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Hans Freudenthal: Perspectives in mathematics education and phenomenological analysis

Adolfo Angulo Romero, Francisco Eduardo Rengifo Silva, Atanacia Santacruz Espinoza,

Jesús Ttito Quispe, Alan Christian López Castillo, Fausto David Berrocal Huarcaya

Espinoza, Jesús Ttito Quispe, Alan Christian López Castillo, Fausto David Berrocal

Huarcaya, 2024

Second edition: September 2024

Edited by:

Editorial Mar Caribe

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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-freudenthal-perspectives-in-

mathematics-education-and-phenomenological-analysis/

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ISBN: 978-9915-9706-7-7

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Non-commercial attribution rights notice: Authors may authorize the general public to reuse their

works for non-profit purposes only, readers may use a work to generate another work as long as

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terms of the license CC BY-NC 4.0.

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Editorial Mar Caribe

Hans Freudenthal: Perspectives in mathematics education and

phenomenological analysis

Colonia, Uruguay

2024

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About the authors and the publication

Institutional identification where the authors work, the research declared by the authors

was carried out at the Instituto Universitario de Tecnología “Los Andes” (IUTLA), Rubio,

Venezuela.

Adolfo Angulo Romero

[email protected].pe

https://orcid.org/0000-0003-2128-843X

Universidad Nacional Intercultural de la

Amazonia, Peru

Francisco Eduardo Rengifo Silva

francisco_rengi[email protected]

https://orcid.org/ 0000-0001-5826-4058

Universidad Nacional de Ucayali, Peru

Atanacia Santacruz Espinoza

asantacruze@unia.edu.pe

https://orcid.org/0000-0002-3103-8947

Universidad Nacional Intercultural de la

Amazonia, Peru

Jesús Ttito Quispe

[email protected]

https://orcid.org/0000-0003-2981-8343

Universidad Nacional Intercultural de la

Amazonia, Peru

Alan Christian López Castillo

alan_lopez@unu.edu.pe

https://orcid.org/0009-0003-7773-2031

Universidad Nacional de Ucayali, Peru

Fausto David Berrocal Huarcaya

fausto_berrocal@unu.edu.pe

https://orcid.org/0009-0000-1224-9741

Universidad Nacional de Ucayali, Peru

Research result book:

Original and unpublished publication, the content of which is the result of a research

process carried out before its publication, has been double-blind peer-reviewed by

external peers, the book has been selected for its scientific quality and because it

contributes significantly to the area of knowledge and illustrates fully developed and

completed research. In addition, the publication has undergone an editorial process that

guarantees its bibliographic standardization and usability.

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Index 
Introduction ..................................................................................................................... 5 
Chapter 1 .......................................................................................................................... 8 
Freudenthal and phenomenological analysis ............................................................. 8 
The nature of mathematics ..................................................................................... 11 
The signs ......................................................................................................... 13 
Objects and the acquisition of concepts ...................................................... 21 
Chapter 2 ........................................................................................................................ 34 
Freudenthal: Fundamentals of Algebra and Geometry .......................................... 34 
The number ..................................................................................................... 34 
Algebra ............................................................................................................ 38 
Geometric objects ........................................................................................... 40 
The movements .............................................................................................. 43 
Statistics ........................................................................................................... 43 
The probability ............................................................................................... 44 
The variables ................................................................................................... 45 
Chapter 3 ........................................................................................................................ 49 
Perspectives on mathematics education .................................................................... 49 
The Platonic idealist conception ............................................................................. 50 
The constructivist conception ................................................................................. 51 
Mathematics and society ......................................................................................... 52 
The purpose of mathematics .................................................................................. 55 
Mathematical culture ............................................................................................... 58 
Language and communication ............................................................................... 60 
Chapter 4 ........................................................................................................................ 64 
A realistic mathematics perspective in early childhood education .................. 64 
The measure.................................................................................................... 66 
Conclusion ..................................................................................................................... 76 
Literature ........................................................................................................................ 78 
 
   

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Introduction

For Freudenthal, the practice of mathematics in the curriculum is not a set

of predetermined theories, goals and means. On the contrary, it is always

associated with positively understood phenomenological processes in

mathematics, as the curriculum is often used in conjunction with the

transformation or development of practice. For Freudenthal (1968), educational

theory was a practical endeavour from which new theoretical ideas could emerge

as a kind of scientific by-product. In his view, curriculum development should

not take place from the academic ivory tower, but in schools, in collaboration with

teachers and students. He expressed similar ideas and, by calling for a practical

curriculum, eloquently questioned curriculum theory in research and

development and innovation.

There are, therefore, similarities between some of the implications of a

colonial approach to curriculum and the concept itself. However, when

phenomenology appears in Freudenthal’s work, it often has the opposite meaning

and he qualified the main direction of the movement as theoretical and top-down,

starting from a vision of curriculum as processes and proposing an alternative of

its own for educational development, which was nothing more than a perspective

of mathematics education, that is, its realist mathematics (Sepúlveda, 2018).

From this perspective, phenomenological, curricular, and ethnographic

real mathematics, or practice-based “mathematization” have similar

characteristics. And this was of fundamental importance to Freudenthal because

the main task of mathematics education should be the mathematization of

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everyday reality. Mathematics cannot be learned in mathematics courses because

initially none of mathematics is empirically real for students, especially in the

early years of life, i.e. in early childhood and primary education. In addition, the

subject of mathematics, based on facts, helps students to adopt a mathematical

approach in everyday situations (Trujillo, 2017). In this book, we can refer to the

realistic mathematical activity, as expressed by Freudenthal (1991), which implies

a mathematical attitude that includes the knowledge of the possibilities and

limitations of learning a mathematical method, that is, the knowledge of when a

mathematical method is appropriate and when it is appropriate or not.

This emphasis on “mathematization” is consistent with the need for

mathematics for all. Freudenthal points out that not all students will become

future mathematicians, since for most, all the mathematical knowledge they use

will be used to solve everyday problems, that is, in the sociocultural context that

surrounds them. Therefore, introducing students to mathematical methods to

solve these types of problems deserves to be considered one of the main priorities

of mathematics education, this is its phenomenology.

The story Freudenthal developed is better known for its critique of

traditional educational research than for its own counter-empirical ideas and

theories (Alsina, 2009). Freudenthal's opposition to much of educational research

arose from his belief that deschooling was necessary, in the authors' view, a

curriculum based on ethnography and lived experience. These breaks can be seen

as method creation or the adoption of perspectives from different angles and, he

argues, from this gap one can tell whether the student has reached a certain level

of understanding or not. To identify these breaches, students must be observed

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individually. This means that group measures and the like are not particularly

useful because they fill individual gaps. Furthermore, the emphasis should be on

observing the learning process rather than examining objective learning

outcomes.

Furthermore, Freudenthal believes that such extensive research cannot

answer educational questions about why and who teaches a given subject.

Freudenthal offers a second set of criticisms of the assessment movement. In

theory, Freudenthal relied on formative rather than punitive education and

focused on unfamiliarity with the subject matter and an overemphasis on

reliability at the expense of validity.

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Chapter 1

Freudenthal and phenomenological analysis

At the beginning of his work, Freudenthal introduces the term

"phenomenology" to describe his method of analyzing mathematical concepts. He

explains that this term is derived from the distinction made in the philosophical

tradition between "phenomenon" and "noumenon." In mathematics, this

distinction is seen between concepts or structures (noumena) and the phenomena

that organize these concepts. For example, geometric figures organize the

phenomena of contours.

Phenomenological analysis of a concept or structure involves describing

the phenomena it organizes and examining the relationship between the concept

or structure and these phenomena. This analysis must consider the development

and current use of mathematics, as well as the original purpose of the concept or

structure and its later extensions (Puig, 1997). The relationship between

phenomena and concepts becomes more complex when mental objects are

introduced, and phenomenological analysis must also consider the relationships

between phenomena and mental objects, and between mental objects and

concepts.

To better understand Freudenthal's ideas, it is necessary to strip the terms

of their philosophical baggage. Instead of using "noumenon," it is more

appropriate to refer to the concept as a "medium of organization." This

emphasizes the function of concepts in relation to phenomena. The term

"phenomenon" can still be used to describe what we experience mathematically,

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but its original meaning of appearance must be set aside. It is important to

recognize that the means of organization of the phenomena we use to understand

our mathematical experience are also an object of experience. The pair of

phenomena and means of organization is defined by their relationship rather than

by their belonging to different worlds.

This understanding allows us to view the series of phenomena and means

of organization as interconnected, with the means of organization of one pair

becoming the phenomena of the next. To engage in phenomenology is thus to

describe one of these series or one of their pairs (Chakravorty, 2010). Although

the terms ‘phenomenon’ and ‘noumenon’ are taken from the philosophical

tradition, Freudenthal does not provide a clear definition of these terms. He

claims that he does not use them in the sense given to them by philosophers such

as Hegel, Husserl, or Heidegger, but he does not align himself with any specific

philosophical system. The ‘noumenon’ is identified as the ‘object of thought’

without further explanation, and the ‘phenomenon’ is described as something we

have experience of. These terms have Greek origins, where ‘noumenon’ means

‘that which is thought by reason’ or ‘that which is intelligible’, and ‘phenomenon’

means ‘that which appears’.

Phenomena are the appearances or what we perceive of things, while

noumena are considered the true reality in the realist philosophical tradition

(Navarro, 1971). Identifying mathematical concepts with noumena places them

outside our realm of experience. However, this contradicts one of the

characteristics of mathematics pointed out by Freudenthal: that mathematical

concepts are part of the field of phenomena that they organize. Mathematical

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concepts are not separate from our experience nor in a different world from that

of the phenomena that they organize.

The phenomenological analysis developed by Freudenthal serves the

purpose of didactics, although it is based on concepts from philosophy and has

implications for the understanding of mathematics. Freudenthal distinguishes

several types of phenomenology, all of which are important for didactics, but only

one is specifically qualified as didactic. These types include phenomenology,

didactic phenomenology, genetic phenomenology, and historical

phenomenology.

Each type of analysis focuses on different phenomena in relation to the

concept being studied. In pure phenomenology, the focus is on the current state

and use of mathematics. In didactic phenomenology, the focus is on phenomena

present in the students' world and those introduced in teaching sequences. In

genetic phenomenology, the focus is on phenomena related to the cognitive

development of students. In historical phenomenology, the focus is on

phenomena that led to the creation and extension of the concept in question.

The description of the relationships between phenomena and the concept

varies depending on the type of phenomenological analysis. In pure

phenomenology, mathematical concepts are treated as cognitive products, while

in didactic phenomenology they are treated as cognitive processes within the

educational system (Zolkower and Bressan, 2012). Freudenthal emphasizes that a

didactic phenomenology should not necessarily be based on a genetic

phenomenology. The correct order for conducting diverse types of

phenomenological analysis begins with pure phenomenology, followed by

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historical phenomenology, then didactic phenomenology, and finally genetic

phenomenology. It is essential that any effective phenomenological analysis in

teaching is supported by a sound analysis of pure phenomenology.

The nature of mathematics

Mathematical concepts are means of organizing phenomena in the world.

However, this characterization lacks specificity unless we define what we mean

by "the world" and identify the phenomena that mathematical concepts organize.

