Editing Order of operations (section)

==Mnemonics==
{{anchor|PEMDAS|PEDMAS|BEDMAS|BEMDAS|BODMAS|BIDMAS|BIMDAS|BOMDAS|GEMDAS|BERDMAS|PERDMAS|BPODMAS}}
[[Mnemonic]] [[Acronym|acronyms]] are often taught in primary schools to help students remember the order of operations.{{r|Mathcentre}}{{r|Ginsburg}} The acronym '''''PEMDAS''''', which stands for '''P'''arentheses, '''E'''xponents, '''M'''ultiplication/'''D'''ivision, '''A'''ddition/'''S'''ubtraction,{{r|Vanderbeek}} is common in the [[Mathematics education in the United States|United States]]{{r|ASAT}} and France.{{r|Micmaths}} Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally".{{r|Ball}} The United Kingdom and other [[Commonwealth of Nations|Commonwealth]] countries may use '''''BODMAS''''' (or sometimes '''''BOMDAS'''''), standing for '''B'''rackets, '''O'''f, '''D'''ivision/'''M'''ultiplication, '''A'''ddition/'''S'''ubtraction, with "of" meaning fraction multiplication.{{r|Davies|Knight}} Sometimes the '''O''' is instead expanded as '''O'''rder, meaning exponent or root,{{r|Knight|NSW syllabus}} or replaced by '''I''' for '''I'''ndices in the alternative mnemonic '''''BIDMAS'''''.{{r|Knight|Foster}} In Canada and New Zealand '''''BEDMAS''''' is common.{{r|Naddor}}

{{anchor|Punkt vor Strich}}In Germany, the convention is simply taught as {{lang|de|[[:de:Punktrechnung vor Strichrechnung|Punktrechnung vor Strichrechnung]]}}, "dot operations before line operations" referring to the graphical shapes of the taught operator signs {{unichar|B7|note=multiplication}}, {{unichar|2236|note=division}}, and {{unichar|2B|note=addition}}, {{unichar|2212|note=subtraction}}.

These mnemonics may be misleading when written this way.{{r|Ball}} For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression{{r|Ball}} <math>a - b + c</math> as <math>a-(b+c)</math>, while the correct evaluation is <math>(a - b) + c</math>. These values are different when <math>c\ne 0</math>.

Mnemonic acronyms have been criticized for not developing a conceptual understanding of the order of operations, and not addressing student questions about its purpose or flexibility.{{r|Ameis}}{{r|Cheng}} Students learning the order of operations via mnemonic acronyms routinely make mistakes,{{r|LLT}} as do some pre-service teachers.{{r|Dupree}} Even when students correctly learn the acronym, a disproportionate focus on memorization of trivia crowds out substantive mathematical content.{{r|Wu}} The acronym's procedural application does not match experts' intuitive understanding of mathematical notation: mathematical notation indicates groupings in ways other than parentheses or brackets and a mathematical expression is a [[binary expression tree|tree-like hierarchy]] rather than a linearly "ordered" structure; furthermore, there is no single order by which mathematical expressions must be simplified or evaluated and no universal canonical simplification for any particular expression, and experts fluently apply valid transformations and substitutions in whatever order is convenient, so learning a rigid procedure can lead students to a misleading and limiting understanding of mathematical notation.{{r|Taff}}