J. Terwel

Hans Freudenthal a mathematician on didactics and curriculum theory.

The main ideas in the work of Hans Freudenthal (1905-1990), the Dutch mathematician and mathematics educator, related to curriculum theory and didactics are described. Freudenthals educational credo, mathematics as a human activity, is explored. From this pedagogical point of departure, Freudenthals criticism of educational research and educational theories is sketched and fleshed out. Freudenthals approaches to mathematics education, developmental research and curriculum development can be seen as alternatives to the mainstream Anglo-Saxon approaches to curriculum theory.

Curriculum differentiation: multiple perspectives and developments in education

This paper examines curriculum differentiation (i.e. streaming or ability‐grouping). After placing curriculum differentiation in an international perspective, it outlines the main conclusions from empirical research on differentiation over several decades. Against this empirical background, it describes and considers the three specific orientations towards curriculum differentiation: a genetic perspective, a cultural perspective, and a sociological perspective. The insights from the various perspectives are integrated and expanded in a framework of curriculum theory, research, and practice.

PREFERENCES FOR VARIOUS LEARNING ENVIRONMENTS: TEACHERS' AND PARENTS' PERCEPTIONS

In the last ten years, a number of innovations, mainly inspired by constructivist notions of learning, have been introduced at various levels of the Dutch educational system. However, constructivist learning environments are rarely implemented. Teachers tend to stick to expository and structured learning environments. This consistent finding requires research in order to gain insight into teachers preferences for learning environments and to determine the factors that support and impede the realization of these learning environments. Regarding the influence of social backgrounds on student learning, is it also important to take stock of parental views on learning environments.This study is focused on teachers preferences for learning environments, their reported teaching behavior, and how these match with parents preferences. Three parallel questionnaires were developed for teachers (n = 285), students (n = 951), and parents (n = 636) to measure preferences and behavior at different levels of education, for three types of learning environments: direct instruction; discovery learning; and authentic pedagogy. The results show that teachers often prefer direct instruction, and seldom promote discovery learning. While teachers sometimes realize authentic pedagogy, constructive learning tasks are seldom used. Teachers reported practice and parents preferences for their children appear to correspond reasonably.Results of multiple regression analyses show that the use of the three types of learning environments yield different predictors. For the use of discovery learning and authentic pedagogy, confidence in students regulative skills is an important predictor. In predicting the use of direct instruction, the teachers own conception of learning turns out to be an important predictor.

Providing or Designing? Constructing Models in Primary Maths Education.

Abstract The goal of this exploratory study was to uncover the construction processes which occur when pupils are taught to work with models in primary maths education. Two approaches were studied: ‘providing models’ versus ‘designing models in co-construction’. A qualitative observational study involved two groups of pupils of a primary school in the Netherlands. A series of lessons involving problem-oriented tasks was given. For this article’s purpose, we studied the learning processes of one pupil per condition in detail. Interpreting the results it is assumed that upper-grade pupils may be able to design models in co-construction as long as they receive sufficient teacher guidance.

Learning for mathematical insight: a longitudinal comparative study on modelling

Abstract This paper reports on a longitudinal study of teaching and learning the subject of fractions in two matched groups of ten 9–10-year-old students. In the experimental group fractions are introduced using the bar and the number line as (mental) models, in the control group the subject is introduced by fair sharing and the circle-model. In the experimental group students are invited to discuss, in the control group students work individually. The groups are compared on several occasions during one year. After one year, the experimental students show more proficiency in fractions than those in the control group.

AUDREY'S ACQUISITION OF FRACTIONS: A CASE STUDY INTO THE LEARNING OF FORMAL MATHEMATICS

National standards for teaching mathematics in primary schools in the Netherlands leave little room for formal fractions. However,a newly developed programme in fractions aims at learning formal fractions. The starting point in the development of this curriculum is the students’ acquisition of `numeracy infractions’. In this case study we describe the growth in reasoning ability with fractions of one student in this newly developed programme of 30 lessons during one whole school year. In the study we found indications that the programme and its teaching stimulated the progress of an average performer in mathematics. Moreover we found arguments as to what extent formal operations with fractions suits as an educational goal.

