Davydov’s mathematics curriculum was designed according to the principles of the Cultural Historical Activity Theory. In this study, we analyzed some developmental effects of its realization in Grade 1, in relation to the children’s school readiness level, and their teacher’s experience. We assessed two groups of developmental effects: some general math abilities ; and some abilities, which are very specific to Davydov’s mathematics curriculum. At the beginning of the Grade 1, we divided all participants into three (...) groups according to their level of motivation, voluntary regulation, and logical preservation in terms of J. Piaget. After 1 year we measured general effects again and also measured specific effects. The results showed that all the children became significantly better in all general math abilities. It was also found that progress in math abilities does not depend on the initial level of school readiness of children. Children with different levels of voluntary regulation, motivational readiness, and the level of logical preservation show improvements in general math abilities. As for the mathematical skills specific to the Davydov’ program, the achieved level of their development is also not related to the initial parameters of readiness for school. Also, there were no differences in improvements in general effects and in specific effects between the classes with experienced and not-so-experienced teacher. So, there are some reasons to believe that the level of the child’s actual development does not play a fundamental role for education, which built in accordance with the principles of CHAT. (shrink)

Roger Gibson; Logic as a Core Curriculum Subject: its case as an alternative to mathematics, Journal of Philosophy of Education, Volume 20, Issue 1, 30 May 2006.

Since the Fall of 1993, at the Centre Interdisciplinaire de Recherche sur l'Apprentissage et le D/span>veloppement en /span>ducation of the Universit/span> du Qu/span>bec /span> Montr/span>al, two mathematicians and one philosopher have collaborated to design and develop a research project involving philosophy, mathematics and sciences. Previous observations in the classroom had led the researchers to realize that, within the school curriculum, children like some subject matters and dislike others. Most of them usually succeed in arts, physical education and language (...) arts, but many have difficulties in succeeding in mathematics. Why? On the one hand, as Matthew Lipman advocates, the school curricula are not sufficiently "meaningful" for children. On the other hand, some studies in the field of mathematics suggest that there are myths and prejudices about mathematics in primary schools and that the school system is partly responsible for this. Indeed, the school system does not invite children to express emotions in class about mathematics nor does it favor creativity. It does not allow dialogue among peers about mathematical concepts and problems, nor the construction of mathematical knowledge by the students themselves. (shrink)

Roger Gibson; Logic as a Core Curriculum Subject: its case as an alternative to mathematics, Journal of Philosophy of Education, Volume 20, Issue 1, 30 May 2006.

This paper presents a qualitative study of four Grade 1 and Grade 2 national mathematics curricula coming from francophone countries: Côte d’Ivoire, Djibouti, Canada (Québec), and France. A comparative analysis was performed to identify differences that potentially lead to inequities among the countries. We identified all the concepts present in the Grade 1 and Grade 2 mathematics curriculum using the mathematical literacy framework developed by the OECD through its PISA assessment: quantity, change and relationship, data and uncertainty, (...) and space and shape. Then we looked across countries to find the major differences among them. The findings show three categories of possible inequities among the different curricula: some mathematical concepts are not presented in culturally relevant ways to students, some concepts present in the mathematics curricula are not mathematical concepts, and some important mathematical concepts are not part of the mathematics curricula. Our results show that these inequities appear in all four national curricula analysed here. Those findings highlight the need to question the political inequities among mathematics curricula around the world in order to give a chance for all students to learn strong and meaningful mathematics. (shrink)

Mathematics has been the most successful and is the most mature of the sciences. Its first great master work – Euclid's ‘Elements’ – which helped to establish the field and demonstrate the power of its methods, was written about 2400 years ago; and it served as a standard text in the mathematics curriculum well into the twentieth century. By contrast, the first comparable master work of physics – Newton's Principia – was written 300 odd years ago. And (...) the juvenile science of biology only got its first master work – Darwin's ‘On the Origin of Species’ – a mere 150 years ago. The development of the subject has also been extraordinarily fertile, particularly in the last three centuries, and it is perhaps only in the last century that the other sciences have begun to approach mathematics in the steady accumulation of knowledge that it has been able to offer. There has, moreover, been almost universal agreement on its methods and how they are to be applied. What we require is proof; and, in practice, there is very little disagreement over whether or not we have it. The other sciences, by contrast, tend to get mired in controversy over the significance of this or that experimental finding or over whether one theory is to be preferred to another. (shrink)

