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Cognition Dec 2022In a series of recently published studies purportedly on the "additive-area heuristic," Yousif & Keil (2019, 2020) argue for a systematic distortion in the perception of...
In a series of recently published studies purportedly on the "additive-area heuristic," Yousif & Keil (2019, 2020) argue for a systematic distortion in the perception of the cumulative area of an item array and further claim that previous findings of numerical cognition and magnitude perception in general are "at risk" (Yousif & Keil, 2021). This commentary describes serious stimulus design flaws present in all of Yousif and colleagues experiments that prevent from making such conclusions. Specifically, item arrays used in those studies demonstrate a skewed correlational structure between selected magnitude dimensions and exhibit unbalanced ranges across different magnitude dimensions of interest. Because the perception of magnitude dimensions interferes one another and because our perceptual system is sensitive to the statistical regularities of the sensory input, such a biased design makes it difficult, if not impossible, to interpret the choice behavior of an observer making magnitude judgments. By re-introducing the mathematical framework for a systematic construction of dot array stimuli (DeWind et al., 2015) and by re-analyzing the data from another recent study on area perception (Tomlinson et al., 2020), this paper explains-and introduces a MATLAB code for-an optimal method for designing and constructing dot arrays for magnitude perception studies.
Topics: Cognition; Heuristics; Humans; Judgment; Mathematics
PubMed: 34625223
DOI: 10.1016/j.cognition.2021.104919 -
Bulletin of Mathematical Biology Jan 2023Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating...
Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction-diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.
Topics: Mathematical Concepts; Computer Simulation; Models, Biological; Nonlinear Dynamics
PubMed: 36637542
DOI: 10.1007/s11538-022-01115-2 -
The British Journal of Educational... Mar 2024When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to...
BACKGROUND
When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles.
AIMS
This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability.
SAMPLE
A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China.
METHODS
A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time).
RESULTS
Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability.
CONCLUSIONS
The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.
Topics: Male; Female; Humans; Young Adult; Adult; Cognition; Language; Reaction Time; Educational Status; Mathematics
PubMed: 37574834
DOI: 10.1111/bjep.12632 -
The British Journal of Developmental... Mar 2022This study investigated the relationships of four executive functioning skills (including verbal working memory, spatial working memory, inhibitory control, and...
This study investigated the relationships of four executive functioning skills (including verbal working memory, spatial working memory, inhibitory control, and cognitive flexibility) with young children's mental computation and applied mathematical problem-solving. Two hundred and twenty-five Chinese kindergarteners were tested with a battery of general cognitive, executive functioning and mathematics skills. Results showed that when children's age, gender, non-verbal intelligence, and listening comprehension skills were controlled, verbal working memory and cognitive flexibility were significant correlates of mental computation, whereas verbal working memory, spatial working memory, and cognitive flexibility were significant correlates of applied mathematical problem-solving. Inhibitory control was not significantly associated with the two domains of mathematics under investigation. The findings highlight the differential roles of different executive functioning skills in early mathematical skills and offer practical implication for helping young children in learning complex mathematical skills.
Topics: Aptitude; Child; Child, Preschool; Executive Function; Humans; Mathematics; Memory, Short-Term; Problem Solving
PubMed: 34580894
DOI: 10.1111/bjdp.12396 -
Bone May 2020The needs of everyday life, such as counting and measuring, are roots of theoretical mathematics. I believe these roots are why mathematical ideas ground research so... (Review)
Review
The needs of everyday life, such as counting and measuring, are roots of theoretical mathematics. I believe these roots are why mathematical ideas ground research so amazingly well within many scientific fields. Initially trained as a theoretical mathematician and having collaborated with non-mathematicians in the field of bone research, I address the advantages and challenges of collaborations across fields of research among investigators trained in different disciplines. I report on the mathematical ideas that have guided my research on the mechanics of bone tissue. I explain how the mathematical ideas of local vs. global properties influence my research. Polarized light microscopy (PLM) is a tool that I use consistently, in association with other microscopy techniques, to investigate bone in its healthy state and in the presence of bone disease, in humans and in animal models. I review the results that I and investigators around the world have obtained with PLM. Applied to thin bone sections, PLM yields extinct (black) and bright (white) signals that are interpreted in terms of the orientation of collagen type I, by means of other microscopy techniques. Collagen type I is an elementary component of bone tissue. Its orientation is important for the mechanical function of bone. Images obtained by PLM at a specific bone site yield big data sets regarding collagen orientation. Multiple data sets in respect of multiple sites are often needed for research because the bone tissue differs by location in response to the distinct forces acting on it. Mathematics, defined by philosophers as the theory of patterns, offers the backdrop for pattern identification in the big data sets regarding collagen orientation. I also discuss the computational aspect of the research, pursuant to which the patterns identified are incorporated in simulations of mechanical behaviors of bone. These mathematical ideas serve to understand the role of collagen orientation in bone fracture risk.
Topics: Animals; Bone Diseases; Bone and Bones; Collagen; Computer Simulation; Humans; Mathematics; Microscopy, Polarization
PubMed: 32088399
DOI: 10.1016/j.bone.2020.115295 -
Journal of Speech, Language, and... Sep 2021Purpose Language is an important skill required for children to succeed in school. Higher language skills are associated with school readiness in young children and...
