"The title Children’s Mathematics aptly describes the purpose of this book, which is to highlight the extent to which children’s mathematical markings can be a window of opportunity. This book attempts to understand how children in the early years (3 – 8 years age range) contemplate and comprehend their mathematical world.

      Over a twelve-year period, Worthington and Carruthers have assembled mathematical evidence from children. They have analysed approximately seven hundred examples of graphics depicting powerful cognition (schemas) which gradually evolve into recognisable forms of mathematics in the early years of children’s development. The examples include an assortment of the authors’ own teaching practices and selections from experiences in other classrooms as consultants, advisors and lecturers. They have also drawn upon their thirteen unpublished research works. Passionate about mathematics, when researching in 1990 they came to the realization that, whilst there was an abundance of literature about literacy, there existed a corresponding lack of material addressing children’s mathematics.

      Children’s Mathematics recognises that teaching and learning practice is shaped by theories past and present and this book is centred upon socio- cultural theory. It commences by focusing upon a sampling of learning theories and examines how they have influenced teachers’ beliefs and practices in respect to children’s expectations and mathematical learning. Mathematics is often viewed from a behaviourist perspective, wherein traditional teaching is the norm. Such a viewpoint warrants scrutiny.

      In order to comprehend children’s mathematical thinking, the authors applied Athey’s (1990) model of ‘schemas’ – which describe children’s repeated behaviours patterns and are based upon children’s self-interest – as they contain parallels to mathematics. Schemas provide insights into children’s development, because through schemas children grasp ideas intuitively; and so as children explore their world they build upon their natural curiosity about maths. Hence schemas form the ‘footstools’ for more complex thinking about maths. The schematic marks that children initiate during this formative stage help bridge the gap between formal and informal mathematical mathematics. To support children’s schemas is to feed their natural curiosity which, in turn, extends to their mathematical thinking.

      In respect of emergent writing, we are currently in the position we were twenty-five years ago, especially in terms of literacy. Educators then were beginning to understand how children’s earliest scribbles evolved into formalized letters, words and sentences. Those whom understand emergent writing will find it easier to help children move from informal maths to more abstract versions. The authors contend there are important links between children’s early literacy and early mathematical graphics. For while younger children do not place formalized marks on paper as readily as older children, they nevertheless make marks, some of which make mathematical meaning. Although not complementing as adult conception of mathematics, these rudimentary marks are children’s representations fo thoughts and mathematical interpretations. Provided these early graphics are nurtured and developed further, then the written method helps bridge the gap between informal ‘home’ mathematics and more formalized abstract versions.

      Comprehension of accepted mathematical terminology to deduce meaning is problematic. It is a situation too often ignored, yet remains central in the mathematics debate. Young children frequently express confusion when translating from home vocabulary to school mathematics terminology; the imposition of an alternative language can present an awesome gulf for some. Worthington and Carruthers therefore, propose countenancing movement between children’s own mathematical understandings / graphics and formal mathematical symbolism. Children are then gradually encouraged to transform their marks into standardised mathematical symbols, the initial transposition metamorphoses into a single strand. This ‘bi-numerate’ process compares with second language learners, whereby children interpret more than one language simultaneously.

      Following analysis of approximately seven hundred children’s examples, the authors identified five common forms of graphical marks: dynamic, pictographic, iconic, written and symbolic. They recognised a development in – and the correlation of – the children’s marks and meanings. Because as the marks were co-constructed and negotiated with their peers, the children’s mathematical ideas were extended. Altogether, five principles of the development of mathematical graphics were identified: early play with objects and exploration of marks, early written numerals, numerals as labels, representations of quantities and counting, and early operations – development of children’s own written methods.

      Chapter eight couldn’t arrive soon enough for this reviewer; given the professed importance bestowed on the graphics, what type of environment would support their implementation? It requires and environment that authorises children to initiate and make sense of their own mathematical understandings, either individually or collaboratively. Either approach should be valued equally, with adult directed or adult led sessions. Case studies from classroom practice begins with observations of children initiating their won learning through play, plus a mixture of teacher directed small groups and whole class teaching. These examples demonstrate children are not adverse in employing their own graphics and ways of working, they understood what they had achieved. The exercise contained meaning for them. When assessing the marks therefore, the authors propose a divergent method with the aim to identify what the learner knows, understands and can manage; it is no good saying what is right or wrong – that leads nowhere for further teaching.

      Children, all too frequently, are influenced to play the ‘mathematical game’ of finding the ‘one right answer’ which coincides with teachers’ expectations. In the struggle for understanding however, when an independent approach to mathematics is encouraged and nurtured, and children are licensed to consider a range of possibilities, they discover and absorb more about mathematical concepts and attain more than their perceived best. This book demonstrates the provocativeness of maths and its potential to inspire, motivate and challenge the youngest minds. Children’s Mathematics will stimulate professional development in early years’ mathematics and is recommended reading for all early educators."

Heather Lawes, Department of Education and Children’s Services (South Australia).