Mathematics Further Reading
We hope you find these suggestions for further reading helpful. We have provided links where appropriate, but in many cases a subscription will be required to access a particular research article. Where an article is available without charge on another website, we have included a link to that for your information. In the case of subscription articles, we do not recommend any particular provider and suggest that you use the information provided below to help you identify the most appropriate research source for you.
Carruthers, E. (2015). Listening to children’s mathematics in school. In B. Perry., A. Gervasoni and A. MacDonald. Eds. Mathematics and Transition to School – International Perspectives. Sydney, Australia: Springer.
This chapter begins by emphasising how the Foundation Stage has resulted in ‘’uneasy pedagogies’ that are vastly at odds with the same curriculum’s emphasis on valuing children’s interests, and resulting assessment scores show poor achievement. This chapter argues that there is clearly a need to make a conceptual shift in the teaching of mathematics in England. The author builds on previous work into children’s mathematical graphics (e.g. Carruthers and Worthington, 2006) highlighting how this differs from children ‘recording’ mathematics. The chapter focuses not only on the children’s graphics, but also on the pedagogy, imaginary play and child-initiated learning. Evidence from a study conducted with 15 teachers in the Foundation Stage highlights a question posed by one teacher in the study, ‘How do teachers question and stand firm with their own views against a political agenda?” It concludes that government bodies and teacher representations need to ‘respectfully and knowingly’ listen and respond to teachers.
Carruthers, E. (2012) Are the children thinking mathematically? The pedagogy of children’s mathematical graphics. In. M. McAteer. Improving Primary Mathematics: Teaching and Learning. Maidenhead: Open University Press, 192-212.
Based on a case study of one teacher, this chapter explores the pedagogy of children’s mathematical graphics as she describes how she started to uncover the children’s graphical communications. Beginning with two government publications popular with teachers and early years practitioners; Mark Making Matters (DCSF 2008) and Children Thinking Mathematically (DCSF 2009), shows how their emphasis on children and their personal graphics is at considerable odds with the current curriculum for mathematics. The chapter aims to uncover the pedagogy to support the children’s mathematical thinking, highlighting significant pedagogical strategies and explaining some of the strategies in the primary school. Drawing on one expert teacher’s reflections as she develops her thinking it includes a number of children’s examples. The teacher in the study concluded that the children in her class were now exceeding expectations for mathematics in the Foundation Stage curriculum, and that the curriculum failed to support teachers’ understanding of children’s mathematics.
Worthington, M. and Van Oers, B. (2016) Pretend play and the cultural foundations of mathematics. European Early Childhood Research Association Journal, 1-16.
The aim of this study is to uncover the emergence of cultural mathematical understandings and communications in young children’s spontaneous pretend play. It is based on Vygotskian cultural-historical perspectives and social-semiotic theory, informed by research into ‘funds of knowledge’ and considers how children’s informal knowledge of family practices enriches their play and cultural mathematical understandings. Longitudinal, ethnographic data were gathered in an inner-city mainstream nursery in the south-west of England. Data include written observation and graphics of seven children aged three to four years of age engaged in social pretend play. The findings reveal that many play episodes included aspects of mathematics and that these increased through the year: they show how the children’s home cultural knowledge underpinned their pretend play and informed their mathematics. Where children are immersed in mathematical- and graphical-rich environments, bridging home and early childhood cultures becomes a natural feature of their pretend play.
Worthington, M. and Van Oers, B. (2016) Children’s social literacies: Meaning making and the emergence of graphical signs and texts in pretence. Journal of Early Childhood Literacy, 1-29.
This study builds on recent research into young children’s pretend play. Social literacy practices and events in which children engaged were investigated to reveal features of their meaning making. Data were collected from case studies of seven children aged three to four years in an inner-city maintained nursery school in southwest England, as part of a larger longitudinal ethnographic study. Data comprise written documentation of the children’s play and their visual representations, and the analysis follows an interpretive, social semiotic multimodal paradigm. The findings make a compelling case for greater appreciation of pretence as a potentially valuable context for the enculturation of literacies, highlighting the diversity and richness of children’s spontaneous meaning making and self-chosen literacy events. Informed by cultural and literacy practices of home and nursery, the children’s communications show how meanings and signs are carried across time, space and contexts. Rich and sustained play supported the children’s self-initiated literacies in which they explored a heterogeneous range of textual genres, revealing their developing semiotic understandings and expanding repertoire.
