On Knowledge and Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Cognitive Abilities ◽  
Historical Development ◽  
Mathematical Knowledge ◽  
General Scheme ◽  
Philosophy Of Mathematics ◽  
Scientific Practices ◽  
Advanced Mathematics ◽  
Mathematical Practices ◽  
Everyday Practices ◽  
Different Levels

This book proposes a novel analysis of mathematical knowledge from a practice-oriented standpoint and within the context of the philosophy of mathematics. The approach it is advocating is a cognitive, pragmatist, historical one. It emphasizes a view of mathematics as knowledge produced by human agents, on the basis of their biological and cognitive abilities, the latter being mediated by culture. It also gives importance to the practical roots of mathematics—that is, its roots in everyday practices, technical practices, mathematical practices themselves, and scientific practices. Finally, the approach stresses the importance of analyzing mathematics' historical development, and of accepting the presence of hypothetical elements in advanced mathematics. The book's main thesis is that several different levels of knowledge and practice are coexistent, and that their links and interplay are crucial to mathematical knowledge. This chapter offers some remarks that may help readers locate the book's arguments within a general scheme.

Mathematical Knowledge and the Interplay of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
Philosophy Of Mathematics ◽  
Advanced Mathematics ◽  
New Approach ◽  
Agent Based ◽  
Epistemology Of Mathematics ◽  
Actual Experience ◽  
Elementary Math ◽  
Mathematical Tradition

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

scholarly journals Which Prior Mathematical Knowledge Is Necessary for Study Success in the University Study Entrance Phase? Results on a New Model of Knowledge Levels Based on a Reanalysis of Data from Existing Studies

2020 ◽  
Vol 6 (3) ◽  
pp. 375-403 ◽  
Author(s):  
Stefanie Rach ◽  
Stefan Ufer
Keyword(s):  
Prior Knowledge ◽  
Mathematical Knowledge ◽  
Mathematics Teacher Education ◽  
Advanced Mathematics ◽  
Formal Representation ◽  
Study Success ◽  
Mathematics Courses ◽  
First Semester ◽  
Four Levels ◽  
Different Levels

Abstract The transition from school to tertiary mathematics courses, which involve advanced mathematics, is a challenge for many students. Prior research has established the central role of prior mathematical knowledge for successfully dealing with challenges in learning processes during the study entrance phase. However, beyond knowing that more prior knowledge is beneficial for study success, especially passing courses, it is not yet known how a level of prior knowledge can be characterized that is sufficient for a successful start into a mathematics program. The aim of this contribution is to specify the appropriate level of mathematical knowledge that predicts study success in the first semester. Based on theoretical analysis of the demands in tertiary mathematics courses, we develop a mathematical test with 17 items in the domain of Analysis. Thereby, we focus on different levels of conceptual understanding by linking between different (in)formal representation formats and different levels of mathematical argumentations. The empirical results are based on a re-analysis of five studies in which in sum 1553 students of bachelor mathematics and mathematics teacher education programs deal with some of these items in each case. By identifying four levels of knowledge, we indicate that linking multiple representations is an important skill at the study entrance phase. With these levels of knowledge, it might be possible to identify students at risk of failing. So, the findings could contribute to more precise study advice and support before and while studying advanced mathematics at university.

The Web of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Current Trend ◽  
Mathematical Practice ◽  
Mathematical Activity ◽  
Scientific Practices ◽  
General Introduction ◽  
Philosophical Work ◽  
Mathematical Practices ◽  
David Hilbert ◽  
Different Levels ◽  
The Web

This chapter is a general introduction to the current trend of studies of mathematical practice, with particular emphasis on historical and philosophical work. It offers a preliminary explanation of the notion of mathematical practice, first by considering the work of historians and philosophers on mathematical practices, from Archimedes and David Hilbert to Jens Høyrup, Penelope Maddy, Marcus Giaquinto, and Philip S. Kitcher. The chapter then characterizes the notion of mathematical practice by successively proposing several constraints. It argues that several different levels of practice and knowledge are coexistent and that their interrelationships are crucial to mathematical knowledge. It shows how the scheme of a web of interrelated practices—counting practices, measuring practices, technical practices, scientific practices—with their systematic links acting as constraint and guide, can be applied in the analysis of very different levels of mathematical activity.

Taking the History of Mathematics Seriously

Conceptus ◽  
2009 ◽  
Vol 38 (94) ◽  
Author(s):  
Adrian Frey
Keyword(s):  
Historical Development ◽  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
History Of Mathematics ◽  
Mathematical Truth ◽  
Historical Account ◽  
Historical Turn ◽  
History Of

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.

Agents and Frameworks

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Cognitive Abilities ◽  
Mathematical Knowledge ◽  
Mathematical Practices

This chapter introduces a “core scheme” for the analysis of mathematical practices that can be presented in a simplified way by means of a couple dubbed Framework–Agent. Mathematics in practice will typically depend on the performance of several Framework–Agent couples, intertwined in several possible ways. The chapter first considers two different kinds of frameworks, “theoretical” and “symbolic.” before discussing the role of exemplars in mathematical knowledge. It then explains how the links between different frameworks and practices are established concretely by an agent. It also examines the cognitive abilities involved in mastering counting practice, the importance of cognition in mathematics, and Philip S. Kitcher's concept of “metamathematics” in relation to agents. The chapter concludes by expounding on the notion of “systematic links.”

