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The curriculum resources used for teaching secondary mathematics vary considerably from school to school. Some schools base their teaching largely on a single published scheme, while others design their own schemes of learning, curating their resources from a range of (often free) online sources. Both approaches seem problematic from the perspective of experiencing the mathematics curriculum as a coherent story, and neither seems likely to take best advantage of the accumulated body of knowledge in the education research literature about effective didactics for mathematics. In this position paper, as we embark on the collaborative, research-informed design of a complete, fully-resourced, free-to-access mathematics curriculum for students aged 11–14, we use the conceptual framework of mathematics curriculum as a story to draw out five key curriculum design principles. A mathematics curriculum should harness and develop the skills and expertise of teachers; balance the teaching of fluency, reasoning and problem solving; give explicit attention to important errors and misconceptions; compare and contrast alternative methods; and engineer coherence through strategic use of consistent representations and contexts. We use these five principles to set out our vision for the next step in research-informed mathematics curriculum design. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate how small-group and whole-class discussions during whole-class teaching contribute to students' learning. The study includes 33 video-recorded small-group discussions in four mathematics lessons on enlarging and reducing two-dimensional geometric figures in Grade 8. The results show that different objects of learning were enacted in the different small-groups' discussions, and that in the majority of the small-group discussions the students did not have the possibility to learn what was intended, since critical aspects were not explored. The combination of group and whole-class discussions contributed to that critical aspects of the specific subject matter came to the fore and thereby were possible to discern. The results indicate that small-group discussions without subsequent whole-class discussions may have less effect on student learning. [ABSTRACT FROM AUTHOR]

Little is known about whether manipulatives can support children's inferences of the mathematical structure of word problems. The objective was to test the effects of using manipulatives during problem solving on students' understanding of the additive relationships in word problems. Third and fifth graders (N = 45) solved one-step addition and subtraction word problems that described either a mathematical action or relation. Children in each grade were randomly assigned to either a manipulatives or a paper-and-pencil condition. Problem structure understanding was assessed by the degree to which students' strategies and verbal justifications reflected the quantitative relationships described in the problem text. Performance of the fifth graders was not impacted by the use of manipulatives, but in the third grade, strategy performance with manipulatives was superior to performance without manipulatives on Relation problems. The degree to which the third graders' justifications were aligned with corresponding problem structure was positively impacted by manipulatives regardless of problem type. [ABSTRACT FROM AUTHOR]

Recently, Chen and Ma [A generalized shift-splitting preconditioner for complex symmetric linear systems, J. Comput. Appl. Math., 344 (2018), pp. 691–700] have presented the generalized shift-splitting (GSS) method to solve complex symmetric linear systems. In this paper, we establish a class of modified GSS (MGSS) method to extend the existing GSS method. We prove the convergence of the proposed method under suitable restrictions on the iteration parameters. Lastly, numerical experiments are carried out, and experimental results show that the new iteration method is feasible and effective. [ABSTRACT FROM AUTHOR]

Moothathu [Colloq. Math. 120 (2010), pp. 127–138] proved that multi-transitivity, weakly mixing and Δ-transitivity are equivalent for minimal homeomorphisms. In this paper, we extend this result to Abelian group actions. Moreover, we point out that a multi-transitive system of Abelian group action is Li–Yorke ε-chaotic for some ε > 0. [ABSTRACT FROM AUTHOR]

Reflective practice is a crucial element of professional growth that is gaining in popularity in teacher education, yet the ability to reflect is a skill that is neither natural nor easy to develop. This paper reports on an investigation that sought to establish if pre-service teachers at a South African university – in the process of learning to reflect on practice through analysing video-recorded lessons – reflected differently when reflecting on their own practice from the way they reflected on the practice of other teachers. Pre-service teachers were introduced to the Six Lens Framework as a tool to aid reflection on mathematics teaching. The authors compared two sets of reflections written by four pre-service teachers (PSTs) who participated in the study. One set was based on a video-recorded lesson of another teacher's practice while the other was based on a video-recording of their own practice. The study found that the PSTs' reflections in both cases were generally at the lowest level of the reflection spectrum. They mostly described classroom events rather than analysing the lessons to provide explanation suggestions, or reflectivity. We found, however, that all four PSTs reflected at substantially greater length on their own lessons than on those of other teachers. Furthermore, three of the four teachers' reflections on their own practice focused on mathematical rather than general aspects and provided more suggestions for improvement than their reflections on the practice of other teachers. [ABSTRACT FROM AUTHOR]

