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This paper reports that synergetics are used to analyze the crazing evolution. On this basis, chaotic effect is explored. The chaos equation is established and verified. The theoretical derivation are consistent with the experimental results. We design a special specimen with a special loading mode, the transient monitoring function of acoustic emission (AE) technology is used to track and detect the crazing inside the PMMA in real time, and the experiments show that synergetics can explain the crazing properties of polymer. Importantly, the mathematical explanation is also given. The AE analysis, synergetics, and craze photo reached a conclusion that the crazing has chaotic behavior. After analyzing the AE events and crazing at different stress levels, the accuracy of synergetic approach for crazing is verified. By studying the course of AE events and crazing, the self-organization effect is proposed. The research results will provide data support for the application of PMMA in ship, aircraft, and precision instruments.

Studies on drawing often focused on spatial aspects of the finished products. Here, the drawing behavior was studied by analyzing its intermittent process, between drawing (i.e. marking a surface) and interruption (i.e. a pause in the marking gesture). To assess how this intermittence develops with age, we collected finger-drawings on a touchscreen by 185 individuals (children and adults). We measured the temporal structure of each drawing sequence to determine its complexity. To do this, we applied temporal fractal estimators to each drawing time series before combining them in a Principal Component Analysis procedure. The youngest children (3-year-old) drew in a more stereotypical way with long-range dependence detected in their alternations between states. Among older children and adults, the complexity of drawing sequences increased showing a less predictable behavior as their drawings become more detailed and figurative. This study improves our understanding of the temporal aspects of drawing behavior, and contributes to an objective understanding of its ontogeny.

My purpose here is to give some background to a series of sculptures I have been creating in which nonorientable surfaces, especially crosscaps, play a mathematical content role. The fundamental circle of theorems involved is on the classification of surfaces in terms of handles and crosscaps. The most basic examples are spheres, tori, möbius bands, projective planes, and klein bottles, cf., [CF1, HF1] for discussion of my earlier sculpture involving these concepts. For an exposition and further references to the mathematical background for the circle of theorems I refer to, see [FG1].

Natural, everyday sensorimotor behaviors, such as rising from sitting, typically have an intrinsic organization of several levels of analysis. Taking this intrinsic organization as key to understanding neural dynamics is neither a top-down nor a bottom-up approach, but rather a meshing of multiple centers and levels of analysis. Motor control requires body dynamics that are consistent with physical dynamics, besides the more microscopic levels of neural dynamics. The dynamics of separate movements have been investigated as if the ends can be capped off, separated from the rest of the individual's life. Is this dynamically correct? Even chaotic behavior is deterministic. However, the mathematics of nonlinear oscillations is not all of dynamics. This paper relates Bloch's dynamical theorem to the modular, conditional approach to sensorimotor and other neural functioning.

Bloch's dynamical theorem lays a foundation for the piecewise study of structurally accurate dynamics in theoretical neurobiology. Piecewise studies can be used as a modeling option complementary to the methods of nonlinear oscillator dynamics. By applying Bloch's theorem, dynamics of movements analyzed piecewise can be extended into a smooth flow on any manifold, either as a whole or conditionally. Conditional dynamics makes dynamical modeling options explicit, often depending on what variables the organism can control, and allows one to take different modeling options at different junctures in analyzing the same phenomenon. For example, this approach allows the study of complex motor control problems to be reduced to modular constructions using singularities and flow lines. Dynamical contingencies are expressed using the mathematics of ordered structures. This paper presents Bloch's dynamical theorem and its relevance to model construction in theoretical neurobiology. Specific examples, integrated into physiological and behavioral context, are cited from the literature.

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs.

Establishing a mathematical supply-chain model is a proposition that has received attention due to its inherent benefits of evolving global supply-chain efficiencies. This paper discusses the prevailing relationships found within apparel supply-chain environments, and contemplates the complex issues indicated for constituting a mathematical model. Principal results identified within the data suggest, that the multifarious nature of global supply-chain activities require a degree of simplification in order to fully dilate the necessary factors which affect, each subsection of the chain. Subsequently, the research findings allowed the division of supply-chain components into subsections, which amassed a coherent method of product development activity. Concurrently, the supply-chain model was found to allow systematic mathematical formulae analysis, of cost and time, within the multiple contexts of each subsection encountered. The paper indicates the supply-chain model structure, the mathematics, and considers how product analysis of cost and time can improve the comprehension of product lifecycle management.