Phenomenology, as a method, aims to investigate phenomena organized by

mathematical concepts through analysis, rather than assuming prior knowledge

of these phenomena.

Therefore, it is not possible to predetermine the type of phenomena

organized by mathematics without conducting specific analyses. Freudenthal's

phenomenological analysis serves as a foundation for the organization of

mathematics teaching, but it does not go into depth in explaining the nature of

mathematics itself. One could use this analysis without committing to any

epistemological or ontological stance towards mathematics. In other words, one

could view mathematical concepts as tools for organizing phenomena in teaching,

without necessarily believing that this organization reflects the true nature of

things. However, it is important to note that students' and teachers' beliefs about

the nature of mathematics influence how they perceive mathematical activities in

the classroom and the knowledge they aim to convey.

On the other hand, one could interpret from the above statement that

mathematics exists in a world separate from the one it organizes, namely the real

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world around us. However, I do not find this interpretation to be the most

adequate. In fact, if we consider the origin or the lowest level, the phenomena

organized by mathematical concepts are those of the real, physical, everyday

world. Our experiences with this physical world involve objects, their properties,

the actions we perform on them, and the properties of those actions. Thus, the

phenomena organized by mathematics are the objects, properties, actions, and

properties of actions in the world, seen as means of organization and considered

in relation to them.

The phenomena that organize mathematical concepts are the phenomena

of the world that encompass human cognitive products, including the products

of mathematical activity (Vergnaud, 1988). These phenomena include the objects

of the world, their properties, the actions we perform on them, and the properties

of those actions, in their relation to the means of organization. The gradual

progression of phenomena and means of organization involves two processes: the

creation of mathematical concepts as means of organization and the

objectification of the means of organization, allowing them to form part of new

pairs in the position of phenomena. This gradual progression illustrates the

production of increasingly abstract and higher-level mathematical objects,

demonstrating that mathematical activity generates its own content.

The first interpretation makes it clear that mathematical concepts do not

exist in an ideal world that we study, nor do they have a pre-existing existence

prior to mathematical activity. Mathematical activity is not about discovering the

geography of a world where objects reside. However, it is also important to note

that Freudenthal's analysis does not simply describe mathematical activity as an

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interplay between world phenomena and mathematical means of organization.

Rather, Freudenthal recognizes that the means of organization themselves

become objects situated in a field of phenomena.

Mathematical objects thus become part of our experiential world, entering

as phenomena into a new relationship with other phenomena and means of

organization. This process of incorporating mathematical objects into our world

leads to the creation of new mathematical concepts, and this cycle continues

repeatedly. Mathematics thus exists within the same world as the phenomena it

organizes, and there is not a separate world but a unique world that expands with

each mathematical product. It was mentioned earlier that the reason

mathematical concepts do not exist as ideal objects in a separate world is due to

the role of the sign systems in which mathematics is expressed or written.

The signs

Mathematical texts, regardless of the reader's experience, often contain

symbols and terms that are not commonly used in everyday language. This

distinction has led to the notion that the language of mathematics is separate from

the vernacular. Furthermore, throughout the history of mathematics, there have

been instances where new concepts and ideas were developed, further

highlighting the unique nature of mathematical discourse.

The language of mathematics plays a crucial role and is often considered

separate from everyday language. Some mathematicians believe that true

mathematics is only written in a fully formalized language, while everyday

mathematical texts are considered to contain "abuses of language" due to the use

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of the vernacular. However, rather than focusing on the types of signs used in

mathematics, it is more important to study the processes of meaning and the

production of meaning. Therefore, there is no need to separate mathematical

signs from the vernacular or other means of representation.

Instead, all signs used in mathematical activity can be considered part of a

mathematical sign system. These signs are not homogeneous and can be

described using terminology borrowed from semiotics, which focuses on the

expression and content of a sign. Mathematical sign systems contain signs with

different expressions, like other human activities such as film or singing.

Mathematical concepts are created through the relationship between

phenomena and the means of organization described by mathematical sign

systems. These mathematical objects have a material existence that is given to

them by the sign systems that describe and create them. As the means of

organization become more abstract, mathematical sign systems also become more

abstract and create corresponding abstract concepts (Streefland, 1991).

In the preceding discussion, I have highlighted the crucial elements of a

mathematical framework that incorporates principles derived from Freudenthal's

phenomenology of mathematics and Filloy's concept of mathematical sign

systems. However, it is important to recognize that this description still lacks

complexity. To provide a more complete understanding, I will now introduce

additional ideas that I believe are essential for a complete description. These ideas

are derived from the works of Lakatos and Kitcher, and while I will not delve into

their complexities here, I encourage readers to consult the corresponding texts in

the bibliography for further exploration.

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In the realm of mathematical activity, the objects of experience consist of

objects in the world, properties of the objects, actions performed on those objects,

and properties of those actions. Keitel (1987) introduces the notion that the actions

mentioned are not necessarily actions that we perform or can perform, but rather

are actions that can be performed by an ideal subject endowed with greater

powers of action than our own.

For example, this ideal subject might run through the sequence of natural

numbers or use Hilbert's choice function. While this idea may suggest that

mathematics can develop based on arbitrary stipulations of these powers, thus

generating mathematical concepts that lack any epistemic or practical purpose,

this danger is avoided in practice through various means.

While mathematical concepts are created in the process of organizing

phenomena, they are not immutable once created. Rather, they are subject to

modification over time due to their use and the new mathematical sign systems

in which they are described. However, these modifications should not be seen as

indicative of errors in the original concepts or as a linear progression toward a

singular truth. This is because we reject the notion that mathematical objects have

an existence prior to the process that creates them.

For Beltrán (1996), a unique perspective on the evolution of concepts in

history is presented by Lakatos, who examines how concepts evolve under the

pressure of proving the theorems in which they are involved. Lakatos highlights

how the establishment of the C + V = A + 2 conjecture for any polyhedron and its

proof by Euler led to the emergence of examples of solids that did not align with

the proof or, more significantly, with the concept itself.

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One way to address this danger is through the role of mathematical sign

systems in the creation of mathematical concepts. These systems not only allow

us to organize phenomena by creating relevant concepts, but they also allow us

to perform new actions on mathematical objects or to appreciate the possibility of

doing so if certain constraints were removed. These new actions are not arbitrary

but are suggested by the more abstract mathematical sign systems, thus extending

actions that we had previously seen ourselves capable of performing or that we

had established as feasible at a lower level. Furthermore, the acceptance of these

new actions by the mathematical community serves as a regulatory mechanism.

However, this acceptance is not without controversy, as seen in the case of

Hilbert's choice function.

According to one concept of mathematical objects, there exists an ideal

object known as a polyhedron, and the purpose of mathematical activity is to

discover its properties. This theorem provides evidence that the solids in question

are not true polyhedral, or alternatively, the proof of their properties is flawed.

However, it is important to note that Lakatos's interpretation of the story differs

from this theorem. Lakatos distinguishes between two types of counterexamples:

local and global counterexamples.

A local counterexample refers to a solid that has features that make the

proof inapplicable, but still satisfies the relation in question. These

counterexamples do not refute the conjecture; rather, they indicate that the proof

is based on a property that was assumed to hold for all polyhedral, but this

assumption is incorrect. Therefore, what is refuted is a slogan that has been used

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implicitly, thereby invalidating the proof. The existence of these counterexamples

introduces a discrepancy in the concepts that was not present before.

The impact of the emergence of global counterexamples is important for

our analysis. When a counterexample refutes a conjecture on a global scale, it is

particularly noteworthy. Lakatos first presents global counterexamples to Euler's

theorem, including a solid formed by a cube with a cubic hole inside it and a solid

formed by two tetrahedra joined by an edge or a vertex. He then introduces an

even more intriguing case of a solid star, the verification or denial of the relation

depending on whether its faces are considered star polygons or not.

The existence of these counterexamples creates tension between the

concept, the theorem, and its proof. Several approaches can resolve this tension

and impact the concept of a polyhedron. The most fundamental approaches

include: 1) Exclusion of monsters. The presented counterexamples are considered

unconventional examples of polyhedral, referred to as monsters. While their

existence is possible, it is not desired. The possibility of their existence is

determined by the definition of a polyhedron that is used, leading to the

development of a new definition that explicitly excludes them to maintain the

theorem. 2) Exclusion of exceptions. The presented counterexamples are

considered valid examples of the concept, even though their existence was not

anticipated when the conjecture was originally posed. To return to safer ground,

the conjecture is modified by introducing a distinction within the concept that

separates these examples. 3) Adjustment of monsters. The objects are examined

from a distinct perspective, causing them to cease being counterexamples. This is

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exemplified by the two diverse ways of perceiving star polyhedral: composed of

star polygons or not.

Regular polygon: For any isometry S, a non-invariant point A0 exhibits an

orbit, which comprises the set of points an into which A0 transforms by powers

Sn, where n spans all integers. Consequently, we can describe the polygon

A0A1A2... generated in this way as regular. Since the possible values of n span

negative integers, the sequence of points extends both forward and backward,

and the polygon A0A1A2... should be more accurately described as: ...A-2A-

1A0A1A2... This example is particularly noteworthy because the definitions of

polygon and polyhedron each incorporate distinct approaches to addressing the

tension we are examining.

It is evident that the definition of a polygon is developed by accepting new

objects, explicitly expanding the content of the concept (which will later require

further classification into several types of polygons). Polyhedron: A polyhedron

is a finite collection of planar polygons, known as faces, together with all their

edges and vertices, which satisfy the following three conditions: i) Each edge

belongs to exactly two faces, and these faces are not in the same plane. ii) The

faces that share a vertex form a single cycle, that is, their intersection by a

sufficiently small sphere, centered on the common vertex, constitutes a single

spherical polygon. iii) No proper subset of the faces satisfies condition i).

Regular polyhedron: For any polyhedron, we define a flag (A, AB, ABC...)

as the geometric configuration composed of a vertex A, an edge AB containing

that vertex, and a face ABC... containing that edge. A polyhedron is considered

regular if its symmetry group exhibits transitivity within its flags. Despite the

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simplicity of these basic methods for dealing with tension, the concept of a

polyhedron is significantly affected in all cases.

Whether counterexamples are recognized or excluded as instances of the

concept, the semantic field is expanded. In one scenario, the content of the

expression increases, effectively broadening the range of phenomena that the

concept was initially intended to encompass; this expansion constitutes the

semantic field. In the other scenario, the concept establishes connections with new

objects that were previously distanced from it in the new definition, thus

becoming an integral part of its content. The entire process is more complex,

involving increasingly intricate or abstract mathematical systems of symbols into

which concepts, originally expressed in simpler or less abstract mathematical

systems, are translated.

As Lakatos states, the concepts generated through this process do not

improve the original concepts; they are not specifications or generalizations of

them. Instead, they transform the original concepts into something completely

different: they create new concepts. This is precisely what I intend to emphasize:

the result of Lakatos' process of tension between concepts, theorems, and proofs

is not the delineation of the true concept of a polyhedron that corresponds to a

preexisting ideal object, but the creation of new concepts.

In the definition of a polyhedron, each condition is intended to exclude

certain objects. For example, condition i) prevents the inclusion of a solid formed

by two tetrahedrons joined by an edge as a polyhedron, thus excluding monsters

from the category. However, this same condition also prohibits the interpretation

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of a star polyhedron whose faces are not the corresponding star polygons, since

this would result in faces that lie in the same plane.

