Open peer commentary on the article “Constructivist Model Building: Empirical Examples From Mathematics Education” by Catherine Ulrich, Erik S. Tillema, Amy J. Hackenberg & Anderson Norton. Upshot: I comment on the nature and exemplification of second-order models in Ulrich et. al. I discuss what I see as the theoretical gap between second-order models and mathematics pedagogy. Finally, I share work we are doing to contribute towards filling that theoretical gap.

Hermann Grassmann's Ausdehnungslehre of 1844 and his Lehrbuch der Arithmetik of 1861 are landmark works in mathematics; the former not only developed new mathematical fields but also both contributed to the setting of modern standards of rigor. Their very modernity, however, may obscure features of Grassmann's view of the foundations of mathematics that were not adopted since. Grassmann gave a key role to the learning of mathematics that affected his method of presentation, including his emphasis on making (...) initial assumptions explicit. In order to better understand this less well-known aspect of his work it will help to examine why some commentators have overlooked his theme of unifying logic, pedagogy and foundations, while others have recognised it. (shrink)

This paper traces the genealogy of a long-enduring controversy in Western philosophy viz, whether philosophic and mathematical methodologies are equal but separate and distinct approaches to rational inquiry, or whether one is superior to the other from the standpoint of epistemology, and, ultimately, a pedagogy which supports and promotes conceptual and critical thinking. With the Socratic teacher in mind, philosophic methodology, viewed by Plato as a dialectical process of free-ranging inquiry, compelled him to distinguish the work of philosophy from (...) that of mathematics, since philosophic methodology could not function unless freed from the constraints contained in a system built on axioms, propositions, and images. In his dialogue Meno, Socrates’ pedagogic demonstration, supported by the unique capacity of mathematics to provide epistemic closure, foreshadowed a slow-growing rift between philosophy and mathematics, one which came to a head in the 17th century rationalism of Descartes. In Descartes’ work on methodology, mathematics, by overriding the seemingly inconclusive meanderings of philosophy, became the supreme pathway to knowledge. Philosophy, thereafter, had to fight for its life, while pedagogy, questing more and more for “completeness” through formalization, ironically moved toward a fragmentation and reductionism which a mathematized Cartesian methodology supported. The curriculum divisions which resulted were presaged in Descartes’ writing on methodology; the isolation of the individual learner from the larger community of discourse was supported by Descartes’ “cogito.” Suggested here is that a pedagogic approach to the teaching of mathematics, which extends beyond the limitations of the “single solution” methodology utilized in Meno, is possible—an approach which allows pedagogy to spread the Socratic use of dialectical thinking even into that sphere of apodicticity. This suggests that an alternative to the monistic rationalism of Descartes’ mathematized methodology is possible for pedagogy in general, now bringing back into play what Descartes found expendable—viz. dialectical philosophy itself, dialogical inquiry, history, literature, and the arts. Had Socrates, who came to philosophy primarily as a teacher fascinated with intellectual paradoxes, explored, in Meno, the pedagogic possibility of a dialectical approach to mathematics, his larger work in dialectical pedagogy might have been less teacher-centered—notwithstanding his ironic “ignorance”; and given his historic significance as the most celebrated teacher in Western culture, modern pedagogy might have been less overwhelmed by the Cartesian drive toward mechanized and reductionist “single-solution,” non-communal teaching and thinking. (shrink)

The paper describes the Czech project in the history of mathematics which was initiated at the Faculty of Mathematics and Physics of Charles University in Prague at the end of the eighties of the 20th century. Its main aim is to map the development of mathematical research in the Czech lands in the second half of the nineteenth and the first half of the twentieth century. The main result of this project is the production of monographs. These chart (...) out the life and work of some prominent Czech personalities-mathematicians who have had substantial influence on the development of Czech mathematics. The aim of the project, the structure of the monographs and the general method are discussed in the paper. (shrink)

A beneficial effect of gesture on learning has been demonstrated in multiple domains, including mathematics, science, and foreign language vocabulary. However, because gesture is known to co-vary with other non-verbal behaviors, including eye gaze and prosody along with face, lip, and body movements, it is possible the beneficial effect of gesture is instead attributable to these other behaviors. We used a computer-generated animated pedagogical agent to control both verbal and non-verbal behavior. Children viewed lessons on mathematical equivalence in which (...) an avatar either gestured or did not gesture, while eye gaze, head position, and lip movements remained identical across gesture conditions. Children who observed the gesturing avatar learned more, and they solved problems more quickly. Moreover, those children who learned were more likely to transfer and generalize their knowledge. These findings provide converging evidence that gesture facilitates math learning, and they reveal the potential for using technology to study non-verbal behavior in controlled experiments. (shrink)

