Constantinos Christou

The development of mental processing : efficiency, working memory, and thinking

This Monograph aims to contribute to the information processing, the differential, and the developmental modeling of the mind, and to work these into an integrated theory. Toward this aim, a longitudinal study is presented that investigates the relations between processing efficiency, working memory, and problem solving from the age of 8 years to to the age of 16 years. The study involved 113 participants, about equally drawn among 8-, 10-, 12-, and 14-year-olds at the first testing; these participants were tested two more times spaced one year apart. Participants were tested with a large array of tasks addressed to processing efficiency (i.e., speed of processing and inhibition), working memory (in terms of Baddeleys model, phonological storage, visual storage, and the central executive of working memory), and problem solving (quantitative, spatial, and verbal reasoning). Confirmatory factor analysis validated the presence of each of the above dimensions and indicated that they are organized in a three-stratum hierarchy. The first stratum includes all of the individual dimensions mentioned above. These dimensions are organized, at the second stratum, in three constructs: processing efficiency, working memory, and problem solving. Finally, all second-order constructs are strongly related to a third-order general factor. This structure was stable in time. Structural equation modeling indicated that the various dimensions are interrelated in a cascade fashion so that more fundamental dimensions are part of more complex dimensions. That is, speed of processing is the most important aspect of processing efficiency, and it perfectly relates to the condition of inhibition, indicating that the more efficient one is in stimulus encoding and identification, the more efficient one is in inhibition. In turn, processing efficiency is strongly related to the condition of executive processes in working memory, which, in turn, is related to the condition of the two modality-specific stores (phonological and visual). Finally, problem solving is related to processing efficiency and working memory, the central executive in particular. All dimensions appear to change systematically with time. Growth modeling suggested that there are significant individual differences in attainment in each of the three aspects of the mind investigated. Moreover, each of the three aspects of the mind as well as their interrelations change differently during development. Mixture growth modeling suggested that there are four types of developing persons, each defined by a different combination of performance in these aspects of the mind. Some types are more efficient and stable developers than others. These analyses indicated that processing efficiency is a factor closely associated with developmental differences in problem solving, whereas working memory is associated with individual differences. Modeling by logistic equations uncovered the rates and form of change in the various dimensions and their reciprocal interactions during development. These findings are discussed from the point of view of information processing, differential, and developmental models of thinking, and an integrative model is proposed.

Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis.

A Study of the Mathematics Teaching Efficacy Beliefs of Primary Teachers

The affective domain has in recent years attracted much attention from the mathematics research community; empirical data seem to increasingly support expert opinion that affect plays a decisive role in the process of cognitive development. One of the less researched dimensions of the affective domain is teachers’ beliefs about the efficacy of their mathematics teaching. Though there are studies examining efficacy-beliefs with respect to mathematics learning, we have not been able to locate any study related to efficacy-beliefs with respect to teaching mathematics. Belief in one’s ability to overcome obstacles and bring about a predetermined outcome may be decisive in motivation to undertake the endeavour, and to put in time and resources. In this chapter, based on a sample in Cyprus, we examine primary school teachers’ efficacy-beliefs with respect mathematics teaching. From an analysis of self-reported questionnaires and interview data, it was found that teachers feel quite competent to teach mathematics, and that the level of efficacy improves, after diminishing during the initial career period. The preservice program seemed to make a difference to beliefs about the efficacy of their mathematics teaching; however, in general the teachers seemed to be critical about the preservice program they passed through. The findings of this study might have important implications for teacher preparation and teacher development in general.

Mapping and development of intuitive proportional thinking

Abstract The purpose of this study was two-fold. First, to find out students’ informal understanding of proportional problems, and discuss their solution strategies. Second, to investigate how the intuitions developed by students influence their strategies to solve proportional problems. To this end, we interviewed 16 students in Grades 4 and 5, while they were solving proportional problems. It was found that students intuitively used the unit-rate strategy indicating an attempt to transfer the knowledge resulted by their experience with solving simple multiplicative problems. Fourth and fifth graders tended to shift from the unit-rate strategy to other strategies if there was no easy way to calculate the unit-value directly from the context of the problems. Since fifth graders were more comfortable than fourth graders in calculating the unit-value, they felt less the need to invent other solution strategies.

Modeling the Stroop phenomenon: Processes, processing flow, and development.

Abstract A model of the Stroop phenomenon is proposed which postulates that the classic effect is an additive function of three parameters, that is, dimension selection (decision making about which dimension to respond to), dimension identification (encoding and identification of the relevant dimension), and interference control (filtering out of interference from non-relevant dimensions). The study used stimuli addressed to three symbol systems (verbal, numerical, and figural), two types of stimulus composition (compatible vs. incompatible), and two types of dimension selection (decision needed about the to-be-identified dimension vs. no decision needed). Participants were 9, 11, 13, and 15 years old and they were tested twice. The model was found to hold under all stimulus and presentation conditions. Moreover, it was found that the three parameters are differentially related to age. The implications of the model for general theories of cognition and cognitive development are discussed.

