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Research has shown the importance of helping students, especially those with mild-to-moderate learning disabilities, to offload information during problem-solving. When students can get their thoughts onto paper, number line strategies can help them develop a firm foundation in mathematical problem-solving while understanding the relationships between mathematical operations. These strategies are helpful for the development of addition, subtraction, multiplication, division, and later, fractional mathematics. In this article, we describe the progression of number lines as a supportive strategy for elementary students and those with developmental delays in mathematics to improve mathematical understanding. This strategy is based on students being able to show their work and think about what they have written on paper or how they have used manipulatives.

This paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number Δ=Δ(E,χ)$\Delta =\Delta (E,\chi)$, where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of E×$E^\times$, can be appropriately extended to a local ε‐factor εΔ=εΔ(E,ρ)$\varepsilon _\Delta =\varepsilon _\Delta (E,\rho)$ for all virtual representations ρ of the corresponding Weil group WE$W_E$. Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of εΔ$\varepsilon _\Delta$ will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role. [ABSTRACT FROM AUTHOR]

The rapid growth of multimedia pedagogy in the education sector has brought about a game-based technological approach that is shaping the learning of children nowadays. The focus of the approach is to encourage active participation from children; the effect can be seen in their interaction, involvement, and engagement throughout the learning content. Active participation in a learning environment would have a substantial impact on children's self-development and academic achievement, particularly for those with learning disabilities (LD). Therefore, the purpose of this paper is to systematically analyze research conducted on Game-Based Learning (GBL) courseware to support the education of children with LD. A systematic literature review was undertaken, following the PRISMA framework for paper selection. A total of 109 articles were retrieved from the Scopus and Science Direct databases by using a specific keywords search. 14 articles were finalized at the end of the screening based on the inclusion and exclusion criteria. Results reveal the trend of publications, approaches used, and research themes of the selected papers, which include the courseware requirement, student adaptation, and impact of the implementation. The findings demonstrate that GBL is one of the effective methods to be applied in preschool education. It has a significant impact on the development of cognitive skills and assists children who have difficulty in reading, writing, and arithmetic. The findings in this paper can be used as a guide in developing GBL courseware that is developmentally appropriate and effective for children with LD.

Background: The Mental Starters Assessment Project (MSAP) seeks to address poor performance in Grade 3 mathematics. The programme focusses on eliminating inefficient counting methods and promoting strategic mathematical skills, including numerical sense, mental calculation and rapid recall skills. Additionally, MSAP supports teachers' professional growth by providing them with a toolkit of effective calculation strategies to bridge the performance gap and enhance mathematical education. Aim: This paper explores the insight gained from reflections of final year Bachelor of Foundation Phase (FP) initial teacher educators (ITE) students in South Africa. Setting: Grade 3 classrooms. Methods: The ITE students were given training and materials to implement the MSAP, and this occurred over a 4-week teaching practicum, after which they completed a reflective task on the implementation. A total of 20 students were selected from a cohort of 138 based on their academic performance. Results: The analysis of the reflections showed that ITE students benefitted from reflecting on their practice and highlighted important elements of their professional learning. The reflections raised issues related to challenges in their pedagogical content knowledge (PCK) as well as their confidence and competence to teach mathematics and manage the classroom context. Conclusion: With a multi-dimensional model of reflection, ITE students can achieve a deeper understanding of mathematics teaching and learning when building learners' mental strategies, fostering professional growth and elevating the overall quality of mathematics education. Contribution: Overall, the findings provide insight into the benefits of reflective practices for ITE students' professional development and the improvement of mathematics education.

This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7-8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student's identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking.

This paper describes a study for teaching number system in a computer arithmetic course and addresses the existing gap in the current course by focusing on the characteristics of applications. A redundant residue number system (RRNS) is an efficient and innovative number system that inherits the interesting properties of both residue number system (RNS) and redundant number systems. It breaks the operands into residues by considering a moduli-set and then signifies the residues by a redundant representation. So, by limiting the carry propagation chain inside and outside the moduli, RRNS proposes more efficient arithmetic units which could be employed in digital signal processing (DSP) applications. Despite the applicability of RRNS, there is not well-organized teaching on RRNS covering the recent achievements. Besides, there is an important question for students and researchers what is the most appropriate number representation for each application category? In this paper, we present a step-by-step education process for RRNS that includes basic concepts of RNS and the redundant number system. Then we address characteristics of different applications that RRNS is suitable to be employed for them. So, the method gives a well-organized proceeding to computer arithmetic designers and students.