RESUMEN Introducción: La dimensión ambiental constituye un proceso esencial en la formación de profesionales, de modo que en el desempeño de su profesión puedan educar hacia el desarrollo sostenible a las futuras generaciones. Objetivo: Representar en un modelo pedagógico la lógica de la dimensión ambiental en la formación laboral investigativa de los profesionales en la universidad. Materiales y métodos: Se utilizaron métodos teóricos, empíricos y matemáticos-estadísticos. Resultados: Elaboración del modelo pedagógico de dimensión ambiental en la formación laboral investigativa de los (...) profesionales en formación. El modelo está estructurado en tres subsistemas: uno teórico, se centra en la integración de saberes pedagógico ambientales a partir de las relaciones que se establecen entre sus componentes, otro metodológico, el cual resume la proyección ambiental de la formación laboral investigativa y un subsistema procedimental, que manifiesta la instrumentación transformadora ambiental que debe ocurrir en los estudiantes. El modelo aporta una cualidad sinérgica que le imprime identidad, totalidad y contribuye a su autodesarrollo. Conclusiones: Con la elaboración del modelo se revelan nuevas interpretaciones que se manifiestan como relaciones teóricas funcionales surgidas entre los componentes de los subsistemas que lo integran y a la vez, entre los subsistemas como macro componentes. La validación del modelo resultó estadísticamente significativa, mediante la instrumentación de una estrategia pedagógica para la dimensión ambiental en la formación laboral investigativa. ABSTRACT Introduction: The environmental component is an essential process in the training of professionals, that enables them to educate future generations towards sustainable development. Objective: To represent in a pedagogical model the logic of the environmental dimension in the research training of professionals. Materials and methods: Theoretical, empirical and mathematical-statistical methods were used. Findings: The development of the pedagogical model of environmental dimension in research and labor training of professionals. The model is structured in three subsystems: the first focuses on the integration of environmental pedagogical knowledge from the relationships established between its components; the second is of a methodological character and summarizes the environmental projection of research training; whereas the third one is procedural, expressing the environmental transformative instrumentation that must occur in students. The model provides a synergistic quality that consolidates identity, totality and contributes to its self-development. Conclusions: With the elaboration of the model new interpretations are revealed that are manifested as functional theoretical relations arising between the components of the subsystems that integrate it and at the same time, between the subsystems as macro components. The evaluation of the model was statistically significant, through the instrumentation of a pedagogical strategy for the environmental dimension in the research and labor training. (shrink)

This paper develops a methodology for using Activity Theory (AT) to investigate pedagogical practices in primary school mathematics classrooms by selecting object-oriented pedagogical activity as the unit of analysis. While an understanding of object-oriented activity is central to Activity Theory (AT), the notion of object is a frequently debated and often misunderstood one. The conceptual confusion surrounding the object arises both from difficulties related to translating the original Russian conceptualisation of object-oriented activity into English as well as from the (...) different interpretations of the object currently in use within two contemporary approaches in activity theory. This paper seeks to clarify understandings of the object by exploring notions of object oriented activity. To this end, the paper traces the historical development of the object through Leontiev (1978; 1981) and Engeström’s (1987; 1999) expansion of Vygotsky’s original triadic understanding of object oriented activity. Drawing on Basil Bernstein’s (1996) notion of evaluative criteria as those rules that transmit the criteria for the production of a legitimate text, the paper goes on to elaborate a methodology for using AT to analyse observational data by developing the notion of “evaluative episodes” as pedagogical moments in which the pedagogical object is made visible. Findings indicate that an evaluative episode can serve as an analytical space in which the dynamism of an activity system is momentarily frozen, enabling one to model human activity in the system under investigation and, hence, in this study, to understand pedagogy in context. (shrink)

Good teaching requires not only an acute sensitivity to one’s students but also a kind of flexibility enabling one to respond appropriately in concrete situations. This paper analyzes the pedagogical strategy that Socrates employs with Callicles and Theaetetus, arguing that Socrates exhibits the kind of flexibility required of a good teacher. In articulating the pedagogical flexibility that Socrates's exhibits, the paper also provides an overview of the divided line, the mathematical curriculum that Socrates proposes in the “Republic,” and a discussion (...) of how mathematical education relates to philosophy. (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)

