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Mathematical frameworks termed anomalous relaxation models are used to characterize the distribution of prime numbers in an unconventional way. Many scholars need clarification on the challenges with prime numbers (PN), because of their mysterious and complicated qualities because it is challenging to build a model that characterizes the relaxed distribution of PN in a collection of all natural numbers. Although certain qualitative methods, like Riemann's Theory, exist to the best of the educational knowledge can characterize the distribution of the prime numbers, the law controlling the distribution of the PN is unclear, and the correct model needs to be established. The distribution of prime numbers is obtained in the current work for the first time using the anomalous relaxation model with the fractional derivative. The new model matches well with the data of PN, according to comparisons with Riemann's Theory and the PN Theory of sustainable development education. [ABSTRACT FROM AUTHOR]

The subject is introduced by a reflection on the different aspects of numbers and a review of a paper written by Andreas Nieder: Representing Something Out of Nothing: The Dawning of Zero. This is followed by a reflection on the precise nature of the transformation of zero from a signifier of quantity to a number. Edmund Husserl called this the constitution (Stiftung) of the zero as a new ideal object and he described the process in The Origin of Geometry. Zero as a number is an element of a self-contained symbolic deductive system that does not refer to the life world and has no sense at all; it is, in the words of the analytic philosophical tradition, nonsense. This implies that no meaning can possibly be transferred from zero as a quantity to zero as a number. The history of zero as a number coincides with the history of abstract mathematics and the development of calculation systems, the tremendous usefulness of which is precisely based on its very meaninglessness. All speculation about any eventual progress of the concept of zero as a quantity to zero as a number is therefore unsubstantiated. Marc Richir showed that the constitution of an ideal object is a leap; it leaves a gap between its new sense and its historical basis. This chapter will conclude with the question whether there is a link between this historical basis of zero as a quantity and the concept of emptiness in Buddhism; this supposed relation has been and still is the pet of quite a few Buddhologists. [ABSTRACT FROM AUTHOR]

This is an exploration of the mathematics, history, and philosophy of zero. It looks at zero and its development from three perspectives: purely as a sign (syntactic); its meaning (semantics); and its social contexts, roles, and uses (pragmatic). Adopting the meaning as use theory, it argues that the meaning of zero was not fully established until the time of Brahmagupta, at around 628 CE, for Brahmagupta established the role of zero within the network of number relations in the integers. Most notably, zero is the additive identity. This synthesis is analogous for number theory with Euclid's geometry. Thus, it is posited that there are stages in the development of the concept of zero. First, its predecessor serves as a placeholder in an empty column. Second, zero is understood as a sign for nothing (nought), resting on the concept of the empty void. Third, its role as additive identity is understood and accepted. The conjecture that the idea of void is a prerequisite for the development of the number zero is confirmed positively in Indian and Mayan cultures. But the case of Ancient Egypt, which has the concept of void but not zero shows that having the concept of void appears, not of itself, sufficient. [ABSTRACT FROM AUTHOR]

Copyright of International Journal of Education Technology & Scientific Researches is the property of Ayse Burcu Ulusoy and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences.

The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences. [ABSTRACT FROM AUTHOR]

The first contact that a pre-service teacher has with didactics of mathematics is the notion of natural number, being still in the university classroom and not having started working a real classroom. Therefore, the main objective of this work is to relate the knowledge of a trainee teacher to the different difficulties developed by the children and to evaluate the learning processes in a real environment. The participants were 20 future teachers and 40 (9-year-old) children. During the experience, six letters closely related to the contents of the university subject were exchanged; as well as two socialisation letters; and four videos, including two presentation and two farewell videos. Among the results obtained, we highlight that the participating university students have been able to reinforce the knowledge learned in class through the analysis of the children's resolutions, specifically the following: a) counting, b) resolution of additive-concrete situations, c) resolution of multiplicative-concrete situations, d) resolution of a monetary problem, and e) being aware of the manipulation of Cuisenaire's rods.

According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey's bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess an exact quantity of the elements in a collection without counting them), or on the ability to approximate quantities (i.e., to assess an approximate quantity of the elements in a collection without counting them), or both, fail to provide a conceptual basis for bootstrapping the concept of an exact natural number. In particular, I argue that none of the existing theories explains one of the key characteristics of the natural number structure: the equidistances between successive elements of the natural numbers progression. I suggest that this regularity could be based on another innate cognitive ability, namely sensitivity to the regularity of rhythm. In the final section, I propose a new position within the core cognition paradigm, inspired by structuralist positions in philosophy of mathematics. [ABSTRACT FROM AUTHOR]

In this research, tools of the Ontosemiotic Approach (OSA) were used to analyse the formative experience of two teachers in Primary Education service, guided towards the design and implementation of tasks for mobile learning, based on the meaning of the natural number and the use of Kahoot. The experience was developed during a workshop on the design of interactive lessons, where teachers were trained on the use of Kahoot for the design of mathematical content and its implementation in a real class context. Didactic engineering in the general sense of OSA was used as a methodological element. Specifically, the case of the design and implementation of an interactive lesson on the meaning of number sequence was analysed, where some results of the retrospective analysis of the formative experience indicated that: 1) the reference meaning of the natural number is a tool that allows to guide the design of tasks towards the partial meanings, the identification of primary objects, and the Ontosemiotic analysis of them; 2) the use of mobile devices and Kahoot together motivated the teachers. With these results, two training actions are proposed, aimed at teachers: 1) the design of mobile content from the reference meaning of the natural number, and 2) the assessment of the didactic suitability of teaching experiences mediated by mobile devices. [ABSTRACT FROM AUTHOR]

Copyright of Educacao e Pesquisa is the property of Faculdade de Educacao da Universidade de Sao Paulo and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper we focus on the cardinal aspect of the natural number, where the number is associated with a set to indicate the amount of elements it has. It is about studying how cardinal thinking is in children in pre-school education. Presenting tasks with the logical-mathematical schemes of the cardinal number, and taking into account that each task corresponds to the child's maturational age, we analyze the strategies followed, as well as the mistakes made by those students who have not overcome this task, achieving all this diagnose the numerical thinking in its cardinal aspect.

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel's incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel's theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas's arithmetic and the semantic inadequacy of Penrose's arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel's theorems: that we cannot prove that we are consistent (Gödel's Unknowability Thesis), and that we cannot fully describe our notion of a natural number. [ABSTRACT FROM AUTHOR]