bluuebunny/crossref_metadata_2025 huggingface dataset:
Cranial Trepanation: Case Studies between the IV Century BC and VI Century in Southern Italy
Authors: Vovlas Alessio , Di Nunno Nunzio , Panzarino Ginevra , Sublimi Saponetti Sandro
Published: January 2021

Cranial trepanation is among the oldest head surgical procedures, creating a hole through the skull. This procedure is the surgical removal of a portion of the skull as a medical treatment to cure a specific disease or as a religious or magical ritual. Cranial trepanation has been reported from very early times and widespread all over the world. This work is based on data from six trepanned skulls from the Hellenistic Age (IV century BC) to the Late Antiquity (VI century). The samples come from Puglia (Southern Italy), from the northern (the peninsula of Gargano) to the central area: three cases come from Vieste (Foggia), Ischitella (Foggia) and Alberona (Foggia), the others three from Gravina in Puglia (Bari) and Canosa (Barletta, Andria, Trani). All skulls have been evaluated via an anthropological, radiological and odontological approach. This study indicates the value of a methodology, which encompasses anthropology, radiology and imaging, and archeo-forensic investigations, highlighting the need for multidisciplinary teamwork in any assessment of human remains. The study supports the hypothesis of the presence of a long tradition of medical centres in the Mediterranean basin area.
Economics of Solid Waste Management: A Review
Authors: Balasubramanian Muniyandi
Published: January 2021

Solid Waste Management is one of the importance environmental issues at many developing countries. There is a lack of studies on economic analysis of solid waste management in the many cities at the national and international level. Most of the Municipal Corporation or city management is the major responsibility for better waste management. However, the local governments has been allocated budget for solid waste management without analysing cost and benefit of solid waste. Although, waste management budget is focusing on collected waste but, uncollected waste has been creating a number of socio, economic and health issues. Therefore, this chapter has presents a details review on economics of solid waste management at the various developing and developed countries. The main policy implication of the paper is to emphasis on better understanding of economic importance of solid waste management to the local policy makers.
Business Process Linguistic Modeling: Theory and Practice Part II: BPLM Business Process Designer
Authors: Stašák Jozef , Kultan Jaroslav , Schmidt Peter , Khawla Bouafia
Published: January 2021

The business activities provided within any firm or company should be checked and controlled continuously, while two principal approaches should be applied: (a) qualitative monitoring, (b) quantitative evaluations and getting to know the rules, which regulate structure and functionality of business processes (BPs) implemented and operated there plays a role of principle importance and they are derived based on actual BP models. Therefore we have designed a conceptual model of application denoted as BPLM Process Designer in form of expert system (ES) operating based on principles closely related to business process linguistic modeling approach, where linguistic sets and PBPL Equation play a role of principle importance. Our contribution contains such application description from qualitative, quantitative and design point of view. The ES qualitative description contains references to appropriate math relations and algorithms postulated within subsequent sections. Those sections are accompanied by the case study, which indicates how the math relations and algorithms might be applied within BPLM Process Designer functionality. However, those sections are accompanied by ES structure and functionality description as well, which represent the BPLM Process Designer mean or facility.
Blister Blight Disease of Tea: An Enigma
Authors: Chaliha Chayanika , Kalita Eeshan
Published: January 2021

Tea is one of the most popular beverages consumed across the world and is also considered a major cash crop in countries with a moderately hot and humid climate. Tea is produced from the leaves of woody, perennial, and monoculture crop tea plants. The tea leaves being the source of production the foliar diseases which may be caused by a variety of bacteria, fungi, and other pests have serious impacts on production. The blister blight disease is one such serious foliar tea disease caused by the obligate biotrophic fungus Exobasidium vexans. E. vexans, belonging to the phylum basidiomycete primarily infects the young succulent harvestable tea leaves and results in ~40% yield crop loss. It reportedly alters the critical biochemical characteristics of tea such as catechin, flavonoid, phenol, as well as the aroma in severely affected plants. The disease is managed, so far, by administering high doses of copper-based chemical fungicides. Although alternate approaches such as the use of biocontrol agents, biotic and abiotic elicitors for inducing systemic acquired resistance, and transgenic resistant varieties have been tested, they are far from being adopted worldwide. As the research on blister blight disease is chiefly focussed towards the evaluation of defense responses in tea plants, during infection very little is yet known about the pathogenesis and the factors contributing to the disease. The purpose of this chapter is to explore blister blight disease and to highlight the current challenges involved in understanding the pathogen and pathogenic mechanism that could significantly contribute to better disease management.
Heat Transfer in a MHD Nanofluid Over a Stretching Sheet
Authors: Poply Vikas
Published: January 2021