Over the past few years we have seen an increasing number of legal proceedings related to inappropriately implemented technology. At the same time career paths have diverged from the foundation of statistics out to Data Scientist, Machine Learning and AI. All of these new branches being fundamentally branches of statistics and mathematics. This has meant that formal training has struggled to keep up with what is required in the plethora of new roles. Mathematics as a taught subject is (...) still based in decades old teaching specifications and has not been updated centrally as a curriculum to include new technologies, coding or ethics. This subject area is firmly split between ICT and Mathematics in secondary school, continuing on to be split between Computer Science and Mathematics at University. As we move forwards with technology we see these once seperate fields becoming increasingly intertwined. We propose that existing provision for concepts such as ethics and societal responsibility in analysis currently exist but have not been incorporated into the mainstream curriculum of School or University. This is partially due to the split between fields in an educational setting but also the speed with which education is able to keep up with Industry and its requirements. Principles and frameworks of socially responsible modelling beginning at school level means that ethics and real-life modelling are introduced much earlier than normal. Integrating these concepts with philosophical principles of society and politics ensures a suitable background for future modellers and users of technology to draw on. Modelling is currently undertaken in technical sciences at University but the Subject Benchmark Statements are not current (Subject Benchmark Statements describe the nature of study and the academic standards expected of graduates in specific subject areas. They show what graduates might reasonably be expected to know, do and understand at the end of their studies). Even in 2019 the UK did not yet have Benchmark Statements that discuss the learning to be done in Higher Education around AI and advanced Machine Learning. Where there is provision for AI and Data Science within degree courses ethics is not generally highlighted as a key concept. As such there can be a lack of focus on the teaching of modelling or ethics as specific skills. The skills required to use a basic statistical model, for example would not be sufficient to start from scratch and build an ethical model reflecting real-world scenarios with which to inform policy or organisational decision making. This is a skill in itself and includes such aspects as awareness of data quality, ethics, user implementation problems, context and an understanding of the environment that the model is being created in. As the field of analytics has progressed so quickly in the last decade modules such as those covering AI have simply been bolted on to Maths or Computing degrees rather than being fully detailed as key areas of study or driving new, more integrated courses of study. This paper posits that gaps at primary, secondary and tertiary educational levels need to be addressed. Implementing and integrating key concepts from school level is essential so that areas such as assumptions, caveats, quality assurance and answering the right questions with constructive challenge become a cultural fixture. This not only helps developers of technology but users of rapidly developing technology also. In addition, leadership and soft skills as part of this education will ensure that a cultural shift can take place and promote continuous improvement in analysis within organisations. The addition of key concepts throughout the educational system and the updating of potentially outdated curriculums is key to ensuring a functional society. A society where every citizen is a user of tech and those that develop it can be ethical and socially responsible in its development. (shrink)

The present study assessed the language teachers' pedagogical beliefs and orientations in integrating technology in the online classroom and its effect on students' motivation and engagement. It utilized a cross-sectional correlational research survey. The study respondents were the randomly sampled 205 language teachers (μ= 437, n= 205) and 317 language students (μ= 1800, n= 317) of select higher educational institutions in the Philippines. The study results revealed that respondents hold positive pedagogical beliefs and orientations using technology-based teaching in their language (...) classroom. Test of difference showed that female teachers manifested a firmer belief in student-centered online language teaching than their male counterparts. However, the utilization and attitude towards technology in the language classroom is favorably associated with the male teachers. As to students' level of language learning motivation and engagement, it was found out that male and female students have high level of language learning engagement. Further, the test of relationship showed that the higher the teachers' belief in utilizing student-centered teaching to integrate technology in the language classroom, the higher the students are motivated and engaged in learning. In like manner, it was also revealed that teacher-centered belief is negatively correlated to student’s motivation and engagement in online language learning. In this regard, the pedagogical assumptions that hold EFL teachers positively to integrate technology in the language classroom. This study generally offers implications for enhancing language teacher's digital literacy to promote motivating, fruitful, and engaging language lessons for 21ts century learning. (shrink)

Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science (italics mine) requires making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role regarding the learning, development and use of all the sciences. Such philosophical principles have not (...) come from speculation but from studying and concentrating the development of actual mathematical subjects such as algebraic geometry, functional analysis, continuum mechanics, combinatorics, etc. The simplifications in the conceptual relations revealed by such philosophical/mathematical advances are of great value, not only in: (1) clarifying and guiding further research in the mathematical sciences, but also in (2) reforming pedagogy and popularization in ways that will not prove deceptive. (shrink)

