An introduction to the theory of complexity:A case study with dynamic systems and fractality (CONNECTIONS AND UNDERSTANDING IN MATHEMATICS EDUCATION: MAKING SENSE OF A COMPLEX WORLD) Proceedings-CIEAEM-71-Braga-Portugal

Abstract

The present work has as its purpose, through a brief reflection on the influence of mathematical education, the changes produced by scientific advances and especially by the philosophy or principles that govern scientific research in a moment determined. The theory
of complexity emerged in the mid-twentieth century as a scientific paradigm whose purpose is "to understand the complexity of life". The author of this work provides some comments on this paradigm from the theoretical point of view necessary to study complex objects in
mathematics education. A case study on dynamic systems and fractality will constitute the experience carried out with some of 14, 15 and 16 years old, and proposes to reflect with this work, from the conviction, that it is necessary to develop an updated version of the milestone of the publication in 1908 of the book by Professor Felix Klein of the University of Göttingen: "Elementary Mathematics from a Higher Point of View". Therefore, it would be necessary to modify the curricula through innovation and take new aspects, new models and
new creativity: there are many mathematical domains almost unexplored in Primary Education and / Secondary Education that organized in an original and creative way, would allow the design of enriching activities in the classroom.

Full text

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WORKING GROUP 1 / GROUP DE TRAVAIL 1
CIEAEM 71
Braga (Portugal)
July, 22 - 26 2019
CONNECTIONS AND UNDERSTANDING IN MATHEMATICS EDUCATION:
MAKING SENSE OF A COMPLEX WORLD
CONNEXIONS ET COMPREHENSION DANS L'ENSEIGNEMENT DES
MATHEMATIQUES:
DONNER UN SENS A UN MONDE COMPLEXE

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An introduction to the theory of complexity:A case
study with dynamic systems and fractality
Sixto Romero Sánchez
Escuela Técnica Superior de Ingeniería
Campus de El Carmen-Universidad de Huelva (Spain)
E-mail: sixto@uhu.es
Abstract. The present work has as its purpose, through a brief reflection on the influence of
mathematical education, the changes produced by scientific advances and especially by the
philosophy or principles that govern scientific research in a moment determined. The theory
of complexity emerged in the mid-twentieth century as a scientific paradigm whose purpose
is "to understand the complexity of life". The author of this work provides some comments
on this paradigm from the theoretical point of view necessary to study complex objects in
mathematics education. A case study on dynamic systems and fractality will constitute the
experience carried out with some of 14, 15 and 16 years old, and proposes to reflect with this
work, from the conviction, that it is necessary to develop an updated version of the milestone
of the publication in 1908 of the book by Professor Felix Klein of the University of
Göttingen: "Elementary Mathematics from a Higher Point of View". Therefore, it would be
necessary to modify the curricula through innovation and take new aspects, new models and
new creativity: there are many mathematical domains almost unexplored in Primary
Education and / Secondary Education that organized in an original and creative way, would
allow the design of enriching activities in the classroom.
Résumé:Le présent travail a pour objet, à travers une brève réflexion sur l'influence de
l'enseignement des mathématiques, les changements produits par les avancées scientifiques
et surtout par la philosophie ou les principes qui régissent la recherche scientifique dans un
moment déterminé. La théorie de la complexité est apparue au milieu du XXe siècle comme
un paradigme scientifique dont le but est de "comprendre la complexité de la vie". L'auteur
de cet ouvrage fournit quelques commentaires sur ce paradigme du point de vue théorique
nécessaire à l'étude des objets complexes dans l'enseignement des mathématiques. Une étude
de cas sur les systèmes dynamiques et la fractalité constituera l'expérience réalisée avec des
élèves de 14, 15 et 16 ans, et se propose de réfléchir à partir de la conviction qu'il est
nécessaire de développer une version actualisée de la publication de 1908 du livre du
professeur Felix Klein de l'Université de Göttingen : "Elementary Mathematicsfrom a Higher
Point of View" (Les mathématiques élémentaires d'un point de vue supérieur). Il serait donc
nécessaire de modifier les programmes d'études par l'innovation et de prendre de nouveaux
aspects, de nouveaux modèles et une nouvelle créativité : il y a de nombreux domaines
mathématiques presque inexplorés dans l'enseignement primaire et/ou secondaire qui,
organisés de manière originale et créative, permettraient de concevoir des activités
enrichissantes en classe.
Keywords: Complexity theory, dynamic systems, fractality, complex objects.
1. Introduction
The 21st century is facing enormous challenges that come from societies, increasingly interrelated

