In questo lavoro vengono sviluppate alcune considerazioni sulla funzione dell’istruzione nel presente contesto storico–culturale, in particolare riguardo alla matematica. Tra i molti aspetti che rendono l’epoca attuale diversa dalle precedenti ve ne sono due che più di altri invocano la necessità di un cambiamento nelle impostazioni didattiche. Il primo è una crisi che, per durata ed estensione, appare non più come dinamica di aggiustamento all’interno di un modello ma come irreversibile degenerazione del modello stesso. Il secondo riguarda la quantità di (...) informazione a cui l’individuo ha accesso e la facilità e velocità di tale accesso. Cambiamenti così profondi provocano una domanda: i modi e i contenuti della didattica devono rimanere gli stessi o occorre ripensare l’istruzione progettando cambiamenti altrettanto irreversibili quanto lo sono quelli dell’economia e della società? Queste considerazioni ci portano a una riflessione sul rapporto tra educazione e società e, più in generale, tra cultura e società. I prodotti scientifici e culturali sono dei codici cristallizzati come idee platoniche o piuttosto acquistano valenza e significato a partire dal contesto in cui nascono? Questa domanda poi ne richiama altre ancora sul rapporto tra individuo, sistema scolastico e società. In che misura il ruolo della scuola di standardizzare la cultura è compatibile con quello di favorire la creatività e le possibilità espressive dell’individuo? E cosa significa oggi dare all’individuo gli strumenti per sviluppare al meglio le proprie possibilità? L’insegnante dovrebbe osservare, ipotizzare, sperimentare, esplorare opportunità senza pregiudizi culturali. Egli sa che molte idee che sulla carta sembrano vincenti potrebbero rivelarsi inefficaci, ma ciò non deve costituire un alibi. Se infatti si possono nutrire dei dubbi su alcune proposte di innovazione, su una cosa vi è certezza: la didattica tradizionale ha ormai poco senso ed è totalmente inadeguata rispetto alle sfide di questi e dei prossimi anni. (shrink)

There is more than one way to think. Most people are familiar with the systematic, rule-based thinking that one finds in a mathematical proof or a computer program. But such thinking does not produce breakthroughs in mathematics and science nor is it the kind of thinking that results in significant learning. Deep thinking is a different and more basic way of using the mind. It results in the discontinuous "aha!" experience, which is the essence of creativity. It is at (...) the heart of every paradigm shift or reframing of a problematic situation. The identification of deep thinking as the default state of the mind has the potential to reframe our current approach to technological change, education, and the nature of mathematics and science. For example, there is an unbridgeable gap between deep thinking and computer simulations of thinking. Many people suspect that such a gap exists, but find it difficult to make this intuition precise. This book identifies the way in which the authentic intelligence of deep thinking differs from the artificial intelligence of "big data" and "analytics." Deep thinking is the essential ingredient in every significant learning experience, which leads to a new way to think about education. It is also essential to the construction of conceptual systems that are at the heart of mathematics and science, and of the technologies that shape the modern world. Deep thinking can be found whenever one conceptual system morphs into another. The sources of this study include the cognitive development of numbers in children, neuropsychology, the study of creativity, and the historical development of mathematics and science. The approach is unusual and original. It comes out of the author's lengthy experience as a mathematician, teacher, and writer of books about mathematics and science, such as How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Spot: Science and the Crisis of Uncertainty. (shrink)

In response to the impending spread of COVID-19, universities worldwide abruptly stopped face-to-face teaching and switched to technology-mediated teaching. As a result, the use of technology in the learning processes of students of different disciplines became essential and the only way to teach, communicate and collaborate for months. In this crisis context, we conducted a longitudinal study in four German universities, in which we collected a total of 875 responses from students of information systems and music (...) and arts at four points in time during the spring–summer 2020 semester. Our study focused on (1) the students’ acceptance of technology-mediated learning, (2) any change in this acceptance during the semester and (3) the differences in acceptance between the two disciplines. We applied the Technology Acceptance Model and were able to validate it for the extreme situation of the COVID-19 pandemic. We extended the model with three new variables (time flexibility, learning flexibility and social isolation) that influenced the construct of perceived usefulness. Furthermore, we detected differences between the disciplines and over time. In this paper, we present and discuss our study’s results and derive short- and long-term implications for science and practice. (shrink)

Ever since the early decades of this century, there have emerged a number of competing schools of ecology that have attempted to weave the concepts underlying natural resource management and natural-historical traditions into a formal theoretical framework. It was widely believed that the discovery of the fundamental mechanisms underlying ecological phenomena would allow ecologists to articulate mathematically rigorous statements whose validity was not predicated on contingent factors. The formulation of such statements would elevate ecology to the standing of a rigorous (...) scientific discipline on a par with physics. However, there was no agreement as to the fundamental units of ecology. Systems ecologists sought to identify the fundamental organization that tied the physical and biological components of ecosystems into an irreducible unit: the ecosystem was their fundamental unit. Population ecologists sought, instead, to identify the biological mechanisms regulating the abundance and distribution of plant and animal species: to these ecologists, the individual organism was the fundamental unit of ecology, and the physical environment was nothing more than a stage upon which the play of individuals in perennial competition took place. As Joel Hagen has pointed out, the two schools were thus dividied by fundamentally different and irreconcilable assumptions about the nature of ecosystems.Notwithstanding these divisive efforts to elevate the image of ecology, the discipline remained in the shadows of American academia until the mid-1960s, when systems ecologists succeeded in projecting ecology onto the national scene. They did so by seeking closer involvement with practical problems: they argued before Congress that their approach to the theoretical problems of ecology was uniquely suited to the solution of the impending “environmental crisis.” With the establishment of the International Biological Program, they succeeded in attracting unprecedented levels of funding for systems ecology research. Theoretical population ecologists, on the other hand, found themselves consigned to the outer regions of this new institutional landscape. The systems ecologists' successful capture of the limelight and the purse brought the divisions between them and population ecologists into sharper relief — hence the hardening of the division of ecology observed by Hagen.45I have argued that the population biologist Richard Levins, prompted by these institutional developments, sought to challenge the social position of systems ecology, and to assert the intellectual priority of theoretical population ecology. He attempted to do so by articulating a nontrivial and rather carefully thought out classification of ecological models that led to the disqualification of systems analysis as a legitimate approach to the study of ecological phenomena. I have suggested that — ultimately —Levins's case against systems analysis in ecology rested on the view that an aspiration to realism and prediction was incompatible with an interest in theoretical issues, a concern that he equated with the search for generality. He sought to reinforce this argument by exploiting the fact that systems ecologists had staked their future on the provision of technical solutions to the problems of the “environmental crisis”: he associated systems ecologists' aspiration to realism and precision with a concern for practical issues, trading on the widely accepted view that practical imperatives are incompatible with the aims of scientific inquiry.46 These are plausible, but nonetheless questionable, claims which have now become an integral part of ecological knowledge. And finally, I hope to have shown how even the most abstract levels of scientific argument are shaped by political considerations, and how discussions of the conceptual development of modern ecology might benefit from a greater consideration of its historical and social dimensions.47. (shrink)

