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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]

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]

In this note we point out and correct a mistake in our paper "GlobalLpestimates for degenerate Ornstein–Uhlenbeck operators with variable coefficients", published in Math. Nachr. 286 (2013), no. 11–12, 1087–1101. [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, 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

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]

In this paper, we study the uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$. Under suitable conditions, we prove that, for any given positive integer k$$ k $$, the problem we considered has at most one solution possessing exactly k−1$$ k-1 $$ nodes. Together with the results presented by Nagasaki [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2): 211–232, 1989] and Tanaka [Proc. Roy. Soc. Edinburgh Sect. A. 138 (6): 1331–1343, 2008], we can prove that more types of nonlinear elliptic equations have the uniqueness of nodal radial solutions. [ABSTRACT FROM AUTHOR]

This paper details a literature review of mathematics vocabulary intervention studies for students with mathematics difficulty. The primary aim was to identify instructional practices that support mathematics vocabulary development. We conducted a database search to identify mathematics intervention studies either focused exclusively on vocabulary or with an embedded vocabulary component. Ultimately, 13 studies with participants from kindergarten to Grade 8 were included in the review. The majority of included studies had dual foci on vocabulary and other mathematics content, including whole number computation, word problem solving, fractions, algebra, or geometry. All studies that measured mathematics vocabulary performance indicated positive student outcomes. In addition, multiple studies indicated positive effects on measures of other mathematics content beyond mathematics vocabulary knowledge. We noted six instructional practices across studies that bolstered the mathematics vocabulary performance of students with mathematics difficulty. From most to least common, these practices included formal vocabulary use, explicit instruction, use of representations, repeated exposures, pre‐teaching, and graphic organizers. Implications are addressed for both researchers and practitioners. [ABSTRACT FROM AUTHOR]

This paper extends some results of Mattila (J. Fractal Geom. 66 (2021) 389–401 and Ann. Acad. Sci. Fenn. A Math. 42 (2017) 611–620), in particular, removing assumptions of positive lower density. We give conditions on a general family Pλ:Rn→Rm,λ∈Λ$P_{\lambda }:\mathbb {R}^n\rightarrow \mathbb {R}^m, \lambda \in \Lambda$, of orthogonal projections which guarantee that the Hausdorff dimension formula dimA∩Pλ−1{u}=s−m$\dim A\cap P_{\lambda }^{-1}\lbrace u\rbrace =s-m$ holds generically for measurable sets A⊂Rn$A\subset \mathbb {R}^{n}$ with positive and finite s$s$‐dimensional Hausdorff measure, s>m$s>m$. As an application we prove for Borel sets A,B⊂Rn$A,B\subset \mathbb {R}^{n}$ with positive s$s$‐ and t-dimensional$t{\text{-dimensional}}$ measures that if s+(n−1)t/n>n$s + (n-1)t/n > n$, then dimA∩(g(B)+z)⩾s+t−n$\dim A\cap (g(B)+z) \geqslant s+t - n$ for almost all rotations g$g$ and for positively many z∈Rn$z\in \mathbb {R}^{n}$. We shall also give an application to the estimates of the dimension of the set of exceptional rotations. [ABSTRACT FROM AUTHOR]

We establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics. [ABSTRACT FROM AUTHOR]

Open science is receiving widespread attention globally, and preprinting offers an important way to implement open science practices in scholarly publishing. To develop a systematic understanding of researchers' adoption of and attitudes toward preprinting, we conducted a survey of authors of research papers published in 2021 and early 2022. Our survey results show that the United States and Europe led the way in the adoption of preprinting. The United States and European respondents reported a higher familiarity with and a stronger commitment to preprinting than their colleagues elsewhere in the world. The adoption of preprinting is much stronger in physics and astronomy as well as mathematics and computer science than in other research areas. Respondents identified free accessibility of preprints and acceleration of research communication as the most important benefits of preprinting. Low reliability and credibility of preprints, sharing results before peer review and premature media coverage are the most significant concerns about preprinting, emphasized in particular by respondents in the life and health sciences. According to respondents, the most crucial strategies to encourage preprinting are integrating preprinting into journal submission workflows and providing recognition for posting preprints. [ABSTRACT FROM AUTHOR]