Le rôle des mathématiques dans la formation des officiers de l’armée néerlandaise, a profondément changé pendant le premier xixe siècle avec la fondation de l’Académie militaire en 1828. Les mathématiques étaient au centre de la formation. L’Académie était un des lieux les plus importants de diffusion des connaissances mathématiques aux Pays-Bas pendant la première moitié du xixe siècle, mais elle a perdu ce rôle pendant les années 1860-1870. Dans cet article, je me propose d’examiner à la fois les programmes de (...) l’Académie et le rôle central des enseignants de mathématiques de l’Académie dans la Société Mathématique néerlandaise et dans l’Académie royale des Sciences. Par ailleurs, l’évolution des mathématiques, visible dans ces deux institutions dans les années 1850, n’a pas entraîné une adaptation des programmes au sein de l’Académie. Ainsi, à partir de 1870, l’Académie a presque complètement disparu de la vie mathématique des Pays-Bas. The role of mathematics in the education of Dutch army officers changed dramatically in the early 19th century. This shift became visible with the founding of the Dutch Royal Military Academy in 1828, which shows the role of mathematics in military education shifting toward a purely pedagogical function. This change in approach, which primarily occurred in 1860’s, affected the standing and perception of an academy once highly respected in the field of mathematics. In the present work, I examine both the curricula at the Academy and the crucial role of the Academy’s mathematics instructors in the Dutch Mathematical Society and the Royal Academy of Sciences. Importantly, these two institutions managed to successfully adapt to the changing character of mathematical sciences that occurred around 185 Mathematicians at the Royal Military Academy, however, appeared to cling to an outdated view of mathematics. Failing to update their approach ultimately caused the Academy to lose its once prominent place in the Dutch mathematical community. (shrink)

Open peer commentary on the article “Radical Constructivist Structural Design Education for Large Cohorts of Chinese Learners” by Christiane M. Herr. Upshot: In this comment, I take Herr’s proposition for a constructivist-informed pedagogy for structural design education to extract initial ideas for a framework for a constructivist pedagogy, a framework focused on the decision-making of a constructivist teacher. I enhance this initial framework with initial findings of a study I conducted with a constructivist mathematics teacher.

Frank Quinn of Jaffe-Quinn fame worked out the basics of his own account of how mathematical practice should be described and analyzed, partly by historical comparisons with 19th century mathematics, partly by an analysis of contemporary mathematics and its pedagogy. Despite his claim that for this task, "professional philosophers seem as irrelevant as Aristotle is to modern physics," this philosophy talk will provide a critical summary of his main observations and arguments. The goal is to inject some (...) of Quinns remarks to the current conversation on mathematical practice. [1] Jaffe, Arthur, Quinn, Frank. "Theoretical Mathematics: Towards a synthesis of mathematics and theoretical physics," Bulletin of the American Mathematical Society NS 29:1, 113. [2] Quinn, Frank. Contributions to a Science of Mathematics, manuscript, 98pp. [3] Quinn, Frank. "A revolution in mathematics? What really happened a century ago and why it matters today," Notices American Mathematical Society 59:1 31-37. (shrink)

There has been a major international focus on the education and care of toddlers. To date, empirical studies on adults’ interactions in play with toddlers have focussed on the proximity of teachers, teachers’ affective responses, and joint attention between adults and children in play. However, less attention has been given to the role of two teachers working together in supporting children’s exploration of concepts in joint play. This paper takes a cultural-historical perspective and draws upon the concepts of play and (...) subjective positioning to investigate how co-teaching promotes toddlers’ joint play and exploration of mathematical concepts. Visual data of a group of toddlers playing with two teachers is analysed to explore how the teachers applied co-teaching pedagogy to support children’s exploration of mathematical concept. The study argues that co-teaching provides potential to support collective play as two teachers can work collaboratively to address the toddlers’ challenges and needs and support their exploration of the rules and mathematical concepts in play. Co-teaching by paired teachers enables toddlers’ joint play to become more complex, helping the children to explore mathematical concepts and, in turn, further enhancing their play. (shrink)

An introduction to Mathematical Logic using a unique pedagogical approach in which the students implement the underlying conceps as well as almost all the mathematical proofs in the Python programming language. The textbook is accompanied by an extensive collection of programming tasks, code skeletons, and unit tests. The covered mathematical material includes Propositional Logic and first-order Predicate Logic, culminating in a proof of Gödel's Completeness Theorem. A "sneak peak" into Gödel's Incompleteness Theorem is also provided.

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)

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)

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)

This second edition encourages the integration of technology into a pedagogically sound learning sequence for primary mathematics.

