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基于GraphSAGE网络的藏文短文本分类研究
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作者 敬容 杨逸民 +3 位作者 万福成 国旗 于洪志 马宁 《中文信息学报》 CSCD 北大核心 2024年第9期58-65,共8页
文本分类是自然语言处理领域的重要研究方向,由于藏文数据的稀缺性、语言学特征抽取的复杂性、篇章结构的多样性等因素导致藏文文本分类任务进展缓慢。因此,该文以图神经作为基础模型进行改进。首先,在“音节-音节”“音节-文档”建模... 文本分类是自然语言处理领域的重要研究方向,由于藏文数据的稀缺性、语言学特征抽取的复杂性、篇章结构的多样性等因素导致藏文文本分类任务进展缓慢。因此,该文以图神经作为基础模型进行改进。首先,在“音节-音节”“音节-文档”建模的基础上,融合文档特征,采用二元分类模型动态网络构建“文档-文档”边,以充分挖掘短文本的全局特征,增加滑动窗口,减少模型的计算复杂度并寻找最优窗口取值。其次,针对藏文短文本的音节稀疏性,首次引入GraphSAGE作为基础模型,并探究不同聚合方式在藏文短文本分类上的性能差异。最后,为捕获节点间关系的异质性,对邻居节点进行特征加权再平均池化以增强模型的特征提取能力。在TNCC标题文本数据集上,该文模型的分类准确率达到了62.50%,与传统GCN、原始GraphSAGE和预训练语言模型CINO相比,该方法在分类准确率上分别提高了2.56%、1%和2.4%。 展开更多
关键词 图神经网络 藏文文本分类 TNCC数据集
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Algorithm for Visualization of Zero Divisor Graphs of the Ring ℤn Using MAPLE Coding
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作者 Nasir Ali 《Open Journal of Discrete Mathematics》 2024年第1期1-8,共8页
This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study ... This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations. 展开更多
关键词 Zero Divisor Graph Ring Theory Maple Algorithm n Modulo n Graph Theory Mathematical Computing
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A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS
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作者 高雅 毛井 吴传喜 《Acta Mathematica Scientia》 SCIE CSCD 2024年第2期474-483,共10页
In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piece... In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation. 展开更多
关键词 mean curvature flow spacelike graphs translating spacelike graphs maximal spacelike graphs constant mean curvature Lorentz manifolds
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A Value for Games Defined on Graphs
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作者 Néstor Bravo 《Applied Mathematics》 2024年第5期331-348,共18页
Given a graph g=( V,A ) , we define a space of subgraphs M with the binary operation of union and the unique decomposition property into blocks. This space allows us to discuss a notion of minimal subgraphs (minimal c... Given a graph g=( V,A ) , we define a space of subgraphs M with the binary operation of union and the unique decomposition property into blocks. This space allows us to discuss a notion of minimal subgraphs (minimal coalitions) that are of interest for the game. Additionally, a partition of the game is defined in terms of the gain of each block, and subsequently, a solution to the game is defined based on distributing to each player (node and edge) present in each block a payment proportional to their contribution to the coalition. 展开更多
关键词 Graph Theory Values for graphs Cooperation Games Potential Function
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BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS
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作者 吴艺婷 《Acta Mathematica Scientia》 SCIE CSCD 2024年第2期609-631,共23页
In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of ... In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133). 展开更多
关键词 semilinear parabolic system on graphs BLOW-UP heat kernel estimate on graphs
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Perfect 1-k Matchings of Bipartite Graphs
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作者 Wenduan Dai Yan Liu Yanfang Wu 《Open Journal of Discrete Mathematics》 2024年第4期43-53,共11页
Let k be a positive integer and G a bipartite graph with bipartition (X,Y). A perfect 1-k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is inc... Let k be a positive integer and G a bipartite graph with bipartition (X,Y). A perfect 1-k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is incident with exactly k edges in M. A perfect 1-k matching is an optimal semi-matching related to the load-balancing problem, where a semi-matching is an edge subset M such that each vertex in Y is incident with exactly one edge in M, and a vertex in X can be incident with an arbitrary number of edges in M. In this paper, we give three sufficient and necessary conditions for the existence of perfect 1-k matchings and for the existence of 1-k matchings covering | X |−dvertices in X, respectively, and characterize k-elementary bipartite graph which is a graph such that the subgraph induced by all k-allowed edges is connected, where an edge is k-allowed if it is contained in a perfect 1-k matching. 展开更多
关键词 Bipartite Graph Semi-Matching Perfect 1-k Matching k-Elementary Graph
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Depth First:Optimal Path Discovery Between Designated Nodes in Random Ring-Based Graphs
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作者 Li Qi Xu Jiasheng +4 位作者 Zhang Haonan Kang Huquan Fu Luoyi Long Fei Wang Xinbing 《China Communications》 SCIE CSCD 2024年第9期225-241,共17页
