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, and specifically the propositions on how these should be taught, in a sample of Brazilian textbooks used in state 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

Notes

1. Of course, algebraic irrational numbers can be introduced through the solution to some polynomial equations; however, since they are not transcendental numbers, the need for these algebraic numbers can be seen as a motivation to go further than the rationals. But it is not enough to justify the need for the set of real numbers as a mathematical structure.

2. In this sense, the anthropological theory recognises the importance of ostensives (e.g. Bosch & Chevallard, 1999), which allow the manipulation of mathematical objects.

3. We use the term ‘density property’ in the following sense: a metric space (E, d) is said to be dense if for every element x ∊ E, and for every neighborhood U of x, there is point y ∊ U, y ≠ x. Our choice of terminology (which is not necessarily standard in mathematical literature) is due to its broad use in Brazilian mathematics textbooks.

4. Briefly, paraphrasing Barbé et al. (2005, pp. 238–239) these six moments are: the first encounter with the MO at stake; the exploration of the type of tasks T and elaboration of a technique τ relative to these tasks; the constitution of the technological–theoretical environment relative to τ; the technical work, which at the same time aims to improve the technique making it more powerful and reliable and develop the mastery of its use; the institutionalisation, the aim of which is to identify what the specific mathematical knowledge to learn exactly is; and, the evaluation of what was learnt.