend a free, multiplatform software application called GeoGebra (www.GeoGebra.org). GeoGebra provides the opportunity to explore a wide variety of algebraic and geometric concepts through hands-on manipulation of graphs, tables, formulas, and shapes. It is also a convenient and easy way to generate graphics and visuals for presentations, test questions, and homework problems. The software is available for Mac, Windows, and Linux operating systems, and a Web applet version is available that works within your Internet browser.
Mathematical Lens: Raise High the Ridge Beam, Carpenters
Authors: Matheson Lauren
Published: January 2021

During a walk around Toronto, Lauren Matheson noticed a shed being built in a very narrow corner of a triangular lot (see photographs 1, 2, and 3). The odd-shaped property made designing and building the shed difficult, but, thanks to a creative construction crew, the shed was built and by all accounts is well used. Lauren was then a preservice teacher candidate in Ron Lancaster's mathematics methods course, and he used images of the shed as the basis for his “Mathematical Lens” assignment. Lauren's photographs and questions are given here; the editors hope that readers agree that his work was worth an A+.
Designing Mathematical Learning Environments for Teachers
Authors: Madden Sandra R.
Published: January 2021

Scaffolding tasks can help increase teachers' comfort and competence with technology
Pathological Examples with Technology
Authors: Lassak Marshall
Published: January 2021

Teaching moments occur when the use of calculators leads to unexpected but mathematically explainable results.
Sound Off!: Students' Beliefs about Mathematics: Lessons Learned from Teaching Note Taking
Authors: Wieman Rob
Published: January 2021

After many years of teaching mathematics, I still fall into the trap of assuming that my students think as I do. Indeed, this failure to recognize my own assumptions and to acknowledge that others may not share them is at the root of most of my teaching problems.
Fixing Ganache: Another Real–Life Use for Algebra
Authors: Kalman Adam M.
Published: January 2021

In a real–world application of proportional reasoning and equation solving, students adjust ingredient amounts in a recipe for chocolate ganache.
Delving Deeper: Patterns in Binomial Coefficients and Repeated Differences
Authors: Cox Roger D.
Published: January 2021

“Delving Deeper” offers a forum for classroom teachers to share the mathematics from their own work with the journal's readership; it appears in every issue of Mathematics Teacher. Manuscripts for the department should be submitted via http://mt.msubmit.net. For more background information on the department and guidelines for submitting a manuscript, visit http://www.nctm.org/publications/content.aspx?id=10440#delving.
The Back Page: My Favorite Lesson: Sinusoids
Authors: Allen Eileen
Published: January 2021

In my precalculus classes, after the unit on graphing sinusoidal functions, I break the class into groups of four students. They are given word problem applications related to a point spinning on a wheel or a buoy bouncing in the bay. Each group is given six problems—from activity sheet A, B, or C—to determine the model for each situation (see activity sheet B in fig. 1).
Inverse Functions: What Our Teachers Didn't Tell Us
Authors: Wilson Frank C. , Adamson Scott , Cox Trey , O'Bryan Alan
Published: January 2021

Our teachers misled us, but we don't blame them. They were only teaching what was in the textbook. And as new teachers—because of our lack of experience and our reliance on the textbook—we continued to teach the procedure we had learned as students. It wasn't until we began writing textbooks ourselves (Wilson 2007; Wilson et al. forthcoming) that we were compelled to confront the inverse function falsehoods in our intellectual past. These contradictions were difficult to detect because they were broadly accepted and perpetuated in widely used textbooks
Taking the Work out of Homework
Authors: Cox Teodora B. , Singer Stacy L.
Published: January 2021

Technology daily impacts students' learning experiences. The increased use of calculators, computer algebra systems, and computer–based and Web–based assessments opens up new opportunities and challenges for teaching and learning mathematics. Students' lives are becoming busier, and they have less time to dedicate to homework outside the classroom. Also, many students admit that they do not enjoy completing homework assignments.
Activities for Students: Rational Functions: A New Perspective
Authors: Davis Anna A. , Zielke Ronald E. , Lickeri Jessica L.
Published: January 2021

