Investigating Proportional Relationships from Two Perspectives (InPReP2)

加强中学数学教育与美国STEM教育面临的任何挑战一样紧迫。比例和比例关系是数学和科学的重要基础，但它们对学生和教师构成了长期的挑战。从两个角度调查比例关系(InPReP2)将考察未来教师如何在他们作为预备课程一部分的数学课程中发展对比例关系的理解。该项目将在教师准备课程中收集数据，并通过访谈收集数据，在此期间，未来的教师将解决比例关系问题。课程和面试将基于一种潜在的变革性方法来处理比率和比例关系，这一点被以往的研究忽视了。该项目还将进行一项研究，比较佐治亚大学(UGA)和肯纳索州立大学(KSU)的职前中学和中学数学教师如何对比例关系进行推理。关于未来教师更容易和更难学习的发现将被用来确定教学的关键目标，以更好地帮助未来教师帮助他们的学生学习关键但困难的中学数学。过去的研究表明，教师在比例关系、协变关系和乘法关系方面的困难可能深深植根于乘法概念领域，但这项研究并没有直接考察这些根源。InPReP2将考察职前中学教师用量来推理乘法和除法的能力如何用比例关系来支持和约束他们的能力。InPReP2还将基于两个互补的视角研究比例关系，一个是通过双数字线使协变在视觉上清晰，另一个是通过条形图使数量之间的乘法关系在视觉上清晰。InPReP2的中心假设是，这两个视角可能既促进了比例关系的关键特征，又开辟了使后续主题更容易理解的新途径。InPReP2将研究在全国大样本中协调详细案例研究与最新进展调查乘法推理的方法。InPReP2还将在临床和实验环境中研究乘法推理的不同初始优势和劣势如何影响未来教师对比例和应用这两种观点的理解。InPReP2关于乘法推理的结果将为改进的教师教育和专业发展内容课程提供参考，这些课程将使全国各地的教师为强调数量理性的标准做好准备。从数量上理解比例关系的教师将更好地准备支持他们的学生关于协变和乘法关系的推理，这是大学准备工作的基础。InPReP2方法将通过调查诊断分类模型如何与推理措施相结合来促进STEM教育研究，以协调对大样本的调查与对小样本的详细案例研究。这些方法对于在复杂的、多方面的领域中产生关于学习的普遍结论具有广泛的影响。InPReP2将把对新的数学教育研究人员和教师教育工作者的培训纳入该项目的所有方面。

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.