Previous research suggests that the Approximate Number System (ANS) allows people to approximate the cardinality of a set. This ability to discern numerical quantities may explain how meaning becomes associated with number symbols. However, recently it has been argued that ANS representations are not directly numerical, but rather are formed by amalgamating perceptual features confounded with the set's cardinality. In this paper, we approach the question of whether ANS representations are numerical by studying the properties they have, rather than how they are formed. Across two pre-registered within-subjects studies, we measured 189 participants' ability to multiply the numbers between 2 and 8. Participants completed symbolic and nonsymbolic versions of the task. Results showed that participants succeeded at above-chance levels when multiplying nonsymbolic representations within the subitizing range (2-4) but were at chance levels when multiplying numbers within the ANS range (5-8). We conclude that, unlike Object Tracking System (OTS) representations, two ANS representations cannot be multiplied together. We suggest that investigating which numerical properties ANS representations possess may advance the debate over whether the ANS is a genuinely numerical system.

This paper reported the design and development of a conventional and digital mathematical board game for use by students in learning arithmetic. At the time of research, there is no significant indication that a mathematical board game is available in scientific and published patent documentation. The availability of mathematical board games for students' drills and practice in arithmetic, especially in mathematical statement construction, would benefit them, as this competency is an essential life skill. This research was conducted through the design and development research method with the procedure of users' need analysis, researcher as developer capability analysis, product design, product development, field testing in its natural setting environment, and the prototype. The board game prototype was developed in both conventional printed and digital versions. The field testing for the conventional printed version was conducted at secondary school classes with 34 and 36 students, respectively, while for the digital by selected participants. The field testing shows that the developed mathematical board game can work as expected in its natural setting environment.

In this paper we direct attention to 5-6-year-olds' learning of arithmetic skills through a thorough analysis of changes in the children's ways of encountering and experiencing numbers. The foundation for our approach is phenomenographic, in that our object of analysis is differences in children's ways of completing an arithmetic task, which are considered to be expressions of their ways of experiencing numbers and what is possible to do with numbers. A qualitative analysis of 103 children's ways of encountering the task gives an outcome space of varying ways of experiencing numbers. This is further analyzed through the lens of variation theory of learning, explaining why differences occur and how observed changes over a prolonged period of time can shed light on how children learn the meaning of numbers, allowing them to solve arithmetic problems. The results show how observed changes are liberating new and powerful problem-solving strategies. Emanating from empirical research, the results of our study contribute to the theoretical understanding of young children's learning of arithmetic skills, taking the starting point in the child's lived experiences rather than cognitive processes. This approach to interpreting learning, we suggest, has pedagogical implications concerning what is fundamental to teach children for their further development in mathematics.

This study explores student flexibility in mathematics by examining the relationship between accuracy and strategy use for solving arithmetic and algebra problems. Core to procedural flexibility is the ability to select and accurately execute the most appropriate strategy for a given problem. Yet the relationship between strategy selection and accurate execution is nuanced and poorly understood. In this paper, this relationship was examined in the context of an assessment where students were asked to complete the same problem twice using different approaches. In particular, we explored (a) the extent to which students were more accurate when selecting standard or better-than-standard strategies, (b) whether this accuracy-strategy use relationship differed depending on whether the student solved a problem for the first time or the second time, and (c) the extent to which students were more accurate when solving algebraic versus arithmetic problems. Our results indicate significant associations between accuracy and all of these aspects--we found differences in accuracy based on strategy, problem type, and a significant interaction effect between strategy and assessment part. These findings have important implications both for researchers investigating procedural flexibility as well as secondary mathematics educators who seek to promote this capacity among their students.

Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that: (1) analogical reasoning predicts students' abilities only in generalized arithmetic; (2) serial reasoning predicts students' abilities only in generalized arithmetic; (3) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (4) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed.

