What are common mistakes students make when applying PEMDAS/BODMAS to mixed operations?
Executive summary
Students repeatedly trip over the equal-ranking of multiplication/division and addition/subtraction, and the ambiguous presentation of PEMDAS/BODMAS; these lead to left-to-right mistakes and sign errors that change answers (sources note left‑to‑right rule and common misinterpretations) [1] [2] [3]. Authorities and educators recommend treating division as multiplication by a reciprocal and subtraction as adding the opposite, and to use parentheses to remove ambiguity [1] [4].
1. Misreading the mnemonic as a strict sequence — “M before D” and “A before S”
Many pupils treat PEMDAS or BODMAS as a strict top‑to‑bottom checklist and therefore perform multiplication always before division (and addition before subtraction), producing wrong results when both appear; educators explicitly warn that multiplication and division are one tier to be done left to right, and the same for addition/subtraction [1] [2] [5].
2. Ignoring left‑to‑right evaluation: the silent order that matters
A frequent error is failing to evaluate multiplication/division or addition/subtraction from left to right. Several teaching guides and tutors list neglecting the left‑to‑right rule as a common mistake that alters final answers [2] [3] [6].
3. Not rewriting subtraction/division as inverse operations
Experts recommend converting subtraction into addition of the opposite and division into multiplication by a reciprocal to avoid procedural mistakes. This reframing removes the false hierarchy implied by PEMDAS and prevents students from automatically privileging one symbol over its inverse [1].
4. Implicit multiplication and ambiguous notation: where punctuation fails
Ambiguities like a ÷ bc or 3/4*5 create linguistic problems: different readers (and software) can interpret them differently unless parentheses are added. Mathematicians and pedagogues argue the safe remedy is to add brackets because the written form, not math, creates the confusion [4].
5. Overlooking signs inside parentheses and minus‑sign distribution errors
Students commonly forget to change signs when a negative sign precedes parentheses, leading to inverted operations inside the bracket. Classroom blogs flag “ignoring the minus sign” as a recurring error that stems from mechanical application of the mnemonic instead of following algebraic sign rules [7].
6. Social‑media puzzles and viral traps that weaponize notation
Viral problems (e.g., the 8 ÷ 2(2+2) debate) exploit notation and mnemonic misunderstandings. Commentators point out that PEMDAS alone is not always sufficient to resolve such puzzles without clearer notation or parentheses, and that calculators/softwares may interpret implicit multiplication differently [8] [4].
7. Pedagogical fixes recommended by multiple sources
Authors and tutors converge on practical mitigations: teach MD and AS as paired, left‑to‑right operations; encourage rewriting division/subtraction as multiplication/addition of inverses; and insist on explicit parentheses to remove ambiguity. Several guides advise visual strategies (underlining brackets/exponents) and habitually adding parentheses when writing expressions [1] [2] [9].
8. Where guidance diverges and what to watch for
Sources agree on the main mistakes, but differ on emphasis: some focus on mnemonic redesign (drop D and S from PEMDAS thinking), while others stress notation hygiene and consistent left‑to‑right rules enforced by worked examples. There is also divergence about how much to blame PEMDAS versus sloppy notation—Harvard commentary frames much of the problem as linguistic ambiguity and urges always using brackets [1] [4].
9. Practical checklist for students and teachers
Follow this compact checklist drawn from the reporting: (a) when M and D (or A and S) both appear, process left to right; (b) convert subtraction/division into inverses when helpful; (c) always add parentheses to make intent explicit; (d) double‑check signs when distributing a leading minus into brackets [1] [7] [4].
Limitations and sources: This analysis draws exclusively on the provided teaching guides, blogs and commentaries about PEMDAS/BODMAS; available sources do not mention controlled classroom trial data comparing mnemonic variants or long‑term learning outcomes for specific interventions. Citations: Harvard pedagogy page (ambiguity) [4]; The Math Doctors & teaching commentary [1]; kid‑guide and tutoring pages noting common misconceptions [2] [9]; sign/parentheses mistakes and blog posts [7] [10]; viral‑puzzle analysis [8]; BODMAS/BEDMAS explanatory resources and common errors [3] [6].