Cognitive processes underlying mathematical concept construction: The missing process of structural abstraction

The purpose of this paper is twofold: On the one hand, this work frames a variety of considerations on cognitive processes underlying mathematical concept construction in two research strands, namely an actions-first strand and an objects-first strand, that mainly shapes past and current approaches on abstraction in learning mathematics. This classification provides the identification of an often overlooked fundamental cognitive process, namely structural abstraction. On the other hand, this work shows a theory-driven and research-based approach illuminating the hidden architecture of cognitive processes involved in structural abstraction that gives new insights into an integrated framework on abstraction in learning mathematics. Based on our findings in empirical investigations, the paper outlines a theoretical framework on the cognitive processes taking place on mental (rather than physical) objects

Emerging insights from the evolving framework of structural abstraction

Only recently ‘abstraction from objects’ has attracted attention in the literature as a form of abstraction that has the potential to take account of the complexity of students’ knowing and learning processes compatible with their strategy of giving meaning. This paper draws attention to several emerging insights from the evolving framework of structural abstraction in students’ knowing and learning of the limit concept of a sequence. Particular ideas are accentuated that we need to understand from a theoretical point of view since they reveal a new way of understanding knowing and learning advanced mathematical concepts

Images of abstraction in mathematics education: Contradictions, controversies, and convergences

In this paper we offer a critical reflection of the mathematics education literature on abstraction. We explore several explicit or implicit basic orientations, or what we call images, about abstraction in knowing and learning mathematics. Our reflection is intended to provide readers with an organized way to discern the contradictions, controversies, and convergences concerning the many images of abstraction. Given the complexity and multidimensionality of the notion of abstraction, we argue that seemingly conflicting views become alternatives to be explored rather than competitors to be eliminated. We suggest considering abstraction as a constructive process that characterizes the development of mathematical thinking and learning and accounts for the contextuality of students’ ideas by acknowledging knowledge as a complex system

The Lotka-Volterra Dynamical System and its Discretization

Dynamical systems are a valuable asset for the study of population dynamics. On this topic, much has been done since Lotka and Volterra presented the very first continuous system to understand how the interaction between two species -- the prey and the predator -- influences the growth of both populations. The definition of time is crucial and, among options, one can have continuous time and discrete time. The choice of a method to proceed with the discretization of a continuous dynamical system is, however, essential, because the qualitative behavior of the system is expected to be identical in both cases, despite being two different temporal spaces. In this work, our main goal is to apply two different discretization methods to the classical Lotka-Volterra dynamical system: the standard progressive Euler's method and the nonstandard Mickens' method. Fixed points and their stability are analyzed in both cases, proving that the first method leads to dynamic inconsistency and numerical instability, while the second is capable of keeping all the properties of the original continuous model.Comment: This is a preprint of a paper whose final form is published at [http://dx.doi.org/10.1201/9781003388678-19

Herding behaviour and sentiment: Evidence in a small European market

AbstractThis work studies herding behaviour in a small European market, by analysing the stocks that constituted the Portuguese stock PSI-20 index, for the period between 2003 and 2011. The two different approaches used to measure herding intensity led to different results, suggesting that measurements of the herding phenomenon are sensitive to the method used. Consequently, there is a need for further research into the methodology used to test this phenomenon. Additionally, the study analyses the relationship between herd behaviour and investor sentiment, an area that has been little explored. In applying causality tests to the impact of sentiment on herd behaviour, only weak evidence is found that sentiment influences herding

Assessing the impact of COVID-19 on the performance of organ transplant services using data envelopment analysis

Organ transplant is one of the best options for many medical conditions, and in many cases, it may be the only treatment option. Recent evidence suggests, however, that the COVID-19 pandemic might have detrimentally affected the provision of this type of healthcare services. The main purpose of this article is to use Data Envelopment Analysis and the Malmquist Index to assess the impact that the pandemic caused by the novel coronavirus SARS-CoV-2 had on the provision of solid organ transplant services. To this purpose, we use three complementary models, each focusing on specific aspects of the organ donation and transplantation process, and data from Brazil, which has one of the most extensive public organ transplant programs in the world. Using data from 17 States plus the Federal District, the results of our analysis show a significant drop in the performance of the services in terms of the organ donation and transplantation process from 2018 to 2020, but the results also indicate that not all aspects of the process and States were equally affected. Furthermore, by using different models, this research also allows us to gain a more comprehensive and informative assessment of the performance of the States in delivering this type of service and identify opportunities for reciprocal learning, expanding our knowledge on this important issue and offering opportunities for further research.info:eu-repo/semantics/publishedVersio

Optical microtopographic characterization of arqueological ceramics

Optics and optics and photonics based inspection tools and methods had expensively proven their invaluable importance in the preservation of cultural heritage and artwork. The non-invasive inspection of the 3D shape of objects and of the micro-relief structure of its surfaces can be of high importance in the characterization process required in most works of restoration or preservation of archeological artwork. In this communication we will report on a method of optical non-invasive microtopographic characterization of the surface of archeological ceramics. The samples used in this study are pre-historical and pre-colonial ceramics and pottery of tribes in the Paranaiba valley in Minas Gerais, Brazil. The pottery found is decorated with incisions with different geometric distributions and levels of complexity corresponding to two periods of indigenous Indian occupations: one from a period dated at 1,095 ± 186 years ago and another of the early nineteenth century dated between 212 ± 19 years and 190 ± 30 years ago seemingly corresponding to the occupation of the territory by southern Kayapós tribes.info:eu-repo/semantics/publishedVersio