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universally. Thus, it is possible to observe significant economic changes of globalization of capital
and markets, both industrial and financial; extremely, dynamic advances, in science and technology,
in mechanical, virtual, spatial spheres, etc. All of which has affected all living beings who live this
reality. Our cultures have absorbed all these new globalized aspects, making them their own and
interchangeable: communications are almost instantaneous, in real time; the mobility of information
is surprising; the warlike conflicts between countries transmitted by television.
Literacy is linked to the computer; virtual reality is almost the new company of children, closer than
the games of manual and collective creation.
We are undoubtedly immersed in a scientific-technical revolution that means a new way of
producing and thinking about our reality. The theoretical-practical needs and problems have
demanded changes and epistemological ruptures, and even of rationality itself. Along with Thomas
Kuhn, physicist, philosopher of science and American historian, known for his contribution to the
change of orientation of philosophy and scientific sociology in the 1960s, one could say that we are
facing paradigm changes, as a result of scientific revolutions.
At the end of the twentieth century, these partial and disciplinary scientific movements, not only
began to interrelate, but also, to be measured as a single movement. In this sense, some authors
began to identify and make deep reflections on the synchronic similarities, in spite of the different
plots and knowledge problems. All of which has resulted in the configuration and naming of a new
generalizing scientific paradigm, capable of encompassing all the sciences in general, and in
mathematics in particular, as is the emergence of this new complex paradigm or complexity.
The scientific paradigm of complexity comes to overcome the historical insufficiency, the classical
paradigm and its corresponding valuation of the notion of simplicity and domination of man
towards nature (Bacon, 1998). It overcomes, therefore, the identification of complexity with
something complicated and on the contrary, transcendence consists in affirming that the complex is
an attribute of reality and that it is, therefore, irreducible to discrete entities. His proposals value the
dialectical units of the simple and the complex, the validation of chance, uncertainty, chaos,
indeterminacy and emergence, non-linearity, etc. (Taeli, 2010). However, this paradigm of
complexity, not only comes to conform with the ontological view of how reality is, but requires that
an epistemological coherence, complex thinking or non-classical rationality, are increasingly
accepted.
In this sense, also, the new dialogues of scientific knowledge, come from the recognition of the
incapacities of the obstinate disciplinary autonomies, to respond to the new requirements of the
complexities of the world. In short, the needs of the world in its economic, social and political
dynamics, such as the magnitude of the Eco-social crisis, have demonstrated the insufficiency of the
simplistic and reductionist paradigms to provide solutions to the changes required by it; hence,
Philosophy and its all-encompassing approach, has collaborated in identifying and promoting the
new complex scientific paradigm or complexity.
1.1. Education in several contexts
a) In the context of a classical paradigm
In its origins, capitalism, as a new economic-social formation, revolutionized everything; the ways
of producing, thinking and feeling reality. According to this, the process of formal or informal
education or training of the subjects that the self-reproduce, is affected, to the same extent as the
rest of their circumstances; they participate in the same modifications of the scientific picture of the
world. That is to say, education will be determinant for the way that the subjects will understand the
world and interact in it, and with it as a society and in nature.
b) In the current context
In recent times there have been changes that have modified our actions, however, the essence of the
contradictory working capital relationship remains.
Indeed, profits are no longer necessarily produced on the basis of industrial production, but the

