This study shows that children up to the age of 7 spontaneously use symmetry in their drawings to express their knowledge about plant life. In the sample analysed, this is a very frequent strategy and becomes more complex with the education level, as highlighted by the researchers in the Faculty of Education -  Bilbao (José Domingo Villarroel and Álvaro Antón) and the Faculty of Science and Technology (María Merino).

“The fact of establishing that, well before the age of 7, children display drawing skills that include the spontaneous depiction of symmetries, should exert a significant influence on infant teaching and learning processes, not only with respect to the sphere of comprehension of biological phenomena but also with respect to the development of geometrical thinking,” said the Professor of the Department of Mathematics and Experimental Sciences Didactics, José Domingo Villarroel.  

The results of this research have been published in the international scientific journal Symmetry that includes articles on symmetry in Mathematics and in the rest of the sciences. This journal is indexed in the Science Citation Index Expanded (Web of Science) with an impact factor of 1.256.

The research method

Prof Villarroel explains that “a highly significant factor related to teaching and learning processes is about being able to determine the pupils’ abilities and knowledge so as to be able to adapt didactic activities to these prior determinants. This is one of the golden rules in education”.

In this respect, the lecturer in the same department, Álvaro Antón, points out that “knowing that children spontaneously use symmetry provides an opportunity to explore graphical expression in childhood and use this resource in the teaching activities relating to biological phenomena and geometrical knowledge”.

To conduct their analysis the research team worked with a sample of 116 drawings produced by 65 girls and 41 boys from three schools in Pre-primary and Primary Education located in the Uribe-Kosta district in Bizkaia during the 2012-2013 academic year. They were spontaneous drawings, without any knowledge or instructions beforehand relating to symmetry.

The researchers selected plant life as the subject for pictorial expression, a subject that a priori has no apparent link with geometry and symmetry. With the help of a puppet, the children were encouraged to produce a drawing that would explain to the puppet what plants are like, where they live and what is good for them. The individual activity took about ten minutes.

These representations are the ones that were analysed by the research team and they found that the girls and boys used two types of symmetry. As the lecturer in the area of Statistics and Operations Research María Merino explained, “they use cyclic symmetry (which presents rotational symmetry around a central point), for example, when they depict the sun; and dihedral symmetry (which includes both rotational symmetry and reflection symmetry) when depicting the human shape, for example. Of the two, the most common is dihedral symmetry which they use to draw the plant world, people and their environment or decorative elements such as stars or hearts”.

At the same time the researchers saw that the complexity in depicting dihedral symmetries is greater than that corresponding to cyclic symmetries, and that at higher levels of education, when children need to express more in-depth knowledge, they also draw more complex symmetrical pictorial items. From the gender perspective, girls use complex symmetries more frequently than boys.

This study is a first step towards finding out the connection existing between children’s knowledge about the plant world and the images they produce; the aim is to find out what relationship exists between geometrical thinking and the graphical expression of this thinking. “The study of these connections is hugely interesting, because,” asserts Villarroel, “scientific activity is always linked to Mathematics. Scientific thinking is inevitably linked to mathematical thinking and that is why it is important to understand how relations begin to be established during childhood between both types of thinking, that relating to the explanation of biological phenomena and geometry”.

Authors of the research

José Domingo Villarroel holds a PhD from the UPV/EHU and is a lecturer and researcher in the Faculty of Education – Bilbao (UPV/EHU) in the Department of Mathematics and Experimental Sciences Didactics. https://orcid.org/0000-0003-4108-6122

Maria Merino-Maestre holds a PhD from the UPV/EHU and is a lecturer and researcher in the Faculty of Science and Technology (UPV/EHU) in the Department of Applied Mathematics and Statistics and Operations Research.

Álvaro Antón-Baranda holds a PhD in Sciences from the UPV/EHU and is a lecturer and researcher in the Faculty of Education - Bilbao (UPV/EHU) in the Department of Applied Mathematics and Statistics and Operations Research. https://orcid.org/0000-0003-4108-6122