Assessing elementary algebra with STACK

This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS Maxima, is taken as a case study with which to test the implementation of the ideas developed in this paper. The general characteristics needed from a CAS for the application of computer aided assessment, or computer based learning is discussed. In particular the need for consistently implemented noun forms of elementary arithmetic operations, together with their traditional verb forms is identified.     

Do-it-yourself computational astronomy

We overview our GRAPE (GRAvity PipE) and GRAPE-DR project to develop dedicated computers for astrophysical N-body simulations. The basic idea of GRAPE is to attach a custom-build computer dedicated to the calculation of gravitational interaction between particles to a general-purpose programmable computer. By this hybrid architecture, we can achieve both a wide range of applications and very high peak performance. GRAPE-6, completed in 2002, achieved the peak speed of 64 Tflops. The next machine, GRAPE-DR, will have the peak speed of 2 Pflops and will be completed in 2008. We discuss the physics of stellar systems, evolution of general-purpose high-performance computers, our GRAPE and GRAPE-DR projects and issues of numerical algorithms.     

On the role of computers and complementary situations for gendering in mathematics classrooms

My study is about the (de)gendering effects of computers on mathematical teaching and learning. I approach the issue by directing attention to the possible intertwining of participants’ relationships to mathematics or computers with their relationships to gender in classroom interaction. According to my analysis, the constitution of gender-neutral relationships to mathematics and computers is subtly interspersed with gendered forms. They emerge from certain conducive situations, and are about mathematically unspecific program manipulations. Thus, findings do not indicate the further gendering of mathematics but the gendering of technology. However, computer algebra systems (CAS) seem to minimize the effects.     

Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.     

Students' attitude towards computer algebra systems (CAS) and their choice of using CAS in problem-solving

We explore students choice of using computer algebra systems (CAS) in problem-solving relative to their self-reported attitude towards learning mathematics with CAS. Our research design is a case study of nine Norwegian upper-secondary mathematics students with a wide range of attitude towards CAS. Our findings on routine problems indicate that (1) students use CAS whenever students perceive the problem as time-consuming regardless of their attitude towards CAS, and (2) students attitude affects their use of CAS whenever students perceive the problem as non-time-consuming. Norway, among other countries, has implemented CAS as an essential digital resource towards learning mathematics in upper-secondary school. Our discussion focuses on the implications of our findings have on local mathematics educators and national policy-makers.     

Teaching spline approximation techniques using maple

Using Computer Algebra Systems (CAS)-such as MAPLE-in teaching and learning mathematical concepts is a great challenge both from a didactical and a scientific point of view. We have to rewrite our traditional paper based teaching materials for interactive and living electronic worksheets, Only few statements and principles have to be acquired by the learner and the teacher from the CAS and after they can visualise, make animations modify quickly the program data, perform symbolic and numeric calculations step by step and in the whole, and verify deductions on their own. The author prepared Maple worksheets for teaching different types of function approximating techniques, such as interpolation-, least square-, spline and uniform approximation methods for post-graduate mechanical engineering students. In this paper we want to demonstrate how can we keep and improve the famous problem solving principles and rules given by G. Pólya and R. Descartes (Pólya 1962), when we use the capabilities of CAS. The education principles active learning, motivations and the successive phases are getting new meaning in the CAS. Our examples are always concerning with spline functions. Handling the formulas, calculating values and giving proofs are always in the form of Maple statements.     

Layout-aware scientific computing: A case study using the MILC code

Nowadays, high performance computers have more cores and nodes than ever before. Computation is spread out among them, leading to more communication cost than before. For this reason, communication can easily become the bottleneck of a system and limit its scalability. The layout of an application on a computer is the key factor to preserve communication locality and reduce its cost. In this paper, we propose a straightforward model to optimize the layout for scientific applications by minimizing inter-node communication cost. The model takes into account the latency and bandwidth of the network and associates them with the dominant layout variables of the application. We take the MILC code as an example and analyze its communication patterns. According to our experimental results, the model developed for the MILC code achieved a satisfactory accuracy for predicting the performance, leading to up to 31% performance improvement.     