Mathematicians often describe the importance of well-developed intuition to productive research and successful learning. But neither education researchers nor philosophers interested in epistemic dimensions of mathematical practice have yet given the topic the sustained attention it deserves. The trouble is partly that intuition in the relevant sense lacks a usefully clear characterization, so we begin by offering one: mature intuition, we say, is the capacity for fast, fluent, reliable and insightful inference with respect to some subject matter. We illustrate the (...) role of mature intuition in mathematical practice with an assortment of examples, including data from a sequence of clinical interviews in which a student improves upon initially misleading covariational intuitions. Finally, we show how the study of intuition can yield insights for philosophers and education theorists. First, it contributes to a longstanding debate in epistemology by undermining epistemicism, the view that an agent’s degree of objectual understanding is determined exclusively by their knowledge, beliefs and credences. We argue on the contrary that intuition can contribute directly and independently to understanding. Second, we identify potential pedagogical avenues towards the development of mature intuition, highlighting strategies including adding imagery, developing associations, establishing confidence and generalizing concepts. (shrink)

Encyclopaedism” is the systematically pursued activity within the Unity of Science Movement, devoted to the collecting of scientific statements of the logic of science logic, mathematics, physics, biology and other sciences, including all the “cross connections” between the given disciplines.

This article sheds new light on Karl Marx’s theoretical legacy to promote educational philosophy and theory in science, technology, engineering, arts, and mathematics (STEAM) education. Accordingly, this article selects Marxian thought as an exemplar case to elaborate critical pedagogy and aesthetics and culminate with an overview of the Marxian approach to the theoretical underpinnings of STEAM education. Hence, the article critically reviews Marx’s original works and earlier Marxian scholars’ contributions to extending critical pedagogy and its applications to (...) the STEAM paradigm and informational and communication technology (ICT) development. This article critiques notable structural problems of STEAM education that produce power differentials such as educational gaps and digital divides in the current age of digital capitalism. Given the context, this article refines art and communication theories through the lens of his biography and writings. Additionally, this article incorporates the Marxian viewpoints about reproduction and aesthetic literacy into STEAM education. Overall, this article reviews the theoretical legacy of Marx and his essential critical and pedagogical theory, calling on scholars to pay close attention to re-orienting STEAM pedagogy as an emerging multidisciplinary field. (shrink)

Recent politicization of mathematics has driven questions about its pedagogy in U.S. schools, but these questions fail to recognize mathematics as a potentially oppressive tool. In this essay, I demonstrate that there are much larger forces at play and that improving the way people think about mathematics and how people regard it in their lives are much more valuable. Here, I briefly explore three distinct eras of mathematical development that yielded three distinct cultural responses. First is (...) Fibonacci, whose work was generally accepted and steered clear of criticism. Next is Galileo, a mathematician who faced significant pushback and whose research and progress were halted and banned. I then examine the United States and highlight a government’s ability to wrest control of mathematics structures absent the ability to completely stifle publication and research. I will draw an arc through these three distinct eras to conclude the most effective way to dismantle mathematically-based power structures: a mutual effort to flatten hierarchies that equally involve those in power and those at the mercy of the structure itself. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...) all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to (...) class='Hi'>mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (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)

The economist Paul Samuelson acknowledged that he was a disciple of Edwin Bidwell Wilson (1879–1964), an American polymath who was a protégé of Josiah Willard Gibbs. Wilson’s influence on the development of sciences in America has been relatively neglected, as he mostly acted behind the scenes of academia at the organizational and pedagogical fronts. At the basis of his activism were original ideas about the foundations of mathematics and science. This essay reconstructs Wilson’s career and foundational discussions, which evolved (...) as he reacted to David Hilbert’s mathematics, Bertrand Russell’s logic, Henri Poincaré’s conventionalism, Karl Pearson’s statistics, Charles Sanders Peirce’s inference theory, and Lawrence Henderson’s work in social sciences. In brief, after having started his professional life as a mathematician at Yale University (1902–1907), Wilson marginalized himself from mathematics and joined other academic communities and fields, working in mathematical physics at MIT (1907–1922) and in statistics, social science, and economics at Harvard University (1922–retirement). He defined mathematics as a language, meaning by this that mathematics and science were reciprocally indispensable, with the former providing structure and the latter meaning. In an American exceptionalist spirit, Wilson also meant to suggest that homegrown mathematical and scientific traditions should prevail over European ones. (shrink)

Teaching Newtonian physics involves the replacement of students’ ideas about physical situations with precise concepts appropriate for mathematical applications. This paper focuses on the concepts of ‘matter’ and ‘mass’. We suggest that students, like some pre-Newtonian scientists we examine, use these terms in a way that conflicts with their Newtonian meaning. Specifically, ‘matter’ and ‘mass’ indicate to them the sorts of things that are tangible, bulky, and take up space. In Newtonian mechanics, however, the terms are defined by Newton’s Second (...) Law: ‘mass’ is simply a measure of the acceleration generated by an impressed force. We examine the relationship between these conceptions as it was discussed by Newton and his editor, Roger Cotes, when analyzing a series of pendulum experiments. We suggest that these experiments, as well as more sophisticated computer simulations, can be used in the classroom to sufficiently differentiate the colloquial and precise meaning of these terms. (shrink)

