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Hans Freudenthal's realistic mathematical theory: Didactics and research paradigms
Juan Carlos Lázaro Guillermo, Erick Guitton Lozano, Juan Luis Pérez Marín, Milton Juan Carlos
Barreda Fachin, Rogger Wagner Peña Pasmiño, Julia Cecilia Yon Delgado
© Juan Carlos Lázaro Guillermo, Erick Guitton Lozano, Juan Luis Pérez Marín, Milton Juan
Carlos Barreda Fachin, Rogger Wagner Peña Pasmiño, Julia Cecilia Yon Delgado, 2024
Second edition: September, 2024
Edited by:
Editorial Mar Caribe
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Av. General Flores 547, Colonia, Colonia-Uruguay.
RUC: 15605646601
Cover design: Yelitza Sanchez Caceres
Translation of the original Spanish edition into English: Ysaelen Josefina Odor Rossel
E-book available at https://editorialmarcaribe.es/hans-freudenthals-realistic-mathematical-
theory-didactics-and-research-paradigms/
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ISBN: 978-9915-9706-6-0
ARK: ark:/10951/isbn.9789915970660
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for non-profit purposes only, readers may use a work to generate another work as long as research credit
is given and they grant the publisher the right to first publish their essay under the terms of the license
CC BY-NC 4.0.
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About the authors and the publication
Juan Carlos Lázaro Guillermo
https://orcid.org/0000-0002-4785-9344
Universidad Nacional Intercultural de la
Amazonia, Perú
Erick Guitton Lozano
https://orcid.org/0000-0001-8819-0555
Universidad Nacional Intercultural de la
Amazonia, Perú
Juan Luis Pérez Marín
https://orcid.org/0000-0002-3671-1782
Universidad Nacional Intercultural de la
Amazonia, Perú
Milton Juan Carlos Barreda Fachin
https://orcid.org/0000-0001-6947-8078
Universidad Nacional de Ucayali, Perú
Rogger Wagner Peña Pasmiño
rowapepas@gmail.com
https://orcid.org/0000-0002-0549-0956
Universidad Nacional Intercultural de la
Amazonia, Perú
Julia Cecilia Yon Delgado
https://orcid.org/0000-0003-4119-2072
Universidad Nacional Intercultural de la
Amazonia, Perú
Book result of research:
Original and unpublished publication, whose content is the result of a research process conducted
prior to publication, has been reviewed by external double-blind peers, the book has been selected
for its scientific quality and because it contributes significantly to the area of knowledge and
illustrates a fully developed and completed research. In addition, the publication has undergone
an editorial process that guarantees its bibliographic standardization and
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Editorial Mar Caribe
Hans Freudenthal's realistic mathematical theory:
Didactics and research paradigms
Uruguay, 2024
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Index
Introduction ...................................................................................................................................6
Chapter 1 ......................................................................................................................................10
Mathematician and Theorist: Hans Freudenthal ....................................................................10
Mathematics: human activity .......................................................................................... 13
The Criticism ...................................................................................................................... 19
The reinvention .................................................................................................................. 22
Phenomenology in didactics ............................................................................................ 24
Research for development................................................................................................ 26
Chapter 2 ......................................................................................................................................33
Realistic mathematics education ...............................................................................................33
The Contexts ...................................................................................................................... 38
The Models ......................................................................................................................... 40
The interaction ................................................................................................................... 42
The theoretical bases: EMR .............................................................................................. 44
Practical scenarios and challenging circumstances................................... 45
The role of the teacher ................................................................................... 49
Phenomenology ................................................................................................................. 51
Chapter 3 ......................................................................................................................................56
The contextualization of realistic mathematics in education ...............................................56
Guiding principles ......................................................................................... 60
The keys ........................................................................................................... 62
The didactic perspectives of mathematics ..................................................................... 66
Mathematics Education: Theory and Philosophy ..................................... 71
The psychology of mathematics education ................................................ 80
Problem solving ............................................................................................. 87
Chapter 4 ......................................................................................................................................98
Fundamental didactics ...............................................................................................................98
Paradigms ..................................................................................................... 109
Research paradigms .....................................................................................114
The consolidation of mathematics teaching ..............................................118
Conclusion .................................................................................................................................123
Literature ....................................................................................................................................125
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Introduction
Freudenthal was an enthusiastic advocate of reform in traditional mathematics
education. His extensive work as a founder and active participant in groups such as the
International Group on Psychology and Mathematics Education (PME) and the
International Commission for the Study and Improvement of the Teaching of
Mathematics (CIEAEM) contributed to his popularity. In these forums he expressed his
opposition to the pedagogical and didactic approaches that prevailed in the mid-
twentieth century, such as operational goal theory, structured assessment tests,
standardized educational research, and the direct application of Piaget's structuralism
and constructivism in the classroom.
He also criticized the separation between educational research, curriculum
development and teaching practice, as well as the introduction of "modern" mathematics
into schools. Freudenthal's publications on Mathematics Education spanned many years,
during which he collaborated with other members of the Institute for the Development
of Mathematics Education (IOWO), which he founded in 1970 at Utrecht University.
Today the institute is known as the Freudenthal Institute. Together, the members of the
group worked in schools, alongside regular teachers, studying students' informal
knowledge and finding ways to connect it to proposed activities and models. They
designed and tested sequences, continually improving them based on analysis of their
implementation. This work laid the foundation for the current approach known as
Realistic Mathematics Education (RME).
Hans Freudenthal, a mathematician and educator of German descent, earned his
doctorate at the University of Berlin. However, due to his Jewish heritage, he was forced
to emigrate from Germany during the rise of the Nazi regime. He found refuge in the
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Netherlands, where he continued his academic career and developed his pedagogical
theories. Unfortunately, he had to remain in hiding during the years of World War II.
Freudenthal believed that the learning process should be based on situations that require
organization.
He criticized Piaget for attempting to impose psychological development on the
system of categories used by mathematicians, using mathematical terminology with
different meanings. Based on his own experiences, Freudenthal argued that learning was
more closely related to linguistic development than to cognitive development. He was
concerned with how Piaget's work influenced teaching methodologists to translate
research findings into instructional guidelines for mathematics education, transforming
an epistemological theory into a pedagogical theory.
He held discussions with Chevallart about his theory of transposition, which he
believed was based on the expert knowledge of mathematicians. Freudenthal argued that
mathematics taught in schools should not reflect any interpretation of philosophical or
scientific ideas, unless they were from a much earlier period.
Freudenthal's opposition to the prevailing psychology, pedagogy, and didactics of
the time was founded. It was rooted in his deep knowledge of the mathematical
discipline, his passion for teaching it, and his first-hand experience in the classroom. He
questioned the artificial nature of the educational goals and learning domains proposed
by Bloom, arguing that they had a negative impact on both school tests and
developmental testing. He accused Bloom of conceiving of learning as a process in which
knowledge is simply poured into students' heads. Similarly, he disagreed with Gagné's
view that learning is a continuous process that progresses from simple to complex
structures.
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Freudenthal believed that learning involved sudden leaps of reinvention,
demonstrated by students experiencing "aha" moments, developing shortcuts in their
strategies, shifting their perspectives, and using models of varying levels of
formalization. He argued that learning actually moves from rich, complex structures of
the real world to the more general, abstract, and formal structures of mathematics.
Although Freudenthal's references to non-mathematical authors were limited, he
acknowledged influences from Decroly, whose interests aligned with his own theory of
learning mathematics in real-life contexts, and Dewey, with whom he saw similarities in
the idea of guided reinvention. He was also inspired by Pierre and Dina Van Hiele,
incorporating their levels of mathematization into his work on the development of
geometric thinking and its didactics. He was also influenced by the phenomenological
pedagogy of Lagenveld, the intuitive didactics of Castelnuovo E., the progressive
education of Petersen, Kry Van Perreren and the sociocultural theories of Eastern Europe.
Realistic Mathematics Education, as presented in this book, does not claim to be a
comprehensive learning theory like constructivism, but is a comprehensive philosophy
(according to Freudenthal) that is implemented through a set of teaching theories specific
to mathematical subjects. The central ideas of this approach are the following: -
Mathematics is considered a human activity (what Freudenthal calls mathematization)
and therefore should be accessible to all. - The development of mathematical
understanding occurs in different stages where contexts and models play an important
role.
This development is facilitated through the process of guided reinvention, within
a diverse cognitive environment. - From a curricular perspective, the guided reinvention
of mathematics as a mathematization activity requires the use of didactic phenomenology
as a research methodology.
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It is about searching for contexts and situations that generate the need for
mathematical organization. The history of mathematics and students' spontaneous
mathematical inventions and productions serve as primary sources for this search. These
concepts, commonly known as the Principles of Realistic Mathematics Education, are
explained in more detail below: The Activity Principle emphasizes that mathematics
should be seen as a human activity that can be accessed and learned through active
participation.
According to Freudenthal, teaching the process of mathematical activity is more
important than teaching the end result. The focus should not be solely on learning
algorithms or concepts, but on the process of algorithmization, algebrization, abstraction,
formalization, and structuring. According to this principle, mathematics should be
accessible to all students, recognizing that not all need to pursue careers in mathematics.
The goal is for students to develop mathematical and critical thinking skills to apply to
everyday problems.
The emphasis is on providing access to knowledge, skills and dispositions through
real-life situations, uncovering the hidden processes within mathematical products.
Freudenthal draws inspiration from the activities of mathematicians, whether in pure or
applied mathematics, who are engaged in problem solving and problem-solving and in
organizing content related to mathematical concepts and real-world information.
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Chapter 1
Mathematician and Theorist: Hans Freudenthal
Throughout his professional career, Hans Freudenthal's perspectives on
educational reform diverged from all contemporary approaches. He questioned the
"new" mathematics, operational objectives, rigid assessment methods, standardized
empirical quantitative research, and strict divisions between curriculum research,
development, and implementation. Interestingly, while his ideas were initially seen as
rebellious, they have now gained widespread acceptance. This suggests the important
role that Hans Freudenthal played not only in mathematics education but also in
curriculum theory and methodological research.
In addition to his reputation as a mathematical researcher, Freudenthal also delved
into the educational and psychological traditions of Europe and the United States,
making his own contributions to mathematics education. Today, he is widely recognized
as one of the most influential mathematics educators of his time. In this paper we aim to
highlight some of Freudenthal's ideas, although it is impossible to cover them all. We will
focus on pedagogy and curriculum theory, exploring aspects of Freudenthal's work and
the theories that are relevant from these perspectives.
There are notable differences between the curriculum theory developed by
educators in the United States and Europe, despite arguments that they address similar
issues. These differences arise from fundamental disparities in cultural, philosophical,
and institutional backgrounds. In Europe, pedagogical theory includes the concept of
Didactics, which is considered a form of humanities.
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This perspective is based on the practice of education, focusing on realist
education and the phenomenological theory of Bildung, which encompasses the
formation of the individual's personality. It goes beyond the mere transmission of
knowledge and also emphasizes the development of norms, values, and skills necessary
to be a "good" citizen or part of an intellectual elite. On the other hand, Ausbildung refers
to vocational and professional training. Didactics in this context is primarily concerned
with theories about the purpose and content of education and instruction.
In the Netherlands, didactics is influenced by the phenomenological pedagogy of
the Geisteswissenschaftliche, as exemplified by the work of Langeveld at Utrecht
University in 1965. Although this perspective lost importance in the 1960s and 1970s,
leading to the gradual replacement of a general didactic perspective with formal models
of learning and teaching popularized by American educational psychologists such as
Robert Glaser, Robert de Cecco and Benjamin Bloom. Despite this shift, the content of
didactics developed in faculties and institutes of mathematics and educational sciences
was not completely overshadowed by this movement.
Despite never mentioning students like Wolfgang Klafki, Freudenthal's questions
about what should be taught in school subjects, for what purpose, and for whom are
similar to those posed by Klafki. Freudenthal's belief in "mathematics as a human
activity" can be seen as a representation of a Geisteswissenschaftliche, a
phenomenological theory of mathematics education that focuses on the practical aspects
of teaching and education rather than simply transmitting pre-existing mathematical
knowledge. Some of Freudenthal's main ideas, such as "reinvention" and his critique of
the "anti-didactic inversion" of traditional deductive instruction, may have been
influenced by the progressive education and pedagogical reform movement. Figures
such as Peter Petersen and Maria Montessori played a role in shaping these ideas.
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According to Freudenthal, curriculum theory is not a fixed set of theories and
purposes but is process-dependent. The term "curriculum" is often used in conjunction
with change or development, such as curriculum development or research development.
For Freudenthal, curriculum theory is a practical endeavour that can lead to the
emergence of new theoretical ideas. He believes that curriculum development should not
be led by academic leaders but should involve collaboration between teachers and
students in schools. Similar ideas are shared by Schwab, who advocates curriculum as
"practice" and challenges the dominant curriculum theory of his time. As a result, there
are similarities between certain branches of the Anglo-Saxon approach to curriculum
theory and Freudenthal's understanding of curriculum (Gravemeijer & Terwel, 2000).
However, in Freudenthal's writings there is often a negative connotation
associated with the term 'curriculum'. He describes the dominant Anglo-Saxon
curriculum movement as a behaviorist and top-down theory, referring to it as 'boxology'.
In contrast, Freudenthal presents his own view of curriculum as a process, which he calls
educational development. Whereas curriculum development focuses on the creation of
curriculum materials, Freudenthal seeks to go a step further by promoting changes in
classroom teaching through educational development.
Educational development therefore goes beyond instructional design and
encompasses comprehensive strategic innovation. This innovation is based on an explicit
educational philosophy and involves the development of diverse materials as part of the
overall strategy. Research plays a crucial role in driving this entire process, aligning well
with the pedagogical tradition. Specifically, qualitative and interpretive research is
employed, drawing on teaching experiences in individual classes. Dialogue between
researchers, curriculum developers and teachers are central to this approach.
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Mathematics: human activity
Freudenthal was known for his opposition to the "new mathematics" of the 1960s,
which was based on modern mathematics and set theory. He believed in traditional
pedagogy and criticized the new approach because he believed it neglected what should
be taught and how it should be taught. He recognized that mathematics is characterized
by its generality and wide applicability, but he also saw abstraction as a problem in
teaching. While abstract mathematics is flexible in an objective sense, it may not be useful
to people who cannot apply this flexibility to their own lives. Freudenthal argued that
mathematics should be taught as a useful tool, but not simply by teaching mathematical
concepts and then applying them. He believed that the order of teaching was important
and that mathematics should be taught by mathematizing. This approach emphasizes the
process of doing mathematics rather than focusing solely on the end result. In traditional
mathematics education, the starting point is often the result of the mathematical activity
of others, which Freudenthal saw as an anti-didactic reversal. He believed that teaching
should begin with the activity itself and not the end result.
The pursuit of mathematics involves both problem solving and the establishment
of a structured discipline. To effectively solve real-world problems, they need to be
organized and addressed using mathematical patterns. Likewise, mathematics itself
requires organization, whether by organizing new or existing results, whether one's own
or those of others, to enhance understanding. This may involve exploring new ideas,
examining broader contexts, or applying an axiomatic approach, as Freudenthal
suggested in 1971.
Freudenthal's approach to mathematization includes both "subjects of reality" and
"mathematical subjects," encompassing both applied mathematics and pure mathematics.
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This sets him apart from other mathematics educators who also emphasize mathematical
activity but base their discourse on the discourse of pure mathematics researchers.
Freudenthal's description of mathematical activity as a model for mathematics education
differs from the above in two respects:
• First, it incorporates applied mathematics or the process of using mathematics to
solve real-world problems.
• Second, it shifts the focus from the structure of the activity to the activity itself and
its results.
• Furthermore, the concept of "discourse" refers to a social practice in which the act
of mathematizing gives significant importance to mental engagement.
Freudenthal's comprehensive definition of mathematics as a human endeavor
aligns most effectively with a practical discourse, such as that found in applied
mathematics. In this type of discourse, there is a greater emphasis on effectiveness and
efficiency, and less focus on speculative guesswork without a clear goal. He employs the
term "mathematize" comprehensively, encompassing both the organization and
application of mathematical principles. By selecting the word "organize," Freudenthal
conveys that mathematizing involves more than simply translating concepts into a
structured system of symbols. Furthermore, the act of organizing the subject matter itself
should lead to the development of a symbolic representation.
Precision is also a key aspect of mathematization. Mathematics is known for its
precision and accuracy, and by applying mathematical reasoning, we can ensure that our
solutions are accurate and error-free. We use mathematical tools, such as formulas,
equations, and calculations, to arrive at accurate and reliable results. By mathematizing,
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we can avoid misconceptions and misinterpretations, leading to a more accurate
mathematical understanding.
When we talk about making something “more mathematically,” we are referring
to the process of applying mathematical principles and concepts in a way that emphasizes
generality, certainty, precision, and brevity. Generality refers to the ability to apply
mathematical ideas in diverse contexts and situations. By mathematizing, we can
recognize patterns, identify relationships, and make connections that extend beyond
specific examples. This allows us to solve a wide range of problems and understand the
underlying principles that govern them. Certainty is another characteristic of
mathematization.
When approaching a problem mathematically, we strive for logical reasoning and
evidence-based solutions. We rely on the rules and principles of mathematics to guide
our thinking and ensure our conclusions are dependable and well-founded. Mathematics
has a unique language and symbolism that allows us to express complex ideas and
concepts concisely. When mathematizing, we aim to use this language effectively, using
symbols, notation, and concise explanations to communicate mathematical ideas
efficiently. This allows for clearer communication and a more streamlined approach to
problem solving:
• By generality: generalizations (observation of analogies, classifications, structures)
• In order to establish certainty, it is crucial to engage in a process of reflection,
justification and testing. This can be achieved by employing a systematic
approach, which involves developing and testing conjectures, hypotheses or
theories. By thoroughly examining the evidence and subjecting it to rigorous
scrutiny, the validity and reliability of the conclusions drawn can be determined.
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By emphasizing the importance of critical thinking and logical reasoning, this
systematic approach ensures that certainty is not simply assumed but
substantiated through sound and rigorous analysis.
• To ensure accuracy, it is important to use multiple methods, such as modeling,
symbolizing, and defining, to limit interpretations and assess the validity of
information.
• To maintain conciseness, it is essential to symbolize and schematize, which
involves creating standardized procedures and notations.
When viewed from this perspective, the act of mathematizing mathematical
objects and the act of mathematizing real-world issues have similar characteristics. This
is a crucial concept for Freudenthal, as he suggests that children's mathematics education
should focus on the application of mathematical principles to everyday situations.
Children are not able to mathematize mathematics itself, as they have no direct
experience with mathematical objects. Similarly, when students mathematize real-world
disciplinary objects, they become more familiar with using mathematical approaches to
solve problems in their daily lives. This also relates to Freudenthal's idea of "problem-
finding," which involves having a mathematical mindset that understands the strengths
and limitations of using mathematics in different situations.
The notion of “mathematizing reality” is a central aspect of the concept of
“mathematics for all.” Freudenthal recognizes that not all students will become
mathematicians in the future but emphasizes that the mathematics they learn must be
applicable to everyday problem solving. It is therefore important to prioritize teaching
students how to approach problem solving using mathematical methods. This objective
can be combined with the goal of students applying mathematical concepts to situations
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that are relevant to their own experiences. From this perspective, it is not surprising that
Freudenthal strongly criticizes the concept of didactic transposition, proposed by
Chevallard (1985), which relies on the expert knowledge of mathematicians. Freudenthal
argues that mathematics taught in schools should not be a mere translation of
philosophical or scientific ideas, unless they are from a much earlier time (Gravemeijer &
Terwel, 2000).
Keitel (1987) argues that the main goal is to develop a mathematics curriculum that
is accessible to all individuals while retaining the essence of mathematics itself. To
achieve this, he suggests that teachers should sometimes move away from real-world
problems and focus on the concepts, structures and systems that have been established
and tested within mathematical science. Building on Freudenthal's concept of
mathematization, the idea of horizontal and vertical mathematization is introduced.
