Prior development and research 1 Developing good practice

We begin by providing some background on our view of good practice in mathematics education.

The contents of good practice were developed as a response to reaction to the problems with the world-wide New Math movement in the 1960s and inspired by Freudenthal’s ideas (1978) about a new approach to mathematics education, embodied in what is now called as Realistic Mathematics Education (RME) (Treffers & Goffree, 1985; Treffers, 1991; Streefland, 1993; Gravemeijer, 1994).

Three late twentieth-century developments provided the foundation for MILE: - developmental research in the Wiskobas project;

- the formulation of national core objectives in the Netherlands;

- the discussions on a new publication about ‘a National Programme for Mathematics Education on Elementary Schools.’

Developmental research

In the Wiskobas project (1970-1980; note 4), a new mathematics curriculum for primary schools was developed with the support of Freudenthal. It resulted in a concrete realistic program that describes a clear image of good practice in five learning-teaching (L-T) principles (Treffers, 1991).

L.1 Construction. Learning mathematics is a constructive activity.

T.1 Concrete basis for orientation. Make mathematics concrete. Create recognizable contexts to which children can assign their own meanings.

L.2 Raising the level. Learning mathematics takes place somewhere between the informal mathematics of the children themselves (intuitive notions and self- invented procedures) and the formal mathematics of adults.

T.2 Models. To be able to achieve the required raising in level during the teaching- learning process, the pupils must have at their disposal the tools for bridging the gap between informal and formal mathematics (Gravemeijer, 1994).

Theory-enriched practical knowledge in mathematics teacher education

L.3 Reflection. Learning mathematics is stimulated by reflection. Reflection is, as it were, the engine for raising the level (Freudenthal, 1991).

T.3 Reflective moments. The teacher finds the right times to bring reflective moments into mathematics teaching. Good occasions for reflection include any cognitive conflicts that might occur and anything the pupil may have thought of independently (‘own productions’) (Streefland, 1991; Selter, 1993).

L.4 The social context. Children learn more often than not in the company of adults or other children. This means that other actors in the learning environment can provide the impulse for learning. As the different actors communicate with each other about mathematical concepts and procedures, they argue about them and come to insights collectively.

T.4 Interactive mathematics lessons. The teacher organizes mathematics education such that interaction becomes a natural part of it. This, in turn, creates a pedagogical climate in which all the pupils can take part in the interaction. The concept of a classroom as a sort of ‘mathematical community’ gives it an extra dimension, as does the Mathematical Conference in the class described by Selter (1993).

L.5 Structuring. If children construct their own meaningful mathematics, then new knowledge and insights become incorporated in what they have already learned. This means that the available mathematical knowledge (think, for example, of cognitive structures) is subject to constant upgrading. The new knowledge is fitted into the existing cognitive structure (assimilation) or the total structure is adjusted to accommodate the new insights (accommodation). Also, one aspect of learning is the task of bringing structure to what is being learned.

T.5 Interweaving the strands of learning. The teacher bases mathematics teaching on real-world situations, both as sources of ideas and as places to apply them. The first case would be an example of ‘horizontal mathematizing.’ Further, the mathematical ideas being used can themselves form the subject matter (vertical mathematizing). This brings connections with other mathematical ideas into the picture, partly as a result of the concrete background.

At the very end of the century (Goffree & Frowijn, 2000) ‘good practice’ had to be defined again, but this time with the intention to create an instrument for self-evaluation in schools. For this goal the principles were elaborated into more refined statements, called ‘indicators’ of realistic mathematics education, used to observe and analyze mathematics teaching in classrooms.

- The teacher is teaching mathematics by problem solving. - Problems are introduced in familiar contexts.

- While exploring the context of a problem the non-mathematical aspects mentioned by students are also considered.

- The context gives meaning to the mathematical activities.

- Introduction, problem setting, problem solving, and subsequent discussion are realized in interaction with the whole class.

- In order to stimulate mathematical activities in cooperative groups, the teacher creates reasons for the students to discuss, to explain, to cooperate, to convince each other, and to distribute tasks properly.

- Sufficient learning time is spent on the introduction and exploration of ‘models’. - The use of concrete models (e.g., schemas such as number line, reckon rack, or

fraction strips) results in the use of mental models.