The entrepreneurial element is one of the aspects emphasized in the primary school mathematics education curriculum in Malaysia. However, previous studies have found that application of entrepreneurial elements in mathematics teaching is still lacking. This study was therefore conducted to identify the real challenges that mathematics teachers face in applying the entrepreneurial element in mathematics teaching. This study is qualitative case study which involved six primary school mathematics teachers. Semi-structured interviews, observation, document analysis (...) and field notes were used for the data collection process in this study. The data obtained was analyzed using the constant comparative analysis method to determine themes and sub-themes. The findings were that a teacher’s knowledge, teaching style, limited time of teaching, and attitudes, and in-service courses and training are among the identified challenges in applying entrepreneurial elements in mathematics teaching. This study expands knowledge and the literature on the challenges faced by school mathematics teachers in applying entrepreneurial elements in mathematics teaching. Various initiatives need to be taken to ensure that all the challenges can be overcome, and ultimately enable the entrepreneurial elements to be applied to students more widely. (shrink)

It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects and (...) theories really describes mathematical reality, as Plato quite clearly expressed his views on math, according to which mathematical concepts objectively exist as distinct entities between the world of ideas and the world of material things. In the context of foundations of mathematics, so called ‘Gödel’s Platonism‘ is of particular interest. It is shown in the article how Platonic objectification of mathematical concepts contributes to the development of modern mathematics by revealing philosophical understanding of the nature of abstraction. To substantiate his point of view, the author draws the works of contemporary experts in the field of philosophy of mathematics. (shrink)

A general introduction to the celebrated foundational crisis, discussing how the characteristic traits of modern mathematics (acceptance of the notion of an “arbitrary” function proposed by Dirichlet; wholehearted acceptance of infinite sets and the higher infinite; a preference “to put thoughts in the place of calculations” and to concentrate on “structures” characterized axiomatically; a reliance on “purely existential” methods of proof) provoked extensive polemics and alternative approaches. Going beyond exclusive concentration on the paradoxes, it also discusses the role (...) of the axiom of Choice, issues of predicativity, and goes on to present the ideas of intuitionism and Hilbert's program. The essay closes with Gödel's contributions and the aftermath. (shrink)

The COVID-19 coronavirus pandemic raises a host of challenging ethical questions at every level of society. However, some of the most acute questions relate to decision making in intensive care. The problem is that a small but significant proportion of patients develop severe viral pneumonitis and respiratory failure. It now seems likely that the number of critically ill patients will overwhelm the capacity of intensive care units (ICUs) within many health systems, including the National Health Service in the UK. The (...) experience of Northern Italy—a couple of weeks ahead of the UK, suggests that it will simply not be possible to provide mechanical ventilation to every patient who might need it. When the crunch comes, the unpalatable question facing clinicians is which patient to save. There are some obvious strategies to avoid or reduce the problem—through measures to increase intensive care capacity, and via society-level interventions to reduce the spread of the virus. These are vitally important, but unfortunately they are unlikely to prevent the problem of demand outstripping supply from occurring. What, then, should clinicians do? How should they allocate the scarce resource of intensive care—particularly over the coming weeks as the crisis escalates? There are different values at stake in triage decisions, but at a basic level the key values are those of benefit and …. (shrink)

ArgumentThis paper studies the evolution of mathematics teaching in France and Germany from 1900 to about 1980. These two countries were leading in the processes of international modernization. We investigate the similarities and differences during the various periods, which showed to constitute significant time units and this in a remarkably parallel manner for the two countries. We argue that the processes of reform concerning the teaching of this major school subject are not understandable from within mathematics (...) education or even within the school system. Rather, the evolution of the processes of reform prove to be intimately tied to changing conceptions of modernity according to respective social and cultural values and to changing epistemological conceptions of mathematics. It is particularly novel that we show the key impact of the changing social status of primary schooling for these modernization processes. (shrink)

The theory of objectification offers a perspective to conceptualize learning as a collective cultural-historical process and to transform classrooms into sites of communal life where students make the experience of an ethics of solidarity, plurality, and inclusivity.

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of (...) these conferred an only marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

J. N. (Jitendra Nath) Mohanty (1928–2023). -/- Professor J. N. Mohanty has characterized his life and philosophy as being both “inside” and “outside” East and West, i.e., inside and outside traditions of India and those of the West, living in both India and United States: geographically, culturally, and philosophically; while also traveling the world: Melbourne to Moscow. Most of his academic time was spent teaching at the University of Oklahoma, The New School Graduate Faculty, and finally Temple University. Yet (...) his preeminent work in Husserlian phenomenology developed alongside his eminent work in Indian philosophy: describing his interests as “a fusion of disparate horizons.” -/- J. N. Mohanty was born September 26, 1928, in Cuttack (Odisha, East India). After graduating from high school, he went on to study both Indian and Western philosophy in Calcutta, earning bachelor’s and master’s degrees. There he read Whitehead and Kant’s First Critique; although he wanted to include mathematics in his curriculum, he was led instead to include Indian philosophy and Sanskrit. On the shelves of his teacher, Ras Vihar Das, he came upon a copy of the English translation (Ideas I, 1931) of Edmund Husserl’s classic Ideen I (1913), which presented Husserl’s ground-breaking conception of phenomenology. In 1952–1954 he left India for the first time, reaching Göttingen to study mathematics and philosophy, which earned him a doctorate in mathematics and “philosophy of mathematical sciences” (in his own words). In Göttingen, Mohanty found the powerful mathematical world that Husserl himself had earlier interacted with, where several Husserl students had formed the Göttingen school of phenomenology. During these years Mohanty studied primarily mathematics, alongside Kant and also Vedic Sanskrit. From his friend Günter Patzig, interpreter of Aristotle and Frege, Mohanty was drawn to Frege in relation to mathematical logic. He attended lectures of Heidegger, intrigued by his ontological thinking. Yet, despite the Husserlian legacy, Mohanty was completely self-taught in his studies of Husserl (as he has reported). With a doctorate in mathematics, and ideas from Kant and Frege in his philosophical background, Mohanty set about crafting his own conception of philosophy grounded in phenomenology, drawing on Husserl’s extensive work, critically sifting through Husserl’s texts and their emerging concepts of intentionality, meaning, subject, intersubjectivity, and world. In between he wrote his first book-length study: on phenomenological insights in Nicolai Hartmann and A. N. Whitehead (1958). -/- Over many decades Mohanty formulated and argued, in analytical detail, for a conception of phenomenology and its place in philosophy, later presented in a clear and concise book titled Transcendental Phenomenology: An Analytic Account (1989). Over these very decades the same scholar explored classical and recent Indian philosophy, thinking through kindred ideas of consciousness, self, and knowledge drawn from the Indian philosophical contexts. While writing on Nyāya theory of truth, he also pondered whether the world and finite individual are illusory or real, and whether Marx, Arendt, Gandhi (whom he heard speak in Calcutta), or Vinoba Bhave (with whom he marched across India for the land-grant movement) could best navigate post-Independent India’s social and svarāj or self-rule reforms. The two Mohantys, thinking through a vision of self and world, turned out to be “non-different” or “non-dual” as they each practiced critical phenomenology from both inside and outside the respective philosophical and cultural traditions. Numerous students, fellows, and colleagues or collaborators have benefited immensely from this infusion and unified approach to diversity in philosophical thought. In the 2000s, moving into retirement, Mohanty wrote two long books devoted to his understanding of Husserl and phenomenology and the calling of philosophy itself. This two-volume study shows Mohanty himself thinking through Husserl, critically, in The Philosophy of Edmund Husserl: A Historical Introduction (2008), and Edmund Husserl’s Freiburg Years: 1916–1938. As Mohanty worked on Husserl, he carefully indicated where he agreed and where he rejected or changed ideas, all part of his practice in phenomenology of “description and interpretation.” It was the same pattern he used in addressing the thought of Husserl vis-à-vis Kant or Frege or even Quine. -/- As Mohanty developed his understanding of phenomenology over the years, he wrote books on theory of meaning and the concept of intentionality, developing a model of ideal meaning and its foundation in intentionality, drawing on Husserl’s results. He followed these with the book Husserl and Frege (1982), linking the thought of those foundational figures for the “continental” and “analytic” traditions, respectively, in twentieth-century Western philosophy. Over his long career Mohanty addressed both traditions in his clear and accessible writing style. While developing his views on phenomenology, Mohanty regularly looked to “Husserl and his others,” evaluating views in Heidegger, Sartre, Merleau-Ponty, Ricoeur, and then Derrida and others in the wake of Husserl. With his background in Kant, Mohanty also looked toward Heidegger and Hegel in relation to Husserl’s later work in the Crisis (1935–38). Similarly, he looked to contemporary analytic philosophers, adjudicating his own, oft-wise Husserlian views in relation to Frege, Nagel, and others. Amid his active scholarly career, Mohanty co-founded the journal Husserl Studies, and was editorial advisor to Philosophy & Phenomenological Research, Philosophy East & West, Journal of Indian Philosophy, Sophia, among others. -/- Yet all this while Mohanty was also thinking and writing about Indian philosophy and its relation to phenomenology. In his own retrospective, Indian philosophy is “the permanent background” of his Husserlian thinking, while Kant is the recurrent Western background of his Husserlian phenomenology. Mohanty’s form of “transcendental phenomenology” evolved, in his own perspective, against the background of his studies of Navya-Nyāya on logic and Vedānta on consciousness, in Indian philosophy, and against the background of Kant’s First Critique on the transcendental, in Western philosophy. Accordingly, Mohanty’s study of logical form and of the intentionality of consciousness seeks a fusion of East and West in the conception of transcendental phenomenology. (Cf. Mohanty’s apt response to critics in The Empirical and the Transcendental (2000).) Even in the context of North American Husserl scholarship, Mohanty has exercised an earnest fusion of East and West. For the so-called East Coast and West Coast interpretations of Husserl’s crucial notion of noema both find a sympathetic spirit in J. N. Mohanty’s careful and nuanced interpretation of Husserlian transcendental phenomenology. With Mohanty the two faces of the noema are the logical (Fregean) and the phenomenal (Kantian), and these views of intentional structure join in consciousness—in a way resonant with Indian thought. -/- - David Woodruff Smith (UC Irvine) and Purushottama Bilimoria (SFSU, San Francisco; University of Melbourne) -/- For photo of Prof Mohanty visit APA online memorial_minutes2023 Courtesy of APA Memorial Minutes (& Proceedings) 2023 . (shrink)

