The paper describes two versions of an inquiry-based activity in geometry, designed as a game between two players. The game is inspired by Hintikka's semantic game, which is a familiar tool in the field of logic to define truth. The activity is designed in a dynamic geometry environment (DGE). The inquiry is initially guided by the game itself and…

This paper offers a typology of forms and uses of abduction that can be exploited to better analyze abduction in proving processes. Based on the work of Peirce and Eco, we describe different kinds of abductions that occur in students' mathematical activity and extend Toulmin's model of an argument as a methodological tool to describe students'…

To explicate certain phenomena, e.g., the possibility of deduction without definition, we hypothesize that an individual is able to understand and appreciate reasoning with a due feeling of its necessity when the concept image of each concept involved in the reasoning has reached a certain level of development; we then speak of "deep intuition".…

This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed…

In recent years, semiotics has become an innovative theoretical framework in mathematics education. The purpose of this article is to show that semiotics can be used to explain learning as a process of experimenting with and communicating about one's own representations (in particular "diagrams") of mathematical problems. As a paradigmatic…

In early algebra students often struggle with equation solving. Modeled on Streefland's studies of students' own productions a prototype pre-algebra learning strand was designed which takes students' informal (arithmetical) strategies as a starting point for solving equations. In order to make available the skills and tools needed for manipulating…