Background: Previous studies indicated that working memory (WM) updating and WM capacity play essential roles in mathematical ability. However, it is unclear whether WM capacity mediates the effect of WM updating on mathematics, and whether the cascading effects vary with different mathematical domains. Aims: The current study aims to explore the longitudinal mediating role of WM capacity between WM updating and mathematical performance, and how the relations change with the age and domains. Sample: A total of 131 Chinese first-graders participated the study. Methods: Participants were required to complete tasks on WM updating and WM capacity in Grade 1 and Grade 2, as well as paper-and-pencil tests on mathematics achievement in Grade 3. The role of WM updating and capacity in the development of pupil's mathematical achievement was examined. Results: Results revealed that verbal WM updating in Grade 1 predicted basic arithmetic and logical-visuospatial ability in Grade 3 via its cascading effect on verbal WM capacity in Grade 2. Moreover, visuospatial WM updating in Grade 1 predicted visuospatial WM capacity in Grade 2. Visuospatial WM capacity in Grade 1 predicted logical-visuospatial ability in Grade 3 instead of basic arithmetic ability in Grade 3. Conclusions: The findings suggested that WM updating exerts effect on pupil's mathematical performance via WM capacity, meanwhile, this effect depends on children's mathematics domain.

Networked computers can potentially support classrooms to be more interactive. It can help students share representations amongst themselves and work together on a shared virtual activity space. In research on the role of shared screens or shared virtual workspace in learning settings, there has been less attention paid to contexts where learners are co-located. This paper looks at the impact of the shared screen in a computational game environment on mathematics learning and practices and the construction of learners' emotions and social status in classroom interactions. We designed two versions of a simple arithmetic game: a solo version in which the student played the game alone and a multi-player version in which the screen was shared, and the players could see the arithmetic moves of the other players. We implemented these two versions of the game in a 4th-grade classroom in a suburban school in a large metropolis in India. Classroom sessions were video recorded, computer logs were collected, and field notes were taken. Focus group sessions were held with the students. We coded a portion of the data to get at patterns of classroom interactions. Then we drew on qualitative video analysis tools to analyze specific episodes to understand the fine timescale dynamics of dominant interaction patterns in each setting. Our analysis shows that the shared screen served as a shared memory device, keeping a record of all the students' posts, and was entangled in the moment-to-moment dynamics of self- and peer- assessments of arithmetic. These findings suggest that thoughtful integration of networked digital tools in computer-supported learning environments can increase student-student interactions and support disciplinary learning.

This paper focuses on determining the opinions of primary school third-grade pupils about the conceptual meaning and use of the numbers "0" and "1". The current study employed the case study design, which is one of the qualitative research methods. In the selection of the sample, the criterion sampling method, one of the purposive sampling methods, was used. The study was conducted with the participation of a total of 114 third-grade pupils (58 girls and 56 boys) attending two state primary schools located in the city of Ankara in the fall term of the 2019-2020 school year. As the data collection tool, a semi-structured interview form developed by the researchers in relation to the concepts of "0" and "1" was used. In the analysis of the data obtained in relation to the concepts of "0" and "1", content analysis was used. The primary school third-grade pupils made sense of the number "0" in association with the categories of four operations, ineffective element, absorbing element, number, natural number, meaningless, valueless, absence, even number, beginning, and letter. The categories in which most thoughts about the meaning of the number "1" are gathered are number, ineffective element, and uniqueness. It was determined that the pupils are more successful in the use of the numbers "0" and "1" in the addition and subtraction operations than in the multiplication and division operations.

Grounded in problem-based learning and with respect to four mathematics domains (arithmetic, random events and counting, number theory, and geometry), we designed a series of programming-based learning tasks for middle school students to co-develop computational thinking (CT) and corresponding mathematical thinking. Various CT concepts and practices articulating the designated mathematical problems were involved in the tasks. In addition to delineating the design of these learning tasks, this paper presents a qualitative study in which we examined 74 students' learning outcomes and characterized their CT and mathematical thinking co-development as they accomplished the tasks. The research results demonstrate the codevelopment of both mathematics- and CT-related concepts and practices in the four mathematics domains. Two types of interactions are identified: (i) applying mathematical knowledge to construct CT artifacts and (ii) generating new mathematical knowledge with CT practice. The new insights provided by the present work are threefold. First, from a mathematical learning perspective, the nature of the solution processes of the designed problems should not be immediately obvious. Second, from a technology-enhanced learning perspective, the dynamic representations and immediate visual feedback afforded by the programming tool are beneficial to student learning. Third, from a pedagogical perspective, the room for customization offered by both the designed problems and programming tools can provide affordances for learning.