Decision Making Modeled as a Theorem Proving Process

2012 ◽  
Vol 4 (3) ◽  
pp. 1-11
Author(s):  
Jacques Calmet ◽  
Marvin Oliver Schneider
Keyword(s):  
Artificial Intelligence ◽  
Decision Making ◽  
Information Technology ◽  
Multiagent Systems ◽  
Theorem Proving ◽  
Mathematical Knowledge ◽  
Knowledge Bases ◽  
Advanced Mathematics ◽  
Knowledge Communities ◽  
Computational Systems

The authors introduce a theoretical framework enabling to process decisions making along some of the lines and methodologies used to mechanize mathematics and more specifically to mechanize the proofs of theorems. An underlying goal of Decision Support Systems is to trust the decision that is designed. This is also the main goal of their framework. Indeed, the proof of a theorem is always trustworthy. By analogy, this implies that a decision validated through theorem proving methodologies brings trust. To reach such a goal the authors have to rely on a series of abstractions enabling to process all of the knowledge involved in decision making. They deal with an Agent Oriented Abstraction for Multiagent Systems, Object Mechanized Computational Systems, Abstraction Based Information Technology, Virtual Knowledge Communities, topological specification of knowledge bases using Logical Fibering. This approach considers some underlying hypothesis such that knowledge is at the heart of any decision making and that trust transcends the concept of belief. This introduces methodologies from Artificial Intelligence. Another overall goal is to build tools using advanced mathematics for users without specific mathematical knowledge.

scholarly journals Social class and wise reasoning about interpersonal conflicts across regions, persons and situations

2017 ◽  
Vol 284 (1869) ◽  
pp. 20171870 ◽  
Author(s):  
Justin P. Brienza ◽  
Igor Grossmann
Keyword(s):  
Social Class ◽  
Cognitive Abilities ◽  
Online Survey ◽  
Levels Of Analysis ◽  
Ecological Framework ◽  
Ecological Constraints ◽  
Interpersonal Conflicts ◽  
Class Differences ◽  
Interpersonal Situations ◽  
Different Levels

We propose that class is inversely related to a propensity for using wise reasoning (recognizing limits of their knowledge, consider world in flux and change, acknowledges and integrate different perspectives) in interpersonal situations, contrary to established class advantage in abstract cognition. Two studies—an online survey from regions differing in economic affluence ( n = 2 145) and a representative in-lab study with stratified sampling of adults from working and middle-class backgrounds ( n = 299)—tested this proposition, indicating that higher social class consistently related to lower levels of wise reasoning across different levels of analysis, including regional and individual differences, and subjective construal of specific situations. The results held across personal and standardized hypothetical situations, across self-reported and observed wise reasoning, and when controlling for fluid and crystallized cognitive abilities. Consistent with an ecological framework, class differences in wise reasoning were specific to interpersonal (versus societal) conflicts. These findings suggest that higher social class weighs individuals down by providing the ecological constraints that undermine wise reasoning about interpersonal affairs.

scholarly journals Critical Thinking in Health Sciences and How It Pertains to Sonography Education: A Review of the Literature

2020 ◽  
Vol 36 (3) ◽  
pp. 244-250
Author(s):  
Brandy Weidman ◽  
Helen Salisbury
Keyword(s):  
Critical Thinking ◽  
Professional Practice ◽  
Cognitive Abilities ◽  
Thinking Skills ◽  
Health Sciences ◽  
Critical Thinking Skills ◽  
Review Of The Literature ◽  
Reasoning Test ◽  
Health Sciences Students ◽  
Different Levels

Objective: Critical thinking is an important skill that sonographers must develop beginning in educational programs and into professional practice. Critical thinking requires students to reflect on information, use judgment skills, and engage in higher levels of thinking, including analysis, interpretation, inference, evaluation, and explanation, to formulate reliable decisions. Methods: Current research related to critical thinking has focused on medicine, nursing, physical therapy, pharmacy, and dental programs, but there has been no description of assessing sonography students. The Dreyfus model has been used as a framework to describe acquired skills that reflects students’ progress from novice to expert clinicians. This model illustrates specific cognitive abilities that students develop as they advance in education. Results: This review of the literature describes critical thinking skills coupled with a framework to understand different levels of cognitive thinking, as well as how it can be assessed. Conclusion: To understand differences between undergraduate sonography students and experts, the Dreyfus model is an excellent model to recognize progression. It can be used with the Health Sciences Reasoning Test, which is a nationally recognized critical thinking examination that can ascertain different levels of health sciences students’ critical thinking skills.

scholarly journals Ethnomathematics and indigenous teacher education: Waka migrations

