The availability of technology in the mathematics classroom challenges the way teachers orchestrate student learning. Using the theory of instrumental orchestration as the main interpretative framework, this study investigates which types of orchestrations teachers develop when using technology and to what extent these are related to teachers' views on mathematics education and the role of technology therein. Data consisted of videotapes of 38 lessons taught by three teachers, who also provided information on their views through questionnaires and interviews. Qualitative analysis of these data led to the identification of orchestration types and teacher profiles. The orchestration preferences of the three teachers proved to be related to their views. A detailed analysis of one exemplary episode suggests how other theoretical perspectives might complement the theory of instrumental orchestration.

ABSTRACT. The concept of function is a central but difficult topic in secondary school mathematics curricula, which encompasses a transition from an operational to a structural view. The question in this paper is how the use of computer tools may foster this transition. With domain-specific pedagogical knowledge on the learning of function as a point of departure and the notions of emergent modeling and instrumentation as design heuristics, a potentially rich technology-intensive learning arrangement for grade 8 students was designed and field-tested. The results suggest that the relationship between tool use and conceptual development benefits from preliminary activities, from tools offering representations that allow for progressively increasing levels of reasoning, and from intertwinement with paper-and-pencil work.

This mixed-methods study investigates the effects of student attitudes and behaviours on the outcomes of learning mathematics with computer tools. A computer tool was used to help students develop the mathematical concept of function. In the whole sample (N = 521), student attitudes could account for a 3.4 point difference in test scores between individuals on a 10-point scale. General attitude towards mathematics positively predicted test scores.However, more able students who were well-disposed towards mathematical computer tools achieved lower scores. Self-reported behaviours were unrelated to test scores. Detailed observation of a small number of students (N = 8) revealed that positive attitudes towards mathematics and mathematical computer tools augmented exhibited learning behaviours, and that both a positive attitude to mathematical computer tools and exhibited learning behaviours benefited tool mastery. Although tool mastery and test scores are intimately related, reflective processes appear to mediate this relationship. Promoting learning with mathematical computer tools needs to take several factors into account, including improving student attitudes, raising levels of learning behaviours, and giving sufficient opportunity for constructing new mathematical knowledge within meaningful mathematical discourse.

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Non-routine word problem solving is an essential feature of the mathematical development of elementary school students worldwide. Many students experience difficulties in solving these problems due to erroneous problem comprehension. These difficulties could be alleviated by instructing students how to use visual representations that clarify the problem structure and the relations between solution-relevant elements (so-called visual-schematic representations). Research shows that instructional effectiveness depends largely on teachers' mathematical knowledge for teaching. Teachers' knowledge of visual representations is therefore essential to instructing word problem comprehension in this way. As there is little to no literature investigating teachers' practices in this area, the goal of the present study is to examine teachers' use of visual representations to support non-routine word problem solving. Eight mainstream elementary school teachers implemented an innovative approach focused on the use of visual-schematic representations. After a short training, teachers were able to produce these representations during instruction. However, some teachers seemed unclear about what these representations comprise and what function they serve within the word problem solving context. Teachers seemed to base their use of representations on personal preferences rather than on an optimal fit with the word problem characteristics. These aspects need to be addressed in teacher training and professional development programs.! This study makes an unique contribution to research in the important and problematic area of word problem solving in regular classrooms. The results of this study are relevant for educational researcher, teachers, and teacher educators who deal with difficulties in instructing mathematical word problems.

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Developing fluency in arithmetic facts is instrumental to mathematics learning. This study compares the effects of two practice conditions on children's fluency in simple multiplication facts. Third and fourth graders in the Netherlands (N = 282) practised in either a conventional recall condition where they produced answers to problems, or a choice condition where they had to choose between competing answers that included common multiplication errors. Practice in the choice condition was faster and as accurate as recall practice but was not more beneficial to performance on speed tests of practised facts. For more experienced students, recall practice led to greater improvement on a conventional recall fluency test. Differential effects of practice conditions on test performance are explained in terms of practice-to-test transfer demands. The relative merits of recall and choice tasks in multiplication fact learning are discussed.

In countries with high differentiation between academic and vocational education, an individual's future prospects are strongly determined by the educational track to which he or she is assigned. This large-scale, cross-sectional study focuses on low-performing students in academic tracks who face being moved to a vocational track. If more is understood about these students, measures could be taken to improve their performance and keep them within academic education. The study investigates performance patterns in academic tracks in the first three years of secondary school in the Netherlands. By identifying patterns that reveal how competence levels in different domains are related at different stages of development and by comparing low performers with other students, the study sheds light on individual and educational aspects that may be amenable to intervention. School grades were analysed for 1596 students. School performance was found to reflect three domains-languages (language of schooling and modern foreign languages), social studies and science and math-that appear to interact in a process of co-construction. General language skills were robustly related to performance in other domains-particularly social studies-throughout the first three years of secondary school. By comparison, proficiency specifically in the language of schooling was less strongly related to social studies and science and math performance after the first year. Suggestions are given as to how educators and curriculum developers could use these insights to accommodate individual and developmental differences and to develop learning materials that may help low performers keep on track. IntroductionIn countries with educational systems that are stratified according to general scholastic ability, an individual's future occupational status and employment prospects are strongly determined by the educational track to which he or she is assigned at school. This is particularly so where there is high differentiation between academic and vocational education. In contrast to those in academic tracks, individuals in vocational tracks often have no direct access to higher education or professional employment and attain lower occupational status (Shavit & M€ uller, 2000;Wolbers, 2007;Andersen & Van de Werfhorst, 2010 Vol. 41, No. 1, February 2015, pp. 48-71 DOI: 10.1002 tracks with lower-level skills face a future with lower wages and higher levels of unemployment (Oesch, 2010;OECD, 2012).In stratified systems, students are often placed in designated educational tracks from early on in secondary school, with initial track allocation depending on performance in the final years of primary school and subsequent movement between tracks depending on on-going performance in the first years of secondary school. In consequence, students who perform poorly in academic tracks may be moved to a vocational track. This considerably impacts their future prospects in an economic climate where higher educational levels are needed to compete in an increasingly in...