Therefore, this condition does not allow the adjustment of that monster,

and star polyhedral are only considered polyhedral when their faces are the

corresponding star polygons. The extension of the concept of polyhedron lies in

the relationships that each condition establishes with other mathematical objects,

as well as its representation within the system of signs used.

Lakatos's idea suggests that mathematical concepts are not fixed in their

original form. They undergo changes, driven by the tensions that arise from being

involved in proving and disproving. However, mathematical activity goes

beyond simple theorem proving. Problem solving is a key factor in the

development of mathematics, which includes theorem proving and other

activities.

Problem solving involves proving theorems in two ways:

• First, it encompasses theorem proving on a global scale, where all

problems are considered as problems and a distinction is made between

finding and proving problems.

• Second, problem solving involves proving theorems in the context of each

problem.

In fact, what characterizes problem solving in mathematics is that the

result obtained must be accompanied by a justification that verifies the conditions

of the problem. This extends the scope in which concepts are influenced and

modified beyond the mere demonstration of theorems, to include problem

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solving. Furthermore, the formulation of recent problems and the study of

families of problems are also important aspects of problem solving that contribute

to the creation of mathematical concepts.

Problem solving is not the only mathematical activity that generates

concepts. The organization of results obtained through problem solving and

theorem proving into a deductive system is another crucial aspect of

mathematics. This systematic organization can take many forms, from local to

global and from axiomatic to formalized. However, it is a fundamental

component of mathematics as mathematicians have moved from accumulating

results and techniques to developing comprehensive frameworks. In this process,

the use of definitions in mathematics has also evolved. In mathematics,

definitions serve not only to explain the meaning of terms to people, but also as

links in deductive chains within the activities of organizing deductive systems.

The process of definition involves organizing the properties of a

mathematical object by deduction. It focuses on identifying properties that can be

used to form a deductive system, either at a local or global scale, in which the

mathematical object can be included. It is important to note that highlighting

specific properties to define a concept is not a neutral or innocent act. On the one

hand, it suggests that the concept was originally created to organize the

corresponding phenomena. On the other hand, it means that the content of the

concept is now determined by the deductions made within the defined system.

Like theorem proving, the process of definition also leads to the creation of new

concepts.

Objects and the acquisition of concepts

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In this section, a new idea proposed by Freudenthal will be presented that

challenges the ideas I have presented so far. This idea revolves around the

distinction between a mental object and a concept. I find this concept particularly

important because it leads Freudenthal to adopt a didactic stance: the primary

goal of education should be the development of mental objects, with the

acquisition of concepts being of secondary importance. The position is especially

significant in the context of compulsory schooling, as it raises questions about

what mathematics can offer to the general population. However, it also has

relevance for the phenomenological analysis of mathematical concepts,

particularly within the framework of didactic phenomenology (Treffers, 1987).

It is crucial to remember that this analysis precedes the organization of

teaching and is done with that goal in mind. The mental object/concept distinction

that Freudenthal presents can be understood as the result of considering the

individuals who conceive or use mathematics in contrast to mathematics as a

discipline or a historically, socially, or culturally established body of knowledge.

In the previous sections, we have focused primarily on mathematical concepts

within the discipline itself, with limited consideration of specific individuals. And

various mathematical concepts and their relationship to phenomena and means

of organization have been analyzed. Reference has also been made to the

objectification of these means of organization and how they are integrated into a

higher-level relationship between phenomena. Furthermore, how concepts

undergo transformations because of mathematical activities such as theorem

proving, problem solving, organization in a deductive system, and the process of

definition has been explored. Throughout all of this I have emphasized that

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mathematical concepts do not exist independently of the mathematical activity

that gives rise to them.

For example, consider the concept of a dog. The mental object associated

with this concept is not capable of barking, as it is merely a representation of the

concept in the mind. In the era of “modern mathematics,” attempts were made to

teach students the concept of numbers through the Cantorian construction of

cardinals. Although, if these students had relied solely on the official curriculum,

they would have left school without acquiring a true understanding of numbers.

Instead, they needed to develop a mental object of numbers outside of what the

educational programs intended to teach them.

The transcript presented was about an exceptional individual who

possesses superior abilities and performs actions that exceed our own capabilities.

To understand this concept, we need to explore the contrast between mental

objects and mathematical concepts. In everyday language, we typically refer to a

person's understanding of a concept as their "concept," or we may use the term

"conception" to emphasize that it is their personal interpretation of the concept.

However, for the purposes of this discussion, we will use the term "mental object"

to highlight that it is a part or perspective of the concept that exists within the

individual's mind.

Thus, mental objects and concepts can differ in their interpretations and

applications. By exploring the various contexts in which numbers are used, we

can gain a deeper understanding of the meanings they have in different

situations. To illustrate the difference between mental objects and concepts, I will

use the complex and multifaceted concept of numbers as an example. Rather than

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following Freudenthal's approach, we will describe it in semiotic terms. Numbers

are not only used in mathematical activities or in the classroom; they are also used

in a variety of everyday contexts. These contexts can include sequences, counting,

cardinals, ordinals, measurements, labels, written digits, magic, and calculations.

Although I will not go into the details of each context, I do want to emphasize that

there are distinct characteristics and meanings associated with numbers in each

situation.

The semantic field of "number" encompasses all uses of numbers in various

contexts, defining their encyclopedic meaning. Understanding the specific

context in which a number is used enables the reader or receiver of a message to

interpret it correctly. However, individuals do not have access to the full range of

number uses in a culture or episteme. Rather, they operate within their own

personal semantic field, which evolves and produces meaning in situations that

require new uses of numbers.

This personal semantic field aligns with what Freudenthal calls “mental

object number.” Freudenthal suggests that educational systems should aim to

develop students’ personal semantic fields to encompass a wide range of number

uses, ensuring that they can correctly interpret any situation involving numbers.

Mundane uses of numbers occur in diverse contexts where phenomena are

organized using the concept of number, both in its original form and in its

extended applications.

The concept of mental objects helps organize these phenomena, allowing

individuals to engage in activities such as numeration. Mental objects are formed

through chains of phenomena and means of organization, like concepts. While

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the mundane contexts of number use mentioned above represent lower levels, the

phenomenological richness of numbers in secondary school requires

consideration of additional contexts, including mathematized ones. This

explanation provides an initial understanding of what a mental object is and how

it is formed. However, Freudenthal's term could simply have been called

something else.

It should be emphasized that this is not the only difference between mental

objects and concepts, and I do not mean to suggest that the relationship between

them is simply a matter of relating a part of the mental object's content to its

whole. Before delving into other aspects of this relationship, however, it is

important to note that this account serves as the basis for Freudenthal's position,

which I mentioned earlier in this section. According to Freudenthal, the

acquisition of concepts is a secondary educational goal that can be postponed

until a solid mental object has been established.

The relationship between mental objects and concepts is more complex

than the number example I just provided, as my explanation has focused solely

on comparing the expansion of the semantic field of number and Peano's

definition, neglecting centuries of history that have shaped both contexts in which

number is used, which retain traces of its organization through number concepts,

and Peano's definition. The concept of number is a topic that requires further

clarification. To introduce a term that distinguishes it, it is important to explain

what other meanings the term "concept" has and how it differs from what we have

called "mental object."

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We have already established that mental objects exist in people's minds,

while concepts exist in the realm of mathematics. However, this distinction alone

would not be sufficient if we believed that mental objects are simply reflections

of concepts in people's minds. The relationship between mental objects and

concepts is more nuanced than that. Let me explain it in terms of semiotics. The

mental object of number has been equated with the personal semantic field, which

is made up of the various uses of numbers in different contexts and the culturally

established meanings associated with them.

Mathematical concepts of natural numbers, such as those developed by

Peano, Cantor or Benacerraf, for example, are the result of a long historical

process. In the previous sections, I have examined the creation and modification

of these concepts. From the semiotic perspective I am employing here, any

mathematical concept of number that we wish to examine appears as the result of

a definitional process that incorporates it into a deductively organized system,

extracting it from the broader semantic field.

For example, Peano's concept of natural numbers, particularly in its more

modern versions, breaks down meaning from the sequence context and presents

it as a series of axioms that comprehensively describe its components. On the

other hand, the concept of natural number derived from Cantor's construction is

tied to the cardinal context, as indicated by the name Cantor originally gave it. In

this account, concepts are related to a portion of the mental object, since the

process of defining them involves selecting specific aspects of the meaning that

the mental object encompasses.

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When considering the processes involved in the creation and modification

of concepts in history, it is important to understand the relationship between the

mental object formed from these contexts and the concept of number defined by

Peano. This relationship goes beyond a simple part/whole dynamic. A well-

constituted mental object is one that can account for all uses and phenomena in

diverse contexts.

The goal of educational systems, as described by Freudenthal, is to

establish these good mental objects. To fully understand a concept, it is necessary

to examine how it has developed in mathematics and how it has been organized

within a deductive system. The specific relationship between a mathematical

concept and its corresponding mental object influences how the mental object is

constituted in relation to the acquisition of the concept. The constituents of a good

mental object are determined through phenomenological analysis of the concept

in question.

The analysis that has led us to differentiate between mental objects and

concepts is a didactic analysis, focused on examining mathematics and its

structures in the context of education. We recognize that students develop mental

objects, which are their own personal understanding and organization of

knowledge, and we also recognize the socially and culturally established content

that we want students to learn, which we refer to as concepts.

Our exploration of the contrast between mental objects and concepts is

done specifically within the educational system. In the context of the story,

mathematical concepts are not something that exists before our experiences but

are created through the mathematical activity of mathematicians. In this sense,

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mathematical concepts are simply crystallizations of mental objects. It is even

possible, as Freudenthal suggests, for mathematicians to work with a mental

object for a long time before formalizing it into a concept, as was the case with the

concept of continuity.

So, what does it mean to “turn a mental object into a concept” in this

context? I now approach the distinction between mental objects and concepts

differently than when I initially examined it. I have associated mental objects with

individuals because they are formed through subjective experiences and serve as

a way for individuals to make sense of and have control over their experiences.

On the other hand, concepts are associated with mathematics as a

discipline, but they also serve as tools for organizing phenomena. In the school

system, concepts are introduced to students before they have direct experiences

with the corresponding phenomena. The goal of the system is for students to

develop mental objects that align with established concepts and to have access to

the valuable means of organizing phenomena that history has provided us.

The topic at hand is the school system and its role in the creation of new

mathematical concepts. This discussion focuses on the analysis of the mental

objects used by mathematicians to organize phenomena and define them

conceptually within the field of mathematics. It is through mathematical activity

that concepts are derived from these mental objects.

This idea is also reflected in Lakatos' work, "Proofs and Refutations,"

where he mentions that the concept of a polyhedron is assumed to be understood,

despite not being explicitly defined. As mathematicians prove theorems and

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explore diverse ways of modifying concepts, they find counterexamples that

challenge and shape their mental objects, leading to the creation of new concepts.

It is common for discrepancies to arise between the mental object and the

concept derived from it. In such cases, one may realize that one's mental object

was not as well-formed as initially believed, prompting one to modify or revise

the conceptual definition. For example, Freudenthal highlights the concept of

continuity as a case in which a mismatch occurs between the mental object and

the concept created.

