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Hans Freudenthal a mathematician on didactics and curriculum theory.

Gravemeijer, K.P.E.; Terwel, J.

published in

Journal of Curriculum Studies

2000

DOI (link to publisher)

10.1080/00220270050167170

Link to publication in VU Research Portal

citation for published version (APA)

Gravemeijer, K. P. E., & Terwel, J. (2000). Hans Freudenthal a mathematician on didactics and curriculum

theory. Journal of Curriculum Studies, 32(6), 777-796. https://doi.org/10.1080/00220270050167170

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Hans Freudenthal: a mathematician on didactics and

curriculum theory

K. GRAVEMEIJER and J. TERWEL

The main ideas in the work of Hans Freudenthal (1905± 1990), the Dutch math-

ematician and mathematics educator, related to curriculum theory and didactics are

described. Freudenthal’ s educational credo, ` mathematics as a human activity’ , is

explored. From this pedagogical point of departure, Freudenthal’ s criticism of

educational research and educational theories is sketched and ¯ eshed out.

Freudenthal’ s approaches to mathematics education, developmental research and

curriculum development can be seen as alternatives to the mainstream ` Anglo-Saxon’

approaches to curriculum theory.

During his professional life, Hans Freudenthal’ s views contradicted almost

every contemporary approach to educational ` reform’ : the ` new’ math-

ematics, operationalized objectives, rigid forms of assessment, standardized

quantitative empirical research, a strict division of labour between cur-

riculum research and development, or between development and imple-

mentation. Looking back from the present, it is of great interest to see how

his ideas, which may at the time have seemed to embody recalcitrance for

its own sake, have now become widely accepted. It would, of course, be far-

fetched to suggest that this correlation implies a causal relationship, but it

does indicate Hans Freudenthal’ s special role, not only in mathematics

education, but also in the development of curriculum theory and research

methodology.

Hans Freudenthal had already earned his spurs as a research mathe-

matician when he developed an interest in mathematics education and

made himself acquainted with educational and psychological traditions in

Europe and the US. Today, he is probably best known as one of the most

in¯ uential mathematics educators of his time.

1

In this paper, we shall try to

highlight some of Freudenthal’ s main ideas, while acknowledging that we

cannot do justice to his wide-ranging work ± even if we were able to.

2

Our point of view will centre on curriculum theory and pedagogy, and we

J.CURRICULUM STUDIES,2000, V OL.32, NO.6, 777± 796

Koeno Gravemeijer is research coordinator in the Freudenthal Institute at the University of

Utrecht, The Netherlands, and a research associate professor at Peabody College of

Vanderbilt University, Nashville, USA. His interests are in symbolizing and modelling in

education theory and in developmental research as a research method.

Jan Terwel is Professor of Educational Psychology in the Faculty of Psychology and

Education at the Vrije University Amsterdam and Professor of Education at the Graduate

School of Teaching and Learning, University of Amsterdam, Van der Boechorststraat, 1081

BT Amsterdam, The Netherlands (e-mail: J.Terwel@psy.vu.nl). His interests are focused

on curriculum studies, social interaction, and learning and cognition, especially in science

and mathematics.

Journal of Curriculum Studies ISSN 0022± 0272 print/ISSN 1366± 5839 online #2000 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

will consider the aspects of Freudenthal’ s work and theories that are

relevant from those perspectives (Blankertz 1973, Hopmann and Riquarts

1995a, b).

We start with a consideration of Freudenthal’ s place within the cur-

riculum-meets-Didaktik discussion and try to clarify the main sources of

Freudenthal’ s theory of mathematics education. We continue by elaborat-

ing Freudenthal’ s philosophy of mathematics education, the cornerstone of

his work. We move on to his criticism of what had become ` tradition’ in

educational research and educational theory. We subsequently discuss

Freudenthal’ s proposed alternative to ` research, development and dif-

fusion’ , i.e. the traditional RD&D model. His alternative incorporates a

philosophy, or theory, of mathematics education, its elaboration in devel-

opmental research, and an understanding of the overarching process of

educational development in which this is embedded. This will be followed

by a short description and analysis of Freudenthal’ s in¯ uence on educa-

tional change in the Netherlands. We conclude with a discussion of

curriculum theory, didactics, and ` mathematics for all’ .

Didaktik, curriculum theory, and Freudenthal

Although it has been suggested that curriculum theory as developed in the

US and European didactics are concerned with the same questions,

there are striking diå erences in approach (Westbury 1995, 2000). These

diå erences derive from more basic diå erences in cultural, philosophical

and institutional backgrounds. In the European context, the concept of

Didaktik is embedded in a pedagogical theory; thus the notion of pedagogy

as a form of Geisteswissenschaftliche along with the phenomenological

theories of Bildung (i.e. ` formation’ ), have their points of departure in the

practice of education, i.e. in educational reality. And Bildung can be seen as

opposed to Ausbildung.Bildung refers to the ideal of personality formation,

and does not entail simply the transmission of knowledge, but also the

development of the knowledge, norms and values associated with ` good’

citizenship and/or a membership of the cultural and intellectual e

Âlite.

Ausbildung, on the other hand, refers to vocational and professional educa-

tion. In this context, Didaktik is primarily concerned with theories of the

aims and content of education and instruction.

3

In the Netherlands, Didaktik was related to the Geisteswissenschaftliche

phenomenological pedagogy, as represented, for example, by Langeveld

(1965) at the University of Utrecht. This position lost its dominance in the

1960s and 1970s and, as a consequence, the concept of a general Didaktik

was (to some extent, and gradually) replaced by formal models of learning

and instruction as seen in the work of US educational psychologists such

as Robert Glaser, Robert de Cecco, and Benjamin Bloom. However,

subject-matter Didaktik ± as developed within faculties and institutes of

mathematics and science education ± was not totally swamped by this

movement.

Although it does not show in his rare references, Freudenthal was well-

acquainted with educational and psychological traditions in Europe and the

778 K.GRAV EMEIJER AND J .TERWEL

US. He made many visits, which he called ` Bildung journeys’ , and was also

in¯ uenced by the pedagogical ideas of his wife Suus Lutter-Freudenthal,

one of the driving forces behind the reception of Peter Petersen’ s Jena-plan

movement in the Netherlands. He was also strongly in¯ uenced by the

reform-pedagogy of the Belgian educationist Ovide Decroly (Freudenthal

1973b) and was an active member of the New Education Fellowship in

which Ovide Decroly participated, where he was also in¯ uenced by Pierre

van Hiele and van Hiele’ s wife, Hieke Geldof-van Hiele (both mathematics

teachers who conducted their doctoral research under Freudenthal and

Langeveld).

4

For example, Decroly’ s educational idea of centre d’ inteÃreÃt,

which could be elaborated in space and time, resembles Freudenthal’ s ideas

on the learning of mathematics in ` real-life’ contexts. Decroly’ s principle of

elaboration in space (how it appears in diå erent countries) and time

(development in history), under the teacher’ s guidance, corresponds to

Freudenthal’ s (and Dewey’ s) idea of guided reinvention.

