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Topic: RE: [math-learn] Order of operations
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Mary Ann Matras

Posts: 12
Registered: 12/6/04
RE: [math-learn] Order of operations
Posted: Feb 6, 2001 10:43 AM
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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


Mary Ann Matras
Mathematics Department
East Stroudsburg University
East Stroudsburg, PA 18301
570-422-3440
mmatras@po-box.esu.edu


-----Original Message-----
From: Barry Kissane [mailto://kissane@central.murdoch.edu.au]
Sent: Tuesday, February 06, 2001 9:41 AM
To: math-learn@yahoogroups.com
Subject: Re: [math-learn] Order of operations


Hi all

Ladnor said:

"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 ÷ without your help."

Worth a read.

Regards



Barry

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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