"This book is interesting and well-written. The research methods were explained clearly and conclusions were summarized nicely. It is a relatively quick read at only 130 pages. Anyone who has been told, or who has told others, that mathematicians make better thinkers should read this book." MAA Reviews "The authors particularly attend to protecting positive correlations against the self-selection interpretation, merely that logical minds elect studying more mathematics. Here, one finds a stimulating survey of the systemic difficulties people have (...) with basic syllogisms and deductions." CHOICE connect "The authors particularly attend to protecting positive correlations against the self-selection interpretation, merely that logical minds elect studying more mathematics. Here, one finds a stimulating survey of the systemic difficulties people have with basic syllogisms and deductions." CHOICE Connect For centuries, educational policymakers have believed that studying mathematics is important, in part because it develops general thinking skills that are useful throughout life. This 'Theory of Formal Discipline' (TFD) has been used as a justification for mathematics education globally. Despite this, few empirical studies have directly investigated the issue, and those which have showed mixed results. Does Mathematical Study Develop Logical Thinking? describes a rigorous investigation of the TFD. It reviews the theory's history and prior research on the topic, followed by reports on a series of recent empirical studies. It argues that, contrary to the position held by sceptics, advanced mathematical study does develop certain general thinking skills, however these are much more restricted than those typically claimed by TFD proponents. Perfect for students, researchers and policymakers in education, further education and mathematics, this book provides much needed insight into the theory and practice of the foundations of modern educational policy. (shrink)

What is argumentation? -- Building a classroom culture of argumentation -- Structuring classroom discussions to focus on argumentation -- Infusing all instruction with argumentation -- Argumentation for all students -- Argumentation and the mathematical practices -- Technology in teaching and learning argumentation -- Assessing argumentation and proof -- Conclusion.

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

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

We study the interconnection between Physics and Mathematics in concrete instances, departing from the usual expression for the Coulomb electric field, produced by a point-like charge. It is scrutinized by means of six epistemology-intensive questions and radical answers are proposed, intended to widen one’s understanding of the subject. Our interventions act along two complementary directions. One of them regards ontology, since questions induce one to look closely at the electric charge, from different perspectives, promoting reflections about its nature (...) and reinforcing the corresponding concept. Formal manipulations rely on the identification of concepts with symbols, and the other direction concerns the spatial extension of mathematical structures. Our questions and their somewhat unusual answers help disclosing information which is not present in many textbooks, and show that Mathematics can be used as an efficient epistemological tool in Physics teaching. (shrink)

Mathematics is an important foundation for the development of science education. In the past, when instructors taught mathematical concepts of geometry shapes, they usually used traditional textbooks and aids to conduct teaching activities, which resulted in students not being able to understand the principles completely. Nowadays, it has become a trend to integrate emerging technologies into mathematics courses and to use digital instructional aids. Emerging technologies can effectively enhance students’ sensory experience while strengthening their impressions and understandings (...) of subject concepts. In this paper, we apply virtual reality immersive technologies to develop a “virtual reality immersive learning mathematics geometry system,” which is used to teach mathematical geometry concepts. Teachers use the system to develop three basic mathematical geometry learning materials: “Triangular pyramid volume = 1/3 prism volume,” “Cone volume calculation,” and “Triangle center of gravity derivation.” In the experimental activity, the teacher uses virtual reality teaching aids to guide students to learn mathematical geometry concepts in a fun way so that they can achieve the effectiveness of immersive learning. This study explores the impact of using the virtual reality immersive learning mathematics geometry system on students’ technology acceptance, learning motivations, and learning performance. The experimental result showed that using the virtual reality immersive learning mathematics geometry system can improve the learning motivation and learning performance of students. The findings indicated that the experimental group had better learning outcomes after completing the learning tasks of three geometric units. The experimental group used the virtual reality immersive learning mathematics geometry system which can lead to better learning outcomes. According to the ARCS questionnaire, students in the experimental group were confident to understand new subjects. At the same time, the mode of completing the game can effectively give students a sense of accomplishment. The use of emerging technologies in the classroom can be an attractive learning mode for students. (shrink)

