Preface

The following writings correspond to some of the assignments for the course of Vocational Educational Teacher (VET) which I attended in Haaga–Helia University of Applied Sciences in Helsinki, Finland in 2017.

Here you will find a few of the many tasks that we were working on, and they refer some to articles of study and some to a part of the Learning Journal that we had to construct. One of the most interesting things about this period of learning and practice has been the use of digitalization, ICT tools, and social media, to build and share learnings. My peers and I have been involved in this process through many different stages and experiencing them in a very active way. I, still believe that my artistic background conveys to a certain point the need for personal interaction and lively presence, proof of this is the wonderful experience in debate and exchange that workshops had given us.

However, the great importance of sharing insight and reflection through videos, photography, and writings is nowadays a great opportunity for exchange, criticism, and meditated findings and outcomes. This also allows us to explore and experiment in diverse directions and fields, which is in a way part of my teaching proposal to innovate and make learnings more holistic and significant. Hereby I also include two Powerpoint presentations on Competences and Assessment that might clarify my approach to education and what I see as a chance for connecting science and art disciplines.

So, even though my Thesis on the Correlation between Theatre and Mathematics is not yet translated from Spanish to English Language I will include it as a footer to be considered by Spanish speakers as a contribution to scholar knowledge.

Thank you to my tutors in Haaga-Helia UAS,  Ms. Irmeli Pietilä and Mr. David Mauffret.

With my best desires,

                                       C. J. V. S.

Orientation Essay

To begin with this essay, I will state answers to the trigger questions referred to in the Vocational Teachers´ Handbook, and comment on them briefly, to continue with my educational environmental experience and my work as a vocational teacher.

·         What kinds of competences are required of the vocational teacher on the basis of the Handbook?

·         What do the competences described in the handbook mean in terms of your teacher work?

      i. As a vocational teacher, my actions are based in knowledge of vocational and professional education.

·         Developing my practical theory using scientific reasoning.

·         To provide pedagogical reasoning for my actions.

Most of my life I have been surrounded by both scientific and artistic environments. My father was an engineer and my uncles ranged from pilots, dentists, lawyers up to musicians and entrepreneurs.

   My parish school, even when it was under the administration of priests and nuns, was called, at that time, experimental and nowadays it may be called an alternative school. From a very early age (8 years old) I was encouraged to teach younger children on Saturday reinforcement sessions since I was regarded as an excellent student. So, teaching was a natural thing for me. Later, I paid my bills by taking classes in English, maths, and other subjects. Choosing my profession was no trouble. I wanted to be an MD, but life took me elsewhere since it was so difficult to get a post in the 70s, so Biology was my first experience at universities, later Agricultural Economics and Forestry Engineering took also, part of my study experience and learnings. 

    Parallel to all this my artistic sense was enhanced by my art teachers and literature took a place amongst my passions. After my British scholarship award, a teaching commitment took me in a new direction, and after some years I completed my pedagogical studies as an Art Teacher with a mention in Drama.

     ii. As a vocational teacher, my actions should be ethically, ecologically, and socially justified.

·         I work according to ethical principles of vocational teachers.

·         In my work, I promote human rights and equality and act according to democratic principles.

·       I recognize and take into account the impact economic conditions have in my work, and the economic consequences of my work.

Because of my scientific background, I regarded ecological and social approaches as one and as very important issues to be considered when working with children. Part of my counseling and projects for children were based on these and encouraged them to include such matters in their considerations. Not only in their plays but also as a  way to influence communities so they can take advantage of their reflections as to build consciousness and to make them appear as a possibility of economical profit to them.

As the book “A force for good,[1] states in its 6th chapter on “Economics as if people mattered”: ‘Doing good while doing well’ was, unknowingly, one of my life mottos, in Spanish the saying is “Do good without looking to whom”.

     iii.     As a vocational teacher, I develop my competence together with workplace

representatives, students, and colleagues through a systematic assessment by

using various documentation means.

·         I recognize and define development areas.