The paper aims to present the extent to which the application of the concrete mathematical and musical play contributes to the child's cognitive development. The research was conducted in two international schools in Belgrade, using the play titled 'Musical Monkeys'. The results of the research have shown that this kind of approach in working with children encourages attention and motivation, activity and engagement, while requiring and encouraging critical thinking in solving the set of mathematical or musical tasks and contributing to cooperation as well as finding a joint strategy in problem-solving. The obtained results confirm that the children established connection between the mathematical and musical segments, reaching a culmination in a moment of interest within the play, and rises inevitable question – how can mathematics and music be connected? [ABSTRACT FROM AUTHOR]

It is indisputable that one of the factors connected to improvements in the quality of education is the implementation of relevant curricular reform that is pertinent to the culture and needs of the given country. This study aimed to comprehensively investigate how the adaptation from an exogenous curriculum was attempted through analysing the three mathematics curricula for the years 1983, 2004, and 2015 that have been applied after independence, reports and policy papers issued by the Ministry of Education (MINED), and interviews with technical officials involved with the curriculum revision in 2015. The results show that Mozambique has been steadily and appropriately moving away from an exogenous curriculum in a cycle of 10 years through the accumulation of their own experiences. The series of observations that we attained could be relevant to other African nations from the viewpoint of transitioning away from an exogenous curriculum. [ABSTRACT FROM AUTHOR]

Summative and formative assessment are a core part of the teaching and learning process. A broad range of assessment methods is desirable to give a true measure of students' understanding and progress. However, research indicates that proctored, closed-book, summative assessments are traditionally favoured by university mathematics lecturers. In March 2020, assessment practices had to be abruptly reconsidered by the majority, midway through semester. An online survey of how mathematics lecturers adapted to emergency remote teaching was conducted in June 2020, with responses from 257 teaching staff in 29 countries. In this paper, the authors detail how assessment in mathematics was undertaken, remotely and virtually. We investigate how lecturers dealt with an enforced move away from traditional closed-book, highly weighted, proctored, in-person examinations, and how satisfied they were with their eventual assessments. We explore lecturers' perceptions of the impact the changes had on grade distribution during this unique teaching period. Finally, we report on whether their approaches to teaching the same modules in the future will change as a result of their experience of emergency remote teaching during the Covid-19 pandemic. [ABSTRACT FROM AUTHOR]

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G := ∥ x → ∥ 2 ∂ 0 + x → ∑ k = 1 3 x k ∂ k , according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory. 2019;13(6):2527–2539; Colombo F, González-Cervantes JO, Sabadini I. A non-constant coefficients differential operator associated to slice monogenic functions. Trans Am Math Soc. 2013;365:303–318]. This association allows to show a behavior of the theory of slice regular functions similar to the well known theories of the hypercomplex analysis. [ABSTRACT FROM AUTHOR]

Although we know that asking questions is an essential aspect of online tutoring, there is limited research on this topic. The aim of this paper was to identify commonly used direct question types and explore the effects of using these question types on conversation intensity, approach to tutoring, perceived satisfaction and perceived learning. The research setting was individual online synchronous tutoring in mathematics. The empirical data was based on 13,317 logged conversations and a questionnaire. The tutors used a mix of open, more student-centred questions, and closed, more teacher-centred questions. In contrast to previous research, this study provides a more positive account indicating that it is indeed possible to train tutors to focus on asking questions, rather than delivering content. Frequent use of many of the question types contributed to increased conversation intensity. However, there were few question types that were associated with statistically significant effects on perceived satisfaction or learning. There are no silver bullet question types that by themselves led to positive effects on perceived satisfaction and learning. The question types could be used by teachers and teacher students when reflecting on what types of questions they are asking, and what kind of questions they could be asking. [ABSTRACT FROM AUTHOR]

This paper investigates some consequences of a mathematical understanding of infinity and connectedness for a panentheistic conception of God. Given the existence of God and an understanding of the world in terms of the finite, countable infinity, or uncountable infinity I argue, (a) that a panentheistic conception of God is supported given a mathematical understanding of the infinite, (b) that by applying the notion of uncountable infinity and the mathematical concept of connectedness a panentheistic God can be seen as unifying irrespective of whether the world includes a finite, countably infinite or an uncountably infinite number of entities. [ABSTRACT FROM AUTHOR]

This paper uses a framework established by one of the author's in a previous study and refines it to develop a rubric for mathematics tutor training in third-level education. The paper further opens the discussion on the necessity of, and practical methods for, providing high quality, efficient tutor training at third-level. Models of mathematics teacher knowledge are unanimous in stating that solely possessing strong subject matter knowledge (SMK) is not sufficient for one to be deemed to be a good teacher. Despite the necessity of high quality tutoring at third-level, past research indicates that though tutors' SMK is usually of high quality, their ability to transform their knowledge so that students will understand the content, as well as their general teaching skills (e.g. board work, speech), often has room to improve. This paper discusses the development of the rubric and its use in a mathematics tutor training programme at third-level. The tutor training programme was found to lead to improvements in multiple elements of the tutors' teaching. [ABSTRACT FROM AUTHOR]

