Multiple perspectives on the concept of conditional probability

Conditional probability is a key to the subjectivist theory of probability; however, it plays a subsidiary role in the usual conception of probability where its counterpart, namely independence is of basic importance. The paper investigates these concepts from various perspectives in order to shed light on their multi-faceted character. We will include the mathematical, philosophical, and educational perspectives. Furthermore, we will inspect conditional probability from the corners of competing ideas and solving strategies. For the comprehension of conditional probability, a wider approach is urgently needed to overcome the well-known problems in learning the concepts, which seem nearly unaffected by teaching

05491 Abstracts Collection -- Spatial Cognition: Specialization and Integration

From 04.12.05 to 09.12.05, the Dagstuhl Seminar 05491 ``Spatial Cognition: Specialization and Integration\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Mathematical Reasoning Required when Students Seek the Original Graph from a Derivative Graph

Background: Finding the original graph when given the derivative graph is not a trivial task for students, even though they are able to find the derivative graph when given the original graph. Objective: In the context of qualitative research, this paper presents and analyses the mathematical reasoning that comes to light when the students seek the original graph from a derivative graph. Design: The research is assigned as a qualitative study, where the analyses of cases aim to extend understanding with respect to some phenomena or theory. Setting and participants: The study was conducted with 86 students of State University in East Java. We conducted clinical interviews and present data highlighting the reasoning participants used when solving tasks. Data collection and analysis: Task-based interviews were used to collect data and data analysis was used to analyse interpretations of the graphs that emerged as mathematical reasoning models. Results: From our data analysis, we found three mathematical reasoning was rooted in students’ conscious of problem situations on graphs we provided, consists of direct reasoning, reversible reasoning, and synthesis of direct-reversible reasoning. Conclusions: From here we suggest that there are different mathematical reasoning in constructed the original graph due to the mental activity in which students use the relation between a function and its derivative. The future project continued this inquiry with rigorous single-subject students’ experiments