The study of asymptotes constitutes one of the earliest and most significant encounters a high school mathematics student can have with infinity. The study of horizontal asymptotes, in particular, contributes to a student's later understanding of limits and the notion of “arbitrarily close to.” Typically, students are exposed to horizontal asymptotes in the contexts of exponential and rational functions. Real–life applications of exponential functions abound, but real–life applications of rational functions do not. In this article, we present a hands–on activity that allows students to explore horizontal asymptotes of graphs of rational functions visually and mathematically and that has been used successfully with intermediate algebra, precalculus, college algebra, and mathematics methods students.
The Back Page: My Favorite Lesson: Which Fits Better?
Authors: Meyer Margaret R.
Published: January 2021

One of my favorite lessons comes from a problem I first heard posed as an open–ended assessment problem by David Clarke at an NCTM conference years ago:
Integrating Mathematical, Technologic, and Pedgogical Knowledge in Teacher Preparation
Authors: Schneiter Kady , Kohler Brynja R. , Watts Brandon J.
Published: January 2021

In Principles and Standards for School Mathematics (NCTM 2000), the authors describe a vision for school mathematics in which “all students have access to high-quality, engaging mathematics instruction” (p. 3). Students deserve teachers who are knowledgeable about mathematical content, who understand and use a variety of teaching strategies appropriately, who can effectively integrate technology into classroom learning, and who continually progress as professionals. But what kinds of experiences lead to such professionalism?
Robotics 101
Authors: Sultan Alan
Published: January 2021

Robots are used in all kinds of industrial settings. They are used to rivet bolts to cars, to move items from one conveyor belt to another, to gather information from other planets, and even to perform some very delicate types of surgery. Anyone who has watched a robot perform its tasks cannot help but be impressed by how it works. Here we offer some insight into the mathematics that goes into designing such a robot.
Technology Tips – Piecewise-Defined Functions on the TI–Nspire
Authors: Ilaria Daniel R.
Published: January 2021

Students generally first encounter piecewise–defined functions in the form of a step function (perhaps the postage stamp function) in an algebra class. Piecewise–defined functions do not play a central role in mathematics before calculus although they can serve as challenging examples in the precalculus curriculum. Before the advent of the TI–Nspire, entering piecewise–defined functions on the calculator was time consuming and not particularly user friendly. That has changed.
Back to Treasure Island
Authors: Shriki Atara
Published: January 2021

Some twenty years ago, when I was a university student, one of my lecturers presented a problem that he called Treasure Island. At first glance, the problem appeared to be unsolvable. After students made some futile attempts, the lecturer presented the surprising solution, without providing any explanation or even a hint. I spent the rest of the lecture thinking about the problem and trying to discover a solution.
Pirates of the Parametric
Authors: Madden Seán P. , Allison Dean
Published: January 2021

Some children enjoy playing with the spring–loaded, plastic toy cannon that accompanies many model pirate ships. This cannon pivots in such a way that it can be fired at any angle from about −15 degrees to 195 degrees when viewed from a position directly in front of the cannon. The spring provides each fired projectile with approximately the same initial velocity. This toy, together with a digital camera, can be used with high school students to demonstrate the influence of angle on parabolic trajectories and, simultaneously, to explore the underlying parametric equations. In particular, students can discover and verify that the vertices of projectile motion paths for a given initial speed and a range of launch angles trace an ellipse.
Exposing the Mathematical Wizard: Approximating Trigonometric Functions
Authors: Gordon Sheldon P.
Published: January 2021

In the climactic scene in The Wizard of Oz, Toto draws back the curtain to expose the Wizard of Oz, and Frank Morgan admits, “I am really a very good man but just a poor wizard.” This statement is reminiscent of Arthur C. Clarke's famous third law: “Any sufficiently advanced technology is indistinguishable from magic” (Clarke 1962, p. 36). For almost all students, what happens when they push buttons on their calculators is essentially magic, and the techniques used are seemingly pure wizardry.
Connecting Research to Teaching: Making Student Thinking Public
Authors: Stockero Shari L. , Van Zoest Laura R.
Published: January 2021