This paper explores the challenge of determining the mathematics curriculum for students with Down syndrome in our digital age where the tools of mathematics including electronic calculators and smartphone applications are readily available. Aspects once considered essential for 'functional mathematics' such as written calculation, using cash and telling time can be undertaken with easily accessible devices. Deeper concepts such as generalising through algebra, making decisions about money, and engaging in mathematical problem solving and applications are now a possibility. This paper explores the challenge through the theoretical perspectives of Educational Quality of Life and Numeracy Development, presenting five principles for mathematics curriculum planning. The possibility exists of using year level appropriate mathematics curriculum to build lifelong numeracy -- 'functional' mathematics for a new age. [ABSTRACT FROM AUTHOR]

Consider a homogeneous two state (failure-success or zero-one) Markov chain of first order. The paper deals with the position and the length of the shortest segment of the first n, , trials of the chain in which all runs of ones of length greater than or equal to a fixed number are concentrated. Accordingly, we define random variables denoting the starting/ending position of the first/last such runs in the chain as well as the implied distance between them. The paper provides exact closed form expressions for the probability mass function of these random variables given that the number of the considered runs in the chain is at least two. An application concerning DNA sequences is discussed. It is accompanied by numerics which exemplify further the theoretical results. [ABSTRACT FROM AUTHOR]

England’s Further Education (FE) sector is in permanent flux with policy interpretations and translations taking place at multiple levels within increasingly large and complex multi-site organisations. Devolved responsibility gives managers considerable influence in policy enactment processes which can lead to within-college tensions between vocational and mathematics teachers. This paper examines two within-college policies affecting students’ mathematics learning opportunities: (1) subject choice, and (2) examination entry levels. These policies have produced inequitable opportunities for students on different vocational study programmes. Given the strategic importance of improving mathematics education, this paper explains how multiple actors and structures interact in the enactment of policy in complex FE college settings. Such understandings are needed to inform better policy design and implementation that in turn can improve mathematics education in Further Education colleges in England. [ABSTRACT FROM AUTHOR]

This paper seeks to explore the nature of dialogue within calculus lecture rooms. This paper reports on part of a larger study, in which the participants were two lecturers lecturing to 198 third year pre-service teachers at a University in South Africa. This qualitative interpretive study was framed within the ambits of social constructivism and aspects of Mortimer and Scott's framework were used to analyse dialogue in the lecture room. Data for this paper were generated through video-recorded lecture observations. Subsequently, a qualitative data analysis of the generated data revealed that dialogue within the calculus lecture rooms under study may be in the form of explanations and questioning. Questioning included questions that assess students' prior knowledge, probing, follow-up and guiding questions. The findings of this study are of value to both lecturers and students within higher education with implications for teaching and learning mathematics within higher education are presented. [ABSTRACT FROM AUTHOR]

The relation between affect-driven feedback and engagement on a given task has been largely investigated. This relation can be used to make personalised instructional decisions and/or modify the affect content within the feedback. However, although it is generally assumed that providing encouraging feedback to students should help them adopt a state of flow, there are instances where those messages might result counterproductive. In this paper, we present a case study with 48 secondary school students using an Intelligent Tutoring System for arithmetical word problem solving. This system, which makes some common assumptions on how to relate affective state with performance, takes into account subjective (user's affective state) and objective information (previous problem performance) to decide the upcoming difficulty levels and the type of affective feedback to be delivered. Surprisingly, results revealed that feedback was more effective when no emotional content was included, and lead to the conclusion that purely instructional and concise help messages are more important than the emotional reinforcement contained therein. This finding shows that this is still an open issue. Different settings present different constraints generating related compounding factors that affect obtained results. This research confirms that new approaches are required to determine when, how and where affect-driven feedback is needed. Affect-driven feedback, engagement and their mutual relation have been largely investigated. Student's interactions combined with their emotional state can be used to make personalised instructional decisions and/or modify the affect content within the feedback, aiming to entice engagement on the task. However, although it is generally assumed that providing encouraging feedback to the students should help them adopt a state of flow, there are instances where those encouraging messages might result counterproductive. In this paper, we analyze these issues in terms of a case study with 48 secondary school students using an Intelligent Tutoring System for arithmetical word problem solving. This system, which makes some common assumptions on how to relate affective state with performance, takes into account subjective (user's affective state) and objective (previous problem performance) information to decide the difficulty level of the next exercise and the type of affective feedback to be delivered. Surprisingly, findings revealed that feedback was more effective when no emotional content was included in the messages, and lead to the conclusion that purely instructional and concise help messages are more important than the emotional reinforcement contained therein. This finding, which coincides with related work, shows that this is still an open issue. Different settings present different constraints and there are related compounding factors that affect obtained results, such as the message's contents and their target, how to measure the effect of the message on engagement through affective variables considering other issues involved, and to what extent engagement can be manipulated solely in terms of affective feedback. The contribution here is that this research confirms that new approaches are needed to determine when, how and where affect-driven feedback is needed. In particular, based on our previous experience in developing educational recommender systems, we suggest the combination of user-centred design methodologies with data mining methods to yield a more effective feedback. [ABSTRACT FROM AUTHOR]