When the first explicit definition of continuity was introduced, numerous

examples of continuous functions that had not previously been considered as

such emerged. However, over time, successive generations of mathematicians

become accustomed to these new and unconventional cases of continuity. They

reshape their primitive mental object to align with the concept defined by the new

functions. It is important to note that the primitive mental object remains essential

to the advancement of mathematics and is not simply replaced by the concept.

Instead, a new mental object is formed that encompasses the concept created by

the definition, or at least is provisionally compatible with it.

The relationship between mental objects and concepts can vary greatly.

Mental objects are used to organize phenomena and precede concepts. However,

concepts do not replace mental objects but contribute to the creation of new

mental objects that include or are compatible with them. Sometimes, there is a

significant gap between a mental object and a concept, as with the mental object

curve and the Jordan curve concept. In the field of topology, mental objects alone

do not provide deep understanding and it is necessary to form concepts that

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involve more than just local organization. These concepts fall into a broader field

of study.

Phenomena can be organized at a higher level by mental objects such as

spaces and manifolds of arbitrary dimension. These mental objects are then

transformed into concepts by further organizational processes and the creation of

more abstract sign systems to describe them. This transformation process is

evident when considering the idea of a mental object and the progressive ascent

through the chain of phenomenon/medium pairs of organization. However, it is

important to note that not all domains of mathematics require concepts to

progress.

Elementary geometry, for example, can be organized using only mental

objects, without the need for concepts. In this case, concepts can be formed

through local organizations, where the distances between mental objects and

concepts are resolved. For example, a rectangle can be understood as a mental

object without the concept of a "square," but a local organization can introduce

the concept of a square without conflict. However, there are also examples in

geometry where the distance between the mental object and the concept is

obvious.

This is the case for elementary concepts like point, line, and surface, which

closely resemble objects in the physical world. Physical objects, such as specks or

marks made with a sharp object, suggest mental objects that need to be delineated

and turned into concepts. This process is evident in Euclid's definitions of point,

line, and surface, where mental objects are transformed into concepts. It is worth

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noting that the path of cognitive development does not necessarily go from one

dimension to three, but rather the opposite.

Surfaces are experienced before lines, and mental objects associated with

lines are constituted through various phenomenological sources, such as surface

edges, arrows, threads, paths, and cuts. The difference between

phenomenological sources and Euclidean definitions highlights the distance

between mental objects and concepts. This distance becomes even greater when

one considers the mental object involved in the transition from one dimension to

another, specifically in the dimension itself. The transformation from dimension

to concept is achieved in topology, where there is a significant distance between

the mental object and the concept.

However, there is also a new type of property that arises when the

properties evident in the mental object are difficult to prove, such as the n-

dimensional Cartesian product of n segments. This creates a significant gap

between the mental object and the concept of dimension. The gap becomes even

more pronounced when fractional dimensions are introduced, forcing us to

discuss dimensions in a non-traditional sense.

When analyzing didactic phenomenology, it is crucial to consider pure

phenomenology, recognizing that there are cases where the distance between

mental object and concept is insurmountable. This is especially true in secondary

school teaching. Understanding the formation of mental objects through teaching

involves recognizing the various forms this distance can take.

Alongside the examples mentioned above, it is important to note that there

are cases where certain components are essential for the concept but not relevant

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for the constitution of the mental object. For example, the comparison of

unstructured sets is crucial for the concept of cardinal numbers, but it plays a

minimal role in the formation of the mental object because real-life situations

rarely involve unstructured sets. Likewise, structure is necessary for making

comparisons rather than being eliminated. Furthermore, didactic

phenomenology reveals that the phenomena organized by a concept can be so

diverse that they constitute different mental objects depending on the field of

phenomena being explored in teaching. To fully understand the concept, it is

necessary to integrate these different mental objects into a unified whole.

When it comes to comparing plane figures in terms of area, there are

different methods available. A direct comparison can be made if one figure is

contained within another, or an indirect comparison can be made using

transformations, congruences, and other area-preserving techniques.

Alternatively, figures can be measured individually. Measurement can be

performed by covering the figure with unit areas or by approximations using

interior and exterior methods. In these cases, additivity of areas is used when

combining disjoint plane figures, or convergence of areas is considered when

using approximations. However, it is not clear whether these approaches lead to

the same measurement result and proving their equivalence is not a simple task.

Lengths, areas, and volumes are fundamental measurements in elementary

geometry.

These concepts are crucial in the study of measurement. The process of

measurement begins by comparing the qualities of objects. This comparison is

achieved by establishing a unit and considering objects as entities that possess the

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quality being measured. For example, if an object can be described as “long,” then

length can be attributed to it. However, understanding length, area, and volume

as concepts presents challenges due to the various approaches to defining the

mental objects of area and volume. In addition, there are mental objects that only

manifest themselves in a mathematical or mathematized context. Analytic

geometry serves as an example in secondary education.

The use of coordinate systems for global localization led to the

algebraization of geometry throughout history. The Cartesian coordinate system

is effective for describing geometric figures, mechanical motions, and functions.

Geometric properties are expressed algebraically through relations between

coordinates, motions in time-dependent functions, and geometric applications

within systems of functions with multiple variables. The characteristic

phenomena of analytic geometry can only be explored within mathematized

contexts, using the sign systems of algebraic expressions and Cartesian

representations.

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Chapter 2

Freudenthal: Fundamentals of Algebra and Geometry

The number

It is important to recognize the multifaceted nature of numbers and their

various interpretations and applications to fully understand their meaning and

implications. The development of numerical concepts in secondary school can

only be achieved through the formation of solid mental objects, traversing the

semantic field of "number." It is important to note that this process of mental

object formation is not a one-time event that results in an immutable mental

object.

The meanings of numbers can be modified by factors such as the use of

negative numbers in the ordinal and sequence context, or decimal expressions in

the measurement context. Furthermore, it is important to recognize that in the

Secondary School numbers are not only used in the contexts mentioned above,

but also in other mathematized contexts. Exploring the concept of "number" in

natural language requires a didactic phenomenology that considers how mental

objects are formed and integrated to account for the diverse uses of "number" in

different contexts (Burkhardt, 1988).

This exploration begins outside of school and continues throughout the

Primary stage. To introduce a phenomenological analysis relevant to the

Secondary stage, it is necessary to make a brief reference to these initial

phenomena and to study how the meaning of "number" is extended and modified

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by the new phenomena that students encounter at this stage. The extension of the

meaning of numbers reaches its peak in what can be called algebraic access to the

concept of number. This involves organizing arithmetic operations and

constructing a higher-level concept of number that is distinct from those formed

in previous contexts.

In this context, a number is defined as something that allows for arithmetic

operations. This new concept of number, which has historical significance,

legitimizes all numbers that were previously considered alien to the true concept

of number and were consequently given different names. As mentioned above,

the mental object of “number” is formed to organize various phenomena, which

can be understood by examining the objects to which numbers refer (individual

objects, sets, words, the numbers themselves). This includes considering whether

the objects are discrete or continuous, whether they are ordered or not, and the

nature of the units and the number that describes the objects. These characteristics

can be categorized into different contexts of number usage, such as cardinal,

ordinal, measurement, sequence, counting, label, magic, and reading contexts.

The combination of all these usages in different contexts forms the semantic field

of “number,” and an individual’s experiences contribute to the formation of his

or her mental object of “number.”

The concept of exclusion, which refers to the idea of considering non-

numeric objects as numbers, has been around since the 11th century. During this

time, Arab algebraists such as al-Karajī were already treating algebraic objects as

numbers. This idea was further developed by Cantor, who aimed to show that

the transfinite numbers he had introduced were in fact legitimate numbers.

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The concept of “number” also applies to all basic arithmetic operations.

Thus, addition and subtraction, their meanings are derived from the actions of

counting, combining sets, and comparing sizes within a field of meaning that is

integrated through corresponding teaching tools such as the number line.

However, multiplication and division have even more diverse meanings. Unlike

“number,” the meanings of these operations are influenced by new phenomena

and diverse types of numbers. For example, when dealing with fractions or

decimal numbers, we need to expand our understanding of “number of times” to

understand multiplication. Furthermore, in more advanced mathematical

contexts, operations are often seen as extensions of algebraic concepts.

A ratio is a mathematical function that relates two numbers or values. It

can be calculated using basic arithmetic operations, but what matters is the value

that the function assigns to each pair, which can be determined using algorithmic

procedures. However, if we interpret a ratio as simply the result of a division, we

lose sight of its true meaning. The importance of a ratio lies not in the process of

assigning a value, but in the ability to compare ratios of equality or inequality,

without knowing their specific magnitudes.

The concept of ratio allows us to express statements like "a is to b as c is to

d" without reducing these comparisons to numerical values. From a

phenomenological perspective, the logical status of ratio can be described in

terms of the equivalence relation "having the same ratio." This is consistent with

Euclid's definition in Book V of the Elements, where he does not define "ratio"

itself, but rather the notion of ratios having the same ratio. The logical status of

ratio is therefore considered to be at a higher level than that of numbers, fractions,

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lengths, and other concepts typically encountered in education. This higher level

is characterized by an intensive property, which organizes the relationships

between objects or sets of objects rather than focusing on their extensive

properties.

The variety of intensive properties of objects organized by reason is

enormous and encompasses a wide range of factors. In the context of teaching, it

is important to consider an important division within these properties: the

relation can be a relation within a single magnitude or between multiple

magnitudes. This division can be represented by two measurement spaces or

magnitudes, with a linear application connecting them.

The relationship within a magnitude is considered internal, while the

relationship between the two magnitudes is considered external. A proportion

involves a linear function that relates these measurement spaces. Linearity

implies that the internal ratios remain constant under the function, and the

external ratios between the elements to which the function corresponds are also

constant.

Linearity is demonstrated through the implicit concept of equal spaces

traversed in equal times for internal ratios, and through the explicit function

f(x)=αx, where α represents a constant, for external ratios. Furthermore, a didactic

phenomenology reveals that the development of the ratio and proportion mental

object involves precursor mental objects. These precursor mental objects are often

qualitative in nature and involve comparisons of ratios, providing a context in

which the concept of equality of ratios, or proportion, can be understood. An

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important precursor mental object, identified by Freudenthal as the "relatively"

mental object, plays a crucial role in this process.

The concept of "relatively" allows for meaningful statements, such as that

one chocolate is sweeter than another because of its higher sugar content. The

term "relatively" refers to a criterion of comparison, which may be implicit or

explicit, such as weight. The mental object of "relatively" is developed in teaching

through several steps, including understanding that dispositions can be

relativized, understanding the meaning of "relatively" as "in relation to...", using

"relatively" and "in relation to" with understanding, providing appropriate

completion of these terms within a given context, operational knowledge of their

meaning, and the ability to explain their meaning to others.

Algebra

Modern algebra is a field of study that categorizes and analyzes the

structural features of collections of various objects, incorporating defined

operations within them. These properties and objects are derived from the way

lower-level phenomena are organized and have evolved over time. A significant

event in the history of algebra is the writing of the Concise Book of al-Jabr and

al-Muqabala by al-Khwarizmi in the ninth century. This marks the birth of

algebra as a distinct discipline within mathematics.