Much of the neuroimaging research has focused on how mathematical operations are performed. Although this body of research has provided insight for the refinement of pedagogy, there are very few neuroimaging studies on how mathematical operations should be taught. In this article, we describe the teaching of algebra in Singapore schools and the imperatives that led us to develop two neuroimaging studies that examined questions of curricular concerns. One of the challenges was to condense issues from classrooms into tasks (...) suitable for neuroimaging studies. Another challenge, not particular to the neuroimaging method, was to draw suitable inferences from the findings and translate them into pedagogical practices. We describe our efforts and outline some continuing challenges. (shrink)

psychiatric diagnoses such as attention deficit hyperactivity disorder is a rapidly growing and globally increasing phenomenon, not least in different educational contexts such as in family and in school. Children and youths labelled as ADHD are challenging normative claims in terms of nurturing and education, whereas those labelled as ADHD are considered a risk for society to handle. The dominant paradigm regarding ADHD is biomedical, where different levels of attention and activity-impulsivity are perceived as neurobiological dys/functions within the brain best (...) managed by means of an individual diagnosis and instrumental pedagogy. The majority of those labelled as having ADHD encounter a dominant educational model in the form of what is referred to in this article as neurobehaviorism, which is based on onto-epistemological violence. As opposed to this act of violence against being—and against the psychiatrized subject—a less violent educational model is proposed, based on French philosopher Alain Badiou’s ontological examination of being and his concept of love as a truth procedure. In terms of the latter, the focus is on the potential of the encounter as a ‘Two scene of love’. Here, the encounter is a place where it is possible to create new truths and subjects, instead of taking the individual diagnosis as an axiom which only leads to individuals having fixed identities codified in a hierarchical order. This argument is drawn from the ‘mathematical’ formula 1 + 1 = ♥, which originates from an online forum for people who have come into contact with ADHD in one way or another. (shrink)

In this paper, we discuss a proposal for reform in the teaching of Euclidean geometry that reveals the symbiotic relationship between axiomatics and pedagogy. We examine the role of intuition in this kind of reform, as expressed by Mario Pieri, a prominent member of the Schools of Peano and Segre at the University of Turin. We are well aware of the centuries of attention paid to the notion of intuition by mathematicians, mathematics educators, philosophers, psychologists, historians, and others. (...) To set a context for Pieri’s proposal, we only seek to open a small window on views of the pedagogical role of intuition, from primary education to university study that may have informed early 20th century efforts to improve the teaching of geometry at the secondary school level. Pieri addressed the topic of intuition in many of his axiomatizations, including those in projective geometry which was his main area of concentration in foundations. We focus here primarily on his axiom systems for elementary geometry, which embraced the transformational approach of Felix Klein’s vision for the subject. Our goal is to convey Pieri’s thoughts on how to integrate two types of intuition, denoted as sensible and rational, in his endeavors to improve the teaching of the geometry of Euclid. We show how Pieri’s views on geometric intuition and pedagogical reform were either ignored or misrepresented in several notable publications at the turn of the 20th century. In particular, we give Pieri a voice in response to specific comments made in the early 1900s by Federigo Enriques, Ugo Amaldi, and Florian Cajori in widely circulated publications inspired by Klein. (shrink)

The 1990s could be called The Decade of Sociology in mathematics education. It was during those years that the sociology of mathematics became a core ingredient of discourse in mathematics education and the philosophy of mathematics and mathematics education. Unresolved questions and uncertainties have emerged out of this discourse that hinge on the key concept of social construction. More generally, what is at issue is the very idea of “the social”. Within the framework of the (...) general problem of “the social”, we want to open a discussion of boundaries and margins in mathematics and mathematics education. By theorizing the divisions of purity and danger, we will be able to better understand the intersection of logic, mathematics, and thinking with gender, race, and class, and morals, ethics, and values in the classroom. The process of transforming the sociology of mathematics and the sociology of mind into pedagogical tools for mathematics educators and philosophers of education has already begun. One of the tasks before us is the development of a more profound and at the same time more practical grasp of “the social”. Our objective in this paper is to move ourselves and our readers in the direction of just such a grasp of the social. (shrink)

