Search Results (14)

This paper presents a novel visual approach to teaching multiplication to elementary school pupils using stick intersections. Within the South African context, where students consistently demonstrate low mathematics achievement, particularly in foundational arithmetic operations, this research explores an alternative pedagogical strategy that transforms abstract multiplication concepts into visual, concrete, countable representations. Building on theories of embodied cognition and visual mathematics, this study implemented and evaluated the stick intersection method with 45 Grade 4 students in Polokwane, Limpopo Province. Using a mixed-methods approach combining quantitative assessments with qualitative observations, the results revealed statistically significant improvements in multiplication performance across all complexity levels, with particularly substantial gains among previously low-performing students (61.3% improvement, d = 1.87). Qualitative findings demonstrated enhanced student engagement, deeper conceptual understanding of place value, and overwhelmingly positive learner perceptions of the method. The visual approach proved especially valuable in the multilingual South African classroom context, where it transcended language barriers by providing direct visual access to mathematical concepts. High retention rates (94.9%) one-month post-intervention suggest the method facilitated lasting conceptual understanding rather than temporary procedural knowledge. This research contributes to mathematics education by demonstrating how visually oriented, culturally responsive pedagogical approaches can address persistent challenges in developing mathematics proficiency, particularly in resource-constrained educational environments. Full article

As an essential medium for the cultural narrative of architectural space, studying the cognitive transformation mechanisms of traditional ethnic decorative patterns is critical for their effective preservation and innovative application. This research focuses on typical decorative motifs found in She ethnic architectural heritage, systematically classifying them into five categories—animal, plant, human figure, totem, and geometric—based on symbolic themes, formal structure, and cultural function. Correspondingly, 20 sets of standardized black-and-white line drawing stimuli were developed for experimental use. Methodologically, this study utilized the EyeLink 1000 eye-tracking system to acquire real-time gaze metrics, including fixation duration and saccadic amplitude, as well as pupil dilation responses from participants engaged in a controlled pattern observation task. Immediately after observation, participants completed a semantic differential assessment using a five-point Likert scale. Data analysis employed descriptive statistics, analysis of variance (ANOVA), Kruskal–Wallis tests, and Bonferroni-adjusted post hoc comparisons (α = 0.05). Attention allocation was further examined through heatmaps and gaze trajectory visualizations to provide comprehensive insight into visual engagement. Two principal findings were identified: first, male participants showed a predominant focus on holistic structural composition and cultural symbol representation, whereas female participants exhibited a processing bias towards fine details; second, concrete symbols imbued with historical significance elicited more pronounced emotional responses, while abstract geometric patterns necessitated formal reconstruction to enhance cognitive accessibility. These findings offer empirical support for gender-inclusive architectural design strategies and inform practical approaches for safeguarding cultural heritage within contemporary architectural environments. Consequently, modern reinterpretation of traditional decorative patterns should balance cultural narrative fidelity with functional adaptation, achieving inclusive expression through contextual reconstruction and interactive design strategies. Future research directions include expanding participant demographics to encompass cross-cultural cohorts and incorporating multimodal neuroimaging techniques to elucidate the underlying cognitive and affective mechanisms, thereby advancing the sustainable transmission and innovation of ethnic cultural heritage. Full article

Background: The embodied cognition approach, as applied to concrete knowledge, is centred on the role of the perceptual and motor aspects of experience. To extend the embodied framework to abstract knowledge, some studies have suggested that further dimensions, such as affective or social experiences, are relevant for the semantic representations of abstract concepts. The objective of this study is to develop a measure that can quantitatively capture the multidimensional nature of abstract concepts. Methods: We used dimension-rating methods, known to be suitable, to account for the semantic representations of abstract concepts, to develop a new database of 964 Italian words, rated by 542 participants. Besides classical psycholinguistic variables (i.e., concreteness, imageability, familiarity, age of acquisition, semantic diversity) and affective norms (i.e., valence, arousal), we collected ratings on selected dimensions characterizing the semantic representations of abstract concepts, i.e., introspective, mental state, quantitative, spatial, social, moral, theoretical, and economic dimensions. The measure of exclusivity was incorporated to quantify the number of dimensions, and the respective relevance, for each concept. Concepts with a high value of exclusivity rely on only one/a few dimension/s with high value on the respective rating scale. Results: A multidimensional representation characterized most abstract concepts, with two robust major clusters. The first was characterized by dense intersections among introspective, mental state, social, and moral dimensions; the second, less interconnected, cluster revolved around quantitative, spatial, theoretical, and economic dimensions. Quantitative, theoretical, and economic concepts obtained higher exclusivity values. Conclusions: The present study contributes to the investigation of the semantic organization of abstract words and supports a controlled selection and definition of stimuli for clinical and research settings. Full article