The teacher is one of the key factors in the transmission of curricular objectives to their students. Their knowledge and positive attitude toward students' ethnomathematics and its pedagogical approach play an important role to make classroom teaching culture friendly. This paper is intended to explore the in-service teachers’ knowledge of ethnomathematics and perception on the integration of the ethnomathematics approach in their teaching with respect to the demographic factors of in-service teachers such as gender, academic status, teaching (...) experience, teacher training, and nature of institution they associated. This study was conducted among 120 in-service mathematics teachers from 32 secondary schools in Kathmandu District. The explanatory sequential mix method was employed to carry out this study. The perception of in-service teachers on ethnomathematics pedagogy in their classroom teaching was measured with survey instruments in the first stage. For further justification of the results obtained from the quantitative data, six teachers with different demographic factors were interviewed in the second stage of the data collection process. The relationship between in-service teachers’ understanding on ethnomathematics and different demographic factors was investigated using the Pearson Chi-square test. From the statistical data analysis, it was found that there is no relationship between gender, academic qualifications, and teachers' training in their understanding of ethnomathematics. However, a significant relationship was found between the teachers' experience and nature of the institutions they were involved, and their understanding on ethnomathematics. In-service teachers have a similar understanding on ethnomathematics and their perception in the integration of the ethnomathematics approach in the teaching of school mathematics. (shrink)

The purpose of this research is to reflect on the role of the context environment of the contextualized situations of teaching school mathematics, which make significant contributions to the initial training of the mathematics teacher at the Universidad Popular del Cesar-Valledupar-Colombia, where the informants were the twelve students enrolled in the subjects of Elective III, Mathematics Didactics, History and Epistemology of Mathematics and Degree Work II, of the VI, VII, VIII and IX semesters respectively, of (...) the Bachelor's Degree in Mathematics program; assigned to the Faculty of Education, of the Universidad Popular del Cesar. The study was situated in the qualitative paradigm, under the epistemological approach of sociocultural constructivism, with the recapitulations of the foundations and orientations of the phenomenological-transcendental methodology. It is concluded that it is feasible to change the mathematics master class for a space where students can interact in the teaching-learning process, with the necessary knowledge to understand the messages of the context of the habitat of the subject who learns; and for this reason, the relationships of the trio: Teacher-School Mathematical Knowledge-Students and these with the context can always be placed in the context of mathematics education. (shrink)

Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from neuroimaging and single cell recording (...) experiments converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts. (shrink)

Metacognitive skills are essential to be developed by the students to help improve their academic performance. This study was conducted with the aim of determining whether the use of problem-solving strategies can increase students' metacognitive skills. Students of the Mathematics in the World course offered at a local college in the Philippines were selected to participate in this study. Using an independent sample t-test, results revealed that students who were the recipients of problem-solving strategies scored statistically higher (...) in formal examinations compared to those students who were taught using conventional strategies. The researchers concluded that students who were taught using the problem-solving strategy enhanced their metacognitive skills based on their scores in the test of metacognitive skills during the post-test. The researchers suggested that teachers should incorporate problem-solving strategies into their lessons since this research demonstrates this to be an effective strategy for enhancing students' metacognitive skills essential to their learning. (shrink)

Word problems in mathematics seem to constantly pose learning difficulties for all kinds of students. Recent work in math education (for example, [Lakoff, G. & Nuñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books]) suggests that the difficulties stem from an inability on the part of students to decipher the metaphorical properties of the language in which such problems are cast. A 2003 pilot study [Danesi, (...) M. (2003a). Semiotica, 145, 71–83] confirmed this hypothesis in an anecdotal way. This paper reviews the implications of that study and of a follow-up one that is described here as well, in the light of how the metaphorical analysis of word problems allows learners to overcome typical difficulties in word problem-solving by teaching them how to flesh out the underlying concepts and convert them into appropriate representations. (shrink)

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

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

A Volume in Cognition, Equity & Society: International Perspectives formally known as International Perspectives on Mathematics Education - Cognition, Equity & Society Series Editor Bharath Sriraman, The University of Montana and Lyn English, Queensland University of Technology The diversity of research in mathematics education has been addressed as both, a problem and a strength. When manifested through adherence to different intellectual roots and theoretical orientations, diversions constitute 'refractions' of mathematics education. The collection and analysis of empirical data (...) in a study are by necessity refracted through the specific analytical lens employed, as well as the aim of the study itself. Refractions can also refer to looking at old phenomena through new lenses. The chapters in this book are refracted through philosophical, political, mathematical and personal lenses by distinguished authors in the field, addressing issues about the elusive experience of doing mathematics, purification of texts, refractions, mathematics and ethnomathematics, political messages in textbook tasks, mathematics education policy debate, the political in mathematics education research, philosophy and mathematics, meanings and representations, identity of mathematical modeling, and dilemmas in the teaching of calculus. An ancient Sanskrit adage states that Knowledge is something that grows when shared, but shrinks when hoarded. Academics engaged in the generation of new Knowledge are blessed with both the time and the freedom to engage in pursuits that allow for intellectual pleasure. As a phenomenon of the Zeitgeist many have succumbed to the increased corporatization of academic work, engaging in activities for monetary and self advancement purposes. Are there any real intellectuals left in academia, a l Adorno, Bourdieu, Chomsky, Foucault, among others? This Festschrift is dedicated to academics that don't bother with self promotion or aggrandizement of themselves or their ideas in simplistic terms. (shrink)

