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Tablet‐based technology has become a conduit for the administration of curriculum‐based measurement (CBM) tools. Using tablets to administer CBM probes has many advantages; however, little is known about how students perform on CBMs when administered via a tablet. The current investigation compared digits correct per minute obtained from 44 third‐grade students on CBM addition fluency probes administered via three different modalities. Students completed the probes using traditional paper and pencil, a tablet using their fingers to write the answer, or a tablet application using a keyboard to type the answer. A within‐subjects group design showed students performed significantly better on the tablet when using their finger to write the answer, even though most students (58%) indicated this was their least preferred modality. The discussion focuses on the implications for educators and the use of different CBM modalities for interindividual and intraindividual comparisons. Practitioner points: Third‐grade students responded to math facts faster when writing with their fingers on an iPad, even compared to paper and pencil.The results are inconsistent with other studies that found students were faster when using paper and pencil.Finally, most third‐grade students preferred answering addition facts with a pencil on paper. [ABSTRACT FROM AUTHOR]

Convex in x domain, eigenfunction, level sets, local rearrangement Erratum to the paper "Convexity in x of the level sets of the first Dirichlet eigenfunction": (Published in Math. Nachr. Keywords: convex in x domain; eigenfunction; level sets; local rearrangement EN convex in x domain eigenfunction level sets local rearrangement 1025 1025 1 03/21/23 20230301 NES 230301 Lemma 3.2 is incorrect. [Extracted from the article]

In this paper, I shall investigate whether Hegel can be considered as a sort of ancestor of McDowell's disjunctivism. If this hypothesis turns out to be plausible, then the paper offers two gains. On the one hand, it offers an innovative interpretation of the way in which Hegel conceives of our sensible epistemic access to the world. On the other hand, McDowell's own claim that his own theoretical proposal has a Hegelian sound is supported by a previously unexplored argument. I organize my analysis into three parts: I sketch McDowell's version of disjunctivism (Section 2); I analyze some passages from Hegel that I believe are important for showing some similarities between his and McDowell's argumentative strategy (Section 3); in the conclusion (Section 4), I highlight a number of core features that Hegel seems to share with McDowell's disjunctivism and I submit that they are sufficient to label Hegel the "grandfather" of McDowell's disjunctivism. [ABSTRACT FROM AUTHOR]

The surface properties of newsprint and other paper qualities are to a great extent determined by the properties of the cellulose fibres. An appropriate description of these fibres as they appear in the paper is therefore important and can be used for quality classification and process monitoring. We suggest a model that considers the fibre geometry and appearance. It is based on a two-dimensional shot-noise process. The model is fit by minimizing a weighted least squares distance between the model-based and estimated covariance functions and this provides estimates of the fibre size, intensity and the non-uniform distribution of the fibre orientation. The model is applied to simulated and real data. [ABSTRACT FROM AUTHOR]

We derive a new representation for the exact convergence factor of classical two-level and two-grid preconditioners. Based on this result, we establish necessary and sufficient conditions for constructing the components of efficient algebraic multigrid (AMG) methods. The relation of the sharp estimate to the classical two-level hierarchical basis methods is discussed as well. Lastly, as an application, we give an optimal two-grid convergence proof of a purely algebraic ‘window’-AMG method. Published in 2005 by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper offers an analytical review of Miller's classic paper of 1956 on 'the magical number 7'. It takes account of that paper's contradictions as well as of its implicit conclusions. Miller's text is by no means clear in its logic or in its terms, and a review might be deemed helpful to those who are trying to understand it. The discussion begins with a brief historical overview. Miller's paper is then reviewed according to the order of the points made by Miller, thus facilitating a lucid comparative reading with that paper. Guidance on interpreting Miller's text is given throughout, leading to a number of technical conclusions. Finally, two general conclusions are offered, one which refers to the decision-making literature, and one which refers more synoptically to science itself. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Background: The sudden growth in online instruction due to COVID‐19 restrictions has given renewed urgency to questions about remote learning that have remained unresolved. Web‐based assessment software provides instructors an array of options for varying testing parameters, but the pedagogical impacts of some of these variations has yet to be examined. Objectives: The current study examines whether question order affects student performance on online college math assessments. Drawing on the literature of testing and math anxiety, we hypothesized that difficult questions near the beginning of an assessment would have a destructive effect on student confidence, which would in turn have a deleterious effect on their performance. Methods: We employed an observational 'big data' methodology, analysing 23,468 final exams completed by students in 10 different courses over eight semesters at a Math Emporium in a large technical university in the eastern United States. Students were freshmen and sophomores enrolled in non‐engineering math courses. We regressed the final score on the difficulty level of the first and second questions, controlling for several other factors. Results and Conclusions: We found that several factors—day of the week, amount of time before the deadline, number of minutes spent on the exam—have more of an impact on score than question order. This pattern was consistent across sexes. Takeaways: Our findings contradict some previous studies, which have found that difficult early questions degrade student performance, and that this affect is more pronounced in females. This work enriches our understanding of how students respond to online assessment. Lay Description: What is already known about this topic: Previous studies have had mixed results about the effect of question difficulty ordering on student performance.Furthermore, few studies have examined this issue for computer assessments. What this paper adds: We employ a novel methodology with a uniquely large data set.Our findings indicate that difficulty ordering has no effect on student performance. Implications for practice: Math instructors can use the random ordering feature of computer assessment generating tools because question order does not affect performance. [ABSTRACT FROM AUTHOR]

This is an example of an original article in the layout expected by The Photogrammetric Record. The abstract is a concise summary of the essential elements of the paper and should not contain detailed results. It should be no longer than 150 words. This template indicates how a paper submitted to the journal should be structured and formatted, including figures, tables, mathematics and references. [ABSTRACT FROM AUTHOR]

Explicit definition of the limits of citation analysis demands additional tests for the validity of citation analysis. The stability of citation rankings over time can be regarded as confirming the validity of evaluative citation analysis. This stability over time was investigated for two sets of citation records from the Web of Science ( Thomson Reuters, Philadelphia, PA) for articles published in journals classified in Journal Citation Reports as Mathematics. These sets are of all such articles for the 1960s and for the 1970s. This study employs only descriptive statistics and draws no inferences to any larger population. The study found a high correlation from one decade to the next of rankings among sets of most highly cited articles. However, the study found a low correlation for rankings among articles whose ranks were the 500 directly below those of the 500 most cited. This perhaps expected result is discussed in terms of the Glänzel- Schubert- Schoepflin stochastic model for citation processes and also in connection with an account of the purposes of evaluative citation analysis. This interpretative context suggests why the limitations of citation analysis may be inherent to citation analysis even when it is done well. [ABSTRACT FROM AUTHOR]