This paper proposes a one-to-one trajectory synthetization method with stable long-term individual mobility behavior based on a generalizable area embedding. Previous methods concentrate on producing highly detailed data on short-term and restricted areas for, e.g., autonomous driving scenarios. Another possibility consists of city-wide and beyond scales that can be used to predict general traffic flows. The now-presented approach takes the tracked mobility behavior of individuals and creates coherent synthetic mobility data. These generated data reflect the person’s long-term mobility behavior, guaranteeing location persistency and sound embedding within the point-of-interest structure of the observed area. After an analysis and clustering step of the original data, the area is distributed into a geospatial grid structure (H3 is used here). The neighborhood relationships between the grids are interpreted as a graph. A feed-forward autoencoder and a graph encoding–decoding network generate a latent space representation of the area. The original clustered data are associated with their respective H3 grids. With a greedy algorithm approach and concerning privacy strategies, new combinations of grids are generated as top-level patterns for individual mobility behavior. Based on the original data, concrete locations within the new grids are found and connected to ways. The goal is to generate a dataset that shows equivalence in aggregated characteristics and distances in comparison with the original data. The described method is applied to a sample of 120 from a study with 1000 participants whose mobility data were generated in the city of Munich in Germany. The results show the applicability of the approach in generating synthetic data, enabling further research on individual mobility behavior and patterns. The result comprises a sharable dataset on the same abstraction level as the input data, which can be beneficial for different applications, particularly for machine learning. Full article

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed $3max(a,b)$ in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation. Full article

Mathematical manipulatives and the concrete–representational–abstract (CRA) instructional approach are common in elementary classrooms, but their use declines significantly by high school. This paper describes a mixed methods study focused on knowledge retention and perceptions of students in a high school Algebra I inclusion class after a lesson on square roots using a novel algebra manipulative. Twenty-five students in a high school Algebra I inclusion class engaged in an interactive lesson on square roots paired with the manipulative to support their conceptual understanding. Participants completed a pretest, a post-treatment questionnaire, and a delayed post-test. The two-sample t test showed a significant difference in students’ pretest–post-test scores. However, conventional content analysis of the questionnaires showed that most students did not believe the CRA instructional approach supported their learning. Implications include increased use of manipulatives to teach abstract algebraic topics to support students’ conceptual understanding and destigmatizing the use of manipulatives in secondary mathematics classrooms. Full article

Adding pictures to instructional materials that are relevant and representational supports meaningful learning. However, it is not always straightforward to generate such pictures, for example, for abstract concepts. It is much easier to make representational pictures of concrete concepts, “table” or “chair”, compared to abstract concepts, “loyalty” or “democracy”. The field of finance is full of abstract or complex financial concepts, such as pension, market value, and asset valuation—to name a few. How do we then make pictures of such financial concepts that can represent them? In this regard, visual metaphors could provide hints as to how complex financial concepts can be presented in the form of pictures. For this purpose, this study analyzed the representation of complex financial concepts in terms of visual metaphors. Visual metaphors of five financial concepts were selected from the financial learning content online. These included: (1) risk diversification, (2) inflation, (3) compound interest, (4) time value of money, and (5) financial risk. Using the content analysis approach, each of the visual metaphors were analyzed to determine how different features of the given financial concept were mapped onto the visual metaphor, making them representational. Results indicate that visual metaphors could be an effective and creative way to present complex financial concepts in the form of representational pictures. Full article