This paper focuses on optimally determining the existence of connected paths between some given nodes in random ring-based graphs.Serving as a fundamental underlying structure in network modeling,ring topology appears... This paper focuses on optimally determining the existence of connected paths between some given nodes in random ring-based graphs.Serving as a fundamental underlying structure in network modeling,ring topology appears as commonplace in many realistic scenarios.Regarding this,we consider graphs composed of rings,with some possible connected paths between them.Without prior knowledge of the exact node permutations on rings,the existence of each edge can be unraveled through edge testing at a unit cost in one step.The problem examined is that of determining whether the given nodes are connected by a path or separated by a cut,with the minimum expected costs involved.Dividing the problem into different cases based on different topologies of the ring-based networks,we propose the corresponding policies that aim to quickly seek the paths between nodes.A common feature shared by all those policies is that we stick to going in the same direction during edge searching,with edge testing in each step only involving the test between the source and the node that has been tested most.The simple searching rule,interestingly,can be interpreted as a delightful property stemming from the neat structure of ring-based networks,which makes the searching process not rely on any sophisticated behaviors.We prove the optimality of the proposed policies by calculating the expected cost incurred and making a comparison with the other class of strategies.The effectiveness of the proposed policies is also verified through extensive simulations,from which we even disclose three extra intriguing findings:i)in a onering network,the cost will grow drastically with the number of designated nodes when the number is small and will grow slightly when that number is large;ii)in ring-based network,Depth First is optimal in detecting the connectivity between designated nodes;iii)the problem of multi-ring networks shares large similarity with that of two-ring networks,and a larger number of ties between rings will not influence the expected cost. 展开更多
关键词 connectivity analysis cost minimization path discover ring-based graph
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Batch Active Learning for Multispectral and Hyperspectral Image Segmentation Using Similarity Graphs
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作者 Bohan Chen Kevin Miller +1 位作者 Andrea L.Bertozzi Jon Schwenk 《Communications on Applied Mathematics and Computation》 EI 2024年第2期1013-1033,共21页
Graph learning,when used as a semi-supervised learning(SSL)method,performs well for classification tasks with a low label rate.We provide a graph-based batch active learning pipeline for pixel/patch neighborhood multi... Graph learning,when used as a semi-supervised learning(SSL)method,performs well for classification tasks with a low label rate.We provide a graph-based batch active learning pipeline for pixel/patch neighborhood multi-or hyperspectral image segmentation.Our batch active learning approach selects a collection of unlabeled pixels that satisfy a graph local maximum constraint for the active learning acquisition function that determines the relative importance of each pixel to the classification.This work builds on recent advances in the design of novel active learning acquisition functions(e.g.,the Model Change approach in arXiv:2110.07739)while adding important further developments including patch-neighborhood image analysis and batch active learning methods to further increase the accuracy and greatly increase the computational efficiency of these methods.In addition to improvements in the accuracy,our approach can greatly reduce the number of labeled pixels needed to achieve the same level of the accuracy based on randomly selected labeled pixels. 展开更多
关键词 Image segmentation Graph learning Batch active learning Hyperspectral image
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GraphSTGAN:Situation understanding network of slow-fast high maneuvering targets for maritime monitor services of IoT data
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作者 Guanlin Wu Haipeng Wang +1 位作者 Yu Liu You He 《Digital Communications and Networks》 SCIE CSCD 2024年第3期620-630,共11页
With the rapid growth of the maritime Internet of Things(IoT)devices for Maritime Monitor Services(MMS),maritime traffic controllers could not handle a massive amount of data in time.For unmanned MMS,one of the key te... With the rapid growth of the maritime Internet of Things(IoT)devices for Maritime Monitor Services(MMS),maritime traffic controllers could not handle a massive amount of data in time.For unmanned MMS,one of the key technologies is situation understanding.However,the presence of slow-fast high maneuvering targets and track breakages due to radar blind zones make modeling the dynamics of marine multi-agents difficult,and pose significant challenges to maritime situation understanding.In order to comprehend the situation accurately and thus offer unmanned MMS,it is crucial to model the complex dynamics of multi-agents using IoT big data.Nevertheless,previous methods typically rely on complex assumptions,are plagued by unstructured data,and disregard the interactions between multiple agents and the spatial-temporal correlations.A deep learning model,Graph Spatial-Temporal Generative Adversarial Network(GraphSTGAN),is proposed in this paper,which uses graph neural network to model unstructured data and uses STGAN to learn the spatial-temporal dependencies and interactions.Extensive experiments show the effectiveness and robustness of the proposed method. 展开更多
