12 HABITS OF MIND
12B) COMPUTATION AND ESTIMATION
Operations with whole numbers. Research suggests using word problems as a basis for teaching addition and subtraction concepts, rather than teaching computational skills first and then applying them to solve problems (Carpenter & Moser, 1983). Students should be exposed to a large variety of addition and subtraction situations (compare, combine, equalize, change add to, and change taken from) and given opportunities to consider different meanings for the +, -, and = marks. For example, when 9 - 3 = 6 represents the situation "John has 3 cars. Andy has 9 cars. How many more cars does Andy have?" the minus sign means compare rather than take away (Fuson, 1992).
Research has identified a developmental progression of concepts and skills that students use for addition and subtraction (Fuson, 1988; Fuson, 1992). There is some evidence that instruction based on this progression can help (Romberg & Carpenter, 1986). For example, after a year-long instruction based on this progression, 2nd-graders could solve almost all add or subtract problems with sums up to 18 (Fuson & Willis, 1989).
Students make a variety of errors in multi-digit addition and subtraction calculations (Brown & Van Lehn, 1982). Given traditional instruction, a substantial number of 4th- and 5th-graders are not able to subtract some whole numbers successfully (Fuson, 1992). Student errors suggest students interpret and treat multi-digit numbers as single-digit numbers placed adjacent to each other, rather than using place-value meanings for the digits in different positions (Fuson, 1992). With specially designed instruction, 2nd-graders are able to understand place value and to add and subtract four-digit numbers more accurately and meaningfully than 3rd-graders receiving traditional instruction (Fuson, 1992). Research also suggests students interpret multiplication of whole numbers mainly as repeated addition. This interpretation is inadequate for many multiplication problems and can lead to restrictive intuitive notions such as "multiplication always makes larger" (Greer, 1992).
Operations with fractions and decimals. Elementary- and middle-school students make several errors when they operate on decimals and fractions (Benander & Clement, 1985; Kouba et al., 1988; Peck & Jencks, 1981; Wearne & Hiebert, 1988). For example, many middle-school students cannot add 4 + 0.3 correctly or 7 1/6 + 3 1/2 (Kouba et al., 1988; Wearne & Hiebert, 1988). These errors are due in part to the fact that students lack essential concepts about decimals and fractions and have memorized procedures that they apply incorrectly. Interventions to improve concept knowledge can lead to increased ability by 5th-graders to add and subtract decimals correctly (Wearne & Hiebert, 1988).
Students of all ages misunderstand multiplication and division (Bell et al., 1984; Graeber & Tirosh, 1988; Greer, 1992). Commonly held misconceptions include "multiplication always makes larger," "division always makes smaller," "the divisor must always be smaller than the dividend." Students may correctly select multiplication as the operation needed to calculate the cost of gasoline when the amount and unit cost are integers, then select division for the same problem when the amount and unit cost are decimal numbers (Bell et al., 1981). Numerous suggestions have been made to improve student concepts of multiplication (Greer, 1992), but further research is needed to determine how effective these suggestions will be in the classroom.
Converting between fractions and decimals. Lower middle-school students may have difficulties understanding the relationship between fractions and decimal numbers (Markovits & Sowder, 1991). They may think that fractions and decimals can occur together in a single expression, like 0.5 + 1/2, or they might believe that they must not change from one representation to the other (from 1/2 to 0.5 and back) within a given problem. Instruction that focuses on the meaning of fractions and decimals forms a basis on which to build a good understanding of the relationship between fractions and decimals. Instruction that merely shows how to translate between the two forms does not provide a conceptual base for understanding the relationship (Markovits & Sowder, 1991).
Number comparison. Lower elementary students do not have procedures to compare the size of whole numbers. By 4th grade, students generally have no difficulty comparing the sizes of whole numbers up to four digits (Sowder, 1988). Students are less successful when the number of digits is much larger or when more than two numbers are to be compared. This might be due to increased memory requirements of working with more or larger numbers (Sowder, 1988). Upper elementary- and middle-school students taught traditionally cannot successfully compare decimal numbers (Sowder, 1988, 1992). Rather they overgeneralize the features of the whole number system to the decimal numbers (Resnick et al., 1989). They apply a "more digits make bigger" rule (according to which .1814 > .385). After specially designed instruction which develops good meanings for decimal symbols, many students are able to compare decimal numbers with understanding by 5th grade (Wearne & Hiebert, 1988). Upper elementary- and middle-school students taught traditionally cannot compare fractions successfully (Sowder, 1988). Students' difficulties here indicate they treat the numerator and the denominator separately. Specially designed instruction to teach meanings for fractions can help to improve ordering fractions by as early as the end of the 5th grade (Behr et al., 1984).
Calculators. The use of calculators in K-12 mathematics does not hinder the development of basic computation skills and frequently improves concept development and paper-and-pencil skills, both in basic operations and in problem solving (Hembree & Dessart, 1986; Kaput, 1992). The use of calculators in testing produces higher scores than paper-and-pencil efforts in problem solving as well as in basic operations (Hembree & Dessart, 1986).
Estimation skills. Good estimators use a variety of estimating tactics and switch easily between them. They have a good understanding of place value and the meaning of operations, and they are skilled in mental computation. Poor estimators rely on algorithms that are more likely to yield the exact answer. They lack an understanding of the notion and value of estimation and often describe it as "guessing" (Sowder, 1992b). Before 6th grade, students develop very few estimation skills from their natural experiences (Case & Sowder, 1990; Sowder, 1992b). As a result, some researchers caution that teaching estimation to young children may have as its single effect that they master specific procedures in a superficial manner (Sowder, 1992b).
Proportional reasoning. Early adolescents and also many adults have difficulty with proportional reasoning (Behr, 1987; Hart, 1988). Difficulty is influenced by the problem format, the particular numbers in the problem, the types of ratios used, and the problem situation (Heller et al., 1989; Karplus et al., 1983; Tournaire & Pulos, 1985; Vergnaud, 1988). Middle-school students can solve problems in proportions that involve simple numbers and simple wordings (Vergnaud, 1988), but troubles appear with more difficult numerical values or problem contexts. Problems using 2:1 ratios are easier than problems using n:1 ratios, and can be solved by elementary-school children (Shayer & Adey, 1981). Problems using n:1 ratios are easier than problems using other integer ratios (e.g., 6/2) which in turn are easier than problems using non-integer ratios (e.g., 6/4) (Tournaire & Pulos, 1985). Different ratio types (e.g., speed, exchange, mixture) appear to give more or less difficulty. For example, speed problems appear to be more difficult than exchange problems (Heller et al., 1989; Vergnaud, 1988). And these difficulties compound one another. Unfamiliarity with the problem situation causes even more difficulty when it occurs with a difficult ratio type (Heller et al., 1989).