Front End Estimation (FEE)

Date: 08/23/2001 at 23:24:29
From: Rebecca Dickens
Subject: Front end estimation/rounding -- 1500 minus 140 est 1900?

My son was asked to estimate 1088 minus 399. His answer was 700.  
Wrong - the "correct" answer was 600. The explanation given was 
1000 minus 400 equals 600.

I can determine NO rule for estimation that could have given the 
"correct" result (600) that could not, if applied to a similar problem 
I created (1520 minus 110) result in an absurd answer, i.e.:

1) If the "correct method" was to round off ONLY to the first number 
   (regardless of column), that gives us 1000 minus 400 equal 600, BUT 
   then if we apply that same principle to MY problem (1502 minus 
   140), do we get 2000 minus 100 for an estimate of 1900?

2) If we apply the original "front end estimating" to my son's 
   original problem (1088 minus 399), using only the first two columns 
   we get 10 - 3 = 7, add two zeroes, 700 (my son's answer). If we
   make "adjustments" to "front end estimating", it makes no 
   difference (99 minus 88 equals 11 which is less than 50, so no 
   adjustment).

3) If we round off to the hundreds, then we get 1100 minus 400 equals 
   700 (my son's reasoning).

4) If we round off to ONLY the first number and then apply front end 
   estimating to the first two numbers, on my son's problem we get 
   1000 minus 400 equals 600. Apply that same principle to MY problem 
   (1502 minus 140) and you get 20 minus 1 = 19, add two zeroes, for 
   an estimate of 1900.

Now, as I understand it, it is appropriate to make an "adjustment" to 
FEE. Had a proper adjustment been made to my son's problem, the 
answer would have ended up at 700 (you could add back 88, or estimate 
90, the rule being: in subtraction, if one figure was lowered and the 
other raised, then you can take the difference, in this case 88 minus 
1 for 87, and ADD it to or SUBTRACT it from the result depending on 
which figures were lowered/raised. And this same adjustment made to MY 
problem would have improved the estimate to 1400 - 500 minus 40 equals 
460 round to 500 subtract from 1900 equals 1400). But since my son's 
answer of 700 was counted wrong, such an adjustment was not made, so 
the answer to my problem stands at 1900.

Am I missing something?  Where can I find the official rules for FEE?

Thanks for your help.

Rebecca Dickens

Date: 08/24/2001 at 12:33:56
From: Doctor Peterson
Subject: Re: Front end estimation/rounding -- 1500 minus 140 est 1900?

Hi, Rebecca.

I'm with you: it's silly to teach a method of estimation that gives 
less accurate results than other methods, and then call a better 
estimate "wrong" because it doesn't follow the rules for the requested 
method. I'm also not sure that there are any "official" rules for FEE.

I should mention that I had never heard of FEE until I joined Ask Dr. 
Math, and until now I have ignored questions about it. In fact, I'm 
not sure any of us have ever answered such questions, because it's 
just a school method, not something mathematicians bother with. I find 
no references to it in our archives. But I've been looking into it, 
and have a few thoughts that may be useful.

Here is a site I found that discusses estimation strategies as taught 
to students:

   http://dimacs.rutgers.edu/nj_math_coalition/framework/ch10/ch10_03-04.html   

This introduces FEE by saying, 

   "A reasonable approximation, then, of a multi-digit sum or 
   difference can always be made by considering only the leftmost 
   places and ignoring the others. This strategy is referred to as 
   front end estimation and is the main estimation strategy that many 
   adults use. In third and fourth grades, it should accompany the 
   traditional rounding strategies." 

In other words, it sounds as if they teach this because people who 
don't know better use it, and perhaps in order to show later that 
there are better ways. In other words, it's just a first guess, and 
not a really sound method. Maybe you will find out that the problem 
you had trouble with was given in order to teach the defects of the 
method. (Somehow I doubt it.)

As to official rules, from what I've seen looking around the Web 
(where most references seem to be in education standards), you are 
just supposed to take one or more digits at the front, and then adjust 
any way you feel like; there's not a lot of consistency in it. 
Sometimes they blindly use the first digits, even though they have 
different place values, and then use the "adjustment" step to correct 
for this foolishness. Other people seem to be able to recognize that 
corresponding digits should be added, talking about adding the 
"front end _column_", not the front digit of each number. The 
adjustment is then nothing more than adding another column.

