Mathematics Commons^™

Learning, Participation And Local School Mathematics Practice, Cristina Frade, Konstantinos Tatsis

The Mathematics Enthusiast

This paper reports on a study whose aim was to examine students’ learning in terms of participation in collective mathematical discussions. Our basic theoretical assumptions are based on a combination of situated learning perspectives and a framework that links social and psychological approaches of mathematical activity and learning. The research was carried out in a Year-8 classroom (students were aged 12 to 13), and the mathematical subject under investigation was area measurement. Data are presented to illustrate possible correspondences between ‘signs’ of learning and ‘local’ changes of participation. We conclude by discussing some pedagogical implications resulting from the study.

If A.B = 0 Then A = 0 Or B = 0?, Cristina Ochoviet, Asuman Oktaç

The Mathematics Enthusiast

We present a study carried out in Uruguay, with secondary school students and tertiary level mathematics students, concerning the zero-product property. In our research we observed that when early secondary and late secondary school students have to solve equations of the form (ax + b)(cx + d) = 0, they do not always apply the property, even when it is the only available tool and have received specific instruction on its application to the resolution of equations of this type. We also detected an error that students make when they have to verify the solutions of this type of equations. …

The Origins Of The Genus Concept In Quadratic Forms, Mark Beintema, Azar Khosravani

The Mathematics Enthusiast

We present an elementary exposition of genus theory for integral binary quadratic forms, placed in a historical context.

The Impact Of Undergraduate Mathematics Courses On College Student’S Geometric Reasoning Stages, Nuh Aydin, Erdogan Halat

The Mathematics Enthusiast

The purpose of this study is to investigate possible effects of different college level mathematics courses on college students’ van Hiele levels of geometric understanding. Particularly, since logical reasoning is an important aspect of geometric understanding, it would be interesting to see whether there are differences in van Hiele levels of students who have taken non-geometry courses that emphasize or focus on logic and proofs (Category I) and those that don’t (Category II). We compared geometric reasoning stages of students from the two categories. One hundred and forty nine college students taking various courses from the two categories have been …

Elementary School Pre-Service Teachers’ Understandings Of Algebraic Generalizations, Jean E. Hallagan, Audrey C. Rule, Lynn F. Carlson

The Mathematics Enthusiast

It is critical for all students to learn algebra, including the ability to generalize, to function in our increasingly complex world. This pretest/intervention/posttest study of preservice elementary teachers (N = 63) in their math methods course assessed their knowledge of writing and applying algebraic generalizations using instructor-made rubrics along with analysis of work samples and reported insights. Initially, although most subjects could solve a specific case, they had considerable difficulty determining an algebraic rule. After a problemsolving- based teaching intervention, students improved in their ability to generalize, however, they encountered more difficulty with determining the algebraic generalization for items arranged …

A Longitudinal Study Of Student’S Representations For Division Of Fractions, Sylvia Bulgar

The Mathematics Enthusiast

The representations that students use as part of their mathematical problem solving can provide us with a window into their grasp of the concepts they are exploring and developing. In this paper, the author indicates how these representations can evolve over time and enrich the understanding of division of fractions, often thought to be the most difficult of elementary school mathematical topics. The results of this research suggest that when appropriate problems are provided for students, in a meaningful context, they can demonstrate understanding of division of fractions that is durable over time, and that they are able to flexibly …

Catch Me If You Can!, Steve Humble

The Mathematics Enthusiast

Learning mathematics outside the classroom is not enrichment, it is at the core of empowering an individuals understanding of the subject.

The three activities described in this article can all be used outside the classroom in a maths lesson. 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 occur in the classroom. By taking maths beyond the classroom, we can more clearly illustrate the connections between the real world and what they are studying …

Fostering Connections Between The Verbal, Algebraic, And Geometric Representations Of Basic Planar Curves For Student’S Success In The Study Of Mathematics, Margo F. Kondratieva, Oana G. Radu

The Mathematics Enthusiast

We discuss the significance of making connections between the verbal, algebraic, and geometric representations of basic mathematical objects for students’ understanding of mathematical instructions. Our survey of 499 students enrolled in a precalculus university course reveals that such connections are not always present, even if the objects themselves are familiar to the students. We stress that the ability of making these connections needs to be specifically addressed in teaching mathematics at various levels. A proper attention to the matter contributes to the formation of students’ mathematical background, which makes a difference for their success in study of calculus, in particular.

How To Increase Mathematical Creativity- An Experiment, Kai Brunkalla

The Mathematics Enthusiast

Creativity is an integral part of mathematics. In this article I examine the increase in awareness of creativity in mathematics using Fröbel’s blocks in a college classroom. A majority of students found the introduction of the “gifts” of the founder of Kindergarten to a college geometry classroom enhancing their interest in mathematics. They judged the wooden blocks helpful in their understanding of geometry. The students showed increased awareness of creativity in mathematics.

