# Introduction

In Chapter 1,1 will briefly introduce two fundamental mathematical ideas and three instructional principles that undergird this book and my NSF project on which this book is based. Using these fundamental ideas and principles as a conceptual framework, I have examined the classroom practices of U.S. and Chinese expert teachers and developed an approach named *Teaching through Example-based Problem Solving* (TEPS), which will be detailed throughout the rest of this book.

## Two Fundamental Mathematical Ideas: Early Algebra

Early algebra in elementary school has drawn increasing attention in the field (Cai *8c* Knuth, 2011; Carpenter et al., 2003; Greenes *8c* Rubenstein, 2008; Howe, 2010,2014; Kaput, Carrahcr, *8c* Blanton, 2008; Kieran, 2018; Lannin, Ellis, *8c* Elliott, 2011; Russell et ah, 2011). In the Common Core State Standards, there is a domain named “Operations *8c* Algebraic Thinking” which appears across grades K-5 (NGAC *8c* CCSSO, 2010). The existence of this domain indicates that the national standards anticipate the development of children’s algebraic thinking during the teaching of arithmetic throughout all elementary grades. In other words, arithmetic concepts should be taught in ways that lay a foundation for students’ future learning of algebra.

Early algebra may be approached from various angles and early algebraic concepts include many topics such as equivalence, patterns, and functional thinking (Blanton, Levi, Crites, *8c* Dougherty, 2011; Carpenter et al., 2003; Schliemann, Carraher, & Brizuela, 2007). In this book, I focus on two fundamental mathematical ideas—inverse relations and the basic properties of operations—that occur repeatedly across early grades in the Common Core. For instance, the Grade 1 (Gl)^{1} and Grade 3 (G3) standards for “Operations *8c* Algebraic Thinking” require that students

Understand and apply properties of operations and the relationship between addition and subtraction. (CCSS.MATH. CONTENT. 1. OA. В. 3).

Understand properties of multiplication and the relationship between multiplication and division. (CCSS.MATH.C0NTENT.3.0A.B.5).

In addition, the “Number & Operations in Base Ten” standards treat inverse relations and the basic properties of operations with the same importance as place value. For instance, the standards expect students to add and subtract “*based on place value, properties of operations, and/ or the relationship between addition and subtraction*” (c.g., CCSS.MATH. CONTENT.1.NBT.C.4). Likewise, they expect students to multiply and divide “*based on place value, the properties of operations, and/or the relationship between multiplication and division*” (e.g., CCSS.MATH. CONTENT.5.NBT.B.6). In fact, early in kindergarten, students are expected to understand number composition and decomposition, which lays a foundation for understanding inverse relations (Ding, 2016).

As indicated above, inverse relations include both additive and multiplicative cases. In U.S. classrooms, additive inverses mainly appear in grades K-2 while multiplicative inverses appear in grades 3-5. In Chapter 2, I will provide detailed examples of inverse tasks. Researchers (e.g., Nunes, Bryant, & Watson, 2009) argue that students cannot fully comprehend the four arithmetic operations without an understanding of inverse relations. Nunes et al. further point out that children’s inverse understanding predicts their future algebraic learning. Despite being a fundamental mathematical idea that should be learned in elementary school (NGAC & CCSSO, 2010), students have often been found to lack understanding of this concept (Baroody, 1987; Nunes et al., 2009; Torbeyns, De Smedt, Ghesquiere, & Verschaffel, 2009).

The “basic properties of operations” mainly refer to the commutative, the associative, and the distributive properties (CP, AP, and DP, respectively). These properties, which are also known as the laws of arithmetic, form the backbone of algebraic understanding (Bruner, 1977; Howe, 2010, 2014; National Research Council [NRC], 2001; Schifter, Monk, Russell, & Bastable, 2008; Wu, 2009). Imagine: when one solves 3.v + 5.v = 16, they have to combine the like-terms and transform the equation into *3x* + 5л; = (3 + 5)л;=16. The DP is inherent in the way these like-terms are added. Even though the basic properties of operations may be formally introduced in later elementary grades, they all can be embedded in earlier learning of number and operations. As introduced by the Common Core (NGAC & CCSSO, 2010), when a first-grader understands that “If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known” (p. 15), they arc using the CP of addition. In the same vein, when adding 2 + 6 + 4, if a first grader knows “the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12” (p. 15), they are using the AP of addition.

Even though studies have reported the potential of enhancing children’s learning with these fundamental ideas (e.g., Nunes et al., 2009; Schifter, Monk et al., 2008), the current literature lacks instruction about how these big ideas may be taught in classrooms to develop children’s sophisticated algebraic thinking. This book aims to address this gap by exploring instructional insights that are aligned with cognitive research assertions and best international practices.