ABSTRACT. 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 example, we apply a Peircean semiotic framework to answer the question of how students develop a notion of 'distribution' in a statistics course by 'diagrammatic reasoning' and by forming 'hypostatic abstractions', that is by forming new mathematical objects which can be used as means for communication and further reasoning. Peirce's semiotic terminology is used as an alternative to concepts such as modeling, symbolizing, and reification. We will show that it is a precise instrument of analysis with regard to the complexity of learning and communicating in mathematics classrooms.

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As a committee of the National Academy of Engineering recognized, ethics education should foster the ability of students to analyze complex decision situations and ill-structured problems. Building on the NAE's insights, we report about an innovative teaching approach that has two main features: first, it places the emphasis on deliberation and on self-directed, problem-based learning in small groups of students; and second, it focuses on understanding ill-structured problems. The first innovation is motivated by an abundance of scholarly research that supports the value of deliberative learning practices. The second results from a critique of the traditional case-study approach in engineering ethics. A key problem with standard cases is that they are usually described in such a fashion that renders the ethical problem as being too obvious and simplistic. The practitioner, by contrast, may face problems that are ill-structured. In the collaborative learning environment described here, groups of students use interactive and web-based argument visualization software called "AGORA-net: Participate - Deliberate!". The function of the software is to structure communication and problem solving in small groups. Students are confronted with the task of identifying possible stakeholder positions and reconstructing their legitimacy by constructing justifications for these positions in the form of graphically represented argument maps. The argument maps are then presented in class so that these stakeholder positions and their respective justifications become visible and can be brought into a reasoned dialogue. Argument mapping provides an opportunity for students to collaborate in teams and to develop critical thinking and argumentation skills.

In the first part of this paper, I delineate Peirce's general concept of diagrammatic reasoning from other usages of the term that focus either on diagrammatic systems as developed in logic and AI or on reasoning with mental models. The main function of Peirce's form of diagrammatic reasoning is to facilitate individual or social thinking processes in situations that are too complex to be coped with exclusively by internal cognitive means. I provide a diagrammatic definition of diagrammatic reasoning that emphasizes the construction of, and experimentation with, external representations based on the rules and conventions of a chosen representation system. The second part starts with a summary of empirical research regarding cognitive effects of working with diagrams and a critique of approaches that use 'mental models' to explain those effects. The main focus of this section is, however, to elaborate the idea that diagrammatic reasoning should be conceptualized as a case of 'distributed cognition.' Using the mathematics lesson described by Plato in his Meno, I analyze those cognitive conditions of diagrammatic reasoning that are relevant in this case.