Connections between approximating irrationals by a sequence of
decimals and introducing the notion of convergence
Hichem Khechine1 and Alejandro González-Martín2
1
2
Université virtuelle, Isefc, Tunis, Tunisia; [email protected]
Université de Montréal, Département Didactique, Canada; [email protected]
The History of mathematics shows that approximating irrationals by sequences of decimals played
an important role in the construction of the set of real numbers. It seems reasonable to conjecture
that activities related to such approximations could prove useful when introducing students to
notions such as limits, convergence, real numbers, and density. In this poster we present a
praxeological analysis of activities used to introduce the set of real numbers – in particular,
approximating irrationals using sequences of decimals – found in the official ninth-level
mathematics textbook for basic education in Tunisia. We compare these praxeologies with those
present in numerical activities that introduce the notion of limit in the third level of secondary
school. Our goal is to identify the common elements of these praxeologies.
Keywords: Approximation, decimals, convergence, irrationals, praxeology.
Introduction and research problem
The shift from rational numbers to irrational and real numbers can prove challenging for students,
and the way irrationals and reals are introduced in secondary textbooks does not seem to promote
the development of ideas that allow pupils to adequately grasp convergence or density later in their
tertiary studies (González-Martín, Giraldo, & Souto, 2013). Difficulties with irrational numbers also
have been reported in university students (Kidron, 2016). However, we believe that certain
activities, such as studying the approximation of irrational numbers by sequences of decimals, could
help students grasp, informally, the ideas behind the formal
definition of convergence at
university, where real numbers are introduced as equivalence classes of Cauchy sequences of
rationals. We have not found works in the literature that address the connections between the
teaching of these two notions. For this reason, we seek to investigate how they are presented in
textbooks and study the connections that are made (or the lack thereof). Our research is guided by
the following questions: 1) how do pre-university textbooks organize and present the notions of
limit and of approximation of a real number by a sequence of decimals?; and 2) what opportunities
do textbook tasks provide for students to develop connections between the two notions?
Theoretical framework
Our research uses tools from Chevallard’s (1999) anthropological theory of the didactic (ATD).
ATD acknowledges that every human activity generates a praxeology or praxeological organisation
identified by the quadruplet [T/ /θ/Θ], where T is a type of task, is a technique used to complete
this task, θ is a discourse (technology) that justifies and explains the technique, and Θ is a theory
that includes and justifies the given discourse. The couple [T/ ] is the practical block (or praxis)
and [θ/Θ] is the theoretical block (or logos). We focus on the institutional relationship with the
notions of real number (and activities of approximation) and limit, as reflected by the textbooks.
Methodology
Our study uses the tools provided by ATD to analyse the textbooks and recommendations of the
Tunisian Ministry of Education’s official programme. Teachers in Tunisia use a different textbook
for each school level, published by the Ministry. We chose two textbooks for our analysis: the first
is used in the ninth level of the basic cycle (14-15 year-old students) and the second is used in the
third level of secondary school (17-18 year-old students), which is the penultimate year before
university. We are currently analysing the first textbook – in particular the content related to real
numbers, paying special attention to tasks concerning the approximation of an irrational by a
sequence of decimals. We are focusing on whether such tasks implicitly use a theoretical block
based on the notion of limit. We plan to analyse the second textbook before the end of 2016,
focusing on the chapter related to limits, to determine whether this chapter develops praxeologies
that use (implicitly or explicitly) any of the elements present in the first textbook.
Data analysis and discussion
At the time this first version of the poster proposal is being submitted, we are undertaking the
analyses of the first textbook. We have identified four types of tasks in the book that are based on
the approximation of irrationals by sequences of decimals, all of which aim to introduce routine
techniques such as the algorithm for the framing of √2 between two decimals. This is presented in a
praxeology where pupils are first asked to accept, through a pictorial task (Pythagoras’ theorem is
presented at the end of the year), that a square whose surface is 2 has a side measuring √2; after
that, the steps of the algorithm of approximation are explicitly presented, to be reproduced for the
cases of √5 and √8. The textbook proposes to make these calculations using a calculator, without
presenting the reasons for this tedious calculation. There is no use of the real line for the type of
tasks of approximation, which could allow students to visualise a framing sequence of an irrational.
The institutional relationship with this notion is marked by an institutional void with regard to the
theoretical block. We conjecture that this void can have an impact later in the third level of
secondary school, during the introduction of the notions of limit and convergence. The analyses that
we plan to conduct on the second textbook will allow us to see whether certain elements used in the
first book are used – and how they are used – by the second. We plan to make some conjectures
about students’ learning based on the elements present and absent at both levels.
References
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique de la
didactique. Recherches en Didactique des Mathématiques, 19 (2), 221-226.
González-Martín, A. S., Giraldo, V., & Souto, A. M. (2013). The introduction of real numbers in
secondary education: an institutional analysis of textbooks. Research in Mathematics Education,
15(3), 230–248. doi: 10.1080/14794802.2013.803778
Kidron, I. (2016). Understanding irrational numbers by means of their representation as nonrepeating decimals. In E. Nardi, C. Winsløw & T. Hausberger (Eds.), Proceedings of the First
Conference of the International Network for Didactic: INDRUM 2016 (pp. 73–82). Montpellier,
France: University of Montpellier and INDRUM.