Background: Game‐based learning can frame problem‐solving as a sense‐making experience with domain‐specific tasks for school students. However, multiple challenges arise when trying to support learners in such a complex, problem‐oriented learning environment. Objectives and Methods: With an architecture‐themed mathematics learning game, we conducted two mixed‐method studies to explore the impact and design of game‐based mathematical experience on the math problem‐solving performance of middle school students. Results and Conclusions: The study findings suggested a positive impact of game‐based math experience on math problem‐solving for middle school students. Problematization‐oriented game‐based math tasks with structuring features enhanced students' reasoning with problems and channelled it to doing mathematics. Takeaways: The current research findings support the initiative to frame learning as a sense‐making experience with domain‐specific tasks and inform the design of game‐based mathematical experience and learning support. Lay Description: What is already known about this topic: Realistic mathematical practices frame learning as a sense‐making experience.Challenges arise when involving learners in realistic mathematical experiences driven by domain‐specific tasks. What this paper adds: Game‐based simulation of math tasks framed learning as sense‐making experience.Game‐based math experiencing promoted math contextual problem‐solving performance.Structuring features enhanced problematized experiencing with math tasks. Implications for practice and/or policy: Teachers can integrate and facilitate game‐based math experiencing in class.Efforts should be made to make game‐based math tasks meaningful for students. [ABSTRACT FROM AUTHOR]

Background: Even though monitoring and control enactment are key aspects of self‐regulated learning (SRL), Adaptive learning technologies (ALTs) may reduce the need for learners to monitor and control their learning. Personalized dashboards are effective in supporting learners' monitoring and can potentially support control behaviour. Allowing learners to enact control over their learning process, seems to hold potential for increasing their motivation. Objectives: Therefore, this study's aim was to investigate the relation between control enactment and motivation. We examined how learners enacted control while learning with an ALT with personalized dashboards and examined the relation between learners' enactment of control and their motivation. Methods: Seventy‐eight primary school learners (Grade 5) participated. During the lesson, learners worked on mathematics in the ALT and concurrently were shown personalized visualizations that supported monitoring and enacting control over their learning process. Learners could enact control to change problems' difficulty to easy, medium, or hard. Motivation was measured before and after learning. Results: The SEM analyses showed that how learners enacted control was related to their motivation. Choosing difficult problems was related to more enjoyment and competence while choosing easy problems related to more pressure and tension. Learners who complied with the suggested difficulty level experienced less choice, but also less pressure/tension and more enjoyment and competence. Conclusions: These results provide avenues to account for broader learner characteristics like SRL and motivation to optimize learning. This way, hybrid systems in which control enactment is a shared responsibility of the system and learner, can be improved to support SRL development. Lay Description: What is currently known: Monitor and control enactment are key aspects of self‐regulated learning (SRL).Adaptive learning technologies may reduce the need for learners to monitor and control their learning.Personalized dashboards support learners' monitoring behaviour and can potentially also support control enactment.Enactment of control could positively affect learners' motivation. What does this paper add: There is a large variation in how learners enact control and whether they comply with the scaffold.Learners who chose more difficult problems experienced more enjoyment, while learners choosing easy problems, experienced more pressure.Learners complied with the scaffold, experienced less choice, but also less pressure and more enjoyment.This study provides avenues to account for broader learner characteristics like SRL and motivation to optimize learning. Practical implications: Support control enactment by actively involve learners in choice during the learning process.Offer monitor and enactment in conjunction, as learners can only make well‐founded choices if they can monitor their learning process.Intrinsically and extrinsically motivated learners showed different choice patterns and learning experience.It is important to have an understanding and monitoring of learners' motivation. [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]

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]

We give a corrected proof of the last theorem (Theorem 4.11) of Regularized Riesz energies of submanifolds, published in Math. Nachr. 291 (2018), no. 8–9, 1356–1373. Namely, we prove that the Riesz energy EΩ(z) of a compact body Ω is Möbius invariant if and only if the dimension of Ω is even and z=−2dimΩ. [ABSTRACT FROM AUTHOR]

Background Study: The effects of the flipped classroom (FC) on students' academic performance remain ambiguous, and the use of pre‐class videos may be one of the main factors hindering students' progress. A software package called GeoGebra has been proposed as a substitute for pre‐class videos to aid students' learning. Objective: This study investigates the effect of the FC using GeoGebra (FCG) on mathematical performance and memory among senior secondary students. Methods: One hundred and sixty students of a senior high school in China were selected. A quasi‐experimental design was implemented in which the classes were split into the intervention group (FCG), control group A (FCV) and control group B (TL) by random assignment. Pretest, Post‐test 1 and Post‐test 2 were introduced. Results and Conclusion: The results indicate that both the FCG and the FC with pre‐class videos (FCV) were more effective than traditional lectures in enhancing students' academic test scores, and that the effect of the FCG was comparable to its video counterpart, F(2, 136) = 3.752, ρ = 0.026, η2 = 0.052. Although their effects were similar, the qualitative results revealed some hidden benefits associated with the replacement of pre‐class video with GeoGebra. Furthermore, although they were equally effective in promoting students' performance in the short‐term, the long‐term effects on academic performance of both forms of FC were non‐significant when compared with traditional lectures. Lay Description: What is already known about this topic: Flipped classroom is an effective strategy in teaching mathematics.Among those challenges, video may be one of the main factors hindering the effect of flipped classroom. What this paper adds: GeoGebra could be an effective alternative to video.Effect between the flipped classroom using GeoGebra and the flipped classroom using video were similar.Both the flipped classroom with GeoGebra and the flipped classroom with videos were more effective than traditional lectures. Implications for practice and/or policy: Students' short‐term performance could be enhanced by visualisation provided by flipped classroom.To enhance students' long‐term performance, an appropriate amount of homework is suggested. [ABSTRACT FROM AUTHOR]

