Chapter 15 Cycles of Generalizing Activities in the Classroom Susanne Strachota, Eric Knuth and Maria Blanton Abstract This study considers classroom situations in which students and the teacher co-contribute to promoting generalization. It specifically focuses on the ways in which students and a teacher in one classroom engage in generalizing arithmetic. Generalized arithmetic is an important route into early algebra (Kaput in Algebra in the Early Grades. Routledge, New York, 2008); its potential as a way to deepen students’ understandings of concepts of school arithmetic makes it an important focus of early algebra research. In the analysis we identified general- izations around properties of arithmetic and the actions that promoted these types of generalizations, and then considered the relationship between these actions. Analysis revealed that generalizations became platforms for further generalization. Keywords Early algebra Á Teachers promoting generalization Á Student learning Generalizing Á Generalization Á Generalized arithmetic 15.1 Introduction For many students, algebra continues to be a gatekeeper to future academic and employment opportunities. In response, initiatives (e.g., Common Core State Standards Initiative 2010; National Council of Teachers of Mathematics [NCTM] S. Strachota (&) University of Wisconsin–Madison, 1025 West Johnson Street, Madison, WI 53706, USA e-mail: sstrachota@wisc.edu E. Knuth University of Texas at Austin, 1912 Speedway STOP D5700, Austin, TX 78712, USA e-mail: eric.knuth@austin.utexas.edu M. Blanton TERC, 2067 Massachusetts Ave., Cambridge, MA 02140, USA e-mail: maria_blanton@terc.edu © Springer International Publishing AG 2018 C. Kieran (ed.), Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds, ICME-13 Monographs, https://doi.org/10.1007/978-3-319-68351-5_15 351