Topic Study Group 1:
New developments and trends in mathematics education at preschool level
B101 and B102 Rooms
  • Terezinha Nunes (United Kingdom)
    terezinha.nunes@education.ox.ac.uk
  • Douglas Clements (USA)
    clements@buffalo.edu
Top of page
Team members:
  • Sue Giffiord (United Kingdom)
    sgifford@dircon.co.uk
  • Clotilde Juárez (Mexico)
    cjuarezh@yahoo.com
  • Catherine Taveau (France)
    catherine.taveau@paris.iufm.fr
Top of page
Aims and focus

Topic Study Group 1: New developments and trends in mathematics education at pre-school level

In Topic Study Group 1 we will discuss contemporary developments in mathematics education at the preschool and initial years of primary level (approximately ages 0 through 7 years). Our aim is to promote cross-fertilization between theory and practice, research and developmental activities. To this end, researchers and educators are strongly encouraged to participate and consider the mutual challenges that stem from their different activities in their own countries.

Our presentations and discussions will focus on a diversity of issues related to young children’s learning. Among these, we want to discuss young children’s intuitive and informal knowledge of mathematics, the cognitive basis and challenges for young children in learning mathematics, their interest in learning mathematics, the needs of young children at-risk for mathematics difficulties, early interventions that promote their understanding and learning of mathematics, what sort of mathematics can young children be taught and what sorts of assessment are appropriate for young learners of mathematics. Papers focusing on smaller or larger samples, using quantitative or qualitative methods are welcome.

Young children’s intuitive and informal knowledge of mathematics is currently of great interest. There are researchers who assert that children have some innate knowledge of mathematics, others suggest that mathematics knowledge is constructed by children in their cultural context. Our group would like to examine the mathematics that children bring to the classroom, what they need to know in order to learn the curriculum that they are taught, and their interest in mathematical ideas and symbols. Some children may be at risk for mathematics difficulties because they have not developed this informal mathematical knowledge when they start school. Whatever the explanations for this, pre-schools could give them this opportunity.

What sort of mathematics can young children be taught is also a focal point for our group. This discussion includes issues such as activities that promote learning and interest in mathematics among young children, language and thinking in mathematics, problem solving, learning with different kinds of tools (concrete materials, the number line, computers, drawings etc.) and in different types of classroom environments. It would also be of interest to discuss monitoring and assessing young children’s mathematical learning.

Speakers will have a limited amount of time for their presentations but most will bring papers for distribution. The sessions will offer participants the opportunity to form networks for further communication and collaboration.

Top of page
Program

Tuesday 8th July 12:00 – 13:00

Welcome to TSG1 – Terezinha Nunes

Presentations

1. The role of logic in young children’s mathematics learning
Terezinha Nunes
Department of Education, University of Oxford, Oxford, UK

2. Preschoolers’ Problem Solving Processes and Strategies Related to Accuracy While Solving Reversibility and Missing Addend Problems
Luz Stella López
Universidad del Norte & Marymount School, Barranquilla, Colombia

3. What is teaching early mathematics? A case study from Big Math for Little Kids
Miriam Amit
University Ben-Gurion, Israel

Wednesday 9th July 12:00 – 13:30

Presentations

1. How do you teach nursery children mathematics?’ In search of a mathematics pedagogy for the early years
Sue Gifford
University of Roehampton, UK

2. Young children counting at home
Rose Griffiths
University of Leicester, UK

3. Avoiding Ten, a Cognitive Bomb
Allan Tarp
The MATHeCADEMY.net, Hornslets Alle 27, DK-8500 Grenaa, Denmark

4. Learning Mathematics in the First Two Years at School. The New Zealand Experience
Pamela Perger
Faculty of Education, The University of Auckland, New Zealand

5. Building Integers in Preschool: A Cooperative Learning Experience
Jose Manuel Serrano
Universidad Complutense de Madrid, Madrid, Spain

6. Pre-school teachers’ mathematics knowledge and its implication for educational policy
Miguel Friz Carrillo, Universidad del Bío-Bío, Chile
Marjorie Sámuel Sánchez, Universidad Católica del Maule, Chile
Susan Sanhueza Henríquez, Departamento Psicología de la Salud, Universidad de Alicante, España

Friday 11th July 12:30 – 13:30

Presentations

1. Discovering geometry at preschool level
Catherine Taveau
Université Paris-Sorbonne – IUFM de Paris – IREM paris 7- France