The intention behind carrying out this research work is to analyze the heat transfer characteristics in a Magnetohydrodynamic (MHD) boundary layer nanofluid flow over a stretching sheet. Two phase representation of nanofluid studied the consequence of Brownian motion along with thermophoresis. The major purpose of study is to investigate the significant role of prominent fluid parameters thermophoresis, Brownian motion, Eckert number, Schmidt number and magnetic parameter on profile of velocity, temperature distribution and concentration. Runge–Kutta Fehlberg (RKF) method was adopted to numerically solve the non-linear governing equations and the linked boundary conditions by use of shooting technique. Over all the consequence of prominent fluid parameters are explained via graphs, whereas distinction of several valuable engineering quantities like skin friction coefficient, local Nusselt number and local Sherwood number are also tabulated. The finding of present study helps to control the rate of heat transportation as well as fluid velocity in any manufacturing processes and industrial applications to make desired quality of final product.
Aptamers: Magic Bullet for Theranostic Applications
Authors: Sett Arghya
Published: January 2021

Aptamers are a short polymer of oligonucleotides (natural or modified) that can bind to its cognate target (small molecules to large macromolecules like proteins, cells, microorganisms etc.) with high affinity and selectivity. They can fold into unique secondary and tertiary conformation in solution (pH, ionic concentration) and bind to their targets in a specific manner (binding constants in sub-nano to pico molar range). They rival the monoclonal antibodies and other specific biological ligands with respect to affinity, stability, robustness, non-immunogenicity and facile to synthesis. Nucleic acid aptamers are selected from an oligonucleotide library by an iterative process called SELEX (Systematic Evolution of Ligands by Exponential Enrichment Analysis). These aptamers are compatible to any kind of chemical modification, conjugation and functionalization. Briefly, this chapter discusses about the diagnostic and therapeutic application of aptamers.
Empirical Test of Fama and French Three-Factor Model in the Egyptian Stock Exchange
Authors: Abd-Alla Mustafa Hussein , Sobh Mahmoud
Published: January 2021

We test the empirical validity of the three-factor model of Fama and French in the Egyptian Stock Exchange (EGX) using monthly excess stock returns of 50 stocks listed on the EGX from January 2014 to December 2018. Our findings do not support Fama and French three-factor model, where the coefficient of the beta was insignificant. The “SBM” coefficient and the “HML” coefficient were equal to zero and insignificant, which confirms the absence of the small firm effect and book-to-market ratio effect in the market. We conclude that there is no relation between expected return and Fama-French risk factors.
The Impact of Herding on the Risk Pricing in the Egyptian Stock Exchange
Authors: Abd-Alla Mustafa Hussein , Sobh Mahmoud
Published: January 2021

We test the impact of herding behaviour on the risk pricing in the Egyptian Stock Exchange (EGX) by adding an additional risk factor reflecting herding behaviour to the Fama and French three-factor model. We construct a portfolio to mimic an additional risk factor related to herding behaviour, in addition to the original risk factors in the Fama and French three-factor model. The three-factor model will be tested in its original form and re-tested after adding the herding behaviour factor. The study is based on Hwang and Salmon methodology, in which the state space approach based on Kaman’s filter was used to measure herding behaviour. We used monthly excess stock returns of 50 stocks listed on the EGX from January 2014 to December 2018. The results do not support Fama and French model before and after adding the herding behaviour factor, therefore, there is no effect of herding behaviour on the risk pricing in the Egyptian Stock Exchange.
Commonalities in Returns in the Stock Markets of the Visegrad Group: A Quantile Coherency Approach
Authors: Łęt Blanka
Published: January 2021

The aim of this paper is to investigate the dependence structure in the frequency domain for the joint distribution of returns from the stock markets in the countries belonging to the V4 countries. We analyze twenty-years of historical daily prices of four main stock indices from the Czech Republic, Hungary, Poland, and Slovakia. Using a quantile coherency measure we found, that linkages between Czech, Hungarian, and Polish stock markets are significantly positive for all considered quantiles and frequencies. These three markets are more strongly dependent during the long downturns and the effect is permanent after the European Union accession. The Slovak stock market is the least connected with other countries in the group. Results of the paper revealed, that Czech, Hungarian and Polish stock market is subject to similar trends in terms of returns for different investment horizons. International market participants should incorporate interdependencies between these markets during the portfolio building process.
Applying Kant’s Ethics to Video Game Business Models
Authors: Roy Nandita ,   Philosophy Documentation Center
Published: January 2021

This article expands on existing models of analyzing business ethics of monetization in video games using the concept of categorical imperatives, as posited by the philosopher Immanuel Kant. A model is advanced to analyze and evaluate the business logics of video game monetization using a Kantian framework, which falls in the deontological category of normative ethics. Using two categorical imperatives, existing models of game monetization are divided into ethical or unethical, and presented using the case example of Star Wars: Battlefront II (2017). This analysis aims to provide video game developers and businesses with ethical guidelines for game monetization which may also be profitable for them in the long term. Within the framework of video game monetization, a deontological analysis is relevant due to the fact that the game developer is engaged in a continuous role of making the game more playable/payable. This article applies Kantian business ethics to the context of a new sector, that of video game businesses, and thereby presents a broader ethical perspective to video game developers, which will help them monetize games in an ethical manner which is also profitable in the long run.
How Plants Live
Authors: Hall Matthew ,   International Association for Environmental Philosophy
Published: January 2021

