Michael H. G. Hoffmann

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 of mathematical problems. As a paradigmatic example, we apply a Peircean semiotic framework to answer the question of how students learned the concept of "distribution" in a statistics course by "diagrammatic reasoning" and by developing "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 for notions 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 of communication in mathematics classroom

Abductive reasoning takes place in forming``hypotheses'''' in order to explain ``facts.'''' Thus, theconcept of abduction promises an understanding ofcreativity in science and learning. It raises,however, also a lot of problems. Some of them will bediscussed in this paper. After analyzing thedifference between induction and abduction (1), Ishall discuss Peirce''s claim that there is a ``logic''''of abduction (2). The thesis is that this claim can beunderstood, if we make a clear distinction between inferential elements and perceptive elements of abductive reasoning. For Peirce, the creative act offorming explanatory hypotheses and the emergence of``new ideas'''' belongs exclusively to the perceptive side of abduction. Thus, it is necessary to study the roleof perception in abductive reasoning (3). A furtherproblem is the question whether there is arelationship between abduction and Peirce''s concept of``theorematic reasoning'''' in mathematics (4). Both forms of reasoning could be connected, because both arebased on perception. The last problem concerns therole of instincts in explaining the success ofabductive reasoning in science, and the question whether the concept of instinct might be replaced bymethods of inquiry (5).

Starting from the observation that small children can count more objects than numbers—a phenomenon that I am calling the “lifeworld dependency of cognition”—and an analysis of finger calculation, the paper shows how learning can be explained as the development of cognitive systems. Parts of those systems are not only an individual’s different forms of knowledge and cognitive abilities, but also other people, things, and signs. The paper argues that cognitive systems are first of all semiotic systems since they are dependent on signs and representations as mediators. The two main questions discussed here are how the external world constrains and promotes the development of cognitive abilities, and how we can move from cognitive abilities that are necessarily connected with concrete situations to abstract knowledge

Signs do not only “represent” something for somebody, as Peirce’s definition goes, but also “mediate” relations between us and our world, including ourselves, as has been elaborated by Vygotsky. We call the first the representational function of a sign and the second the epistemological function since in using signs we make distinctions, specify objects and relations, structure our observations, and organize societal and cognitive activity. The goal of this paper is, on the one hand, to develop a model in which both these functions appear as complementary and, on the other, to show that this complementarity is essential for the dynamics of scientific activity, causing a dialectical process of generating new epistemological and representational means. This will be demonstrated with an example of how two scientists with different background knowledge analyze educational data collaboratively

Discussions concerning belief revision, theorydevelopment, and ``creativity'' in philosophy andAI, reveal a growing interest in Peirce'sconcept of abduction. Peirce introducedabduction in an attempt to providetheoretical dignity and clarification to thedifficult problem of knowledge generation. Hewrote that ``An Abduction is Originary inrespect to being the only kind of argumentwhich starts a new idea'' (Peirce, CP 2.26).These discussions, however, led to considerabledebates about the precise way in which Peirce'sabduction can be used to explain knowledgegeneration (cf. Magnani, 1999; Hoffmann, 1999).The crucial question is that of understandinghow we can get the new elements capableof enlarging our theories. Under thesecircumstances, it might be helpful to step outof the entanglement and reconsider the basis ofthe problem that originally triggered Peirce'sinterest in abduction. This will lead us toanother Peircean concept, that of ``diagrammaticreasoning,'' which I discuss here in the contextof his ``pragmatism.'' In this way, I hope toreach a better understanding of thecontribution of ``abduction'' to the knowledgegeneration process.

Among the many problems posed by Peirce's concept of abduction is how to determine the scope of this form of inference, and how to distinguish different types of abduction. This problem can be illustrated by taking a look at one of his best known definitions of the term:Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis.The second half of this quote is not part of the definition, but an explanation of it. However, it adds something to this definition because it says implicitly that there are only three logical..