2. Keys for using picture books to support kindergartners’ learning of mathematics
Sylvia van den Boogaard
FIsme, Utrecht University, the Netherlands
Marja van den Heuvel-Panhuizen,
FIsme, Utrecht University, the Netherlands & IQB, Humboldt University Berlin, Germany

3. Technology use and geometry development in the early years- an analysis based on the human-media-geometry unit
Anna Chronaki
Department of Early Childhood Education, University of Thessaly, Volos, Greece

4. Mathematics Within Early Childhood – A New Zealand Case Study
Shiree Lee
School of Sciences, Mathematics and Technology Education, The Faculty of Education, The University of Auckland, New Zealand

Saturday 12th July 12:00 – 13:30

Presentations

1. Scaling Up TRIAD: Teaching Preschool Math with Trajectories and Technologies
Douglas H. Clements & Julie Sarama
University at Buffalo, The State University of New York, Department of Learning and Instruction, Graduate School of Education

2. Relating Spatial Structures to the Development of Kindergartner’s Number Sense
Fenna van Nes & Jan de Lange
Freudenthal Institute for Science and Mathematics Education, Utrecht University, The Netherlands

3. Exploring children’s informal knowledge in the teaching of fractions
Ema Mamede
University of Minho, Braga, Portugal

4. Teaching Mathematics at the beginning of the school
Esther Grossi
GEEMPA, Porto Alegre, Brazil

5. The Division Process at Kindergarden
Hugo Rodriguez Carmona
Mexican Society of Geography and Statistics
Email: hugo_rodriguezc@yahoo.com.mx

6. Adapting a School Numeracy Programme for Early Childhood Play-based Learning
Shiree Lee, Gregor Lomas, Pamela Perger
Faculty of Education, The University of Auckland, New Zealand

Closing remarks: Douglas Clements

Top of page
Abstracts

Tuesday 8th July

The role of logic in young children’s mathematics learning
Terezinha Nunes
Department of Education, University of Oxford, Oxford, UK

Abstract
It has often been claimed that children’s understanding of mathematics is based on their ability to reason logically. This presentation will provide evidence for this claim, which has been so far accepted by many but not supported by research. Two studies will be described. One was a longitudinal study in which we showed (a) that measures of 6-year-old children’s logical abilities and of their working memory both predict their mathematical achievement over a period of 16 months even after controls for differences in age and intelligence, and (b) that the logical scores continued to predict mathematical levels after appropriate controls had been taken into account. In the second study we improved the logical reasoning of a group of children at risk for underachieving in mathematics and showed that they made more progress in mathematics than a similar group of children from the same schools who were not given this teaching. Together these studies showed a strong link between logical reasoning and mathematical development and also established that it is possible to improve mathematics learning in children at risk by improving their logical reasoning in their first year of school.

Preschoolers’ Problem Solving Processes and Strategies Related to Accuracy While Solving Reversibility and Missing Addend Problems
Luz Stella López
Universidad del Norte & Marymount School, Barranquilla, Colombia.

Abstract
This presentation will examine research on the effects of the use of processes and arithmetic strategies on accuracy, in 162 children, ages 4, 5, and 6, while solving Reversibility and Missing Addend problems. The research questions were: Which are the processes and strategies most used? Are different types of processes and strategies used depending upon the type of problem solved? Are children likely to succeed when using different types of processes and strategies? Do the processes and strategies that predict accuracy vary in relationship to the problem solved? A Semi- Structured Clinical Interview was designed and validated as a method to gather data. Results show that there is a tendency for the reports on the use of processes to vary in increasing mean percentages by age. The reports of the use of arithmetic strategies also vary by age, reflecting changes from the more concrete to the more mental ones, as age increases. The processes and strategies accessed do not vary by problem type, yet these do vary when predicting accuracy. Certain processes and strategies interact to predict success. Results will be analyzed in relationship to the literature on problem solving processes and arithmetic strategies.