Teachers often have students publicly share their mathematical thinking as part of classroom instruction. Before reading further, we invite you to stop and think about this practice by writing down your responses to the following two questions:
The Back Page: My Favorite Lesson – Slopes of Parallel and Perpendicular Lines: A Visual Justification
Authors: Wallace-Gomez Patricia
Published: January 2021

When teaching slopes of parallel and perpendicular lines, I want students to have a visual image of the lines, not just memorize a formula. A simple exercise with parallel lines can get the message across.
Activities for Students: Designing the Dynamic Domino Race
Authors: Bremigan Elizabeth George
Published: January 2021

Students of all ages are enthusiastic about the Olympic Games. Many mathematics teachers use this context as an opportunity for students to examine numerical data while they display and discuss the results of different events and the success of various countries during the Olympic Games. These discussions allow teachers to address several aspects of the Data Analysis and Probability Standard from Principles and Standards for School Mathematics (NCTM 2000).
Volumes and Cube Dissections
Authors: Mihaila Ioana
Published: January 2021

This article began with the 1998 mathematics contest at Coastal Carolina University in Conway, South Carolina.
Sharing Teaching Ideas: Motivating Students to Learn Mathematics Using Resources from Outside the Classroom
Authors: Howe R. Michael , Deitte Jennifer M.
Published: January 2021

Teaching required mathematics courses poses a challenge to teachers at all levels. Many of these difficulties are the result of cultural attitudes in the United States that make it socially acceptable, even trendy, to lack mathematical knowledge. Most of our students are aware of the inherent value of mathematics; but because mathematics is a subject that requires hard work, they choose to deny its importance. Our challenge is to somehow motivate students to take responsibility for their own learning. This goal is most effectively accomplished if students convince themselves that mathematics is interesting and useful.
Where Is the Highest Point on the Roof of a Shed?
Authors: Bruyn Ysbrand de
Published: January 2021

Some time ago, a freshman engineering student returned to the high school where I taught and described to me some of his struggles with university mathematics. The simplex method of linear programming was one topic that we discussed. This student could perform the mechanics of the simplex method—finding pivot columns, pivot rows, and row reductions; however, he confessed that he had no idea why the simplex method worked to find the maximum point. Moreover, he could not connect the theorems that his professor proved in the lectures with the simplex method. So I asked him this question: “Where is the highest point on the roof, excluding the overhang, of a shed with a plane, but sloping, roof?” He looked at me and was clearly wondering why sheds had anything to do with linear programming.
Participatory Research Methods for Investigating Digital Health Literacy
Authors: De Souza Ruth , Butt Danny , Jethani Suneel , Marmo Chris
Published: January 2021

DIGITAL TECHNOLOGIES AND PRE/PERI-NATAL APPS ARE TRANSFORMING MATERNITY CARE AS WOMEN USE COMMUNICATIONS TECHNOLOGIES TO OBTAIN INFORMATION AND SUPPORT. WOMEN FROM CULTURALLY AND LINGUISTICALLY DIVERSE (CALD) BACKGROUNDS EXPERIENCE INFORMATION AND SUPPORT GAPS, HOWEVER, RESEARCH ON NEW HEALTH COMMUNICATIONS HAS RARELY ENGAGED WITH THIS GROUP, RAISING CONCERNS ABOUT THE APPLICABILITY OF THESE TECHNOLOGIES WITH HIGHLY DIVERSE COMMUNITIES. THE GOAL OF THIS PILOT STUDY LOCATED IN IN NORTH RICHMOND, A CULTURALLY DIVERSE NEIGHBOURHOOD OF MELBOURNE, WAS TO TEST THE VALUE OF “DESIGN PROBES” IN ENGAGING WOMEN FROM CALD BACKGROUNDS IN DISCUSSIONS ABOUT HEALTH TRACKING AND WEARABLE HEALTH TECHNOLOGIES IN THE CONTEXT OF PREGNANCY AND PARENTING. THIS STUDY FOUND THAT DESIGN PROBES WERE EFFECTIVE IN ENABLING PARTICIPANTS TO PARTICIPATE IN EXTENDED DISCUSSIONS ON THE IMPLICATIONS OF DIGITAL HEALTH TECHNOLOGIES. PARTICIPANTS WERE AWARE OF THE WAYS THAT HEALTH INFORMATION SERVED THE NEEDS OF MULTIPLE ENTITIES, AND SOUGHT MORE CONTROL OVER WHERE AND WITH WHOM THEIR DATA WAS SHARED WITH.
More Inclusion Through Participation?
Authors: Nikolova Karina , Stelzmann Daniela
Published: January 2021

