What would proving, that is, age appropriate mathematical reasoning, look like for 8- and 9-year olds? Through actual classroom observations and interventions, initially gathered for research, the author describes “what one should look for and expect of young children engaged in proving activities” (Forward). The eight main classroom episodes come from a Year 4 class in England (8- and 9-year olds) taught by Mrs. Howard (a pseudonym) or from a third-grade class in the U.S. taught by Deborah Ball. The proving activities are mainly arithmetical or combinatorial, but one can imagine similar geometrical tasks. A key question that the author asks, and attempts to answer, is: What would it take for elementary teachers to productively engage their students in proving? (p. 36).

Chapter 1, the Introduction, sets the stage for the rest of the book. It begins with an actual classroom episode involving Vicky Zack’s fifth-grade Canadian class. The children were pondering: How many different squares of various sizes are there in a 4-by-4 square, made up of 16 1-by-1 “little squares”. That is, how many different 1-by-1, 2-by-2, 3-by-3, and 4-by-4 squares are there? After noticing a pattern, namely that the number of squares for the 4-by-4 case is 1^{2} + 2^{2 }+ 3^{2 }+ 4^{2 }= 30, the children considered the problem for 10-by-10 and 60-by-60 squares. A few children conjectured that the number of “little squares” for the 60-by-60 square would be obtained by multiplying the total number of 10-by-10 squares by 6, that is, 385 x 6 = 2310. But other children challenged this, offering the argument that considering only the number of 1-by-1 squares would already result in 60 x 60 =3600, which settled the matter. The point of the episode is to show that children themselves are capable of resolving their mathematical disagreements, not by appealing to the authority of the teacher, but by means of mathematical reasoning. This episode, like all the others in the book, are taken from actual research either already reported in the literature or undertaken for the writing of the book.

Chapter 2 discusses the importance of proving and the role of mathematical tasks. While mathematicians do not need to be convinced of the importance of proving, this chapter considers the relevant literature on the importance of proving in elementary school, which is somewhat different. The author states that for him “proving” is defined broadly to denote the mathematical activity associated with the search for a proof, where proof is defined as follows.

Proof is a mathematical argument, a connected sequence of assertions for or against a mathematical claim, with the following characteristics:

1. It uses statements accepted by the classroom community (set of accepted statements) that are true and available without further justification;

2. It employs forms of reasoning (modes of argumentation) that are valid and know to, or within the conceptual reach of, the classroom community; and

3. It is communicated with forms of expression (modes of argument representation) that are appropriate and known to, or within the conceptual reach of, the classroom community. (Stylianides, 2007b, p. 291)

After this in Chapter 2, the author reconsiders the introductory classroom episode from Vicky Zack’s fifth-grade class and indicates why he considers the children’s argument as a proof in light of the above definition. (1). It used true statements already accepted by the class, namely that there are 60 x 60 = 3600 1-by-1“little squares” in the 60-by-60 square. (2). It employed reasoning understandable to the class, namely, that accepting the conjecture leads to a contradiction as the total number of squares is definitely bigger than 3600, and hence, the total number cannot be 2310. (3). The oral argument was presented in language that the students in the class could follow.

Chapter 3 discusses the book’s categorization of the proving tasks, whose description and implementation constitutes the bulk of the book (Chapters 4–7, on pages 41–152). There are eight detailed classroom episodes, A through H, with Chapter 4 devoted to tasks with ambiguous conditions. Chapters 5, 6, and 7 discuss tasks involving a single case, finitely many cases, and infinitely many cases, respectively. Episode A in Chapter 4 comes from Mrs. Howard’s class: *How many ways do you think there are to make the number 10*? (*Prove your answer*)*. *The children’s responses, both classroom excerpts and written work, are described and exhibited. The teacher and children together gradually refine the question several times. One of these is: *How many ways do you think there are to make the number 10 if we just use addition and commutative pairs of number sentences do not count as different?* (*Prove your answer*)*. *As the children engaged with this question, they made observations, asked questions, illustrated their ideas on paper, and justified their claims. For example, one child’s observation was that allowing negative numbers would lead to an infinite number of ways to make 10. Not only did the task engage students in exploring, conjecturing, and justifying, it gave them opportunities to solidify and deepen their knowledge of arithmetical properties.

The classroom episodes in Chapters 4–7, together with children’s responses and work, can be very helpful to readers, especially preservice elementary teachers and beginning elementary teachers, who may not have many ideas about, or very much experience with, the variety of tasks they might pose and the variety of responses 8- and 9-year old children can give. At the end of each chapter after the classroom episodes have been described and illustrated, there is a general discussion with considerations and suggestions regarding the role of the teacher. These include clarifying assumptions and definitions, as well as the kinds of implementation considerations a teacher might want to make. For example, a teacher might want to consider whether an ambiguous task, or a more clearly articulated proving task, would be more appropriate for a given class of children.

Chapter 8 is a short concluding chapter. Table 8.1 succinctly summarize the eight main episodes in Chapters 4–7, indicating the purposes of the proving tasks in terms of justifying or refuting statements. For example, for the task in Episode A discussed above, the aim is to have students learn to systematically consider all of the finitely many possibilities. In addition, the role of the teacher in selecting or designing proving tasks is highlighted. Along the way, the author points out that while most of the curriculum in the early grades is devoted to calculation that this work, “or work in arithmetic more broadly, need not be separated from proving”. Further, “Proving can afford students with an opportunity to deepen their existing knowledge of calculation algorithms.” (p. 86).

I have a few minor quibbles with the book. The author, after the Preface, refers to Mrs. Howard as “Howard”, which could be the name of one of the children, which initially makes some of the excerpts a bit harder to follow than necessary. Referring to Deborah Ball as “Ball” is less disturbing as it is less likely to be a child’s name. Another quibble is the use of British terms like “secondment” (p. 37) unknown to most U.S readers. I had never encountered the term “secondment” until I taught at a Nigerian university which had several professors from the U.K. who were on secondment, that is, essentially on loan to the Nigerian university for a period of five years. In Chapter 5, which contains a Missing Digits problem, the reader needs to decipher that the “carry digits” are below, rather than above, the calculation, in contrast to how American children are taught to do it. No explanation is given. (In elementary arithmetic, a *carry digit* is a digit that is transferred from one column of digits to another column of more significant digits.) While this can be figured out, writing the calculation this way without explanation adds an unnecessary complication for a U.S. reader.

Who is this book for? While it is clear it could be very useful for preservice and beginning inservice elementary teachers, the research upon which the book is based is well-documented. So masters’ students in a teacher education program focusing on mathematics for elementary education would also benefit. Also, aspiring researchers in elementary mathematics education will find the many references (pp. 167–180) useful. Also, the numerous text citations are given in *APA Style*; for example “Zaslavsky (2005),” so an interested reader can easily consult the reference list in the back and go to the original source for more details about a particular research study. Indeed, anyone teaching a methods course for preservice elementary teachers would find this book a good source of proving tasks/activities, together with actual children’s words and work. I can even imagine a mathematics for elementary education course, taught in a mathematics department by someone who favors active learning, using some of these activities with preservice elementary teachers.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.