Cal loved mathematics word problems. He delighted in new types of problems, exclaiming "This is fun!" when a problem was presented. At the end of third grade, he successfully solved word problems involving addition, subtraction, multiplication, division, multiple steps, and extraneous information.
Cal had not always loved mathematics. At the beginning of third grade, he was identified as learning disabled (LD) in language, reading, and mathematics. Although he had strengths in visualizing situations, his weaknesses in memorizing facts and procedures put him at a disadvantage during mathematics instruction.
Evan, a second grader, was identified as LD in mathematics and reading. His classroom teacher stated that he had limited mathematics skills and concepts. Tests revealed that although his verbal reasoning was in the superior range, he had difficulty with mathematical reasoning and computation. When given a mathematics problem, his primary strategy was to guess the answer.
An Instructional Approach That Makes Sense
Some teachers would argue that the best instructional approach for Cal and Evan is to focus on remediating their deficits. Some would recommend teaching the basic facts and computational procedures and providing extensive practice on these skills. Instead, Cal and Evan's instruction built on their prior knowledge and strengths. This instruction encouraged Cal's love of mathematics and helped Evan think more and guess less.
Cal, Evan, and three other LD students participated in a study designed to assess and encourage their natural problem-solving strategies (Behrend 1994). Previous research with non-LD children has revealed children's natural abilities to solve word problems by modeling the relationships within the problems (Carpenter 1985; Carpenter et al. 1993; Carpenter, Fennema, and Franke 1996). Individual interviews conducted before the eight small-group instructional sessions showed that these LD students also had natural strategies to solve problems, although they did not use them consistently. Instruction revolved around posing a word problem, allowing students time to solve the problem in any way they chose, sharing their strategies, and discussing similarities and differences among the strategies. During the thirty-minute sessions, students usually solved and discussed three to four word problems that represented a variety of mathematical operations. Despite the short duration of the study, individual interviews con ducted after the instructional sessions showed that the students were able to solve a variety of word problems by making sense of the problem situation instead of applying a rote procedure.
The following examples from Cal's and Evan's work highlight this strategy. Because both Cal and Evan had difficulty remembering procedures, instruction that focused on understanding the problem and solving it in a way that made sense built on their natural problem-solving strategies. Word problems provided a context for the numbers, and students could attach meaning to the numbers and operations.
Examples of Students' Work
Encouraging Evan, who generally guessed the answer, to think about the problem changed his perception of mathematics. He initially believed that problems did not make sense and any answer was as good as any other. Explaining how he solved the problem forced him to think about how he got his answer. During an early session, he was asked to solve this problem: "There are 6 children and 9 cookies. How many more cookies are there than children?" Evan guessed eight. The problem was repeated and Evan was asked to try it again. He said three and explained, "So how I got it is, if each of them had one cookie, and they ate that one cookie, then there would be three left." Evan made sense of the problem by visualizing the situation. Guessing became an occasional strategy rather than a primary strategy as he …