The wavelet method based on the Meyer wavelet function and scaling function is a rather effective regularization method for solving some illposed problems. Recently, there are many works on this method limited to the a priori choice rule. The typical paper [H. Cheng and C.L. Fu, Wavelets and numerical pseudodifferential operator, Appl. Math. Model. 40 (2016), pp. 1776-1787] has systematically considered the a priori choice rule in the framework of the pseudodifferential operator (ψDO). In this paper, we will systematically consider the a posteriori choice rule for two kinds of ill-posed problems in the framework of the ψDO, and construct the convergence error estimates between the exact solution and its regularized approximation. [ABSTRACT FROM AUTHOR]

In this paper, we present two main results about best proximity points for multivalued mappings. First, taking into account Klim and Wardowski's approach in fixed-point theory, we obtain a new result for multivalued mappings which includes the main theorem of Kamran [Kamranm, T. (2009). Mizoguchi–Takahashi's type fixed point theorem. Comput. Math. Appl. 57(3):507–511]. Second, combining this approach with cyclic contraction, we give a general best proximity point result, and so we obtain many well-known fixed-point theorems such as Mizoguchi–Takahashi, Feng–Liu and Klim–Wardowski [Mizoguchi, N., Takahashi, W. (1989). Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1):177–188; Feng, Y., Liu, S. (2006). Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317(1):103–112; Klim, D., Wardowski, D. (2007). Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334(1):132–139]. Also, we support our results by providing some nontrivial, illustrative and comparative examples. [ABSTRACT FROM AUTHOR]

In this paper, a new conjugate gradient-based projection method for solving a system of nonlinear monotone equations is proposed. The method can be viewed as an extension of a family of conjugate gradient methods for unconstrained optimization by Li et al. [A new family of conjugate gradient methods for unconstrained optimization, J. Appl. Math. Comput. 58 (2018), pp. 219–234]. The proposed method is derivative-free which makes it suitable for large-scale nonlinear monotone equations. We show that the method satisfies the descent condition independent of line searches and that the method is globally convergent. Numerical results indicate that the proposed method is efficient. [ABSTRACT FROM AUTHOR]

The aim of this paper is to reveal the relationship between the topological entropy of a free semigroup action and that of its induced transformations. More precisely, we prove that the topological entropy of a free semigroup action (X , F) vanishes if and only if the topological entropy of its induced system (M (X) , F) is zero; if the topological entropy of (X , F) is positive, then that of its induced systems (M (X) , F) , (K (X) , F) are infinite. This in particular extends the results for classical topological entropy done by Bauer and Sigmund (1975 Monatsh. Math.79 81–92), Glasner and Weiss (1995 J. Amer. Math. Soc.8 665–686). [ABSTRACT FROM AUTHOR]

This paper reports on mathematics remediation efforts at a public 4-year institution. Specifically, it describes redesign efforts that led to the implementation of co-requisite mathematics remediation. Data on this program were collected and analyzed with respect to completion rates, length of time to earn college mathematics credit, and factors contributing to student success. Results include that, even with the co-requisite model, many students are still getting lost in the pipeline to earning college-level mathematics credit. Results also suggest high school GPA as being a significant factor between passing and non-passing students. [ABSTRACT FROM AUTHOR]

The percentage of college students receiving unproductive grades in introductory mathematics courses is a concern for post-secondary institutions across the country. Many factors can be targeted as potential explanations for this lack of success, yet none of these issues are more noteworthy than the fact that many students enter college mathematics courses with significant gaps in their fundamental mathematical background. In this paper, the authors discuss a way to implement an online remedial program to help students overcome their deficiencies following a "just-in-time teaching" model. Relevant data are presented supporting a positive outcome. [ABSTRACT FROM AUTHOR]

In this paper, we present two nonmonotone versions of adaptive cubic regularized (ARC) method for unconstrained optimization problems. The proposed methods are a combination of the ARC algorithm with the nonmonotone line search methods introduced by Zhang and Hager [A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), pp. 1043–1056] and Ahookhosh et al. [A nonmonotone trust-region line search method for large-scale unconstrained optimization, Appl. Math. Model. 36 (2012), pp. 478–487]. The global convergence analysis for these iterative algorithms is established under suitable conditions. Several numerical examples are given to illustrate the efficiency and robustness of the newly suggested methods. The obtained results show the satisfactory performance of the proposed algorithms when compared to the basic ARC algorithm. [ABSTRACT FROM AUTHOR]

