Areti Panaoura

Geometric and algebraic approaches in the concept of complex numbers

This study explores pupils’ performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from high schools in Greece (17–18 years old). Results shed light on pupils’ use of two distinct approaches to solve complex number tasks: the geometric and the algebraic approach. The geometric approach was used more frequently, while the pupils used the algebraic approach more consistently and in a more persistent way. The phenomenon of compartmentalization indicating a fragmental understanding of complex numbers was revealed among pupils who implemented the geometric approach. A common phenomenon was pupils’ difficulty in complex number problem solving, irrespective of their preferred type of approach.

Exploring Different Aspects of the Understanding of Function: Toward a Four-Facet Model

Based on a synthesis of the relevant literature, this study explored students’ display of behavior in four aspects of the understanding of function: effectiveness in solving a word problem, concept definition, examples of function, recognizing functions in graphic form, and transferring function from one mode of representation to another. A main concern was to examine problem-solving in relation to the other types of displayed behavior. Data were obtained from students in grades 11 and 12. Findings indicated that students were more capable in giving examples of function rather than providing an appropriate definition of the concept. The lowest level of success was observed in problem-solving on functions. Students’ problem-solving effectiveness was found to have a predictive role in whether they would successfully employ the concept in various forms of representation, in giving a definition and examples of function.RésuméÀ partir d’une synthèse de la littérature pertinente, cette étude analyse les comportements des étudiants pour ce qui est de quatre aspects de la compréhension de la fonction: l’efficacité lorsqu’il s’agit de résoudre un problème de mots, la définition des concepts, la présentation d’exemples de fonction, et enfin la reconnaissance des fonctions sous leur forme graphique et leur transposition d’un mode de représentation à l’autre. Nous nous sommes souciés tout particulièrement d’analyser la résolution de problèmes en relation avec les autres types de comportements des étudiants. Les données analysées proviennent d’étudiants de 11e et de 12e années, et les résultats indiquent que les élèves étaient mieux en mesure de fournir des exemples de fonction comparativement à leur capacité de donner une définition juste des concepts. C’est dans la résolution de problèmes regardant les fonctions que le taux de succès a été le moins élevé. Il ressort que l’efficacité des étudiants pour ce qui est de la résolution de problèmes permet de prédire leur capacité d’utiliser avec succès le concept dans ses différentes formes de représentation, et de fournir une définition et des exemples de fonctions.

The structure of students’ beliefs about the use of representations and their performance on the learning of fractions

Cognitive development of any concept is related with affective development. The present study investigates students’ beliefs about the use of different types of representation in understanding the concept of fractions and their self‐efficacy beliefs about their ability to transfer information between different types of representation, in relation to their performance on understanding the concept. Data were collected from 1701 students in Grade Five to Grade Eight. Results revealed that multiple‐representation flexibility, ability on solving problems with various modes of representation, beliefs about the use of representations and self‐efficacy beliefs about using them constructed an integrated model with strong interrelations in the different educational levels. Confirmatory factor analysis affirmed the existence of differential effects of multiple‐representation flexibility and problem‐solving ability in respect to cognitive performance and the existence of general beliefs and self‐efficacy beliefs about the use and the role of representations. Results suggested the invariance of this structure across primary (Grades Five and Six) and secondary education (Grades Seven and Eight). However, there are interesting differences concerning the interrelations among those cognitive and affective factors between primary and secondary education.

A model on the cognitive and affective factors for the use of representations at the learning of decimals

In a previous article of the same journal, we have discussed the interrelations of students’ beliefs and self‐efficacy beliefs for the use of representations and their respective cognitive performance on the learning of fraction addition. In the present paper, we confirm a similar structure of cognitive and affective factors on using representations for the concept of decimals and mainly we discuss the various interrelations among those factors. Data were collected from 1701 students in Grades 5–8 (11–14‐years‐old). Results revealed that multiple‐representation flexibility, ability on solving problems with various modes of representation, beliefs about the use of representations and self‐efficacy beliefs about using them constructed an integrated model with strong interrelations that has differences and similarities with the respective model concerning the concept of fractions.

A multidimensional approach to explore the understanding of the notion of absolute value

The study aimed to investigative students’ conceptions on the notion of absolute value and their abilities in applying the specific notion in routine and non-routine situations. A questionnaire was constructed and administered to 17-year-old students. Data were analysed using the hierarchical clustering of variables and the implicative method, while the qualitative approach was used to identify and analyse students’ errors. The results revealed students’ strong tendency to use algorithmic processes even in situations in which this kind of reasoning was not suited. This tendency and students’ errors in the tasks were assumed to occur primarily due to obstacles of didactic origin concerning the didactic contract and subsequently due to epistemological obstacles grounded in the history of mathematics.

Using representations in geometry: a model of students’ cognitive and affective performance

Self-efficacy beliefs in mathematics, as a dimension of the affective domain, are related with students’ performance on solving tasks and mainly on overcoming cognitive obstacles. The present study investigated the interrelations of cognitive performance on geometry and young students’ self-efficacy beliefs about using representations for solving geometrical tasks. The emphasis was on confirming a theoretical model for the primary-school and secondary-school students and identifying the differences and similarities for the two ages. A quantitative study was developed and data were collected from 1086 students in Grades 5–8. Confirmatory factor analysis affirmed the existence of a coherent model of affective dimensions about the use of representations for understanding the geometrical concepts, which becomes more stable across the educational levels.