DUE TO THEIR HIGH USER NUMBER, SOCIAL NETWORK SITES CAN BE REGARDED AS A PUBLIC SPHERE AND SHOULD BE ACCESSIBLE TO ALL DEMOCRATIC CITIZENS. NEVERTHELESS, THERE ARE SOME UNDERREPRESENTED SOCIAL GROUPS SUCH AS PEOPLE WITH MILD INTELLECTUAL DISABILITIES. THIS QUALITATIVE STUDY INVESTIGATES HOW YOUNG PEOPLE WITH MILD INTELLECTUAL DISABILITIES IN GERMANY ACCESS, INTERACT AND PARTICIPATE ON SOCIAL NETWORK SITES. THEREFORE, WE CONDUCTED 8 QUALITATIVE INTERVIEWS WITH YOUNG PEOPLE WITH MILD INTELLECTUAL DISABILITIES (AND THEIR CAREGIVERS, N=7) AND ANALYZED THE TRANSCRIBED INTERVIEWS WITH AN INDUCTIVE QUALITATIVE CONTENT ANALYSIS TECHNIQUE. OUR PARTICIPANTS DESCRIBED VARIOUS WAYS OF ACCESSING, INTERACTING AND PARTICIPATING ONLINE (E.G., SHARING THEIR OWN CONTENT). FURTHERMORE, OUR PARTICIPANTS GATHERED BOTH POSITIVE (E.G., GAINING NEW FRIENDS) AND NEGATIVE (E.G., HATE SPEECH) EXPERIENCES THERE. MOREOVER, OUR RESULTS GIVE INFORMATION ABOUT ASPECTS OF SOCIAL NETWORK SITES THAT SHOULD BE ALTERED SO THAT A POSITIVE SOCIAL CHANGE BASED ON THE INCLUSION PRINCIPLE CAN TAKE PLACE.
Editorial
Authors: Klausen Maja , Grønning Anette , Stage Carsten
Published: January 2021

THIS SPECIAL ISSUE, ENTITLED “HEALTH, MEDIA AND PARTICIPATION”, CONSISTS OF SEVEN ARTICLES THAT EXPLORE SOME OF THE DIFFERENT WAYS THAT MEDIA PARTICIPATION AND HEALTH PARTICIPATION INTERTWINE IN CONTEMPORARY MEDIA CULTURE. IN THESE SEVEN ARTICLES, PARTICIPATION IN HEALTH AND MEDIA IS ADDRESSED IN THE ANALYSIS OF A VARIETY OF PRACTICES: PATIENTS THAT USE MEDIA TO BECOME PARTICIPANTS IN CO-DECISION AND SELF-CARE PROCESSES, PATIENTS AND CITIZENS BEING MORE OR LESS ABLE TO USE MEDIA TO ENGAGE IN (PATIENT) COMMUNITIES, PATIENTS COMMUNICATING WITH (AND AFFECTING) INSTITUTIONS AND AUTHORITIES IN NEW MEDIATED WAYS, HEALTH PROFESSIONALS USING SOCIAL MEDIA TO CREATE PUBLIC AWARENESS ABOUT POLITICALLY URGENT ISSUES AND HEALTH PROFESSIONALS CO-DESIGNING DIGITAL LEARNING TECHNOLOGIES. ALL THE CONTRIBUTIONS ARE IN THIS WAY UNITED BY TAKING AN INTEREST IN THE DEMOCRATIC POTENTIALS AND DILEMMAS OF PARTICIPATING IN HEALTH THROUGH MEDIA PARTICIPATION. THE ISSUE ALSO INCLUDES ONE OPEN SECTION ARTICLE. THE SPECIAL ISSUE IS EDITED BY MAJA KLAUSEN, ANETTE GRØNNING AND CARSTEN STAGE.
My Throat “Tickles”
Authors: Klausen Maja , Grønning Anette
Published: January 2021

