Idea of Estimation

• Students need to recognize the difference between estimation and approximation in order to select and use the appropriate tool in a computational or measurement situation. Estimation is an educated guess subject to “ballpark” error constraints while approximation is an attempt to procedurally determine the actual value within small error constraints (J. Sowder, 1992a).

• Good estimators tend to have strong self-concepts relative to mathematics, attribute their success in estimation to their ability rather than mere effort, and believe that estimation is an important tool. In contrast, poor estimators tend to have a weak self-concept relative to mathematics, attribute the success of others to effort, and believe that estimation is neither important nor useful (J. Sowder, 1989).

• The inability to use estimation skills is a direct consequence of student focus on mechanical manipulations of numbers, ignoring operational meaning, number sense, or concept of quantity/magnitude (Reys, 1984). • The ability to multiply and divide by powers of ten is “fundamental” to the development and use of estimation skills (Rubenstein, 1985).

• Three estimation processes are used by “good” estimators in Grades 7 through adult. First, reformulation massages the numbers into a more mentally friendly form using related skills such as rounding, truncating, and compatible numbers (e.g., using 6+8+4 to estimate 632+879+453 or using 7200 60 to estimate 7431 58). Second, translation alters the mathematical structure into an easier form (e.g., using the multiplication 4x80 to estimate the sum 78+82+77+79). And third, compensation involves adjustments made either before or after a mental calculation to bring the estimate closer to the exact answer. In this study, the less skilled students “felt bound” to make estimates using the rounding techniques they had been taught even if the result was not optimal for use in a subsequent calculation (e.g., use of compatible numbers) (Reys et al., 1982).

• Student improvement in computational estimation depends on several skills and conceptual understandings. Students need to be flexible in their thinking and have a good understanding of place value, basic facts, operation properties, and number comparisons. In contrast, students who do not improve as estimators seem “tied” to the mental replication of their pencil-and-paper algorithms and fail to see any purpose for doing estimation, often equating it to guessing (Reys et al., 1982; Rubenstein, 1985; J. Sowder, 1992b). Also, good estimators tended to be selfconfident, tolerant of errors, and flexible while using a variety of strategies (Reys et al., 1982).

• Teacher emphasis on place value concepts, decomposing and recomposing numbers, the invention of appropriate algorithms, and other rational number sense skills have a long-term impact on middle school students’ abilities using computational estimation. Rather than learning new concepts, the students seemed to be reorganizing their number understandings and creating new ways of using their existing knowledge as “intuitive notions of number were called to the surface and new connections were formed” (Markovits and Sowder, 1994).

• Students prefer the use of informal mental computational strategies over formal written algorithms and are also more proficient and consistent in their use (Carraher and Schliemann, 1985).

• Students’ acquisition of mental computation and estimation skills enhances the related development of number sense; the key seems to be the intervening focus on the search for computational shortcuts based on number properties (J. Sowder, 1988). 

• Experiences with mental computation improve students’ understanding of number and flexibility as they work with numbers. The instructional key was students’ discussions of potential strategies rather than the presentation and practice of rules (Markovits and Sowder, 1988).

• Mental computation becomes efficient when it involves algorithms different from the standard algorithms done using pencil and paper. Also, mental computational strategies are quite personal, being dependent on a student’s creativity, flexibility, and understanding of number concepts and properties. For example, consider the skills and thinking involved in computing the sum 74+29 by mentally representing the problem as 70+(29+1)+3 = 103 (J. Sowder, 1988).

• The “heart” of flexible mental computation is the ability to decompose and recompose numbers (Resnick, 1989).

• The use of a context enhances students’ ability to estimate in two ways. First, a context for an estimation helps students overcome difficulties in conceptualizing the operations needed in that context (e.g., the need to multiply by a number less than one producing a “smaller” answer). Second, a context for an estimation helps

students bypass an algorithmic response (e.g., being able to truncate digits after a decimal point as being basically insignificant when using decimal numbers) (Morgan, 1988).

• Young students tend to use good estimation strategies on addition problems slightly above their ability level. When given more difficult problems in addition, students get discouraged and resort to wild guessing (Dowker, 1989). 

• Students have a difficult time accepting either the use of more than one estimation strategy or more than one estimation result as being appropriate, perhaps because of an emphasis on the “one right answer” in mathematics classrooms. These difficulties lessened as the students progressed from the elementary grades into the middle school (Sowder and Wheeler, 1989).

• Students need to be able to produce reasonable estimates for computations involving decimals or fractions prior to instruction on the standard computational algorithms (Mack, 1988; Owens, 1987).

• Students estimating in percent situations need to use benchmarks such as 10 percent, 25 percent, 33 percent, 50 percent, 75 percent, and 100 percent, especially if they can associate a pictorial image. Also, student success seems to depend on a flexible understanding of equivalent representations of percents as decimals or fractions (Lembke and Reys, 1994).