Introduction of an historical and anthropological perspective in mathematics : an example in secondary school in France (original) (raw)
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Irrational Numbers A Constructive approach at Elementary and High School
Author's Edition, 2013
Irrational numbers, subject of this ebook, represent a sophisticated mathematical idea, focused on little intuitive theoretical aspects, with few connections to sensorial world. A crucial point lies on the fact that talking about irrational numbers necessarily leads to discuss the tenuous and inherent connection with real numbers. As an initial resource, we highlight a literature review from Arcavi et al. (1987), Fischbein; Jehian; Cohen (1995), Soares; Ferreira; Moreira (1999), Ripoll (2001), Rezende (2003), Leviathan (2004), Zazkis;Sirotic (2004), Sirotic;Zazkis (2007), Barthel (2010), Silva (2011) and Voskoglou e Kosyvyas (2011), among some others, researchers who conducted studies with respect to irrational and real numbers. For this composition, we initiated a research considering didactical, epistemological and historical resources to discuss the problem of introducing this numeric field in a significant way at Elementary Mathematics. Facing the assessments made through the argumentations pointed out, we highlight the limitations of operative, deterministic and exact aspects to present irrational numbers presentation, a feature that is common in Mathematics’ teaching. The relationship of tension and interaction between some axes – finite&infinite, exact&approximate and discrete&continuous – allowed us to conceive a metaphoric and suggestive place that we call 'the Space of Meanings'. We consider ‘the Space of Meanings’ was a metaphoric field that allows understanding and guiding an irrational numbers approach at Elementary Mathematics, in a more comprehensive and significant context, subject which will be discussed throughout this text.
WHAT IS MATHEMATICS? Perspectives inspired by anthropology
2015
The paper discusses the question "what is mathematics" from a point of view inspired by anthropology. In this perspective, the character of mathematical thinking and argument is strongly affected-almost essentially determined, indeed-by the dynamics of the specific social, mostly professional environments by which it is carried. Environments where future practitioners are taught as apprentices produce an approach different from that resulting from teaching in a school-the latter inviting to intra-mathematical explanation in a way the former does not. Moreover, once the interaction with the early classical Greek philosophical quest for causes and general explanations had caused mathematical explanation to become an autonomous endeavour in the shape of explicit proof and deductivity, proof and deductivity presented themselves as optionssometimes exploited, sometimes not-even in the teaching of mathematics for practitioners.
On Teaching and Learning Mathematics from a Cultural-Historical Perspective
Teaching and Learning Secondary School Mathematics. Advances in Mathematics Education. , 2018
In this chapter, we discuss some ideas of a cultural-historical theory of the teaching and learning of mathematics. The basic ideas emerged from, and have evolved during, an ongoing long-term collaboration between researchers and teachers. Since its inception, this long-term collaboration has sought to offer an alternative to child-centred individualist educational perspectives. It endeavours to understand and foster mathematics thinking, teaching, and learning conceived of as cultural-historical phenomena. This collaboration has led to what has been termed the theory of objectification. We illustrate the basic ideas through the discussion of a classroom episode where what is at stake is the production and understanding of graphs in a grade 10 mathematics class.
Mathematics and its ideologies. (An anthropologist's observations)
Semiotic Review, 2020
Starting from the profound impact of Kenneth Arrow's Impossibility Theorem on the social sciences of the postwar twentieth century, this essay engages with the ways in which mathematics can be seen as a language-ideologically inflated notational system. In the mid-twentieth century, a profound belief in mathematics as a purely objective and non-ideological organization of knowledge took hold, and mathematical proof became the most authoritative type of statement on reality. When something was ruled 'logically impossible', real-world occurences could be seen as transgressions and exceptions. Hidden inside this belief is a set of irrational, metaphysical assumptions about humans and social behavior that can be laid bare by means of linguistic-anthropological analysis.
Anthropological Approaches in Mathematics Education, French Perspectives
Encyclopedia of Mathematics Education, 2014
This entry encompasses two interrelated though distinct approaches to mathematics education: the Anthropological Theory of the Didactic (ATD for short) and the Joint Action Theory in Didactics (JATD). Historically, the germs of ATD are to be found in the theory of didactic transposition (Chevallard 1991), whose scope was at first limited to the genesis and the ensuing peculiarities of the (mathematical) "contents" studied at school; from this perspective, ATD should be regarded as the result of a definite effort to go further by providing a unitary theory of didactic phenomena as defined in what follows. As for JATD, it has emerged from the theory of didactic situations (Brousseau 1997) and the anthropological theory of the didactic by focusing on the very nature of the communicational epistemic process within didactic transactions. ATD and JATD share a common conception of knowledge as a practice and a discourse on practice together-i.e. as a praxeology-along with a pragmatist epistemology which gives a prominent place to praxis. Their well thought-out anthropological stance leads the researcher to study didactic facts wherever they are located in social practices. Although these theorizations are by necessity expounded tersely, we hope their forthright presentation will allow the reader to catch the gist of them.
Acta Didactica Universitatis Comenianae. Mathematics, 2010
A quick glance at contemporary mathematics education makes plain that we are living in a time of important changes. The ideas conveyed by classical theories in our field, including learning as a mental adaptive construction and the conception of the teacher as a mere learning facilitator, are now questioned. There has been an important shift provoked by a profound need to regain contact with the realm of the social and the cultural. However, this shift, produced by what I term here “the anthropological turn in Mathematics Education”, is not without its problems. It requires a re-conceptualization of Mathematics Education and, more specifically, of the learner, the teacher and the knowledge to be learned. In this article, I present an overview of what mathematical activity and classroom practice look like from an emerging sociocultural perspective – the theory of knowledge objectification.
A study on the comprehension of irrational numbers
math.unipa.it
Following a theoretical introduction concerning the difficulties that people face for understanding the structure of the basic sets of numbers, we present a classroom experiment on the comprehension of the irrational numbers by students that took place at the 1 st Pilot High School of lower level (Gymnasium) of Athens and at the Graduate Technological Educational Institute of Patras, Greece. The outcomes of our experiment seem to validate our basic hypothesis that the main intuitive difficulty for students towards the understanding of irrational numbers has to do with their semiotic representations (i.e. the ways in which we describe and we write them down). Other conclusions include the degree of affect of age, of the width of mathematical knowledge, of geometric representations, etc, for the comprehension of the irrational numbers.
The use of History into Mathematics Education links teaching-learning processes with historical elements. In this paper we discuss some epistemological issues related with the historical analysis of a mathematical topic, in order to achieve an effective and correct use of historical data into Mathematics Education. In particular we present some theoretical frameworks and underline the primary importance of the correct social and cultural contextualisation. Finally, we propose the comparison of some different strategies used by mathematicians in different historical periods in order to prove a theorem, with reference to presented theoretical frameworks.