Epistemology Of Mathematics Research Articles

Misconceptions about randomisation harm validity of randomised controlled trials.

The coherence theory of truth, the epistemology of evidence-based medicine, mathematical statistics, and axiomatic mathematics. To explore mathematical misconceptions inhering in randomised controlled trial designs, suggest improvements, encourage meta-methodological discussions and call for further interdisciplinary studies. Mathematical-statistical analyses and science-philosophical considerations. Randomisation does not (necessarily) generate equal samples, ergo, outcomes of usual RCTs are not as reliable as they claim. Moreover, ignoring initial sample discrepancies may cause inaccuracies similar to type I and type II errors. Insufficient awareness of these flaws harms final RCT statements about significance and evidence levels, hence their loss of trustworthiness. Statistical parameters such as the standard error of the mean may help to estimate the expected distinction between random samples. Researchers in EBM should be aware of systemic misconceptions in RCT standards. Pre-measurement can reduce shortcomings, e.g. through calculation how sample differences impact on usual RCT processing, or randomisation is given up in favour of mathematical minimisation of sample differences, i.e. optimising statistical sample equality. Moreover, the promising future of dynamic simulation models is highlighted.

Context of Contextualized Teaching Situations in the Initial Training of Mathematics Teachers at the Popular University of Cesar

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.

Philosophy of Mathematics in Primary Education Mathematics Learning: Ontological, Epistemological, and Methodological

The purpose of this study is to examine the relationship, position, and role of the philosophy of mathematics in learning mathematics in basic education through the perspective of the scope of philosophy, namely ontological mathematics, epistemological mathematics, and methodological mathematics. This research is a qualitative descriptive research with a literature study approach. Data collection techniques are carried out through the documentation method of articles originating from the Google Scholar database, as well as books related to the philosophy of mathematics. Data analysis techniques consist of data reduction, data presentation, and conclusion. The results highlight three main aspects within the scope of the philosophy of mathematics: ontological, epistemological, and methodological. Ontological mathematics addresses the ontological foundations of mathematics for teaching, including the role of mathematics in various contexts. The epistemology of mathematics emphasizes the importance of understanding the discipline from multiple perspectives to enhance learning and curriculum design. Meanwhile, the methodology of mathematics highlights the role of specific methods in determining mathematical solutions. In conclusion, understanding the philosophy of mathematics in these three aspects is important for educators to improve the learning process of mathematics at the primary level comprehensively and optimally.

The Relation Between Mathematics Philosophy And Culture-Based Learning Model

The aim of this study is to study the correlation of mathematical philosophy with cultural-based learning models. The method of research used is the method of library research, i.e. conducting studies related to mathematical philosophy, philosophical mathematics learning, culture-based mathematic learning, the connection of mathematically philosopher with ethnomatematics, the relationship of mathematic philosophies with cultural-based learning models. The results of the research have been obtained that the philosofy of maths, the model of learning based on culture, and ethnomathematics have close connections in the context of understanding and teaching of Mathematics. Mathematical philosophy deals with the foundations of mathematical epistemology and ontology, while a culture-based learning model emphasizes the integration of values and cultural contexts in mathematics learning. Ethnomatematics refers to the study of mathematical ways of thinking in a particular culture. By understanding the cultures, traditions, and world views of a particular society, ethnomathematics can help develop more culturally relevant and meaningful learning models. Therefore, the interrelationship between mathematical philosophy, culture-based learning models, and ethnomathematics can create a more holistic and contextual approach to mathematics learning.
 Keywords: Mathematical Philosophy, Cultural Based Learning Model, Ethnomathematics

The relevance of salience for the epistemology of mathematics

The relevance of salience for the epistemology of mathematics

Structuralism and informal provability

Mathematical structuralism can be understood as a theory of mathematical ontology, of the objects that mathematics is about. It can also be understood as a theory of the semantics for mathematical discourse, of how and to what mathematical terms refer. In this paper we propose an epistemological interpretation of mathematical structuralism. According to this interpretation, the main epistemological claim is that mathematical knowledge is purely structural in character; mathematical statements contain purely structural information. To make this more precise, we invoke a notion that is central to mathematical epistemology, the notion of (informal) proof. Appealing to the notion of proof, an epistemological version of the structuralist thesis can be formulated as: Every mathematical statement that is provable expresses purely structural information. We introduce a bi-modal framework that formalizes the notions of structural information and informal provability in order to draw connections between them and confirm that the epistemological structuralist thesis holds.

Towards a critical epistemology of mathematics

This essay addresses a critical epistemology of mathematics as an investigation into the epistemic limitations of mathematical thinking. After arguing for the relevance of a critical epistemology of mathematics, I discuss assumptions underlying standard arithmetic and assumptions underlying standard logic as examples for such epistemic limitations of mathematical thinking. Looking into the work of philosophically inte­res­ted scholars in mathematics education such as Alan Bishop and Ole Skovsmose, I discuss some early insights for a critical epistemology of mathematics. I conclude that these insights can only be the beginning, that we are yet far away from a proper under­standing of the epistemic limitations of mathematics, and that more research is needed.