What is teaching early mathematics? A case study from Big Math for Little Kids
Miriam Amit
University Ben Gurion, Israel

Abstract
There now appears to be widespread agreement that early childhood mathematics education (ECME) should be implemented on a wide scale, particularly for disadvantaged children. Yet little is known about the teaching of early mathematics. The goal of this paper is to demystify the process. We analyze one early childhood teacher’s work as she attempts to implement an extended activity on mapping. Her efforts reveal the myriad activities involved in teaching mathematics to young children. Her teaching entails all the processes involved in teaching mathematics to older children, including profound knowledge of the subject matter, pedagogical content knowledge, lecturing, introduction of symbolism, and connecting everyday experience to abstract ideas. Teaching early mathematics to young children is essentially the same as teaching it to older children. The implications of this observation for professional development are enormous: extensive pre-service and in-service education is necessary to train a new generation of effective teachers of mathematics.

Wednesday, 9th July

How do you teach nursery children mathematics?’ In search of a mathematics pedagogy for the early years
Sue Gifford
University of Roehampton, UK

Abstract
Recent concerns with underachievement have focused attention on mathematics education in pre-school. Evidence points to the effectiveness of mathematics focused, adult led activities, yet little is known about the nature of effective mathematics ‘teaching’ for such young children. With a view to establishing principles for appropriate pedagogy, this paper reviews current research on how young children learn, including evidence from neuroscience, pre-school interventions and home learning. This evidence points to the importance of planning cumulative experience, building on children’s spatial memory, combining kinaesthetic, visual, and verbal learning, and providing opportunities for problem solving and discussion. Close relationships with teachers and working in friendship pairs are important for learning. Children’s views of themselves as mathematical learners are crucial, alongside their interests and mathematical purposes, home learning environments and parental pedagogy. The variation in children’s experience and attitudes indicates the importance of working with families and of helping children make connections. Recommended activities include repetitive routines, rhymes, games and stories which involve problem solving. Resources include a range of computer software and apparatus providing schematic visual images. Interactive adult strategies include modelling, indirect questions and challenges, using errors, puppets and humour and praising effort over performance. Effective staff development includes involvement in mathematics focused activities to foster confidence and the use of learning trajectories in formative assessment. Current developments point to the need to create mathematical communities of practice in pre-school with raised teacher expectations for young children’s mathematical learning.

Young children counting at home
Rose Griffiths
University of Leicester, UK

Abstract
This presentation examines the contribution made by family members to children’s early learning about counting, and discusses the ways in which children develop their understanding of counting in everyday situations. It describes a research project where children’s counting practices at home were filmed in twelve families with children aged two, three or four. Five of the families were multilingual, and children were learning to count in two (or more) languages. It explores the variety of contexts in which children’s skills were extended, as they counted toys, apples, fingers and toes, stairs and how many times they swung on a swing. The families were able to provide children with frequent, meaningful and enjoyable opportunities to count over a long period, often only for a few minutes at a time but with a high level of individual attention, in a way that is more difficult to manage in an educational setting. The challenge for practitioners is to value what families achieve, to use good ideas from home where appropriate in our settings, and to provide support and advice to families, including helping parents and carers to share good ideas from common family practice. A selection of film clips from the project have been used to make a DVD for parents, carers and practitioners.

Avoiding Ten, a Cognitive Bomb
Allan Tarp
The MATHeCADEMY.net, Hornslets Alle 27, DK-8500 Grenaa, Denmark, +45 8632 1899

Abstract
Being the only number with a name but without an icon, ten easily becomes a cognitive bomb to young brains. While ten is the follower of nine by nature, 10 is the follower of 9 by a pastoral choice hiding its alternatives: with 8 as bundle-size 10 is the follower of 7, and the follower of nine is 12.
Using anti-pastoral sophist research searching for alternatives to pastoral choices presented as nature, this paper shows that the root of mathematics is the study of multiplicity; that the root of numbers is counting by bundling & stacking using cup-writing and decimal-numbers to report the resulting stacks of bundles and unbundled; that the root of operation is the bundling & stacking process where’7-2’ means ‘from 7 take away 2’, ‘7/2’ means ‘take away 2s’, ‘3x2’ means ‘3 2-bundles’, and ‘5+2’ means ‘juxtaposing 2 1s to 1 5s’; that the root of formulas is the recount formula T = (T/b)x b telling that a total T is counted by taking away bs T/b times, thus predicting the counting result with numbers entered: T = 9 = (9/4)x 4 = 2.1 4s; that the nature of equations is reversed calculations performed by reversing the calculation sign of the numbers, so if zx3+2 = 8 then z = (8-2)/3; that the nature of calculus is adding stacks by uniting their bundles-sizes asking 3 4s + 2 5s = ? 9s.
Finally the paper shows how the core of mathematics can be learned by using 1digit numbers alone.