With democracy in mind, promoting students’ cognitive, personal, and social development can inform and shape the mathematics curriculum and classroom practice with the goal of their becoming more capable, self-reflective, and socially aware human beings. Toward that realization, their mathematics experience could include: heuristics, as it provides a natural language for problem solving; habits of mind, so students can think and act with a more developed “reflective intelligence”; and multiple-centers investigations, where collaborations based on shared mathematical interest (...) can be pursued. In this way, their education would be “a fostering, a nurturing, a cultivating process” that supports the development of democratic society. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Humanist History of Mathematics?Regiomontanus's Padua Oration in ContextJames Steven ByrneIn the spring of 1464, the German astronomer, astrologer, and mathematician Johannes Müller (1436–76), known as Regiomontanus (a Latinization of the name of his hometown, Königsberg in Franconia), offered a course of lectures on the Arabic astronomer al-Farghani at the University of Padua. The only one of these to survive is his inaugural oration on the history and (...) utility of the mathematical arts.1 Regiomontanus tells his audience that the purpose of the oration is torelate first the origin of our arts, and among which nations they first began to be cultivated, in what way they were at last translated from various foreign tongues into Latin, which of our ancestors were famed in these disciplines, and to whom in our lifetimes recognition should be granted.2To this end, he offers a history of the quadrivial arts (arithmetic, geometry, music, and astronomy) and other important mathematical disciplines from [End Page 41] antiquity to his own time, praises their utility, and exhorts his audience to revive the languishing study of mathematics at Padua. Astrology, a discipline with which Regiomontanus himself was closely associated, is singled out for particular praise.Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenth-and-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled [End Page 42] after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various "algorisms" (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). Advanced students could read numerous more specific treatises by Arabic and Latin authors. For optics, the Perspectiva communis of John Peckham was the most common basic text, with texts... (shrink)

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated (...) to the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history. (shrink)

This comprehensive text on African Mathematics addresses some of the problematic issues in the field, such as attitudes, curriculum development, educational change, academic achievement, standardized and other tests, performance factors, student characteristics, cross-cultural differences and studies, literacy, native speakers, social class and differences, equal education, teaching methods, and more.

June 1985 B.S. in Mathematics, California Institute of Technology (Pasadena, California) Senior thesis: Elementary induction on finite abelian groups Supervisor: Alexander S. Kechris..

Philosophy of Education and the Curriculum is a six-chapter book that first elucidates the forms of knowledge argument and religious education. Subsequent chapters detail the mathematics, natural science, and history forms of knowledge. Moral philosophy and moral education are then explained. The last chapter provides an explanation of learning.

Epistemology is a conceptual template for how we think about the world, and the study of how we come to know the world around us. The world does not dictate unequivocally how to interpret it. This article will explore this position on the fluidity of epistemic constructs through two prominent philosophical perspectives, those being derived from the works of Ludwig Wittgenstein and Michael Foucault, respectively. These insights will be used to more deeply unfold the current situation for Aboriginal students within (...) dominant approaches to mathematics curriculum in Australia, and the subsequent approaches to the inclusion of Aboriginal knowledge and epistemologies. It is suggested that the epistemic constructs most valued and thus credited as conveyors of ‘truth’, and therefore positioned as powerful forms of knowledge within dominant curriculum and education policy, are those derived from Western, Eurocentric origins. This privileging of particular epistemological constructs over others is reinforced unconsciously through the articulation of educational goals deemed most appropriate, or ‘socially just’, for the Aboriginal student population. The place of Aboriginal knowledge within such constructs is therefore reflective of broader ideation around the role of education within society and its failure to challenge existing structures of power and injustice. (shrink)

The field of quantitative analysis is often mistaken to be a discipline free from ethical burdens. The quantitative financial analyst or “quant” profession holds a position of significant responsibility as the keeper of mathematical models used in complex derivative security pricing and risk management. Despite this responsibility very few postgraduate programs address the teaching of ethics and professional standards in their curriculum, and the credibility of the profession has suffered as a result of several high-profile financial losses. Some of (...) these failures could have been avoided and their impacts diminished if ethical considerations were integrated with quantitative method. Appropriate development in ethics education for quants is needed to identify points in the decision-making process where ethical questions can arise, and to explain how quants can protect stakeholders from the costs of unethical behaviour. An approach to ethics education needs to be flexible and allow for different methods to infuse ethical coverage into the course. Such an approach will go some way towards aligning the profession with other specialisations in banking and avoid the need for complex and unnecessary regulation. (shrink)

This article traces the development of the theory and practice of what is known as ‘community of inquiry’ as an ideal of classroom praxis. The concept has ancient and uncertain origins, but was seized upon as a form of pedagogy by the originators of the Philosophy for Children program in the 1970s. Its location at the intersection of the discourses of argumentation theory, communications theory, semiotics, systems theory, dialogue theory, learning theory and group psychodynamics makes of it a rich site (...) for the dialogue between theory and practice in education. This article is an exploration of those intersections, and a prospectus of its possible role in the formation and reformulation of school curriculum. It will be argued here that, when formulated as community of philosophical inquiry in particular, it offers the possibility of ‘philosophising’ the school curriculum in general, by extending the concept-work that doing philosophy entails to all of the disciplines. The article begins with an attempt at an operational definition of the term as, move to an analysis of its dynamics, offers an example of its use in a mathematics classroom, and finishes with a schematic view of its whole-curriculum and whole-school possibilities. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture (...) on the very same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