Developing Artificial Human-Like Arithmetical Intelligence (and Why)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics.

School mathematics and teaching authorship in Brazilian curriculum production (1929-2019)

The purpose of this text is to present an institutional research and discuss the partial results of its development. The proposal is situated in the context of the “Mathematics in the curriculum of Basic Education, its epistemologies and policies”, which aims to map and analyze the curricular proposals and/or the prescribed curriculum in Brazil, from 1929 to 2019, so that it is possible to understand how the epistemology of mathematics supported each movement, as well as its implications for the teaching of the discipline in each historical period. Furthermore, it seeks to understand and discuss the production/formulation contexts of each curricular proposal in terms of showing whether there was teacher participation/authorship in its elaboration. The preliminary results achieved by the investigation are part of the search to subsidize a critical look at the field of curriculum in its interface with Mathematics Education.

Craftland is Mathland: Mathematical insight and the generative role of fiber crafts in maker education

There is tremendous excitement around makerspaces for deepening and enriching curricula across subjects, as well as engaging traditionally marginalized learners in new ways. To address the lack of translation of maker education projects to mathematics learning, we propose that educators aspire to create a “Mathland” when designing maker educational activities. Mathlands are environments envisioned by Seymour Papert where mathematics are learned alongside ways of doing mathematics in self-selected contexts, leading to an epistemology and natural language of mathematics that pervades all experiences. To imagine a Mathland where women’s participation in mathematics is lifelong and lifewide, we explore traditionally female-dominated fiber crafts where long-term engagement, mathematics, and heritage intersect. As part of a longitudinal embedded multi-year ethnographic study, we conducted cohort analyses as well as grounded, iterative, and thematic coding of semi-structured interview data, augmented with crafting artifacts from 65 adult fiber crafters. Using qualitative analytical techniques, we asked: How does math occur in craft? How do crafters observe the intersection between math and craft in process? Fiber crafts were found to present a “Mathland,” a lifelong context for immersive math engagement. We present crafters’ math insights in the craft, as well as multiple aspects of the crafts and surrounding communities that supported the crafters in sustaining their engagement with mathematics throughout their lifetime. This study has implications for the design of inclusive and lifelong maker educational environments for mathematics learning.

On the Epistemological Relevance of Social Power and Justice in Mathematics

In this paper we argue that questions about which mathematical ideas mathematicians are exposed to and choose to pay attention to are epistemologically relevant and entangled with power dynamics and social justice concerns. There is a considerable body of literature that discusses the dissemination and uptake of ideas as social justice issues. We argue that these insights are also relevant for the epistemology of mathematics. We make this visible by a journalistic exploration of relevant cases and embed our insights into the larger question how mathematical ideas are taken up in mathematical practices. We argue that epistemologies of mathematics ought to account for questions of exposure to and choice of attention to mathematical ideas, and remark on the political relevance of such epistemologies.

The Impact of Incorporating Indigenous and Other Nontraditional Ways of Mathematical Knowing into a University-Level Geometry Course

During the Fall 2021 semester, the author taught a university-level geometry course into which they incorporated texts and discussions on mathematics and mathematical epistemology from outside of the “Western” tradition typically centered in college math curricula. Analysis of student survey responses and students’ reflections on their work offer some evidence that even minimal engagement with these nontraditional perspectives, facilitated intentionally, led to increases in students’ appreciation of other epistemic traditions. Though the smallness of the sample size prohibits drawing broader conclusions, the significance of some findings suggests a critical need for further study of these pedagogical practices.

The Quasi-Empirical Epistemology of Mathematics

Abstract This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that (1) Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; (2) what can be called the quasi-empirical epistemology of mathematics is not correct; (3) analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.

Frege's Curiously Two-Dimensional Concept-Script

In this paper I argue that the two-dimensional character of Frege’s Begriffsschrift plays an epistemological role in his argument for the analyticity of arithmetic. First, I motivate the claim that its two-dimensional character needs a historical explanation. Then, to set the stage, I discuss Frege’s notion of a Begriffsschrift and Kant’s epistemology of mathematics as synthetic a priori and partly grounded in intuition, canvassing Frege’s sharp disagreement on these points. Finally, I argue that the two-dimensional character of Frege’s notations play the epistemological role of facilitating our grasp of logical truths (foundational and derived) independently of intuition. The rest of this paper critically evaluates Frege’s view and discusses Macbeth’s (2005) account.

Thin Objects Are Not Transparent.

In this short paper, we analyse whether assuming that mathematical objects are "thin" in Linnebo's sense simplifies the epistemology of mathematics. Towards this end, we introduce the notion of transparency and show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparency of some thin objects is a fruitful characteristic mark of abstract mathematics.

