Absence or Entrenchment? Belief Revision, Diagnostic Error and the Design of a Trial on Misconception Interventions in Mathematics
DOI:
https://doi.org/10.5281/zenodo.22712852Keywords:
belief revision, conceptual change, misconceptions, certainty of response index, aptitude-by-treatment interaction, simulation studyAbstract
Not knowing a mathematical relation and firmly believing its negation are distinct epistemic states that, in a ranked belief-base model, require different belief-change operations; grading both as one kind of error hides that difference from intervention studies. We model instruction as semi-revision by input of given credibility. Absence is revised by any input that delivers the target; a misconception yields only to input more credible than its strongest minimal support. Operators whose acceptance ignores either the base or the credibility predict no state-by-intervention interaction; this one predicts an interaction with no effect in absence. Under non-differential diagnosis the interaction is scaled by PPV + NPV − 1 (0.70 at sensitivity and specificity 0.85 and prevalence 0.53, 0.39 at prevalence 0.89). In published certainty tables for 93 students, high-certainty answers made up 98 of 110, 48 of 91 and 40 of 115 wrong answers in three groups. No intervention data exist. An ADEMP simulation reports operating characteristics of a proposed, not yet preregistered, 2 × 2 trial with classrooms allocated within schools. Under stipulated parameters, 16 schools of four classrooms with 40 students tested per classroom give power 0.96 for a latent-state interaction of 0.30, attenuated in the diagnosed strata, if about half are analysed (0.86 if a third are); type I error of the primary test stayed within its declared tolerance in every null condition explored, and so did a secondary profile test with joint variance in the seven conditions examined. These are properties of a design under assumptions, not evidence about learning.
Received: 1 August 2026 | Peer review completed: 10 September 2026 | Accepted: 12 September 2026
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