Teachers in transition: Moving towards CAS-supported classrooms

In the near future many teachers may be required to incorporate CAS into their teaching practices. Based on classroom observations and interviews over two years, this paper reports how two teachers made the transition from using graphics calculators to CAS calculators while teaching differential calculus to upper secondary school students. Both teachers taught with CAS in ways that were consistent with their beliefs about learning and teaching. Over two years, the teachers' teaching approaches and purpose for use of technology were stable and seemed to be underpinned by their beliefs about learning. In contrast, both teachers made changes to the content they taught (and thus what they used technology for) in response to new institutional knowledge. Content choice seemed to be underpinned by the teachers' purpose for teaching. Other influences impacted on what the teachers taught and how they taught it: the teachers' content knowledge, their pedagogical content knowledge, and the lack of legitimacy of CAS as a tool for learning and during examinations in the trial school and wider educational community. The extent of differences noted between the responses of just two teachers indicates that there will be many responses to using CAS in classrooms, as teachers aim to achieve different learning goals and interpret their responsibilities to students in different ways.     

Influences of CAS and GC in early algebra

In the last years, several studies have investigated the role of technology in teaching and learning mathematics. However, the specific role of computer algebra systems (CAS) in early algebra in contrast to graphic calculators (GC) is still unclear. The CAYEN project is researching this field by comparing 13-year-old pupils—one GC class and two CAS classes have been observed while acquiring elementary algebraic competences with nearly the same teaching sequence. The field of algebraic competences is split into syntactic abilities and symbol sense. The results of this explorative case study show that the development of symbol sense is influenced by the adoption of CAS in the learning process. Especially when transitioning from arithmetic to algebra, the pupils’ views of algebra as well as their conceptions of algebraic objects seem to be affected by the availability of CAS.     

Toward Understanding the Role of Technological Tools in Statistical Learning

This article begins with some context setting on new views of statistics and statistical education. These views are reflected, in particular, in the introduction of exploratory data analysis (EDA) into the statistics curriculum. Then, a detailed example of EDA learning activity in the middle school is introduced, which makes use of the power of the spreadsheet to mediate students' construction of meanings for statistical conceptions. Through this example, I endeavor to illustrate how an attempt at serious integration of computers in teaching and learning statistics brings about a cascade of changes in curriculum materials, classroom praxis, and students' ways of learning. A theoretical discussion follows that underpins the impact of technological tools on teaching and learning statistics by emphasizing how the computer lends itself to supporting cognitive and sociocultural processes. Subsequently, I present a sample of educational technologies, which represents the sorts of software that have typically been used in statistics instruction: statistical packages (tools), microworlds, tutorials, resources (including Internet resources), and teachers' metatools. Finally, certain implications and recommendations for the use of computers in the statistical educational milieu are suggested.     

Appropriate use of a CAS in the teaching and learning of mathematics

If the use of a computer algebra system (CAS) is to be meaningful and have an impact on students, then it must be grounded in good pedagogy and have some clearly defined goals. It is the authors' belief that an important goal for teaching mathematics with the CAS is that courses be designed so that students can become active participants in their learning experience, planning the problem-solving strategies and carrying them out. The CAS becomes an important tool and a partner in this learning process. To this end, here the authors' have linked the use of the CAS to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.     

Toward Understanding the Role of Technological Tools in Statistical Learning

This article begins with some context setting on new views of statistics and statistical education. These views are reflected, in particular, in the introduction of exploratory data analysis (EDA) into the statistics curriculum. Then, a detailed example of EDA learning activity in the middle school is introduced, which makes use of the power of the spreadsheet to mediate students' construction of meanings for statistical conceptions. Through this example, I endeavor to illustrate how an attempt at serious integration of computers in teaching and learning statistics brings about a cascade of changes in curriculum materials, classroom praxis, and students' ways of learning. A theoretical discussion follows that underpins the impact of technological tools on teaching and learning statistics by emphasizing how the computer lends itself to supporting cognitive and sociocultural processes. Subsequently, I present a sample of educational technologies, which represents the sorts of software that have typically been used in statistics instruction: statistical packages (tools), microworlds, tutorials, resources (including Internet resources), and teachers' metatools. Finally, certain implications and recommendations for the use of computers in the statistical educational milieu are suggested.     

The power of various real-valued quantum queries

The computation of combinatorial and numerical problems on quantum computers is often much faster than on a classical computer in numbers of queries. A query is a procedure by which the quantum computer gains information about the specific problem. Different query definitions were given and our aim is to review them and to show that these definitions are not equivalent. To achieve this result we will study the simulation and approximation of one query type by another. While approximation is “easy” in one direction, we will show that it is “hard” in the other direction by a lower bound for the numbers of queries needed in the simulation. The main tool in this lower bound proof is a relationship between quantum algorithms and trigonometric polynomials that we will establish.     

