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Computations in Mathematics, Mathematical Education, and Science

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A special issue of Computation (ISSN 2079-3197).

Deadline for manuscript submissions: closed (30 September 2024) | Viewed by 14066

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Special Issue Editor

Prof. Dr. Sergei Abramovich

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Guest Editor

School of Education and Professional Studies, State University of New York, Potsdam, NY, USA
Interests: mathematics education; differential equations; control theory

Special Issue Information

Dear Colleagues,

Just as advances in mathematics often depend on the methods of computation available, the effectiveness of applications of mathematics to education and science depends on our knowledge and understanding of how computers can support advances in areas that use mathematics. The aim of this Special Issue is to collect scholarly reports on the effective use of computations within the wide range of experiences, grade levels, and topics. Of special interest are submissions that demonstrate the duality of mathematical and computational methods in the sense that whereas computations facilitate access to mathematical knowledge, mathematics itself can be used to improve the efficiency of computations, which, in turn, enable advancements in various applications of mathematics to education and science.

At the pre-college level of mathematics education, the Special Issue seeks to identify successful experiences in using computations to communicate the presence of big ideas within seemingly mundane curricular topics and, by the same token, in enabling the study of traditionally difficult and conceptually rich topics through the use of computations. At the college level of mathematics education, the Special Issue invites articles that demonstrate how experimental approaches to mathematics that draw on the power of software to perform numerical and symbolic computations as well as graphical and geometric constructions make it possible to balance informal and formal learning of mathematical ideas. In applications of mathematics to science, this Special Issue invites submissions demonstrating how the availability of symbolic computations enables transition from results based on informal experiments to formal justifications of the results using methods of formal mathematics. Recommended topics to be considered may center on the following questions:

How does the use of computations affect mathematics research?
How are computations used in the preparation of PK-12 teachers of mathematics?
How does the use of computations enable the revision of undergraduate mathematics curricula?
How does the use of computations facilitate the transition from high school mathematics to university mathematics?
How does the growth of online degree programs affect the use of digital technology within mathematics courses of such programs?
How does the use or computations affect research in science?

Articles are expected to include a theoretical discussion of educational, mathematical, and epistemological issues associated with the use of computations in mathematics and their applications to education and science.

Prof. Dr. Sergei Abramovich
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Computation is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

mathematics
education
science
research
digital computation
curriculum development
online programs
teacher preparation

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Published Papers (6 papers)

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23 pages, 1730 KiB

Open AccessArticle

Teaching K–3 Multi-Digit Arithmetic Computation to Students with Slow Language Processing

by Richard M. Oldrieve

Computation 2024, 12(6), 128; https://doi.org/10.3390/computation12060128 - 19 Jun 2024

Cited by 1 | Viewed by 900 | Correction

Abstract

The purpose of this article is to present three related studies that build on each other to demonstrate first the need and then the efficacy of the Blended Arithmetic Curriculum (BAC) to help students overcome both slow language processing and the environmental effects of being a student in an urban school district. The author’s underlying theory is that K–3 students with slow language processing may be good at complex reasoning, but still struggle with retrieving basic computational facts. Nonetheless, if they did not learn their facts, these students would struggle with K–3 multi-digit arithmetic computation, and ultimately struggle with their hypothesized strength: seeing numeric patterns as would be needed in university level computation. To teach arithmetic facts conceptually, the author developed a paper and pencil curriculum that first teaches complex multi-digit addition with regrouping using a limited number of facts such as 5 + 5, 9 + 1, 1 + 9; 7 + 7, 7 + 8, 8 + 7, and 8 + 8 in problems such as 197 + 108 = 305 so that fact retrieval and computation are fast and accurate. At the end of 2nd grade, urban students with learning disabilities solved 42 two-digit by two-digit problems with 92 percent accuracy in an average of 7 min. The results matched those of suburban students and were significantly faster and more accurate than general education students in the same urban school. Full article