关键词 Internet of things MULTI-AGENTS Graph neural network Maritime monitoring services
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On an Invariant of Tournament Digraphs
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作者 Boris F. Melnikov Bowen Liu 《Journal of Applied Mathematics and Physics》 2024年第7期2711-2722,共12页
To date, it is unknown whether it is possible to construct a complete graph invariant in polynomial time, so fast algorithms for checking non-isomorphism are important, including heuristic algorithms, and for successf... To date, it is unknown whether it is possible to construct a complete graph invariant in polynomial time, so fast algorithms for checking non-isomorphism are important, including heuristic algorithms, and for successful implementations of such heuristics, both the tasks of some modification of previously described graph invariants and the description of new invariants remain relevant. Many of the described invariants make it possible to distinguish a larger number of graphs in the real time of a computer program. In this paper, we propose an invariant for a special kind of directed graphs, namely, for tournaments. The last ones, from our point of view, are interesting because when fixing the order of vertices, the number of different tournaments is exactly equal to the number of undirected graphs, also with fixing the order of vertices. In the invariant we are considering, all possible tournaments consisting of a subset of vertices of a given digraph with the same set of arcs are iterated over. For such subset tournaments, the places are calculated in the usual way, which are summed up to obtain the final values of the points of the vertices;these points form the proposed invariant. As we expected, calculations of the new invariant showed that it does not coincide with the most natural invariant for tournaments, in which the number of points is calculated for each participant. So far, we have conducted a small number of computational experiments, and the minimum value of the pair correlation between the sequences representing these two invariants that we found is for dimension 15. 展开更多
关键词 GRAPH Directed Graph TOURNAMENT ?nvariant
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Maximal Resonance of{(3,4),4}-Fullerene Graphs
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作者 YANG Rui MA Yan-fei 《Chinese Quarterly Journal of Mathematics》 2024年第1期1-17,共17页
A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S i... A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating.Let k be a positive integer,S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern.Moreover,if graph S is k-resonant for any integer k,then S is called maximally resonant.In this paper,we show that the maximally resonant{(3,4),4}-fullerene graphs are S_6,S_8,S_(10)^(2),S_(12)^(2),S_(12)^(4),S_(12)^(5),S_(14)^(3),S_(14)^(5),S_(16)^(3),S_(18)^(5),S_(24)as shown in Fig.1.As a corollary,it is shown that if a{(3,4),4}-fullerene graph is 4-resonant,then it is also maximally resonant. 展开更多
关键词 {(3 4) 4}-Fullerene graph k-Resonant Maximally resonant
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Weak External Bisection of Some Graphs
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作者 Yumin Liu 《Journal of Applied Mathematics and Physics》 2024年第1期91-97,共7页
Let G be a graph. A bipartition of G is a bipartition of V (G) with V (G) = V<sub>1</sub> ∪ V<sub>2</sub> and V<sub>1</sub> ∩ V<sub>2</sub> = ∅. If a bipartition satis... Let G be a graph. A bipartition of G is a bipartition of V (G) with V (G) = V<sub>1</sub> ∪ V<sub>2</sub> and V<sub>1</sub> ∩ V<sub>2</sub> = ∅. If a bipartition satisfies ∥V<sub>1</sub>∣ - ∣V<sub>2</sub>∥ ≤ 1, we call it a bisection. The research in this paper is mainly based on a conjecture proposed by Bollobás and Scott. The conjecture is that every graph G has a bisection (V<sub>1</sub>, V<sub>2</sub>) such that ∀v ∈ V<sub>1</sub>, at least half minuses one of the neighbors of v are in the V<sub>2</sub>;∀v ∈ V<sub>2</sub>, at least half minuses one of the neighbors of v are in the V<sub>1</sub>. In this paper, we confirm this conjecture for some bipartite graphs, crown graphs and windmill graphs. 展开更多
关键词 Weak External Bisection Bipartite Graph Windmill Graph
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On the (Δ + 2)-Total-Colorability of Planar Graphs with 7-Cycles Containing at Most Two Chords
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作者 Jian Chang Jingru Liu Fan Zhang 《Journal of Applied Mathematics and Physics》 2024年第7期2702-2710,共9页
The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove tha... The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords. 展开更多
关键词 Planar Graph 7-Cycle 8-Totally-Colorable Maximum Degree
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The Past and Future of Aesthetics in Mathematics
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作者 Xiaoyu Chen 《Journal of Contemporary Educational Research》 2024年第7期126-132,共7页
