Asian option as a fixed-point

We characterize the price of an Asian option, a financial contract, as a fixed-point of a non-linear operator. In recent years, there has been interest in incorporating changes of regime into the parameters describing the evolution of the underlying asset price, namely the interest rate and the volatility, to model sudden exogenous events in the economy. Asian options are particularly interesting because the payoff depends on the integrated asset price. We study the case of both floating- and fixed-strike Asian call options with arithmetic averaging when the asset follows a regime-switching geometric Brownian motion with coefficients that depend on a Markov chain. The typical approach to finding the value of a financial option is to solve an associated system of coupled partial differential equations. Alternatively, we propose an iterative procedure that converges to the value of this contract with geometric rate using a classical fixed-point theorem.     

Undergraduate mathematics achievement in the emerging ethnic engineers programme

This paper addresses the issue of collegiate mathematics achievement of underrepresented minority students as it investigates the impact of a cooperative learning calculus programme on the first-year calculus experience of non-Asian ethnic minority engineering students. The Emerging Ethnic Engineers Programme in the College of Engineering at the University of Cincinnati is a successful, comprehensive programme that focuses on the recruitment, retention, academic success, professional development, and timely graduation of underrepresented coloured students. The objectives of the programme are accomplished through three interrelated phases: pre-college science and mathematics programmes, first-year collegiate programmes, and upper-division programmes. The underlying principles of the first-year programme include academic achievement and establishing a strong sense of community among the cohort. This report will focus on the cooperative learning calculus programme that has been successful in improving retention and academic success rates for coloured freshmen engineering students.     

Modeling of certain problems in financial mathematics: Spread option pricing

The problem of valuating exotic options, namely, the option on the spread between two forward interest rates is considered. The price of the option is derived under the assumption that the dynamics of debt instruments and the interest rates are described by the Heath-Jarrow-Morton model. The parameters of the model are estimated, and the price of the option is numerically computed based on Russian bond market data.     

Capital renewal as a real option

We consider the timing of replacement of obsolete subsystems within an extensive, complex infrastructure. Such replacement action, known as capital renewal, must balance uncertainty about future profitability against uncertainty about future renewal costs. Treating renewal investments as real options, we derive an optimal solution to the infinite horizon version of this problem and determine the total present value of an institution’s capital renewal options. We investigate the sensitivity of the infinite horizon solution to variations in key problem parameters and highlight the system scenarios in which timely renewal activity is most profitable. For finite horizon renewal planning, we show that our solution performs better than a policy of constant periodic renewals if more than two renewal cycles are completed.     

Hereditary stress as a cultural force in mathematics

Among the forces affecting the course of mathematical evolution is what the author has called “hereditary stress.” The term designates a cultural, not a psychological force, and it is internal to mathematics, not environmental. It appears to be synonymous with what A. L. Kroeber called “potentialities” and G. Sarton termed “growth pressure.” Neither of these scholars gave any analysis of it. The present article attempts to do this in the restricted context of mathematics.Its chief components seem to be: (A) Capacity. The quantity and intrinsic interest of the results that the basic theory and methodology of a field are capable of yielding. (B) Significance. The field's promise of yielding results significant for the advancement of mathematics or related fields. (C) Challenge. The emergence of problems whose solutions require an ingenuity and/or methodology which distinguish them from those problems whose solutions are of a more routine character. (D) Status. The esteem in which the field is held. (E) Conceptual Stress. The stresses created by the need for new conceptual materials to furnish a logical basis for explaining phenomena; outstanding among these is symbolic stress. (F) Paradox. Emergence of paradoxes or inconsistencies.These are discussed individually. Analysis of Component (E), for example, shows that it evidently stems from several sources; e.g., the necessity for a new concept which will afford means of solving problems previously inaccessible; stresses created by the need for introducing order into a chaos of materials recognizably related; and the need for new attitudes toward mathematical existence and mathematical “reality.”     

Aesthetics as a liberating force in mathematics education?