·         I introduce experimental solutions to development areas and justify them theoretically.

·         I share the results of the development work with fellow students, work community and networks.

In my experience, the different places that I have worked in define diverse educational templates or assessment plans and these vary through time and cases. However, most of them adjust to a certain curriculum agreed upon consensus, previous learnings, or new data shared by schools all over the globe. A clear example of this is the IBO community which I formed part of. Through forums and constant feedback on different countries and the exchange that goes on and on, teachers around the world share standardized assessment means and apply them depending on their own settings. This exchange has proven to be effective, into a certain point, but at least gives teachers the opportunity  to find advice from peers that may be facing the same questions on the other side of the globe with cases that if no alike, similar into same degree.

     iv.     As a vocational teacher, I use and develop learning-centred tutoring, teaching and assessment methods collaboratively with industry representatives, students and colleagues.

·         My actions are determined by the competence students need in the world of work. 

·    I am familiar with the concept and practice of vocational education and design of study processes which are personalised and negotiable.

·         I assess my students through tutoring, and in the basis of their competence.

·         I am proficient in tutoring, teaching and assessment in different types of learning environments.

Planning is one of the most important issues when teaching, therefore it must be considered exhaustively and each student should take the best of it by knowing what is expected from them at the end of their training for work or scholar competence. The collaborative method proved to be one of the great ways to plan what works effectively. Just as important should be the effects of the learnings and if they can be applied in each societal case. So, serious surveys must be carried out previous to this planning regarding the needs of the communities and their social components: the person, the families, the associations and other means of gatherings that may be practise by communities. And last, but never least to be considered part of them as a VET.

      v. As a vocational teacher, I develop learning environments and processes by contributing to networks and communities.

·         I develop the personal and collaborative learning processes of students in cooperation of industry representatives, students and colleagues.

·         I use digital tools and justify the use pedagogically.

As I mentioned before this experience has been part of my last twenty years of teaching through projects. These projects and its planning have encouraged students and their communities to take advantage of knowledge and training pointing out problems, identifying possible solutions and through active output and with the use of all kind of means answer to demanding problems.

As when in the community in the Andes we identified the urge to save the river and carried on with the children of our school an action towards this goal raising conscience in the community and the surroundings of the river to make profit of this possibility to save it and make a business on trout farming.

    vi.     As a vocational teacher, I develop my interactions skills with an open minded and creative attitude and the value of diversity.

·         I give students time, attention, and respect in each encounter.

·         I create possibilities of dialogue as part of teaching and tutoring.

·         I take the varied cultural backgrounds of my students into account in personal and peer tutoring situations.

In my experience if there is no respect for diversity you are wasting one of the best opportunities of growth you can come across with. Being myself an artist there is no way I may miss this vision. Observation and register of reality has been a cornerstone on my artistic development and a solution in my way to face and encourage divergent intelligence in students. The result of this unpredictable problem-solving system is that there are many answers or ways to answer a problem if the student or teacher stays in an active creative state.

   vii. As a vocational teacher, I am familiar with the laws and regulations related to   vocational teacher education and the world of work, applying this knowledge in   my work. Being able to recognise and define development areas.

·     I am able to act and justify actions according to laws and regulations that guide my line of work, and also to be familiar with the strategy of my organisation and to act accordingly.

·        To be familiar with the collective labour agreements of my field and respect them in my work. To share the results of the development work with my fellow students, my work community, and my networks.

Now I am more familiar with the ones applied in Finland, which are included in the site of “Ethical principles for the teaching profession” mentioned currently here below:

http://www.oaj.fi/cs/oaj/ethical%20principles%20for%20the%20teaching%20profession . As for the development areas, I have read the articles referring the history of VET and UAS’ on RDI.[2]

  viii.     As a vocational teacher, I am an active member of society and to lay the ground  work for future vocational education.

·         I follow societal discussions on the role of education.

·         I bring in views on future competencies to my work.

