Mathematical Cognition Research Articles

Kant’s Prize Essay and Nineteenth Century Formalism

Abstract Kant’s Prize Essay of 1764 emphasizes the importance for mathematical cognition of manipulating signs according to rules, which has led some recent commentators to ask whether Kant’s position there is a species of mathematical formalism. While most have hesitated to find formalism in the Prize Essay, this hesitation derives from misconceptions about what formalists actually believe. I therefore examine some nineteenth century formalists who were in dialogue with Kant, using their views as a model against which to compare the Prize Essay. I argue that Kant’s view in the Prize Essay is continuous with their formalism, since the Prize Essay shares the essential aspects of their views about the sign-signified relation in mathematics and the role of signs in mathematical reasoning.

Partial Meanings that a Teacher Mobilizes for Learning Limits at the School Level

Objective: The objective of this study is to analyze the partial meanings of the function limit object in a variable that a teacher mobilizes when teaching the Calculus unit at the school level. Theoretical Framework: For the analysis of the partial meanings of the mathematical object, the epistemic configurations proposed by the OntoSemiotic Approach to Mathematical Cognition and Instruction (OSA) were used. Method: The study is qualitative in nature and its methodological design is an instrumental-descriptive case study.. To carry out the analysis of the meanings mobilized by the teacher, a semi-structured interview, teaching material, field notes from class observation and the notes of two students of the course were used. Results and Discussion: The results obtained revealed that the teacher preferentially mobilizes the last two epistemic configurations of the mathematical object and to a limited extent some primary objects of other configurations. Research Implications: The research provides a reflection for the pedagogical practices of teachers regarding the epistemic richness of the mathematical object, mainly to Chilean teachers, who must teach said object promoting all its partial meanings according to the requirements of the Chilean school curriculum. Originality/Value: The research lies in the importance of the pedagogical practices that teachers should consider when teaching said object, in particular, the mathematical richness of the partial meanings of the object.

Dimensional versus categorical approach: A comparative study of mathematical cognition

BackgroundResearchers have employed two distinct methods to understand the cognitive underpinnings of mathematical ability: categorical and dimensional. These two methods have different underlying assumptions. However, to the best of our knowledge, research to date has not empirically tested which method can better predict variance in mathematical ability. Method104 children from Indian public schools in the 3rd and 4th grades completed a mathematical ability test. For the categorical approach, participants were categorized into two groups: mathematical learning difficulty and high math achieving. For the dimensional approach, the data of all participants were considered. The cognitive abilities measured included approximate number system, working memory, inhibitory control, and spatial ability. ResultsMixed factorial ANOVA and hierarchical regressions revealed that the dimensional approach demonstrated better predictive power for mathematical ability than the categorical approach. ConclusionsThe dimensional approach offers a more comprehensive insight into mathematical cognition, enabling greater control over the predictors.

Development of a Cognitive Assessment Checklist for First-Grade Mathematics: Utilizing Hierarchical Cognitive Diagnostic Modeling in Elementary Education

This study aimed to address the need for a comprehensive assessment tool to evaluate the mathematical abilities of first-grade students through cognitive diagnostic assessment (CDA). The primary challenge involved in this endeavor was to delineate the specific cognitive skills and sub-skills pertinent to first-grade mathematics (FG-M) and to determine the most suitable model for assessment. Employing a mixed-methods approach, the research identified nine cognitive attributes essential for FG-M and developed a Q-matrix delineating the relationship between these attributes. A preliminary version of the FG-M checklist comprising 74 items was formulated. Subsequently, the assessment tool was administered to 1018 first-grade students, with their teachers utilizing the FG-M checklist for evaluation. Results indicated a commendable accuracy in classifying the nine cognitive attributes, ranging from 0.68 to 0.92, with an average accuracy of 0.76. Furthermore, the validity of the scores was corroborated by a substantial correlation (r = 0.78) between scores from the FG-M checklist and the arithmetic subtest of the Wechsler Intelligence Scale for Children, Fourth Edition (WISC-IV). Overall, the study concluded that the scores from the FG-M checklist are reliable for assessing the mathematical skills of first-grade students. The findings underscored the students’ proficiency in fundamental mathematical concepts such as cardinality, addition, subtraction, weight, and statistics. However, areas of weakness were identified in concepts related to multiplication, time, symmetry, and geometry. This research contributes significantly to the advancement of CDA in educational contexts, providing educators with a valuable tool for identifying students’ strengths and weaknesses in mathematical cognition.

