In this work, an interdisciplinary approach has been adopted for the study of: • teaching strategies of an Intelligent Tutoring System, in the paradigm of multiple teaching strategies, and • the use of Computer Algebra Systems (CAS) in teaching problem solving in university mathematics. As a result, the SIMTA (Styles Implemented by Methods Tactics Actions) theoretical framework has been developed to support and sustain teaching strategies in the paradigm of multiple teaching strategies. TeLoDe (TEaching Linear Ordinary Differential Equations), is a prototype Intelligent Tutoring System, teaching the solution of linear second order differential equations with constant coefficients in a novel way. This novel way, which has been empirically tested, has been achieved by augmenting Maple and represents an alternative use of CASs where the human lecturer and Maple are interlocked in a symbiotic and interdependent manner. In SIMTA, the contemporary concept of teaching strategy is rethought and proposed to be viewed at two fundamental levels: • the organisational level • and the operational level. The organisational level deals with the structure of the teaching strategy whereas the operational level deals with the manifestation of that structure. In SIMTA the organisational level is represented by a triple generic structure, method, tactic(s), action(s). A method is a mechanism for structuring the subject matter (e.g. analogy, examples, generalisation, specialisation). Likewise, a tactic is a mechanism for facilitating the interaction (e.g. explicit interaction, implicit interaction). An action is a low level activity such as display this message, ask this question. In SIMTA, the exact manifestation of the above generic structures (analogies, examples, implicit interaction, explicit interaction) depends on the concept of style: different styles result in different manifestations of the same generic structures. Thus, in SIMTA the concept of multiple teaching strategies is seen as merely a collection of teaching strategies manifested under the same style. These strategies operate with the aim of offering alternative representations of the same task at hand and ensuring that the lea~er is active by activating, directing and maintaining exploration. To help demonstrate the feasibility of SIMTA, two styles, the expository style and the , guided discovery style have been formed. The expository style draws on Ausubel's theory of meaningfulleaming, whereas, the guided discovery style draws on Bruner's work. These styles have been implemented in TeLoDe. TeLoDe, incorporates a teaching strategy module, based on a style, and declarative knowledge. Its purpose is threefold: (i) to serve as a research tool for the SIMTA framework, (ii) to serve as a prototype, demonstrating clearly how a 'second generation' CAS which undertakes the procedural aspect of mathematics allowing the human tutor to concentrate on its conceptual aspect, could be developed, (iii) to demonstrate how Maple and human lecturers are given clear roles which are, nevertheless, interdependent in carrying out the teaching of university mathematics. Two small-scale empirical studies were carried out in order to test SIMTA and TeLoDe respectively. The first study involved lecturers whereas the second study was carried out in a classroom environment. The results found from these studies demonstrate that TeLoDe has a potential as a teaching tool for problem solving in university mathematics in a novel way.

<p> Partial Evaluation (PE) is a program transformation technique that generates a specialized version of a program with respect to a subset of its inputs. PE is an automatic approach to program generation and meta-programming. This thesis presents a method of partially evaluating Maple programs using a fully online methodology.</p> <p> We present an implementation called MapleMIX, and use it towards two goals. Firstly we show how MapleMIX can be used to generate optimized versions of generic programs written in Maple. Secondly we use MapleMIX to mine symbolic computation code for residual theorems, which we present as precise solutions to parametric problems encountered in Computer Algebra Systems.</p> <p> The implementation of MapleMIX has been modularized using a high-level intermediate language called M-form. Several syntax transformations from Maple to M-form make it an ideal representation for performing program specialization. Many specialization techniques have been explored including a novel online approach to handle partially-static data structures and an on-the-fly syntax transformation technique that propagates loop context into the body of dynamic conditionals.</p> / Thesis / Master of Science (MSc)

At a time of transition, when the increasing availability and affordability of Computer Algebra Systems (CAS) presents mathematics educators with new challenges, this thesis explores two facets of students’ abilities and understanding that impact on the use of CAS in teaching and learning mathematics. In this thesis, these are called ‘Algebraic Insight’ and ‘Effective Use of CAS’. A framework is presented and described for each construct and then the frameworks are explored within the context of a course in introductory calculus, taught by the researcher to a class of 21 undergraduate tertiary students. Algebraic Insight is the subset of Symbol Sense required when using CAS for the mathematical solution phase of problem solving. The framework breaks Algebraic Insight into two aspects: ability to Link Representations (symbolic, numeric, graphical); and Algebraic Expectation, the cognitive skill required to monitor symbolic work (comparable to arithmetic estimation for monitoring numeric work). The framework of Effective Use of CAS is also divided into two aspects: Technical, using syntax and program features; and Personal, the willingness to use CAS in a judicious manner.