Horizontal mathematization involves transforming a contextual problem into a
mathematical problem, while vertical mathematization involves taking the mathematical
discipline to a higher level. Vertical mathematization can be fostered by presenting
problems that have mathematical solutions at various levels of complexity. Freudenthal
(1991) describes this distinction by explaining that horizontal mathematization bridges
the gap between the real world and the realm of symbols.
In the real world, individuals live, act, and experience various emotions, while in
the symbolic world symbols are created, manipulated, and understood through
mechanical, integral, and reflexive processes. The real world represents what is perceived
as reality, while the symbolic world represents abstraction. However, the boundaries
between these two worlds are not clearly defined and can fluctuate. Freudenthal
emphasizes that the distinction between horizontal and vertical mathematization is not
rigid, as the perception of reality varies from person to person. He defines reality as a
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combination of interpretation and sensory experience, suggesting that mathematics can
also be part of an individual's reality. The concept of reality and what is considered
common sense is not fixed but is influenced by personal learning processes. Therefore,
Freudenthal's statement that "mathematics begins and remains in reality" should be
interpreted as an acknowledgement of the dynamic nature of reality and its relationship
to mathematics.
In Freudenthal's perspective, the concepts of "common sense" and "reality" are
subjective and depend on the individual's point of view. This means that the distinction
between vertical and horizontal mathematization must also be assessed from the
individual's perspective. Whether a specific mathematical activity is considered "vertical"
or "horizontal" depends on the nature of the activity and the person's understanding of
mathematics. For example, a symbolic activity may be routine for a student, categorizing
it as horizontal mathematization.
However, if the same symbolic activity involves a new invention for another
student, it would be considered vertical mathematization. The latter is most evident when
a student replaces his or her method of solving or way of describing with a more
sophisticated, organized, and mathematical approach. These changes can be fostered by
reflecting on the methods of solving and deepening understanding. Engaging in whole-
class discussions that explore different solution methods, interpretations, and ideas can
contribute to these changes. Thus, during such discussions, students may discover
alternative solution methods that are more advantageous than the current ones. This
highlights the importance of dialogue in mathematization, emphasizing that it is not
solely an individual activity.
Likewise, Freudenthal also emphasizes the importance of group work in
mathematics education. He first introduced the concept of small group learning in 1945
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and later advocated mathematics education in diverse groups. According to Freudenthal,
both hardworking and lazy students can benefit from collaborative learning.
Surprisingly, reviewing his works from the 1940s onwards, Freudenthal found that he
had consistently advocated cooperative learning in small and diverse groups.
The Criticism
Freudenthal's reputation is not only based on his own theoretical ideas, but also
on his criticism of "traditional" research. In the educational research community in the
Netherlands, he encountered strong opposition for his stance against those who relied on
an empiricist methodology and complicated statistical analyses. Drawing on his
experience as a mathematician, Freudenthal skillfully exposed the significant
shortcomings in the application of mathematics and statistics in numerous cases of
supposedly "high" empirical research.
Freudenthal's stance against much of educational research stems from his belief
that interruptions in the learning process are crucial. These disruptions can be seen as
shortcuts or opportunities to gain different perspectives. According to Freudenthal, it is
through these disruptions that it can be determined whether a student has reached a
certain level of understanding. To identify these disruptions, individual students must
be closely monitored. This approach ignores the importance of groups and the removal
of individual disruptions. Furthermore, the focus should be on observing the learning
process rather than testing the achievement of learning objectives. Overall, Freudenthal
argued that traditional research methods could not adequately address educational
questions about the purpose and target audience of a particular topic.
Freudenthal expressed further concerns and objections towards the testing
movement and offered a second round of criticism. His skepticism revolved around the
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methods employed in testing and he harshly criticized the detrimental impact that
examinations and testing techniques had on the field of education. The crux of his
criticism centered on the lack of understanding of the subject matter being tested and the
excessive emphasis placed on reliability, while ignoring the importance of validity.
Freudenthal clearly did not share the same positive outlook and enthusiasm as the
proponents of objective testing.
In a broader sense, Freudenthal's critique of educational research focuses on
methodologists who possess extensive knowledge about research methods but lack an
understanding of education itself. He vehemently opposes the division between content
and form, arguing that this approach results in empty models that require experts to fill
them with educational substance. These models do not consider whether the content
actually aligns with educational principles (Gravemeijer & Terwel, 2000). Furthermore,
he voices comparable objections toward comprehensive educational theories.
According to Freudenthal, general educational theories do not align with the
specific needs of mathematics education and may even be detrimental to the type of
education they are intended to support. He specifically criticizes the educational theories
proposed by Bloom, Gagné, and Piaget. Freudenthal argues that Bloom's Taxonomy of
Educational Objectives is not suitable for mathematics education because it focuses on
classification rather than the active process of structuring reality. He believes that
students gain control over reality through this structuring process, and the artificial
categories of Bloom's Taxonomy have a negative impact on both school and
developmental testing.
He also rejects Bloom's mastery learning strategy, accusing it of treating learning
as a passive process of knowledge transfer. Similarly, he disagrees with Gagné's concept
of task analysis, as it does not align with his view of mathematics as a human activity.
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Freudenthal questions whether mathematics is really so different from other disciplines
and expresses the wish that someone with a background in both mathematics and
psychology would bridge this gap.
While Gagné sees learning as a continuous progression from simple to complex
structures, Freudenthal sees it as a discontinuous process from the rich and complex
structures of everyday life to the abstract structures of symbolic mathematics. He believes
that the starting points for learning should be situations that require organization and
that learners should develop their own categories based on their needs.
Freudenthal also criticizes Piaget for his approach to mathematics and his
experiments. However, what he is more concerned about is how Piaget's work influences
teaching methodologies by basing their practices on theories they have learned from a
psychologist. He argues that these methodologies often misinterpret or misinterpret
Piaget's mathematical presuppositions rather than being based on the actual findings of
his experiments.
In his work, Freudenthal delves deeper into the concept of constructivism and
offers both criticism and support for this epistemology. While he criticizes constructivist
epistemology as an observer, he argues that his own perspective as an actor aligns with
this epistemology. Specifically, he views mathematics from the perspective of a practicing
mathematician and characterizes it as a well-developed form of common sense, which is
intricately linked to his idea of an “extended reality.” In terms of education, Freudenthal
aims to ensure that students’ experiences help them internalize mathematical knowledge
and view it as a seamless extension of their everyday life experiences. Based on this, it
can be inferred that Freudenthal is actually more aligned with constructivism than it
initially appears, despite his criticisms of it.
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Freudenthal’s perspective on mathematics education emphasizes the importance
of viewing mathematics as more than just a series of steps or procedures, but as a dynamic
human endeavor. While it is crucial to recognize that engaging in this activity also
produces mathematical knowledge and concepts, this raises the question of how to
design mathematics education that effectively combines these two aspects. To address
this, Freudenthal proposed several concepts, including the concepts of “guided
invention,” “levels of learning processes,” and “didactic phenomenology,” all of which
offer valuable insights into addressing this challenge.
The reinvention
According to the principle of reinvention, the learning process can be structured
in a way that allows students to encounter and understand mathematics. Curriculum
development begins with an idea or concept, and through experimentation and personal
problem solving, students can arrive at their own solutions. The study of the history of
mathematics can serve as a useful tool in this process, guiding students along the learning
journey.
This approach, known as “guided reinvention,” emphasizes the importance of the
learning process itself rather than simply acquiring knowledge. It encourages students to
take ownership of the knowledge they acquire and to feel responsible for it. To facilitate
this, students should be given the opportunity to build their own mathematical
knowledge bases based on their learning experiences. Freudenthal suggests that the
history of mathematics can be a source of inspiration for students and that the principle
of reinvention can also be influenced by informal solution methods. Often, students’
informal strategies can be seen as anticipations or precursors of formal processes. This
process of mathematization, similar to finding solutions, is a form of reinvention. When
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selecting contextual problems for students, it is important to choose those that allow for
a variety of solution methods, preferably those that reflect a learning path.
Freudenthal believes that the inventive approach to teaching is an expansion of the
Socratic method. He refers to “thought experiments” as a way of illustrating this, where
textbook authors imagine themselves interacting with students and imagining their
reactions and outcomes. These planned experiments involve anticipating students’
reactions and devising strategies to address them. The goal is for students to reinvent the
teaching topic through interaction and engagement. Freudenthal comments that while
the student activity in the Socratic method is fictional, they should feel that their
understanding and ideas are developing during the teaching process, with the teacher
serving as a facilitator.
For Freudenthal, the Socratic method gives students a more active role in the
process of constructing their own knowledge. However, there is a similarity between both
approaches when it comes to anticipating and planning learning paths. This idea of
anticipation and planning is discussed in relation to various challenges that need to be
addressed, such as the mental activity of students and the necessary actions that need to
occur for the process of reinvention to be feasible.
Freudenthal expands the idea of reinvention by introducing the concept of
“progressive mathematization.” This concept involves both the observer’s perspective of
reinvention and the student’s perspective of experiencing “progressive
mathematization” as an actor. Students begin by mathematizing a real-world topic and
then move on to analyzing their own mathematical activity. This step is crucial as it
includes a vertical component, explained by Freudenthal in relation to Van Hiele’s theory,
which states that activity at one level becomes the object of analysis at the next level.
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The shift from “operator” to “object” means the transition from a procedure-based
approach to a focus on the object itself, as observed by Sfard (1995) in the historical
development of mathematics and the materialization described by Ernest (1991).
Freudenthal’s level theory forms the basis of realistic mathematics education (RME),
which emphasizes the emergence of operational models in situational problem solving
and their gradual transformation into entities that serve as models for formal
mathematical reasoning (Gravemeijer & Terwel, 2000).
Phenomenology in didactics
Freudenthal emphasizes the importance of matching mathematical objects to the
real-world phenomena they represent. In contrast to the concept acquisition approach,
which involves using tangible materials to embody concepts, he suggests using
phenomenologically rich situations: situations that are organized in a systematic way. In
this approach, the selection of situations must be made carefully to ensure that they can
be organized and understood using the mathematical objects that students are learning
to construct.
The ultimate goal is to explore how the “object of thought” (nooumenon) describes
and analyzes the “phenomenon” in a way that makes it accessible to calculation and
thinking activities. This type of phenomenological analysis forms the basis of a didactic
phenomenology that deepens the educational perspective of phenomenological analysis.
For example, in order for students to understand the concept of length as a mathematical
object, they must be confronted with situations in which length is an organizing principle.
Within the framework of phenomenological didactics, it is necessary to investigate
the suitability of situations where a particular mathematical topic is applied, in order to
determine its potential impact on the process of progressive mathematization. If we
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understand mathematics as a practical means of problem solving, it is reasonable to
expect that current applications of mathematics involve problems that highlight these
processes. Therefore, formal mathematics can be seen as a process of generalization and
formalization of concepts and problem-solving procedures in various situations. The goal
of phenomenological research, therefore, is to identify problematic situations that can be
generalized and to discover situations that provoke paradigmatic solution procedures,
which serve as a basis for vertical mathematization. By identifying phenomena that can
be mathematized, we can better understand how they were originally conceived.
When considering research, Hans Freudenthal often asked himself what its
purpose was, and he always concluded that the purpose was to bring about change.
Education must continually adapt to the ever-changing society it serves. Therefore, the
concept of “change” is more preferable than that of “reform,” since what constitutes
better education depends on the needs and priorities of society at a given time and how
society evolves.
Education must change accordingly. In this sense, an important function of the
researcher is to chart the path of change. Freudenthal believed that research should not
be disconnected from the classroom, unlike traditional research. Rather, the search for the
path of change should begin in the classroom. This philosophy of the goals and functions
of research guided the approach to research at the Institute for the Development of
Mathematics Education (IOWO), which Freudenthal directed.
At the time of the IOWO's creation, the predominant model in the German
educational community was the R&D model. This model emphasized a separation
between curriculum development and implementation, which contradicted
Freudenthal's approach to "educational development." The concept of educational
development, as Freudenthal saw it, encompassed not only curriculum development but
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also the ultimate goal of changing educational practice. Thus, educational development
involved anticipating curriculum implementation from the outset as well as choosing a
comprehensive approach to change that encompassed teacher training, counseling,
developmental testing, and opinion formation, all based on the same educational
philosophy.
In contrast to the curriculum movement, Freudenthal integrated research,
development, implementation, and dissemination. As a result of this approach, he
advocated the involvement of all stakeholders from the outset, under the banner of
“educational development in dialogue with the field.” The kind of change Freudenthal
advocated was rooted in his belief that mathematics is a human activity. At the time
IOWO was launched, however, little research had been done on this type of mathematics
education. Therefore, questions about how to develop instruction had to be addressed
during the development process itself.
Research for development
At first, our mathematician was reluctant to label IOWO's work as research. He
believed they were observing as engineers, not as researchers. However, he later realized
that this perspective separated research from educational development and failed to
capture the interconnected nature of development in "developmental research."
According to him, new knowledge must be justified by the process by which it was
acquired.
The essence of developmental research lies in making the cyclical process of
development and research consciously experienced and clearly reported. This enables
others, such as teachers, to retrace the steps of the researcher in the learning process.
Freudenthal emphasizes the importance of being constantly aware of the developmental
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process to ensure “traceability.” In order for the results of developmental research to be
credible and transferable, reflection on the developmental process must be informed.
The researcher should conduct thought experiments to understand how teaching
and learning processes progress and then find evidence in teaching experiments to
validate his or her expectations. Feedback from practical experience should drive an
interaction between development and research. Ideas developed on paper should be put
into practice immediately and classroom events should be consistently analysed and
applied to further develop the work.
This process of deliberation and testing should result in a product that is both
theoretically and empirically grounded. According to Freudenthal, developmental
research can provide teachers with a framework to inform their own decisions. Within
this framework, teachers can develop hypothetical learning trajectories that consider the
current classroom situation as well as their own goals and values. Teachers can use this
framework as a starting point, firmly rooted in the European teaching tradition, to guide
their teaching.
Research conducted on national assessments has revealed that students in the
Netherlands in the later years of primary school tend to achieve higher levels of success
when interacting with modern texts compared to traditional ones. However, it is
important to note that this trend does not apply to topics such as measurements and
written algorithms. These findings suggest that the strategic approach of incorporating
educational development in dialogue with the field, as implemented in the introduction
of the Dutch curriculum and school textbooks, is the driving force behind this positive
outcome. Indeed, retrospective studies examining innovations in mathematics education
in both primary and secondary school have identified several key factors contributing to
this success.
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A key aspect of this plan would involve the review and renewal of the textbooks
used in mathematics education. These textbooks would be carefully reviewed and
updated to align with the new philosophy of mathematics education, incorporating
innovative teaching approaches and engaging content. In addition, a thorough review of
examinations would be undertaken to ensure that they accurately assess students’
understanding and mastery of mathematical concepts. This would involve revising the
format and content of examinations, as well as incorporating more open-ended and
problem-solving questions that promote critical thinking and the application of
mathematical knowledge. Finally, research and development would play a crucial role in
driving innovation in mathematics education.
By continuously refining and improving the field of mathematics education, a
dynamic and forward-thinking approach to teaching and learning would be ensured. To
achieve significant and lasting improvements in mathematics education, a
comprehensive and ambitious plan needs to be implemented. The plan would encompass
several key components, including establishing a robust and transformative philosophy
of mathematics education that empowers and inspires both students and teachers. It
would also involve creating and refining a wide range of instructional sequences,
examples, and prototypes that effectively engage students and facilitate their
understanding of mathematical concepts.
These educational materials would be continuously developed and updated to
reflect the latest advances in pedagogical techniques and educational research. And, the
establishment of a mathematics education community would serve as a vital mediating
infrastructure, facilitating the exchange of best practices, resources, and ideas among
educators. The community would provide a platform for dialogue and collaboration,
promoting the dissemination of innovative teaching strategies and approaches. To
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support the implementation of this plan, professionalization activities would be
organized to enhance the skills and knowledge of mathematics educators, providing
opportunities for ongoing professional development, as well as fostering collaboration
and networking within the mathematics education community. To ensure widespread
adoption and implementation of these improvements, efforts would be made to increase
the accessibility and availability of high-quality mathematics education resources,
providing comprehensive training and support to already in-service teachers, as well as
developing and disseminating publications highlighting effective teaching methods and
strategies.
Research development plays a crucial role in driving innovation strategies. Its
main objective is to generate prototypes and theories that serve as valuable resources for
teacher educators, textbook authors, and school consultants. These intermediaries, in
turn, facilitate effective communication between researchers and teachers. The
fundamental principle guiding educational development is the concept of engaging in
meaningful dialogue with practical applications. Meaning, the Institute places great
emphasis on involving diverse stakeholders, including teacher educators, consultants,
textbook authors, researchers, test designers, and teachers themselves, in the research and
development process from the very beginning. Rather than isolating itself in an ivory
tower, the Institute recognizes the importance of incorporating real-world knowledge
and experience into its innovative efforts.
It is not an easy task to locate Freudenthal's work in the contexts of didactics and
curriculum studies because of his unique writing style, which lacks references to the
authors who have influenced him. When it comes to didactics, Freudenthal often uses
this term to describe the correct teaching and learning processes, which he believes
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should be rooted in reality. He firmly rejects the deductive approach, which he calls "the
anti-didactic conversion."
According to Freudenthal, didactics is concerned with the processes involved in
education. This aligns with Klafki’s use of the same term, as draw inspiration from the
phenomenological theory of Bildung as a pedagogical reform. Both start from the practice
of education as a foundation and strive to overcome the exclusionary and elitist aspects
of Bildung theory at certain points in their professional lives. Both emphasize the practical
side of education and advocate comprehensive schooling as a necessary reform.
However, Klafki focuses primarily on planned lessons and lesson preparation, where the
learning process may not be entirely real. Klafki’s fundamental questions revolve around
the content of Bildung, while he pays less attention to teaching methods and processes.
Freudenthal mentions the term "curriculum," although he does not use it as
frequently as the word "didactics." In regard to his perspective on curriculum
development and the role of theory, there is a remarkable resemblance to the work of
Joseph Schwab, who occupies an important position in American curriculum theory. In a
similar vein, but without any influence from Schwab, Freudenthal stresses the unique
nature of curriculum work and the importance of dialogue between curriculum experts
and teachers.
He strongly opposes the idea of a rigid curriculum system and firmly rejects the
concept of packaging and organizing content into predetermined structures. This view is
particularly noteworthy at a time when curriculum theory was influenced by a
behavioural approach, and the R&D (Research, Development and Diffusion) method was
hailed as the ultimate solution in Germany and the Netherlands.
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Freudenthal also advocates that mathematics be viewed as a human activity and
encourages guided reinvention. This humanistic, practical, process-oriented,
phenomenological, and pedagogical reform philosophy, which is widely discussed in the
context of curriculum development, distinguishes Freudenthal's stance from that of many
of his contemporaries in the field of mathematics education. His beliefs often clashed with
those of behaviorist-oriented psychologists such as Bloom and proponents of the "new
mathematics" movement, who proposed the development of a mathematics curriculum
based on an abstract deductive system.
Freudenthal, who was educated in and influenced by the German Bildung
tradition, rejects the idea of an exclusive form of education reserved for an elite group
separated from the masses. Instead, he strongly advocates “mathematics for all” and
strives to make mathematics accessible to all individuals. He condemns any form of
conforming to societal norms and aligning oneself with the inevitable effects of
mathematical concepts.
He strongly believes that students with different ability levels in the early years of
secondary education, which is typically between 12 and 15 years old in the Dutch context,
should not only be in the same class but also follow the same curriculum. Consistent with
his pedagogical beliefs, he emphasizes the importance of forming diverse learning
groups. Many of Freudenthal's ideas remain the subject of ongoing debate.
Psychologists, who view learning as an informational process, strongly oppose
educational theories of this nature. Similarly, there are instances of opposition within the
mathematics education community to the fundamental idea that students must make the
transition from the real world to the world of mathematics. Critics argue that drawing
from real-life experiences and reinventing mathematical concepts is a waste of time.
However, it is important to note that those who oppose Freudenthal's ideas have limited
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empirical evidence to support their view. Several teaching experiences have
demonstrated the value of the Realistic Mathematics Education (RME) approach.