Background: Students often show unfavourable attribution: they attribute poor school performance to stable factors such as lack of ability and good school performance to variable factors such as effort. However, attribution can be influenced by individualized digital re‐attributional feedback leading to positive motivational effects and higher learning outcomes. This is very promising, but it still is unclear, whether this digital re‐attributional feedback can also be successfully integrated in everyday classroom activities. Objectives: The present field experiment investigated how integrating digital re‐attributional feedback into classroom instruction affects student attribution, motivation and learning outcomes. Methods: In the experiment, 8th–10th grade high school students (N = 322) worked with a digital mathematics learning program which was integrated in a three‐week teaching unit. Half the students in each classroom received only standard feedback after each practice task (SF group); the other half received additionally an individual re‐attributional feedback (RF group) after every third task. Attribution, mathematics self‐concept, and self‐efficacy were measured by an online questionnaire twice a week; learning outcomes in mathematics were measured weekly. Results and Conclusion: Hierarchical analyses showed that re‐attributional feedback led to a more favourable attribution in case of success on stable factors. Especially low‐performing students benefited from this feedback. No effects on attribution in case of failure, self‐efficacy or learning outcomes could be found. Further research could investigate if certain adaptations to the digital re‐attributional feedback is more effective in a real classroom setting and has a broader impact on different students. Lay Description: What is already known about this topic: Working with computers in the classroom and at home is becoming more common.Re‐attributional feedback can increase student's motivation when given directly after solving a task and at a high density.Digital re‐attributive feedback can be given in a high density.Re‐attributional training sessions have been shown to be effective in individual training sessions or outside the students' classroom. What this paper adds: The study took place within a real teaching environment during mathematics class for 4 weeks.Digital re‐attributional feedback sessions were directly integrated into the students' online learning which was part of the teaching plan. The implications of study findings for practitioners: Computer‐based re‐attributional training show some limited effects in regular classroom instruction and could be well integrated in student online learning—when adapted accordingly.Low‐performing students show less favourable attribution than better‐performing students in case of success and benefit the most from re‐attributional training. [ABSTRACT FROM AUTHOR]

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

Background: Providing students with worked out problem solutions is a beneficial instructional technique in STEM disciplines, and studying examples that have been worked out incorrectly may be especially helpful for reducing misconceptions in students with low prior content knowledge. However, past results are inconclusive and the effects of incorrect worked examples alone or in combination with correct examples remains unclear. Objectives: We aim to address whether studying incorrect examples alone or in combination with correct examples can support the reduction of students' fraction misconceptions, operationalized as errors made with high confidence. Methods: After incorrectly solving a sampling problem, 130 students in 4th through 11th grade in the U.S. were randomly assigned to a condition in an online problem set focused on fraction equivalence. Students studied either single‐type worked examples (i.e., correct or incorrect; n = 49) or combination‐type worked examples (correct and incorrect; n = 41) or engaged in a problem‐solving control (n = 50). Results: Studying a combination of correct and incorrect worked examples was as effective as the problem‐solving control with feedback at improving fraction equivalence knowledge and reducing the rate of high‐confidence errors. Students in both the combination condition and the problem‐solving with feedback condition outperformed those who studied either correct or incorrect worked examples alone. Conclusions: Results support the inclusion of a combination of correct and incorrect worked examples when teaching students with low prior content knowledge. Studying a combination of example types within an online tutor helps to reduce misconceptions about fractions, a topic students commonly struggle with. A problem‐solving task with corrective feedback worked equally well. Lay Description: What is already known about this topic: Studying worked examples of mathematical solutions improves student learning.Incorrect examples, either alone or in combination with correct examples, are particularly effective for improving learning for students with low prior knowledge in the target content.Comparing correct and incorrect examples is also effective at improving problem‐solving but this effect may be specific to those with high prior knowledge.Still unknown are the impacts of these different combinations of examples on misconceptions in particular. What this paper adds: We show that studying a combination of correct and incorrect examples reduces math misconceptions.Problem‐solving with corrective feedback within an online tutor also has this effect. Implications for practice and/or policy: Using a combination of example types will help reduce students' misconceptions about fractions. [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]

