Students' rules of operation for signed-number arithmetic are applied inconsistently by non-masters and stably by masters¶

Claim · All claims
Evidence · 1 study · associational r? · q2 · n=161 eighth graders
Currency · newest study 1989 · checked against its sources 2026-10-21 (automated, 1 of 1 entry) · not reviewed by a person

The rule a non-mastering student uses on one occasion is a weak predictor of the rule the same student uses on an identical item later in the same test.

Subclaims¶

Reading the evidence codes

q is the design tier, i the impact, n the sample, and the kind with its r the rigour judged within that kind: what each code means.

This claim: no standardized effect is recorded in observations/ for it yet, so no benchmark reading is possible.

q2 i? On two identical 32-item subtests of signed-number operations presented in different random orders, inconsistent rule application was common among students who had not mastered signed-number arithmetic, while students at mastery, who used the right rules, showed a stable pattern. → Tatsuoka 1989

Evidence¶

Tatsuoka 1989¶

Tatsuoka, K. K., Birenbaum, M., & Arnold, J. (1989). On the stability of students' rules of operation for solving arithmetic problems. Journal of Educational Measurement, 26(4), 351–361. doi:10.1111/j.1745-3984.1989.tb00339.x

q2 · within-student repeated-measures comparison · i? · no effect size reported in the abstract · n=161 eighth graders · associational · r?

A test of 64 open-ended items, made of two 32-item subtests containing identical items in random order, was given to 161 eighth graders. The rules of operation underlying each student's answers were identified separately for each subtest and compared. Non-masters' rule use was often inconsistent across the two subtests; masters' was stable. The authors discuss the result in terms of a hypothesis-testing account of learning, in which a learner still trying out rules switches between them. Coded from the published abstract; the full text was not read.

Discussion¶

A diagnosis of a non-mastering student's error rule from one pass through a test is unstable even within a single sitting, so diagnostic tests that infer a "bug" from one response pattern need repeated items. The finding fits the knowledge-in-pieces view of errors as context-dependent below mastery, and parallels the instability of algebra mal-rules (Payne & Squibb 1990).

Learner Variables¶

Prior Knowledge — predictor: learners who differ on it differ in outcomes

Students' algebra mal-rules are used very irregularly, are severely skewed in frequency, and differ in explanatory power from school to school — a parallel finding in algebra
Student misconceptions can be read as knowledge elements that serve in some contexts and are refined and reorganised during learning, rather than as strongly held flawed ideas that instruction must confront and replace — the theory this finding fits
Test A contains 28 errors and Tests B and C contain 29 each, yet scores for rights are practically the same — related

Page metadata

Field	Value
`type`	`claim`
`id`	`arithmetic-rule-use-inconsistent-below-mastery-stable-at-mastery`
`title`	`Students' rules of operation for signed-number arithmetic are applied inconsistently by non-masters and stably by masters`
`description`	`When 161 eighth graders answered the same 32 signed-number items twice in different random orders, students who had not mastered the operations commonly switched the rules they used, while masters applied rules stably.`
`status`	`review`
`evidence_strength`	`weak`
`newest_evidence`	`1989`
`source_checked`	`2026-10-21`
`generated`	by `claude/unspecified` on `2026-10-21`
`trust`	unverified — no one has confirmed this page's content
`sources`	1 — `tatsuoka-1989`

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