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Very recently, P. Saha et al. [A global optimality result in probabilistic spaces using control function. Optimization. 2021;70:2387–2400] studied the existence of best proximity points for Banach type ϕ-proximal contractions in the framework of Menger probabilistic metric spaces. In this paper, we present a simple proof of their main existence result which is based on a fixed point theorem for ϕ-contractive self-mappings in complete Menger PM-spaces due to B.S. Choudhury and K. Das [A new contraction principle in Menger spaces. Acta Math Sin. 2008;24:13791386]. [ABSTRACT FROM AUTHOR]

This paper focuses on the Southern African Association for Research in Mathematics, Science and Technology Education's (SAARMSTE's) role as a platform supporting research dissemination and connecting researchers in early grades mathematics (EGM) in the Southern African region. A review of the Long Papers in SAARMSTE over the last 20 years supports the finding of the other review papers in this Special Issue: that there has been substantial growth of attention to EGM since 2013. However, two distinctions are marked when looking at conference papers rather than journal papers. Firstly, there is a particularly large expansion of work in the last 5 years, with a broadening base of participation in this work. Second, looking across all the formats of conference presentations indicates SAARMSTE's role in supporting and building EGM as a key focus of research attention, and bringing together regional and international groups with interests in this area. Given that conference proceedings usually offer a broader picture of emerging interests than journal papers, we reflect on the range of foci of attention within EGM in the SAARMSTE Proceedings, and trends within this. These trends also help us to point to areas that are likely to be of key interest in the next decade. [ABSTRACT FROM AUTHOR]

Curriculum-based measurement (CBM) tools are increasingly administered through technology-based modalities such as computers and tablets. Two studies were conducted to examine whether students perform similarly on paper-based and tablet-based math fact probes. Ten students completed 1-min addition or multiplication math probes using a single-case multielement design. Students completed the probes using traditional paper and pencil, a tablet using a stylus to write the answer, or a tablet application using a keyboard to type the answer. Visual analysis of time series graphs showed that the majority of students performed better on the paper-based probes than on either the keyboard or stylus probes. Nonoverlap of all pairs (NAP) effect size measures indicated medium to large differences between paper and stylus probes and paper and keyboard probes but weak effects between stylus and keyboard probes. Discussion focuses on the implications for educators and the use of different CBM modalities for interindividual and intraindividual comparisons. [ABSTRACT FROM AUTHOR]

The application of STEM (Science, Technology, Engineering, and Math) education in solving real-world problems is challenging. To tackle this challenge, a project-based learning approach that integrates robots with cost-effective self-made toys to problem-solving was adopted to assist rural primary school students to apply STEM skills. The purpose of this study was to evaluate the effects of the project-based learning approach which includes lecturing and hands-on activities on primary school students' STEM learning outcomes and attitudes. An experiment was conducted with 25 primary school students who voluntarily participated in a STEM summer-camp programme. Students were asked to assemble a paper house equipped with an IoT control module and LED light switches, and write and edit robot scripts to produce a robot-based storytelling narrative using the paper house they made as a context. The results show that the project-based learning approach was an effective approach for cultivating primary school students' STEM knowledge and skills as evidenced from the post-written test. The relationships between the students' STEM learning outcomes and attitudes were also confirmed by a clustering analysis. Students who had higher learning attitudes also achieved higher STEM learning outcomes; the findings are also supported by the feedback from the open-ended questionnaire items. This study suggests that integrating low-cost self-made toys and robots in project-based learning activities is an effective and practical approach to enhance primary school students' learning outcomes and learning attitudes in STEM education. [ABSTRACT FROM AUTHOR]

In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [ABSTRACT FROM AUTHOR]

The connections between critical pedagogy, living theory and participatory action research (PAR) are discussed to explore their combined strength for empowering students, positively impacting on their attitudes towards their mathematics learning and creating social change in their primary-secondary mathematics transitions. This transition is recognised as creating social inequalities which existing transition research has failed to resolve. The interpretation of critical pedagogy, living theory and PAR are described before a summary of their application in a small scale, two-year study in a British curriculum school in Muscat, Oman. Critical pedagogy combined with living theory and PAR provides the theoretical and methodological framework to empower the students epistemologically. This paper gives an example of how PAR with students was conducted within the framework of critical pedagogy theory and living theory methodology. This account provides a valuable reference for participatory action researchers. The paper concludes that the combination of critical pedagogy, living theory and PAR can empower students to create social change. [ABSTRACT FROM AUTHOR]