Entertainment and entrepreneurship in the arts are two facts that match perfectly well.  Usually the choice of the subject to stage follows an interdisciplinary discussion on relevance and pertinence. Collaborative planning always enhances the opportunity to discuss what is the best for the time, the community and to bring forward problems or conflicts to be solved. It is also a great opportunity to share specific problems on students and questions that raise from group managing situations.

    ix. As a vocational teacher, I guide my students in such a way to enable to graduate  and to find employment and to strengthen their possibilities for entrepreneurship.

·  To promote my student’s employability and entrepreneurial skills.

·  To support my student’ study processes according to the tutoring plan of the educational institution and the workplace.

Arts in a wide scope have their own joy and their own way to deal with success, frustration and restarting processes. On experiencing staging their labour, art students have always to justify the importance of it in many practical ways. Finding an adequate place for their show or exposition, setting it, and advertising are a few things they learn as soon as they start. Other more sophisticated details as design in lighting, scenery, costumes, hand programmes, posters, flyers and media advertising are rarely put aside or given to a producer in the student stage. So, most of the tutorial is a practical doing and at the same time a creative process. These give art students the tools and instruments to bring forward questions, solve problems on their own and learn to trust and delegate responsibilities to be part of the whole.

·         What thoughts, questions or comments do the competence criteria raise?

1    What would be the mainstream theories that will carry our competence criteria?

This question raises because there are many subjects to the choice of competence criteria but mainly the ones that adjust to the VET are surrounded by the capability to work in harmony with entrepreneurs, students and colleagues. To make sure that a flexible curriculum may answer to the expectations of the community, the needs of society and those referred to the changing world through social media, work nets and

ICT opportunities, I want to be sure of using updated criteria. Because my speciality is set in a face to face basis practice and on-site interaction to develop accordingly.

2   Is there a new source or handbook or related document that we can get familiar with to fill these criteria raise?

The book Curriculum, Plans and Processes in Instructional Design[3] has been for me a source of inspiration in this sense, when consulting the new criteria to design, as this book is conceived as a guide for teachers of science and to relate to “constructivist learning environments and to suggest ways to scaffold their epistemological development” (Chap. 3 pp. 86).

As in my personal experience there are many other books of reference on methods and pedagogics insight that can be considered and that I will mention in the bibliography of this essay

3    Are competences the same number for science and art matters?

So far that is what I have been trying to prove through my Thesis on the correlation in the practice and learning of Theatre discipline and their influence in the learning of Maths. Scholars separate arts from science prejudiced and even Education has not been seen for many years as a scientific discipline. This proves that not every human practice embodies scientific support, avoiding seeing that the technical skills of the arts can be a superb opportunity for the learning of other subjects. So, in a way what I visualize as a future glance is to prevent our students from separating science from arts when learning, moreover taking a holistic opportunity to match these to subjects to improve their learnings, using the tools they handle to them to deepen into their emotions with a sense of certainty and meaning, beholding the world with renewal and creativity to understand and grasp the meaning of their lives into service. Feeling that their skills will be fully used by others and that their effort will be rewarded by the whole mankind as far as they act being honest to their own selves and to the necessity of their communities.

4    Which are the competences considered for arts in Finland?

In Peru, Educational assessment system takes care of only two competences for arts, this are: expresses and appreciates (…subject or matter of study in the form X of art).

5 Being our peer group multidisciplinary are we supposed to share projects or make individual ones for our PDP?

        Me

·         Your current work sector and job description.

My current work as teacher depends now on taking workshops, performance and supporting associations projects in education through the arts. To meet this aims I am one of the founders of a group of multidisciplinary artist Association based in Helsinki: Kulkijat ry.

This association main activity is to support Arts for Education, to develop persons and communities that are multicultural and based in Finland. To soften ways to create their own posts of work, to relate to other artist and educators and to carry on projects that will help them to fit in their new environment and communities by using their art skills and talents and to share them with others. As a theatre director and entertainer, I had been working in Helsinki for the past year and a half, writing, producing and directing plays of mime, puppets and classical theatre. Participating, as well, in performative ensembles with a multidisciplinary approach.

