Nature Of Mathematical Knowledge Research Articles

Studying mathematics at university level: a sequential cohort study for investigating connotative aspects of epistemological beliefs

In modern information societies, evaluating information, data, or knowledge claims is crucial. As these activities are influenced by epistemological beliefs, such beliefs are a key element of education in the sense that educational institutions intend to prepare students for professional and social life. Hence, this study aims to examine the development of students' epistemological beliefs about mathematics in the course of studying mathematics at university level. In a sequential cohort study, 1774 responses of mathematics students were surveyed in three consecutive years. The participants rated claims about the nature of mathematical knowledge in a spontaneous manner - a research procedure stimulating their connotative aspects of epistemological beliefs. Analyses showed that students hold different epistemological beliefs regarding ‘mathematics as a scientific discipline’ and ‘mathematics as a school subject’ - with mathematics major students and future upper secondary school teachers rating ‘mathematics as a school subject’ to be relatively simple and superficial compared to future primary school and lower secondary school teachers. Nonetheless, all student groups find ‘mathematics as a scientific discipline’ to be significantly more complex and profound than ‘mathematics as a school subject’. Interestingly, the domain-specific beliefs remain rather stable in the course of studies regardless of the students’ study programme.

Mathematical Epistemic Beliefs: Through the Gender Lens

This large survey study set out to ascertain whether gender differences in teachers’ mathematical epistemic beliefs exist, and the extent to which teachers’ gender as well as their teaching experience level, education level, and socio-economic setting of the schools can predict these beliefs. Specifically, the study examined teachers’ beliefs concerning the structure, stability, and source of mathematical knowledge. This is crucial given that previous research have demonstrated the potential role of teachers’ mathematical epistemic beliefs in shaping their mathematics teaching, and hence their students’ mathematics learning outcomes and perceptions of the subject. The current study drew on survey data collected from 745 primary teachers using items from the Thai Teachers’ Mathematics Education-related Beliefs (TTMEB) questionnaire. The data were analysed using Independent-samples t-test and multiple regression. The findings reveal a very limited to no effect of gender on teachers’ mathematical epistemic beliefs, and also alert us to the fact that when it comes to exploring factors that shape one’s beliefs, their exogenous characteristics, such as the socio-economic setting of where they live and work, should too be taken into account. Overall, the study helps to dispel any myths about gender differences in the way male and female teachers perceive mathematics, specifically the nature of mathematical knowledge.

“It does not Mean that They Cannot Do Mathematics”: Beliefs about Mathematics Learning Difficulties

Namibian preservice high school mathematics teachers’ (N=4) and teacher educators’ (N=3) beliefs about mathematics learning difficulties (MLD) were investigated to document the beliefs developed at the end of the teacher education program, the views and practices that might be emphasized in the program, and possible changes in these beliefs during the first year teaching. Preservice teachers were interviewed before they graduated from the teacher education program and during their first-year teaching. Teacher educators were interviewed once. Participants believed that the most important factor causing MLD at the high school was students’ knowledge and beliefs. Teacher educators stated former unqualified teachers and preservice teachers stated curriculum and teachers’ workload as other causes of MLD. Participants suggested individual support and mixed ability grouping to reduce students’ MLD. The partial alignment of beliefs among the participants showed that preservice teachers might have developed some beliefs through the views and practices emphasized in the program. Preservice teachers elaborated more on MLD when they became teachers and stated similar beliefs with some stress on their MLD-related practices. Preservice teachers did not comment on how teachers’ knowledge and practices might influence students’ MLD in both times. Findings pointed that teacher education programs should focus on increasing preservice teachers’ awareness of how their knowledge, practices and beliefs about MLD might affect students to improve their future practices. It is possible that teachers’ beliefs about MLD could be related to their beliefs about the nature of mathematical knowledge and indicators of high performance in mathematics.

Exploring Teachers’ Experiences on the Nature of Mathematics based on their Curricular and Pedagogical Practices: A Phenomenological Inquiry

The school mathematics curriculum development process in Nepal is still conventional. Contextualized mathematics teaching and learning has not been the priority of school education of Nepal. Majority of the school mathematics teachers in Nepal still conceptualize the mathematics and mathematics curriculum from the conformist approach. Thus, this paper aims to explore mathematics teachers’ experiences on the nature of mathematics focusing on their curricular and pedagogical practices based on the research question how mathematics teachers experience the nature of mathematical knowledge during their curricular and pedagogical practices. Ernest’s and Luitel’s nature of mathematical knowledge has enabled me to evaluate teachers’ experiences towards the nature of mathematical knowledge and Habermas’s theory of knowledge constitutive interest has guided the research for knowledge and implied others’ concept. The study has relied on the phenomenological approach and reflected the experiences of four mathematics teachers. The paper concludes that mathematics curriculum guided by emancipatory interest as well as practical interest promotes counter-hegemonic vision of teaching and learning mathematics and might develop critical consciousness. Critical consciousness embedded mathematics curriculum helps students and teachers to unpack the cultural nature of mathematics education and taken-for-granted assumptions about teaching and learning through critical pedagogy.

