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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]

This paper reports a study designed to explore prospective teachers‘ learning to teach mathematics in the complex context of a university-based curriculum and instruction course. The course, taught with a focus on inquiry, used prospective teachers‘ weekly interactions with students as springboards for investigations of mathematics, teaching, and learning. A framework of questioning, listening, and responding is used to analyze prospective teachers‘ understandings and the tensions they experienced. The difficulties and tensions prospective teachers face in turn pose challenges and tensions for teacher educators. [ABSTRACT FROM AUTHOR]

There rarely seems to be any disagreement that teaching should build on children‘s mathematics, but there are many possible ways to put this demand in concrete terms. In this sense, the present paper describes a teaching experiment in grade three based on Treffers‘ idea of vertical mathematisation. Central components of the developed course are discussed, with focus on the use of the empty number line and of maths diaries as well as on the idea of conducting maths conferences. The teaching experiment is illustrated by means of children‘s own productions and finally put into a broader context as contributing to the design science approach of mathematics education. [ABSTRACT FROM AUTHOR]

This paper analyzes the interpretations of equality and the equal symbol of the third-grade children who participated in a year long whole-class socio constructivist teaching experiment. These children initially interpreted the equal symbol as a command to perform an arithmetical operation; it was less natural to them to interpret it as a relational symbol to compare two quantities. By the end of the school year, children were able to conceptualize the ’quantitative sameness‘ of two numerical expressions and describe it by using the phrase ’is the same as‘, the words ’equal‘ or ’equals‘, or the symbols ’=‘ or ’=s‘. These children expanded their conceptualizations of equality due to their active role in class discussions, the arithmetical tasks that took into account children‘s difficulties, and the teacher‘s intellectual sensitivity to ’strike a delicate balance between the force of teaching and the freedom of learning‘ (Freudenthal, 1991, p. 55). [ABSTRACT FROM AUTHOR]

This paper reports on important learning processes emerging during adult mathematics classes that used a teaching approach compatible with a social constructivist theory of knowing. A cyclical teaching model encouraged much discussion about the mathematical problems presented to paired groups. Whole-class student-led sharing sessions allowed students to continue their interactive constitution of meaning and solutions. Comparisons of pre- and post-course scores on attitude and belief questionnaires indicated significant increases in positive attitudes and beliefs about mathematics and about themselves doing mathematics. The overall positive changes in scores on questionnaires are explained in terms of recurrent patterns in thinking inferred from data obtained by observation, interview, and reflective comments. Changes are explained in terms of the interactive effects of affective and heuristic processing during problem solving. Affective processes precipitated students‘ responsiveness, modifying the immediate learning context which, in turn, influenced the students‘ thinking, creating a ’snowballing‘ effect on learning and affect. Changes were clearly facilitated by the interaction between students and by their shared understandings. [ABSTRACT FROM AUTHOR]

This paper presents two case studies of learners attempting to make sense of the concept of normal distribution – in particular, why physical phenomena such as height fall into normal distributions. The notion of epistemological anxiety is advanced as being the source of the difficulties learners have in making sense of normal distributions. The framework of Connected Mathematics is used to analyze the learners‘ coming to understanding of normal distributions and as a source of therapeutic intervention for the epistemological anxiety. As both a symbolizing medium and an aid to analysis, one learner is provided with an object-based parallel modeling language (OBPML) with which the learner can formulate computational hypotheses about the local probability rules that lead to the emergence of a global distribution. Through the investigation, supported by the OBPML, the learner makes connections between the micro-rules of probability and the resultant (macro-) statistical distributions. Conclusions are then drawn about the features of a Connected Mathematics learning environment that enable confrontation with epistemological anxiety and the features of modeling languages that enable learners to conduct a successful probability investigation. [ABSTRACT FROM AUTHOR]

In this paper, we explore the relationship between learners‘ actions, visualisations and the means by which these are articulated. We describe a microworld, Mathsticks, designed to help students construct mathematical meanings by forging links between the rhythms of their actions and the visual and corresponding symbolic representations they developed. Through a case study of two students interacting with Mathsticks, we illustrate a view of mathematics learning which places at its core the medium of expression, and the building of connections between different mathematisations rather than ascending to hierarchies of decontextualisation. [ABSTRACT FROM AUTHOR]

A correction is presented to the article "Generating and Using Examples in the Proving Process" by J. Sandefur, J. Mason, G. J. Stylianides, and A. Watson from the September 13, 2013 online issue, which misquoted another paper on pedagogical knowledge of mathematics.