The graphics calculator in mathematics education

This article covers a project conducted by the Freudenthal Institute from August 1991 to September 1994 entitled “The graphics calculator in mathematics education.” The theory of realistic mathematics education was taken as the point of departure for formulating the hypotheses. The developmental research design was used. Observation of the students' behavior during the experimental lessons supports the premise that the graphics calculator can stimulate the use of realistic contexts, the exploratory and dynamic approach to mathematics, a more integrated view of mathematics, and a more flexible behavior in problem solving.     

Experimental mathematics and proofs in the classroom

Experimental mathematics is a serious branch of mathematics that starts gaining attention both in mathematics education and research. We given examples of using experimental techniques (not only) on the classroom. At first sight it seems that introducing experiments will weaken the formal rules and the abstractness of mathematics that are considered a valuable contribution to education as a whole. By putting proof and experiment side by side we show how this can be avoided. We also highlight consequences of experimentation for educational computer software.     

Developing mathematical modelling skills: The role of CAS

Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters This is an important part of validating a mathematical model formulated to describe, a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.     

数学实验——数学方法、数学软件和数学应用的融合

论述了数学方法、数学软件和数学应用在数学实验课中的作用,表述了数学方法、数学软件和数学应用之间的关系.     

随机波动率模型下技术分析指标的一些性质(英文)

一些流行的技术指标(例如布林带,RSI,ROC等)被股市交易者广为使用.交易者将每日(小时,周,……)的实际股价作为计算某个技术指标的样本,通过观察相关频率来指导投资.技术指标的有效性已在广泛的应用中得到了验证.我们已经证明在Black-Scholes模型下,某些技术指标有许多有用的统计性质.作为更一般的情况,随机波动率模型在金融数学中得到了广泛的讨论.本文基于随机波动率模型对技术指标的统计性质进行了研究.研究结果表明,如果股票价格服从随机波动率模型,则技术指标的合理性可以得到有力的证明,从这个角度我们为技术分析奠定理论基础.     

Determinants of allocative and technical inefficiency in stochastic frontier models: An analysis of Norwegian electricity distribution firms

Recent econometric advances have made it possible to distinguish between persistent and transient technical inefficiency along with allocative inefficiency in stochastic frontier models for panel data. Kumbhakar et al. (2020) and Lai and Kumbhakar (2019) introduce a methodology that allows for the estimation of these inefficiency components and costs therefrom, while including determinants of both components of technical inefficiency. We extend these models to include technical change and determinants of allocative inefficiency (input misallocation). Including a set of variables that influence input misallocation, we are able to determine the effects of these variables on the cost of allocative inefficiency. We provide empirical evidence on the costs of all three types of inefficiency using data on 149 Norwegian electricity distribution firms between 2000 and 2016. We find that the cost of input misallocation is only slightly lower than that of technical inefficiency. Our results reject a commonly imposed modeling assumption that firms are fully allocatively efficient.     

The quality of non-textual elements in mathematics textbooks: an exploratory comparison between South Korea and the United States

As an exploratory investigation, this study aims to analyze non-textual elements in some Korean and US mathematics textbooks using a conceptual framework whose components include accuracy, connectivity, contextuality, and conciseness. By analyzing three US textbooks and three Korean ones, the study not only shows patterns in the use of non-textual elements in mathematics textbooks in different contexts but also provides insights into how to assess the quality of non-textual elements in mathematics textbooks, which I hope will contribute to the provision of more meaningful and productive learning opportunities to school children. Overall, the results from this study show that there is significant difference across topics and textbooks, which implies different opportunities to learn through non-textual elements. This study makes a unique contribution to the conceptualization of non-textual elements in mathematics education and has implications for textbook analysis and curriculum development.     

Symmetry and the golden ratio in the analysis of a regular pentagon

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.     

Variations on play with interactive computer simulations: balancing competing priorities

U.S. mathematics teachers face considerable pressures to keep up with pacing guides and to prepare students for standardized tests. At the same time, they are called upon to engage students in innovative exploratory activities and to incorporate new technologies into their lessons. These competing priorities pose considerable challenges. Against this backdrop, we investigated how middle-school mathematics teachers incorporated play into lessons involving interactive computer simulations (sims). The teachers used PhET sims in a variety of lessons. Following general guidelines for teaching with PhET sims, these lessons included a short period of play prior to more structured work with the sim. Our analysis of 15 mathematics lessons involving play led to the identification of four characteristics that distinguish the play phases of these lessons. Based on combinations of these characteristics, we identified three specific profiles of play, which lie at different points along a continuum of priorities from foregrounding students’ ideas to keeping pace. We discuss the implications associated with each profile of the play phase, and we begin to articulate a theory that frames teaching with play as a matter of balancing divergent and convergent modes of activity.     

“LeActiveMath”—a new innovative European elearning system for calculus contents

This contribution gives, an overview of the project “LeActiveMath”. Within this project a new mathematics learning software has been developed. LeActiveMath is an innovative eLearning system for high school and college or university level classrooms which can also be used in informal contexts for self-learning, since it is adaptive to the learner and his or her learning context in many respects. Topics cover elements of basic knowledge like ‘linear equations’ as well as more sophisticated contents like ‘differential calculus’. This article describes some of the innovative components of the software that are meant to support the students' self-regulated learning. We conclude by reporting on the first evaluations in math classorooms in fall 2005.     

Assessing views of coaching via a video-based tool

Mathematics coaches represent a unique group of didacticians, or individuals who work with practicing teachers. Twenty-eight mathematics coaches participated in this exploratory study, which used video viewing to examine the coach–teacher dynamic. To gather data about participants’ views of effective coaching practices, we developed the Video Assessment of Coaching instrument, which provided coaches with opportunities to express their views of effective practice and implementation. The participants expressed views of effective coaching that often did not align with those of coaching authors. The significance of this research lies in its efforts to document the views that mathematics coaches develop as practitioners, as an early step in the examination of the relationships between the views of coaches and coaches’ effectiveness in improving teacher practice, knowledge, and attitudes.     