Furthermore, numerous studies investigating the effects of mathematics curriculum
influenced by Freudenthal's ideas have shown that learning mathematics in real-life
contexts and within diverse groups is feasible and effective.
The impact of Freudenthal's ideas is evident in all the Dutch texts. Moreover, there
is practical and empirical evidence supporting the feasibility and efficiency of the EMR
approach. One of Freudenthal's most convincing arguments in favour of EMR is that not
all students will become mathematicians in the future; instead, they will primarily need
mathematical skills that help them solve problems in everyday life situations.
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Chapter 2
Realistic mathematics education
Mathematics is interconnected with the world and the image of a mathematician
is shaped by societal perceptions. The perspective is now reflected in many countries’
curricula and in the Programme for International Student Assessment (PISA)
assessments. Mathematical literacy, according to PISA, is an individual’s ability to
recognise and understand the role of mathematics in the world, make informed
mathematical judgements and use mathematics in a way that meets their present and
future needs as a responsible and reflective citizen.
This view emphasizes the importance of mathematics in society and its practical
application in diverse contexts. Freudenthal's influence extends beyond his academic
career. He played a major role in the International Group on Psychology and
Mathematics Education, the journal Educational Studies in Mathematics, and the
International Commission for the Study and Improvement of Mathematics Teaching.
Through his numerous writings, he expressed his opposition to the pedagogical and
teaching approaches that emerged in the mid-20th century, including Bloom's taxonomy,
structured assessments, quantitative methods in educational research, the direct
application of Piaget's ideas in the classroom, the separation of educational research,
curriculum development, and teaching practice, and the introduction of modern or
established mathematics into schools.
Realistic Mathematics Education (RME) is an educational approach developed
since the late 1960s by Hans Freudenthal and his colleagues at the Freudenthal Institute
for Mathematics and Science Education at Utrecht University in the Netherlands. Hans
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Freudenthal, a leading mathematician specializing in topology, algebra, and the history
of mathematics, was forced to emigrate from Germany due to the rise of the Nazis. In the
Netherlands he set about promoting changes in mathematics education not only within
the country but also in other European nations.
Freudenthal has been publishing on mathematics education since 1948. Over the
years he worked with the Institute for the Development of Mathematics Education
(IOWO), which he founded in 1970 at Utrecht University, together with other
collaborators. This institute has laid the foundation for the current development and
expansion of the Educational Materials and Resources (EMR) programme. Treffers (1987)
describes the principles on which EMR is based:
• Principles include a focus on phenomenological exploration, where students are
exposed to rich and meaningful phenomena to develop an intuitive understanding
of mathematical concepts.
• The use of models and symbols is also emphasized, as students move from
informal, context-bound notions to more formal mathematical ideas.
• Students' own constructions and productions are valued and used in the teaching
process, since their personal experiences contribute to meaningful learning.
• Interaction is another key aspect, as students can compare and contrast their
contributions, reflecting on the process of mathematization.
• It is also important to link curricular themes and axes since the connections
between different areas of mathematics are considered when teaching specific
topics. For example, when teaching statistics, the necessary algebraic or scientific
knowledge is considered, and when introducing the notion of distribution, it is
linked to other statistical concepts.
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During a presentation addressed to educators in the field of mathematics,
Freudenthal states that mathematics involves problem solving and the organization of
objects of study, which may be real-world phenomena that require the organization of
mathematical patterns to solve problems. Alternatively, they may be mathematical issues,
whether new or old, one's own or someone else's, that need to be organized with new
ideas to achieve better understanding in a broader context or through an axiomatic
approach. He goes on to discuss how children are initially taught mathematics as an
activity, but as they mature, they are often presented with a pre-constructed, well-
organized mathematical system under the assumption that rational individuals will
understand deductive systems. However, this approach is not effective.
For Freudenthal, transmitting ready-made mathematics, which is the product of
mathematicians or textbook authors, is counterproductive in terms of teaching. Instead,
he suggests teaching students to mathematize. Treffers (1987) further expands this
concept by differentiating between two dimensions of mathematization: horizontal and
vertical. Horizontal mathematization involves transforming a real-world problem into a
mathematical problem using common sense, intuition, observation, empirical
approximation, and inductive experimentation.
On the other hand, vertical mathematization involves navigating within the realm
of mathematical reality through schematization, generalization, proof, rigor, and
symbolization. Horizontal mathematization leads from the world of life to the world of
symbols, where individuals live, act, and experience, while vertical mathematization
involves the creation, recreation, and manipulation of symbols in a mechanical,
comprehensive, and reflexive manner (Zolkower & Bressan, 2012). It is important to note
that the boundaries between these two worlds are not clearly defined and can expand or
contract depending on several factors.
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To teach students how to apply mathematical concepts to real-life situations, it is
important to engage them in guided activities that involve organizing realistic problems.
The terms “realistic” and “reality” are used in this context to refer to situations that align
with common sense and are perceived as genuine within a given scenario.
In the early grades, we focus on familiar everyday contexts and situations
involving numbers, such as people getting on and off a bus. As students become more
familiar with numbers and their relationships, their understanding of what is real or
meaningful to them expands. It is important to note that the term “realistic” is often
misunderstood in a narrow sense, which is due to the choice of this name. In Dutch, “zich
realis-eren” means to imagine. Therefore, in a broader sense, a situation is considered
realistic as long as it is presented to the individual as feasible, reasonable, or imaginable.
For example, when we teach geometry and measurement, estimation, ratios, and
proportions, we can draw inspiration from works of fiction such as “Gulliver’s Travels.”
The goal of mathematics education, according to Freudenthal, is to develop in
students a mathematical disposition that includes a variety of skills and abilities. This
includes the ability to identify the essential aspects of a situation, problem, procedure,
algorithm, symbolization, or axiomatic system. It also involves recognizing common
features, analogies, and isomorphisms, as well as providing examples of general ideas
and discovering new objects and operations.
Students should be encouraged to find shortcuts, develop new strategies, invent
new symbolizations, and reflect on their own thinking by considering different
perspectives or points of view. In addition, mathematical readiness includes using
functional language and conventional variables, determining the appropriate level of
precision for a given problem, identifying mathematical structures in a context, and
recognizing when it is not relevant or appropriate to use mathematics. Therefore,
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students should consider their own activity as an object of reflection in order to advance
their understanding to a higher level.
To develop this mindset it is necessary to go through a teaching-learning process
that involves a guided reinvention, as described by Freudenthal (1991). The goal of this
process is not simply to teach mathematics, but rather to teach students how to think
mathematically, how to abstract concepts, how to create schemes, how to formalize
formulas, how to algorithmize procedures, and how to express mathematical ideas in
verbal form.
This approach to teaching, known as guided reinvention, is based on the principles
of instructional phenomenology, which involves looking for real-life contexts and
problem situations that foster mathematical thinking. By examining the ways
mathematical objects are used and understood in everyday language and situations,
educators can develop localized theories for teaching these concepts.
Didactic phenomenology is based both on the History of Mathematics, considering
the crucial moments in the development of mathematical ideas and their evolution over
time, and on the unique thoughts and creations of the students themselves. Thus, the
EMR (Teaching and Mathematization of Reality) approach considers learning as a non-
linear process involving progressively higher levels of organization, abstraction,
generalization and formalization.
The transition from one level of learning to another, which usually occurs
suddenly and means a break in learning, involves the use of a model to symbolize a
situation. Gradually, this model becomes detached from the original situation and
becomes a tool for organizing similar situations. There are four levels involved in this
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distinction between a model of and a model for: situational, referential, generalization,
and formal.
• At the situational level, strategies develop spontaneously to organize the
problematic situation.
• The reference level introduces graphic models, notations and procedures that
represent the problem but are still connected to the specific situation.
• The general level is reached through exploration, reflection and generalization,
which moves away from any reference to the context.
• Finally, the formal level involves working with general and conventional
procedures and notations that are disconnected from their original contexts.
To foster these processes, it is important to work on problems that can be solved
using different tools and to encourage the use of multiple strategies and procedures.
Thus, students' work on these problems can reveal their understanding and arithmetic
skills at a particular moment in time, which is valuable for making instructional decisions.
This information not only helps make small-scale decisions but also guides larger-scale
decisions. The class's collective understanding and problem-solving strategies provide a
snapshot of its learning trajectory. The strategies used by individual students offer insight
into the long journey the class will take. What is happening in the classroom at any given
moment provides a glimpse of what is to come and what is to come.
The Contexts
Context refers to a specific aspect of reality that is mathematized during a learning
process; they are not artificial disguises for mathematical content, but rather real-life
situations that curriculum designers and teachers present to students to encourage them
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to apply mathematical concepts. Freudenthal argues that viewing context as a distraction
from the mathematical message is a mistake, since context itself is the message and
mathematics is the tool used to understand it.
When a context is meaningful to a student, it serves as a starting point for his or
her mathematical activity, drawing on his or her common sense and suggesting the use
of informal strategies relevant to the situation. It is important to note that the realism of
a context depends on the student's prior experience and ability to imagine or visualize it.
For example, a first-grade student may find it as "real" to work with situations involving
changes in the number of passengers on a bus during different routes as he or she would
later find it to work with arrows as symbols representing such changes in later years.
Such contexts pave the way for higher-level mathematical concepts such as
operators and equations. Streefland (1991) supports this idea by describing a research
project on teaching fractions that begins with the concept of fraction and ratio
simultaneously by mathematizing situations involving equal distribution, such as
distributing 5 chocolate bars among 6 children.
Realistic contexts serve two functions: first, as a resource for generating
mathematical ideas and second, as a domain for applying mathematical concepts. By
using meaningful real-life situations as a starting point, students can bridge the gap
between reality and mathematics through interactions with peers, teacher guidance, and
the use of appropriate models that emerge from their own thinking. This approach allows
students to develop skills such as structuring, organizing, symbolizing, visualizing, and
schematizing. They can also progress in their mathematical understanding by improving
the efficiency of procedures, using shortcuts, and making the transition from colloquial
language to the conventional language of symbols and variables.
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From this perspective, it is also argued that in order to improve students'
mathematical thinking skills and, consequently, to improve the general mathematical
competence of individuals, it is imperative to critically analyze and explore the
connections between mathematics and its applications (both positive and negative) in
various areas such as science (including social, natural and exact sciences) and
technology.
The Models
Numerous models have been developed within the work carried out at the EMR.
These include models such as money, the rekenrek (a two-coloured abacus with 20 balls
arranged in two identical rows) and paradigmatic situations such as the collective, which
is represented by arrows to symbolise dynamic situations before and after. Other models
include the "pancake house", the parents' meeting at school and the candy factory with
10-packs.
Models such as the circular model, the double or percentage bar, and the ratio table
have also been explored. Two-coloured necklaces structured in groups of 10 have also
been used, which led to the development of the "open" number line as an arithmetic
model. In addition, the number line has been used as a model for solving linear equations,
and the notebook notation and the combination table have been used to work with
systems of two equations with two unknowns.
The use of these models, among others, is essential to counter one of the greatest
challenges in mathematics teaching, which is the tendency towards algorithmization and
premature formalization. Thus, models play a crucial role in simplifying complex realities
or theories, allowing for mathematical treatment. They emerge and develop through a
guided reinvention process and can be applied to diverse contexts.
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EMR has worked on numerous models, including money, rekenrek, paradigmatic
situations, and various arithmetic and equation-solving models. The use of these models
is important to combat the negative effects of algorithmization and premature
formalization in mathematics teaching. According to Freudenthal, a model serves as a
means of simplifying and idealizing a complex reality or theory, making it more
amenable to mathematical treatment. It is not a pre-existing artifact or representation, but
rather an entity that emerges and evolves through a process of guided reinvention.
Initially, models are closely tied to the specific contexts and situations in which they arise,
but over time they become decoupled and take on characteristics of formal, general
models that can be applied to a variety of contexts, both within and outside of
mathematics. This transition involves moving from being a “model of” a particular
situation to being a “model of” mathematical reasoning in a variety of situations.
Models advocate for an increase in the use of mathematical concepts in a way that
is relatable and understandable to students; they should be flexible enough to be applied
in more advanced or broader contexts, while still allowing students to understand their
initial meaning and purpose. It is important that models support both vertical
progression in mathematical understanding and the ability to connect to the original
context or situation. This allows students to fully understand the meaning and
significance of their actions within the model. Models should behave in a natural and
obvious way, aligning with informal strategies and being applicable to a wide range of
scenarios. As an example, the percentage bar initially emerges from a specific context
such as parking lots or movie theaters, where it represents the occupation of spaces
through shading. Over time, the percentage bar is detached from its original context and
transformed into a formal tool that can be used to work with and reflect on percentages.
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The ratio table, the bar model, and the double number line are schematic models
that differ from traditional algorithms because they keep important aspects of context
visible. These models allow for recording intermediate steps and are easily adapted to
each student's level. They also suggest the use of shortcuts and multiple strategies for
problem solving. By using these three models simultaneously, we can examine the
advantages of each for different types of problems and explore mathematical
relationships within them. In addition, the combined table and notebook notation are
notable tools for algebraization proposed by EMR.
These tools give students the ability to understand traditional methods, such as
understanding the components of a system of equations, identifying what they are
looking for, recognizing equivalent equations, understanding why certain systems may
have one, multiple, or no solutions, and determining the most appropriate method for
finding these solutions. In general, the transition to working with pure systems is not a
challenge for these students, and in case they encounter difficulties, they can use these
models to recall the typical situations that led to their creation. This allows them to
redefine the operations they perform at a formal algebraic level.
The interaction
In RME, reflecting and mathematizing are closely related. According to
Freudenthal, students need to be able to reflect on their own activity in order to reach the
highest level of understanding. In guided reinvention processes, interaction between
teacher and students is crucial to promote reflection and the exchange of ideas. The
classroom should provide a space for individual, group, and collective action and
reflection, where students not only answer questions and solve problems, but also
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formulate their own mathematical questions, share and evaluate ideas and solution
methods, and symbolize and generalize mathematical relationships.
Typically, sharing in a mathematics class occurs after the problem has been solved,
but under the guidance of a trained teacher, sharing can take the form of “thinking
together out loud” in the present tense and subjunctive and conditional moods, allowing
for the sharing of ideas in the process of development. The question arises as to how these
types of conversations can help students fully understand and engage in the task of
mathematizing.
In the field of EMR, there is a significant emphasis on problem formulation and
solution. However, the focus is not solely on teaching students how to solve specific
problems, but rather on cultivating their ability and inclination to apply mathematical
concepts and methods in a variety of contexts (such as arithmetic, geometry, algebra, and
formalization). To achieve this goal, teachers should present open-ended questions that
are within the reach or imaginable of their students, and they should strive to understand
students' thoughts and reasoning processes.
The teacher must value and take a genuine interest in student input, encouraging
interactive situations in both whole-class and small-group settings. It is crucial that the
teacher builds on student ideas, guiding them through reflective processes that promote
higher levels of mathematical thinking and understanding for each individual student
and for the class as a whole. This requires a teacher who can anticipate key developmental
milestones along the path of progressive mathematization.
If the main activity of students is to engage in mathematics, then what is the main
activity of teachers and professors? According to Freudenthal, their main activity is to
organize and structure the teaching process, which has both a horizontal and a vertical
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aspect. The process of teaching mathematics is parallel to the process of mathematizing.
It involves becoming aware of the didactic reality and creating a framework for teaching,
on the one hand, and developing a deeper understanding and generalizing from teaching
situations, on the other. Horizontally, teachers focus on the teaching and learning
phenomena that occur in their classrooms and in other classrooms. Vertically, they reflect
on these situations and use them to enhance their own teaching strategies and techniques
to support the mathematization process.
The theoretical bases: EMR
Realist Mathematics Education, as an international movement, was founded by
Hans Freudenthal, a German mathematician and educator. The movement emerged in
the 1960s as a response to the mechanistic approach to teaching arithmetic and the use of
"modern" or "conjunctitarian" mathematics in classrooms. Today, many of Freudenthal's
original ideas are adopted and discussed in current educational theories and have
influenced the curricula of a number of countries, including the United States, Japan,
Indonesia, Great Britain, Germany, Denmark, Spain, Portugal, South Africa, Brazil, and
Puerto Rico.
A fundamental principle of EMR is that mathematics education must be grounded
in reality, relevant to students, and meaningful to society in order for it to be valuable to
human development. According to Freudenthal, the perception of mathematics is
intertwined with our perception of the world, the role of mathematicians is linked to our
understanding of humanity, and the teaching of mathematics is connected to society as a
whole. In his view, a critical consideration during his time was whether mathematics
should be seen as a subject for a select minority or as a subject for all individuals. He
believed that it is crucial for all students to have some level of engagement with
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mathematical work, which he defined as the act of organizing reality using mathematical
concepts and tools, including mathematics itself.
Mathematizing is a step-by-step process that involves various actions such as:
• Identify important features in different situations, problems, algorithms, formulas,
symbols and systems based on axioms.
• Find common points, similarities, analogies and isomorphisms between these
elements.
• It also involves providing concrete examples to illustrate general ideas and
concepts.
• It requires approaching difficult situations in a systematic and exemplary manner.
In addition, mathematization involves the sudden appearance of new objects and
mental operations that help in problem solving.
• It involves searching for efficient strategies and finding ways to simplify initial
approaches and symbolizations to create formal diagrams, algorithms, symbols
and systems.
• Finally, mathematizing involves reflecting on the entire mathematical process,
considering the various phenomena involved from multiple perspectives.
Practical scenarios and challenging circumstances
A context refers to a specific domain of reality that is revealed to students during
the learning process in order to be mathematized. Mathematics evolved as a means of
mathematizing real-life situations in the natural and social environment, and therefore
its teaching should also be based on the organization of such situations. However, this
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does not imply focusing solely on perceptual phenomena, as this would restrict students'
opportunities to gain experience and engage with mathematics itself.
The goal is for students, who may initially lack sufficient mathematical skills, to
reinvent these tools by tackling problems presented in realistic contexts and situations. A
context can take the form of an event, proposition, or situation derived from reality that
is meaningful to students or can be imagined, prompting the use of mathematical
methods based on their own experiences. It provides concrete meaning and support for
relationships and operations that are relevant to mathematics.
These situations can be drawn from everyday experiences, such as bus routes or
shopping and money management. In addition to contexts derived from daily life,
mathematics itself offers contexts within the realm of problems involving pure numbers
and numerical relations, such as the context of prime numbers. There are several types of
contexts, including real, artificial (fantasy), mathematical, and virtual, each of which
originates in reality but incorporates non-real elements for purposes of simplification or
simulation.
In the field of mathematics education, it is crucial to recognize the important role
that realistic contexts play in the student learning process:
• Realistic contexts serve as a foundation for teaching and learning, allowing
students to develop mathematical concepts and apply them across a variety of
domains.
• When carefully selected, these contexts capture students' interest, fostering
engagement and motivation.
• Realistic contexts serve as tangible objects of study, facilitating the accessibility of
mathematical content for students at different levels of understanding.
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• By incorporating real-world scenarios, students are encouraged to use their
common sense and leverage their informal knowledge to build mathematical
models.
• The openness of these contexts, allowing for multiple strategies and solutions,
fosters meaningful mathematical discussions among students.
• These realistic contexts are explored in a comprehensive and in-depth manner,
ensuring a deep understanding of the mathematical concepts involved.
While it is important to consider the relative nature of the realistic context to avoid
over-generalizations and over-simplifications, the realism of a context depends on
students' prior experiences and their ability to imagine or visualize it; it is beneficial to
use the models that emerge from students' own mathematical activities as tools to
represent and organize these contexts and situations.
These models serve as intermediaries through which complex realities or theories
are idealized or simplified for formal mathematical treatment. It is crucial to note that in
the context of EMR, the term “model” does not refer to pre-existing models imposed from
formal mathematics, but to emergent models that develop during the teaching-learning
process. And they are formed through the organization and reorganization of activities
that arise from problematic situations. Initially, these models are intricately linked to the
specific contexts and situations from which they arise, but over time they become
decoupled and take on characteristics of formal and general models. As a result, they can
be applied to diverse contexts and situations, moving from being a “model of” a
particular situation to a “model for” mathematical reasoning in both mathematical and
non-mathematical settings.