Grasha–Riechmann Teaching Styles have a high potential to be applied in Mathematics especially to help increase teacher educators’ knowledge. However, very little attention has been paid to the study of identifying the teaching style patterns of Mathematics teachers at the primary school National-Type Chinese Primary Schools or Sekolah Jenis Kebangsaan Cina SJKC. There is increasing concern about how this teaching style related to the teaching experience. This study aims to identify the patterns (...) of Grasha–Riechmann Teaching Styles among primary school Mathematics teachers and the relationship between Grasha–Riechmann Teaching Styles with teaching experience. The quantitative approach through a survey was applied to 97 Mathematics teachers of SJKC Kepong, Kuala Lumpur using the simple random sampling method. The instrument was adapted from the Grasha–Riechmann Teaching Styles Questionnaire, which measures five teaching styles such as Personal Model Teaching Style, Expert Teaching Style, Formal Authority Teaching Style, Delegator Teaching Style, and Facilitator Teaching Style. The patterns showed that the Personal Model Teaching Style is the most dominant, and the Facilitator Teaching as the least dominant style. The Spearman’s Rho Correlation also reported a very weak significant correlation between Grasha–Riechmann Teaching Styles with the teachers’ Mathematics teaching experience, specifically for Expert, Formal Authority, and Facilitator Teaching Styles. The study provides practical implications for educators’ professional development to diversify the training of teachers by experience and adapt them to the needs of student learning in primary school. These findings trigger ideas to get a better understanding by other demographic variables such as gender, age, and complexity of Mathematics subject. (shrink)

Self-efficacy is an integral part of personal factors that contributes substantially to students' success in mathematics. This review draws on previous intervention studies to identify, describe, and expose underlying mechanisms of interventions that foster mathematics self-efficacy. The findings show that effective mathematics self-efficacy interventions can be categorized into three categories using their underlying mechanisms: those that directly manipulate sources of self-efficacy to foster the construct, and those that either embed self-efficacy features in teaching methods or in (...) learning strategies. Specific examples of interventions that fall in each of these three categories are described including their features and the underlying mechanisms that improve students' mathematics self-efficacy. I argue for the two “most effective” interventions that foster mathematics self-efficacy and their relevance to either pre-university or university students with implications for teaching and learning of mathematics. (shrink)

This guide is designed as a teaching tool, aimed at both teachers and student teachers. Its main purpose is to enhance the value of "The Mathematical Experience" as a textbook. It includes a sample syllabus, outlines for group work, sample exams and hints for grading essays.

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.

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)

In these days, there is an increasing technological development in intelligent tutoring systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that help students understand the basics of math and that helps a lot of students of all ages to understand the topic because it's important for students of adding and subtracting. Through which the student will be able to study the course and solve related (...) problems. An evaluation of the intelligent tutoring systems was carried out and the results were encouraging. (shrink)

While Garfinkel’s early work, captured in Studies in Ethnomethodology, has received a lot of attention and discussion, this has not been the case for his later work since the 1970s. In this paper, we critically examine the aims of Garfinkel’s later ethnomethodological studies of work programme and evaluate key ideas such as the ‘missing what’ in the sociology of work, ‘the unique adequacy requirements of methods’, and the notion of ‘hybrid studies’. We do so through a detailed engagement with a (...) study that has frequently been singled out as exemplary by Garfinkel for his studies of work programme, namely Livingston’s The Ethnomethodological Foundations of Mathematics. We show how Livingston uses the proof of Gödel’s Incompleteness Theorem as a way to exhibit the work involved in understanding mathematical proofs. We then discuss how Livingston uses this example to introduce a distinction between the written ‘proof account’ and the associated ‘lived work’ of working through that proof, which we argue allows Livingston to provide a powerful critique of a formalist understanding of the objectivity of mathematics. We then discuss three aspects that we find problematic in Livingston’s and Garfinkel’s claims about Livingston’s study. Firstly, we question whether written proofs are best conceived of as descriptions or accounts. Secondly, we interrogate whether an ethnomethodological study could teach mathematicians how to make discoveries. Thirdly, we throw doubt on Garfinkel’s claim that Livingston’s results are results in mathematics. We conclude this paper with a discussion of how Livingston’s study Gödel’s proof highlights key tensions in Garfinkel’s later work. Firstly, we argue that there exists an ambiguity in Garfinkel’s treatment of texts as ‘incompetent’. Secondly, we show that Garfinkel’s attempt to extend his idea of classical studies from sociology to other professions and disciplines is problematic. Finally, we question Garfinkel’s proposals to reorient ethnomethodological studies away from sociological audiences and ask whether ethnomethodological studies promise the delivery of a ‘large prize’ or, rather, provide something like ‘helpful therapy’. (shrink)