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accumulation of profit has produced financial markets that act as anarchically and chaotically as the
dynamics of goods; but, in the same way, we must recognize the novelty and complexity of the
accelerated dynamics, with a greater degree of sensitivity to changes in the initial conditions; this
has given rise to the emergence of productive forces not previously thought of, both mechanical and
virtual.
- The complex as an attribute is inherent in reality, and affirms that the systems of nature are not
given in advance, nor immutable; so much so that it is recognized that the systems can vary
completely if the initial conditions are changed to a minimum. Hence the difficulty of propitiating
the duality of hard and soft or natural sciences and of the spirit.
- It is no longer possible to continue with the Cartesian duality of separate subjects and objects, they
are interrelated. As also the subject is no longer considered as an atom, but as a system.
- And therefore, for this complex reality the only thought that can be understood, understand and
formulate is dialectical, hermeneutical, contextualized and complex thinking. The simple world is
no longer real, it is a category that does not express it. The truth, and knowledge, will depend on the
relationships that emerge from the interactions ; it is not given, that is why it cannot be delivered,
transmitted as knowledge, and it must be built from practice (this makes a greater internal-historical
coherence with a pedagogical model, be it with more constructivist, critical and relational
proposals).
Therefore, educational models based on the complex scientific paradigm among others, should be
consistent with the new scientific picture of the world. Hence, the teaching-learning process must
contemplate: For this paradigm, the systems of ideas, theories and knowledge, dynamic and
emergent, given that, the only possible thing to do, is to allow the student to build his/her own,
without falling into a solipsism or subjective idealism, that is why it is given, from its
contextualized construction within a social structure.
1.2. Iideas about complexity theory in education
Interest in complexity theory, a relative of chaos theory, has become well established in the business
and scientific communities in recent years. Complexity theory argues that systems are dynamically
evolving interactions of many parts which cannot be predicted easily. In the book, School
Leadership and Complexity, Keith Morrison (2002) introduces complexity theory to the world of
education, drawing out its implications for school leadership.
In this book, he suggests that schools are complex, nonlinear and unpredictable systems, and that
this impacts significantly on leadership, relationships and communication within them. As schools
race to keep up with change and innovation, this book suggests that it is possible to find order
without control and to lead without coercion. Key areas are:
• Schools and self-organization.
• Leadership for self-organization.
• Supporting emergence through the learning organization.
• Schools and their environments.
• Communication.
• Fitness landscapes.
This book will be of interest to headteachers and middle managers, and those on higher level
courses in educational leadership and management.
2. Complexity theory
The complexity theory, as a complete theory, was developed from the 1980s, particularly in the
work of the Santa Fe Institute in the United States. In some ways, the old regime of chaos theory
has given way to the study of complexity as "life on the edge of chaos

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Fig.1. Complexity Theory
It is an attempt to explain how open systems work, as seen through holistic shows. In complexity
theory, a system can be described as a collection of interactive parts that, together, function as a
whole; It has limits and properties.
This interaction is so complex that the behavior of the system cannot be understood only as an
"emerging consequence" of the sum of the constituent elements. The key elements of complexity
theory can be seen reflected in the figure 2, with components:
« …In the physical sciences, Laplacian and Newtonian theories of a deterministic universe have
collapsed and have been replaced by theories of chaos and complexity in explaining natural
processes and phenomena, the impact of which is being felt in the social sciences (e.g. McPherson,
1995). For Laplace and Newton, the universe was a rationalistic, deterministic and clockwork
order; effects were functions of causes, small causes (minimal initial conditions) produced small
effects (minimal and predictable) and large causes (multiple initial conditions) produced large
(multiple) effects. Predictability, causality, patterning, universality and 'grand' overarching
theories, linearity, continuity, stability, objectivity – all contributed to the view of the universe as an
ordered and internally harmonistic mechanism in an albeit complex equilibrium; a rational, closed
and deterministic system susceptible to comparatively straightforward scientific discoveries and
laws… ». (Morrison, 2002).
The differential characteristics of the theory of complexity can be observed in the following table-
summary :
Fig. 2. Tabla-Summary. Complexity Theory-Conventional Sapience (Morrison, 2002)
Conventional Sapience (Wisdom)
Complexity Theory
1. Small changes produce small effects
1. Small changes can produce huge effects