Why do we formulate arguments? Usually, things such as persuading opponents, finding consensus, and justifying knowledge are listed as functions of arguments. But arguments can also be used to stimulate reflection on one’s own reasoning. Since this cognitive function of arguments should be important to improve the quality of people’s arguments and reasoning, for learning processes, for coping with “wicked problems,” and for the resolution of conflicts, it deserves to be studied in its own right. This contribution develops first steps towards a theory of reflective argumentation. It provides a definition of reflective argumentation, justifies its importance, delineates it from other cognitive functions of argumentation in a new classification of argument functions, and it discusses how reflection on one’s own reasoning can be stimulated by arguments.

One of the first things President Obama did after coming to office was the establishment of the Office of Public Engagement. As described on its Web site, this office "is the embodiment of the President's goal of making government inclusive, transparent, accountable and responsible." The Office of Public Engagement is supposed to "create and coordinate opportunities for direct dialogue between the Obama Administration and the American public, while bringing new voices to the table and ensuring that everyone can participate and inform the work of the President."1As the president explained in his memorandum on transparency and open government, "Public engagement enhances the Government's effectiveness and improves ..

Technology is not only an object of philosophical reflection but also something that can change this reflection. This paper discusses the potential of computer-supported argument visualization tools for coping with the complexity of philosophical arguments. I will show, in particular, how the interactive and web-based argument mapping software “AGORA-net” can change the practice of philosophical reflection, communication, and collaboration. AGORA-net allows the graphical representation of complex argumentations in logical form and the synchronous and asynchronous collaboration on those “argument maps” on the internet. Web-based argument mapping can overcome limits of space, time, and access, and it can empower users from all over the world to clarify their reasoning and to participate in deliberation and debate. Collaborative and web-based argument mapping tools such as AGORA-net can change the practice of arguing in two dimensions. First, arguing on web-based argument maps in both collaborative and adversarial form can lead to a fundamental shift in the way arguments are produced and debated. It can provide an alternative to the traditional four-step process of writing, publishing, debating, and responding in new writing with its clear distinction between individual and social activities by a process in which these four steps happen virtually simultaneously, and individual and social activities become more closely intertwined. Second, by replacing the linear form of arguments through graphical representations of networks of inferential relations which can grow over time in an infinite space, these tools do not only allow a clear visualization of structures and relations, but also forms of collaboration in which, for example, participants work on different “construction zones” of larger argument maps, or debates are performed at specific points of disagreement on those maps. I introduce the term synergetic logosymphysis to describe a practice that combines these two dimensions of collaborative- and web-based argument mapping

A large body of research in cognitive science differentiates human reasoning into two types: fast, intuitive, and emotional “System 1” thinking, and slower, more reflective “System 2” reasoning. According to this research, human reasoning is by default fast and intuitive, but that means that it is prone to error and biases that cloud our judgments and decision making. To improve the quality of reasoning, critical thinking education should develop strategies to slow it down and to become more reflective. The goal of such education should be to enable and motivate students to identify weaknesses, gaps, biases, and limiting perspectives in their own reasoning and to correct them. This contribution discusses how this goal could be achieved with regard to reasoning that involves the construction of arguments; or more precisely: how computer-supported argument visualization tools could be designed that support reflection on the quality of arguments and their improvement. Three types of CSAV approaches are distinguished that focus on reflection and self-correcting reasoning. The first one is to trigger reflection by confronting the user with specific questions that direct attention to critical points. The second approach uses templates that, on the one hand, provide a particular structure to reason about an issue by means of arguments and, on the other, include prompts to enter specific items. And a third approach is realized in specifically designed user guidance that attempts to trigger reflection and self-correction. These types of approaches are currently realized only in very few CSAV tools. In order to inform the future development of what I call reflection tools, this article discusses the potential and limitations of these types and tools with regard to five explanations of the observation that students hardly ever engage in substantial revisions of what they wrote: a lack of strategies how to do it; cognitive overload; certain epistemic beliefs; myside bias; and over-confidence in the quality of one’s own reasoning. The question is: To what degree can each of the CSAV approaches and tools address these five potential obstacles to reflection and self-correction?

