Using CAS to solve a mathematics task: A deconstruction

I investigate how and whether a heterogeneous group of first-year university mathematics students in South Africa harness the potential power of a computer algebra system (CAS) when doing a specific mathematics task. In order to do this, I develop a framework for deconstructing a mathematics task requiring the use of CAS, into its primary components. This framework is based on the semiotic notion of diagrammatic reasoning whereby reasoning consists of construction of signs, transformation of signs, and observation and interpretation of signs. I use the framework to distinguish between the activities of students who were computer literate on entry to university and those who were not computer literate. The analysis suggests that formerly non-computer literate students are no worse than computer literate students in using CAS to construct various representations of signs, but that they are less able to interpret these signs. I propose that, in the South African context, this is largely due to inequities in prior mathematical education, rather than a lack of computer literacy per se.     

Experiences of using computer assisted assessment in engineering mathematics

This article discusses issues associated with large-scale computer assisted assessment (CAA) of the mathematical skills of first-year undergraduate engineering students. It explains the background to this testing as it relates to a wider open-learning project in engineering mathematics. Details of issues specifically associated with mathematics testing are given. Practical issues of constructing appropriate tests, their implementation and their delivery are described. Finally, the results of a survey of students' perceptions of CAA and the influence of these on future developments are presented.     

Using computer algebra systems in mathematical classrooms

Abstract This paper describes a research whose main focus is the use of Computer Algebra Systems (CAS) in mathematical classrooms and the didactical possibilities linked with its use. The possibilities of integrating Self-Regulated Learning (SRL) within the CAS environment are brought into focus. Forty-three Israeli students (mean age 13.3) were assigned to two learning algebraic groups. The first group received explicit meta-cognitive SRL with CAS (CAS + SRL); the second group was exposed to CAS without SRL (CAS). Empirical results from the experimental and case study designs revealed that (CAS + SRL) students outperformed (CAS) students on algebraic thinking and that (CAS + SRL) students regulated their learning more effectively.     

A Computer Algebra Primer and Homework Exercises for use in an Intermediate Macroeconomics Course – A Student/Teacher Collaboration

We discuss using computer algebras systems (CAS) in an undergraduate intermediate macroeconomics class. The criteria used in choosing a CAS are considered. One criterion was that students would bear little or no financial. The lessons learned in implementing a CAS – things that worked well and those that did not work so well – are noted. We also discuss the need for communications with information technology staff for successful implementation. An erratum to this article can be found at     

Student Proof Exercises Using MathsTiles and Isabelle/HOL in an Intelligent Book

The Intelligent Book project aims to improve online education by designing materials that can model the subject matter they teach, in the manner of a reactive learning environment. In this paper, we investigate using an automated proof assistant, particularly Isabelle/HOL, as the model supporting first year undergraduate exercises in which students write proofs in number theory. Automated proof assistants are generally considered to be difficult for novices to learn. We examine whether, by providing a very specialized interface, it is possible to build something that is usable enough to be of educational value. To ensure students cannot “game the system” the exercise avoids tactic-choosing interaction styles but asks the student to write out the proof. Proofs are written using MathsTiles: composable tiles that resemble written mathematics. Unlike traditional syntax-directed editors, MathsTiles allows students to keep many answer fragments on the canvas at the same time and does not constrain the order in which an answer is written. Also, the tile syntax does not need to match the underlying Isar syntax exactly, and different tiles can be used for different questions. The exercises take place within the context of an Intelligent Book. We performed a user study and qualitative analysis of the system. Some users were able to complete proofs with much less training than is usual for the automated proof assistant itself, but there remain significant usability issues to overcome.     

面向自然交互的多通道人机对话系统中答句自动生成方法

多通道自然人机对话系统要求计算机能够对用户的语句产生智能应答,传统的人机对话系统由于知识库的限制以及用户话语的随意性,当对话内容超出知识库范围时,系统将无法应答或产生与用户期望不符的回答,这在一定程度上影响了人机对话系统用户的体验感.为了解决该问题,提出了一种融合多模态历史交互信息和面向数据的句法分析(data-oriented parsing,简称DOP)模型的最优答句生成方法:首先从大规模句法树库中提取上下文无关文法的语法规则,然后结合对话过程中用户呈现的表情、姿态等多模态历史交互信息,融合DOP模型对上下文无关文法生成的汉语句子进行过滤,最终生成一个符合语法规则且符合语义的答句返回给用户,让计算机在无法获得知识库支撑时,根据交互历史信息生成应对当前对话的语句,有效地提升了多通道自然人机交互系统用户的体验感.该方法应用于交通信息查询以及咖啡厅的多主题多模态人机自由对话系统.用户的体验表明,该方法能够有效提高用户交互的自然度和体验感.     

基于Maple系统的交巡警服务平台的设置与调度实验

基于计算机代数系统Maple,结合2011全国大学生数学建模竞赛B题需要解决的问题,通过建立数学模型,采用数学实验技术,给出了满意的解答,对于计算机相关领域的教学工作具有一定的指导意义。     

基于Maple系统的灾情巡视路线

基于计算机代数系统Maple,结合1998年全国大学生数学建模竞赛B题需要解决的问题,通过建立数学模型,采用数学实验技术,给出了满意的解答,对于计算机相关领域的教学工作具有一定的指导意义。     

Schoenfeld’s problem solving theory in a student controlled learning environment

This paper evaluates the effectiveness of a student controlled computer program for high school mathematics based on instruction principles derived from Schoenfeld’s theory of problem solving. The computer program allows students to choose problems and to make use of hints during different episodes of solving problems. Crucial episodes are: analyzing the problem, selecting appropriate mathematical knowledge, making a plan, carrying it out, and checking the answer against the question asked.     