Technology is an essential aspect of everyday life, thus the growth and integrationof information and communication technology (ICT) in learning and teaching isexpected. Beginning with the origins of technology based on mathematics as a science,we narrow our focus in this scholarly study on the use of ICT in the teaching andlearning of mathematics in elementary education. We focused on studying the impactof ICT on the teaching and learning process, as well as the usage of ICT (...) tools in thedevelopment of knowledge in mathematics. The goal of this systematic analysis is toshow the benefits of integrating ICT into mathematics teaching and learning, as well asthe constraints and challenges that elementary school teachers and students encounter.Our comprehensive analysis of pertinent theoretical and empirical findings ledus to the general conclusion that teacher proficiency with the new technology plays amajor role in the integration of ICT. The positive impact of ICT on mathematics teachingand learning is undeniable, because technology reduces logistics in the planningand implementation of instruction, increases interaction in the classroom, and encouragesmathematics learning through the visualization of abstract concepts and processes,while also allowing for independent study. (shrink)

Feedback encompasses different aspects of learning and enables students tofocus on the goal of the lesson while being conscious of their responsibility fortheir success. It should neither be strictly criticism, nor mere praise; rather, itshould contain a set of information that will guide them in correcting their mistakesand learning. Feedback has a positive impact on student learning (Voerman et al.,2012) and is a key factor in increasing mathematical achievement (Clarke, 2001).The purpose of this action research is to improve elementary (...) class='Hi'>mathematics teachingthrough the provision of feedback. Analysis of the video recordings from the lessonsrevealed the prevalence of feedback aimed at assessing students’ comprehensionand achievement. Additionally, students received feedback on the assignment. Asthe planned activities were implemented, an increase in student motivation, controlover their own learning and student satisfaction was also observed. The findingsdemonstrate that it is possible to improve mathematics education by introducingchanges that align with for the students’ abilities and interests. In doing so, it iscrucial for teachers to be knowledgeable about effective teaching methods thatmust be adapted to the educational context in which they work. (shrink)

Argument“The problem of university courses on infinitesimal calculus and their demarcation from infinitesimal calculus in high schools” is the published version of an address Otto Toeplitz delivered at a meeting of the German Mathematical Society held in Düsseldorf in 1926. It contains the most detailed exposition of Toeplitz's ideas about mathematics education, particularly his thinking about the role of the history of mathematics in mathematics education, which he called the “genetic method” to teaching mathematics. The (...) tensions and assumptions about mathematics, history of mathematics, and historiography revealed in this piece dedicated to educational ideas are what make Toeplitz's text interesting in the study of historiography of mathematics. In general, the ways historiography of mathematics and teaching of mathematics, even without an immediate concern for history, are deeply entangled and, in our view, worth attention both in historical and educational research. (shrink)

In Faculty Misconduct in Collegiate Teaching, higher education researchers John Braxton and Alan Bayer address issues of impropriety and misconduct in the teaching role at the postsecondary level. Braxton and Bayer define and examine norms of teaching behavior: what they are, how they come to exist, and how transgressions are detected and addressed. Do faculty members across various collegiate settings, for example, share views about appropriate and inappropriate teaching behaviors, as they share expectations regarding actions related (...) to research? And what mechanisms are utilized to correct inappropriate behavior on the part of college and university teachers? The authors' work is based on survey results obtained from faculty members at research universities, liberal arts colleges, and two-year community, junior, and technical colleges. Braxton and Bayer's focus is on undergraduate teaching in four disciplines: biology, history, mathematics, and psychology. In their analyses, the authors examine how individual, disciplinary, and institutional differences influence professorial behavior. In contrast to the more explicitly understood and enforced rules of conduct in research, the authors find that teaching norms are informally defined and observed. They argue that a formal code of ethics for undergraduate teaching would serve the dual purpose of improving undergraduate education and elevating the status of college teaching. A groundbreaking study of contemporary academe, Faculty Misconduct in Collegiate Teaching is required reading for all university and college instructors and administrators. (shrink)

The healthcare crisis that has developed in the last two decades during China's economic reform has caused healthcare and hospital financing reforms to be largely experienced by patients as a crisis in the patient–healthcare professional relationship (PPR) at the bedside. The nature and magnitude of this crisis were epitomized by the "Harbin Scandal"—an incident that took place in August 2005 in a Harbin teaching hospital in which the family of an elderly patient hospitalized in the intensive (...) care unit (ICU) for 66 days paid over RMB ¥6 million. The news was publicized globally and ended in the firing of six top hospital administrators including the hospital president and the ICU director. This paper seeks to show that the Chinese healthcare crisis is ultimately linked to a conflict of interests between patients and healthcare professionals (HCPs), which is inherent in the reformed healthcare system of China. Hence the crisis is, at its core, a crisis of fidelity and confidence that must be restored to the PPR. At the "macro" level, it is simplistic to blame the crisis on the failure of the market system, and at the "micro" level, it is naïve to expect that a contractual understanding of the PPR will effectively restore the confidence of patients. This paper will show that the fiduciary relationship and medical professionalism share similar attributes, with fidelity being the core value of both. It concludes that the loss of medical fidelity implies the dissolution of the PPR and the demise of the medical profession and challenges Chinese HCPs to keep their fidelity as a means to both protect their patients’ interests and to preserve their profession's survival. (shrink)

This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".