The recent proliferation of human-plant (or plant-human) studies are informed by understandings of how plants live. Philosopher Michael Marder has developed a philosophy of plant ontology, founded on notions of modular independence, radical openness and ontological indifference. This paper critiques, and ultimately rejects, Marder’s key concepts, using a swathe of empirical evidence and theory from the plant sciences and evolutionary ecology. It posits a number of positive statements about these aspects of plant being that better align with the scientific evidence base. The implications for plant ethics are also briefly explored.
Introduction
Authors: Lappan Glenda
Published: January 2021

NCTM President Glenda Lappan discusses her choice of an article to include in a celebration of 100 years of Mathematics Teacher.
“Editorial”
Authors: Metzler W. H.
Published: January 2021

Modern psychology has done much during the last decade or two in correcting false notions concerning various phases of education. The old idea that the study of some subjects furnished an all-round mental discipline entirely available for the handling of any other subject has been much modified and replaced by the more rational idea of the study of those subjects furnishing somewhat less general and some more special disciplines. It would seem, however, that there has been much misunderstanding and misinterpretation of the results of recent psychological study especially as stated by some writers, if indeed it is not true that some enthusiastic.
Sharing Teaching Ideas: A Deeper Look at Related Rates in Calculus
Authors: Santulli Thomas V.
Published: January 2021

There are numerous places in the calculus curriculum where students can discover important concepts. Some of these ideas are counterintuitive, while many others are very reasonable. Whenever possible, our students should go a little further than just settling for an answer. All too often they obtain an answer and simply leave it at that. or they might check to see if their answer matches the back of the book and then move on to the next problem. Instead, we must encourage them to analyze the result, checking for reasonableness. Is the answer in line with their expectations, or is it so counterintuitive that further exploration is required to determine why? In calculus, where many of the concepts are truly profound, this extra step is necessary for real understanding. Moreover, sometimes a secondary lesson can be learned, or at least verified, by looking at the results of a problem from several different perspectives.
Sharing Teaching Ideas: Demystifying Systems
Authors: King Sybrina L.
Published: January 2021

It has been twenty-five years since I taught my first beginning algebra course. For many years, classes in the private school at which I taught typically comprised highly motivated, mathematically gifted students. However, my class rosters changed dramatically six years ago when I first started teaching at a public, urban high school with a focus on the performing arts. It did not take long for me to realize that this was not going to be my father's algebra class, to paraphrase a well-known advertising slogan. Many of my students lacked fundamental mathematical skills. Furthermore, the overwhelming majority of my students were far more interested in dance, sculpture, and dramatic readings than in algorithms, graphs, and linear equations. For the first time in nearly two decades, I was truly challenged to create classes that would appeal to the nontraditional algebra student. After considerable trial and error, I discovered that my students responded well to lessons that included hands-on activities, incorporated hand-held technology, and connected to their experiential base. The following multiday lesson on the topic of solving linear systems of equations illustrates these three strategies.
Con-fusing Pairs: An Intriguing Investigation of LCMs
Authors: Bennett Curtis D. , DeYoung Mary J. , Rutledge James J. , Young Elaine
Published: January 2021

As today's high school students and undergraduates contemplate the conic sections known to the Greeks or learn the calculus dating back to Newton, they rarely stop to realize that mathematics is continually being discovered and invented. A great deal of current mathematical knowledge has been established since these young students were born. What is the driving force behind this new knowledge? The simple answer is curiosity.
Sound Off!: Placement Tests: The Shaky Bridge Connecting School and College Mathematics
Authors: Gordon Sheldon P.
Published: January 2021

The earth's crust is composed of a series of large plates floating on underlying molten magma. These plates are constantly shifting, and as they bump into one another, one plate often rides up on top of the other near its edges. The interfaces between plates form fault lines in the earth, and the resultant pressures that build up along the interfaces eventually release to form earthquakes, as we are too often reminded in vivid news reports from around the world.
Delving Deeper: The Ubiquitous Catalan Numbers
Authors: Koshy Thomas
Published: January 2021

Just as Fibonacci and Lucas numbers are a great source of fun and excitement for both amateurs and professionals alike (Askey 2005, Koshy 2002), so are the less well-known Catalan numbers. They too are excellent candidates for mathematical activities such as experimentation, exploration, and conjecture. I was surprised to see that, like Fibonacci and Lucas numbers, Catalan numbers seemed to show up in several problems I had assigned to students over the years.
Technology Tips: Visualizing DeMorgan's Laws Using Logic Simulation Software
Authors: Edwards Michael Todd
Published: January 2021