Mental brackets constitute an idiosyncratic use of brackets sometimes used to evaluate arithmetic expressions and are closely connected with students' structure sense. The relevant literature describes the use of mental brackets focusing on primary school students and in the context of arithmetic. Using 181 high school students' solutions to seven tasks, this paper attempts to broaden the scope of mental brackets. It investigates high school students' mental brackets use and their presence in different mathematical domains. The findings show that high school students use mental brackets in the context of arithmetic while evaluating arithmetic expressions, and in the context of algebra, mainly when substituting numerical values in the variables. Moreover, the analysis provides further insights into the use of mental brackets. Sometimes they are applied to the whole expression (global mental brackets) and other times just to parts of it (local mental brackets).

Children's multiplicative thinking as the recognition of equal group structures and the enumeration of the composite units was the subject of this research. In this paper, we provide an overview of the Multiplication and Division Investigations project. The results were obtained from a small sample of Australian children (n = 21) in their first year of school (mean age 5 years 6 months) who participated in a teaching experiment of five lessons taught by their classroom teacher. The tasks introduced children to the "equal groups" aspect of multiplication. A theoretical framework of constructivist learning, together with research literature underpinning early multiplicative thinking, tasks, and children's thinking, was used to design the research. Our findings indicate that young children could imagine equal group structures and, in doing so, recognise and enumerate composite units. As the children came to these tasks without any prior formal instruction, it seemed that they had intuitive understandings of equal group structures based on their life experiences. We argue that the implications for teaching include creating learning provocations that elicit children's early ideas of multiplication, visualisation, and abstraction. The research has also shown the importance of observing children, listening to their explanations of their thinking, and using insights provided by their drawings.

In current teaching materials, when using Dedekind cuts to construct real numbers, the definition of a Dedekind cut is always involved in defining addition and multiplication. In this paper, as it is done in many current textbooks, Dedekind cuts are used to construct the set of real numbers. Then the order in it is defined, and the least-upper-bound property, as well as the density properties of rational cuts in the set of real numbers, is established. Later, when defining addition and multiplication and establishing relative properties, the definition of cut will not be used again. This makes the process simpler and easier to understand.

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic.

Programming is an interdisciplinary practice with applications in both mathematics and computer science. Mathematics concerns rigor, abstraction, and generalization. Computer science predominantly concerns efficiency, concreteness, and physicality. This makes programming a medium for problem solving that mediates between mathematics and computer science in intriguing ways. Behind programming practices is computational thinking (CT), a mode of thinking involved in formulating and solving problems so that the solutions could be represented and carried out by computing means. In this paper, CT is seen as a boundary object connecting mathematics and computer science in a school problem-solving context. In particular, we examine and analyse middle school students' work upon engaging in mathematical problem solving-in a programming environment, taking CT as a boundary object embedded in the block-based programming environment, Scratch. The analysis is guided by observing boundary crossing features of CT in the students' artefacts produced in Scratch while solving mathematical problems related to symmetry and arithmetic sequence. The findings of this study open up new dimensions to explore CT as a boundary object in integrated STEM pedagogy.

In this paper, we describe an experiment in using history to work on problem-solving and the relationship between arithmetic and algebra. The students involved attended the first year of the Italian upper secondary school (grade 9). The original sources we used are problems from Italian treatises on arithmetic and algebra that appeared in the Middle Ages and the Renaissance. At the beginning, we present these treatises, which belong to a widespread mathematical tradition. We then describe the classroom experiment: we administered 14 problems from ancient treatises accompanied by the request to solve them and write impressions and comments. Data were collected through written protocols and interviews. The analysis of the results focused on the strategies implemented in solving the problems, with particular reference to the use of arithmetic and algebra, and on the perception of the cultural aspect introduced by these problems in the students' approach to mathematics. The findings show the difficulties of some students on topics, such as fractions and direct proportionality, which should have been acquired in earlier school years. Combining the use of history with aspects of research in mathematics education allowed us to outline some teaching implications linked to the use of history.

