The Genesis and Controversy of the Order of Operations: Unraveling PEMDAS
The order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), has been a cornerstone of mathematical education for decades. However, like many mathematical concepts, this rule did not emerge from a single inventor but evolved gradually over time. In this article, we delve into the history of the order of operations, its significance in modern mathematics, and the ongoing debate surrounding the educational tool PEMDAS.
The Evolution of the Order of Operations
Understanding the complexity and importance of operations in arithmetic and algebra became increasingly crucial as mathematical expressions grew in intricacy. The concept of order of operations was not the work of a lone inventor. Instead, it developed as a collective effort to clarify the sequence of calculations, especially with the introduction of algebraic expressions.
By the 19th century, educators and mathematicians began to formalize these rules to avoid ambiguity in mathematical expressions. Acronyms like PEMDAS, BIDMAS, or BODMAS became popular in the 20th century as mnemonic devices to teach these rules effectively. However, these tools are not without controversy, as some educators argue that they can perpetuate misconceptions about mathematical operations.
Understanding the Order of Operations
C.S., a math educator, offers a straightforward method to approach the order of operations:
Perform the operations inside any grouping symbols (parentheses, brackets, braces), starting with the innermost symbol. Perform all multiplications and divisions in the order they appear from left to right. Perform all additions and subtractions in the order they appear from left to right.This method emphasizes the inherent logic of the operations without the burden of an additional mnemonic. Jerome E. Kaufmann supports this approach in his book, Algebra with Trigonometry for College Students, where he outlines these fundamental steps for solving mathematical expressions.
The Controversy Surrounding PEMDAS
Despite its widespread use, PEMDAS has faced criticism from some educators. Thomas, a passionate math advocate, argues that the mnemonic could mislead students into treating addition and subtraction, and multiplication and division, as separate and distinct operations. He suggests that, from an early age, students should be taught that these operations are inverses of each other.
Thomas also criticizes the inclusion of parentheses in the order of operations, believing that the focus should be solely on the operations themselves. He proposes a revised acronym, ELMDAS (Exponents, Liquation [meaning operations], Multiplication and Division from left to right, Addition and Subtraction from left to right), to better reflect the inverse nature of these operations.
Thomas further argues that logarithms are often introduced without a proper foundation in inverse operations, a result of the limited understanding that PEMDAS can foster. He believes that teaching logarithms as the inverse of exponentials could simplify the introduction of this concept to younger students.
The Future of Order of Operations Instruction
The debate around the order of operations and PEMDAS underscores the ongoing need for educators to refine their teaching methods. By addressing the limitations of mnemonic devices and emphasizing the fundamental principles of mathematics, such as the inverse nature of operations, educators can better prepare students for more advanced mathematical concepts.
As the education system continues to evolve, it is essential to incorporate innovative teaching strategies that not only clarify the order of operations but also foster a deeper understanding of mathematical principles.
References:
Kaufmann, J.E. (1992). Algebra with Trigonometry for College Students (Third Edition). PWS-KENT Publishing Company, Boston, MA. Thomas, personal communication.