Enhancing M Enhancing mathematics problem-solving skills in AI-driven environment: Integrated SEM-neural network approach

This study explores the nexus of gamification, artificial intelligence (AI), and mathematics cognition. Sample size of 71 responded in an intervention using game-based learning (GBL) approach. The purpose of designing the GBL was to enhance computational thinking and mathematical skills. The research employed multigroup partial least squares structural equation modelling (MGA-PLS-SEM) and artificial neural networks (ANN) through multilayer perceptron (MLP) as data analysis technique. The findings showed significant positive influence on class engagement, attitudes toward mathematics, as well as student performance. The analysis also revealed gender-related variations, which affirmed the model's consistency across diverse groups. The study validated the hypothesis and consequently advocated for the transformative potential of gamification, in preparation of 21st-century learners for AI-driven digital landscape. The implications are to ensure the integration of gamified elements into educational strategies, benefiting educators, curriculum developers, and policymakers resonating strongly for educators, curriculum developers, and policymakers.

Sociocognitive configuration: meanings and creations in the mathematical learning of middle school students

The study explores the socio-cognitive configuration in the mathematical learning of middle school students, relating cognitive processes to the social interactions that shape perceptions and performance in mathematics. A qualitative approach was adopted to delve into students' subjective experiences. Meaningful interactions revealed in the classroom enhance mathematical understanding and foster critical and problem-solving skills. The socio-cognitive configuration achieved evidence that mathematical cognition is deeply affected and defined by its social context and educational practices. This study underscores the importance of integrating students' personal and collective meanings in mathematical learning to foster a holistic pedagogical approach to both the transmission of knowledge and its social and personal relevance, which transforms mathematics teaching by making it more applicable to students' real-life challenges and contexts

Creating and sharing linear algebra metaphors as an assessment for engaging students beyond the cognitive domain

In this research, we investigated how a summative assessment of creating and sharing metaphors for linear algebra concepts supported undergraduate students’ affective, behavioral, and cognitive engagement. All seven research participants were enrolled in an introductory linear algebra course designed to develop students’ geometric understanding of linear algebra concepts in R2 and R3. Using embodied cognition and an engagement framework, we analyzed students’ written responses and video-taped their focus group discussion. Our findings suggest that this summative assessment (a) privileged mathematical aesthetics and the affective domain of learning, (b) engaged students in binding formal aspects of linear algebra concepts with metaphors that they enacted via embodiment, and (c) was an opportunity to demonstrate learning and higher-order cognition. Thus, illustrating that assessments can focus on aesthetic and affective domains of mathematics while simultaneously integrating serious mathematical cognition. We conclude by offering adaptations of this assessment for other mathematics courses.

Forty years of research on mathematical cognition: Summing it up.

In this article, I describe my academic journey as a Canadian researcher in the field of mathematical cognition. The article is a version of the talk I gave when I was presented with the Donald O. Hebb Award from the Canadian Society for Brain, Behaviour and Cognitive Science in July 2023 at the University of Guelph. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Number and domain both affect the relation between executive function and mathematics achievement: A study of children's executive function with and without numbers.

Magnitude processing and executive functions (EFs) have emerged as robust predictors of mathematics achievement. However, the nature of these associations is still unclear. For example, it is uncertain if EFs applied in the context of domain-specific mathematical cognition (i.e., EFs applied while processing numbers) are more closely related to mathematics achievement than EFs applied in nonnumerical, domain-general contexts. Also, how distinct EF domains-that is, working memory, inhibitory control, and cognitive flexibility-and contents-that is, numerical versus nonnumerical-moderate the association between magnitude processing and mathematics achievement has not been fully understood. To address these issues, we investigated how magnitude processing, EFs applied to nonnumerical and numerical task stimuli, and their interactions were associated with mathematics achievement. Three hundred fifty-nine Brazilian third- to fifth-grade (8-10 years old) students completed measures of working memory, inhibitory control, and cognitive flexibility with numerical and nonnumerical task versions, nonsymbolic and symbolic magnitude comparison, and mathematics achievement. A series of regression models indicated that nonsymbolic and symbolic magnitude processing are consistently associated with mathematics achievement, even when controlling for working memory, inhibitory control, and cognitive flexibility measured with both numerical and nonnumerical contents. All EF measures were associated with mathematics achievement. However, cognitive flexibility measured with numerical content showed the strongest association. Results support the hypothesis that magnitude processing and EFs are uniquely associated with mathematics achievement. Furthermore, EFs measured with nonnumerical and numerical contents related differently to mathematics achievement, even when controlling for symbolic and nonsymbolic magnitude processing, suggesting they encompass somewhat distinct cognitive processes. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Changes in Brain Functional Connectivity Underlying the Space-Number Association.

