Dialogic Developmental Movement in Mathematical Modelling: A Bakhtinian Vygotskian Analysis of Consolidation in Fermi Estimation

Abstract Although dialogic interaction is widely regarded as beneficial for mathematical learning, research offers limited analytic means for determining when such interaction reorganises learners’ reasoning rather than merely sustaining participation. This paper examines how dialogue becomes developmentally consequential within contextualised modelling activity, drawing on Bakhtinian dialogism and Vygotskian sociocultural theory. Through microanalysis of two Critical Mathematical Incidents (CMIs) from a single 90-minute Core Mathematics lesson on Fermi estimation involving three learners in an English Further Education college, the study traces shifts in learners’ contextual anchoring , stance , tool appropriation , and epistemic authority across pre-dialogue, in-dialogue , and post-dialogue phases. The analysis introduces the phase-contrast indicator (ΔΔ) as an interpretive heuristic , not a quantitative measure, for distinguishing activation (the initial emergence of contextual and representational resources in dialogue) from consolidation (the subsequent reorganisation of those resources into structured, learner-owned modelling reasoning). Findings show that consolidation depends not on dialogue alone but on the intersection of dialogic challenge with access to cultural tools and representational structure. A counterexample demonstrates that engagement without representational structure produces activation but not consolidation. The analytic approach offers transferable conceptual tools for researchers examining how classroom dialogue contributes to the development of mathematical reasoning, within the interpretive scope appropriate to a single-case microanalytic design.