Facilitating Pre-service Mathematics Teachers’ Understanding of Directly and Inversely Proportional Relationships using Hands-on and Real-World Problems

Abstract

The purpose of this study was to examine four pre-service middle and high school mathematics teachers’ understanding of the directly and inversely proportional relationships presented through hands-on and real-world missing-value word problems. The problems included single and multiple proportions, and the PSTs were provided with plastic gears and a mini number balance system in the hands-on problems. A case study methodology with multiple cases was used in designing this study. Analysis of the semi-structured interviews revealed that for the PSTs, identifying directly proportional relationships was easier than identifying inversely proportional relationships. The PSTs recognized inversely proportional relationships in the hands-on problems but had difficulty recognizing them in the missing-value word problems. Therefore, the inclusion of the hands-on problems appeared to facilitate the PSTs’ understanding of the inversely proportional relationships. When forming multiplicative relationships between quantities, they mostly reasoned about within measure spaces. Furthermore, when determining directly and inversely proportional relationships, they usually attended to the qualitative relationships and constancy of the rate of change. In the multiple proportions problems, the PSTs had difficulty finding a mathematical formula that expressed relationships between quantities compared. This result suggested possible constraints in their understanding of the multiplicative relationships when more than two quantities were present.