Le but de cet article est d’étudier l’enseignement des mathématiques au Collège d’Artillerie espagnol de 1764 à 1842, et de repérer les influences étrangères sur elle. Après quelques hésitations, un programme standard est adopté commençant par l’arithmétique et finissant par le calcul différentiel et intégral et la mécanique. L’unique changement important fut en 1819 quand la mécanique devint une matière indépendante. Pendant les premières décennies, avec Giannini comme premier professeur, l’emprise italienne était importante, néanmoins il existait une influence française et (...) anglaise. Avec le temps l’inspiration française augmenta. Les manuels employés pendant la plupart du temps étaient ceux de Giannini ou Odriozola, professeurs du Collège; mais ils ont utilisé aussi des livres de mathématiciens français comme Lacroix, Legendre ou Monge. The present article studies the impact of foreign influence on mathematics education at the Spanish Royal College of Artillery from 1764 to 1842. After some vacillation, the academy’s standard teaching program included from arithmetic to calculus and mechanics. The most noteworthy change occurred in 1819 when mechanics would be considered as an independent subject. During the first decades during which time Giannini was first professor, the academy’s approach to teaching mathematics was mostly influenced by methods being used in Italy and, to a certain degree, in England and France. French influence increased considerably with time. While most of the textbooks used at the academy were authored by two of its professors, Giannini and Odriozola, the curriculum would be expanded to include works by French mathematicians such as Lacroix, Monge and Legendre. (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.

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.1 (2003) 1-12 [Access article in PDF] Art and Mathematics in Education Richard Hickman and Peter Huckstep We begin by asking a simple question: To what extent can art education be related to mathematics education? One reason for asking this is that there is, on the one hand, a significant body of claims that assert that mathematics is an art, and, (...) on the other, work in art that has a mathematical basis. Observations of these kinds are not trivial. They have significant implications for the teaching of these areas of the curriculum in at least two ways. First, there is the methodological issue of the extent to which we should teach mathematics and art separately, and second, the teleological question of why they appear in the curriculum at all. So the relationship between the nature of mathematics and art, perceived or real, bears down on questions of the individuation and the justification of these disciplines, or, in other words, upon pedagogy and purpose. Although in principle both the pedagogy and the purpose of any discipline are distinct, there are important connections between them, as we shall draw out.As far as mathematics goes, it has been stressed that the purposes of even rudimentary aspects of mathematics such as counting do need to be made explicit to pupils. 1 What have been seen as errors in understanding are internally linked with pupils' ideas of the purposes of counting. Anna Sierpinska discusses a similar situation within the broader aspect of what she calls "theory" in higher mathematics. 2 More generally, John Passmore, in his analysis of pupils' understanding includes the case where a pupil sees no need to understand since he or she cannot understand the point of learning the subject. 3 If the purposes of art and mathematics are strikingly similar then there is reason to suppose that certain pedagogical approaches that [End Page 1] follow from this will improve understanding of both mathematics and art, and therefore understanding of reasons for engaging in math-based and art-based activities.In schools we see what appears to be some evidence of connections between art and mathematics in cross-curricular work, such as that described in Jacqueline Cossentino and David Schaffer. 4 Apart from providing much-needed motivation, an important value of this approach toward mathematics is that it can illuminate pupils' understanding of some of its purpose by, at the very least, allowing them to recognize instances of mathematics at work, often in unsuspecting contexts. In particular, by facilitating some transference of knowledge pupils can appreciate the so-called "power" of mathematics in its widespread application to the world of artefacts produced by themselves and others.Can we go further than displaying connections between these disciplines? Can we treat mathematics itself as art? In particular, can we suppose that pupils can make mathematics just as they can make artefacts? Some writers seem to imply this by invoking the notion of creativity as a common property between the two disciplines. 5 Leaving aside the ironic sense of creativity in, for example, the case of "creative accountancy" pupils must indeed "make" sense of mathematics and they must originate mathematical assertions of their own that have not been directly taught. In both respects mathematics is similar to the learning of a language. Further than this, we value pupils' inventiveness in their approaches to areas of mathematics, calculations for example.The linking of mathematics and art via creativity has its most persuasive force in the change of attitude toward axiomatic foundations. 6 The story is familiar: for centuries Euclidean geometry was (wrongly) supposed to offer truths of the actual space in which we lived, since its axioms were thought to be confirmed by our intuitions of space. However, when it was found that consistent and useful axiomatic systems could be constructed by making certain alterations to these axioms, it became clear that mathematics was not simply discovered. It involved a significant measure of creativity. Yet one thing is clear: the sense that children make of mathematics, the original utterances they make... (shrink)

This article reclaims mathematics from the measures of profit and control by first presenting an anarchist analysis of mathematics? status quo societal uses and pedagogic activities. From this analysis, a vision for an anarchist math education is developed, as well as suggestions for how government school practitioners sympathetic to anarchism can insert this vision into their current work. Aspects to this vision include teacher autonomy, freedom from hierarchical curriculum structure and math class as a non-coercive, happy place. (...) Finally, mathematics is argued to be essential knowledge for anarchistic society for three potentialities: in solving social and technological problems through application, as an analytic technology and for increasing individual happiness via the aesthetic dimension. (shrink)

