PEMDAS and other lies :)

We use ‘correct answers’ as a visible indication of knowledge.  If the learning environment focuses on homework systems, correct answers may be the only measure used.  However, even when we ‘look at the work’, we may confuse following a procedure with knowing what to do.

PEMDAS may be the most commonly used tool in the teaching of mathematics experienced by our students.  I have seen PEMDAS written on work papers and notebooks; I have heard students say PEMDAS when explaining how to ‘do a problem’ … and I’ve heard instructors say that PEMDAS describes what to do with an expression.

The problem is that PEMDAS is a lie.  PEMDAS only provides a memory tool (a mnemonic) for steps that might apply to some expressions in some situations.  Previously, I have written about the issues with the “P” (parentheses) component of this tool (see https://www.devmathrevival.net/?p=301).  Today, I am thinking about some of the ways in which PEMDAS is false or incomplete.

Take a simple expression like -4².  PEMDAS does not give any interpretation of this expression.  The issue here is that the memory aid only deals with exponents and the 4 binary operations; the negation (opposite) involved here is outside of the rule.  If we established mathematical truth based on an agreement among students passing a course, the truth would be at risk on this expression — whether “16” or “-16” would win a majority would vary by semester.

PEMDAS is incomplete about operations in general, such as the negation above … or absolute value.  Given the visual similarity with parentheses, most students see that the ‘inside’ of an absolute value is simplified first.  However, what to do with an expression like  3|x – 2|?  Is there a choice to distribute?  As we know, and students are confused about, the order of operations provides one possible procedure … properties of numbers and expressions completes the story, and these properties are more important in mathematics.  Getting the correct answer to 8 + 5(2) in a pre-algebra course has nothing to do with being ready to succeed in algebra, or math in general.  Basic expressions like 8 + 5x are a challenge for many students, partially due to how strong the PEMDAS link is.

Another example:  what does PEMDAS tell us about mixed numbers?  This is a special case of the ‘parentheses problem’, where there is no symbol of grouping.  Fractions, in general, are an area of weakness.  We tell students that “you need a common denominator” or “cross multiply” — both of which appear to violate PEMDAS (we would divide left to right).  Properties are the important thing here as well; adding requires similar objects.  We focus so much on correct answers and perhaps ‘correct steps’ that we miss opportunities to address the mathematics behind the visible work.

The meaning of an expression with mixed operations is based on the priority of each operation; mathematically, the level of abstraction of an operation determines the priority.  Multiplying is abstracted from the concept of repeated adding, so multiplying carries a higher priority; exponentiation is abstracted from the concept of repeated multiplying, and has a higher priority.  Lowest abstractions are the basic concepts — add, subtract, negation.  For those of you involved with programming, this approach should sound familiar — computing environments are based on a detailed list of these levels of abstractions.  In mathematics, our world is defined by properties which provide necessary choices for types of expressions where equivalent forms can be created without using the prioritization.

The big lie in PEMDAS is that those 6 words say something important about mathematics.  Those 6 words do not say anything important about mathematics, only about an oversimplification that produces some correct answers to some expressions without understanding the mathematics.  Properties and relationships are the important building blocks of mathematics; a student starting from PEMDAS has to unlearn that material before understanding mathematics.   If our goal is to have students compute correct answers for any expression, then we would never use PEMDAS — it is woefully incomplete, and we would need the prioritization list like a computer program uses.  If our goal is to have students understand mathematics, we would deal with the concepts that determine the order along with the properties that provide choices; a focus would be on the correct reading and interpretation of expressions.

Do your students a favor; avoid using PEMDAS.  Use mathematics instead.

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