·         Vocational and working life competence

My competence include teaching and organizing educators and students to convey holistic knowledge in art and science with practical results in their daily life. Producing a change and sharing it within professional life.

To orient students to commitment and to set targets as to produce events that show care of relevant issues related to environment, social and economic matters.

To raise personal and social development and self-esteem through education.

Counselling is a very important mission and one of the many virtues that a teacher must demonstrate. This is a constant in growth and only achieved by constant practice of critical thinking.

Ethical practice; being a model in our lives will prove worth to be followed.

Share art knowledge; many people, amongst them entrepreneurs and other professionals, have taken advantage of the techniques developed by theatre people, and artists from all disciplines. These techniques have improved their way of life, suggesting that to put them into practice have demonstrated a group spirit and a win-win thinking that have made their lives more entertaining.

·         Your development path as a teacher/trainer and expert

Firstly, I entered university to study science Biology and Economics later, almost at the same time I started to practice corporal mime as an artistic hobby at the beginning and because of it went through ballet, contemporary dance, and theatre acting disciplines. Puppeteering and playwriting came later as a source of work and amusement, meanwhile I was practicing Yoga, Tai chi and other martial arts, later I worked also as artistic assistant director. So, my practice took me to teach and develop into a pedagogical teacher. In 1988, I was awarded a British Council Scholarship to study Classical Theatre in London and left university studies for that reason. But they demanded from me a period of teaching my learnings on my return to my homeland.

From that moment on I reckon teaching was a way to pay bills at the beginning and later on a mean to keep discipline and teach others my discoveries and my joy on it.

Because my expertise was so appreciated in schools in my country, being art teachers in Drama so few, a law was enacted to encourage artists to embrace pedagogic studies, so they could be able to teach at educational institutions, so I applied to get my degree and succeeded on it. 

My identity

As to answer who am I now as a Teacher?

I may answer that in many ways a Vocational Teacher that has been teaching for around 31 years, my experience has widened and I have taught people and groups from all ages. Being diverse disciplines part of my teaching experience: mime, contemporary dance and ballet, puppets, classical acting and Direction. Besides I have taught playwriting and technical stage matters on lighting and costumes, masks and make-up. Attended scholarships of specialization workshops on Prosthetics and puppets for TV in Spain. Whilst this are part of my artistic background I have also taught languages and worked a Thesis on the Correlation between Drama and Mathematics to point out the holistic way of understanding of human being according to the Paideia concept. 

Actually, I broadcast a live music radio programme once a week in FM in Helsinki, and during the last two years have been awarded a couple of scholarships to playwright and perform puppets, have been invited to teach in several art venues and planning a short film with other artist in Helsinki. Taught and choreographed stage fight last year in Savonlinna Opera festival and would be participating this year in august there. Presently, will be performing in two theatrical and circus festivals in Helsinki and in Karjalohja as mime.

·         My strengths, weaknesses, opportunities and challenges as a vocational teacher

Strengths are: thirty years of teaching practice, this include schools from early years all the way to upper school, universities of education and arts, other educational institutions for teachers, workshops and theatrical productions.

Weaknesses: The main one, now, is my lack of Finnish language command even though I manage to take classes in my own language and English.

Opportunities: Since I have a broad experience in the arts and an innovating project in design and maths I have been invited to several venues to teach and lately to perform in mime and to direct in my mother tongue.

Challenges as a Vocational Teacher: to innovate education regarding the holistic point of view, integrating creativity to science matters and to encourage science students to relate to the arts as a tool to experience integration. This new vision we call “Arts for Education.”

·         My teacher identity and direction of development in the near future

I define myself as a Holistic Art Teacher with a bias on science and nature approach.

·         My motives, learning outcomes and expectations regarding study and learning.

My main motivation is to integrate all people and make them conscious of their mission and the joy of finding it.

·         I want to learn and know more about:

Art and science in education and the links between them.