Mathematics Conceptions by Teachers from an Ethnomathematical Perspective

Abstract The ethnomathematical perspective implies substantial epistemological changes in mathematics conception with respect to the positivist tradition. This research focuses on a workshop for pre-service teachers, designed and developed under the ethnomathematical perspective, and which promotes reflection on the nature of mathematical knowledge. To that end, we analyzed the teachers’ answers about the nature of the mathematics described after the participation in this workshop. First, we identified some characteristic approaches of mathematics from the ethnomathematical perspective - the practical, social, and cultural approaches - and then used them to analyze the participants’ observations, which are considered as evidence of their conceptions about the nature of mathematical knowledge. Later, we grouped participants in profiles defined in relation to the incorporation of mathematics approaches according to Ethnomathematics. In conclusion, the workshop is shown as an environment conducive to reflection.

Can Digital Computers Support Ancient Mathematical Consciousness?

This paper poses, discusses, but does not definitively answer, the following questions: What sorts of reasoning machinery could the ancient mathematicians, and other intelligent animals, be using for spatial reasoning, before the discovery of modern logical mechanisms? “Diagrams in minds” perhaps? How and why did natural selection produce such machinery? Is there a single package of biological abilities for spatial reasoning, or did different sorts of mathematical competence evolve at different times, forming a “layered” system? Do the layers develop in individuals at different stages? Which components are shared with other intelligent species? Does some or all of the machinery exist at or before birth in humans and if not how and when does it develop, and what is the role of experience in its development? How do brains implement such machinery? Could similar machines be implemented as virtual machines on digital computers, and if not what sorts of non-digital “Super Turing” mechanisms could replicate the required functionality, including discovery of impossibility and necessity? How are impossibility and necessity represented in brains? Are chemical mechanisms required? How could such mechanisms be specified in a genome? Are some not specified in the genome but products of interaction between genome and environment? Does Turing’s work on chemical morphogenesis published shortly before he died indicate that he was interested in this problem? Will the answers to these questions vindicate Immanuel Kant’s claims about the nature of mathematical knowledge, including his claim that mathematical truths are non-empirical, synthetic and necessary? Perhaps it’s time for discussions of consciousness to return to the nature of ancient mathematical consciousness, and related aspects of everyday human and non-human intelligence, usually ignored by consciousness theorists.

Matemáticas y lenguaje: concepciones de los profesores desde una perspectiva etnomatemática

El objetivo de la investigación es describir y analizar las concepciones sobre la naturaleza de las matemáticas de un grupo de docentes en formación y en activo de una universidad argentina, tras la participación en un taller sobre trenzado artesanal. En el taller se promueve una visión de las matemáticas bajo una perspectiva sociocultural, y en particular etnomatemática, a través del lenguaje matemático utilizado por los artesanos. Tras el taller los participantes responden unas preguntas abiertas sobre las implicaciones de la actividad realizada en la naturaleza del conocimiento matemático. Se realiza un análisis descriptivo e interpretativo de las respuestas de los docentes. A partir de la teoría etnomatemática se definen las dimensiones para analizar las concepciones de los participantes sobre la naturaleza de las matemáticas bajo esta perspectiva sociocultural. Así mismo, se proponen unas etapas que constituyen una hipótesis de progresión del desarrollo de las concepciones sobre las matemáticas.

Cultural Immersion and Mathematics Teacher Education: Explorations in Morocco and South Africa

This article examines the first author's personal experiences as a mathematics educator studying ethnomathematics and indigenous knowledge systems (IKS) through immersion in indigenous cultures. In describing these encounters, we highlight the importance and impact of cultural immersion experiences on educators' perspectives regarding the nature of mathematical knowledge. We argue that cultural immersion not only builds a well-rounded individual, but also provides educators with the necessary tools to maintain relevance in the diverse and constantly evolving landscape that is the classroom. Insights gained from this study have direct implications in our teaching practices as they remind us to be mindful of the fact that children from diverse backgrounds have different modes of thinking, possess diverse perceptual abilities, and spend differential efforts on tasks depending on personal criteria which they deem useful.

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Are beliefs believable? An investigation of college students’ epistemological beliefs and behavior in mathematics

College students’ epistemological belief in their academic performance of mathematics has been documented and is receiving increased attention. However, to what extent and in what ways problem solvers’ beliefs about the nature of mathematical knowledge and thinking impact their performances and behavior is not clear and deserves further investigation. The present study investigated how Taiwanese college students espousing unlike epistemological beliefs in mathematics performed differently within different contexts, and in what contexts these college students’ epistemological beliefs were consistent with their performances and behavior. Results yielded from the survey of students’ performances on standardized tests, semi-open problems, and their behaviors on pattern-finding tasks, suggest mixed consequences. It appears that beliefs played a more reliable role within the well-structured context but lost its credibility in non-standardized tasks.