In this paper, we answer the question why statistics journals get lower impact factors than journals in other disciplines. We analyze diffusion patterns of papers in several journals in various academic fields. To obtain insights into the diffusion of the citation counts of the papers, the data are analysed with the Bass model, leading to values for the time-to-peak that can be used to compare the speeds of diffusion paper citations of the different disciplines. Estimation results show that for statistics journals, it takes significantly more years to reach their peak. To further investigate diffusion, we also compute the percentages of the total number of citations a paper has after 2 or 3 years. Again, it appears that statistics journals have slower citation diffusion than journals in other disciplines. We conclude with some suggestions to reduce this disparity. [ABSTRACT FROM AUTHOR]

This paper aims to trace the origins of dual entry beyond a Eurocentric boundary. The aim can be refined further by asking: did the Chinese use dual entry and how did dual entry appear in Europe. Double entry is more advanced as it includes the location of the account in which the contra entry is made. Dual entry approaches accounting, like mathematics, as a tautology, whereby an equation is constructed from a simple axiom. Defined as such, this paper argues that the concept of dual entry had its earliest origins in China. [ABSTRACT FROM AUTHOR]

This is an example of an original article in the layout expected by The Photogrammetric Record. The abstract is a concise summary of the essential elements of the paper and should not contain detailed results. It should be no longer than 150 words. This template indicates how a paper submitted to the journal should be structured and formatted, including figures, tables, mathematics and references. [ABSTRACT FROM AUTHOR]

Benford's law is an empirical observation, first reported by Simon Newcomb in 1881 and then independently by Frank Benford in 1938: the first significant digits of numbers in large data are often distributed according to a logarithmically decreasing function. Being contrary to intuition, the law was forgotten as a mere curious observation. However, in the last two decades relevant literature has grown exponentially—an evolution typical of “Sleeping Beauties” (SBs) publications that go unnoticed (sleep) for a long time and then suddenly become the center of attention (are awakened). Thus, in the present study, we show that the two papers, Newcomb ( ) and Benford ( ), Newcomb ( , American Journal of Mathematics, 4, 39–40) and Benford (1938, Proc. Am. Phil. Soc., 78, 551–572) papers are clearly SBs. The former was in a deep sleep for 110 years, whereas the latter was in a deep sleep for a comparatively lesser period of 31 years up to 1968, and in a state of less deep sleep for another 27 years, up to 1995. Both SBs were awakened in the year 1995 by Hill ( , Statistical Science, 10, 354–363). In so doing, we show that the waking prince (Hill, ) is more often quoted than the SB whom he kissed—in this Benford's law case, wondering whether this is a general effect—to be usefully studied. [ABSTRACT FROM AUTHOR]

This paper explores how the twin processes of neoliberalism and neoconservatism work together on, and through, curricula and their associated pedagogies. It bridges the gap between policy and classroom practice, focusing on the particular example of the school subject of mathematics and the notion of mastery, operationalised in the English education system as Teaching for Mastery (TfM). From this context, it develops a theoretical argument using Dean's analytics of government as part of a broader Foucauldian frame, to analyse how TfM is constructed as a particular policy truth. It then shifts the analysis from a wide, social one to the individual classroom level using a psychological argument to critique TfM in its own terms, examining the onto‐epistemological nature of mathematics as a subject. In doing so, it explores ways in which mastery might be problematic in classrooms, even whilst appearing to offer a solution at policy level to long‐standing problems in English schooling. The aim is not to suggest that TfM has nothing to offer, but to point to ways in which it draws on the psychology of teaching and learning in a very particular manner, inscribing pupils with very specific mathematical subjectivities. By providing this insight into how neoliberal policy positions play out at practitioner level via curricula and pedagogies, the paper raises questions which are philosophical, political, and ethical, regarding the potential effect of TfM on teachers' and pupils' experiences of mathematics in schools, including implications for equity of this experience amongst the latter. [ABSTRACT FROM AUTHOR]

Both second-order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second-order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first-order set theory as a very high-order logic. [ABSTRACT FROM AUTHOR]

In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's 'Why a diagram is (sometimes) worth 10,000 words', between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's 'characteristic visual shape' in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry. [ABSTRACT FROM AUTHOR]

This paper examines the relationship between higher education structure and firm innovation. By drawing on and expanding Schumpeterian growth theory, we analyze this underlying mechanism and conduct empirical tests using a cross‐national survey database of 9,230 firms from 30 countries. The empirical results show that a higher education structure, exhibiting a heightened inclination towards fundamental disciplines such as mathematics and natural sciences (the percentage of graduates in these disciplines), is more conducive to fostering firm innovation. This innovation‐promoting effect is more noticeable in domestic and established enterprises and varies in accordance with a country's institutional conditions. This paper provides new insights into the study of the impact of higher education structure on firm innovation. [ABSTRACT FROM AUTHOR]

This paper concerns the bubbling phenomena for the L2$L^2$‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non‐linear Schrödinger equations (NLS). However, unlike the single‐bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders |t|2+$|t|^{2+}$ and |t|3−$|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure. [ABSTRACT FROM AUTHOR]

Considering the fact that authorship order plays such a significant role as a basis for scientific merit, this paper looks into the practices of authorship order from a research ethical perspective. We conclude that there is a wide variety of practices and no common understanding of what the different authorship positions signify. Authorship guidelines do not provide much help. We recognize that, regardless of what system for valuing authorship positions is used, it will be misleading and unfair in most applications because relative contributions vary in ways that are not captured by fixed value assignments to authorship positions. In theory, assigning percentage figures reflecting the relative contributions of the authors would solve that problem, but we argue that such a scheme is not likely to work in practice. It can also be questioned whether relative, rather than absolute, contributions should be the basis for scientific merit. Contributorship is discussed as an alternative, but is recognized to be insufficient both in communicating absolute and relative contributions, as standardly used. However, there may be a way forward with contributorship, but then, the level of detail needs to increase considerably and its application be standardized. [ABSTRACT FROM AUTHOR]

The article provides information on a planned issue focusing on the development of mathematical cognition and provides information on how to submit papers on the topic for publication.

