Perspective and Purpose

Recent investigations by Steffe, Richards, and von Glasersfeld (1979) have indicated that addition and subtraction problem-solving competencies are developmental in nature and that these competencies build upon counting abilities. They postulate that, in beginning addition and subtraction, a type of problem-solving strategy termed counting-all develops prior to another kind of strategy termed counting-on (for addition) and counting-back (for subtraction).

If these tasks are developmental, one may assume that students approach the tasks in qualitatively different ways based upon their developmental levels. Neo-Piagetian researchers have postulated that a quantitative measure of development explains the qualitatively different ways in which children react to the same cognitive task at different stages of development. The measure, termed mental space or M-Space, describes the number of schemes which may be coordinated at one time. First graders, the majority of whom have an M-Space of a+2 or a+3, are capable of solving addition and subtraction word problems by utilizing the counting-all and/or the counting-on (back) strategies. Given this information, the purpose of this study was to determine what effect M-Space level has on the strategy a subject uses to solve problems when he is trained on a strategy which either matches or mismatches his M-Space level.

Design

To determine whether a match between M-Space and strategy demand is necessary or whether instruction will facilitate the chunking of schemes which allows the developmental task to be solved by a strategy which would otherwise be above the subject's M-Space level, the following steps occurred: one hundred thirty-nine first graders were pretested to identify those who could count to sixteen, perform numeral/number correspondence to sixteen, but could not solve addition and subtraction number fact problems to sixteen. One hundred fifteen subjects meeting these criteria were given the Cucumber Test and Backward Digit Span Test to assess their M-Space levels. After eliminating subjects before and during training, 50 subjects remained. Twenty-six subjects with an a+2 M-Space were divided into two training groups. Approximately half of the group was trained to use an a+2 strategy (the count-all strategy) to solve addition and subtraction word problems and the other half of the group was trained to use an a+3 strategy (the count-on (back) strategy). The same training procedure was used for the twenty-four subjects with an M-Space of a+3. Four to five weeks later, a delayed posttest consisting of four addition and four subtraction problems and one each of three types of transfer problems was presented.

Results

Mann-Whitney test results indicated that there were significantly fewer a+3 responses by the subjects with an a+2 M-Space who were trained to use an a+3 strategy than there were for subjects with an a+3 M-Space trained to use an a+3 strategy. However, there was no significant difference between those with an a+2 M-Space trained on an a+2 strategy and those with an a+2 M-Space trained on an a+3 strategy. Results of other research questions indicated that subjects gave similar responses to transfer problems which varied by material or additional variable; for subjects with an a+3 M-Space trained on an a+3 strategy, there were significantly more a+3 addition responses than subtraction responses; the implied comparison subtraction problem was answered incorrectly more often than straight take-away subtraction problems, and students tended to devise simple addition and subtraction problems and solve them by using memorized number facts.

Discussion

The findings indicate that more study is warranted for the application of the M-Space construct to a theory of how mathematical knowledge develops sequentially, the different ways in which addition and subtraction tasks can be conceptualized, and the instructional implications of applying a developmentally based theory of instruction to mathematics problem-solving.