A Dosage × Precision Framework for Instructional Effectiveness: Propositions, Boundary Conditions, and an Illustrative Case from Solid-State Physics
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 (Pt), representational match (Pr), and scope restriction (Ps)---and to argue that only representational match acts directly on extraneous load. This decomposition yields a compact model in which the effective dosage is Deff=D Ps, the germane processing is G=D Psη(Pr) with η 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, dD/dPr=−η/η', 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),
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BibTeX
@misc{tao2026dosage,
title = {{A Dosage $\times$ Precision Framework for Instructional Effectiveness: Propositions, Boundary Conditions, and an Illustrative Case from Solid-State Physics}},
author = {Kun Tao and Chenglong Jia and Desheng Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28516},
eprint = {2609.28516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28516},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Dosage × Precision Framework for Instructional Effectiveness: Propositions, Boundary Conditions, and an Illustrative Case from Solid-State Physics
AU - Kun Tao
AU - Chenglong Jia
AU - Desheng Xue
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.28516
UR - https://arxiv.org/abs/2609.28516
AB - 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_eff=D P_s, the germane processing is G=D P_sη(P_r) with η 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, dD/dP_r=−η/η', 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),
ER -