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]

There exists a mistake in the proof of Theorem 4.2. We present a new proof of this theorem, which shows that the main results of the paper are still true. [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]

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]

This paper is concerned with the noncommutative analog of the normal subgroup theorem for certain groups. Inspired by Kalantar and Panagopoulos (arXiv:2108.02928, 2021, 16), we show that all Γ$\Gamma$‐invariant subalgebras of LΓ$L\Gamma$ and Cr∗(Γ)$C^*_r(\Gamma)$ are (Γ$\Gamma$‐)coamenable. The groups we work with satisfy a singularity phenomenon described by Bader et al. (Invent. Math. 229 (2022), 929–985). The setup of singularity allows us to obtain a description of Γ$\Gamma$‐invariant intermediate von Neumann subalgebras L∞(X,ξ)⊂M⊂L∞(X,ξ)⋊Γ$L^{\infty }(X,\xi)\subset \mathcal {M}\subset L^{\infty }(X,\xi)\rtimes \Gamma$ in terms of the normal subgroups of Γ$\Gamma$. [ABSTRACT FROM AUTHOR]

In 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions u(·,t)$ \bm{u}(\cdot ,t)$ to the incompressible Navier–Stokes in Rn$ \mathbb {R}^{n}$: if the associated Stokes flows had their L2$L^{2}$ norms bounded by O(1+t)−α$ O(1 + t)^{-\;\!\alpha }$ for some 0<α⩽(n+2)/4$ 0 < \alpha \leqslant (n+2)/4$, then the same would be true of ∥u(·,t)∥L2(Rn)$ \Vert \hspace{0.56917pt} \bm{u}(\cdot ,t) \hspace{0.56917pt} \Vert _{L^{2}(\mathbb {R}^{n})}$. The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form (1+t)−α≲∥u(·,t)∥L2(Rn)≲(1+t)−α$(1+t)^{-\alpha }\lesssim \Vert \bm{u}(\cdot ,t)\Vert _{L^2(\mathbb {R}^n)} \lesssim (1+t)^{-\alpha }$. [ABSTRACT FROM AUTHOR]

Although the educational potential of social media has been widely acknowledged, it remains unclear whether social media use might present opportunities for the development of math identity. This study drew on the socialization framework of math identity to examine the relationships between social media use for math learning, bicultural identity integration, ethnic‐math beliefs and math identity among ethnic minority high school students in China. Structural equation modelling results showed that (a) social media use was positively related to math identity; (b) high bicultural identity integration was positively associated with math identity, while biased ethnic‐math beliefs were negatively related to math identity and (c) social media use suppressed the socialization process of math identity, weakening the maladaptive roles of low bicultural identity integration and biased ethnic‐math beliefs in shaping math identity. Our findings highlight the beneficial role of social media use in student math learning. Theoretical and practical implications are discussed. Practitioner notesWhat is already known about this topicSocial media use for learning is associated with students' academic motivation, engagement and academic achievement.Social media use may have affordances for identity development, especially for ethnic minorities.The formation of ethnic minority students' math identity is intertwined with their math and ethnic socialization.What this paper addsSocial media use for math learning was directly and positively related to ethnic minorities' math identity.This study quantitatively confirmed the associations of bicultural identity integration and ethnic‐math beliefs with math identity.Social media use suppressed the relations of bicultural identity integration and ethnic‐math beliefs with math identity.Implications for practice and/or policyPromoting social media use for math learning benefits ethnic minority students' math identity.Social media use for math learning could alleviate the maladaptive role of low bicultural identity integration and biased ethnic‐math beliefs in math identity development.Ethnic minority students' math identity socialization processes merit attention. [ABSTRACT FROM AUTHOR]

In this paper, we study the non‐Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky (Selecta Math. (N.S.) 4 (1998) 279–320). [ABSTRACT FROM AUTHOR]

Given a collection of hypergraphs H=(H1,...,Hm)${{\bf H}}=(H_1, \ldots , H_m)$ with the same vertex set, an m$m$‐edge graph F⊂∪i∈[m]Hi$F\subset \cup _{i\in [m]}H_i$ is a transversal if there is a bijection ϕ:E(F)→[m]$\phi :E(F)\rightarrow [m]$ such that e∈E(Hϕ(e))$e\in E(H_{\phi (e)})$ for each e∈E(F)$e\in E(F)$. How large does the minimum degree of each Hi$H_i$ need to be so that H${{\bf H}}$ necessarily contains a copy of F$F$ that is a transversal? Each Hi$H_i$ in the collection could be the same hypergraph, hence the minimum degree of each Hi$H_i$ needs to be large enough to ensure that F⊆Hi$F\subseteq H_i$. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac‐type results for (powers of) Hamilton cycles. For example, we derive that any collection of rn$rn$ graphs on an n$n$‐vertex set, each with minimum degree at least (r/(r+1)+o(1))n$(r/(r+1)+o(1))n$, contains a transversal copy of the r$r$th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture. [ABSTRACT FROM AUTHOR]

