Using the Magma Computer Algebra System in Abstract Algebra Courses

Students meeting their first serious course in abstract algebra commonly experience difficulty understanding many of the basic concepts. This makes it very hard for them to interpret and generate proofs. The first step in reaching an understanding of such concepts is often best achieved by constructing and manipulating instances in particular algebraic structures. Computer algebra systems open up the possibility of students being able to experiment rapidly and conveniently with such concepts in a variety of structures having non-trivial size and complexity. The new Magma computer algebra system has a syntax and semantics based directly on fundamental algebraic notions, and consequently should provide an appropriate learning environment for those branches of mathematics that are heavily algebraic in nature. This paper describes the development of Magma-based exercises and a course methodology that utilizes Magma as a key learning tool in a Pass-level Rings and Fields course given at the University of Sydney.     

Technology practices: Confirmatory factor analysis and exploration of teachers' technology integration in subject areas

It is thought that gains in student learning from the use of digital technologies are more likely to be related to teachers' practice than the technology itself. In secondary schooling, a key aspect of this is concerned with understanding how digital technologies are used to support teaching and learning in specific subject areas. Subject areas have their own conventions and expectations for learning that will influence teachers' technology use and technology-supported student tasks. The aim of this paper is to present confirmatory factor analysis of a scale considering common technology-related tasks in three subject areas. Data included in the current analysis are teacher questionnaires collected in 2010 (N = 3624), as part of a large-scale one-to-one laptop initiative in Australia. Results from the 2010 data confirm a five-factor structure revealing significant differences in teachers' professional and instructional uses of digital technologies among three core subject areas: English, Mathematics and Science. Trends are confirmed through a second teacher data set collected in 2011 for Wave 2 of the same one-on-one laptop initiative. Implications of these findings in relation to understanding and supporting effective technology integration and areas of future research are discussed.     

TPACK-in-Action: Unpacking the contextual influences of teachers' construction of technological pedagogical content knowledge (TPACK)

In school-based environments, teachers need to consider contextual factors such as the availability of technology and school policies as they apply TPACK to address instructional problems. However, qualitative TPACK studies tend to exemplify the seven TPACK constructs whereas its contextual influences have not been articulated in detail. This study describes TPACK-in-Action, a framework that can be used to visualize the interplay between TPACK and four contextual factors (Physical/Technological, Cultural/Institutional, Interpersonal, and Intrapersonal) that influence teachers' design of ICT lessons. This was used to analyze the lesson design discussions of 24 school teachers from a Singapore primary school who were teaching the levels of Primary 1, 4, and 5. Content analysis of the transcribed audio-recordings of teachers' discussions and chi-square analysis of coding frequencies found that when the logistics of lesson implementation as per the Cultural/Institutional category dominated group discussions, it curtailed the emergence of TPACK. When Intrapersonal factors such as beliefs of teaching and students were articulated and its pedagogical implications considered, it facilitated TPACK. Furthermore, the team facilitated by an experienced educational technologist also demonstrated higher occurrences of TPACK. These results suggest that for ICT innovation to be effective, the composition of design teams need to be carefully considered. Teachers also need to develop competencies to facilitate and discourse about design such that contextual concerns can be turned into opportunities to support pedagogical improvement.     

基于MatLab的线性代数教学应用

文章针对学生学习线性代数困难原因的分析,说明在线性代数中引入Matlab进行教学改革创新的必要性。通过Matlab程序在初等矩阵教学中演示和练习,使学生能够直观形象地理解和掌握线性代数知识,为后续线性代数的学习打下牢固的基础。     

