The notion of practical argumentation is rooted in the broader tradition of practical reasoning, which dates back to Aristotle’s Nicomachean Ethics, specifically Books VI and VII [1]. Unlike theoretical reasoning, which is primarily oriented towards establishing what is true or acceptable as knowledge, practical reasoning is directed towards action, choice, and conduct. For Audi, Aristotle’s notion of practical reasoning is not easy to express in clear and precise language. More specifically, this type of reasoning “provides a way to understand and explain actions [and] makes actions that are based on practical reasoning intelligible as conduct in accordance with one’s practical judgment” [2] (p. 28). For Aristotle, the notion of practical reasoning is expressed in the form of a syllogism, also known as a practical syllogism, which serves to illustrate both the process of practical reasoning and the formalisation of that process [1]. In contemporary argumentation theory, this tradition has been developed as the study of arguments whose conclusions recommend, justify, or evaluate action. Thus, the historical development of practical argumentation can be understood as a shift from the practical syllogism, as a formalisation of practical reasoning, to the practical argument, as an analytical structure for examining the justification of actions.

This understanding of the practical argument is consistent with the work of Green [3], Fenstermacher [4], and Walton [5], among others, who have approached practical argumentation from different perspectives. Green [3] related practical arguments to teaching competence, arguing that instruction involves acting upon the premises that guide students’ practical reasoning. Fenstermacher [4] extended this idea to the relationship between research and teaching practice, suggesting that research contributes to practice when it improves the practical arguments that inform teachers’ decisions. Walton [5] characterised practical reasoning as goal-driven, knowledge-based, and action-guiding argumentation, thereby connecting agents’ goals, available means, circumstances, and possible consequences.

The educational relevance of practical argumentation can be better understood in relation to the development of reflective approaches to professional practice. Schön’s work [6] on the reflective practitioner highlighted that professionals often act in situations that are uncertain, unique, and marked by conflicting demands, and that their knowledge-in-action is not always fully explicit. Teaching can be understood as one such professional practice, since teachers constantly interpret classroom events, select possible courses of action, and justify their decisions in relation to learning objectives, students’ responses, curricular demands, available resources, and professional values.

More recently, research on teacher professional reasoning has developed through several related trajectories, including professional noticing, video-supported reflection, lesson study, instructional coaching, and digitally mediated professional learning. In mathematics education, professional noticing has been conceptualised as teachers’ capacity to attend to students’ mathematical thinking, interpret students’ understandings, and decide how to respond instructionally [7]. Research on mathematics teacher noticing has also highlighted the importance of analysing how teachers make sense of classroom events and how this sense-making informs instructional decisions [8]. Similarly, lesson study has become a prominent collaborative professional development approach in mathematics education, generally organised around cycles of study, planning, teaching, and reflection, with a strong focus on students’ thinking and learning [9]. Recent work on AI-supported and video-based reflection further indicates that digital tools can make classroom activity more visible and support teachers’ noticing, interpretation, and reflection [10]. However, these approaches do not always provide a detailed account of the argumentative structure through which teachers justify their pedagogical decisions, where practical argumentation contributes to this landscape.

Given the focus of this entry, it is important to distinguish practical argumentation from two other types of argumentation. On the one hand, between mathematical argumentation and practical argumentation: the former arises within mathematical practices and is concerned with the production, justification, and validation of mathematical claims through inductive, abductive, analogical, and deductive forms of reasoning; the latter, when examined in teaching contexts, is concerned with the justification, assessment, or reconsideration of pedagogical actions planned or carried out in the classroom [11]. On the other hand, between theoretical argumentation and practical argumentation: the former enables answering questions such as ‘is p true?’ (where p is a description of the world); the latter enables answering questions such as ‘what should A do in situation x?’ (where A is the name or description of an agent, and x is the description of a problem situation) [12].

In the context of mathematics education, argumentation has traditionally been studied in relation to mathematical reasoning, proof, justification, and the validation of mathematical claims. Research on proof and argumentation has shown the centrality of these processes in students’ mathematical activity and in mathematics education research more broadly [13,14]. In fact, in the specialised literature, the development of mathematical argumentation is regarded as a fundamental aspect of learning this discipline [15,16,17], that is, as a competence to be developed in the classroom [18]. Recent work in the philosophy of mathematical practice has also shown that mathematical proof activity is not reducible to a mere sequence of deductive steps. Hamami and Morris [19,20] argue that proofs are connected to the plans and intentions of the agents who produce them, and that proof activities may be analysed in terms of planning agency and practical reasoning. This perspective is relevant because it shows that action-oriented reasoning may also play a role within mathematical practice itself. However, practical argumentation introduces a different but complementary focus: instead of analysing how mathematical claims are justified, it examines how pedagogical actions in the mathematics classroom are justified. This shift is particularly relevant for studying episodes of teaching reflection on practice, where prospective and practising mathematics teachers justify, assess, weigh up, or reconsider their pedagogical decisions [11,21,22].

This entry presents practical argumentation as a conceptual and analytical tool for studying teachers’ pedagogical decision-making when reflecting on their practice. The first section traces the general development of practical argumentation and situates it in relation to mathematics education. The second section presents applications of practical argumentation in different fields of knowledge and outlines proposed models for its analysis. The third section explains a scheme for structuring and representing practical arguments in connection with didactical criteria for analysing mathematics teaching and learning. Finally, the entry discusses the current status, limitations, and future prospects of practical argumentation in research on teaching reflection on practice, emphasising its value for making visible the reasons underlying mathematics teachers’ decisions in situations characterised by mathematical, didactical, and classroom-management tensions.