Iliada Elia

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.

Developing a framework for the evaluation of picturebooks that support kindergartners’ learning of mathematics

The purpose of this study was to investigate what experts in the use of picturebooks in mathematics education consider powerful characteristics of such books in the support of young childrens learning of mathematics. The study started by investigating experts’ views of such characteristics, as reflected in academic and professional publications on the use of picturebooks in mathematics education. This resulted in a first version of a framework of learning-supportive characteristics of picturebooks. In the second part of the study the framework was refined, and its tenability was tested through a four-round Delphi method, in which seven experts were asked to comment on, and work with, the framework when evaluating three picturebooks. The experts’ evaluations of these books showed that a larger number of learning-supportive characteristics were identified when using the framework than when not using it.

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.

Effects of Reading Picture Books on Kindergartners' Mathematics Performance.

This article describes a field experiment with a pretest–posttest control group design which investigated the potential of reading picture books to children for supporting their mathematical understanding. The study involved 384 children from 18 kindergarten classes in 18 schools in the Netherlands. During three months, the children in the nine experimental classes were read picture books. Data analysis revealed that, when controlled for relevant covariates, the picture book reading programme had a positive effect (d = .13) on kindergartners’ mathematics performance as measured by a project test containing items on number, measurement and geometry. Compared to the increase from pretest to posttest in the control group, the increase in the experimental group was 22% larger. No significant differential intervention effects were found between subgroups based on kindergarten year, age, home language, socio-economic status and mathematics and language ability, but a significant intervention effect was found for girls and not for boys.

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.

Pupils’ visual representations in standard and problematic problem solving in mathematics: their role in the breach of the didactical contract

ABSTRACT This study investigated the modes of representations generated by kindergarteners and first graders while solving standard and problematic problems in mathematics. Furthermore, it examined the influence of pupils’ visual representations on the breach of the didactical contract rules in problem solving. The sample of the study consisted of 38 kindergarteners (age 5–6) and 34 first graders (age 6–7). Two standard problems (addition and subtraction) and two problematic problems were given to the participants. The majority of kindergarteners used a variety of spontaneous visual representations in order to solve both types of problems. In contrast, first graders mainly used symbolic representations corresponding to the numbers involved in the text of the problems. Results also suggested that the visual representations prevented kindergarteners from obeying the didactical contract rules. In fact, many kindergarteners were inclined to draw descriptive pictures about the meaning of the problems, without persevering in giving a symbolic answer in the standard problems or providing a stereotyped solution for the problematic problems. First graders, however, gave a routine solution, that is, a symbolic answer to the two problematic problems, complying with the didactical contract rule that every problem given to them has an answer. RESUME: L’étude présente examine les modes de représentation générés par les élèves de l’École Maternelle et de la première année de l’École Primaire dans le processus de solution de problèmes standards et non ordinaires en mathématiques. De plus, elle étudie l’influence des représentations visuelles des élèves sur la ‘rupture’ des règles du contrat didactique dans la résolution de problèmes. L’échantillon de la recherche était constitué de 38 élèves de l’École Maternelle (âgés de 5–6 ans) et de 34 élèves de la première année de l’École Primaire (âgés de 6–7 ans). Deux problèmes standards (addition et soustraction) et deux problèmes non ordinaires étaient donnés aux participants. La majorité des élèves de l’École Maternelle avaient employé une variété de représentations visuelles spontanées afin de résoudre les deux types de problèmes. En revanche, les élèves de la première année de l’École Primaire avaient notamment employé des représentations symboliques correspondant aux chiffres inclus dans le texte des problèmes. Les résultats suggèrent aussi que les représentations visuelles avaient dissuadé les élèves de l’École Maternelle d’obéir aux règles du contrat didactique. En fait, plusieurs élèves de l’École Maternelle étaient enclins de dessiner des images descriptives sur la signification des problèmes sans insister pour donner une réponse symbolique aux problèmes standards ou pour fournir une solution stéréotypée aux problèmes non ordinaires. Toutefois, les élèves de la première année de l’École Primaire avaient donné une solution typique, à savoir, une réponse symbolique aux deux problèmes non ordinaires en s’alignant sur la règle du contrat didactique qui prévoit que tout problème donné a une réponse. ZUSAMMENFASSUNG: Die Studie befasst sich mit Deutungsweisen durch Schüler in Kindergarten und erstem Schuljahr beim Lösen von standard‐ und problematischen Aufgaben in Mathematik. Ferner untersucht sie den Einfluss bildlicher Deutungsmuster durch Schüler auf die Verletzung didaktischer Regeln beim Lösen von Aufgaben. Die Stichprobe bestand aus 38 Kindergarten‐Kindern (Alter 5–6) und 34 Erstklaesslern (Alter 6–7). Zwei Standardaufgaben (Addition und Subtraktion) sowie zwei problematische Aufgaben wurden gestellt. Die Mehrheit der Kindergarten‐Kinder verwendete eine Vielfalt spontaner Deutungsmuster um beide Arten von Aufgaben zu lösen. Im Gegensatz dazu griffen Erstklässler vorwiegend auf in der Aufgabenstellung enthaltene symbolische Darstellungen zurück. Die Ergebnisse der Studie legen nahe, dass bildliche Darstellungen Kindergarten‐Kinder daran hindern, didaktischen Regeln zu folgen. In der Tat waren viele Kindergarten‐Kinder geneigt, bildliche Beschreibungen der Problemstellung anstelle von symbolischen oder stereotypen Lösungen zu erstellen. Erstklässler hingegen gaben eine Routine‐Lösung, also eine symbolische Antwort, auf die beiden problematischen Aufgaben und folgten der didaktischen Regel, derzufolge jede ihnen gestellte Aufgabe eine Antwort erhält. RESUMEN: El presente estudio investiga los modos de representación generados por los alumnos de Educación Preescolar y del primer curso de Educación Primaria en el proceso de solución de problemas estándar y complejos en matemáticas. Además, examina la influencia de las representaciones visuales de los alumnos sobre la ruptura de las reglas del contrato didáctico en la solución de problemas. La muestra del estudio consistía en 38 alumnos de Educación Preescolar (edades entre 5 y 6 años) y en 34 alumnos del primer curso de educación primaria (edades entre 6 y 7 años). Dos problemas estándar (sumar y restar) y dos problemas complejos se entregaron a los participantes. La mayoría de los alumnos de Educación Preescolar utilizó una variedad de representaciones visuales espontáneas para resolver los dos tipos de problemas. En cambio, los alumnos del primer curso de Educación Primaria usaron principalmente representaciones simbólicas correspondientes a los números incluidos en el texto de los problemas. Los resultados sugieren también que las representaciones visuales disuadieron a los alumnos de Educación Preescolar de obedecer al contrato didáctico. De hecho, varios alumnos de Educación Preescolar se inclinaron a dibujar imágenes descriptivas del significado de los problemas sin insistir en dar una respuesta simbólica a los problemas estándar o en entregar una solución estereotipada a los problemas complejos. De todas maneras, los alumnos del primer curso de Educación Primaria dieron una solución típica, es decir, una respuesta simbólica a los dos problemas complejos aferrándose a la regla del contrato didáctico que implica que todo problema planteado tiene una respuesta.