Teaching introductory statistics is a service requirement for many mathematics faculty, and the audience is often anxious about the subject and not always excited about the material. Making class fun, relevant, and visually stimulating is key to keeping students engaged in the material and also in promoting student learning. Collecting data from the students, be it demographic information or experimental data, can achieve this, and immediately invites the students to be interested in the data and motivated to answer-related research questions. To facilitate and speed up this process, we developed the free data collection and analysis platform Classroom Stats which is designed to make class more fun and to improve student learning through straightforward visual analysis of the data. In this paper, we will discuss the importance of using student generated data in class and how to use Classroom Stats to make the process easy and efficient. [ABSTRACT FROM AUTHOR]

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: ], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis. [ABSTRACT FROM AUTHOR]

The problem of 'applicability' of mathematics to modern physical sciences has been labeled as an 'unreasonably effective' and unexplainable 'miracle' by prominent physicists such as Eugene Wigner and Paul Dirac. Philosophers of science from contending traditions have also contributed to the debates at large. While some have tried to trivialize the problem, others, in particular those with a phenomenological orientation, have attempted to eliminate this riddle, claiming that the two domains are 'identical'. This paper, while challenging some of these interpretations, proposes a solution from a dialectical materialist position. Drawing on Evald Ilyenkov's concept of the 'ideal' as the objectivized form of human activity in social nature, it is suggested that the two realms are necessarily interconnected as schemata of human activity which aim for a highly quantified conceptualization of reality. [ABSTRACT FROM AUTHOR]

In this paper we discuss the important Abel's summation formula, which is a very powerful tool for analysing series of real or complex numbers. We derive from it an integral test which may be useful in cases where the classical integral test may not be applied. We also discuss how this new integral test may be used when one is dealing with alternating series of real or complex numbers. The main goal of this note is to provide a supplementary material for courses dealing with infinite series of real or complex numbers. [ABSTRACT FROM AUTHOR]

This paper reports on themes that arose in an investigation of university lecturers' views on the teaching of linear algebra. This focus on themes was the initial part of a study concentrating on four areas: What is important to teach in a first course in linear algebra? Are there teaching methods which are particularly suited for such a course? Are there tools that should/should not be used; and do the answers to these questions vary according to the degree (Engineering or Mathematics) the students follow? Questionnaire data was coded using thematic analysis which generated 11 themes related to the four questions above. The Results section presents the themes. The Discussion section considers the themes as a whole; splits – dualities – in teaching linear algebra; students' challenges with abstraction; aspects of doing mathematics; and pedagogical issues. [ABSTRACT FROM AUTHOR]

This paper examines how seven teachers working in contexts of limited resources used the prescribed textbook for teaching, and the kinds of teacher–textbook relationships forged in the interactions. The study employs a sociocultural perspective to explore the processes by which teachers mobilise the affordances of the textbook to the teacher's practice, thereby advancing a particular way for studying and understanding better the teacher–textbook relationships in particular contexts. A methodological approach aggregating results for all teachers and looking for patterns of mobilisation across teachers allowed for the analysis of patterns of mobilisation regardless of the teacher. Findings point to generally tacit use of the textbook and a need for intervention on textbook use by teachers. The study makes recommendations for the production of educative guides as well as further research on the perceived role of the textbook in the teacher's practice. [ABSTRACT FROM AUTHOR]

Background: There are frequent calls in the literature for school science and mathematics departments to collaborate, largely in response to perceived overlaps between the two subjects in the context of Science, Technology, Engineering and Mathematics (STEM). Yet few studies explore how such collaborations might work. This paper is unusual both in its focus on mathematics/science collaborations which have not arisen from a specific short-term intervention and in its focus on the views of practising teachers rather than policy-makers or curriculum developers. Purpose: We ask how and why collaborations get started and explore how mathematics and science departments actually work together in secondary (high) schools in England. We ask what some of the affordances and challenges are in both initiating and sustaining collaborative practice. Design and methods: Six schools were identified and visited and semi-structured interviews carried out with the 15 teachers most closely involved in collaborating, to explore their perspectives and insights. Results: The findings show that collaborations are possible, though they are challenging to sustain, and they can be approached in a number of ways. Mathematics/science collaboration can be a key site of professional learning for teachers, particularly about the 'other' curriculum. Informal conversations across departments were highly valued but tended to be between those with a well-established pre-existing relationship. While physical structures can promote collaboration, it needs strong support from senior leadership teams to both begin and continue. Conclusion: Contrary to the commonly espoused view that there are many overlaps between mathematics and science in school, it can be a significant challenge for teachers to find them. Collaboration is neither straightforward to begin nor to sustain. Researchers and policy-makers should thus be cautious about recommending collaboration as straightforward for science and mathematics teachers to adopt. These conclusions provide a major challenge to simplistic advocacy of STEM in schools. [ABSTRACT FROM AUTHOR]