Worthington, M. (2015) Draft title of doctorate: The genesis of mathematical semiosis in early childhood.
This doctoral research analyses data from case studies of seven 3-4 year olds attending the maintained nursery at Redcliffe Children’s Centre in Bristol. It considers the children’s cultural mathematical knowledge of home and ways in which they adapt and extend this knowledge within their rich pretend play in the nursery. The research focuses particularly on the children’s own graphical communications of mathematics, highlighting ways in which the children make meaningful beginnings with the abstract symbolic language of mathematics and how these connect to the mathematical notations of ‘school’ mathematics. The study argues that meaningful social and cultural contexts such as pretend play provide potentially rich and sustained contexts for young children’s mathematics. The findings are likely to challenge current narrow ‘basic skills’ approaches in England.
Abstracts for three doctoral papers for this ongoing research are attached below.
Hughes, M. (1986) Children and Number: Difficulties in Learning Mathematics. Oxford: John Wiley and Sons.
In Children and Number Martin Hughes proposes a new perspective on children’s early attempts to understand mathematics. He describes the surprisingly substantial knowledge about number that children acquire naturally before they start school, contrasting this with the difficulties presented by the formal written symbolism of mathematics in the classroom. He argues that children need to build links between their informal and their formal understanding of number, and shows what happens when these links are not made.
Children and Number describes many novel ways in which young children can be helped to learn about number. The author shows that the written symbols children often invent for themselves are more meaningful to them than the symbols that they are taught.
Cook, D. 2006. Mathematical sense making and role play in the nursery. Early Child Development and Care, 121: 1, 55-66.
The question about “bridging the gap” between learning at home and learning in nursery or school is an important one for teachers and others to consider.
Firstly how is this body of knowledge to be assessed and secondly how might it be exploited in a manner suitable for young children. Given the number of individual children concerned and the curriculum imperatives existing in many countries the question of moving children’s learning from the intimate immediate, socially contextualised informal and functionally relevant literacy and mathematics of home and family to the decontextualised abstract, procedurally driven world of school learning is of great importance. Drawing upon Literacy studies this study recognised a number of dominant strands, the author of this article observed and added numerate materials to the children’s themed pretend play areas to stimulate mathematical talk and representations.
Björklund, C. (2008) Toddlers opportunities to learn mathematics. International Journal of Early Childhood, Vol. 40, No. 1, 81-95.
The author of this paper emphasises how children will strive to understand their surrounding world, through their interactions with the world and with people in which they grow, when mathematics is seen as a social and cultural phenomenon. Their environment therefore plays an important role in what children experience and their opportunities for learning. This article describes opportunities for toddlers to experience and learn basic mathematical concepts. Analysis of critical conditions of learning that may be discerned in toddlers’ daily activities is explored and four authentic examples are described and discussed. Results from the qualitatively analysed videographic study of toddlers’ experiences of mathematics in day-care show that variation, simultaneity and reasonableness seem to be critical conditions of learning, as well as the opportunity to focus on important aspects in a phenomenon. Adults working with toddlers therefore play a very important role in setting perimeters for toddlers’ experiences and opportunities to explore mathematical phenomena.
Björklund, C. (2010) Broadening the horizon: toddlers’ strategies for learning mathematics, International Journal of Early Years Education, 18:1, 71-84.
This study analyses and discusses those strategies for learning that are essential for toddlers’ development of an understanding of basic aspects of mathematics. Analysis focuses on authentic episodes where toddlers aged one to three are interacting with other people. Critical aspects that a learner has not been previously able to focus on may emerge during interaction with others since such settings provide an opportunity for the problemisation of differing aspects and perspectives. This study seeks to discern toddlers’ strategies for learning through the observation of those strategies that the toddlers themselves initiate, which may reveal the competence for learning that young children possess. The author concludes that toddlers, when given the opportunity to develop their interaction on their own terms, use the ensuing change in perspective as a strategy for learning. For young children, the ability to set goals, express one’s understanding, and take the other’s perspective and see it as an opportunity to deepen one’s own understanding then becomes essential.
Björklund, C. (2012) What counts when working with mathematics in a toddler-group? Early Years, 32(2), 215-228
This study investigated an early years educator working in a toddler-group (children aged 1–3) who took part in an in- service program. The purpose of this study is in the context of explorative play and describes how the educator develops strategies where toddlers are given relevant opportunities to explore mathematical concepts. Results of the analyses highlight the effects of a heightened awareness of mathematics that enhance the opportunities to explore mathematical concepts and principles. It also shows how this can lead to an awareness demonstrated in interaction with the children, enabling the educator to take the children’s initiatives as starting points for planned education, and the need to use subtle and adequate mathematical language with very young children.
