In America they say "PEMDAS" and the British often use "BODMAS" and some decry any mnemonics at all, but everyone teaches essentially the same rules of evaluation.
Consider the poor Algebra I or II student, who has seen and heard the following in his first few weeks of school:
"Multiplication and Division are handled as equals, from left to right."
"When we say 6a we mean 6 times a, where a just represents any number that we might choose to be use in place of it"
And then later we write on the board 6a2 / 3a = ?
and so the student says, "I get it, that means 6 times a2 then I divide by 3 and multiply by a. It must be 3a3."
Then, wanting to follow instructions, he types the expression into his Ti-84 + silver edition calculator after storing a value for x, and sure enough, he gets the value of 3x 3.
But of course, you explain, that's wrong. when we divide like this we mean that the 3a is intended to be a monomial term, taken as a single unit.
Your student is compliant, so he/she just nods and murmurs "Ok" and as you walk back to the front of the room, turns to their neighbor and says "You understand that?" and gets a shrug and a side to side head-shake... all just part of the mystery of mathematics.
If you realize what is confusing them, you may become very conscientious about writing problems with the fractions written out with horizontal fraction bars $\frac{6x^{2}}{2x}$ and talk about "implied parentheses". Unfortunatly many of your students have only heard the term "implied" used in prejudicial situations;"He implied I had stolen it!" or similar, and really have no idea what you mean. So maybe you get REALLY conscientious and write every fractional expression using actual parentheses $\frac{(6x^{2})}{(2x)}$.... and then that night on the homework, the book does NOT use them.
The repercussions of this order of operations confusion leads to students unsure of whether to write the linear term in an equation as $\frac{5x}{2}$, or 5$\frac{x}{2}$ or $\frac{5}{2}$ x.. but sometimes that even leads to $\frac{5}{2x}$ .
5 comments:
In typical Canadian fashion we (generally) combine the American and UK acronyms and teach 'BEDMAS.' Does anyone use "index" for exponents and teach PIDMAS or BIDMAS? :)
Pointing out these issues is really useful - it is unfortunate that some teachers hard-headedly insist when teaching this stuff that it is completely consistent and free from any contradictions. Far better to realize (and point out) that, just with natural language, context is important. The notation we use is a shorthand for what could be made completely rigorous - thanks to convention we can use our notation more freely. The struggle to learn these conventions is the struggle to attain fluency, a challenge in any language.
We (mathematicians) have tried to use visual cues, so that multiplication and division group more tightly on the page than plus and minus, and exponents more tightly yet.
And then we have stupid textbooks that use the bar and dots symbol for division in order of operations problems, which I have *never* seen otherwise.
Sue,
I don't think it will ever go away... in my web page notes I have "The symbol "÷" is called an obelus, and was first used for a division symbol around 1650. The invention is often credited to British Mathematician John Pell but I have also seen credit given to J H Zahn, Teutsche Algebra (1659). The colon, ":", was used as a fraction symbol, and later as a division symbol by Liebnitz around 1685 in much the same fashion as the obelus, "8:4=2". " and later in the same article, "Cajori remarks that De Morgan recommended the use of the / in 1843, and although he continued to use : in his subsequent works, his advice was taken up by Stokes from 1880 and several others. Some years later the National Committee on Mathematical Requirements (1923) opined, "Since neither ÷ nor :, as signs of division, plays any part in business life, it seems proper to consider only the needs of algebra, and to make more use of the fractional form and (where the meaning is clear) of the symbol /, and to drop the symbol ÷ in writing algebraic expressions." and yet here we are in the 21st century and it is still around.
Poor handwriting is another factor to consider. I don't know how many times I have pointed out to students that "1/2a" just doesn't cut it when what they really mean is "(1/2)a". They can't even read their own work from one line of a problem to the next.
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