9 THE MATHEMATICAL WORLD
Students' understanding of the Mathematical World has been extensively researched. The Handbook of Research in Mathematics Teaching and Learning. (Grouws, 1992) as well as the papers presented at the "research agenda conferences" on number concepts in the middle grades (Hiebert & Behr, 1988) and on the learning and teaching of algebra (Wagner & Kieran, 1989), reveal that there is a rich and growing research base related to Numbers, Symbolic Relationships, Shapes, and Uncertainty. There is still little research related to Reasoning, however. As in other domains, research has focused on what students understand about mathematical concepts at isolated points in time or on how this understanding evolves naturally in students. Research on instructional interventions that improve student understanding has received less attention.
9A) NUMBERS
Whole numbers. During preschool and elementary-school years, children develop meanings for number words in which sequence, count, and cardinal meanings of number words become increasingly integrated (Fuson et. al., 1982; Fuson, 1988). Students' own meanings for number words determine to some extent their strategies for adding and subtracting and the complexity of problems they can solve. Elementary- and middle-school students may have limited ability with place value (Sowder, 1992a). Sowder reports that middle-school students are able to identify the place values of the digits that appear in a number, but they cannot use the knowledge confidently in context (for example, students have trouble determining how many boxes of 100 candy bars could be packed from 48,638 candy bars).
Rational numbers. Upper elementary- and middle-school students often do not understand that decimal fractions represent concrete objects that can be measured by units, tenths of units, hundredths of units, and so on (Hiebert, 1992). For example, students have trouble writing decimals for shaded parts of rectangular regions divided into 10 or 100 equal parts (Hiebert & Wearne, 1986). Other students have little understanding of the value represented by each of the digits of a decimal number or know the value of the number is the sum of the value of its digits. Students of all ages have problems choosing the largest or the smallest in a set of decimals with different numbers of digits to the right of the decimal points (Carpenter et al., 1981; Hiebert & Wearne, 1986; Resnick et al., 1989). Upper elementary-school students can establish rich meanings for decimal symbols and do a variety of decimal tasks well after specially designed instruction using base-10 blocks (Wearne & Hiebert, 1988, 1989).
Upper elementary- and middle-school students may exhibit limited understanding of the meaning of fractional numbers (Kieren, 1992). For example, many 7th-graders do not recognize that 5 1/4 is the same as 5 + 1/4 (Kouba et al., 1988). In addition, elementary-school students may have difficulties perceiving a fraction as a single quantity (Sowder, 1988), but rather see it as a pair of whole numbers. An intuitive basis for developing the concept of fractional number is provid-ed by partitioning (Kieren, 1992) and by seeing fractions as multiples of basic units--for example, 3/4 is 1/4 and 1/4 and 1/4 rather than 3 of 4 parts (Behr et. al., 1983).
Estimation. Middle-school and even high-school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so that they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).
Number symbols. There is very little research into student understanding of number symbols as arbitrary conventions. It does indicate that not until 11 years of age do most children consider that correct counting with nonstandard symbols is as adequate as correct counting with standard symbols (Saxe et. al, 1989).