A Dosage $\times$ Precision Framework for Instructional Effectiveness: Propositions, Boundary Conditions, and an Illustrative Case from Solid-State Physics

arXiv:2609.28516v1 Announce Type: new Abstract: We propose that instructional effectiveness may depend on an interaction between two design dimensions: dosage---the total duration of student-centered active learning allocated to a topic---and precision---the degree to which an intervention targets a specific, documented cognitive bottleneck with a mechanism-matched representational tool. This paper formalizes the framework, anchors it in cognitive load theory, derives its falsifiable predictions, and clarifies its incremental contribution relative to existing instructional-design literature. The central theoretical move is to decompose precision into three components---targeting ($P_t$), representational match ($P_r$), and scope restriction ($P_s$)---and to argue that only representational match acts directly on extraneous load. This decomposition yields a compact model in which the effective dosage is $D_{\mathrm{eff}}=D P_s$, the germane processing is $G=D P_s\eta(P_r)$ with $\eta$ the representational efficiency, and learning is $E=g(G)$ with $g$ increasing and concave \rev{(here $E$ is the cumulative learning outcome on the target construct, not a rate)}. The framework's core proposition (P3)---that precision can compensate for dosage---then becomes a quantifiable substitutability claim, $\mathrm{d}D/\mathrm{d}P_r=-\eta/\eta'$, rather than a mere existence statement; its scope is bounded by a precision ceiling and a minimum dosage threshold. We distinguish the framework from Carroll's classic time-needed model by the directional character of precision and by the estimability of the substitution rate, and from the ICAP framework by distinguishing ``matching of participation content to the bottleneck'' from ``mode of participation.'' As an illustrative case (not a test of the framework),