Al-Khwārizmī's approach to algebra is unique in that he establishes the

diverse types of numbers needed for calculations, such as treasures, roots, and

simple numbers. He then explores the combinations of these types and develops

algorithms for solving each type. These types serve as canonical forms to which

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any problem can be reduced. Al-Khwārizmī's contribution lies not in the methods

of solution, but in the establishment of a comprehensive set of solvable canonical

forms and the organization of their application to problem solving.

Another significant advance in algebra is attributed to Galois, who moved

from searching for novel solutions to studying the solvability conditions of

equations. The history of algebra since then can be seen as a series of advances

resulting from the objectification of earlier level organizational methods.

However, this historical perspective is not particularly relevant to algebra taught

in today's high school curriculum, as it has abandoned the modern algebra

introduced in the 1970s.

Instead, it is more important to analyse the characteristics of natural

language and the sign systems of school arithmetic, as they provide the basis for

students to acquire the language of algebra. To carry out this analysis,

Freudenthal examines various aspects, such as transformation rules in natural

language, arithmetic language, language as action, formalization as a means and

as an end, the algorithmic construction of proper names, punctuation rules, and

the use of variables in everyday language.

In the language of mathematics, variables play a significant role. The equal

sign is another crucial element that helps establish equality between different

mathematical expressions. Algebraic strategies and tactics are employed to solve

equations and manipulate mathematical expressions. Formal substitution is a

technique used to replace variables with specific values.

The algebraic principle of permanence states that if two expressions are

equal and one is modified, the other expression must also be modified in the same

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way to maintain the equality. Algebraic translation involves converting posed

problems or real-life situations into algebraic expressions or equations. Moving

on to linear algebra, it is considered a practical tool rather than a historical study

of its origins.

Rather than exploring the historical context of linear algebra, the focus is

on its applications and how it can organize and make sense of phenomena in

various fields. The concept of linear algebra can be understood by examining its

meaning in specific application contexts. For example, the product of matrices can

be given significance by applying it to graph connectivity matrices. This

understanding can then be extended to other contexts without relying on the

specific content but on the expression and manipulation of matrices.

Geometric objects

The concept of space, both as a mental construct and as a mathematical

idea, did not originate as a foundation of geometry. Rather, it is the result of a

prolonged process of development. According to Freudenthal, while geometric

objects exist within space as concepts, the corresponding mental objects

associated with these concepts are situated within a geometric context.

Our starting point, both in the historical sense and in the personal history

of everyone, is not phenomena that can only be experienced at a level already

defined by mathematics and organized by the concept of space. Rather, it is other

phenomena and contexts that come first. Initially, organized phenomena are

shapes and configurations that are observed in a visual context, such as contours

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and lines of sight. These are closely related to human creation itself, as humans

produce “geometric” shapes.

Geometrical objects, as concepts, develop from mental objects that serve as

tools for organizing the "geometrical" figures that are observed or drawn on

Earth. In defining these objects, their definitions must be separated from the

sensory properties of the figures they are intended to organize. For example,

Euclid defines a point as having no parts and a line as a length without width,

using properties that highlight deficiencies and detachments. This creates a

concept separate from the mental object that organizes the corresponding

phenomena. Therefore, any action or observation made with these figures must

be analyzed to be accepted as a geometrical object. For example, if a circle is

drawn on the ground or on paper and a tangent line is added to it, in the drawing

they do not intersect at a single point but "along their entire length," as Protagoras

argues.

To learn geometry effectively, drawings must be placed in geometric

contexts. Furthermore, the relationships between the drawing and the geometric

object cannot be learned without proper instruction. Colette Laborde proposes a

solution to this problem in the context of the Cabri-geometer. Students are

exposed to problem situations involving drawings, where geometry becomes a

useful tool for modelling and problem solving. For example, geometry allows the

creation of drawings that meet specific constraints in a more efficient way than

trial and error. The accuracy of the results can be ensured by geometric principles,

such as the tangency of a line to a circle when it is perpendicular to the radius.

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Likewise, students are also presented with geometry situations where drawing

and experimentation help to avoid lengthy theoretical solutions.

The very nature of the drawing also influences its interpretation. A

drawing can only refer to theoretical objects of geometry to the extent that the

reader chooses to interpret it as such. The interpretation is influenced by the

reader's chosen theory and his or her knowledge. And the context in which the

drawing is presented is crucial in determining the type of interpretation. A well-

developed mental object must incorporate an analysis of the elements of the

figure and the relationships between them. Also, the relationship between the

drawing and the geometric object is complex because it requires interpretation by

a human subject.

Consequently, a geometric drawing may not always be interpreted as

representing a geometric object, and interpretations of the same drawing may

vary depending on the reader, their knowledge, and the context. During Euclid's

time, mathematicians discovered that there was a significant gap between the

concept constructed by Euclid and the primitive mental object. However, they

also realized that geometric figures drawn on paper, known as geometric

drawings, were used to represent geometric objects. This relationship between

figure, drawing, and geometric object is crucial in the formation of corresponding

mental objects and the understanding of concepts.

In contrast, when a computer environment such as Cabri-geometer is

considered, the dynamics between drawing and geometric objects are altered, as

we have just discussed. Drawings created with Cabri on a computer screen,

known as Cabri-drawings, exhibit different behaviors compared to drawings

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made with pencil on paper. This is because Cabri drawings are not drawn

manually but are defined by program primitives that always align with geometric

properties.

The modification of a Cabri drawing resulting from the movement of its

elements disrupts certain interpretations of the drawing based on its spatial

properties. Moreover, in the experience of students, the scope of phenomena

organized by geometric objects in Secondary School is incredibly diverse and can

be found in various aspects of nature, art and human creations. Although we will

not make an exhaustive list, it should be noted that this abundance of geometric

phenomena has significantly expanded in recent times with the emergence of

infographic products such as video games, digital images used as covers of

television programs, music videos and computer games.

The movements

Geometric transformations involve the physical motion of geometric

figures. However, there is a complex and conflicting relationship between the

geometric characteristics of these transformations and the spatial properties of the

motions. This leads to difficulties in recognizing that different paths can result in

the same transformation, accepting certain transformations as identities. It is

important to consider this gap between real-world phenomena and mathematical

concepts of geometric transformations. There are many phenomena in the

students' environment that can be explored and that are relevant to

understanding geometric transformations.

Statistics

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Descriptive statistics has been developed for the purpose of organizing

numerical data provided and covers a wide range of social, political and economic

phenomena. These phenomena serve as contexts in which statistical concepts are

applied and it is important to consider them as educational opportunities, as they

allow individuals to experiment and form mental objects related to statistics.

Statistics concepts are primarily concerned with quantitative information

contained in data and aim to summarize, characterize, and organize it in a way

that facilitates comparison with other massive data sets. In everyday life,

statistical concepts can be observed in various media such as newspapers and

television, where they are used to describe different topics or predict voter

behavior during political campaigns.

These real-life applications contribute to the understanding and formation

of mental objects related to statistics. However, within the realm of statistical

inference, phenomena become more complex and abstract as they involve

deriving knowledge from observing features of cases. Ian Hacking has pointed

out the challenges of establishing statistical inference within the framework of

Galilean science as it requires navigating the concept of “acceptable evidence” in

different historical periods and social practices. Fisher’s contribution to this field

is significant as it introduces the idea that rejecting the null hypothesis is not

equivalent to completely refuting it. Instead, the alternative is to reject or err in

rejecting the null hypothesis.

The probability

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Probability has its roots in the realm of gambling and uncertain events,

such as lottery drawings or weather patterns. In everyday language, the term

"likely" can have two meanings: it suggests that something may happen, but it

also implies a personal belief that it will happen. The distinction between these

interpretations is often conveyed through the speaker's emphasis. Thus, the terms

"likely" and "possible" have a closely related meaning. Various perspectives,

including logicism, subjectivist, and frequentist views, have been proposed to

explain the concept of probability. These theories shed light on the diverse range

of phenomena that fall under the umbrella of probability.

Challenges are often encountered in defining the fundamental ideas of

probability and randomness. A solid understanding of randomness and

probability can be built using elements derived from Kolmogorov specifications:

An event may have multiple outcomes. The outcome of the event cannot be

predicted with certainty, even with the knowledge of those observing it. The

event may be repeated under identical conditions a substantial number of times,

allowing generalizable conclusions to be drawn. The sequence of outcomes

obtained during repetition does not show any discernible pattern that can be

predicted by the observer. As the number of repetitions increases, fluctuations in

the relative frequencies of outcomes become more stable and gradually decrease

in magnitude in a predictable manner.

The variables

The current practice in mathematics of referring to "variables" as means of

expressing general propositions, which are what Freudenthal calls "multi-

purpose nouns," is a recent development. Traditionally, the term "variable" has

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always denoted something that fluctuates or changes, whether in the physical,

social, mental, or mathematical realm. Initially, the concept of variables

encompassed observable phenomena in the physical, social, and mental realms,

but it was eventually expanded to include mathematical objects such as numbers,

magnitudes, and points, which are also considered variables.

The concept of function originates from the recognition of a relationship or

dependency between variables. This dependency may be established, postulated,

generated, or replicated within the physical, social, or mental domains, as well as

between mathematical variables that may also have connections to variables in

other domains. As this dependency is further explored, it can be objectified and

treated as a mental construct. Before this objectification can occur, however, the

dependency must first be experienced, used, stimulated, made conscious,

considered as an object, given a name, and situated within a larger framework of

interdependencies:

• Functions arose as relationships between magnitudes of variables whose

variability was compared in infinitesimal quantities.

• The freedom to change variables from dependent to independent and

between independents led to a new kind of operation with functions:

composition and inversion. It was this new operational richness that led to

the success of the function concept.

• The need to distinguish between dependent and independent variables has

led to an emphasis on functions rather than relationships. Despite what

algebraic and analytical expressions suggest, development has tended

toward univalent functions.

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• A change in perspective led from describing visual data through

analytically expressed functions to visualizing functions through graphs.

• An arbitrary function appears in the calculus of variations and solving

differential equations. This "arbitrariness" refers not only to the nature of

the functional dependence, but also to the nature of the variables, which

can be numbers, points, curves, functions, elements of arbitrary sets.

• Analysis functions, geometric transformations, permutations of finite sets,

and applications between arbitrary sets combine to create the general

concept of a function.

• This concept, in turn, is used to organize a wide variety of objects, from

algebraic operations to logical predicates.

This latter wealth of such diverse phenomena integrated into the general

concept of function, phenomena which, moreover, many of them belong to the

world of mathematics itself, makes function as a mental object much more

complex than the number of objects, geometric or even reason. The concept of

function can only be mastered at advanced stages of schooling when students

may already have had experience with many these phenomena. What can really

be constituted as a mental object in secondary education is the idea of variable

and functional dependence, and it is difficult to overcome even the geometric

transformations that are also being experienced.

By delving exhaustively into these concepts, which unfortunately cannot

be fully explored within the confines of this discussion, we realize the vast gulf

that exists between these concepts themselves and their origins in the initial

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phenomena and first mental constructs that emerge in the realm of mathematics

and in the firsthand experiences of individuals. This is reminiscent of Cantor's

famous comment to Dedekind, in which he confessed, intentionally, that he was

able to perceive the validity of a certain concept, but that he struggled to fully

embrace its reality.