The introduction of computers into a school

This paper briefly describes the authors' research into the introduction of computers in a secondary school. The research consists of an in-depth study of the implementation of computers in a comprehensive school (i.e. a case study). The school had recently purchased seventeen microcomputers. Funds for these were obtained through a wide range of sources. Twelve of the microcomputers were for the school's newly formed Computer Studies Department and the other five were for use within five major departments in the school, where they were intended to be used specifically for CAL applications. The research looks at the different factors, at various levels, which influenced the introduction of computers into the school. This entailed investigating the organisation of, and decision-making within the school. the views and opinions of the teachers about computers in general, the microcomputer in their department, and an analysis of the possible use of microcomputers in their teaching.The focus of this paper however is on the teacher, and attempts to outline the analysis that was carried out in order to get a clearer understanding of 15 of the teachers' views and opinions towards computers in general and in teaching. The first level of analysis was through a series of criteria which gave us a profile of each of the teachers. Having built up these profiles, this led us to draw a spectrum of teachers' views and opinions towards the use of computers in teaching, ranging from those who were favourable to those who were antagonistic towards them. We then looked at the spectrum of teachers' views which led us to analyse at a deeper level the changing roles that teachers perceived would be brought about by using computers in their teaching. We give two examples which illustrate these different models of analysis.     

科学计算软件在线性代数中地审美教学策略

线性代数是一门抽象性和理论性较强的学科,在教学中所涉及的例子较少能与实际生活密切相连,让人感觉到学习的内容与生产实际脱节,体现不了数学的应用价值。科学计算软件就是将抽象知识转换为形象,由理论转化为应用的一种工具,从而在线性代数的教学中融入审美教学成为可能。研究与实践表明,审美教学的实施对培养人的创新思维及数学应用意识,提高教学质量以及学生的数学素质方面起着重要的作用。     

Hypermatrix Algebra: Theory

Algebraic manipulation of two-dimensional data structures—typical in image processing—requires operations more powerful than those afforded by classical matrix algebra. To this end, hypermatrix algebra provides a compact treatment of multidimensional objects and associated operations. Initially presented as array algebra, it uses a notation derived from tensor analysis. By extending matrix algebra with Kronecker products and matrix representations of shuffle permutations, a duality between the two algebras is shown. This serves as a means of transferring into the domain that is most convenient for a particular type of operation or representation. Hypermatrix algebra can be a powerful design tool for multidimensional massively parallel computing structures. A mathematical foundation for such an algebra is presented in this paper as a supplement to separate application notes.     

Kleene Algebra and Bytecode Verification

Most standard approaches to the static analysis of programs, such as the popular worklist method, are first-order methods that inductively annotate program points with abstract values. In [6] we introduced a second-order approach based on Kleene algebra. In this approach, the primary objects of interest are not the abstract data values, but the transfer functions that manipulate them. These elements form a left-handed Kleene algebra. The dataflow labeling is not achieved by inductively labeling the program with abstract values, but rather by computing the star (Kleene closure) of a matrix of transfer functions. In this paper we show how this general framework applies to the problem of Java bytecode verification. We show how to specify transfer functions arising in Java bytecode verification in such a way that the Kleene algebra operations (join, composition, star) can be computed efficiently. We also give a hybrid dataflow analysis algorithm that computes the closure of a matrix on a cutset of the control flow graph, thereby avoiding the recalculation of dataflow information when there are cycles in the graph. This method could potentially improve the performance over the standard worklist algorithm when a small cutset can be found.     

关系代数中除法运算的教学探讨

从关系代数中除法运算的定义出发,以实例分析和图表表示为工具,通过类比整数的除法运算阐明关系除法运算的本质含义,并给出关系代数中除法运算的基本运算表达式。解决以往除法运算的抽象定义难以被学生理解的不足,促进学生对知识点的透彻理解,取得较好的教学效果。     

Learning About Ourselves: A Review of The Mathematics Teacher in the Digital Era

ABSTRACT

Mathematics Education in the Digital Era is a title that implies the ubiquity of digital technology in our everyday lives. In fact, it can be argued that digital technology is already ubiquitous for many stakeholders in mathematics education. Volume 2 of this series is titled The Mathematics Teacher in the Digital Era, which implies an exploration of the mathematics teacher as individual agent or actor in the experience of teaching and learning mathematics. What is exciting is the book editors' potentially unintended insights into teachers' affective sense, their beliefs, and their subsequent decisions for future pedagogical effort.     