In this paper, we present the analysis of the students' achievements in learning calculus in a dynamic software environment during the Covid19 crisis. Two groups of students, the experimental and the control one, were monitored. Blended learning was applied to the students in the experimental group, with the help of Microsoft Teams and dynamic software GeoGebra, in autumn 2020. All students in the control group learned in the classroom without using GeoGebra in 2019. The comparison between these two groups of first-year students, regarding their calculus test results, is described in this paper. It is interesting that the results of the experimental group were significantly better than the results of students in the control group, despite the fact that the students from the experimental group learned during the Covid19 crisis. [ABSTRACT FROM AUTHOR]

Emerging digital technologies offer a transformative potential to redefine learning tasks and many examples of this potential are now available. The scaling of the innovative pedagogies emerging from the research into widespread and sustainable practice, however, remains problematic. This paper addresses the issue of scaling by using Design Based Research (DBR), also known as Educational Design Research, within teacher professional development to reposition teachers' thinking about the place of digital tools in their teaching. Using a project seeking to support the use of new digital technologies to develop children's spatial reasoning as a 'worked example', this paper highlights how the bringing together of the knowledge of educational research and knowledge of teaching practice in DBR can provide a catalyst for epistemic change. The paper will argue that DBR positions the knowledge and practice objects of both research and teaching as 'epistemic' or 'not yet known' objects and, therefore, the legitimate focus of experimentation and reflection. [ABSTRACT FROM AUTHOR]

This paper employs two scalar parameters to extend auxiliary function-based integral inequality into a new formulations, say a parametrised auxiliary function-based integral inequality. Using these formulations, the constituent signals are utilised more efficiently by exploiting the interaction between them to reduce conservatism. Numerical example for stability analysis of linear systems with time-varying delay shows the improved performance of the proposed new formulations in terms of maximum delay bounds and decision variables. [ABSTRACT FROM AUTHOR]

In this paper, we explore Norwegian teachers’ perspectives on inclusive teaching practices in the mathematics classroom, defined as practices which maximise every pupil’s potential regardless of prior attainment. As previous research suggests, both mathematics teachers’ perspectives in general and the conceptualisation of inclusion, inclusive education, and inclusive practices are culturally situated, varying significantly across countries and educational systems. We draw on data from a large project in Norway focusing on the use of grouping by attainment and its relation to policy and pedagogical practices around inclusion in mathematics. Participants were 13 primary and lower-secondary mathematics teachers from six schools in the Oslo area. Analysis of semi-structured interviews focusing on strategies for inclusion of all pupils in mathematics classrooms reveals the cultural particularities of mathematics teachers’ perspectives on inclusive practices, highlighting the value of similar investigations in other cultural contexts. [ABSTRACT FROM AUTHOR]

Teaching music often focuses on developing musical concepts, through comparisons and discriminations. This paper contributes to the discussion on concept development in music through recognition of common elements in a situation and a shift from a local to a general level, considering theoretical approaches to mathematical generalisation. It sets to explore preschool children's understanding of tempo and its use in new musical situations. The research involved a musical intervention in six preschool classrooms. Critical incidences from focus group discussions and non-participant observation recorded children's understanding of tempo from their musical actions, drawings, responses and reflection to teacher questions. The findings highlight that children could go beyond the specific content of the activities and focus on the basic characteristics of tempo, integrating them into their song creations. A teaching approach oriented to generalisation through concept development can assist teachers to elaborate musical content and realise students' level of understanding. [ABSTRACT FROM AUTHOR]

In this paper, we study the centralizer of a separating continuous flow without fixed points. We show that if M is a compact metric space and $ \phi _t:M\to M $ ϕ t : M → M is a separating flow without fixed points, then $ \phi _t $ ϕ t has a quasi-trivial centralizer, that is, if a continuous flow $ \psi _t $ ψ t commutes with $ \phi _t $ ϕ t , then there exists a continuous function $ A: M\to \mathbb {R} $ A : M → R which is invariant along the orbit of $ \phi _t $ ϕ t such that $ \psi _t(x)=\phi _{A(x)t}(x) $ ψ t (x) = ϕ A (x) t (x) holds for all $ x\in M $ x ∈ M. We also show that if M is a compact Riemannian manifold without boundary and $ \Phi _u $ Φ u is a homogenous separating $ C^1 $ C 1 $ \mathbb {R}^m $ R m -action on M, then $ \Phi _u $ Φ u has a quasi-trivial centralizer, that is, if $ \Psi _u $ Ψ u is a $ \mathbb {R}^{ m} $ R m -action on M commuting with $ \Phi _u $ Φ u , then there is a continuous map $ A: M\to \mathcal {M}_{m\times m}(\mathbb {R}) $ A : M → M m × m (R) which is invariant along orbit of $ \Phi _u $ Φ u such that $ \Psi _{u}(x)=\Phi _{A(x)u}(x) $ Ψ u (x) = Φ A (x) u (x) for all $ x\in M $ x ∈ M. These improve Theorem 1 of [M. Oka, Expansive flows and their centralizers, Nagoya Math. J. 64 (1976), pp. 1–15.] and Theorem 2 of [W. Bonomo, J. Rocha, and P. Varandas, The centralizer of Komuro-expansive flows and expansive $ \mathbb {R}^d $ R d -actions, Math. Z. 289(3–4) (2018), pp. 1059–1088.] respectively. [ABSTRACT FROM AUTHOR]