One hopes from the philosophy of education a general theory of the curriculum together with a deductively related batch of specific theories for designing each portion of a curriculum. This anthology of nineteen reprints sheds little light on the general problem, but it does gather under one roof a handy collection of articles relevant to an understanding of some of the problems of specific curriculum decision-making. The authors are concerned with the intellectual content of the curriculum, (...) which they reasonably divide into the general disciplines of the formal sciences, the physical and natural sciences, history and the social sciences, literature and the arts, and philosophy. Within each discipline we get a string of philosophical discussions of the structure and methodology of that discipline. Thus, within one volume we have a bit of the philosophy of logic and mathematics--e.g., a paper on necessary truth, a paper on foundations; a bit of the philosophy of science--e.g., Ernest Nagel's "Teleological Explanation and Teleological Systems"; a bit of aesthetics; a bit of metaphilosophy--e.g., Friedrich Waismann's "How I See Philosophy"; and so on. But this by itself can only give us a philosophy of the intellectual disciplines. It cannot by itself provide us with a philosophy of the curriculum. The authors do provide us with two papers on the curriculum, including a very stimulating piece by G. C. Fields--"The Teaching of Philosophy." One could hope for more. Sadly there are few such papers to choose from.--J. F. (shrink)

Le but de cet article est d’étudier l’enseignement des mathématiques au Collège d’Artillerie espagnol de 1764 à 1842, et de repérer les influences étrangères sur elle. Après quelques hésitations, un programme standard est adopté commençant par l’arithmétique et finissant par le calcul différentiel et intégral et la mécanique. L’unique changement important fut en 1819 quand la mécanique devint une matière indépendante. Pendant les premières décennies, avec Giannini comme premier professeur, l’emprise italienne était importante, néanmoins il existait une influence française et (...) anglaise. Avec le temps l’inspiration française augmenta. Les manuels employés pendant la plupart du temps étaient ceux de Giannini ou Odriozola, professeurs du Collège; mais ils ont utilisé aussi des livres de mathématiciens français comme Lacroix, Legendre ou Monge. The present article studies the impact of foreign influence on mathematics education at the Spanish Royal College of Artillery from 1764 to 1842. After some vacillation, the academy’s standard teaching program included from arithmetic to calculus and mechanics. The most noteworthy change occurred in 1819 when mechanics would be considered as an independent subject. During the first decades during which time Giannini was first professor, the academy’s approach to teaching mathematics was mostly influenced by methods being used in Italy and, to a certain degree, in England and France. French influence increased considerably with time. While most of the textbooks used at the academy were authored by two of its professors, Giannini and Odriozola, the curriculum would be expanded to include works by French mathematicians such as Lacroix, Monge and Legendre. (shrink)

Le but de cet article est d’étudier l’enseignement des mathématiques au Collège d’Artillerie espagnol de 1764 à 1842, et de repérer les influences étrangères sur elle. Après quelques hésitations, un programme standard est adopté commençant par l’arithmétique et finissant par le calcul différentiel et intégral et la mécanique. L’unique changement important fut en 1819 quand la mécanique devint une matière indépendante. Pendant les premières décennies, avec Giannini comme premier professeur, l’emprise italienne était importante, néanmoins il existait une influence française et (...) anglaise. Avec le temps l’inspiration française augmenta. Les manuels employés pendant la plupart du temps étaient ceux de Giannini ou Odriozola, professeurs du Collège; mais ils ont utilisé aussi des livres de mathématiciens français comme Lacroix, Legendre ou Monge. The present article studies the impact of foreign influence on mathematics education at the Spanish Royal College of Artillery from 1764 to 1842. After some vacillation, the academy’s standard teaching program included from arithmetic to calculus and mechanics. The most noteworthy change occurred in 1819 when mechanics would be considered as an independent subject. During the first decades during which time Giannini was first professor, the academy’s approach to teaching mathematics was mostly influenced by methods being used in Italy and, to a certain degree, in England and France. French influence increased considerably with time. While most of the textbooks used at the academy were authored by two of its professors, Giannini and Odriozola, the curriculum would be expanded to include works by French mathematicians such as Lacroix, Monge and Legendre. (shrink)

Theoretically, critical mathematics education (CME) differs markedly from traditional, teacher-centered mathematics teaching approaches. In addition to meeting mathematical goals, a main focus of CME lies on fostering students’ critical consciousness and positioning them as active and informed actors in the classroom and beyond. However, only few empirical studies explicitly focus on how students experience CME instruction and curriculum. In this contribution, we draw on interview data to examine how K-12 students in one Austrian and one American school (...) setting respond to and perceive CME lessons. The analysis revealed that, within both classroom settings, students’ perceptions vary greatly, with some perceiving CME lessons as being completely different from prior experiences in school mathematics to some seeing little-to-no difference. Additionally features of CME instruction that mathematics teachers and researchers might consider to be the most salient are not necessarily the ones that students report attending to. Reasons for that might be manifold, and students might need more time with CME instruction and especially with reflecting on such lessons to identify the rationale behind the instructional approach. Moreover, researchers should be aware that students might not share their understanding of critical instructional reforms and might have diverse or even divergent perceptions. (shrink)

