Doug Clarke

Supporting Teachers in the Development of Young Children's Mathematical Thinking: Three Large Scale Cases.

Recognition of the importance of the early childhood years in the development of numeracy is a significant characteristic of the New Zealand Numeracy Development Project, the Victorian Early Numeracy Research Project and the Count Me In Too program in New South Wales, Australia. This article outlines the background, key components and major impacts of these three innovative and successful professional development and research initiatives. Juxtaposing the three projects highlights important commonalities—research-based frameworks, diagnostic interviews, and whole-school approaches to professional development. Each program has been significant in rethinking what mathematics and how mathematics is taught to young children.

The Mathematical Knowledge and Understanding Young Children Bring to School.

As part of the Victorian Early Numeracy Research Project, over 1400 Victorian children in the first (Preparatory) year of school were assessed in mathematics by their classroom teachers. Using a task-based, one-to-one interview, administered during the first and last month of the school year, a picture emerged of the mathematical knowledge and understanding that young children bring to school, and the changes in this knowledge and understanding during the first year of school. A major feature of this research was that high quality, robust information on young children’s mathematical understanding was collected for so many children. An important finding was that much of what has traditionally formed the mathematics curriculum for the first year of school was already understood clearly by many children on arrival at school. In this article, data on children’s understanding are shared, and some implications for classroom practice are discussed.

The Mathematics Teacher and Curriculum Development

The teacher is the key to worthwhile mathematical experiences for children. In this chapter, we offer an appreciation of the crucial role played by teachers in any meaningful curriculum; a recognition that teachers need to be supplied with appropriate resources (text or student materials, teacher support material, relevant technology, and an appropriate physical environment); and an acknowledgment that teachers need time: Time to plan, time to meet together, time to assimilate new content and pedagogy into their repertoire, and sufficient hours timetabled for mathematics. The views of teachers as either irrelevant or merely agents of change are examined, and these are contrasted with a view of the teacher as curriculum maker. Some factors which constrain the teacher’s role in mathematics curriculum development are considered, and a range of curriculum projects and approaches to curriculum policy are discussed. During the discussion, reference will be made to experiences in a variety of countries and contexts, illustrative of the points we are making. We also report in greater detail on three specific examples of curriculum development (in Papua New Guinea, a DutchlU.S. joint initiative, and Australia), and reflect on the implications of these examples for those seeking to maximise the role of the teacher in curriculum development.

Understanding, Assessing and Developing Children's Mathematical Thinking: The Power of a One-to-One Interview for Preservice Teachers in Providing Insights into Appropriate Pedagogical Practices.

Abstract At Australian Catholic University and Monash University, preservice mathematics teachers are required to conduct and analyse one-to-one mathematics assessment interviews with primary-aged children. The assessment tool is drawn from the Early Numeracy Research Project, where it was used with over 11,000 children in Victorian schools. The interview assesses content from Number, Measurement and Geometry, in an interactive, hands-on format, with childrens responses and strategies determining the path through the interview protocol. Follow-up discussion in class enabled preservice teachers to explore appropriate pedagogies that build upon what had been learned from the interviews. The research described in this chapter sought to investigate the effectiveness of this process. Interviews and written questionnaires were the data sources. Analysis of the data suggested that teachers were more aware of the kinds of strategies that children use including their variety and relative level of sophistication, and that the interview and subsequent discussion stimulated preservice teachers to reflect upon appropriate classroom experiences for young mathematics learners.

Mathematische Kompetenzen von Vorschulkindern: Ergebnisse eines Ländervergleichs zwischen Australien und Deutschland

