Question

Dear Mitch,

As a teacher who is new to teaching elementary school math, I don't get why my district's textbooks put so much emphasis on "estimating". Is this really considered a major math skill? Isn't it the kind of thing that people use to avoid having to do real math?

-- Mrs. Goates

Washington D.C.

Answer

Dear Mrs. Goates,

Before I respond in a way that you might think is harsh, I want you to know that your question is one of the most common ones we receive. People often don't think of estimation as having much relevance in the exacting world of math calculations with its purported exactness of procedures leading to a set of specific results. But, as I have written in response to similar questions in the past, estimation on its own is not only an authentic area of mathematics, but in the opinion of many (including me) is among the most important topics taught in the standard U.S. Math curriculum.

Why?

Well, beyond school, in everyday life of real people, an ability to estimate not only requires having developed a real feeling for numbers (officially called 'number sense'), but is the one skill that can be used daily (and typically MUST be used daily) to help make the kind of decisions that are most likely to improve the quality of a person's life. It is almost always more important to be able to look at the gas meter on one's dashboard to determine the likelihood of having enough fuel to take a particular trip without having to refuel than it is to be able to calculate the average number of miles the vehicle gets per gallon of gas. In short, the ability to make a reasonably safe estimate can make the difference between getting stuck without gas on a lonely road at night and making one's way safely home.

There are plenty of ways the transition can be made from the world of calculating to the skills one develops to make increasingly helpful estimations.

Here is one classroom activity I recommend:

Bring a set of three or four matching jars into school (with lids to properly close them.)

I have found that the short jars typically used to package salsa works well.

Have one of the jars filled with pennies, one filled with nickels, one filled with quarters, and one with dimes.

First: Have the class estimate how many coins are in each jar.

Next: Have them estimate the VALUE of each jar's contents.

Next: Have them weigh each jar.

Next: Have the students take out the coins and add up each collection to determine the exact value.

Next: Give students an opportunity to adjust their estimations

Repeat with larger jars or the same jars only half-filled.

There is a relationship between each jar's weight and the coin inside and the jar's contents' value (Somewhere along the way, don't forget to include a chance for each group to subtract the weight of the empty jar...).

An exact ratio can be determined so that on the following day they can determine the value from the weight of a somewhat filled jar and the coin it contains, or they can determine the coin from a covered jar's contained value and weight, etc.

A chart can be made, and constants discovered.

That, incidentally, is how many useful and well known formulas and constants have been developed. In fact, one of my all-time favorite lessons involves using such a procedure to have students arrive at the famous constant 'pi' on their own. That lesson is explained in detail in another Q&A of long ago, so I won't repeat it here except to say that it involves having students use tape-measures to measure various circumferences around campus -- from jars of pencils and the openings of trash cans to tires in the parking lot. Their diameter and/or radius is also measured and recorded, and part of the relevant formula may be provided. From those pieces of information and several measurements, the students begin working on estimations for the value of pi. And they keep refining it until, magically, they use it to make accurate predictions about circles they have not yet approached.

There is an even more obvious link between the ability to estimate and the ability to calculate: Checking one's calculations for "reasonableness". The difference between a mathematician using a calculator and a 'non-math' person relying on a calculator is that members of the first category can spot when he/she has pressed a wrong button or misplaced a decimal.

Hope this helps,

Mitch