Mathematics Commons^™

Early Intervention In College Mathematics Courses: A Component Of The Stem Rrg Program Funded By The Us Department Of Education, Rohitha Goonatilake, Eduardo Chappa

The Mathematics Enthusiast

Student preparedness in entry level college mathematics courses has decreased in the past. In order for students to be successful in these courses, early intervention is required. Texas A&M International University (TAMIU) has implemented an early intervention program, the Mathematics Enrichment Program (MEP), which will target freshman students entering College Algebra courses in Spring 2009. This project is part of the STEM Recruitment, Retention, and Graduation program, recently funded by the US Department of Education. This paper will highlight the components of the project, most importantly the extent of the intervention, how the project is planned, and preliminary results.

New Year Tidings, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Leading Learning Within A Plc: Implementing New Mathematics Content, Ann Heirdsfield, Janeen Lamb, Gayle Spry

The Mathematics Enthusiast

This paper does two things. Firstly, it examines the literature that coalesces around theoretical models of teacher professional development (PD) within a professional learning community (PLC). Secondly, these models are used to analyse support provided to two year 3 teachers, while implementing the draft Queensland mathematics syllabus. The findings from this study suggest that the development of this small PLC extended the teachers’ Zone of Enactment which in turn led to teacher action and reflection. This was demonstrated by the teachers leading their own learning as well as that of their students.

Tme Volume 7, Number 1

The Mathematics Enthusiast

No abstract provided.

Mathematical Reasoning In Service Courses: Why Students Need Mathematical Modeling Problems, Kris H. Green, Allen Emerson

The Mathematics Enthusiast

In this paper we argue that conventional mathematics word problems are not aligned with the typical learning goals and expectations partner disciplines, especially business, have in requiring that their students take mathematics courses. Using the taxonomy of educational objectives presented by Anderson and Krathwohl (2001) we show how mathematical modeling problems can be used to promote the needed alignment and contrast two examples to illustrate the differences. We then demonstrate how the more conventional word problem can be rewritten as a modeling problem. Sample assessment materials and instructional activities are included to support teachers in making the transition to the …

Gender And Mathematics Education In Pakistan: A Situation Analysis, Anjum Halai

The Mathematics Enthusiast

This paper reports from a situation analysis of gender issues in mathematics education in Pakistan. It was undertaken at the initiation of a large scale project which aimed to understand how curriculum change in mathematics and science education may be implemented in ways that contribute to poverty alleviation and promote gender equity in disadvantaged rural settings. The paper posits that issues of access, achievement and quality of mathematics education are integrally linked with questions of equity, in this case gender equity. It identifies several questions and arenas for further research and makes recommendations for policy and practice in mathematics education.

High School Teachers Use Of Dynamic Software To Generate Serendipitous Mathematical Relations, Manuel Santos-Trigo, Hugo Espinosa-Pérez

The Mathematics Enthusiast

In this study, we document and analyse problem-solving approaches that high school teachers exhibited as a result of using dynamic software (Cabri-Geometry) to construct and examine geometric configurations. What type of questions do teachers pose and pursue while representing and exploring mathematical tasks or objects dynamically? To what extent their initial problem solving strategies are enhanced with the use of the tools? Results indicate that the use of the tool offered the participants the opportunity of constructing geometric configuration (formed by simple mathematical objects) that led them to identify and explore key mathematical relations.

“What Was Really Accomplished Today?” Mathematics Content Specialists Observe A Class For Prospective K–8 Teachers, Andrew M. Tyminksi, Sarah Ledford, Dennis Hembree

The Mathematics Enthusiast

One of the important activities mathematics teacher educators engage in is the development of teachers at both the in-service and pre-service levels. Also of importance is the professional development of these professional developers. In the summer of 2004, a summer institute was held that allowed mathematics teacher educators watch the teaching of a mathematics content course for prospective K–8 teachers. This paper examines the manner in which a specific group of mathematics content specialists experienced this professional development.

Randomness: Developing An Understanding Of Mathematical Order, Steve Humble

The Mathematics Enthusiast

Is randomness merely the human inability to recognise a pattern that may in fact exist?