A Model of Mathematical Giftedness: Integrating Natural, Creative, and Mathematical Abilities

The present study aims to examine the structure and relationships among the components of mathematical giftedness and to identify groups of students that differ across these components. The proposed model is innovative in terms of integrating natural/cognitive, creative, and mathematical abilities leading to a new conceptualization of mathematical giftedness. In our view, mathematical giftedness consists of mathematical ability and mathematical creativity, whereas natural/cognitive abilities predict mathematical giftedness. To verify our model, data were collected from 239 elementary school students through four instruments. Data analysis verified the structure of the model, indicating that mathematical ability contributes more than mathematical creativity to the construct of mathematical giftedness. Furthermore, the natural/cognitive abilities (fluid intelligence and working memory) predict mathematical giftedness. Three different groups of students were identified, which reflect three distinct levels of performance, namely, low-, average-,and high-ability students. Among the high-ability students a group of 9-year-olds was identified as gifted.RésuméLa présente étude a pour but d’analyser la structure des éléments et des relations qui entrent en jeu dans le talent mathématique, et de déterminer quels sont les étudiants qui se distinguent dans tous ces éléments. Le modèle proposé est innovateur en ce sens qu’il intègre les habiletés cognitives/naturelles, créatives et mathématiques, pour arriver à une nouvelle conceptualisation du talent mathématique. À notre avis, on peut définir le talent mathématique comme un mélange d’habileté et de créativité, alors que les habiletés cognitives/naturelles sont un prédicteur du talent mathématique. Pour vérifier ce modèle, nous avons recueilli des données provenant de 239 élèves de niveau primaire, au moyen de quatre instruments. L’analyse des données confirme la structure du modèle et indique que l’habileté mathématique contribue en plus grande mesure au construit de talent mathématique comparativement à la créativité mathématique. Par ailleurs, les habiletés cognitives/naturelles (intelligence fluide et mémoire dynamique) s’avèrent un prédicteur du talent mathématique. Trois groupes d’étudiants ont été identifiés, qui reflètent trois niveaux de performance distincts: étudiants d’habileté inférieure, moyenne et supérieure. Parmi les élèves du groupe de niveau supérieur, un groupe d’élèves de 9 ans a été identifié comme particulièrement doué.

Cypriot and Greek Primary Teachers’ Conceptions about Mathematical Assessment∗

ABSTRACT The purpose of this study, which is part of a larger research program, was to investigate teachers’ conceptions about assessment. A mailed questionnaire sent to 5th and 6th grade teachers in Cyprus and Greece produced a rather optimistic picture about the investigated issues. However, some semi‐structured interviews, which followed, showed that teachers in both countries seemed to have a rather vague understanding of relevant key concepts. ∗We thank the reviewers for their constructive comments, which resulted to improve the initial version of this paper.

Family parameters of achievement: A structural equation model

The aim of the present study was to investigate the effects of several structural and functional characteristics of the family on the child’s actual school achievement. A structural equation model was constructed and its ability to fit the data was tested. It was found that the child’s achievement is directly influenced by the socio-economic status of the family, which is a structural characteristic. The child’s achievement is also directly influenced by the child self-image and by various indications of the family function, such as parental involvement activities and expectations. Furthermore, the results show that the child’s self image is the strongest factor that explains school achievement. Among the many implications that this study offers for educational policy and practice is that children’s achievement may be significantly influenced by altering factors that are outside of school and outside of the students themselves; namely, parental behavior and family functioning.RésuméLe but de la recherche était d’étudier les effets de différentes caractéristiques structurales et fonctionnelles des familles sur la réussite scolaire actuelle des enfants. Um modèle d’équation structurale a été construit et sa capacité à s’ajuster aux donnée a été testée. Les résultats montrent que la réussite scolaire des enfants est directement influencée par le statut socio-économique de la famille (caractéristique structurale). La réussite scolaire des enfants est également directement influencée par l’image qu’ils, ont d’eux-mêmes, et par différentes caractéristiques fonctionnelles familiales telles que les activités d’investissement des parents et leurs expectations. Par ailleurs, les résultats montrent que l’image de soi des enfants est le facteur qui intervient le plus pour expliquer la réussite scolaire. Les auteurs en tirent quelques implications d’ordre éducatif.

Tracing Students’ Modeling Processes in School

In this study, we report on an analysis of the mathematization processes of one 6th and one 8th grade group, with emphasis on the similarities and differences between the two groups in solving a modeling problem. Results provide evidence that all students developed the necessary mathematical constructs and processes to actively solve the problem through meaningful problem solving. Eighth graders who were involved in a higher level of understanding the problem presented in the activity employed more sophisticated mathematical concepts and operations, better validated and communicated their results and reached more efficient models. Finally, a reflection on the differences in the diversity and sophistication of the constructed models and mathematization processes between the two groups raises issues regarding the design and implementation of modeling activities in elementary and lower secondary school level.

Cognitive styles, task presentation mode and mathematical performance

This paper reports the outcomes of an empirical study undertaken to investigate the relationship of prospective teachers’ cognitive styles and levels of performance in measurement and spatial tasks. A total of 116 prospective kindergarten school teachers were tested using the VICS and the extended CSA-WA tests (Peterson 2005) in order to place them along the Verbal/Imagery and the Wholistic/Analytic cognitive style continua. The same prospective teachers were also administered a mathematical test with 6 measurement and 6 spatial tasks. The results suggest that there were no significant differences between Verbalisers-Imagers and Wholistic-Analytic prospective teachers in their performance on the spatial pictorial and textual tasks, and on the measurement textual tasks. However, there were differences between Verbalisers-Imagers and Wholistic-Analytic prospective teachers in their performance on the measurement pictorial tasks. This difference was attributed to the performance of low achievers. High achievers performed in the same way independently of their cognitive styles.