2,810 research outputs found
Active Learning in Sophomore Mathematics: A Cautionary Tale
Math 245: Multivariate Calculus, Linear Algebra, and Differential Equations with Computer I is the first half of a year-long sophomore sequence that emphasizes the subjects\u27 interconnections and grounding in real-world applications. The sequence is aimed primarily at students from physical and mathematical sciences and engineering. In Fall, 1998, as a result of my affiliation with the Science, Technology, Engineering, and Mathematics Teacher Education Collaborative (STEMTEC), I continued and extended previously-introduced reforms in Math 245, including: motivating mathematical ideas with real-world phenomena; student use of computer technology; and, learning by discovery and experimentation. I also introduced additional pedagogical strategies for more actively involving the students in their own learning—a collaborative exam component and in-class problem-solving exercises. The in-class exercises were well received and usually productive; two were especially effective at revealing normally unarticulated thinking. The collaborative exam component was of questionable benefit and was subsequently abandoned. Overall student performance, as measured by traditional means, was disappointing. Among the plausible reasons for this result is that too much material was covered in too short a time. Experience here suggests that active-learning strategies can be useful, but are unlikely to succeed unless one sets realistic limits to content coverage
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
Symbolic computation: systems and applications
The article presents an overview of symbolic computation systems, their classification-in-history, the most popular CAS, examples of systems and some of their applications. Symbolics versus numeric, enhancement in mathematics, computing nature of CAS, related projects, networks, references are discussed
SMT-Solving Induction Proofs of Inequalities
This paper accompanies a new dataset of non-linear real arithmetic problems
for the SMT-LIB benchmark collection. The problems come from an automated proof
procedure of Gerhold--Kauers, which is well suited for solution by SMT. The
problems of this type have not been tackled by SMT-solvers before. We describe
the proof technique and give one new such proof to illustrate it. We then
describe the dataset and the results of benchmarking. The benchmarks on the new
dataset are quite different to the existing ones. The benchmarking also brings
forward some interesting debate on the use/inclusion of rational functions and
algebraic numbers in the SMT-LIB.Comment: Presented at the 2022 SC-Square Worksho
Making Presentation Math Computable
This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book
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