The most common course for me to teach is ‘intermediate algebra’, and I’ve been thinking of the many issues with that course as part of the college curriculum. However, my interest today is in poking at PEMDAS … and the poor way we often teach the order of operations. As you know, understanding the order of operations concept is one key part of understanding basic algebraic notation.
An easy poke at PEMDAS is the “P” (parentheses for us, ‘B’ bracket in some other countries). The problem below is actually from our beginning algebra curriculum:
Operator precedence usually places products and quotients at the same level, with the normal parsing from left to right (answer: 8). Of course this ‘tie breaking’ rule is arbitrary; however, a convention about this is necessary for all machine calculation … and our students interact with these machines.
I’ve seen people say that this is a silly point, without merit … and they suggest including sufficient grouping to avoid any “ambiguity” from the expression. I’ve also seen people say that there is no such thing as implicit multiplication (as in the problem above, or as in an algebraic term like -3x). What they mean is that implicit multiplication has the same priority as explicit multiplication; some programming environments do not allow implicit products in order to avoid issues with that precedence.
If we state the problem algebraically, it might be:
16÷4k, where k=2
We, of course, prefer fraction notation for quotients due to the ‘confusion’ created by the divided by symbol (which our students write as a slash):
One discussion site has a comment that we should use those grouping symbols to be clear, and concludes with a comment that the answer changes when we use algebraic notation for the same quotient & product expression. (see http://math.stackexchange.com/questions/33215/what-is-48%C3%B7293 ) This ‘changing answer’ feature should bother all of us!
In the original problem above, the product involves parentheses … so our PEMDAS-based students always calculate that product first. They have no idea that there is an issue with implied products when variables are involved; I’m okay with that at the time (we get to it later). In all of my years of reviewing missed problems like that one, I’ve never heard a student justify their answer by ‘implied products have a higher priority’. They always say “parentheses first”.
If we could say “GEMDAS” (for “grouping”) we would be more honest. I’m not sure what “G” means for my poor aunt Sally … but, then, having a sentence for an mnemonic with no connected meaning is likely to be a bad thing. When we continually talk about ‘remember my dear aunt Sally’, we encourage students to process information at the lowest possible level — instead of a beginning understanding, all they get is a memorized rule which is fundamentally flawed.
The role of mnemonics in ‘remembering’ has been studied. The book Cognitive Psychology and Instruction, 4th edition Bruning et al has a review of research on this on pages 72-73 (it’s also in their 5th edition though I don’t have that page reference). The basic conclusion was that mnemonics help students remember when mnemonics help students remember … and can interfere with remembering when the student does not find them helpful. That means the some students can use them to remember, some students get confused … and (in my view) all students have negative consequences for using poor aunt sally.
I think the emphasis on PEMDAS also creates a mental ‘twist’ in our students’ minds. They take expressions which do not have stated grouping and insert parentheses so that the basic meaning is changed:
5x² is mistakenly processed as (5x)²
In the intermediate algebra course, some strange things happen relative to parentheses.
(3x² – 5) + (4x + 3) is treated as a product
A good portion of my class time is spent on un-learning PEMDAS and building some understanding of notation with order of operations. The biggest problem … grouping that is done with other symbols besides parentheses (fraction bars, radical symbols, absolute value, etc).
Because I’ve been teaching so long, I’m occasionally asked about any changes I notice. Folks expect me to report that students are less prepared now compared to 30 or 40 years ago. Actually, there have been improvements in the mathematics preparation of our students. However, these improvements are not uniformly distributed both in terms of students and in terms of mathematics. In particular, students struggle more now with order of operations; some of that degradation seems to be due to the over-use of PEMDAS.
We should avoid books that build in PEMDAS, and we should avoid the mnemonic in our classes. Understanding something is much better than memorizing an erroneous rule.
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