The three activities described show how order can be found in seemly random activities. The author has found that by using these activities on randomness, his students have developed a greater understanding of mathematical pattern and sequence. The teaching mathematical concepts in this way, engages and reinforces learning. It puts the ideas learnt into a setting and allows time for those ideas to be developed without any of the maths hang-ups which can sometimes occur in the classroom. By taking the maths beyond the classroom, …

The Constructs Of Phd Students About Infinity: An Application Of Repertory Grids, Serdar Aztekin, Ahmet Arikan, Bharath Sriraman

The Mathematics Enthusiast

Infinity has been one of the more difficult concepts for humanity to grasp. A major component of the research on mathematics education related to infinity has been the study of student’s conceptions and reasoning about calculus subjects, particularly limits and series. Some related studies are about Cantor’s ordinal and cardinal infinity. However since most students at the high school and college level are unfamiliar with symbolic representations and terminology, such as a set theoretic approach, a context (generally geometric) is used for investigating notions of infinity indirectly. In this paper we report on a study on the constructs of PhD …

When Is .999... Less Than 1?, Karin Usadi Katz, Mikhail G. Katz

The Mathematics Enthusiast

We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999... be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.

The Journal (Wheel) Keeps On Turning, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Mathematical Beauty And Its Characteristics- A Study On The Student's Point Of View, Astrid Brinkmann

The Mathematics Enthusiast

Based on the statement, that the experience of mathematical beauty has a positive influence on students’ motivations and attitudes towards mathematics and its study, the focus of this paper is the aesthetic component of mathematics. First, the role of aesthetics for perception and education is addressed. The appreciation of the beauty of mathematics is one of the wellsprings of this subject, not only in research but also in school education. This should have implications for the teaching of mathematics. However the beauty making elements have not been very well analysed. In particular, it is not clear to what extent the …

From Trapezoids To The Fundamental Theorem Of Calculus, William Gratzer, Srilal Krishnan

The Mathematics Enthusiast

The philosophy of Mathematics Education undergoes changes from the school to college level and students generally have a tough time coping with the transition. It is our endeavor to impress the importance of introducing college level topics at an early stage, so that students are not lost in the transition. Keeping this in mind, we suggest an early exposure to an important topic from Calculus; approximating the area of a planar region. Traditionally this topic is introduced using Riemann Sums but in this paper we try to follow a student’s natural inclination in approximating areas and explain how this approach …

Graph Isomorphisms And Matrix Similarity: Switching Between Representations, Thierry Dana-Picard

The Mathematics Enthusiast

A proof whether two graphs (possibly oriented graphs or multigraphs, etc.) are isomorphic or not can be derived by various methods. Some of them are reasonable for small numbers of vertices and/or edges, but not for larger numbers. Switching from iconic representation to a matrix representation transforms the problem of Graph Theory into a problem in Linear Algebra. The support provided by a Computer Algebra System is analyzed, in particular with regard to the building of new mathematical knowledge through a transition from graphical to algebraic representation. Moreover two important issues are discussed: a. the need for more than one …

Two Applications Of Art To Geometry, Viktor Blåsjö

The Mathematics Enthusiast

Geometry and art exploit the same source of human pleasure: the exercise of our spatial intuition. It is not surprising, then, that interconnections between them abound. Applications of geometry to art, of which we shall indicate a few, go back at least to Alberti’s De Pictura (1435). But although geometry started out, as it so often does, as a most courteous suitor in its relationship with art, it was soon to be affectionately rewarded. We shall study two of these rewards.

Helping Teachers Un-Structure: A Promising Approach, Eric Hsu, Judy Kysh, Katherine Ramage, Diane Resek

The Mathematics Enthusiast

The amount of overt structure in the presentation of a task affects students’ engagement, creativity, and willingness to tolerate frustration. In a professional development project, with algebra teachers from nine American schools, we tried to help teachers make judicious decisions in their use of structure by having them facilitate low-structure tasks, remove structure from overly structured tasks, and observe “at-risk” students engaged in learning through low-structure tasks. Project schools that worked on structuring generally improved their algebra passing rates, both overall and for African-American students.

Who Can Solve 2x=1? An Analysis Of Cognitive Load Related To Learning Linear Equation Solving, Timo Tossavainen

The Mathematics Enthusiast

Using 2x = 1 as an example, we discuss the cognitive load related to learning linear equation solving. In the framework of the Cognitive Load Theory we consider especially the intrinsic cognitive load needed in arithmetical, geometrical and real analytical approach to linear equation solving. This will be done e.g. from the point of view of the conceptual and procedural knowledge of mathematics and the APOS Theory. Basing on our observations, in the end of the paper we design a setting for teaching linear equation solving.

