Physical Sciences and Mathematics Commons^™

Reasoning-And-Proving Within Ireland’S Reform-Oriented National Syllabi, Jon Davis

The Mathematics Enthusiast

As educational systems around the world attempt to reform their mathematics programs to increase students’ opportunities to engage in processes central to the practice of mathematics such as proof, it is important to understand how this mathematical act is portrayed in national curriculum documents that drive that change. This study examined the presence of reasoning-and-proving (RP) in Ireland’s national reform-oriented secondary syllabi for junior cycle (ages 12-15) and senior cycle (ages 15-18) students. The analyses reveal that there were no differences among direct and indirect RP learning outcomes within each syllabus, but statistically significant differences did exist across syllabi in …

Generalizing Cantor-Schroeder-Bernstein: Counterexamples In Standard Settings, Tien Chih

The Mathematics Enthusiast

The Cantor-Schroeder-Bernstein theorem states that any two sets that have injections into each other have the same cardinality, i.e. there is a bijection between them. Another way to phrase this is if two sets A;B have monomorphisms from A to B and B to A, then they are isomorphic in the setting of sets. One naturally wonders if this may be extended to other commonly studied systems of sets with structure and functions which preserve that structure. Given two objects with injective structure preserving maps between them are the structures of these objects the same? In other words, would these …

Editorial: Is Every Tme Issue Special?, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Bhaskara’S Approximation For The Sine, Karel Stroethoff

The Mathematics Enthusiast

The 7th century Indian mathematician Bhaskara (c.600 – c.680) obtained a remarkable approximation for the sine function. Many subsequent ancient authors have given versions of this rule, but none provided a proof or described how the result was obtained. Grover [1] provides a possible explanation, but I think the rule can be explained more clearly. Rather than give the rule first, we will derive it, and then discuss its accuracy, and explore some alternative approximations. Our derivation is simply an exercise in modeling.

Pursuing Coherence Among Proportionality, Linearity, And Similarity: Two Pathways From Preservice Teachers’ Geometric Representations, Hyung Sook Lee, Jaehoon Yim

The Mathematics Enthusiast

The importance of using multiple representations of a mathematical concept and connecting the representations has been discussed in learning and teaching mathematics. The Common Core State Standards further the discussion with an emphasis on focus and coherence in teaching mathematical concepts across grades. Preservice teachers in our problem solving class were asked to use geometric representations to solve a problem that required proportional reasoning. They were also asked to sequence the works of their peers as well as their own from a developmental perspective. Sequencing geometric representations with various levels was challenging because it required showing a coherent understanding of …

The Development Of Calculus In The Kerala School, Phoebe Webb

The Mathematics Enthusiast

The Kerala School of mathematics, founded by Madhava in Southern India, produced many great works in the area of trigonometry during the fifteenth through eighteenth centuries. This paper focuses on Madhava's derivation of the power series for sine and cosine, as well as a series similar to the well-known Taylor Series. The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the basics of calculus long before it was developed in Europe. Other evidence from Indian mathematics up to this point such as interest in …

Development Of The Binary Number System And The Foundations Of Computer Science, Daniel R. Lande

The Mathematics Enthusiast

This paper discusses the formalization of the binary number system and the groundwork that was laid for the future of digital circuitry, computers, and the field of computer science. The goal of this paper is to show how Gottfried Leibniz formalized the binary number system and solidified his thoughts through an analysis of the Chinese I Ching. In addition, Leib-niz’s work in logic and with computing machines is presented. This work laid the foundation for Boolean algebra and digital circuitry which was continued by George Boole, Augustus De Mor-gan, and Claude Shannon in the centuries following. Some have coined Leibniz …

Numerically Integrating Irregularly-Spaced (X, Y) Data, B. Cameron Reed

The Mathematics Enthusiast

This article describes a computer program for numerically integrating under a y(x) curve of experimental data where the abscissa values are not equally-spaced. This technique is based on fitting parabolas to successive groups of three data points, and may be regarded as a generalization of Simpson’s rule.