科学计算软件在线性代数中地审美教学策略

线性代数是一门抽象性和理论性较强的学科，在教学中所涉及的例子较少能与实际生活密切相连，让人感觉到学习的内容与生产实际脱节，体现不了数学的应用价值。科学计算软件就是将抽象知识转换为形象，由理论转化为应用的一种工具，从而在线性代数的教学中融入审美教学成为可能。研究与实践表明，审美教学的实施对培养人的创新思维及数学应用意识，提高教学质量以及学生的数学素质方面起着重要的作用。     

How teachers integrate a math computer game: Professional development use,teaching practices,and student achievement

As more attention is placed on designing digital educational games to align with schools' academic aims (e.g., Common Core), questions arise regarding how professional development (PD) may support teachers' using games for instruction and how such integration might impact students' achievement. This study seeks to (a) understand how teachers use PD resources (e.g., technology personnel and game‐use workshops) for integration; (b) determine how teachers integrate games into their instruction; and (c) examine how those teaching practices are associated with student achievement. This mixed method study used survey and interview responses from elementary school teachers (n = 863) with access to PD resources for implementing a math game intervention and standardized math‐test scores from their second‐ through sixth‐grade students (n = 10,715). Findings showed few teachers sought PD assistance for integration, but many desired such support. Some reported using integrative practices (i.e., referencing game and using game‐generated progress reports) to identify struggling students, whereas several found integration challenging. Teachers' reordering of game objectives to align with lessons and viewing of game‐based PD videos were associated with increased student math achievement in our OLS‐analysis. However, this result was no longer statistically significant within a school fixed‐effects model, suggesting school differences may influence how strongly teachers' practices are associated with student achievement.     

Social capital,empowerment and educational change: a scenario of permeation of one‐to‐one technology in school

The standard model of technology diffusion argues that changing teachers' perceptions and epistemological beliefs about the value of an innovation is conducive to successful implementation. It is believed that teachers' receptiveness to technology and a re‐conceptualization of its role in teaching and learning can drive the development of new pedagogies and curricula and eventually, bring about a new learning experience. In this paper, we argued that changing teachers' conceptions does not necessarily effect change in their practices and student learning. The results of our case study indicated that the impetus for change comes from the social capital and informal social forces in the school, and that parental support, permeation of technology in student learning and teacher empowerment are pivotal to the success of the implementation.     

What influences geography teachers' usage of geographic information systems? A structural equation analysis

Understanding the usage of the geographic information system (GIS) among geography teachers is a crucial step in evaluating the current dissemination of GIS knowledge and skills in Taiwan's educational system. The primary contribution of this research is to further our understanding of the factors that affect teachers' GIS usage. The structural equation model was employed to analyze the data collected from 725 senior high school geography teachers. This was done using a survey questionnaire inspired by the Technology Acceptance Model (TAM), which postulates the importance of how teachers perceived the usefulness and ease of use of GIS. Further, this study investigates the direct effect of GIS workshop attendance on actual GIS usage and assesses whether GIS workshop attendance mediates the relationship between perceived ease of use, perceived usefulness, and actual usage. Structural equation modeling results suggest that the perceived usefulness of adopting GIS is vital as it directly affects teachers' attendance at GIS training, and can further prompt their application of GIS in lectures. The perceived ease of GIS use does not influence actual usage directly, but does affect teachers' GIS usage in teaching through perceived usefulness and workshop attendance. Finally, workshop attendance can increase teachers' usage of GIS and mediate the association between perceived usefulness and actual usage.     

Teachers' pedagogical change mechanism – Pattern of structural relations between teachers' pedagogical characteristics and teachers' perceptions of transactional distance (TTD) in different teaching environments

In this study we examined how the variables: teaching autonomy, teaching goals teaching style and teachers' perceptions of transactional distance (TTD) work together within the pedagogical decision-making process in different teaching environments. The independent variable was the teaching environment, which was divided into three: entirely distance environment (Distance-E); blended distance environment (Distance-B); and the traditional environment (Traditional). The dependent variables were teaching autonomy, teaching goals, teaching style and TTD, i.e. teachers' evaluation of their ability to communicate with the students, explain the content and their satisfaction from the teaching environment. One hundred and sixty faculty members who teach in distance environments and one hundred and sixty faculty members who teach in a traditional environment answered a five-part study questionnaire. The finding indicated that teaching autonomy, teaching goal and teaching style are significant predictors of TTD. Furthermore, teaching style serves as a mediator variable connecting teaching autonomy and the teaching goal to the TTD. This pattern was repeated in both structural and measurement models in all three teaching environments, indicating the centrality of the teaching style as well as teaching goal and teaching autonomy in the teachers' pedagogical decision-making process and teaching outputs. We concluded that affecting the teachers' pedagogical characteristics, which serve as mechanisms that influence the results of teaching, is an important component in the process of the technological change in 21st century education.     

