The gap between theory and practice in mathematics education, particularly in primary-teacher education, necessitates innovative teaching methodologies. This paper explores the implementation of academic portfolios as a teaching innovation in Algebra and Number Systems I and II courses within the primary teacher education programme at Pontificia Universidad Católica de Chile. The methodology involved integrating academic portfolios to align course content with essential learning outcomes for future teaching roles. Implementation begins with a negotiation between students and teachers to establish a learning contract, followed by an overview of course rules, content, objectives, materials, and grading rubrics. Preliminary findings indicate that this innovative method enhances engagement with mathematical concepts, improves assessment efficacy in teacher training, and may contribute to enhanced preparation of primary mathematics teachers. The study highlights the role of portfolios in making students active participants in their learning, significantly enhancing the educational experience of teacher candidates. These findings suggest a promising avenue for future educational assessme
This paper identifies several different interconnected challenges preventing the move towards more ethical and sustainable mathematics education: the entrenched belief in mathematical neutrality, the difficulty of simultaneously reforming mathematics and its pedagogy, the gap between academic theory and classroom practice, and the need for epistemic decolonisation. In this context, we look at both bottom-up and top-down approaches, and argue that globalised frameworks such as the United Nations' Sustainable Development Goals are insufficient for this transformation, and that ethical and sustainable forms of mathematics ought not to be built using these as their (philosophical) foundation. These frameworks are often rooted in a Western-centric development paradigm that can perpetuate colonial hierarchies and fails to resolve inherent conflicts between economic growth and ecological integrity. As an alternative, this paper advocates for embracing localised, culturally-situated mathematical practices. Using the Ethics in Mathematics Project as a case study within a Western, Global North institution, this paper illustrates a critical-pragmatic, multi-level strategy for fostering ethica
Many problems in industry --- and in the social, natural, information, and medical sciences --- involve discrete data and benefit from approaches from subjects such as network science, information theory, optimization, probability, and statistics. The study of networks is concerned explicitly with connectivity between different entities, and it has become very prominent in industrial settings, an importance that has intensified amidst the modern data deluge. In this commentary, we discuss the role of network analysis in industrial and applied mathematics, and we give several examples of network science in industry. We focus, in particular, on discussing a physical-applied-mathematics approach to the study of networks. We also discuss several of our own collaborations with industry on projects in network analysis.
"Phase-locking" is a fundamental phenomenon in which coupled or periodically forced oscillators synchronise. The Arnold family of circle maps, which describes a forced oscillator, is the simplest mathematical model of phase-locking and has been studied intensively since its introduction in the 1960s. The family exhibits regions of parameter space where phase-locking phenomena can be observed. A long-standing question asked whether "hyperbolic" parameters~-- those whose behaviour is dominated by periodic attractors, and which are therefore stable under perturbation~-- are dense within the family. A positive answer was given in 2015 by van Strien and the author, which implies that, no matter how chaotic a map within the family may behave, there are always systems with stable behaviour nearby. This research was a focal point of a pioneering collaboration with composer Emily Howard, commencing with Howard's residency in Liverpool's mathematics department in 2015. The collaboration generated impacts on creativity, culture and society, including several musical works by Howard, and lasting influence on artistic practice through a first-of-its-kind centre for science and music. We describ
A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other "good" operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent and semirings are examined. In this case, basic algorithms are polynomial as well as their interval versions. This situation is very different from the traditional linear algebra, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.
Origami is the art of paper folding, and it borrows its name from two Japanese words \emph{ori} and \emph{kami}. In Japanese, {ori} means folding, and the paper is called {kami}. While origami is just a hobby to most, there is a lot more to it. If you fold a square sheet of paper into any of the traditional origami model (for example the flapping bird) and unfold it, you can see crease patterns. These crease patterns tell us that there is a lot of geometry hidden behind the folds. In this article, we investigate the symbiotic relationship between mathematics and origami. The first part of this article explores the utility of origami in education. We will see how origami could become an effective way of teaching methods of geometry, mainly because of its experiential nature. Complex origami patterns cannot be created out of thin air. They usually involve understanding deep mathematical theories and the ability to apply them to paper folding. In the second part of the article, we attempt to provide a glimpse of this beautiful connection between origami and mathematics.