Mathematics is an art as old as civilization. Most of the time hidden and respected, sometimes appearing in bright light, mathematicians have always had a privileged role in society, as problem solvers, guardians of an art, deeply attached to values of intellectual freedom and opinion challenge. “The essence of mathematics lies in its freedom”, said Georg Cantor. But mathematicians are also accountable to society, which is in need of keeping a link to its most singular and respected science, especially at a time of algorithmic transformation. I was lucky enough to experience the role of mathematician as a public spokesperson, advocating for mathematical sciences as both an art and a technology creator. Later, as a member of Parliament, then head of the Scientific Parliamentary Office, I experienced the intensity and complexity of science in politics, at a time when public action needs to rest on science and when human factors are more challenging than ever.

The Plenary Panel 3 at ICME-14 was especially devoted to the COVID-19 pandemic. Its goal was to review and reflect on the challenges raised by the pandemic, the responsibilities and roles for mathematicians and mathematics educators in this context, and to draw some lessons for the future. This text begins with the presentation of the panel and the four invited panelists. Then each panelist synthesizes her/his contribution to the panel, and we end by some lessons drawn from these contributions and the exchanges between the panelists and with the audience.

This report presents preliminary results from a survey commissioned for ICME-14 on teaching and learning of mathematical modelling and interdisciplinary mathematics educations. The systematic literature review focusses on how a well-understood relation between mathematics and the real world underpins interdisciplinary work, interdisciplinarity in research and teaching teams, issues and challenges in the relationships among mathematical modelling, mathematics, the real world and interdisciplinarity, and mathematical modelling and a well-understood relation to the real world ensuring mathematical depth in STEM integration.

Persistent differences in mathematics performance between general and special education students underscore the need for improving teachers’ preparation to better serve the needs of students with different strengths and needs. Math for All is a research-based, intensive professional learning program designed to help K–5th grade teachers provide accessible, high-quality mathematics instruction to ALL students, including students with disabilities. Using a neurodevelopmental framework (NDF), we analyze the lesson, understand the strengths and challenges of a student, and build adaptations that support the student’s access to the lesson. Multiple research studies have demonstrated the efficacy of the Math for All approach.

This short paper outlines the main aims and objectives of the CAPTeaM project and the activities that took place during the CAPTeaM workshop at ICME14 on Wednesday 14 July 2021.

This session article unpacks mathematics teaching and learning focused on racial equity and social justice. Specifically, the session will explore the intersection of mathematics teaching and learning with racial equity and social justice across four critical reasons: a) Building an informed society; b) Connecting mathematics to cultural and community histories as valuable resources; c) Confronting and solving real-world mathematics as a tool to confront inequitable and unjust contexts; d) Use mathematics as a tool for democracy and creating a more just society.

There have been reviews or meta-analysis showing that using manipulatives is an effective intervention for learning mathematics for students with disabilities, including autism spectrum disorders (ASD), without concentrating on the effects on generalization and maintenance. We conducted a meta-analysis to evaluate the effect of manipulatives on generalizing and/or maintaining mathematical skills for individuals with ASD and whether the effect varies with different participant characteristics, study design, intervention characteristics and mathematical content, focusing on the single-case studies. After application of the What Works Clearinghouse design standards, a total of 11 studies were included in the review: three studies collected data points during generalization phases, five studies collected data points during maintenance phases, the other three studies collected both generalization and maintenance data. Aggregate Tau-U and non-overlap of all pairs effect sizes (NAP) were calculated for each study and conducted moderator analyses. Overall, effect size scores ranged from small to significant effects across all comparisons. On average, most comparisons from the baseline to generalization and maintenance produced medium to large effects. Whereas, minor effects were found in most of the intervention of generalization and maintenance comparisons. Further moderator analysis regard to generalization and maintenance revealed that out of seven variables analyzed, only manipulatives types served as a moderator for maintenance. The findings suggest that manipulatives interventions were likely to result in mixed effects on mathematical skill generalization and maintenance within children with ASD, especially virtual manipulatives. Limitations and implications for future research and practice are discussed.