·         The competence areas that I would like to deepen in knowledge are: 

Education, society and culture and integration of ancient traditions.

·         My own learning outcomes match the competence areas described in the handbook:

Through constant practice, they have been found in one stage or another. Projects of students and some suggested by me have oriented discovery.

Each and all the competences appeared in some stage of my educational experience suggesting they are appropriate to follow as a guideline. Work with peers and community to improve ways of learning.

Maybe take a competence and add a competence could be an interesting exercise for the group.

·         In your teaching practice, in what kinds of tutoring and teaching situations can you demonstrate your competencies and develop them?

In an everyday teaching practice oriented to various flexible goals, the competences question the teacher and assess answers. It is also essential, in art to have a wide background that enables the VET to give advice worth to each case, personalizing it so that it allows change and discovery to happen.

Other competences are found in the meetings with peers. Most of the time the unexpected mistakes are a way to learn flexibility then experience takes place to find new and creative answers. This may come from the peers, the colleagues or the exchange in the institution itself or with entrepreneurs that you work with.

·         I intend to assess your learning in the studies, from results.

Assessment is a constant process with a no ending point in a way just like learning.

Though we have to clarify what is to be assessed, so, what is expected from the student. This process is under construction for the time being and I hope will take me to some new stage where I myself would be satisfied with an outstanding result or with new discoveries.

·         What kind of competence portfolio do you want to make?

I would like suggestions on this matter, but I have a webpage in construction for my PDP.  Maybe a blog to support it too.

·         What are your resources and time management constraints in the studies?

My studies are my main goal now.

·         What wishes do you have in terms of the role of your tutor?

I would like to know if there have been art students before in this course and if they have found subjects for working in UAS.

Vocational teacher's work

·         What are the current topics of discussion in your school or the target school you have chosen?

The schools where I will be teaching are set in Helsinki area and are getting their practices from Caisa Cultural Centre.  I had also applied to Laurea and Metropolia Universities of Applied Sciences (Ammattikorkeakoulu) in Espoo.

·         How are current development challenges brought up and how are they addressed?

Practice has shown the best way to improve teaching techniques, so flexible curricula is one of the more interesting subjects when coming up to teaching the arts.

The refining, adjusting and assimilation techniques work very well here, even though they may sound behaviourist. This redefining aims and goals and building confidence in the students has proven to be rewarding, since they allow you to know your students and their interests better.

·    What kinds of opportunities does your work and operational environment offer for the development of vocational teacher competences?

My project is innovating the view of Art and Science in a broad multicultural sense, this vision that is now being applied in different places in the world can turn global, so there is a broad scope to develop in this matter and project.

·    How do you combine your own development needs as a vocational teacher with those of your work and operational environment?

Integrating my professional skills with solving difficulties in learning, because there are many techniques of the arts that are being used by other professions to improve their standards. Interdisciplinary group-work, peer criticism, creativity, improvisation techniques, critical thinking and many others are practices common to artist in all stages of their work.

BIBLIOGRAPHY

· Aaltonen et al. 2013. Practical skills, and education and development – Vocational education and training in Finland. Haaga-Helia University of Applied Sciences. (pdf)

·  Laukia Jari 2013. Teacher education in the area of vocational education and training – the Finnish perspective. P.     29.

·    Nissilä Säde-Pirkko 2013. The vocational teacher’s changing role and identity in changing contexts. P. 95.

·    Juuti Sini and Raehalme Outi 2013. Identities in transition – from an expert to a vocational teacher. P. 118.

