PEMDAS may be a good mechanism for teaching grade school students but it has serious drawbacks especially when moving on to more complex expressions. One teacher, Jason Taff at the Burroughs School in St. Louis points out some of the shortcomings of the mnemonic and has come up with a method for teaching the order of operations based on identifying terms and factors (mathematics Teacher, Vol III, No 2, Oct 2017). Let me quote from that article, "PEMDAS fares poorly because it never was an order of operations to begin with. It has always been a hierarchy of operations. PEMDA tells us what comes first or second or third, when the more relevant concept involves which operations more closely bind numbers and expressions into their nested structure..." So here is where the real foundational differences appear between the sides with those seeing implied multiplication and juxtaposition of variables as more closely binding than explicit multiplication. The one proponents see the sub-expression 2(1+2) as a product in itself with a unique value as if it was surrounded by another set of grouping symbols. Why is this? Let's examine a few areas where grouping is implied though not shown overtly. Take for example the expression 8÷2^3 (8 divided by 2 cubed). One could not simply write as a step to calculate this 8÷2*2*2 since PEMDAS (using the strict left-to-right convention) would evaluate this as 16. So, the indicated exponentiation has to be enclosed in parentheses. Also, take 6÷3! (6 divided by 3 factorial). Again, one could not simply write 6÷3*2*1 as PEMDAS would solve this incorrectly. So, again, parentheses are implied. Now, let's talk about juxtaposition and implied multiplication. My college text, written by a Ph.D. Professor, clearly states that a÷bc is a÷(bc) or a over bc but not (a÷b)*c. For one thing, bc should be seen as a product before division takes place since we are allowed to use commutation and the associative properties of multiplication (in this case variables) to move things around so that bc=cb, and substituting this in the expression would result in different PEEMDAS solutions. that is problematic. Also, what about implied multiplication especially when it comes to factored expressions? Do we now just forget about the distributive property? After all, one could make the case that 2(1+2) is just the factored form of (2+4). Or in its expanded form, (2*1+2*2) using distribution. In each case, the viral expression would resolve to 1. Yes, not every expression showing implied multiplication is a factored version of some group, but even if not, do they not have a unique value in that the multiplier outside the group is the coefficient of that group (or in some cases as factoring also is a GCF). Further, many assert that the P in PEMDAS simply means doing the operation in the grouping. So, in the sub-expression, 2(1+2) they simply add (1+2) and assert that the group is resolved. One could take issue with that especially if it is a factored expression and the GCF is essential in computing the original value of the expression. One could go on about simplifying expressions especially when variables are involved, but let's leave things here and hopefully get some enlightening comments.
PEMDAS may be a good mechanism for teaching grade school students but it has serious drawbacks especially when moving on to more complex expressions. One teacher, Jason Taff at the Burroughs School in St. Louis points out some of the shortcomings of the mnemonic and has come up with a method for teaching the order of operations based on identifying terms and factors (mathematics Teacher, Vol III, No 2, Oct 2017). Let me quote from that article, "PEMDAS fares poorly because it never was an order of operations to begin with. It has always been a hierarchy of operations. PEMDA tells us what comes first or second or third, when the more relevant concept involves which operations more closely bind numbers and expressions into their nested structure..." So here is where the real foundational differences appear between the sides with those seeing implied multiplication and juxtaposition of variables as more closely binding than explicit multiplication. The one proponents see the sub-expression 2(1+2) as a product in itself with a unique value as if it was surrounded by another set of grouping symbols. Why is this? Let's examine a few areas where grouping is implied though not shown overtly. Take for example the expression 8÷2^3 (8 divided by 2 cubed). One could not simply write as a step to calculate this 8÷2*2*2 since PEMDAS (using the strict left-to-right convention) would evaluate this as 16. So, the indicated exponentiation has to be enclosed in parentheses. Also, take 6÷3! (6 divided by 3 factorial). Again, one could not simply write 6÷3*2*1 as PEMDAS would solve this incorrectly. So, again, parentheses are implied. Now, let's talk about juxtaposition and implied multiplication. My college text, written by a Ph.D. Professor, clearly states that a÷bc is a÷(bc) or a over bc but not (a÷b)*c. For one thing, bc should be seen as a product before division takes place since we are allowed to use commutation and the associative properties of multiplication (in this case variables) to move things around so that bc=cb, and substituting this in the expression would result in different PEEMDAS solutions. that is problematic. Also, what about implied multiplication especially when it comes to factored expressions? Do we now just forget about the distributive property? After all, one could make the case that 2(1+2) is just the factored form of (2+4). Or in its expanded form, (2*1+2*2) using distribution. In each case, the viral expression would resolve to 1. Yes, not every expression showing implied multiplication is a factored version of some group, but even if not, do they not have a unique value in that the multiplier outside the group is the coefficient of that group (or in some cases as factoring also is a GCF). Further, many assert that the P in PEMDAS simply means doing the operation in the grouping. So, in the sub-expression, 2(1+2) they simply add (1+2) and assert that the group is resolved. One could take issue with that especially if it is a factored expression and the GCF is essential in computing the original value of the expression. One could go on about simplifying expressions especially when variables are involved, but let's leave things here and hopefully get some enlightening comments.