Numerous studies affirm the importance of students' attitudes for mathematical education. This study uses mathematical autobiographical essays to examine the mathematical experiences and attitudes of undergraduate students enrolled in mathematics classes at a Midwestern University in the United States. The essays of 41 students (20 men, 21 women) out of 170 enrolled in College Algebra, Precalculus, and Calculus were analyzed using the categories of (1) confidence in mathematics ability, (2) value of mathematics, and (3) liking of mathematics. This paper provides context and theoretical depth to student reports of their attitudes in these categories. Statements at the intersections of both confidence and liking, and value and liking provide additional insights. First, students who perceive mathematics as valuable are more likely to like it. Second, the vast majority of students talk about confidence and liking together, and with a positive correlation. Further, these students tend to talk about liking or not liking mathematics because of their perceived ability in mathematics. Our study suggests the importance of proactive teaching strategies to improve students' perceptions of the value, liking of, and their confidence in, mathematics, and also suggests self-efficacy as a potential theoretical basis for this work. [ABSTRACT FROM AUTHOR]

This paper describes the implementation of a STEM scholarship program which utilized a holistic approach to providing a multi-dimensional student support system. The program has been successful in encouraging and supporting women in Applied Mathematics and Computer Science by offering a diverse suite of extracurricular opportunities, actively engaging them in organized events, research projects, and participation in STEM communities, and helping them achieve higher GPAs and shorter times to graduation. The supported women also benefitted from close mentoring relationships with the faculty mentors. The program emphasized the development of empowering settings for women's engagement and achievement, which act to sustain and expand interest in mathematics and computing, and thereby help them to see themselves as future professionals in the field. [ABSTRACT FROM AUTHOR]

The Mathematical Sciences Department at Shenandoah University focused on increasing the participation of women in the mathematics program by fostering a welcoming learning community and promoting double majors. Data from the past decade is presented. During this time, the program saw an increase in women undergraduate mathematics majors from 11% to as high as 71%. This paper describes the initiatives taken to make these changes. [ABSTRACT FROM AUTHOR]

The University of Minnesota Talented Youth Mathematics Program (UMTYMP) is a selective, five-year accelerated mathematics program for students in grades 6–12. During the program, students take college mathematics courses on University of Minnesota campuses, starting with algebra and continuing through logic and proofs, linear algebra, and multivariable calculus. The majority of UMTYMP students come from two demographic groups: White and Asian/Asian American. In 2020, we surveyed UMTYMP students to understand the impact of model minority stereotypes (MMS) on Asian/Asian American students, particularly girls, who are labeled as "gifted" and/or "talented." In this paper, we reveal the preliminary results of this study, discuss their implications, and provide recommendations for addressing the MMS in mathematics programs. [ABSTRACT FROM AUTHOR]

We classify two of Bertrand Russell's theories of events within the point-free ontology. The first of such approaches was presented informally by Russell in 'The World of Physics and the World of Sense' (Lecture IV in Our Knowledge of the External World of 1914). Based on this theory, Russell sketched ways to construct instants as collections of events. This paper formalizes Russell's approach from 1914. We will also show that in such a reconstructed theory, we obtain all axioms of Russell's second theory from 1936 and all axioms of Thomason's theory of events from 1989. Russell's work certainly influenced the works of Stanisław Leśniewski, his student Alfred Tarski, and Czesław Lejewski – prominent members of the Lvov-Warsaw School (LWS). We see our work in the tradition of the research of Leśniewski and Tarski. Building on the technical tools developed in this environment and in the spirit of the traditional research of the LWS, we engage here, in particular, with two classic works by Russell on fundamental ontology. [ABSTRACT FROM AUTHOR]

The problem of precise characterisation of traditional forms of reasoning applied in mathematics was independently investigated and successfully resolved by Jaśkowski and Gentzen in 1934. However, there are traces of earlier interests in this field exhibited by the members of the Lvov-Warsaw School. We focus on the results obtained by Jaśkowski and Leśniewski. Jaśkowski provided the first formal system of natural deduction in 1926. Leśniewski also demonstrated in some of his papers how to construct proofs in accordance with intuitively correct principles. [ABSTRACT FROM AUTHOR]