Revemop ◽  
2020 ◽  
Vol 2 ◽  
pp. e202008
Author(s):  
Tony Trinick ◽  
Tamsin Tamsin Meaney
Keyword(s):  
Teacher Education ◽  
Preservice Teacher ◽  
Indigenous Peoples ◽  
Mathematical Knowledge ◽  
Preservice Teacher Education ◽  
Traditional Methods ◽  
Mathematical Practices ◽  
Language And Culture ◽  
Primary Identification

In order to assist Indigenous peoples to revive their language and culture, teachers need strategies to enhance both cultural and mathematical knowledge for students. This paper presents findings from a project in which pre-service teachers investigated ethnomathematical practices using the context of ancestral ocean voyages by canoes. This context was chosen because a primary identification marker for Māori are their ancestral canoes. The results indicated that these pre-service teachers did not generally associate these ancestral voyages with mathematical practices, indicating that more work is needed to increase their understandings of ethnomathematics. Their understandings about the knowledge and practices connected to traditional methods of navigation were disrupted by myths perpetuated by European colonists. Despite this, a renaissance in canoe building and interest in traditional navigation practices provided the pre-service teachers with valuable information.Keywords: Traditional navigation. Ethnomathematics. Preservice teacher education. Cultural symmetry.Etnomatemáticas y formación de professores indígenas: migraciones de canoas WakaPara ayudar a los pueblos indígenas a revivir su idioma y su cultura, los maestros necesitan estrategias para mejorar el conocimiento cultural y matemático de los estudiantes. Este artículo presenta los hallazgos de un proyecto en el cual los maestros de pre-servicio investigaron las prácticas etnomatemáticas utilizando el contexto de viajes oceánicos ancestrales en canoas. Este contexto fue elegido porque un marcador de identificación principal para los Māori son sus canoas ancestrales. Los resultados indicaron que estos maestros de pre-servicio generalmente no asociaron estos viajes ancestrales con las prácticas matemáticas, lo que indica que se necesita más trabajo para aumentar su comprensión de las etnomatemáticas. Su comprensión sobre el conocimiento y las prácticas relacionadas con los métodos tradicionales de navegación fueron interrumpidos por los mitos perpetuados por los colonos europeos. A pesar de esto, un renacimiento en la construcción de canoas y el interés en las prácticas tradicionales de navegación proporcionaron a los maestros de pre-servicio informaciones valiosas.Palabras clave: Navegaciones tradicionales. Etnomatemáticas. Formación de profesores. Simetría cultural.Etnomatemática e formação de professores indígenas: migrações de canoas WakaBuscando auxiliar os povos indígenas a reavivar a sua língua e cultura, os professores precisam utilizar estratégias para aprimorar os conhecimentos culturais e matemáticos dos alunos. Esse trabalho apresenta os resultados de um projeto em que os professores em formação docente investigaram as práticas etnomatemáticas utilizando o contexto de viagens oceânicas ancestrais por canoas. Essa conjuntura foi escolhida porque uma das principais marcas identitárias dos Māori são as suas canoas ancestrais. Os resultados indicaram que, geralmente, esses professores não associaram essas jornadas ancestrais com as práticas matemáticas, sinalizando ser necessário um maior treinamento para aumentar o seu entendimento da etnomatemática. A compreensão sobre os conhecimentos e as práticas ligadas aos métodos tradicionais de navegação foram prejudicados por mitos perpetuados pelos colonizadores europeus. Contudo, um ressurgimento da construção de canoas e do interesse pelas práticas tradicionais de navegação propiciou valiosas informações para os professores em formação docente.Palavras-chave: Navegações tradicionais. Etnomatemática. Formação de professores. Simetria cultural.

scholarly journals Exploring Mathematical Concepts and Philosophical Values in Jember Batik

Abjadia ◽  
2021 ◽  
Vol 6 (2) ◽  
pp. 144-159
Author(s):  
Devita Amalia ◽  
Dwi Noviani ◽  
M. Fadil Djamali ◽  
Imam Rofiki
Keyword(s):  
Data Collection ◽  
Mathematical Knowledge ◽  
Geometric Transformation ◽  
Mathematical Concepts ◽  
Design Data ◽  
Mathematical Practices ◽  
Different Cultures

Ethnomathematics are different ways of doing mathematics taking into account the academic mathematical knowledge developed by different sectors of society as well as taking into account the different modes in which different cultures negotiate their mathematical practices (ways of grouping, counting, measuring, designing tools, or playing). Based on this research, this study aims to describe the results of ethnomathematics exploration in Jember batik motifs. The method of analysis used in this research was a qualitative approac with an ethnographic design. Data collection techniques were observation, documentation, and interviews. This research was conducted at Rumah Batik Rolla Jember and Rezti'z Batik Tegalsari Ambulu Jember. The research was conducted for one week. The results of this study indicate that the ethnomathematics in the Jember batik motif has a philosophical value that describes the natural wealth of Jember Regency in each of its motifs, and there are mathematical concepts in the form of geometric transformation concepts (reflection, translation, rotation, and dilation) along with the concept of number patterns.

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