Learning Mathematics in the First Two Years at School. The New Zealand Experience
Pamela Perger
Faculty of Education, The University of Auckland, New Zealand

Abstract
By the time a child in New Zealand reaches their seventh birthday they have experienced at least two mathematical learning environments. One where mathematical concepts are learnt as the child makes sense of the world around them, the other a more formal school environment. The New Zealand curriculum document states that children should learn mathematics through the use of meaningful contexts (Ministry of Education, 2007) yet these contexts are very different to the contexts the child has experienced prior to starting school.
With the introduction of the Numeracy Development Project [NDP] the teaching of number has become more separated from other subject areas. The basis of the NDP is a framework of strategies and knowledge that children are facilitated through to increase their numeracy capacity. The framework is divided into two categories, knowledge (numeral identification, counting sequence [forward and backwards], place value, basic facts) and strategy (addition and subtraction, multiplication and division, and proportion and ratio). As well as the frameworks a diagnostic interview was developed enabling teachers to make more specific analysis of children’s mathematical knowledge and strategy. Support material was developed and collated to aid teachers in planning for the specific needs identified through the diagnostic interview. Young children viewed many of these specially developed activities as games, thus providing a meaningful context for learning mathematics.
This presentation explores the diagnostic interview (NumPa), the specifically developed activities and the Strategy Teaching Model that supports the teaching and learning of knowledge and strategies identified within the New Zealand Numeracy Frameworks.

Building Integers in Preschool: A Cooperative Learning Experience
Jose Manuel Serrano
Universidad Complutense de Madrid, Madrid, Spain

Abstract
The present work is an experience conducted in a classroom of Infantile Education (preschool)with children of five years old in order they managed to construct the set of integer numbers and were able to operate with the law of composition (+) in the interval [- 9, +9] because notion of ten is not included in the curricular contents of these students. The experience took place in the center’s gymnasium. Purpose was having a larger space, where we could obtain a suitable representation of the numerical straight line for the cognitive structure of the students, so the classroom was insufficient for the correct development of the activities. A train was used as material. The train was made with panels (two wagons) whose wheels can be connected. As well, two set of piston rod with a flag (stations) that took numbers from one to nine. And a drawer as the point always the train left. The drawer represented the origin so it was designated with letter “O” (origin). The activity was a game that demanded the displacement of the train. Children knew the only rule was the train only could go forwards or backwards, that is to say, the train could change of sense, but not of direction. The methodology was cooperative, with interdependence of objectives, division of the task, and differentiation and alternation of rolls. The initial task consisted of differentiating what stations went in front of the drawer and what stations went behind the drawer. It was established, for mutual agreement, to mark the flags of the numbers. The following tasks consisted of problems situations of displacements towards ahead or backwards of both wagons that composed the train. Purpose was the dominion of those displacements. Next a group of tasks was developed to reach a consensus about the representation of the displacements (numbers and marks) in order that they had an exact and equivalent knowledge of which it became at every moment. Finally, inverse activities were made. Students had to translate a representation (numbers and signs) into the real situation. All the students ended up solving situations such as (3-7), that is to say, they went three stations towards ahead (3) and soon they went backwards seven stations (-7), so the train located four station behind the origin. With these activities it was obtained children had a representation, of the content and the task, which are necessary condition for the construction of the meaning and the attribution of sense.

A study of preschool teachers’ the mathematical knowledge and its implications for educational policy
Miguel Friz Carrillo, Universidad del Bío-Bío, Chile
Marjorie Sámuel Sánchez, Universidad Católica del Maule, Chile
Susan Sanhueza Henríquez, Departamento Psicología de la Salud, Universidad de Alicante, España

Abstract
The research aims to assess the mathematical knowledge that preschool teachers have. The methodology is determined by a quantitative approach, non-experimental descriptive, survey type. The results indicate that high level of knowledge and skills are found in elementary logic (notion of time, symbolic function, similarities and differences) whereas numbers and space concepts and their functions need to be worked in further depth by teachers. One major obstacle is the gap that stands between the informal mathematical language and the discipline systematized language. Although there are no significant differences in the responses provided by teachers from private and state-supported schools, it is necessary to consider the training models of teachers, since the evidence shows that these programs do not impact significantly on the way in which teachers approach mathematical knowledge.

