"The authors of this book explore the ways in which children make their own marks on paper to signify their thinking in early mathematics. The writers contend that children’s own ‘mathematical graphics’ enable them to make links between the informal mathematics that they confront in the classroom. They view learning school mathematics as being akin to learning a language and argue that children’s mathematical graphics help them to grasp the new language of formal mathematics, allowing them to become ‘bi-numerate’. The book contains many examples of mathematical graphics gathered from British children aged three to eight years of age. The writers discuss young children’s behavioural schemas and the ways in which they use their own writing to explore mathematical ideas. They emphasize the active role that children play in their own learning. They also give practical suggestions for teachers on providing children with environments that stimulate their mark-making and helping children to develop their own written methods for mathematics. In addition, the authors tackle the challenging issue of assessing young children’s representations and provide useful advice on what we can learn about young children from studying their mathematical marks. Throughout the book, the authors emphasize the importance of mathematics in the home and suggest ways of involving parents and families in helping young children to develop their mathematical thinking.

     The writers make a very convincing case for the usefulness of exploring children’s marks in order to understand their mathematical cognition. The examples of children’s written representations provide fascinating insights into how different children think about mathematics. One child represents a set of objects by covering a page with dots and another draws a set of objects and then a hand taking away a subset in order to solve a subtraction problem. The authors favour eliciting and recording verbal explanations from children when possible, and many of the examples they give require careful consideration of both the written marks and children’s verbal descriptions in order to understand children’s thinking. Indeed, in many cases, the verbal explanations that children give are as illuminating as their written marks. Children use a wide range of strategies to tackle the mathematical problems that they encounter inside and outside the classroom, and their marks reflect this. Moreover, the encouragement to use paper and pens appears to help young children invent even more problem-solving strategies. The descriptions of young children’s marks and explanations in the book help us to understand how rich and varied their mathematical thinking (e.g. the correctness of their responses to a worksheet). We need to explore the processes by which individual children arrive at their answers and not just focus on the products of their thinking. Taking children’s mathematical marks seriously helps them to achieve this aim.  

     The authors argue that children's mark-making provides them with a developmental stepping-stone between their own mathematics based on knowledge of the everyday world and the abstract symbolism of school mathematics. This is an important claim. It is important because a key challenge for both educators and researchers in children's mathematics is the gap between the mathematical concepts and relations that children understand in the real worlds of home and play, and the abstract language of school mathematics. There are numerous reports in the psychological and educational literature of children entering formal schooling with rich and powerful ways of approaching mathematics in daily life but failing to make connections between this strong knowledge base and school mathematics. Some researchers have advocated the use of physical aids as a way of helping children bridge the gap. Yet, as the authors point out, the role of concrete manipulatives in helping children make connections between their own informal mathematics and formal school mathematics has not been clearly established.

    Currently though, it also seems unclear whether children's mark-making facilitates or merely reflects developmental changes in their mathematical cognition. Children's marks are certainly worthy of consideration as windows on their thinking, but further research is needed to establish the precise role that mark-making plays in their cognitive development. Nonetheless, the book contains some very strong pointers to the potential developmental significance of mark-making. In particular, many of the examples of children's written representation seem to capture a mixture of formal and informal mathematical thinking by combining children's idiosyncratic written representation with their use of formal symbols. These examples suggest that at least some children use their marks on paper to help them reflect on the links between formal and informal mathemaitcs. Such examples support the authors' claims for the centrality of children's mark-making and underscore the need for researchers and educators to consider carefully the role of children's own 'written' methods in their mathematical development." Kate Canobi, The Department of Psychology, University of Melbourne. Australia.