Difficulties In Solving Context-Based Pisa Mathematics Tasks: An Analysis Of Students’ Errors, Ariyadi Wijaya, Marja Van Den Heuvel-Panhuizen, Michiel Doorman, Alexander Robitzsch

The Mathematics Enthusiast

The intention of this study was to clarify students’ difficulties in solving context-based mathematics tasks as used in the Programme for International Student Assessment (PISA). The study was carried out with 362 Indonesian ninth- and tenth-grade students. In the study we used 34 released PISA mathematics tasks including three task types: reproduction, connection, and reflection. Students’ difficulties were identified by using Newman’s error categories, which were connected to the modeling process described by Blum and Leiss and to the PISA stages of mathematization, including (1) comprehending a task, (2) transforming the task into a mathematical problem, (3) processing mathematical procedures, …

Math As A Tool Of Anti-Semitism, Jay Egenhoff

The Mathematics Enthusiast

At Moscow State University’s Department of Mathematics during the 1970’s and 1980’s, there was rampant discrimination against Jewish and other unwanted students. The professors at the math department made a strong effort to keep Jewish students out of the department. They designed "killer" or "coffin" problems and Jewish students had to answer them during an oral exam. These problems have simple solutions, but require a clever strategy to solve them. This paper explores some of the context of this episode and provides several problems with detailed solutions.

The Miracle Of Applied Mathematics, Frank Blume

The Mathematics Enthusiast

We provide an outline of a college-level lecture on systems of differential equations that is intended to offer an alternative, integrative view of the subject. Particular emphasis is placed on the fact that very elementary concepts and insights can be employed to gain access to a very wide range of important applications, including the Schr¨odinger equation, the Klein-Gordon equation, the Laplace equation, and the Poisson equation.

A Study On Malaysian Mathematicians’ Way Of Knowing, Lim Chap Sam

The Mathematics Enthusiast

“Can you name me a mathematician?” “Einstein??”

“Do you wish to become a mathematician one day?”

If these questions were asked to the public or any school students, the most likely answer to both questions might be a big ‘No!’. Why is this so?

The director of the Public Understanding of Mathematics Forum, Gene Kloz (1996) claims that the mathematics profession is the most misunderstood in all of academia. According to him, the public thinks that mathematicians contemplate ancient proofs and work as lonely recluses. Moreover, the most common public image of a mathematician has been furnished by a physicist …

An Examination Of Pre-Service Secondary Mathematics Teachers’ Conceptions Of Angles, Melike Yigit

The Mathematics Enthusiast

The concept of angles is one of the foundational concepts to develop of geometric knowledge, but it remains a difficult concept for students and teachers to grasp. Exiting studies claimed that students’ difficulties in learning of the concept of angles are based on learning of the multiple definitions of an angle, describing angles measuring the size of angles, and conceiving different types of angles such as 0-line angles, 1- line angles, and 2-line angles. This study was designed to gain better insight into pre-service secondary mathematics teachers’ (PSMTs) mental constructions of the concept of angles from the perspective of Action-Process-Object-Schema …

Common Sense About The Common Core, Alan H. Schoenfeld

The Mathematics Enthusiast

Is the Common Core the best thing since sliced bread, or the work of the devil? Is it brand new, or a rehash of old ideas? Is it anything more than a brand name, or is there substance? Can it work, given the implementation challenges in our political and school systems? Opinions about the Common Core are everywhere, but the op-eds I’ve seen are often short on facts, and equally short on common sense. A mathematician by training, I’ve worked for nearly 40 years as an education researcher, curriculum materials developer, test developer, standards writer, and teacher. What follows is …

Book Review Of The Tower Of Hanoi: Myths And Maths (Birkhäuser), Cory Palmer

The Mathematics Enthusiast

As the title of the book suggests, the central topic of “The Tower of Hanoi – Myths and Maths” by Hinz, Klavˇzar, Milutinovi´c, and Petr is the famous puzzle of the same name. The classic Tower of Hanoi puzzle along with a number of its variations and related puzzles are examined in a rigorous mathematical framework.

