From Addition to Multiplication … and Back: The Development of Students’ Additive and Multiplicative Reasoning Skills

Abstract This study builds on two lines of research that have so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. We investigated the development with age of both kinds of erroneous solution methods. We gave a test containing missing-value problems to 325 third, fourth, fifth, and sixth graders. Half of the problems had an additive structure and half had a proportional structure. Moreover, in half of the problems the internal and external ratios between the given numbers were integer, while in the other cases numbers were chosen so that these ratios were noninteger. The results indicate a development from applying additive methods “anywhere” in the early years of primary school to applying proportional methods “anywhere” in the later years. Between these two stages many students went through an intermediate stage where they simultaneously applied additive methods to proportional problems and proportional methods to additive problems, switching between them based on the numbers given in the problem. ACKNOWLEDGMENTS This research was partially supported by Grant GOA 2006/01 “Developing adaptive expertise in mathematics education” from the Research Fund K.U.Leuven, Belgium. Notes Disagreements between our categorization and the cluster analysis categorization mostly concerned students that we identified as correct reasoners (a category not identified and used by the cluster analysis) and minor disagreement on the number-sensitive and correct/number-sensitive group when students also gave one other answer.