This month's tips illustrate an interactive, inquiry-based approach for exploring logic operators and equivalence of various Boolean expressions using technology. Specifically, I illustrate the use of several logic simulation tools to perform the following tasks: (1) build and “run” a virtual logic circuit illustrating x ∧ y and (2) verify the validity of DeMorgan's laws.
Linear and Quadratic Change: A Problem from Japan
Authors: Peterson Blake E.
Published: January 2021

n the fall of 2003, I had the opportunity to conduct some research on the student teaching process in Japan. During my seven weeks of research at the junior high school affiliated with Ehime University in Matsuyama, Japan, I observed mathematics lessons taught by student teachers as well as many more lessons taught by experienced teachers. The basis for most of these lessons was wonderfully rich mathematics problems. In these lessons a problem was posed to students, time was given for them to explore it, and then a discussion of the solutions to the problem took place. A detailed description of similar problem-based lessons can be found in The Teaching Gap (Stigler and Hiebert 1999) and The Open-Ended Approach: A New Proposal for Teaching Mathematics (Becker and Shimada 1997).
Country Data Project
Authors: Kranendonk Henry
Published: January 2021

The article describes a six week program developed for summer school class of 8th grade students with low achievements. The program targeted deficiencies in interpreting information from graphs and tables, working with percents and constructing responses to open-ended tasks. Activities were designed around data obtained from the United States Census Bureau. Analyzing, interpreting and comparing population data from 16 countries students develop understanding of targeted topics and connections to an understanding of selected countries of the world.
Building a Career Mathematics File: Challenging Students to Find the Importance of Mathematics in a Variety of Occupations
Authors: Keleher Lori A.
Published: January 2021

The article describes a system of occupational problems designed to help students discover connections between mathematics and life environment. It provides recommendations on what, how, and why regarding to building of own classroom career mathematics file. Sample problems and assessment rubrics are offered and discussed. Student's activities could be span through the academic year.
From the 1980s: What Should Not Be in the Algebra and Geometry Curricula of Average College-Bound Students?
Authors: Usiskin Zalman
Published: January 2021

In this September 1980 article the author argues that curriculum is overcrowded and some topics should be deleted or replaced by more modern ones. He proposes for deletion several topics from algebra and geometry based on pragmatic criteria of whether the topic has real life application clearly understood by student and whether it is used in later courses. The author suggests removing word problems from algebra and some proof from geometry asserting among other that “A computer program is much like a proof”.
Happy Integers
Authors: Duncan Donald C.
Published: January 2021

The favorite article of a past NCTM president chosen for reprise as part of the celebration of the journal's 100th volume year. This article is about cheery sequences and happy numbers, with an opportunity to discuss wording of conjectures and to explore proofs. Includes a computer program in BASIC.
The Internet: Problem Solving Friend or Foe?
Authors: Wanko Jeffrey J.
Published: January 2021

Teaching problem solving to today's students requires teachers to be aware of the ways their students may use the internet as both a resource and as a tool for solving problems. In this article, I describe some of my own experiences in teaching problem solving to preservice teachers and how the existence of the internet has affected the ways in which I design and pose problems to my students.
Shifting from Traditional to Nontraditional Teaching Practices Using Multiple Representations
Authors: Rider Robin
Published: January 2021

Algebraic reasoning skills can be enhanced by building connections among symbolic, tabular, and graphical representations. But how do classroom teachers change traditional teaching practices to foster the use of different representational forms? This article explores how I learned to incorporate representational fluency in teaching and assessment. It builds on ideas from classroom practice and advocates making small changes in teaching practices to foster connections between representations.
Activities for Students: Monod's Nightmare Problem
Authors: Collins R. Lee
Published: January 2021

Through problems that arise in contexts outside of mathematics, students are provided with rich opportunities to hypothesize, investigate, and analyze real-life phenomena. The particular connection in this activity is from the biological sciences to mathematics. Students will consider what would happen if Escherichia coli (E. coli) bacteria were allowed to grow unchecked in the absence of death. Starting with one E. coli cell, students will calculate the time, in hours, it would take to fill a room with the bacteria.
Investigating the Relationship between Latitude and Temperature
Authors: McGivney-Burelle Jean , McGivney Raymond J. , McGivney Katherine G.
Published: January 2021

This article describes an engaging datagathering activity that involves exploring relationships between latitude and average monthly temperatures of cities in the Western Hemisphere. Our extensive work with middle school and high school students has confirmed this activity's attractive features: It is elastic (i.e., it can be adapted for use in either algebra 1 or algebra 2); it is interdisciplinary and can be taught through team teaching; and it requires wise and effective use of technology.
Vertex-Edge Graphs: An Essential Topic in High School Geometry
Authors: Hart Eric W.
Published: January 2021