A scientometric-based assessment of the literature on geopolymers was conducted in this study to determine its critical aspects. Typical review studies are restricted in their capability to link disparate segments of the literature in a systematic and exact way. Knowledge mapping, co-citation, and co-occurrence are very difficult components of creative research. This study adopted an advanced strategy of data mining, data processing and analysis, visualization and presentation, and interpretation of the bibliographic data on geopolymers. The Scopus database was used to search for and retrieve the data needed to complete the study’s objectives. The relevant sources of publications, keyword assessment, productive authors based on publications and citations, top papers based on citations received, and areas actively engaged in the research of geopolymers are recognized during the data assessment. The VOSviewer (VOS: visualization of similarities) software application was employed to analyze the literature data comprising citation, bibliographic, abstract, keywords, funding, and other information from 7468 relevant publications. In addition, the applications and restrictions associated with the use of geopolymers in the construction sector are discussed, as well as possible solutions to overcome these restrictions. The scientometric analysis revealed that the leading publication source (journal) in terms of articles and citations is “Construction and building materials”; the mostly employed keywords are geopolymer, fly ash, and compressive strength; and the top active and contributing countries based on publications are China, India, and Australia. Because of the quantitative and graphical representation of participating nations and researchers, this study can help academics to create collaborative efforts and exchange creative ideas and approaches. In addition, this study concluded that the large-scale usage of geopolymer concrete is constrained by factors such as curing regime, activator solution scarcity and expense, efflorescence, and alkali–silica reaction. However, embracing the potential solutions outlined in this study might assist in boosting the building industry’s adoption of geopolymer concrete. Full article

The group theoretical description of the periodic system of elements in the framework of the Rumer–Fet model is considered. We introduce the concept of a single quantum system, the generating core of which is an abstract $C^{*}$-algebra. It is shown that various concrete implementations of the operator algebra depend on the structure of the generators of the fundamental symmetry group attached to the energy operator. In the case of the generators of the complex shell of a group algebra of a conformal group, the spectrum of states of a single quantum system is given in the framework of the basic representation of the Rumer–Fet group, which leads to a group-theoretic interpretation of the Mendeleev’s periodic system of elements. A mass formula is introduced that allows giving the termwise mass splitting for the main multiplet of the Rumer–Fet group. The masses of elements of the Seaborg table (eight-periodic extension of the Mendeleev table) are calculated starting from the atomic number $Z=3$ and going to $Z=220$. The continuation of the Seaborg homology between lanthanides and actinides is established with the group of superactinides. A 10-periodic extension of the periodic table is introduced in the framework of the group-theoretic approach. The multiplet structure of the extended table’s periods is considered in detail. It is shown that the period lengths of the system of elements are determined by the structure of the basic representation of the Rumer–Fet group. The theoretical masses of the elements of 10th and 11th periods are calculated starting from $Z=221$ and going to to $Z=364$. The concept of hypertwistor is introduced. Full article

We find ourselves situated within a world that can be experienced visually, for the first time, in its wholeness. Using conceptual analysis, we intend to show that notions born within the practice of habitation, such as the sense of place, place attachment, and hearth, can help us evaluate the psychological implications of the images of Earth taken from space. We chose a phenomenological approach to human habitation because it allows concepts pertaining to connected and inherently interdisciplinary fields, for instance environmental psychology or human geography, to be reunited under the umbrella of an anthropological interpretation. The sensory and imaginary connotations of the notion of place may be noticed starting from the distinction between space as mathematical abstraction and concrete places being experienced directly. An analysis of the nature of this connection leads to the finding that we actively imagine and reimagine the surrounding world as an unfolding space in which we are constantly attempting to dwell. What is of particular interest for us is the manner in which technologically-mediated visual experience may inspire cognitive representations or may generate profound emotions, such as the attachment to a particular place. Therefore, the value of imagination for the anthropology of habitation is not rendered by its compensatory role, but by its link to ontogenesis. Familiar places, which continue to attract us, are capable of triggering unique imaginary processes, reveries which refer us to the primordial steps of ontogenesis with outmost intensity. The process of subjective appropriation of the world begins with that privileged space of origin specific to each of us, the space which we identify with most intensely. Thus, the psychological impact of the image of Earth from space: we become intensely aware that this planet is our Place within a hostile universe. Full article

