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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.