“The order of operations is an iconic mathematics topic that seems untouchable by time, reform, or mathematical discoveries,” say Jennifer Bay-Williams (University of Louisville) and Sherri Martinie (Kansas State University) in this article in Teaching Children Mathematics. Yet there are a number of myths and misconceptions that can throw students off track:
• Myth #1: The order of operations was arbitrarily designed long ago. On the contrary, the order in which calculations are made has a firm mathematical basis in any era, say Bay-Williams and Martinie. Students should understand why multiplication precedes addition in expressions like this one: 4 + 3 x 5. If the four is added to the three before multiplication, the answer is quite different – and incorrect.
• Myth #2: The order of operations must always be the same. “Teaching the order of operations as a rigid set of rules is mathematically misguided and misses the opportunity to consider when we can and cannot apply the properties of the operations and preserve equivalence,” say Bay-Williams and Martinie. In the expression 53+ 4 x 16 + 24 x 4, the simplifying and multiplication can be done in any order as long as they are done before the addition. “In this example,” say Bay-Williams and Martinie, “mathematically proficient students should first look at the problem holistically and decide how to most efficiently find the solution, applying what they know about the properties of the operations and the order of operations.”
• Myth #3: The order of operations cannot be taught conceptually. “Our goal to help students become mathematically proficient requires that we try to make the connection among concepts, procedures, and facts,” say Bay-Williams and Martinie. By using real-world problems, students can understand the underpinnings of order of operations and not just follow procedural steps. For example, a sixth-grade teacher has students read Two of Everything (Hong, 1993), in which Mr. Haktak discovers a large magic pot that doubles everything that goes into it. The teacher then poses the problem 8 + 3 x 5 + 7 and says, “The Haktaks have one stack of eight coins, three stacks of five coins, and one stack of seven coins. Tell me how many coins they have.” By trial and error, students figure out that they need to do the multiplication first, but it doesn’t matter in what order they do the addition.
• Myth #4: Order of operations is best taught using memory triggers. The problem with the popular Please Excuse My Dear Aunt Sally and PEMDAS/PEDMAS prompts is that they can lead to confusion because they imply that there are six steps and that multiplication precedes division and addition precedes subtraction. In the expression 45 ÷ 5 x 9, if the multiplication is done first, the answer will be wrong. Better for students to learn order of operations conceptually and then use a simple visual representation of the hierarchy of operations in a pyramid: Exponents on top, then division and multiplication (solved left to right), then addition and subtraction (solved left to right).
• Myth #5: Four operation steps are in the order of operations. The order is often listed as Parenthesis, Exponents, Multiplication and Division, and Addition and Subtraction.
“Parentheses are grouping symbols, not operation symbols,” say Bay-Williams and Martinie. “Therefore, there are only three operation steps. This suggests that the pyramid visual described just above is the best teaching tool since it clarifies that multiplication and division are at the same level in the order of operations. For elementary students, the space for exponents at the top of the pyramid can be left blank.
When teaching order of operations in the middle grades, say Bay-Williams and Martinie, it’s interesting to note that there are differences between countries. For example, in Kenya, students learn that division comes before multiplication. In the expression 100 x 20 ÷ 5, Americans do the multiplication first, Kenyans do it second – and the answer is the same! In the U.K. and other English-speaking countries, the acronym is often BEDMAS – Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
Bay-Williams and Martinie conclude that “we must teach the order of operations through meaningful tasks that use context (e.g., stacking coins) and engage students in problems that are focused on finding equivalent expressions (like comparing the Kenya explanation to the U.S. explanation for the order of multiplication and division). This approach is much more likely to help students become flexible, accurate, and efficient in simplifying expressions – in other words, procedurally fluent.”
“Order of Operations: The Myth and the Math” by Jennifer Bay-Williams and Sherri Martinie in Teaching Children Mathematics, August 2015 (Vol. 22, #1, p. 20-27), available for purchase at http://bit.ly/1IYPyNk; Bay-Williams can be reached at j.baywilliams@louisville.edu.
