Eleni Deliyianni

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.

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.

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.

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.