Reikerås, E., Moser, T. and Egil Tønnessen, F. (2015) Mathematical skills and motor life skills in toddlers: do differences in mathematical skills reflect differences in motor skills? European Early Childhood Education Research Journal, 1-17.
This Norwegian study focuses on children’s mathematical and motor life skills, which were assessed by structured observation in the natural environments. The children’s mathematical skills were assessed using the observation material consisting of six sections: counting and series of numbers; enumeration; shape and space; pattern and order; mathematical language and logical reasoning. Consistent with recent studies of older children, the results revealed a significant relationship between motor skills and mathematical skills at an early age (two years and nine months) and that the relation between these two developmental domains can be measured by authentic assessment conducted by kindergarten teachers.
Guy Roberts-Holmes (2015) The ‘datafication’ of early years pedagogy: ‘if the teaching is good, the data should be good and if there’s bad teaching, there is bad data’, Journal of Education Policy, 30:3, 302-315.
This English article argues that early years high-stakes national assessments act as a ‘meta-policy’, ‘steering’ early years pedagogy ‘from a distance’ and have the power to challenge, disrupt and constrain early years teachers’ deeply held child- centred pedagogical values. Roberts-Holmes argues that the current narrowing of early years assessment, along with increased inspection and surveillance, operates as a policy technology leading to an intensification of ‘school readiness’ pressures upon the earliest stage of education. Focusing on literacy and maths, the paper suggests that this has encouraged a functional ‘datafication’ of early years pedagogy so that early years teachers’ work is increasingly constrained by performativity demands to produce ‘appropriate’ data that results in more formal schooling. This data ‘delivery chain’ may well start earlier and become stronger from September 2016 when the English Government plans to impose a Baseline Check on four-year-old children in Reception class.
NCETM (2009) Researching Effective CPD for Mathematics Education (RECME). National Centre for Excellence in Teaching Mathematics.
This 2-year, nationally funded project is the largest UK research project into CPD in mathematics education to date. It was led by Project Director Dr. Els De Geest from the University of Oxford with researchers form University of Bristol, Kings College London and University of Birmingham, and supported by the National Centre for Excellence in Teaching Mathematics (NCETM). The CPD initiative – a local Children’s Mathematics Network group in Bristol – is featured in the final report (with a focus on the Bristol CMN group) as one of six successful national CPD initiatives.
Back, J. and Joubert, M. (2009) Reflecting on practice in early years’ settings: developing teachers’ understandings of children’s early mathematics. Proceedings of the British Society for Research into Learning Mathematics 29(1) March 2009, 14-18.
This article presents some of the findings of the Researching Effective Ontinuing Professional Development in Mathematics Education (RECME) Project. It focuses on a Children’s Mathematics Network Group in Bristol, a grassroots group that was owned and developed by teachers and practitioners. The group’s informal meetings impacted on the teachers’ and practitioners’ thinking and practice over time and changed their understanding of young children’s mathematical thinking and representations. Elizabeth Carruthers and Maulfry Worthington promote Early Years Children’s Mathematics Network Groups through the Children’s Mathematics Network.
NOTE: The pseudonyms ‘Melanie’ and ‘Lizzy’ in the article refer to Maulfry Worthington and Elizabeth Carruthers.
Carruthers, E. and Worthington. M. (2009) An early years CPD initiative for mathematics: the power of ‘grassroots’ learning. Proceedings of the British Society for Research into Learning Mathematics, 29 (1) March 2009, 1-6.
Local ‘grassroots’ Children’s Mathematics Network groups are initiated and ‘owned’ by teachers and practitioners who explore and develop their understanding of children’s mathematical graphics (Carruthers & Worthington, 2005; 2006; DCSF, 2008) in their own ways. New research findings from Bristol reveal the effectiveness of this form of ‘continuing professional development’ (CPD) and its impact on children’s mathematical thinking (NCETM, 2009). This paper explores the philosophy underpinning these groups, and their inter-connectedness with teachers’ and practitioners developing pedagogy and children’s mathematics. The research on which this article draws was a DfES funded research study in mathematics education was conducted as part of the Researching Effective CPD for Mathematics Education (RECME).
