{
  "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 $G=D P_sη(P_r)$ 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, $\\mathrm{d}D/\\mathrm{d}P_r=-η/η'$, 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),",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28516",
  "author_total": 3,
  "authors": [
    {
      "name": "Kun Tao"
    },
    {
      "name": "Chenglong Jia"
    },
    {
      "name": "Desheng Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6gspxgkahmwtx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28516",
  "fulltext_url": null,
  "id": "6gspxgkahmwtx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28516",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "physics.soc-ph"
    ],
    [
      "openalex",
      "3205|Visual and Cognitive Learning Processes"
    ]
  ],
  "openalex": "W7214305370",
  "pdf_url": "https://arxiv.org/pdf/2609.28516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6gspxgkahmwtx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.28516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28516"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.soc-ph",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "A Dosage $\\times$ Precision Framework for Instructional Effectiveness: Propositions, Boundary Conditions, and an Illustrative Case from Solid-State Physics",
  "title_text": "A Dosage × Precision Framework for Instructional Effectiveness: Propositions, Boundary Conditions, and an Illustrative Case from Solid-State Physics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:11Z",
  "url": "https://research.draconitic.com/paper/6gspxgkahmwtx",
  "venue": "arXiv",
  "versions": []
}