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2. Effects are straightforward functions
of causes
2. Effects are not straightforward functions
of causes
3. Systems are deterministic, linear and
stable
3. Systems are indeterministic, nonlinear
and
unstable
4. Certainty and closure are possible
4. Uncertainty and openness prevail
5. The universe is regular, uniform,
controllable and predictable
5. The universe is irregular, diverse,
uncontrollable and unpredictable
6. Systems are fixed and finite
6. Systems evolve, emerge and are infinite
7.Universal, all-encompassing theories
can account for phenomena
7 Local, situationally specific theories
account
for phenomena
8. A system can be understood by
analyzing its component elements
(fragmentation and atomization)
8. A system can only be understood
holistically,
by examining its relationship to its
environments (however defined)
9. Change is reversible
9. Change is irreversible - there is a
unidirectional arrow or time
10. Similar initial conditions produce
similar outcomes
10.Similar initial conditions produce
dissimilar outcome
The previous summary indicates the transformation that indicates the change of prototype or
paradigm revealed by the theory of complexity. The link between a deterministic choice of the
universe and modernism is not complicated to distinguish; both have the same principles as basis
for progress, and yet they have real difficulties in showing the changes that occur or should occur in
school (Riley, 2000: 35).
The emergence of the concept of complexity theory appropriates the modernist spirit of change and
uncertainty that begins in the last century with the Heisenberg uncertainty principle, quantum
physics and the theory of relativity.
3. "New" arguments
“Man cannot discover new oceans unless he has
Courage to lose sight of the coast". André Gide
Our proposal, based on the ideas cited ut-supra can be, among others: theory and optimization of
graphics, dynamic systems, fractals, topology, information processing, code theory and
cryptography, modelling ...
Our idea is to face, provoke and challenge through the tasks for our students that allow them to use
mathematical models (Duperret, 2009) for the description and analysis of aspects close to their daily
tasks and purposes, in order to motivate and consolidate their mathematics. The goals are
knowledge, on the one hand, and on the other, to establish and implant solid cognitive principles for

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its conception. In addition, it is important to test the descriptions, analyses and dilations of nearby
states in their everyday environments as "experimental mathematics".
Fig.3. New arguments against the immobility
3.1. Cases studies
There are infinite case studies that we could enumerate: a) the evolution of the population of a
species; b) the Malthus growth model; c) related to the economy: the banks, working with the
concept of debit, etc., that is mathematical models for processes that evolve over time, that is,
discrete DSs that often involve the iteration process. But due to space limitation I will present a case
related to an ecological model,
3.1.1. Related with a Model of Ecology.
We are going to present, as a summary, the steps that must be taken to get an idea of what a
dynamic system is, focusing on graphic visualization, fundamentally.
I. Case Presentation
“In Spain, we have 1250 individuals of a protected species of birds. Experts believe that the
existing bird population decreases by 7% each year either by natural causes, or by poachers. There
is also a captive breeding programme which increases the bird population by 5 individuals each
year. Some questions that we can present to our students could be:
a) Write the relation of recurrence that relates the existing population in year k, xk, with which there
was year k-1, xk-1
.
b) Determine a formula that allows to obtain directly xk as a function of k.
c) If the conditions do not change, will this species end up in danger of extinction? (It is established
that a species is in danger of extinction if the number of its individuals is less than 100)”.
II. Ideas for the solution

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1. We take a cartesian representation system for each point (x, y), where x is the number of
individuals at time k-1 and y is the number of individuals at time k.
2. Check that the raised question above responds to the dynamic system that is expressed by the
equation:
1
0.93 5
kk
xx
−
=+
3. Represent the dynamic growth curve.
4. Given an amount x of birds this year, graphically represents the amount that will be there next
year and in 2 years.
5. What number of birds can there be this year so that next year there will be more?
6. What number of birds can there be this year so that next year there will be less?
III. Graphical visualization of the dynamic.
Fig. 4. Visualization of the dynamic
IV. Modelling and formalization: Dynamical of one-dimensional linear applications
Let L: R → R be a linear application, that is, L (x) = a.x with a  R.
Also, through the Graphic Analysis we will see the behaviour of certain dynamics from linear
one-dimensional applications:
- If | a | <1, all the orbits converge to point 0. A first way to investigate the dynamical generated
by this application would be to take a starting point, for example 1, and calculate its orbit
(projections on the axis OX of the points obtained.
1, a2, a3, a4, a5, ...
The orbit converges to point 0.
Fig. 5. Orbits converge to point 0
x
0
x1
x1
x
2
x2
x
0
(0,5
)
?
(500/7, 500/7)
Attractive
equilibrium
point

