Koeno Gravemeijer

• Doctor of Education
• Professor Emeritus at Eindhoven University of Technology

Our society is changing rapidly under the influence of globalization, robotization and digitalization. In order to prepare students for their future, education will have to change as well. In this chapter we will discuss three changes which are advocated for. The first concerns the well-known call for 21st century skills, the second, a strong empha...

The aim of this article is to offer an operational elaboration of the emergent modeling instructional design heuristic. This is one of the three instructional design heuristics with which the theory of realistic mathematics education (RME) may be characterized. First some background is given and it is elucidated how this heuristic was construed as...

Als je gaat nadenken over de actualisering van een curriculum, zou je eerst moeten kijken naar wat de maatschappij en het vervolgonderwijs vragen. Waarbij dat laatste ook niet in steen is gegoten, maar kan veranderen onder invloed van wat de maatschappij vraagt. Ik beperk me hier echter tot het tweede doel, voorbereiding op maatschappij en werk. He...

De aanbeveling in de NRO/Kennisrotonde-publicatie, ‘Differentiatie in de klas: wat werkt?’, om mastery learning te gebruiken lijkt ongenuanceerd en niet gefundeerd. Er zullen vast situaties zijn waar mastery learning ‘werkt’, maar een disclaimer was hier op zijn plaats geweest.

This chapter describes a socio-constructivist elaboration of Realistic Mathematics Education (RME) that emerged from my collaboration with Paul Cobb and Erna Yackel. It is argued that RME and socio-constructivism are compatible and complement each other. Socio-constructivism points to the critical role of the classroom culture, while RME offers a t...

This study was conducted among 269 student teachers at 11 primary teacher training colleges in the Netherlands. To investigate their competence in integrating theory and practice in their reflections on mathematics teaching, a learning environment was designed to evoke theory use in reflections on practice. To be able to systematically describe the...

Design research has become a powerful research methodology of increasing relevance in mathematics education research. This chapter provides an overview and selected insights for novice researchers who want to find out if this research methodology is suitable for their own projects, and what possible research outcomes can look like. As topic-specifi...

The gist of this article is that a shift is needed towards a mathematics curriculum in which teaching for understanding is the main objective. Before elaborating on what such a shift might entail, two compelling arguments for making such a change are presented. One is based on research, which showed that a one-sided emphasis on skills induced the t...

Research on solving mathematical word problems suggests that students may perform better on problems with a close to real-life representation of the problem situation than on word problems. In this study we pursued real-life representation by a mainly depictive representation of the problem situation, mostly by photographs. The prediction that stud...

This correction stands to correct the reference found in the reference list of the introductory paper above.

This correction stands to correct the reference found in the reference list of the closing paper above.

This article reports on a post hoc study using a randomised controlled trial with 31,842 students in the Netherlands and an instrument consisting of 21 paired problems. The trial showed a variability in the differences of students’ results in solving contextual mathematical problems with either a descriptive or a depictive representation of the pro...

full text: http://www.educationaldesigner.org/ed/volume3/issue10/article39/

The goal of this chapter is to draw attention to the need to think about learning environments and their design as a way of considering how sustainable change in the learning and teaching of statistics can be supported. The goal is not to advocate one particular approach to the design of learning environments but rather to raise awareness of the ne...

This paper attempts to engage the field in a discussion about what mathematics is needed for students to engage in society, especially with an increase in technology and digitalization. In this respect, mathematics holds a special place in STEM as machines do most of the calculations that students are taught in K-12. We raise questions about what m...

In answer to a call for innovative science and technology education in primary education, we started a design research project to explore how to teach instantaneous speed in grade five. In this article we present the results of a series of teaching experiments that were conducted to design, try out, and improve a local instruction theory on teachin...

The aim of this study is to contribute to the body of knowledge on the use of contextual mathematical problems. Word problems are a predominant genre in mathematics classrooms in assessing students' ability to solve problems from everyday life. Research on word problems, however, reveals a range of difficulties in their use in mathematics education...

