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

Student: JE (just enough information).

Researcher: What about the second sentence? Would you still say JE?

Student: Yeah.

Researcher: Did you use that second sentence?

Student: Ymm ... oh . . . NN. You wouldn't need that. Before she sold them she had 43 comic books.

DISCUSSION

The group analyses and the interview data are discussed in the context of the literature on early mathematics learning and mathematics disabilities. It is recognized that the present study was limited to the understanding of elementary mathematics word problems with a rather circumscribed focus on the consistent/inconsistent and adequate/inadequate language information and that early mathematics learning is both complex and many-faceted. However, the emphasis on "mathematics thinking" is in keeping with notions put forward by Ginsburg (1977), Geary (1993, 1996), and Goldman and her colleagues at the Cognition and Technology Group at Vanderbilt (CTGV) (Goldman et al., 1997).

On both kinds of language information tasks there was the expected grade differences in performance. It would appear that Grade 3 seems to be the sensitive stage where children learn more complex mathematical problems and have to bring their declarative, procedural, and conceptual knowledge to bear on the solution of these problems. The significant difference between the more able and less able students on the basis of the BAS Matrix task for only the adequacy of language information task may be explained by the more complex and varied demand of this task placed on the students. The considerably larger standard deviation in relation to the mean for the inconsistent than the consistent items may be due to the children's difficulty in grasping the inverse relationship between language and mathematics reasoning. Similarly, there are larger individual differences for the "not needed" items in terms of adequacy of language information as compared with the "not enough" and the "just enough" items. It seems some children have not picked up certain principles about logical relationship and the needed information to arrive at solutions of problems.

The interview data complement those from the group analyses. The excerpts of exchanges show that problem-solvers will do well to follow the suggestions of Montague (1997) and NOTE (1989) in taking broader views rather than narrowly focusing on computation. Montague (1997) suggests that the interrelated steps of read, translate, transform (e.g., using diagrams and visuals), estimate, compute, and verify should serve all students well. This step-by-step approach is particularly useful to those with disabilities as they can see how one step feeds into another. Moreover, students should be encouraged to discuss in class the nature of a problem and to justify their procedures in arriving at a solution (Thornton et al., 1997). They need to separate relevant from irrelevant information and to use multiple ways to represent and communicate mathematics information (Goldman et al., 1997). Goldman et al. also point out that when both declarative and procedural knowledge is in place, students with disabilities often fail to apply their knowledge in meaningful ways. These researchers have further shown how the use of multimedia technology can provide both concrete and meaningful contexts for problem solution. In a recent paper, Bransford and his colleagues at CTGV discuss their current thinking about designing learning environments that invite and sustain mathematical thinking in middle school years (Bransford, Zech, Schwartz, Barron, Vye, & the Cognition and Technology Group at Vanderbilt [CTGV], 2000). These learning environments are organized around a series of videodiscs and CD-ROM-based adventures known as Jasper Woodbury Problem Solving Series. The aim is to help students focus on mathematical modeling as a way to think visually as well as symbolically, and to create solutions.