·             Beijaard D., Meijer, P. & Verloop N. 2004. Reconsidering research on teachers’ professional identity. Teaching and Teacher                       

                    Education 20, 107-128. (pdf)

·        Finnish VET in nutshell. (pdf)

· ALEXANDER, Frederick Mathias Man's Supreme Inheritance, Methuen (London, 1910), revised and enlarged (New York, 1918), later editions 1941, 1946, 1957, Mouritz (UK, 1996), reprinted 2002. Constructive Conscious Control of the Individual, Centerline Press (USA,1923), revised 1946, Mouritz (UK, 2004) The Use of the Self, E. P. Dutton (New York, 1932), republished by Orion Publishing, 2001, The Universal Constant In Living, Dutton (New York, 1941), Chaterson (London, 1942), later editions 1943, 1946, Centerline Press (USA, 1941, 1986), Mouritz (UK,2000)

·       ARMSTRONG, Thomas. Inteligencias múltiples. Cómo descubrirlas y estimularlas en sus hijos. Bogotá, Grupo Editorial Norma, 2001.

·        ARTAUD, Antonin, El Teatro y su Doble. Grupo Editorial Tomo México, 2003.

·        BERRY, Cicely. Voice and the Actor, Ed. Harrad Ltd.1973 y por  Mackays and Chatham. London 1988.

·      CASTRONUOVO, Julio.   Lecciones de Pantomima, Ed. Fundamentos Ensayos y Manuales RESAD Madrid, España, 2006.

 ·       DIAZ BARRIGA ARCEO, Frida y HERNÁNDEZ ROJAS, Gerardo. Estrategias docentes para un aprendizaje 

significativo. México, McGraw-Hill / Interamericana Editores, 1998.

·       FELDENKRAIS, Moshe, (1980). Autoconciencia por el movimiento ejercicios para el desarrollo personal. Buenos Aires, Editorial Paidós

·         FLORES VELAZCO, Marco Hernán. Creatividad & Educación. Técnicas para el desarrollo de capacidades creativas.

·        GARDNER, Howard. Arte, mente y cerebro. Una aproximación cognitiva a la creatividad. Barcelona, Ed. Paidós, 2001.

Estructuras de la Mente. La teoría de las Inteligencias Múltiples, Fondo de Cultura Económica, México, 2009

Estructuras de la mente. La teoría de las inteligencias múltiples México,Fondo de Cultura Económica, 2009, 7ma reimpresión.

Inteligencias múltiples. La teoría en la práctica. Barcelona, ed. Paidós, 1999.

·       GOLEMAN, Daniel. La inteligencia emocional. Buenos Aires, Javier Vergara Editor, 1996.

        A force for good, London Bloomsbury Publishing Plc.

·        HEIDEGGER, Martin. La Pregunta por la Cosa, Ed. Alta Argentina 1979

·         JAEGER, Werner. Paideia: los ideales de la cultura Griega. Fondo de Cultura Económica México, 2009

·         JOHNSTONE, Keith. IMPRO, Ed. Routledge NY Methuen Drama 1989

·         LAYTON, William. ¿Por qué? Trampolín del Actor, Fundamentos Madrid 2010

·         LE BOULCH, Jean (1995), Educación por el movimiento en la edad escolar. Perú, Editorial  Colegio Jean Le   Boulch

·         PIAGET, Jean. Psicología de la Inteligencia, Editorial Psique, Bs. As., 1976

·        PIERCE JONES, Frank, A Study of The Alexander Technique: “ Body Awareness in Action” Ed. Schoecken Books 1976 NY USA.

·        RICCI, Bitti Pío E. / CORTESI, Santa. Comportamiento no Verbal y comunicación. Editorial Gustavo Gili S. A. España, 1980.

·          RUSSELL, Bertrand La perspectiva Científica. , Ed. Sarpe Madrid 1985

·        UNESCO. La educación encierra un tesoro. Informe a la UNESCO de la Comisión Internacional sobre la educación para el siglo XXI, presidida por Jacques Delors. Edit. UNESCO, 1996.

·     STOKOE, Patricia (1988). La expresión corporal. Guía didáctica para el docente. Buenos Aires, Editorial Ricordi Americana S.A.

·           VILAR, Jean, De la tradición teatral. Editorial La Pléyade 1972

·           VILLIERS, André. El Arte del Comediante Ed. Universitaria de Buenos Aires 1959

·            WIENER, Joseph (1985). Movimiento Creativo para Niños. Barcelona, Editorial Grafinosa.

