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Advances in Experimental Medicine and... 2023Recent research in educational neuroscience has established the correlation between the way the human brain works and the process of perceiving and learning mathematical...
Recent research in educational neuroscience has established the correlation between the way the human brain works and the process of perceiving and learning mathematical concepts. In this chapter, a research approach is proposed, based on the principles of educational neuroscience, and focuses on the way students deal with new knowledge in mathematics. Initially, using neuroscientific techniques and a multidimensional approach to new knowledge, data will be collected from students. By collecting neurophysiological measurements and analyzing the data, an attempt will be made to formulate learning paths for a better understanding of fractional concepts, based on the needs of each student.
Topics: Humans; Learning; Brain; Students; Neurosciences; Mathematics
PubMed: 37486483
DOI: 10.1007/978-3-031-31982-2_10 -
PloS One 2022Number transcoding is the cognitive task of converting between different numerical codes (i.e. visual "42", verbal "forty-two"). Visual symbolic to verbal transcoding...
Number transcoding is the cognitive task of converting between different numerical codes (i.e. visual "42", verbal "forty-two"). Visual symbolic to verbal transcoding and vice versa strongly relies on language proficiency. We evaluated transcoding of German-French bilinguals from Luxembourg in 5th, 8th, 11th graders and adults. In the Luxembourgish educational system, children acquire mathematics in German (LM1) until the 7th grade, and then the language of learning mathematic switches to French (LM2). French `70s `80s `90s are less transparent than `30s `40s `50s numbers, since they have a base-20 structure, which is not the case in German. Transcoding was evaluated with a reading aloud and a verbal-visual number matching task. Results of both tasks show a cognitive cost for transcoding numbers having a base-20 structure (i.e. `70s, `80s and `90s), such that response times were slower in all age groups. Furthermore, considering only base-10 numbers (i.e. `30s `40s `50s), it appeared that transcoding in LM2 (French) also entailed a cost. While participants across age groups tended to read numbers slower in LM2, this effect was limited to the youngest age group in the matching task. In addition, participants made more errors when reading LM2 numbers. In conclusion, we observed an age-independent language effect with numbers having a base-20 structure in French, reflecting their reduced transparency with respect to the decimal system. Moreover, we find an effect of language of math acquisition such that transcoding is less well mastered in LM2. This effect tended to persist until adulthood in the reading aloud task, while in the matching task performance both languages become similar in older adolescents and young adults. This study supports the link between numbers and language, especially highlighting the impact of language on reading numbers aloud from childhood to adulthood.
Topics: Adolescent; Adult; Child; Humans; Language; Learning; Luxembourg; Mathematics; Reading; Young Adult
PubMed: 36037234
DOI: 10.1371/journal.pone.0273391 -
Philosophical Transactions. Series A,... Dec 2021Virtually all forms of life, from single-cell eukaryotes to complex, highly differentiated multicellular organisms, exhibit a property referred to as symmetry. However,...
Virtually all forms of life, from single-cell eukaryotes to complex, highly differentiated multicellular organisms, exhibit a property referred to as symmetry. However, precise measures of symmetry are often difficult to formulate and apply in a meaningful way to biological systems, where symmetries and asymmetries can be dynamic and transient, or be visually apparent but not reliably quantifiable using standard measures from mathematics and physics. Here, we present and illustrate a novel measure that draws on concepts from information theory to quantify the degree of symmetry, enabling the identification of approximate symmetries that may be present in a pattern or a biological image. We apply the measure to rotation, reflection and translation symmetries in patterns produced by a Turing model, as well as natural objects (algae, flowers and leaves). This method of symmetry quantification is unbiased and rigorous, and requires minimal manual processing compared to alternative measures. The proposed method is therefore a useful tool for comparison and identification of symmetries in biological systems, with potential future applications to symmetries that arise during development, as observed or as produced by mathematical models. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
Topics: Mathematics; Models, Biological; Models, Theoretical; Morphogenesis; Physics; Plants
PubMed: 34743597
DOI: 10.1098/rsta.2020.0273 -
PloS One 2022In opinion dynamics, as in general usage, polarisation is subjective. To understand polarisation, we need to develop more precise methods to measure the agreement in...
In opinion dynamics, as in general usage, polarisation is subjective. To understand polarisation, we need to develop more precise methods to measure the agreement in society. This paper presents four mathematical measures of polarisation derived from graph and network representations of societies and information-theoretic divergences or distance metrics. Two of the methods, min-max flow and spectral radius, rely on graph theory and define polarisation in terms of the structural characteristics of networks. The other two methods represent opinions as probability density functions and use the Kullback-Leibler divergence and the Hellinger distance as polarisation measures. We present a series of opinion dynamics simulations from two common models to test the effectiveness of the methods. Results show that the four measures provide insight into the different aspects of polarisation and allow real-time monitoring of social networks for indicators of polarisation. The three measures, the spectral radius, Kullback-Leibler divergence and Hellinger distance, smoothly delineated between different amounts of polarisation, i.e. how many cluster there were in the simulation, while also measuring with more granularity how close simulations were to consensus. Min-max flow failed to accomplish such nuance.