Although Freudenthal never referred to scholars like the German

Wolfgang Klafki, the basic questions that Klafki addressed were also

Freudenthal’ s questions (see Freudenthal 1973a): What is to be taught in

a school subject? for what purpose? and to whom? His credo ` mathematics

as a human activity’ can be seen as an expression of a Geisteswissen-

schaftliche, phenomenological theory of mathematics education which has

its point of departure in the practice of education and teaching, and not

in the transmission of mathematics as a pre-formed system. Some of

his main ideas (such as ` reinvention’ ) and his criticism of the ` antididactic

inversion’ of traditional (deductive) instruction were probably inspired by

the reform-pedaogy movement, i.e. by progressive education in which the

ideas of, among others, Peter Petersen and Maria Montessori were im-

portant.

As viewed by Freudenthal, curriculum theory is not a ® xed, pre-stated

set of theories, aims and means, contents, and methods. Rather, it is always

related to processes. Understood positively, the word ` curriculum’ is more

often than not used in combination with change or development, for

example, as in curriculum development or developmental research. For

Freudenthal, curriculum theory was a practical endeavour from which new

theoretical ideas might arise as a kind of scienti® c by-product. For him,

curriculum development was not to be conducted from academic ivory

towers, but in schools, in collaboration with teachers and students (Freu-

denthal 1973a). Similar ideas are expressed by Schwab (1970: see also

Walker 1990), who, in his plea for curriculum as ` practical’ , eloquently

challenged the mainstream RD&D curriculum theory of his time. Thus,

there are similarities between some branches of the Anglo-Saxon approach

to curriculum theory and Freudenthal’ s understanding of curriculum.

But, when the word ` curriculum’ appears in the work of Freudenthal, it

usually has a negative connotation. He writes about the mainstream of the

Anglo-Saxon curriculum movement as a theory-driven, top-down endeav-

our, and referred to this approach as ` boxology’ . As we have suggested, the

most striking feature of Freudenthal’ s position is his view of curricula as

processes and he proposed his own alternative to curriculum development

which he called educational development. Whereas curriculum development

HANS FREU DENTH AL 779

centres on the development of curriculum materials, Freudenthal wanted to

go one step further: educational development should seek to foster actual

change in on-going classroom teaching. Consequently, such educational

development is much more than instructional design; it is an all-embracing

innovation strategy, based, on the one hand, on an explicit educational

philosophy and, on the other hand, incorporates developments in all sorts

of educational materials as part of its strategy. The engine of this whole

process is developmental research, an approach which ® ts the pedagogical

tradition very well; it is a qualitative/interpretative research tied to teaching

experiments in individual classrooms. A central role is given to dialogue

between researchers, curriculum developers, and teachers.

Mathematics as a human activity

Freudenthal was an outspoken opponent of the ` new mathematics’ of the

1960s, which took its starting point as the attainment of modern math-

ematics, especially set theory. With this criticism, he showed himself as an

exponent of the pedagogical tradition in the sense that his criticisms were

grounded in a discussion about what was to be taught, and why. Thus, he

acknowledged generality and wide applicability as special characteristics of

mathematics, and he also acknowledged that modern mathematics

abstracted mathematics even further while at the same time enhancing

¯ exibility. However, in his view, abstracting was the source of the peda-

gogical problem.

In an objective sense the most abstract mathematics is without doubt also the

most ¯ exible. But not subjectively, since it is wasted on individuals who are

not able to avail themselves of this ¯ exibility (Freudenthal 1968: 5).

Since the applicability of mathematics was also often problematic, he

concluded that mathematics had to be taught in order to be useful. He

observed that this could not be accomplished by simply teaching a ` useful

mathematics’ ; that would inevitably result in a kind of mathematics that

was useful only in a limited set of contexts. However, he also rejected the

alternative: ` If this means teaching pure mathematics and afterwards

showing how to apply it, I’ m afraid we shall be no better oå . I think this

is just the wrong order’ (Freudenthal 1968: 5). Instead, mathematics should

be taught as mathematizing. This view of the task of school mathematics

was not only motivated by its importance for usefulness; for Freudenthal

mathematics was ® rst and foremost an activity, a human activity, as he often

emphasized. As a research mathematician, doing mathematics was more

important to Freudenthal than mathematics as a ready-made product. In

his view, the same should hold true for mathematics education: math-

ematics education was a process of doing mathematics that led to a result,

mathematics-as-a-product. In traditional mathematics education, the result

of the mathematical activities of others was taken as a starting point for

instruction, and Freudenthal (1973b) characterized this as an anti-didac-

tical inversion. Things were upside down if one started by teaching the

result of an activity rather than by teaching the activity itself.

5

780 K.GRAV EMEIJER AND J .TERWEL

[Mathematics as a human activity] is an activity of solving problems, of

looking for problems, but it is also an activity of organizing a subject matter.

This can be a matter from reality which has to be organized according to

mathematical patterns if problems from reality have to be solved. It can also

be a mathematical matter, new or old results, of your own or others, which

have to be organized according to new ideas, to be better understood, in a

broader context, or by an axiomatic approach (Freudenthal 1971: 413± 414).

He termed this organizing activity ` mathematizing’ in other publica-

tions and it should be emphasized that it involves both ` matter from reality’

and ` mathematical matter’ . In other words, Freudenthal included both

applied mathematics and pure mathematics in his conception of mathema-

tizing. In this sense, his starting point diå ered from other mathematics

educators who also emphasized mathematical activity but focused on a

mathematical discourse that was modelled on the discourse of pure research

mathematicians ± as this was reconstructed, e.g. by Lakatos (1976).

The image of mathematical activity that Freudenthal elected as a

paradigm for mathematics education diå ered from this in two ways.

First, it included, as mentioned earlier, applied mathematics, or to be

more precise, ` mathematizing matter from reality’ . Secondly, the focus was

not on the form of the activity, but on the activity itself, as well as on its

eå ect. Moreover, the notion of ` discourse’ referred to a social practice,

whereas the idea of mathematizing put a stronger emphasis on mental

activity. Freudenthal’ s broader de® nition of mathematics as a human

activity ® tted in better with a more pragmatic discourse, such as one

might expect in applied mathematics. In such discourse there would be

more emphasis on adequacy and eæ ciency, and less on goal-free conjectur-

ing, for instance.

Freudenthal used the word ` mathematizing’ in a broad sense: it was a

form of organizing that also incorporated mathematical matter. By choosing

the word ` organizing’ , Freudenthal also indicated that, for him, mathema-

tizing was not just a translation into a ready-made symbol system. Instead,

a way of symbolizing might emerge in the process of organizing the subject

matter. It was the organizing activity itself that was central to Freu-

denthal’ s conception.

Mathematizing literally stands for ` making more mathematical’ . To

clarify what ` more mathematical’ means, one may think of such character-

istics of mathematics as generality, certainty, exactness, and brevity. To

clarify what is to be understood by mathematizing we may look at the

following speci® c strategies within these characteristics (Gravemeijer 1994;

see also Treå ers 1987):

.for generality: generalizing (looking for analogies, classifying,

structuring);

.for certainty: re¯ ecting, justifying, proving (using a systematic

approach, elaborating and testing conjectures, etc.);

.for exactness: modelling, symbolizing, de® ning (limiting interpret-

ations and validity); and

.for brevity: symbolizing and schematizing (developing standard

procedures and notations).

HANS FREU DENTH AL 781

Viewed from this angle, mathematizing subject matter from mathematics

and mathematizing matter from reality share the same characteristics. And,

this was fundamental for Freudenthal, since, in his view, mathematics

education for young children should aim above all at mathematizing

everyday reality. Young children cannot mathematize mathematical

matter, since, at the beginning, there is no mathematical matter that is

experientially real to them. Moreover, mathematizing subject matter from

reality also familiarizes the students with a mathematical approach to

everyday-life situations. We may also refer here to the mathematical

activity of ` looking for problems’ , mentioned by Freudenthal, which

implies a mathematical attitude that encompasses knowledge of the poss-

ibilities and the limitations of a mathematical approach, i.e. knowing when

a mathematical approach is appropriate and when it is not.