Nowadays, typing is considered as one of the highly important aspects of accessibility that the visually impaired and blinds struggle with the most as existing virtual keyboards are complex and slow. This paper proposes a new text entry method named SwingBoard for visually impaired and blind smartphone users to solve their accessibility problem. It supports a–z, 0–9, 7 punctuations, 12 symbols, and eight keyboard functionalities that are arranged in 8 zones (specific range of angles), four segments, two modes, and different gestures. The proposed keyboard is suitable for the either single-handed or both-handed operation that tracks swipe angle and length to trigger any of the 66 key events. The key triggering process is based on only swiping the finger at different angles with different lengths. Typing speed of SwingBoard is increased by including some effective features such as the quick alphabet and number mode shifting, haptic feedback feature, talkback on swipe to learn the map quickly, and customizable swipe length feature. At the end of 150 one-minute tests, seven blind participants reached an average of 19.89 words per minute (WPM) with an 88% accuracy rate which is one of the fastest-ever recorded average typing speeds for the blind. Almost all users found SwingBoard effective, easy to learn and want to keep using it. SwingBoard is a handy virtual keyboard for visually impaired people with amazing typing speed and accuracy. Rehabilitation would be easier for a visually impaired/disabled person who can adapt the current technology-oriented world and use communication tools easily. As visually impaired and blind people faced a lot of problems in their regular life because of their inability to see things, coping with the evolving world's demands from a person to do a certain task using smart devices would be easier for them with the proposed virtual keyboard topology. As the number of visually impaired and blind users of smartphones rises, faster typing becomes a vital aspect of the smartphone experience. Deaf-blind communities are expanding but solutions for them are not increasing at the same pace because of the limitation of implication. Research on a virtual keyboard with the proposed eyes-free swipe-based typing operation and ears-free reliability on haptic feedback would enable others to create new solutions. [ABSTRACT FROM AUTHOR]

The majority of children in mainstream (no fee, poorly resourced, African language dominant) public schools in South Africa fall significantly behind curricular norms in mathematics by Grade 3; the early learning gap grows across time. The provision of quality instructional materials is a well-recognised component of effective strategies to improve early grade mathematics in low- and middle-income contexts. However, there has been little explication of what constitutes quality instructional materials, nor long-term design work to contribute theory to this instructional design challenge. Across 15 years, the author has been working in an education design hub with teacher educators and teachers in the rural Eastern Cape, with the goal of improving early grade mathematics. The gains in mathematics performance have been among the most significant reported in research literature. This paper presents the design principles emerging from the work. Engaging the theoretical layers discussed by diSessa and Cobb, the paper contributes to design theory by emphasising the relationship between explicit instructional assumptions and design principles, integrating four ontological categories (motive, pedagogy, language use and mathematical meaning-making). Since 2011, the Department of Basic Education has provided learner workbooks to support teaching and learning in foundation phase mathematics at system scale. It is hoped that this paper will contribute to the ongoing development of this vital national resource. The paper makes explicit instructional assumptions and design principles (available for critique, adaptation, and refinement), and provides a more nuanced framework through which to assess and improve materials across time. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

This article describes the outcomes of a faculty partnership between a math professor and a composition professor to improve the final paper in a research-focused senior capstone math course. Much attention has been paid in recent years to the importance of undergraduate research experiences and how to best create and support such endeavors. However, little space has been dedicated to helping math faculty design and scaffold writing assignments to support their students' development as scholars. We describe how our partnership took place, the changes we made to existing assignments and the creation of new ones, and an assessment of our efforts completed by an interdisciplinary group of faculty and students, which showed marked improvement in the areas of information literacy, critical thinking, and quality of writing. [ABSTRACT FROM AUTHOR]

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some questions instructors can ask themselves to design novel and engaging activities when constrained to teaching from a particular curriculum. [ABSTRACT FROM AUTHOR]

Among the problems identified at Technical and Vocational Education and Training (TVET) colleges, low achievement in mathematical subjects is the most prominent one. This paper documents a qualitative case study undertaken at TVET College in Gauteng with the purpose of exploring the National Certificate Vocational (NC(V)) Level 4 students' errors and associated misconceptions when answering optimisation questions in differential calculus. The participants were 60 students who were registered for a course in mathematics. Data were generated from the written student responses to two non-routine test items followed by interviews. Using the Newman error hierarchical model to analyse the data, it was discovered that students errors were conceptual and procedural as well as systematic and non-systematic. The literature also alluded to instructional approaches as some of the causes of students' misconceptions and the errors and that interventions should target students as well as the instructors. [ABSTRACT FROM AUTHOR]

If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

With constantly changing political landscapes affecting the ability of college students with minoritized identities of sexuality and/or gender (MIoSG; Vaccaro et al., 2015) to thrive on campus, higher educators need to understand student reported stressors to design more inclusive learning environments. Building from minority stress theory and using data from a grounded theory study with 56 collegiate STEM students with MIoSG, this paper documents stressors that students reported as contributing to diminished wellbeing. We used constant comparative grounded theory analysis to identify stressors shared by all participants, which included lack of belonging and invisibility in competitive STEM cultures as well as exclusionary interactions with STEM peers. Recommendations include the design and delivery of holistic education and support services on campus. [ABSTRACT FROM AUTHOR]