This study explored and compared the perspectives of Taiwanese in-service and pre-service high school mathematics teachers regarding ideal teaching behaviours; the perspectives of a nationwide sample of students were taken as the baseline. Fourteen factors contributing to ideal teaching behaviours were identified through exploratory factor analyses. Nine factors, including idea explanation and speedy lecture, were rooted in traditional Chinese culture; five factors, including concrete representation and student activities, were influenced by Western cultures. Three teacher profiles were identified (...) through k-means clustering analysis. The perspectives of in-service teachers were dominated by a painless meaning-emphasised profile; these teachers emphasised meaningful learning for students and avoided the fast pace and demanding requirements that can cause distress in students, whereas pre-service teachers were dominated by an all-round profile, revealing their openness to all factors. Compared with... (shrink)

The emergence of the COVID-19 pandemic initiated major disruptions to higher education systems. Physical spaces that previously supported interpersonal interaction and community were abruptly inactivated, and faculty largely took on the responsibility of accommodating classroom structures in rapidly changing situations. This study employed interviews to examine how undergraduate Science, Technology, Engineering, and Mathematics (STEM) instructors adapted instruction to accommodate the mandated transition to virtual learning and how these accommodations supported or hindered community and belonging during the onset of the (...) pandemic. Interviews with 25 STEM faculty at an undergraduate Hispanic Serving Institution revealed a wide range of accommodations they made to their courses and how they managed communication with students. Faculty strived to support student belonging with responses ranging from caring to crisis management, though some faculty expressed feelings of powerlessness when unable to accommodate certain challenges. The case of a responsive and flexible instructor is presented to highlight a productive response to a crisis. These retrospective findings point to strategies to support faculty teaching in virtual learning environments in the future; increasing opportunities for student–student and student–faculty interaction, supporting faculty in learning technologies that support these interactions and addressing faculty’s feelings of powerlessness. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " (...) Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

Upshot: In this response to the open peer commentaries on our target article, we address two emerging themes: the need to explicate further the nature of learning processes from a radical constructivist perspective, and the need to investigate further the implications of our research for classroom teaching.

En 1741, la Royal Military Academy de Woolwich est créée par le Board of Ordnance afin d’instruire les futurs artilleurs et ingénieurs militaires. Cette instruction s’appuie dès le départ sur les mathématiques. Dans cet article, nous présentons et étudions les différents programmes sur la longue période (entre 1741 et les années 1860). Les évolutions, les changements mais aussi les constances sont évalués et nous donnons les raisons de ceux-ci. L’âge de recrutement, le poids du Board of Ordnance ou encore les (...) diverses guerres ont aussi une influence importante sur la place des mathématiques dans les programmes. Par ailleurs, le peu de turn-over des professeurs entraîne aussi une inertie de l’enseignement des mathématiques. In 1741, the Royal Military Academy at Woolwich was created by the Board of Ordnance in order to instruct artillery officers and military engineers. From the beginning, the teaching was mainly based on mathematics. In this paper, we study different curricula and point out the long-term (between 1741 and the 1860s) evolution as well as continuity in mathematical learning and we give reasons for them. The age of recruitment, the authority of the Board of Ordnance and several wars had an influence on the role of mathematics in the syllabus. Furthermore, the low turnover of professors led to inertia in the teaching of mathematics. (shrink)

The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in (...) Germany were getting popular. The works by G. Frege and B. Russell should be considered more progressive towards the development of mathematical logic. The perspective of mathematical logic in solving the crisis of mathematics in Ukraine was noticed by Professor of Mathematics of Novorossiysk University Ivan Vladislavovich Sleshynsky. Sleshynsky is a Doctor of Mathematical Sciences, Professor of Novorossiysk University. After studying at the University for two years he was a Fellow at the Department of Mathematics of Novorossiysk University, defended his master’s thesis and was sent to a scientific internship in Berlin, where he listened to the lectures by K. Weierstrass, L. Kronecker, E. Kummer, G. Bruns. Under the direction of K. Weierstrass he prepared a doctoral dissertation for defense. He returned to his native university in 1882, and at the same time he was a teacher of mathematics in the seminary, Odesa high schools, and taught mathematics at the Odesa Higher Women’s Courses. Having considerable achievements in the field of mathematics, in particular, Pringsheim’s Theorem proved by Sleshinsky on the conditions of convergence of continuous fractions, I. Sleshynsky drew attention to a new direction of logical science. The most significant work for the development of national mathematical logic is the translation by I. Sleshynsky from the French language “Algebra of Logic” by L. Couturat. Among the most famous students of I. Sleshynsky, who studied and worked at Novorossiysk University and influenced the development of mathematical logic, one should mention E. Bunitsky and S. Shatunovsky. The second period of scientific work of I. Sleshynsky is connected with Poland. In 1911 he was invited to teach mathematical disciplines at Jagiellonian University and focused on mathematical logic. I. Sleshynsky’s report “On Traditional Logic”, delivered at the meeting of the Philosophical Society in Krakow. He developed the common belief among mathematicians that logic was not necessary for mathematics. His own experience of teaching one of the most difficult topics in higher mathematics – differential calculus, pushed him to explore logic, since the requirement of perfect mathematical proof required this. In one of his further works of this period, he noted the promising development of mathematical logic and its importance for mathematics. He claimed that for the mathematics of future he needed a new logic, which he saw in the “Principles of Mathematics” by A. Whitehead and B. Russell. Works on mathematical logic by I. Sleszynski prompted many of his students in Poland to undertake in-depth studies in this field, including T. Kotarbiński, S. Jaśkowski, V. Boreyko, and S. Zaremba. Thanks to S. Zaremba, I. Sleshynsky managed to complete the long-planned concept, a two-volume work “Theory of Proof”, the basis of which were lectures of Professor. The crisis period in mathematics of the early twentieth century, marked by the search for greater clarity in the very foundations of mathematical reasoning, led to the transition from the study of mathematical objects to the study of structures. The most successful means of doing this were proposed by mathematical logic. Thanks to Professor I. Sleshynsky, who succeeded in making Novorossiysk University a center of popularization of mathematical logic in the beginning of the twentieth century the ideas of mathematical logic in scientific environment became more popular. However, historical events prevented the ideas of mathematical logic in the domestic scientific space from the further development. (shrink)

ABSTRACT: The Smith Report has generated central questions about the mathematics education of UK adolescents. This paper highlights the close match between the goals of school mathematics, adolescence and exploratory pedagogy. This is contrasted with the prescriptive nature of current regimes. In particular, without careful attention to pedagogy it is possible that the introduction of different pathways may lead to a failure to achieve the outcomes desired by employers and universities, and to inequity in provision for students.