This emphasis on ` mathematizing reality’ ® ts in with the call for

` mathematics for all’ (see Damerow and Westbury 1985, Keitel 1987).

Freudenthal stressed that not all students are future mathematicians: for

the majority, all the mathematics they will ever use will be to solve

problems in everyday-life situations. Therefore, familiarizing students

with a mathematical approach to this type of problem-solving deserved

to be a highest priority in mathematics education. This goal could be

combined with the objective of having students mathematize problem

situations that would be experientially real to them.

6

In this light, it will not come as a surprise that Freudenthal forcefully

attacked the transposition didactique, espoused by the French mathematics

educator Chevallard (1985), who took the expert knowledge of the math-

ematician as his point of departure:

The mathematics that the vast majority of our future citizens learn in school

does not re¯ ect any kind of rendering ± for didactic purposes or otherwise ±

of philosophical or scienti® c insights, unless they are those of an epoch long

past (Freudenthal 1986: 326; our translation).

According to Keitel (1987), the central question is to realize a ` math-

ematics for all’ that remains ` mathematics’ . Consequently, she argues, it

may be necessary at times for the teacher to leave behind everyday-life

problems and refer to the science of mathematics ± in order to show the

constellations of concepts, structures, and systems which have been

invented and tested there. Elaborating Freudenthal’ s idea of math-

ematizing, Treå ers (1987) made a distinction between horizontal and

vertical mathematization. The former involves converting a contextual

problem into a mathematical problem, the latter involves taking math-

ematical matter onto a higher plane. Vertical mathematization can be

induced by setting problems which admit solutions on diå erent math-

ematical levels.

Freudenthal (1991: 41, 42) characterized this distinction as follows:

Horizontal mathematization leads from the world of life to the world of

symbols. In the world of life one lives, acts (and suå ers); in the other one

symbols are shaped, reshaped, and manipulated, mechanically, comprehend-

ingly, re¯ ectingly: this is vertical mathematization. The world of life is what

is experienced as reality (in the sense I used the word before), as is a symbol

782 K.GRAV EMEIJER AND J .TERWEL

world with regard to abstraction. To be sure the frontiers of these worlds are

vaguely marked. The worlds can expand and shrink ± also at one another’ s

expense.

As Freudenthal indicates, the boundaries between what is to be denoted

as ` horizontal mathematization’ and ` vertical mathematization’ are not

clear-cut. The crux lies in what is to be understood as ` reality’ and he

(1991: 17) provided the following elucidation: ` I prefer to apply the term

reality to what common sense experiences as real at a certain stage’ . Reality

is understood as a mixture of interpretation and sensual experience, which

implies that mathematics, too, can become part of a person’ s reality. Reality

and what a person counts as common sense are not static but grow, and are

aå ected by the individual’ s learning process. This is also how Freu-

denthal’ s (1991: 18) statement ` Mathematics starting at, and staying

within, reality’ must be understood.

It will be clear that, in Freudenthal’ s view, ` common sense’ and ` reality’

were construed from the viewpoint of the actor. This implies that the

boundary between horizontal and vertical mathematization has to be

assessed from the actor’ s point of view as well. Whether a certain aspect

of a person’ s mathematical activity is to be called ` vertical’ or ` horizontal’

depends on the question as to whether the activity involves some extension

of that person’ s mathematical reality. A symbolizing activity, for instance,

could be a routine activity for a student. This would be a case of horizontal

mathematizing. However, if the same manner of symbolizing were a new

invention for another student, then this would involve vertical mathema-

tization. Vertical mathematization is the most clearly visible if a student

explicitly replaces his or her solution method by one on a higher level. This

could be a shift to a solution method, or a way of describing that is more

sophisticated, better organized, or, in short, more mathematical (in accord-

ance with the characteristics we laid out earlier).

Such shifts can be induced by re¯ ecting upon solution methods and

underlying understanding. Whole-class discussions of solution methods,

interpretations, and insights will enhance the likelihood of those shifts;

especially if the problem at hand gives rise to a variety of solution methods

on diå erent levels.

7

When comparing and discussing their solution

methods, for instance, some students may realize that other solution

methods have advantages over their current method. This crucial role of

dialogue as applied to interpretations, ideas and methods once more shows

that an emphasis on mathematizing does not only imply solitary activity on

the part of the individual student.

But, the dialogue need not only take the form of whole-class discus-

sions. Freudenthal also espoused group work. His ® rst plea for learning in

small groups was in 1945, during a symposium of the New Educational

Fellowship. Later, he advocated mathematics education in heterogeneous

groups (Freudenthal 1987, 1991). In his opinion, both weaker and stronger

students would pro® t from collaboration. And, as Freudenthal (1987: 338)

noted, on re-reading the work he had produced from 1945 onwards, he

realized, to his own surprise, how consistently, and for more than 40 years,

HANS FREU DENTH AL 783

he had been a protagonist of co-operative learning in small heterogeneous

groups.

8

Criticism of educational research

To some, Freudenthal is perhaps better known for his criticism of ` tradi-

tional’ educational research than for his own ideas and theories. In the

Netherlands, he was a dreaded opponent of anyone in the educational

research community who used an empiricist methodology and over-sophis-

ticated statistics. He used his powers as a mathematician to show the many

¯ aws in the manner in which mathematics (i.e. statistics) was used in many

examples of ` hard’ empirical research.

Freudenthal’ s opposition to much educational research was related to

his conviction that discontinuities in the learning process are essential.

Such discontinuities may be seen as creating shortcuts, or taking diå erent

perspectives (Freudenthal 1991; see also van den Heuvel-Panhuizen 1996).

It is in such discontinuity, he argued, that one can perceive whether a

student has achieved a certain level of comprehension. To be able to identify

these discontinuities, students must be followed individually. This implies that

group means and the like are not particularly useful, since means wash out

the individual discontinuities. Moreover, the emphasis should be on

observing learning processes, not on testing ` objective’ learning outcomes.

In addition, Freudenthal believed that such ` hard’ research could not

answer the educational questions of for what purpose a subject is being

taught, and to whom (Freudenthal 1973a, b, 1988).

Freudenthal directed a second set of criticisms towards the testing

movement. He was skeptical of objective testing methods and condemned

the negative in¯ uence of examinations and testing techniques on education.

The hard core of his criticisms centred on ignorance of subject matter and

the overestimation of reliability at the expense of validity (Freudenthal

1980, 1991) and he did not share the optimism of the objective testing

movement.

9

More generally, Freudenthal’ s criticisms of educational research

focused on methodologists whose strength consisted in ` . . . knowing every-

thing about research, but nothing about education’ (Freudenthal 1991:

151). He ® ercely opposed the separation of content and form. In his view,

this leads to empty models that have to be ® lled by content experts: ` They

gladly leave to the educational researcher the responsibility of his own to ® ll

empty vessels with educational contents, but they are unconcerned about

whether these ® t or not’ (Freudenthal 1991: 151). He oå ered similar

criticisms of general educational theories.