This article offers a reflection on the findings of three PhD studies, in the domains of, respectively, subtraction under 100, fractions, and algebra, which independently of each other showed that Dutch students' proficiency fell short of what might be expected of reform in mathematics education aiming at conceptual understanding. In all three case...

Context-based approaches in science education, such as context-based chemistry or physics are gaining ground. For example, in The Netherlands new context-based curricula have been developed for the natural sciences,1 which will be implemented in the years to come (SLO, 2010). However, in order to create effective contextbased education, teachers wi...

Pupils can interpret sketches, floor plans and maps that vary regarding the aspects of scale preservation, detail and orientation. Pupils can indicate the position of people or objects in space. During this process, they make use of sketches, floor plans and maps, and of informal and formal coordinate systems.

In this chapter we describe the learning-teaching trajectories in geometry for each of the three terrains ‘Spatial sense’, ‘Plane and solid figures’, and ‘Visualization and representation’. Every learning-teaching trajectory description is connected to an example activity that is described, analyzed and discussed based on the core insights.

Babies come into contact with geometry while they are still in the cradle. This happens sooner than with any other aspect of mathematics. Freudenthal, among other authors, has written beautifully about this topic in his book ‘ Mathematics as an Educational Task’. He is one of the founders of Realistic Mathematics Education and writes ‘about the tru...

We see graphs everywhere: on television and in newspapers and periodicals. Graphs - which term we use here to include also diagrams and pie charts - allow you to present information in a compact and comprehensible fashion. An example is shown in the adjacent figure, where McDonalds explains the nutritional value of their French fries (medium). The...

Arithmetic, measurement and geometry are closely related. You could even say that measurement and geometry build a bridge between everyday reality on one side and mathematics on the other. Measurement is what we do when we quantify reality, i.e. when we allocate numbers in order to acquire a grip on reality. With these numbers, we can calculate and...

Measurement is a way to gain control of reality. For example, we talk about how big something is or how heavy it is. Or we wonder how far away something is, how much it costs, how sweet it is, how hot it is, or how long something lasts. Measurement is a specific mathematical approach to reality. If we want pupils to learn to look at reality in a si...

In this chapter we will provide a concise summary of the learning-teaching trajectories and targets for measurement. In very broad terms, during the upper grades of primary school, measurement education focuses on flexible manipulation of units and reasoning with units. Activities therefore do not focus on imprinting relationships and calculation r...

"This book is the fourth – and final – publication in the TAL project series. This TAL project was initiated by the Dutch Ministry of Education, Culture and Science, with the aim to improve the quality of mathematics education by providing a perspective on didactic goals and learning-teaching trajectories, and on the relationship between them. The...

This study concentrated on the theory–practice problem in mathematics teacher education. We examined 13 student teachers’ use of theory when they reflected on teaching practice in a class specifically designed to optimize the chance for theory use. We developed a Reflection Analysis Instrument with which the student teachers’ use of theory could be...

Design research continues to gain prominence as a significant methodology in the mathematics education research community. This overview summarizes the origins and the current state of design research practices focusing on methodological requirements and processes of theorizing. While recognizing the rich variations in the foci and scale of design...

To prepare students for participation in our society, where interpreting, representing, and manipulating of dynamic phenomena are becoming key activities, we believe that one should start developing a mathematical understanding of change at an early age. We therefore started a design research project to teach the concept of instantaneous speed in...

The purpose of TSG 32 was to gather congress participants who are interested in research, policy or design that focuses on mathematics curriculum development. The TSG aimed at including presentations and discussions of the state-of-the-art in this topic area and new trends and developments in research and practice in mathematics education. Curricul...

It remains difficult to foster problem-solving and mathematical-reasoning capabilities in classrooms where students and teachers are accustomed to the more traditional forms of education. Several studies suggest that this difficulty might be related to the kind of knowledge students acquire in such environments, which could be fragmented and superf...

In this chapter, we discuss the question of how we can encourage mathematics education to shift towards more inquiry-oriented practices in schools and what role textbooks and teachers play in such a reform. The stage is set by an exposition on the need for curriculum innovation in light of the demands of the twenty-first century. This points to a n...