continent. 1.4 (2011): 242—252. Introduction The following two works were produced by visual artist Jonas Staal and writer Vincent W.J. van Gerven Oei during a visit as artists in residence at The Bag Factory, Johannesburg, South Africa during the summer of 2010. Both works were produced in situ and comprised in both cases a public intervention conceived by Staal and a textual work conceived by Van Gerven Oei. It was their aim, in both cases, to produce complementary works that could (...) be read “through each other,” in which the movement of artistic construction would be imitated by textual deconstruction and vice versa. Both works deal with the way in which capital, apartheid, and monumentality are interwoven in South-African society. The Missing Link addresses a monument for democracy, erected on the premises of a private corporation running both an amusement park and the Apartheid Museum franchise. The textual intervention accompanying this public intervention investigates the limits of the inclusiveness of the anti-discrimination section in the South-African constitution, itself a monumental work of democracy. The Monument for the Distribution of Wealth deals with the history and eventual dissolution of a monumental square, commemorating the Soweto Uprising, in one of the poorest townships of Johannesburg. The history accompanying this public dismemberment of memory is equally fragmented, which is expressed by the many voices recounting uncertain and perhaps even irrelevant “facts” about the genius loci, the way in which the memorial space has actually entered into the memory of the inhabitants surrounding it. Next to their individual practices, Staal and Van Gerven Oei have worked together on a number of projects ever since 2007, including several art residencies. Their work involves an investigation of the different interfaces between art, politics, and public space in media ranging from theater and public interventions, to video installations and (co-authored) textual works. The Missing Link Missing Link (1) Missing Link (2) Missing Link (3) Intervention on the monument entitled The Seven Pillars of the Constitution , part of the Apartheid Museum in Johannesburg, South Africa. The Seven Pillars represent the constitutional values propagated in South Africa since the abolition of the apartheid regime. The monument and museum were built by the Gold Reef Resorts corporation, which is also responsible for the theme park and casino next to it. By placing the word “capitalism” on the wall surrounding the museum, the capitalist system and the constant social divisions that it implies are interpreted as sophistic continuations of apartheid politics. Through the capitalist system, apartheid is still operational within South African society. Whereas during the apartheid regime, the separation clearly ran along race divisions, in the current, “democratic” system, the same actual separation is sustained without it being an explicit element of the foundations of the country, i.e. the constitution. This intervention foregrounds the constant role of capitalism is both periods. At the same time the intervention acknowledges that contemporary Western artisthood is a mirror of the privileges that are offered by the capitalist system, and the type of artist that it produces. The intervention initiates a critical discourse concerning the capitalist system situated within the disaster of capitalism itself, in other words: through the desire to break with the presumption that art would be able to operate outside the capitalist system and to confirm that art—and the artist himself—is in fact modeled on this system. The public intervention by Staal was supplemented by a textual intervention by Van Gerven Oei: a proposition to alter the seventeenth amendment of the South African constitution. Even though the anti-discrimination legislation in South-Africa is one of the most stringent and inclusive in the world, one factor—wealth—remains outside its scope, thus continuing the schisms along racial lines produced during the apartheid regime: §9.3. The state may not unfairly discriminate directly or indirectly against anyone on one or more grounds, including race, gender, sex, pregnancy, marital status, ethnic or social origin, colour, sexual orientation, age, disability, religion, conscience, belief, culture, language, birth, and wealth. Monument for the Distribution of Wealth Intervention concerning the June 16 Memorial Acre in Central Western Jabavu, one of the poorest townships of Johannesburg. The monument comprises a park on which different are placed recalling an important protest the black population against the former apartheid regime. In 1976, from a school adjacent to the park, they started a massive protest against the introduction of Afrikaans in the school curriculum. The police reacted violently, and shot several hundreds of students. The construction of the June 16 Memorial Acre was started in 2005, and from the start was the paragon of corruption. Coordinated by local politicians, some family members of the protest leader from 1976 gained control of the realization of the monument, outside the regulation through external institutions. The available budget of 41 million rand (at that time well over 5 million euro) was largely embezzled. In the meantime, the monument has become fully dilapidated and defaced. The park has become overgrown with weeds and covered in a layer of dirt, and the local population is slowly plundering the square to use the material for the construction and decoration of their own houses. The Monument for the Distribution of Wealth develops the dynamics already existent around the June 16 Memorial Acre. Without obtaining official permission in advance, several local inhabitants were hired to break down the monument, sort the materials, and offer it to the neighborhood. Thus, the redistribution of wealth after the fall of the apartheid regime is finally taking place, albeit from the mere remains of the capital that was once invested in the community. The words “monument” and “for free” are spray-painted on the stacks of material, both in English and Zulu, as these are locally the most common languages. Van Gerven Oei supplemented Staal’s public intervention with an account of the history of the monument based on a series of interviews. The account clarifies how the different interests within the protest 1976 are reflected in the exceeding decay of the park, and how in the end those interests were represented by the June 16 Memorial Acre. Monument for the Distribution of Wealth 1: Removing stones Monument for the Distribution of Wealth 3: Pulling down statues Monument for the Distribution of Wealth 4: Offering the material Monument for the Distribution of Wealth 5: Offering the material Monument for the Distribution of Wealth 6 The next day A Fragmentary History of the Monument for the Distribution of Wealth, Formerly Known as the June 16 Memorial Acre in Central Western Jabavu, Soweto Vincent W.J. van Gerven Oei The following text, based on interviews and online research aims to provide parts of a history of the park in front of Morris Isaacson High School in Central Western Jabavu, Soweto. The idea for the transformation of the park into a memorial site has its source in the events of June 16, 1976: the start of the the student uprising in Soweto. The development of the park was started in the early 1980s, and the actual transformation into a memorial site, the June 16 Memorial Acre, was initiated in 2005. Over the last few years, several monumental additions have been made to the park: A marble monument with three pillars was revealed on June 16, 2006. A sculpture of a book and several billboards on June 16, 2008. A sculpture of student leader Tsietsi Mashinini on June 16, 2010. The park was transformed into the Monument for the Distribution of Wealth on August 3, 2010. According to the entry “Youth Struggle” on the website South African History Online, the Bantu Education Act was introduced in 1953 . In 1954 , Dr Verwoerd, Minister of Native Affairs, stated: “What is the use of teaching the Bantu child mathematics when it cannot use it in practice? That is quite absurd.” According to the entry “Soweto uprising” on Wikipedia, the Afrikaans Medium Decree was issued in 1974 , forcing all black schools to use Afrikaans and English in a 50-50 mix as languages of instruction. The Regional Director of Bantu Education (Northern Transvaal Region), J.G. Erasmus, told Circuit Inspectors and Principals of Schools that from January 1, 1975 , Afrikaans had to be used for mathematics, arithmetic, and social studies from standard five (7th grade). English would be the medium for general science and practical subjects. Indigenous languages would be used for religion instruction, music, and physical culture. According the entry “Soweto uprising” on Wikipedia, on April 30, 1976 , students from the Orlando West Junior School in Soweto went on strike, refusing to go to school. Their example was followed by other schools in Soweto. A student from Morris Isaacson High School, Toboho “Tsietsi” Mashinini, proposed a meeting on June 13, 1976 to discuss further action. According to Weizmann Hamilton’s article “The Soweto Uprising 1976,” which appeared in the September 1986 edition of Inqaba Ya Basebenzi (Fortress of the Revolution), on June 13, 1976 , the South African Students’ Movement called a meeting at the Donaldson Community Center in Orlando. 