Eyes on Math: A Visual Approach to Teaching Math Conceptsreviewed by Wan Park  March 17, 2014 Title: Eyes on Math: A Visual Approach to Teaching Math Concepts Author(s): Marian Small Publisher: Teachers College Press, New York ISBN: 0807753912, Pages: 240, Year: 2012 Search for book at Amazon.com INTRODUCTION Picking Up the Book When faced with Eyes on Math: A Visual Approach to Teaching Math Concepts by Marian Small, and illustrated by Amy Lin, it is hard not to be confused initially, but the confusion is of the good kind. The duo of authors, who have already published More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction clearly have researched the important and effective role that inquiry plays in mathematical teaching and learning. A photo of a child inquisitively starring at what could only be described as friendly quadruplets (and some pairs) of penguins on floating ice graces the cover. The illustration of the penguins comes from one of the lessons in the book about multiplication for K2 graders. The question that accompanies the illustration is only halfshown on the photograph, and the incompleteness of the inquiry draws you into following the penguins inside of the book. A Need for a Hook as a Teacher For a math teacher, however, providing an illustration with an accompanying prompt only consists of a hook, however good that may be. Planning, implementing and reflecting on a lesson plan may require more substance than the material provided in the book. Moreover, the static nature of the visual and a prompt may not be enough to sustain the interest of students entrenched in today’s culture of dynamic multimedia. INTENDED OUTCOME VS MY EXPERIENCE Complementary Source of Material Luckily, the authors never intended the book to be either an edutainment piece or a textbook replacement. It is clear from scanning the table of contents that the book serves as a source of complementary lesson material that teachers can study ahead of the time, and adapt varying prompts that are most apt for particular classes of students. CONTENT OF THE BOOK Within the book, there are four chapters. The first chapter lays the foundational theories that ground the book in psychological and educational studies that have described the process and importance of visualizing in math, focusing on “important” math, and communicating mathematical ideas. Chapters Two, Three and Four, which make up the bulk of the book, list numerous visuals and the corresponding mathematical concepts for grade bands K2, 35, and 68, respectively. CHAPTER ONE  FOUNDING THEORIES Visualization with Math Ironically, the section in Chapter One that grounds the visualization in math may represent the weakest link. Some studies that Small cites are either too broad to be used for this kind of specific manner of implementation (Adam & Victor, 1993; Rowan & Bourne, 1994) or indirectly related to teaching with visualization in K8 mathematics classrooms (Nelson, 2000; Sadoski & Paivio, 2001; Tufte, 2001). The one research citation that directly relates the theory and the book’s intention aligns the type of learning with the method of learning (Murphy, 2007). Focus on Math Sections focusing on “important” math and mathematical communication in the rest of the Chapter One provide a solid foundation for the importance of providing opportunities for students to delve deeply into mathematical concepts by developing their ideas and communicating those ideas through interactions with others while learning. CHAPTER TWO  K2 GRADES Starting with Chapter One, the reader, presumably a math teacher, encounters a practical list of mathematical concepts and corresponding Common Core Standards, which can be crossreferenced in the Appendix to look up the lessons for a particular standard. Passing this list, the reader is finally invited to the actual illustrations and prompting questions. For K2 grades, the following topics are presented: counting, comparing quantities, adding, subtracting, place values, benchmark fractions, and shapes in 2D and 3D. One of the better visuals for kindergarten students involves writing “mathematical sentences,” given the visual of a bookcase with 3 shelves, each shelf with different number of books already on them. Next to the bookshelf is a pile of books. One of the math sentences you are expected to make is 7+6=13, since one of the shelves has 7 books and the pile has 6 books. This process of creating mathematical sentences out of a visual enacts a mental visualization of moving the books from the pile to the shelf. Since the act of moving or putting books on a shelf is familiar (or easy enough to imagine) to most students, transforming a phenomenon to a precise explanatory sentence allows learners to practice this type of articulation. In addition, one of the supplementary questions, which relates thinking about an addition strategy, asks students, “to figure out why 3+4 is, why can you say 5, 6, 7?” (p. 31) which refers to the 'counting on' strategy from the bigger addend. CHAPTER THREE  35 GRADES For higher elementary school grades, Chapter Three presents the following topics: multiplication, division, place values, fractions, operations with fractions, measurement, graphs, shapes, lines, and algebraic thinking. Similar to Chapter Two, 36 common core standards are linked to the visuals and questions. Among these indicators, operations and fractions are heavily emphasized. One of the better examples involves the introduction to multiplication concepts, which shows groups of 2 or 4 penguins floating on the ice. The authors believe there will be lively discussions in the class since the number of members in each group must be decided before the situation can be described as a multiplication sentence. CHAPTER FOUR  68 GRADES Chapter Four contains illustrations that introduce and/or explore the following topics: factors, ratios, rates, integers, algebraic properties, areas, Pythagorean theorem, pi, statistical ideas, probability, transformational geometry, and algebra. In these topics, number sense, expressions and equations, ratios and proportions, and geometry are addressed in similar numbers. Many visuals and questions lead students into the root of the concepts, as they progress into more sophisticated concepts. CONCLUSION The Quality of The Questions Overall, the authors provide numerous examples for math teachers to introduce or explore traditional lessons visually. The questions accompanying the illustrations present valuable prompts for inquiring about the math topics. The link to the Common Core standards makes it convenient for teachers to apply the lessons, and the supplementary questions are, in my opinion, not supplementary at all, but necessary for students to encounter. Depth of understanding may dramatically improve with these questions. Succeeding in Achieving the Intended Goal In the end, the authors achieve their goal of providing math teachers with complementary and invaluable materials for lessons. In practice, more likely than not, there always will be an assigned mathematics curriculum for the school, and since this book is not intended to replace the existing curriculum, instructors of elementary and middle school mathematics can rest easy in exploring options for presenting the lessons using this book.
