The roles of tools and models in a prospective elementary teachers’ developing understanding of multidigit multiplication

Highlights

• We present a case study of a PST’s developing reasoning about products.

• We frame the analysis primarily in terms of models and tools.

• We illuminate her reasoning and key supports for the progress she made.

• We discuss implications for mathematics teacher education.

Abstract

It is important for prospective elementary teachers to understand multidigit multiplication deeply; however, the development of such understanding presents challenges. We document the development of a prospective elementary teacher’s reasoning about multidigit multiplication during a Number and Operations course. We present evidence of profound progress in Valerie’s understanding of multidigit multiplication, and we highlight the roles of particular tools and models in her developing reasoning. In this way, we contribute an illuminating case study that can inform the work of mathematics teacher educators. We discuss specific instructional implications that derive from this case.

Introduction

Elementary teachers today are expected to engage students in mathematical practices that involve reasoning and conceptual understanding, not merely to train students to recall facts and procedures (National Governors Assoc., 2010; National Research Council [NRC], 2001). Engaging students in these mathematical practices requires attunement to children’s mathematical thinking (Jacobs, Lamb, & Philipp, 2010). If students are to have opportunities to work conceptually and be given the freedom to solve problems using their own strategies, then the teacher must be able to readily make sense of those strategies to inform instructional decisions (Lampert, Beasley, Ghousseini, Kazemi, & Franke, 2010). Therefore, elementary teachers need to have a deep and flexible understanding of elementary mathematics (Ball, Thames, & Phelps, 2008; Conference Board of Mathematical Sciences [CBMS], 2012). However, the K-12 education of prospective elementary teachers (PTs) does not prepare them sufficiently for these challenges. PTs tend to rely on the mathematical facts, rules, and procedures that they learned in school (Ball, 1990; Ma, 1999; Thanheiser, 2009). Thus, effective, conceptually focused elementary mathematics teacher preparation is vital.

In order to help improve PTs’ mathematical knowledge and prepare them for the challenges of teaching for conceptual understanding and flexible reasoning, mathematics teacher educators would benefit from a body of research concerning how PTs’ content knowledge can improve by building upon their prior knowledge. Currently, such a literature base is lacking. Few articles provide illuminating accounts of PTs’ mathematical conceptions or document successful learning trajectories (Thanheiser et al., 2014). Noteworthy contributions in this vein include the work of McClain (2003), Simon and Blume (1994), and Thanheiser (2010).

In the domain of whole-number operations, multidigit multiplication is an especially challenging topic for PTs to understand conceptually (Harkness & Thomas, 2008; Simon & Blume, 1994; Southwell & Penglase, 2005). This topic is important both within elementary arithmetic and for its connections to applications of the distributive property in algebra (Kaput, Blanton, & Moreno, 2008). In order to support their students’ understanding of multiplication and to prepare them for a successful transition from arithmetic to algebra, PTs need to learn to reason more meaningfully and flexibly about multidigit products than they typically do. We view this reasoning as a microcosm of number sense.

Greeno (1991) described number sense in terms of situated knowing in a conceptual domain. In his environment metaphor, people learn to navigate the domain of numbers and quantities with the aid of mental models. Number sense development takes the form of the refinement of these models with sensitivity to the constraints and affordances of the environment. In adopting this perspective, learners' mental models become of great interest. In particular, we are concerned with PTs’ models of multidigit multiplication and how these can develop with instructional support.

In order to better understand how a PT can develop an improved understanding of multidigit multiplication, we present a case study involving a particular PT whom we call Valerie.¹ We provide a rich account of Valerie’s models of multiplication, and we trace the evolution of her reasoning as it became both more flexible and better attuned to the properties of multiplication. This study is situated in a Number and Operations course. We document progress over time in Valerie’s reasoning, we relate those changes to significant classroom events, and we identify particular tools and models that supported Valerie’s progress.² Such detailed accounts of PTs’ learning, which are scarce in the literature, have the power to better equip mathematics teacher educators to support PTs in understanding elementary mathematics deeply (Mewborn, 2001).