Athanasios Gagatsis

Ability to Translate from One Representation of the Concept of Function to Another and Mathematical Problem Solving

Representations are used extensively in mathematics and translation ability is highly correlated with success in mathematics education. The authors investigate the translation ability of university students as far as the concept of function is concerned. The research focuses on the relationship between success in, solving direct translation tasks and success in solving problems by articulating different representations of the concept of function. Furthermore, it examines the relationship between student performance and the nature of the representation included in the translation task. The ability to pass from one representation to another was associated with success in problem solving. These results indicate that translation ability should be considered as an important factor in problem solving. Percentages are lower when an iconic representation is included in the translation task. This could be partly attributed to the holistic nature of iconic representations and to the way the concept of function is taught at secondary schools.

Using geometrical models in a process of reflective thinking in learning and teaching mathematics

In this paper we investigate how geometrical models can be used in learning and teaching mathematics, in connection with the development of a process of reflective thinking, which we study first in general. Some more specific questions — arising from the use of geometrical models in the classroom — have led us to an experimental study, the results of which are presented and discussed in the paper.

Cognitive and Metacognitive Aspects of Proportional Reasoning

In this study we attempt to propose a new model of proportional reasoning based both on bibliographical and research data. This is impelled with the help of three written tests involving analogical, proportional, and non-proportional situations that were administered to pupils from grade 7 to 9. The results suggest the existence of a multi-component model of proportional reasoning, contributing in this way to the reformulation of the concept. In this model, the components of analogical reasoning, routine proportionality, and meta-analogical awareness take a constitutive part. Thus, proportional reasoning does not coincide exclusively with success in solving a certain restricted range of proportional problems, as routine missing-value and comparison problems (routine proportionality), but it also involves handling verbal and arithmetical analogies (analogical reasoning) as well as the awareness of discerning non-proportional situations (meta-analogical awareness), which is metacognitive in nature.

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.

Young children's understanding of geometric shapes: The role of geometric models

SUMMARY This study explores the role of polygonal shapes as geometrical models in teaching mathematics by eliciting and interpreting young childrens geometric conceptions and understanding about shapes, through their responses while being involved in relevant activities. More specifically, we examined the cases of polygons or “polytopes” of dimensions 0, 1 and 2 (0-polytopes are points, 1-polytopes are line segments and 2-polytopes are (convex) polygons), by asking children of 4–7 years of age to draw a stairway of figures (triangles, squares and rectangles) with each shape being bigger than its preceding one. Our ultimate aim was to investigate the implications the findings have for advancing childrens geometric thinking and understanding, and thus for teaching geometry more efficiently, in early childhood. For the analysis of the collected data, Grass Implicative Statistical Model was used. Results showed that children were mainly using two strategies while solving the problems: (a) conservation of shape, by increasing both dimensions of the figure and (b), increasing one dimension of the figure. Each strategy seems to reflect a different way of reasoning and understanding, possibly corresponding to a different level of development, as far as geometric thinking, is concerned. Also, children appear to work relatively more flexibly with tasks using rectangles than tasks using squares. This finding suggests that geometry instruction needs to introduce geometric shapes in a mathematically correct manner by using accurate definitions and explanations of relative properties and characteristics, hierarchical commonalities and differences among shapes.

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.

Assessing Student Problem-Solving Skills

This article examines the structure of several HIV risk behaviors in an ethnically and geographically diverse sample of 8,251 clients from 10 innovative demonstration projects intended for adolescents living with, or at risk for, HIV. Exploratory and confirmatory factor analyses identified 2 risk factors for men (sexual intercourse with men and a general risk factor) and 3 factors for women (sexual intercourse with men, substance abuse, and a high risky sex behavior factor). All factors except women engaging in risky sex with men strongly predicted known HIV status of clients for men and women. The findings from this investigation highlight the use of structural equation modeling for applied problems involving overlapping and complex sets of risk behaviors in youth who present at community health programs.

Tracing the development of representational flexibility and problem solving in fraction addition: a longitudinal study

The study models the development of students’ multiple-representation flexibility and the use of problem-solving strategies and representations in fraction addition. A test administered three times, with breaks of 3–4 months between successive measurements to 108 students at a transition within primary school (Grade 5–6), 132 students at a transition from primary to secondary education (Grade 6–7) and 148 students at a transition within secondary school (Grade 7–8). Multivariate analysis of variance for repeated measures and dynamic structural equation modelling were carried out in order to analyse the data. Findings suggested that students’ performance improve through measurements. Dynamic modelling provided evidence for the strong interrelation between representational flexibility and problem solving at the three measurements. The results indicated the students’ established pre-existent knowledge and the important role the initial state of the aforementioned cognitive parameters plays on their advancement. Didactical implications are discussed.