Criticisms of general education theories

Freudenthal believed that general education theories not only do not ® t the

situation of mathematics education, but in many cases are detrimental to

784 K.GRAV EMEIJER AND J .TERWEL

the kind of education he endorsed. We may see this in his criticisms of

Bloom, Gagne

Â, and Piaget. Thus, he judged Bloom’ s Taxonomy of Educa-

tional Objectives to be inappropriate for mathematics education. Instead of

aiming for a classi® cation (resulting in taxonomies), he proposed that the

activity of reality-structuring should be looked at. It is by structuring that

students get a grip on reality; the arti® cial character of the categories of

educational goals in the Taxonomy have a negative eå ect on both schooling

and test development (Freudenthal 1979). Bloom’ s strategy of mastery

learning was also vigorously rejected by Freudenthal (1980); he accused

Bloom of conceiving of learning as a process in which knowledge is poured

into the heads of the students.

Robert Gagne

Âalso came under ® re from Freudenthal. He found the

idea of task analyses, as presented in The Conditions of Learning (Gagne

Â

1977), to be completely incompatible with the idea of mathematics as

an human activity. ` A feeling of loneliness seized me: is mathematics really

so diå erent? I wish that someone who profoundly understands both

mathematics and psychology would show us the bridge’ (Freudenthal

1973b: vi).

Gagne

Âconceived of the learning process as a continuous process that

moved from the acquisition of simple to complex structures. Freudenthal

saw educational processes as discontinuous: from rich, complex structures

of the world of everyday-life to the abstract structures of the world of

symbols ± and not the other way around. Starting points should be found in

situations that ` beg to be organized’ where, as Freudenthal (1991: 30) put it,

categories are not pre-de® ned but are developed by the learners themselves,

and need to be accommodated to their needs.

Freudenthal also criticized Piaget for his mathematics and his experi-

ments. What worried Freudenthal the most, however, was that Piaget’ s

work seduced teaching methodologists into translating its research ® ndings

into instructional settings for mathematics education:

It is a sad story to see didacticians founding their practice on theories they

learned from a psychologist; what they borrow from Piaget are not the results

of his experiments but the wrong, or at least misunderstood, mathematical

presuppositions (Freudenthal 1973b: 193).

10

Freudenthal (1991) also addressed constructivism. But, although he

criticizes the constructivist epistemology from an observer’ s point of view,

it can be argued that the way he sees mathematics from an actor’ s point of

view is compatible with this epistemology. Thus, from the perspective of an

active mathematician, he characterizes mathematics as a form of (well-

developed) common sense ± a notion that is strongly tied to his idea of

` expanding reality’ . Moreover, his educational goal is to make sure that the

students experience ` objective mathematical knowledge’ as a seamless

extension of their everyday-life experience. This leads us to conclude

that Freudenthal stands much closer to constructivism then one might

gather from his attack on it.

HANS FREU DENTH AL 785

Realistic mathematics education

We can summarize Freudenthal’ s view on mathematics education as

follows. Mathematics must be seen foremost as a process, a human activity.

However, at the same time, this activity has to result in mathematics as a

product. This leads to the (design) question of how to shape a mathematics

education that integrates both goals. Freudenthal’ s work was based on a

number of ideas about how to deal with these questions. These ideas can be

discussed under the headings of ` guided reinvention’ , ` levels in the learning

process’ , and ` didactical phenomenology’ .

Guided reinvention

According to the reinvention principle, a route to learning along which a

student is able, in principle, to ® nd the intended mathematics by himself

or herself has to be mapped out (Freudenthal 1973b). To do so, the

curriculum developer starts with a thought experiment, imagining a

route by which he or she could have arrived at a personal solution.

Knowledge of the history of mathematics can be used as a heuristic

device in this process.

Freudenthal (1991) spoke of ` guided reinvention’ with an emphasis on

the character of the learning process rather than on inventing as such. The

idea was to allow learners to come to regard the knowledge they acquire as

their own, personal knowledge, knowledge for which they themselves are

responsible. On the teaching side, students should be given the opportunity

to build their own mathematical knowledge-store on the basis of such a

learning process.

Freudenthal acknowledged the history of mathematics as a source of his

inspiration. Subsequently, it has become clear that the reinvention prin-

ciple can also be inspired by informal solution procedures (Stree¯ and 1990,

Gravemeijer 1994). More often than not, students’ informal strategies can

be interpreted as anticipating more formal procedures. In such cases,

mathematizing similar solution procedures constitutes the reinvention

process.

In general, contextual problems that allow for a wide variety of solution

procedures will be selected, preferably solution procedures that in them-

selves re¯ ect a possible learning route. Freudenthal saw the reinvention

approach as an elaboration of the Socratic method and to illustrate the

Socratic method, he spoke of ` thought experiments’ , i.e. the thought-

experiment of teachers or textbook authors who imagine they are teaching

students while interacting with them and dealing with their probable

reactions. One part of the thought-experiment, therefore, lies in anticipat-

ing student reactions. The other part consists in the design of a course of

action that ® ts anticipated student reactions. More precisely, the idea is that

teaching matter is re-invented by students in such interaction. Freudenthal

(1991: 100± 101) comments:

786 K.GRAV EMEIJER AND J .TERWEL

Though the student’ s own activity is a ® ction in the Socratic method, the

student should be left with the feeling that it [i.e. understanding and insight]

arose during the teaching process; that it was born during the lesson, with the

teacher only acting as midwife.

Freudenthal did not subscribe to the Socratic method as such. He gave a

much more active role to students in the process of constructing their own

knowledge. The similarity, however, lies in the anticipation, in the planning

of possible learning trajectories (Simon 1995). Such conjectured learning

trajectories encompass the various problems that should be posed, the

anticipated mental activities of students, and the actions that should be

taken to make the reinvention process possible.

Levels in the learning process

Freudenthal complemented the concept of reinvention with what Treå ers

(1987) called ` progressive mathematization’ . What could be seen as rein-

vention from an observer’ s point of view, should be experienced by the

student as ` progressive mathematization’ ± from an actor’ s point of view.

Students should begin by mathematizing subject matter from reality. Next,

they should switch to analysing their own mathematical activity. This latter

procedure is essential since it contains a vertical component, which

Freudenthal (1971: 417), with reference to Van Hiele, described in the

following manner: ` The activity on one level is subjected to analysis on the

next, the operational matter on one level becomes subject matter on

the next level’ .

This shift from ` operational’ to ` subject matter’ relates to the shift from

procedures to objects, which Sfard (1995) observed in the history of

mathematics. It also relates to what Ernest (1991: 78) has called ` rei® ca-

tion’ . Freudenthal’ s level-theory shaped the RME-view on educational

models. Instead of ready-made models, RME looks for models that emerge

® rst as operational models of situated solution procedures, and then

gradually evolve into entities of their own to function as models for

formal mathematical reasoning (Gravemeijer 1999).

Didactical phenomenology

Freudenthal emphasized the importance of a phenomenological embedding

of mathematical objects. In opposition to the concept-attainment approach,

which implies the embodiment of concepts in concrete materials, Freu-

denthal proposed the use of phenomenologically rich situations: situations

that are begging to be organized. In such a didactical phenomenology

(Freudenthal 1983), situations should be selected in such a way that they

can be organized by the mathematical objects which the students are

supposed to construct. The objective is to ® gure out how the ` thought-

matter’ (nooumenon) describes and analyses the ` phenomenon’ . How it

would make the phenomenon accessible for calculation and thinking

HANS FREU DENTH AL 787

activity. Such phenomenological analysis lays the basis for a didactical

phenomenology which also incorporates a discussion of what phenomen-

ological analysis meant from an educational perspective. For example, to

construct length as a mathematical object students should be confronted

with situations where phenomena have to be organized by length.