Korean Teachers’ Perceptions Of Student Success In Mathematics: Concept Versus Procedure, Insook Chung

The Mathematics Enthusiast

This article examines the Korean classroom teachers’ beliefs about mathematics education in elementary schools. Their perceptions about contributing factors to Korean students’ high achievement scores in international comparative studies in the area of mathematics are explored. Elementary classroom teachers were surveyed using the researchermade questionnaire (Teacher Perception about Mathematics Curriculum) and 141 teachers completed the questionnaire. The data collected was analyzed by a descriptive analysis. The results reveal that the majority of classroom teachers agreed that real life applications, processing skills, using concrete instructional manipulatives, and conceptual knowledge are very important in teaching children mathematics. Most of teachers participating in …

Book X Of The Elements: Ordering Irrationals, Jade Roskam

The Mathematics Enthusiast

Book X from The Elements contains more than three times the number of propositions in any of the other Books of Euclid. With length as a factor, anyone attempting to understand Euclidean geometry may be hoping for a manageable subject matter, something comparable to Book VII’s investigation of number theory. They are instead faced with a dizzying array of new terminology aimed at the understanding of irrational magnitudes without a numerical analogue to aid understanding. The true beauty of Book X is seen in its systematic examination and labeling of irrational lines. This paper investigates the early theory of irrationals, …

Tme Volume 6, Numbers 1 And 2

The Mathematics Enthusiast

No abstract provided.

The Brachistochrone Problem: Mathematics For A Broad Audience Via A Large Context Problem, Jeff Babb, James Currie

The Mathematics Enthusiast

Large context problems (LCP) are useful in teaching the history of science. In this article we consider the brachistochrone problem in a context stretching from Euclid through the Bernoullis. We highlight a variety of results understandable by students without a background in analytic geometry. By a judicious choice of methods and themes, large parts of the history of calculus can be made accessible to students in Humanities or Education.

How Can Science History Contribute To The Development Of New Proposals In The Teaching Of The Notion Of Derivatives?, Arnaud Mayrargue

The Mathematics Enthusiast

The 18th century was a milestone for the incorporation of mathematics into physics. By this time already seen in Newton’s work, we know that a great deal of progress had found the light of day concerning the relationship between physics and mathematics, particularly as the latter began to deal with universal gravitation and optics. Based on his theory of monochromatic light, which used mathematics to describe optical phenomena, Newton became interested in the behavior of the various colors which, according to him, compose white light. Furthermore, toward the end of the 17th century, scientists had a remarkable mathematical tool at …

"Obeying A Rule": Ludwig Wittgenstein And The Foundations Of Set Theory, Giorgio T. Bagni

The Mathematics Enthusiast

In this paper we propose some reflections on Wittgenstein’s ideas about grammar and rules; then we shall consider some consequences of these for the foundations of set theory and, in particular, for the introduction of major concepts of set theory in education. For instance, a community of practice can decide to follow a particular rule that forbids the derivation of arbitrary sentences from a contradiction: since, according to Radford’s perspective, knowledge is the result of thinking, and thinking is a cognitive social praxis, the mentioned choice can be considered as a form of real and effective knowledge.

Introduction To Thinking As Communicating, Anna Sfard

The Mathematics Enthusiast

This book is a result of years-long attempts to change my own thinking about thinking, a task seemingly as improbable as breaking a hammer by hitting it with itself. In this unlikely undertaking, I have been inspired by Lev Vygotsky, the Byelorussian psychologist who devoted his life to “characterizing the uniquely human aspects of behavior,”2 and by Ludwig Wittgenstein, the Austrian-British philosopher who insisted that no substantial progress can be made in this kind of endeavor unless the ways we talk, and thus think, about uniquely human “forms of life” undergo extensive revisions.

Mathematics For Middle School Teachers: Choices, Successes, And Challenges, Linda Martin, Kristin Umland

The Mathematics Enthusiast

The opportunities for mathematics department faculty from institutions of higher education (IHEs) to work with middle school mathematics teachers are on the rise; the U.S. Department of Education has allocated almost $800 million for mathematics and science partnerships since 2004 that require collaboration between faculty from IHEs departments of arts and sciences and school districts. Changes in credentialing requirements due to the No Child Left Behind Act of 2001 mean that middle school mathematics teachers must take more mathematics content courses, yet current offerings for teachers in many mathematics departments typically focus on elementary or high school mathematics and often …

Main Points Of Objections To The "No Child Left Behind", Mike O'Lear, Bettina Dahl

The Mathematics Enthusiast

Adequate Yearly Progress (AYP):

--The guidelines fail to distinguish between successful schools and unsuccessful schools

--Between schools under-performing in just one area vs. schools likely to need a complete overhaul

--Does not give credit for student growth toward a high standard

--Need to establish AYP levels which make a distinction between struggling schools and those needing only limited assistance

Early Insurance Mechanisms And Their Mathematical Foundations, Amy Minto

The Mathematics Enthusiast

This article gives a historical survey of early insurance mathematics and its development in relation to the maritime industry and the origins of the life insurance industry. The history of the probability and statistics and its intricate connection to the actuarial sciences and modern insurance can be traced to the time period described in this paper.