- The teacher continuously anticipates students’ reactions during interactive class discussion.

- The pedagogical climate allows children to make mistakes and the teacher to overtly discuss these errors and their possible causes.

- The teacher takes time for reflective moments during the mathematics class. - Students are stimulated to create mathematical problems themselves (e.g., for

peers) and also to solve these problems reflectively.

- Teacher and students have an open mind for other people’s solutions. - Frequently asked questions are “Why?” and “Are you sure?”.

The provoking character of ‘cognitive conflicts’ is used to challenge children’s thinking.

The formulation of national core objectives

Increasing attention to quality management in primary education is the second development to consider. The National Institute for Curriculum Development in the Netherlands (SLO) published, after a national debate in the different domains, a list of core objectives for the school subjects (SLO, 1993; Treffers, De Moor & Feijs, 1989).

National Programme for Mathematics Education on Elementary Schools

A subsequent publication showed how to teach ‘in the spirit of Wiskobas’ in order to realize the core objectives: Standards for primary mathematics education (Treffers et al., 1989; Treffers & De Moor, 1990). These standards fueled a broad debate about ‘realistic mathematics education,’ that resulted in widely accepted and theoretically founded views of ‘good practice in realistic mathematics education.’

3.2.2 Good practice for teacher education

During the Wiskobas project, teacher educators participated in the research and development. Freudenthal supported these activities; he participated in field tests and increasingly viewed student teachers’ learning processes as an emerging outcome of

Theory-enriched practical knowledge in mathematics teacher education

mathematizing and didactisizing (Freudenthal, 1991). Thus in the years of Wiskobas a new approach to primary mathematics teacher education was designed, in close connection to the creation of realistic mathematics education (Goffree, 1979).

Following the Standards for primary mathematics education and the Standards for mathematics evaluation and teaching (NCTM 1989, 1992), the Dutch Association of Primary Mathematics Educators (NVORWO) submitted a request to the National Institute of Curriculum Development (SLO) to draft a similar publication specifically for Dutch teacher education. In 1990, a group comprising ten mathematics educators started developing national standards and presented the results to colleagues as a handbook for teacher educators (Goffree & Dolk, 1995).

The philosophy of teacher education elaborated in the handbook is founded on three pillars: a teacher education adaptation of the socio-constructivist vision of knowledge acquisition, reflection as the main driving force of the professionalization of teachers (Schön, 1983, 1987) and the interpretation of practical knowledge as a way of narrative knowing (Gudmundsdottir, 1995). The statement “Real teaching practice has to be the starting point of teacher education” is emphasized. In the attempt to elaborate this principle into concrete curriculum materials for student teachers, an essential question still remained: How can curriculum designers give a learning environment a ‘natural’ aura? And next: what do we mean by ‘natural’? Student teachers’ fieldwork practice is natural by definition, but when they discuss this practice, they often stick to a superficial interchange of ideas and opinions (Verloop, 2001). Rarely do these discussions reach a level of theoretical reflections.

Learning in practice is mostly a solo task because student teachers not often have the opportunity to discuss common experiences and observations. Moreover, they usually focus on fulfilling responsibilities and on survival issues, so talk about actions dominates their reflections on the profession. As a result, they do not acquire practical knowledge that can be generalized across situations or organize their narratives of teaching into a broader framework.

The group of ten Dutch teacher educators got a new perspective on this problem when they visited the School of Education of the University of Michigan. They were introduced to the Student Learning Environment (SLE), created by Lampert & Loewenberg Ball (1998), which became a source of inspiration for the making of MILE. Using the records of real teaching practice the Michigan student teachers could access a whole year of mathematics teaching, with options to observe teaching and learning from different points of view (teacher, students, subject matter, curriculum, classroom climate, et cetera.).

Although MILE would become a quite different learning environment than SLE was in 1995 (Goffree & Oonk, 2001), MILE is based on a similar philosophy about the

presentation of real teaching practice in teacher education: “good practice for student teachers learning about teaching primary mathematics in the spirit of realistic mathematics education.” Knowing this, it was important to make video recordings of practice without losing quality.

3.3 The making of MILE