The author's argument for the existence of infinitely different cardinals

using the diagonal method serves as a clear example of the challenges posed by

mathematical concepts such as continuity and infinity. These concepts have no

direct correspondences in our physical experiences, and it is unclear whether

infinity is a singular concept. Mathematicians developed these concepts to

understand and organize phenomena that occur within the realm of mathematics,

where objects are produced or occur in highly mathematized contexts. However,

teaching these concepts requires considering that a deep understanding of them

can only be achieved through experience of the phenomena they organize.

While the acquisition of these concepts may present difficulties, it does not

mean that they should be abandoned. Instead, a teaching approach should

involve creating a variety of experiences that encompass the various relevant

phenomena and organizing instruction in a way that allows for the formation of

a mental object capable of dealing with those phenomena. Furthermore, it is

important to recognize that many of the phenomena crucial to forming strong

mental objects are intrinsic to the mathematical means of organization

themselves, which can be studied at a higher level.

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Chapter 3

Perspectives on mathematics education

When considering one's ideas about mathematics, a variety of opinions

and beliefs about the subject, the mathematical activity, and the ability to learn

mathematics may emerge. This discussion may seem irrelevant to teachers who

are primarily concerned with improving the effectiveness of their teaching. The

concept of knowledge belongs to the branch of philosophy known as

epistemology, which explores theories of knowledge. However, beliefs about the

nature of mathematics can influence a teacher's actions in the classroom.

For example, imagine a teacher who believes that mathematical objects

have an existence of their own, albeit immaterial. From this teacher's point of

view, objects such as triangles, sums, fractions, and probabilities exist

independently of the people who use them or the problems to which they are

applied, and even independently of culture. According to this belief, the best way

to teach mathematics would be to present these objects to students, much as one

would show a child an elephant at the zoo or through a video.

How can we demonstrate what a circle or other mathematical object is?

This teacher would say that the best approach is to teach the definitions and

properties of these objects, as this is what constitutes “knowing math.” Applying

concepts or solving mathematical problems would be considered secondary and

would be taught once the student has mastered the mathematical objects

themselves. On the other hand, some teachers view mathematics as a product of

human ingenuity and activity, much like music or literature. They view

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mathematics as something that was invented because of human curiosity and the

need to solve a variety of problems, such as trading goods, construction,

engineering, and astronomy.

For these professors, the fixed nature of mathematical objects today, or in

earlier historical periods, is the result of social negotiation. The individuals who

created these objects had to agree on their rules of operation, ensuring that each

new concept fit coherently with existing ones. Moreover, the history of

mathematics shows that the definitions, properties, and theorems established by

renowned mathematicians are not infallible and are subject to evolution.

With this perspective in mind, learning and teaching mathematics must

recognize that students naturally encounter difficulties and make mistakes

during the learning process, and that these mistakes can serve as valuable

learning opportunities. This aligns with constructivist psychological theories

about mathematics learning, which are based on the philosophical view of

mathematics known as social constructivism.

The Platonic idealist conception

Within the wide range of beliefs about the relationship between

mathematics and its practical applications, as well as its importance in the field of

teaching and learning, there are two distinct and contrasting conceptions. One of

these conceptions, which prevailed among many professional mathematicians

until recently, asserts that students must first understand the fundamental

structures of mathematics in an axiomatic way.

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It is assumed that once this foundation is firmly established, the student

will find it easy to solve the various applications and problems that arise.

According to this perspective, without a solid mathematical foundation, it will be

difficult to apply mathematical principles except in extremely simplistic

scenarios. Consequently, pure and applied mathematics are perceived as separate

disciplines, with abstract mathematical structures taking priority over their

applications in the natural and social realms.

Practical applications of mathematics are considered an "appendix" to the

study of mathematics, and thus ignoring this appendix would not cause any harm

to the student's understanding. Those who subscribe to this belief view

mathematics as an autonomous discipline, capable of developing without

considering its applications to other sciences, based solely on internal

mathematical problems. This conception of mathematics is known as the "idealist-

platonic" perspective. This perspective allows for a simple curriculum, as there is

no need to incorporate applications from other areas (Gravemeijer, 1994). Instead,

these applications are "filtered" to abstract away mathematical concepts,

properties, and theorems, resulting in a "pure" mathematical domain.

The constructivist conception

Many mathematicians and math teachers argue that math should have a

strong connection to its real-world applications in various subjects. They believe

that it is crucial to demonstrate to students the practicality and relevance of each

math concept before introducing it. The idea is to show students how distinct

parts of math serve specific purposes. For example, by placing children in

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situations where they need to exchange items, we create a need for them to

compare, count, and organize collections of objects.

This gradually introduces them to the concept of natural numbers.

According to this perspective, applications of mathematics, both external and

internal, should come before and after the development of mathematical

concepts. These applications should emerge naturally in response to problems

encountered in the physical, biological, and social environments in which

humans live. Students should be able to recognize that axiomatization,

generalization, and abstraction in mathematics are necessary to understand and

solve real-world problems.

Proponents of this approach to mathematics and its teaching would prefer

to start with problems from nature and society and use them as a basis for

building the fundamental structures of mathematics. In this way, students would

gain a deeper understanding of the close relationship between mathematics and

its applications. However, developing a curriculum based on this constructivist

approach is complex because it requires knowledge not only of mathematics but

also of other fields.

Structures found in the physical, biological, and social sciences are often

more complex than those found in mathematics and do not always align perfectly

with purely mathematical structures. Although there is a wealth of material on

the applications of mathematics in a variety of areas, the task of selecting,

organizing, and integrating this material is not easy.

Mathematics and society

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Teaching mathematics involves not only imparting knowledge and skills,

but also fostering understanding and appreciation for the role of mathematics in

society and the power and limitations of the mathematical method. Mathematics

has a rich history of evolution to solve practical problems and has played a crucial

role in the development of various fields. Thus, concepts within mathematics

have also undergone changes over time, adapting to novel approaches and

advances in technology.

By recognizing these aspects, educators can provide students with a

comprehensive understanding of mathematics and its importance in society. This

pattern of mathematics evolving in response to various problems is not unique to

statistics. Geometry, for example, emerged from the need to solve agricultural

and architectural problems. Different numbering systems evolved along with the

need for faster arithmetic calculations.

Probability theory was developed to address problems related to

gambling. Mathematics serves as a framework upon which scientific models are

built, aiding in the process of modeling reality and often validating these models.

Mathematical calculations, for example, allowed the discovery of the existence of

distant planets in our solar system long before they could be observed. From a

historical perspective, mathematics is a constantly evolving body of knowledge.

Throughout its evolution, the need to solve practical problems (both

within mathematics itself and in relation to other disciplines) has played a

significant role. For example, the origins of statistics can be traced back to ancient

civilizations such as the Chinese, Sumerian and Egyptian, which collected data

on population, goods and production. Even the Bible refers to counting Israelites

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of military service age in the book of Numbers. Censuses themselves were already

established in the Roman Empire in the fourth century BC. However, it is only

recently that statistics has acquired the status of a science.

In the 17th century, political arithmetic emerged from the German school

of Conring. Achenwall, a disciple of Conring, focused his work on the collection

and analysis of numerical data for specific purposes, laying the foundations of

the statistical method. This example demonstrates how mathematics, including

statistics, has developed in response to various problems.

When considering the teaching of mathematics, it is crucial to reflect on

two important objectives:

• First, our goal is for students to understand and appreciate the role of

mathematics in society, including its diverse applications and

contributions to social development.

• Second, we strive to ensure that students understand and value the

mathematical method, which encompasses the types of questions that

mathematics can help answer, the fundamental ways of reasoning and

working mathematically, as well as their strengths and limitations.

However, the evolution of mathematics is not only the accumulation of

knowledge or the development of new fields of application. The concepts

themselves within mathematics have undergone changes in meaning over time,

being expanded, specified, revised, gaining relevance or, at times, being relegated

to the background. For example, the calculation of probabilities underwent a

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significant transformation with the incorporation of concepts from set theory into

Kolmogorov's axiomatics.

This innovative approach allowed the application of mathematical analysis

to probability, leading to advances in the theory and its practical applications in

the last century. Similarly, the manual calculation of logarithms and circular

functions (e.g., sines, cosines) used to be widely taught, with students spending

hours learning related algorithms. However, with the advent of calculators and

computers, these functions can now be calculated directly, making manual

calculation obsolete. The same trend is occurring today with the calculation of

square roots.

The purpose of mathematics

Mathematical applications play a vital role in our everyday lives, and it is

critical that students recognize and appreciate this. To achieve this, it is important

for us to provide comprehensive examples and situations in class that

demonstrate the wide range of phenomena that mathematics allows us to

organize and understand (Higginson, 1980).

Beyond the biological and physical context of the individual, we live in a

society filled with mathematical situations. Family, school, work and leisure

activities all involve mathematical elements. Numerical or statistical studies can

be conducted to quantify the number of children in a family, the age of parents at

marriage, the types of work people do and the different beliefs and hobbies within

different families. Public transport is relied upon to get to school or go on holiday,

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and we can estimate factors such as travel time, distance and the number of

passengers using buses.

During our free time, we participate in games of chance such as billiards

or lotteries, and we attend sporting events whose outcomes are uncertain, and we

may have to queue to buy tickets. Buying insurance policies, investing in the stock

market and making financial decisions are examples where statistics and

probability are essential tools.

One area where mathematics is applicable is biology. Students can be

reminded that many inherited characteristics, such as sex, hair color, and birth

weight, cannot be predicted in advance. Also, traits such as height, heart rate, and

red blood cell count can vary depending on when they are measured. Probability

allows us to describe and analyze these characteristics.

In the field of medicine, statistical epidemiological studies are performed

to quantify a patient's condition and track its progression using charts and

graphs. These conditions are then compared to average values in a healthy

individual. Determining the red blood cell count from a blood sample is an

example of situations involving proportional reasoning and the concept of

sampling.

Mathematical models of population growth are also used to make

predictions about world population, the possibility of whale extinction, the

spread of diseases, and even the life expectancy of individuals. Nature provides

us with numerous examples of geometric concepts, which can be abstracted and

studied. As students grow older, measurement activities can be proposed to help

them differentiate different magnitudes and estimate quantities such as weight

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and length. Governments at various levels need information to make informed

decisions. That is why censuses and surveys are conducted. From election results

to population censuses, a wide range of statistics influence government decisions.

Consumer price indices, labor force rates, immigration and emigration,

demographic statistics, and the production of different goods and trade are all

examples of ratios and proportions that we hear about daily in the news.

In the economic world, national and corporate accounting, as well as the

control and forecasting of production processes, are based on mathematical

methods and models. In our complex economy, it is essential to have a basic

understanding of financial mathematics. Everyday operations such as opening a

current account, subscribing to a pension plan or obtaining a mortgage loan

require this type of mathematics.

Beyond the biological context, our daily lives are also immersed in a

physical environment. Measuring magnitudes such as temperature and speed is

essential. The constructions that surround us, such as buildings, roads, bridges

and squares, present opportunities to analyse geometric shapes. The

development of these structures requires calculations, measurements, estimations

and statistical analysis.

Weather phenomena provide excellent examples of random events. The

duration, intensity, and extent of rain, thunderstorms, and hail, as well as

maximum and minimum temperatures and wind intensity and direction, are all

random variables. The consequences of these phenomena, such as the volume of

water in a swamp or the extent of damage caused by flooding or hail, also offer

opportunities to study statistics and probability.

