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.

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Statistics is a challenging subject for many university students. In addition to dedicated methods of didactics of statistics, adaptive educational technologies can also offer a promising approach to target this challenge. Inspectable student models provide students with information about their mastery of the domain, thus triggering reflection and supporting the planning of subsequent study steps. In this article, we investigate the question of whether insights from didactics of statistics can be combined with inspectable student models and examine if the two can reinforce each other. Five inspectable student models were implemented within five didactically grounded online statistics modules, which were offered to 160 Social Sciences students as part of their first-year university statistics course. The student models were evaluated using several methods. Learning curve analysis and predictive validity analysis examined the quality of the student models from the technical point of view, while a questionnaire and a task analysis provided a didactical perspective. The results suggest that students appreciated the overall design, but the learning curve analysis revealed several weaknesses in the implemented domain structure. The task analysis revealed four underlying problems that help to explain these weaknesses. Addressing these problems improved both the predictive validity of the adjusted student models and the quality of the instructional modules themselves. These results provide insight into how inspectable student models and didactics of statistics can augment each other in the design of rich instructional modules for statistics.

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