Skip to page content


Funded by a four-year, $1.3 million grant from the National Science Foundation, the project, "Investigating Proportional Relationships from Two Perspectives," investigates how future mathematics teachers make connections among multiplication and division, fractions, ratios and proportional relationships, linear functions and statistical samples.

By learning to solve mathematical problems in multiple ways, future teachers can gain deeper knowledge of content they teach and the flexibility to help diverse students learn mathematics. The project will look at ways UGA students' understanding of certain concepts, such as ratios and proportional relationships, can build upon their understanding of multiplication and division. This "interconnected" knowledge can help both teachers and students remember what they learn.

[Read about us in UGA Columns]( grant-to-train-future-teachers-of-middle-grade-math/)


Investigating Proportional Relationships from Two Perspectives (InPReP2; Izsák, mathematics education; Beckmann, mathematics/mathematics education) is a Middle Stage Research Proposal to investigate how future mathematics teachers in preparation programs at the University of Georgia develop facility reasoning about proportional relationships in terms of quantities. InPReP2 will collect and analyze whole-class video data and written artifacts from 5 cohorts of future mathematics teachers (approximately 125 teachers total) and will conduct a series of 6 hour-long cognitive interviews with 16 pairs of these future teachers (32 total), selected to reflect different initial strengths and weaknesses with core components of multiplicative reasoning.

Intellectual Merit

InPReP2 will make three significant contributions to teaching and research about proportional relationships—arguably the most central mathematics in the middle grades. First, InPReP2 will investigate a new, potentially transformative approach to proportional relationships that is based on two complementary perspectives, one that makes co-variation 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, taken together, the two perspectives may facilitate both key features of proportional relationships and open new pathways that make subsequent topics, like linear equations and statistical samples, more accessible. Second, InPReP2 will investigate new, potentially transformative methods that coordinate detailed case studies with recent advances surveying multiplicative reasoning in large, national samples. Third, the few studies that have examined teachers' reasoning about proportional relationships have emphasized deficits. InPReP2 will examine how different initial strengths and weaknesses with multiplicative reasoning affect how future teachers make sense of the two perspectives on proportions and applications. InPReP2 will investigate these research questions about future middle grades teachers:

  1. How do future teachers' reasoning about multiplication, division, and fractions support and constrain their learning of the two perspectives on ratios and proportional relationships?
  2. How do future teachers use number lines and strip (tape) diagrams to support reasoning about ratios and proportional relationships from the two perspectives?
  3. How do future teachers use the two perspectives on ratios and proportional relationships to support reasoning about linear equations in two variables and statistical samples?

Broader Impact

InPReP2 will contribute new insights about multiplicative reasoning and new research methods, both of which could be applied in future studies of students and teachers learning other STEM content. By examining a potentially transformative approach to ratio and proportional relationships, the project will generate results about multiplicative reasoning that can inform improved teacher education content courses and professional development that prepare teachers cross the nation for standards that emphasize sense making with quantities, including the Common Core State Standards. 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. Furthermore, the project will lay a foundation for future studies of middle grades students' reasoning with the two perspectives on proportional relationships. Thus, the project has high potential for improving teaching and learning middle grades mathematics fundamental to subsequent college readiness. InPReP2 methods will advance STEM education research by demonstrating 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. Furthermore, the surveys used by InPReP2 are available for researchers interested in using similar mixed methods in future studies of teachers' multiplicative reasoning. Finally, training of new mathematics education researchers and teacher educators will be integrated into all aspects of the project.


National Science Foundation

Principal Investigator

Andrew G. Izsak


Professor, Mathematics

Laine P. Bradshaw

Assistant Professor, Educational Psychology

Active Since

September 2014