This book discusses both the philosophy of mathematics and of mathematics education. The first part is a critique of existing approaches and a new philosophy of mathematics. Chapters include: (1) "A Critique of Absolutist Philosophies of Mathematics," (2) "The Philosophy of Mathematics Reconceptualized," (3) "Social Constructivism as a Philosophy of Mathematics," (4) "Social Constructivism and Subjective Knowledge," and (5) "The Parallels of Social Constructivism." The second part of the book explores the philosophy of (...) class='Hi'>mathematics education and shows that many aspects of mathematics education rest on underlying, usually implicit, philosophical assumptions. Making these assumptions explicit puts a critical tool into the hands of teachers and researchers. Chapters in this section are: (6) "Aims and Ideologies of Mathematics Education," (7) "Groups with Utilitarian Ideologies," (8) "Groups with Purist Ideologies," (9) "The Social Change Ideology of the Public Educators," (10) "Critical Review of Cockcroft and the National Curriculum," (11) "Hierarchy in Mathematics, Learning, Ability and Society," (12) "Mathematics, Values and Equal Opportunities," and (13) "Investigation, Problem Solving and Pedagogy." Each chapter contains notes. (Contains 654 references.) (MKR). (shrink)

Upshot: This is a book for thoughtful science and mathematics teachers and curriculum developers and educational philosophers. Quale helps us to challenge pernicious received “truths‘ and offers us intriguing perspectives, valuable discourse ventures and practical paedagogic strategies to engage the youth of today who are turning away from science in droves, to their and our cost.

This study aimed to investigate the longitudinal trends in mathematical competencies of Grade 8 students in a public high school located in Zamboanga del Sur, Philippines. The study collected data over a period of six academic years, allowing for a comprehensive analysis of students' performance in 16 distinct mathematical competences of basic education curriculum. These topics include, but are not limited to, special products and factors, factoring, and basic concepts of probability. Using a quantitative research design, the study analyzed (...) both numerical and secondary data to identify patterns or trends in students' mathematical competencies. Trends were analyzed through regression analysis using slopes. The significance of these trends was then evaluated using t-tests for the standard error of slopes. The statistical analysis revealed significant upward trends in certain mathematical topics, indicating an improvement in the students' competencies over the six academic years. However, certain topics displayed downward or stable trends, indicating areas for improvement in the teaching methods used for these topics. The study highlights the importance of flexible and adaptable approaches to mathematics education. It emphasizes the need for constant evaluation and improvement of teaching methods to enhance student learning outcomes. The findings of this study provide valuable insights for educators and policymakers to guide evidence-based decision-making and improve the quality of mathematics education in the new normal setup. Implications and future research directions are incorporated in the study. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.1 (2003) 1-12 [Access article in PDF] Art and Mathematics in Education Richard Hickman and Peter Huckstep We begin by asking a simple question: To what extent can art education be related to mathematics education? One reason for asking this is that there is, on the one hand, a significant body of claims that assert that mathematics is an art, and, (...) on the other, work in art that has a mathematical basis. Observations of these kinds are not trivial. They have significant implications for the teaching of these areas of the curriculum in at least two ways. First, there is the methodological issue of the extent to which we should teach mathematics and art separately, and second, the teleological question of why they appear in the curriculum at all. So the relationship between the nature of mathematics and art, perceived or real, bears down on questions of the individuation and the justification of these disciplines, or, in other words, upon pedagogy and purpose. Although in principle both the pedagogy and the purpose of any discipline are distinct, there are important connections between them, as we shall draw out.As far as mathematics goes, it has been stressed that the purposes of even rudimentary aspects of mathematics such as counting do need to be made explicit to pupils. 1 What have been seen as errors in understanding are internally linked with pupils' ideas of the purposes of counting. Anna Sierpinska discusses a similar situation within the broader aspect of what she calls "theory" in higher mathematics. 2 More generally, John Passmore, in his analysis of pupils' understanding includes the case where a pupil sees no need to understand since he or she cannot understand the point of learning the subject. 3 If the purposes of art and mathematics are strikingly similar then there is reason to suppose that certain pedagogical approaches that [End Page 1] follow from this will improve understanding of both mathematics and art, and therefore understanding of reasons for engaging in math-based and art-based activities.In schools we see what appears to be some evidence of connections between art and mathematics in cross-curricular work, such as that described in Jacqueline Cossentino and David Schaffer. 4 Apart from providing much-needed motivation, an important value of this approach toward mathematics is that it can illuminate pupils' understanding of some of its purpose by, at the very least, allowing them to recognize instances of mathematics at work, often in unsuspecting contexts. In particular, by facilitating some transference of knowledge pupils can appreciate the so-called "power" of mathematics in its widespread application to the world of artefacts produced by themselves and others.Can we go further than displaying connections between these disciplines? Can we treat mathematics itself as art? In particular, can we suppose that pupils can make mathematics just as they can make artefacts? Some writers seem to imply this by invoking the notion of creativity as a common property between the two disciplines. 5 Leaving aside the ironic sense of creativity in, for example, the case of "creative accountancy" pupils must indeed "make" sense of mathematics and they must originate mathematical assertions of their own that have not been directly taught. In both respects mathematics is similar to the learning of a language. Further than this, we value pupils' inventiveness in their approaches to areas of mathematics, calculations for example.The linking of mathematics and art via creativity has its most persuasive force in the change of attitude toward axiomatic foundations. 6 The story is familiar: for centuries Euclidean geometry was (wrongly) supposed to offer truths of the actual space in which we lived, since its axioms were thought to be confirmed by our intuitions of space. However, when it was found that consistent and useful axiomatic systems could be constructed by making certain alterations to these axioms, it became clear that mathematics was not simply discovered. It involved a significant measure of creativity. Yet one thing is clear: the sense that children make of mathematics, the original utterances they make... (shrink)