This paper explores the connotations of mathematical aesthetics and its connections with art,facilitated by collaboration with Ester,an individual with an artistic professional background.It begins by tracing the hist... This paper explores the connotations of mathematical aesthetics and its connections with art,facilitated by collaboration with Ester,an individual with an artistic professional background.It begins by tracing the historical evolution of aesthetics from the classical pursuit of authenticity to the modern shift toward self-expression in art.The discussion then highlights the similarities in the pursuit of truth between mathematics and art,despite their methodological differences.Through an analysis of aesthetic elements in mathematics,such as lines and function graphs,the article illustrates that the beauty of mathematics is not only manifested in cognitive processes but can also be intuitively expressed through visual arts.The paper further examines the influence of mathematics on the development of art,particularly how Leonardo da Vinci applied mathematical principles to his artworks.Additionally,the article addresses art students’perceptions of mathematics,proposes the customization of math courses for art students,and discusses future trends in the integration of mathematics and art,emphasizing the significance of art therapy and the altruistic direction of art.Lastly,the authors use a poster to visually convey the idea that the beauty of mathematics can be experienced through the senses. 展开更多
关键词 Mathematical aesthetics Integration of mathematics and art Applications of mathematics in art
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Integration of Mathematics History into Junior Middle School Education: A Pedagogical Approach to Enhance Mathematical Literacy
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作者 Cheng Cheng 《Journal of Contemporary Educational Research》 2024年第3期69-72,共4页
The integration of the history of mathematics into junior middle school mathematics education represents a significant focus of international research in mathematics and education.The mathematics curriculum standards ... The integration of the history of mathematics into junior middle school mathematics education represents a significant focus of international research in mathematics and education.The mathematics curriculum standards for compulsory education have emphasized the importance of incorporating the history of mathematics,aiming to gradually integrate it into the mathematics classroom.However,in the practical implementation of junior middle school mathematics education,the effective combination of the history of mathematics with teaching methodologies remains largely unexplored.This article explores the integration of junior middle school mathematics education and the history of mathematics,aiming to provide classroom teaching recommendations for teachers and promote the formation of students’mathematical literacy. 展开更多
关键词 History of mathematics Junior middle school mathematics education Teaching methodologies
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Enhancing Learners’Performance in Grade 7 Mathematics Through 50-30-20 Exercise
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作者 Joniel S.Cerbito 《Journal of Contemporary Educational Research》 2024年第1期233-245,共13页
Assessment exercises constitute a crucial component of the teaching and learning process,serving the purpose of gauging the degree to which learning objectives have been accomplished.This study aims to assess the math... Assessment exercises constitute a crucial component of the teaching and learning process,serving the purpose of gauging the degree to which learning objectives have been accomplished.This study aims to assess the mathematics performance of Grade 7 learners using the 50-30-20 exercise.Specifically,this study seeks to determine the learners’pre-test and post-test mean scores,identify significant differences between the pre-test and post-test results,evaluate learners’exercises,and propose enhanced exercises.The research employs a quasi-experimental design,with 40 Grade 7 learners in the school year 2023-2024 as participants,selected through purposive non-random sampling.Statistical data analysis involves the use of mean,standard deviation,paired t-test,and Cohen’s D effect size.Ethical considerations were paramount,as evidenced by a letter of authorization from the school head outlining the strict adherence to voluntary participation,informed parental consent,anonymity,confidentiality,risk mitigation,results-sharing protocols,and the commitment to keeping research data confidential.The data yielded a remarkable outcome:the experimental group exhibited improvement in both the pre-test and post-test.This result substantiates the initial objective of the study,showcasing a noteworthy and favorable performance among the participants.Consequently,it suggests that a majority of the participants strongly agree that the 50-30-20 exercises contribute to enhancing their understanding and problem-solving skills,as well as their ability to grasp mathematical concepts and improve their overall performance in mathematics.Therefore,the 50-30-20 exercises not only facilitated students in understanding mathematics lessons but were also aligned with the Department of Education’s development plan. 展开更多
关键词 Mathematical assessment exercises Learners’mathematical performance 50-30-20 exercises
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Measurement Analysis and Evaluation of Twenty-Five Years of Mathematics Education Research in China:Visual Econometric Analysis Based on CiteSpace Knowledge Mapping(1999-2024)