(This article belongs to the Special Issue Computations in Mathematics, Mathematical Education, and Science)

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15 pages, 6176 KiB

Open AccessArticle

Exploring Polygonal Number Sieves through Computational Triangulation

by Sergei Abramovich

Computation 2023, 11(12), 251; https://doi.org/10.3390/computation11120251 - 10 Dec 2023

Viewed by 1896

Abstract

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured. Full article

(This article belongs to the Special Issue Computations in Mathematics, Mathematical Education, and Science)

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16 pages, 938 KiB

Open AccessArticle

The Accuracy of Computational Results from Wolfram Mathematica in the Context of Summation in Trigonometry

by David Nocar, George Grossman, Jiří Vaško and Tomáš Zdráhal

Computation 2023, 11(11), 222; https://doi.org/10.3390/computation11110222 - 6 Nov 2023

Viewed by 1784

Abstract

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation [...] Read more.

\sum_{n = 0}^{44} {\tan (1 + 4 n)}^{°} .

This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics. Full article

(This article belongs to the Special Issue Computations in Mathematics, Mathematical Education, and Science)

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15 pages, 1902 KiB

Open AccessArticle

Computational “Accompaniment” of the Introduction of New Mathematical Concepts

by Andrey Lavrenov, Elena Tolkacheva and Sergei Pozdniakov

Computation 2023, 11(10), 194; https://doi.org/10.3390/computation11100194 - 2 Oct 2023

Cited by 1 | Viewed by 1484

Abstract

The computational capabilities of computer tools expand the student’s search capabilities. Conducting computational experiments in the classroom is no longer an organizational problem. This raises the “black box” problem, when the student perceives the computational module as a magician’s box and loses conceptual control over the computational process. This article analyses the use of various computer tools, both existing and specially created for “key” computational experiments, that aim at revealing the essential aspects of the introduced concepts using specific examples. This article deals with a number of topics of algebra and calculus that are transitional from school to university, and it shows how computational experiments in the form of a “transparent” box can be used. Full article

(This article belongs to the Special Issue Computations in Mathematics, Mathematical Education, and Science)

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28 pages, 416 KiB

Open AccessArticle

Revealing the Genetic Code Symmetries through Computations Involving Fibonacci-like Sequences and Their Properties

by Tidjani Négadi

Computation 2023, 11(8), 154; https://doi.org/10.3390/computation11080154 - 7 Aug 2023

Cited by 5 | Viewed by 5976

Abstract

In this work, we present a new way of studying the mathematical structure of the genetic code. This study relies on the use of mathematical computations involving five Fibonacci-like sequences; a few of their “seeds” or “initial conditions” are chosen according to the chemical and physical data of the three amino acids serine, arginine and leucine, playing a prominent role in a recent symmetry classification scheme of the genetic code. It appears that these mathematical sequences, of the same kind as the famous Fibonacci series, apart from their usual recurrence relations, are highly intertwined by many useful linear relationships. Using these sequences and also various sums or linear combinations of them, we derive several physical and chemical quantities of interest, such as the number of total coding codons, 61, obeying various degeneracy patterns, the detailed number of H/CNOS atoms and the integer molecular mass (or nucleon number), in the side chains of the coded amino acids and also in various degeneracy patterns, in agreement with those described in the literature. We also discover, as a by-product, an accurate description of the very chemical structure of the four ribonucleotides uridine monophosphate (UMP), cytidine monophosphate (CMP), adenosine monophosphate (AMP) and guanosine monophosphate (GMP), the building blocks of RNA whose groupings, in three units, constitute the triplet codons. In summary, we find a full mathematical and chemical connection with the “ideal sextet’s classification scheme”, which we alluded to above, as well as with others—notably, the Findley–Findley–McGlynn and Rumer’s symmetrical classifications. Full article

(This article belongs to the Special Issue Computations in Mathematics, Mathematical Education, and Science)

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