In this paper, we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in Bonnans and Ramírez [Perturbation analysis of second-order cone programming problems, Math Program. 2005;104(2–30):205–227] for conic optimization problems, and in Peña and Roshchina [A complementarity partition theorem for multifold conic systems, Math Program. 2013;142(1–2):579–589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular, that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions. [ABSTRACT FROM AUTHOR]

This paper sheds new light on Samuelson’s early methodology as presented in his Foundations of Economic Analysis (1947) by reflecting on the similarity between his mathematical economics and Edwin B. Wilson’s mathematics. Wilson was Samuelson’s professor of advanced mathematical and statistical economics; he was also a protégé of Josiah Willard Gibbs. Wilson defined mathematics as a language that consisted of three interconnected aspects: postulational, axiomatic, and operational. In his Foundations, in a Wilsonian style, Samuelson wrote in the opening page, ‘Mathematics is a Language’ and claimed that he offered operationally meaningful theorems. In this paper, it is argued that these maxims embodied Wilson’s approach, which framed Samuelson’s mathematical and statistical thinking around 1940 and which led him to present his work as being mathematically, theoretically, and empirically well founded. Wilson’s and Percy Bridgman’s operational methodologies are also compared and Wilson is presented as a mediator between Bridgman and Samuelson. [ABSTRACT FROM AUTHOR]

Adaptive designs for multi-armed clinical trials have become increasingly popular recently because of their potential to shorten development times and to increase patient response. However, developing response-adaptive designs that offer patient-benefit while ensuring the resulting trial provides a statistically rigorous and unbiased comparison of the different treatments included is highly challenging. In this paper, the theory of Multi-Armed Bandit Problems is used to define near optimal adaptive designs in the context of a clinical trial with a normally distributed endpoint with known variance. We report the operating characteristics (type I error, power, bias) and patient-benefit of these approaches and alternative designs using simulation studies based on an ongoing trial. These results are then compared to those recently published in the context of Bernoulli endpoints. Many limitations and advantages are similar in both cases but there are also important differences, specially with respect to type I error control. This paper proposes a simulation-based testing procedure to correct for the observed type I error inflation that bandit-based and adaptive rules can induce. [ABSTRACT FROM AUTHOR]

Purpose: The home-based rehabilitation of elderly patients improves their autonomy, independence and reintegration into society. Hence, a suitable environment plays an important role in rehabilitation, as do different assistance technologies. The majority of accidents at home involving elderly people occur in the bathroom. Therefore, the planning of the layout of facilities is important in this potentially dangerous area. This paper proposes an approach towards designing and optimizing the layout of facilities in the bathroom, based on logistical and nonlogistical relationships.Methodology: A fuzzy-based analytical hierarchical process (fuzzy-AHP) is then proposed for a comprehensive evaluation of the alternatives for this layout plan. This approach was applied to the home of a 71 years old female patient, who was experiencing home-based rehabilitation. After the initial designing and optimizing of the layout of the facilities in her bathroom, a plan could then be created for her particular needs.Findings: The results of this research could then enable the home-based rehabilitation of elderly patients to be more effective.Value: This paper develops a new approach to design and optimize the layout of facilities in bathroom for the elderly. Implications for Rehabilitation Develop a new approach to design and optimize the layout of facilities in bathroom.Provide a mathematical and more scientific approach to home layout design for home-based rehabilitation.Provide new opportunities for research, for both the therapist and the patient to analyse the home facility layout. [ABSTRACT FROM AUTHOR]

Ainsworth et al.’s paper “Sources of Bias in Outcome Assessment in Randomised Controlled Trials: A Case Study” examines alternative accounts for a large difference in effect size between 2 outcomes in the same intervention evaluation. It argues that the probable explanation relates to masking: Only one outcome measure was administered by those aware of participants’ treatment assignment. This paper shows this conclusion is not substantiated by the evidence: The original paper fails to exclude alternative explanations, and what it takes as positive evidence for the preferred explanation is actually negative. While accepting the importance of masking in randomised controlled trials, this paper concludes that the original question was based on a misconception about effect sizes: Seen correctly as a measure of whole study design, the question of effect size difference between different outcome measures does not need asking. [ABSTRACT FROM AUTHOR]