Friday 11th July

Discovering geometry at preschool level
Catherine Taveau
Université Paris-Sorbonne – IUFM de Paris – IREM de Paris 7 – France

Abstract
In France, all three-year-old children are at preschool. During their schooling, the French young children have to learn the following knowledge: number sense, space and time sense, shapes and magnitudes. At preschool, the syllabus give a key and crucial role to the use of language by placing it at the center of the overall acquisition.
In this paper, I will mainly focus on geometry which is more difficult to teach at this school level. Some geometrical activities have been worked out based on didactic tools (Duval’s representation registers and Houdement-Kuzniak’s geometrical paradigms).
These activities have been used by teachers and their implementation have been analyzed in teachers’ training with video recordings. Then we have improved these activities to make them available to the teachers’ community via booklets, multimedia package (DVD, CD-ROM) or professional websites. In our proposal, some examples of such activities are given and students’ difficulties are analyzed.

Keys for using picture books to support kindergartners’ learning of mathematics
Sylvia van den Boogaard
FIsme, Utrecht University, the Netherlands
Marja van den Heuvel-Panhuizen,
FIsme, Utrecht University, the Netherlands & IQB, Humboldt University Berlin, Germany

Abstract
Picture books can offer children a meaningful context for learning mathematics, and provide them with an informal experience base for mathematical operations, objects and structures that can be a springboard for a more formal level of understanding. The “PIcture books and COncept development in mathematics” (PICO-ma) project intends to generate more knowledge on the working and the effect of picture books on the learning of mathematics by young children. Among other things, the project is aimed at identifying what characteristics of picture books that have not been written to teach mathematics, can contribute to the development of mathematical concepts in kindergartners, and at finding out how to read the books to children so that these learning-supporting characteristics of picture books are strengthened.
In the presentation, we zoom in on the keys we have developed to elicit children’s thinking and talking about the mathematical concepts that appear in the books. These keys include teacher behavior such as (1) asking oneself a question, (2) playing ignorant, and (3) showing an inquiring expression. Using these keys in reading sessions, we found that they make children cognitively engaged during the reading session. We will illustrate our findings with video clips addressing examples from number (with special attention to “structuring numbers”), measurement (in particular the theme “growth”), and geometry (with the focus on “taking a point of view”).

Technology use and geometry development in the early years- an analysis based on the human-media-geometry unit
Anna Chronaki
Department of Early Childhood Education, University of Thessaly, Volos, Greece

Abstract
Lev Semonovich Vygotsky has repeatedly claimed through his writings that understanding the development of scientific concepts in childhood supports our attempt for devising successful teaching methods. Within this realm, the interrelation of scientific and spontaneous concepts is linked with school instructional practices, child’s development and social interactions. With particular reference to mathematical concepts, he points out, that when a child learns some mathematical operations, ‘the development of that operation or concept has only begun’, and continues arguing that ‘…the curve of development does not coincide with the curve of school instruction; by and large, instruction, precedes development’ (p. 102). Τhe present paper attempts to address the above, by discussing the particular case of geometry teaching and learning as mediated by technology-based tools and interactions. A series of episodes are analyzed with an eye to trace and interpret scientific development as part of evolving interactions amongst educator-children-technology. This perspective relates the studying of the development of scientific concepts with a type of analysis that is based on units. The term ‘unit’ is used here to emphasize that the product of analysis is not about the single elements of a situation but mainly about its wholeness. For example, a word meaning can be the unit of verbal thought and it is in ‘word meaning’ that thought and speech unite into verbal thought. Taking the above into consideration, the present paper takes as a unit of analysis the system children-technology-maths activity. This unit is related to the construct of subject-mediated tool-object as proposed by Leont’ev (1978) and discussed later by Tikhomirov (1981). It enables to interpret child’s mathematical activity and technology use as part of a complex whole and not as isolated and separate actions or operations. And, in parallel, assists to explore how technology (i.e. a dynamic geometry microworld) mediates the development of mathematical concepts (i.e. abstracting geometric properties) by creating an instructional context based on ‘word meanings’.