The Role of Picture Books in Young Children’s Mathematics Learning

In this chapter we address the role of picture books in kindergartners’ learning of mathematics. The chapter is based on various studies we carried out on this topic from different perspectives. All studies sought to provide insight into the power of picture books to contribute to the development of mathematical understanding by young children. We start the chapter with some background information about picture books as a didactical tool in mathematics education. Then, we discuss a framework of picture book characteristics that support young children’s learning of mathematics. In the next section, we give a short impression of children’s spontaneous mathematics-related utterances that occur during the reading of a picture book. This section is followed by a deeper look at the influence of the pictures in a picture book. Hereafter, the reading itself is the focus. Three book reading techniques are discussed and illustrated by classroom vignettes. Finally, based on an intervention program in which kindergartners were read a series of picture books, we report what we learned about the effectiveness of picture book reading on kindergartners’ performance in mathematics.

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.

Gesture in a Kindergarten Mathematics Classroom.

ABSTRACT Recent studies have advocated that mathematical meaning is mediated by gestures. This case study explores the gestures kindergarten children produce when learning spatial concepts in a mathematics classroom setting. Based on a video study of a mathematical lesson in a kindergarten class, we concentrated on the verbal and non-verbal behavior of one kindergartner who produced a great amount of gestures during instruction. The microgenetic approach was used for the analysis of the data. The results showed that the kindergartner used gestures throughout the whole instruction. For all the spatial concepts that were addressed (‘in’ and ‘out’, ‘on’ and ‘under’, ‘up’ and ‘down’), he produced mainly deictic gestures referring either to existing or virtual objects. The child was found to produce different types of gestures in different spatial contexts. Our analysis revealed the occurrence of a gesture–speech match and a gesture–speech mismatch. In the latter case, the childs gestures were found to complement and enrich his verbal utterances. The childs gestures along with his speech acted as semiotic means of objectification of specific spatial relations that were rather abstract for the child and were not represented adequately by speech. Besides the phenomenon of the coordination between oral speech and gesture, enacted by the child himself, evidence was found for the coordination between the two semiotic systems activated by different people, that is, the child under study and the other children or the teacher of the class. Furthermore, the teachers gestures were found to influence the childs gestures in different ways. These findings are discussed and suggestions for further research on the role of gestures in the development of spatial thinking in the early years are drawn.