What are common errors students make applying order of operations and how do teachers correct them?
Executive summary
Students commonly misapply the order of operations by treating the mnemonic PEMDAS as strictly sequential instead of recognizing equal-precedence pairs and left-to-right associativity, mixing up subtraction/addition or multiplication/division and ignoring grouping symbols; teachers correct this through explicit instruction on left-to-right rules, targeted error-analysis tasks, and scaffolded practice that highlights associativity and negative signs [1] [2] [3]. Research shows these misconceptions can become deeply held and persist without deliberate remediation, so correction requires more than rote repetition of acronyms—it demands conceptual interventions and diagnostic measurement [4].
1. Common mistake: Reading PEMDAS too literally and forcing M before D or A before S
A pervasive student error is assuming the letters in PEMDAS indicate a strict first-to-last priority—so multiplication always precedes division and addition always precedes subtraction—when in fact multiplication/division and addition/subtraction are equal-precedence operations done left to right; teachers report this misunderstanding causes errors on expressions like 8 ÷ 2 × 4 and 2 + 3 − 1 [1] [5] [2]. Classroom guides and blogs stress that emphasizing the left-to-right rule corrects many mistakes and that alternative acronyms or rewordings can reduce confusion by pairing operations instead of ordering letters [6] [7].
2. Common mistake: Ignoring grouping symbols and signs (parentheses and leading minus signs)
Students often neglect nested grouping symbols or mishandle a minus sign in front of parentheses—failing to invert signs when distributing a leading negative or misreading nested brackets—leading to systematic errors in multi-step problems; teachers counter this by drilling parentheses-first practice and explicit sign-distribution lessons because the oversight is a frequent source of wrong answers [8] [3]. Resources designed for classroom use include task cards and worked examples that force students to unpack nested symbols slowly and check sign changes at every step [3] [9].
3. Common mistake: Doing operations “in the order they look easiest” or prematurely simplifying
Many learners try to perform friendly operations first (e.g., add small numbers before following precedence) or replace sub-expressions with equivalent standalone forms without considering context, which violates order rules and yields incorrect results; expert commentaries warn against substituting expressions as if they were independent and recommend stepwise rewriting tied to the convention [10]. Teachers correct this by modeling the correct rewrite-and-evaluate sequence and by designing activities that require students to justify each transformation to expose false equivalences [10] [6].
4. Why mistakes persist: misconceptions, oversimplified mnemonics, and lack of measurement
Empirical work finds that misconceptions produced by oversimplified mnemonics are robust and can persist into adulthood unless specifically measured and addressed; a recent study developed instruments to quantify how people interpret order of operations and showed reminders alone don’t always change deep misunderstandings around associativity and precedence [4]. This implies teachers need diagnostic assessments and targeted interventions—error analysis tasks, spaced practice, and associativity shortcuts taught explicitly—rather than assuming a one-time mnemonic will suffice [9] [4].
5. How teachers correct the errors: concrete strategies and alternative framings
Successful corrective practices reported across teacher resources include: replacing or reframing PEMDAS to highlight paired operations, using error-identification activities where students find and fix mistakes, slow stepwise modeling with nested-symbol practice, and targeted assessments that reveal persistent misconceptions; these strategies are recommended because they move learners from memorization to reasoning and because curated materials (task cards, worksheets, interactive problems) scaffold understanding [2] [9] [3] [5]. Alternative viewpoints exist—some educators prefer GEMS or GEMA acronyms or insist on strict PEMDAS—but the consensus in instructional materials is that left-to-right associativity and careful attention to grouping and signs are the critical fixes [6] [11].