Whether small number magnitudes are inherently represented as lying to the left of larger ones, the space-number association (SNA), is an important issue in mathematical cognition. In this fMRI study, we used a go/no-go implicit association task to investigate the brain activity and functional connectivity underlying the SNA. Arabic digits lower or higher than 5 and left- or right-pointing arrows were alternated as central targets. In a single-code task condition, participants responded to a specific number magnitude and to all arrows or to a specific arrow direction and to all number magnitudes. In a joint-code (JC) condition, responses were provided after congruent, for example, "go when a number is lower than 5 or an arrow points left," or incongruent, for example, "go when a number is lower than 5 or an arrow points right," SNAs. The SNA was only found in the JC condition, where responses were faster with congruent instructions. Analyses of fMRI functional connectivity showed that the SNA was matched with enhanced excitatory inputs from ACC, the left TPJ, and the left inferior frontal gyrus to the left and right intraparietal sulcus (IPS). Incongruent JC trials were associated with enhanced excitatory modulation from ACC to the left and right IPS. These results show that the SNA is associated with enhanced activation of top-down brain control and changes in the functional interaction between the left and right IPS. We conclude that the SNA does not depend on an inherent and bottom-up spatial coding of number magnitudes.

Eye tracking study in children to assess mental calculation and eye movements

Eye tracking technology is a high-potential tool for different mathematic cognition research areas. Moreover, there is a dire need for more studies that provide detailed information on the quality of registered eye data. This study aimed to illustrate the applicability of eye tracking in the examination of mathematical cognition, focusing specifically on primary school students completing a computerized mental arithmetic task. Results suggested that the eye tracking device effectively captured high-quality eye movement data when primary school children engaged in this specific task. Furthermore, significant negative correlations have been found between task performance and number of eye fixations. Finally, eye movements distinctions between “Areas of Interest” have been found, indicating different visual tracking associated with different components of arithmetic calculations. This study underscores the extensive possibilities for future research employing eye tracking devices during computerized calculation tasks as assessment tools to explore the complex visual and cognitive processes.

Enhancing Early Mathematical Learning through Multimedia Technologies: A GeoGebra-Based Approach in Preschool Education

Multimedia technologies, encompassing images, animations, and sounds, offer innovative possibilities for preschool educators, particularly in the context of presenting mathematical content and fostering children's mathematical cognition. This study is driven by a dual objective: firstly, to formulate a GeoGebra-based model for the assimilation of fundamental mathematical concepts related to numbers and sets; secondly, to execute this model in practical educational settings. The research conducted for this paper spanned five years (2018-2023) as part of a comprehensive five-year project, comprising three distinct phases, each featuring specialized workshops. A cohort of 380 preschool teachers completed GeoGebra training, which was followed by the integration of the proposed model into the educational curriculum across preschool institutions in two districts within the Republic of Serbia. Preschool educators independently implemented two activities aligned with their project themes. The study's findings emphasize the effectiveness of implementing an interactive project-based learning approach in preschool settings, where well-integrated GeoGebra applets, particularly when harmonized with activities involving physical and tangible elements, can yield positive outcomes.

Numbers in Context: Cardinals, Ordinals, and Nominals in American English.

There are three main types of number used in modern, industrialized societies. Cardinals count sets (e.g., people, objects) and quantify elements of conventional scales (e.g., money, distance), ordinals index positions in ordered sequences (e.g., years, pages), and nominals serve as unique identifiers (e.g., telephone numbers, player numbers). Many studies that have cited number frequencies in support of claims about numerical cognition and mathematical cognition hinge on the assumption that most numbers analyzed are cardinal. This paper is the first to investigate the relative frequencies of different number types, presenting a corpus analysis of morphologically unmarked numbers (not, e.g., "eighth" or "21st") in which we manually annotated 3,600 concordances in the Corpus of Contemporary American English. Overall, cardinals are dominant-both pure cardinals (sets) and measurements (scales)-except in the range 1,000-10,000, which is dominated by ordinal years, like 1996 and 2004. Ordinals occur less often overall, and nominals even less so. Only for cardinals do round numbers, associated with approximation, dominate overall and increase with magnitude. In comparison with other registers, academic writing contains a lower proportion of measurements as well as a higher proportion of ordinals and, to some extent, nominals. In writing, pure cardinals and measurements are usually represented as number words, but measurements-especially larger, unround ones-are more likely to be numerals. Ordinals and nominals are mostly represented as numerals. Altogether, this paper reveals how numbers are used in American English, establishing an initial baseline for any analyses of number frequencies and shedding new light on the cognitive and psychological study of number.