Topics: Computer Simulation; Mathematics; Social Segregation
PubMed: 36194573
DOI: 10.1371/journal.pone.0275283 -
Nature Dec 2021
Topics: Artificial Intelligence; Intuition; Machine Learning; Mathematics
PubMed: 34853454
DOI: 10.1038/d41586-021-03512-4 -
Philosophical Transactions. Series A,... Jun 2020George Gabriel Stokes won the coveted title of Senior Wrangler in 1841, a year in which the examination papers for the Cambridge Mathematical Tripos were notoriously...
George Gabriel Stokes won the coveted title of Senior Wrangler in 1841, a year in which the examination papers for the Cambridge Mathematical Tripos were notoriously difficult. Coming top in the Mathematical Tripos was a notable achievement, but for Stokes it was a prize hard won after several years of preparation, and not only years spent at Cambridge. When Stokes arrived at Pembroke College, he had spent the previous two years at Bristol College, a school which prided itself on its success in preparing students for Oxford and Cambridge. This article follows Stokes' path to the senior wranglership, tracing his mathematical journey from his arrival in Bristol to the end of his final year of undergraduate study at Cambridge. This article is part of the theme issue 'Stokes at 200 (Part 1)'.
Topics: History, 19th Century; History, 20th Century; Mathematics; United Kingdom
PubMed: 32507086
DOI: 10.1098/rsta.2019.0506 -
Philosophical Transactions. Series A,... Dec 2021In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and... (Review)
Review
In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
Topics: Diffusion; Mathematics; Models, Biological; Morphogenesis
PubMed: 34743603
DOI: 10.1098/rsta.2020.0268 -
Current Opinion in Neurobiology Jun 2023Learning is a multi-faceted phenomenon of critical importance and hence attracted a great deal of research, both experimental and theoretical. In this review, we will... (Review)
Review
Learning is a multi-faceted phenomenon of critical importance and hence attracted a great deal of research, both experimental and theoretical. In this review, we will consider some of the paradigmatic examples of learning and discuss the common themes in theoretical learning research, such as levels of modeling and their corresponding relation to experimental observations and mathematical ideas common to different types of learning.
Topics: Models, Theoretical; Learning; Mathematics
PubMed: 37043892
DOI: 10.1016/j.conb.2023.102721 -
Molecules (Basel, Switzerland) Jun 2023Since chemistry, materials science, and crystallography deal with three-dimensional structures, they use mathematics such as geometry and symmetry. In recent years, the...
Since chemistry, materials science, and crystallography deal with three-dimensional structures, they use mathematics such as geometry and symmetry. In recent years, the application of topology and mathematics to material design has yielded remarkable results. It can also be seen that differential geometry has been applied to various fields of chemistry for a relatively long time. There is also the possibility of using new mathematics, such as the crystal structure database, which represents big data, for computational chemistry (Hirshfeld surface analysis). On the other hand, group theory (space group and point group) is useful for crystal structures, including determining their electronic properties and the symmetries of molecules with relatively high symmetry. However, these strengths are not exhibited in the low-symmetry molecules that are actually handled. A new use of mathematics for chemical research is required that is suitable for the age when computational chemistry and artificial intelligence can be used.
Topics: Artificial Intelligence; Mathematics; Crystallography
PubMed: 37298985
DOI: 10.3390/molecules28114509 -
Psychological Research Jul 2022There is a notion that mathematical equations can be considered aesthetic objects. However, whereas some aesthetic experiences are triggered primarily by the sensory...
There is a notion that mathematical equations can be considered aesthetic objects. However, whereas some aesthetic experiences are triggered primarily by the sensory properties of objects, for mathematical equations aesthetic judgments extend beyond their sensory qualities and are also informed by semantics and knowledge. Therefore, to the extent that expertise in mathematics represents the accumulation of domain knowledge, it should influence aesthetic judgments of equations. In a between-groups study design involving university students who majored in mathematics (i.e., experts) or not (i.e., laypeople), we found support for the hypothesis that mathematics majors exhibit more agreement in their aesthetic judgments of equations-reflecting a greater degree of shared variance driven by formal training in the domain. Furthermore, their judgments were driven more strongly by familiarity and meaning than was the case for laypeople. These results suggest that expertise via advanced training in mathematics alters (and sharpens) aesthetic judgments of mathematical equations.
Topics: Esthetics; Humans; Judgment; Mathematics; Semantics
PubMed: 34495389
DOI: 10.1007/s00426-021-01592-5