Spatial cognitive abilities, including mental rotation and visuo-spatial working memory are correlated with mathematical performance, and several studies have shown that training of these abilities can enhance mathematical performance. Here, we investigated the behavioral and neural correlates of MR and vsWM training combined with number line training. Fifty-seven children, aged 6–7, performed 25 days of NL training combined with either vsWM or MR and participated in an Electroencephalography -session in school to measure resting state activity and steady-state visual evoked potentials (...) during a vsWM task before and after training. Fifty children, aged 6–7, received usual teaching and acted as a control group. Compared to the control group, both training groups improved on a combined measure of mathematics. Cognitive improvement was specific to the training. Significant pre-post changes in resting state-EEG, common to both training groups, were found for power as well as for coherence, with no significant differences in rs-EEG-changes between the vsWM and MR groups. Two of the common rs-EEG changes were correlated with mathematical improvement: an increase in coherence between the central frontal lobe and the right parietal lobe in frequencies ranging from 16 to 25 Hz, and an increase in coherence between the left frontal lobe and the right parietal lobe ranging from 23 to 25 Hz. These results indicate that changes in fronto-parietal coherence are related to an increase in mathematical performance, which thus might be a useful measure in further investigations of mathematical interventions in children. (shrink)

This book has been designed specifically to support the student through the IB Diploma Programme in Mathematical Studies. It includes worked examples and numerous opportunities for practice. In addition the book will provide students with features integrated with study and learning approaches, TOK and the IB learner profile. Examples and activities drawn from around the world will encourage students to develop an international perspective.

This paper considers an approach to teaching ethics in bioengineering based on the How People Learn (HPL) framework. Curricula based on this framework have been effective in mathematics and science instruction from the kindergarten to the college levels. This framework is well suited to teaching bioengineering ethics because it helps learners develop “adaptive expertise”. Adaptive expertise refers to the ability to use knowledge and experience in a domain to learn in unanticipated situations. It differs from routine expertise, (...) which requires using knowledge appropriately to solve routine problems. Adaptive expertise is an important educational objective for bioengineers because the regulations and knowledge base in the discipline are likely to change significantly over the course of their careers. This study compares the performance of undergraduate bioengineering students who learned about ethics for stem cell research using the HPL method of instruction to the performance of students who learned following a standard lecture sequence. Both groups learned the factual material equally well, but the HPL group was more prepared to act adaptively when presented with a novel situation. (shrink)

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue (...) to exist there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science. (shrink)

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

Many results in fuzzy logic depend on the mathematical structure the truth value set obeys. In this textbook the algebraic foundations of many-valued and fuzzy reasoning are introduced. The book is self-contained, thus no previous knowledge in algebra or in logic is required. It contains 134 exercises with complete answers, and can therefore be used as teaching material at universities for both undergraduated and post-graduated courses. Chapter 1 starts from such basic concepts as order, lattice, equivalence and residuated lattice. (...) It contains a full section on BL-algebras. Chapter 2 concerns MV-algebra and its basic properties. Chapter 3 applies these mathematical results on Lukasiewicz-Pavelka style fuzzy logic, which is studied in details; besides semantics, syntax and completeness of this logic, a lot of examples are given. Chapter 4 shows the connection between fuzzy relations, approximate reasoning and fuzzy IF-THEN rules to residuated lattices. (shrink)

Mathematical argument in the elementary grades : what and why? -- Elementary students as mathematicians -- The teaching model -- Using the lesson sequences : what the teacher does -- Mathematical argument in the elementary classroom : impact on students and teachers.