Secondary school geometry is perhaps most succinctly described as the study of shape. Many aspects of shape are studied, such as properties of and relationships among shapes, location of shapes, transformations of shapes, and reasoning about shape. Consider an important counterpoint to this shape story or perhaps chapter zero in the story—the study of vertex-edge graphs, which are geometric objects for which shape is not an essential characteristic.
Lesson Study with Preservice Teachers
Authors: McMahon Mary T. , Hines Ellen
Published: January 2021

The value of collaboration and reflection with peers to improving instructional practices is well known (e.g., Lieberman 1992; Little 1982; Little and McLaughlin 1993; Romberg 1988). However, practicing mathematics teachers are often challenged to find time in their busy schedules to collaborate with peers. Recently, during the implementation of a lesson study experience with a small group of preservice secondary mathematics teachers, we observed firsthand how lesson study could be used to encourage collaborative reflection among preservice teaching peers and how it potentially could be used to support ongoing professional development of in-service teachers while respecting their time constraints.
Fractals in the Classroom
Authors: Fraboni Michael , Moller Trisha
Published: January 2021

What exactly is a fractal? Traditionally, students learn about the familiar forms of symmetry: reflection, translation, and rotation. Intuitively, fractals are symmetric with respect to magnification. A magnification of a small part of the fractal looks essentially the same as the entire picture. More formally, fractals have the property of self-similarity—that is, a fractal is any shape that is made up of smaller copies of itself. Self-similarity is what distinguishes fractals from most conventional Euclidean figures and makes them appealing. Do fractals hold the same characteristics as other Euclidean objects? Fractals offer much to explore for even very young students.
Can You Fathom This? Connecting Data Analysis, Algebra, and Geometry with Probability Simulation
Authors: Edwards Michael Todd , Phelps Steve
Published: January 2021

Without question, students live in an increasingly data-driven world. Data analysis plays a prominent role in various facets of modern life: Schools evaluate and revise programs on the basis of test scores; policymakers make decisions on the basis of information gleaned from polling data; supermarkets stock shelves on the basis of data collected at checkout lanes. Not surprisingly, this increased reliance on data has significantly influenced the teaching and learning of school mathematics. The study of data analysis and probability—a rarity in the secondary school curriculum twenty years ago (Boland and Nicholson 1996)—is now commonplace.
A Pretty Good Fit
Authors: Erickson Tim
Published: January 2021

Data analysis is not just for statistics class. Students can look at real data that arise from whatever they are studying. This is certainly true for students studying functions. Starting in algebra courses, data analysis can show students contexts in which various functions make sense, and students can use functions to model real phenomena. Although students can do this by hand, using functions to model real phenomena is practical on a regular basis only when computers and high-powered handheld technological tools take over repetitive calculations and graphing.
Connecting Research to Teaching: Mathematics: A Second Language
Authors: Jones Bradley r. , hopper peggy F. , Franz dana pomykal
Published: January 2021

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Illuminating the Mathematics of Lamp Shades
Authors: Matthews Michael E. , Gross Greg
Published: January 2021

Creating lamp shades to specific design parameters allows rich explorations in the mathematics of circles and triangles.
Developing Reasoning through Proof in High School Calculus
Authors: Perrin John Robert
Published: January 2021

Students develop reasoning skills by constructing mathematical proofs of some nontypical calculus results.
Area by Dissection
Authors: Mihaila Ioana , Barger Ellen
Published: January 2021

Building sets of problems rooted in the same concept allows students to develop understanding and take on increasingly complex problems.
Mathematical Lens: Tent Mathematics
Authors: Turton Roger W.
Published: January 2021

Most people go camping to escape the responsibilities of their professional lives. However, for a mathematician, even something as recreational as a tent contains some interesting reminders of mathematical functions. Photograph 1 shows an interior view of one of the zippered flaps of a tent used by the author and his wife for camping.
Thoughts on Why (−1)(−1) = +1
Authors: Rapke Tina
Published: January 2021

Do two wrongs make a right? Using the distributive property avoids pseudoreasoning.
Standards for High School Mathematics: Why, What, How?
Authors: Hart Eric W. , Martin W. Gary
Published: January 2021

Compare your own experience of learning algebra with Bertrand Russell's recollection: I was made to learn by heart: “The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.” I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.”
Examining Cylindrical Dice
Authors: Jones Dustin L.
Published: January 2021

Prospective mathematics teachers try to determine the dimensions of a cylindrical die that ensure that the probability of landing on the lateral surface is equal to that of landing on an end.
Mathematics Skills and NAEP Results over a Generation
Authors: Rutledge Zachary , Kloosterman Peter , Kenney Patricia Ann
Published: January 2021

Analysis of U.S. students' performance on the National Assessment of Educational Progress's Long-Term Trend program focuses on some questions on which student performance changed significantly between 1982 and 2004.
Technology Tips: TI-Nspire Calculators: Better but Still Not Perfect
Authors: Bossé Michael J. , Adu-Gyamfi Kwaku
Published: January 2021