Scholastic Standards In The United States – The Discussion Concerning The ‘Common Core’, Alan H. Schoenfeld, Günter Törner

The Mathematics Enthusiast

Preface: This article has been developed based on a personal discussion between the German author Günter Törner and Alan Schoenfeld, who is an expert in the field of mathematical didactics. Basically there are three reasons for us to share our insights with the public:

(1) Readers, having subscribed to Jerry Becker’s e-mail information network, have received numerous messages over the past few months; what do we need to know about this fact in Germany?

(2) Scholastic standards – a keyword that sounds very familiar to us in terms of educational policy… But it is also a hot topic in other …

Tme Volume 11, Number 3

The Mathematics Enthusiast

No abstract provided.

Mathematical Content Knowledge For Teaching Elementary Mathematics: A Focus On Fractions, Dana Olanoff, Jane-Jane Lo, Jennifer Tobias

The Mathematics Enthusiast

This article presents a research summary of prospective elementary teachers’ (PTs’) mathematical content knowledge in the area of fractions. The authors conducted an extensive review of the research literature and present the findings across three time frames: a historical look (pre-­‐1998), a current perspective (1998–2011), and a look at the horizon (2011–2013). We discuss 43 articles written across these time frames that focus on PTs’ fraction knowledge. Consistent across these papers is that PTs’ fraction knowledge is relatively strong when it comes to performing procedures, but that they generally lack flexibility in moving away from procedures and using “fraction number …

Tme Volume 11, Number 2

The Mathematics Enthusiast

No abstract provided.

Foreword, Lynn Hart

The Mathematics Enthusiast

The authors in this Special Issue of The Mathematics Enthusiast make an important contribution to the knowledge base in mathematics education. They examine a body of research on a significant issue. They review what we know and make suggestions about what we need to know. They move the field forward by taking the time to look back and learn.

Mathematical Content Knowledge For Teaching Elementary Mathematics: A Focus On Whole-Number Concepts And Operations, Eva Thanheiser, Ian Whitacre, George Roy

The Mathematics Enthusiast

This report represents part of a recent effort to summarize the state of knowledge of prospective elementary teachers’ (PTs’) mathematics content knowledge and the development thereof. Extensive reviews of the research literature were conducted by a recent PME-­‐NA Working Group across various content areas. This report focuses on whole number and operations. Research in this area is scarce. What we do know from the literature is that PTs’ knowledge of whole number and operations is insufficient and in need of improvement. PTs reason about whole numbers and operations in ways that are tied to the standard algorithms. At the same …

Preface, Eva Thanheiser, Christine Browning

The Mathematics Enthusiast

Bharath Sriraman noted in his Editorial for Vol. 10, nos. 1–2 that the first issue of The Mathematics Enthusiast (known then as The Montana Mathematics Enthusiast) published in April 2004 was “the result of four idealistic elementary school teachers believing in the mission of this journal and writing about their attempts to reconcile the mathematics content they were learning in a mathematics for elementary school teachers course with existing mathematics education research found in practitioners’ journals as well as standards imposed by institutions’ framing policy” (p. 2). Ten years later we return to a similar focus.

Prospective Elementary Teacher Mathematics Content Knowledge: An Introduction, Christine Browning, Eva Thanheiser, Alden J. Edson, Patrick Kimani, Dana Olanoff, Jennifer Tobias, Ian Whitacre

The Mathematics Enthusiast

This Special Issue on the mathematical content knowledge of prospective elementary teachers (PTs) provides summaries of the extant peer-­‐reviewed research literature from 1978 to 2012 on PTs’ content knowledge across several mathematical topics, specifically whole number and operations, fractions, decimals, geometry and measurement, and algebra. Each topic-­‐specific summary of the literature is presented in a self-­‐contained paper, written by a subgroup of a larger Working Group that has collaborated across several years, resulting in this Special Issue sharing the final work. The authors hope this summative look at prospective teacher content knowledge will be of interest to the mathematics education …

Mathematical Content Knowledge For Teaching Elementary Mathematics: A Focus On Decimals, Signe Kastberg, Crystal Morton

The Mathematics Enthusiast

In the last 25 years a small collection of reports of studies focused on gaining insight into PTs’ knowledge of decimals has been published. Three themes are used to frame findings from papers published prior to 1998. Additional findings from papers published between 1998 and 2011 are discussed. Direction for future research that can contribute to the development of curriculum and instruction in mathematics teacher education is shared.