Motivation: This research investigates the transformative impact of integrating AI into mathematics education, aiming to enhance students' conceptual understanding and self‐efficacy. It addresses the crucial need for innovative teaching methods in response to contemporary challenges in education and aims to fill gaps in understanding the potential drawbacks and benefits of AI implementation. Objectives: The primary goal of this study is to investigate the effects of AI tools, such as GeoGebra and ChatGPT, on students' conceptual understanding and self‐efficacy in a mathematics classroom setting, with a focus on collaborative learning, teacher‐led discussions, and problem‐solving. Methods: Employing a mixed‐methods approach, the research involves pre‐ and post‐implementation surveys, supplemented by qualitative data from interviews, focus groups, and classroom observations. The study emphasises the utilisation of AI in collaborative learning environments and explores the challenges and opportunities associated with its integration. Results and Conclusions: The study reveals a noteworthy enhancement in students' conceptual understanding and self‐efficacy belief when AI tools are incorporated into mathematics education. However, it highlights concerns such as technological challenges, potential teacher replacement fears, biases in AI systems, and difficulties related to pacing and social interaction. To optimise the use of AI, the paper suggests strategies like adaptive learning platforms, real‐world simulations, intelligent tutoring systems, and data‐driven instruction, positioning AI as a complement rather than a substitute for traditional teaching methods. The study recommends addressing challenges through additional support resources and transparent communication, emphasising the need for a thoughtful integration of AI in education. Lay Description: What is currently known about the subject matter?: AI is increasingly being integrated into educational settings.AI offers potential for personalised learning experiences.Concerns exist regarding AI replacing human educators and ethical issues such as data privacy and bias.Effectiveness of AI in improving student performance varies depending on the implementation and context. What this paper adds?: Provides empirical evidence on how GeoGebra and ChatGPT enhance students' conceptual understanding and confidence in mathematics.Identifies specific challenges and concerns from students and teachers regarding AI use.Demonstrates how integrating GeoGebra and ChatGPT boosts student engagement and motivation, leading to improved understanding of mathematical concepts.Offers insights into student engagement levels and interaction patterns with AI tools. Implications of the findings for practitioners: Emphasise the need for careful selection and validation of AI to ensure they meet educational objectives.Address ethical concerns and ensure data privacy when implementing AI in education.Encourage ongoing evaluation and refinement of practices related to the use of AI, ensuring that they meet ethical standards and address data privacy concerns.Advocate for the establishment of acceptable criteria and conventions for AI use in education, improving their effectiveness and usability based on student and educator feedback. [ABSTRACT FROM AUTHOR]

The attainment of unit status or I AM is a crucial stage in Winnicott's theory of development and arguably ranks amongst Winnicott's more well known ideas, such as transitional object, true and false self, and primary maternal preoccupation. Using Winnicott's 1968 paper 'Sum, I am', the author traces the dangers inherent in attaining I AM and what might hamper this achievement, the personal significance of numbers and divisibility, and how this links with his subsequent ideas on monotheism and a radical revision of Freud's dual drive theory. Returning to the 'Sum, I am' paper, the author briefly explores the importance Winnicott places on play in pedagogy and the significance of an intermediate space followed by Winnicott's thoughts about death, dissociation, and the importance of a mouse! Four clinical examples are used to illustrate difficulties in achieving I AM. Finally, the author suggests the spatula game in 'Observation of infants in a set situation' published over a quarter of a century earlier is a prime example of an I AM experience, with a beginning, middle and end. [ABSTRACT FROM AUTHOR]

Previous studies have often demonstrated that educational television can have a positive effect on learning outcomes in low‐income countries when delivered in controlled settings. However, existing research in low‐resource contexts has scarcely considered the association between child outcomes and viewing in usual environments (ie, at their home, a friend's home or a relative's home). This lack of research is striking, as evidence from controlled settings might provide limited information on the effects of normal television exposure. This paper, therefore, investigates the relationship between normal exposure to a popular Tanzania‐produced cartoon, Ubongo Kids and mathematics capability, as represented by plausible values derived from an item response theory model applied to children's test responses. Cross‐sectional investigation of a sample of 38 682 Tanzanian children suggested normal educational television exposure to be significantly associated with mathematics capability, when controlling for age, sex, school enrolment, Kiswahili attainment and household fixed effects. While cross‐sectional results are not necessarily causal, the findings in this paper broadly correspond with those from previous designs using repeated observations. What is more, considering association results alongside cost and viewership estimates suggests television‐based interventions to be highly cost effective. Practitioner NotesWhat is already known about this topic National data for Tanzania suggests the country to have both low levels of mathematics proficiency and considerable access to television technology.Educational television‐based interventions might, therefore, be worthy of consideration by educational policymakers, especially given that studies conducted in controlled settings indicate that television can deliver learning benefits.There is, however, little evidence concerning educational television exposure outside of controlled settings. Only one such study in a low‐income context that concerned the association between learning outcomes and viewing among primary‐age children has been identified.What this paper adds This paper addresses this dearth of research by investigating the association between normal exposure to a Tanzanian cartoon, Ubongo Kids and mathematics capability, as derived from an item response theory model applied to the test responses of 38 682 children.Cross‐sectional findings suggest the association between normal television exposure and mathematics capability to be significant.Further, a cost‐effectiveness comparison with alternate interventions in comparable contexts indicates that educational television is highly cost‐effective.Implications for practice and/or policy The findings presented in this paper concerning educational television viewership in usual environments act to triangulate those from prior research conducted in controlled settings. As such, policymakers in low‐income contexts now possess more convincing evidence on the potential influence of educational television interventions.Additionally, the cost‐effectiveness comparison made suggests that educational television should be considered a viable option by policymakers seeking to address learning outcomes with limited resources. [ABSTRACT FROM AUTHOR]

Research on fostering learning about percentages within intelligent tutoring systems (ITSs) is limited. Additionally, there is a lack of data‐driven approaches for improving the design of ITS to facilitate learning about percentages. To address these gaps, we first investigated whether students' understanding of basic mathematical skills (eg, arithmetic, measurement units and geometry) and fractions within an ITS predicts their understanding of percentages. We then applied a psychological network analysis to evaluate interdependencies within the data on 44 subtopics of basic mathematical concepts, fractions and percentages. We leveraged a large‐scale dataset consisting of 2798 students using the ITS bettermarks and working on approximately 4.1 million mathematical problems. We found that advanced arithmetic, measurement units, geometry and fraction understanding significantly predicted percentage understanding. Closer inspection indicated that percentage understanding was best predicted by problems sharing similar features, such as fraction word problems and fraction/natural number multiplication/division problems. Our findings suggest that practitioners and software developers may consider revising specific subtopics which share features with percentage problems for students struggling with percentages. More broadly, our study demonstrates how evaluating interdependencies between subtopics covered within an ITS as a data‐driven approach can provide practical insights for improving the design of ITSs. Practitioner notes What is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics. What is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics. [ABSTRACT FROM AUTHOR]