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Models in the field of EMR serve not only as representations, but also as tools for
analysis and reflection. They are used to perform various actions and operations, as well
as to visualize, explain, compare, contrast, and verify relationships. To serve these
purposes, these models must meet a number of crucial criteria:
• First, they must be based on realistic and imaginable contexts.
• They must have enough flexibility to be applicable to more advanced or general
levels.
• Unlike traditional teaching methods, where models are fixed, these models are
subject to change over time. This dynamic nature allows for progression in the
mathematization process, while also allowing students the ability to revisit the
original situations from which the strategies were derived. This ability to move
between levels is what makes these models particularly powerful.
• Finally, these models must be viable, that is, they must behave in a natural and
obvious way. They must align with students' informal strategies, as if the students
themselves could have discovered them independently, and they must also be
easily adaptable to different situations.
It is important to note that informal solutions and independent creations by
students play a central role in the teaching and learning process. By working on problems
that can be solved in multiple ways, students' levels of understanding and computational
skills at a given time can be revealed. This information is crucial not only for making
small-scale teaching decisions but also for guiding larger-scale educational decisions.
A snapshot of the classroom, with its various levels of understanding, provides
insight into the trajectory of learning and teaching. The solution strategies employed by
individual students collectively expose essential elements of the long-term path that
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students will undertake. Thus, what is observed in the classroom today anticipates what
is to come and beyond. Instructional phenomenology involves initially studying the
different ways in which a mathematical concept, such as fractions, ratios, functions,
proportions, and angles, manifests itself in real life. This includes considering how these
concepts are commonly referred to in everyday language. From this understanding, the
didactics of the topic can be built.
The EMR has experimented with several classroom models that are easily
presented through contextual situations and can be recreated by students. These models
include manipulative teaching materials such as tokens, money, and necklaces with two-
colored balls arranged in groups of ten. In addition, paradigmatic situations such as the
bus, the pancake restaurant, the parent meeting, the 10-unit candy factory, and the
location of a fire have been used.
Diagrams such as the circle model, the double bar or percentage model, and the
ratio table have also been used, as well as diagrams such as the tree and path diagrams.
Notational forms such as arrow language, notebook notation, and the combination table
have also been used to solve systems of equations with two unknowns, and symbolically
expressed procedures such as algorithms or column formulas. The exploration of
contexts and models that naturally lead to the use of mathematics is known as didactic
phenomenology, a concept coined by Freudenthal. This approach is strongly influenced
by the history of mathematics and the ideas and creations of students that arise during
the teaching process.
The role of the teacher
In the context of EMR, mathematics instruction should involve a guided
reinvention approach. Students should have opportunities to independently discover
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mathematical concepts and skills by organizing and structuring real-life problems.
During this process, students interact with their peers and receive guidance from the
teacher. Effective learning in this approach involves explicit negotiation, intervention,
discussion, cooperation, and assessment.
Informal methods serve as a foundation for students to eventually understand
formal mathematical concepts. This interactive teaching method requires students to
explain, justify, agree or disagree, question alternatives, and reflect on their thinking. The
teacher plays a crucial role as a mediator, facilitating communication between students
and the problems they encounter, as well as facilitating communication between students
themselves. Furthermore, the teacher bridges the gap between students' informal
problem-solving approaches and the established formal tools of mathematics.
The act of learning mathematics is considered a social process in which individuals
come together to reflect collectively, resulting in deeper understanding. Both vertical
interactions between teachers and students, and horizontal interactions between
students, play a crucial role in this process. How the teacher manages these interactions
is key to maximizing opportunities for students to generate, exchange, and understand
ideas.
It is important to note that a class is not considered a homogeneous entity, but
rather a group of individuals who follow their own unique learning paths. However, this
does not mean that the class is divided into groups with similar processes. Instead, the
class remains together as an organizational unit or engages in cooperative work in diverse
groups, as Freudenthal advocates.
By selecting problems that suit different levels of understanding, all students can
work on them. Additionally, there is a strong emphasis on integrating the various strands
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or units of the mathematics curriculum. Solving real-life problems often requires making
connections and using a wide range of mathematical concepts and tools.
The EMR curriculum avoids strict distinctions between curricular strands, creating
a more cohesive approach to teaching and allowing for different methods of
mathematizing situations using various models and languages. This ensures a high level
of coherence across the curriculum, rather than teaching each strand in isolation and
ignoring the connections between them. In practical applications, problem solving
typically requires more than just knowledge of arithmetic, algebra, or geometry.
Phenomenology
Freudenthal's perspective on mathematical objects differs from traditional
mathematical philosophies such as realism and Platonism. These philosophies believe
that mathematical concepts exist independently of human activity and are discovered
through mathematical exploration. Freudenthal, however, maintains that mathematical
concepts are created and constructed through mathematical practice. He suggests that
mathematical objects are not just tools for organization, but real objects with their own
properties and actions.
As these mathematical objects are brought into the world, the world itself expands
and grows. Mathematical concepts and ideas are used to organize phenomena of both
the real world and mathematics. On the other hand, mental objects are creations of
individuals from their experiences and serve as a means to organize and understand their
own experiences. Freudenthal also recognizes the challenge of teaching mathematical
concepts as they require instilling the corresponding concepts in the minds of students.
This analysis aims to explore the scope and method proposed by Freudenthal in his
theory of Didactic Phenomenology of Mathematical Structures for teaching and learning
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mathematical concepts. It involves examining the relevant literature and understanding
the key ideas put forward by Freudenthal.
The term phenomenology, as used here, does not refer to the interpretations given
by philosophers such as Husserl, Hegel, or Heidegger. Rather, it belongs to the Greek
origins of the word, where "phainomeno" means "that which appears." In this context,
phenomena are appearances or how things appear to us. In the realist philosophical
tradition, the world of noumena is considered the real world.
The contrast between phenomenon and noumenon represents a contrast between
two worlds: the world of appearance and experience (phenomenon) and the world of the
sensible and intelligible (noumenon). Some philosophers hold that mathematical
concepts are noumena, which places them outside the realm of our experience. However,
this contradicts the ideas of Freudenthal, who sees mathematical concepts as a means of
organizing phenomena. According to this view, mathematical concepts are part of the
field of phenomena that are organized by new mathematical ideas.
Mathematical concepts are therefore not separate from our experiences or in a
separate world from the phenomena they organize. They are in fact objects of our
mathematical experience. To engage in phenomenology, one must describe the
relationship between these series or pairs: the phenomenon and the means of
organization. The process of creating mathematical objects involves the means of
organization becoming objects that appear in the field of phenomena.
Thus, mathematical objects become incorporated into our experiences and become
part of a new relationship between phenomenon and means of organization. This
iterative process continues and leads to the creation of new mathematical concepts and
the generation of increasingly abstract mathematical objects. The phenomenology of a
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mathematical concept, structure, or idea involves describing the noumenon (the concept
itself) in relation to the phenomena it organizes. This includes identifying the phenomena
for whose organization and extension the concept was created, understanding how it acts
as a means of organization for these phenomena, and recognizing the power it gives us
over these phenomena.
Concepts, known as noumena, are intricately connected to the phainomenon.
When we look at the didactic element in this relationship, specifically how the concept R
phenomena are acquired in the teaching and learning process, we enter the realm of
didactic phenomenology. This field explores the phenomena that exist in the world of
students and those that occur in teaching sequences, particularly in the context of
mathematics.
By examining the fRc relationship in terms of students’ cognitive growth, we
engage in genetic phenomenology. Here, the focus is on how phenomena are perceived
and understood in relation to students’ cognitive development. Furthermore, if we
explore the historical acquisition of this fRc relationship, we enter the realm of historical
phenomenology. In this case, we investigate the phenomena for which the concept was
originally created and how it was subsequently expanded to encompass other
phenomena.
The suggested order for studying these phenomenologies begins with pure
phenomenology, gaining knowledge of mathematics and its practical applications. This
is followed by historical phenomenology, which provides information on the formation
of these relationships throughout history. Next, we delve into didactic phenomenology,
understanding the teaching and learning process, and finally, genetic phenomenology
examines the cognitive growth of students. It is important to note that the description of
the relationships between the phenomenon and the concept considers both the
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relationships established in the first case and how these relationships were developed,
acquired or formed in the educational system, cognitively or historically in the other three
cases.
Phenomenology, which can be defined as a method of analyzing mathematical
content, involves the phenomenological analysis of mathematical concepts or objects.
This analysis is carried out with a didactic intention, that is, it is carried out prior to any
curricular design or development and is considered a component of didactic analysis.
The purpose of phenomenological analysis is to serve as a basis for organizing the
teaching of mathematics, rather than attempting to provide an explanation of the nature
of mathematics.
One of the main tasks of phenomenology is to investigate phenomena that
organize mathematical concepts by analyzing them. These phenomena are assumed not
to have existed before. In contrast to the typical approach to teaching mathematics,
Didactic Phenomenology proposes a different approach. It suggests starting with
phenomena that require organization by a concept and then teaching students how to
manipulate these means of organization.
Instructional phenomenology should be used to develop plans with this type of
approach. For example, when teaching about Groups, instead of starting with the concept
of Group and trying to materialize it, the focus would be on examining the phenomena
that could lead the student to form the mental object that is being mathematized by the
concept of group. If the necessary phenomena are not available at a certain age, attempts
to inspire the concept are abandoned.
In the school system, concepts are often introduced to students before they have
any experience with the phenomena involved. The educational system aims to help
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students form mental objects as a means of organizing these phenomena, and it also
provides access to the means of organization that history has provided, which are
concepts.
In the context of history, mathematical concepts do not exist prior to our
experience with them. It is the activity of mathematicians that creates these concepts.
Throughout history, mathematical concepts have emerged as consolidations of mental
objects. Mathematical activity generates concepts from mental objects. The relationship
between concepts and mental objects is complex, as both serve as means of organizing
phenomena. Mental objects preexist concepts, and concepts do not replace mental objects,
but rather allow the formation of new mental objects that contain or are compatible with
them. The distance between the initial mental object and the concept can be significant.
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Chapter 3
The contextualization of realistic mathematics in education
The demand to establish a connection between mathematics taught in educational
institutions and the lives of students is a call from society, coming from both the academic
and professional worlds. This demand is not isolated; it is part of a broader request to the
school system itself, where society as a whole expects that what is taught in our schools
will enable students to function effectively in their lives. There have been swift responses
to these demands.
At the international level, the Organization for Economic Co-operation and
Development (OECD) has highlighted, through the PISA study, the importance of
developing mathematical skills that enable individuals to recognize and understand the
role of mathematics in the world, to reason well-foundedly and to use mathematics
according to their vital needs as constructive, engaged and reflective citizens. More
recently, the same study emphasises that the development of a mathematical culture in
schools should help individuals to identify and understand the role of mathematics in
the world, providing them with the necessary judgement to make decisions based on
becoming constructive, engaged and reflective citizens.
The National Council of Teachers of Mathematics (NCTM) also emphasizes the
importance of connecting the mathematics taught with the current and future lives of
students. In the Latin American context, this social demand is also evident. The curricular
guidelines established by the Colombian Ministry of Education state that the main
objective of mathematics education is to help people make sense of the world around
them and understand the meanings constructed by others.
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By learning mathematics, students not only develop their capacity for logical
thinking and reflection, but they also acquire powerful tools to explore, represent, explain
and predict reality, allowing them to act in and for it (Ministry of National Education,
1998). Venezuela's proposal for teaching all subjects in the curriculum also emphasizes
the importance of linking education to the lives of students, making it relevant both
individually and socially. In addition to these demands from society, there is also a
demand from the academic community. Within the field of mathematics education, the
need to establish this connection is defended from different epistemic perspectives, such
as Critical Mathematics Education and Realistic Mathematics Education. These demands
are justified by the desire for mathematics education to contribute to the formation of
conscious and participatory citizens, promoting inclusive mathematics and avoiding the
exclusive nature of traditional approaches.
When focusing on educational institutions, it is increasingly common for people,
particularly teachers, to emphasise that “mathematics is everywhere”. This is a response
to the growing societal demand for school mathematics to be relevant and applicable to
students’ present and future lives. However, while there is widespread agreement on the
importance of connecting mathematics to students’ lives, it has proven difficult to meet
this expectation. It is common and well-established that when both pre-service and in-
service teachers are asked to provide specific examples, they often struggle to do so. Their
answers often revolve around basic concepts related to buying and selling, but there are
many other phenomena and life situations in which mathematics plays a role.
In other words, mathematics must be meaningful to the students who learn it by
relating it to their needs and interests in their own life experiences. To achieve this,
teachers must create opportunities for ongoing communication with their students, and
the discourse they employ plays a crucial role in this process. When we refer to teacher
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discourse, we are considering it in a pragmatic sense, considering the contextual
relationships that guide the communicative interaction between the teacher and his or
her students. What we are advocating is the need to endow mathematics education with
meaning that resonates with the students who learn it.
To ensure that mathematics is meaningful to students, it is essential to connect it
to their everyday lives. This connection must be both personal and social. Therefore, we
propose to prioritize the contextualization of mathematics teaching. This idea is not new,
as several researchers and educators have emphasized the need for this approach. They
argue that mathematics should be seen as a human activity and should be taught in
relation to students’ reality. Instead of seeing mathematics as a deductive system, these
scholars suggest that students should engage with mathematics through real-life
experiences that help them see it as a tool to organize and understand their present and
future realities.
This perspective remains relevant in current debates about curriculum design, as
well as in the views of organizations such as the NCTM and the OECD. From an
individual point of view, each person learns best when knowledge is meaningful to his
or her own life. From a societal perspective, mathematics education should have practical
applications that enable individuals to integrate into society.
The information collected provides evidence of educational practices in
mathematics that incorporate real-life situations into the teaching processes. However,
these practices can be seen as deviations that should be avoided in our classrooms. We
will discuss three specific deviations: teachers' understanding of reality, the types of
relationships that are established between mathematical concepts and real-life situations,
and the depth of the study of mathematical objects in the classroom.
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It is important to reconsider certain educational practices in mathematics that
incorporate real-life situations. Teachers need to have a broader understanding of reality,
not just limiting themselves to everyday situations, and they should prioritize the
integration of mathematics with students' life experiences from the beginning. By
adopting a more comprehensive approach that combines theory and practice,
mathematics can become more meaningful and relevant to students, regardless of their
socioeconomic background.
The first deviation concerns teachers’ understanding of reality and how it
influences the integration of mathematics with real-life situations. In interviews with
mathematics teachers, both pre-service and practicum, they were asked to provide
examples of real-life situations that could be linked to mathematics. Without exception,
these teachers gave examples that were perceptible to the senses and closely related to
the students’ everyday lives. This finding is consistent with other studies that have also
observed this tendency among mathematics teachers. However, limiting the context to
only everyday situations have ethical and pedagogical implications. It primarily
disadvantages students who have limited life experiences due to their socioeconomic
disadvantage.
The context can be geographically and chronologically close to the student, but it
should not be limited exclusively to his or her immediate environment. For example,
studying Mayan culture and its contributions to mathematics may not be an everyday
context for students, but it can still be meaningful and interesting to them if they are
properly motivated. Exploring mathematics in different cultural contexts can broaden
students' horizons, regardless of their socioeconomic background.
The second deviation relates to the types of relationships that are established
between mathematical concepts and real-life situations. Many teachers tend to present
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mathematical theory first and then try to show its application in specific contexts. This
deductive approach to learning assumes that students can only understand and learn
theory if it is disconnected from their life experiences. However, presenting mathematics
in a theoretical way without relating it to students’ real-life experiences can make it less
interesting for them and difficult to understand the subject. We believe in a teaching
approach that combines theory and practice, where mathematics is integrated into
students’ lives based on their previous experiences. This approach enables students to
see the relevance and applicability of mathematics in their daily lives and broadens their
worldview.
Furthermore, we want to highlight another aspect that deviates from our
expectations. This concerns the degree to which mathematical concepts are explored in
the classroom, specifically through the incorporation of real-life situations that are
relevant to students. Unfortunately, what we often witness is an oversimplification of
mathematical content. When teachers try to connect mathematics to students’ everyday
lives, they often do so in a superficial and inconsistent manner. For example, we have
encountered cases where teachers try to contextualize weight measurements by asking
students to use them only when following a recipe for a class project. It is clear that there
is no in-depth exploration of the mathematical concepts involved, nor any expansion of
the practical applications of weight measurements.
Guiding principles
The keys to contextualizing mathematics are based on principles laid out in what
is known as realist mathematics education. This approach, developed by Freudenthal,
Gravemeijer, Puig, and Goffre, emphasizes viewing mathematics as a human activity.
According to this perspective, teaching mathematics involves creating a connection
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between mathematical concepts and the student's real-world experiences. The goal is for
students to view mathematics as a tool for organizing, understanding, and transforming
the world around them.
From an epistemological point of view, this means moving from the current
approach of presenting mathematics as a discipline with a fixed and unattainable
deductive system, to a vision of mathematics as a continuous construction process. In this
new approach, the interaction of students with their environment becomes a process of
reinvention guided by the teacher. This type of contextualization extends beyond the
human body to other phenomena or contexts (f1, f2, f3,...). For example, architecture and
the visual arts provide ample examples of symmetry and rotation that can be studied
using mathematical concepts. By incorporating these mathematical contents into the
study of different contexts, the horizon of mathematical understanding is broadened.
This approach to mathematics education involves two simultaneous processes:
horizontal mathematization and vertical mathematization. Horizontal mathematization
is relating a set of non-mathematical situations to mathematical concepts. For example,
when we are presented with the human body, we may not immediately see the
mathematics that can be derived from it. However, mathematics can help us better
understand the human body and vice versa. It is the teacher's role to discover and
highlight the mathematical elements inherent to the human body. In this way, both
mathematics and the human body are perceived in a new light, the so-called Didactic
Phenomenology. In the case of the example of the human body, mathematics allows us
to explore concepts such as symmetry, rotation, proportions and the golden ratio, using
both conventional and non-conventional length measurement systems.
At the same time, while we focus on the process of horizontal mathematization in
the classroom, it is important to address vertical mathematization as well. Vertical
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mathematization involves broadening the mathematical processes that are derived from
the examination of various phenomena. In other words, it is not only about broadening
the scope of different phenomena or contexts in relation to horizontal mathematization
processes; but it also involves going deeper into the study of mathematical objects. This
concept is known as vertical mathematization, as put forward by Treffers (1987). To better
illustrate this, let us consider the example of the study of the human body. By exploring
the human body, we can explore the interconnection of different mathematical objects
and gain a deeper understanding of them. For example, by examining measurements and
proportions, we can go deeper into the generalization of unit conversion and explore
Thales’ theorem.
The keys
The task of bringing the phrase “mathematics is everywhere” to life in the
classroom is not easy, as they point out. Typically, there is a lack of diversity in the
phenomena or contexts through which we teach mathematics, often relying on examples
related to basic arithmetic and geometry. This situation can be attributed to differences
in social practices and codes within the school system compared to those outside it.
The key, therefore, is to find ways to incorporate into our educational practices the
social practices and communicative codes of mathematics that exist beyond the school
environment. To achieve this incorporation of the mathematical world into our
classrooms, there are various areas and people that can be approached. In this discussion,
we will focus specifically on the role of the teacher.
An important aspect of teaching mathematics in a contextualized manner is for the
mathematics educator to have a deep understanding of the mathematical concept itself,
including its foundations, history, and real-world applications. By knowing the origins
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of a mathematical topic, educators can understand the issues that led to its development
and find similar situations that can be adapted for learning purposes. For example, the
discovery of the number pi by ancient civilizations such as the Greeks and Egyptians,
who used it in measurement problems involving circles, can be replicated today if
students manipulate circular objects to arrive at the same conclusions. In addition to
historical context, it is also crucial to understand modern applications of mathematical
concepts.
Educators should ask questions about the usefulness and relevance of the topic in
different contexts and the problems it can solve. It is also important to recognize the
connection between the problems being studied and the conceptual structure of the
mathematical concept being taught. This involves identifying a set of situations that share
a common underlying theme, which may be natural, social, or cultural. For example,
when teaching about pi, educators can present relatable situations to students, such as
designing flowerpots at school or solving industrial design problems that require
choosing between a cylindrical container or a parallelepiped based on their respective
capacities. In such cases, the value of pi plays a fundamental role in making the decision.
