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In this paper, we examine sixth grade students' degree of conceptualization of fractions. A specially developed test aimed to measure students' understanding of fractions along the three stages proposed by Sfard (1991) was administered to 321 sixth grade students. The Rasch model was applied to specify the reliability of the test across the sample and cluster analysis to locate groups by facility level. The analysis revealed six such levels. The characteristics of each level were specified according to Sfard's framework and the results of the fraction test. Based on our findings, we draw implications for the learning and teaching of fractions and provide suggestions for future research.

The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems. Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems' structure and information (Novick & Hurley, 2001;Diezmann, 2005). Novick and Hurley were the first to introduce three well-defined types of diagrams, that is, network, hierarchy, and matrix, which represent different problematic situations. In the present study, we investigated the effects of these types of diagrams in non-routine mathematical problem solving by contrasting students' abilities to solve problems with and without the presence of diagrams. Structural equation modeling affirmed the existence of two first-order factors indicating the differential effects of the problems' representation, i.e., text with diagrams and without diagrams, and a second-order factor representing general non-routine problem solving ability in mathematics. Implicative analysis showed the influence of the presence of diagrams in the problems' hierarchical ordering. Furthermore, results provided support for other studies (e.g. Diezman & English, 2001) which documented some students' difficulties to use diagrams efficiently for the solution of problems. We discuss the findings and provide suggestions for the efficient use of diagrams in the problem solving situation.

The purpose of this publication is to record the current state of the art in research on mathematics-related affect. Research on mathematics-related affect is varied in theories and concepts. Rather than trying to address all perspectives in one chapter, we have identified significant strands of research and invited colleagues from these strands to each write a short section summarizing the state of the art in that strand.The concepts and theories pertaining to the affective domain can be mapped along three dimensions (Hannula 2012). The first dimension identifies three broad categories of affect: motivation, emotions, and beliefs. In this Topical Survey, motivation is covered in Sect. 2.5 (Middleton, Jansen, and Goldin), which also discusses how emotions and beliefs relate to motivation; Sects. 2.2 (Pantziara) and 1.2.3 (Zhang and Morselli) are on beliefs; and Sect. 2.1 (Di Martino) on attitude more or less cross-cuts through all these categories. The second dimension is movement from rapidly fluctuating state to more stable trait. All of the sections in this chapter focus on trait-type affect while only Sect. 2.5 (by Middleton, Jansen, and Goldin) discusses both of these dimensions (referred to as "in the moment" and "long term"). The last dimension covers the theorizing level, which has three main levels in mathematics-related affect: physiological (embodied), psychological (individual), and social. Mathematics-related affect has mainly been studied using psychological theories and consequently most sections discuss only such research. The so-called social turn (Lerman 2000) in mathematics education is in this Topical Survey mainly reflected in Sect. 2.4 (Heyd-Metzuyanim, Lutovac, and Kaasila) on identity, but Sect. 2.5 (Middleton, Jansen, and Goldin) on motivation also has both a section which discusses the social level and how it interplays with the individual level and a section on self-efficacy which highlights the emerging research on the collective efficacy of collaborative groups. The physiological level

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