Demetra Pitta-Pantazi

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.

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.

The embodied, proceptual, and formal worlds in the context of functions

Abstract: In this study we use Tall et al.’s (2000) theory on mathematical concept development, which describes three worlds of operations, the embodied, the proceptual and the formal (Tall, 2004; Tall, 2003; Watson, Spirou, & Tall, 2002). The purpose of the study is threefold: first, to identify mathematical tasks in the context of function that reflect the three worlds of operations; second, to investigate whether students’ thinking corresponds to the embodied, the proceptual, and the formal modes of thinking; and third, to reveal the structure of and relationships among the three worlds of operations as these unfold through students’ responses. The study was conducted with first‐year university students. The results suggested that mathematical tasks can be categorized on the basis of Tall et al.’s (2000) theory and indicated that students exhibit different kinds of thinking, which reflect to a large extent the three worlds of operations. Three classes of students were identified in terms of the difficu...

Mathematical Creativity: Product, Person, Process and Press

In this chapter we provide an overview of the state-of-the-art in mathematical creativity. To do so, we will use as a road map the 4Ps theory proposed by Rhodes in which four strands are used to capture the definition of creativity. In particular, (1) product: the communication of a unique, novel and useful idea or concept; (2) person: cognitive abilities, personality traits and biographical experiences; (3) process: the methodology that produces a creative product; and (4) press: the environment where creative ideas are produced. In this chapter we will first discuss the four strands in the framework of general creativity and then transfer and adapt these considerations to the field of mathematics education. In an attempt to define and describe mathematical creativity we will present several examples drawn from various research studies, and highlight some of the main findings, hoping to offer a springboard for further developments. We suggest that although these strands can be studied in isolation, it is only when their overlap and interconnections are considered that we may get a clearer picture of the complex concept of creativity.

CERME7 Working Group 7: Mathematical potential, creativity and talent

WG7 was a new CERME WG, whose purpose to draw the attention of the mathematics education community to the field of mathematical potential, creativity and talent, and to encourage empirical research that will contribute to the development of our understanding of the field. The first central theme was theoretical foundations of mathematical talent and creativity. Research from the University of Cyprus aimed to clarify the construct of mathematical giftedness. The study by Kattou et al. demonstrated that mathematical ability may be a predictor of mathematical creativity and, vice versa, that the level of mathematical ability depends on the level of mathematical creativity. Kontoyianni et al. used instruments measuring mathematical ability, mathematical creativity, self-perceptions of mathematical behaviour and fluid intelligence. The study results revealed that mathematical giftedness can be described in terms of mathematical ability and mathematical creativity. Research from the University of Haifa used Leikin’s (2009) model for evaluation of creativity as a compound of fluency, flexibility and originality, employing Multiple Solution Tasks (MSTs). Leikin and Kloss demonstrated that correctness in problem solving is highly correlated with fluency and flexibility, whereas originality is shown as a special mental quality. Leikin, Levav-Waynberg and Guberman demonstrated that development of mathematical creativity is significant not only in groups of high achievers but also in groups of middle achievers. They concluded with a hypothesis that in the fluency-flexibility-originality triad, fluency and flexibility are of a dynamic nature, whereas originality is a ‘gift’. Levenson suggested combining theories related to collective learning and theories related to mathematical creativity in order to investigate the notion of collective mathematical creativity. She explored the possible relationship between individual and collective mathematical creativity in elementary school mathematics classrooms. Using a conceptual approach from systems theory to design a model for the causality of giftedness, Brandl argued that being gifted in mathematics does not necessarily lead to high attainment in this subject, while being high-attaining in mathematics does not necessarily mean being mathematically gifted.