Theory of quantum computation and philosophy of mathematics. Part I

The aim of this paper is to present some basic notions of the theory of quantum computing and to compare them with the basic notions of the classical theory of computation. I am convinced, that the results of quantum computation theory (QCT) are not only interesting in themselves, but also should be taken into account in discussions concerning the nature of mathematical knowledge. The philosophical discussion will however be postponed to another paper. QCT seems not to be well-known among philosophers (at least not to the degree it deserves), so the aim of this paper is to provide the necessary technical preliminaries presented in a way accessible to the general philosophical audience.

Taking the History of Mathematics Seriously

SummaryKitcher’s philosophy of mathematics rests on the idea that a philosopher who tries to understand mathematical knowledge ought to take its historical development into consideration. In this paper, I take a closer look at Kitcher’s reasons for proposing such a historical turn. I argue that, whereas a historical account is indeed an essential part of the standpoint advanced in The Nature of Mathematical Knowledge, this is no longer the case for the position defended in the manuscript Mathematical Truth? The Wittgensteinian account of mathematics advocated in that manuscript does not force us to take a historical turn.

Understanding the structure of the set of rational numbers: a conceptual change approach

In the present article, we argue that the conceptual change approach to learning can apply in the case of mathematics, taking into consideration the particular nature of mathematical knowledge and the neurobiological bases of mathematical cognition. In the empirical study that is reported in this article, we investigated ninth graders’ understanding of algebraic and structural properties of rational numbers, from a conceptual change perspective. We make the point that understanding rational numbers is not indiscriminately difficult. We show that prior knowledge about natural numbers supports students dealing with algebraic properties of rational numbers, while the idea of discreteness is a fundamental presupposition, which constrains students’ understanding of density.

Teaching Mathematics: Ritual, Principle and Practice

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of mathematics as a community of practice and the teacher's role as epistemological authority in inducting pupils into such practices.

Social Epistemology Meets the Invisible Hand: Kitcher on the Advancement of Science

Philip Kitcher's new book is very ambitious in scope, more ambitious, certainly, than any of his previous books. Unlike Kitcher's Abusing Science: The Case against Creationism (1982), this is not a book that focusses on just one single issue in contemporary science or educational policy; nor is it, like his The Nature of Mathematical Knowledge (1983) and Vaulting Ambition: Sociobiology and the Quest for Human Nature (1985), a book that emphasizes just one relatively narrow philosophical topic. The Advancement of Science presents a complete, and quite general, position within the philosophy of natural science. Its purpose is much closer to Popper's The Logic of Scientific Discovery (1959) or even to Nagel's The Structure of Science (1961) than it is to any of Kitcher's earlier books. Kitcher starts off with an explicit epistemology, moves to a “goal” for science based on that epistemology, and then uses this epistemic goal as the target for the various methodological and social or institutional strategies that he offers.

Solver's structures of a problem situation and their global restructing

In this article, mathematical problem-solving processes are investigated from the viewpoint of “solver's structures representing a problem situation.” First, this concept is introduced in order to describe the progress of problem-solving processes in the terms which differ from Pólya's four steps. Second, I consider the convergence of the problem-solver's structures (which might be described as “the development of these representations”), and one direction is established theoretically, based on recent discussions about the nature of mathematical knowledge. Third, the problem-solving process by an expert solver is closely analyzed from the viewpoint of “solver's structures.” This analysis shows that the solver's structures were modified during the solving process in what will be called global restructuring. After finding a characteristic of the inner state of the problem situation, the solver revised his or her structure by using that characteristic as the basic principle. This transition of the property from one of characteristics into a basic principle could facilitate the convergence to the desirable direction. This possibility is discussed in the final section.

A structural analysis of students' epistemic views

Student responses to a structured questionnaire concerned with views on mathematical knowledge, activity and learning, were analysed and interpreted using factorial techniques. The constructs which emerge from the analysis may provide heuristically useful for understanding student beliefs. The findings suggest that there is no simple systematic relationship between beliefs about the nature of mathematical knowledge and activity and about the teaching and learning of mathematics.

Mathematics as Empirical: The Nature of Mathematical Knowledge.

Mathematics as Empirical: The Nature of Mathematical Knowledge.

Mathematics as Empirical: The Nature of Mathematical Knowledge.

Mathematics as Empirical: The Nature of Mathematical Knowledge.

The Nature of Mathematical Knowledge by Philip Kitcher

The Nature of Mathematical Knowledge by Philip Kitcher