300-400 Students representing 55 schools decided to hold a mass demonstration on June 16. According to Brian Mokhele, member of the Joint Community Safety Forum, Dr Edelstein was the first victim of the Soweto uprising and killed the day before the march on June 15, 1976 . Edelstein was an administrator at the pass office in Jabavu and gave golf courses to the local community. Edelstein was put in a garbage bin and pierced by pickaxes. The garbage bin was left at the very spot of the murder for many years. A few years ago, a child was beheaded at the same spot, and the basketball court next to it has been abandoned since. According to Marcus Neustetter, founder of the Trinity Session, this story is untrue. According the entry “Soweto uprising” on Wikipedia, on June 16 , Tsietsi Mashinini led students from Morris Isaacson High School to join up with others who walked from Naledi High School. A crowd of between 3,000 and 10,000 eventually ended up near Orlando High School. According to Raymond Marlowe, a local photographer, Tsietsi Mashinini was heading the march. According to the entry “Hector Pieterson” on Wikipedia, Dr Edelstein died on June 16, 1976 , stoned to death by mob and left with a sign around his neck proclaiming “Beware Afrikaaners.” The first child to die that day was called Hastings Ndlovu. According to Pat Motsiri, Orlando West is claiming struggle heritage through the Hector Pieterson Museum, while Hector Pieterson was from Jabavu. According to Pat Motsiri, his generation effectively struggled between 1980 and 1991, forcing the release of Nelson Mandela and negotiations with the apartheid regime while the 1976-generation was safely in exile. Nevertheless, this has not been recognized in any monument. After the abolition of apartheid, the generation from 1976 returned from exile, occupied important government and ANC positions, creating an abundance of 1976 memorials and refusing to acknowledge that this was only possible because of the younger generation’s struggle. He calls this a generational conflict. According to Archibald Dlamini, the park officer responsible for the Memorial Acre, he started working for the municipality in 1978. In 1981/82 , Isaac Makhele from Pimville, who used to work in the cemetery business, was the first developer of the park. It used to be just a normal park until City Parks decided to develop the Memorial Acre in 2006. In 2007 the work was stopped by the community. According to Brian Mokhele, he left the country in 1989 after he participated in the riots of 1986. But when he returned in 1999 he found that nothing had changed. He says that they were promised to be protected by the Constitution, but that reality is different. The police uses fear to suppress them so that they don’t come out to talk openly. He has been arrested twice, both times harassed and tortured by the police, but in the end always released without indictment. He says that this is their way to threaten communities to back off from politics. According to Moses, who is sitting outside rolling a joint, Brian knows everything. He tells Brian to tell me everything he knows. According to Brian Mokhele, Tsietsi Mashinini was possibly murdered in 1990 during his exile in New York. Two weeks before he was supposed to return to South Africa, his papers in order, he was found dead under mysterious circumstances. His coffin was sealed when he was buried. According to Pat Motsiri, he came up with the idea for the Memorial Acre in 2003 . He submitted the documents for the proposal to the council, which sidelined him as soon as the budget came out in 2005 . According to Brian Mokhele, there was on estimation R 41,000,000 spent to redesign the park and turn it into the Memorial Acre. The millions were divided by Amos Masondo, the mayor of Soweto, the local councilor Bongani D. Zondi, and the director of the city of Johannesburg, Pat Lephunya. They were dividing the money between several contractors: Tsietsi Mashinini’s brothers were involved in the development of the park, they got the tender to do the green areas, the landscaping. Construction was done by other companies, some did the paving, the toilets, etc. EMBA, a private company appointed by the municipality was in control of the money flow, but the money was quickly gone. According to Raymond Marlowe, the contractor bought a BMW with the money. According to Archibald Dlamini, the Mashinini brothers got the tender, so the space would look more like the other places around in Soweto. It was agreed that after they were done, they would return the property to the municipality. They did whatever they could do. According to Mafaisa, a member of the Jabavu business community, he was one of the contractors for the landscaping and the pavement under Mpho Mashinini, one of the brothers. He says that I should contact Mavi for information on Mpho. According to Poi Stuurman, a local youth worker, Mafaisa is one of the guys who ran away with the money. According to Mavi, Mpho Mashinini was never a contractor. The contracts were organized by Sbu Butelezi, the former head of the Gauteng Department for Public Works. The June 16 Foundation and the Mashinini brothers will be the beneficiaries of the park when it is finished. According to Brian Mokhele, City Parks did not accept the Memorial Acre because it was not finished. The rest of the year, the unfinished park is not maintained, as should have happened. This was done deliberately so that in the end they can just clean the whole thing up and have a reason to redo the whole park. According to Brian Mokhele, the Mashinini brothers now work for the government. People that manipulate for money purposes always come from the government’s side. Because the park was left unfinished, the people from the neighborhood are taking away the stones to decorate their own homes with. According to Archibald Dlamini, because City Parks doesn’t accept responsibility of the park, he officially has nothing the guard, except for his cottage, which is municipal property. The thieves come at night and destroy the park, but he cannot do anything because he is sleeping. According to the website of the Thanda Foundation on June 16, 2006 a bronze statue of Hector Pieterson, the first child to die in the 1976 protests, made by Kobus Hattingh and Jacob Maponyane was unveiled in the Maponya Mall in Soweto. The statue is sculpted after the famous image shot by Sam Nzima of Mbuyisa Makhubo carrying the dead body of the boy. The sculpture was sponsored by the Thanda Foundation, founded by the Swedish entrepreneur Dan Olofsson and South-African entrepreneur Matthews Phosa. According to the official website of the City of Johannesburg, the Memorial Acre and Artwork were unveiled in 2006. According to a blog post on sowetouprisings.com, the Memorial Acre was still under development on July 24, 2006. According to Archibald Dlamini, City Parks only cleans up the park once a year just before the June 16 celebrations. Everybody is waiting for the Mashinini brothers to finish their job. The last time he talked with them was in 2007 . According to a sign on the school grounds of the Morris Isaacson High School, the June 16 Trail will be finished in 2008 . According to a blog post on sowetouprisings.com, the Memorial Acre contains another monument erected in Tsietsi’s honor. The monument was created as part of the Sunday Times Heritage Public Art program. Its physical form resembles a giant book which symbolizes the crisis in education experienced in 1976. On the face of the book is the map of the route taken by the students from Morris Isaacson High School in Central Western Jabavu to Phefeni Junior Secondary in Orlando West (currently the site of the Hector Pieterson Museum), while the back cover of the ‘book’ is inscribed with a tribute to Tsietsi Mashinini. The monument was revealed on June 16, 2008 . According to Marcus Neustetter, the billboards on the Memorial Acre were part of a school project realized in 2008. Following several workshops, the students from different high schools along the June 16 Trail were invited to work with artists on the billboards, while the neighborhood community was invited to watch the process during the festivities on June 15 and 16, 2008 . The billboards were supposed to be removed because of construction works on the Memorial Acre, which never ended up happening. According to Brian Mokhele, the former toilet facilities were converted into a house for the park officer. This park officer has been working for city parks for more than 12-15 years, but does nothing here, because the park, including the new toilet buildings, is not finished. The government is now moving around looking for people to take this job because they stopped it. They confronted everybody who was going in and chased them away. According to John, in 2009 , some girls, around 16 or 17 years old, were raped by four men who had been drinking in a local shebeen. When the bar quit they said that they would go home by car, but instead raped the girls on the Memorial Acre nearby. This happened in the unfinished toilets, because the doors couldn’t be closed. According to Brian Mokhele, there used to be some fences around the park because of the construction work that was eventually stopped, but these were also stolen. According to Archibald Dlamini, people from the neighborhood started about two and a half years ago, and the last piece was stolen near the end of 2009 . Sometimes he would catch someone with a roll of fence, and then use it for his own cottage. According to the official website of the City of Johannesburg, a statue of Tsietsi Mashinini by Johannes Pokhela was revealed on June 16, 2010 , “Youth Day.” According to Shirley Makutoane, deputy principal of Isaac Morrison High School, the statue of Tsietsi Mashinini, funded by the June 16 Foundation, has temporarily been placed within the school perimeter. The statue will be moved to the Memorial Acre when it will be finished, in 2011 . According to Brian Mokhele, beside the June 16 Foundation, there is also a June 16 Memorial Acre Foundation. Both foundations are quarreling about the money involved in the Memorial Acre project. Nobody knows who’s involved in them. According to Marcus Neustetter, the June 16 Foundation consists of people that were part of 1976 protest movement, local government officials, representatives of the Hector Pieterson Museum, and the council. According to students from the Isaac Morrison High School, the statue of Tsietsi Mashinini is on the school ground because on the Memorial Acre it would be vandalized by youths from White City, an adjacent neighborhood. Accorcing to Brian Mokhele, the statue of Tsietsi should eventually be mounted on the Memorial Acre. It is wrong that the statue is in the school at the moment, because it is not a public school. He wants the statues to depict the massiveness of the force that was coming into Soweto after the protests. According to Brian Mokhele, City Parks, City of Jo’burg, City Lights, and SAPS are making some sort of plan to take the plan back. They want to remove the trees from the Memorial Acre, and redesign the Memorial Acre into a relaxing park, without political content. They want to depoliticize the square. In doing so,they will have their own employment and not use local workforces. According to Brian Mokhele, the community wants to remove the monuments, amphitheater, and sculptures from the Memorial Acre because they do not resemble anything. The sculptures should be depicting the truth of what happened, because the Memorial Acre is a political heritage site. He wants to involve the people that actually participated in the struggle to make the monument so that everyone can enjoy it and get a better understanding of the struggle heritage. Therefore, he proposes collective ideology in which everyone has a say. This would prevent future vandalization of the monument. According to Pat Motsiri, the sculptures must depict the event around June 16, 1976. Like the story of Dr Edelstein, who was pierced by pickaxes, forced into a garbage bin and burned alive. According to Jonas Staal, the Memorial Acre should be destroyed, its elements stacked on pallets, thus forming the Monument for the Distribution of Wealth . According to Mafaisa, his men can do the work quickly. He has about twenty men working under him. (shrink)

