**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).