There is evidence that problem-based learning (PBL) is an effective approach to teach team and problem-solving skills, but also to acquire content knowledge. However, there is hardly any literature about using PBL in philosophy classes. One problem is that PBL is resource intensive because a facilitator is needed for each group of students to support learning efforts and monitor group dynamics. In order to establish more PBL classes, the question is whether PBL can be provided without the need for facilitators. We present a combination of five strategies—among them the collaborative argument visualization software AGORA-net—to replace facilitators. Additionally, we present evidence that these strategies are sufficient to provide a PBL experience that achieves intended learning goals in an ethics class and is satisfying for students without facilitators.

Although there is increasing interest in philosophy of science in transcendental reasoning, there is hardly any discussion about transcendental arguments. Since this might be related to the dominant understanding of transcendental arguments as a tool to defeat epistemological skepticism, and since the power of transcendental arguments to achieve this goal has convincingly been disputed by Barry Stroud, this contribution proposes, first, a new definition of the transcendental argument which allows its presentation in a simple modus ponens and, second, a pragmatist re-interpretation of this argument form that leaves it to the scientific community to debate, criticize, refine, or reaffirm its core claim: a premise which claims that the truth of a certain assumption is a necessary condition for something that is generally accepted. The proposed “logico-pragmatist interpretation” highlights the role of transcendental arguments as a methodological step to move science forward, just as abduction and inference to the best explanation do.

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

We all seem to have a sense of what good and bad arguments are, and there is a long history—focusing on fallacies—of trying to provide objective standards that would allow a clear separation of good and bad arguments. This contribution discusses the limits of attempts to determine the quality of arguments. It begins with defining bad arguments as those that deviate from an established standard of good arguments. Since there are different conceptualizations of “argument”—as controversy, as debate, and as justification—and since arguments in each of these senses can be used for different purposes, a first problem is that we would need a large variety of standards for “good” arguments. After this, the contribution focuses in particular on proposals made in the literature on how to assess the quality of arguments in the sense of justification. It distinguishes three problems of assessment: How to determine whether reasons are acceptable, whether reasons are sufficient to justify the conclusion, and how to identify arguments in real-world speech acts and texts? It is argued that limitations of argument assessment result from unavoidable relativism: The assessment of many—if not most—arguments depends on the epistemic situation of the evaluator.

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

Most of the epistemological debate on disagreement tries to develop standards that describe which actions or beliefs would be rational under specific circumstances in a controversy. To build things on a firm foundation, much work starts from certain idealizations—for example the assumption that parties in a disagreement share all the evidence that is relevant and are equal with regard to their abilities and dispositions. This contribution, by contrast, focuses on a different question and takes a different route. The question is: What should people actually do who find themselves in deep disagreement with others? And instead of building theory on some “firm foundation,” the paper starts from a specific goal—building consensus by creating new proposals—and asks, first, which actions are suitable to achieve this goal and, second, what are the epistemic conditions of these actions. With regard to the latter, the paper focuses on what has been called framing and reframing in conflict research, and argues that both metaphors need and deserve a suitable epistemological conceptualization.

Different situations — like school and workplace — demand different forms of knowledge. Even more important, in particular for lifelong learning, are forms of knowledge we need for managing movements between those situations. To develop a better understanding of how to ‘navigate’ knowledge boundaries, this paper analyzes, firstly, interviews with scientists interpreting familiar and unfamiliar graphs. Our goal is to identify those forms of knowledge that should receive special attention in education. Secondly, the article elaborates — based on Peirce’s semiotics — an epistemologically reflected semiotic model to describe the role and conditions of knowledge necessary for crossing knowledge boundaries.