The paper addresses the potential relationships between shared authority in mathematics classrooms and students' mathematical reasoning. Even though tensions and challenges related to shared authority are explicated in the literature, there are few examples of how these issues play out in mathematics teaching. We investigate the case of a mathematics teacher attempting to share authority as well as applying several moves recognized as supporting meaningful student learning. Data has been collected in a fourth-grade Norwegian classroom and is analyzed by means of open coding, inspired by literature. We identify the moves used by the teacher, and we rank these moves along two dimensions: (1) their potential to support mathematical reasoning and (2) their potential for sharing authority. From this, we uncover how a teacher's work of orchestrating mathematical discussions involves moves in all four quadrants, and we discuss how the interplay of moves affects the authority structures and the collaborative reasoning in the classroom. [ABSTRACT FROM AUTHOR]

With this paper, we share an activity for the undergraduate mathematics classroom called The Harp Project that leverages the aesthetic nature of both the mathematical and the musical arts. This project was conceived as a STEAM/PBL project with the added feature that it was carried out in pieces by an entire class. Anecdotal evidence suggests that the uncommonly high level of students' interest and engagement in the experience and the communal feel of their participation can be attributed to this collective quality of the project's design. For this reason, we invite others to consider this curricular experience as it appears viable for supporting efforts to cultivate a broader population of students with an affinity for mathematics. [ABSTRACT FROM AUTHOR]

Let $ \beta (z) $ β (z) be the Dirichlet beta function. Series for $ \beta (4) $ β (4) are very rare in the historical development of mathematics. With the help of the operator method and a transformation formula for hypergeometric series, we prove a surprising conjectural series of Z.-W Sun for the mathematical constant $ \beta (4) $ β (4). Furthermore, we find five new series for the same constant in this paper. [ABSTRACT FROM AUTHOR]

Owing to its recognition as a 21st-century skill, computational thinking (CT) is currently being introduced into school curricula around the world. However, in-service teachers are largely unprepared for this implementation, which, in turn, makes teacher educators (TEds) important stakeholders in preparing prospective teachers to integrate CT into their classroom practices. In this regard, TEds are charged with a twofold responsibility: they must develop not only their own CT skills and digital competence but also a way of teaching these to the next generation of teachers who will facilitate future pupils’ learning. In this paper, we report on 17 TEds’ experience regarding the challenges and opportunities of integrating CT into Norway’s primary teacher education mathematics and science courses two years after CT’s introduction into Norwegian primary schools. A data-driven thematic analysis of semi-structured interviews was conducted. Our analysis suggests that it is challenging to integrate CT into existing courses. Such challenges, as well as opportunities, seem to apply at four levels: the systemic, teacher educator, student teacher, and subject levels. The results provide valuable insights for key stakeholders into the challenges and opportunities of integrating CT into teacher education, thus contributing to the body of research on professional digital competence. [ABSTRACT FROM AUTHOR]

What is the "right way" to define dimension? Mathematicians working in the early and middle 20th-century formalized three intuitive definitions of dimension that all turned out to be equivalent on separable metric spaces. But were these definitions the "right" ones? What would it mean to have the "right" definition of dimension? In this paper we attempt to inspire thought about these questions by introducing Abbott dimension, a geometrically intuitive definition of dimension based on Edwin Abbott's 1884 novella Flatland. We show that while Abbott dimension has intuitive appeal, it does not always agree with the classical definitions of dimension on separable metric spaces. [ABSTRACT FROM AUTHOR]

I revisit my 1993 paper on Putnam and mathematical realism focusing on the indispensability argument and how it has fared over the years. This argument starts from the claim that mathematics is an indispensable part of science and draws the conclusion, from holistic considerations about confirmation, that the ontology of science includes abstract objects as well as the physical entities science deals with. [ABSTRACT FROM AUTHOR]

Inclusive teaching practices can be characterized as recurrent ways how teachers work with their students' diverse abilities, but how exactly are they enacted in subject matter classrooms? The paper proposes a conceptual framework to unpack inclusive practices according to the student ability to which they refer, in five typical jobs for teachers: (a) identifying the demands for the ability, (b) differentiating learning goals, (c) compensating for low abilities, (d) enhancing abilities, and (e) addressing the abilities in joint learning. The proposed job-ability framework for inclusive teaching practices is substantiated in a video study of 25 mathematics lessons on percentages with the same curriculum material. In total, rather than 50, 133 different inclusive teaching practices were identified in 3862 sequences and structured into 20 cells. They address four abilities (from most often to least often): (1) selective attention/working memory, (2) mathematical pre-knowledge, (3) language proficiency, and (4) metacognitive regulation. The large variance of enacted practices identified within and between lessons calls for professional development that elicits, leverages, and extends the repertoire of practices. While the reported frequencies are specific to the chosen teaching unit, the job-ability framework can be transferred to other subject-matter classrooms and used in professional development programs. [ABSTRACT FROM AUTHOR]

