Estimation: it's more than a guess.
A teacher is holding up a jar full of marbles for her 2ndgrade
class to see. "Today, we are going to be doing estimating,"
she says. "Guess how many marbles are in the jar." One child
enthusiastically calls out "25." "Okay," says the
teacher. Another volunteers "100." "Great!" says the
teacher. A third student calls out "10,000." Diplomatically,
the teacher responds, "Do you really think that there are 10,000
marbles in this little jar?" The student sheepishly smiles and
shakes her head. The teacher calls for more guesses, but none of the
students respond. Such strategies, unfortunately, do not give the
students the experiences they need to construct the true meaning of
estimation.
What Is Estimation? Estimation is nothing more than quickly and reasonably developing an idea about the quantity or size of something without actually counting or measuring it. Many times, an estimate will suffice when answering such questions as "How many?," "How high?," or "How hot?" (Linn, 1970). It is a skill that should be introduced in prekindergarten and kindergarten and reinforced in all the later grades, especially grades 1 through 3. In fact, students will need this skill all through their mathematics careers. Young children's estimation experiences usually involve measuring or counting through visual perception (Charlesworth & Radeloff, 1991). As the students gain experience with estimation, their accuracy increases (Althouse, 1994). What Are the Values of Estimation Skills? The National Council of Teachers of Mathematics (NCTM) has developed five goals for students (1989). Students of mathematics should: 1) value mathematics, 2) become confident in their ability, 3) become a math problem solver, 4) learn to communicate mathematically, and 5) learn to reason. Having the ability to estimate can help students reach all of these goals. Knowing when and how to estimate gives students more tools and strategies with which to solve problems. Being able to reason and communicate better mathematically improves students' confidence, and they come to value mathematics as a distinct way of thinking, instead of viewing it as a collection of unconnected rules and formulas. Our daily lives are filled with situations that require estimation: comparing prices at a store, changing the amounts of ingredients used in a recipe, determining the best routes when driving, and verifying calculator computations. One of the main objectives of estimation is determining if answers or measurements are indeed reasonable. In fact, many times we estimate without even realizing it. Reys (1992) suggests that over 80 percent of all mathematical applications call for estimation, rather than exact computation  throwing into question the emphasis in elementary school mathematics upon exact computation. How Can Estimation Skills Be Developed? One cannot assume that students who can compute numbers can estimate equally well. The skills and objectives of the two domains are totally different. Just because students can add 12 + 13 does not mean that they also understand that the answer is greater than 20 and less than 30. Young children should have many varied experiences in counting sets of objects and estimating in order to develop a better sense of numbers' values. It also is very important to have students make estimates, not guesses (Murray, 1993). Guesses can get out of control, as the opening scene of this article illustrates. The students' answers of "25," "100," or "10,000" can all be considered guesses, but not all of them were reasonable estimates. If given a small package of M&M's[R] and asked to estimate the number of candies it contained, a person could do this by thinking of the size of one candy and judging how many might fit into the package. On the other hand, one could only hazard a guess if asked how many of each color the package contained, because there is little or no information to work with, and no true way of knowing the correct amount until the package is opened. The language of estimation also is important. Words and phrases such as "about," "close to," "just about," "a little more (less) than," and "between" can help students construct the concept of estimation, as well as related skills. Students should understand that the goal is to predict a quantity or amount that is as accurate as possible, by using quick and easy methods (Van de Walle, 1994). Estimation, therefore, needs to be integrated into the entire mathematics curriculum, and not be taught as a standalone concept. If students are taught estimation as an arbitrary set of rules, they will not see the connections to problemsolving, numeration, place value, and other mathematical concepts. Students must be free to take risks when learning estimation, as they would with any other skill. The teacher should encourage this environment by accepting and not judging the students' responses, no matter how far off the mark they might be. After some help and practice, students can begin to determine whether or not their estimates are reasonable. Students must understand that the objective is not to find one correct answer or solution to an estimation exercise, but rather to find a range of acceptable answers or solutions. By listening to other students' estimates and participating in discussions as to why some were "good" or "acceptable," students can continue to construct the meaning of estimation. Estimation skills and strategies are based on "nice" numbers that are easy to use. As an example, students trying to determine if they can afford items that cost $.93 and $1.06 would find it easier to estimate the cost of each item as being a dollar. In general, multiples of 10 are usually easy to work with, and constitute "nice" numbers (as are numbers such as 25, 55, 75, etc.). The way that a question is phrased also helps further define the true meaning of estimation. If asked to estimate how many children are in their class, most students would immediately begin counting their classmates. This is not really estimation, because it encourages the idea of producing an exact answer; counting their classmates is something that the students have probably done on many occasions because the answer is easily obtainable. A followup question, such as "Estimate the number of children in our school," can lead students to a range of responses to answer "about how many," which is truly an estimate. Constructing the Concept of Estimation Keeping these ideas in mind, let's take another look at the opening scene. The teacher gathers those students she feels are ready to explore estimation further. These students have counted sets of objects many times and have had a number of experiences in estimating quantities of objects up to 35 or 40. The teacher wants to see if they can transfer their knowledge to a more complex situation. She tells the class, "Today we are going to continue working with estimation. Sometimes we do not need to know the total number exactly. A number that is 'just about' the total or 'close to' the total is good enough. This is called estimation. If we are having a big party we can estimate that we are going to have about 50 people. For now it is not important whether the total will be 48 or 53 or 55. Estimation is not a guess. To make a guess you do not have to think about how many there are. Any number can be a guess. To make an estimate you have to think." The teacher picks up a handful of marbles from a container and begins to place them into a jar, counting out loud: ". . . 