Background: Mindfulness practices enhance executive function skills and academic achievement, spurring interest in integrating mindfulness interventions into education. Embedding mindfulness practice into a digital math game may provide a low‐cost, scalable way to induce mindfulness and boost game‐based learning, yet this approach remains unexplored. Objectives: We investigated the learning benefits of integrating mindfulness exercises in a digital math learning game and examined how students' trait mindfulness might moderate the outcomes. Methods: Two classroom studies were conducted with 404 5th and 6th grade students from six public schools in the U.S. (nStudy 1 = 227, nStudy 2 = 177). The two randomized controlled experiments assigned students to one of the three conditions: passive control (playing the digital learning game Decimal Point), story‐enriched active control, or mindfulness‐enriched condition. Trait mindfulness, learning gains, and in‐game problem‐solving (including problem‐solving duration, error count and correctness after reminder) were assessed. Study 2 included a manipulation check to better understand the effects of the mindfulness intervention. Results: Findings showed no significant differences in learning gains, problem‐solving duration or error count among the conditions. Students' trait mindfulness did not moderate these outcomes. Mindfulness reminders in the mindfulness‐enriched game led to more correct answers after errors than jokes in the story‐enriched game. Study 2 revealed that we failed to induce higher state mindfulness through the mindfulness inductions. Conclusions: Mindfulness prompts could be especially beneficial for students experiencing frustration during gameplay, warranting more exploration for digital game‐based instruction. We highlight barriers and future directions for fostering mindfulness through computer‐based instruction in classrooms. Lay Description: What is currently known about this topic?: Researchers focus on digital games as productive learning environments for math, with potential for higher learning gains compared with traditional methods. However, results are mixed, and not all games lead to improved math outcomes.Executive function (EF) skills are crucial for math learning, and mindfulness‐based interventions show promise in enhancing EF skills in school‐aged students.Embedding mindfulness practice into a digital math game may provide a low‐cost, scalable way to induce mindfulness and, in turn, boost EF skills and game‐based learning, yet this approach remains unexplored What does this paper add?: This paper presents the results of two randomized control trials investigating the feasibility and benefits of incorporating mindfulness exercises into a digital math game designed for middle school students. Benefits are compared to those of a story‐enriched and regular version of the same digital math learning game.The paper also explores variations in the effects of the mindfulness‐enriched game based on students' initial trait mindfulness levels.We observe that listening to mindfulness inductions at the beginning of game sessions do not induce mindfulness, and therefore does not benefit math learning.We find that mindfulness prompts received after recurrent errors can be beneficial for students' problem solving. Implications of study findings for practitioners: Our study provides important information on how digital learning game designers should attempt to induce mindfulness in a digital game to promote learning.Digital learning game designers should consider incorporating mindfulness exercises into their games strategically. Presenting mindfulness inductions in audio format at the beginning of game sessions may not induce mindfulness or benefit math learning. Instead, designers should focus on integrating mindfulness prompts at moments when students encounter frustration within the learning game.Beyond embedding audio mindfulness exercises in the game, learning designers should also consider alternative technological and game features to induce mindfulness within a learning game. [ABSTRACT FROM AUTHOR]

Background: With the growing integration of technology in education, the adoption of blended learning (b‐learning) has gained attention. B‐learning combines traditional classroom teaching with online components, holding potential to enhance student outcomes and educational efficiency. Yet, current research predominantly concentrates on higher education institutions in urban areas, creating a void in understanding its impact on K‐12 education, particularly in rural settings. Objectives: This qualitative study aims to propose recommendations for successful implementation of blended learning in rural K‐12 areas by addressing the key stakeholders influencing its adoption and identifying the main factors affecting its success. Methods: We performed a content analysis of grey literature documents detailing the implementation of b‐learning in K‐12 education. Additionally, interviews with crucial stakeholders such as teachers, principals, and experts in rural schools in South America provide insights into the challenges and prospects of b‐learning adoption in these contexts. Results and Conclusions: The study identifies pivotal stakeholders for effective b‐learning implementation, outlining their roles and addressing challenges inherent in rural settings. Recommendations for enhancing b‐learning's implementation in developing countries are also proposed. The research underscores the significance of involving diverse stakeholders such as governmental bodies, school leaders, educators, students, and families to ensure a holistic and efficient approach to blended learning. Lay Description: What is currently known about this topic: Blended Learning Definition and Benefits: Blended learning combines face‐to‐face and online instruction, enhancing student engagement and personalization by providing diverse learning pathways.Growth in K‐12 Implementation: Blended learning gained traction in K‐12 education, boosting academic outcomes, driven by recognition of learner diversity and potential in meeting students' needs.Contextual Challenges: Implementation challenges arise in low‐income households and rural areas, including limited access to technology and internet connectivity, impacting student engagement and success. What does this paper add: Focusing on Rural Dynamics: This study helps to fill a gap by investigating blended learning in rural K‐12 settings, spotlighting key players and success drivers in these unique contexts.Recommendations from Various Sources: By combining interviews and grey literature, this research generates comprehensive, practical recommendations grounded in both empirical insights and real‐world perspectives.Insights into Rural Hurdles: This paper extends the discussion on blended learning by addressing specific rural barriers, proposing family involvement, private sector partnerships for training, and tailored pedagogical strategies. Implications for practice or policy: Leadership and Collaboration: Effective school leadership and collaboration with local governments are pivotal for fruitful blended learning implementation, ensuring defined roles, responsibility, and accountability.Adaptable Curriculum Design: An adaptable, centralized curriculum aligns with recommendations, fostering efficient tracking of student progress and enabling customized learning.Engagement‐Centric Pedagogies: Recommendations spotlight interactive, student‐centered teaching methods, tailored to diverse student needs, nurturing active participation and elevated learning outcomes. [ABSTRACT FROM AUTHOR]

Background: Evidence‐based practice is the principle governing a range of healthcare practices and beyond. However, it has suffered from a lack of philosophical rigour. This paper sets out to analyse the epistemological basis of evidence‐based practice. Method: The paper uses a conceptual analysis. First, it describes the implicit epistemology at work in evidence‐based practice. Second, it evaluates the implicit epistemological basis. Results: The analysis indicates that evidence‐based practice lacks an explicit epistemological basis. It shows, moreover, that the implicit epistemological basis is untenable. Conclusion: There is a need to re‐think the epistemological basis for evidence‐based practice. Evidence‐based practice is out of touch with developments within philosophy of science. [ABSTRACT FROM AUTHOR]

