ABSTRACT

Strengthening middle school mathematics education is as urgent as any challenge facing STEM education in the United States. Ratios and proportional relationships are a critical foundation for mathematics and science, yet they pose perennial challenges for students and teachers. Investigating Proportional Relationships from Two Perspectives (InPReP2) will examine how future teachers develop understandings of proportional relationships in mathematics courses they take as part of their preparation programs. The project will collect data in teacher preparation courses and through interviews during which future teachers solve problems about proportional relationships. The courses and interviews will be based on a potentially transformative approach to ratio and proportional relationships, which has been overlooked by past research. The project will also conduct a study comparing how preservice middle grades and secondary grades mathematics teachers at the University of Georgia (UGA) and Kennesaw State University (KSU) reason about proportional relationships. Discoveries about what is easier and what is harder for future teachers to learn will be used to identify key targets for instruction that better prepare future teachers to help their students with critical, yet difficult, middle grades mathematics.

Past research suggests teachers' difficulties with proportional relationships, covariation, and multiplicative relationships are likely to have roots deep in the multiplicative conceptual field, yet this same research has not examined those roots directly. InPReP2 will examine how preservice middle school teachers' capacities to reason about multiplication and division in terms of quantities support and constrain their facility with proportional relationships. InPReP2 will also investigate proportional relationships based on two complementary perspectives, one that makes covariation visually explicit with double number lines and one that makes multiplicative relationships between quantities visually explicit with strip diagrams. The central hypothesis for InPReP2 is that the two perspectives may facilitate both key features of proportional relationships and open new pathways that make subsequent topics more accessible. InPReP2 will investigate methods that coordinate detailed case studies with recent advances surveying multiplicative reasoning in large, national samples. InPReP2 will also examine in both clinical and experimental settings how different initial strengths and weaknesses with multiplicative reasoning affect how future teachers make sense of the two perspectives on proportions and applications. InPReP2 results about multiplicative reasoning will inform improved teacher education and professional development content courses that prepare teachers across the nation for standards that emphasize sense making with quantities. Teachers with an understanding of proportional relationships in terms of quantities will be better prepared to support their students' reasoning about co-variation and multiplicative relationships, which is fundamental to college readiness. InPReP2 methods will advance STEM education research by investigating how Diagnostic Classification Models, in combination with measures of reasoning, can be used to coordinate surveys of large samples with detailed case-studies of small samples. Such methods have broad implications for generating generalizable findings about learning in complex, multi-faceted domains. InPReP2 will integrate training of new mathematics education researchers and teacher educators into all aspects of the project.

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