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Mathematical culture

Placing problem solving and modelling at the forefront of mathematics

education has important educational implications. It would be contradictory to

present mathematics as something closed, disconnected from reality and

complete, considering its historical origins and current applications. It is crucial

to recognise that mathematical knowledge enables us to model and solve

problems from different fields, while problems originating in non-mathematical

contexts often form the intuitive basis for the development of new mathematical

knowledge.

Education aims at developing well-rounded citizens, and with the

evolution of the concept of culture in modern society, the role of mathematics is

gaining recognition. Mathematics education aims to cultivate this cultural aspect.

However, the aim is not to transform future citizens into “amateur

mathematicians” or to train them in complex calculations, as computers can now

perform such tasks (Gravemeijer, 1997). Instead, the aim is to equip people with

the ability to critically interpret and evaluate mathematical information and

arguments supported by data found in various contexts, such as the media or

their professional work.

Furthermore, mathematics education aims to improve communication

skills, enabling people to discuss and transmit mathematical information

effectively, as well as to solve mathematical problems encountered in daily life or

in professional settings. From a pedagogical and epistemological perspective, it

is crucial to differentiate between the process of constructing mathematical

knowledge and the characteristics of well-developed mathematical knowledge.

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Formalization, precision, and lack of ambiguity are traits associated with

mathematics as a mature science. However, the construction of mathematical

knowledge, both historically and in the process of individual learning, is

inseparable from concrete activities, intuition, and inductive approaches

activated through tasks and problem solving. Gaining experience and

understanding of mathematical notions, properties, and relationships through

real-world applications is not only a prerequisite for formalization, but also a

necessary condition for correctly interpreting and utilizing the potential within

formal mathematical structures (Mosterín, 1987).

When it comes to teaching mathematics, it is crucial to adapt the approach

to the age and knowledge level of the students. Different individuals have

dissimilar needs and perceptions of the physical and social environment, as well

as different interests compared to adults. Therefore, mechanically transferring

"real" situations, even if meaningful to adults, may not capture students' interest.

The historical construction of mathematics highlights the importance of

empirical-inductive reasoning, which often plays a more active role in developing

new concepts compared to deductive reasoning. Mathematicians do not

formulate theorems on the first attempt. Instead, they rely on previous trials,

examples and counterexamples, specific case solutions, and the ability to modify

initial conditions to investigate outcomes. These intuitive processes provide

invaluable clues for developing propositions and theories. Neglecting these

intuitive procedures, as observed in some curricular proposals, deprives students

of a powerful tool for exploring and constructing mathematical knowledge

(English, 2008).

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Language and communication

Mathematics, like other scientific disciplines, encompasses a body of

knowledge that has its own unique characteristics and internal organization.

What distinguishes mathematical knowledge is its incredible ability to be

communicated concisely and unambiguously. This is made possible by using

various systems of symbolic notation, such as numbers, letters, tables, and graphs.

By using these tools, mathematics can accurately represent a wide range of

information, drawing attention to aspects and relationships that may not be easily

observable.

Thus, mathematics enables us to anticipate and predict future events,

situations, or outcomes that have not yet occurred. Take, for example, the

expression “2n” to represent an even number. This simple equation is equivalent

to (n+1) +(n-1), which reveals that every even number can be expressed as the sum

of two consecutive odd numbers. However, it would be a mistake to assume that

the power of mathematical knowledge lies solely in its precise and unambiguous

symbolic notations. The ability of these notations to represent, explain, and

predict is deeply rooted in mathematical knowledge itself, and the notations

function as a supporting framework.

The emphasis on the need for constructive activity should not lead to

ignoring the internal structure of mathematics, which serves to connect and

organize its various components. Indeed, the structure of mathematics is

particularly complex and meaningful. This structure has a vertical aspect, where

certain concepts build on others, creating a temporal sequence in learning.

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Sometimes it requires working on certain aspects only to integrate others,

which are considered more important from an educational point of view.

However, it is important to note that there is rarely a single or clearly superior

path, and if there is, it is usually based on pedagogy rather than epistemology.

On the contrary, certain conceptions about the internal structure of mathematics

can hinder learning, as demonstrated by the attempt to base all school

mathematics on set theory.

Another implication of the relational nature of mathematics is the existence

of general strategies or procedures that can be applied in different fields and for

different purposes. For example, concepts such as number, counting, ordering,

classifying, symbolizing, inferring; are equally useful in geometry and statistics.

To help students recognize the similarity and usefulness of these strategies and

procedures from different perspectives, it is important to carefully select the

teaching content and pay special attention to it.

The foundation of logical-mathematical knowledge lies in the human

being's ability to establish relationships between objects or situations based on

their interaction with them. This ability includes the ability to abstract and

consider these relationships above others that may also be present. For example,

in the statements "A is larger than B", "A is three centimeters larger than B", "B is

three centimeters shorter than A", we are not simply describing properties of the

objects A and B themselves.

Instead, we are expressing the relationship between a shared property -

size - which is the result of comparing objects in terms of this property, ignoring

many other properties such as color, shape, mass, density, volume. Things such

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as "larger than", "smaller than", "three centimeters more than", "three centimeters

less than", are mental constructs rather than direct observations of the object's

properties. Even describing objects, A and B as large or small involves comparing

them to other similar objects one has encountered in the past.

This simple example illustrates how mathematical knowledge involves the

construction of relationships through interaction with objects. Mathematics is

therefore more about construction than deduction in terms of its development

and acquisition (English & Sriraman, 2010). If we separate mathematical

knowledge from its constructive origins, we risk reducing it to pure formalism,

losing its potential as a tool for representation, explanation and prediction.

Throughout its historical development, mathematics has revealed an

additional characteristic: its ability to provide us with a dual perspective of

reality. On the one hand, mathematics is considered an "exact science" where the

results of operations and transformations are unambiguous. However, when we

compare mathematical models with real-world situations, they are always

approximate due to their inability to perfectly represent reality.

While some aspects of this duality may be apparent in students’ first

encounters with mathematics, others may become apparent later. Unfortunately,

many educational curricula tend to prioritize one side of this coin, favoring the

traditional view of mathematics as an exact science. This means that concepts

such as certainty (“yes” or “no,” “true” or “false”) are often emphasized over

probability (“it is possible that…”, “with a significance level of…”) and precision

(“the diagonal measures 2,” “the area of a circle is πr^2”) is prioritized over

estimation (“I am wrong by at most one-tenth,” “the golden ratio is

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approximately 5/3”). It is crucial that mathematics education embrace both

approaches, not only because they provide a wealth of intrinsic value, but also

because the neglected side of the dual perspective has significant implications for

the practical applications of mathematics in today’s world.

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Chapter 4

A realistic mathematics perspective in early childhood education

Traditionally, mathematics education has focused primarily on

memorization to get students to solve exercises and pass tests. Unfortunately, this

approach has led to a lack of transferability of mathematical knowledge to real-

life situations, leaving many people struggling to apply what they learned in

school to everyday scenarios where mathematical understanding is crucial.

The consequences of this teaching model are evident in international

assessments of mathematical performance. For example, in the TIMMS 2015

study, Spanish students in the fourth grade of Primary Education obtained 505

points, while the OECD average was 525 points. Similarly, in the PISA 2015 study,

Spanish results were four points below the OECD average (490 points), according

to data from the Ministry of Education, Culture and Sport (2016a, 2016b).

The poor mathematical performance of primary and secondary school

students is believed to be partly attributable to the way mathematics is taught in

early childhood education. In many cases, the predominant approach to teaching

mathematics at an early age still revolves around memorizing basic skills through

repetition. To a lesser extent, the conceptual approach is also used, which

emphasizes learning procedures with understanding through manipulation and

experimentation with materials.

While this approach may occasionally present activities without a clear

context or purpose, there is an effort to promote meaningful learning. However,

this learning often fails to materialize due to inadequate teacher management. In

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addition to these approaches, current curriculum guidelines suggest that drawing

knowledge from one's own experiences and considering the contexts of daily life

is crucial for effective mathematics teaching and learning, especially at an early

age.

In this sense, current curricular guidelines indicate that the objective is to

involve children in the process of discovering and representing the various

contexts that make up their environment, facilitating their integration and

participation in them (Bressan et al., 2016). In the field of mathematics education,

this perspective aligns closely with the principles of Realistic Mathematics

Education (RME) developed by Freudenthal. RME advocates learning

mathematics by actively participating in real or realistic contexts, such as

everyday life situations, which resonate with students' experiences and

understanding.

Realistic mathematics education is based on the belief that mathematics

should have practical value, be relatable to children and relevant to society.

According to Freudenthal, although not all children become mathematicians, all

adults use mathematics to solve everyday problems.

In line with these principles, several authors have described EMR based on

the following ideas:

• Activity: Mathematics is a human activity, and its main objective is to

organize the world around us through mathematical concepts.

• Reality: Mathematics is best learned in real-world or realistic contexts.

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• Levels: Understanding mathematics involves progressing through diverse

levels, including the situational level (understanding within a specific

situation), the referential level (using models and descriptions), the general

level (exploration, reflection, and generalization), and the formal level

(using standard procedures and notation).

• Guided Reinvention: Formal mathematical knowledge is developed

through the guidance and mediation of the teacher.

• Interaction: Teaching mathematics is considered a social activity, where

interaction between students and between students and teachers can

enhance understanding.

• Interconnection: Different mathematical topics should not be taught in

isolation but rather integrated.

Children should learn mathematics in real and meaningful contexts, which

allow them to develop concepts and apply rules. This approach emphasizes the

need to transfer problems from everyday life to the realm of mathematics, solve

them, and then transfer the solutions back to the real world, thus familiarizing

students with the mathematical world.

The measure

From the process of comparison and using an object or one's own body as

a unit of measurement, one can begin to quantify a given measurement. This

involves the use of familiar and, eventually, conventional anthropometric units.

Estimation is also a key aspect of the quantification process. Contrary to

widespread belief, estimation is not simply a guess. It involves making value

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judgments and approximations based on previous information or experience. It

requires mental operations and the use of quick and simple numbers.

An estimate may not provide an exact value, but it allows for decision

making and people can interpret it differently. According to Alsina (2006), the

process of learning about magnitudes with understanding can be divided into

three phases:

• The first phase involves preparing for measurement, during which

students identify magnitudes through activities that involve comparisons

using quantifiers such as "more than," "less than," and "equal to."

• The second phase is quantification of measurement, where units are

introduced. Initially, children use units related to their own body, such as

spans and steps, before moving on to conventional units of the decimal

metric system, such as meters, grams and liters.

• The final phase is the decimal measurement system, which is not normally

addressed in early childhood education. In this phase, multiples and

submultiples of reference units for different magnitudes are introduced.

Measurement is a branch of mathematics that encompasses knowledge

and activities related to continuous quantities and measurable attributes

commonly found in everyday life, such as length, area, volume, capacity, mass,

and time. It is closely related to geometry as it involves understanding space, as

well as numbers and operations, since measurements are expressed using

numerical values. Measurement is also strongly linked to our understanding of

the natural world.

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Clements and Sarrama (2014) propose learning trajectories for early

childhood measurement that help define what mathematical concepts children

can understand, how they understand them, and how adults can support their

understanding. These trajectories describe the developmental progression and

the age at which ideas are typically acquired. For example, children can recognize

lengths by age 3, make direct and indirect comparisons of lengths by age 4, order

lengths sequentially by age 5, and finally measure lengths by age 6.