How can we deal with the diversity of theories in mathematics education? This was the main question that led the authors of this book to found the Networking Theories Group. Starting from the shared assumption that the existence of different theories is a resource for mathematics education research, the authors have explored the possibilities of interactions between theories, such as contrasting, coordinating, and locally integrating them. The book explains and illustrates what it means to network theories; it presents (...) networking as a challenging but fruitful research practice and shows how the Group dealt with this challenge considering five theoretical approaches, namely the approach of Action, Production, and Communication (APC), the Theory of Didactical Situations (TDS), the Anthropological Theory of the Didactic (ATD), the approach of Abstraction in Context (AiC), and the Theory of Interest-Dense Situations (IDS). A synthetic presentation of each theory and their connections shows how the activity of networking generates questions at the theoretical, methodological and practical levels and how the work on these questions leads to both theoretical and practical progress. The core of the book consists of four new networking case studies which illustrate what exactly can be gained by this approach and what kind of difficulties might arise. (shrink)

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

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)

A large portion of mathematics education centers heavily around imitative reasoning and rote learning, raising concerns about students’ lack of deeper and conceptual understanding of mathematics. To address these concerns, there has been a growing focus on students learning and teachers teaching methods that aim to enhance conceptual understanding and problem-solving skills. One suggestion is allowing students to construct their own solution methods using creative mathematical reasoning, a method that in previous studies has been contrasted against algorithmic (...) reasoning with positive effects on test tasks. Although previous studies have evaluated the effects of CMR, they have ignored if and to what extent intrinsic cognitive motivation play a role. This study investigated the effects of intrinsic cognitive motivation to engage in cognitive strenuous mathematical tasks, operationalized through Need for Cognition, and working memory capacity. Two independent groups, consisting of upper secondary students, practiced non-routine mathematical problem solving with CMR and AR tasks and were tested 1 week later. An initial t-test confirmed that the CMR group outperformed the AR group. Structural equation modeling revealed that NFC was a significant predictor of math performance for the CMR group but not for the AR group. The results also showed that WMC was a strong predictor of math performance independent of group. These results are discussed in terms of allowing for time and opportunities for struggle with constructing own solution methods using CMR, thereby enhancing students conceptual understanding. (shrink)

Although many agree that all teaching rests on a theory of knowledge, this is an in-depth exploration of the philosophy of mathematics for education, building on the work of Lakatos and Wittgenstein.

The purpose of this cross-sectional study was to examine pre-service teachers’ perceptions of their mathematical pedagogical content knowledge and to determine effect of demographic variables (Gender, year of study) on their mathematical pedagogical content knowledge. A Likert scale questionnaire was used to collect data from 104 pre-service teachers. Descriptive statistics and Mann-Whitney U-test were used to examine pre-service teachers’ perceived knowledge of teaching strategies, mathematical language and symbols, misconceptions, curriculum, and their perceived knowledge of learners. Results show (...) that in general, majority pre-service teachers have strong perceptions of their mathematical pedagogical content knowledge. They consider lesson planning, purpose of the mathematics curriculum and assessment tools used in a mathematics classroom as important aspects of their perceived knowledge of the curriculum. It was also revealed that their perceived knowledge of misconception would enable them to know and anticipate their learners’ misconceptions. However, results also indicated that there was no statistically significant difference in pre-service teachers’ mathematical pedagogical content knowledge based on gender and year of study. Study implications are also discussed. (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 article discusses the role of mathematics during physics lessons in upper-secondary school. Mathematics is an inherent part of theoretical models in physics and makes powerful predictions of natural phenomena possible. Ability to use both theoretical models and mathematics is central in physics. This paper takes as a starting point that the relations made during physics lessons between the three entities Reality, Theoretical models and Mathematics are of the outmost importance. A framework has been developed to (...) sustain analyses of the communication during physics lessons. The study described in this article has explored the role of mathematics for physics teaching and learning in upper-secondary school during different kinds of physics lessons. Observations are from three physics classes led by one teacher. The developed analytical framework is described together with results from the analysis of the 7 lessons. The results show that there are some relations made by students and teacher between theoretical models and reality, but the bulk of the discussion in the classroom is concerning the relation between theoretical models and mathematics. The results reported on here indicate that this also holds true for all the investigated organizational forms lectures, problem solving in groups and labwork. (shrink)

In a long-term study concerning mathematical tasks in primary schools, it was noted that teachers had difficulty in discussing mathematical processes and many lacked the vocabulary to do this. However, certain words and phrases such as 'pattern' or 'looking for pattern' were used with more confidence. With this in mind, discussions with teachers about commercial mathematics tasks were analysed based on mentions of pattern. It was found that the word was used frequently, but that some teachers had a (...) wider and more sophisticated view of 'pattern' than others. There were also variations in their views about teaching pattern, with some making it clear that they had strategies for helping children to see patterns. There was a feeling amongst some teachers that not all children could see patterns, and there was a tendency for this group to regard 'pattern spotting' as a top set activity. (shrink)