Online distance mathematics education in Brazil: research, practice and policy

In this article, we address online distance mathematics education research and practice in Brazil, which are relative newcomers to the educational scene. We present the national context of education in Brazil, highlighting the organization of the educational system, and also a summary of national legislation on distance education and an overview of digital inclusion in the country. We outline the potential and relevance of distance education for the Brazilian educational system and show how it could intervene in the system. With respect to research and practice in online mathematics education, we present support for research, examples of studies and highlight different aspects being addressed, including its essential components. In addition, we discuss the synergy between distance education and teacher education, and mathematics distance education and modeling, as well as other initiatives in the national scenario.     

Optimal component selection based on cohesion & coupling for component based software system under build-or-buy scheme

Component based software system approach is concerned with the system development by integrating components. The component based software construction primarily focuses on the view that software systems can be built up in modular fashion. The modular design is a logical collection of several independent developed components that are assembled with well defined software architecture. These components can be developed in-house or can be obtained commercially from outside market making build versus buy decision an important consideration in development process. Cohesion and coupling (C&C) plays a major role in determining the system quality in terms of reliability, maintainability and availability. Cohesion is defined as the internal interaction of components within the module. On the other hand, coupling is the external interaction of the module with other modules i.e. interaction of components amongst the modules of the software system. High cohesion and low coupling is one of the important criteria for good software design. Intra-modular coupling density (ICD) is a measure that describes the relationship between cohesion and coupling of modules in a modular software system and its value lies between zero and one. This paper deals with the selection of right mix of components for a modular software system using build-or-buy strategy. In this paper, fuzzy bi-criteria optimization model is formulated for component selection under build-or-buy scheme. The model simultaneously maximizes intra-modular coupling density (ICD) and functionality within the limitation of budget, reliability and delivery time. The model is further extended by incorporating the issue of compatibility amongst the components of the modules. A case study is devised to explain the formulated model.     

Quality mesh generation for molecular skin surfaces using restricted union of balls

Quality surface meshes for molecular models are desirable in the studies of protein shapes and functionalities. However, there is still no robust software that is capable to generate such meshes with good quality. In this paper, we present a Delaunay-based surface triangulation algorithm generating quality surface meshes for the molecular skin model. We expand the restricted union of balls along the surface and generate an ε-sampling of the skin surface incrementally. At the same time, a quality surface mesh is extracted from the Delaunay triangulation of the sample points. The algorithm supports robust and efficient implementation and guarantees the mesh quality and topology as well. Our results facilitate molecular visualization and have made a contribution towards generating quality volumetric tetrahedral meshes for the macromolecules.     

Use of Technology for Science and Mathematics Collaborative Learning

This article presents an example of how computer software can be used to facilitate collaborative learning and the integration of mathematics and science. “Snap shots” from a pilot project, with thirty high school juniors who were involved in a university summer program, reveal how student-centered learning is facilitated by technology. This exploratory trial provides a glimpse of what the “classroom after next” might look like utilizing groupware in instructional settings.     

Using Metaphors to Unpack Student Beliefs About Mathematics

This paper reports on an exploratory study of the mathematical beliefs of a group of ninth and tenth grade students at a large, college preparatory, private school in the Southeastern United States. These beliefs were revealed using contemporary metaphor theory. A thematic analysis of the students' metaphors for mathematics indicated that students had well developed and complex views about mathematics including math as: an Interconnected Structure, a Hierarchical Structure, a Journey of Discovery, an Uncertain Journey, and a Tool. Another prevalent theme revealed by the metaphors was that students believe perseverance is needed for success in mathematics. The data also suggest an impact of gender and tracking on students beliefs about mathematics. Creating metaphors for mathematics provided a catalyst for student reflection, class discussion, and qualitative data, which could aid program evaluation. Several areas for future research were identified through this exploratory study.     

Visualizing geometric algorithms over the Web

The visual nature of geometry applications makes them a natural area where visualization can be an effective tool for demonstrating algorithms. In this paper we propose a new model, called Mocha, for interactive visualization of algorithms over the World Wide Web. Mocha is a distributed model with a client-server architecture that optimally partitions the software components of a typical algorithm execution and visualization system, and leverages the power of the Java language, which has become the standard for distributing interactive platform-independent applications across the Web. Mocha provides high levels of security, protects the algorithm code, places a light communication load on the Internet, and allows users with limited computing resources to access executions of computationally expensive algorithms. The user interface combines fast responsiveness with the powerful authoring capabilities of hypertext narratives.We describe the architecture of Mocha, show its advantages over previous methods, and present a prototype that can be accessed by any user with a Java-enabled Web browser. The Mocha prototype has been widely accessed over the Web, as demonstrated by the statistics that we have collected, and the Mocha model has been adopted by other research groups. Mocha is currently part of a broader system, called GeomNet, which performs distributed geometric computing over the Internet.     

Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

A computational model for interfaces

Decompositions of the plane into disjoint components separated by curves occur frequently. We describe a package of subroutines which provides facilities for defining, building, and modifying such decompositions and for efficiently solving various point and area location problems. Beyond the point that the specification of this package may be useful to others, we reach the broader conclusion that well-designed data structures and support routines allow the use of more conceptual or non-numerical portions of mathematics in the computational process, thereby extending greatly the potential scope of the use of computers in scientific problem solving. Ideas from conceptual mathematics, symbolic computation, and computer science can be utilized within the framework of scientific computing and have an important role to play in that area.