The Contributions Of Comprehension Tests To Cognitive And Affective Development Of Prospective Teachers: A Case Study, Yüksel Dede

The Mathematics Enthusiast

The aim of this study was to investigate the role of comprehension tests while teaching algebra and its effects on students’ success. This study was carried out with 108 third year undergraduate students enrolled in math education in faculty of education. Several data collection instruments were used for gathering data from the participants such as; comprehension test, written documents, semi-structured interviews schedule, and participant and nonparticipant observations sheets. Collected data were subjected to content analysis and triangulation among the data was ensured. Results indicated that three different major categories emerged from the content analysis of the data: (1) measure of …

On The Use Of Realistic Fermi Problems For Introducing Mathematical Modelling In School, Jonas Bergman Ärlebäck

The Mathematics Enthusiast

In this paper uses an analytical tool refereed to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld’s parsing protocol coding scheme to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups’ solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extramathematical knowledge used …

Intuitions Of "Infinite Numbers": Infinite Magnitude Vs. Infinite Representation, Ami Mamolo

The Mathematics Enthusiast

This study examines undergraduate students’ emerging conceptions of infinity as manifested in their engagement with geometric tasks. Students’ attempts to reduce the level of abstraction of infinity and properties of infinite quantities are described. Their arguments revealed they perceive infinity as an ongoing process, rather than a completed one, and fail to notice conflicting ideas. In particular, confusion between the infinite magnitude of points on a line segment and the infinite representation of real numbers was observed. Furthermore, students struggled to draw a connection between real numbers and their representation on a number line.

Small Change - Big Difference, Ilana Lavy, Atara Shriki

The Mathematics Enthusiast

Starting in a well known theorem concerning medians of triangle and using the ‘What If Not?’ strategy, we describe an example of an activity in which some relations among segments and areas in triangle were revealed. Some of the relations were proved by means of Affine Geometry.

An Application Of Gröbner Bases, Shengxiang Xia, Gaoxiang Xia

The Mathematics Enthusiast

In this paper, we program a procedure using Maple's packages, with it we can realize mechanical proving of some theorems in elementary geometry.

If Mathematics Is A Language, How Do You Swear In It?, Dave Wagner

The Mathematics Enthusiast

Swears are words that are considered rude or offensive. Like most other words, they are arbitrary symbols that index meaning: there is nothing inherently wrong with the letters that spell a swear word, but strung together they conjure strong meaning. This reminds us that language has power. This is true in mathematics classrooms too, where language practices structure the way participants understand mathematics and where teachers and students can use language powerfully to shape their own mathematical experience and the experiences of others.

Mathematical Curiosities About Division Of Integers, Jérôme Proulx, Mary Beisiegel

The Mathematics Enthusiast

As mathematics educators, our focus of attention is mainly placed on the learning and teaching of mathematics. But, as we study phenomena of mathematical learning and teaching, we often come across intriguing mathematical phenomena that capture our interest. We find ourselves often bouncing mathematical ideas back and forth, not just looking for (new/better) ways of teaching or presenting a mathematical concept, but also of uncovering and discovering potential understandings of the concept. These mathematical issues we encounter represent for us a significant aspect of our work, and are also very stimulating. One of these issues arose for us as we …

Sum Of "N" Consecutive Integers, Steve Humble

The Mathematics Enthusiast

For all n , it is always possible to find at least one sum of n consecutive numbers with an equivalent sum of n - 1 consecutive numbers?

Cubism And The Fourth Dimension, Elijah Bodish

The Mathematics Enthusiast

When one looks into the subject of geometries that attempt to explain fourthdimensional space, it is inevitable that one encounters references to Cubism. The purpose of this paper is to find what the similarities between this mathematical concept and cubism are. There are many historical arguments as to how the cubists encountered literature about the fourth-dimension, and whether they were exposed to it at all, which I will for the most part omit and instead let the art speak for itself. It is important to see how two fields are interrelated in order to gain a better understanding of both …

Book Review: What's All The Commotion Over Commognition? A Review Of Anna Sfard's Thinking As Communicating, Bharath Sriraman

The Mathematics Enthusiast

If straight edge and compass constructions are the so-called “atoms” of Euclidean geometry, if sequences are the “atoms” of Analysis, then what are the “atoms” (if any) of mathematics education? Arguably mathematics education is a much wider field than Euclidean Geometry or Elementary Analysis, however there are several fundamental things that the field purports to study, chief among which is mathematical thinking or more generally “thinking”. The book under review, though it appears in a Cambridge University Press series entitled Learning in Doing: Social, Cognitive, and Computational Perspectives, is in my view situated at the intersection of Consciousness Studies, …

Tme Volume 6, Number 3

The Mathematics Enthusiast

No abstract provided.

A Conceptual Framework For Cross-Curricular Teaching, Astrid Beckmann

The Mathematics Enthusiast

The Montana Mathematics Enthusiast is pleased to present a special supplemental issue this year focused on the issue of interdisciplinarity in the curriculum and classroom. This book is based on the longitudinal research carried out by Prof. Astrid Beckmann of the University of Education, Schwäbisch Gmünd- Germany, who is one of the co-founders of the Mathematics and its Connections to the Arts and Sciences international group (MACAS).