THIS ARTICLE INVESTIGATES THE MULTIPLE WAYS IN WHICH BODIES PRESENT THEMSELVES IN EMAIL CONSULTATIONS (ECONS) BETWEEN PATIENTS AND GENERAL PRACTITIONERS (GPs). THE DATA STEM FROM A LARGER QUALITATIVE RESEARCH PROJECT ON DIGITAL CONSULTATION IN DENMARK, FOCUSING ON THE 65+ AGE GROUP. OUR ANALYSIS DEPARTS FROM A SOCIOTECHNICAL PERSPECTIVE BY CONSIDERING THE INTRODUCTION OF THE TECHNOLOGY IN QUESTION – THE ECON BETWEEN THE PATIENT AND GP – AS CREATING NEW CHALLENGES AND OPPORTUNITIES FOR PARTICIPATION IN AND THE EXCHANGE OF COMMUNICATION WITHIN THIS RELATIONSHIP. OUR ANALYSIS LEADS TO A SIX-CATEGORY TYPOLOGY OF BODIES IN AFFECTIVE DISCOURSE IN ECONS: SENSATIONS, EMOTIONS, COUNTINGS, MEDICATION, VISUALS AND MOVEMENTS. THE ANALYSIS BRINGS TOGETHER A THEORETICAL PERSPECTIVE, THE MUTUAL SHAPINGS OF THE SOCIAL AND TECHNOLOGICAL, WITH AN AFFECTIVE PRACTICE-BASED APPROACH. THE ECON PRODUCES WHAT WE CALL AN “UNRULY DATA VITALITY”, AS THE DATA ARE ACCESSED AND RECOMBINED IN THE LARGER DIGITAL ECOSYSTEM OF THE DANISH PRIMARY CARE SECTOR.
Digitalization of Healthcare Practice
Authors: Marchetti Emanuela , Petersen Camilla Kølsen
Published: January 2021

THE HEALTHCARE SECTOR HAS ENTERED AN ECOLOGICAL EVOLUTION, A SYSTEMIC CHANGE CAUSED BY THE INTRODUCTION OF NEW TECHNOLOGY AND FINANCIAL CUTS, WHICH IS DEEPLY TRANSFORMING EXISTING PRACTICES IN RELATION TO SKILLS AND VALUES (NARDI & O’DAY, 1999). CONDUCTING A PARTICIPATORY DESIGN PROCESS, AIMED AT THE DEVELOPMENT OF A DIGITAL SIMULATION FOR THE EDUCATION OF OCCUPATIONAL THERAPY, WE FOUND THAT THE ONGOING DIGITALIZATION IS CHANGING EDUCATIONAL AND WORKING PRACTICES, CHALLENGING THE PROFESSIONAL IDENTITY OF THERAPISTS AND GRADUATE STUDENTS, BUT ALSO INTRODUCING NEW PROFESSIONAL OPPORTUNITIES.

WE USE A PRAGMATIC CONSTRUCTIVIST PERSPECTIVE ON LANGUAGE GAMES DEFINED AS A COLLABORATIVE CONSTRUCTION OF OUR WORLD THROUGH ACTION AND COMMUNICATION THAT FULFIL OUR PURPOSES IN HUMAN PRACTICE. OCCUPATIONAL THERAPY EMERGES AS A PROFESSION IN FLUX AS THERAPISTS MUST MOVE FROM THEIR TRADITIONAL ROLE AS HEALERS TO RETHINK THEMSELVES AS FREELANCERS OR EMPLOYED IN THE PRIVATE SECTOR. WE IDENTIFIED FOUR DIFFERENT LANGUAGE GAMES, WHICH ARTICULATE HOW THERAPISTS MUST RETHINK THEIR PROFESSIONAL IDENTITY AND PURPOSEFUL PRACTICE IN RELATION TO EMERGING WORKING CONTEXTS AND RELATIONSHIP TO THE PATIENTS.
Histoire sociale de la télévision en Espagne
Authors: De la Fuente Manuel
Published: January 2021

Manuel Palacio, Histoire sociale de la télévision en Espagne. Lyon: Grimh, 2019, 171 págs.