While there is increased attention to power, privilege, and access in mathematics education, conversations around race and disability are often left out of the conversation. Disability in mathematics continues to be studied with a lens that focuses on behavior, rather than attending to the situated and sociopolitical context in which teaching and learning takes place. This paper specifically calls on the importance of an intersectional analysis of ability and the need for explicit conversation on the interwoven nature of race and disability to uncover exclusionary practices of hyper‐labeling, hyper‐surveilling, and hyper‐punishing for those outside notions of normalcy. Drawing on sociopolitical perspectives informed by Disability Critical Race Theory (DisCrit) and DisCrit Classroom Ecology, I share a framework of DisCrit noticing to consider the interwoven nature of racism and ableism in mathematics classrooms, provide a context to develop preservice teacher DisCrit noticing during mathematics methods, and a case example to highlight implications and transformative possibilities for (re)organizing classrooms through the DisCrit noticing framework. [ABSTRACT FROM AUTHOR]

Large‐scale standardized tests are regularly used to measure student achievement overall and for student subgroups. These uses assume tests provide comparable measures of outcomes across student subgroups, but prior research suggests score comparisons across gender groups may be complicated by the type of test items used. This paper presents evidence that among nationally representative samples of 15‐year‐olds in the United States participating in the 2009, 2012, and 2015 PISA math and reading tests, there are consistent item format by gender differences. On average, male students answer multiple‐choice items correctly relatively more often and female students answer constructed‐response items correctly relatively more often. These patterns were consistent across 34 additional participating PISA jurisdictions, although the size of the format differences varied and were larger on average in reading than math. The average magnitude of the format differences is not large enough to be flagged in routine differential item functioning analyses intended to detect test bias but is large enough to raise questions about the validity of inferences based on comparisons of scores across gender groups. Researchers and other test users should account for test item format, particularly when comparing scores across gender groups. [ABSTRACT FROM AUTHOR]

Background: Measurement of students' self‐regulation skills is an active topic in education research, as effective assessment helps devising support interventions to foster academic achievement. Measures based on event tracing usually require large amounts of data (e.g., MOOCs and large courses), while aptitude measures are often qualitative and need careful interpretation. Precise and interpretable evaluation of self‐regulation skills in a normal K‐12 classroom thus poses a challenge. Objectives: The present study proposes and explores a learning analytics method of combining aptitude and event measures to evaluate student's self‐regulation skills. Methods: An explorative learning analytics study was conducted in a junior high school mathematics class (N=20 students), using a three‐lesson intervention with digital learning materials. Students first assessed their self‐regulation skills with a self‐report questionnaire, after which trace logs and observations of student behaviour were collected. Learning sessions were extracted from trace logs, clustered, and linked to learning strategies. Students were clustered by the self‐report results and learning behaviour profiles. Session clusters, student behaviour clusters and assignment grades were also tested for association. Results and Conclusions: The detected session and student behaviour types were linked to learning tactics and strategies found in prior studies. Additionally, association was found between self‐reported self‐regulation skills and the student behaviour obtained from trace logs. Implications: The results demonstrate the feasibility of concurrently using aptitude and event measures on a classroom scale, providing teachers with a tool to evaluate and support self‐regulated learning. Combined with further measures like predictive learning analytics, teachers can obtain an early and highly interpretable picture of at‐risk students in their classes. Lay Description: What is already known about this topic: Active monitoring of self‐regulated learning skills is primarily standard in large courses and MOOCs.Aptitude measures for self‐regulated learning are commonplace but considered subjective.Event measures for self‐regulated learning are precise but require additional interpretation to link behaviour to learning.Prior studies demonstrated approaches to analysing student behaviour from trace logs. What this paper adds: Both measure types can be used to monitor student behaviour in a classroom actively over a low number of sessions.Event and aptitude measures complement each other by exposing discrepancies between students' behaviours and actions.Measuring self‐regulated learning actively in class can yield valuable information to select appropriate interventions. Implications for practice: Integrating aptitude and event measures in digital learning materials gives teachers quickly actionable information about student behaviour.Approaches to integrating learning analytics for self‐regulated learning in digital learning materials should be further developed.Data‐informed choice of self‐regulated learning interventions should be explored further. [ABSTRACT FROM AUTHOR]

