Exploring preservice middle and high school mathematics teachers’ understanding of directly and inversely proportional relationships
MetadataShow full item record
This study used hands-on and missing-value word problems to examine preservice middle and high school teachers’ knowledge resources when inferring directly and inversely proportional relationships between quantities. Additionally, the study examined preservice teachers’ solution strategies and their difficulties when solving single and multiple proportion problems. An explanatory case study with multiple cases was used to make comparisons within and across cases. This study used the knowledge-in-pieces perspective in reporting preservice teachers’ reasoning about ratios and proportional relationships. It appeared that the extent to which the preservice teachers were successful in coordinating the directly and inversely proportional relationships hinged on their attention to the specific features of the context. Although the preservice teachers accurately inferred the relationships between two covarying quantities as directly proportional or inversely proportional, their inferences were mainly based on attending to qualitative relationships—two quantities are increasing together—and the constancy of the rate of change. Thus, preservice teachers who relied heavily on the qualitative relationships and the constancy of the rate of change often judged nonproportional relationships that consisted of a constant difference or constant sum to be proportional, even after identifying correct nonproportional relationships. The results showed that the contexts of the hands-on problems facilitated the preservice teachers’ coordination of the directly and inversely proportional relationships more than the contexts of the missing-value word problems.