Within the framework of a didactical phenomenology, situations where

a given mathematical topic is applied are to be investigated in order to

assess their suitability as points of impact for a process of progressive

mathematization. If we viewed mathematics as having evolved historically

from practical problem solving, it would be reasonable to expect to ® nd the

problems which gave rise to this process in present-day applications. Next

we could imagine that formal mathematics came into being in a process of

generalizing and formalizing situation-speci® c problem solving procedures

and concepts about a variety of situations. The goal of a phenomenological

investigation is, therefore, to ® nd problem-situations from which situation-

speci® c approaches can be generalized, and to ® nd situations that can evoke

paradigmatic solution-procedures as the basis for vertical mathematization.

To ® nd phenomena which can be mathematized, we can seek to understand

how they were invented.

Research for the sake of educational change

When Freudenthal (1991) thought about research he typically asked

himself ` What is the use of it?’ and he invariably gave the answer

` Change’ . Education had to be constantly adapted to a changing society.

Therefore, ` change’ as a concept was to be preferred over the notion of

` improvement’ , inasmuch as what is regarded as a better education is

dependent on the needs and priorities of society at a given moment in

time ± and as society changes, education would need to change also. In this

light, an important task of the researcher is trail-blazing. Freudenthal saw

the chain from research to the classroom as too long in traditional research:

trail-blazing should not start in armchairs or the laboratory, but in the

classroom. It was this philosophy of the aims and function of research that

guided the approach to research adopted by the Institute for Development

of Mathematics Education (IOWO), of which Freudenthal became the

director.

At the time IOWO was founded, the RD&D model was in vogue within

the Dutch educational research community. In this model, curriculum

development, and what was called ` implementation’ , were completely

separate and it was in opposition to this approach that Freudenthal

(1991) set out his concept of ` educational development’ . This concept

meant more than just curriculum development but also contained the

end-goal of changing educational practice. Moreover, educational devel-

opment not only implied that the implementation of the curriculum was

anticipated from the outset, it also implied choice of a broad change

approach, comprising teacher education, counselling, test-development,

and opinion-shaping ± all based on the same educational philosophy. In

contrast to the curriculum movement, Freudenthal integrated research,

788 K.GRAV EMEIJER AND J .TERWEL

development, implementation, and dissemination. As a consequence of his

orientation on educational practice, he proposed to involve all participants

from the start under the slogan, ` educational development in dialogue with

the ® eld’ .

The type of change Freudenthal pursued was guided by his (1973b)

idea of mathematics as a human activity. However, when IOWO was

launched little research was available about this kind of mathematics

education. Consequently, questions of how to develop instruction had to

be answered during the process of development itself.

Developmental research

Freudenthal was initially reluctant to call to what was done at IOWO

research. ` At our institute we regard ourselves not as researchers but

engineers’ . In addition, he (1973a) regarded theory as a by-product of

educational development. Later, however, he (1988) argued that this

metaphor separates research from educational development, and, thus,

cannot do justice to the intertwined character of development and research

in ` developmental research’ .

1 1

New knowledge had to be legitimated by the

process by which it is gained: to bring the developmental process to

consciousness and to explicate it was the essence of developmental research.

Experiencing the cyclic process of development and research so consciously,

and reporting on it so candidly that it justi® es itself, and that this experience

can be transmitted to others to become like their own experience (Freu-

denthal 1991: 161).

To put it diå erently, the aim of developmental research to Freudenthal

was to create the opportunity for outsiders, e.g. teachers, to retrace the

learning process of the researcher, what Smaling (1987) called ` track-

ability’ . To ensure such trackability, Freudenthal demanded a constant

awareness of the developmental process. And, if its result were to become

credible and transferable, as much as possible of this re¯ ection needed to be

reported.

At the heart of this re¯ ection lay the ` thought experiments’ of the

researcher. The developer would envision how teaching± learning processes

proceed and would subsequently try to ® nd evidence in a teaching experi-

ment to show whether these expectations were right or wrong. The feed-

back from practical experience to (new) thought-experiments would induce

an iteration of development and research: What was invented behind the

desk would be put into practice immediately; what happened in the class-

room would be analysed in a consistent manner and the results used to

continue the developmental work. This process of deliberating and testing

would result in a product that was theoretically and empirically founded,

well-considered, and well-tried.

In this view, developmental research can oå er teachers a frame of

reference which can provide a basis for their own decisions. Against the

backdrop of this framework, teachers can develop hypothetical learning

trajectories (Simon 1995) that take into account both the actual situation of

HANS FREU DENTH AL 789

their classroom and their own goals and values. Teachers are given

arguments and guidelines to enable them to shape their own instruction,

a starting point that is ® rmly built into the European Didaktik tradition.

Developmental research has a double output-channel: one at the level

of theories, and one at the level of curriculum products. Thus, the

developmental research in RME, carried out in and outside the IOWO

and its successor, OWandOC (the present Freudenthal Institute), has

resulted in a wealth of prototypical instructional sequences and other

practical publications. In the Netherlands, these publications have had a

strong in¯ uence on mathematics in schools. Over time, ¹

80% of Dutch

primary schools voluntarily switched to so-called ` realistic’ textbooks. At

the secondary level, curriculum changes initiated by the government gave

the Freudenthal Institute several commissions for the development of new

curricula. As a consequence, all curricula in the Netherlands were, or are

being, exchanged for curricula based on the RME philosophy.

National educational assessment studies have shown that, in the last

year of primary school, Dutch students working with modern textbooks

were in general more successful than students working with traditional

textbooks ± with the exception of the topics of written algorithms and

measurement (Bokhove et al. 1996).

1 2

It seems reasonable to attribute this

success to the innovation strategy ± ` educational development in dialogue

with the ® eld’ ± used in the introduction of these curricula and texts into

Dutch schools. In empirical and retrospective studies on innovation in

mathematics in primary and secondary schools the following agents of

success have been identi® ed (Gravemeijer and Ruinaard 1995, Vermeulen

et al. 1997):

.a powerful and inspiring philosophy of mathematics education;

.the development of examples, and prototypical instructional

sequences;

.professionalization activities;

.the constitution of a mathematics-educational community as a

mediating infrastructure;

.teacher enhancement via in-service teacher training and journals;

.textbook review;

.revision of examinations; and

.developmental research as the engine of innovation.

Developmental research lies at the heart of the innovation strategy.

This work produces the prototypes and the theories that inform teacher-

trainers, textbook authors, and school counsellors. These, in turn, function

as mediators between the developers and the teachers. In accordance with

the concept of educational development in dialogue with the practitioners,

these information streams are bi-directional. To put it diå erently, the core

idea of the Institute was the concept of ` educational development in

dialogue with practitioners’ ; rather than innovations being developed in

ivory towers, practitioners, such as teacher-trainers, counsellors, textbook-

authors, researchers, test-constructors, and the teachers themselves, were

involved in research and development from early on.

1 3

790 K.GRAV EMEIJER AND J .TERWEL

Conclusions and discussion

To place the work of Freudenthal in the contexts of didactics and cur-

riculum studies is not an easy task, due to his eclectic style of writing which

is, for example, almost without references to the authors by whom he was

inspired. In this ® nal section, we shall focus on three main aspects of his

work: didactics, curriculum theory, and ` mathematics for all’ .

