What are common pitfalls when computing expressions with multiplication and addition?
Executive summary
Computing expressions that mix multiplication and addition commonly trips people up because of order-of-operations rules and misleading mnemonics; multiplication has precedence over addition, but multiplication/division and addition/subtraction are each evaluated left-to-right, which students often misapply [1] [2]. Errors typically come from skipping or misreading parentheses, treating the letters of PEMDAS/BODMAS literally, or rushing—problems well documented in education guides and research [3] [4] [5].
1. PEMDAS and what it actually means
Acronyms like PEMDAS, BODMAS or BEDMAS are memory aids, not literal step-by-step orders that put addition always before subtraction or multiplication always before division; the correct convention is parentheses first, then exponents, then multiplication and division (evaluated left-to-right), and finally addition and subtraction (evaluated left-to-right) [2] [1]. Teaching the letters without emphasizing the left-to-right parity creates predictable misconceptions—students can mistakenly do all multiplication before any division or all addition before any subtraction [4] [6].
2. Why multiplication before addition still causes mistakes
Because multiplication has higher precedence than addition, expressions like 2 + 3 × 4 must be evaluated as 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20, yet educational resources repeatedly flag that students add first out of habit or misreading the mnemonic [7] [8]. The problem is compounded when learners default to reading left-to-right for everything; that left-to-right intuition only applies within the tied groups (multiplication/division and addition/subtraction), not across precedence levels [9] [1].
3. Parentheses, exponents and the most costly oversights
Skipping or incompletely solving expressions inside parentheses, and overlooking exponents, are among the most common and consequential mistakes because they change which operations get priority—examples and worksheets routinely warn that neglecting these steps yields wrong answers [3] [10]. Calculators and software may handle some constructs differently (for instance some systems differ on exponent associativity), so blind trust in a device without understanding the rules can also mislead students [6].
4. Subtraction, negatives and the associative shortcut
Rewriting subtraction as addition of a negative (a − b = a + (−b)) is a useful algebraic tactic that clarifies associativity and can simplify computation, but it must be taught explicitly if students are to use it reliably; without that framing they may misorder terms and produce mistakes [11]. Similarly, unary operators like negation interact with multiplication and exponents in ways that often surprise learners unless addressed directly [11] [1].
5. Common classroom behaviors that produce errors
Rushing, wanting to “do one operation before another” based on intuition, and reverting to pure left-to-right evaluation inside parentheses are typical behaviors teachers report; these habits lead to predictable wrong answers and require explicit correction with examples and pacing strategies [5] [3]. Educational blogs and tutoring sites repeatedly recommend slowing down, annotating steps, and checking intermediate results to catch order-of-operations errors [9] [7].
6. Practical fixes and alternate framings
Practical remedies include enforcing the left-to-right rule within tied operation groups, using parentheses to force intended evaluation, teaching subtraction-as-addition-of-negative, and showing counterexamples that reveal mnemonic pitfalls—these approaches are advised across classroom guides and tutoring resources [1] [7] [11]. Because mnemonics can mislead, instructors and learners should pair them with explicit statements about associativity and software quirks so that the rule set is robust in both hand calculations and computational tools [4] [6].