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: My commentary has two general goals. First, I investigate how basic principles of radical constructivism might be used in constructing models of mathematics teaching. Toward that end, I found that I was not in complete intersubjective agreement with Borg et al.’s use of some (...) basic terms. Second, I explore what mathematics teaching might look like in what is construed as constructivist mathematics teaching. Toward that end, I comment on Borg et al.’s use of “negotiation” and explain how a constructivist teacher can establish experiential models of students’ mathematics. (shrink)

Objective: To assess the impact of Khan Academy on the Mathematics educational process for UNDAC students and determine the relevance and efficacy of its content and strategies within the Mathematics-Physics program. Methodology: This research is framed within the positivist paradigm and takes on a quantitative approach. An experimental pre-experimental design was used with pretest and posttest in a single group. The study is applied in nature and is grounded in the contributions of expert researchers and methodologists on the (...) subject. Analytical tools were developed with a reliability coefficient of 0.90, and the investigated population consisted of students from the Mathematics-Physics program at UNDAC. Results: Students who incorporated Khan Academy exhibited significant improvements in their mathematical performance. Persistence in using the platform led to significant enhancements in academic achievement. 70 % of the students reached high levels of mathematical skill development. The majority expressed high satisfaction with the platform. Conclusions: Khan Academy has a meaningful and positive impact on mathematics learning among university students at UNDAC. The platform not only promotes content mastery but also fosters learning autonomy. These findings advocate for the adoption of innovative educational technologies in teaching, especially within STEM disciplines. (shrink)

One challenge posed by climate change education is that, despite the scientific consensus on human induced climate change, the issue is controversial and politicised. A recent poll conducted in the USA revealed that 45% of respondents did not believe that human activity is a key cause of climate change, while 8.3% denied that climate change was occurring at all. The poll also found that those with conservative political beliefs were far more likely to deny anthropogenic climate change. The controversial nature (...) of climate change is a double-edged sword for educators—offering both benefits and risks. Controversial topics can be engaging for students, allowing them to critically examine competing perspectives, justify opinions and, thus, develop and practice critical thinking skills. On the other hand, they may provoke classroom conflict and expose teachers to backlash from students, parents, colleagues or even members of the public, including accusations of indoctrination. This risk has resulted in some teachers adopting problematic approaches to teaching about climate change, such as presenting students with an impartial account of ‘both sides’ of the debate. Some climate change sceptics groups have even developed their own curriculum materials and lobbied for their implementation in schools in order to provide students with a ‘balanced’ account of the issue. These problems, and suggestions for overcoming them, are explored by several papers in this special issue. (shrink)

This study aims to improve the effectiveness of English grammar teaching in high school. Firstly, the Science, Technology, Engineering, Art, and Mathematics (STEAM) educational model is comprehensively discussed. Then, the current situation and difficulties of English grammar teaching in high school are analyzed. Finally, based on the traditional and the STEAM teaching mode, a comprehensive study on the psychological motivation of students is carried out in high school English grammar teaching. The traditional teaching model (...) had little effect on students’ psychological motivation in the process of English grammar teaching. Therefore, students’ satisfaction with English grammar teaching under the traditional teaching model is very low, especially in innovation and practice. The STEAM teaching model has greatly improved the psychological motivation of high school students in the English grammar teaching process. The overall satisfaction rate of the students with the English grammar teaching in the STEAM teaching model is 70%. This teaching model can greatly improve students’ requirements for innovation and practice. This study not only provides a reference for the improvement of English teaching in high schools but also contributes to the reform and development of education. (shrink)

Examining the nature of teachers’ emotions and how they are managed and regulated in the act of teaching is crucial to assess the quality of teachers’ instruction. Despite the essential role emotions play in teachers’ lives and instruction, research on teachers’ emotions has not paid much attention on teachers’ emotions in the context of daily teaching. This paper explored elementary teachers’ emotions while preparing for teaching and during teaching mathematics, reasons that underlie these emotions, and (...) the relationship between their emotions and the quality of their mathematics instruction. Participants were seven elementary teachers working in the U.S. who participated in Holistic Individualized Coaching (HIC) professional development that consisted of five cycles of coaching over an year. For each coaching cycle, pre-coaching conversation and post-coaching conversation data were collected regarding emotions teachers felt in anticipation of teaching and during teaching retrospectively. In order to compare teachers’ emotions with instructional quality, coaching sessions were video recorded and analyzed to determine the quality of instruction. Findings of this study showed that teachers reported six categories of emotions (positive, negative, neutral, blended-positive, blended-negative, and mixed), described emotions often in non-typical ways (e.g., “not nervous”, “anxious but in a positive way”), and experienced mixed emotions (co-occcurence of positive and negative emotions) as the most dominant emotion. Teachers also had more positive emotions anticipating teaching than actually teaching the lesson. The reason teachers felt mixed emotions reflected the complex and context-specific nature of teaching, a phenonemenon not currently described in the teacher emotion literature. There were no clear relationships between emotional experiences and instructional quality. This study allowed participants to freely describe their authentic, complex, overlapping, and ambiguous emotions in the context of active teaching, which contributes opening up the possibilities of diversifying teacher emotion research and shows the significance and usefulness of understanding teachers’ emotions related to active instruction. (shrink)