Another important aspect of contextualization involves the ability to search for
information and analyze it from a classroom perspective. To understand the origins and
applications of certain mathematical topics, it is necessary to explore different sources of
information. One valuable source is information technologies, which provide access to a
wealth of information through web 2.0 platforms. This wealth of information removes
the excuse of not having knowledge about a topic.
Teachers must actively seek out these sources and, more importantly, learn to
discern which ones are dependable and relevant to the specific situation at hand. This
emphasizes the importance of being able to analyze the information provided to us.
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Students can also play a role in this search for information, as they can be guided to
become allies in this research. However, there are other sources that are often overlooked
and neglected in the current educational landscape, such as bibliographic sources and
key informants.
Key informants are people who possess specialized knowledge and insights but
are rarely considered by schools. They could be instrumental in incorporating
mathematical knowledge used outside the school context. For example, if we want to
teach students the concept and calculation of area, we can find numerous resources on
web 2.0 platforms and in school textbooks that provide formal knowledge on the topic.
However, we have a limited understanding of how bricklayers, engineers, and architects
use this mathematical concept in their professions. Their approaches may not always
align with what is taught in school, but their methods are validated in their daily practice.
So, why not invite them to share their knowledge and experiences with students? The
goal is not to replace institutional knowledge with mathematical knowledge derived
from social practices, but rather to complement both worlds and achieve a better
understanding of their potentialities and limitations. By doing so, we can bridge the gap
between mathematics taught in school and mathematics used outside of school.
To utterly understand students and connect with them on a deeper level, engaging
in meaningful conversations and establishing emotional connections is essential. This
involves not only seeking insights from experts and established knowledge, but also
actively communicating with our students to learn about their interests, needs, and
preferred methods of communication. By fostering this teacher-student connection, we
can identify which contexts are most conducive to integrating mathematics into our
students’ lives. However, this requires the teacher to possess strong communication skills
and a willingness to engage in dialogue.
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Creating an environment that supports dialogue is crucial to its development. This
means going beyond the traditional classroom dynamic and allowing for the presentation
and discussion of mathematical ideas. Classrooms should be transformed into forums
where everyone can freely express their opinions and share their findings on
mathematical learning situations. However, the role of the teacher extends beyond the
confines of the classroom. It is important for them to actively participate in other spaces
within the school or even outside of it, where they can engage in honest and ongoing
conversations with their students, the educational community, and society as a whole. In
this way, we can gain a comprehensive understanding of the people we teach, including
their cognitive abilities and the environment in which they operate, such as their social
background, family dynamics, and community influences.
Teaching mathematics in a way that relates to students’ everyday lives involves
integrating its concepts into real-life situations. This means that the content taught in our
classrooms must have practical meaning for students. From our perspective, we believe
that this approach to teaching mathematics can help shape students into citizens who
understand and transform the world in which they live, all within a framework of respect
and freedom.
We strongly believe that contextualization is valid and relevant in today’s
educational system, but it is crucial to note that this contextualization should not be
arbitrary. In order to effectively contextualize mathematics, there are three key factors to
consider. First, the teacher must have a deep understanding of mathematical concepts,
their origins, and their applications. Second, the teacher must be aware of the interests,
needs, and context in which his or her students typically operate. Finally, the teacher
must possess the ability to seek and analyze information in order to expand his or her
own knowledge of mathematics, including its foundations and applications. By doing so,
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the teacher can create learning situations in which mathematics becomes a tool to explain
the realities of students’ lives, both present and future. School mathematics should go
beyond the traditional image of a set of deductively defined theories and rules that
remain unchanged over time. Rather, it should be seen as a constantly developing subject,
where the educator’s guidance is crucial.
Teachers must constantly reinvent their lessons in collaboration with students,
exploring diverse contexts that contribute to a deeper understanding of the mathematical
concepts being studied. The goal is for students to learn mathematics by actively
participating in mathematical activities, and for this to happen, the content being taught
must be meaningful and relevant.
All of these ideas presented here raise important questions about the training and
professional development of mathematics teachers. These questions challenge us to
consider what mathematics future teachers should learn, as well as what theoretical and
methodological tools they should acquire to recognize mathematics in different contexts
and design learning situations accordingly. These questions, among others, could be the
focus of future research. As teacher educators, it is our responsibility to address these
challenges and contribute to rectifying the long-standing gap between teachers and
students, which has been perpetuated by teaching mathematics in isolation from real-
world contexts.
The didactic perspectives of mathematics
It should be noted that there is a distinction between education and didactics.
Education covers a broader scope than didactics, allowing us to differentiate between
Mathematics Education and Mathematics Didactics. By adopting this approach,
mathematics education is defined as "the whole system of knowledge, institutions,
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training plans and training purposes" that constitute a complex and diverse social activity
related to the teaching and learning of mathematics.
Mathematics Education refers to the discipline that studies and investigates the
challenges that arise in mathematics education and proposes well-founded actions for its
transformation. However, in the English-speaking world, the term “Mathematics
Education” is used to refer to the field of knowledge known as Mathematics Education
in countries such as France, Germany and Spain. Mathematics Education is also identified
as a scientific discipline and an interactive social system that encompasses theory,
development and practice.
In his diagram, Steiner (1990) represents Mathematics Education (ME) as a
discipline that is connected, as part of it, to another complex social system called the
Mathematics Teaching System (MTS) - which Steiner refers to as "Education, Mathematics
and Teaching", which is represented as the thicker circle outside of MTS. Within this
system exist several subsystems, including the mathematics classroom (MC) itself,
teacher education (TE), curriculum development (CD) and Mathematics Education (ME)
as an institution that is part of MTS. Steiner further extends the diagram by including the
whole social system concerned with the communication of mathematics, which
encompasses new areas of interest for Mathematics Education, such as the issue of "new
learning in society" (NS) brought about by the use of computers as a medium for teaching
mathematical ideas and skills outside the school context.
It also includes the study of the interrelations between Mathematics Education and
Experimental Science Education (ECE) within this field. Steiner considers theorizing
activity (TEM) as a component of mathematics education and, therefore, of the broader
system we refer to as SEM, which constitutes the system of teaching mathematics. TEM
is positioned externally to consider and analyze the integral global system as a whole.
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Another model proposing the relationships between Mathematics Education and
other disciplines is that presented by Higginson (1980), who identifies mathematics,
psychology, sociology and philosophy as the four fundamental disciplines. Higginson
visualizes Mathematics Education in terms of the interactions between these four
disciplines, represented by the faces of a tetrahedron.
These different dimensions of mathematics education encompass the fundamental
questions that arise in our field, such as what to teach (mathematics), why (philosophy),
to whom and where (sociology), and when and how (psychology). In Higginson's work,
she also explores the applications of this model to clarify essential issues such as
understanding traditional perspectives on mathematics teaching and learning,
understanding the factors that have led to curricular changes in the past and predicting
future changes, and examining the evolution of conceptions about research and teacher
education.
The study of epistemological currents reveals that scientific theories cannot exist
in isolation or be the product of individual efforts. Instead, there must be a community
of researchers who share common interests and agree on appropriate methods for
addressing research problems. It is important to strike a balance between personal
autonomy in developing new ideas and the need for these ideas to be shared and tested
within a community. Theories are therefore the result of collaborative research efforts
within a specific field.
For a field of research to be considered “normal science” according to Kuhn’s
criteria, certain conditions must be met:
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• First, there must be a group of researchers who share common interests and focus
on studying the interrelationships between different aspects of a complex real-
world phenomenon.
• Second, the explanations provided by the theory must be causal and allow
predictions to be made about the phenomenon.
• Finally, the group of researchers must agree on a common vocabulary, syntax, and
procedures for testing and evaluating the theory.
Scientific concepts, propositions and theories are distinguished from non-scientific
constructs by their adherence to the scientific method and logical reasoning, as well as by
their acceptance by the scientific community. However, the requirement of a single
paradigm or a unified community of specialists, as defined by Kuhn, may be too
restrictive. In the social and human sciences, including Mathematics Education, it is
natural and beneficial to have competing schools of thought, as they encourage the
development of diverse research strategies and the exploration of problems from
different perspectives.
The complexity of the phenomena studied may require the coexistence of multiple
research programs, each supported by different paradigms taken from various
disciplines. Bunge's (1985) epistemological approach, which considers scientific fields as
sets of competing lines of research, seems more appropriate to understand the current
state of Mathematics Education.
Some authors have categorized certain didactics as mere technical knowledge or,
at most, technological knowledge, rather than recognizing them as educational sciences
in their own right. However, when considering the relationship between general theory
and specific theory, as Bunge explains, it becomes clear that special didactics are not
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simply subfields or chapters within general didactics or educational psychology. Rather,
they represent specific theories that cover particular aspects of the broader field.
Each specific theory includes the general theory and subsidiary hypotheses that
describe the unique characteristics of the objects being studied. While it is commonly
assumed that the general theory includes all specific theories, Bunge argues that it is
actually the other way around. The general theory can be derived from specific theories
by eliminating the specific premises and focusing only on the assumptions common to
all theories.
This distinction is important when considering the phenomena of learning and
teaching. It is necessary to ask: learning from what? Teaching from what? The nature of
the knowledge being taught, as well as psychopedagogical, social, and cultural factors,
play a role in explaining and predicting learning and teaching phenomena. Therefore, the
practice of Mathematics Education, including programming, curriculum development,
and instructional strategies, must consider the specificity of the knowledge being taught.
The limitations of existing general educational theories result in the development
of new theories that are better suited to explaining and predicting the phenomena they
seek to understand. Indeed, these new theories may even introduce bold and innovative
ideas that challenge the very foundations of established theories. The narrow framework
of traditional teaching techniques, including the use of technology, is insufficient for
theories that are built within certain branches of Mathematics Education research.
Mathematicians, in contemplating the processes of creating and transmitting
mathematical concepts, are forced to take on the role of epistemologists, psychologists,
sociologists, and educators; in other words, they must also become didactic practitioners.
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After carefully considering the criteria set by various authors to define a scientific
discipline, we are left to ask whether the field of Mathematics Education meets these
requirements. Specifically, we question whether there is a community of researchers
within this field who are actively engaged in developing one or multiple research
programs that can generate a comprehensive theory or theories of Mathematics
Education. The following section aims to provide an overview of the current state of
research in this area, with a particular focus on the contributions made by prominent
research groups such as the Theory of Mathematics Education (TME) and Psychology of
Mathematics Education (PME) groups. In addition, we will describe several key
perspectives and approaches in the field, including problem solving and modeling,
sociocultural frameworks, the French school of mathematics education, symbolic
interactionism, the sociocritical point of view, and H. Freudenthal's didactic
phenomenology.
Mathematics Education: Theory and Philosophy
During the 1990s, mathematics education research in the United States lacked a
solid theoretical foundation and did not focus on building theoretical models. However,
in the last two decades, there has been a significant change in this trend. Today, when
publishing articles in peer-reviewed journals, it is mandatory to provide a clear reference
to the theoretical framework that underpins the studies. This change is evident in the
increasing number of publications that discuss and analyze various theoretical and
philosophical approaches to mathematics education.
In 1984, Professor Steiner intended to form a research group at the 5th
International Congress on Mathematics Education (ICME) that would focus on the
development of a Theory of Mathematics Education. This led to the creation of a Thematic
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Area called "Theory of Mathematics Education" at the Congress, which had four
dedicated sessions. After the Congress, discussions continued at subsequent meetings
and a Working Group called TME (Theory of Mathematics Education) was formed.
The TME conferences that have been held since then have shown that there is a
community interested in building the theoretical foundations of Mathematics Education
as a science. This community is made up of researchers in Mathematics Education,
mathematicians, teachers, educational psychologists, educational sociologists and
teacher trainers, among others. Steiner (1985) proposes that Mathematics Education
should serve as a link between mathematics and society. This can be achieved by
exploring forgotten dimensions of mathematics, such as the philosophical, historical,
human, social and didactic dimensions.
By analysing the issues raised within the TME Group, which has attracted a
majority of researchers interested in the theoretical foundations of Mathematics
Education, we can begin to understand the central concepts of Mathematics Didactics as
a scientific discipline. The formation of this scientific community is driven by professional
interests and has fostered an academic orientation in its work. For example, in Germany,
between 1960 and 1975, more than 100 professorships were created in teacher training
colleges specifically for mathematics departments.
A similar phenomenon has been taking place in Spain since 1985 with the
recognition of Mathematics Education as an area of knowledge and the creation of
university departments for teachers in this area. However, there is a risk that Mathematics
Education will become disconnected from social reality due to this academic approach.
In general, the TME Group and its conferences aim to advance the theoretical foundations
of Mathematics Education and promote its integration with other disciplines, while
considering its practical implications and social relevance.
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To address these components, the TME Group explores various topics such as the
definition of Mathematics Education as a discipline, the use of models and theories in
research, the role of macromodels and micromodels, the debate between specific theories
and interdisciplinary approaches, the relationships between Mathematics Education and
its reference fields, and the ethical, social and political aspects of Mathematics Education.
The group also emphasizes the importance of systems theory, particularly social
systems theories, in understanding Mathematics Education as an interactive system. The
TME Group's development program focuses on the current state and future prospects of
Mathematics Education as an academic field and as an intersection between research,
development and practice. This program consists of three components:
• (A) identify and formulate fundamental problems in the orientation, foundation,
methodology and organization of Mathematics Education,
• (B) develop a comprehensive approach to Mathematics Education as an interactive
system, and
• (C) examine the interdependencies and conditions in Mathematics Education,
including the analysis of complementarities.
The Second Conference of the TME Group, which took place in 1985 at the Institut
fur Didaktik der Mathematik (IDM) at Bielefeld University, focused on the broader topic
of "Foundations and Methodology of the Discipline Mathematics Education (Didactics)
of Mathematics." This conference primarily highlighted the role of theory and theorizing
in specific domains within Mathematics Education. Some of the specific topics discussed
included theories about teaching, the theory of teaching situations, interactionist theory
of learning and teaching, the role of metaphors in developmental theory, empirical
theories in mathematics teaching, fundamental mathematical theories, theoretical
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concepts for the teaching of applied mathematics, representation theory for
understanding mathematical learning, and historical studies on the theoretical
development of mathematics education as a discipline.
The conference working groups were engaged in analysing the use of models,
methods, theories, paradigms and other research tools within different research domains.
Despite the diversity of topics discussed at the TME conferences, there is still no clear
consensus on the central issues and fundamental concepts within Mathematics
Education. While the conferences have generated many partial results and practical
guidance for the classroom, progress towards establishing a cohesive academic discipline
with its own theoretical foundations is still lacking.
The theme of the Third Conference, held in 1988 in Antwerp, Belgium, focused on
the role and implications of research in Mathematics Education for teacher education.
This conference aimed to address the significant gap that exists between teaching and
learning in this field. Some of the specific issues discussed included the gap between
teaching and learning in mathematics classrooms, the gap between research on teaching
and research on learning, models for teaching design based on research on learning, the
need for theory and research in development work and projects, the role of content and
different perspectives on mathematics in bridging the gap between research and learning,
the role of social interaction in the classroom, and the implications of the conference
theme for teacher education. In addition, the conference explored the role of computers
as a third component in teaching-learning interaction.
The fourth Conference, held in Oaxtepec, Mexico in 1990, focused on two main
themes: the relationships between theoretical orientations and empirical research
methods in Mathematics Education, and the role of holistic and systemic aspects and
approaches in Mathematics Education. This conference also marked the beginning of the
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presentation of various training programmes for researchers in Mathematics Education
at different universities, both at the PhD and master’s level. As part of this initiative, a
questionnaire was distributed to universities around the world to collect information on
research training, with the aim of establishing a network for the exchange of information
and discussion on the topic.
The organization of research in Mathematics Education is a discipline that fulfills
two main purposes:
• First, it provides information and data on the current status, problems and needs
of Mathematics Education, considering national and regional differences.
• Secondly, it contributes to the development of meta-knowledge and a self-
reflective attitude, which serves as a basis for the establishment and
implementation of development programs in Mathematics Education.
At the fifth Conference, held in 1991 in Paderno del Grappa, Italy, a preliminary
report on the results of the survey on the training of researchers was presented. In
addition, a number of papers were presented on the role of metaphors and metonymies
in mathematics, mathematics education and in the mathematics classroom, as well as the
role of social interaction and knowledge development from a Vygotsky perspective.
These conferences demonstrate the wide range of topics studied within the field of
mathematics education, including mathematics itself, curriculum design, students'
construction of mathematical meaning, teacher-student interactions, teacher preparation
and alternative research methods.
The aims of this network (TME) include exploring current developments in the
philosophy of mathematics, such as Lakatos' fallibilism, and other humanistic
perspectives. They also aim to delve deeper into the philosophical aspects of mathematics
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education, ensuring that philosophical reflection receives the same consideration as other
disciplines in this field. Furthermore, it aims to establish an open international network
of people interested in these topics and to provide opportunities for the exchange and
advancement of ideas and perspectives. The network seeks to foster informal
communication, dialogue and international cooperation between teachers, researchers
and others involved in theoretical and philosophical research on mathematics and
mathematics education.
Interest in the theoretical and philosophical foundations of mathematics education
has grown significantly since 2005, particularly after a research forum dedicated to this
topic was held at the PME Annual Meeting in Melbourne. Since then, numerous
researchers have published various papers in the ZDM journal and the topic has also
attracted attention in one of the working groups of CERME (European Congress on
Research in Mathematics Education).
This growing recognition of interest in mathematics education theory can also be
seen in research handbooks in the field. For example, Silver and Herbst (2007) provide an
overview of the state of theory in mathematics education research in the “Second
Handbook of Research on Mathematics Teaching and Learning” edited by Lester (2007).
Furthermore, Coob (2007) explores the topic “Putting Philosophy to Work: Confronting
Multiple Theoretical Perspectives” in the same handbook.
The importance of theory development is also highlighted in the first and second
editions of English's "Handbook of International Research in Mathematics Education"
(2002) and (2008), respectively. These efforts have culminated in the publication of
"Theories of Mathematics Education. Searching for New Frontiers" edited by Sriramn and
English (Springer, 2010), which contains 19 main chapters along with prefaces and
commentaries prepared by various authors. Topics covered in this book include
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perspectives on theories and philosophies of mathematics education, reflections on
learning theories, theoretical and philosophical foundations of mathematics education
research, the plurality of mathematics education theories, the reconceptualization of
mathematics education as a design science, the fundamental cycle of concept construction
underlying different theoretical frameworks, symbols, and mediations in mathematics
education, and more.
The importance of constructing theories is evident, as they serve as a guide to
formulating research problems and interpreting their findings. Theoretical frameworks
allow the organization of knowledge within a specific field, which is an initial step
towards a comprehensive understanding of the connections that exist in our perceptions.
Theorization is a prerequisite for an area of knowledge to achieve scientific status and
fulfill its function of explaining and predicting phenomena. Indeed, it can be argued that
significant scientific research is always guided by a theory, even if it is not explicitly
stated.
As Mosterín (1987) suggests, theories allow us to bring conceptual order to the
chaotic and formless world, allowing us to reduce complexity to a formula. They provide
us with tools for extrapolation, explanation, and a means of understanding and exerting
control over the world, even if this understanding and control is always uncertain and
problematic.
Mosterín also offers a compelling metaphor, comparing theories to the spider webs
that we, like spiders, use to capture and make sense of the world. These webs should not
be confused with reality itself, but without them, how much further away would we be
from being able to grasp and appreciate the world around us? According to Lester (2010),
employing a theoretical framework to conceptualize and guide research offers several
important advantages:
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• A framework provides a structure for conceptualizing and designing research
studies. Specifically, it helps determine the nature of the questions being asked,
the formulation of these questions, the definition of concepts, constructs, and
processes within the research, as well as acceptable research methods for
discovering and justifying new knowledge about the topic being studied.
• Without a framework, data is meaningless. Whether a data set can be considered
evidence of something is determined by assumptions, beliefs, and the context in
which the data was collected. An important aspect of a researcher's beliefs is the
framework, whether theory-based or not, that they are using. This framework
enables interpretation of the data set.