Model eliciting activities in mathematics learning: Epistemological beliefs

This study aims to determine epistemological beliefs, beliefs in learning mathematics with eliciting activities model, and the relationship between the two on prospective mathematics teachers. Quantitative research method using a survey, which was conducted on 52 students of prospective mathematics teachers. Data collection is done by using a questionnaire. This study uses a questionnaire from TEDS-M which investigates the beliefs of prospective mathematics teachers about teaching mathematics. Data analysis will be carried out quantitatively, both descriptively and by correlation. The results showed that 1) mathematics teacher candidates had mixed beliefs, namely static and dynamic, but more static beliefs about the dominant mathematical epistemology: 2) The application of eliciting activities model learning, mathematics teacher candidates had mixed beliefs, namely conventional and constructive, but more on dominant constructivist beliefs and 3) practical implications for prospective mathematics teachers and the application of learning models must begin with testing or understanding the beliefs of prospective mathematics teachers regarding the epistemology of mathematics.

The relationship between pre-service elementary school mathematics teachers’ beliefs about epistemology of mathematics, teaching and learning, and mathematics assessment

<div><p>This study aims to examine the relationship between epistemological beliefs, teaching-learning beliefs and assessment beliefs in mathematics education. This research is a quantitative study with a correlational study. Data collection using the survey method with a cross-sectional design. The participants were 71 pre-service elementary school , mathematics teachers. The data on beliefs were collected through means of a questionnaire. The data collected from the questionnaire were then analyzed quantitatively through descriptive and inferential statistics. Descriptive statistics utilizes the mean value, maximum value, and standard deviation values. Inferential statistics use the product-moment correlation as well as path analysis. The research results show that there is a positive and significant correlation between static and dynamic beliefs on epistemology of mathematics, and the constructivist beliefs on mathematics teaching and learning, with the productive beliefs on mathematics assessment. In addition, there is seen to be a functional influence between both epistimological beliefs (both static and dynamic), as well as beliefs on teaching and learning (constructivist) and beliefs about mathematic assessment (productive). The results of this research signify the importance of considering one’s beliefs about the epistemology of mathematics and mathematics teaching and learning when constructing their beliefs regarding mathematics assessment.</p></div>

Why Had Russell Not Written Any Books on Aesthetics?

Bertrand Russell, the great philosopher, was extremely prolific in various fields of philosophy, such as metaphysics, mathematical logic and mathematical philosophy, linguistic philosophy, ethics, epistemology, and social and political philosophy, but left little legacy in aesthetics. Some scholars regretted that “If the 20th-century had seen any polymath, Russell is the one. The only branch of philosophy he did not write on is aesthetics.” Although Russell did not write a book or article specifically on aesthetics, discussions on the subject can be found in his various writings, for example, the most prominent proposal of “the beauty of mathematics.” In addition, he also talked about how agnostics explain the beauty and harmony of nature from the perspective of epistemology. Views on aesthetics are scattered in Russell's writings, and no consistent views have been formed; some fragmented ideas about aesthetics are scattered in different philosophical writings. Compared with his profound contribution to other fields of philosophy, Russell did not deal with aesthetic issues systematically, nor did he attempt to answer the basic questions of aesthetics. In addition to personal qualities, there were certain constraints of the greater social environment. One of the main reasons is that in the English world of the 20th century, the analytical philosophy of language occupied the central stage, so that the space reserved for ethics, aesthetics, and political philosophy was quite limited. Furthermore, aesthetics was indeed not a popular philosophical topic culturally acceptable at the time.

From axiomatic systems to the Dogmatic gene and beyond

The positivistic views that dominated the early debate on the foundations of mathematics, at the beginning of the 20th century, survived the “negative results” that have shown the limits of the axiomatic approach since the 1930s. Rigour, abstraction and symbolism have been confused with formalism, based on finite strings of signs, pre-given axioms, and potentially mechanisable rewriting rules. This contributed to major clarifications in the mathematical praxes but obscured the limits of formalisms due to the exclusion of the historical creation of sense proper to any science. We expand on this sometimes fruitful confusion with some case studies. We then hint to the historical creation of sense as a component of an epistemology of mathematics. We continue with an analogy with genocentric approaches in biology, as similar positivistic views resurfaced there fifty years later. Finite sequences of letters in the DNA would completely determine ontogenesis and phylogenesis, according to the Central Dogma of molecular biology. Limits and “negative evidence” have been disregarded while searching for the “gene for” everything. Alternative perspectives require a reconstruction of the sense of history as locus for the constitution of any object of biological knowledge. In particular, the historicity of biological evolution will be understood in terms of changing phase spaces and of the role of rare events in all phylogenetic trajectories. The analysis of the evolutionary production of variability, adaptivity and ecosystemic diversity is a key component of the project we hint to, as part of a renewed relation to the biological environment.

La importancia de la intuición matemática en los procesos de enseñanza

El presente documento muestra cómo elementos constitutivos de la intuición pueden llevar a un docente de matemáticas a innovar en su práctica educativa. Teniendo en cuenta los procesos de enseñanza en matemáticas, se propone a partir del método de sustitución para resolver integrales, evidenciar las características comunes a las definiciones sobre intuición brindadas tanto por filósofos como por matemáticos. Concluimos que la intuición, no sólo tiene un rol importante en la epistemología de las matemáticas, sino que puede jugar un papel fundamental tanto en el aprendizaje de los estudiantes como en las estrategias de enseñanza de los docentes de matemáticas.