Children's strategy choice and success in solving algebra word problems: Interplay between cognitive variables and knowledge

Across cultures and curricula, it is commonly observed that many beginning learne rs of formal algebra revert to familiar methods rooted in arithmetic reasoning to solve algebra problems. Though a prevalent cause for frustration in teachers trying to get stud nts to practice and master letter-symbolic algebra, some have argued that such dis plays of “flexibility” in strategy choice should not be discouraged. Some even found children t o be more successful in solving algebra problems when they use arithmetic methods ( .g., Nathan and Koedinger, 2000a). Are arithmetic methods more effective than algebraic m ethods in solving algebra word problems? Are students truly being “flexible” when the y us arithmetic methods in situations calling for algebraic methods? Is it a case of flexi bility or inflexibility when students persist in using arithmetic methods when they are no longer appropriate or effective? Are students’ strategy choices and their resulting success i n solving algebra word problems contingent upon their cognitive capabilities and their understanding of algebra ? We examined these questions by giving 157 Secondary 2 students a set of algebra word probl ems under specific instructions to use letter-symbolic algebra, as well as tes ts of their algebraic knowledge, intellectual ability, working memory, and inhibitory ability. Results r evealed that (i) despite the specific instructions, a substantial number of students persiste d in using arithmetic methods, suggesting a reluctance or inability to use letter-sy mbolic algebra; (ii) students were not more successful in solving algebra word problems when they used arithmetic methods; (iii) algebraic knowledge, intellectual ability, wo rking memory, and inhibitory abilities contribute in both unique and overlapping ways to both strategy choic e and success in solving algebra word problems. Redesigning Pedagogy 2009 3 CHILDREN'S STRATEGY CHOICE AND SUCCESS IN SOLVING ALGEBRA WORD PROBLEMS: INTERPLAY BETWEEN COGNITIVE VARIABLES AND KNOWLEDGE In most mathematics curricula, children would have had many years of experie nce with arithmetic by the time they are introduced to formal algebra. It is a common observation that beginning learners of algebra frequently revert to familiar arithmet ic thods of problem solving when solving algebra problems (e.g., Nathan & Koedinger, 2000a; Ng, 2003; Stacey & MacGregor, 1999). Some have argued that children are merely employing strate ic flexibility in using arithmetic methods to solve algebra problems, finding them o be more successful when they used arithmetic methods (e.g., Nathan & Koedinger, 2000a). Howe ver, are students truly being flexible when they revert to arithmetic methods? Are they choosing arithmetic methods because they are truly more effective or appropriate ? What are some of the factors influencing children’s strategy choices and their consequent s ucces in solving algebra word problems? Algebra Word Problems There are two general classes of word or story problems – arithmetic problem s involving result-unknown quantities and algebra problems involving start-unknown quantities. Consider the question “A cow weighs 230 kg. A dog weighs 150 kg less than the cow. A goat weighs 130 kg less than the cow. How much do the three animals weigh altogether? ” Questions of this type are considered arithmetic as the unknown values (weight of the dog, weight of the goat, and total weight of the three animals) can easi ly be derived by directly applying arithmetic operations described in the problem on the known value ( weight of the cow). The start-state is given (weight of the cow) while the result-s tate i unknown Redesigning Pedagogy 2009 4 (total weight). Conversely, consider the question “A cow weighs 150 kg more than a dog. A goat weighs 130 kg less than the cow. Altogether the three animals weigh 410 kg. How much does the cow weigh? ” The result-state (total weight) is now known while the start-state (weight of the cow) is unknown. The known value is a combination of the three unknowns and the unknown(s) cannot be easily solved for by directly applying the arithmet ic op rations described in the problem on the known value. Such start-unknown problems are considered algebraic as letter-symbolic algebra “or more sophisticated modeling” are usually required to solve them (Nathan & Koedinger, 2000a; see also Bednarz & Janvier, 1996). Children in Singapore start solving word problems in primary or elementary school (for 6 – 12 year-olds), starting with end-unknown arithmetic word problems. In their la st two years of primary school, they learn to solve start-unknown algebra word problems wit h arithmetic methods of problem solving. When they progress to secondary school (for 13 17 year-olds) they are taught to solve structurally identical problems using a lgebr ic methods. Arithmetic versus Algebraic Problem Solving Strategies Students in the local curricula are taught arithmetic methods such as “listing ”, “guessand-check”, “unwinding” (“working backwards”), “grouping” and the “model” (Ng, Lee , Ang & Khng, 2006; see Table 1) in primary school and letter-symbolic algebra in s eco dary school. With letter-symbolic algebra, one works directly with unknown quantities. A n equation comprising both known and unknown values is first formed using “forward operations” (Usiskin, 1988). To solve for the unknown, one then operates on the equation in a way that maintains its symmetric equivalence – terms are transposed while keeping the lefthand side of the equation balanced with the right-hand side (see Table 1vi). In contrast , when arithmetic strategies are used, one works only with known quantities. The unknowns are Redesigning Pedagogy 2009 5 found by the direct application of arithmetic operations on known values, often involving the inversion of operators described in a problem (see Table 1i-v). -----------------------------------------Insert Table 1 about here -----------------------------------------By the time our students are in their second year of secondary school, they will ha ve a repertoire of both arithmetic and algebraic strategies at their disposal. Se condary school students are required to use only letter-symbolic algebra to solve algebra word pr oblems. However, in a local study, Ng (2003) found a substantial proportion of secondary school students still relied on arithmetic methods to solve algebra word problems (see Na than and Koedinger, 2000a for US-based findings). Persistent use of such methods poses a problem as it can hinder the acquisition of algebraic strategies. Algebra word problems ar given to beginning algebra students to help them learn letter-symbolic algebra which i s not only a more sophisticated method, but also essential for progression to higher mathemati cs (e.g., National Mathematics Advisory Panel, 2008). The more children persist in using ari thmetic strategies, the less chance they will have to practice and master letter -symbolic algebra.