As indicated in the K-12 Overview, students' ability to use estimation appropriately in their daily lives develops as they focus on the reasonableness of answers, explore and construct estimation strategies, and estimate measurements, quantities, and the results of computation.

One of the estimation emphases for very young children is the development of the idea that guessing is an important and exciting part of mathematics. The teacher must employ sound management practices which ensure that everyone's guess is important and which encourage student risk-taking and sharing of ideas about how their guesses were determined. When first asked to guess an answer, many students will give nonsense responses until they establish appropriate experiences, build their sense of numbers, and develop informal strategies for creating a guess. Children begin to make reasonable estimates when the situations involved are relevant to their immediate world. Building on comparisons of common objects and using personal items to build a sense of lengths, weights, or quantities helps children gain confidence in their guessing. As children communicate with each other about how guesses are formulated they begin to develop informal strategies for estimation.

Estimation with computation is as important at these early grade levels as it is at all the other grade levels. Estimation of sums and differences should be a part of the computational process from the very first activity with any sort of computation. Children should regularly be asked About how many do you think there will be in all? or About what do you think the difference is? or About how many do you think will be left? in the standard addition and subtraction settings. These questions are appropriate whether or not exact computations will be done. Children should understand that, sometimes, the estimate will be an accurate enough number to serve as an answer. At other times, an exact computation will need to be done, either mentally, with paper-and-pencil, or with a calculator to arrive at a more precise answer. The particular procedure to be used is dependent on the setting and the problem.

One of the most useful computational estimation strategies at these grade levels also reinforces an important place value idea. Students should understand that in two-digit numbers the tens digit is much more meaningful than the ones digit in contributing to the overall value of the number. A reasonable approximation, then, of a two-digit sum or difference can always be made by considering only the tens digits and ignoring the ones. This strategy is referred to as front end estimation and is used with larger numbers as well, although then the first two digits may be used. It is the main estimation strategy that many adults use.

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in kindergarten and in grades 1 and 2.

Experiences will be such that all students in grades K-2:

1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

• Students place various amounts of counters or other small objects in individual plastic bags. Working in groups of four students, the children choose one bag to be the reference set and judge whether each of the other bags has more than, less than, or the same as the reference set. Initially, they should try to make the judgments without counting. The teacher observes the groups as they work, making notes about the students' progress.

• Young children benefit from frequently comparing sets of objects to some given number. For example, given sets of colored chips arranged on a table, they should name which sets have more than five and which have less than five.

• Students play the card game War with a set of cards without numerals (i.e., cards which only show sets of hearts, clubs, diamonds, or spades). Students will easily distinguish between the 7's and the 3's, but will be reluctant to make judgments about closer numbers like the 4's and 5's without counting. As they play more often, however, their ability to distinguish will visibly improve. They may also begin to notice patterns involving even and odd numbers on their own.

• Students learn to recognize certain arrangements of dots or stars as representing certain numbers. Using flashcards, they estimate the number of dots or stars, and then count to check their estimates.

• After reading Ten Black Dots by Donald Crews, students make up their own uses for 1-10 black dots. They use adhesive dots to create their own books that include uses for each of the numbers from 1 through 10. They then estimate and count the total number of dots they actually use for their book. (The total might surprise them: 55.)

• As an assessment of students' ability to judge without counting, the teacher puts some counters (more than five) on the overhead projector, turns it on for a few seconds, and then asks the students to write whether the number of counters shown is closer to 10 or to 20.

2. Use personal referents, such as the width of a finger as one centimeter, for estimations with measurement.

• Students estimate lengths of pieces of spaghetti, yarn, paper, pencils, paper clips, etc., using suggested non-standard personal units such as width of thumb, length of a foot, and so on. They note that different students get different "right" answers.

• As standard units like foot and centimeter are introduced, students are challenged to findsome part of their body or some personal action that is about that size at this point in their growth. For instance, they may decide that the width of their little finger is almost exactly one centimeter or the length of one baby step is one foot.

• Students use their self-discovered personal body referents to estimate the measures of various classroom objects like the length of the blackboard or the width of a piece of paper. They compare their answers, noting that when larger units are used, the estimated answer is a smaller number; and when smaller units are used, the estimated answer is a larger number.

3. Visually estimate length, area, volume, or angle measure.

• Students look at a quantity of sand, salt, flour, water, macaroni, corn, or popcorn and estimate how many times it could fill up a specified container.

• Students estimate how many pieces of notebook paper it would take to cover a given area such as the blackboard, or a portion of the classroom floor.

• Students regularly estimate lengths using a variety of non-standard units such as my feet, Unifix cubes, paper clips, and orange Cuisenaire Rods. They then measure to verify or revise their estimates.

• Students begin to develop an understanding of angle measure by making right-hand (or left-hand) turns repeatedly to turn completely around. They also compare angles to right-angle "corners," and decide whether an angle is more than or less than a "corner."

• Students note that there are 12 numbers on a clock face and discuss how far each hand moves in an hour. They note that each hand moves in a circle, but that the hour hand moves much more slowly.