We access most of our most cherished beliefs via testimony. Philosophy is no exception. We treat spoken and written philosophical testimony as evidence for philosophical claims. Nonetheless, this paper argues that philosophical testimony is unable to justify philosophical beliefs. If testimony is the only evidence we have to justify philosophical beliefs, this entails skepticism about philosophy. Call this the testimony challenge. First, the paper argues that philosophical testimony does not meet the conditions under which evidence can justify our beliefs. Second, it shows that philosophical testimony cannot provide preemptive epistemic reasons as science and mathematics do. Finally, it answers the self‐defeat objection and a set of objections aiming to block the thesis that philosophical testimony does not justify philosophical beliefs. The paper aims to rethink the role of philosophical testimony in philosophy and reinforce the skeptical worries raised by methodological and disagreement challenges. [ABSTRACT FROM AUTHOR]

This paper considers a specific autonomous Schwarzian differential equation given by Sf=c(f−σ1)(f−σ2)/[(f−τ1)(f−τ2)]$$\begin{equation*} \hspace*{35pt} S_f=c(f-\sigma _1)(f-\sigma _2)/[(f-\tau _1)(f-\tau _2)] \end{equation*}$$and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case. [ABSTRACT FROM AUTHOR]

The recently proposed fractional scoring scheme is used to attribute publications to percentile rank classes. It is shown that in this way uncertainties and ambiguities in the evaluation of specific quantile values and percentile ranks do not occur. Using the fractional scoring the total score of all papers exactly reproduces the theoretical value. [ABSTRACT FROM AUTHOR]

The author briefly reviews studies on the math teaching of secondary school students with mild intellectual disabilities. Then, the author demonstrates ways to teach secondary‐level mathematics to students with mild intellectual disabilities. In this article, readers will learn about how to use manipulatives, diagrams, and gestures to support students' thinking. Readers will also learn how to make connections between new and challenging math content to students' experiences inside and outside of school to support them as they think through mathematics. [ABSTRACT FROM AUTHOR]

The paper investigates semiotics as an integrative, foundational framework for operational research (OR). To this end, the semiotics of the pragmatist C.S. Peirce is explained and its application in OR research, and consulting practice is explored. It is the fourth in a series of papers each seeking to ground OR in a different logic: philosophic/mathematical logic, pragmatic logic, economic logic and now semiotic logic. For Peirce, semiotics is a logic, and as such it is argued in the paper that it can embrace philosophical/mathematical, pragmatic and economic logics. The comprehensive nature of semiotics is such that it brings to a close the research programme aimed at establishing foundations for OR. However, the closure of one programme opens up another—the application of semiotics in OR practice and academic research. Possible avenues for further research include learning from, and collaboration with, neighbouring disciplines such as information systems and computer science. [ABSTRACT FROM AUTHOR]

Background: The substantial growth in gamification research has connected gamified learning to enhanced engagement, improved performance, and greater motivation. Similar to gamification, personalized learning analytics dashboards can enhance student engagement. Objectives: This study explores the student experiences and academic achievements using a gamified dashboard in a large, introductory STEM course. Methods: We examined two groups of students enrolled in different sections of a one‐semester‐long physical geology course with a total enrollment of 223 students. The only difference between the groups was that one had access to the dashboard. The data collection included students' assignments, overall performances, and exam scores. Students in both sections completed a Science Literacy Concept Inventory survey at the beginning and end of the term. Additionally, students completed an end‐of‐term survey containing open‐ended questions on their experience and interactions with specific elements. Results: Students shared mostly positive comments about their experience with the dashboard, and the final grade of students with access to the dashboard was 13% higher, on average, compared to their peers in the non‐dashboard section. Conclusion: With low costs and little time invested, gamified dashboards could have a significant impact on student performance in large STEM lecture courses. Lay Description: What is already known about this topic: Leaner dashboards that provide personalized feedback are linked to improved student performance in undergraduate STEM courses. However, students may be reticent to adopt dashboards, especially students in the middle range of the performance distribution, for whom the impact of the dashboard may be most significant.Prior research on gamification in STEM courses suggests that adding gameful elements like progress trackers, badges, and rewards can increase students' engagement in undergraduate coursework.The combination of personalized learning in dashboards and the motivational experience of gamification should boost student engagement, dashboard adoption, and ultimately foster academic achievement. What this paper adds: In this paper, we examine the differences in academic performance between students with and without access to a gamified learning dashboard in an undergraduate Geology course.We observe that students with access to the dashboard (the treatment group), outperform students without access (the control group). The greatest benefit is observed for students who use the dashboard more frequently.Students reported that the progress tracker, the badges, and the rewards all incentivized their use and provided useful feedback about their performance. Students who did not adopt the dashboard expressed a need for more detail about the tool and its purpose. Implications for practice and/or policy: Instructors and instructional designers for large enrollment undergraduate STEM courses should consider the affordances of gamified learner dashboards for increasing engagement and for providing synchronous feedback about learners' strategies in the course.Aligning the dashboard to the content of the curriculum—by ensuring that badges reflect key concepts‐ encouraged participation and helped students identify the utility of the tool. Dashboards should be personalized to individuals and to the course.Adding gameful elements in a low‐cost intervention for increasing adoption and engagement of dashboards in large enrollment introductory courses. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Choquard equation with weight (−Δ)su=1|x|N−α∗h(x)|u|ph(x)|u|p−2uinℝN,$$ {\left(-\Delta \right)}&#x0005E;su&#x0003D;\left(\frac{1}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{N-\alpha }}\ast h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;p\right)h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}&#x0005E;N, $$where 02s,p>2,α>0$$ 02s,p>2,\alpha >0 $$ and h$$ h $$ is a positive weight function satisfying h(x)≥C|x|a$$ h(x)\ge C{\left&#x0007C;x\right&#x0007C;}&#x0005E;a $$ at infinity, for some a≥0$$ a\ge 0 $$. We establish, in this paper, a Liouville type theorem saying that if maxN−4s−2a,0<α

