Semiotic representations have been an important topic of study in mathematics education

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic.

In this paper we analyse the introduction of irrational and real numbers in secondary textbooks, as they are proposed to be taught, from a sample of Brazilian textbooks used in public schools and approved by the Ministry of Education. The analyses discussed in this paper follow an institutional perspective (using Chevallard's Anthropological Theory of Didactics). Our results indicate that the notion of irrational number is generally introduced on the basis of the decimal representation of numbers, and that the mathematical need for the construction of the field of real numbers remains unclear in the textbooks.It seems that textbooks used in secondary teaching institutions develop mathematical organisations which focus on the practical block.Keywords: real numbers; irrational numbers; textbooks analysis; praxeology; mathematical organisation Introduction and rationaleTraditionally, in elementary school, the introduction of the set of integers is based on the algebraic limitations of the set of natural numbers. This construction is motivated by some "daily life" problems in which it is necessary to find the difference between two natural numbers, the first being smaller than the second. Similarly, the extension from integers to rational numbers involves the limitation of the operation of division, illustrated by some practical problems. Even the construction of complex numbers at 1 higher educational levels is grounded in the algebraic structure of the field of real numbers (which is assumed to be previously known), namely the impossibility of finding roots for some polynomial equations. Therefore, the learning of different sets of numbers can be seen as a progressive extension of the initial perception of numbers through the algebraic structure of nested number sets, from the primitive notion of counting, to the ideas of comparing, measuring and solving equations.The case for the extension from rational to real numbers is particularly dramatic.Unlike the previous extensions, this is not an algebraic jump, as it formally requires theoretical properties such as convergence and completeness. This has proved to be a crucial obstacle, which began with the incommensurable magnitudes debate in Greek mathematics (e.g. Katz 1992, pp. 73-74). Moreover, only a discrete set of numbers is enough to deal with the empirical problem of measurement, whilst the system of real numbers accounts for the construction of a theory of measure consistent with classical geometry. Therefore, the need to construct the set of real numbers can hardly be established or concretized upon empirical motivations. As research has shown (see section 3), these epistemological obstacles and theoretical constraints have repercussions both in teaching and in learning. On the one hand, a formal definition of real number is surely incompatible with elementary and secondary school. On the other hand, the set of real numbers cannot be built upon empirical or algebraic demands i . In our opinion, the main difficulties in motivating the introduction of real num...

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