ZusammenfassungIm Rahmen des australischen Early Numeracy Research Project wurden im Bundesstaat Victoria über 1400 Kinder (Fünfj ährige) von ihren Lehrerinnen und Lehrern hinsichtlich ihrer mathematischen Kenntnisse und Fertigkeiten befragt. Durch materialgestützte aufgabenbasierte Einzelinterviews zu Beginn und am Ende der für alle Kinder verbindlichen Eingangsklasse, dem sog. Preparatory Grade, entstand ein genaueres Bild dahingehend, welches Wissen, Verständnis und welche Fertigkeiten Schulanfänger mitbringen und wie sich dieses Wissen bis zum Ende des Vorschuljahres entwickelt. Mit der deutschen Fassung des ENRP-Interviewleitfadens wurden in einer Region im Nordwesten Deutschlands rund 850 Kindergartenkinder von Studierenden der Universität Oldenburg knapp ein Jahr und unmittelbar vor der Einschulung befragt. In diesem Beitrag werden ausgehend von einer umfassenden Literaturauswertung, die der Entwicklung des Interviewleitfadens zu Grunde lag, die Befunde der australischen und deutschen Erhebungen vorgestellt und vergleichend ausgewertet. Abschließend werden Implikationen für die Gestaltung der vorschulischen Förderung und des Anfangsunterrichts entfaltet.AbstractAs part of the Victorian Early Numeracy Research Project in Australia, over 1400 children in the first (Preparatory) year of school in the state of Victoria were assessed in mathematics by their classroom teachers. Using a task-based, one-to-one interview administered during the first and last month of the school year, a picture emerged of the mathematical knowledge and understanding that young children bring to school and how this develops during the first year of school. The same interview was conducted after translation into German with around 850 kindergarten children (five-year-olds) in the north-western region of Germany by preservice teachers from the University of Oldenburg. In this paper, based on an extensive international literature review that guided the development of the interview protocol, the data on children’s mathematical understanding and its development during the preparatory grade (Australia) and the final year in Kindergarten (Germany) are shared and analysed comparatively. Finally, implications for classroom practice are discussed.

Classroom reform five years down the track: The experiences of two teachers

This case study research investigated changing teacher roles associated with two teachers’ use of innovative mathematics materials at Grade Six level, in a setting which contained all the ideal ingredients for professional growth. Participant observation and interviews with the teachers over a seven-month period early in the use of the innovative materials and for a brief time five years later provided a picture of changing teacher roles, but also a sense of issues that had emerged. or persisted in the longer term. The greatest changes in these teachers’ roles (in the short and long term) related to increasing comfort with posing non-routine problems to students and allowing them to struggle together, and the provision of structured opportunities for student reflection upon activities and learning. However, little change was evident over the five year period in the teachers’ use of assessment practices or in their articulation of the “big ideas” of mathematics in the middle school years.

Using Content-Specific Open-Ended Tasks

This chapter explains the ways that open-ended tasks might contribute to learning, it gives the details of a specific open-ended task and how it might be used in a “lesson”, it indicates the challenges that teachers may experience when using such tasks, it presents a range of examples of this type of task to illustrate the scope and nature of the tasks, and it summarises some research on teachers’ reactions to the tasks. The fundamental argument is that such tasks are accessible by students, able to be used readily by teachers, foster a range of mathematical actions, and contribute to some of the important goals of learning mathematics.

Supporting Mathematics Teachers’ Diagnostic Competence Through the Use of One-to-One, Task-Based Assessment Interviews

In this chapter, the important role that one-to-one, task-based assessment interviews can play in developing inservice and preservice mathematics teachers’ diagnostic competence is presented. We argue that the use of such interviews builds competence through enhancing teachers’ knowledge of individual and group understanding of mathematics, including misconceptions and preferred strategies, while providing an understanding of the typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children in classrooms, building both pedagogical content knowledge and subject matter knowledge.

Constructing a Sequence of Lessons

While tasks are the basic building block, teachers need to create lessons to ensure that the potential of tasks are realised, and also need to plan sequences of such lessons to build coherent learning experiences for students. The following presents some details of the development process and tasks used in one of the sequences created by project teachers. It includes information about the tasks and the instructions given to other teachers and some comments on the intent of the tasks. Some items from the suggested assessment instrument are presented as examples of the type of observation notes that were gathered.

Moving from the Task to the Lesson: Pedagogical Practices and Other Issues

We alluded in earlier chapters to the many challenges and considerations faced by teachers as they seek to take a task and build a coherent lesson around it. We talked about research on task choice, the role of teachers’ content and pedagogical content knowledge, the need to consider extending and enabling prompts to use alongside the task, and the subtle but important differences in the use of different kinds of tasks. Put simply, the task of itself is not sufficient to guarantee student learning, irrespective of its quality. This chapter is an attempt to articulate some key teacher actions which have the potential to maximise learning opportunities presented by a given task. The seven actions which we highlight in this chapter are the following: being clear on the mathematical focus and the goals of the lesson for students; considering the background knowledge which students are likely to bring to the task, how to establish this, and likely responses students will make to the tasks; considering ways in which students who have difficulty making a start on the task and students who solve the task quickly might best be supported; monitoring students’ responses to tasks as they work individually or in small groups on the tasks; selecting students who will be invited to share during discussion time; focusing on connections, generalisation and transfer; and considering what the next lesson might look like.