Prospective teachers' attention on geometrical tasks

This study investigates early childhood prospective teachers' attention to geometrical tasks while designing and using them in the classroom. This is explored in the context of the teaching practice of 11 prospective teachers who taught geometry in early childhood classrooms during the last semester of their university studies. The teaching practice was organized into four stages: design of a lesson plan; classroom implementation; discussion of the lesson with the school practice instructor; and self-assessment report and revision of the lesson. Analysis of data using the Teaching Triad framework (Jaworski, 1994) shows that although the prospective teachers attended to issues of mathematical challenge, sensitivity to students, and management of learning in their planning, in their actual teaching and after class reflection, their attention was focused mainly on management issues. The findings also show that prospective teachers' attention on geometrical tasks can be developed through a process of reflection on their teaching. © 2013 Springer Science+Business Media Dordrecht

Secondary school students' levels of understanding in computing exponents

The aim of this study is to describe and analyze students' levels of understanding of exponents within the context of procedural and conceptual learning via the conceptual change and prototypes' theory. The study was conducted with 202 secondary school students with the use of a questionnaire and semi-structured interviews. The results suggest that three levels of understanding can be identified. At the first level students' interpretation of exponents is based upon exponents that symbolize natural numbers. At Level 2, students' knowledge acquisition process is a process of enrichment of the existing conceptual structures. Students at this level are able to compute exponents with negative numbers by extending the application of prototype examples. Finally, at Level 3 students not only extend the prototype examples but also reorganize their thinking in order to compute and compare exponents with roots, a concept which is quite different from the concept of exponents with natural numbers. © 2007 Elsevier Inc. All rights reserved

Reflective, systemic and analytic thinking in real numbers

The aim of this paper is to propose a theoretical model to analyze prospective teachers' reasoning and knowledge of real numbers, and to provide an empirical verification of it. The model is based on Sierpinska's theory of theoretical thinking. Data were collected from 59 prospective teachers through a written test and interviews. The data indicated that mathematical tasks on real numbers, based on Sierpinska's theory, could be categorized according to whether they require reflective, systemic or analytic thinking. Analysis of the data identified three different groups of prospective teachers reflecting different types of theoretical thinking about real numbers. The interviews confirmed the empirical data from the written test, and provided a better insight into the thinking and characteristic features of the prospective teachers in each group. The analysis also indicated that the participants were more successful in tasks requiring systemic and analytic thinking, and only when this was achieved were they able to solve problems which required reflective thinking. Implications for teaching related to the findings of the study are discussed. © 2012 Springer Science+Business Media B.V

Enhancing creative problem solving in an integrated visual art and geometry program: A pilot study

This article describes a new pedagogical method, an integrated visual art and geometry program, which has the aim to increase primary school students' creative problem solving and geometrical ability. This paper presents the rationale for integrating visual art and geometry education. Furthermore the MathArt pedagogy and program is described and it is explained how the MathArt program intends to increase students' creative thinking and geometrical ability. Additionally initial results of the pilot study are presented, which investigates the effects of the MathArt program

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

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, and 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 difficulty level of the tasks: (a) the proceptual, (b) the proceptual‐emhodied, and (c) the formal. According to Tail’s theory, embodied, proceptual, and formal thinking develop in sequence in an individual’s life. This study indicates that freshmen university students, who have had mathematics as a major in higher secondary school, are only able to deal with the embodied tasks once they had been successful with the proceptual ones. The leap to formal thinking could only be achieved when proceptual manipulations were enhanced with competence in embodied tasks. © 2005 Taylor and Francis Group, LLC

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

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, and 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 difficulty level of the tasks: (a) the proceptual, (b) the proceptual‐emhodied, and (c) the formal. According to Tail’s theory, embodied, proceptual, and formal thinking develop in sequence in an individual’s life. This study indicates that freshmen university students, who have had mathematics as a major in higher secondary school, are only able to deal with the embodied tasks once they had been successful with the proceptual ones. The leap to formal thinking could only be achieved when proceptual manipulations were enhanced with competence in embodied tasks. © 2005 Taylor and Francis Group, LLC

A clustering method for multiple-answer questions on pre-service primary teachers’ views of mathematics

In the last years, research has paid strong attention to pre-service primary teachers’ views of mathematics. Interviews and questionnaires to pre-service teachers during their academic studies are the mainly used tools for collecting data. Qualitative and quantitative approaches may give different insights. In this paper, after a review of the different methods used in the literature to face the topic of pre-service primary teachers’ views of mathematics, we propose a new method. A clustering technique is applied to data collected with multiple-answer questions about pre-service primary teachers’ views of mathematical ability. Obtained clusters are interpreted and compared