Constantly on the lookout for Canadian mathematics education matters, because if Canadian mathematics education matters then Canadian mathematics education matters, three young university bookstore employees, university students, unable to make proper change when I handed them a five dollar bill for a sticker I was purchasing for my laptop, absolutely crushed my spirit. As they say though, it is not what happens to you, rather it is how you react to what happened to you. As such, rising from what I am now calling "the sticker incident," this article is a many part investigation into the oft-heard phrase "Damn kids these days cannot make change without a calculator." Under examination: the subtraction skills of a particular subset of adult employees not often asked to prove their arithmetic resolve; detailing a personal mistake I made while making change in a big spot; recounting a retelling of a mistake that has haunted someone for many years; detailing which particular customers cashiers need to be worried about when making change (old men with coin purses); wonderings as to why stories about making change do not reflect our now penniless country; and, a brief look at the future responsibilities of cashless immigrants. With this particular matter now in the rearview mirror, I am back in the wild looking for other Canadian mathematics education matters. Stay tuned.

"They are the same" is a phrase that teachers often hear from their students in arithmetic and algebra. But what do students mean when they say this? The present paper researches the notion of sameness within algebraic thinking in the context of generating equivalent numerical equalities. A group of Grade 6 Mexican students (10- to 12-year-olds) was presented with tasks that required transforming the given numerical equalities in such a way as to show their truth-value. The students initially indicated this by calculating and demonstrating that the total was the same on both sides. When asked not to calculate, their approaches evolved into more structural transformations involving decomposition so as to arrive at an equality with the same expression on each side. Students used the language of sameness--both visible and invisible--to describe the truth-value of their transformed expressions and equalities. The visible sameness referred to the resulting identical form of each side of the equality and the invisible sameness to the top-down equivalences that they had generated by their decomposing transformations--both types of sameness being characterized by their hidden numerical values. These findings suggest implications for transitioning to the algebraic domain of equations with their similarly hidden values.

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved.

Mathematical giftedness (MG) is an intriguing phenomenon, the nature of which has yet to be sufficiently explored. This study goes a step further in understanding how MG is related to expertise in mathematics (EM) and general giftedness (G). Cognitive testing was conducted among 197 high school students with different levels of G and of EM. Based on our previous studies, we perceive MG as a combination of G and EM. Exploratory factor analysis of test results revealed five main cognitive factors: visual-serial processing (VSP); arithmetic abilities (AA); pattern recognition (PR); auditory working memory (AWM); visual-spatial working memory (VSWM); and Structural equation modeling (SEM) based on the factor analysis revealed clear differences in the role of cognitive abilities as predictors of EM, G, and MG. The study demonstrates that visual components are especially important for the development of EM and that G students are less dependent on their visual cognitive processing. Based on the study results, we argue that EM, G, and MG, which are often considered equivalent characteristics, are interrelated but different in nature. The paper presents a research-based justification that not any gifted is an expert in mathematics and not any expert in mathematics is gifted.

Introducing the distributivity property of multiplication over addition is a well-known challenge in mathematics education, especially in primary school. As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation and evaluation of a modelling activity in which 2nd-grade students are introduced to the key concept of distributivity of multiplication over addition. The results show that modelling activities designed through the heuristics of didactical phenomenology, guided reinvention and emergent modelling may support the accessibility of distributivity, as these students were able to reinvent this concept solving an experientially significant problem. This result can be attributed to a combination of several factors: the choice of a realistic and rich problem, that stimulated students to elaborate formal mathematical concepts mathematizing their informal solving strategies, rooting new understandings in experientially real phenomena; the use of a suitable artifact, that presented mathematics as a means of interpreting and understanding reality and increasing the opportunities for observing mathematics outside of the school context; the role of the teacher, who guided students in reinventing mathematics in an active way.