Strategic Learning in Primary Mathematics Education: Effects of an Experimental Program in Modelling

In most strategy research the focus is on ready-made models provided by the teacher or textbook. However, in this research project the effects are described of an experimental program in primary math education, concerning the construction and use of models by pupils in guided co-construction. This field experiment with an experimental group and a control group involved 238 grade-5 pupils. In a series of experimental lessons, pupils were taught to design models as a tool in the learning of percentages. The results of the experimental program were compared with the outcomes of a program in the control group, based on the teachers strategy of “directly providing models” to the pupils. The conclusion, then, is that children in the experimental condition significantly outperform children in the control condition.

Observation in Teaching: Toward a Practice of Objectivity

Because most informal classroom assessment is based on the observation of students by teachers, the purpose of the present study is to construct a view of this assessment that is consistent with the practitioners’ perspective. To find out how teachers frame observation, we analyzed stories of 25 Dutch teachers about teaching diverse learners and focused on what they said about the observation of students. Most attempts to improve observation have recommended the separation of subject and object in an effort to eliminate teachers’ personal frameworks. The present study, however, shows that observation is embedded in the action of teaching. Thus, the realities of the classroom do not allow teachers to separate themselves from what they are observing. An alternative view of objectivity is presented. This transactional view shows that the quality of teachers’ personal frameworks is important and should not be eliminated: The supposition of being neutral and detached hampers the teaching of diverse learners. To prevent self-fulfilling prophecies from influencing student achievement in a negative way, pedagogical craftsmanship is essential. Starting from this view, the reliability and the validity of observation for classroom assessment are reframed in a nonstatistical way. According to this view, social practice plays an important role in deciding whether observation deserves a quality warrant. Apart from an alternative view of concepts such as objectivity, a practice of objectivity requires new ways of constructing theories and mentoring teachers to sharpen their perceptive faculty.

Schematising in Early Childhood Mathematics Education: Why, When and How?

SUMMARY Discussions about the beginnings of mathematics education in early childhood are often caught up in a dilemma: should we stimulate spontaneous actions or provide direct instruction about elementary number-related actions? In this article we argue that either approach is problematic and neither is an optimal way of promoting the development of mathematical thinking. Using a socio-cultural perspective, we propose an emergent approach that integrates both the childs personal constructions and the educators pedagogical responsibilities. From this stance, we conceptualise mathematical thinking as a form of semiotic activity. Early semiotic activity can be identified in schematising activities in early childhood play. Educationally important questions are then: What are our arguments for introducing schematising activities in early childhood education? And how can educators organise schematising activities in early childhood education? This article gives an overview of some theoretical and empirical arguments drawn from learning theory and developmental theory. It also provides theory-driven descriptions of exemplary practices that can be seen as promising starting points for early mathematics education.

A Social Perspective on New Learning

INTRODUCTIONAccording to Simons (1997), schools need ‘process-oriented instruction’ to letpupils learn in a new learning way. He defines this instruction as: “(…) focussingon the further development of the thinking, learning and self-processes of regulation integrated in regular domain-specific instruction (…) it also tries to handover responsibility for learning and teaching to the learner gradually.” (o.c., p. 12).With young pupils, teachers initially act as external monitors, but gradualscaffolding and metacognitive guidance help pupils to self-regulators.become Moreover, teachers should organize positive self-evaluation and reflection by pupils(o.c.).A change towards new learning will also influence the social characteristicsinherent in learning and teaching. Compared to traditional learning, qualities of thesocial behavior of pupils and teachers, but also the social conditions within theteaching and learning situation, will look process-orienteddifferently in instruction. This will be true in school but also outside school, i.e. at home and inworking places.In this contribution we want to elaborate on social characteristics of, and socialconditions relevant to, the new learning approach in educational practice. First, weintroduce some theoretical terms to order potentially relevant social characteristicsof new learning processes and outcomes. Second, we present different researchexamples, from different kinds of educational practice. We successively analyze theexamples, to illustrate meaningful empirical varieties in the general significance ofthe social perspective on new learning.THEORIZING ON SOCIAL ASPECTS OF NEW LEARNINGLearning is embedded in a