Didactics

Freudenthal often used the term didactics to mean correct teaching and

learning processes, starting with, and staying within, ` reality’ . He referred

to the converse as ` the anti-didactical conversion’ (the deductive approach),

which he ® ercely rejected. According to Freudenthal, didactics should be

concerned with processes. So, there is a commonality and a striking

diå erence between Freudenthal’ s didactics and Klafki’ s use of the term.

Both are in¯ uenced by the phenomenological theory of Bildung and

reform-pedagogy. Both take their point of departure from the practice of

education (educational reality) and endeavoured at certain points in their

professional lives to overcome the exalted and e

Âlistist aspects of the

Bildung-theory. Both stressed the practical side of education and advocated

the comprehensive school as a necessary educational reform. Klafki,

however, mainly focused on lesson-planning or the preparation of lessons

where the process of learning is not real. Klafki’ s fundamental questions

primarily concerned the content of Bildung, while more or less ignoring

teaching methods and processes.

Curriculum theory

Freudenthal used the word ` curriculum’ , but less often than ` didactics’ . In

his views on curriculum development and the role of theory, there is a

striking similarity with the curriculum work of Joseph Schwab, who has

occupied a special position in curriculum theory in the US. Along the same

lines as, but independently from, Schwab, Freudenthal stressed the prac-

tical character of curriculum work and the process of dialogue between

curriculum specialists and teachers. Freudenthal was against any ® xed

curricular system and he ® ercely opposed content being bottled and

funnelled into schemes and structures. This was a remarkable position at

a time when curriculum theory was dominated by behavouristic orienta-

tions and RD&D approaches, and was seen in Germany and the Nether-

lands as the long-expected cure-all for everything (Hopmann and Riquarts

1995a, b). Freudenthal proposed, instead, mathematics as a human activity

and as guided reinvention. It was this humanistic, practical, process-

oriented, phenomenological, and reform-pedagogical credo, elaborated in

the context of curriculum development, that made Freudenthal’ s position

unique among most of his fellow mathematics educators of his time. His

credo inevitably brought Freudenthal into con¯ ict with, for example,

HANS FREU DENTH AL 791

behaviouristically-oriented psychologists such as Bloom and proponents of

the ` new math’ movement who advocated the development of a curriculum

for mathematics as an abstract deductive system.

Mathematics for all

We ® nally turn to Freudenthal’ s position in the debate concerning ` math-

ematics for all’ and the common curriculum (see Damerow and Westbury

1985, Keitel 1987, Dekker 1991). Although Freudenthal was educated in,

and in¯ uenced by, the traditional German Bildung tradition in a dual

school system, he rejected an exclusive form of Bildung for an e

Âlite as

separate from schooling for the masses. He strongly advocated ` math-

ematics for all’ and tried to make mathematics accessible to everybody. He

condemned all forms of streaming and setting by referring to the inevitable

` Matthew eå ects’ . Freudenthal (1973a) was convinced that students from

diå erent ability levels in the ® rst years of secondary education (which in the

Dutch context concerned the 12± 15-year-olds) should not only stay in the

same classroom, but should also follow a common curriculum. His plea for

heterogeneous learning groups built on the other main aspects of his

pedagogical credo.

Several aspects of Freudenthal’ s ideas are still under discussion. Thus,

there is a strong movement against educational theories of his kind from

psychologists, who look at learning from an information-processing point

of view (Anderson et al. 1996). But, also there is sometimes opposition from

inside the mathematics and the mathematics-education communities to the

basic idea that students should proceed from the real world to the math-

ematical world. The main criticism of the RME approach is that it is often

impossible to proceed from experrentially real situations to ` mathematics’ .

Reinvention, in this view, is a waste of time (Verstappen 1991, Keune

1998).

These criticisms have to be mentioned, but it must also be noted that

the opponents of Freudenthal’ s ideas are short on empirical evidence for

their point of view. While various teaching experiments have shown the

value of the RME approach (de Lange 1987, Nelissen 1987, van den Brink

1989, Stree¯ and 1990), the outcomes of several research studies into the

eå ects of mathematics curricula inspired by the ideas of Freudenthal, show

clearly that learning mathematics in real-life contexts in heterogeneous

learning groups is feasible and eå ective (Terwel 1990, 1999, Dekker 1991,

Terwel et al. 1994, Perrenet 1995, Hoek et al. 1997, 1999, Hoek 1998,

Roelofs and Terwel 1999). Also, there is still broad support for RME

among practioners, educators, theorists and curriculum developers. Almost

all Dutch mathematics textbooks show the impact of Freudenthal’ s ideas.

But, although there is empirical and practical evidence for the feasibility

and eå ectiveness of RME, Freudenthal’ s most convincing argument for

RME is that not all students are future mathematicians but, rather that, for

the majority, the mathematics that they will use will be to solve problems in

everyday-life situations.

792 K.GRAV EMEIJER AND J .TERWEL

Notes

1. He was President of the International Committee on Mathematics Instruction (ICMI),

founding editor of Educational Studies in Mathematics, and one of the founders of the

International Group for Psychology and Mathematics Education (PME) established to

overcome the dominating behaviourist approach in educational psychology. He was also

founder and president of the Commission Internationale pour l’ E

Â

tude et l’ AmeÂlioration de

l’ Enseignement des MatheÂmatiques (CIEAEM), which celebrates more than 50 years of

activity. In the Netherlands, Freudenthal was the founder and director of the former

institute for the development of mathematics education, IOWO, later called OW&OC,

and now named the Freudenthal Institute.

2. In this context, we want to draw the attention to The Legacy of Hans Freudenthal

(Stree¯ and 1993).

3. The main representatives of this concept of a Didaktik are Erich Weniger and Wolfgang

Klafki (Blankertz 1973). Klafki (1995, 2000) sought to elaborate the philosophical

notion of Bildung into a more practical theory of lesson-planning; at the heart of his

Didaktik is a set of basic questions concerning, for example, the selection of content,

structuring, meaningfulness, and methods.

4. This international fellowship, which was founded in 1920 with sections in many

countries, was more or less the practical counterpart of the Geisteswissenschaftliche

theory of pedagogy and Didaktik. Freudenthal also explored eastern European socio-

cultural theory. He had a long-lasting debate with Van Parreren, who, in collaboration

with Carpay, introduced Vygotsky’ s ideas into the Netherlands.

5. In this respect, we may observe an aæ nity with Vygotsky’ s (1978: 64) point of

departure: ` . . . we need to concentrate not on the product of de velopment, but on the

very p rocess. . .’ .

6. We stress that the point of departure is not that everyday-life problems will, by

de® nition, be experientially-real for the students, nor that experientially-real problem

situations necessarily have to deal with real-life situations. This is a common

misunderstanding, evoked by the term ` realistic mathematics education’ . Here, realistic

is to be interpreted as referring to experientially-real, not to everyday-life reality.

7. We may note that the quality of this discussion largely depends on what is called the

` didactical contract’ . However, Freudenthal’ s elaborations of mathematizing take on a

psychological perspective at the expense of a social perspective. Although his arguments

for working in heterogeneous groups clearly re¯ ect his acknowledgement of learning as

a social process.