The paper is the result of extensive research carried out among Czech students and focuses on a conception of infinity. The questionnaire survey was taken by 861 students ranging from grades 7 to 13. The aim of the research was to describe the development of students' conceptions of infinity. These conceptions are built on the intuitive phenomenon of the horizon. We monitor the proportional representation of these conceptions in four combinations of views (into the distance and in depth) and contexts (arithmetical and geometrical). It can safely be maintained that the development of the proportional representation of the earliest conception comprehension of the concept of infinity, the so-called natural infinity, is not concurrent with the students' age. The development of the proportional representation of the conception of actual infinity is non-decreasing, at least in the view into the distance, in both contexts with respect to age. In general, it can be stated that the proportional representation of individual conceptions of infinity is strongly dependent on both context and view.

This paper explores how student performance across the full set of multiple modular assessments of individual standards, which we refer to as mini-assessments, from a large scale, operational program of interim assessment can be summarized using Bayesian networks. We follow a completely data-driven approach in which no constraints are imposed to best reflect the empirical relationships between these assessments, and a learning trajectory approach in which constraints are imposed to mirror the stages of a mathematic learning trajectory to provide insight into student learning. Under both approaches, we aim to draw a holistic picture of performance across all of the mini-assessments that provides additional information for students, educators, and administrators. In particular, the graphical structure of the network and the conditional probabilities of mastery provide information above and beyond an overall score on a single mini-assessment. Uses and implications of our work are discussed.

We investigated understanding of the linear independence concept based on the type and nature of connections displayed in seven non-mathematics majors' interview responses to a set of open-ended questions. Through a qualitative analysis, we identified six categories of frequently displayed connections. There were also recognizable differences in the way the connections were applied by the participants. Overall, our findings pointed to an understanding in the form of two main clusters of connections. The two clusters were connected only by linear combination ideas. Each cluster, furthermore, was distinguishable via representation types. The first cluster contained arithmetic/algebraic modes and the second cluster included, mostly, geometric ideas. This paper discusses similarities and differences within and between clusters supported by participant responses. In light of the findings, we provide suggestions for the improvement of linear algebra education of non-mathematics student population.

Open math problem solving is critical to help students deepen the understanding and promote transfer of mathematics knowledge. However, the cognitive mechanism for open math problem solving, particularly the role of spatial abilities, has not been paid enough attention. This study recruited 192 junior middle school students (14.30 ± 0.48 years old). Results showed that both spatial visualisation (measured by paper folding) and spatial working memory (measured by spatial 2-back) significantly associated with open math problem solving with controlled variables including age, sex, nonverbal matrix reasoning, and arithmetic principles. Moreover, spatial working memory was more associated with easy open math problem solving, while spatial visualisation more associated with difficult open math problem solving. These findings suggested that as the difficulty of open math problems increased, the construction of problem space mattered more than retaining and updating problem space in open math problem solving.

Rational numbers can be represented in multiple formats (e.g., fractions, decimals, and percentages), and a rational number notation can be used to express different concepts in different contexts. The present study investigated the distribution of the multiple concepts expressed by these different rational number notations in real-world contexts as well as the semantic alignment between entity type (discrete vs. continuous) and rational number format (decimal vs. fraction) in the Chinese context. Textbook analysis and two paper-and-pencil experiments yielded the following four major findings: (1) Decimals were more likely used to represent numerical magnitudes, while fractions were more likely used to represent relations between two numerical magnitudes. (2) Decimals were more often used to represent continuous entities while fractions were preferred to represent discrete entities. (3) The strength of the association between different formats of rational numbers and their preferred conceptual meanings seemed more pronounced than the semantic alignment between number type and entity type. (4) Percentages were used in a way more similar to fractions than decimals in terms of the concepts they express. These findings indicate that different formats of rational numbers differ dramatically in their use in real-world contexts both in terms of the conceptual meanings they express and the entities they model. Educational implications of this study are discussed.