• Students work through the Will a Dinosaur Fit? lesson that is described in the First Four Standards of this Framework. They determine the size of the room, hear their classmates' presentations about the dinosaurs, and then as a whole class activity estimate which dinosaurs, and how many of them, might fit into the room.

4. Explore, construct, and use a variety of estimation strategies.

• Students are asked if a sixty-seat bus will be adequate to take the two first grade classes on their field trip. After it is known that there are 23 children in one class and 27 in the other, individuals volunteer their answers and give a rationale to support their thinking; front end estimation should lead to the conclusion that the total number of students is between 40 and 60. A discussion might be directed to the question of whether an exact answer to the computation was needed for the problem.

• Students are shown a glass jar filled with about eighty marbles and asked to estimate the number in the jar. In small groups, they discuss various approaches to the problem and strategies they can use. Each group shares one strategy with the class, and the estimate that resulted. The teacher makes notes about students' work throughout the activity.

• Second grade students can be challenged to estimate the total number of students in the school. They will need to talk informally about the average number of students in each class, the number of classes in a grade level, and the number of grade levels in the school. They might then use calculators to get an answer, but the result, even though the exact answer to a computation, is still an estimate to the original problem. They discuss why that is so.

• Primary-grade students explore the meanings of comparison words by listening to How Many is Many? by Margaret Tuten. They compare big and small, long and short, a lot and a few. They list how many pieces of candy would be a few and how many pieces would be many, eventually reaching general agreement, perhaps on 5 as a few. Then they consider whether 5 teaspoons of medicine would be a few.

5. Recognize when estimation is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

• Given a pair of real-life situations, students determine which situation in the pair is the one for which estimation is a good approach and which is the one that probably requires an exact answer. One such pair, for example, might be: sharing a bag of peanuts among 3 friends and paying for 3 tickets at the movie theater.

• Given a set of cartoons with home-made mathematical captions, first graders decide which of the cartoon characters arrived at exact answers and which got estimates. Two of the cartoons might show an adult and a child looking at a jar of jellybeans and the captions might read: Susie guessed that there were 18 jellybeans left in the jar and Susie's mom counted the 14 jellybeans left in the jar.

• Students read or listen to newspaper headlines and discuss which involve exact numbers and which might be estimates.

6. Determine the reasonableness of an answer by estimating the result of operations.

• Students are regularly asked if their answer makes sense in the context of the problem they were solving. They respond with full sentences explaining what they were asked to find and why the numerical answer they found fits the context reasonably, that is, why it could be the answer. For example, first graders might be asked to decide if their answer to the following problem makes sense: Mary made 27 cookies and Jose made 15. How many cookies did they make in all? Some responses might indicate that the answer should be more than 20 + 10 = 30 and less than 30 + 20 = 50. Other students might say that they know that 25 and 15 is 40, so the answer should be a little more.

• Students estimate reasonable numbers of times that particular physical feats can be performed in one minute. For example: How many times can you bounce a basketball in a minute? How many times can you hop on one foot in a minute? How many times can you say the alphabet in a minute? and so on. Other students judge whether the estimates are reasonable or unreasonable and then the tasks are performed and the actual counts made.

• Second-grade students are given a set of thirty cards with two-digit addition problems on them. In one minute, they must sort the cards into two piles: those problems whose answers are greater than 100 and those less than 100. The correct answers can be on the backs of the cards to allow self-checking after the task is completed.

• Second-grade students are given a page of addition or subtraction problems in a multiple choice format with 4 possible answers for each problem. Within some time period whichis much too short for them to do the computations, students are asked to choose the most reasonable answer from each set of four.

7. Apply estimation in working with quantities, measurement, time, computation, and problem solving.

• Students have small pieces of yarn of slightly different lengths ranging from 2 to 6 inches. Each student first estimates the number of his or her pieces it would take to match a much longer piece - about 30 inches long - and then actually counts how many. Then they use their individual pieces to measure other objects in the room. Each child is responsible for estimating the lengths in terms of his or her own yarn, but they can use evidence from other children's measuring to help make their own estimates.

• Students regularly estimate in situations involving classroom routines. For example, at snack time, they may guess how many cups can be filled by each can of juice or how many crackers each student will get if all of the crackers in the box are given out.

• Kindergartners always have fun deciding which color is best represented in a group of multi-colored objects. Good examples of such an activity would be choosing the color that shows up most often (or least often) in bags of M&M's, in handfuls of small squares of colored paper, or in a jar full of marbles. After everyone has committed to a guess, the children can sort the objects and count each color. They can then make bar graphs to show the distribution of the different colors.

• Students use Tana Hoban's photographs in Is It Larger? Is It Smaller? as a starting point for investigating and comparing quantities and measures in their classroom. For example, on one page, three vases are shown filled with three different kinds of flowers. The reader must decide which objects to compare, such as the vases, before ordering them - from tallest to shortest and/or by volume.

References

Crews, Donald. Ten Black Dots. New York: Greenwillow, 1986.

Hoban, Tana. Is It Larger? Is It Smaller? New York: Greenwillow, 1985.

Tuten, Margaret. How Many is Many? Chicago: Children's Press, 1970.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.