• A solid framework allows us to go beyond common sense. Deep understanding,
derived from a commitment to theory building, is often crucial to addressing truly
important problems.
• The goal is to achieve deep understanding. As researchers, we should strive to gain
a comprehensive understanding of the phenomena we are studying, focusing on
important questions rather than seeking only solutions to immediate problems
and dilemmas.
The research framework helps to develop this deep understanding by providing a
structure for designing research studies, interpreting the resulting data, and drawing
conclusions. Lester (2010) distinguishes between three types of research frameworks:
• Theoretical frameworks, which guide research activities based on a formal theory
that offers a coherent and established explanation of certain phenomena and
relationships. Examples of relevant theories used in the study of learning include
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Piaget's theory of intellectual development and Vygotsky's theory of
sociohistorical constructivism.
• Practical frameworks, which are based on the practical knowledge accumulated
by practitioners and managers, previous research findings, and often insights from
public opinion. These frameworks guide research based on what has been shown
to work in practice. Research questions are derived from this knowledge base, and
research findings are used to support, extend, or revise existing practices.
• Conceptual frameworks, which are local theoretical models that justify the choice
of concepts and their relationships in a particular research problem.
Like theoretical frameworks, conceptual frameworks are based on previous
research, but are constructed using a variety of sources, both common and diverse. The
framework used may draw on different theories and aspects of practical knowledge,
depending on the researcher's argument about what is relevant and important to the
research problem.
Burkhardt (1988) distinguishes between two types of theories:
"phenomenological" theories and "fundamental theories." Phenomenological theories
emerge directly from the data and provide a descriptive model of specific phenomena.
They are characterized by limited applicability but are detailed and specific in their
descriptions and predictions. They can be useful in curriculum design and in
understanding phenomena because of their proximity to reality.
A fundamental type of theory is a conceptual framework that encompasses
variables and their relationships, capturing the essential elements of a set of phenomena.
It possesses both descriptive and predictive qualities and is comprehensive within a
clearly defined domain. Such theories serve as analytical models with the aim of
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explaining a wide range of phenomena using a small number of fundamental concepts.
The definition is particularly applicable to fields such as physics and biology, where
theories such as Newtonian mechanics and Mendel's genetic theory align with this
framework. However, when examining theories in the realm of human sciences, such as
"behaviorism," "constructivism," and "developmental theories," Burkhardt raises
questions about their nature and scope.
While these theories offer frameworks for understanding phenomena, they lack
integrity within a limited domain. Consequently, they must be used with the
understanding that they lack established mechanisms for reliable integration into a
predictive model. Burkhardt regards them as overly simplistic descriptions of complex
systems, which can potentially prove problematic. In the context of the physical sciences,
these theories cannot be classified as comprehensive theories or even as models; rather,
they are descriptions of "effects"—important aspects of a behavioral system that must be
considered. However, each of these descriptions, on its own, is inadequate and can lead
to misunderstandings.
The psychology of mathematics education
In the field of Mathematics Education, there is also a significant influence from a
psychological perspective in the study of teaching and learning processes. However, this
predominance of the psychological approach overlooks the importance of balance and
complementarity with the other fundamental disciplines of Mathematics Education. This
influence is evident through the prominence of the International PME Group (Psychology
of Mathematics Education), which was established during the Second International
Congress on Mathematics Education (ICME) and continues to hold annual meetings.
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Among the main objectives of this group, as stated in its statutes, are to promote
collaboration and the international exchange of scientific information related to the
Psychology of Mathematics Education, to encourage interdisciplinary research in this
area involving psychologists, mathematicians and mathematics teachers, and to deepen
their knowledge of the psychological aspects of the teaching and learning of mathematics
and its implications.
Review of the research reports presented at the PME annual meetings reveals that
they encompass both empirical and theoretical research, covering a wide range of topics
that extend beyond the strict boundaries of psychology. While it is not possible to provide
a detailed account of the discussions held at these conferences due to their breadth, the
classification scheme for the research reports is worth mentioning as it broadly represents
current areas of focus within the field.
Cognitive interaction refers to instructional theories that emphasize the exchange
of information between teachers and students, with the goal of facilitating students'
assimilation of accurate information. This perspective includes theories proposed by
Piaget, Bruner, and Ausubel, as well as those that highlight the interaction between
instructional content and students' cognitive processes and skills. Social interaction, on
the other hand, prioritizes the role of individuals involved in instruction as facilitators of
learning. This perspective is represented by Vygotsky and Bandura.
Finally, contextual interaction theories, advocated by Skinner, Gagné, and
Cronbach, among others, emphasize the interaction between individuals and contextual
variables in the instructional process. Educational psychology is a field of study that
focuses on the scientific examination of teaching and learning processes, as well as the
challenges that may arise within these contexts. According to Gimeno Sacristán (1986),
there are various perspectives that view teaching as a technique derived directly from a
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psychological theory of learning, which serves as its foundation. However, this
dependence on psychology is considered detrimental to the development of a unique
theoretical field for both General Didactics and Special Didactics, as it restricts their
ability to create their own theories.
Consequently, educational psychology has the potential to dominate the study of
human behavior in teaching situations, limiting the scope of Didactics. Within
educational psychology there is a branch known as instructional psychology, which is
defined as a "scientific and applied discipline that emerged from educational psychology
and focuses on the study of psychological variables and their interaction with the
components of the teaching and learning processes, taught by specific subjects, with the
aim of teaching specific content or skills to equally specific individuals, within a specific
context."
Researchers analyze and classify different instructional theories and models from
an interactionist perspective into three types: cognitive, social, and contextual interaction.
In considering the essential issues in Mathematics Education that can benefit from a
psychological approach, Vergnaud (1988) identifies the analysis of students' behavior,
their representations, and the unconscious phenomena that occur in their minds, as well
as focusing on the behaviors, representations, and unconscious phenomena of teachers,
parents, and other participants. In addition, he highlights four types of phenomena that
can be fruitfully studied from a psychological perspective: the hierarchical organization
of students' competencies and conceptions, the short-term evolution of concepts and
skills in the classroom, social interactions and unconscious phenomena, and the
identification of real theorems, schemata, and symbols.
Within the psychological approach, one of the key challenges is to identify theories
about mathematical learning that can serve as a basis for teaching. Research on learning
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has provided limited insight into many central issues in instruction, and research on
teaching often makes implicit assumptions about children's learning that are not
consistent with current cognitive theories of learning.
Attempts have been made to apply general theories of learning to derive principles
that can guide instruction. However, behaviorism-based instruction tends to fragment
the curriculum into isolated parts that can be learned through reinforcement, which is
not conducive to effective mathematics instruction that requires an understanding of
fundamental mathematical concepts. Similarly, learning theories derived from Piaget's
genetic epistemology have not adequately explained children's ability to learn
mathematical concepts and skills.
This expansion of the field of interest of PME has led some, such as Fischbein
(1990), to suggest that the psychology of mathematics education is becoming the
paradigm for mathematics education as a whole. Fischbein argues that simply adopting
issues, concepts, theories, and methodologies from general psychology has not yielded
the expected results. He explains that psychology is not a deductive discipline, so the
application of general principles to a specific domain does not usually lead to significant
discoveries.
Even domains of psychology closely related to mathematics education, such as
problem solving, memory, reasoning strategies, creativity, representation, and
imagination, do not directly provide useful and practical recommendations for
mathematics education and may not be the main source of problems in this field. Thus,
the dynamics of mathematical symbolism requires a specific system of concepts beyond
those inspired by general psychology.
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Similarly, familiar psychological concepts take on new meanings in the context of
mathematics and mathematics education. A fundamental assumption underlying current
research on learning is derived from cognitive studies, which suggest that children
actively construct knowledge through their interaction with the environment and the
organization of their own mental constructs. While instruction certainly influences what
a child learns, it does not determine his or her learning.
The child is an active participant in the process of knowledge acquisition,
interpreting, structuring and assimilating information from his or her own mental
frameworks. As Vergnaud points out, most psychologists interested in mathematics
education today can be considered constructivists in some sense, since they believe that
students themselves construct competencies and conceptions.
According to Kilpatrick (1987), the constructivist viewpoint involves two
principles: knowledge is actively constructed by the learner and not passively received
from the environment, and the process of knowledge acquisition organizes one's own
experiential world rather than discovering an independent, pre-existing world external
to the learner's mind. However, it should be noted that not all research in the field aligns
with this perspective. In addition to the initial psychological problems faced by the PME
group, the debate surrounding the research has highlighted the need to consider new
aspects.
Two notable aspects include the specificity of mathematical knowledge and the
social dimension. To study the learning of algebra, geometry or calculus it is necessary to
carry out a deep epistemological analysis of the mathematical concepts involved. It is also
important to recognize that the meaning of these concepts is not based solely on their
formal definition, but rather on the processes involved in their operation. Therefore,
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attention should focus on studying students' cognitive processes rather than their current
abilities or productions.
The social dimension is another crucial factor to consider in research on the
psychology of mathematics education. The social status of the knowledge being taught
and the role of social interactions in the teaching process require careful consideration.
Moving from child-centered studies to studies centered on the student as a learner in the
classroom is a significant step in the development of research in this field.
The student is a child engaged in a learning process within a specific environment,
where social interactions with peers and the teacher play a vital role. This evolution of
the research problem requires the development of more systematic classroom
observations and the organization of specific teaching processes. It also requires the use
of new theoretical and methodological tools to produce solid results that have both
theoretical and practical significance.
However, the lack of specificity among researchers regarding the physical and
social conditions under which knowledge is acquired allows for a wide range of
epistemological viewpoints. These range from simple constructivism, which recognizes
only one principle, to radical constructivism, which accepts both principles and denies
the mind's ability to reflect objective aspects of reality. There is also social constructivism,
which emphasizes the importance of cognitive conflict in the construction of objectivity.
According to Vergnaud, the solution to this epistemological dilemma is quite
simple: knowledge construction involves gradually forming mental representations that
are homomorphic to reality in some respects but not in others. From a methodological
perspective, cognitive scientists observe individuals' problem-solving processes in detail,
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looking for patterns in their behavior and attempting to characterize these patterns with
sufficient precision for students to use as guides for problem solving.
Its aim is to build "process models" of students' understanding, which are then
tested using computer programs that simulate the solver's behavior. As mathematics
educators, we must question whether the computer metaphor adequately explains the
processes of teaching and learning mathematics and what implications information
processing theories have for mathematics teaching.
Kilpatrick cautions against over-reliance on the information metaphor, reminding
us that education should not be solely about transmitting information. While the
information metaphor can be useful, it is important to recognize that there are different
types of information and that something is lost when education is defined solely in terms
of information acquisition. Some authors propose a different approach to problem-
solving and teaching-learning processes, one that assigns a more active role to the solver
and considers the specificities of mathematical content as well as the role of the solver.
When it comes to mathematics learning and information processing, there is
currently no widely accepted theory that covers all the necessary details. Two main
research approaches are identified in this field: constructivism, as mentioned above, and
the cognitive science – information processing approach, which has had a significant
impact on the study of mathematics learning.
Schoenfeld (1987) states that the underlying hypothesis of cognitive science is that
mental structures and cognitive processes are complex but can be understood, leading to
a better understanding of thinking and learning. The main goal is to explain what
constitutes "productive thinking" or the ability to solve meaningful problems. Cognitive
science uses the metaphor of the mind as a computer to understand cognition as
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information processing and, consequently, to understand teaching and learning
processes.
The brain and mind are compared to the computer and its program, where
cognition is carried out by a central processing mechanism controlled by an executive
system that maintains awareness of its actions. Mental models are considered similar to
general-purpose computer models with a central processor capable of storing and
executing programs. In these models, the mind is considered unitary, with mental
structures and operations invariant across different contents. A single mechanism is
thought to underlie the ability to solve a particular class of problems.
Problem solving
Despite the attention paid to research on problem solving, there are doubts about
its relevance to school practice. Some argue that teaching students problem-solving
strategies and phases has little impact on their ability to solve general mathematical
problems. This raises the question of why problem solving is so difficult for most people
in mathematics. From our perspective, problem solving is not just a goal of mathematics
teaching, but the essential means to achieve learning.
Students should have regular opportunities to engage in challenging problem-
solving tasks, which will help them develop critical thinking skills, perseverance,
curiosity, and confidence in unfamiliar situations. Problem-solving should be integrated
into the mathematics curriculum and not treated as a separate component. It should be
connected to the study of different mathematical content areas and incorporate contexts
that are relevant to students' lives and other disciplines. However, there is a lack of
studies exploring the conceptual development that arises from problem-solving and its
interaction with the development of problem-solving competencies.
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Problem solving has emerged in recent years as an important area of research in
mathematics education. This research was initially spurred by the influential work of
Polya in 1945, which led to a large body of research on topics such as computer-simulated
problem solving, expert problem solving, strategies, heuristics, metacognitive processes,
and problem posing. More recently, there has been an increasing emphasis on
mathematical modeling in the elementary and secondary school grades, as well as on
interdisciplinary problem solving. Many of the early studies focused on typical word
problems found in school textbooks and tests.
These problems may be routine, requiring standard computational methods, or
nonroutine, involving finding a solution when the path is not obvious. Nonroutine
problems are particularly challenging for students. The importance given to problem
solving in the curriculum and educational research arises from the belief that problem
solving is at the core of mathematics. Authors such as Lakatos and Polya have contributed
to this perspective, and Polya describes four phases of problem solving: understanding
the problem, conceiving a plan, executing the plan, and examining the solution.
Polya's book has been highly regarded by mathematics educators as a valuable
resource for improving students' nonroutine critical thinking skills and addressing the
common question of what to do when stuck on a problem. However, while Polya's work
describes the ideal analytical person, Schoenfeld's research focuses on the actual behavior
of real problem solvers. Schoenfeld suggests that problem-solving instruction should
help students develop a repertoire of strategies specific to different types of problems,
promote metacognitive strategies for self-regulation, and work to improve students'
beliefs about mathematics and problem solving.
The sociocultural perspectives discussed by Sierpinska and Lerman highlight the
importance of considering the social and cultural context in mathematics education
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research. Understanding the role of social factors, the mediation of tools, and the
development of consciousness can contribute to a more comprehensive and effective
approach to mathematics teaching and learning. One research approach that has been
developed along these lines is Activity Theory. This theory emphasizes the role of the
acting person and the mediation of meaning between the individual and the world. For
the child, society and culture are mediated through tools, particularly cultural tools.
Thought and language are considered dialectically related, as language provides
the child with inherited historical-cultural meanings, but these meanings are continually
reconfigured through intersubjective communication and action. In recent years, there
has been a growing interest in the social context of the mathematics classroom in
mathematics education research.
The role of social context in the development of individuals or groups has been
theorized in a variety of ways. However, the current focus has shifted from identifying
social factors in the affective domain to understanding the impact of the broader social
and cultural environment on child development. In their 1996 review of epistemologies
in mathematics education, Sierpinska and Lerman discuss sociocultural views that have
been applied to the field of research. The term "sociocultural" refers to epistemologies
that view individuals as situated within cultures and social situations, so that context and
activity need to be considered when discussing knowledge or individuals.
Knowledge is seen as culturally produced, subject to change, and influenced by
social values and regulations. Vygotsky and his followers, on the other hand, were
primarily concerned with learning and teaching. Vygotsky did not delve deeply into the
nature of mathematics or other forms of knowledge, except in psychology, which he
sought to redefine as a materialist science. His main focus was the development of
consciousness, which he believed was driven by communication and learning.
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Vygotsky identified two types of thinking: ordinary or spontaneous thinking,
which occurs informally through interactions with peers and adults, and scientific or
theoretical thinking, which consciously aims at teaching and learning through the child's
appropriation of cultural knowledge. An important aspect of Vygotsky's approach is the
recognition that individuals and the world they inhabit are products of their time and
place. An individual's psychology, expressed as consciousness, is shaped by the
mediation of tools, which are influenced by social, historical and cultural context.
This perspective challenges Cartesian dualism and emphasizes the
interconnectedness of subject and object. Based on this understanding, it is argued that
there is no parallel between epistemological obstacles in mathematics and cognitive
obstacles in learning. For example, the concept of negative numbers faced
epistemological obstacles in the development of mathematics in the West, but today
children can learn about negative numbers without recreating that historical struggle.
This suggests that there is no inherent reason to assume a similar parallel between
epistemological and cognitive obstacles. Vygotsky introduced the concept of the zone of
proximal development, which refers to the difference between what a child can do
independently and what he or she can achieve with the help of a peer or experienced
adult. This concept highlights the importance of learning with others and suggests that
learning leads to development.
The perspective contradicts Piaget's belief that development, represented by
stages of child development, drives learning. Vygotsky also emphasized the process of
internalization, which involves the formation of consciousness through the mediation of
tools that are expressions of the social, historical, and cultural situation. This view
integrates teaching and learning at the school level.
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Lave introduced the concept of knowledge in action, which contrasts with a
cognitive perspective and emphasizes the role of context in mathematical practices. His
studies focused primarily on the application of mathematical skills in everyday life and
work situations. Lave criticized the traditional approach to school mathematics, which
prioritizes generalizable techniques and skills, arguing that it should be more relevant to
everyday life.
The concept of socioepistemology is used primarily in the Latin American
educational mathematics community. It is a theoretical framework that suggests
examining the production and dissemination of mathematical knowledge from different
perspectives. This framework originated from research conducted by Cantoral, Farfán
and other academics from the Higher Education Section of the Department of
Educational Mathematics at CINVESTAV (IPN, Mexico).
Socioepistemology not only offers an expanded understanding of epistemology,
emphasizing the socio-epistemic relativity of the meanings of mathematical objects in line
with other sociocultural viewpoints, but also provides a systematic approach to studying
the interactions between this mathematical understanding and cognitive and
instructional aspects. It proposes the examination of mathematical knowledge
considering its social, historical and cultural context, exploring how it was constructed
and disseminated. In addition to recognizing problem solving as a fundamental aspect of
mathematics, the need to explain the sociocultural factors involved in the construction of
mathematical knowledge, the role of the tools used and the various interpretations of
mathematical objects is also recognized.
Current research in Mathematics Education significantly focuses on the idea that
teaching and learning processes should aim to empower individuals and achieve social
transformation. To achieve this, it is necessary to promote strategies that encourage
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reflection on practice by the individuals involved, which can lead to significant changes
in teaching approaches. An example of a research program that aligns with this
perspective is known as “Critical Mathematics Education.” This approach presents an
agenda for studying the relationship between mathematics education and democracy.
Some of the key aspects emphasized by critical theory include:
• preparing students to be active citizens;
• use mathematics as a tool to critically analyze socially relevant issues;
• consider students' interests and perspectives;
• consider cultural conflicts that may arise in the instruction process;
• draw on prior experiences in teaching and learning mathematics to develop critical
thinking skills; and
• give importance to communication within the classroom since it forms the basis of
democratic interactions.
Another area of concern within critical mathematics education is the intersection
between mathematics and technology, which, while solving problems, also creates new
challenges. From a socio-critical perspective, teachers are encouraged to shift their role
from mere facilitators to active constructors of knowledge. It is argued that teachers have
the capacity and should be involved in developing pedagogical theory based on
educational research, bridging the gap that has traditionally separated theory and
practice, where theory is usually left to researchers and practice to teachers in their daily
work.
The researcher becomes an actor committed to achieving change. Participatory
action research is often used as a research methodology in this context. Action research,
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when applied in the school setting, involves studying a social situation in which teachers
and students actively participate in order to improve the quality of their actions. This is
done through a cyclical process of problem identification, planning, implementation,
reflection and evaluation of the results.
Increasing attention has been paid to using semiotics, the "science of signs", to
describe and understand the teaching and learning of mathematics. This interest is driven
by several factors:
• First, there is a growing recognition that mathematical activity is fundamentally a
symbolic activity because of the generality of mathematical objects.
• Second, understanding classroom communication has emphasized the importance
of understanding the nature of mathematical discourse for researchers and
teachers.
• Semiotics provides an adequate theory to account for the complexity of
communication.
• Furthermore, the increasing use of technology in mathematics education has led
to the exploration of semiotics as a means of understanding the cognitive role of
artifacts.