Utilizing culturally-conscious teaching promotes instruction that values the cultural knowledge and experiences of ethnically diverse students. This qualitative study centers culturally responsive teaching (Gay, 2002) and community cultural wealth (Yosso, 2005) to contextualize mathematics learning for an after-school tutoring program in North Minneapolis, Prepare2Npsire. The aim of this research was to use a community-based participatory action research approach to: 1) explore the culture wealth of North Minneapolis, 2) create culturally responsive mathematics activities for Prepare2Nspire. Two activities were developed (...) to center community assets and youth empowerment. The goal of these activities was to strengthen community, curriculum, and classrooms by intentionally connecting students’ multidimensional identities with mathematical learning. Specially, Muhammad’s (2020) culturally and historically responsive literacy framework was adapted using a mathematical lens to: 1) Create tasks that build mathematical proficiencies and 2) Create tasks that cultivate mathematical identities. The emphasis on participatory, community-based, and action-oriented practices in this study encouraged the disruption of deficit-based assumptions of communities while also creating joy-filled learning opportunities for Prepare2Nspire attendees. (shrink)

Context: Constructivist teachers who find themselves working within an educational system that adopts a realist epistemology, may find themselves at odds with their own beliefs when they catch themselves paying closer attention to the knowledge authorities intend them to teach rather than the knowledge being constructed by their learners. Method: In the preliminary analysis of the mathematical learning of six low-performing Year 7 boys in a Maltese secondary school, whom one of us taught during the scholastic year 2014-15, we constructed (...) a conceptual framework which would help us analyze the extent to which he managed to be sensitive to constructivism in a typical classroom setting. We describe the development of the framework M-N-L as a viable analytical tool to search for significant moments in the lessons in which the teacher appeared to engage in what we define as “constructivist teaching” during mathematics lessons. The development of M-N-L is part of a research program investigating the way low-performing students make mathematical sense of new notation with the help of the software Grid Algebra. Results: M-N-L was found to be an effective instrument which helped to determine the extent to which the teacher was sensitive to his own constructivist beliefs while trying to negotiate a balance between the mathematical concepts he was expected to teach and the conceptual constructions of his students. Implications: One major implication is that it is indeed possible for mathematics teachers to be sensitive to the individual constructions of their learners without losing sight of the concepts that society, represented by curriculum planners, deems necessary for students to learn. The other is that researchers in the field of education may find M-N-L a helpful tool to analyze CT during typical didactical situations established in classroom settings. (shrink)

The purpose of the present study was to investigate the effect and use of distributed practice in the context of self-regulated mathematical learning in high school. With distributed practice, a fixed learning duration is spread over several sessions, whereas with massed practice, the same time is spent learning in one session. Distributed practice has been proven to be an effective tool for improving long-term retention of verbal material and simple procedural knowledge in mathematics, at least when the practice schedule (...) is externally guided. In the present study, distributed practice was investigated in a context that required a higher degree of self-regulation. In total, 158 secondary school students were invited to participate. After motivational and cognitive characteristics of the students were assessed, the students were introduced to basic statistics, a topic of their regular curriculum. At the end of the introduction, the students could sign up for the study to further practice this content. Eighty-seven students did so and were randomly assigned either to the distributed or to the massed practice condition. In the distributed practice condition, they received three practice sets on three different days. In the massed practice condition, they received the same three sets, but all on one day. All exercises were worked in the context of self-regulated learning at home. Performance was tested two weeks after the last practice set. Only 44 students finished the study, which hampered the analysis of the effect of distributed practice. The characteristics of the students who completed the exercises were analyzed exploratory: The proportion of students who finished all exercises was significantly higher in the massed than in the distributed practice condition. Within the distributed practice condition, a significantly larger proportion of female students completed the exercises compared to male students. Additionally, among these female students, a larger proportion showed lower concentration difficulty. No such differential effects were revealed in the massed practice condition. Our results suggest that the use of distributed practice in the context of self-regulated learning might depend on learner characteristics. Accordingly, distributed practice might obtain more reliable effects in more externally guided learning contexts. (shrink)

In this report on an interview-based school case study undertaken with seven school leaders using component theory analysis and the hermeneutic method, we reveal the relational essence of learning design at the Australian Science and Mathematics School. The phenomenon of learning togetherness presents, forged by deliberately practised notions of contributive leadership within open learning spaces and ongoing attention to new interdisciplinary curriculum forms. This case study highlights the phenomenological nature of a school that has been deliberately purposed for (...) deep collaborative learning forms, respecting student and teacher ideas in the process, and marginalising habitual industrial school design forms that constrain effective student and teacher learning. The study has relevance for school leaders and teachers wishing to pursue new school design forms within enabling learning cultures that attend more closely to the learning needs of young people poised to enter the Third and Fourth Industrial Revolutions. (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)