Innovative educational approaches in the sciences have emphasized inquiry in the classroom but it is not self-evident that inquiry instruction leads to conceptual understanding. A design research cycle was conducted to investigate how physics instruction aimed at creating a classroom culture of inquiry can contribute to Grade 9 students’ understand...

Integrated curricula seem promising for the increase of attention on science and technology in primary education. A clear picture of the advantages and disadvantages of integration efforts could help curriculum innovation. This review has focused on integrated curricula in primary education from 1994 to 2011. The integrated curricula were categoris...

Starting point for this chapter is that societal changes ask for adaptations of a foundational mathematics curriculum for all. This chapter especially looks at the effects of information technology and globalization on the job market and employability with an eye on its consequences for the goals of mathematics education. It shows that next to the...

The book "Educational design research" (http://international.slo.nl/edr/) comprises two parts. Part A provides an introduction into educational design research as a suitable research approach either to address complex problems in educational practice or to develop or validate theories (e.g.) about learning processes, learning environments and the l...

Als onderdeel van een design research project ontwikkelden we een ICT-rijke lessenserie over snelheden waarin leerlingen het fenomeen momentane stijgsnelheid onderzochten en mathematiseerden. De op getranscribeerde video's van de klassediscussies aangevuld met leerlingproducten vormde de basis van een ingedikte beschrijving van de ontwikkeling van...

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often...

Students' algebraic proficiency is debated worldwide. To investigate the development of algebraic proficiency in Dutch secondary education, we set up a study, in which 1020 students in grades 8 - 12 took four algebra tests over a period of one year. Rasch analysis of the results shows that the students do make progress throughout the assessment, bu...

Based on insights into the nature of vocational mathematical knowledge, we designed a computer tool with which students in laboratory schools at senior secondary vocational school level could develop a better proficiency in the proportional reasoning involved in dilution. We did so because we had identified computations of concentrations of chemica...

Our society is deeply influenced and shaped by artefacts, ideas and values of technology. Technological developments sometimes replace human work, but can also inform human work by making information more accessible and usable (Levy & Murnane, 2005; Kent, Noss, Guile, Hoyles, & Bakker, 2007).

The concept of function is a central but difficult topic in secondary school mathematics curricula, which encompasses a transition from an operational to a structural view. The question in this paper is how the use of computer tools may foster this transition. With domain-specific pedagogical knowledge on the learning of function as a point of depa...

This study investigates what pupils aged 10–12 can learn from working with robots, assuming that understanding robotics is a sign of technological literacy. We conducted cognitive and conceptual analysis to develop a frame of reference for determining pupils’ understanding of robotics. Four perspectives were distinguished with increasing sophistica...

De wereld verandert continue. Van kinds af aan maakt verandering deel uit van ons leven. Het is daarom belangrijk dat kinderen situaties waarin verandering een rol speelt beter leren begrijpen. We noemen dit soort situaties waarin grootheden veranderen bij verstrijken van tijd dynamische systemen. Bij het begrijpen van een dynamisch systeem speelt...

Two research questions are answered: how did teacher instructional skills develop during a whole school year? What is the influence of this development on the interactions between students during the co-operative learning moments? From the analysis, it appeared that the teachers’ instruction changed from direct instruction to a more process and gro...

If we want to make something concrete in mathematics education, we are inclined introduce, what we call, ‘manipulatives’, in the form of tactile objects or visual representations. If we want to make something concrete in a everyday-life conversation, we look for an example. In the former, we try to make a concrete model of our own, abstract, knowle...

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a samp...

The chapter reprinted in this part of the book outlines an analytical approach for teachers develop and revise their instructional practices. The institutional setting of mathematics teaching as we conceptualize it encompasses district1 and school policies for mathematics instruction. It therefore includes both the adoption of curriculum materials...

I completed my dissertation studies with Les Steffe and Ernst von Glasersfeld at the University of Georgia in 1983 and then accepted a faculty position at Purdue University in Indiana. The first study that I conducted at Purdue University was built on my dissertation work and focused on the psychological contexts within which young children interpr...

