What common mistakes occur when people ignore order of operations?
Executive summary
Ignoring the order of operations commonly produces wrong numerical results such as turning 2 + 3 × 4 into 20 instead of 14, or 3 + 5 × 2 into 16 instead of 13; guides and tutors consistently flag “doing addition before multiplication” and misreading PEMDAS/BODMAS as top-to-bottom rules as the chief errors [1] [2] [3]. Pedagogical pieces and research note recurring confusions: treating M before D (or A before S) as absolute instead of left-to-right associativity, neglecting exponents and nested grouping, and over-relying on mnemonics that mask conceptual understanding [4] [5] [3] [6].
1. “Add first” — the single most visible mistake
Students frequently perform addition before multiplication in mixed expressions, producing plainly wrong answers (for example computing 2 + 3 × 4 as (2+3)×4 = 20 rather than 2 + (3×4) = 14). Multiple teaching sites and tutoring services list this as the most common concrete error and use examples like 3 + 5 × 2 to demonstrate the consequence [1] [2] [3].
2. Misreading PEMDAS/BODMAS as strict linear order
Many learners treat acronyms such as PEMDAS or BODMAS as a rigid sequence—“do everything labeled earlier before anything later”—which leads to the false idea that multiplication always comes before division, or addition always before subtraction. Educators warn that multiplication and division share the same priority and must be resolved left to right; the same holds for addition and subtraction [4] [3] [5].
3. Left-to-right associativity errors: M vs D, A vs S
A subtle but frequent mistake is ignoring the left-to-right rule for operations of equal precedence. Pupils who think “M before D” or “A before S” introduce errors when an expression requires division before multiplication or subtraction before addition simply because of its position. Teachers explicitly emphasize the left-to-right approach to fix this [4] [7] [5].
4. Exponents and nested grouping get overlooked
Exponents (orders) and nested parentheses are common stumbling blocks. Guides point out that learners sometimes skip or misplace exponents—eg., failing to square before multiplying—or mishandle multiple levels of brackets, producing major miscalculations in multi-step expressions [1] [8] [6].
5. Mnemonics help but can hide conceptual understanding
Wikipedia and educational commentators note that mnemonic acronyms are useful memory aids but can discourage deeper understanding of why precedence exists; reliance on them correlates with persistent mistakes among students and even some teachers [6]. Research into misconceptions underscores that remembering the letters is not the same as grasping associativity and notation ambiguity [9].
6. Notation and calculator/program differences create traps
The formal rules aren’t uniform across all tools and notations. For example, exponent associativity varies between systems—some compute a^b^c as (a^b)^c while others parse it as a^(b^c)—so blindly applying a mental rule can lead to different numeric outcomes depending on the environment. This ambiguity explains why precise notation and parentheses matter [6].
7. Classroom strategies to reduce these errors
Teachers and tutors recommend emphasizing left-to-right processing for equal-precedence operations, multiple worked examples, deliberate practice with nested symbols, and teaching subtraction as addition of negatives to clarify associativity. Task cards and “find-the-mistake” activities are common classroom interventions [7] [10] [3].
8. Limits of current reporting and where sources disagree
Sources agree on the basic catalog of errors—addition-before-multiplication, misusing PEMDAS, neglecting exponents and grouping—but differ in emphasis: some stress mnemonic misinterpretation [6], others emphasize classroom strategies and pacing [7]. Available sources do not mention whether particular age groups or demographic cohorts make these errors more often, nor do they provide large-scale prevalence numbers beyond pedagogical anecdotes [4] [9].
9. Practical takeaways for learners and teachers
When in doubt, insert parentheses to make intent explicit; do exponents and bracketed operations first; resolve multiplication and division left to right; then addition and subtraction left to right. Use worked examples that flip conventional traps (e.g., expressions where division precedes multiplication) to expose misconceptions, as recommended by tutors and math-education writers [7] [1] [3].