Texas Instruments calculators are ubiquitous in today's K–16 mathematics classrooms, so mathematics instructors must understand the necessary programming limitations of each calculator. Previous generations of TI graphing calculators (the TI-89 and TI-92 calculators) produced some graphical and symbolic errors (documented in BossÉ and Nandakumar [2003, 2004] and Marchand and others [2007]). Fortunately, TI has now remedied most of these errors in its new graphing calculators, the TI-Nspire and the TI-Nspire CAS. Although the TI-Nspire still incorrectly graphs some functions seen only in graduate mathematical studies, it now renders correctly almost all functions that mathematics students encounter through precalculus. Overcoming these programming difficulties was no small matter, and the corrected results are a testament to the more sophisticated programming in these new TI-Nspire calculators. In this article, we will look at some functions that previously did not graph correctly on the TI-89 and TI-92 because of programming constraints, some functions that still do not graph correctly on the TI-Nspire, and improvements in how the TI-Nspire CAS handles the rewriting of symbolic expressions.
Rare and Exotic Probability Bugs
Authors: Kennis James R.
Published: January 2021

In studying probability, students often arrive at the correct answer through incorrect reasoning.
Technology Tips: Investigating Bricard's Proof of Morley's Theorem with The Geometer's Sketchpad
Authors: Althoen Steven C. , Brandell Joseph L.
Published: January 2021

The Geometer's Sketchpad®(GSP), the dynamic geometry software in use in mathematics classrooms for many years, provides a means of investigating classic mathematical theorems. In this article, we will describe our use of GSP to examine Bricard's (1922) proof of Morley's theorem as presented by Coxeter (1969).
Mathematical Lens: Building Mathematically
Authors: Li Aihua , Munakata Mika
Published: January 2021

In January 2008, seventeen participants in the Montclair State University (MSU) GK–12 Fellows in the Middle Program visited China for two weeks. Our group included two mathematics graduate students, four science graduate students, two middle school mathematics teachers, one middle school science teacher, one superintendent, and six MSU mathematics and science faculty members. While in China, we visited several middle and high schools Munakatain Beijing and Xi'an and saw many historical and cultural sites. On our way to the Forbidden City along the Beijing highway known as Ring 3, we passed these three buildings, located at Xihuan Plaza (photograph 1), which seemed to be challenging passersby to describe them mathematically.
Three Ways to Break a Stick
Authors: Bannon Thomas J.
Published: January 2021

The classic problem of finding the probability that the pieces of a stick randomly broken in two places will form a triangle is analyzed anew.
Mathematics: The Universal Language?
Authors: Hoffert Sharon B.
Published: January 2021

A teacher enumerates the challenges, strategies, and rewards of teaching mathematics to English Language Learners.
Connecting Research to Teaching: Visual and Analytic Thinking in Calculus
Authors: Haciomeroglu Erhan Selcuk , Aspinwall Leslie , Presmeg Norma C.
Published: January 2021

A frequent message in mathematics education focuses on the benefits of multiple representations of mathematical concepts (Aspinwall and Shaw 2002). The National Council of Teachers of Mathematics, for instance, claims that “different representations support different ways of thinking about and manipulating mathematical objects” (NCTM 2000, p. 360). A recommendation conveyed in the ongoing calculus reform movement is that students should use multiple representations and make connections among them so that they can develop deeper and more robust understanding of the concepts.
Delving Deeper: Converging on the Eye of God
Authors: Seppala-Holtzman David N. , Rangel Francisco R.
Published: January 2021

The eye of God—the point at which the golden spiral appears to vanish—has long captivated the imagination of mathematicians. While preparing a paper on the Fibonacci numbers and the golden ratio, Francisco Rangel, a student of David Seppala-Holtzman's, decided to engage in some numeric experimentation. He came up with some surprising results that ultimately led to a conjecture about the eye of God. Our purpose here is to describe what he found, prove that his suspicions were indeed correct, and give a geometric interpretation of these results.
Activities for Students: Inquiry into Fractals
Authors: Siegrist Raymond
Published: January 2021

The exotic images of fractals often pique the interest of high school mathematics students, and this interest presents an opportunity for geometry teachers to draw students into an investigation of transformations and patterns. By using a simple building block and fractals' self-imaging characteristic (as the figure grows, it retains the pattern established by the building block), teachers can bring construction of fractals into the high school geometry curriculum. The three activities described in this article engage students in constructing a fractal, searching a fractal for patterns, and using transformations to build different fractals. Students gain insight into patterns as their fractals grow; they flip or rotate fractal pieces by following a design and translating the pieces into place.
Technology Tips: Creating a Mathematical “B” Movie: The Effect of b on the Graph of a Quadratic
Authors: Edwards Thomas G. , Özgün-Koca Asli
Published: January 2021