Purpose Language is an important skill required for children to succeed in school. Higher language skills are associated with school readiness in young children and general mathematics performance. However, many students with mathematics difficulty (MD) may be more likely to present difficulties with language skills than their peers. The purpose of this report was to compare the language performance of children with and without MD. Method We compared child vocabulary, morphology, and syntax between first- and second-grade children ( = 247) classified as with or without MD, controlling for child working memory. Results Children with MD ( = 119) significantly underperformed compared with their peers ( = 155) on all language measures. The largest difference between children with and without MD was in syntax. Conclusions Children with MD present poorer language skills than their peers, which aligns with previous research linking the importance of syntax with mathematics learning. More research is needed to better understand the complex links between language skills and mathematical development.
Topics: Aptitude; Child; Child, Preschool; Humans; Language; Mathematics; Memory, Short-Term; Vocabulary
PubMed: 34310199
DOI: 10.1044/2021_JSLHR-20-00378 -
Psychological Bulletin Jul 2020This study presents a meta-analysis of the relation between language and mathematics. A moderate relation between language and mathematics was found in 344 studies with... (Meta-Analysis)
Meta-Analysis
This study presents a meta-analysis of the relation between language and mathematics. A moderate relation between language and mathematics was found in 344 studies with 393 independent samples and more than 360,000 participants, = .42, 95% CI [.40, .44]. Moderation and partial correlation analyses revealed the following: (a) more complicated language and mathematics skills were associated with stronger relations between language and mathematics; after partialing out working memory and intelligence, rapid automatized naming showed the strongest relation to numerical knowledge; (b) the relation between language and mathematics was stronger among native language speakers than among second-language learners, but this difference was not found after partialing out working memory and intelligence; (c) working memory and intelligence together explained over 50% of the variance in the relation between language and mathematics and explained more variance in such relations involving complex mathematics skills; (d) language and mathematics predicted the development of one another even after controlling for initial performance. These findings suggest that we may use language as a medium to communicate, represent, and retrieve mathematics knowledge as well as to facilitate working memory and reasoning during mathematics performance and learning. With development, the use of language to retrieve mathematics knowledge may be more important for foundational mathematics skills, which in turn further strengthens linguistic thought processes for performing more advanced mathematics tasks. Such use of language may boost the mutual effects of cognition and mathematics across development. (PsycInfo Database Record (c) 2020 APA, all rights reserved).
Topics: Age Factors; Female; Humans; Intelligence; Language; Male; Mathematics; Memory, Short-Term
PubMed: 32297751
DOI: 10.1037/bul0000231 -
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 -
Brain Structure & Function Jan 2023Since the pioneering work of the early 20th century neuropsychologists, the angular gyrus (AG), particularly in the left hemisphere, has been associated with numerical... (Review)
Review
Since the pioneering work of the early 20th century neuropsychologists, the angular gyrus (AG), particularly in the left hemisphere, has been associated with numerical and mathematical processing. The association between the AG and numerical and mathematical processing has been substantiated by neuroimaging research. In the present review article, we will examine what is currently known about the role of the AG in numerical and mathematical processing with a particular focus on arithmetic. Specifically, we will examine the role of the AG in the retrieval of arithmetic facts in both typically developing children and adults. The review article will consider alternative accounts that posit that the involvement of the AG is not specific to arithmetic processing and will consider how numerical and mathematical processing and their association with the AG overlap with other neurocognitive processes. The review closes with a discussion of future directions to further characterize the relationship between the angular gyrus and arithmetic processing.
Topics: Adult; Child; Humans; Magnetic Resonance Imaging; Parietal Lobe; Neuroimaging; Mathematics; Brain Mapping
PubMed: 36376522
DOI: 10.1007/s00429-022-02594-8 -
Scientific Reports Jun 2023Even though, nowadays, cancer is one of the leading causes of death, too little is known about the behavior of this disease due to its unpredictability from one patient...
Even though, nowadays, cancer is one of the leading causes of death, too little is known about the behavior of this disease due to its unpredictability from one patient to another. Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, improvements are still necessary on these models. The primary objective of the present research is to prove the efficiency of fractional order calculus in mathematical oncology, more specifically in tumor growth modeling. For this, a generalization of the four most used differential equation models in tumor volume measurements fitting is realized, using the corresponding fractional order equivalent. Are established the fractional order Exponential, Logistic, Gompertz, General Bertalanffy-Pütter and Classical Bertalanffy-Pütter models for a treated and untreated dataset. The obtained results are compared by Mean Squared Error (MSE) with the integer order correspondent of each model. The results prove the superiority of the fractional order models. The MSE of fractional order models are reduced at least at half in comparison with the MSE of the integer order equivalent. It is demonstrated in this way that fractional order deterministic models can offer a good starting point in finding a proper mathematical model for tumor evolution prediction. Fractional calculus is a suitable method in this case due to its memory property, aspect that particularly characterizes biological processes.
Topics: Humans; Models, Biological; Mathematics; Models, Theoretical; Neoplasms; Medical Oncology
PubMed: 37344605
DOI: 10.1038/s41598-023-37196-9