The author's last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers. [ABSTRACT FROM AUTHOR]

Although the role played by finger use in children's numerical development has been widely investigated, their benefit in arithmetical contexts is still debated today. This scoping review aimed to systematically identify and summarize all studies that have investigated the relation between fingers and arithmetic skills in children. An extensive search on Ovid PsycINFO and Ovid Eric was performed. The reference lists of included articles were also searched for relevant articles. Two reviewers engaged in study selection and data extraction independently, based on the eligibility criteria. Discrepancies were resolved through discussion. Of the 4707 identified studies, 68 met the inclusion criteria and 7 additional papers were added from the reference lists of included studies. A total of 75 studies were included in this review. They came from two main research areas and were conducted with different aims and methods. Studies published in the mathematical education field (n = 29) aimed to determine what finger strategies are used during development and how they support computation skills. Studies published in cognitive psychology and neuroscience (n = 45) specified the cognitive processes and neurobiological mechanisms underlying the fingers/arithmetic relation. Only one study combined issues raised in both research areas. More studies are needed to determine which finger strategy is the most effective, how finger sensorimotor skills mediate the finger strategies/arithmetic relation, and how they should be integrated into educational practice.

Solving arithmetic word problems that entail deep implicit relations is still a challenging problem. However, significant progress has been made in solving Arithmetic Word Problems (AWP) over the past six decades. This paper proposes to discover deep implicit relations by qualia inference to solve Arithmetic Word Problems entailing Deep Implicit Relations (DIR-AWP), such as entailing commonsense or subject-domain knowledge involved in the problem-solving process. This paper proposes to take three steps to solve DIR-AWPs, in which the first three steps are used to conduct the qualia inference process. The first step uses the prepared set of qualia-quantity models to identify qualia scenes from the explicit relations extracted by the Syntax-Semantic (S²) method from the given problem. The second step adds missing entities and deep implicit relations in order using the identified qualia scenes and the qualia-quantity models, respectively. The third step distills the relations for solving the given problem by pruning the spare branches of the qualia dependency graph of all the acquired relations. The research contributes to the field by presenting a comprehensive approach combining explicit and implicit knowledge to enhance reasoning abilities. The experimental results on Math23K demonstrate hat the proposed algorithm is superior to the baseline algorithms in solving AWPs requiring deep implicit relations. [ABSTRACT FROM AUTHOR]

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

Assume that we are given a sequence F of k homogeneous polynomials in n variables of degree at most d and the ideal ℐ generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity d O (n) , whether F is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if F is regular then we can transform ℐ into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity d O (n 2) . Finally, under the same assumption, we establish the new upper bound 2 (d k / 2) 2 n − k − 1 for the maximum degree of the elements of any reduced Gröbner basis of ℐ in the case that n > k. [ABSTRACT FROM AUTHOR]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

In this paper, a stability analysis and the controller improvement of T-S fuzzy Descriptor system are studied. Firstly, by making full use of the related theory of fuzzy affiliation function and combining the design method of fuzzy Lyapunov function with the method of inequality deflation, a stability condition with wider admissibility and less system conservatism is proposed. The advantage of this method is that it is not necessary to ensure that each fuzzy subsystem is progressively stable. We also maximise the boundary of the derivatives of the affiliation function mined. Secondly, a PDC controller and a Non-PDC controller are designed, and the deflation conditions for the linear matrix inequalities of the two controllers are constructed. Finally, some arithmetic simulations and practical examples are given to demonstrate the effectiveness of the method studied in this paper, and the results obtained are less conservative and have larger feasible domains than previous methods. [ABSTRACT FROM AUTHOR]

A binary operation # on Z + is said to be an associative arithmetic if both # and its iteration — the binary operation ∗ defined recursively by: x∗1 = x and x∗y = [x ∗ (y −1)]#x — are associative. E. Rosinger [6] showed that under reasonable conditions an associative arithmetic must be ordinary addition. However, in the general case, there are associative arithmetics that are not ordinary addition. This paper gives examples of these as well as results towards a structure theorem for associative arithmetics. The paper also describes the role that this particular math problem has played in my mathematical life. [ABSTRACT FROM AUTHOR]

In this paper we analyze the stability of the problem of performing a rational QZ step with a shift that is an eigenvalue of a given regular pencil H - λ K in unreduced Hessenberg–Hessenberg form. In exact arithmetic, the backward rational QZ step moves the eigenvalue to the top of the pencil, while the rest of the pencil is maintained in Hessenberg–Hessenberg form, which then yields a deflation of the given shift. But in finite-precision the rational QZ step gets "blurred" and precludes the deflation of the given shift at the top of the pencil. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the rational QZ step can be constructed using this eigenvector, so that the exact deflation is also obtained in finite-precision. [ABSTRACT FROM AUTHOR]