Making the Case for Estimation

Estimation is a funny thing. At first glance it would seem that it’s far easier to arrive at a reasonable estimate than to obtain a precise answer, and for some types of problems this may be true. After all, isn’t it easier to be close than to be exact? However, here’s a simple illustration of how it can be more challenging to find a good estimate than a correct answer:

Estimate the product of 213 x 897.

Many people competent at basic arithmetic can multiply a pair of 3-digit numbers accurately. (I know, most would reach for their cell phone and use the calculator on that, but they could do it on paper!) However, how many of them could arrive at a decent ballpark estimate? Could you? Try it!

It turns out that estimation is often much more difficult than obtaining a precise answer. Students can learn an algorithm, a formal procedure for multiplying a pair of 3-digit numbers, and use this to get the answer, but if they make a mistake somewhere along the way they often have no idea that their answer is incorrect, and often the incorrect answer makes no sense at all. For example, if a student lines up the digits incorrectly in the standard US method of multiplication, they may end up with an answer with the wrong number of digits, and their answer will be very far off. If they don’t take a moment to estimate what it ought to be, they’ll never know. There’s a simple way to estimate the product in the example above: Round 213 down a bit to 200, round 897 up a bit to 900, and multiply 200 by 900. In order to do that efficiently, remove the zeroes, multiply 2 by 9, and then put the zeros back on. So, a reasonable estimate would be:

 

2 x 9 = 18                             180,000

 

The actual answer is 191,061. Taking a few seconds to get this estimate would help you realize if your answer is substantially off, which certainly does happen.

Obtaining reasonable estimates isn’t only useful for identifying answers in math class that can’t possibly be right. According to a study by Suydam and Weaver, “Research of adult usage shows that more than 80 percent of all mathematical computations in daily life involve mental manipulation of numerical quantities rather than written computations. Ironically, research also shows that 70 to 90 percent of the time in elementary school mathematics instruction in the United States is directed toward written computation (Suydam and Weaver, 1981).” If most of the mathematical work we do as adults is done mentally, wouldn’t developing our estimation skills seem to be very useful?

There is a very clear disconnect here. Why do teachers spend so much time focusing on written calculations and so little time on non-written work? That’s an excellent question! Since you’re not likely to change the way your child’s teacher teaches math, or at least to affect this particular imbalance, one thing you can do is help your child learn to make reasonable estimates as well as accurate mental computations. (Please refer to some of my other posts for recommendations on how to help your child improve his/her mental calculation skills, as opposed to estimation skills.)

Math textbooks at the elementary and middle school levels frequently have a few lessons here and there on estimating. However, developing effective estimation strategies and skills comes from frequent practice and guidance, not from practicing it for a day or two each school year. If your child’s teacher doesn’t include estimation in the curriculum on a regular basis, then it’s up to you to do it. Fortunately, it’s very, very easy, and if done correctly, can be framed as a game. My son asks to play mental math games when we’re in the car, in a theater waiting for a movie to start, standing in line, and at a whole host of other times, because I’ve been doing this with him for years, and you can do it as well!

For specific ideas for helping your child develop estimation strategies and skills, at a number of different levels, please refer to my post, Specific Strategies for Developing Estimation Skills.