The introduction of real numbers in secondary education: an institutional analysis of textbooks

Abstract

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...

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

“…Middle school students' estimates of integers in the negative range À10,000 -0 and the combined range À1,000 -1,000 are linear, though with lower accuracy for negative numbers (Young & Booth, 2015). Adults' estimates of integers in the combined range À100 -100 are linear and show comparable accuracy for positive and negative numbers (Ganor-Stern & Tzelgov, 2008). Performance on NLE tasks like these depends partly on recognizing the symmetry of the integers around the zero point (Saxe et al, 2010).…”

“…Mathematics teachers, in an attempt to facilitate understanding, encourage students to think of irrational numbers like p by using rational number approximations like 3.14. Unfortunately, this can lead to conceptual confusions (Gonz alez-Mart ın, Giraldo, & Souto, 2013;Sirotic & Zazkis, 2007).…”

Full issue

“…In fact, our methodological approach has similarities to the one employed in the study by González-Martín et al (2013), especially the use of the notion of praxeology to study tasks in textbooks; but the two methodologies also different, as we shall now explain. These authors investigated the case of the introduction of real numbers in Brazilian textbooks, based on a model which has, at its basis, rather broad classes of tasks for the students, such as : "Classifying a given number as rational or irrational".…”

“…Since a praxeology is a model, it depends on the purpose of modelling what kind of human action it will be useful or feasible to distinguish as a technique; the theory does not provide any strict definition of what would count as a technique (and thereby, as a type of task). We note here that the main difference between our approach and the uses of ATD for textbook analysis provided by Lundberg (2011) and González-Martín et al (2013) is the explicit definition of techniques (presented in Section 4), which enable us to work with types of tasks (in the proper sense of ATD, that is, defined by one technique) rather than the informal use of the term type of task as "a collection of tasks with a similar form and content".…”

Picturing Textbook on Exponent Equations Based on Praxeology Organization

“…Research on mathematics textbooks analysis using praxeology organization, proposed by Chevallard has become a major attention among researchers [6]. For example, Barbe et al [7] and González-Martín et al [8] describe a very detail analysis both textbook and how students learn in a class using the notion of praxeology organization. Furthermore, a research using praxeology organization can also use to capture how a sector in mathematics (e.g proportion) changes from time to time [9,10].…”