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- If | a |> 1, any orbit other than that of point 0 diverges in modulus to infinity,
Fig. 6. Orbits diverge to infinity
V. Ideas-Others Dynamical Systems
It is also interesting to propose to our students that they deepen in other dynamics: for example,
the quadratic forms.
a) The quadratic family
It is the one formed by the applications: f(x) = x 2 + p, pR
b) The logistic family
It is formed by the applications: fc: [0,1] → [0,1] of the form f(x) = cx (1-x), c R
Example c=2
The orbit of any point of (0,1) tends to 0.5 which is now superactractive.
Fig. 7. Point superactractive
Example: Graphic analysis of other dynamics
Fig. 8. Other dynamics
4. Ideas about fractality

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The study of nature as a clear example of complexity. In it, there are abundant examples of forms
belonging to Euclidean geometry (hexagons, cubes, tetrahedrons, squares, triangles, etc.) but their
vast diversity also produces objects that elude Euclidean descriptions. In those cases, fractals
provide us with a better means of explanation.
Fig. 9. Fractals
4.1 Drawn shapes
When we travel by plane we have the opportunity to observe the different ways that nature and
man have generated change on the skin of the Earth's surface. Also if we get such a viewpoint we
notice how nature and humans “conceive" in different ways the infinite elements that make up the
landscape.
Where is the difference? You have to find it in geometry. On the one hand the Euclidean geometry
drawn as if a strip-line was treated by the machines created by the man; and on the other, the
geometry of the curve. We can say that it is a fight of titans between two different styles. We have
shapes drawn by: Earth, life, man, nature, ...The need to measure gives rise to Euclidean geometry
• Why has mankind turned its back on the sinuous and branched forms of nature and opted for
the straight line, the circle and the sphere?
• Why have we broken the natural pattern that had been drawing the skin of the earth since its
formation over thousands of years?
“One answer: to measure”
Definitely, Euclidean geometry is very useful for the description of objects such as crystals or
hives, but we do not find in it objects that can describe popcorn, baked goods, the bark of a tree,
clouds, certain roots or coastal lines. Fractals allow you to model, for example, objects such as a
cauliflower, rivers, clouds, trees (in the snow), a fern leaf or a snowflake. With the incorporation of
chance in programming it is possible, through the computer, to obtain fractals that describe lava
flows and mountainous terrain.
The dynamic systems that we have considered are based on iteration (Romero, 2018); that is, on
the repetition of a process (a calculation) that allows obtaining each term from others previously
calculated. A fractal is a figure obtained through the iteration of a simple geometric process that
gives rise to a structure that can be extraordinarily complicated.
Euclid's geometry is one of the milestones of deductive thinking that, based on five axioms,
created a system of description of the world that met the needs of the natural sciences, of natural
history until well into the 19th century.
4.2 The fractal dimension
The word fractal comes from the Latin adjective fractus which means interrupted, and some form
of nature objects are fragmented, irregular, rough. One fractal is a geometric figure in which a motif
(pattern) is repeated but always decreasing its scale by the same percentage. Take, as an example
the Sierspinski triangle. The Sierpinski triangle is a geometric object of infinite length, although it is
in a finite region of the plane, which implies a dimension greater than one. But at the same time it
has a null area, which indicates dimension less than 2. But so what size does it have?

