Avoiding Careless Mistakes

Date: 01/30/2002 at 03:04:15
From: David
Subject: How to avoid careless mistakes

How to avoid careless mistakes? I have tried do as many problems as 
possible, but mistakes are constantly made just because of 
carelessness!

I would REALLY appreciate your helping me solve this problem.

Date: 01/30/2002 at 11:10:05
From: Doctor Ian
Subject: Re: how to avoid careless mistakes

Hi David,

I used to have this problem, and in my case the first step toward 
getting over it was to read _The Gift of Fire_ and _Less Than Words 
Can Say_ by Richard Mitchell.  You can find them online here:

  The UnderGrammarian - Richard Mitchell
  http://www.sourcetext.com/grammarian/   

If you'd like to own them - and I highly recommend this, since it's a 
good idea to read them once a year - you can order them from 
www.commonreader.com.  

Self-help guru Anthony Robbins teaches that the hard part about 
changing your behavior isn't learning the new behavior, but getting a 
very clear understanding of why you really _need_ to change the old 
behavior! That kind of understanding is what I got from reading 
Mitchell's books. Once I became convinced of the importance of what he 
calls 'the habit of correctness and precision', I found that I started 
adopting it quite naturally, without much effort at all. 

Having said that, it seems to me that many, if not most, of the 
careless mistakes that we see here at Ask Dr. Math are caused by 
trying to do too many steps at once.  For example, 

   3(x - 5) = 18

     3x - 5 = 18


         3x = 23

          x = 23/3         Wrong!

Here, the multiplication wasn't fully distributed over the addition in 
the first step. This is an easy mistake to make, and the way I avoid 
making it is to insert an extra step, in which I leave the parentheses 
in place:

     3(x - 5) = 18

  (3*x - 3*5) = 18

      3x - 15 = 18

           3x = 33

            x = 11         Check:  3(11 - 5) = 3(6) = 18

These 'extra steps' can be especially useful when dealing with 
negative numbers, e.g., 

            3(x - 5) - 4(x - 6) = 

      (3*x - 3*5) - (4*x - 4*6) = 

          (3x - 15) - (4x - 24) = 

        (3x - 15) + -1(4x - 24) = 

    (3x - 15) + (-1*4x - -1*24) = 

        (3x - 15) + (-4x - -24) = 

         (3x - 15) + (-4x + 24) = 

             3x - 15 + -4x + 24 = 

Now I can be pretty sure that I have the signs correct.  This is more 
work than something like 

            3(x - 5) - 4(x - 6) = 
  
          (3x - 15) - (4x - 24) = 

              3x - 15 - 4x - 24 =            

which is much faster, but which also happens to be wrong.  A good rule 
to keep in mind that you can't make mistakes fast enough to get a 
correct answer.  :^D

I also make it a point _always_ to check my answer, after I think I've 
found it. At first this is something you have to remember to do, but 
after a while it becomes natural.

A second kind of careless error, which sort of falls into the same 
category, is caused by translating story problems too quickly into 
equations.  For example, here is the kind of thing we see a lot:

  The sum of half of a number and 8 less than the number is 25.

  (1/2)N - 8 = 25

      (1/2)N = 33

           N = 66         Wrong!

The mistake can be found with a careful step-by-step translation from 
English:

  The sum of half of a number and 8 less than the number is 25.
                     \______/                 \________/
                    
  The sum of half of    N     and 8 less than     N      is 25.
             \___________/   

  The sum of      (N/2)       and 8 less than     N      is 25.
                                  \________________/
  
  The sum of      (N/2)       and      (N - 8)           is 25.
  \__________________________________________/ 

               (N/2) + (N - 8)                           is 25
                                                         \/             

               (N/2) + (N - 8)                            = 25

which leads to 

     (N/2) + (N - 8) = 25

       (N/2) + N - 8 = 25                

           (N/2) + N = 33
  
              (3/2)N = 33

                   N = (2/3) * 33

                   N = 22

   Check:  The sum of half of 22 and 8 less than 22 is 25.

           The sum of 11 and (22 - 8) is 25.

           The sum of 11 and 14 is 25.  Yep!

At each step, I identified one small piece of the sentence to be 
translated into mathematical symbols. I made the translation, and then 
continued as before. If you were doing this on paper, you might do it 
more quickly by crossing things out and writing their replacements 
above or below, e.g., 

  The sum of half of xxxxxxxx and 8 less than xxxxxxxxxx is 25.
                         N                        N     


  The sum of xxxxxxxxxxxxxxxx and xxxxxxxxxxxxxxxxxxxxxx is 25.
                 (1/2)N                   N - 8     
                             

  xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx is 25.
                 (1/2)N       +           N - 8     
     
and so on. (Here, 'xxxxx' would really look like the original words 
with a line drawn through them to show that they no longer need to be 
considered.)

On the one hand, working 'in place' is an easy way to get sucked into 
making careless mistakes; but on the other hand, the more times you 
copy something, the greater the error that you're going to copy it 
incorrectly. With a computer, you can simply copy the old line and 
change it, which is what I've done here. Without a computer, you have 
to use your judgment about whether working in place or copying is more 
likely to cause a problem.  To make that judgment, you have to have 
some idea about the kinds of mistakes that you tend to make, and how 
often you make them. 

Which leads to my final recommendation, which is that you might want 
to keep a notebook of the careless mistakes you make.  Keeping track 
of them would allow you to observe patterns, and figuring out what 
you're doing is the first step towards changing _any_ kind of 
behavior. 

If you find that there are certain _kinds_ of mistakes that you're 
making over and over, feel free to write back and tell us what they 
are.  That would help us come up with techniques that you could use to 
avoid making them in the future. 

I hope this helps. 

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