This text prepares undergraduate mathematics students to meet two challenges in the study of mathematics, namely, to read mathematics independently and to understand and write proofs. The book begins by teaching how to read mathematics actively, constructing examples, extreme cases, and non-examples to aid in understanding an unfamiliar theorem or definition (a technique famililar to any mathematician, but rarely taught); it provides practice by indicating explicitly where work with pencil and paper must interrupt reading. (...) The book then turns to proofs, showing in detail how to discover the structure of a potential proof from the form of the theorem (especially the conclusion). It shows the logical structure behind proof forms (especially quantifier arguments), and analyzes, thoroughly, the often sketchy coding of these forms in proofs as they are ordinarily written. The common introductroy material (such as sets and functions) is used for the numerous exercises, and the book concludes with a set of "Laboratories" on these topics in which the student can practice the skills learned in the earlier chapters. Intended for use as a supplementary text in courses on introductory real analysis, advanced calculus, abstract algebra, or topology, the book may also be used as the main text for a "transitions" course bridging the gab between calculus and higher mathematics. (shrink)

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 (...) class='Hi'>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)

The process of the mathematization of physical situations through differential calculus requires an understanding of the justification for and the meaning of the differential in the context of physics. In this work, four different conceptions about the differential in physics are identified and assessed according to their utility for the mathematization process. We also present an empirical study to probe the conceptions about the differential that are used by students in physics, as well students’ perceptions of how they are (...) expected to use differential calculus in physics. The results support the claim that students have a quasi-exclusive conception of the differential as an infinitesimal increment and that they perceive that their teachers only expect them to use differential calculus in an algorithmic way, without a sound understanding of what are they doing and why. These results are related to the lack of attention paid by conventional physics teaching to the mathematization process. Finally, some proposals for action are put forward. (shrink)

Mathematical proofs generally allow for various levels of detail and conciseness, such that they can be adapted for a particular audience or purpose. Using automated reasoning approaches for teaching proof construction in mathematics presupposes that the step size of proofs in such a system is appropriate within the teaching context. This work proposes a framework that supports the granularity analysis of mathematical proofs, to be used in the automated assessment of students' proof attempts and for the presentation (...) of hints and solutions at a suitable pace. Models for granularity are represented by classifiers, which can be generated by hand or inferred from a corpus of sample judgments via machine-learning techniques. This latter procedure is studied by modeling granularity judgments from four experts. The results provide support for the granularity of assertion-level proofs but also illustrate a degree of subjectivity in assessing step size. (shrink)