Background: A strong knowledge of mathematics, beginning at the elementary level, is critical for participation in today's complex world. The home may be one way to facilitate individualized mathematics instruction, given that children spend more time at home than in an academic institution. Therefore, researchers are interested to see whether the home numeracy environment (HNE) can provide a solid foundation for children's mathematics understanding. Further, children's digital mathematics exploration at home is increasingly common (e.g., using math apps). Objectives: The present study evaluates the digital home numeracy practice (DHNP) model and explores its effect on children's mathematics knowledge across five domains (numeration, number operation, pattern recognition, spatial sense, and applied problem‐solving). Methods: To conduct this study, 117 Canadian parents and their children from Grade 1 through 5 completed a DHNP survey and a range of in‐person mathematics measures. Results and Conclusions: The results identified significant relations between parents and children's implicit mathematics factors (e.g., math anxiety, motivation). Children's mathematics anxiety and parents' academic estimations and expectations for their children positively predicted children's mathematics knowledge. In terms of DHNP components, parental involvement in DHNP predicted children's numeration and applied problem‐solving knowledge. Implications: Taken together, the results detail the contribution of parental and child factors to children's mathematics knowledge and suggest that parents adjust their role in DHNP according to their children's mathematics ability. Lay Description: What is (not) known about the subject matter?: Research is less developed on the impact of home technology in the mathematics learning processExisting research did not use a multi‐measure approach to assess the relationships between a variety of home numeracy factors and fundamental mathematics concepts and processes What is the contribution of this paper?: The paper highlights the importance of considering different home related factors to understand the role of home numeracy practices in children's mathematics developmentIt provides a holistic view of digital home numeracy practice (DHNP) by examining several implicit and explicit components of home numeracy environmentChildren's mathematics anxiety predicts their mathematics knowledgeParental estimation and expectation about their children's mathematics performance predicts their mathematics knowledge across several domains whereas parental role on DHNP impacts children's numeration only What are the implications of the findings?: The findings provide an understanding of the significant value of including digital mathematics practices in the study of the home mathematics environmentThe outcome of the study provides guidance for educators and researchers to better understand and leverage DHNP, and its impact on children's mathematics educationFuture research should investigate to what extent digital features should be included in the DHNP model. This will provide insights into the impact of using well‐designed apps on children's mathematics learning [ABSTRACT FROM AUTHOR]

Background: People with aphasia (PWA) can experience functional numeracy difficulties, that is, problems understanding or using numbers in everyday life, which can have numerous negative impacts on their daily lives. There is growing interest in designing functional numeracy interventions for PWA; however, there are limited suitable assessments available to monitor the impact of these interventions. Existing functional numeracy assessments lack breadth and are not designed to be accessible for PWA, potentially confounding their performance. Additionally, they do not include real‐life demands, such as time pressure, which may affect their ecological validity. Thus, there is a crucial need for a new assessment to facilitate further research of PWA's functional numeracy. Aims: To develop, validate and pilot a wide‐ranging, aphasia‐friendly functional numeracy assessment to investigate how functional numeracy is impacted by aphasia severity and time pressure demands, and to explore predictors of PWA's functional numeracy. Methods & Procedures: To develop the Functional Numeracy Assessment (FNA), 38 items inspired by the General Health Numeracy Test (GHNT) and Excellence Gateway were adapted for suitability for PWA and entered in a computerized psychometric‐style test. The final 23 items (FNA23) were selected based on 213 neurotypical controls' performance, and controlled for difficulty, response modality and required numeracy skills. Aphasia‐friendly adaptations of the GHNT and Subjective Numeracy Scale were used to examine the FNA23's concurrent validity. Internal consistency reliability and interrater reliability (for spoken responses) were also examined. A novel Time Pressure Task was created by slight adaptation of seven FNA23 questions to explore the effects of time pressure on functional numeracy performance. A total of 20 PWA and 102 controls completed all measures on an online testing platform. Outcomes & Results: The FNA23 demonstrated acceptable internal consistency reliability (KR‐20 = 0.81) and perfect interrater reliability (for spoken responses). FNA23 and GHNT scores were positively associated, suggesting satisfactory concurrent validity. PWA demonstrated poorer functional numeracy than controls and took longer to complete assessments, indicating that aphasia impacts functional numeracy. Time pressure did not significantly impact performance. PWA demonstrated a wide range of functional numeracy abilities, with some performing similarly to controls. Conclusions & Implications: The FNA23 is a wide‐ranging, valid and reliable assessment which, with further development, will be a useful tool to identify and monitor PWA's functional numeracy difficulties in research and clinical practice. Considering PWA's widespread functional numeracy difficulties evidenced by this study, all PWA would likely benefit from routine evaluation for functional numeracy difficulties as part of their neurorehabilitation journeys. WHAT THIS PAPER ADDS: What is already known on this subject: Few studies have investigated functional numeracy difficulties in PWA. No published functional numeracy assessments exist that have been specifically designed to be accessible for PWA. What this paper adds to existing knowledge: The newly developed FNA23 is a valid and reliable tool to extensively assess PWA's functional numeracy. This study confirmed previous findings of widespread functional numeracy difficulties in PWA that are related to their aphasia severity. What are the potential or actual clinical implications of this work?: The FNA23 can be used to assess PWA's functional numeracy to inform areas of strengths and difficulties to target in intervention, and to monitor progress towards achieving intervention objectives. All PWA should be routinely evaluated for functional numeracy difficulties. [ABSTRACT FROM AUTHOR]

In this short paper, we analyse whether assuming that mathematical objects are "thin" in Linnebo's sense simplifies the epistemology of mathematics. Towards this end, we introduce the notion of transparency and show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparency of some thin objects is a fruitful characteristic mark of abstract mathematics. [ABSTRACT FROM AUTHOR]

Background: Teachers often rely on the use of open‐ended questions to assess students' conceptual understanding of assigned content. Particularly in the context of mathematics; teachers use these types of questions to gain insight into the processes and strategies adopted by students in solving mathematical problems beyond what is possible through more close‐ended problem types. While these types of problems are valuable to teachers, the variation in student responses to these questions makes it difficult, and time‐consuming, to evaluate and provide directed feedback. It is a well‐studied concept that feedback, both in terms of a numeric score but more importantly in the form of teacher‐authored comments, can help guide students as to how to improve, leading to increased learning. It is for this reason that teachers need better support not only for assessing students' work but also in providing meaningful and directed feedback to students. Objectives: In this paper, we seek to develop, evaluate, and examine machine learning models that support automated open response assessment and feedback. Methods: We build upon the prior research in the automatic assessment of student responses to open‐ended problems and introduce a novel approach that leverages student log data combined with machine learning and natural language processing methods. Utilizing sentence‐level semantic representations of student responses to open‐ended questions, we propose a collaborative filtering‐based approach to both predict student scores as well as recommend appropriate feedback messages for teachers to send to their students. Results and Conclusion: We find that our method outperforms previously published benchmarks across three different metrics for the task of predicting student performance. Through an error analysis, we identify several areas where future works may be able to improve upon our approach. Lay Description: What is already known about this topic: Open‐ended questions are used by teachers in the domain of mathematics to assess their students understanding but automated support for these types of questions are limited in online learning platforms. Recent advancements in areas of machine learning and natural language processing have led to promising results in a range of domains and applications. What this paper adds: Emulating how teachers identify similar student answers can be used to build tools that support teachers in assessing and providing feedback to student open‐ended work. Implications for practice: Developing better automated supports can increase the amount of direct feedback students receive to guide their learning. [ABSTRACT FROM AUTHOR]