一种基于数学表达式的二义文法分析算法

随着计算机科学领域智能化要求的提高,文法分析也需要提供更强壮的功能,要求能对二义文法提供可靠分析。该文提出了一种基于数学表达式的二义文法分析算法,很好地判别了某些数学表达式的正确性。     

On mathematical models of adaptive systems

The aim of this paper is to present a mathematical model for a system to control an object favourably which is unknown for the system—an adaptive system. We make the concept of adaptiveness precise first and consider systems as stochastic sequential machines (SSM's) to answer the question : what kind of logical structure is necessary and sufficient for a SSM to be adaptive. Some necessary conditions for a SSM to be adaptive are shown, and for a specified class of SSM's, the necessary and sufficient conditions to be adaptive are given. Furthermore, some examples of adaptive SSM's are shown.     

Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

We present a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, à la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL (T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order \(4\times 42=168\)), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.     

MathPartner computer algebra

In this paper, we describe general characteristics of the MathPartner computer algebra system (CAS) and Mathpar programming language thereof. MathPartner can be used for scientific and engineering calculations, as well as in high schools and universities. It allows one to carry out both simple calculations (acting as a scientific calculator) and complex calculations with large-scale mathematical objects. Mathpar is a procedural language; it supports a large number of elementary and special functions, as well as matrix and polynomial operators. This service allows one to build function images and animate them. MathPartner also makes it possible to solve some symbolic computation problems on supercomputers with distributed memory. We highlight main differences of MathPartner from other CASs and describe the Mathpar language along with the user service provided.     

Do mathematicians integrate computer algebra systems in university teaching? Comparing a literature review to an international survey study

We present a comparative study of a literature review of 326 selected contributions (Buteau, Marshall, Jarvis & Lavicza, 2010) to an international (US, UK, Hungary) survey of mathematicians (Lavicza, 2008) regarding the use of Computer Algebra Systems (CAS) in post-secondary mathematics education. The comparison results are organized with respect to four emerging themes: Issues in CAS integration and mathematical learning, the notion of mathematical literacy, diverse uses of CAS by practitioners, and, potential benefits of CAS integration. Our analysis suggests that the results of the literature review strongly support the findings and conclusions of Lavicza’s international survey. We contend that Lavicza’s concluding statement about the need to more holistically examine technology integration in post-secondary mathematics departments was significantly realized through both of the compared studies.     

Compensating for noise and perturbances in linear object control

We consider the problem of constructing a robust control system for a linear stationary object under unmeasurable perturbances acting on the object. The perturbances are unbounded on the input of the system and bounded on the output. Based on the system inclusion technique, we consider necessary and sufficient conditions for the linear stationary systems’ invariance to perturbances. With the auxiliary contour method, we compensate for the undesirable effects of perturbances on controlled variables. We give an illustrative example and computer modeling results.     

A new approach of clustering based machine-learning algorithm

Machine-learning research is to study and apply the computer modeling of learning processes in their multiple manifestations, which facilitate the development of intelligent system. In this paper, we have introduced a clustering based machine-learning algorithm called clustering algorithm system (CAS). The CAS algorithm is tested to evaluate its performance and find fruitful results. We have been presented some heuristics to facilitate machine-learning authors to boost up their research works. The InfoBase of the Ministry of Civil Services is used to analyze the CAS algorithm. The CAS algorithm is compared with other machine-learning algorithms like UNIMEM, COBWEB, and CLASSIT, and was found to have some strong points over them. The proposed algorithm combined advantages of two different approaches to machine learning. The first approach is learning from Examples, CAS supports Single and Multiple Inheritance and Exceptions. CAS also avoids probability assumptions which are well understood in concept formation. The second approach is learning by Observation. CAS applies a set of operators that have proven to be effective in conceptual clustering. We have shown how CAS builds and searches through a clusters hierarchy to incorporate or characterize an object.     

Modeling Biological Networks by Action Languages via Answer Set Programming

We describe an approach to modeling biological networks by action languages via answer set programming. To this end, we propose an action language for modeling biological networks, building on previous work by Baral et al. We introduce its syntax and semantics along with a translation into answer set programming, an efficient Boolean Constraint Programming Paradigm. Finally, we describe one of its applications, namely, the sulfur starvation response-pathway of the model plant Arabidopsis thaliana and sketch the functionality of our system and its usage.     

A web based collaborative testing environment

This paper presents a computer supported collaborative testing system built upon the Siette web-based assessment environment. The application poses the same set of questions to a group of students. Each student in the group should answer the same question twice. An initial response is given individually, without knowing the answers of others. Then the system provides some tools to show the other partners' responses, to support distance collaboration. Finally a second individual answer is requested. In this way assessment and collaboration activities are interlaced. At the end of a collaborative testing session, each student will have two scores: the initial score and the final score. Three sets of experiments have been carried out: (1) a set of experiments designed to evaluate and fine tune the application, improve usability, and to collect users' feelings and opinions about the system; (2) a second set of experiments to analyze the impact of collaboration in test results, comparing individual and group performance, and analyzing the factors that correlate to those results; and (3) a set of experiments designed to measure individual short-term learning directly related to the collaborative testing activity. We study whether the use of the system is associated with actual learning, and whether this learning is directly related to collaboration between students. Our studies confirm previous results and provide the following evidence (1) the performance increase is directly related to the access to other partners' answers; (2) a student tends to reach a common answer in most cases; and (3) the consensus is highly correlated with the correct response. Moreover, we have found evidence indicating that most of the students really do learn from collaborative testing. High-performing students improve by self-reflection, regardless the composition of the group, but low-performing students need to be in a group with higher-performing students in order to improve.