Mathematics and music in practice and performance, and in learning and teaching, share many characteristics, such as beauty and harmony, memory and intuition and mind or intellect. These raise the principles of processing information in mathematics and music and, by implication, the role of an acquaintance with the essentials of perception, abstraction and affective connaturality in teacher education. This paper compares mathematics and music and considers the acquisition of knowledge and skills through the external and internal senses (...) and emotions, utilizing the role of knowledge through multiple intelligences. In doing so it does not canvas the utilities of mathematics and music as fields of human endeavour so much as their role in the cultivation of serenity and knowledge in the cultured mind. This is a theoretical paper but it is based on nearly a century of teaching from the combined work of the two authors in the teaching of music and mathematics. The paper highlights the importance of inspiration in teaching, inspiration built on a thorough basis of the foundations of anthropology to include the emotions as well as the intellect. While teacher education programs rightly concern themselves with knowledge of the field of study, knowledge of pedagogy, they do not always consider the ability to inspire which is at the heart of managing and mentoring people. (shrink)

_Published by Taylor & Francis Group for the American Association of Colleges for Teacher Education_ This _Handbook_ addresses the concept and implementation of technological pedagogical content knowledge -- the knowledge and skills that teachers need in order to integrate technology meaningfully into instruction in specific content areas. Recognizing, for example, that effective uses of technology in mathematics are quite different from effective uses of technology in social studies, teachers need specific preparation in using technology in each content area they (...) will be teaching. Offering a series of chapters by scholars in different content areas who apply the technological pedagogical content knowledge framework to their individual content areas, the volume is structured around three themes: What is Technological Pedagogical Content Knowledge? Integrating Technological Pedagogical Content Knowledge into Specific Subject Areas Integrating Technological Pedagogical Content Knowledge into Teacher Education and Professional Development The _Handbook of Technological Pedagogical Content Knowledge for Educators_ is simultaneously a mandate and a manifesto on the engagement of technology in classrooms based on consensus standards and rubrics for effectiveness. As the title of the concluding chapter declares, "It’s about time!" The American Association of Colleges for Teacher Education is a national, voluntary association of higher education institutions and related organizations. Our mission is to promote the learning of all PK-12 students through high-quality, evidence-based preparation and continuing education for all school personnel. For more information on our publications, visit our website at: www.aacte.org. (shrink)

This paper draws attention to the tendency of a range of technologies to reduce pedagogical interactions to a series of datafied transactions of information. This is problematic because such transactions are always by definition reducible to finite possibilities. As the ability to gather and analyse data becomes increasingly fine-grained, the threat that these datafied approaches over-determine the pedagogical space increases. Drawing on the work of Hegel, as interpreted by twentieth century French radical philosopher Alexandre Kojève, this paper develops a model (...) of relational pedagogy which highlights three points of incompatibility with a datafied learning environment reduced to finite measures. Firstly: Kojève’s accont of authority in Hegel posits two aspects to the mimetic relation between teacher and student: recognition and realisation, which belong to the ipseity or about-self-ness of the subject, and are incompatible with a general definition of data. Secondly, the Hegelian approach to human historical time, in particular the assertion that time and desire are begun in the future, not the past, renders it incompatible with mathematical time as used in data processing. Finally, from these it is possible to derive a distinctive notion of the work of pedagogy, grounded in Kojève’s realist reading of Hegel, irreducible to information processing. In consequence of this threefold irreducibility, the paper draws attention to a need for relations of human pedagogical work to be inherent in the design of educational technologies and highlights the dangers of presuming a machine intelligence model in the design of learning environments. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Eclecticism and the Technologies of Discernment in Pietist PedagogyKelly J. WhitmerWhile the Franckesche Stiftungen (the Francke Foundations) of Halle/Saale are perhaps best known today as the institutional centre of German Pietism, throughout much of the eighteenth century they were widely regarded as a pedagogically innovative Schulstadt (or city of schools). The founder of this Schulstadt, August Hermann Francke (1663–1727), was many things to many people: Pietist, radical Lutheran, theologian, (...) pedagogue, professor of Greek and Oriental languages, preacher, member of the Berlin Academy of Sciences and proponent of a Protestant mission. Like most participants in what was initially a grassroots movement oriented around the personality of Philipp Jakob Spener (1635–1705), a minister active in Frankfurt am Main around 1670 who sought to breathe new life into the Lutheran church,2 Francke rarely referred to himself as a Pietist. From his post at the University of Halle (founded in 1694), this Spenerian prote´gé set out to create a new "universal seminar" that would make the assimilation of many diverse points of view, identities, and competing strands of knowledge its [End Page 545] defining mandate. "My purpose," Francke wrote in a letter to a Pastor Ring from Grünebartau (October 29, 1712), "is to develop a universal institution, useful for bringing into a good and holy order all that is important for the Christianity of all orders of society and relates, in some way, to the healing power of the soul."3 His Schulstadt would "bring into order" a myriad of relationships around a single point and relied, quite deliberately I propose, on an eclectic methodology to do this.In recent years, the relationship of Francke's Schulstadt to the implementation of Brandenburg-Prussia's first compulsory schooling system and teacher training institutes has been better documented than in the past.4 Scholars of education have paid closer attention to the methodological innovations of Pietist pedagogy, and its points of convergence with the ideas of John Locke (1632–1704), Jean-Jacques Rousseau (1712–1778) and Johann Heinrich Pestalozzi (1746–1827) have become subjects of mounting interest.5 Francke's significance for the history of education has been assessed mainly in terms of his ability to realize the pedagogically inclined reform projects of seventeenth-century figures, such as Johann Amos Comenius (1592–1670).6 We know that the thousands of children who participated in the life of the Schulstadt very often left Halle to become actively involved as teachers and ministers throughout the German states. Several played pivotal roles in the development of the late eighteenth-century educational reform movement known as Philanthropismus and its associated practices and institutions.7 [End Page 546]Perhaps the most striking feature of Francke's Schulstadt was its curriculum, which featured innovative training regimens in applied mathematics for children from diverse social backgrounds.8 The scale and rigor of these programs were to be found nowhere else at the time and can be linked to the overtly pedagogical mandate of the Berlin Academy of Sciences (f. 1701). These regimens deliberately juxtaposed artisanal and learned knowledge and stressed the importance of the eye as a conciliatory medium.9 Despite having acquired largely anti-empiricist reputations over the years due to their lingering interest in alchemy and their roles in securing the banishment of the "rationalist" mathematician and philosopher, Christian Wolff (1679–1754), from their community in 1723, careful consideration of what Halle Pietists were teaching in their schools suggests a different view. These historical actors reconciled categories of knowledge production and transmission often framed as oppositional—despite what several scholars have recently shown to be their clear interconnectedness.10I argue here for the predominance of eclecticism in the Schulstadt, including a pronounced emphasis on the "middling" mathematical and mechanical sciences as tools of conciliation and self-improvement. I seek to illustrate how eclecticism was much more than a Philosophiebegriff that was well received by some Pietists, as Walter Sparn has argued, but that it is better understood as a set of conciliatory practices, traces of which resonate in Pietist lesson plans.11 Teachers in Francke's schools integrated experimental philosophy into an educational program that was introducing children to the exercises of... (shrink)