Professional bodies expect engineers to show competence in both mathematics and engineering topics such as mechanics, using their abilities in both of these to solve problems. Yet within engineering programmes there is a phenomenon known as ‘The Mathematics Problem’, with students not demonstrating understanding of the subject. This paper will suggest that students are constructing different concept images in engineering and mathematics, based on their perception of either the use or exchange-value for the topics. Using a mixed methods approach, the paper compares 10 different types of concept image constructed by students, which suggests that familiar procedural images are preferred in mathematics. In contrast strategic and conceptual images develop for mechanics throughout the years of the programme, implying that different forms of competence are being constructed by students between the two subjects. The paper argues that this difference is attributed to the perceived use-value of mechanics in the career of the engineer, compared to the exchange-value associated with mathematics. Questions are raised about the relevance of current definitions of competence given that some routine mathematical operations previously performed by engineers are now being replaced by technology, in the new world of work. [ABSTRACT FROM PUBLISHER]

This paper considers the policy ‘roles’ adopted by teachers enacting policy in a department. It draws on a longitudinal study of two secondary mathematics departments endeavouring to make deep change aligned with a demanding curriculum policy. The study validates aspects of an existing typology, demonstrates the existence of a variety of policy ‘critics’ and adds a category of ‘survivor’ teacher, showing the role can have considerable impact on enactment. The paper argues for an extension to a group construct of ‘policy role’, here at a department level, and shows that as teachers struggle to marry the constraints of the range of policies to which they are subject with the time and effort needed to maintain deeply espoused professional values, an adopted group role can serve either to support or constrain individual teacher efforts. [ABSTRACT FROM PUBLISHER]

A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G + uv) < k for any pair of non-adjacent vertices u and v of G. The problem that arises is that of characterizing k-i critical graphs. In this paper, we characterize connected 3-i-critical graphs with minimum vertex cutset of size 2. More specifically, we show that if G is a connected 3-i-critical graph with minimum vertex cutset S of size 2 and the number of components of G−S is exactly two, then G is isomorphic to a graph in one of nine classes of connected 3-i-critical graphs. The results in this paper together with results in [1] and [2] provide a complete characterization of connected 3-i-critical graphs with a minimum cutset of size at most 3. [ABSTRACT FROM AUTHOR]

This paper concerns a robust variable selection method in multiple linear regression: the robust S-nonnegative garrote variable selection method. In this paper the consistency of the method, both in terms of estimation and in terms of variable selection, is established. Moreover, the robustness properties of the method are further investigated by providing a lower bound for the breakdown point, and by deriving the influence function. The provided expressions nicely reveal the impact that the choice of an initial estimator has on the robustness properties of the variable selection method. Illustrative examples of influence functions for the S-nonnegative garrote as well as for the original (non-robust) nonnegative garrote variable selection method are provided. [ABSTRACT FROM AUTHOR]

There is currently a lack of instrumentation with sufficient technical quality focused on the implementation of classroom formative assessment at a grain-size appropriate for the provision of feedback to teachers and program developers. This paper details the development of a validity argument for the High-Impact Classroom Assessment Practices observation protocol (HI-CAP), and examines how to begin evaluating one inference in the interpretative argument. We present a conceptual framework for the HI-CAP and then articulate the interpretive argument. Finally we present evidence to evaluate one part of this argument, the scoring inference, using independent ratings of lessons from pairs of observers across 65 lessons in ninth-grade ELA and mathematics which suggest modest evidence for appropriateness, consistency, and preliminary evidence supporting the scoring model. We conclude with a discussion of the strengths and limitations of the current protocol and training procedures and implications for developing a validity argument for other similar protocols. [ABSTRACT FROM AUTHOR]

Mathematics is frequently considered as value-free, centring entirely on abstract reasoning. However, more often than not, the teaching of values in mathematics classrooms takes place implicitly through the actions of the teacher. The role values play in mathematics learning is also a growing area of interest and is now viewed by many as a contributing factor in the advancement of the quality of mathematics performance. This paper reports on part of a larger study aiming to establish the components that Grade 8–10 learners from a mathematics-and-science-focus school in Johannesburg value as important in the learning of mathematics. A quantitative survey was used to collect data from 249 participants. Principal component analysis was employed to analyse the data. The main findings revealed seven components that learners appeared to value in the learning of mathematics. Knowledge about what learners value in the learning of mathematics can inform teachers' pedagogical practices and the inculcation of values in mathematics teaching, which may, in turn, improve the quality of learning of mathematics. [ABSTRACT FROM AUTHOR]