By tracing roughly 200 years of the formation of American school subjects, this paper complicates some of the self-evidence for calls to adapt school subjects according to complex health concerns, more recently amplified by COVID-19. To do so, the paper diagrams a counter-memory of three key amalgams of health related to the makings of school mathematics and reading-as-literacy: balancing mind–body-spirit-matter-nation networks, scientizing a hygiene of instruction for 'ethnic' minds, and reconfiguring bio-psycho-social adjustment – all pursued as problems of duration, intensity, and distance from differently dynamic and/or racializing norms. Throughout, we draw attention to how both universalizing and ethno-specific orthodoxies and their proposed alternatives have produced school subjects as self-evident strategic sites for addressing health concerns that invite underappreciated dangers today. [ABSTRACT FROM AUTHOR]

This paper set out to contribute to mathematics education research by elaborating on a methodology developed during a study, trying to understand, view and follow shifts in prospective teachers' discursive patterns. The methodology aims to illustrate and describe how prospective teachers adapt to the context of teaching through a flexible process. This flexible process is then described in the result as a narrative. It is argued that the methodology can be used in relation to different theoretical directions, such as research about beliefs, knowledge, or identity. Another contribution is that the methodology presented gives insights into bridging the gap between different analytical levels, micro and macro. With a theoretical foundation in 'Cultural Worlds' [Holland, D., Skinner, D., Lachicotte, W., & Cain, C. (1998). Identity and agency in cultural worlds. Harvard University Press.] the Social Semiotic approach of Systemic Functional Linguistics, SFL [Halliday, M., & Hasan, R. (1989). Language, context, and text: Aspects of language in a social-semiotic perspective (2nd ed.). Oxford University Press] is used as a methodological tool. SFL offers a toolkit that allows the analysis of meaning at the clause level to uncover how and why a speaker produces a particular wording rather than any other in a specific social practice. The paper aims to illustrate and describe how to go beyond findings in the micro-analysis and then present the result as a narrative case. [ABSTRACT FROM AUTHOR]

This paper presents a series of basic computational problems that are mathematically and/or graphically appealing, and provides an idea of places one might go in trying to understand what is happening, integrating mathematics, computation, and graphics. The real point of this paper is to make a case, through those examples, for computation as an early and integral part in the undergraduate mathematics curriculum. By computation we mean the ability to rapidly and easily program exploratory calculations, and as well display the results with a versatile suite of graphics capabilities. The problems used here, and the philosophy behind them, derive from a MATLAB-based course in computation, programming, and mathematical applications that the author has taught for over 10 years. [ABSTRACT FROM AUTHOR]

Let $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function $ f(z)=z+\sum _{n=2}^\infty a_nz^n $ f (z) = z + ∑ n = 2 ∞ a n z n in the class $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) , then $$\begin{align*} & |a_{3}-\lambda a_{2}^2|\\ & \quad \leq (1-\alpha)\cos{\beta}\max\{1,|1-4(1-\alpha)(1-\lambda)\cos{\beta}\,{\rm e}^{{\rm i}\beta}|\},\quad \lambda\in \mathbb{C}. \end{align*} $$ | a 3 − λ a 2 2 | ≤ (1 − α) cos ⁡ β max { 1 , | 1 − 4 (1 − α) (1 − λ) cos ⁡ β e i β | } , λ ∈ C. The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in $ \mathbb {C}^{n} $ C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in $ \mathbb {C}^n $ C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in $ \mathbb {C}^n $ C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated. [ABSTRACT FROM AUTHOR]

The paper proposes EXPLODE, a new model of exploratory learning environment for teaching and learning neural networks. The EXPLODE model is about pedagogically instrumenting a software development environment to transform it into an exploratory learning environment for neural networks. Such an environment is particularly aimed for students who are skilled in programming and meets typical challenges in teaching and learning neural networks, related to the lack of prerequisite knowledge in mathematics. By providing such students with a familiar learning environment and allowing them to programmatically experiment with neural networks, the EXPLODE model aims at improving the students' learning outcomes and learning experience. The effectiveness of the model was evaluated in an experimental study with 77 final-year IT students. The results have shown that the students from the experimental group, that is, those who were exposed to the EXPLODE model, scored higher on the knowledge test. Furthermore, the perceived learning experience of the experimental group was better than that of the control group. The results have also suggested that the proposed model helps students better understand learning topics that require practical experience and facilitates a deeper understanding of the internal operations of neural networks. [ABSTRACT FROM AUTHOR]

