What are common mistakes students make when evaluating mixed addition and multiplication problems?
Executive summary
Students frequently trip on order-of-operations rules when problems mix multiplication and addition: many confuse operation priority (multiplication before addition) or apply rules left-to-right incorrectly, and errors often stem from weak basic facts, parenthesis misuse, or rote rule-memorization that leads to rule‑mixup (examples and studies note these patterns) [1] [2] [3].
1. Misreading the rule: “Do addition first” — a persistent operation-order error
A common, concrete mistake is treating the list of operations as simply a checklist rather than an ordered hierarchy and doing addition before multiplication when addition appears earlier in the written expression; guidance sources stress that multiplication precedes addition (PEMDAS/BODMAS), and learners who ignore that get wrong answers even on otherwise simple expressions [1] [2].
2. Left‑to‑right confusion inside pairs: wrong associativity for M/D or A/S
Students often misunderstand that multiplication and division are processed left to right as a pair, and the same for addition and subtraction; failing to work left-to-right within those pairs (for example doing all multiplication mentally out of sequence) leads to inconsistencies and errors — educators recommend explicit left‑to‑right practice to fix this [4] [5].
3. Parentheses and implicit grouping: forgetting or misusing grouping symbols
Many errors come from ignoring or misreading parentheses and other grouping symbols. The qualitative literature shows students frequently “just ignore the function of parenthesis,” which changes intended order and produces incorrect outcomes; teachers report difficulties explaining parentheses which amplifies student mistakes [3].
4. Poor basic arithmetic fluency causes downstream mistakes
Research and practitioner reports highlight that weak basic skills — slow or inaccurate multiplication/division facts, errors with negative signs, or shaky number sense — produce procedural mistakes when students face mixed problems. Studies link these foundational gaps to larger rule-mixups and procedural failures [3] [6].
5. Rule‑mixup from rote learning: memorized rules without understanding
When students memorize acronyms like PEMDAS without conceptual understanding, they can mix rules or apply them mechanically in the wrong order; the literature describes “rule mix-up” as a source of error where rote learning causes students to apply multiplication/division rules to addition/subtraction improperly [3] [4].
6. Carelessness and off‑task mistakes: not all errors are conceptual
Not every wrong answer signals a misunderstanding. Instructional guides and case studies list carelessness — skipping steps, arithmetic slips, misalignment on paper — as common error types in mixed operations problems, reminding teachers that some mistakes come from distraction or haste, not faulty concepts [6].
7. Specific tricky cases: negatives, exponents, and implicit multiplication
Advanced pitfalls arise when mixed operations involve negatives, exponents, or implicit multiplication (e.g., 2(3+4)). Sources note many students mis-handle negation with exponents and are unsure how implicit multiplication interacts with the order of operations; such historical and notation subtleties produce recurring misunderstandings [4].
8. Instructional remedies: focus on concept, not just acronyms
Practitioners recommend moving beyond acronyms to explicit modeling, lots of varied examples, and slow demonstration of each step (even allowing students to check after each operation). Teaching students to re-group expressions with parentheses and to verbalize why one operation comes before another reduces common mistakes [5] [1].
9. Assessment and teaching implications: diagnose the root cause
Assessment should distinguish procedural slips from conceptual gaps: if a student consistently misorders operations despite correct basic arithmetic, the issue is likely conceptual (rule understanding); if errors are arithmetic or alignment-based, the intervention is practice and neatness. The IRIS case study categorizes errors (wrong operation, miscalculation, procedural, visual‑spatial) to guide targeted remediation [6].
Limitations and gaps in available reporting
The provided sources document common error patterns, classroom observations, and practical fixes, but available sources do not mention exact prevalence percentages across grade levels, nor do they provide large-scale randomized trials that compare specific interventions’ effectiveness; those data are not found in the current reporting [7] [3].