The subject of the study is Friedrich Nietzsche’s concept of nihilism as an integral part of his socio-philosophical views.The relevance of addressing the concept of nihilism in the context of Friedrich Nietzsche’s reflections on society is due to the fact that, in the philosopher’s view, nihilism is the main concept for substantiating the idea of the crisis nature of modern Western civilization. It is because of nihilism, Friedrich Nietzsche believed, that Western society in the historical perspective is doomed to (...) decline and death.So, Friedrich Nietzsche’s critical approach to the problems of society is conceptual in nature, in a historical-philosophical sense, Nietzschean criticism contributed to the formation of the problem of the formation of a mass society and the idea of a crisis of Western culture in Western philosophical and social thought.In accordance with the purpose of the research, the article analyzes the Nietzschean criticism of society in the context of the reception and reflection of this criticism in the thoughts of famous representatives of non-classical philosophy, which is important for the formation of the modern discourse of Western culture.The paper analyzes the concept of nihilism as a prerequisite for Friedrich Nietzsche’s negative attitude towards the development prospects of Western society. The close connection between the idea of nihilism and rationalism is noted, it is rationalism, according to the philosopher, that is a prerequisite for the moral decline of society.It is emphasized that Friedrich Nietzsche’s fundamental criticism of rationalism at the level of general philosophical ideas has certain deviations when the philosopher considers specific socio-political processes.It is concluded that it is precisely in the socio-political aspect in the views of Friedrich Nietzsche that the complexity of the relationship between the concepts of rational and irrational is most characteristically manifested. This complexity actually goes beyond purely Nietzschean philosophy and becomes a subject of reflection for many representatives of non-classical philosophy.The article, based on the statements of a number of researchers of the philosophy of Friedrich Nietzsche, proves the opinion that the philosopher’s anti-irrationalism can be understood in a constructive way, as a supplement and deepening of modern ideas about the nature of the rational, and not always the positive influence of the rational on man and society.Thus, the philosophy of Friedrich Nietzsche acts as a prerequisite for the further deepening of ideas in Western culture about the contradictory nature of society in which rational and irrational elements are inextricably intertwined. (shrink)

Many scholars have looked for similarities between the recent crisis in the Eurozone and the crises that have occurred in the past in developing countries and particularly in Latin America. Problems of balance of payments, public debt, overvaluation of the exchange rate and unregulated capital inflows are frequently mentioned to compare common features of different crisis events. Additionally, Continental European countries are following similar processes of welfare state retrenchment and labour market segmentation to Latin American cases in the (...) past, but in different institutional contexts and with different levels of economic development. This article will discuss common features of past Latin American crises and the current crisis in the Eurozone in order to show how such a comparison could help to manage (and avoid) crises in an integrated financial world as well as to teach how a faulty institutional design can arise from a defective understanding of how financial capitalism works in a complex internationally integrated economic system. (shrink)

Preservice mathematics teachers’ accurate understanding of mathematical problem solving and its teaching is key to the performance of their professional quality. This study aims to investigate preservice mathematics teachers’ understanding of problem solving and its teaching and compares it with the understanding of in-service mathematics teachers. After surveying 326 in-service mathematics teachers, this study constructs a reliable and valid tool for the cognition of mathematical problem solving and its teaching and conducts a questionnaire (...) survey on 26 preservice mathematics teachers. Survey results reveal that preservice mathematics teachers have a good understanding of mathematical problem solving and its teaching and are more confident in the transfer value of problem solving ability. By contrast, in-service teachers are more optimistic that problem solving requires exploration, continuous thinking, and the participation of metacognition. This article concludes that preservice mathematics teachers should focus more on the initiative and creativity of students and put students at the center of education. In addition, teacher educators should provide more teaching practice opportunities for preservice teachers. The findings also show that in-service teachers’ understanding of problem solving and its teaching is inferior to that of preservice teachers on some indicators, implying the importance of post-service training for in-service teachers. (shrink)

Though the Christian faith has in recent years increasingly shown itself to be a truly African religion, a variety of African authors such as Kä Mana, George Kinoti, Hannah Kinoti, August Shutte and Efoé Julien Penoukou have noted that sub-Saharan Africa is facing a moral crisis. This article explores this crisis in as far as it is caused by difficulties in the reception of the Christian ethic by African Christian communities. It points out that this crisis is (...) visible in double morality, immorality and legalism. It shows that it is both caused by rapid social change in contemporary Africa and by the way the Christian ethic was introduced with a lack of attention for the relationship between worldview and ethics, the social impact of changing cultural practices and the importance of virtue ethics. In this way it also points to the shape Christian moral education for present-day Africa should take. (shrink)

It is often thought that there is a “crisis” or something akin to this in the field of medical malpractice in the USA and from time to time, as will be shown, there are suggestions that a similar situation could exist in the UK. This paper will examine what might be meant by the expressions “malpractice” and “crisis” in relation to the UK. It will be argued that there is no evidence to suggest that anything as dramatic as (...) a crisis exists, and that what current concerns exist about medical malpractice and litigation are not based upon an existing or impending crisis. Further, it will suggest that evidence points to medical litigation had reached a plateau, and that, due to a number of factors, if there were a serious problem it is likely to be on the wane. (shrink)

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in (...) the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed. (shrink)

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs (...) have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction. (shrink)

Although progress, no matter how small, has been made by scholars who examined different aspects of the Mkpuru Mmiri [methamphetamine or crystal meth] drug crisis in Nigerian Igbo communities, literature is yet to approach the study from the perspective of Proverbs 22 of the Old Testament. In this study, literature was extended to examining the Mkpuru Mmiri crisis from the lens of Proverbs 22. Today, many youths in Igbo communities are addicted to Mkpuru Mmiri, a stimulant drug. As (...) part of the findings, it was discovered that the effects of Mkpuru Mmiri include paranoia, hallucinations, delinquency and other behaviours inimical to social well-being and even death. Addiction to Mkpuru Mmiri is seen in the increasing number of unstable young people on the streets of the various states in Igbo communities. The author of the book of Proverbs teaches how youths are to live responsible and happy lives. According to Proverbs, evil conduct will always result in divine vengeance and punishment during a person’s earthly life. The themes include ‘home and personal training (vv. 1–6)’, ‘consequences of one’s actions (vv. 4–23)’ and the ‘reasons to avoid bad company (vv. 24–29)’.Contribution: Addiction to Mkpuru Mmiri has no recognised cure. This may be seen in the growing number of unruly young people on the streets of Nigeria’s south-east states. Practical theology is one of the disciplines implicated. (shrink)

The article is devoted to identifying the value of the phenomenon of aesthetic value and beauty of mathematical knowledge and the beauty of mathematical theory of teaching mathematics. The aesthetic potential of mathematical knowledge allows the use of theater technology in the educational process with the active dialogic interaction between teacher and students. The criteria of beauty in mathematical theories are distinguished: the realization of beauty as the unity of the whole, and in the disclosure of the complex (...) through the elementary; methodological interpretation of the beauty in the community of mathematical structures and optimal information content of the meta-language of mathematics; the practical embodiment of beauty in the formalization of the infinite through the finite. The beauty of mathematics is the force that permeates all the ‘layers of knowledge‘ not along, and across, although the effectiveness of mathematical activity due to aesthetic laws, which do not always lend themselves to unambiguous interpretation. In the article it is stated that, depending on the educational goals of communicative impact on the audience, in fact, ‘mathematical lectures theatricality‘ can have different characteristics, the most important of which are teachers artistry and artistic director’s work of a teacher. This cultural phenomenon that includes the theatrical talent, helps create an atmosphere of cooperation needed in varying degrees of activity for pedagogical interaction. The author believes that such approach, developed on the basis of the Stanislavsky system, allows university professors of mathematics significantly improve mathematical lectures. (shrink)