No description available.

We report on some experiences with the general purpose Computer Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce solving systems of polynomial equations and the way they present their solutions. This snapshot (taken in the spring 1996) of the current power of the different systems in a special area concentrates both on CPU-times and the quality of the output.

No description available.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

Informacinių ir komunikacinių technologijų (IKT) įtakota ugdymo technologijų raida pakeitė visų sričių, tarp jų ir matematikos ugdymo turinį. Konstruktyvistinę mokymosi paradigmą, pagrindinius šiuolaikinius didaktinius matematikos mokymo principus – suprantamumą, sąmoningą ir aktyvų žinių perėmimą, vaizdumą, mokymo diferencijavimą, individualizavimą ir kt. – gali įgyvendinti tik atitinkamos kompetencijos matematikos mokytojai. Šalies pedagogų kompiuterinio raštingumo standarto, numatančio mokytojų gebėjimus naudoti IKT ugdymo procese, vienas iš reikalavimų yra „Žinoti pagrindinių ugdymui naudojamų kompiuterinių programų tipus, gebėti analizuoti jų privalumus ir trūkumus. Mokėti pritaikyti svarbiausias bendrosios paskirties ir mokomąsias kompiuterines programas ugdymo procese“. Šio bendro reikalavimo taikymas matematikos mokymui – tai ne tik sudėtingų matematikai taikytinų technologijų žinojimo ir įvaldymo, bet ir gebėjimų tinkamai taikyti jas matematikos mokyme reikalavimas. Informacinių technologijų (IT) ištakos yra pirmosiose skaičiavimo mašinose, kurios buvo kurtos spręsti taikomuosius matematikos uždavinius. Ir asmeninių kompiuterių (AK) pirmoji paskirtis buvo matematinių uždavinių sprendimas. Todėl matematikai skirtų šiuolaikinių technologijų yra žymiai daugiau, negu taikytinų kituose moksluose. Dar daugiau, yra programų sistemų, kuriose, atrodo, realizuota visa matematika: jos ne tik skaičiuoja, pertvarko, prastina, vaizduoja, bet ir sprendžia specifinius bene visų... [toliau žr. visą tekstą] / The survey summarizes investigations of the author on the technological tools and their educational features applicable to mathematics. The purpose of the investigations is enhancement of the technological and educational competence of mathematics teachers. The main problems of the investigation are: characterization of the main technological tools applicable in mathematics at the university level, description which abilities, knowledge, and skills of application in teaching comprise the technological and educational ICT competence of a contemporary mathematician. Development of educational technologies, influenced by information and communication technologies (ICT), has changed the content of all spheres of education, and the content of mathematical education among them. A constructivist paradigm of learning, the main didactic principles of mathematics teaching: understanding, conscious and active acquiring of knowledge, visualization, differentiation and individualization of learning, etc., can realize only mathematics teachers of the adequate competence. The Standard of Computer Literacy of educators in Lithuania provides abilities of teachers to apply ICT in educational process. One of the requirements is “To know the main types of computer programmes useful for education, to have the ability to analyze their merits and demerits, to apply them in the educational process”. The application of the general requirement in teaching mathematics means not only the knowledge and... [to full text]

Computer Algebra Systems (CAS) have been changing the mathematics instruction requirements for many years. Since the tendency of using CAS in mathematics instruction has been rising for decades and reports have often been positive, the implementation of notebook classes seems to be the consequent next step of mathematics instruction supported by computers. Experiences that have been made with the use of CAS in PC-rooms can be transformed directly into the classroom. Hence the use of CAS is no longer limited to certain rooms. The permanent availability of the notebook with installed CAS offers the chance to realize these concepts that have already been approved with the use of CAS so far. The following speech shall show what these concepts could look like and that the use of notebooks is not only the further development of teaching in PC-classes. Examples from personal experience in teaching will especially show meanders and thought-provoking impulses in order to support teachers finding their way into teaching mathematics instruction in notebook classes successfully. Please allow me to point out two things in the beginning: (1) Yes, I am a vehement supporter of the use of notebooks (and the use of CAS in particular) in mathematics instruction. (2) No, I do not believe that teachers who have chosen another path (or at least partly) are teaching badly.