Informal Complexity Intuition at the Consolidation Stage: A Pedagogical Synthesis

The teaching of algorithmic complexity in introductory programming is conventionally deferred — the formal apparatus of big-O notation, asymptotic upper and lower bounds, master theorem, amortised analysis, and recurrence solving sits in a discrete-mathematics or design-and-analysis-of-algorithms course at the university level, and the K-12 informatics curriculum (Vietnam's MOET 2018 informatics syllabus included) either omits it entirely or sketches it at the THPT specialisation stage for the chuyên Tin / IOI track. The deferred-formal-complexity position rests on the implicit claim that complexity reasoning requires the formal apparatus, that informal predecessors of the apparatus are not pedagogically robust, and that introducing such predecessors risks installing misconceptions that the formal stage will then have to overwrite. This paper challenges the deferred-formal position. It synthesises four threads of literature — the foundational design-and-analysis texts [1], [2], [3], [4], the empirical-curve and doubling-test pedagogy of Sedgewick and Wayne [5], [6], the algorithm-visualisation effectiveness meta-analytic work [7], [8], [9], and the misconceptions and complexity-pedagogy CS-education line [10], [11], [12], [13], [14] — under the umbrella of Wing's computational-thinking framing [15] operationalised by Selby-Woollard's developing-definition framework [16]. The paper contributes four results. First, a literature verdict: informal complexity intuition is instillable at the consolidation stage (THCS Volume-3 audience, ages 13–15, after a full pass through loops, sequences, functions, and the standard data structures) under four conditions — prerequisitebase, granularity-floor, error-tolerance, and transfer-to-formalbig-O. Second, an instillation framework consisting of an informal complexity language (grows-fast / grows-slow / scales-linearly / scales-quadratically) and four named pedagogical moves: empirical timing comparison, doubling test, growth-table reading, and structural inspection. Third, a staged sequence specifying which move runs at which point in the consolidation chapters (the Python Pathway Volume 3 sequence on complexity, recursion, sorting, and search-and-graph foundations). Fourth, a transfer prediction: each piece of informal vocabulary corresponds to a specific formal-big-O construct that the specialisation stage introduces, and the correspondence is monotone (no informal claim contradicts the formal claim it maps to). The paper's strongest novel claim is that the deferred-formal position has historically conflated the formal apparatus with the reasoning capacity, and that the consolidation stage can build the reasoning capacity using informal, empirically-anchored moves without taking on the formal apparatus, provided the four conditions hold. Per the elix-researches no-cohort scope, the framework is described and theoretically validated; deployment is identified as a downstream cohort node responsibility and is out of scope here.