This chapter is framed both within the Kantean notions of sensible and intellectual intuitions and within the Peircean notion of collateral knowledge and classification of inferential reasoning into abductive, inductive, and deductive. An overview of the Peircean notion of abduction is followed by a sub-classification of abductions according to Thagard and Eco. The constructive nature of the process of proving seems to involve not only deductive reasoning but also abductive reasoning. The later plays an essential role both in the anticipation of auxiliary constructions and in the construction of geometric arguments. The chapter presents a summary of Kant’s classification of the proposition “the angles in any triangle add 180°” as a synthetic proposition. It also presents a deconstruction of the three classical proofs of this proposition—the Pythagorean proof, Euclid’s proof, and Proclus’ proof. This deconstruction discloses both the Greek analysis-synthesis method of proving and the role of abduction in the analysis phase. It also argues that the deconstruction of classical proofs has pedagogical and epistemological value in the teaching-learning of geometry.
During the last two decades, semiotics has been attaining an important explanatory role within mathematics education. This is partly due to its wide range of applicability. In particular, the success of semiotics in mathematics education may be also a consequence of the iconicity and indexicality embedded in symbols, in general, and mathematical symbols, in particular. The introductory chapter discusses issues of signs, sign use, and communication. On the one hand, it shows how semiotics elucidates the way knowledge and experience of mathematics students can co-construct each other. On the other, it shows how students’ construction of mathematical knowledge is linked to successful communication mediated by visible signs with their rule-like transformations. In this sense, the systems of signs and communication through them are closely tied when students send and receive mathematical messages.
Semiotic reality is a fundamental part of our common reality. Where we stand in this chapter looks upon the teaching-learning of mathematics as a double semiotic process of interpretation. It takes place within the socio-mathematical semiotic reality that teachers and students inherit and jointly activate in the classroom. We argue that, during interpretation, the formation of students’ mathematical conceptions and the attainment of their mathematical Concepts is constructed not only with the guidance of teachers. It also follows a progressive and corrective process of inter-intra interpretation. We emphasize that teachers’ awareness of the evolving nature and refinement of their own processes of interpretation and, especially, their awareness of the interpretations that takes place in the students, is essential to maintain a collaborative and dynamic teaching-learning signifying practice. Our understanding of the Person-Object relation agrees with Vygotsky when we claim that objectification is a special case of internalization. This objectification takes place during Self-Other external activity aided by Self-Self internal activity. Taking a Peircean perspective not only puts a special emphasis on intra-placed mathematical sign-interpretant formation, but it also puts a high focus on intra-abstracting-objectification that takes place in each and every student.
Semiotics as a Tool for Learning Mathematics is a collection of ten theoretical and empirical chapters, from researchers all over the world, who are interested in semiotic notions and their practical uses in mathematics classrooms. Collectively, they present a semiotic contribution to enhance pedagogical aspects both for the teaching of school mathematics and for the preparation of pre-service teachers. This enhancement involves the use of diagrams to visualize implicit or explicit mathematical relations and the use of mathematical discourse to facilitate the emergence of inferential reasoning in the process of argumentation. It will also facilitate the construction of proofs and solutions of mathematical problems as well as the progressive construction of mathematical conceptions that, eventually, will approximate the concept(s) encoded in mathematical symbols. These symbols hinge not only of mental operations but also on indexical and iconic aspects; aspects which often are not taken into account when working on the meaning of mathematical symbols. For such an enhancement to happen, it is necessary to transform basic notions of semiotic theories to make them usable for mathematics education. In addition, it is also necessary to back theoretical claims with empirical data. This anthology attempts to deal with such a conjunction. Overall, this book can be used as a theoretical basis for further semiotic considerations as well as for the design of different ways of teaching mathematical concepts.
Using a semiotic perspective based on Peirce’s triadic sign theory, we try to capture part of the complexity that teacher and students encounter during the transition from an empiric procedure used to solve a geometric problem to a mathematical procedure needed to validate the construction, within a theoretic system for Euclidean geometry. Such a step implies substituting the use of a circle as a tool to transfer length measurements for the postulate that permits establishing a bijective correspondence between the points of a line and the real numbers, and through it, determine a point with a certain distance condition. We analyze a class episode that took place in a geometry course of a pre-service mathematics teacher program.