Mathematics Within Early Childhood – A New Zealand Case Study
Shiree Lee
School of Sciences, Mathematics and Technology Education, The Faculty of Education, The University of Auckland, New Zealand

Abstract
This presentation will discuss examples from a recent case study observing children aged between 12 months and five years as evidence that foundational mathematical learning occurs in children’s play. These examples support the holistic, child-centred and integrated approach to early childhood mathematics promoted within early childhood education in New Zealand. The data collected showed evidence of very young children’s skill and knowledge in rote counting, one to one counting, measuring, addition and subtraction, division, spatial rearrangement, concepts of time, shape recognition, and patterning.
Te Whãriki (Ministry of Education [MoE], 1996) New Zealand’s early childhood curriculum document, describes an holistic and open-ended framework for the education of young children from birth to school entry. Learning outcomes for individual subject content areas are stated within this document although the strong emphasis on child-centred and holistic play-based learning remains the fundamental philosophy in implementing these.
The pedagogical approaches and understanding that teachers within New Zealand early childhood settings hold, is an essential influence on the mathematical experiences that are offered to children in early childhood settings. Early childhood teachers who understand and articulate the importance of children’s play and create an environment that has potential for hands-on discovery, provide opportunities for children to lay the foundations of their mathematical understanding. This notion of children constructing their own knowledge is espoused clearly within Te Whãriki “To grow up as competent and confident learners and communicators, healthy in mind, body and spirit, secure in their sense of belonging and in the knowledge that they make a valued contribution to society.” (MoE, 1996, p. 9).

Saturday 12th July

Scaling Up TRIAD: Teaching Preschool Math with Trajectories and Technologies Douglas H. Clements & Julie Sarama
University at Buffalo, The State University of New York, Department of Learning and Instruction, Graduate School of Education

Abstract
Some research-based educational practices have been successful. Unfortunately, there is a “deep, systemic incapacity of U.S. schools, and practitioners Š to develop, incorporate, and extend new ideas about teaching and learning” (Elmore, 1996, p. 1). Fortunately, research provides guidelines to scale up interventions (http://UBTRIAD.org). We used these guidelines to create and test our TRIAD intervention for low-income preschoolers. TRIAD stands for Technology-enhanced, Research-based, Instruction, Assessment, and professional Development. Technology benefits students, teachers, and researchers. Research is the basis for all aspects of TRIAD: The instruction, the assessments, and the professional development. The goal of the TRIAD intervention is to avoid the dilution and pollution that usually plagues efforts to achieve broad success. This presentation will briefly discuss three studies of the internal workings and the empirical evaluations of each of the three components of TRIAD-instruction, assessment professional development. Each one is build around a central core of learning trajectories (descriptions of children’s thinking and learningŠand a related, conjectured route through a set of instructional tasks). Learning trajectories are at the heart of both TRIAD’s math curriculum and its professional development. Learning trajectories help teachers focus on the “conceptual storyline” of the curriculum, a critical element that is often missed. They facilitate teachers’ learning about math, how children think about and learn this math, and how such learning is supported by the curriculum and its teaching strategies. By illuminating potential developmental progressions, they bring coherence and consistency to TRIAD’s three components of assessment, professional development, and instruction.

Relating Spatial Structures to the Development of Kindergartner’s Number Sense Fenna van Nes & Jan de Lange
Freudenthal Institute for Science and Mathematics Education, Utrecht University, The Netherlands

Abstract
This study is part of a larger interdisciplinary project bridging research from mathematics education with neurosciences to gain insight into how kindergartners’ early spatial and numerical abilities may best be fostered. Regarding the mathematics education component, we are investigating how kindergartners’ spatial structuring abilities may stimulate their number sense and help them to attain more sophisticated numerical strategies. Examples of relevant spatial structures are finger patterns and domino-dot configurations. For this purpose, we developed interactive tasks and classroom activities with which we conducted a teaching experiment in a kindergarten of an elementary school in the Netherlands. The goal of the teaching experiment was (a) to gain insight into kindergartners’ early spatial structuring abilities, and (b) to design an educational setting that can support kindergartners in learning to apply spatial structures as a means to abbreviate numerical procedures.
In this presentation, we will share several video clips of activities from the teaching experiment and discuss how they triggered learning opportunities for the children. The significance of such an educational setting is that it may help identify children who have difficulty recognizing spatial structures and who are falling behind in counting ability at such an early stage in formal schooling. It may support children with instructional activities that are tailored to appeal to their interests, their intrinsic motivation and their mathematical strengths rather than just their weaknesses. In this way, the research may contribute to methods for intercepting problems in the development of early, yet fundamental, mathematical abilities of young children.