Summary The National Curriculum was introduced into British primary schools in 1989 to raise standards of attainment, especially in the basic skills of English and mathematics. Has this expensive innovation succeeded? This paper analyses the mathematics standards of eight cohorts of Year 6 children from five randomly selected primary schools within one Local Education Authority (n=1503) who had all done Mathematics 11 from 1989 to 1996. Examination of the means of the standardised mathematics scores for (...) each cohort reveals no evidence of rising standards in attainment over the eight years. There are fewer over? and underachievers than one would expect with a tendency for compression of scores around the mean. This paper questions the efficacy of the National Curriculum, assessment procedures and the publication of school performance tables in raising standards in mathematics at Key Stage 2. The need for an effective system of monitoring educational standards throughout the UK is reiterated. (shrink)

The various papers and short "discussions" contained in this latest addition to the "Studies in Logic" series were presented at the 1965 International Colloquium in the Philosophy of Science, in London. Of the nine "problems" considered in this symposium, seven have directly to do with philosophy, one is an historical study of the origins of Euclid's axiomatics, and the last is an interesting—if one-sided—discussion of the "new math" controversy in the pre-college curriculum. Happily, this book demonstrates that the important (...) issues in the philosophy of mathematics somewhat outrun the usual ism-mongering debate between representatives of various schools. Unhappily, it also demonstrates that a good deal of undeserving material seems to find its way between the covers of scholarly books. After reading certain of the articles in this volume one is left with the feeling that the authors have not come to grips with the issues which they raise, that a lot of effort has been expended on peripheral matters, and that, in some instances, a sledge hammer has been brought down upon an egg. Abraham Robinson's article, "The Metaphysics of the Calculus," deals with a series of important linguistic and ontological problems concerning the foundations of analysis. Moreover, these are problems which can no longer be ignored, especially in view of the evolution of non-standard analysis. But, unfortunately, the interested reader would do far better to turn to Robinson's recent book on the subject, and pay special attention to the second, third, and last chapters therein. "On a Fregean Dogma" by Fred Sommers gives a rather curious non-quantificational algorithm for the monadic fragment of syllogistic. This Sommers does in the interest of "a conservative effort to restore categorical statements to predicative status." Despite the heat and the occasional interesting remarks which the discussion generated, one is inclined to ask, "So what?" The sequence of three papers by Mostowski, Bernays, and Körner is best considered as a group, since each of these is concerned with the philosophical significance of recent results in set theory and logic. Students of the philosophy of mathematics would do well to read this part of the book very carefully, although one wishes that the authors would elaborate a bit on their specific philosophical views. Finally, there is Georg Kreisel's difficult and amazing paper, "Informal Rigor and Completeness Proofs," in which is presented a mathematical treatment of the problem of selecting the primitive rules and concepts of mathematical systems. Kreisel's ideas are refreshing, unorthodox, dubious, and a genuine contribution to the philosophy of mathematics. In sum, this book is a mixed bag, uneven in the quality of its contents, but worth reading.—H. P. K. (shrink)

The aim of this book is to present mathematical logic to students who are interested in what this field is but have no intention of specializing in it. The point of view is to treat logic on an equal footing to any other topic in the mathematical curriculum. The book starts with a presentation of naive set theory, the theory of sets that mathematicians use on a daily basis. Each subsequent chapter presents one of the main areas of mathematical (...) logic: first order logic and formal proofs, model theory, recursion theory, Gödel's incompleteness theorem, and, finally, the axiomatic set theory. Each chapter includes several interesting highlights—outside of logic when possible—either in the main text, or as exercises or appendices. Exercises are an essential component of the book, and a good number of them are designed to provide an opening to additional topics of interest. (shrink)

The paper that follows is a report written in 1988 describing a city-wide program that was in effect in New York City from 1985 until 1988. It was intended for publication in a journal, The New York Supervisor, which unfortunately ceased to exist during that year. The program described included twelve school districts at its initiation and resulted in a variety of projects which included: A Philosophy for Children project in three districts; a three year effort to intregrate critical thinking (...) in the reading, social studies, and bilingual curricula in one district; a two year program to integrate thinking skills into a gifted and talented program in yet another; infusion of thinking skills into the reading program in two other districts; a mathematics curriculum for gifted middle school students, a science curriculum for junior high school students, a curriculum based on local history; a program for drop-out prevention; and a prgram to enhance critical thinking through a media center in five other districts. (shrink)