The suggestion to engage students in studying the effects on the graph of the quadratic function y = ax2 + bx + c that result from varying the parameters a, b, and c is certainly not a new idea. In this article, we first review a few of the specific suggestions that have occurred over the past fifty years. Then we propose an activity of our own that makes use of the new generation of handheld technologies.
Japanese Logic Puzzles and Proof
Authors: Wanko Jeffrey J.
Published: January 2021

Hashiwokakero, also known as Bridges, is one of several types of Japanese logic puzzles that help students establish proof–readiness skills.
Learning Deductive Reasoning through Games of Logic
Authors: Janelle McFeetors P. , Mason Ralph T.
Published: January 2021

By exploring, analyzing, and convincing, students can develop deductive reasoning strategies during preproof experiences.
Connecting Research to Teaching: Count on Number Theory to Inspire Proof
Authors: Larson Quinn Anne
Published: January 2021

Many students find proofs frustrating, and teachers struggle with how to help students write proofs. In fact, it is well documented that most students who have studied proofs in high school geometry courses do not master them and do not understand their function (Battista 2007; Harel and Sowder 2007). And yet, according to NCTM's Principles and Standards for School Mathematics(2000), “By the end of secondary school, students should be able to understand and produce mathematical proofs … and should appreciate the value of such arguments” (p. 56).
Delving Deeper: One Cut, Two Halves, Three Questions
Authors: Ren Guanshen
Published: January 2021

The popular radio show Car Talk, hosted by Click and Clack, the Tappet brothers, features a weekly puzzler. One week the puzzler asked how to divide a rectangular brownie into two equal halves with one cut. Audience members familiar with basic geometry knew that a cut along a diagonal of a rectangle will divide the rectangle into two equal parts. Click and Clack suggested that any cut through the center of the rectangle, the intersection point of two diagonals, would also work. Why?
Algebra, Home Mortgages, and Recessions
Authors: Mariner Jean A. Miller , Miller Richard A.
Published: January 2021

The 2008–2009 U.S. economic crisis stimulated this exploration of the algebra of mortgages, the bubble in house prices, and how to reduce foreclosure rates.
Investigating the Randomness of Numbers
Authors: Pendleton Kenn
Published: January 2021

If one claims that a sample has been randomly selected from a population, can the merits of the claim be assessed using statistical tests? Are tests alone sufficient?
Technology Tips: Simulation with the TI-Nspire
Authors: Rudolph H eidi J.
Published: January 2021

Simulation is an important learning tool that allows students to grasp probability concepts, especially when the actual scenario does not need to be replicated entirely. For instance, it may be too expensive or impractical to set up an actual experiment. In the cases of tossing coins and rolling dice, gathering the data before analyzing them can be laborious and might be a waste of precious class time—time that might be better spent discussing different possible outcomes or comparing theoretical with experimental results. The TI-Nspire can generate hundreds of data values, through the randInt function, in a spreadsheet for graphing and analysis. The instructions here were based on TI-Nspire's OS 1.7 version (the June 2009 update); commands and screens may differ for calculators with earlier or later operating systems.
Sound Off!: Can We Make High School More Relevant?
Authors: Kranendonk Henry
Published: January 2021

In “Standards for High School Mathematics: Why, What, How?” in the December 2008/January 2009 issue of Mathematics Teacher, Eric W. Hart and W. Gary Martin summarized a perplexing problem in the U.S. education system—namely, the challenge of reforming high school mathematics. The article referenced a 2007 conference sponsored by the Center for the Study of Mathematics Curriculum that addressed two important questions concerning mathematics achievement: What should students learn, and when should they learn it? Presenters from Achieve, Inc., the American Statistical Association, the College Board, the Mathematics Association of America (MAA), and NCTM openly discussed and debated the topic questions. The conference provided an excellent format for dialogue about the nuts and bolts of what students should be taught and when these topics should be taught in the K–12 timeline. Several participants acknowledged, however, that additional factors complicate our challenges with high school students.
Common Errors in Counting Problems
Authors: Annin Scott A. , Lai Kevin S.
Published: January 2021

Analyzing combinatorics problems involving flags and playing cards presents common pitfalls for students—pitfalls that can be avoided.
What Mathematics Do High School Teachers Need to Know?
Authors: Gilbert Michael J. , Coomes Jacqueline
Published: January 2021

The MC3 project defines, describes, and characterizes the mathematics knowledge needed for teaching high school mathematics.
Technology Tips: Scrambling Data with Fathom to Simulate the Hypothesis
Authors: Shafer Kathryn
Published: January 2021

Most students fare quite well with descriptive statistics, but hypothesis testing is often an unfamiliar and frustrating topic for many. As a student, I can recall having to memorize something about alpha and a P-value, but at the time it was just a rule to follow. As an instructor, I now realize that I lacked the conceptual knowledge necessary to understand the process I had memorized. Over the past few years, I have taught statistics with Fathom? Dynamic Data software, version 2 (Key Curriculum Press 2006), and today I cannot envision teaching hypothesis testing without this technological tool.
Celebrating Mathematics All Year 'Round
Authors: Daire Sandra Argüelles
Published: January 2021

