Developing the notions of variation and covariation through patterns : An institutional analysis of primary school textbooks

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Developing the notions of variation and covariation through patterns : An institutional analysis of primary

school textbooks

Eudes Libert Mamas Mavoungou, Alejandro González-Martín

To cite this version:

Eudes Libert Mamas Mavoungou, Alejandro González-Martín. Developing the notions of variation

and covariation through patterns : An institutional analysis of primary school textbooks. CERME 9 -

Ninth Congress of the European Society for Research in Mathematics Education, Charles University in

Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.500-501. �hal-01286973�

500 CERME9 (2015) – TWG03

Developing the notions of variation and

covariation through patterns: An institutional analysis of primary school textbooks

Eudes Libert Mamas Mavoungou and Alejandro S. González-Martín

Université de Montréal, Département de Didactique, Montréal, Canada, [email protected]

We examine how pattern-based activities presented in textbooks for primary education allow developing the notions of variation and covariation, using the con- ceptual tool of institutional relationship of Chevallard (2003). Our sample is formed by the textbooks currently approved by the Quebec Ministry of Education. Ours results reveal a didactic void in the textbooks.

Keywords: Variation and covariation, textbook analysis, institutional relationship.

INTRODUCTION AND RESEARCH PROBLEM Researchers have identified numerous difficulties faced by students in learning functions. Because el- ementary school students are able to grasp pre-alge- braic concepts, researchers recommend introducing pre-algebra at the primary level to facilitate the sub- sequent learning of algebra and functions at higher levels. The notions of variation and covariation have been highlighted for their role in fostering a better understanding of functions at the secondary level (Hitt & González-Martín, 2015). In primary education, the notions of variation and covariation can be taught using pictorial growth patterns, which foster two modes of reasoning: figural and numerical. Becker and Rivera (2005) found that prospective elementary school teachers predominantly use numerical reason- ing to analyse patterns with their pupils. Because both modes of reasoning should be present in the class- room, and considering the tendency of teachers and students to favour numerical reasoning, we are inter- ested in examining how pattern-based activities are presented in elementary school textbooks and how this reveals the institutional relationship with these notions. Specifically, our research questions are: 1) how are notions related to pictorial growth patterns

organised in elementary school textbooks and what type of reasoning do they tend to favour, and, 2) do the textbooks’ activities effectively allow students to develop the notions of variation and covariation?

THEORETICAL FRAMEWORK

Our research leans heavily on Chevallard’s anthropo- logical theory of the didactic (ATD, 2003). In particular, we employ the notion of institutional relationship to study institutional choices concerning the use of the mathematical objects of variation and covariation in textbooks. We also use the notion of praxeology to characterise the institutional relationship with the notions of variation and covariation.

METHODOLOGY

The content analysis applied to textbooks in this study

was performed with tools provided by ATD. We ana-

lysed the three current elementary school textbooks

approved by the Quebec Ministry of Education for the

final years of primary school (10- and 11-year-old stu-

dents), focusing on the following elements: the nature

of symbols used (numerical or pictorial), the types

of tasks and the praxeologies they define, the goal of

these tasks, the independent and dependent variables

at play in the task, the regularity from one position to

subsequent positions, the mode of reasoning favoured,

and the relationships between variables. We paid par-

ticular attention to how the textbooks lead students

to use the information presented to accomplish tasks

and develop an awareness of relationships between

two given quantities.

Developing the notions of variation and covariation through patterns (Eudes Libert Mamas Mavoungou and Alejandro S. González-Martín)

501 DATA ANALYSIS AND DISCUSSION

We found only 18 cases, in two of the three textbooks, in which a task makes use of patterns: 11 tasks using numerical patterns, six tasks using patterns pre- sented both numerically and pictorially, and one task using pictorial patterns. We identified three different types of tasks: 1) introducing the definition of a pat- tern, 2) predicting subsequent terms, and, 3) gener- alising through an expression. Each of these types of task defines a different mathematical organisation:

the first attempts to familiarise students with the idea of change or variation, the second aims to teach stu- dents the idea of covariation, and the third focuses on dependence relations through a rule. The relationship between variables is not specified in either of the two textbooks and both use the symbol n in formulae with- out explicitly giving any meaning to it. All 18 tasks can be solved using just two standard techniques, which favour the use of numerical reasoning. Moreover, the technology that could reinforce and give meaning to the notions of variation and covariation is absent, and therefore students are not provided with optimal con- ditions for developing these notions. The institutional relationship with these notions is characterised by a didactic void, especially with regard to the theoretical block. We are aware that this research has been con- ducted on a small scale and that further study is neces- sary to investigate whether children are given better opportunities to develop the notions of variation and covariation in primary school. For this reason, we are planning to examine potential difficulties students may encounter when presented with patterns and other generalisation activities. We also wish to study teachers’ practices and professional teacher educa- tion in the effective use of patterns.

REFERENCES

Becker, J. R., & Rivera, F. (2005). Generalization schemes in algebra of beginning high school students. In C. Chick &

J. Vincent (Eds.), Proceedings of the 29

Conference of the International Group for Psychology of Mathematics Education (Vol. 4, pp.121–128). Melbourne (Australia): PME.

Chevallard, Y. (2003). Approche anthropologique du rapport au savoir et Didactique des Mathématiques. In S. Maury & M.

Callot (Eds.), Rapport au savoir et didactique (pp. 81–104).

Paris: Éditions Fabert.

Hitt, F., & González-Martín, A.S. (2015). Covariation between variables in a modelling process: The ACODESA (collabo- rative learning, scientific debate and self-reflection) meth-

od. Educational Studies in Mathematics, 88(2), 201–219.

Doi10.1007/s10649-014-9578-7