Here are examples of each approach. First, using only the front digit:

    http://www.mcesc.k12.oh.us/math/Math_pdf/GlossaryForTeachers.pdf   

    FRONT END ESTIMATION: Rounding to the first, or front end, digit
    to make estimation.
    Example: Using front end estimation, 594 + 32 becomes 500 + 30,
    giving an estimate of 530. An adjustment for 94 would give 630.

Next, using the front column:

    http://staff.wssd.k12.pa.us/dberra/level%20one/WholeNum-L1/frontee-L1.html   

    Front End Estimation:
    1. Add or Subtract the front end (leftmost column) digits.
    2. Adjust this estimate by adding or subtracting the the digits
       to the right of the front end digits.
    3. Add the values from steps 1 & 2.
    In other words, pretend everything after the first two digits in
    a number is a zero, then add or subtract.

In my own mind, the core of any study of estimation should involve 
getting a feel for how estimation works. That would include picturing 
what is happening on a number line, thinking about the effect of place 
value, considering the cumulative effects of rounding errors, and 
being able to see when a simplistic strategy can be compensated for by 
a glance at the unused numbers. None of this should involve fixed 
rules. An estimation is simply a "reasonable guess," and may take many 
factors into account.

Unfortunately, teachers need a way to tell whether the students are 
doing "the right thing" and learning what they are taught, rather than 
just judging the effectiveness of whatever strategy they develop with 
experience. That can lead to requiring a specific method and counting 
other methods as wrong. I'm not at all happy with this.

This reference, speaking of assessment, says that "test items should 
_not_ require the use of a specific estimation strategy":

    Estimation - Jeff LeMieux, syzygy-matrix
    http://syzygy.virtualave.net/buch/10ns04.html   

Finally, here is a thread from Teacher2Teacher that you will find 
interesting:

    Front End Estimation
    http://mathforum.org/t2t/thread.taco?thread=1154   

A relevant comment from an answer there:

    Front end estimation can be considered a precursor to rounding,
    since it uses the leading digits, and doesn't involve any
    changing of amounts.  The numerals are right there to be seen
    and used.  It is a great way to introduce estimation, and as
    students become more proficient with using just the leading
    digits, the skill of making adjustments should be introduced.
    It is just another way to find a reasonable answer.

    And, having said that, I would also like to remind you that
    there is not really any such thing as a "wrong" estimate...
    some estimates are less useful than others...   but any
    estimate made using the original problem is a valid estimate.
    And when we are talking about teaching students to estimate,
    the goal is not to find the one correct "estimate" (and prove
    you can regurgitate the teacher's exact method), but to have
    the skill to reason about the numbers being used, and to be
    able to come up with a range that is suitable for using to
    predict the answer, to have a quick and easy-to-do method for
    checking to see what a reasonable answer would be.

Note that FEE is just an introductory method, quick and dirty, to be 
improved by other methods when needed. I suspect that it is really 
meant to be a catch-all description of the ad-hoc methods anyone with 
good number sense would use, looking at the most important digits 
first, then making adjustments of any sort based on the rest of the 
digits. Nothing can be wrong about such an estimate.

Finally, let me comment on the specific problem you asked about. In 
the explanation given, they have not used front-end estimation at all, 
but rounding (since my understanding is that the digits are to be used 
as they stand, with the rounding handled by the adjustment). Secondly, 
they are rounding the first number to one significant digit, when the 
proper thing to do would be to round both numbers to the hundreds, 
giving 1100 - 400 = 700. In any case, it appears that the intent was 
to teach only unadjusted estimation, and the answer is not meant to be 
particularly accurate. I suppose it's valid to ask a student to use a 
specific method, and then call it wrong if that method is not followed 
(even if it's a silly method); but it certainly isn't clear exactly 
what rules they are following! On the other hand, your complaint that 
whatever method they are using would be absurd in an extreme case is 
not a strong argument. Any method can go wrong - the essence of 
estimation is that you are ignoring some details, and what you ignore 
may gang up on you and make you wrong. The problem is simply that they 
don't seem to have made clear what the method being taught really is.

So what do you do? Teach your son to ignore foolish judgments, and 
just do what's right even when it's not appreciated. Follow whatever 
rules you're given when you have to, but don't let them restrict you 
when you're on your own. And encourage him: it sounds as if he has a 
good sense of numbers.

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