ABSTRACTWhile pedagogical content knowledge has been studied across a wide range of research areas, including science, language teaching and mathematics, there is limited research on the role...

Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of (...) Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton’s method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced. (shrink)

Modern technology offers an increasing number of tools for teaching mathematics, but technology adoption in schools encounters many barriers. The Technology Acceptance Model explains that technology usage is dependent on intentions, which rest on perceived ease of use and perceived usefulness. Less is known about the relationship between intentions and actual behaviour. In the current study we show that the level of cognitive investment on the part of the teachers, captured by the construct of Need for Cognition (NC), is (...) crucial in the use of technology in mathematical instruction, while controlling for a variety of background factors. Furthermore NC moderates the relationship between intentions and technology use, such that high NC weakens the relationship between the perceived usefulness of technology in pedagogy and its actual use. (shrink)

This paper has two specific goals. The first is to demonstrate that the theorem in MetaphysicsΘ 9, 1051a24-27 is not equiva-lent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I ar-gue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosoph-ical reason for the Aristotelian theorem being shorter than the Euclidean (...) one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theo-rems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amend-ments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge. (shrink)

This book positions itself at the intersection of the key areas of the modern humanities. Different authors from a variety of countries take innovative approaches to investigating multimodal communication, adapting pedagogical design to digital environments and enhancing cognitive skills through transformations in teaching and learning practices. The eclectic forms under study require eclectic approaches and methodologies, and the authors cross disciplinary boundaries drawing on philosophy, linguistics, semiotics, computational linguistics, mathematics, cognitive studies and neuroaesthetics. Part I presents methods of analysing (...) multimodal communication in its different displays, covering promotional video in crowdfunding project presentations, multimodal public signs of prohibition and visuals as arguments. Part II explores varied teaching methodologies that have emerged as a result of and in response to modern technological changes and contains some practical hints for educators. It demonstrates the pedagogical potential of video games, virtual worlds, linguistic corpora and online dictionaries. Part III focuses on psychological and cognitive factors influencing success in the classroom, primarily, ways of developing students’ and teachers’ personalities. The volume sits at the intersection between Communication Studies, Digital Humanities, Discourse Analysis, Education Theory and Cognitive Studies and is useful to scholars and students of communication, languages, education and other areas of the humanities. This book should trigger scholarly discussions as well as stimulating practitioners’ interest in these fields. (shrink)