In this short review, I will summarize my research experience in three fields in applied mathematics: mathematical biology, applied probability, and applied discrete mathematics. Specifically, I will show how each project was initiated, and what wrong approaches were applied. Such details are important in learning how to do research, but they cannot be read out from research papers. I wish that students and junior researchers in applied mathematics could learn a lesson from this summary.
Mathematics can help analyze the arts and inspire new artwork. Mathematics can also help make transformations from one artistic medium to another, considering exceptions and choices, as well as artists' individual and unique contributions. We propose a method based on diagrammatic thinking and quantum formalism. We exploit decompositions of complex forms into a set of simple shapes, discretization of complex images, and Dirac notation, imagining a world of "prototypes" that can be connected to obtain a fine or coarse-graining approximation of a given visual image. Visual prototypes are exchanged with auditory ones, and the information (position, size) characterizing visual prototypes is connected with the information (onset, duration, loudness, pitch range) characterizing auditory prototypes. The topic is contextualized within a philosophical debate (discreteness and comparison of apparently unrelated objects), it develops through mathematical formalism, and it leads to programming, to spark interdisciplinary thinking and ignite creativity within STEAM.
We analyze a mathematical model to understand the dynamics of bullying in schools. The model considers a population divided into four groups: susceptible individuals, bullies, individuals exposed to bullying, and violent individuals. Transitions between these states occur at rates designed to capture the complex interactions among students, influenced by factors such as romantic rejection, conflicts with peers and teachers, and other school-related challenges. These interactions can escalate into bullying and violent behavior. The model also incorporates the role of parents and school administrators in mitigating bullying through intervention strategies. The results suggest that bullying can be effectively controlled if anti-bullying programs implemented by schools are sufficiently robust. Additionally, the conditions under which bullying persists are explored.
We correct the statements and proofs of the (auxiliary) Propositions 4.1 and 4.2 of our paper `Evaluation of motivic functions, non-nullity, and integrability in fibers' in Advances in Mathematics, Vol. 409, Part A, Paper No. 108635, 29 pages (2022), and we explain how the proofs of the main results can be adapted to work with those corrected propositions.
The old lie of mathematical inadequacy of Indigenous communities has been curiously persistent despite increasing evidence shows that many Indigenous communities practiced mathematics. Attempts to study and teach Indigenous mathematical knowledge have always been questioned and even denied validity. The Aboriginal and Torres Strait Islander Histories and Cultures cross-curriculum priority in the F-10 Australian schools curriculum, from 2022 onwards, includes content elaborations related to Indigenous mathematics, which have been developed and refined by expert Indigenous advisers. We celebrate this initiative, but experience also tells us to expect some resistance from sectors of the education communities who hold to an exclusively Anglo-European provenance of mathematics. Through this review article we seek to constructively forestall potential pushback and address concerns regarding the legitimacy and pedagogical value of Indigenous mathematics, by countering with evidence some published claims of mathematical inadequacies of Australian First Nations cultures.
In this paper, we situate the educational movement of "Ethics in Mathematics," as outlined by the Cambridge University Ethics in Mathematics Project, in the wider area of mathematics ethics education. By focusing on the core message coming out of Ethics in Mathematics, its target group, and educational philosophy, we set it into relation with "Mathematics for Social Justice" and Paul Ernest's recent work on ethics of mathematics. We conclude that, although both Ethics in Mathematics and Mathematics for Social Justice appear antagonistic at first glance, they can be understood as complementary rather than competing educational strategies.
The selection of topics in this text has formed the core of a one semester course in applied mathematics at the Arctic University of Norway that has been running continuously since the 1970s. The class has, during its existence, drawn participants from both applied mathematics and physics, and also to some extent from pure mathematics, analysis in particular. The material in these lecture notes can be covered by one semester's worth of five lecture hours a week. The work requirements for the students consists of seven obligatory projects whose content are taken from exercises and computational projects included in the text.
