Undergraduates’ Collective Argumentation Regarding Integration of Complex Functions Within Three Worlds of Mathematics

Although undergraduate complex variables courses often do not emphasize formal proofs, many widely-used integration theorems contain nuanced hypotheses. Accordingly, students invoking such theorems must verify and attend to these hypotheses via a blend of symbolic, embodied, and formal reasoning. Using Tall’s three worlds of mathematics as a theoretical lens, this research explores undergraduate student pairs’ collective argumentation about integration of complex functions, with emphasis placed on students’ attention to the hypotheses of integration theorems. Data consisted of videotaped, semistructured interviews with two pairs of undergraduates, during which they collectively reasoned about thirteen integration tasks. Videotaped classroom observations were also conducted during the integration unit of the course in which these students were enrolled. Interview data were analyzed by categorizing participants’ responses according to Toulmin’s argumentation scheme, as well as classifying each statement as embodied, symbolic, formal, or blends of the three worlds. The student pairs’ responses were further coded according to Levinson’s four speaker roles in order to document how individuals contributed socially to the collective arguments, and backing statements were identified as either supporting a warrant’s validity, correctness, or field. Findings revealed that participants’ nonverbal modal qualifiers and explicit challenges to each other’s assertions catalyzed new arguments allowing students to reach consensus, verify conjectures, or revisit prior assertions. Hence, while existing frameworks identify two types of participation in collective argumentation, the aforementioned challenges suggest an important third type of participation. Although participants occasionally conflated certain formal hypotheses from the integration theorems, their arguments married traditional integral symbolism with dynamic gestures and clever embodied diagrams. Participants also attended to a phenomenon, referred to in the literature as thinking real, doing complex, in three distinct manners. First, they took care to avoid invoking attributes of real numbers that no longer apply to the complex setting. Second, they intermittently extended their real intuition to the complex setting erroneously. Third, they deliberately called upon attributes of the real numbers that were productive in describing analogous complex number operations. This three-tiered attention to the thinking real, doing complex phenomenon is notable because only the second type is currently documented in existing literature. Collectively, the findings suggest that instructors of complex analysis courses might wish to heavily underscore the importance of geometric interpretations of complex arithmetic early in the course and avoid utilizing acronyms that de-emphasize individual theorem hypotheses. The results also indicate that a more multimodal stance is needed when studying collective argumentation in order to capture covert aspects of students’ communication

Computer algebra techniques in object-oriented mathematical modelling.

This thesis proposes a rigorous object-oriented methodology, supported by computer algebra software, to generate and relate features in a mathematical model. Evidence shows that there is little heuristic or theoretical research into this problem from any of the three principal modelling methodologies: 'case study’, ‘scenario’ and ‘generic’. This thesis comprises two other major strands: applications of computer algebra software and the efficacy of symbolic computation in teaching and learning. Developing the principal algorithms in computer algebra has sometimes been done at the expense of ease of use. Developers have also not concentrated on integrating an algebra engine into other software. A thorough review of quantitative studies in teaching and learning mathematics highlights a serious difficulty in measuring the effect of using computer algebra. This arises because of the disparate nature of learning with and without a computer. This research tackles relationship formulation by casting the problem domain into object-oriented terms and building an appropriate class hierarchy. This capitalises on the fact that specific problems are instances of generic problems involving prototype physical objects. The computer algebra facilitates calculus operations and algebraic manipulation. In conjunction, I develop an object-oriented design methodology applicable to small-scale mathematical modelling. An object model modifies the generic modelling cycle. This allows relationships between features in the mathematical model to be generated automatically. The software is validated by quantifying the benefits of using the object-oriented techniques, and the results are statistically significant. The principal problem domain considered is Newtonian particle mechanics. Although the Newtonian axioms form a firm basis for a mathematical description of interactions between physical objects, applying them within a particular modelling context can cause problems. The goal is to produce an equation of motion. Applications to other contexts are also demonstrated. This research is significant because it formalises feature and equation-generation in a novel way. A new modelling methodology ensures that this crucial stage in the modelling cycle is given priority and automated

On the factorization of polynomials over algebraic fields

SIGLEAvailable from British Library Document Supply Centre- DSC:DX86869 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Plasma Dynamics

Contains research objectives and summary of research on twenty-one projects split into three sections, with four sub-sections in the second section and reports on twelve research projects.National Science Foundation (Grant ENG75-06242)U.S. Energy Research and Development Administration (Contract E(11-1)-2766)U.S. Energy Research and Development Agency (Contract E(11-1)-3070)U.S. Energy Research and Development Administration (Contract E(11-1)-3070)Research Laboratory of Electronics, M.I.T. Industrial Fellowshi

Identities for the gamma and hypergeometric functions: an overview from Euler to the present

A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013Equations involving the gamma and hypergeometric functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions to differential and integral equations. In this work we trace a brief history of the development of the gamma and hypergeometric functions, illustrate the close relationship between them and present a range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Our literature review will show that while continued research into hypergeometric identities has generated many new results, some of these can be shown to be variations of known identities. Hence, we will also discuss computer based methods that have been developed for creating and analysing such identities, in order to check for originality and for numerical validity

Core foundations, algorithms, and language design for symbolic computation in physics

This thesis presents three contributions to the field of symbolic computation, followed by their application to symbolic physics computations. The first contribution is to interfacing systems. The Notation package, which is developed in this thesis, allows the entry and the creation of advanced notations in the Mathematica symbolic computation system. In particular, a complete and functioning notation for both Dirac's BraKet notation as well as a full tensorial notation, are given herein. The second part of the thesis introduces a prototype based rule inheritance language paradigm that is applicable to certain advanced pattern matching rewrite rule language models. In particular, an implementation is presented for Mathematica. After detailing this language extension, it is adopted throughout the rest of the thesis. Finally, the third major contribution is a highly efficient algorithm to canonicalize tensorial expressions. By an innovative technique this algorithm avoids the dummy index relabeling problem. Further algorithmic optimizations are then presented. The complete algorithm handles linear symmetries such as the Bianchi identities. It also fully accommodates partial derivatives as well as mixed index classes. These advances in language and notations are extensively demonstrated on problems in quantum mechanics, angular momentum, general relativity, and quasi-spin. It is shown that the developments in this thesis lead to an extremely flexible, extensible, and powerful working environment for the expression and ensuing calculation of symbolic physics computations

Internal modes in high temperature plasmas

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1983.MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCEVita.Includes bibliographical references.by Geoffrey Bennett Crew.Ph.D