8. Furthermore, he did not con® ne himself to theorizing about co-operative learning, but

was highly involved and supportive as a supervisor of the research project ` Mixed-

ability grouping in mathematics for 12± 16 year-olds’ (Terwel 1984, 1990, Freudenthal

1987).

9. As an example of his stance, we may take his attack on the studies of the International

Association for the Evaluation of Educational Achievement (IEA): his criticism of IEA

focused, among other things, on the validity of the instruments used in those studies

and the lack of ® t between the national curricula and the testing instruments. He

observed that the only actual check carried out concerned only one country, and even in

that case the correlation was not good (Freudenthal 1975: 134). Freudenthal observed

that the lack of correspondence between the national curricula and the test items

resurfaced in the IEA report under the variable ` opportunity-to-learn’ . Here, according

to Freudenthal, things were turned upside-down; the lack of concurrence of the test

items and the curricula was now presented as an important explanatory variable, which,

in turn, was explained by a lack of implementation of the oæ cial curriculum plans.

Freudenthal attributed the diå erences between countries to a lack of correspondence of

the national curricula and the IEA instruments.

10. Freudenthal also criticized Piaget’ s experiments for their lack of validity. He pointed to

two interrelated ¯ aws: the arti® cial character of the questions and the pre-determined

interpretations of students’ reactions. In an appendix to his book Mathematics as an

Educational Task, Freudenthal (1973b, 662± 677) supports his critique on Piaget’ s

mathematics, with extensive translations from Piaget’ s work.

HANS FREU DENTH AL 793

11. He set himself against the ideal of educational research as modelled on research in the

natural sciences. In the natural sciences, he argued, it is easy to present new knowledge

as the result of experiments, since experiments are easily replicated. In education,

replication is impossible in the strict sense of the word. An educational experiment can

never be repeated in an identical manner, under identical conditions.

12. The most recent research of the IEA Third International Mathematics and Science

Study (TIMSS) on 13± 14 year olds has shown that Dutch students do very well in

international competitions, although the TIMSS test items only poorly ® t the Dutch

curriculum and probably would not withstand Freudenthal’ s (1975) critique.

13. Admittedly, this dialogue was split into two levels: the dialogue between the researchers

and the teacher-trainers, etc., on the one hand, and the dialogue between this group and

(a number of) teachers on the other. And, in practice, commercial textbooks were the

main carriers of innovation, so that the actual innovation assumed the character of a

curriculum-as-a-document to a much greater extent than ever intended. The lack of the

® nancial resources required for greater involvement by teachers played a major role in

the strategies that were adopted. Not surprisingly, research has shown that the

instructional practices in school have diå ered signi® cantly from the form of practice

envisioned by the innovators. In an empirical study of primary grades 1± 3, the

characteristics of realistic mathematics education were only partly found (Gravemeijer

et al. 1993 ). And, in an empirical study by Kuiper (1994), it was concluded that

Freudenthal’ s ideas are far from being enacted in everyday classroom practices in

secondary education.

References

ANDERSON,J.R., REDER,L.M.and SIMON,H.A.(1996) Situated learning and education.

Educational Researcher, 25 (5), 5± 11.

BLANKERTZ,H.(1973) Didactiek, theorieeÈn en modellen (Utrecht, The Netherlands/Antwerp,

Belgium: Het Spectrum).

BOKHOVE,J., VAN DER SCHOOT,F.and EGGEN,T.(1996) Balans van het rekenonderwijs aan het

einde van de basisschool 2 (Arnhem, The Netherlands: Cito).

CHEVALLARD,Y.(1985) La Transposition Didactique du Savoir Savant au Savoir EnseigneÂ

(Grenoble, France: Editions La Pense

Âe Sauvage).

DAMEROW,P.and WESTBURY,I.(1985) M athematics for all. Journal of Curriculum Studies, 17

(2), 175± 184.

DEKKER,R.(1991) Wiskunde leren in kleine heterogene groepen [Learning mathematics in

small heterogeneous groups]. Doctoral dissertation, University of Utrecht (De Lier,

The Netherlands: Academisch Boeken Centrum).

DE LANGE,J.(1987) Mathematics, Insight and Meaning: Teaching, Learning and Testing of

Mathematics for the Life and Social Sciences (Utrecht, The Netherlands:

Rijksuniversiteit Utrecht).

ERNEST,P.(1991) The Philosophy of Mathematics Education (Basingstoke: Falmer)

FREUDENTHAL,H.(1968) Why to teach mathematics as to be useful? Educational Studies in

Mathematics, 1 (1 ), 3± 8.

FREUDENTHAL,H.(1971) Geometry between the devil and the deep sea. Educational Studies

in Mathematics, 3 (3/4), 413± 435.

FREUDENTHAL,H.(1973a) De niveaus in het leerproces en de heterogene leergroep met het

oog op de middenschool. In L. J. van Eijk (ed.) Gesamtschule conferentie 1973

(Amsterdam and Purmerend, The Netherlands: APS/ Muuses), 88± 98.

FREUDENTHAL,H.(1973b) Mathematics as an Educational Task (Dordrecht, The

Netherlands: Reidel).

FREUDENTHAL,H.(1975) Pupil’ s achievements internationally compared: the IEA.

Educational Studies in Mathematics, 6 (2), 127± 186.

FREUDENTHAL,H.(1979) Invullen ± vervullen. Euclides, 55, 61± 65.

FREUDENTHAL,H.(1980) Weeding and Sowing: Preface to a Science of Mathematics Education

(Dordrecht, The Netherlands/Boston, MA: Reidel).

794 K.GRAV EMEIJER AND J .TERWEL

FREUDENTHAL,H.(1983) Didactical Phenomenology of Mathematical Structures (Dordrecht,

The Netherlands: Reidel).

FREUDENTHAL,H.(1986) Review of Yves Chevallard, La Transposition Didactique du Savoir

Savant au Savoir EnseigneÂ.Educational Studies in Mathematics, 17 (3), 323± 327.

FREUDENTHAL,H.(1987) Schrijf dat op, Hans. Knipsels uit een leven (Amsterdam, The

Netherlands: Meulenhoå ).

FREUDENTHAL,H.(1988) Ontwikkelingsonderzoek [Developmental research]. In K.

Gravemeijer and K. Koster (eds), Onderzoek, ontwikkeling en ontwikkelingsonderzoek

(Utrecht, The Netherlands: OW en OC), 49± 54.

FREUDENTHAL,H.(1991) Revisiting Mathematics Education: China Lectures (Dordrecht, The

Netherlands: Kluwer).

GAGNE

Â,R.M.(1977) The Conditions of Learning, 3rd edn (New York: Holt, Rinehart &

Winston).

GRAVEMEIJER,K.P.E.(1994) Developing Realistic Mathematics Education. Doctoral

dissertation, Utrecht University (Utrecht: CdBe

Áta Press).

GRAVEMEIJER,K.(1997) Instructional design for reform in mathematics education. In M.

Beishuizen, K. P. E. Gravemeijer and E. C. D. M. van Lieshout (eds), The Role of

Contexts and Models in the Development of Mathematical Strategies and Procedures

(Utrecht: CdBeÁta Press), 13± 34.

GRAVEMEIJER,K.(1999) How emergent models may foster the constitution of formal

mathematics. Mathematical Thinking and Learning, 1(2), 155± 177.

GRAVEMEIJER,K., VAN DEN HEUVEL-PANHUIZEN,M., VAN DONSELAAR,G., RUESINK,N.,

STREEFLAND,L., VERMEULEN,W., TE WOERD,E.and VAN DER PLOEG,D.(1993)

Methoden in het reken-wiskundeonderwijs, een rijke context voor vergelijkend onderzoek

(Utrecht, The Netherlands: CdBe

Áta Press).

GRAVEMEIJER,K.and RUINAARD,M.(1995) Expertise in leren. Over de bevordering van de

vakdidactische deskundigheid van docenten in het funderend onderwijs (Utrecht, The

Netherlands: Adviesraad voor het onderwijs).

HOEK,D.(1998) Social and Cognitive Strategies in Co-operative Groups. Doctoral

dissertation, Graduate School of Teaching and Learning, University of Amsterdam.

HOEK,D., TERWEL,J.and VAN DEN EEDEN,P.(1997) Eå ects of training in the use of social

and cognitive strategies: an intervention study in secondary mathematics in

cooperative groups. Educational Research and Evaluation, 3 (4), 364± 389.

HOEK,D., VAN DEN EEDEN,P.and TERWEL,J.(1999) The eå ects of integrated social and

cognitive strategy instruction on the mathematics achievement in secondary

education. Learning and Instruction, 9 (5), 427± 448.

HOPMANN,S.and RIQUARTS,K.(1995a) Starting a dialogue: issues in a beginning

conversation between Didaktik and the curriculum traditions. Journal of Curriculum

Studies, 27 (1), 3± 12.

HOPMANN,S.and RIQUARTS,K.(1995b) Didaktik and/or Curriculum (Kiel, Germany:

Institut fu

Èr die PaÈdagogik der Naturwissenschaften an der UniversitaÈt Kiel).

KEITEL,C.(1987) What are the goals of mathematics for all? Journal of Curriculum Studies,

19 (5), 393± 407.

KEUNE,F.(1998) Naar de knoppen, Inaugural lecture, University of Nijmegen.

KLAFKI,W.(1995) Didactic analysis as the core of preparation for instruction (Didaktische

Analyse als Kern der Unterrichtsvorbereitung ). Journal of Curriculum Studies, 27 (1),

13± 30.

KLAFKI,W.(2000) Didaktik analysis as the core of preparation for instruction. In I.

Westbury, S. Hopmann and K. Riquarts (eds), Teaching as a Re¯ ective Practice: The

German Didaktik Tradition (Mahwah, NJ: Lawrence Erlbaum Associates), 139± 159.

KUIPER,W.A.J.M.(1994) Curriculumvernieuwing en lespraktijk, Een beschrijvend onderzoek

op het terrein van de natuurwetenschappelijke vakken in het perspectief van de

basisvorming [Curriculum Innovation in Science and Classroom Practices] (Enschede,

The Netherlands: Universiteit Twente).

LAKATOS,I.(1976) Proofs and Refutations: The Logic of Mathematical Discovery

(Cambridge: Cambridge University Press).

LANGEVELD,M.J.(1965) Beknopte Theoretische Pedagogiek [Fundamentals of Educational

Philosophy] (Groningen, The Netherlands: Wolters).

HANS FREU DENTH AL 795

NELISSEN,J.M.C.(1987) Kinderen leren Wiskunde [Children Learn Mathematics]

(Gorinchem, The Netherlands: De Ruiter).

PERRENET,J.C.(1995) Leren probleemoplossen in het wiskunde-onderwijs: samen of alleen

[Problem solving in mathematics]. Doctoral dissertation, University of Amsterdam.

ROELOFS,E.and TERWEL,J.(1999) Constructivism and authentic pedagogy: state of the art

and recent developments in the Dutch national curriculum in secondary education.

Journal of Curriculum Studies, 31 (2), 201± 227.

SCHWAB,J.J.(1970) The Practical: A Language for Curriculum (Washington, DC: National

Educational Association ).

SFARD,A.(1995) Symbolizing mathematical reality into being. Paper presented at the

Symposium on ` Symbolizing, communication, and modelling’ , Vanderbilt

University, Nashville.

SIMON,M.A.(1995) Reconstructing mathematics pedagogy from a constructivist

perpective. Journal for Research in Mathematics Education, 26 (2), 114± 145.

SMALING,A.(1987) Methodologische objectiviteit en kwalitatief onderzoek [Methodological

objectivity and qualitative research]. (Lisse, The Netherlands: Swets and Zeitlinger).

STREEFLAND,L.(1990) Fractions in Realistic Mathematics Education: A Paradigm of

Developmental Research (Dordrecht, The Netherlands: Kluwer).

STREE

AND,L.(ed.) (1993) The Legacy of Hans Freudenthal (Dordrecht, The Netherlands:

Kluwer).

TERWEL,J.(1984) Onderwijs maken. Naar ander onderwijs voor 12± 16-jarigen [Curriculum

development in secondary education]. National Institute for Educational Research

(SVO), SVO-series in Educational Research, No. 77. (Utrecht and Harlingen, The

Netherlands: Flevodruk ).

TERWEL,J.(1990) Real maths in cooperative groups. In N. Davidson (ed.), Cooperative

Learning in Mathematics (Menlo Park, CA: Addison-Wesley), 228± 264.

TERWEL,J.(1999). Constructivism and its implications for curriculum theory and practice.

Journal of Curriculum Studies, 31 (2), 195± 200.

TERWEL,J., HERFS,P.G.P., MERTENS,E.H.M.and PERRENET,J.C.(1994) Cooperative

learning and adaptive instruction in a mathematics curriculum. Journal of Curriculum

Studies, 26 (2), 217± 233.

TREFFERS,A.(1987) Three Dimensions: A Model of Goal and Theory Description in

Mathematics: The Wiskobas Project (Dordrecht, The Netherlands: Reidel).

VAN DEN BRINK,F.J.(1989) Realistisch rekenonderwijs aan jonge kinderen (Utrecht, The

Netherlands: Freudenthal Institute).

VAN DEN HEUVEL-PANHUIZEN,M.(1996) Assessment and Realistic Mathematics Education

(Utrecht, The Netherlands: CdBe

Áta Press).

VERMEULEN,A., VOLMAN,M.and TERWEL,J.(1997) Success factors in curriculum

innovation: mathematics and science. Curriculum and Teaching, 12 (2), 15± 28.

VERSTAPPEN,P.(1991) Easier Theorized Than Done (Enschede, The Netherlands: Dutch

National Institute for Curriculum Developmen).

VYGOTSKY,L.S.(1978) In M. Cole, V. John-Steiner, S. Scribner and E. Souberman (eds),

Mind in Society: The Development of Higher Psychological Processes (Cambridge, MA:

Harvard University Press).

WALKER,D.F.(1990) Fundamentals of Curriculum (San Diego, CA: Harcourt Brace

Jovanovich).

WESTBURY,I.(1995) Didaktik and curriculum theory: are they the two sides of the same

coin? In S. Hopmann and K. Riquarts (eds), Didaktik and/or Curriculum (Kiel:

Institut fu

Èr die Pa

Èdagogik der Naturwissenschaften an der Universita

Èt Kiel), 233± 263.

WESTBURY,I.(2000) Teaching as a re¯ ective practice: what might Didaktik teach

curriculum? In I. Westbury, S. Hopmann and K. Riquarts (eds), Teaching as a

Re¯ ective Practice: The German Didaktik Tradition (Mahwah, NJ: Lawrence Erlbaum

Associates), 15± 39.

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