In 1911, Alfred North Whitehead published a short book "Introduction to Mathematics" (IM) intended for students wanting an explanation of the fundamental ideas of mathematics. Whitehead's IM has enduring value because it was written not long after he and Bertrand Russell published their monumental three-volume work "Principia Mathematica" (PM) -- a publication of immense historical significance for mathematics. IM sheds light on Whitehead's view of mathematics at that time. Whitehead's book places proofs in predicate logic as the mythical starting point of mathematics, although Whitehead himself was slow to understand the significance of symbolic predicate logic.
The main drawback of using generative AI models for advanced mathematics is that these models are not primarily logical reasoning engines. However, Large Language Models, and their refinements, can pick up on patterns in higher mathematics that are difficult for humans to see. By putting the design of generative AI models to their advantage, mathematicians may use them as powerful interactive assistants that can carry out laborious tasks, generate and debug code, check examples, formulate conjectures and more. We discuss how generative AI models can be used to advance mathematics research. We also discuss their integration with neuro-symbolic solvers, Computer Algebra Systems and formal proof assistants such as Lean.
We investigate the Gerver-Ramsey collinearity problem of determining the maximum number of points in a north-east lattice path without $k$ collinear points. Using a satisfiability solver, up to isomorphism we enumerate all north-east lattice paths avoiding $k$ collinear points for $k \leq 6$. We also find a north-east lattice path avoiding $k = 7$ collinear points with 327 steps, improving on the previous best length of 260 steps found by Shallit.
Recent years have witnessed the rapid development of Neuro-Symbolic (NeSy) AI systems, which integrate symbolic reasoning into deep neural networks. However, most of the existing benchmarks for NeSy AI fail to provide long-horizon reasoning tasks with complex multi-agent interactions. Furthermore, they are usually constrained by fixed and simplistic logical rules over limited entities, making them far from real-world complexities. To address these crucial gaps, we introduce LogiCity, the first simulator based on customizable first-order logic (FOL) for an urban-like environment with multiple dynamic agents. LogiCity models diverse urban elements using semantic and spatial concepts, such as IsAmbulance(X) and IsClose(X, Y). These concepts are used to define FOL rules that govern the behavior of various agents. Since the concepts and rules are abstractions, they can be universally applied to cities with any agent compositions, facilitating the instantiation of diverse scenarios. Besides, a key feature of LogiCity is its support for user-configurable abstractions, enabling customizable simulation complexities for logical reasoning. To explore various aspects of NeSy AI, LogiCity intro
In this article we will use Minecraft to experimentally approximate the values of four different mathematical constants. The mathematical constants that we will approximate are $\sqrt{2}, π$, Euler's number $e$, and Apéry's constant $ζ(3)$. We will begin each section with a brief history of the number being approximated and describe where it appears in mathematics. We then explain how we used Minecraft mechanics to approximate the constant. At the end of each section, we provide some ideas for how to apply our techniques to the approximation of other mathematical constants in Minecraft or elsewhere. This article is a proof of concept that Minecraft can be used in higher education. We should note that the goal of this article is not to have the most accurate approximations possible, the goal is to inspire people to have fun while learning about various mathematical topics. We hope you learn something new in this article and feel inspired to try some of these techniques on your own.
Mathematical Program with Complementarity Constraints (MPCC) plays a very important role in many fields such as engineering design, economic equilibrium, multilevel game, and mathematical programming theory itself. In theory its constraints fail to satisfy a standard constraint qualification such as the linear independence constraint qualification (LICQ) or the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. As a result, the developed nonlinear programming theory may not be applied to MPCC class directly. Nowadays, a natural and popular approach is try to find some suitable approximations of an MPCC so that it can be solved by solving a sequence of nonlinear programs. This work aims to solve the MPCC using nonlinear programming techniques, namely the SQP and the regularization scheme. Some algorithms with two iterative processes, the inner and the external, were developed. A set of AMPL problems from MacMPEC database \cite{MacMPEC} were tested. The algorithms performance comparative analysis was carried out.
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of p