Any study of the 'Scientific Revolution' and particularly Descartes' role in the debates surrounding the conception of nature (atoms and the void v. plenum theory, the role of mathematics and experiment in natural knowledge, the status and derivation of the laws of nature, the eternality and necessity of eternal truths, etc.) should be placed in the philosophical, scientific, theological, and sociological context of its time. Seventeenth-century debates concerning the nature of the eternal truths such as '2 + 2 (...) = 4' or the law of inertia turn on the question of whether these truths were created along with nature, or were uncreated and subsisting in God's mind. One's answer to that question has direct consequences for conceptions of the necessity/contingency of mathematical and natural knowledge, how knowledge of such truths is accomplished by humans, and what grounds these truths. In this paper, I review the positions of four successors to Descartes' philosophy on the question of the eternal truths to illustrate how in specific ways that question with its theological, metaphysical, modal, and epistemological dimensions concerned the objectivity and certainty of the discoveries of the new science. Author Recommends: Clarke, Desmond. Descartes' Philosophy of Science . University Park, Penn State Press, 1982. This work provides an account of Descartes as a practicing scientist whose rationalism is mitigated by reliance on experiment and experience. Author re-examines Descartes' philosophical and scientific works in this new light. Dear, Peter. Revolutionizing the Sciences: European Knowledge and its Ambitions, 1500–1700 . Princeton, Princeton University Press, 2001. This work provides a useful overview of the issues and thinkers of the Scientific Revolution. Of particular relevance is chapter 8 on Cartesian and Newtonian science. Funkenstein, Amos. Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century . Princeton, Princeton University Press, 1986. This work is an advanced study of the theological and metaphysical foundations of early modern science. Discussions include questions of God's nature, God's knowledge in relation to human knowledge, providence, the laws of nature, and the truths of mathematics. In particular, chapter 3 discusses Descartes' account of the eternal truths and divine omnipotence. Garber, Daniel. Descartes' Metaphysical Physics . Chicago, University of Chicago Press, 1992. This work examines how Descartes' metaphysical doctrines of God, soul, and body set the groundwork for his physics. It includes a study of God and the grounds for the laws of physics (chapter 9). Henry, John. The Scientific Revolution and the Origins of Modern Science . 3rd ed. New York, Palgrave, Macmillan Press, 2008. This work provides a brief, general, and informative overview of the Scientific Revolution, including the themes of method, magic, religion, and culture. Osler, Margaret J. Divine Will and the Mechanical Philosophy: Gassendi and Descartes on Contingency and Necessity in the Created World . Cambridge, Cambridge University Press, 1994. This work is an examination and comparison of the mechanical philosophies of Gassendi and Descartes. It offers in-depth discussion of the issue of voluntarism and intellectualism in the period and how that related to conceptions of laws of nature and the eternal truths. Shapin, Steven. The Scientific Revolution . Chicago, University of Chicago Press, 1996. This work provides a critical synthesis of as well as a guide to recent scholarship in the history of science for a general readership. Online Materials Dr. Robert A. Hatch's Scientific Revolution Website: http://web.clas.ufl.edu/users/rhatch/pages/03-Sci-Rev/SCI-REV-Home/ A compendium of resources for the study of Scientific Revolution. Early English Books Online: http://eebo.chadwyck.com/home Early English Books Online (EEBO) contains digital facsimile page images of virtually every work printed in England, Ireland, Scotland, Wales and British North America and works in English printed elsewhere from 1473 to 1700. Early Modern Resources: http://www.earlymodernweb.org.uk/emr/ Early Modern Resources is a gateway for all those interested in finding electronic resources relating to the early modern period in history. Gallica, the Digital Library of the Bibliothèque Nationale de France: http://gallica.bnf.fr/ An ever-growing digital library which includes numerous primary and secondary texts of relevance to Descartes and his role in Scientific Revolution. Hatfield, Gary, 'René Descartes', The Stanford Encyclopedia of Philosophy. Spring 2009 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/spr2009/entries/descartes/ Slowik, Edward, 'Descartes' Physics', The Stanford Encyclopedia of Philosophy. Winter 2008 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/win2008/entries/descartes-physics/ Syllabus Sample Syllabus: Cartesian Science The following is five weeks covering Cartesian Science in a course on Descartes or the Scientific Revolution, or 17th-century theories of matter, or related themes on early modern truth and method, especially on the continent. This material is best suited to a graduate level audience, but it could be modified to suit an upper-division undergraduate course, as the readings are basically primary texts whose context and background can be explained in lectures. Week 1: Cartesian Revolution in France • Scientific method • Role of mathematics and experiment • Certainty of scientific knowledge Readings: Hatfield, Gary, 'René Descartes', The Stanford Encyclopedia of Philosophy. Spring 2009 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/spr2009/entries/descartes/ Descartes, Discourse on Method , Parts 1–3 Descartes, Meditations on First Philosophy , First Meditation. Week 2: Descartes' Scientific Treatises • Mechanization and mathematization of nature • Primary–secondary quality distinction Readings: Discourse on Method, Parts 4–6 Selections from Descartes' Scientific Essays: The World or Treatise on Light (ATXI 3–48); Treatise on Man (ATXI 119–202); Optics (ATVI 82–147). Slowik, Edward, 'Descartes' Physics', The Stanford Encyclopedia of Philosophy. Winter 2008 ed. Ed. Edward N. Zalta; URL: http://plato.stanford.edu/archives/win2008/entries/descartes-physics/ Henry, John, 'The Mechanical Philosophy,' chapter 5. The Scientific Revolution and the Origins of Modern Science . 3rd ed. Macmillan, 2008. Week 3: Descartes' Theory of Nature • Descartes' derivation of the law of conservation and the three laws of motion • God's role in the metaphysics and physics of nature Readings: Selections from Principles of Philosophy, Preface (all); Letter to Elizabeth; Part I: 1–8; Part II: 1–45, 55, 64; Part III: 1–4, 15–19, 45–47; Part IV: 187–207. John Henry, 'Religion and Science,' chapter 6. The Scientific Revolution and the Origins of Modern Science . 3rd ed. Macmillan, 2008. Week 4: Post-1650 Cartesian Science: Necessity and Contingency in Nature • Debates on God, Creation, and Causes Readings: Easton, Patricia, 'What is at Stake in the Cartesian Debates on the Eternal Truths?' Philosophy Compass 4.2 (2009): 348–62. Malebranche, Nicolas, 'Elucidation 10', from The Search after Truth (1674). Note: All selections available in Nicolas Malebranche (1992). Philosophical Selections , edited by S. Nadler, Hackett. Gottfried Leibniz (1714) Monadology . Week 5: Causes in Nature and Morals • Theodicy as an explanation of defect and evil in a lawful universe: Malebranche v. Leibniz Readings: Nicolas Malebranche, Elucidation XVI (on occasionalism), and Treatise on Nature and Grace, Discourse One, Part 1. Gottfried Leibniz (1706), Theodicy. Focus Questions Weekly questions can be used to focus the readings. This can be done in a web or e-mail discussion thread, as a weekly assignment, or for in class discussion. I require students to post a short paragraph in response to the question or some posting by a classmate on the question. Students are required to post by 10 a.m. the day before we meet for class on a course website. Week 1: According to Descartes, what role does skepticism play in scientific reasoning? Week 2: Comment on the following: 'But I am supposing this machine to be made by the hands of God, and so I think you may reasonably think it capable of a greater variety of movements than I could possibly imagine in it, and of exhibiting more artistry than I could possibly ascribe to it' [ Treatise on Man ; ATXI 120]. Week 3: What is Descartes' conception of the relation between the metaphysics and physics of nature? Week 4: Critically discuss the positions of Descartes, Malebranche, and Leibniz on what provides the foundation for the certitude of natural knowledge? Week 5: Explain why both Malebranche and Leibniz consider moral sin to be analogous to natural defect? Seminar/Project Idea Hold a debate on the question of the status of the eternal truths. The proposition will be Descartes' position: 'Eternal truths must be both created and necessary if certainty in science is to be possible'. Format: 1. At the beginning of the 5-week module, students will be assigned to one of three roles: Team A, Team B, and judge's panel. Students will be given the debate proposition, but will not be told which team will take the affirmative and which team the negative until the time of the debate. 2. Recommend a variation on the Classic Debate Format to encourage the development of argument: sequence begins with affirmative construction (8 minutes), negative construction (8 minutes), second affirmative construction (8 minutes), second negative construction (8 minutes), first negative rebuttal (4 minutes), first affirmative rebuttal (4 minutes), final negative rebuttal (4 minutes) and final affirmative rebuttal (4 minutes). 3. Judges Panel: will consist of 3–4 judges who will assess the performance of Teams A and B. Judgment should be based on the persuasiveness of the team position. 4. Debate will be held at the end of the fifth week, or semester, whichever makes most sense given the course length and structure. Acknowledgements The author gratefully acknowledges the immensely helpful comments and suggestions by the participants in her graduate seminar on the Scientific Revolution: Benjamin Chicka, Sarah Jacques-Ross, Richard Ross, Marcella Stockstill, and Zohra Wolters. (shrink)