“In Tomorrow's Office” — Networks and Microelectronics

Advances in micro-electronics and networking will bring steadily increasing computing power into tomorrow's office. There are differing approaches to the way in which local area networks and intelligent workstations will use a balance of local and remote computers and network sophistication. Emerging standards will assist the user in choosing amongst them. Improvements in microelectronics are changing the computer itself as new architectures to exploit parallelism are designed and new programming languages to instruct them are developed.     

Elastic waves in regions with multiple interfaces

The purpose of this article is to illustrate how the study of partial differential equations (PDEs) can be made more accessible to undergraduates through the use of applications and computers. The analysis initially concentrates on travelling waves on strings and their reflections from boundaries. Then wave behaviour on strings with multiple interfaces, where incident waves are repeatedly reflected and transmitted, is investigated. Finally, the results are extended to waves in three-dimensional bodies in which longitudinal and transverse waves occur simultaneously. The effectiveness of a computer algebra system (CAS) for visualizing and understanding solutions is demonstrated. In particular, snapshots and sample code for Mathematica animations are provided to demonstrate the physical nature of the solutions. Although intended for PDE students, the contents of this paper are accessible to undergraduates who have studied ordinary differential equations (ODEs) and would be appealing to anyone with a natural curiousity about waves.     

Parent Beliefs about Technology and Innovative Mathematics Instruction

Parent beliefs about roles of education, teachers, computers, and innovative mathematics instruction were examined through factor analysis. Strong relationships between parent beliefs regarding teacher and computer roles were found. The beliefs of parents about the similar roles of teachers and computers in education may impact the implementation of innovations in mathematics education and the uses of computers in education. Reciprocally, the ways computers are implemented in education may impact the beliefs parents have about the purposes of education.     

DP/MIS and OR: what was,what is,what could be

It has been three decades since electronic computers were first introduced into the business community. The application of computers to business functions and business information problems has led to profound changes in the way that businesses operate. OR began to penetrate the business community at roughly the same time that the first large-scale DP applications began to appear.The computer has had a significant impact on OR as an “enabling device”, at least. The OR community has not, however, paid particular attention to the evolution of the computer, its use in business information applications, nor the management problems associated with effective use. The explosive growth of computer technology, the utilization of the computer in new business areas such as office automation, and the evolving awareness of concepts such as Information Resources Management provide a renewed opportunity for OR to contribute substantially.The objectives of this review paper are to describe this new opportunity area for OR, to place it in an appropriate historical and technological context, and to suggest ways in which the OR community might take up this important challenge.     

Euklides dynamic geometry

The spreading of computers is becoming more and more inevitable in education as well. Promising pathfinders are the computerised algebraic systems (CAS) and the dynamic geometrical systems (DGS). These tools are very important for students to be able to play, experiment, recognise the problems and ask the questions themselves. If we would give way to more experimenting activities, we could bring those number of students already derailed by the educational system back to the joy of gaining knowledge on their own.This article introduces the Euklides Dynamic Geometry software, comparing to concurrent softwares and suggesting conceptional questions of dynamic geometry.Important special abilities are animation, track of a point (locus) and tools for examination of structure.     

CAS in general mathematics education

A rational discussion of the use of Computer algebra systems (CAS) in mathematics teaching in general education needs an explicit image of (general) mathematics education, an explication of global perspectives and goals on mathematics teaching focusing on general education (chapter 1). The conception of general education according to the «ability of communication with experts» described in chapter 2 can be such an orientation for analysing, considering, classifying and assessing the didactical possibilities of using CAS. CAS are materialised mathematics allowing for more or less exhaustive outsourcing of operative (also symbolically) knowledge and skills to the machine. This frees up space of time as well as mental space for the development of those competences being in our view relevant for general mathematics education. In chapter 3 the idea of outsourcing and the role of CAS for it is discussed more detailed as well as consequences being possible for the CAS-supported teaching of mathematics. Beyond, CAS can be didactically used and reflected as a model of communication between (mathematical) experts and lay-persons (chapter 4). Chapter 5 outlines some research perspectives.     

Discrete mathematics course supported by CAS MATHEMATICA

In this paper, we discuss examples of assignments for a course in discrete mathematics for undergraduate students majoring in business informatics. We consider several problems with computer-based solutions and discuss general strategies for using computers in teaching mathematics and its applications. In order to evaluate the effectiveness of our approach, we conducted an anonymous survey. The results of the survey provide evidence that our approach contributes to high outcomes and aligns with the course aims and objectives.