Subject matter reconstruction in probability: identifying core knowledge elements for secondary mathematics teachers through the laws of large numbers

Teachers’ mathematical content knowledge in probability often oscillates between two poles: empirically grounded classroom reasoning and formally rigorous probability theory. This tension becomes particularly visible for the Laws of Large Numbers (LLN), where “stabilization” appears in epistemically distinct forms: as an empirically grounded modelling heuristic, and as mathematical theorems formulated within axiomatic probability theory. The present paper uses subject matter reconstruction as a theory-informed, criteria-guided analytic procedure to make explicit which content-related knowledge elements are warranted for coherent teaching of the LLN and how these elements can be connected. These outcomes are presented as theoretically warranted normative propositions rather than empirically validated requirements for effective teaching. The study conducts a structured walkthrough across three perspectives on the LLN—empirical, weak, and strong—following five didacticization steps (mathematical core; reference systems including applications and historical genesis; prerequisite structures; representation levels and basic mental models; and topic-specific didactic concepts) and a subsequent reconstruction phase that consolidates results via the categories peculiarities, commonalities, diversities, and limitations. Outcomes are reported as (a) a topic-specific set of reconstructed knowledge elements with documented allocations to school knowledge (SK), school-related content knowledge (SRCK), and academic knowledge (AK), and (b) an exemplary knowledge network that connects elements through the unifying idea of stabilization using explicit edge types (e.g., shifts in representation, justification, and formalization). The results contribute (1) a transparent, inspectable specification of elementarized academic knowledge for the LLN, (2) a worked example of subject matter reconstruction as a replicable content-analytic method for warranted normative claims, and (3) transferable artefacts (tables and a network representation) that can inform teacher education design and subsequent empirical validation.