For decades, reformers have emphasized discussion over recitation and lecture. Yet, traditional communication patterns are still dominant in mathematics classrooms internationally. In an effort to better understand this challenge, the present study investigates patterns and contributions of research on discussion in mathematics teaching. Based on systematic search in the Eric database, and in selected journals of mathematics education, 72 studies were reviewed. Based on analysis and discussion of the reviewed studies, it is suggested to develop conceptual clarity and include definitions of core terms like discussion, to consider alternative methods for studying discussion in teaching, and to consider shifting the focus from teacher actions to the entailments of the work of leading mathematical discussions.

In this talk, I will introduce mathematics as a code for interdisciplinary dialogues through a story on infinity.

Some interesting examples are given in this article to show how one can learn and investigate mathematics through experiments with the help of computers. Each example begins with a problem. Trying to solve the problem one experiences exploration, invention and application of mathematics in the process.

We show here some of our results on intuitionistic fuzzy topological spaces. In 1983, K.T. Atanassov proposed a generalization of the notion of fuzzy set: the concept of intuitionistic fuzzy set. D. Çoker constructed the fundamental theory on intuitionistic fuzzy topological spaces, and D. Çoker and other mathematicians studied compactness, connectedness, continuity, separation, convergence and paracompactness in intuitionistic fuzzy topological spaces. Finally, G.-J Wang and Y.Y. He showed that every intuitionistic fuzzy set may be regarded as an L-fuzzy set for some appropriate lattice L. Nevertheless, the results obtained by above authors are not redundant with other for ordinary fuzzy sense. Recently, Smarandache defined and studied neutrosophic sets (NSs) which generalize IFSs. This author defined also the notion of neutrosophic topology. We proved that neutrosophic topology does not generalize the concept of intuitionistic fuzzy topology.

People always need more information than they have. They can get some of this information by themselves but a good deal of information remains inaccessible. This situation, which always existed in human civilization, brought forth Oracles. The idea of an Oracle reflected interactions between systems, such as people and states, with less information and systems with more indispensable information. In the 20^th century, it was proved that some information is intrinsically inaccessible and then the concept of an Oracle naturally came to computer science becoming popular in the realm of algorithms and computations. At first, Turing machines with an Oracle were introduced and studied. Later inductive Turing machines with Oracles, limit Turing machines with Oracles and evolutionary Turing machines with Oracles were established and explored. Here we create a theoretical background for the concept of an Oracle. In the first part of this work, we contemplate Oracles in human culture. In the second part, we contemplate Oracles in computer science and mathematics. In the third part, the variety of Oracles is analyzed and classified. In the fourth part, we develop a mathematical theory of Oracles, where the concepts of an Oracle and its brands are formalized and studied in the mathematical framework.

This paper explains how I implement technology in my cross-disciplinary honors course on mathematics and music to strengthen student engagement and learning. I encourage my students to use technical computing software, online tools, and personal computing devices to explore solutions and develop expertise. Students learn to apply technology and mathematics to analyze, manipulate, and create music.

The quantity notions in mathematics and metrology and their relation and interaction are considered. The quantity in mathematics belong to the modelling field and is an ideal object while the quantity in metrology has an experimental character and so is an uncertain object. Every metrological model object including measurement aim, measurand, metrological characteristic of a measuring instrument, are expressed by mathematical quantities. When the model object is evaluated by using measurement, so the experimental quantity is obtained, that is metrological quantity. Because of the principal uncertain character of metrological quantities, measuring data and relating metrological quantities have to be processed using firstly non-classic mathematical means but of special type, taking into account the above-mentioned character. There are approximate linear equations theory, interval arithmetic, fuzzy set, and so on. There is not wide use of these means. The reasons are traditions, and absence of data structure analysis, and special place of stochastic tools. The latter is conditioned by some peculiarities of probabilistic-stochastic models. Main, and wide spread, mistakes or faults in use of these models are discussed. Indirect measurement is considered as the field of most complicated interaction of experimental and mathematical quantities.