Translating Word Sentences to Math Sentences

Date: 03/10/2004 at 15:37:52
From: Elizabeth
Subject: Translating from words to symbols

The sentence "Six more than three times x is four times two less than
x" was translated in our textbook to 6 + 3x = 4(x - 2).  Some of my
students felt it should be 6 + 3x = x - (4*2).  They cited order of
operations and said that they must do the multiplication first, then
the addition/subtraction, which makes sense.

Is there a true rule or law of mathematics or some definite tradition
as to the interpretation of sentences?  Of course I know that if it 
had been written "...four times the quantity two less than x" there 
would be no ambiguity.  My question is with the sentence as written.

Date: 03/11/2004 at 08:53:28
From: Doctor Peterson
Subject: Re: Translating from words to symbols

Hi, Elizabeth.

Although I would probably choose the officially "correct" answer over
the students' version, I think the statement as given is ambiguous. 
English (and any other human language) tends to be ambiguous, and to
depend on contextual clues to guide the hearer; the reason we use
symbols in math is that we want to eliminate the ambiguity of
language.  Especially, _written_ language is ambiguous; I'm sure you
can see how you can read the sentence aloud in different ways to imply
the different orders of operations you noted:

  four times ... two less than x

  four times two ... less than x

The problem is that the order of operations rules apply to algebra, 
not to English; and English grammar does not have definite rules that 
disambiguate this sentence, though there may be expectations that 
would lead us to favor one interpretation over the other.  (These are 
a broad sort of contextual cue, such as the fact that no one in his 
right mind would say "four times two" rather than "eight" in this 
situation.)

So although the author knew what he meant, and we, with experience of 
how math writers think, may tend to see it the same way, I don't 
think we can fault the students for seeing it differently.  In fact, 
if I were working on a real life project and were given those 
instructions, I would repeat it back with added words, or show the 
equation I'd written, in order to check that I understood it 
correctly.  We tend to forget that proper communication requires 
interaction, and there is nothing wrong with this sort of checking!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/