Similar progressions exist for area, volume and angles. Interestingly, it is

often wrongly assumed that the acquisition of measurement knowledge occurs

outside of school, as it is often delegated to the family and social environment.

However, this learning often does not occur. Unfortunately, measurement is often

introduced and taught algorithmically at later stages of primary education,

focusing on unit transformations and losing the true meaning and significance of

measurement.

Therefore, it is of utmost importance to incorporate systematic instruction

in measurement from an early age, encouraging students to interpret and interact

with their environment and real-life situations. This includes observing,

comparing, and evaluating results to give measurement knowledge true

importance and meaning. In addition to these ideas, it is also valuable to consider

international curricular guidelines, such as those provided by the National

Council of Teachers of Mathematics, the Common Core State Standards Initiative,

and current curricular policy guidelines in Spain, to achieve a comprehensive

understanding of measurement concepts in early childhood education (Hurford,

2010).

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According to Alsina, physical exploration of the attributes of objects is

prioritized primarily. This includes activities such as discriminating between

different attributes of objects and materials, identifying qualities. These activities

help children ages 3 to 6 develop an understanding of attributes in general,

particularly measurable attributes. This understanding is primarily achieved

through direct experiences in comparing objects, counting units, and making

connections between spatial concepts and numbers.

However, these contents focus on qualitative relationships between objects

and do not explicitly address magnitudes. Developing interest and curiosity

about measuring instruments. In general terms, the curriculum guidelines cover

the main aspects that should be considered when teaching measurement to young

children. However, they place little emphasis on the measurement process itself,

which is the same for any measurable attribute: selecting a unit, comparing it to

the object, and recording the number of units. In relation to the examination of

these contents, Alsina asserts that it is crucial to recognize the measurable

attributes that young children can understand at an early age. These attributes

include length, volume, weight, area, and time. It is important to involve children

in various comparison activities, introduce them to the concept of quantification

in measurement, and gradually expose them to several types of units, starting

with anthropomorphic units and then progressing to conventional units. In

addition, it is essential to encourage the practice of measurement through both

direct and indirect measurement techniques, using appropriate instruments.

The European Higher Education Area has established a system that aims

to improve the quality of universities in Europe. In this system, students are the

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focal point and take responsibility for their own learning. They actively build

knowledge by connecting latest ideas with existing ones, while the teacher acts as

a facilitator.

This shift in the approach to teaching and learning in universities is not

simply the result of discussions between ministers about what students should

learn and how, but has its roots in educational research, including research in

mathematics education. One specific model that has emerged from mathematics

education research is realist learning.

The model, based on Realistic Mathematics Education (RME), aims to

provide an active and practical training experience. The theoretical foundations

of realistic learning were established through the Comenius Project 2003-2005, led

by Professor Ko Melief of Utrecht University. From this project, Esteve, Melief

and Alsina (2009) coordinated a book that not only describes the theoretical

foundations of realistic learning, but also provides practical tools and techniques

for implementing it in university classrooms. The book also includes accounts of

teacher training experiences in various fields, including mathematics education.

Research in mathematics education, such as the work of Kilpatrick (1992), has

played a key role in shaping this original approach. Mathematics education is

often considered important in the school curriculum because of its relative

independence from external influences, its hierarchical and cumulative nature, its

abstract and arbitrary concepts, and the range of complexity it offers in learning

tasks.

This theory, however, does not claim to be a comprehensive learning

theory like constructivism. It was developed at the Institute for the Development

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of Mathematics Education at Utrecht University, which is now called the

Freudenthal Institute. The most noteworthy features of EMR can be summarized

as follows: it uses everyday life situations or contextual problems as a starting

point for learning mathematics. The situations are gradually mathematized using

models, which act as mediators between the abstract and the concrete, ultimately

forming more formal and abstract structures (Gravemeijer & Terwel, 2000).

Thus, EMR emphasizes classroom interaction, both among students and

between teacher and students. This intense interaction allows teachers to

incorporate students' productions into their teaching. Another key idea of EMR is

to give students the opportunity to reinvent mathematics under adult guidance,

rather than simply being presented with pre-constructed mathematical concepts.

During its initial stage, EMR was characterized using contexts to bridge

the gap between concrete and abstract concepts, the use of models to facilitate

progress, and the incorporation of students' free constructions and concepts into

the teaching and learning process. The various axes of the mathematics

curriculum were also intertwined. EMR initially consisted of basic ideas focused

on the how and what of mathematics teaching. Over time, these ideas were

accumulated and revised, leading to the formation of EMR as we know it today.

The book “Realistic Learning in Initial Teacher Training” by Esteve, Melief

and Alsina, proposes that pre-service teachers should be exposed to diverse ways

of acting and actively practice them. They should develop criteria to determine

when, what and why certain actions are appropriate, and should engage in

systematic reflection. According to this perspective, experiences and practice

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serve as a basis for professional learning and are essential for pre-service teachers

to align theory, classroom practice and their own personal characteristics.

The first principle emphasizes the importance of starting with real

classroom experiences and problems faced by preservice teachers. Instead of

introducing theoretical concepts from textbooks, training begins by actively

engaging preservice teachers in specific classroom situations and encouraging

them to reflect on their thoughts, feelings, needs, and interests. This inductive

approach aims to establish a connection between these experiences and preservice

teachers' future professional roles (Parra, 2013).

The second principle focuses on promoting systematic reflection. It

suggests that learning from experiences is a natural process, but it must be guided

intentionally. Here, five phases are identified in this reflective process: action or

experience, looking back at the action, becoming aware of important aspects of

one's own performance, seeking alternative behaviors, and testing their

effectiveness in new situations. Each cycle of reflection begins with a new

experience and contributes to professional growth.

The third principle highlights the social and interactive nature of learning.

Group discussions and interactions among preservice teachers are crucial to

promoting reflection. By sharing their experiences, preservice teachers are

encouraged to structure their thoughts, explore different perspectives, and

receive feedback from their peers. These reflective interactions deepen the

intentional purpose of professional learning and contribute to collective reflection

and construction.

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The fourth principle distinguishes three levels of learning in teacher

education: representation, schema, and theory. At the representation level,

preservice teachers react spontaneously based on their needs, values, opinions,

feelings, and unconscious tendencies. At the schema level, reflection occurs

during or after a situation, leading to the development of concepts, characteristics,

and principles to describe practice. The theoretical level involves constructing a

logical order and analyzing conceptual relationships within an individual schema

or connecting multiple schemas to form a coherent theory. Realistic learning in

teacher education involves co-construction of knowledge by incorporating

preservice teachers' existing knowledge and experiences with new knowledge

and skills provided by teachers, colleagues, and other resources. It promotes

systematic reflection, social interaction, and the development of a personal and

professional identity.

The fifth principle emphasizes the importance of considering pre-service

teachers as individuals with their own identity. Autonomy and self-regulated

professional development are crucial for their growth. Developing self-awareness

and fostering interest in one's own identity is essential for pre-service teachers to

develop their potential and transfer it to others. This perspective highlights the

need to integrate a moral foundation into training by helping pre-service teachers

develop their own identity.

The development of professional skills goes beyond the mere display of

knowledge and abilities. A competent professional also recognizes the

importance of aligning their actions with their knowledge, skills, motivations, and

values. They approach problem-solving with flexibility, dedication, and

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perseverance. Therefore, the acquisition of reflective competence during initial

teacher education is of utmost importance. Students must cultivate a deep

understanding of the need for and commitment to act in accordance with their

professional expertise to effectively address challenges that arise in their future

practice.

Both individual and group reflection are crucial in the process of training

new teachers. Rather than prescribing solutions, collaborative support aims to

help student teachers navigate their own learning journey. This involves

encouraging them to express their concerns and needs, listening to their

perspectives, drawing on their existing knowledge, and providing them with

guidance and advice. This type of support also ensures a gradual transfer of

control over the learning process to the students themselves.

Peer collaboration has been recognized as a powerful tool for enhancing

and developing higher cognitive processes in learning. Recently, there has been a

growing interest in Vygotskian approaches that focus on collective scaffolding,

which involves the co-construction of knowledge among teachers from each

member's contributions through interactions within a group of students or novice

teachers. This collaborative work, when well guided and supported, facilitates

reflective processes as each learner, including pre-service teachers, verbalizes

their internal thoughts and ideas about the world and their environment.

Self-regulation is another important aspect of teacher training. Students

must learn to reflect on their own actions, confront their own realities, and

identify and solve their own problems. By engaging in ongoing reflection on their

daily work, they develop the ability to independently identify areas for

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improvement and find solutions. This dimension of self-regulation is crucial to

fostering autonomous learning and is based on observation, critical analysis, and

self-assessment. To help students achieve this level of autonomy, they must be

equipped with appropriate tools, such as portfolios and metacognitive prompts.

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Conclusion

In his work, Freudenthal introduces the term "phenomenology" to describe

his method of analyzing mathematical concepts. This method involves analyzing

the phenomena that organize mathematical concepts, considering their

development and current use. Freudenthal distinguishes several types of

phenomenology, such as pure, didactic, genetic, and historical phenomenology,

each focusing on various aspects of the concept studied. A solid analysis of pure

phenomenology is crucial to support any effective phenomenological analysis in

mathematics teaching.

Mathematical concepts are means of organizing phenomena in the

physical, everyday world, and are created through the relationship between

phenomena and the means of organization described by mathematical sign

systems. These mathematical signs are part of a mathematical sign system

containing diverse expressions, and they create abstract concepts as the means of

organization become more abstract. Mathematical objects are incorporated into

our experiential world, becoming phenomena in a new relationship with other

phenomena and means of organization.

Mathematics therefore exists within the same world as the phenomena it

organizes, and not in a separate world. Mathematical concepts do not exist as pre-

existing ideal objects but are created through mathematical activity and

mathematical sign systems. Finally, mathematical concepts are subject to

modification over time due to their use and to the new mathematical sign systems

in which they are described, but these modifications do not indicate errors in the

original concepts or a linear progression toward a single truth.

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Problem solving is not the only mathematical activity that creates concepts.

Organizing the results obtained from solving problems and proving theorems

into a deductive system is another important aspect of mathematics. This

systematic organization can take many forms: from local to global, from axiomatic

to formalized. However, it is a fundamental element of mathematics because

mathematicians have moved from collecting results and methods to developing

complex structures. During this process, the use of definitions in mathematics

also developed. In mathematics, definitions are not only used to explain to people

the meaning of terms, but they are also a link in the chain of inference in

organizing the inference system.

The process of definition involves organizing the properties of a

mathematical object by inference. The focus is on identifying properties that can

be used to create a deductive system, both at a local and global scale, in which a

mathematical object can be embedded. It is important to note that isolating

specific properties to define a concept is not a neutral or harmless action. In

conclusion, this suggests that concepts were originally created to organize related

phenomena and means that the content of the concept is now determined by the

conclusions drawn within the defined system. Just like theorem proving, the

process of definition also leads to the formation of new concepts.

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This edition of “Hans Freudenthal: Perspectives in Mathematics Education

and Phenomenological Analysis” was completed in the city of Colonia del

Sacramento in September 2024.

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