The volume entitled *Teaching and Learning Algebraic Thinking with 5- to 12-Year Olds* is an “ICME 13 Monograph,” which means a sort of proceedings volume from the Topic Study Group 10 (TSG 10) on the Teaching and Learning of Early Algebra, which held a conference in Hamburg Germany in July 2016. Before the group met, there was a pre-conference study presented by the volume’s editor, Carolyn Kieran, which focused on research on early algebra teaching. As pointed out by Kieran:

The core of recent research in early algebra has been a focus on mathematical relations, patterns, and arithmetical structures, with detailed attention to the reasoning processes used by young students, aged from about 6 to 12 years, as they come to construct there relations, patterns, and structures — processes such as noticing conjecturing, generalizing, representing, and justifying. Intertwined with the study of the ways in which these processes are engaged in are the two main mathematical content areas of generalized arithmetic (i.e., number/quantity, operations, properties) and functions.

This study stressed how the field of early algebra has gradually become more delineated since the early 2000s. The contents set the stage for TSG 10 researchers to contribute their research with more recent findings in early algebra best practices and to further their work.

It should be noted that the Hamburg Conference was the first of a quadrennial series of ICME conferences to include the theme of early algebra. The diversity of papers and posters presented attested to the fact that there is a strong international interest in early algebra as an important field of research in mathematics education.

The book includes 17 chapters (papers) representing works of both experienced and younger researchers from 13 countries: Australia, Canada, Germany, Ireland, Italy, Korea, New Zealand, Singapore, South Africa, Spain, Turkey, UK, and USA. Due to the strong international presence, the volume carries the subtitle *The Global Evolution of an Emerging Field of Research and Practice*.

The general focus of this volume is the development of algebraic thinking at the primary lower middle grades. Papers stem from both longitudinal programs of research as well as from single shorter-term studies. As pointed out in the introduction, there are six key aspects presented in the book:

- Theoretical perspectives such as the structural, the linguistic, the analytic, and the expression of generality.
- The emergence of symbolic algebraic thinking.
- Children’s algebraic thinking within current curricular content and the potential of those curricula to address algebraic thinking.
- Functional approaches focusing on the use of story problems, patterning, and function-machine tasks.
- Generalized arithmetic approaches involving work with fractions, operations as objects, and equality concepts.
- The development of practicing and pre-service teachers’ actions to promote algebraic thinking.

There are three parts of the book: Part I (Chapters 1–5) focus on the theoretical perspectives, Part II (Chapters 6–11) is on learning, and Part III (Chapters 12–17) on teaching.

In the first paper, *The Emergence of Symbolic Algebraic Thinking in Primary School*, Luis Radford references Mason et al. (1985) and Kaput et al. (2008a), who offer a conception of algebra that is linked to the idea of generalization. These two authors do not ascribe the same role to signs in algebra. For Mason the alphanumeric symbolism is not a condition for thinking algebraically, but for Kaput in order for a symbolic activity to be called algebraic full symbolization is required. Radford presents his work on a longitudinal investigation of early algebraic thinking on Grades 2 and 6. Even more importantly, his article sets the stage for the in-depth research presented in the entire book. For example, in the paper by JeongSuk Pang and JeongWon Kim, *Characteristics of Korean Students’ Early Algebraic Thinking: A Generalized Arithmetic Perspective*, a study was conducted on how students understand equations by looking at the equation 3n + 2 = 8 and 3☐ + 2 = 8. Letter variables are not taught in Korea until Grade 6.

In Chapter 15, *Cycles of Generalizing Activities in the Classroom*, Susanne Strachota, Eric Knuth, and Maria Blanton point out that many students must take algebra, as it is gatekeeper to future academic and employment opportunities. As pointed out by NCTM as well as several mathematics education scholars, generalization is the core of algebra, but students continue to struggle to generalize and to understand algebraic skills and concepts. In the early algebra intervention, this study focused on the commutative property of addition in grade 3. Table 15.2 on page 359 evaluates students’ algebraic understanding using a one-hour written assessment at the start and end of grades 3, 4, and 5. This was a three-year study and offers many teachable moments that can be used in your classroom. Figures 15.2-15.4 on pages 360–361 offers further intervention in the student’s response from the grade 4 post-test.

In this research on the commutative property of addition, classroom dialog is also presented, which leads to further generalizations as pointed out on page 370:

**Teacher:** Let’s think about this. If you have 3 + 2 like we had, 3 + 2, and 2 + 3, so you have 3 +…you’re adding 3 + 2 on this side and you have the same number on this side, what happens?

**Kyle:** Um, it’s a true equation.

**Teacher: **It’s a true equation. That’s good. Why is it a true equation?

**Kyle: **Because there’s a, there’s the same amount of the equal sign, and if there’s the same amount on the equal sign it’s true.

**Teacher:** All right. I think I know what you’re saying. Tell me if this is it. “Then it is a true equation because you have the same amount on both sides.” Right?

**Kyle:** Mmhmm.

Here, Kyle’s generalization becomes an object of generalization itself since students are able to relate to Kyle’s generalization to other generalizations or equations the represent the commutative property. This is also an example of relating particulars. Figure 15.7 presents the framework of interacting with generalizing-promoting actions.

This book is an excellent resource in the teaching of early algebra. As math educators, we know that algebra is one of the key subjects students need for higher mathematics courses. Many studies have been conducted and presented to help younger students get better introduction to algebra. Over the past decade, Mathematical Knowledge for Teaching (MKT) has been developed to help teachers have a better understanding of the subject matter and a better pedagogical content knowledge. The reader can certainly have a broad and international perspectives on teaching early algebra by reading this book and can be motivated to try out and conduct their own research study. I’m certainly motivated by this book.

Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His research fields are mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at pto2@psu.edu. Webpage: www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.