Effects of consistency and adequacy of language information on understanding elementary mathematics word problems

Annals of Dyslexia, 2001 by Leong, Che Kan, Jerred, Wendy D

Two types of elementary mathematics word problems involving different linguistic structures were devised to examine the understanding and solution of these problems by 91 Grade 3, 4, and 5 children divided into "more able" and "less able" subgroups. One task consisted of 12 consistent and 12 inconsistent language problems on the basic processes of addition, subtraction, multiplication and division. Another task consisted of a total of 36 word problems with 12 items each containing adequate, inadequate, and redundant information, respectively, for problem solution. Subsidiary tasks of general ability, vocabulary, reading comprehension, mathematics concepts, reflection on mathematics learning, and working memory were also administered to provide estimates of the contribution of these "nonmathematics" tasks to the solution of elementary mathematics problems. Analyses of variance and covariance of group data showed significant main effects of grade, consistency, and adequacy of linguistic information in problem solution. Word problems containing inconsistent information were more difficult than those with consistent information. Further, word problems containing inadequate and redundant information were more difficult to classify, and for the children to explain, than those items with just enough information. Interviews with 12 individual children provided further insight into their strategies for problem solutions. Both cognitive and developmental perspectives are important for mathematics learning and teaching for children with or without learning disabilities.

Of the overlapping subgroups of learning disabilities, it is generally agreed that those with developmental dyslexia constitute the largest subgroup and are relatively well researched, whereas those with mathematics disabilities are less well researched. The situation is changing, however, because of important studies on children's mathematical development and learning by such researchers as Geary (1993, 1996), Ginsburg (1977), McCloskey (1992), Miles and Miles (1992), Nunes and Bryant (1996), Rivera (1997), Schoenfeld (1985), and Verschaffel (1994), among others.

ENGAGING STUDENTS IN "MATHEMATICS THINKING"

More than 20 years ago, Ginsburg (1977) explained that mathematics learning and teaching should emphasize building relationships among knowledge among declarative knowledge (facts about mathematics), procedural knowledge (rules, algorithms, procedures to solve mathematics tasks), and conceptual knowledge (connected web of information). Students must understand the nature of the mathematical problems and use their knowledge of facts such as numbers, units of lengths, and money to operate on the different pieces of information. Each of the interrelated aspects of knowledge is critical to developing "mathematical literacy" (Goldman, Hasselbring, & the Cognition and Technology Group at Vanderbilt [CTGV], 1997). The CTGV group envisions engaging students in "mathematics thinking" in solving numerical problems, especially with the help of multimedia technology. These cognitive perspectives are consistent with the goals and directions of the National Council of Teachers of Mathematics (NCTM) (1989) in emphasizing broader mathematical reasoning away from the narrow focus of computation. This broader approach to instruction and curriculum development by engaging students in problem-solving strategies applies to students with or without learning disabilities (Thornton, Langrall, & Jones, 1997).

The present study was motivated by the cognitive information approach (e.g., McCloskey, 1992) and elementary school children's reasoning of mathematical relations and systems (e.g., Geary, 1993, 1996; Nunes & Bryant, 1996). McCloskey's modular functional architecture in representing numeral and verbal information has been influential in both cognitive and neuropsychological studies. In essence, abstract internal representation of numbers constitutes the core and is connected with input and output modules. The input consists of both numeral comprehension (e.g., five times four) and verbal comprehension (e.g., five times four, and the output consists of numeral production (e.g., 20) and verbal production (e.g., twenty). Mediating between the input and the output modules and acting on the abstract semantic quantity representation are the calculating procedures of using number facts to act on mathematical concepts. It would appear that McCloskey's (1992) formal modular model interfaces the declarative, procedural, and conceptual knowledge discussed in mathematics learning more than 20 years ago and reemphasized in both the recent learning and learning disabilities literature. What is also important is that the McCloskey model explains well several types of number processing disabilities and the dissociation between number facts and number procedures in different number processing modules. This modular approach should serve well as a theoretical framework in working with children with or without disabilities in mathematics learning.