The aim of this study was to examine if--and to what extent--a sensory modality's conceptual congruency influences learning in digital environments. Building on previous work in multimedia and embodied learning, the study used custom software to systematically vary the conceptual congruency of two sensory modalities (aural and bodilykinesthetic). One hundred seventy-nine (N = 179) elementary students with prior knowledge of addition used the software as an introduction to multiplication. Results found that the participants who experienced more conceptual congruency in the aural and bodily-kinesthetic modalities demonstrated greater understanding of the target concept as measured by paper-based tests. In addition, the findings revealed the two modalities worked in a combinatorial fashion, suggesting a more-isbetter impact on participant understanding. The study concludes with a discussion of the findings, along with an exploration of the theoretical and practical implications for designing sensory experiences that influence learning. [ABSTRACT FROM AUTHOR]

In this paper the author looks at the letters of two renowned women mathematicians and scientists of the Victorian period, Mary Somerville and Ada Lovelace, while also considering the imperceptibility of Sophie Germain, an important French mathematician and philosopher in their epistolary exchanges and philosophical writings. Drawing on the importance of mathematical correspondences and epistolary education in the creation, circulation and dissemination of knowledge, as well as in processes of formal and informal learning, the author argues that Lovelace’s and Somerville’s letters leave traces of a remarkable genealogical line of women’s mentorship and personal relations in the nineteenth century world of British mathematics in the backdrop of contradictory discourses around gender, mathematics, and science education. [ABSTRACT FROM AUTHOR]

This paper examines the conceptions of 'number sense' as promoted in pre-service primary mathematics education courses at 11 South African Higher Education Institutions through the texts used by academics or prescribed for students. While all the participating institutions agree that the development of primary school learners' 'number sense' is central to their mathematics methodology courses and that there is an overwhelming amount of research and literature on 'number sense' nationally and internationally, their conceptualisations of the nature of 'number sense' vary. Teacher educators, who develop pre-service teacher education courses, were asked to provide the texts, used to underpin the 11 universities' mathematics education modules in the Bachelor of Education (Foundation and Intermediate Phases) and Post Graduate Certificate in Education (Foundation and Intermediate Phases) programmes. These texts were analysed drawing on Whitacre et al.'s emphasis on three 'number sense constructs' identified as Innate Number Sense, Early Number Sense and Mature Number Sense. The results show that there is no common language of description for 'number sense' across the 11 universities. This research implies that there is a need to develop a consistent understanding of 'number sense' and how it is developed across institutions. [ABSTRACT FROM AUTHOR]

Teacher quality in primary schools in South Africa is a serious concern, and how best to prepare teachers in initial teacher education—with regard to mathematics—remains an open and vexing question. There is a confluence of challenges facing the mathematics education community in South Africa: poor conceptual understanding of mathematics amongst teachers; poor attainment in Primary Teacher Education mathematics tests at both first and fourth year levels; and few initial teacher education programmes having evidence of improving knowledge for teaching mathematics. In addition, few lecturers in initial teacher education make explicit the theoretical tenets informing their course design for mathematics. In response to the latter challenge, this paper makes explicit the design choices made in the Emergent Number Sense first year mathematics module in Bachelor of Education degrees, used across six universities in South Africa. It then theorises the choice to commence the B.Ed mathematics programmes with two pathways into number knowledge: (1) a counting pathway, making use of discrete objects or events which can be quantified by counting in ones; and (2) a measurement pathway, making use of continuous extents which can be quantified by introducing a unit of measurement. Each has a distinct metaphor, strategy for the reification of the 10 and supporting representation. This offers the lecturers using the course in their own design trials, and initial teacher educators beyond the Maths4Primary Teachers collective, a conceptual framework for this design choice, which can now be critiqued and further improved. [ABSTRACT FROM AUTHOR]

Research on mathematical equivalence, and to some extent, on number structure in early grade mathematics, has shown that most students at primary and secondary levels understand these concepts instrumentally rather than relationally. The study reported in this paper uses a design research approach to explore both mathematical equivalence and number structure in a primary pre-service teacher education programme at a university in South Africa and reports on an intervention strategy intended to culminate in developing design principles comprising guidelines for the effective teaching of number structure and mathematics equivalence in primary pre-service teacher education programmes. The mixed method study comprises 70 first-year primary pre-service teachers (PSTs) and a teacher educator. The PSTs wrote pre- and post-tests, and were interviewed following the pre-test. Skemp's notion of relational and instrumental understanding guided the design of the intervention and the analysis of the data. The results from the study indicates that PSTs' understanding of the notion of equivalence is similar to what has been reported for students in schools, and that there were some learning gains in the post-test results. Based on the findings from the study in general, some emergent design principles are proposed. [ABSTRACT FROM AUTHOR]