Exploring children’s informal knowledge in the teaching of fractions
Ema Mamede
University of Minho, Braga, Portugal

Abstract
This paper describes a teaching experiment in which children were introduced to fractions using quotient, part-whole or operator situations. The effects of each of these situations were analysed. Children’s informal knowledge of quantities that are represented by fractions has not been systematically analysed across situations. Differences between situations are analysed by comparing what children learn about this type of quantities and their representations in each of these situations, and whether they transfer this learning across situations. The study involved first-graders (N=37), aged 6 to 7 years, from two primary schools from the city of Braga, in Portugal who had not been taught about fractions before. The children were assigned to work in groups. Two types of tasks were presented to the children: tasks involving the equivalence and ordering of quantities represented by fractions, referred to here as “logical tasks”; and tasks where the children were asked to provide a symbolic representation, referred to as “labelling tasks”. We investigated whether the situation in which the concept of fractions was presented to the children influenced their learning of logical and labelling aspects of fractions. Quantitative analyses showed that children developed a better understanding of equivalence and order of fractional quantities when they were introduced to fractions in quotient situations, but there was no transfer of learning to part-whole or to operator situations. When part-whole and operator situations were used to introduce children to fractions, they only learned how to label fractions and were able to transfer this learning across these two situations.

Teaching Mathematics at the beginning of school
Esther Grossi
GEEMPA, Porto Alegre, Brazil

Abstract
The central idea of this presentation at the Eleventh ICME is based on the Theory of Conceptual Fields of Gérard Vergnaud. The idea that we don’t learn step by step, a set of concepts, but we consider three vectors at the same time. These three vectors are: (1) a set of situations; (2) a set of concepts; and (3) a set of symbolic representation.
The set of concepts at the beginning of school are related to five axles: logic, space, structure of numbers, structure of addition and structure of multiplication. In this presentation, it will be argued that the point of departure for children’s understanding of multiplicative structures is division. Division is at the center of mathematics learning for children because it is culturally meaningful: children use partition to solve problems in everyday life more than addition. There is also an affective investment by children in the relation between partition and division: in the majority of real life situations when the need to divide appears, the division is not in equal parts. This is partition and not division. This makes the concept of division attractive for children and adult learners in the initial stages of learning: in mathematics, division is fair.

The Division Process at Kindergarten
Hugo Rodriguez Carmona
Mexican Society of Geography and Statistics
Email: hugo_rodriguezc@yahoo.com.mx

Abstract
Fractions are a big issue for students and for teachers as well. At elementary and secondary schools, most teachers see it as a very big challenge to teach their students how to divide fractions and to find word problems that require the use of the division algorithm. It is also difficult to explain to students which kind of situations in real life requires the division of fractions. This presentation will describe manipulative materials called “desquebra/2”, which represent a very easy way for helping teachers to deal with the division process at kindergarten level and presents activities that can help kids understand the division of fractions process.

Adapting a School Numeracy Programme for Early Childhood Play-based Learning
Shiree Lee, Gregor Lomas, Pamela Perger
Faculty of Education, The University of Auckland, New Zealand

Abstract
The New Zealand Early Childhood curriculum is based on a play-based holistic approach to learning centred around supporting, and building on, the child’s interests. This is in contrast to the more formal classroom instruction prevalent in New Zealand primary schools centred on a curriculum that sets out subject specific goals. For school mathematics the development of a research based numeracy teaching/learning programme led to the construction of an oral diagnostic assessment instrument and frameworks setting out detailed progression stages in the knowledge and strategies necessary for children’s numeracy learning. School entry (age five) data gathered using the instrument as part of the implementation of the programme revealed variable levels of knowledge and skills in aspects of numeracy, but more importantly that these were often far more advanced than anticipated by the school curriculum. For example, in the Numeral identification progression 17% of children recognise numbers to 100 and a further 15% numbers to 1000. This data provides clear evidence that current early childhood experiences can and do impact on children’s readiness for further numeracy learning in school environments. The detail gained from the data can be used to indicate particular aspects of numeracy learning that could be enhanced within early childhood settings; firstly, by the linking of the numeracy programme terminology to the mathematics language of the early childhood curriculum for teachers, and secondly, by the use of the activities (eg. games) and the child-centred approaches designed for use within schools which would not compromise the play-based holistic approaches of the early childhood curriculum.

Top of page
Top of page