High‐dimensional metastable molecular dynamics (MD) can often be characterised by a few features of the system, that is, collective variables (CVs). Thanks to the rapid advance in the area of machine learning and deep learning, various deep learning‐based CV identification techniques have been developed in recent years, allowing accurate modelling and efficient simulation of complex molecular systems. In this paper, we look at two different categories of deep learning‐based approaches for finding CVs, either by computing leading eigenfunctions of transfer operator associated to the underlying dynamics, or by learning an autoencoder via minimisation of reconstruction error. We present a concise overview of the mathematics behind these two approaches and conduct a comparative numerical study of these two approaches on illustrative examples. [ABSTRACT FROM AUTHOR]

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow‐up time are established. Finally, for a simplified version of the model, some blow‐up criteria are proved. More precisely, we analyze a zero‐flux chemotaxis system essentially described as ⋄ut=∇·((u+1)m1−1∇u−χu(u+1)m2−1∇v+ξu(u+1)m3−1∇w)+λu−μukinΩ×(0,Tmax),0=Δv−1|Ω|∫Ωuα+uα=Δw−1|Ω|∫Ωuβ+uβinΩ×(0,Tmax).$$\begin{equation} {\begin{cases} u_t= \nabla \cdot ((u+1)^{m_1-1}\nabla u -\chi u(u+1)^{m_2-1}\nabla v & {}\\ \qquad +\; \xi u(u+1)^{m_3-1}\nabla w) +\lambda u -\mu u^k & \text{ in } \Omega \times (0,T_{max}),\\ 0= \Delta v -\frac{1}{\vert {\Omega }\vert }\int _\Omega u^\alpha + u^\alpha = \Delta w - \frac{1}{\vert {\Omega }\vert }\int _\Omega u^\beta + u^\beta & \text{ in } \Omega \times (0,T_{max}). \end{cases}} \end{equation}$$The problem is formulated in a bounded and smooth domain Ω of Rn$\mathbb {R}^n$, with n≥1$n\ge 1$, for some m1,m2,m3∈R$m_1,m_2,m_3\in \mathbb {R}$, χ,ξ,α,β,λ,μ>0$\chi , \xi , \alpha ,\beta , \lambda ,\mu >0$, k>1$k >1$, and with Tmax∈(0,∞]$T_{max}\in (0,\infty ]$. A sufficiently regular initial data u0≥0$u_0\ge 0$ is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on m2+α$m_2+\alpha$, (i)we prove that any given solution to (⋄$\Diamond$), blowing up at some finite time Tmax$T_{max}$ becomes also unbounded in Lp(Ω)$L^{\mathfrak {p}}(\Omega)$‐norm, for all p>n2(m2−m1+α)${\mathfrak {p}}>\frac{n}{2}(m_2-m_1+\alpha)$;(ii)we give lower bounds T (depending on ∫Ωu0p¯$\int _\Omega u_0^{\bar{p}}$) of Tmax$T_{max}$ for the aforementioned solutions in some Lp¯(Ω)$L^{\bar{p}}(\Omega)$‐norm, being p¯=p¯(n,m1,m2,m3,α,β)≥p$\bar{p}=\bar{p}(n,m_1,m_2,m_3,\alpha ,\beta)\ge \mathfrak {p}$;(iii)whenever m2=m3$m_2=m_3$, we establish sufficient conditions on the parameters ensuring that for some u0 solutions to (⋄$\Diamond$) effectively are unbounded at some finite time. Within the context of blow‐up phenomena connected to problem (⋄$\Diamond$), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic. [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]

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]

In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has 'not yet made the role of miscalculating clear' and that 'the role of the proposition: "I must have miscalculated"...is really the key to an understanding of the "foundations" of mathematics.' In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein's understanding of a 'foundation' for mathematics. Then, by showing how the proposition 'I must have miscalculated' differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics. [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]

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. [ABSTRACT FROM AUTHOR]

