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The introduction of the concept of the variable represents a critical point in the arithmetic–algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of “unknown” interferes with the interpretation of the variable “in functional relation” and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16–19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16–17 years. [ABSTRACT FROM AUTHOR]

The study reported here is the third in a series of research articles (Harkness, S. S., D’Ambrosio, B., & Morrone, A. S.,in Educational Studies in Mathematics 65:235–254, ; Morrone, A. S., Harkness, S. S., D’Ambrosio, B., & Caulfield, R. in Educational Studies in Mathematics 56:19–38, ) about the teaching practices of the same university professor and the mathematics course, Problem Solving, she taught for preservice elementary teachers. The preservice teachers in Problem Solving reported that they were motivated and that Sheila made learning goals salient. For the present study, additional data were collected and analyzed within a qualitative methodology and emergent conceptual framework, not within a motivation goal theory framework as in the two previous studies. This paper explores how Sheila’s “trying to believe,” rather than a focus on “doubting” (Elbow, P., Embracing contraries, Oxford University Press, New York, ), played out in her practice and the implications it had for both classroom conversations about mathematics and her own mathematical thinking. [ABSTRACT FROM AUTHOR]

As a first step of a project on learning difficulties in elementary set theory, we carried out a study aimed at identifying some of the difficulties encountered by a group of 21 mathematics majors taking a course on logic and set theory. In this paper, we shall discuss one of the difficulties that we observed, namely a difficulty in conceiving a set as an object distinct from its elements, i.e. failing to fully grant sets the status of objects. This difficulty may be related to the following three types of errors that we observed in the students' first assignment: (1) confusing belonging and inclusion, (2) confusing the union of sets A, B, C, ... with the set whose elements are A, B, C, ... and (3) adding or deleting curly brackets. We shall present excerpts from the students' work illustrating these three types of errors and explain how they may be related to a difficulty in conceiving a set as an object distinct from its elements. [ABSTRACT FROM AUTHOR]

In this paper we describe an ontological and semiotic model for mathematical knowledge, using elementary combinatorics as an example. We then apply this model to analyze the solving process of some combinatorial problems by students with high mathematical training, and show its utility in providing a semiotic explanation for the difficulty of combinatorial reasoning. We finally analyze the implications of the theoretical model and type of analysis presented for mathematics education research and practice. [ABSTRACT FROM AUTHOR]

From time to time, the International Commission on Mathematical Instruction (ICMI) mounts studies to investigate in depth and detail particular fields of interest in mathematics education. This paper is the Discussion Document of the forthcoming ICMI Study 16, Challenging Mathematics in and beyond the Classroom. [ABSTRACT FROM AUTHOR]

This paper reports on a qualitative study in which three third grade students were presented with a mathematical challenge related to the volume of a cuboid. The task required the construction of containers and the enumeration of the multi-link cubes held by each of the containers. The study videotaped and investigated the students working as a group through potentially seven different problem solving categories, and how they dealt with the student-generated mathematical dilemmas that surfaced during their exploration of the original problem. The tapes were examined for the problem solving actions that the students demonstrated, and an analysis of the students' counting strategies and solutions explored the connection between the children's spatial structuring and their use of numerical operations in enumerating 3-D rectangular arrays of cubes. The notions of perseverance and control are considered as they emerge during autonomous problem solving and the mathematical residue that results from the developing understanding of volume is discussed. [ABSTRACT FROM AUTHOR]

The paper we present is a case study which has two objectives: the identification of the interactions between pairs of 16 and 17-year-old students related to problem solving and the influence of such interactions in their cognitive development. To achieve these aims, we consider problems associated with a definite conceptual structure – problems comparing flat surface areas –, and we focus the analysis of the interactions on the point of view of the thematic and interlocutive dimensions of the discourse. Such analysis has allowed us to identify a wide typology of exchanges, and, from this, to outline four models of interaction - alternative, guided, relaunching, and co-operative. In the cases analysed, these interactions significantly influence the individual development of cognitive and heuristic abilities in the problem solving process. [ABSTRACT FROM AUTHOR]