Johnsen Høines, M. (2004) Do we welcome children’s mathematics? Who are we? Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004, 1, (scroll down to pages 84-89 of the conference report to read this article).
In this article Norwegian researcher Marit Johnsen Høines considers how planning for mathematics play an important role in embracing informal mathematics learning. She discusses the contradictions in teaching the subject of mathematics and how including children’s perspectives and understanding allow a significant perspective on differentiation, inclusion and diversity.
Anthony, G. and Walshaw, M. (2009) Mathematics Education in the Early Years: building bridges, Contemporary Issues in Early Childhood, 10(2), 107-121.
The development of mathematical competencies begins at birth. This article argues that all students, irrespective of age, have the capacity to become powerful mathematics learners, and that young children’s developing understandings are most appropriately situated within ‘social and cultural contexts that make sense to the children involved’. The authors emphasise that policy makers and mathematics educators need to be increasingly informed by research that bridges the early years divide, if calls for greater harmony of approaches to the teaching of mathematics across early years education are to be realised.
MacDonald, A. (2012) Young children’s photographs of measurement in the home. Early Years: An International Research Journal, 32:1, 71-85.
This study features digital photographs taken by five- and six-year old children, taken in order to provide insight into the types of experiences children have with measurement in the home context. It highlights the ways that children’s engagement in cultural practices outside of school have a profound impact on the knowledge that children bring to the classroom, and that may provide a base for school instruction. The study shows how the children were able to identify attributes, and comparison featured strongly among the photographs, with images including length, height, volume, and mass.
Franzén, K. (2015) Under threes’ mathematical learning. European Early Childhood Education Research Journal, 2015, 23(1), 43-54.
The article focuses on mathematics for young children in preschool in Sweden. It offers a new perspective on learning, showing how that children often use their bodies as a tool for understanding mathematical concepts.
Maclellan, E. (1993) The significance of counting. Education 3-13, 21(3), p. 18-22.
This article highlights the importance of counting in the early years and how we should be counting more and reflecting upon the ‘matching’ and ‘sorting’ activities introduced by Jean Piaget over 40 years ago.
Aubrey, C. School of Education, University of Durham: An investigation of children’s knowledge of mathematics at school entry and the knowledge their teachers hold about teaching and learning mathematics, about young learners and mathematical subject knowledge. British Educational Research Journal, 20 (1), pp. 105-120.
Van Oers, B. Is it play? (2013) Towards a reconceptualisation of role play from an activity theory perspective. European Early Childhood Education Research Journal, 21(2), 185-196).
Seo, K-H. and Ginsburg, H. (2011) What is developmentally appropriate in early childhood mathematics education? In D. Clements., J. Sarama. And DiBiase, A-M. (Eds.) Engaging Young Children In Mathematics. London: Routledge.
For an interesting interview with Dr Herbert Ginsburg, visit the Scholastic website www.scholastic.com
Haylock, D. (2007) Key Concepts in Teaching Primary Mathematics. London: Sage Publications.
Covering the key principles and concepts in the teaching and learning of mathematics in primary schools, this text provides trainee and practising teachers with a quick and easy reference to what they need to know for their course, and in the classroom. The entries are arranged alphabetically, and each contains a brief definition, followed by an explanation and discussion, practical examples and annotated suggestions for further reading.
K Young-Hye, S. and Ginsberg, H. (2003) What is developmentally appropriate in early childhood mathematics? Lessons from new research. In Clements, D. and Sarama, J. (eds.) Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education. London: Routledge, 91-104
Boaler, J. (2010) The Elephant in the Classroom: Helping Children Learn and Love Maths. London: Souvenir Press Ltd.
Why do millions struggle with mathematics and what can teachers do to change that? Jo Boaler has followed the progress of thousands of pupils in two countries (the UK and USA), monitoring how they learn maths through their school careers and then following them into adult life. This remarkable research is the foundation of her investigations into the impact that differing maths experiences can have on an entire generation. Jo Boaler outlines what has gone wrong, identifying the problems facing children in mathematics classrooms today and offers concrete solutions for parents and teachers that will revolutionise children s experiences with maths. The Elephant in the Maths Classroom offers concrete suggestions on ways to teach maths well, and ways to help children in the home, that will offer new and more effective ways of learning maths. This is an exciting way forward, a new approach that teaches children to reason and problem solve; helping all children, even those who think that they are maths failures and that they could never enjoy maths. An indispensable guide and resource for parents, teachers and educationalists, that inspires and enthuses as much as it teaches.
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