This paper presents the idea, structure, history, goals, and accomplishments of mathematics and science laboratories as they have been organized and taught at Trinity College. The laboratories are designed to develop specific science and mathematics problem-solving skills, presenting them within the context of humanities-related inquiry (e.g. neural network theory within the context of philosophy of mind). These laboratories are especially valuable in providing humanities students with literacy in advanced science and mathematics materials that, since they are not (...) requisite for humanities majors, humanities students would not be exposed to otherwise. Especially in the case of philosophy, laboratories bear the additional benefit of dissolving insularity, opening up study onto directly relevant fields and enriching and informing philosophical inquiry. The author concludes by considering philosophy’s relationship to science and mathematics, what these relationships imply for how a philosophy education should be structured, and the important role that science and mathematics laboratories play in that education. (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)

Three clusters of philosophically significant issues arise from Frege’s discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege’s philosophical discussion and technical execution of logicism is the so‐called Hume’s Principle, considered in The Foundations of Arithmetic . Some varieties of neo‐Fregean logicism are based on taking this principle as a contextual definition of the operator ‘the number of …’, and criticisms of such neo‐Fregean programs sometimes appeal (...) to Frege’s objections to contextual definitions in later writings. Finally, a critical question about the definitions on which Frege’s proofs of the laws of arithmetic depend is whether the logical structures of the definientia reflect our pre‐Fregean understanding of arithmetical terms. It seems that unless they do, it is unclear how Frege’s proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes the definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre‐definitional understanding. One or more of these topics may be studied in a survey course in the philosophy of mathematics or a course on Frege’s philosophy. The latter two topics are obviously central in a seminar in the philosophy of mathematics in general or more specialized seminars on logicism, or on mathematical definitions and concept formation. (shrink)

Machines were introduced as calculating devices to simulate operations carried out by human computors following fixed algorithms: this is true for the early mechanical calculators devised by Pascal and Leibniz, for the analytical engine built by Babbage, and the theoretical machines introduced by Turing. The distinguishing feature of the latter is their universality: They are claimed to be able to capture any algorithm whatsoever and, conversely, any procedure they can carry out is evidently algorithmic. The study of such "paper (...) machines" by mathematical means is the topic of our contribution. This is not only in accord with its usual understanding in computer science, but conceptually and historically right, when we recall the purpose for which Turing machines were introduced. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- (...) Blending Knowledge Structures in the Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)