What can be even more confusing is that different countries apparently use different abbreviations. My students come to the university knowing PPMDAS (which stands for parentheses, powers, multiplication, division, addition, subtraction and which students claim to remember as Pretty please my dear aunt Sally)) or PEMDAS (parentheses, exponents ,multiplication, division, addition, subtraction also remembered as Please excuse my dear Aunt Sally). I personally never learned either until I started teaching at the university level. I often send my math ed students to the stores to look at calculators and to find one or more that does not use order of operations. Radio Shack tends to have quite an array of cute ones usually shaped as rulers or whatever. Some of these also report that anything divided by zero is zero although they may put a small E somewhere on the display to indicate error.
Mary Ann Matras Mathematics Department East Stroudsburg University East Stroudsburg, PA 18301 570-422-3440 firstname.lastname@example.org
-----Original Message----- From: Barry Kissane [mailto://email@example.com] Sent: Tuesday, February 06, 2001 9:41 AM To: firstname.lastname@example.org Subject: Re: [math-learn] Order of operations
"Order of Operations has only to do with various versions of conventional laziness when writing math expressions."
I agree that the premature dropping of the multiplication sign is an error, albeit a very common one. [Eg, in Access to Algebra, an Australian junior secondary school algebra textbook series (Years 7-10), we kept using it throughout Book 1 (of four books)] It seems that algebra texts typically drop the sign very early - I have even seem some that START by using expressions such as 2b and 5g, which is particularly problematic.
But I disagree that there is nothing mathematically important in order of operations. It is crucial that children realise that there is an issue at stake: written expressions are ambiguous and we need to decide on a way of removing the ambiguity or everyone will be confused. This is not an issue about calculators - at least not initially. Nor is there a lot of point in teaching students the "correct" way (ie the conventional way, agreed to by people), unless they have some idea that there are other reasonable ways of interpreting expressions and thus we need to decide which to accept. Common as it might be, teaching children to follow a rule blindly is hardly educational - and it certainly isn't mathematical - but we need to make sure our children understand that there is something at stake.
In response to Carol's question (which I don't have in front of me at the moment ...) about whether or not it is important for calculators to have the conventions built in, I think I'd argue that it is useful if they are built in, but it is also useful for student to ALSO see and use calculators for which this is not the case. (These are easy to find in many households and surprisingly many shops.) Otherwise, they will not see that there is a problem here that is worth solving (by using a convention).
Incidentally, mnemonics such as BODMAS (or variations on the theme, such as BIMDAS - and there are several others, some differing across national borders!) do not entirely resolve problems. The actual mathematical conventions are that multiplication and division are at a higher level than addition and subtraction, and thus are performed first. But the very fact that there is a BODMAS and a BIMDAS makes clear that there is no intended order for these two: they are performed in sequence from left to right. The same is true for addition and subtraction (although I haven't seen a BODMSA or BIMDSA, perhaps because they are too hard to pronounce!).
Thus, although a strict reading of BODMAS would suggest that 15 - 4 + 2 = 9 (doing the A before the S), in fact we by convention interpret this as 13 (doing the S before the A because it appears first reading left to right). [BTW, is this a problem of a kind for children from cultural groups whose language (eg Arabic) is written right to left??] Small wonder that kids trying to follow a rule blindly get confused!
By happy chance, aspects of this matter is discussed in the most recent edition of the wonderful journal (Mathematics Teaching, the journal of the Association of Teachers of Mathematics in the UK), which arrived in my mailbox only yesterday. [IMHO, there is no better journal for teachers of mathematics anywhere in the English-speaking world.] Ruth Forrester & John Searl describe their adventures with Year 7's in a short article entitled 'Uncle BODMAS and friends' (Number 173, December 2000, pp 34-5). They conclude with the following:
"Thank you Primary Seven, you showed us how important calculators are in understanding basic arithmetic. We would not have thought properly about the non-associativity problem of - and ÃÂ· without your help."
Worth a read.
Barry Kissane Senior Lecturer, The Australian Institute of Education Murdoch University, Murdoch, Western Australia 6150 Telephone +618-9360-2677 (International) 08-9360-2677 (within Australia) Home Telephone +618-9474-3278 (International) 08-9474-3278 (within Australia) Facsimile +618-9360-6296 (International) 08-9360-6296 (within Australia) Internet: http://wwwstaff.murdoch.edu.au/~kissane
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