Three Strikes, Tom O'Brien, Marianne Smith

The Mathematics Enthusiast

The first part of the paper describes and comments upon three aspects of the back-to-basics movement: the make-up and mindset of the National Mathematics Advisory Panel (NMP), the movement’s history in California, and recent “grassroots” activities of the movement in the state of Washington. The second part reports and comments on the principal findings of the NMP report.

Tme Volume 5, Numbers 2 And 3

The Mathematics Enthusiast

No abstract provided.

Chess And Problem Solving Involving Patterns, Dores Ferreira, Pedro Palhares

The Mathematics Enthusiast

In this paper we present the context and results from a study, with 3rd to 6th grades children, about the relationship between chess and problem solving involving geometric and numeric patterns. The main result of this study is the existence of a relation between strength of play and patterns involving problem solving. We have included in the beginning an analysis of chess as a context for elementary mathematics problems, also showing its richness historically.

A Case Study Using Soft Systems Methodology In The Evolution Of A Mathematics Module, Jon Warwick

The Mathematics Enthusiast

This paper describes the application of Soft Systems Methodology as a tool for facilitating the review of a taught mathematics module so that the views of those engaged with the module could be captured and conflicting expectations and views highlighted. Checkland’s Soft Systems Methodology is used since it enables the capture of stakeholder views and addresses both ‘hard’ and ‘soft’ aspects of the learning experience. Stages in the application of Soft Systems Methodology are illustrated including the development of a rich picture and conceptual models and the work was conducted using a stakeholder group that included students taking the module …

History Of Mathematics In Mathematics Education: A Saussurean Perspective, Michael Fried

The Mathematics Enthusiast

It is not only because of a certain eclecticism in mathematics education research that semiotic ideas have begun to take root there: it is also because of the dawning recognition that key areas of interest in mathematics education genuinely have a semiotic nature. For this, one need only point to research focused on meaning, communication, language, and culture (e.g., Presmeg, 1997; Ernest,1997; Radford, 2001; Brown, 2001). The ways in which semiotics informs cultural aspects of mathematics education, particularly those connected with the history of mathematics, were highlighted in a discussion session at the 2003 meeting of the International Group for …

Comparison Of Geometric Figures, Spyros Glenis

The Mathematics Enthusiast

Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein’s view for the Euclidean Geometry, we defined a relation “≤ ” as: S1 ≤ S2 whenever there is a Euclidean isometry f :R² →R² , so that . This relation is not an order because there are …

Incoherence Of A Concept Image And Erroneous Conclusions In The Case Of Differentiability, Antti Viholainen

The Mathematics Enthusiast

The level of the coherence of a concept image conveys how well the cognitive structure concerning the concept is organized. This study considers the relationship between deficiencies in the coherence of the concept image and erroneous conclusions in the case of differentiability. The study is based on an interview where the student made conclusions contradictory to the formal theory of mathematics. He used an erroneous method to study the differentiability of piecewise defined functions. This method became the key factor which maintained the internal coherence of the concept image. It made it possible to build a cognitive structure whose basis …

Editorial: 2 To 3- An Omnibus Of Features & Policy Issues In Mathematics Education, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Inverses - Why We Teach And Why We Need Talk More About It More Often!, Woong Lim

The Mathematics Enthusiast

This article examines the key role that the notion of inverses plays in numerous mathematical concepts.

Chopping Logs: A Look At The History And Uses Of Logarithms, Rafael Villarreal-Calderon

The Mathematics Enthusiast

Logarithms are an integral part of many forms of technology, and their history and development help to see their importance and relevance. This paper surveys the origins of logarithms and their usefulness both in ancient and modern times.

Mathematics And The World: What Do Teachers Recognize As Mathematics In Real World Practice?, Barbara Garii, Lillian Okumu

The Mathematics Enthusiast

Elementary school teachers are encouraged to better integrate appropriate mathematics pedagogy with deeper, more relevant mathematics content. However, many teach a mathematics they do not fully understand to students who see, recognize, and use less mathematics than ever before. Both teachers and students struggle to articulate the role mathematics plays in society as mathematics becomes more embedded into our technology. In this study, we asked teachers to record the mathematics they used on a daily basis during a 1-week period. Their responses indicate that they do not recognize that mathematics plays any important role in technological and professional practices. This …