16, 17, 18. I just placed 18 marbles into this jar." The class watches her intently as she picks up a second similar handful of marbles and places them in the jar without counting them. This is followed by a third handful and a fourth handful. The jar cannot hold another handful so she picks up a smaller amount and places them in the jar so it is almost full. "Now, I want you to think and give me an estimate of about how many marbles are in the jar," the teacher says. "Don't guess. Think about what I just did with the marbles and give me an estimate." Child 1: "I know, 53." Teacher: "All right. However, remember the other day we were talking about 'nice numbers,' those numbers that are easy to use? Would you like to give me your estimate again?" Child 1: "50!" Teacher: "Okay." Child 2: "100." Teacher: "Okay." Child 3: "1,000." Teacher: "Okay." This continues until several estimates, ranging from 20 to 1,000, are written on the board. "Any more?" the teacher asks. "Let's all look at the estimates and talk about which of these might tell us about how many marbles are in the jar. Remember, I put 18 marbles in the jar the first time and then put in about that many again, then again, then again. Then I added a small amount." Child 4: "Well if you put in 18 marbles and then more handfuls, there would be more than 20 marbles." Child 5: "And there are not 1,000 marbles. That would be a lot of marbles and the jar is too small. 18 is close to 20, and 20 and 20 are 40, and 40 and 40 are 80 and a few more maybe make 90 marbles." Teacher: "So you think there are about 90 marbles in the jar? Anyone with another idea?" Child 6: "Maybe there were more, maybe it is 100." Teacher: "So, some of you think the total could be close to 90 or 100. Estimates that are close to the total are called reasonable estimates. based on your reasoning, we can say that 90 and 100 are reasonable estimates." Incorporating these ideas into lessons or presentations clearly gives children the chance to construct the meaning of estimation. They offer opportunities for original estimates, discussion, and thinking. This allows the students to see that estimation is not the same as guessing; rather, it is a method of thinking that is used to solve real problems. Some Activities To Develop Estimation Skills * Children's Literature David Whitin (1994) favors using children's literature to explore estimation, believing that books can demonstrate estimation across such mathematical concepts as volume, length, area, and time. Most important, Whitin says that literature dynamically demonstrates how mathematics is a concrete way of thinking. He cites Counting on Frank by Rod Clement (1991) as a children's book that best incorporates estimation strategies. The story is about a young boy who likes to calculate or estimate things: how long of a line can be drawn with a ballpoint pen before it runs out of ink, or how long it would take to fill the entire bathroom with water, if both faucets were running. After reading or listening to the story, students can make their own calculations and estimates. * Focusing on the Range Place sets of like objects in paper bags. Have students (individually or in pairs) examine the closed bags and estimate a range of numbers that describe how many are in the bag. After estimating that there are, for example, between 50 and 75 blocks in the bag, the students should open the bag and count the objects to verify their estimate. As the students become more proficient in their estimates, encourage them to reduce the size of their range, in order to become more accurate ("There are between 60 and 70 blocks"). * More Or Less This activity can be used with many different objects and measurements. Provide a 1pound weight and several grocery store objects that are commonly sold by the pound (fruits, vegetables, nuts, etc.). After getting a feeling for the weight, have the students hold each of the objects and estimate if it weighs more or less than a pound. Provide a balance scale to verify the estimates. Including different weights and different objects will extend the activity. * How Long? Put three or four ranges of time on the board. [less than] 1 minute 1 to 12 hours 1 day to 1 week [greater than] 1 month Then, have the students generate activities that they estimate would fall under each range (e.g., driving from New York to San Francisco, brushing your teeth, or watching television on Saturday). The ranges can be modified to extend students' thinking. * Would You Rather . . .? Ask students such questions as "To have more money, would you rather have your height measured in stacked pennies or measured in dollar bills placed endtoend?" The students should first estimate which measure would be more beneficial to them, then they should verify their estimate by solving the problem with whatever materials are at hand. Other questions, such as, "Would you rather have your height measured in stacked quarters or your weight measured in dimes?," can extend their thinking and add to their estimation experiences. Conclusion Estimation is crucial to becoming a good problem solver, and experience and practice are critical to becoming a good estimator. The sooner we expose students to estimation and to related skills, the more they will value estimation and understand when and how to apply it effectively. References Althouse, R. (1994). Investigating mathematics with young children. New York: Teachers College Press. Charlesworth, R., & Radeloff, D. J. (1991). Experiences in mathematics for young children (2nd ed.). New York: Delmar. Clement, R. (1991). Counting on Frank. Milwaukee, WI: Gareth Stevens Children's Books. Linn, C.F. (1970). Estimation. New York: Thomas Y. Company. Murray, T. (1993). Estimation explorations. Hayward, CA: Activity Resources. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Reys, B. (1992). Estimation. In T. R. Post (Ed.), Teaching mathematics in grades K8 (pp. 279301). Needham Heights, MA: Allyn & Bacon. Van de Walle, J. (1994). Elementary school mathematics. White Plains, NY: Longman. Whitin, D. J. (1994). Exploring estimation through children's literature. Teaching Children Mathematics, 41(8), 436441. Stephen J. Micklo is Associate Professor, College of Education, University of South Florida St. Petersburg. 

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