Standards-based grading (SBG) is an assessment/grading system that attaches scores to the mastery level of specific learning objectives. This methodology is gaining momentum in the K-12 environment but has been largely overlooked at the post-secondary level. In this paper, I will discuss the implementation of this practice in a College Algebra course. I will also discuss the motivation in attempting an alternative assessment system, the development and implementation process and describe the resulting system. Comparisons will be made between a College Algebra course utilizing the traditional points-based system and the same course in which SBG was incorporated. These comparisons will include observed student behaviors, student perception, the administrative efforts required of the instructor, and student performance in subsequent courses. [ABSTRACT FROM AUTHOR]

A Mathematics Department Chair plays many roles within the department and within the university as a whole. Facilitating the development and teaching of mathematics services courses requires working with mathematics faculty as well as members of other departments on campus. In this paper, we discuss many issues related to the Department Chair's role in mathematics service courses such as determining course content needs, placing students in appropriate courses, scheduling classes, assessing service courses, and looking to the future of mathematics services courses. Central to all of these issues is ongoing communication with colleagues in and out of the department. [ABSTRACT FROM AUTHOR]

This paper presents findings from a two-year qualitative study examining integration of computer science (CS) and computational thinking (CT) into elementary mathematics instruction. Integrated units were developed by elementary teachers and CS/CT coaches with support from university faculty with expertise in CS/CT and elementary mathematics. CS/CT instruction primarily relied on the Scratch environment, although some lessons made use of Code.org materials. This research primarily relied on two theories of integration (i.e. Kiray, 2012. A new model for the integration of science and mathematics: The balance model. Energy Education Science and Technology Part B: Social and Educational Studies, 4(3), 1181–1196) that provided insight into the level of interconnection between the disciplines and the relative amount of instructional time spent within each discipline. Findings revealed that cross-grade CS/CT concepts included sequencing, looping, and conditional logic. Within each category: (a) concepts were taught with increasing complexity across the grades, (b) the mathematics was dominant and CS/CT was important but secondary, and (c) three types of lessons emerged: No integration, partial integration, and full integration. Lastly, lessons generally included a transition from less integrated to more integrated activities with an initial focus on discipline-specific conceptual understanding prior to integrated activities. [ABSTRACT FROM AUTHOR]

A key debate in computer science education is whether and how computational thinking (CT) is used within disciplines other than computer science. Broad definitions provide many avenues for developing integrated instruction, as practices within existing activities can simply be reframed in terms of CT. But such general use of the term CT may confuse its meaning and dilute its power as a tool for bringing CS to all. In this paper, we take the viewpoint that mathematical thinking and computational thinking share common practices, such as using repetition to accomplish tasks, but these shared ideas may develop differently in mathematics and computer science. We use document analysis to analyze the K-5 Common Core State Standards for Mathematics for the presence of these common practices and describe the specific ways they develop in mathematics. We compare these elements of mathematical thinking to related elements of CT to identify synergies and differences between CT and mathematical thinking. We argue that these synergies and differences have implications for the development of integrated instruction designed to bring CS to all. [ABSTRACT FROM AUTHOR]

Introductory and remedial courses in college mathematics often serve as a gateway to student success. However, these university courses have seen little innovation despite historically low success rates. We sought to address deficits of the traditional lecture model and enhance student learning in an introductory college algebra course through the implementation of a flipped classroom. In this paper, we fully describe the course design, and discuss the interplay between online and in-class activities. We then present data regarding student self-efficacy and success rates in subsequent quantitative courses from our work across five semesters of implementing and researching the course. [ABSTRACT FROM AUTHOR]

In this paper, the convergence for improved multi-point Chebyshev-like methods is considered. Compared to the results of Chebyshev method with second derivative in reference [M.A. Hernández, M.A. Salanova, J. Comput. Appl. Math. 126 (2000), pp. 131–143], the R-order is maximised and the Hölder continuity condition is also relaxed. Moreover, these improved methods do not need second derivatives, when the computation of second derivative is large, these improved methods will be more efficient than Chebyshev method. Under the relaxed condition, an existence-uniqueness theorem is proved. The R-order of convergence for these improved methods is also analysed. [ABSTRACT FROM AUTHOR]

Students' attainment in mathematics is associated with several personal and contextual factors, but little research has been carried out to explore the intersectionality between them. The current paper aims to fill this gap. Census assessment data collected in Italy at Grade 10 (on average, 15-year-old students) by the Italian national institute for the evaluation of the educational system were analysed via a multilevel regression model to account for data hierarchy (422,865 students in 24,279 classrooms, in 3,950 schools). Census data in educational research allowed exploration of the intersectionality between independent variables at different levels of the model's hierarchy. Results showed that reading skills mediate the relationship between students' attainment in mathematics and their gender, citizenship status, and socioeconomic status (SES). Implications for policy and practice are discussed. [ABSTRACT FROM AUTHOR]