Chapter 1: The Essence of Mathematical Conventions

In the realm of mathematics education, clarity is key. It’s vital to eliminate unnecessary jargon when instructing young learners.

This piece is inspired by Bella L and her insightful articles on math education. Recently, she highlighted a crucial aspect of teaching mathematics: the sequence in which operations are performed in expressions, commonly known as the "order of operations." Specifically, multiplication and division share equal precedence, which is higher than that of addition and subtraction, which also share equal precedence. Parentheses and function arguments, such as those in the notation y=f(x), take precedence above all.

To visualize this, one might consider that parentheses and function arguments indicate the beginnings and ends of distinct mathematical expressions, which should be resolved first. Following that, multiplication and division are addressed, with addition and subtraction tackled last. The reason multiplication and division, as well as addition and subtraction, are of equal priority lies in their inverse relationship: division is the inverse of multiplication, while subtraction is the inverse of addition. However, inverses are not commutative, unlike multiplication and addition.

Through careful reasoning, this noncommutativity becomes clearer: when viewing subtraction as an inverse operation, the expression is not symmetrical. It matters whether we write (—a) + b or a + (—b). Similarly, with division, the order of inverting matters, hence x*y⁻¹ ≠ y/x.

Bella encapsulates these priority rules with the acronym PEMDAS—Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. In my youth, we referred to it as "BODMAS," a term I’ll elaborate on shortly.

As someone who has engaged with mathematics and physics throughout my career, I often overlook how bewildering these conventions can be for children. They are merely conventions—nothing more. If a child discovers the joy of understanding the "why" behind mathematics, they soon realize that conventions like PEMDAS lack intrinsic justification. They are constructs people created to foster a shared understanding.

Yet, it’s essential to explain that these conventions facilitate a common linguistic framework. Ultimately, there is as little "why" in PEMDAS as there is in the silent final -e in many English words. This is simply how language has evolved. The same can be observed in German, where the final -r is often ignored in pronunciation, but society doesn’t crumble due to such linguistic quirks. Conventions exist and can change over time.

However, unlike the silent letters in language, the PEMDAS convention addresses real ambiguity that must be resolved.

Bella’s recent article stirred memories of my own education, leading to the realization, "So that’s what they call it these days!" Fifty years ago, we used "BODMAS" or "BOMDAS," and chanted "Brackets, Orders, Division, Multiplication, Addition, Subtraction." The "O" now stands for "Operations," which is broader than "Exponents." My math teacher’s confusion led her to use "ordination," a term that occasionally appeared in older texts referring to exponentiation.

This confusion sparked amusing exchanges, especially since my uncle was an Anglican priest. I would often question how priests related to math, which perplexed my teacher and did little to enhance our rapport. My uncle would humorously clarify these terms, recalling his own experiences with them in the 1930s.

Reflecting on these experiences, I realize I eventually internalized the order of operations, likely around the age of 13 or 14, as I began exploring algebra. Here, multiplication often loses its symbol, appearing as mere juxtaposition or a dot, while division is expressed with a slash or as x^(-1).

This shift in notation reflects a more intimate relationship with the operations. During this period, I delved into John Fraleigh’s "A First Course in Abstract Algebra," where multiplication was positioned as the "fundamental" group operation. Yet, children typically view multiplication as repeated addition and exponentiation as repeated multiplication—a reverse learning order.

For the inquisitive, there's an infinite series of complex operations defined by Knuth's Up-Arrows. A single arrow, a↑b, denotes exponentiation (a^b). As the number of arrows increases, so does the complexity of operations, which can extend to roots and rational numbers. However, this complexity often exceeds human comprehension and is primarily relevant to computer science—Donald Knuth’s area of expertise.

For me, the relationship with calculators has been more tumultuous. My first was a Polish suffix calculator, requiring entry of operands followed by the operation. This design instinctively encourages proper nesting of operations in one’s mind. Without an equals sign, beginning a calculation necessitates adherence to order rules, which may explain the disappearance of bracket keys over time.

This experience solidified my understanding of operational order. My fondness for my old calculator, named after the Voyager space mission, stems from my admiration for NASA’s endeavors. Despite my disinterest in gadgets, I cherish this device, as I have never relied on calculators in my professional life. After my undergraduate exams, I bid farewell to them.

For any complex calculations, it’s essential to set them up in systems like Mathematica or as custom code, allowing for thorough testing and debugging, especially when stakes are high.

Two years ago, when my son mentioned "BODMAS," I was taken aback. I inquired if they clarified what "O" meant. To my astonishment, he believed it referred to "Oxponentiation" after his teacher's vague explanation. This prompted humorous thoughts about "oxponentiation" being related to ordaining priests at Oxford or perhaps a critique of convoluted math terminology.

My daughter, on the other hand, is taught to refer to it as AOOC—Algebraic Operation Order Convention. While clunky, it emphasizes that PEMDAS, BODMAS, AOOC, or whatever term is used, is merely a linguistic convention, not a mathematical principle.

This distinction is crucial. While it may seem arbitrary, conventions are necessary to ensure clear communication of concepts. As children learn these rules, understanding their conventional nature can alleviate confusion, underscoring the importance of agreement in resolving ambiguity.

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