This article investigates different meanings associated with contemporary scholarship on the aesthetic dimension of inquiry and experience, and uses them to suggest possibilities for challenging widely held beliefs about the elitist and/or frivolous nature of aesthetic concerns in mathematics education. By relating aesthetics to emerging areas of interest in mathematics education such as affect, embodiment and enculturation, as well as to issues of power and discourse, this article argues for aesthetic awareness as a liberating, and also connective force in mathematics education.     

Combining theories in research in mathematics teacher education

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.     

A research journey through mathematics coaching

Classroom coaching in mathematics is flexible in its definition, complex in its enactment, variable in its outcomes, and dependent on setting and circumstances. Multiple lines of inquiry are required to navigate this subjective terrain: research on coaching encompasses understanding perceptions of coaching held by coaches, teachers, and administrators, measuring the effectiveness of coaching in terms of teachers’ content knowledge and instructional practices, and exploring the nature of coaching within an educational ecosystem. This paper describes a cumulative sequence of research studies that inform current understanding of classroom coaching in mathematics, highlighting methodological decisions made at various crossroads and elaborating on the populations, methods, and instruments used to investigate coaching. A presentation of findings related to the three domains of perception, effectiveness, and nature is followed by reflections on features of coaching that pose particular challenges, questions that remain to be answered, and promising avenues of future inquiry.     

The onto-semiotic approach to research in mathematics education

In this paper we synthesize the theoretical model about mathematical cognition and instruction that we have been developing in the past years, which provides conceptual and methodological tools to pose and deal with research problems in mathematics education. Following Steiner’s Theory of Mathematics Education Programme, this theoretical framework is based on elements taken from diverse disciplines such as anthropology, semiotics and ecology. We also assume complementary elements from different theoretical models used in mathematics education to develop a unified approach to didactic phenomena that takes into account their epistemological, cognitive, socio cultural and instructional dimensions.     

Enhancing elementary-school mathematics teachers' efficacy beliefs: a qualitative action research

Individuals and societies that can use mathematics effectively in this period of rapid changes will have a voice on increasing the opportunities and potentials which can shape their future. This has brought affective characteristics, such as self-efficacy, that affect mathematics achievement into focus of the research. Teacher efficacy refers to the extent to which a teacher feels capable to help students learn, influence students’ performance and commitment, and thus plays a crucial role in developing the student in all aspects. In this study, we used two sources of efficacy beliefs, mastery experiences and physiological and emotional states, in an interesting and challenging seven month workshop, as tools to foster teacher efficacy for six elementary-school teachers who were frustrated and wanted to leave their job. Our aim was to study the nature of these teachers’ efficacy in order to change it. In this qualitative action research, we used open interviews, non-participant observations and field notes. Results show that these teachers became efficacious, their students’ achievements and motivation were enhanced, and the school climate was changed. Qualitative inquiry of this construct sheds light on efficacy beliefs of mathematics teachers. Nurturing teacher efficacy has borne much fruit in the field of mathematics in school.     

Problem solving as a challenge for mathematics education in The Netherlands

This paper deals with the challenge to establish problem solving as a living domain in mathematics education in The Netherlands. While serious attempts are made to implement a problem-oriented curriculum based on principles of realistic mathematics education with room for modelling and with integrated use of technology, the PISA 2003 results suggest that this has been successful in educational practice only to a limited extent. The main difficulties encountered include institutional factors such as national examinations and textbooks, and issues concerning design and training. One of the main challenges is the design of good problem solving tasks that are original, non-routine and new to the students. It is recommended to pay attention to problem solving in primary education and in textbook series, to exploit the benefits of technology for problem solving activities and to use the schools’ freedom to organize school-based examinations for types of assessment that are more appropriate for problem solving.     

Interviewing in mathematics education research: Choosing the questions

With the development of qualitative methodologies, interviewing has become one of the main tools in mathematics education research. As the first step in analyzing interviewing in mathematics education we focus here on the stage of planning, specifically, on designing the interview questions. We attempt to outline several features of interview questions and understand what guides researchers in choosing the interview questions. Our observations and conclusions are based on examining research in mathematics education that uses interviews as a data-collection tool and on interviews with practicing researchers reflecting on their practice.