Educational applications (apps) are ubiquitous within children's learning environments and emerging evidence has demonstrated their efficacy. However, it remains unclear what the active ingredients (ie, mechanisms), or combination of ingredients, of successful maths apps are. The current study developed a new, open‐access, three‐step framework for assessing the educational value of maths apps, comprised of type of app, mathematical content and app design features. When applied to a selection of available maths apps previously evaluated with children in the first 3 years of school (the final sample included 23 apps), results showed that practice‐based apps were the most common app type tested (n = 15). Basic number skills, such as number representation and relationships, were the most common area of mathematics targeted by apps (n = 21). A follow‐up qualitative comparative analysis showed observed learning outcomes with maths apps were enhanced when apps combined the following: a scaffolded and personalised learning journey (programmatic levelling) and explanations of why answers were right or wrong (explanatory feedback), as well as praise, such as 'Great job!' (motivational feedback). This novel evidence stresses the significance of feedback and levelling design features that teaching practitioners and other stakeholders should consider when deciding which apps to use with young children. Directions for future research are discussed. Practitioner notesWhat is already known about this topic Educational apps have been shown to support maths attainment in the first 3 years of school.Several existing frameworks have attempted to assess the educational value of some of these maths apps.Emerging experimental evidence also demonstrates the benefits of specific app design features, including feedback and levelling.What this paper adds Practice‐based maths apps are the most common type of app previously evaluated with young children.These evaluated maths apps have mostly focused on basic number skills.The combination of explanatory and motivational feedback, with programmatic levelling (either dynamic or static), was a necessary condition for enhancing learning outcomes with maths apps.Implications for practice and policy The inclusion of feedback and levelling in maths apps should be considered by app developers when designing apps, and by educational practitioners and parents when deciding which apps to use with their children.Further consideration is also needed for the development of educational apps that include a broad range of maths skills. [ABSTRACT FROM AUTHOR]

This is an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author's memoir L. Positselski, Memoirs of the American Math. Society 212 (2011), no. 996, vi+133. The paper is intended to serve as a pedagogical introduction and a summary of the covariant duality between DG‐algebras and curved DG‐coalgebras, as well as the triality between DG‐modules, CDG‐comodules, and CDG‐contramodules. Some personal reminiscences are included as a part of historical discussion. [ABSTRACT FROM AUTHOR]

Background: Prior work has shown that middle school students struggle with algebra and that game‐based educational technologies, such as DragonBox and From Here to There!, are effective at improving students' algebraic performance. However, it remains unclear which aspects of algebraic knowledge shift as a result of playing these games and what game design features support algebraic learning. Objective: Using the data from a randomized controlled trial conducted in the 2020–2021 academic year, we (a) examined students' relative performance on assessment items measuring conceptual knowledge, procedural knowledge and procedural flexibility in algebraic equation solving, and (b) identified changes in these aspects of algebraic knowledge after playing DragonBox or From Here to There!. Methods: Eight hundred eighty‐seven seventh‐graders were randomly assigned to playing DragonBox or From Here to There! for nine 30‐min sessions throughout the school year. Students also completed a pretest, midtest and posttest measuring their algebraic knowledge. Results and Conclusion: First, prior to the intervention, students scored the highest on procedural knowledge and lowest on conceptual knowledge. Second, students significantly improved and maintained learning gains on conceptual knowledge throughout the intervention. Their performance on procedural knowledge and procedural flexibility items increased at midtest but decreased at posttest. Third, the pattern of results was consistent for students in the DragonBox and From Here to There! conditions, suggesting that both games may support students' conceptual understanding of algebra. The findings have implications for research and practice on supporting algebraic learning through game‐based technologies during and beyond educational disruptions. Lay Description: What is already known about this topic: Middle school students struggle with aspects of algebra.DragonBox and From Here to There! are two game‐based technologies designed to support students' algebraic learning.Both games implement point rewards and dynamic interactions; they differ in the uses of algebraic notation and problem structures. What this paper adds: Both games improved seventh‐graders' conceptual knowledge of algebraic equivalence.Students also improved on procedural knowledge and procedural flexibility, but the gains did not last to the end of the intervention. Implications for practice and/or policy: The games and their shared features may provide an engaging way to support students' conceptual understanding of algebra. [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]

This paper studies the nonlocal fractional analog of the famous paper of Brezis and Nirenberg [Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477]. Namely, we focus on the following model: P−Δsu−λu=α|u|p−2u+β|u|2s∗−2uinΩ,u=0inRN∖Ω,$$\begin{align*}\hskip5pc {\left(\mathcal{P}\right)} {\left\{ \def\eqcellsep{&}\begin{array}{l} {\left(-\Delta \right)}^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s-2}u \quad \mbox{in}\quad \Omega ,\\ u=0\quad \mbox{in}\quad \mathbb {R}^N\setminus \Omega , \end{array} \right.}\hskip-5pc \end{align*}$$where (−Δ)s$(-\Delta)^s$ is the fractional Laplace operator, s∈(0,1)$s \in (0,1)$, with N>2s$N > 2s$, 20,λ,α∈R$\beta >0,\, \lambda , \alpha \in \mathbb {R}$, and establish the existence of nontrivial solutions and sign‐changing solutions for the problem (P)$(\mathcal{P})$. [ABSTRACT FROM AUTHOR]

This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics. [ABSTRACT FROM AUTHOR]