AlephZero and mathematical experience

This essay explores the impact of automated proof construction on three key areas of mathematical cognition: on how we judge the role one piece of mathematics plays in another, on how we make mistakes in reasoning about mathematical objects, and on how we understand what our theorems are truly about. It concludes by speculating on a new form of mathematical experience that these methods could make possible: “glitching”, a game-like search for uncanny consequences of our definitions.

Means End Analysis (MEA): A Learning Model to Improve Mathematics Cognitive Learning Outcomes in the Post-Pandemic COVID-19

This research is motivated by the low cognitive learning outcomes of students in solving math problems. Alternative learning to the low cognitive learning outcomes of students is to apply the Means-End Analysis learning model. This model is designed using a quasi-experimental method. This study aims to determine the difference in the average cognitive learning outcomes of class VI students in learning mathematics using the Means-End Analysis model with conventional learning using the Direct Instruction model. The population of this research is all sixth-grade students of SDN 267 Bandung for the academic year 2021/2022. The samples taken were class VI A, and class VI B using a purpose sampling technique. Data collection techniques used are tests and observations. The data analysis technique used an independent t-test with a significance level of 5%. The results obtained that: there is a difference in the average cognitive learning outcomes of students between those taught with the Means End Analysis learning model and conventional learning using the Direct Instruction model. So that the Means-End Analysis model can be used as a learning model that can improve cognitive mathematics learning outcomes in the post-pandemic COVID-19. Keywords: mathematics cognitive learning outcomes, means end analysis (MEA): a learning model, post-pandemic covid-19

A brief, multiple-choice assessment of mature number sense is strongly correlated with more resource-intensive measures

Students who exhibit mature number sense make sense of numbers and operations, use reasoning to notice patterns, and flexibly choose effective problem-solving strategies (McIntosh et al., 1997, https://ro.ecu.edu.au/ecuworks/6819). Due to its dispositional nature, mature number sense is typically measured through in-depth interviews or tests of strategy usage. Yet, the lack of an efficient, rigorously developed measure has made it difficult to collect systematic, replicable evidence on students’ mature number sense. To address this, we developed a brief assessment of mature number sense. The present study provides additional convergent evidence of validity for this measure with US students in grades 3-8 (8–14 years old). We compared middle school (N = 40) and upper elementary school (N = 41) scores from the brief assessment with an established, time-intensive measure (Yang, 2019, https://doi.org/10.1007/s10649-018-9874-8) and an in-depth interview of student strategy usage (Markovits & Sowder, 1994, https://doi.org/10.2307/749290). We found strong correlations (r > 0.7) across all three measures, and this held even when controlling for students’ arithmetic scores (pr > 0.6). Researchers and educators can now use the brief assessment to investigate students’ mathematical thinking and advance knowledge of a key aspect of mathematical cognition.

Withdrawn: Mathematical cognition related to the large numbers in early societies: A study based on 5th-century Buddhist Commentaries in Sri Lanka.

Universal presentations and interpretations of the history of mathematics have now been challenged, identifying the need of resurrecting hitherto marginalized contributions. In such studies, cognitive factors influencing the related development trajectories play a crucial role. Different societies would have had different mathematical cognitions contributing to the development of distinct forms of mathematics. This study attempts to surface the related cognitive contents of using large numbers. The principal source used for the study—the Buddhist Commentaries—determines the period and region covered. As such, it explores the conditions prevailed in 5th century CE (approximately) in Sri Lanka.2000 Mathematics Subject Classification (MSC 2000) code: 01A07

EEG alpha and theta time-frequency structure during a written mathematical task.