• Semiotics is well suited to this task because of its focus on the cultural conventions
and meanings associated with signs and artifacts.
The unique insights that a semiotic perspective brings to the understanding of
communication and learning in mathematics aim to model the role of mathematical sign
systems, meaning structures, mathematical rules, and the motivations behind
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mathematical activity within a coherent framework. The use of semiotics in the study of
mathematical activity is justified given the essential role of signs in mathematics.
Signs, symbols and notations play a similar role in communicating mathematical
ideas in both educational contexts and learning processes. The semiotic perspective
differs from psychological perspectives by focusing on signs and their use rather than
solely on mental structures and functions. It encompasses both the individual and social
dimensions of mathematical activity, teaching and learning by considering mathematics
as a communicative act.
Semiotic systems, consisting of signs, rules of sign production, and the
relationships between signs and meanings, are considered integral to understanding the
use of signs in mathematics. Godino and his collaborators have developed an
"ontosemiotic approach" to mathematics education that recognizes the fundamental role
of language, semiotics, and ontological issues in describing and understanding the
processes of teaching and learning mathematics. They view mathematical objects as
emerging from the systems of practices used to solve specific problems, and this
perspective complements existing semiotic perspectives in mathematics education.
A significant body of research in the field of mathematics education focuses on
examining the connections between teachers, students, and mathematical tasks in
mathematics classes. The goal is to find well-founded answers to questions such as how
teachers and students develop a shared understanding of mathematical concepts to
ensure a smooth flow of the class.
Researchers also investigate how students understand and respond to teacher
interventions. To address these questions, it is essential to develop theoretical
perspectives that can effectively interpret and analyze the intricate nature of mathematics
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lessons. Some of the key questions that interactionism addresses in mathematics
education include: how mathematical meanings are interactively formed in different
classroom cultures, how these meanings are stabilized, and how they are influenced by
the type of classroom culture in which they evolve.
The interactionist program introduces concepts such as domains of subjective
experience, patterns of interaction, and sociomathematical norms. The notion of domains
of subjective experience, developed by Bauersfeld, Krummheuer, and Voigt (1988),
adapts psychological concepts such as "script," "frame," "expert system," and
"microworld" to the study of mathematical learning. According to this model, individuals
form experiences within specific contexts and situations, incorporating cognitive,
emotional, and motor aspects. These experiences are then stored in memory as distinct
domains of subjective experience, reflecting the complexity and relevance of the situation
as perceived by the individual.
Symbolic interactionism (SI) is a theoretical perspective that has been used to
examine these relationships and has analytical implications. It asserts that cultural and
social dimensions are not peripheral to mathematics learning but rather intrinsic to it.
According to Sierpinska and Lerman (1996), who synthesized the interactionist program
applied to mathematics education, interactionism is an approach that promotes a
sociocultural understanding of the sources and development of knowledge.
The focus of study is on the interactions between individuals within a culture, with
an emphasis on the subjective construction of knowledge through interaction. This
perspective assumes that cultural and social processes are integral to mathematical
activity. One approach, as suggested by Bauersfeld (1994), is to use theoretical constructs
from sociology and linguistics, such as ethnomethodology, social interactionism, and
discourse analysis. However, since these disciplines do not directly address the teaching
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and learning of curricular content, some translation is required to address the specific
issues within mathematics education.
This approach is based on the premise that different practices emerge in the
classroom depending on whether mathematics is seen as a collection of objective truths
or as a process of shared mathematization. The latter perspective emphasizes the
importance of the "interactive constitution" of meaning in classrooms, highlighting the
relationships between the social characteristics of interaction processes and the thinking
of both teachers and students.
The foundations of the interactionist perspective can be summarized as follows:
teacher and students interactively shape classroom culture, conventions and agreements
emerge through interactive processes, and communication is based on negotiation and
shared meanings. The goals of research within the interactionist program in mathematics
education, as stated by Sierpinska and Lerman (1996), are to achieve a better
understanding of teaching and learning phenomena in ordinary school contexts. The
main goal is not to develop theories for action or to design teaching actions, but rather to
describe and discuss different possibilities. The research does not aim to improve the
microculture of individual classrooms in the same way that it can influence the
mathematics curriculum or the macroculture characterized by general principles and
teaching strategies.
Negotiating the meaning of a particular situation can be fragile and prone to
different interpretations due to ambiguity. Even if there is a shared context, there is
always a risk of breakdown and disorganization during the interactive process. To
minimize this risk, interaction patterns are formed. These patterns are considered
regularities that are created through the interaction between teacher and students,
seeking to make human interactions more predictable and less risky in their organization
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and development. Furthermore, interactions between teachers and students are often
guided by implicit norms or obligations. From an interactionist perspective, the use of
language is crucial, emphasizing the importance of negotiating meanings in the
development of students' understanding of mathematical concepts and their beliefs and
attitudes towards mathematics.
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Chapter 4
Fundamental didactics
In recent years, there has been a remarkable growth in interest and research
around Mathematics Education. A group of researchers, including notable figures such
as Brousseau, Chevallard and Vergnaud, have been working to develop a theoretical
understanding of mathematics didactics. The approach, known as the "fundamental"
conception of Didactics, distinguishes itself from other approaches by emphasizing a
global view of teaching, a strong connection to mathematics, specific learning theories,
and a search for unique research paradigms.
This line of research aims to establish an original theoretical framework,
developing its own concepts and methods, and considering teaching-learning situations
in a comprehensive manner. The investigation of these issues requires a methodological
approach that involves experimentation in a dialectical interaction with theory.
Experimental observations are compared with the theoretical framework and
adjustments can be made based on the coherence of the concepts developed and their
comprehensiveness in relation to the relevant phenomena.
The Mathematics Didactics approach is based on a systemic vision, considering
the overall functioning of teaching-learning phenomena. It recognizes that the separate
study of individual components cannot fully explain the overall functioning, just as it
cannot explain economic or social phenomena. Chevallard and Johsua describe the
Didactic System as being composed of three main subsystems: the teacher, the student,
and the knowledge taught. In addition, the system is influenced by the world outside the
school, including society, parents, and mathematicians. The intermediate zone, known as
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the noosphere, is a place of conflict and transactions that facilitates the articulation
between the system and its environment. It encompasses all individuals in society who
reflect on the content and methods of teaching. The media, which consist of the materials,
games, and teaching situations with which the student interacts, are also included as a
component.
This line of research in Mathematics Education seeks to understand the production
and communication of mathematical knowledge, focusing on the specific characteristics
of these processes. It considers teaching and learning phenomena from a systemic
perspective and emphasizes the interaction between the teacher, the student and the
knowledge taught. The development of an original theoretical framework, the use of
experimentation alongside theory and the exploration of diverse concepts and methods
are key aspects of this approach.
The models that have been developed include exploration of epistemological,
social, and cognitive dimensions. They strive to understand the complex interactions
between knowledge, students, and teachers within the classroom context. One researcher,
Laborde, has raised two important questions in relation to the study of teaching and
learning in mathematics. First, how can the conditions for effective teaching be
characterized to facilitate specific types of learning? And second, what elements should
be included in the description of a teaching process to ensure that it can be replicated in
terms of the learning it induces in students? These questions guide research and
emphasize the importance of determining the mathematical knowledge that students
wish to construct and comparing it to what is actually achieved during the teaching
process.
The theory we are discussing encompasses its own perspective on mathematical
learning, based on a Piagetian approach that emphasizes the construction of knowledge
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through the continuous interaction between the student and the subject matter. However,
this theory is distinguished from other constructivist theories by its particular focus on
the relationship between the student and knowledge. While content serves as the basis
for the development of mental structures, the didactic point of view adds another layer
of importance to the study of the student-knowledge relationship.
The main concern of research is the exploration of the conditions under which
knowledge is formed, with the ultimate goal of optimizing, controlling, and reproducing
it in educational settings. This requires paying particular attention to the object of
interaction between the student and knowledge, i.e. the problem-solving situation, and
how teachers manage this interaction. Recognition of the crucial role that situational
aspects, context, and culture play in shaping students’ cognitive behaviors is highlighted
in the field of Mathematics Education Psychology, although this situational dimension is
often overlooked as a separate area of research.
However, G. Brousseau's Theory of Didactic Situations stands as an initiative that
addresses this gap. The relationship with knowledge is examined from a perspective of
relativity, considering that knowledge can vary depending on the institutional context.
For example, someone may be considered to have knowledge of probability within the
scope of school education, but not within the academic sphere, and even within the
academic world there are further distinctions based on the different levels of expertise
required.
It is therefore necessary to differentiate between the institutional relationship to
knowledge (what is considered acceptable within a particular institution) and the
personal relationship to knowledge (an individual's understanding of a given topic),
which may or may not align with the institutional perspective. Two fundamental
questions arise from these concepts:
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• What conditions ensure the successful integration of a specific element of
knowledge and its institutional and personal relationships?
• What restrictions could hinder compliance with these conditions?
The study of the institutional relationship with knowledge, its conditions and its
effects are considered the central problem of Didactics. While the study of personal
relationships with knowledge is crucial in practice, it is considered epistemologically
secondary. However, this study program cannot be successful without considering the
various conditioning factors (cognitive, cultural, social, unconscious, physiological, etc.)
that may influence or affect a student's personal relationship with the knowledge in
question.
The relativity of knowledge within different institutions gives rise to the concept
of didactic transposition, which refers to the process of adapting mathematical
knowledge to make it suitable for teaching. In the initial phase of transposition,
mathematical knowledge is transformed into pedagogical knowledge. This involves
moving from describing the uses of a concept to describing the concept itself and the
organizational advantages it offers. The process of didactic transposition involves
decontextualizing the concept and removing its historical context, thus presenting it as a
timeless reality detached from its origin, utility or relevance.
Once the concept is introduced, the didactic operation takes over, using it for
educational purposes that do not necessarily align with the original intentions of its
creators. As the concept is integrated into the knowledge taught, it undergoes a process
of recontextualization. However, at the first educational levels, this recontextualization
may not completely restore the original mode of existence of the concept or fulfill all the
functions intended for its introduction.
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To go deeper into the topic of conditional probability, it is worth mentioning that
high school textbooks often introduce a concept called a “conditional event,” which is not
typically found in academic probability calculus. This concept refers to the event where
B occurs given that A has already occurred, and is denoted as B/A. However, it is
important to note that the event algebra is always isomorphic to a set algebra, meaning
that the available operations are limited to union, intersection, and difference.
The study of didactic transposition focuses on identifying and analyzing these
differences and understanding the reasons behind them, in order to rectify any
misconceptions and ensure that mathematical objects are correctly understood in
teaching. The brief description we have provided of some theoretical notions developed
by French didacticians serves as an example of how the French School of Mathematics
Didactics is establishing a solid foundation of theoretical concepts.
These concepts form the basis of a research programme similar to Lakatos'
approach. The ability of researchers in this field to pose new research problems and offer
new perspectives on classic problems is evident in their scientific production. Terms such
as didactic transposition, didactic contract and obstacle are increasingly used in
publications and international conferences focused on Mathematics Education. It is
undeniable that France has a distinct line of research in this field, as demonstrated by
Balachef's work, which represents an epistemological advance for this scientific
discipline. It remains to be seen whether this line of research will end up becoming the
predominant paradigm in the future.
Hans Freudenthal is an esteemed author in the field of mathematics education
who has made important contributions to the topic. His book, "Didactic Phenomenology
of Mathematical Structures," is widely regarded as a valuable resource for didactic
research, curriculum development, and the practice of mathematics teaching. Two key
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concepts introduced by Freudenthal continue to generate interest and reflection: "didactic
phenomenology" and the "constitution of mental objects."
Freudenthal criticizes the concept acquisition approach, which he believes views
mathematics as conceptual structures separated from their cultural and problem-solving
origins. In traditional teaching methods, the emphasis is on students learning
mathematics as a finished product, devoid of its practical application. Freudenthal
advocates prioritizing phenomenology, the problem situations that drive mathematical
action, and the development of problem-solving strategies. These problem situations
allow students to begin to constitute “mental objects,” which are personal cognitive
structures that can then be enriched by a discursive and cultural understanding of
mathematics.
The constitution of mental objects, as analysed by Freudenthal, challenges the
conventional approach of trying to instil abstract mathematical concepts in students
without providing concrete examples or experiences. Freudenthal argues that attempting
to materialise bare concepts through concretisation often proves insufficient, as
concretisations are often inadequate representations of the essential features of concepts.
Instead, Freudenthal suggests starting with phenomena that demand organization
and teaching students how to manipulate the means of organization from that starting
point. This approach reverses the traditional method of teaching abstractions by making
them concrete. To implement this approach effectively, the assistance of didactic
phenomenology is necessary to develop plans and strategies. Didactic phenomenology,
as defined by Freudenthal, involves using mathematical concepts, structures, and ideas
to organize phenomena both in the real world and in mathematics itself.
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For example, geometric figures such as triangles, parallelograms, rhombuses, and
squares help us organize boundary phenomena, while numbers organize quantity
phenomena. At a higher level, geometric constructions and demonstrations organize the
phenomenon of geometric figures, and the decimal system organizes the phenomenon of
numbers. The phenomenology of a mathematical concept or structure, according to
Freudenthal, involves describing its relationship to the phenomena it organizes,
identifying the phenomena for which it was created and those to which it can be
extended, understanding how it acts as a means of organization, and recognizing the
power it gives us over those phenomena.
When the focus is on how this relationship is acquired in a teaching and learning
process, we speak of didactic phenomenology of that concept or structure. Accordingly,
the work of Hans Freudenthal highlights the importance of didactic phenomenology and
the constitution of mental objects in mathematics education. By understanding the
relationship between mathematical concepts and the phenomena they organize, and by
starting with problem situations to develop cognitive structures, students can gain a
deeper and more meaningful understanding of mathematics.
The field of research on mathematics teaching and curriculum in Mathematics
Education is highly intriguing. At the practical level, curriculum and instruction play a
central role in improving school mathematics programs and raise important research
questions. By incorporating findings from other areas of Mathematics Education,
particularly learning theories, research on curriculum and instruction aims to
systematically understand and improve several aspects:
• the selection and organization of mathematical ideas to be taught;
• presenting these ideas to students; and
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• evaluating program effectiveness and student performance.
It seeks to determine the most effective combinations of content, sequencing,
strategies and delivery systems for different student skill profiles.
The complexity of research on curriculum and teaching is a notable feature.
Consequently, designers of curriculum materials and instructional procedures often rely
on personal creativity, intuitive judgments, and informal testing. Limited research is
available explaining how the system can transform a combination of needs, interests, and
values into a scientifically sound curriculum. As a result, topic selection in school
mathematics is determined by factors such as the internal structure of the discipline
(without rigorous epistemological analysis), public interest (measured informally),
recommendations from respected experts, and sometimes textbooks prepared with little
scientific basis.
Therefore, there is currently no consistent theoretical and experimental basis for
research on curriculum and instruction. The search for a theory of instruction as a priority
topic for future research merits designing theoretical models that establish relationships
between key curricular and instructional variables. While the primary goal in this field
has been to find the best method of instruction, efforts to identify general procedures,
sequencing strategies, or appropriate presentation formats have been unproductive.
Consequently, research now focuses on microscopic analyses of the curricular process
and on exploring the expected effects of specific approaches in particular situations and
content areas.
Another area of curriculum and instruction research investigates general
questions independent of specific content. Thus, most research on teaching has not
directly addressed mathematics, and the few studies that have focused on mathematics
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teaching have aimed to improve traditional methods rather than align with cognitive
research perspectives. As a result, these studies may have irrelevant or potentially
harmful findings.
In many research studies on teaching, the content being taught is often overlooked
or considered peripheral. Therefore, the need for research that considers specific content
and teaching techniques appropriate to that content is recognized. In general, studies
conducted within the process-product paradigm for teaching mathematics have not
provided teachers with a comprehensive list of observable behaviors that would enhance
their competence and ensure student learning.
This reflects the early stages of what Kuhn (1969) called "normal science," where a
paradigm or set of organizing principles that make all facts potentially relevant is lacking.
Studies of mathematics teaching conducted under an interpretivist paradigm, although
less common than positivist approaches, offer valuable insights into different aspects of
mathematics teaching through different conceptual lenses. For example, research into a
teacher's thinking about and teaching mathematics, and the impact of these beliefs on
their teaching practices, is gaining increasing interest.
Is it simply a matter of practical knowledge, a technology that is based on and
depends on other sciences, or is it that there are problems that require a level of theoretical
analysis and a methodology proper to true scientific knowledge? This epistemological
reflection is crucial to effectively guide didactic research, as it influences the formulation
of its central questions. However, there has been limited discussion on this topic in the
literature. The extreme complexity of the problems of Mathematics Education leads to
two extreme reactions: those who claim that Mathematics Didactics cannot be based on
scientific foundations and, therefore, teaching mathematics is essentially an art; and those
who believe that Didactics can be a science, but only focus on a partial aspect of the
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problems, such as content analysis, curriculum construction, teaching methods,
development of skills in students and classroom interaction.
This reductionist approach leads to different definitions and perspectives.
Mathematics Didactics can be seen as the art of teaching: a set of means and procedures
for making mathematics known. However, two scientific conceptions are distinguished,
which are called the applied multidisciplinary conception and the autonomous
conception (also called fundamental or mathematical). As a bridge between these two
groups there is also a technical conception, which considers didactics as a teaching
technique.
From the perspective of the multidisciplinary conception, which is aligned with
Steiner's second tendency, didactics becomes a convenient label for the teachings
necessary for the technical and professional training of teachers. Didactics, as a field of
scientific knowledge, would imply research on teaching within established scientific
disciplines such as psychology, semiotics, sociology, linguistics, epistemology, logic,
neurophysiology, pedagogy, pediatrics and psychoanalysis. In this case, didactic
knowledge would be a technology based on other sciences. The autonomous conception
seeks to integrate all the aforementioned meanings and assign them a place in relation to
a unifying theory of the didactic fact, with specific foundations and methods that point
to an endogenous justification.
This conception can be the starting point to address the need for a theoretical basis
that allows a better understanding and identification of the various positions, aspects and
intentions underlying the different definitions of Mathematics Education, and to analyze
the relationships between these positions in a dialectical understanding of the entire field.
The French School of Didactics aims to build its own scientific field of study, which is not
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limited and dependent on the development of other scientific fields, which may not
always be consistent.
This aim contrasts with the position of those who do not advocate the search for
internal theories (household theories) because of the risk of inappropriate restrictions.
The nature of the subject and its problems demand an interdisciplinary approach, and it
is believed that it would be a mistake not to make significant use of the knowledge that
other disciplines have already produced on specific aspects of those problems.
Mathematics Education should strive for transdisciplinarity, as defined by Piaget, which
encompasses not only interactions or reciprocities between specialized research projects,
but also locates these relationships within a total system without fixed boundaries
between disciplines.
The nature of mathematics education research is also considered, questioning
whether mathematics educators should see themselves as applied educational
psychologists, applied cognitive psychologists, or applied social scientists. Alternatively,
should they be considered scientists in the field of physics or other pure sciences? Or
should they be seen as engineers or other design-oriented scientists, whose research
draws on multiple practical and disciplinary perspectives, guided by the need to solve
real problems and develop relevant theories?
Brousseau's 1988 analysis examines how his conception of Mathematics Didactics,
as a theory for communicating mathematical knowledge, compares with other
perspectives and orientations. He argues that there is no conflict between his theory and
others, but that his theory encourages the integration of ideas from different domains and
their application to teaching. His theory promotes a healthy relationship between science
and technology, rather than focusing on prescriptions and reproductions. Brousseau does
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not categorically condemn any educational action, but he warns against expecting
didactics to fulfil functions that it does not have to fulfil.
He believes that it is a mistake to impose didactics on all educational action, as this
can create challenges that may be beyond their capabilities. At worst, this can result in
experts in the field taking on responsibilities for which they are not prepared, leading to
errors similar to those seen in other disciplines such as economics. As Godino argued in
1990, the improvement of mathematics education depends on factors outside of didactic
research itself, such as curricular guidelines, assessment procedures and teaching
materials. Therefore, it is essential to facilitate communication between those responsible
for these factors and researchers, as well as to promote didactic research. While didactic
research cannot provide teachers with model situations to imitate, it can provide them
with valuable knowledge to address the challenging nature of teaching mathematics in
the classroom.