My career as a mathematics education researcher began in 1978 when I entered the master’s program in mathematics education at the University of Georgia. Prior to moving to Georgia with my wife Jenny, I had completed my undergraduate degree in mathematics at Bristol University in England and was a secondary math teacher for 2 years. Our primary reas...

The emergent perspective described in Chapter 4 serves to orient the analysis of the teacher’s and students’ actions and interactions in particular mathematics classrooms. As noted in Chapter 4, this framework was developed while analyzing the data generated in the course of two year-long classroom design experiments (an initial experiment conducte...

The notion of a classroom mathematical practice was introduced in the reprinted chapter in the last part. It is fair to say in retrospect that my initial use of this construct was largely intuitive: I had not defined a classroom mathematical practice with any precision or clarified how other researchers might identify the mathematical practices int...

The professionalization program VTB-Pro (elaborating technology in primary education, professional) was founded in the year 2007 and enabled 5000 teachers and 5000 student teachers to professionalize themselves in the field of science and technology. This chapter describes the investigation of three professionalization programs developed by the kno...

Our current society is deeply influenced and shaped by artefacts, ideas and values of science and technology, for example in health care, energy, transportation and communication. Also, issues such as pollution and nuclear energy become objects of public debate. In their jobs, professionals are confronted with an increased use of information and co...

Science and technology play an important part in present-day society. People need some degree of scientific and technological literacy to cope with the challenges and possibilities of the many products, devices and processes available. Robotics is a good example of this. For many people it is a black box. The way it functions is hidden under its su...

The last quarter of the twentieth century has seen fundamental shifts in educational theories. These shifts were initiated by the research of Fullan and Pomfret (1977), which revealed fundamental flaws in the curriculum-innovation model of “research, development, and diffusion,” known as the RDD model. These flaws showed that one of the key pillars...

Dynamic systems are everywhere; key to comprehend dynamic systems is the concept of a compound variable quantity. To understand the world around them, children need a good understanding of compound variable quantities. Furthermore better knowledge of this concept gives children a better foundation for the learning of calculus related concepts. The...

The availability of technology in the mathematics classroom challenges the way teachers orchestrate student learning. Using the theory of instrumental orchestration as the main interpretative framework, this study investigates which types of orchestrations teachers develop when using technology and to what extent these are related to teachers’ view...

Building on the claim that teachers need to know how innovative instructional approaches work to be able to adapt them to their own classrooms, design research is presented as a research method that aims to offer exactly that kind of information. We elaborate design research that aims to develop a local instruction theory-a theory about the process...

In this paper we focus on an instructional sequence that aims at supporting students in their learning of the basic principles of rate of change and velocity. The conjectured process of teaching and learning is supposed to ensure that the mathematical and physical concepts will be rooted in students’ understanding of everyday-life situations. Stude...

This chapter discusses the relation between ‘emergent modelling’ and ‘mathematical modelling’. The former that has its roots in RME theory constitutes the main theme of this chapter. It is argued that mathematical modelling requires a preceding learning process, since it requires abstract mathematical knowledge to construe a mathematical model. The...

Op zoek naar concretiseringen die het kinderlijk denken en ondersteunen

Using Freudenthal’s method of historical phenomenology, the history of statistics was investigated as a source of inspiration for instructional design. Based on systematically selected historical examples, hypotheses were formulated about how students could be supported in learning to reason with particular statistical concepts and graphs. Such a h...

In this chapter, we investigate the relationship between computer algebra use and algebraic thinking from the perspective of the instrumental approach to learning mathematics in a technological environment, which was addressed in the previous chapter. Data comes from a research study on use of computer algebra for developing algebraic insights. Tea...

In dit artikel wordt gereageerd op de wiskundige Opmeer (2005), die in het vorige nummer van dit tijdschrift een kritische bespreking wijdde aan het discussiestuk ‘De kern van breuken, verhoudingen, procenten en kommagetallen’ (Keijzer, Figueiredo, Van Galen, Gravemeijer & Van Herpen, 2005). In deze reactie wordt betoogd dat Opmeers kritiek voortko...

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjus...