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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. 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 fleshed 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.
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Hans Freudenthal a mathematician on didactics and curriculum theory.
Gravemeijer, K.P.E.; Terwel, J.
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Journal of Curriculum Studies
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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.
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Download date: 11. Sep. 2022
Hans Freudenthal: a mathematician on didactics and
curriculum theory
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
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.
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.
Our point of view will centre on curriculum theory and pedagogy, and we
J.CURRICULUM STUDIES,2000, V OL.32, NO.6, 77 796
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: 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
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
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.
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
Although it does not show in his rare references, Freudenthal was well-
acquainted with educational and psychological traditions in Europe and the
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 i 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
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-
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
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.
[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
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 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,
.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).
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.
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
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
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
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.
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 pr 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,
he had been a protagonist of co-operative learning in small heterogeneous
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
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
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 art cial character of the categories of
educational goals in the Taxonomy have a negative 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
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).
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).
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.
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
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:
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
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
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,
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
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 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
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 i 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
.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
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’ .
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,
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 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
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
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 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 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.
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
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 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 co 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
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, 66 677) supports his critique on Piaget’ s
mathematics, with extensive translations from Piaget’ s work.
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 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
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... The approach to teaching mathematics through RME starts with a contextual problem situated in real life and continues with activities and assignments that are situated more within the world of mathematics (Gravemeijer & Terwel, 2000;van den Heuvel-Panhuizen, 2020). In this environment, students use their informal knowledge to create formal mathematical knowledge through the mathematization process under the guidance of the teacher (i.e., guided reinvention). ...
... Non-routine mathematical problems, even when related to real life, might cause challenges for students or cause them to feel bored (Gravemeijer & Terwel, 2000;van den Heuvel-Panhuizen & Drijivers, 2014). In this sense, virtual learning environments can help students to construct knowledge by integrating mathematics with real life (Mart ın-Guti errez et al., 2017; Pasqualotti & Freitas, 2002). ...
... Step 2: Solving problems individually or in a group As students engage with a problem, Gravemeijer and Terwel (2000) suggest that teachers should assist them in developing their model from context dependent models. Considering the potential of learning by doing in mathematical understanding (Freudenthal, 1983), we used various virtual tools, such as the NVLM (National Library of Virtual Manipulatives) and Toy Theater to help students in constructing their own cognitive models. ...
Although mathematical literacy skills have been emphasized in education, standardized test results show that students still struggle in this regard. Hypothesizing that both virtual and realistic mathematics education approaches may be useful, this article combines two approaches as Virtual Realistic Mathematics Education (VRME) to enhance mathematical literacy. A one-group experimental research design was adopted with the participation of 20 6th-grade middle school students. The qualitative data were also used to explain the improvements in mathematical literacy. The results revealed that all of the dimensions of mathematical literacy skills; employ, interpret and formulate were improved while the improvements in interpret dimension was the lowest. The results also indicated the importance of knowledge co-construction through discussion for the development of mathematical literacy, especially for the interpret dimension. Finally, the study demonstrated that virtual tools can be used to improve mathematical literacy by expanding RME.
... Matematik i förskolans lek, aktiviteter och processer har studerats med hjälp av perspektivet matematisera (Björklund m.fl., 2018;Gejard, 2018;Reis, 2011). Perspektivet på matematik som matematisera kan beskrivas som ett processinriktat förhållningssätt nära förbundet med matematik sprungen ur omvärlden (Freudenthal, 1972(Freudenthal, , 2002Gravemeijer & Terwel, 2000). Gravemeijer och Terwel (2000, s. 780) skriver att "Freudenthal mathematics was first and foremost an activity, a human activity, as he often emphasized". ...
... Gravemeijer och Terwel (2000, s. 780) skriver att "Freudenthal mathematics was first and foremost an activity, a human activity, as he often emphasized". Freudenthals tankar om begreppet matematisera vände sig till det matematikdidaktiska fältet med tankar om att vidga och utveckla skolmatematiken (Gravemeijer & Terwel, 2000). Begreppet relaterar till att vara aktiv och på så sätt antas eleven kunna utveckla färdigheter och förståelse för matematik i sin omvärld (Freudenthal, 1968). ...
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Den här artikeln avser att skapa kunskap om fenomenet, matematik i fritidshemmet. Sedan 2016 uttrycks explicit i nationella styrdokument att matematik ska vara en del av fritidshemmets utbildning. I studien undersöks hur matematik samskapas i fritidshemmets praktik i möte med materialiteter, mänskliga och ickemänskliga. Artikelns empiriska data består av deltagande observationer, producerad med barn och personal i två fritidshem i en medelstor kommun i Sverige. Den teoretiska ramen omfattar agentisk realism och ett kulturhistoriskt perspektiv på matematik, vilken inkluderar matematiska aktiviteter och inommatematiska värderingar. I analysen samläses empirisk data, tidigare forskning inom matematikdidaktik för yngre barn med agentisk realism och det kulturhistoriska perspektivet på matematik. Resultatet synliggör en variant av matematik som händelse i fritidshemmet, vilken här benämns matematik-a. Att matematik-a kan liknas med att lek-a, att skapa tillsammans och ”bli till” med materiellt-diskursiva praktiker. Att matematik-a handlar om att ”vara i” matematiska händelser utan att aktiviteten relaterar till specifika och uttalade kunskapskrav. Här möjliggörs för matematiska möten i olika sammanhang, upplevelser och skapanden tillsammans med material, mänskliga och icke mänskliga.
... Culture is a special way that human beings develop and have (a certain society) to adjust to the environment and be passed on from generation to generation, while mathematics is realized due to human activity. This is in line with Freudenthal's statement that "mathematics as a human activity" (Gravemeijer & Terwel, 2000). Furthermore, (Suherman, 2011) defines mathematics as the science of forms, structures, quantities and other related concepts, and divided into three areas, namely algebra, analysis and geometry. ...
... Polya (Shirali, 2014), and Hans Freudenthal (Gravemeijer & Terwel, 2000) (Pangeni, 2016). There are ample possibilities of use of ICT embedded technology of education in overall education system management and particular subject teaching and learning. ...
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My teaching carrier at Tribhuvan University (TU) Nepal, as a faculty, started twelve years ago, mostly lecturing and writing on a blackboard using chalk while teaching my students. After a few years, the classroom environment changed. Blackboards were replaced by whiteboards, and chalk by board markers, and the innovation of information and communication technology in education appeared in prominence. With interest, enthusiasm, and growing competence in me, I started to use a laptop, projector, touch-board, e-pen, and other different digital tools as teaching/learning objects. I also use Mathematics specific digital tools like GeoGebra, Mathematica, MATLAB, and Maple, and pedagogical tools like Moodle (modular object-oriented dynamic learning environment) to provide Virtual Learning Environment (VLE) for students’ learning. So, this Dissertation entitled “Virtual Learning Environment for Engaged and Interactive Learning of Higher Mathematics” is focused on introducing ICT-enhanced pedagogy in higher Mathematics education named M-VLE. This study, therefore, was carried out using design-based experimental research to analyze the effectiveness of M-VLE in Nepalese context. Action research was carried out to formulate M-VLE as a digital pedagogy using Moodle learning management system for the experimental intervention.
... In other words, this philosophical reflection is based on the contrast between the objects constructed in concepts, which are called objects of thought, and which will be called nooumenon, and the situations that these mathematical objects organise, when one has acquired experience, which will be the phenomena. Other authors, such as Gravemeijer and Terwel (2000), determine that the situations must be selected in such a way that they can be organised by the mathematical objects that the students have had to construct. The object to be considered will be the nooumenon, and this describes and analyses the phenomenon. ...
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In this paper we aim to characterise and define the phenomena of the infinite limit of a function at infinity. Based on the intuitive and formal approaches, we obtain as results five phenomena organised by a definition of this limit: intuitive unlimited growth of a function, for plus and minus infinity, and intuitive unlimited decrease of a function, for plus and minus infinity (intuitive approach), and the round-trip phenomenon of infinite limit functions (formal approach). All this is intended to help overcome the difficulties that pre-university students have with the concept of limit, contributing from phenomenology, Advanced and Elementary Mathematical Thinking, and APOS theory. Keywords: limit, infinity, functions, phenomenology, Advanced Mathematical Thinking, APOS
... 30), and in Freudenthal's view, learners can only mathematize that which is experientially real to them. Gravemeijer and Terwel (2000) clarify that Freudenthal here views "reality" as including the whole of a person's lived, embodied experiences and interpretive perspectives. Dienes (1960), designer of Dienes blocks, similarly describes the construction of mathematics as a "crystallizing" or "distilling" of experience and emphasizes the dynamic relation between an idea and its embodiments. ...
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Introduction This article illustrates a pedagogical approach to integrating models and modeling in Geometry with mathematics teacher-learners (MTLs). It analyzes the work of MTLs in a course titled “Computers, Teaching, and Mathematical Visualization” (or “MathViz”), which is designed to engage MTLs in making mathematics together. They use a range of both physical and virtual models of 2-manifolds to formulate and investigate geometric conjectures of their own. Objectives The article articulates the theoretical basis and design rationale of MathViz; it analyzes illustrative examples of the discourse produced in collaborative investigations; and it describes the impact of this approach in the students’ own voices. Methods MathViz has been iteratively refined and researched over the past 6 years. This study focuses on one iteration, aiming to capture the phenomenological experience of the MTLs as they structured and pursued their own mathematical investigations. Video data from two class sessions of the Fall 2021 iteration of the course are analyzed to illustrate the discourse of collaborating students and the nature of their shared inquiry. Excerpts from this class’s Learning Journals are then analyzed to capture themes across students’ experience of the course and their perspectives on its impact. Results Analysis of students’ discourse (while investigating cones) shows how they used models and gesture to make sense of geometric phenomena; forged connections with investigations they had conducted throughout the course on different surfaces; and articulated and proved mathematical conjectures of their own. Analysis of students’ Learning Journals illustrates how experiences in MathViz contributed to their conceptualization of making mathematics together, using a variety of models and technologies, and developing a set of practices that that they could introduce with their future students. Discussion An argument is made that this approach to collective mathematical investigation is not only viable and valuable for MTLs, but is also relevant to philosophical reflections about the nature of mathematical knowledge-creation.
... Mathematization in Realistic Mathematics Education was expressed by Hans Freudenthdal in two ways, horizontal and vertical. Horizontal mathematization is expressed as turning a real life problem into a mathematical formula and vertical mathematization is expressed as reaching a solution by establishing communication and relationships between mathematical expressions (Gravemeijer & Terwel, 2000). With horizontal mathematization, the targeted concept is reached while with vertical mathematization, this concept is used to progress onto more general concepts, the obtained concept is generalized and formulas are reached by working with symbols and establishing relationships between concepts (Van den Heuvel-Panhuizen & Drijvers, 2014). ...
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The purpose of the study is to investigate how the error based activities improve the mathematization competency of preservice mathematics teachers. The study was designed as a case study which is one of the qualitative study design. The sample consists of 38 third grade elementary mathematics teacher candidates studying in a university in Turkey. The study group consisted of 20 pre-service teachers; the comparison group consisted of 18 pre-service teachers. Data were collected through PISA questions consisting of 11 questions total. Data were analysed through descriptive analysis. The findings of the study indicates that the study group performed better than the comparison group on getting the full score for most of the questions. Both of the groups performed mostly on getting full and zero scores and rarely getting partial scores. The study group performed mostly at level 3 when the comparison group performed mostly at level 2 and level 1.
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This article focuses on curriculum innovation in the subjects of mathematics, physics, chemistry and biology in Dutch secondary education over the last four decades. The impetus for this study was a supposition expressed by the Dutch Ministry of Education to the effect that curriculum innovation in physics, chemistry and biology may not have been as successful as in mathematics. In order to test this supposition a study was designed with the purpose of exploring the state of the innovations in mathematics and science, and to gain insight into its causes. The article describes innovations in the four subjects under study and discusses factors contributing to successful curriculum innovations in secondary education in these subjects. Three historical stages are distinguished in innovations in mathematics and science. It is suggested that a fourth stage is developing which may overcome some of the limitations of the former stages and which may be characterized as ‘strategic learning in contexts’.
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The AGO 12 to 16 Project (the acronym AGO stands for the Dutch equivalent of 'Adaptive Instruction and Co-operative Learning') seeks to develop and evaluate a mathematics curriculum which is suitable for mixed-ability groups in secondary education. The research questions we will address here are, first, whether this curriculum is feasible and effective, and, second, what effects, if any, the context variables time and mean cognitive level of the class have on learning. Many mathematics programmes make insufficient allowance for the differences in intellectual ability that exist in mixed-ability classes. In order to change this situation we developed a mathematics curriculum with adaptive qualities. The evaluation of the experimental curriculum was carried out in two stages. During the first stage the curriculum was used at two schools with the aim of investigating the feasibility of the programme. Experience with the implementation of the programme led to some improvements in the experimental materials. By and large the AGO model appeared to be feasible in secondary classrooms. In the second stage, which was on a large scale, the focus was on the effectiveness of the programme. Six hundred students, 13 teachers and six schools were involved in the research. Teachers in the experimental group were trained in AGO methods and in implementing the new AGO curriculum. Teachers in the control groups worked with the existing programme following their usual methods of teaching. The main conclusion of the study is positive. The AGO model as a whole proved to be practical and effective in learning mathematics. The AGO model has a positive effect on the intercept, which means that the mean scores of AGO classes are higher than the mean scores of non-AGO classes. It may be concluded that, on the average, students benefit from learning in AGO classes as compared with non-AGO classes. AGO does not increase or decrease the differences between students in the same class. As expected, positive effects of two context variables were found: (1) the total amount of time spent in class covering the mathematical content and (2) class composition as indicated by the mean pretest score (aptitude) of the class.
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This chapter describes author’s first encounters with cooperative learning and his personal viewpoint on education. In this view, education should be inclusive, adaptive and cooperative. This is followed by a rationale for a realistic mathematics curriculum and its main characteristics. The theory of ‘levels in the learning process’ - developed by European scholars and researchers like Piaget, Selz, Kohnstamm, Van Hiele and Freudenthal – is described and elucidated by assignments and examples. The next part describes the role of the teacher who guides the problem solving processes of students in small heterogeneous groups of 2-4 students in secondary mathematics education. The chapter contains various sample materials and assignments for cooperative learning in mathematics. In addition, descriptions and analyses of small group interaction processes are presented. The chapter closes with some reflections and recommendations with regards to implementing cooperative learning in heterogeneous classrooms.
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The recently introduced national curriculum for the first stage of Dutch secondary education requires not only a change in educational content but also a change in educational processes. The knowledge students acquire is expected to be related to everyday life, and to be meaningfully embedded in society. Furthermore, the student is expected to use social and cognitive strategies such as researching, collaborating, and expressing opinions. The accompanying learning environment necessitates active and interactive learners as well as teachers who use various strategies to promote 'authentic learning'. To what extent do Dutch teachers use teaching strategies to foster authentic learning? From 1993 to 1996, three large Dutch secondary schools (between 1000-1400 students) were subjected to an in-depth inquiry. These schools were expected to implement the state-mandated innovations in the 1993-1994 core curriculum. The results show that none of the schools scores highly on the characteristics of authentic pedagogy. Authentic pedagogy demands a major change in the teacher's role, including a change in the use of curricular materials and the development of new teaching strategies embedded in a supporting school organization. The results are viewed in the context of the recent discussion on information-processing theory versus radical constructivism. Implications for curriculum and classroom practice are suggested.
Under the banner of constructivism, a world-wide change in the orientation to school learning has taken place. In the context of the constructivist movement an important question is how curriculum studies should view such concepts as 'development' and 'implementation'. If students and teachers together construct or enact their own curricula, what are the consequences in terms of curriculum theory and practice? What is the state of practice with respect to teaching and learning from a constructivist point of view?