This paper examines the hitherto unknown scientific collaboration between the siblings Elli Heesch (1904–1993) and Heinrich Heesch (1906–1995). Heinrich Heesch, a well-known mathematician, was spearheading the early development of the computer-aided proof of the four-colour theorem. Much less is known about his sister Elli Heesch, a philosopher and logician. Together with her brother she investigated tiling problems and worked out a solution of Hilbert's 18th problem. In 1944, Elli and Heinrich Heesch wrote a joint treatise on the industrial application of the tessellation method, which was of great interest to the German war and armaments industry. The collaboration of the Heesch siblings illustrates individual, disciplinary, cultural, and political aspects of knowledge production. The common interplay of close family relations and socio-political conditions that we find here underlines the fact that women's contributions to solving mathematical problems often remained invisible. [ABSTRACT FROM AUTHOR]

In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors (1) give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers; (2) establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers. The instant proof simplifies discussions in the journal papers: College Math. J. 43 (2012), no. 2, 141-146; Amer. Math. Monthly 121 (2014), no. 3, 267-267; Amer. Math. Monthly 123 (2016), no. 4, 405-406; Elem. Math. 71 (2016), no. 3, 109-121; and Mathematics 5 (2017), no. 3, Article 40, 31 pages. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new nonlinear optimization model to solve semidefinite optimization problems (SDPs), providing some properties related to local optimal solutions. The proposed model is based on another nonlinear optimization model given by [S. Burer and R. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program. Ser. B 95 (2003), pp. 329–357], but it has several nice properties not seen in the existing one. Firstly, the decision variable of the proposed model is a triangular low-rank matrix. Secondly, the existence of a strict local optimum of the proposed model is guaranteed under some conditions, whereas the existing model has no strict local optimum. In other words, it is difficult to construct solution methods equipped with fast convergence using the existing model. We also present some numerical results, showing that the use of the proposed model allows to deliver highly accurate solutions. [ABSTRACT FROM AUTHOR]

Following the seminal work of Gloria Ladson-Billings, research on culturally responsive pedagogy (CRP) has, in recent years, expanded significantly. Ladson-Billings proposes three elements of CRP: academic achievement, cultural competence, and sociopolitical consciousness. Nevertheless, in mathematics education research on CRP, the sociopolitical consciousness element is often under-explored or even absent. This paper began as an investigation of how LadsonBillings’ three elements could be used to examine prospective and practicing teachers’ (PPTs') perspectives on CRP, prior to their participation in a professional development course on CRP in the mathematics classroom. Thirty-one participants from three separate offerings of the course responded in writing to a set of open questions about CRP (in general and in mathematics). Thematic data analysis pointed to the complexities of categorising the data based primarily on Ladson-Billings’ three elements. In addition, our analysis indicated five components underpinning participants’ responses: challenges; opportunities; fears; resistance; insights. We conclude with a revised conceptualisation of CRP for mathematics teacher education programmes aimed at supporting PPTs’ development of their CRP-related knowledge, skills, and dispositions. [ABSTRACT FROM AUTHOR]

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect based on finite samples and using TDA methods that ignore the probabilistic mechanism that generates the data. In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behaviour of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results. [ABSTRACT FROM AUTHOR]

Engaging high school students experiencing a mild to moderate Intellectual Disability (ID) in conversations about their math learning experiences can provide valuable insights for enhancing inclusive classroom teaching and learning. This paper reports on the pilot study of a broader research project that focuses on the voice of high school students experiencing ID in relation to their mathematics education. The study aims to analyse students’ experiences and perspectives on how they are taught mathematics based upon direct input from the students themselves.This research is significant because students experiencing ID have not been provided with the same opportunities to contribute to their learning as most other education stakeholders (parents, school leaders, teachers, and other student groups). This discrepancy is exemplified by the exclusion of students with ID from the descriptions of academic success in the NSW Curriculum Review (2020).The current research study addresses an important gap by acknowledging that students with ID have been underrepresented in the collection of student voices recognised in education. The study confirms that students with ID can actively participate in discussions about their learning experiences, offering unique and insightful recommendations for improving how mathematics is taught to them. [ABSTRACT FROM AUTHOR]

This paper considers a continuous-time bidimensional renewal risk model with subexponential claims. In this model, the claim size vectors and their inter-arrival times form a sequence of independent and identically distributed random vectors following some general dependence structures introduced by [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.]. We not only extend the results of [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.] under some weak conditions, but we also obtain an asymptotic estimate of the finite-time sum-ruin probability. Furthermore, when the distributions of claim sizes have nonzero lower Karamata indices, some explicit asymptotic formulas are established for the infinite-time ruin probabilities. [ABSTRACT FROM AUTHOR]