Paradigms
The fundamental or mathematical conception aims to integrate all the above-
mentioned meanings and assign them a place in relation to a unifying theory of the
didactic phenomenon. This theory would have specific and endogenous justifications
and methods. This conception could potentially address the need highlighted by Steiner
for a theoretical basis that would improve understanding and identify the various
positions, aspects and intentions underlying the different definitions of mathematics
education. It would also analyse the relationships between these positions and bring
them together in a dialectical understanding of the entire field.
When mathematics educators or a group of teachers embark on research in their
field, they are immediately faced with the epistemological problem of understanding the
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nature of Mathematics Education and the corresponding methodological paradigms.
These issues influence the formulation of research problems and the determination of
their significance. In our case, where a research tradition and established paradigms are
lacking in the field, it becomes even more crucial to clarify the principles that have shaped
Mathematics Education Theory and the potential research methods, as they dictate the
types of research that can be conducted.
A literature review and synthesis by Hurford (2010) on the application of
theoretical insights from complex and dynamic systems theory to understanding learning
processes convincingly supports Steiner's views on the systems approach to mathematics
education. Hurford suggests that educational researchers now have the tools and
opportunity to build learning models that encompass inherent complexity in ways that
were not previously feasible.
It is time to move beyond simplistic models that reduce learning to basic stimulus-
response pairs or to static collections of isolated scenes of student learning. The
perspectives and models offered by systems theory for understanding learning are
preparing us to take that important step forward. The complexity of Mathematics
Education is its defining characteristic.
As described by Steiner, mathematics encompasses the intricate phenomenon of
mathematics in its historical and contemporary development, its interrelationship with
other sciences, practical areas, technology and culture. It also encompasses the complex
structure of teaching and schooling within our society, as well as the diverse conditions
and factors that influence the cognitive and social development of students.
This complexity has led many authors to adopt a Systems Theory approach in their
theoretical considerations. The interdisciplinary notion of system, which is embraced by
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all social sciences, becomes necessary when it is understood that the overall functioning
of a set of elements cannot be explained solely by their individual contributions.
In fact, the behaviour of these elements can even be influenced by their inclusion
in the system. In the case of mathematics teaching, a systemic approach is essential. It not
only considers the mathematics teaching system as a whole and the conceptual systems
that comprise it, but also considers the teaching systems that manifest themselves in a
classroom.
The main subsystems in this context are the teacher, the students and the
knowledge being taught. Adopting a systemic approach to teaching problems is
fundamental because it highlights that Mathematics Didactics is at the centre of multiple
interactions and, therefore, must develop its own problems and methodologies.
However, this does not mean ignoring the contributions of related disciplines,
particularly psychology and epistemology.
Furthermore, a systemic approach reveals the common structure that connects the
didactics of various disciplines, but also recognizes the unique challenges posed by
different domains of knowledge. Steiner further emphasizes that the systemic view of
mathematics didactics is self-referential, as it includes mathematics education as one of
its own subsystems. This self-referentiality necessitates a systemic approach as an
organizational metaparadigm for mathematics education, not only to manage the
complexity of the field as a whole but also because the systemic character is evident in
each specific problem within the field.
From the discussion of these conceptions it emerges that there is a dialectical
debate between the production of theoretical knowledge and practical knowledge in
didactics. To clarify this distinction the labels "Theoretical Didactics" and "Technical (or
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Practical) Didactics" can be used. The first refers to the academic discipline that aims to
describe and explain the states and evolution of didactic and cognitive systems, while the
second focuses on the problems of decision-making in the classroom and reflective action
in specific contexts.
The theoretical perspective prioritizes understanding how the system works and
discovering general laws that explain its dynamics, since the application of these
principles can lead to the solution of specific problems. On the other hand, the practical
perspective, adopted by researchers and applied professionals, recognizes the urgency of
solving immediate problems without waiting for theoretical science to discover general
principles. This theory-practice debate is not exclusive to Didactics but is observed in
various sciences, including medicine and economics.
In Mathematics Education, both the technical and multidisciplinary conception
adopt an applied science point of view, relying on general theoretical principles from
other disciplines such as psychology, pedagogy and sociology. Special mathematics
education then applies these principles to the specific domain of mathematical concepts
and skills, with the aim of providing solutions for teaching mathematics.
In the mathematical or fundamental conception, didactics is presented as a science
that deals with the production and communication of knowledge, focusing specifically
on the unique aspects of this production and communication. The objects of study in this
conception are the essential operations of the diffusion of knowledge, the conditions of
this diffusion and the transformations it causes both in knowledge and in its users. In
addition, this conception examines the institutions and activities that aim to facilitate
these operations.
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Research problems arising from the fundamental conception tend to be more
theoretical in nature, often involving model building. The ultimate goal of didactics,
according to this conception, is to build a theory of teaching processes that provides a
practical mastery of classroom phenomena. Research in the field of Mathematics
Education, like other fields such as medicine, agriculture and management, requires a
combination of theoretical and practical developments. This involves studying the
foundations of cognitive development and individual differences in mathematics
learning, as well as addressing decision-making problems in classrooms, schools and
teacher training programmes.
Research in this field covers a spectrum from pure research that may not have
immediate applicability to technological research and development, to the development
of educational materials that are tested and evaluated in laboratories and classrooms.
Each of the different concepts within Mathematics Education is characterized by the types
of problems they address.
Mathematics Didactics challenges reductionism by highlighting the limitations of
general psychopedagogical theories such as behaviourism and constructivism when
applied to teaching specific content. It emphasises the importance of the knowledge that
is transmitted and suggests the need for content-specific theories that explain the
functioning of the educational system from a knowledge-based perspective. This view is
shared by Freudenthal, who expresses scepticism towards general learning theories and
emphasises the uniqueness of mathematics in terms of pedagogical approaches.
The French school, still in the early stages of developing its theoretical framework,
prioritizes theoretical issues over technical ones due to the lack of secure reference points
for the proposals. However, considering the complexity of the teaching system,
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optimizing its functioning requires a collaborative effort between different research
perspectives, both theoretical and applied.
The fundamental conception of Mathematics Education, with its mathematical
perspective, plays a significant role in identifying theoretical concepts and didactic
phenomena that contribute to the dissemination of mathematical knowledge. The
connection between theory and practice and the social change that theoretical research
demands require the creation of an “interface” that is currently underdeveloped. This
interface could potentially be formed through explicit recognition of action research,
which aims to achieve social change and empowerment. Research conducted with the
active participation of teachers in research teams can serve as an interface within the
teaching system. Kilpatrick advocates closer collaboration between researchers and
teachers, emphasizing the need for joint efforts in research and implementation. This
aligns with a sociocritical perspective of action research, which seeks to optimize the
functioning of the entire system.
Research paradigms
When attempting to critically evaluate research findings in Mathematics
Education, it becomes evident that their nature is relative to the specific circumstances of
the participants (teachers and students) and the context in which they occur. Thus, it is
noteworthy that empirical findings in mathematics education not only lack universality
across different contexts, but their validity may also change over time due to the ever-
changing society in which mathematics education is conducted. Another factor that
significantly influences the validity and significance of research findings is the
perspective from which the research is conducted, known as the research paradigm.
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There are two extremes of this spectrum: the positivist or process-product
approach, which aims to discover laws and confirm hypotheses about behaviors and
procedures associated with student achievement, and the interpretive approach, which
seeks to understand the personal meaning of events, to study the interactions between
individuals and their environment, and to explore the thoughts, attitudes, and
perceptions of participants.
The positivist or process-product program employs quantitative methods, often
using systematic measurement, experimental designs, and mathematical modeling,
whereas the interpretive program (including ecological and ethnographic approaches) is
associated with naturalistic observations, case studies, ethnography, and narrative
reporting. Several distinctive features are found and highlighted between these two
approaches: the limited involvement of positivist researchers in the lives or activities of
their subjects as compared to ethnographers, the lack of interest among positivist
researchers in the intersubjective meanings that may arise in the schools or classrooms
studied, the infrequent use of sociocultural theories by positivist researchers to interpret
their findings, and the limited attention paid by educational anthropologists within the
interpretive approach to cognitive abilities, theories of cognitive development and
information processing, the reluctance to manipulate variables and force natural events,
and the rare attempt to address educational problems.
These disparate programs coexist and have coexisted in the field of teaching and
learning, including mathematics, particularly in research conducted from a
multidisciplinary perspective. However, much of current educational research, especially
the most innovative designs, can be classified as occupying an intermediate position
between these paradigms. A research model is therefore proposed that comprises four
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dimensions or suppositional modes: deductive-inductive, generative-verifying,
constructive-enumerative, and subjective-objective.
The deductive-inductive dimension refers to the reliance on existing theories or
the generation of new theories through the research process. The generative-verifying
dimension relates to the degree to which the results of one group can be generalized to
others, and verifiable research aims to establish generalizations beyond a single group.
The modes of formulation and design of variables and categories of analysis define the
constructive-enumerative dimension, while the subjective-objective dimension refers to
the constructs that are studied in relation to the participants involved. In addition to these
paradigms, there is a third socio-critical paradigm, which advocates connecting research
with practice to promote greater freedom and autonomy among participants. Mere
observation of educational encounters in a classroom is insufficient; it is also necessary to
provide direct guidance to practice, which requires greater collaboration between
teachers and researchers.
An example of how several paradigms can be integrated is demonstrated by
research conducted by the French School of Mathematics Education. This research
focuses on the study of how knowledge is formed, controlled and reproduced in the
school environment. An important aspect of this research is the exploration of the
relationship between the two subsystems involved - knowledge and students -
particularly through the problematic situation and the management of this interaction by
the teacher.
The methodology employed in this research program is guided by certain
assumptions, including the need for a holistic and case-study approach due to the
complexity of the phenomenon under investigation, as well as the use of multiple data
collection techniques. Furthermore, the specificity of mathematical knowledge allows for
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the generation of hypotheses from the study of this knowledge and its epistemological
origins. As a result, this research program incorporates elements from different
paradigms. For example, features of the positivist-experimental paradigm are evident in
the careful preparation of lessons, the formulation of hypotheses based on a general
theory, and the use of statistical methods for data analysis.
On the other hand, the ecological-ethnographic paradigm is reflected in the holistic
and qualitative approach to the study of the phenomenon, the interest in the variables
and interrelations of the process, the possibility of generating new hypotheses during the
research and the use of multiple data collection techniques, including ethnographic
methods such as observation. In general, the research paradigm adopted by the
mathematical conception of Mathematics Education is situated between deductive and
inductive reasoning, as well as between generative and enumerative approaches,
combining elements from both ends of the spectrum.
Adopting a systemic perspective can help resolve any conflicts between different
ideas and models. To achieve this, we need an integrative approach that considers theory,
development and practice, and embraces positivism, interpretivism and critique. These
different viewpoints should be seen as complementary and part of a broader
understanding. According to Steiner (1985), the concept of complementarity is a useful
tool for understanding the relationships between various types and levels of knowledge
and activity.
Interdisciplinary and fundamental perspectives are compatible and can work
together. By considering Mathematics Education as part of mathematics, we can establish
a "mathematical didactics" of mathematics, similar to mathematical logic or
metamathematics. However, this science cannot replace the contributions made by other
sciences. Teaching situations involve multiple aspects and phenomena, and Didactics (in
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its fundamental sense) has not yet fully explored and explained these phenomena with
specific concepts and methods.
On the other hand, the incorporation of external knowledge is crucial and must be
done under the guidance of a specific theory. This approach allows for a healthy
relationship between science and technique in teaching, rather than a relationship based
on prescription and reproduction. Kilpatrick (1981) also advocates eclecticism with
regard to methods. We should not abandon quantitative statistical techniques, which are
only just beginning to be applied, in favour of exclusively ethnographic methods.
Exploratory data analysis can complement quantitative methods in the field of
mathematics education. Kilpatrick also suggests that researchers should adopt a
convergent approach, where studies explore a topic from multiple perspectives using
various methods, rather than focusing on replication studies. In summary, the questions
raised in this discussion are essential aspects of the development program proposed by
Steiner (1985) for the Theory of Mathematics Education. These aspects include identifying
and addressing key issues in the orientation, foundation, methodology, and organization
of Mathematics Education as a discipline, and developing a comprehensive approach to
Mathematics Education as a whole, considering it as an interactive system encompassing
research, development, and practice, and emphasizing the dynamic role of theory-
practice exchange and interdisciplinary cooperation.
The consolidation of mathematics teaching
The recognition of Mathematics Didactics as an "area of knowledge" by the
Council of Universities in 1984, together with the implementation of the University
Reform Law (LRU) in the same year, has paved the way for the creation of university
departments dedicated to this field in Spain. These departments have played a crucial
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role in the advancement of mathematics education, as they are entrusted with teaching
and research responsibilities in the relevant areas of knowledge.
The departments have access to important research resources, including more than
200 permanent professors dedicated to research and specific bibliographic collections.
Institutional consolidation is also evidenced by the existence of doctoral programs and
the defense of doctoral theses on the teaching and learning of mathematics, as well as the
financing of research projects in competition with other areas of knowledge.
In 1997, the Society for Research in Mathematics Education (SEIEM) was formed,
demonstrating the growing awareness of the specific interests and needs of the
mathematics education research community. The realm of practical action is primarily
the domain of the teacher, who is responsible for instructing one or more groups of
students in mathematics. A teacher's primary goal is to enhance student learning, so his
or her primary interest is to obtain information that can have an immediate impact on his
or her teaching.
On the other hand, the technological component, also known as applied research,
is more focused on prescribing solutions and developing action devices. This field is
inhabited by curriculum designers, authors of school textbooks and creators of teaching
materials. Finally, scientific research, which encompasses basic, analytical and
descriptive studies, is concerned with the development of theories. This type of research
is usually carried out in university institutions. Mathematics education is a complex and
diverse system consisting of three distinct components or fields:
• Firstly, there is a practical and reflective action, which implies that teachers
actively participate in the teaching and learning processes related to mathematics.
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• Secondly, there is teaching technology, which focuses on the development of
materials and resources using scientific knowledge.
• Finally, there is scientific research, which aims to understand the general
functioning of mathematics teaching, as well as specific teaching systems
involving teachers, students and mathematical knowledge.
Despite their shared interest in improving mathematics education, these three
fields have different perspectives, goals, available resources, operational rules, and
constraints. Internationally, mathematics education has also experienced a consolidation
with the existence of similar research institutions and institutes in countries such as
Mexico and Germany. In addition, there are several research journals and handbooks
dedicated to the field, as well as international conferences that provide avenues for
researchers to share their findings and collaborate. The ICMI, an international
commission on mathematics instruction, has played an important role in promoting
research in mathematics education throughout the 20th century. Its study conducted in
Washington in 1994 highlighted the maturity of mathematics education as a scientific
discipline with its own goals and methods, further solidifying its status as a distinct field
of study.
In terms of research programmes and methods, there has been a shift from the use
of a primarily psychostatistical approach in the 1970s and 1980s, which focused on tests
and their reliability. There is now a proliferation of methods, the exploration of different
research agendas and the adoption of eclectic positions. This does not mean that the
psychological approach has lost importance, as demonstrated by the vitality of the
international PME group.
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Research is currently being conducted using a variety of approaches, including
interpretive, ethnographic, anthropological, and sociocritical methods. Some argue that
this diversity is beneficial, as it allows for different perspectives to be considered.
However, I believe that it can lead to confusion among research communities and make
efforts less productive. The multitude of approaches, theories, and methods in
mathematics education research calls for a more structured and organized approach,
similar to the philosophy of science.
Although mathematics education can be considered a mature discipline
sociologically, it may not be so philosophically or methodologically. The problem of
diversity in theories has been addressed by the European Congress of Mathematics
Education (CERME) in its working group, which has led to the publication of several
papers in conference proceedings and in the journal ZDM.
These challenges include difficulties in communication due to different
assumptions and languages, discrepancies in empirical results due to different
perspectives, and obstacles to scientific progress. It is argued that for the diversity of
theories to be fruitful, different approaches and traditions must interact. To address these
challenges, strategies that connect theories and theoretical approaches must be actively
sought. This can be done through empirical studies that combine different theoretical
approaches, developing theories as part of a connected set to reduce their number and
clarify their strengths and weaknesses, and fostering a discourse on theory development
and its qualities in mathematics education research, which also considers metatheoretical
and methodological considerations.
When discussing the aspect of mathematics education known as reflective
practice, it is important to acknowledge the important role played by mathematics
teachers' associations at various levels: regional, national and international. This is
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evidenced by the existence of organisations such as the Spanish Federation of
Associations of Mathematics Teachers, which is made up of 12 regional societies, as well
as their respective journals and conferences aimed at teachers.
At the international level we see the influence of powerful institutions such as the
NCTM in the USA, the ICME and the FISEM, together with its journal UNIÓN. However,
it is crucial to recognise that these activities often have limited connections with the
scientific and academic component of mathematics education. This is evident through the
existence of independent professional societies and separate journals for “teachers” and
“researchers” in countries such as Spain, France and Portugal.
This disconnect is evident in the development of mathematics curricula, which
have traditionally been prepared by commissions that overlook the expertise of
specialized university departments. The separation between academia and practice is
most pronounced in the initial training and continuing professional development of
secondary school mathematics teachers, where there is limited involvement of
mathematics education specialists. In conclusion, while mathematics education has made
significant progress as an academic discipline on the international stage over the past
three decades, its development has been uneven in different aspects and particularly in
the integration between them.
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Conclusion
Freudenthal was a strong advocate of reforming traditional mathematics
education. His extensive work as a founder and active participant in groups such as the
International Council on Psychological and Mathematical Education (PME) and the
International Commission for Research and Improvement of Mathematics Education
(CIEAEM) contributed to his fame. In these forums, he expressed his opposition to the
dominant pedagogical and didactic approaches of the mid-twentieth century, such as
performance goal theory, structured assessment tests and standardized educational
surveys, and the direct application of Piaget's structuralism and constructivism in the
classroom.
Hans Freudenthal, a German-born mathematician and educator, earned his PhD
at the University of Berlin. However, due to his Jewish background, he was forced to
emigrate from Germany during the rise of the Nazi regime. He sought refuge in the
Netherlands, where he continued his studies and developed pedagogical theories.
Unfortunately, he had to go into hiding during World War II. Freudenthal believed that
the learning process should be based on situations that require organization. He criticized
Piaget for trying to impose psychological development on the system of categories used
by mathematicians, using mathematical terms with different meanings.
Drawing on his own experience, Freudenthal argued that learning is more closely
related to language development than to cognitive development. He was concerned
about how Piaget's work influenced teachers who turned research findings into
guidelines for mathematics education, turning an epistemological theory into a violation
of pedagogical theory.
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He discussed with Chevallard his theory of transposition, which he believed was
based on the expert knowledge of mathematicians. Freudenthal argued that mathematics
taught in schools should not reflect any interpretation of philosophical or scientific ideas
unless they were much older. Freudenthal's opposition to the then-prevailing
psychology, pedagogy, and teaching methods was founded. This was based on his
extensive knowledge of mathematics, his passion for teaching mathematics, and his direct
experience in the classroom. He questioned the artificial nature of Bloom's educational
goals and fields of study, arguing that they had a negative impact on both academic and
developmental tests. He accused Bloom of viewing learning as a process in which
knowledge is simply transmitted into the student's head. Similarly, he disagreed with
Gagne's view that learning is a continuous process, developing from simple to complex
structures.
In conclusion, Freudenthal believes that learning involves sudden leaps in
rethinking, demonstrated by students finding shortcuts in their strategies, changing
perspectives, and using models with varying degrees of formality. However,
Freudenthal's references to non-mathematical authors are limited; he acknowledges the
influence of Decroli, whose interests coincided with his own theories on learning
mathematics in everyday contexts, and Dewey, in whom he sees similarities in the idea
of guided rethinking and was influenced by Lagenveld's phenomenological pedagogy.
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This edition of “Hans Freudenthal's Realistic Mathematical Theory: Didactics and
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September 2024.
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