A Longitudinal Study Revisiting the Notion of Early Number Sense: Algebraic Arithmetic AS a Catalyst for Number Sense Development

ABSTRACT The aim of this study was to propose a new conceptualization of early number sense. Six-year-old students’ (n = 204) number sense was tracked from the beginning of Grade 1 through the beginning of Grade 2. Data analysis suggested that elementary arithmetic, conventional arithmetic, and algebraic arithmetic contributed to the latent construct early number sense, and the invariance of the model over time was validated empirically. Algebraic arithmetic represents the dimension of early number sense that moves beyond conventional arithmetic and encompasses an abstract understanding of the relations between numbers. A parallel process growth model showed that the three components of number sense adopt a linear growth rate. A structural model showed that the growth rate of the algebraic arithmetic component has a direct effect on the growth rate of conventional arithmetic, and subsequently the growth rate of conventional arithmetic predicts the growth rate of elementary arithmetic.

What Have We Learned About Giftedness and Creativity? An Overview of a Five Years Journey

The aim of this chapter is to offer an overview of a series of studies conducted at the University of Cyprus, regarding the definition and identification of mathematically gifted students, the relation between mathematical creativity, practical and analytical abilities, as well as the relation between giftedness, creativity and other cognitive factors such as, intelligence and cognitive styles. During our research in the field of giftedness and creativity we developed material for nurturing primary school mathematically gifted students and also explored the possibilities that technology may offer in the development of mathematical creativity. Although our research is still evolving, this chapter offers a glimpse of some of our most important findings.

Topic Study Group No. 29: Mathematics and Creativity

The activities of the TSG started with the co-chairs and team members of the TSG offering a brief overview of the main topics that the group would address. During the first session, three articles were presented. The first two presentations proposed two new tools and methods for the measurement of mathematical creativity. In their article “Creativity-in-progress rubric on proving: Enhancing students creativity,” Karakok, El Turkey, Savic, Tang, Naccarato, and Plaxco presented a new formative assessment tool, the Creativity-in-Progress, which can be used to measure individuals’ creativity while engaged in mathematical proof. The researchers described the development of this tool and its categories. Joklitschke, Rott, and Schindler, in their article “Revisiting the identification of mathematical creativity: Validity concerns regarding the correctness of solutions,” suggested that with the existing methods for the measurement of creativity, students’ potential is not sufficiently assessed and valued. Thus, they suggested modifications. One of the modifications they suggested was that students’ erroneous or unfinished solutions may also be used for the assessment of mathematical creativity. A third study presented during the first session by Pitta-Pantazi and Sophocleous, entitled “Higher order thinking in mathematics: A theoretical formulation and its empirical validation,” went beyond to the identification and measurement of mathematical creativity and extended towards the assessment and measurement of higher order thinking in mathematics. The researchers suggested that higher order thinking is constituted by several subcomponents: basic, critical, and complex mathematical thinking. In their article, the researchers empirically validated a model of higher order thinking and presented tools that they used for its measurement.

Parsing the notion of algebraic thinking within a cognitive perspective

Abstract There is a growing consensus that algebra is an important aspect of mathematics teaching and learning and several abilities are required in order students to have successful performance in algebra. The present study uses insights from the domain of psychology to enrich what is currently known in the domain of mathematics education about the relationship of algebraic thinking with abilities involved in fundamental cognitive processes. In total, 190 students between the ages of 13–17 years old were tested through two tests. The first test addressed four types of cognitive systems which are responsible for the representation and processing of different types of relations in the environment: the spatial-imaginal, the causal-experimental, the qualitative-analytic and the verbal-propositional. The second test addressed algebraic thinking. The results support the key role of the four types of cognitive processes in students’ algebraic thinking. The results also suggest that abilities involved in the four types of cognitive processes predict algebraic thinking abilities, irrespective of the age of the students.

Developing and Enhancing Elementary School Students’ Higher Order Mathematical Thinking with SimCalc

A fundamental question regarding the use of technology in mathematics education is the way in which technology supports and promotes higher order thinking in mathematics. In this chapter, we try to describe and analyze the way in which SimCalc might develop and enhance elementary school students’ higher order mathematical thinking. To this end, we use an adaptation of the Integrated Thinking Model of Iowa Department of Education (1989). Specifically, we argue that SimCalc offers elementary school students the opportunity to develop not only content knowledge, but also critical, creative, and complex thinking skills. We justify the above argument by providing examples of mathematical activities using SimCalc, and students’ responses.