Students' algebra mal-rules are used very irregularly, are severely skewed in frequency, and differ in explanatory power from school to school¶

Claim · All claims
Evidence · 1 study · associational r? · q2 · n=students in three schools (number not stated in the abstract)
Currency · newest study 1990 · not yet checked against its sources · not reviewed by a person

A mal-rule is a buggy rewriting rule (for example, an incorrect way of moving a term across an equation) used to explain a student's error. This claim is about how stable and how general such rules are.

Subclaims¶

q2 i? Mal-rules are very unstable: students typically use them very irregularly rather than applying a buggy rule every time it could apply. → Payne 1990

q2 i? Mal-rule frequency is severely skewed, with a few frequent mal-rules and many infrequent ones, and different mal-rules have explanatory power in different schools; many of the most powerful mal-rules found had not been reported before. → Payne 1990

q2 i? Mal-rule diagnosis was more successful with more skilled students, and every mal-rule found was associated with high confidence ratings by at least one student, so confidence did not separate slips from mistakes. → Payne 1990

Evidence¶

Payne 1990¶

Payne, S. J., & Squibb, H. R. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14(3), 445–481. doi:10.1207/s15516709cog1403_4

q2 · observational error-analysis study · i? · no effect size reported in the abstract · n=students in three schools (number not stated in the abstract) · associational · r?

An empirical study of elementary algebra errors in three separate schools, with errors diagnosed as mal-rules in the sense proposed by Sleeman. The authors report the instability, skew and school-specificity above, and argue that slips and mistakes cannot be cleanly separated: most errors depend on properties of the knowledge base and the cognitive architecture, so errors in a procedural skill cannot be assumed to be purely syntactic in origin. Coded from the published abstract; the full text was not read.

Discussion¶

For diagnostic assessment the finding cuts two ways. A catalogue of mal-rules built in one population may not transfer to another, and a single wrong answer attributed to a mal-rule is weak evidence that the student will make the same error next time. Both fit the knowledge-in-pieces view of errors as context-dependent, and both bear on systems that model or simulate a student by a fixed set of buggy rules. The same pattern, inconsistent rule use below mastery, was reported for signed-number arithmetic (Tatsuoka et al. 1989).

Related Claims¶

• Students' rules of operation for signed-number arithmetic are applied inconsistently by non-masters and stably by masters — a parallel finding in arithmetic
• Student errors on underestimated fraction items reflect systematic misconception patterns rather than random error — related; reads errors as systematic
• 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