In this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative Lp$L_p$‐spaces for a fixed 1

The sequence spaces ℓp(Cn)$$ {\ell}_p\left({C}&amp;#x0005E;n\right) $$, c0(Cn)$$ {c}_0\left({C}&amp;#x0005E;n\right) $$ and c(Cn)$$ c\left({C}&amp;#x0005E;n\right) $$ are introduced by Roopaei and Başar [8] as the domain of the Cesàro matrix Cn$$ {C}&amp;#x0005E;n $$ of order n$$ n $$ in the spaces ℓp$$ {\ell}_p $$, c0$$ {c}_0 $$ and c$$ c $$ of absolutely p$$ p $$‐summable, null and convergent sequences, respectively, where 0

Background: Although research on mathematics learning programs has taken off in recent years, little is known about how different person characteristics are related to practice behaviour with such programs. When implementing a mathematics learning program in the classroom, it might be important to know whether students with specific characteristics need more encouragement to make use of this learning opportunity. Objectives: Using a supply‐use model, we analysed the predictive power of students' cognitive characteristics (prior mathematics knowledge), personality traits (conscientiousness), motivational‐affective characteristics (mathematics self‐concept and mathematics anxiety), and family background characteristics (socioeconomic status and migration background) for practising with an adaptive arithmetic learning program. Methods: We used longitudinal data from 203 fifth graders from seven non‐academic‐track schools in Germany. Practice behaviour, measured by completed tasks in the learning program, was recorded weekly for every student over a period of 22 weeks. Results and Conclusions: The results of our multilevel analyses showed that mathematics anxious students practised less with the program. We did not find any relationship with the other characteristics. Takeaways: Our results suggest that mathematics anxious students need more encouragement when practising with a mathematics learning program; otherwise, they do not get the chance to benefit from the use. Lay Description: What is already known about this topic: The use of mathematics learning programmes in mathematics education has recently intensified.It is important that students practice with such learning programmes regularly over a longer period of time in order for them to achieve learning success.Students differ in their mathematics learning behaviour. What this paper adds: Little is known about how person characteristics are related to practice behaviour with mathematics learning programmes.Students may differ in their use of a mathematics learning programme, which is why cognitive characteristics, personality traits, motivational‐affective characteristics, and family background characteristics may affect students' practice behaviour. Implications for practice: Mathematics anxious students practiced less with a mathematics learning program, and need more encouragement to benefit equally from the implementation in school.Teachers should keep in mind that after the initial enthusiasm, practice with a programme may decrease over time, especially after school holidays. [ABSTRACT FROM AUTHOR]

Background: Readability metrics provide us with an objective and efficient way to assess the quality of educational texts. We can use the readability measures for finding assessment items that are difficult to read for a given grade level. Hard‐to‐read math word problems can put some students at a disadvantage if they are behind in their literacy learning. Despite their math abilities, these students can perform poorly on difficult‐to‐read word problems because of their poor reading skills. Less readable math tests can create equity issues for students who are relatively new to the language of assessment. Less readable test items can also affect the assessment's construct validity by partially measuring reading comprehension. Objectives: This study shows how large language models help us improve the readability of math assessment items. Methods: We analysed 250 test items from grades 3 to 5 of EngageNY, an open‐source curriculum. We used the GPT‐3 AI system to simplify the text of these math word problems. We used text prompts and the few‐shot learning method for the simplification task. Results and Conclusions: On average, GPT‐3 AI produced output passages that showed improvements in readability metrics, but the outputs had a large amount of noise and were often unrelated to the input. We used thresholds over text similarity metrics and changes in readability measures to filter out the noise. We found meaningful simplifications that can be given to item authors as suggestions for improvement. Takeaways: GPT‐3 AI is capable of simplifying hard‐to‐read math word problems. The model generates noisy simplifications using text prompts or few‐shot learning methods. The noise can be filtered using text similarity and readability measures. The meaningful simplifications AI produces are sound but not ready to be used as a direct replacement for the original items. To improve test quality, simplifications can be suggested to item authors at the time of digital question authoring. Lay Description: What is Known About the Subject: Difficult to read math assessment items cause measurement and equity issuesReadability of math word problems negatively correlated with outcomes What Our Paper Adds: GPT‐3 AI system is capable to simplify math word problemsPrompt‐based and few‐shot learning‐based approaches are able to create meaningful simplifications, but with a very low accuracy rateText similarity and readability measures can be used to filter out noisy outputs and discover interesting simplifications What are the Implications for Practitioners: GPT‐3 AI is capable of generating relevant math world problem simplificationsSimplified text can be used as suggestions to question authors during the digital item authoring process [ABSTRACT FROM AUTHOR]