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……… ……….
Fig. 10. Sierspinki Triangle
4.2.1 Definition of self-similarity.
a) Be a segment of length L = 1.
………
Fig. 11. Segment
We can cover it, for example, with:
2 segments of size 1/2: N = 2, R = 1/2; (1/2)-1 = 2 ______ 4 segments of size 1/4: N = 4, R = 1/4;
(1/4)-1 = 4 _______ 8 segments of size 1/8: N = 8, R = 1/8; (1/8) 1 = 8
Note that the exponent -1 changed sign coincides with the dimension 1 of a line.
b) Be a square of length L=1.
…….
Fig. 11. Square
We can for, 4 squares of size 1/2: N = 4, R = 1/2; (1/2) -2 = 4,
16 squares of size 1/4: N = 16, R = 1/4; (1/4) -2 = 16___________64 squares of size 1/8: N = 64,
R = 1/8; (1/8) -2 = 64
Note that the exponent -2 changed sign matches the dimension 2 of a plane.
This leads us to the Definition of self-similarity, D of an object, made of N exact copies to itself
and reduced by a factor R:
The relationship
D
NR
−
=
determine the dimension D of the geometric object.
What exponent D do we find when applying this method to the Sierpinski triangle?
3 triangles of side 1/2: N = 3, R = 1/2; (1/2) -D = 3
9 triangles of side1/4: N = 9, R = 1/4; (1/4) -D = 9
27 triangles of side 1/8: N = 27, R = 1/8; (1/8) –D = 27
……………………………………………….
3n triangles of side 1 / 2n: N = 3n, R = (1/2n); (1/ 2n) -D = 3n .
a) For the line:
ln 2 1
1
ln
2
D=−=
⎛⎞
⎜⎟
⎝⎠
, b) For the square:
ln 4 2
1
ln
2
D=−=
⎛⎞
⎜⎟
⎝⎠
; c) For the cube:
ln 8 3
1
ln
2
D=−=
⎛⎞
⎜⎟
⎝⎠
; d) For the Triángulo de Sierpinski:
ln 3 1, 5894 9 6
1
ln 2
D=−=
⎛⎞
⎜⎟
⎝⎠
5. Conclusion

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With this work I wanted to emphasize that there are certain (mathematical) models and other
abstract systems (Romero, 2015) that are, in addition, dynamic systems and explain many complex
real life phenomena: It is available for our high school students!
With the cases presented, we give answers to the complex questions that can be posed at the
teacher's level: teaching and learning science in general, and mathematics in particular, should be an
activity that allows the individual to build (Izquierdo, et al., 1999, 2004) their way of feeling,
thinking, speaking and acting on the world around us, choosing those complex scientific models as
one of the possible points of reference. The use of these complex models by the student should be
guided by the research path (Bonil, et al, 2010) and the contrast of the information obtained from
the existing models. It is a way of exercising the imagination to understand the complexity in the
approach, for example, in ecology: do mathematical models respond to certain influences in the
fundamental conditions of the life of certain species?
One way to measure the length of a curve is to approximate it to the length of a series of small
lines that make it. We call that procedure rectification. The smaller the lines chosen for the
approximating, the more accurate will be our measure. But ... what happens if we try to measure the
total length of a square? Not its perimeter, but the length of the square by this method of
rectification. Does that question even make sense? When we have repeated this tedious operation
infinitely, we can say that we have covered the square with lines. There will not be a single point
through which a line does not pass, nor will any of them pass more than one at a time. To find
mathematically the value of the length of the line that makes squares we use the limit:
But ... what happens if we try to measure the volume of a geometric object? we would arrive with
a process analogous to that “¡So the length of a square is infinite and the volume is zero!”
The well-known case of the Sierpinski triangle has infinite length and zero area. Surprise?
There are many almost unexplored mathematical domains in Primary and / or Secondary
Education that, organized in an original and creative way, would allow the design of enriching
classroom activities and carry out activities such as the above and introduce everyone, from the
conviction, into the world of Mathematics, "Should he use his courage as used by the mythological
hero Ulysses?"
6. References
Alligood, K.T., Sauer, T.D., & Yorke, J. A . (2009). Chaos. An Introduction to Dynamical
Systems. Springer. NY.pp.5,6, 44 y 45.
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