The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?

Guardeño, Antonio José Durán; Riera, Mario Pérez; Malumbres, Juan

doi:10.1090/noti1173

Cited by 39 publications

(23 citation statements)

References 5 publications

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“…It is an area of Mathematics perceived as the main source of failure in the undergraduate level because of its nature which involves abstract and complex ideas and the way it is being taught to the students (Sahin, Cavlazoglu & Zeytuncu, 2015). With these, initiatives around the world have introduced a range of innovative and interactive learning technologies such as graphic software (Robutti, 2010;Lavizca, 2010) and computer algebra system (Özgün-Koca, 2010;Mignotte, 2012;Durán, Pérez & Varona, 2014) to explore Calculus concepts. The use of these technologies offer new ways to learn and teach Calculus that help deepen students' understanding of abstract and complex ideas (Arango, Gaviria & Valencia, 2015;Šumonja, Veličković & Šubarević, 2015;Zakaria & Salleh, 2015) which include conceptual understanding (Bartell, Webel, Bowen & Dyson, 2013;Richland, Stigler & Holyoak, 2012) and procedural skills (Rittle-Johnson & Schneider, 2014;Cragg & Gilmore, 2014) and also increases positive attitude of students towards the subject (Sang, Valcke, Van Braak & Tondeur, 2010;Yuan & Chun-Yi, 2012).…”

Section: Introductionmentioning

confidence: 99%

Improving students' attitude, conceptual understanding and procedural skills in differential calculus through Microsoft mathematics

Mendezabal

Tindowen

2018

J. Technol. Sci. Educ.

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This study examined the effects of using Microsoft Mathematics on students' attitude, conceptual understanding, and procedural skills in Differential Calculus. A quasi-experimental research design was used in which two different learning environments were compared. The participants of the study were two classes of Electrical Engineering students enrolled in Differential Calculus course, assigned randomly as control and experimental groups with 30 students in each group. The control group was taught using the traditional approach of teaching Differential Calculus while the experimental group was taught the same lessons using the Microsoft Mathematics embedded activity sheets. The experimental group learned through exploration and discovery of various concepts. The findings indicated that the participants had little understanding of the concepts and processes of Calculus prior to the conduct of the study. A significant improvement in their performances was noted after the experimentation. This suggests that the use of Microsoft Mathematics in teaching and learning Differential Calculus improves students' conceptual understanding and procedural skills. It is also found that the use of Microsoft Mathematics in teaching and learning calculus is equally effective as the traditional approach. In terms of attitude, the experimental group demonstrated a "favorable" to "very highly favorable" attitude along the five (5) domains of the MTAS. A significant difference exists between the pretest and posttest attitude of the participants on the domain "learning Mathematics with technology".

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Section: Introductionmentioning

confidence: 99%

Improving students' attitude, conceptual understanding and procedural skills in differential calculus through Microsoft mathematics

Mendezabal

Tindowen

2018

J. Technol. Sci. Educ.

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show abstract

“…For example, they presented an example involving a 14 × 14 matrix of pseudorandom integers between −99 and 99, which was then multiplied by a certain diagonal matrix with large entries, and then added to another matrix of pseudorandom integers between −999 and 999. When they then attempted to compute the determinant of the resulting fixed matrix using Mathematica version 9, the found that the results were not consistent -they often obtained different answers for the same problem [27]. This problem now seems to have been resolved in Mathematica version 10, according to some tests by the present authors.…”

Section: Reproducibility In Symbolic Computingmentioning

confidence: 81%

Facilitating Reproducibility in Scientific Computing: Principles and Practice

Bailey

Borwein²,

Stodden³

2016

Reproducibility: Principles, Problems, Practices, and Prospects

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“…Computer algebra systems, such as Mathematica and Maple, have also been used to solve differential equations symbolically and to overcome the inaccuracies introduced by computer arithmetic based computations and numerical methods. However, the algorithms used by these systems are not rigorously verified and thus can produce error-prone results [29]. These flaws in the traditional techniques are tremendously undesirable in case of the high safety-critical domain of transportation, as ignoring some corner cases may lead to dire consequences, such as frequent traffic congestions, road accidents and loss of human lives in worst cases.…”

Section: Related Workmentioning

confidence: 99%

Toward the Formalization of Macroscopic Models of Traffic Flow Using Higher-Order-Logic Theorem Proving

et al. 2020

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Next-generation transportation will be integrated, interconnected and highly autonomous. One key challenge in traffic management is ensuring safety while maintaining the required level of service quality. In such future mobility systems, rigorous formalization and validation will thus become critical to ensure that the transportation network operates as intended, traffic properties are reliably maintained, and services resume in a timely manner after potential disruptions. Formal methods have recently gained considerable attention as a modeling and verification paradigm capable of addressing many challenges associated with next-generation transportation systems. In this regard, we propose to use higher-orderlogic theorem proving for formally analyzing transportation systems. As a first step towards this direction, we present a logical framework for the formal analysis of macroscopic models of traffic flow. Leveraging upon the high expressiveness of the underlying logic, we formally model the continuous components of macroscopic models while capturing their real behavior. In particular, we present a formalization of some foundation concepts of macroscopic models, namely density, flow rate, mean speed, relative occupancy, and shockwave using the higher-order-logic theorem prover HOL Light. This choice is primarily motivated by the fact that the macroscopic models deal with the traffic flow dynamics and thus play a vital role in planning strategies in allocating resources for implementing optimized and balanced transportation systems. For illustration, we perform the formal input-output and shockwave analysis of a German freeway. The case study demonstrated the practicability of this formal approach due to the high expressiveness of the underlying logic. The proposed research is first step towards formalizing the foundational mathematical theories and core concepts of traffic flow theory. This accomplishment will open new ways to plan and model various components of the transportation systems such as highway links, ramp metering, merging behavior and eventually address the problem of routing vehicles in a network of automated vehicles. INDEX TERMS Input-output analysis, macroscopic model, shockwave analysis, theorem proving, transportation modeling, validation methods.

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The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?

Cited by 39 publications

References 5 publications

Improving students' attitude, conceptual understanding and procedural skills in differential calculus through Microsoft mathematics

Improving students' attitude, conceptual understanding and procedural skills in differential calculus through Microsoft mathematics

Facilitating Reproducibility in Scientific Computing: Principles and Practice

Toward the Formalization of Macroscopic Models of Traffic Flow Using Higher-Order-Logic Theorem Proving

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