Five Myths About Teaching Order of Operations
by Michael Keany
Sep 21, 2015
Five Myths About Teaching Order of Operations
“The order of operations is an iconic mathematics topic that seems untouchable by time, reform, or mathematical discoveries,” say Jennifer Bay-Williams (University of Louisville) and Sherri Martinie (Kansas State University) in this article in Teaching Children Mathematics. Yet there are a number of myths and misconceptions that can throw students off track:
• Myth #1: The order of operations was arbitrarily designed long ago. On the contrary, the order in which calculations are made has a firm mathematical basis in any era, say Bay-Williams and Martinie. Students should understand why multiplication precedes addition in expressions like this one: 4 + 3 x 5. If the four is added to the three before multiplication, the answer is quite different – and incorrect.
• Myth #2: The order of operations must always be the same. “Teaching the order of operations as a rigid set of rules is mathematically misguided and misses the opportunity to consider when we can and cannot apply the properties of the operations and preserve equivalence,” say Bay-Williams and Martinie. In the expression 53 + 4 x 16 + 24 x 4, the simplifying and multiplication can be done in any order as long as they are done before the addition. “In this example,” say Bay-Williams and Martinie, “mathematically proficient students should first look at the problem holistically and decide how to most efficiently find the solution, applying what they know about the properties of the operations and the order of operations.”
• Myth #3: The order of operations cannot be taught conceptually. “Our goal to help students become mathematically proficient requires that we try to make the connection among concepts, procedures, and facts,” say Bay-Williams and Martinie. By using real-world problems, students can understand the underpinnings of order of operations and not just follow procedural steps. For example, a sixth-grade teacher has students read Two of Everything (Hong, 1993), in which Mr. Haktak discovers a large magic pot that doubles everything that goes into it. The teacher then poses the problem 8 + 3 x 5 + 7 and says, “The Haktaks have one stack of eight coins, three stacks of five coins, and one stack of seven coins. Tell me how many coins they have.” By trial and error, students figure out that they need to do the multiplication first, but it doesn’t matter in what order they do the addition.
• Myth #4: Order of operations is best taught using memory triggers. The problem with the popular Please Excuse My Dear Aunt Sally and PEMDAS/PEDMAS prompts is that they can lead to confusion because they imply that there are six steps and that multiplication precedes division and addition precedes subtraction. In the expression 45 ÷ 5 x 9, if the multiplication is done first, the answer will be wrong. Better for students to learn order of operations conceptually and then use a simple visual representation of the hierarchy of operations in a pyramid: Exponents on top, then division and multiplication (solved left to right), then addition and subtraction (solved left to right).
• Myth #5: Four operation steps are in the order of operations. The order is often listed as Parenthesis, Exponents, Multiplication and Division, and Addition and Subtraction.
“Parentheses are grouping symbols, not operation symbols,” say Bay-Williams and Martinie. “Therefore, there are only three operation steps. This suggests that the pyramid visual described just above is the best teaching tool since it clarifies that multiplication and division are at the same level in the order of operations. For elementary students, the space for exponents at the top of the pyramid can be left blank.
When teaching order of operations in the middle grades, say Bay-Williams and Martinie, it’s interesting to note that there are differences between countries. For example, in Kenya, students learn that division comes before multiplication. In the expression 100 x 20 ÷ 5, Americans do the multiplication first, Kenyans do it second – and the answer is the same! In the U.K. and other English-speaking countries, the acronym is often BEDMAS – Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
Bay-Williams and Martinie conclude that “we must teach the order of operations through meaningful tasks that use context (e.g., stacking coins) and engage students in problems that are focused on finding equivalent expressions (like comparing the Kenya explanation to the U.S. explanation for the order of multiplication and division). This approach is much more likely to help students become flexible, accurate, and efficient in simplifying expressions – in other words, procedurally fluent.”
“Order of Operations: The Myth and the Math” by Jennifer Bay-Williams and Sherri Martinie in Teaching Children Mathematics, August 2015 (Vol. 22, #1, p. 20-27), available for purchase at http://bit.ly/1IYPyNk; Bay-Williams can be reached at j.baywilliams@louisville.edu.
From the Marshall Memo #599