Pi Day and Pi Week festivities–a good place to start–can be a springboard to mathematics celebrations throughout the year.
Connected Representations: From Proportion to Linear Functions
Authors: Baltus Christopher
Published: January 2021

Building on mutually reinforcing representations, students can trace a path from proportion to linear functions.
Proof and Prealgebra
Authors: Barnes Rachael , Hamon Suzanne
Published: January 2021

The prealgebra curriculum presents many opportunities for encouraging students to justify their inferences. Requiring students to communicate clearly the reasoning behind their solutions, with appropriate mathematical language and notation, helps lay the groundwork for future, proof-based mathematics courses. Prealgebra students can even be introduced to formal proof in several contexts. Once they have a basic concept of what it means to “prove”something, conversations naturally arise about what steps must be included in a rigorous proof and the difference between conjecture and proof.
Technology Tips: The TI-Nspire and Lissajous Figures
Authors: Hanna William , Davis Michael S.
Published: January 2021

Lissajous figures are not typically included in most secondary school mathematics curricula, but they provide a fertile ground for students to explore real-world mathematics and make connections with some of the mathematics they have previously learned. When studied with the help of the TI-Nspire, Lissajous figures present opportunities for students to make conjectures and share mathematical thinking.
Doubling Time for Nonexponential Families of Functions
Authors: Gordon Sheldon P.
Published: January 2021

The notions of doubling time and half-life are investigated for families of functions other than exponential, including linear, power, logarithmic, trigonometric, and logistic.
Using Portfolio Assignments to Assess Students' Mathematical Thinking
Authors: Fukawa-Connelly Timothy , Buck Stephen
Published: January 2021

A model in use at Prospect Hill Academy in Cambridge, Massachusetts, provides a skills-based portfolio assessment for mathematics classes.
Technology Tips: Why Are Shot Puts Thrown at 31 degrees? Using Autograph for Applications of the Parabola
Authors: Butler Douglas
Published: January 2021

Autograph is a two- and threedimensional dynamic statistics and graphing utility, developed in England, that has grown out of direct classroom experience. A simple select– and-right-click interface, together with tools such as Autograph's unique Slow Plot, Scribble Tool, and dynamic Constant Controller help make the classroom experience interactive.
The Back Page: My Favorite Lesson: Transforming Functions
Authors: Daire Sandra Argüelles
Published: January 2021

The day I held a graphing calculator for the first time, I decided to use it as a tool to teach transformations of functions. After years of revisions and many calculators later, this lesson is still one of my favorites. Whether I am teaching algebra, trigonometry, or calculus, this topic is universal and as necessary to students struggling with high school mathematics as to more advanced students.
Lesson Study: Still a Work in Progress in America
Authors: Tolle Penelope P.
Published: January 2021

The postlesson discussion in Japanese lesson study—arguably the most important component—is underused in American teaching.
Apportionment in the Democratic Primary Process
Authors: Geist Kristi , Jones Michael A. , Wilson Jennifer
Published: January 2021

How the delegates for the 2008 New Jersey Democratic presidential primary were divided may have influenced the number of delegates the candidates received—and possibly the primary's outcome.
Delving Deeper: Spicing Up Counting through Geometry
Authors: Rivera F. D.
Published: January 2021

An analysis of the combinatorics problems in many algebra textbooks for high school students reveals that time-honored classic problems are valued. These problems often involve finding combinations and arrangements of numbers and letters on license plates for fictitious states (with and without repetition); digits in an n-digit number that is either even or odd; or males and females who are either combined to form a committee of a certain size with known restrictions or arranged to sit or stand together according to some specified configuration. Examples of such problems are shown in figure 1.
The Back Page: My Favorite Lesson: Angling for Enthusiasm Hooks Students
Authors: Reisch Christopher P.
Published: January 2021

Western New York State is known for many things, most notably snowfall. Less well known are its resources for year-round world-class fishing. In the Great Lakes tributaries that vein the landscape, warm-water species can be caught throughout the summer months. During the fall, winter, and spring months, these tributaries support large lake-runs of trout and salmon. If you live in western New York, as my students and I do, it is likely that you have angled.
Delving Deeper: Visualizing Pythagorean Triples and Beyond
Authors: Booze David A.
Published: January 2021

Exploring the algebraic properties that form the framework for analytical generation of Pythagorean triples and quadruples is a challenging topic for high school algebra students. I have had considerable success in motivating my students to explore these properties by using a visual method. Students can use this visual method to find Pythagorean triples and quadruples easily and can capably and colorfully supply as many integer solutions as they desire to these well-known equations. Here I will present a visual method for producing integer solutions to the equations a2 + b2 = c2 and a2 + b2 + c2 = d2 and then develop an algebraic representation of this method.