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Finite Fields
Published in Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Emerging Security Algorithms and Techniques, 2019
The word “algebra” is used to mean many different things. This word is used in the title of al-Khwarizmi’s book 1,200 years ago. But as a subject it is studied 4,000 years ago by ancient Babylonia and Egypt. Modern algebra, also called as Abstract algebra, is a branch of mathematics which deals with the algebraic structure of various sets such as real numbers, complex numbers, set of integers under modulo n (Zn), matrices, modules, vector spaces, lattices, and algebras (Hazewinkel, Gubareni, & Kiričenko, 2005). Knonecker (1888) claimed that the study of modern algebra began with his first paper of Vandermonde. Cauhy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory (Connor & Robertson, 2001).
Toward a unified account of definitions in mathematics education research: a systematic literature review
Published in International Journal of Mathematical Education in Science and Technology, 2023
Hermund André Torkildsen, Tore Alexander Forbregd, Eivind Kaspersen, Trygve Solstad
Definitions may therefore support the development of new definitions, both equivalent and non-equivalent. For example, the definition ‘a number of unifix cubes is an even number if we can pair two and two cubes such that no cube is left over’ does not extend to negative numbers, but the definition ‘an even number is a multiple of two’ easily does (the example space is extended). In abstract algebra a vector space is often defined with a (rather long) list of properties. However, we can define a vector space V to be an -module, where is a ring, by simply requiring that is a field (the example space becomes properly smaller, i.e. all vector spaces are modules, but not all modules are vector spaces).
Roadmap to glory: scaffolding real analysis for deeper learning
Published in International Journal of Mathematical Education in Science and Technology, 2023
Timothy Huber, Josef Sifuentes, Aaron T. Wilson
Adaptation of active learning methods is an effective evidence-based strategy to increase student learning (Theobald et al., 2020). We plan on mirroring the approach discussed here in other core upper-level mathematics courses such as Abstract Algebra, Advanced Linear Algebra, Topology, Number Theory, and other ‘gateway’ courses to advanced mathematics. There is an initial time investment required to adapt a course to this format. Animated and still illustrations were created using Matlab’s coding environment. The lecture notes were typeset with LaTex, and the lecture videos were recorded and edited using Camtasia video editing software. Some elements of the approach taken in Real Analysis may need to be altered to optimize learning in other courses. Furthermore, significant faculty involvement is needed to create lasting change in the way these other courses are taught. Faculty involved in the Real Analysis revamp speak to colleagues informally and formally about their work in teaching seminars.
Mathematicians’ beliefs, instruction, and students’ beliefs: how related are they?
Published in International Journal of Mathematical Education in Science and Technology, 2021
Furthermore, abstract algebra instruction specifically is important because it is a class required for graduation for 89% of mathematics departments training secondary pre-service teachers, and either required or generally taken in 95% of programs in the United States (Blair et al., 2013, p. 54). For mathematics majors in general, Abstract Algebra is required by more than 85% of mathematics departments across types of institutions (Blair et al., 2013, p. 53). Not only is Abstract Algebra important, but it might also be considered a ‘best case scenario’ for implementing and studying a variety of types of instruction. Many research-based reform materials, including the Inquiry-Oriented Abstract Algebra (IOAA) curriculum (Larsen et al., 2013), have been developed for Abstract Algebra, providing more materials for active instruction in this course than for many other upper-level mathematics courses (Johnson et al., 2018). Additionally, Abstract Algebra is often a terminal course, which potentially allows more flexibility with the course material.