Research on Learning Fractions and Relational Thinking: Guiding Teacher’s Thinking About Children’s Learning

• In Event: Current Research on the Use of Learning Progressions in Mathematics Formative Assessment: Explorations and Synthesis

Abstract

Research on children’s thinking about rational number and related constructs such as ratio and proportion has a long, rich history in mathematics education and appears to support several possible learning progressions. In this session, the author presents a new framework that portrays the development of children’s thinking about fractions and decimals (author, 2011) and considers the role of professional development in supporting teachers to use this framework to make instructional decisions. In particular, the framework supports a form of formative assessment that Erickson (2008) termed “proximal,” because it involves a “continual taking stock” (p. 187) as teachers interact with students during instruction. It also supports teachers’ day-to-day decision making on the basis of students’ mathematical thinking as it is revealed during instruction and students’ engagement in tasks. Putting this framework into practice for use by teachers is a complex endeavor whose success depends on offering professional development that addresses teachers’ integrated knowledge of children’s thinking and mathematics and teachers’ use of this knowledge in practice.
The framework describes how children’s early understanding of the fundamental properties of operations and equality and children’s intuitive ideas of partitioning quantities support the progressive development of understanding of fractions and decimals. Learning fractions is characterized as foundational to the study of algebra because it offers students the opportunity to grapple with the fundamental mathematical relationships that constitute the core of algebra. These relationships govern how addition, subtraction, multiplication, and division work in algebra as well as arithmetic; allowing students to grapple with them as they study fractions helps students to develop a tightly integrated understanding of number and operation.
Pilot data on the design and implementation of professional development that supports teachers to use this framework to inform instruction are presented. Drawing on a combination of surveys, interviews, and existing research, the researcher outlines theoretical considerations in the design of the professional development and how it supports teachers to understand and use information about children’s thinking as a basis for instructional decision making. For example, the professional development design is based in part on a “pedagogy of enactment” as urged by Grossman and MacDonald (2008). Conclusions focus on how teachers could use this information in their interactions with students.

Author

• Susan B. Empson, The University of Texas - Austin