Mathematical Content Knowledge For Teaching Elementary Mathematics: A Focus On Geometry And Measurement, Christine Browning, Alden J. Edson, Patrick Kimani, Fatma Aslan-Tutak

The Mathematics Enthusiast

This paper summarizes the extant peer-­‐reviewed research on PTs’ understanding of geometry and measurement, focusing on a wide variety of topics within these content domains. When looking across the 26 studies reviewed, findings span a variety of content topics, providing little depth in either the geometry or measurement content domain. However, collective findings do indicate PTs’ overall conceptions in geometry and measurement to be limited and weak, with PTs relying on memorized procedural processes. Some evidence indicates that cognitive development, along with spatial visualization skills, plays a greater role in learning geometry than memory skills. In addition, the van Hiele …

Mathematical Content Knowledge For Teaching Elementary Mathematics: A Focus On Algebra, Krista Strand, Briana Mills

The Mathematics Enthusiast

As part of a recent effort to summarize research-­‐based knowledge of prospective elementary school teachers’ (PTs) mathematical content knowledge, this paper summarizes research literature on PTs’ knowledge of algebra, focusing on the range of years from 1998 through 2012. The 21 papers included in this summary focus on a broad range of topics within algebra, such as (a) producing, representing, and justifying generalizations; (b) interpreting and using algebraic symbols; (c) solving algebraic word problems; and (d) understanding functions. Looking across this body of research, three themes are identified: (1) PTs generally have strong procedural skills and can make mathematically sound …

Prospective Elementary Mathematics Teacher Content Knowledge: What Do We Know, What Do We Not Know, And Where Do We Go?, Eva Thanheiser, Christine Browning, Alden J. Edson, Jane-Jane Lo, Ian Whitacre, Dana Olanoff, Crystal Morton

The Mathematics Enthusiast

In this Special Issue, the authors reviewed 112 research studies from 1978 to 2012 on prospective elementary teachers’ content knowledge in five content areas: whole numbers and operations, fractions, decimals, geometry and measurement, and algebra. Looking across these studies, this final paper identifies the trends and common themes in terms of the counts and types of studies and commonalities among findings. Analyses of the counts show that the number of articles published each year focusing on prospective teacher (PT) content knowledge is increasing. Most articles across the content areas show that PTs tend to rely on procedures rather than concepts. …

Appendix: References By Content Area

The Mathematics Enthusiast

No abstract provided.

Approaching Professional Learning: What Teachers Want, Peter Liljedahl

The Mathematics Enthusiast

Teachers do not come to professional learning opportunities as blank slates. Instead, they come to these settings with a complex collection of wants and needs. The research presented here takes a closer look at these wants across five different professional learning settings distilling form the data a taxonomy of five categories of wants that teachers may approach professional learning with. The resultant taxonomy, as well as teachers behaviours vis-à-vis this taxonomy indicate that we need to rethink our role as facilitators within these settings as well as the role that single workshops can play in the professional learning of teachers.

The Teachers Institute Approach To Professional Development, Roger Howe

The Mathematics Enthusiast

The Yale New Haven Teachers Institute (YNHTI) provides a distinctive, perhaps nearly unique, approach to professional development. It originated in the 1978 as an outreach activity of Yale University to the New Haven Public Schools. For 20 years, it operated almost exclusively in New Haven. In 1998, under the leadership of its founder, James Vivian, YNHTI conducted a National Demonstration Project, and since 2004 has promoted a National Initiative, to spread the Teachers Institute model to other cities, with a focus on school districts with low income demographics. Currently there is a League of Teachers Institutes with Institutes operating in …