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作者 Jie Wang Songlin Qi 《Journal of Contemporary Educational Research》 2024年第5期128-140,共13页
Since the new century,China’s mathematics curriculum reform in basic education has continued to move forward in attempts and explorations,presenting many new changes,trends,movements,and developments.Sorting out,anal... Since the new century,China’s mathematics curriculum reform in basic education has continued to move forward in attempts and explorations,presenting many new changes,trends,movements,and developments.Sorting out,analyzing,and summarizing the achievements,experiences,problems,and challenges in this journey are conducive to providing insights for the reform and development of the Chinese basic education mathematics curriculum in the new era.This paper analyses the research on mathematics education in China(1999-2024)using the visual measurement of CiteSpace knowledge mapping,hoping to provide directions for the future of mathematics education in China. 展开更多
关键词 mathematics education Visual econometric analysis CITESPACE
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A Study on the Application of Task-Driven Teaching Method in High School Mathematics Teaching: Taking “The Determination of Vertical Line and Plane” as an Example
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作者 Xiaowei Fu 《Journal of Contemporary Educational Research》 2024年第8期1-7,共7页
The“Ordinary High School Mathematics Curriculum Standards(2017 Edition,2020 Revision)”clearly stated in“Teaching Suggestions”that“teaching activities based on the core literacy of mathematics should grasp the ess... The“Ordinary High School Mathematics Curriculum Standards(2017 Edition,2020 Revision)”clearly stated in“Teaching Suggestions”that“teaching activities based on the core literacy of mathematics should grasp the essence of mathematics,create appropriate teaching situations,put forward appropriate mathematical questions,stimulate students to think and communicate,and form and develop the core literacy of mathematics.”The task-driven teaching model is a new type of teaching method that takes tasks as the main line,teachers as the guide,and students as the main body,which can enable students to engage deeply in classroom discussions and think actively.Based on the characteristics and principles of the task-driven teaching method,this paper designs a high school mathematics classroom teaching based on the task-driven teaching method,hoping to provide a reference for the majority of front-line teachers. 展开更多
关键词 TASK-DRIVEN High school mathematics Instructional design
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Measurement Analysis and Evaluation of Twenty- Five Years of Chinese Mathematics Textbooks: Visual Analysis Based on CiteSpace Knowledge Graph (1999-2024)
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作者 Jie Wang 《Journal of Contemporary Educational Research》 2024年第6期243-254,共12页
At present,textbooks based on core literacy have become the inevitable demands of China’s curriculum reform,and the literacy of textbook goal construction is the key to the implementation of core literacy requirement... At present,textbooks based on core literacy have become the inevitable demands of China’s curriculum reform,and the literacy of textbook goal construction is the key to the implementation of core literacy requirements,which is a huge challenge for textbook compilers.In this paper,we use the visual metrology of the CiteSpace knowledge graph to analyze Chinese mathematics textbooks(1999-2024),hoping to guide the future direction of Chinese mathematics textbook research. 展开更多
关键词 Visualization mathematics textbooks Future outlook Keywords clustering Time zone
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Curriculum Reform of Higher Vocational Mathematics Under the Background of Digital Transformation
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作者 Tianci Zhao 《Journal of Contemporary Educational Research》 2024年第4期215-220,共6页
Under the background of digital transformation,the reform of the higher vocational mathematics curriculum faces urgent challenges and opportunities.This article explores the impact of digital transformation on the ref... Under the background of digital transformation,the reform of the higher vocational mathematics curriculum faces urgent challenges and opportunities.This article explores the impact of digital transformation on the reform of higher vocational mathematics curriculum and emphasizes the importance of improving teaching methods centered on learners.The article proposes specific reform methods and discusses the practical application of digital technology in the reform process.By combining digital technology with specific reform methods,further conducting innovative practice research,and continuously exploring the path of reform,we can effectively improve the quality of higher vocational mathematics classroom teaching and provide strong support for the cultivation of comprehensive qualities and employment abilities. 展开更多
关键词 Digital transformation LEARNER-CENTERED Higher vocational mathematics Personalized development
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