Physical Sciences and Mathematics Commons^™

"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.

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 …

New Titles In Mathematics Education

The Mathematics Enthusiast

No abstract provided.

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

The Political Context Of The National Mathematics Advisory Panel, Eric Rico Gutstein

The Mathematics Enthusiast

The National Mathematics Advisory Panel needs to be situated in its broader political context to more fully understand it. Who created it, for what purpose, and who will (and will not) benefit from it are key questions I address in this article. My argument is that the NMAP, as part of a larger initiative undertaken by the Bush Administration and US financial/corporate elites, serves capital’s efforts to shore up the US’s weakening economic global position and does not benefit the majority of the US people—particularly marginalized and excluded students of color and low-income students.

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 …

Associative Operations On A Three-Element Set, Friðrik Diego, Kristín Halla Jónsdóttir

The Mathematics Enthusiast

A set with a binary operation is a fundamental concept in algebra and one of the most fundamental properties of a binary operation is associativity. In this paper the authors discuss binary operations on a three-element set and show, by an inclusion-exclusion argument, that exactly 113 operations out of the 19,683 existing operations on the set are associative. Moreover these 113 associative operations are accounted for by means of their operation tables.

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 …

E(Race)Ing Race From A National Conversation On Mathematics Teaching And Learning: The National Mathematics Advisory Panel As White Institutional Space, Danny Bernard Martin

The Mathematics Enthusiast

In this paper, I wish to argue that several factors support a characterization of the National Mathematics Advisory Panel as an instantiation of the white institutional space (Moore, 2008) that characterizes mathematics education research and policy contexts more generally. In particular, my analysis highlights that mathematics education research and policy contexts such as the National Mathematics Advisory Panel are not immune to the structural and institutional racism that characterize many other areas of U.S. society.

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

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.

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 …

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 …

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 …

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.

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.

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 …

Magic Math Cards, Steve Humble

The Mathematics Enthusiast

Starting lessons with a good story, captures students imagination. The story then leads nicely into a magic card experiment which can be used to help teach the topic. I have used magic with a variety of age groups and found that they ask different questions of the task, and search for different levels of answers.

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.

Guest Editorial: Reaction To The Final Report Of The National Mathematics Advisory Panel, Brian Greer

The Mathematics Enthusiast

In 2006, President Bush appointed the National Mathematics Advisory Panel (NMAP henceforth) which issued its Final Report on March 13, 2008. The Final Report (summary), plus reports of three subcommittees and five Task Groups, together with two appendices containing the Presidential Executive Order setting up the panel, and details of the panel membership and other personnel, can be downloaded from www.ed.gov/about/bdscomm/list/mathpanel/index.html

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.

Algebra For All?, Brian Greer

The Mathematics Enthusiast

In 1957, the director of one of the major math curriculum projects in the UK was quoted in a newspaper as saying: "Up went Sputnik and down came all the pure mathematicians saying we must do sets and be saved". The corresponding message from the National Math Panel Final Report is "Up went the economic performance of China and India and down came all the pure mathematicians saying we must do algebra and be saved".

Mathematical Cognition And The Final Report Of The National Mathematics Advisory Panel: A Critical, Cultural-Historical Activity Theoretic Analysis, Wolff-Michael Roth

The Mathematics Enthusiast

In its Final Report, the National Mathematics Advisory Panel has depicted a stark image of mathematical competencies and achievement among U.S. students. The Advisory Panel notes the lack of research with “truly scientific” rigor and, mentioning Vygotsky’s cultural-historical activity theory in passing, suggests its utility to be untested. In reading the report, I noted the limited understanding of the mathematics education literature it articulates and a complete failure to draw on established, “tried and proven,” theory and practice in mathematics education founded upon an encompassing culturalhistorical activity theory. This theory is comprehensive and encompassing, because it retains activity in its …

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.

Meet The Authors

The Mathematics Enthusiast

No abstract provided.

Tme Volume 5, Numbers 2 And 3

The Mathematics Enthusiast

No abstract provided.

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.

Critique On Eisenberg's Article, Renuka Vithal

The Mathematics Enthusiast

I enjoyed reading the article and learned some interesting (and disturbing) information about well known mathematicians that is not so well known. However in its current form it is more appropriate for the popular media rather than as an academic or scholarly article.

Having said that it does raise some serious questions of ethics and values that all mathematics educators should be engaging. The increasingly popular view that mathematics teaching be socially contextualised means that this kind of historical information may be communicated in lecture rooms and classroom with little understanding or awareness of the "hidden curriculum" being enacted.

Final Comments On Eisenberg's Paper, Brian Greer

The Mathematics Enthusiast

I must begin by expressing appreciation for the spirit in which Ted accepted my comments as intended to be intellectually and not personally provocative. Let me also assure him that I was not at all offended by the paper – it would be more accurate to say that I was stimulated by it to voice some strong disagreements. Beyond the particular aspects on which we have different perspectives, I applaud his decision to raise ethical issues, too often ignored in writing about mathematics education.

The Action Map As A Tool For Assessing Situated Mathematical Problem Solving Performance, Murad Jurdak

The Mathematics Enthusiast

The aim of this paper is to investigate the appropriateness, concurrent and construct validity of action map as a tool for assessing situated problem solving performance. Action map is rooted in activity theory whose stipulations are compatible with situated problem solving. Thirty-one last year secondary students were given three tasks with real –world context. Based on the analysis of the written solutions and interviews, evidence is presented on the appropriateness and validity of action map as an instrument to assess situated problem solving performance.