Several independent authors reported a high share of uncited publications, which include those produced by top scientists. This share was repeatedly reported to exceed 10% of the total papers produced, without any explanation of this phenomenon and the lack of difference in uncitedness between average and successful researchers. In this report, we analyze the uncitedness among two independent groups of highly visible scientists (mathematicians represented by Fields medalists, and researchers in physiology or medicine represented by Nobel Prize laureates in the respective field). Analysis of both groups led to the identical conclusion: over 90% of the uncited database records of highly visible scientists can be explained by the inclusion of editorial materials progress reports presented at international meetings (meeting abstracts), discussion items (letters to the editor, discussion), personalia (biographic items), and by errors of omission and commission of the Web of Science ( Wo S) database and of the citing documents. Only a marginal amount of original articles and reviews were found to be uncited (0.9 and 0.3%, respectively), which is in strong contrast with the previously reported data, which never addressed the document types among the uncited records. [ABSTRACT FROM AUTHOR]

Despite extensive research on attainment grouping, the impact of attainment grouping on pupil attainment remains poorly understood and contested. This paper presents evidence from a study conducted with 2944 12–13 year olds, from 76 schools in England, who were allocated to between‐class attainment groups ('setting') in English and mathematics over the first 2 years of secondary schooling. After controlling for prior attainment, pupils in the top set performed significantly better than pupils in the middle and bottom sets in both English and mathematics. The findings indicate a widening gap in attainment, especially in the case of English. Findings, especially in the case of mathematics, provide more evidence of a relative benefit for pupils placed in top sets than a relative detriment for those in bottom sets. [ABSTRACT FROM AUTHOR]

Summary: Luis Raúl Pericchi Guerra was born in Caracas, Venezuela, on 11 March 1952. He completed a B.S. in Mathematics in 1975 at the Universidad Simón Bolívar in Caracas, an M.S. in Statistics at the University of California Berkeley in 1978 and a Ph.D. in Statistics at Imperial College London in 1981. After graduating from Imperial College, Luis Raúl went back to Universidad Simón Bolívar. There, he played a key role in the developing of graduate programmes in Statistics and single handedly built an internationally recognised group focused on Bayesian statistics. In 2001, he moved to the Universidad de Puerto Rico in Rio Piedras to become the Chair of the Mathematics Department. At Universidad de Puerto Rico, he was instrumental in the establishment of a Ph.D. track in Computational Mathematics and Statistics. Luis Raúl has published over 120 papers in statistical and domain‐specific journals, making significant contributions to several areas of Bayesian statistics (especially in the areas of model selection and Bayesian robustness) and their application (especially in hydrology). He is a Fellow of the American Statistical Association, the International Society for Bayesian Analysis, the John Simon Guggenheim Memorial Foundation and an Elected Member of the International Statistical Institute. This conversation took place over multiple sessions during the 2022 O'Bayes meeting in Santa Cruz, California, and the months that followed. [ABSTRACT FROM AUTHOR]

Background: As one of the mainstream forms of programming education, educational robotics (ER) have been a crucial way to develop K‐12 students' programming ability. Objectives: The purpose of this study is to clarify the content of programming ability, to verify the effectiveness of ER as a teaching method to improve students' programming abilities and the conditions for more effective instructional design. Methods: The method of meta‐analysis was adopted to conduct this study. 4180 K‐12 students from 36 studies with a total of 85 effect sizes were captured. Based on the contents of included literature, an ER programming ability model was put forward, which was composed of essential ability and generative ability. The meta‐analysis of essential ability (N = 22) and generative ability (N = 64) was carried out respectively. Results and conclusion: The results showed that ER activity has a positive effect on the essential ability (Hedges' g = 0.539 CI [0.327, 0.752], p < 0.001) and generative ability (Hedges' g = 0.535 CI [0.426, 0.643], p < 0.001). Additionally, the results of moderator analysis showed that utilizing ER to solve mathematics problems is able to more effectively develop students' essential ability, while the STEM interdisciplinary ER instruction had the largest effect on generative ability. Simultaneously, the individual programming form is conducive to students' essential ability, while the generative ability is significantly affected by the cooperative programming form. No matter for the evaluation of essential ability or generative ability, ER had the greatest effect on kindergarten children' programming ability. Moreover, the intervention duration ranging from 1 to 5 weeks, the samples size less than 50 and the choice of measurement scale can effectively improve students' programming ability. Implications: This research enriched the theoretical basis of programming ability, and provided reference and guidance for K‐12 programming teaching practitioners and researchers in ER teaching design and practise. Lay Description: What is (not) known about the subject matter?: Educational robotics (ER) is one of the mainstream form of K‐12 programming education.K‐12 students develop various abilities by coding ER.Different instruction conditions in ER education may affect the teaching effect. What is the contribution of this paper?: This study sorted out the specific content of programming ability in ER activities, and classified programming ability into essential ability and generative ability.This study respectively validated the effectiveness of ER activity on improving essential ability and generative ability of K‐12 students.This study considered discipline integration, cooperation degree, grade level, intervention duration, sample size and assessment tools as moderator variables, and analyzed their effects sizes on ER developing programming ability. What are the implications for practitioners?: This paper clarified the specific content of programming ability in K‐12 ER education, and enriched the theoretical basis of programming ability.It provided evidence for the effectiveness of K‐12 ER education.This study provided an reference and guidance for ER instructional design for K‐12 programming teachers and researchers. [ABSTRACT FROM AUTHOR]

Presents a list of articles about statistics and mathematics that will be published in the April 11, 2003 issue of 'Australian and New Zealand Journal of Statistics.'

We show that if G$G$ is a finitely generated group such that its profinite completion Ĝ$\widehat{G}$ is "far from being projective" (i.e., the kernel of the universal Frattini cover of Ĝ$\widehat{G}$ is not a small profinite group), then the class of existentially closed G$G$‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is "far from being projective," the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper. [ABSTRACT FROM AUTHOR]