Inquiry-based mobile learning in secondary school science education: A systematic review

Recent years have seen a growing call for inquiry-based learning in science education, and mobile technologies are perceived as increasingly valuable tools to support this approach. However, there is a lack of understanding of mobile technology-supported inquiry-based learning (mIBL) in secondary science education. More evidence-based, nuanced insights are needed into how using mobile technologies might facilitate students' engagement with various levels of inquiry and enhance their science learning. We, therefore, conducted a robust systematic literature review (SLR) of the research articles on mIBL in secondary school science education that have been published from 2000 to 2019. We reviewed and analysed 31 empirical studies (34 articles) to explore the types of mIBL, and the benefits and constraints of mIBL in secondary school science education. The findings of this SLR suggest new research areas for further exploration and provide implications for science teachers' selection, use and design of mIBL approaches in their teaching.     

Matlab在线性代数实践教学中的应用

文中论述了在线性代数教学中培养学生科学计算能力的必要性和重要性,通过把Matlab引入到定理验证、含符号变量行列式和线性方程组的教学实践,说明将线性代数理论、应用及Matlab有机结合,可以帮助学生直观地理解掌握理论知识,通过Matlab的应用掌握科学计算方法,提高学生的科学计算能力。     

The effect of SDLR and self‐efficacy in preschool teachers by using WS learning

This study examined whether self‐directed learning readiness (SDLR) moderates the association between Internet self‐efficacy and approaches to learning by web searching (ALWS). A total of 329 valid questionnaires were used for the correlation and path analysis. The results revealed that preschool teachers' SDLR significantly moderated the influence of Internet self‐efficacy and ALWS. Thus, this result suggests that spending more time on SDLR may increase preschool teachers' ALWS, which may enhance their teaching practice via web‐based professional development. Findings from this study may have implications for preschool teachers who aim to improve their teaching practice by targeting web‐searching environment and approaches to learning. The findings suggest that the positive mediation role of SDLR between Internet self‐efficacy and ALWS, especially effective learning, active learning and independent learning of SDLR. Because SDLR has the great influence on the online learning domain, preparing teacher's self‐directed learning ability is important for educational authorities to facilitate teachers' better performance when learning online.     

Action to process: constructing functions from algebra word problems

Abstract

The ability to build a mathematical model of a situation described in ordinary language has always been an important goal of school mathematics teaching. This paper gives one explanation for the difficulties many students experience with certain stylized modeling tasks (solving algebra word problems) and puts these difficulties in the context of a theory about the acquisition of the concept of mathematical function. It also describes a visual programming language that can act as a bridge, and in some sense an alternative, to formal algebraic notation as a language for describing situations. Tracing the author's classroom experiments and his reflections on those experiments, the paper suggests a method for helping students.     

Integrating Computer Algebra into Proof Planning

Mechanized reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two different tasks proving and calculating. Even more important, proof and computation are often interwoven and not easily separable.In this article we advocate an integration of computer algebra into mechanized reasoning systems at the proof plan level. This approach allows us to view the computer algebra algorithms as methods, that is, declarative representations of the problem-solving knowledge specific to a certain mathematical domain. Automation can be achieved in many cases by searching for a hierarchic proof plan at the method level by using suitable domain-specific control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows us to solve a large class of problems that are not automatically solvable by separate systems.Our approach also gives an answer to the correctness problems inherent in such an integration. We advocate an approach where the computer algebra system produces high-level protocol information that can be processed by an interface to derive proof plans. Such a proof plan in turn can be expanded to proofs at different levels of abstraction, so the approach is well suited for producing a high-level verbalized explication as well as for a low-level, machine-checkable, calculus-level proof.We present an implementation of our ideas and exemplify them using an automatically solved example.Changes in the criterion of rigor of the proof' engender major revolutions in mathematics. H. Poincaré, 1905