Since the first electroencephalogram (EEG) was obtained, there have been many possibilities to use it as a tool to access brain cognitive dynamics. Mathematical (Math) problem solving is one of the most important cortical processes, but it is still far from being well understood. EEG is an inexpensive and simple indirect measure of brain operation, but only recently has low-cost equipment (mobile EEG) allowed sophisticated analyses in non-clinical settings. The main purpose of this work is to study EEG activation during a Math task in a realistic environment, using mobile EEG. A matching pursuit (MP)-based signal analysis technique was employed, since MP properties render it a priori suitable to study induced EEG activity over long time sequences, when it is not tightly locked to a given stimulus. The study sample comprised sixty healthy volunteers. Unlike the majority of previous studies, subjects were studied in a sitting position with their eyes open. They completed a written Math task outside the EEG lab, wearing a mobile EEG device (EPOC+). Theta [4 Hz-7.5 Hz], alpha (7.5 Hz-13 Hz] and 0.5 Hz micro-bands in the [0.5 Hz-20 Hz] range were studied with a low-density stochastic MP dictionary. Over 1-min windows, ongoing EEG alpha and theta activity was decomposed into numerous MP atoms with median duration around 3s, similar to the duration of induced, time-locked activity obtained with event-related (des)synchronization (ERS/ERD) studies. Relative to Rest, there was lower right-side and posterior MP alpha atom/min during Math, whereas MP theta atom/min was significantly higher on anteriorly located electrodes, especially on the left side. MP alpha findings were particularly significant on a narrow range around 10 Hz-10.5 Hz, consistent with FFT alpha peak findings from ERS/ERD studies. With a streamlined protocol, these results replicate previous findings of EEG alpha and theta activation obtained during Math tasks with different signal analysis techniques and in different time frames. The efficient application to real-world, noisy EEG data with a low-resolution stochastic MP dictionary shows that this technique is very encouraging. These results provide support for studies of mathematical cognition with mobile EEG and matching pursuit.

Profiles of mathematical deficits in children with dyslexia

Despite a high rate of concurrent mathematical difficulties among children with dyslexia, we still have limited information regarding the prevalence and severity of mathematical deficits in this population. To address this gap, we developed a comprehensive battery of cognitive tests, known as the UCSF Mathematical Cognition Battery (MCB), with the aim of identifying deficits in four distinct mathematical domains: number processing, arithmetical procedures, arithmetic facts retrieval, and geometrical abilities. The mathematical abilities of a cohort of 75 children referred to the UCSF Dyslexia Center with a diagnosis of dyslexia, along with 18 typically developing controls aged 7 to 16, were initially evaluated using a behavioral neurology approach. A team of professional clinicians classified the 75 children with dyslexia into five groups, based on parents’ and teachers’ reported symptoms and clinical history. These groups included children with no mathematical deficits and children with mathematical deficits in number processing, arithmetical procedures, arithmetic facts retrieval, or geometrical abilities. Subsequently, the children underwent evaluation using the MCB to determine concordance with the clinicians’ impressions. Additionally, neuropsychological and cognitive standardized tests were administered. Our study reveals that within a cohort of children with dyslexia, 66% exhibit mathematical deficits, and among those with mathematical deficits, there is heterogeneity in the nature of these deficits. If these findings are confirmed in larger samples, they can potentially pave the way for new diagnostic approaches, consistent subtype classification, and, ultimately personalized interventions.

Math on cortex-enhanced delta phase synchrony in math experts during long and complex math demonstrations.

Neural oscillations are important for working memory and reasoning and they are modulated during cognitively challenging tasks, like mathematics. Previous work has examined local cortical synchrony on theta (4-8Hz) and alpha (8-13Hz) bands over frontal and parietal electrodes during short mathematical tasks when sitting. However, it is unknown whether processing of long and complex math stimuli evokes inter-regional functional connectivity. We recorded cortical activity with EEG while math experts and novices watched long (13-68seconds) and complex (bachelor-level) math demonstrations when sitting and standing. Fronto-parietal connectivity over the left hemisphere was stronger in math experts than novices reflected by enhanced delta (0.5-4Hz) phase synchrony in experts. Processing of complex math tasks when standing extended the difference to right hemisphere, suggesting that other cognitive processes, such as maintenance of body balance when standing, may interfere with novice's internal concentration required during complex math tasks more than in experts. There were no groups differences in phase synchrony over theta or alpha frequencies. These results suggest that low-frequency oscillations modulate inter-regional connectivity during long and complex mathematical cognition and demonstrate one way in which the brain functions of math experts differ from those of novices: through enhanced fronto-parietal functional connectivity.