Background: Despite its potentially significant functional and emotional impact, acalculia is still too rarely assessed and managed by speech and language therapists. Research on the rehabilitation of numerical transcoding remains scarce in the literature and, despite positive results, presents a low level of evidence. Aims: The present study aims to evaluate the effectiveness of a targeted rehabilitation of numerical transcoding in two patients suffering from a chronic secondary acalculia. Methods & Procedures: Two post‐brain injury females with secondary acalculia took part in a single‐case experimental design with multiple baseline across subjects according to a three‐phase experimental protocol: baseline involving global cognitive rehabilitation (5–7 measurements with randomized sequential introduction); targeted intervention (10 measurements); follow‐up (2 immediate measurements and 1 month after the end of the intervention). Repeated outcome measures consisted of six lists composed of numbers of equivalent difficulty that were used alternately to assess numerical transcoding. We used a reverse digit span as a control measure to assess the specificity of the intervention. Rehabilitation lasted 5 weeks and consisted of errorless learning with colour cues, tables and number–words cards. Outcomes & Results: During baseline period involving global cognitive rehabilitation, transcoding scores remained unchanged. In contrast, there was a significant improvement in scores for both patients during the intervention phase targeting transcoding and maintenance of benefits 1‐month post‐intervention. Conclusions & Implications: This study demonstrates that a specific rehabilitation targeting numerical transcoding following chronic secondary acalculia can be effective in improving transcoding skills. WHAT THIS PAPER ADDS: What is already known on the subject: Transcoding difficulties in patients with acalculia can cause a significant disability in everyday life activities. In secondary acalculia, rehabilitation of cognitive functions associated with number processing (attention, working memory, language) is not sufficient for improvement of transcoding. What this paper adds to existing knowledge: An intervention specifically targeting numerical transcoding significantly and durably improves the skills of patients with chronic secondary acalculia. What are the potential or actual clinical implications of this work?: Procedural error‐free intervention using colour cueing, tables, cards with number–words, copy and repetition seems effective to improve transcoding skills in chronic acalculia. [ABSTRACT FROM AUTHOR]

Background: Digital game‐based learning is gaining increased attention from both researchers and educators for improving mathematics instruction. However, the evidence for game‐based learning is mixed and research with rigorous research design and analyses are limited. Objective: Here, in a pre‐registered randomized controlled study, we investigated whether a fraction game designed collaboratively by educational experts and professional game developers may serve as a useful tool to enhance students' fraction knowledge. Methods: We assigned French fifth graders to either an experimental group who used the game (n = 110) or a control group (n = 78) who received traditional instruction on fractions. Fraction knowledge was assessed pre‐ and post‐intervention. Results: Results show that students in the active control group had superior overall fraction performance than students in the experimental group at the end of intervention. However, the game had a positive effect on decimal learning. We also found a positive relation between game performance and overall fraction knowledge scores at post‐test. Contribution: The study highlights the importance of game metrics as indicators of personalized assessment tools. Given the increased usage of games in learning mathematics and the equivocal results on the effectiveness of these games, our study also highlights the importance of pre‐registration and randomized controlled studies. Lay Description: What is already known about this topic: Digital game‐based learning (DGBL) is gaining popularity in mathematics teaching.DGBL shows potential for teaching difficult abstract concepts such as fractions.However, research evaluating DGBL for fractions is limited. What this paper adds: The study examines a fraction game designed by education experts and game designers.Students fraction knowledge is assessed using paper‐based test.Challenges associated with DGBL in the classroom are discussed in the paper. Implications for practice: The results will guide educators in designing effective fraction instruction in classroom.The study highlights the critical role of instructional support in DGBL for fractions.Game based metrics can be used to individualize instruction in the classroom. [ABSTRACT FROM AUTHOR]

For any ring A$A$ and a small, pre‐additive, Hom‐finite, and locally bounded category Q$Q$ that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors Q→AMod$Q \rightarrow {}_{A} \operatorname{Mod}$ has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the Q$Q$‐shaped derived category of A$A$. The usual derived category of A$A$ is one example; more general examples arise by taking Q$Q$ to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and García‐Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and Šťovíček (Math. Z. 287 (2017), no. 3‐4, 1109–1155). [ABSTRACT FROM AUTHOR]

The aim of this paper is to revisit propagation of chaos for a Langevin‐type interacting particle system recently proposed as a method to sample probability measures. The interacting particle system we consider coincides, in the setting of a log‐quadratic target distribution, with the ensemble Kalman sampler [SIAM J. Appl. Dyn. Syst. 19 (2020), no. 1, 412–441], for which propagation of chaos was first proved by Ding and Li in [SIAM J. Math. Anal. 53 (2021), no. 2, 1546–1578]. Like these authors, we prove propagation of chaos with an approach based on a synchronous coupling, as in Sznitman's classical argument. Instead of relying on a boostrapping argument, however, we use a technique based on stopping times in order to handle the presence of the empirical covariance in the coefficients of the dynamics. The use of stopping times to handle the lack of global Lipschitz continuity in the coefficients of stochastic dynamics originates from numerical analysis [SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063] and was recently employed to prove mean‐field limits for consensus‐based optimization and related interacting particle systems [arXiv:2312.07373, 2023; Math. Models Methods Appl. Sci. 33 (2023), no. 2, 289–339]. In the context of ensemble Langevin sampling, this technique enables proving pathwise propagation of chaos with optimal rate, whereas previous results were optimal only up to a positive ε$\varepsilon$. [ABSTRACT FROM AUTHOR]

We consider the graphical mean curvature flow of maps f:Rm→Rn$\mathbf {f}:{\mathbb {R}^{m}}\rightarrow {\mathbb {R}^{n}}$, m⩾2$m\geqslant 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well‐known maximum principle of Ecker and Huisken derived in their seminal paper [Ann. of Math. (2) 130:3(1989), 453–471]. In the case of uniformly area decreasing maps f:Rm→R2$\mathbf {f}:{\mathbb {R}^{m}} \rightarrow {\mathbb {R}^{2}}$, m⩾2$m\geqslant 2$, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self‐expander. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the well‐posedness, blow‐up phenomena, and continuity of the data‐to‐solution map of the Cauchy problem for a two‐component higher order Camassa–Holm (CH) system. The local well‐posedness is established in Besov spaces Bp,11p×Bp,12+1p$B_{p,1}^{\frac{1}{p}} \times B_{p,1}^{2+\frac{1}{p}}$ with 1≤p<∞$1 \le p < \infty$, which improves the local well‐posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution‐to‐data map, that is, the ill‐posedness is derived in Besov space Bp,∞s−2×Bp,∞s$B_{p,\infty }^{s - 2} \times B_{p,\infty }^s$ with 1≤p≤∞$1 \le p \le \infty$ and s>max{2+1p,52}$s>\max \lbrace 2+\frac{1}{p},\frac{5}{2}\rbrace$. Then, the nonuniform continuous and Hölder continuous dependence on initial data for this system are also presented in Besov spaces Bp,rs−2×Bp,rs$B_{p,r}^{s - 2} \times B_{p,r}^s$ with 1≤p,r<∞$1 \le p,r < \infty$ and s>max{2+1p,52}$s > \max \lbrace {2+\frac{1}{p},\frac{5}{2}}\rbrace$. Finally, the precise blow‐up criteria for the strong solutions of the two‐component higher order CH system is determined in the lowest Sobolev space Hs−2×Hs$H^{s-2}\times H^s$ with s>52$s>\frac{5}{2}$, which improves the blow‐up criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509–1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. [ABSTRACT FROM AUTHOR]