In a clear, nontechnical account, Jack Goldstein tells the story of this entrepreneurial American scientist who played an essential part in experiments important to the development of quantum mechanics, who later became an advisor to the government during much of the Cold War period, and whose leadership in educational reform resulted in the restructuring of the entire American high school science curriculum. Jerrold Zacharias was a physicist well placed by historical circumstance to take a central part in the development (...) of American science, science policy, and science education. In a clear, nontechnical account, Jack Goldstein tells the story of this entrepreneurial American scientist who played an essential part in experiments important to the development of quantum mechanics, who later became an advisor to the government during much of the Cold War period, and whose leadership in educational reform resulted in the restructuring of the entire American high school science curriculum. Zacharias lived at a time when an individual with imagination and courage could make a difference, whether at the forefront of science or in matters of public policy. He believed that every citizen, even those with modest scientific sophistication and knowledge, could learn to think like a scientist. Now, at a time when the issues of science education and science literacy are again of compelling national interest, his ideas merit close attention.Goldstein describes Zacharias's coming of scientific age in the early 1930s, as a member of 1. 1. Rabi's group at Columbia, and examines the leading role he played during World War II at MIT's Radiation Laboratory and at the Manhattan Project. From about 1955 on, Goldstein observes, Zacharias made significant contributions to science education in physics, chemistry, biology, and mathematics at the primary, secondary, and college levels. As a result of his initiatives, science and mathematics curriculum development flourished in a number of third-world countries. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Editor’s NoteJessica HeybachThis issue of education and culture offers readers theoretical in-sights and clarifications to social dilemmas as well as the concerns of the classroom. The authors contained in this issue take up questions of political literacy, moral judgment, the mathematics curriculum and classroom, and the social studies curriculum and classroom. If I had to offer a throughline within these articles, it is the pragmatist conception (...) of judgment that should be cultivated through social relations and classroom practices. These authors are concerned with citizens’ capacity for intelligent judgment within communities and society at large. Pushing back against prevailing logics, these authors offer corrective arguments by positioning Dewey’s thought and pragmatism in contrast to the current logics operating in classrooms and society.Charles Lowery takes up the consequences of modern society’s “conflation of an associated way of living with government, and government with politics, and politics with partisanship” on conceptions of community in his article, “The Role of Education in The Public and Its Problems: A Deweyan Perspective on Political Literacy.” Building on key authors that have previously advanced this line of inquiry central to Dewey’s The Public and Its Problems, Lowery attends to how contemporary education and the development of a politically literate citizenry through democratic habits rather than knowledge accumulation should be centered as the central task of schooling. Lowery highlights current obstacles to cultivating political literacy in the schools such as dogma, disinformation, divisiveness, and de-democratization, and concludes by underscoring the work of school leaders as central to the task of political literacy.Davin Carr-Chellman puts forth a provocative argument which attends to the consequences of instrumental rationality and identity thinking on moral judgment in his article, “Education for Moral Judgment: Situational Creativity and Dewey’s Aesthetics.” Drawing from critical theory, Carr-Chellman emphasizes how dehumanization, moral indifference, and alienation are byproducts of instrumental rationality, measurability, and identity thinking. He then engages Dewey’s aesthetics and Joas’s concept of situational creativity as possible resources to reconstruct human experience and renew the human capacity for moral judgment. According to Carr-Chellman, small acts of creativity and relationship-building become essential to resisting human relations that have become “abstract, impersonal, vague, and competitive.” Thus, an education for moral judgment must humanize through authentic relationships and meaning-making learning opportunities that resist the alienation now culturally rampant. [End Page 1]Marshall Gordon argues that mathematics curriculum and classroom should be understood as a site for developing habits of mind that foster democracy in his article, “Teaching Mathematics with Democracy in Mind.” Gordon elaborates on the problematic persistence of abstracted, procedural mathematics which valorizes memorization and calls for mathematics to capitalize on the unrealized democratic potentials in the mathematics classroom. Drawing on Dewey’s claim in Democracy and Education that “Education is not an affair of ‘telling’ and being told, but an active and constructive process,” Gordon describes a mathematics for democracy. Gordon describes mathematics as a powerful force in the development of students’ reflective intelligence and their capacity for problem-solving by problem-clarifying strategies rather than problem-solving procedures (for example, “take things apart, make the problem simpler, reason by analogy, and tinker”) and collaboration through “multiple-centers” which capitalizes on shared-interest investigations.Dave Powell explores the relationship between inquiry and Dewey’s pragmatic epistemology to advance democratic citizenship education in the social studies classroom in his article, “Has Anybody Here Seen My Old Friend John? Making the Case for a More Pragmatic Social Studies.” In particular, Powell attends to educators’ stance on truth and how “reframing their epistemological orientations” opens up possibilities for different future actions. He begins by describing how pragmatist thinking and Dewey’s philosophical commitments of child-centeredness could shape conceptions of inquiry and allow a reflective practitioner to ask different kinds of questions associated with social studies. Through a rich discussion of what Deweyan thought makes possible in social studies, Powell advances the promise of a citizenry able to think pragmatically in a democracy. He argues that social studies can reject fixed notions of truth, foster trust among students as they exchange ideas, reject opportunities for reverse indoctrination, and cultivate intelligence, as practical... (shrink)

This is the BJPS Short Read version of the article When Fields Are Not Degrees of Freedom. In our article, Vera Hartenstein and I show that the world of classical electromagnetism might differ radically from the one we see in physics textbooks and experience day-to-day. First, light may not exist; second, the laws of electromagnetism are either incomplete or completely different; and, third, the mathematics needed to make exact calculations with these novel laws is in early development and not (...) part of the general mathematics curriculum at colleges and universities. (shrink)