Research on multidigit addition and subtraction is now sufficient to question some present textbook practices and suggest alternatives. These practices revolve around the organization and placement of topics within the curriculum and around teaching/learning methods. These questions are being raised because the evidence indicates that U.S. children do not learn place-value concepts or multidigit addition and subtraction adequately and even many children who calculate correctly show little understanding of the procedures they are using (e.g., Cauley, 1988; Karnii & Joseph, 1988; Kouba et al., 1988; Labinowicz, 1985; Lindquist, 1989; Resnick, 1983; Resnick & Omanson, 1987; Ross, 1989; Stigler, Lee, & Stevenson, in press; Tougher. 1981).
Four ways in which subtraction is more difficult than addition are discussed: Verbal solutions are not always parallel to object solutions. Two correct methods exist for counting down a certain number of words, and these methods may interfere with each other. Special problems exist with subtraction on the number line. And subtraction is not just take away but has multiple situational interpretations. These points, along with some recent research results, suggest the research question: Should children be taught to solve subtraction statements such as 8 − 5 =? by counting up from 5 to 8?
Karen C. Fuson
Symbolic subtraction problems such as 14 − 8 =? were interpreted to children as story problems with dot pictures that supported count-up solutions. Children in first-grade mathematics classes were taught with considerable success to solve such symbolic subtraction problems by counting up from the smaller number to the larger (“8, 9, 10, 11, 12, 13, 14; 6 more make 14”) while keeping track of the number counted up by using one-handed finger panerns. The children improved quire considerably on a rimed test of the more difficult subtraction combinations, and this improvement held up over a month. Interviews indicated that almost all children could count up to solve subtraction combinations they did not know. Many used counting up to solve subtraction story problems with different semantic structures: performance on compare, separate (take-away), and equalize (how many more to make the same?) story problems was similar and good.
Karen C. Fuson
Children's Mathematical Thinking is an excellent book for preservice or in-service preschool, primary, and special education teachers. Much recent research on children's mathematics learning is presented in an interesting and assimilable manner. Two features used throughout the book contribute to its effectiveness with its intended audience: Short case studies are described, and two general theories of learning, absorption theory and cognitive theory, are contrasted.
This study examines some effects on preservice elementary teachers of a combined mathematics and mathematics methods course that used manipulative materials as the primary means of learning. The effects investigated were changes in trainee desire to use, ability to use, and actual use of manipulative materials in teaching; changes in trainee desire or actual behavior with respect to teaching in a learner-focused manner; changes in trainee understanding of elementary mathematics; differences between learning in a concrete, physical way and learning in a symbolic, abstract way; and changes in trainee attitudes of enjoyment of and feelings of competence in teaching mathematics.
Karen C. Fuson
Traditionally in the United States and Canada, students have first learned how to compute with whole numbers and then have applied that kind of computation. This approach presents several problems. First, less-advanced students sometimes never reach the application phase, so their learning is greatly limited. Second, word problems usually appear at the end of each section or chapter on computation, so sensible students do not read the problems carefully: They simply perform the operation that they have just practiced on the numbers in the problem. This practice, plus the emphasis on teaching students to focus on key words in problems rather than to build a complete mental model of the problem situation, leads to poor problem solving because students never learn to read and model the problems themselves. Third, seeing problem situations only after learning the mathematical operations keeps students from linking those operations with aspects of the problem situations. This isolation limits the meaningfulness of the operations and the ability of children to use the operations in a variety of situations.
Karen C. Fuson and Adrienne M. Fuson
Children in the United States ordinarily invent a series of increasingly abbreviated and abstract strategies to solve addition and subtraction problems during their first 4 years in school (Carpenter & Moser, 1984; Fuson, 1988, in press–a, in press–b; Steffe & Cobb, 1988). Several studies have shown that instruction can help children learn specific strategies in this developmental sequence. Fuson (1986), Fuson and Secada (1986), and Fuson and Willis (1988) demonstrated that by the end of first grade children of all achievement levels could add and subtract single-digit sums and differences (sums to 18) by sequence counting on and sequence counting up. Sequence counting on and counting up are abbreviated counting strategies in which the number words present the addends and the sum. In both strategies the counting begins by saying the number word of the first addend. For example, to count on to add 8 + 6, a child would say, “8 (pause), 9, 10, 11, 12, 13, 14.” The same sequence of number words is used to find 14–8 by counting up, but the answer is the number of words said after the first addend word rather than the last word in the sequence. When the second addend is larger than 2 or 3, some method of keeping track of the words said for the second addend is required. In the studies above this method was one-handed finger panems that showed quantities l through 9 (the thumb is 5) so that children could hold their pencil in their writing hand all of the time. The counting-on and counting-up instruction related the counting words to objects showing the addends and the sum, thus focusing on conceptual prerequisites for these abbreviated counting procedures and enabling children to relate counting and cardinal meanings of number words (Secada, Fuson, & Hall, 1983). The countingup instruction provided interpretations of subtraction and the“–” symbol as adding on, as well as the usual take-away interpretation that leads children to count down for subtraction.
Karen C. Fuson and Youngshim Kwon
Korean children's ability to solve addition problems with sums of 10, single-digit addition problems with sums between 10 and 18, and single-digit subtraction problems with minuends between 10 and 18 was assessed in interviews given at the end of the first semester of first grade, before children had studied problems involving numbers larger than 10 in school. These children showed considerable competence with all three kinds of problems, solving correctly 95%, 85%, and 75% of these problems, respectively. Almost two-thirds of the solutions of the problems above 10 were addition or subtraction recomposition methods structured around ten or known facts. Korean children demonstrated two finger methods that allowed fingers to be reused to show sums over 10. These methods and the regular named-ten Korean number words for numbers between 10 and 18 (“ten, ten one, ten two, ten three,…, ten eight”) support Korean children's learning of three efficient recomposition methods structured around ten.
Glenda Lappan and Karen C. Fuson
Karen C. Fuson
Edited by Glenda Lappan
Teachers have succcssfully used a conceptually based unit to teach subtraction with minuends to eighteen (e.g., 13–8) to all ability levels of first graders (Fuson 1986a; Fuson and Willis, in press). Children subtracted by counting up from the smaller to the larger number and kept track of how many words they had counted up by using one-handed finger patterns (see fig. 1). Children had already learned the finger patterns and how to use the finger patterns to add single-digit sums to eighteen by counting on; this procedure and directions for teaching counting on with finger patterns are described by Fuson (1987). Subtraction was initially introduced as the operation done in three different situations: equalizing, comparing, and “taking away” (see the section “Teaching Counting Up with One-handed Finger Patterns” and Carpenter and Moser ).