Over the last few years, there has been an increase in the studies that consider experiential (visual) information by building multi-modal language models and representations. It is shown by several studies that language acquisition in humans starts with learning concrete concepts through images and then continues with learning abstract ideas through the text. In this work, the curriculum learning method is used to teach the model concrete/abstract concepts through images and their corresponding captions to accomplish multi-modal language modeling/representation. We use the BERT and Resnet-152 models on each modality and combine them using attentive pooling to perform pre-training on the newly constructed dataset, which is collected from the Wikimedia Commons based on concrete/abstract words. To show the performance of the proposed model, downstream tasks and ablation studies are performed. The contribution of this work is two-fold: A new dataset is constructed from Wikimedia Commons based on concrete/abstract words, and a new multi-modal pre-training approach based on curriculum learning is proposed. The results show that the proposed multi-modal pre-training approach contributes to the success of the model. Full article

Representation and abstraction are two of the fundamental concepts of computer science. Together they enable “high-level” programming: without abstraction programming would be tied to machine code; without a machine representation, it would be a pure mathematical exercise. Representation begins with an abstract structure and seeks to find a more concrete one. Abstraction does the reverse: it starts with concrete structures and abstracts away. While formal accounts of representation are easy to find, abstraction is a different matter. In this paper, we provide an analysis of data abstraction based upon some contemporary work in the philosophy of mathematics. The paper contains a mathematical account of how Frege’s approach to abstraction may be interpreted, modified, extended and imported into type theory. We argue that representation and abstraction, while mathematical siblings, are philosophically quite different. A case of special interest concerns the abstract/physical interface which houses both the physical representation of abstract structures and the abstraction of physical systems. Full article

The term concept has been a prominent part of investigations in psychology and neurobiology where, mostly, it is mathematically or theoretically represented. Concepts are also studied in the computational domain through their symbolic, distributed and hybrid representations. The majority of these approaches focused on addressing concrete concepts notion, but the view of the abstract concept is rarely explored. Moreover, most computational approaches have a predefined structure or configurations. The proposed method, Regulated Activation Network (RAN), has an evolving topology and learns representations of abstract concepts by exploiting the geometrical view of concepts, without supervision. In the article, first, a Toy-data problem was used to demonstrate the RANs modeling. Secondly, we demonstrate the liberty of concept identifier choice in RANs modeling and deep hierarchy generation using the IRIS dataset. Thirdly, data from the IoT’s human activity recognition problem is used to show automatic identification of alike classes as abstract concepts. The evaluation of RAN with eight UCI benchmarks and the comparisons with five Machine Learning models establishes the RANs credibility as a classifier. The classification operation also proved the RANs hypothesis of abstract concept representation. The experiments demonstrate the RANs ability to simulate psychological processes (like concept creation and learning) and carry out effective classification irrespective of training data size. Full article

Automated source code documentation approaches often describe methods in abstract terms, using the words contained in the static source code or code excerpts from repositories. In this paper, we describe DynamiDoc: a simple automated documentation generator based on dynamic analysis. Our representation-based approach traces the program being executed and records string representations of concrete argument values, a return value and a target object state before and after each method execution. Then, for each method, it generates documentation sentences with examples, such as “When called on [3, 1.2] with element = 3, the object changed to [1.2]”. Advantages and shortcomings of the approach are listed. We also found out that the generated sentences are substantially shorter than the methods they describe. According to our small-scale study, the majority of objects in the generated documentation have their string representations overridden, which further confirms the potential usefulness of our approach. Finally, we propose an alternative, variable-based approach that describes the values of individual member variables, rather than the state of an object as a whole. Full article