[1] GOLEMAN, Daniel. Ed. Bloonsbury, London, GB 2015

[2] This means that UAS will develop their research, developmental and innovation (RDI) activities and RDI is taken into

  account in the job descriptions of the teaching personnel. Also, the innovation competencies of vocational teachers will be

  developed. HH Practical Skills pp. 20

[3] Ed. SEEL,Norbert and DIJKSTRA

Study Circle

Summary on the book Curriculum, Plans, and Processes in Instructional Design[1]

Introduction and Chapter I

Introduction and Curriculum Development

According to this reference, constructing a theory that serves both explanation and discovery, is a slow process of step by step rather than sudden decisive changes and shifts of paradigms. We can see the story of education and its disciplines as a continues process of evolution where only theories that are theoretically sound and can be effectively situated in educational practice can survive.

The authors mention the first works on curriculum in history and how from these two movements emerged: reform pedagogy and the approach of experiential education and didactics. While some did not survive there still exists a Journal of experimental education and others in which an Empirical approach to education is the guiding principal. Nevertheless, we can see attempts to combine multimedia learning environments with reform pedagogy. As many as 30 different didactic approaches have emerged and some have criticized this saying that dozens of them do not specify the field of interest in a unique and comprehensive manner but rather centered around scientific desiderata (sic).

Some authors classify the huge number of didactics into several main categories that are oriented towards the concepts Bildung (German term for a philosophy of self-cultivation, understood as a process of both cultural and personal maturation) learning, and interaction.

In consequence, there is a distinction between:

a) Didactics as a culture-related education (Bildung, Weniger, 1956)

b) Didactics as a theory of teaching and learning in general. (Dolch, 1965)

c) Didactics as a theory of instruction with an emphasis on interactions and curriculum development (Heimann,1962; Schulz, 1980).

To the authors these different didactics are characterized by the dichotomy of Bildung and learning. On one hand, a culture-related education emphasizes on the individual constructive interactions with culture in the attempt to understand its sense and meaning. On the other hand, didactics is related to learning and teaching for which specific situations are constructed. The contents of learning are determined through the well- grounded selections and sequence of cultural contents, e.g. in the form of curriculum or syllabus, whereas the intent of learning is related to different personality traits of the subjects, e. g., at the cognitive affective, social, and behavioral levels. Both categories have an accent in different aspects so they are not homogenous. For example, the second one includes various approaches of cybernetic didactics as well as psychological didactics of Aebli (1963), Ausubel and Robinson (!969), and Bruner (1966) who were influenced by Piaget.

We can summarize the field of didactics as follows: In a literal sense, didactics (διδασκɛιʋ) means teaching and instructing, but also learning and being taught.

Consequently, both concepts of “didactics as a theory of teaching and learning” (in the sense of Dolch, 1965) and of “didactics as a theory of the contents of Bildung” corresponds best with the mission of this volume. However, this leads immediately to the status of ID… It is mainly characterized by a strong relatedness to information and communication technologies, especially computer technology…as to consider it an engineering discipline. So, the development of instructional systems and support tools for instructional designers and developers as somewhat similar to software engineering systems and support tools for software engineers.

The Current Status of ID

…As a science ID aimed at detailed and theoretically well-founded specifications for the development, implementation, evaluation and maintenance of effective learning situations. (cf. Dick & Reiser, 1989; Richey, 1986). From a technological point of view, ID was mainly concerned with creating concrete instructional materials and environments in which students could learn effectively.

Thus, ID is considered a science of planning whose scope range from the prognosis of global developments to the binding determination of instructional activities.

The term planning, here denotes both a process (i.e., the preparation and development of a plan) and a result (i.e., a particular plan). Accordingly, conventional models of ID usually entailed all steps of planning—starting with needs assessment and the construction and sequencing of learning tasks, and leading to the formative evaluation of the effectiveness of instructional materials. In sum, for decades ID has been recognized generally as a discipline that aims at the integration of different educational activities, involving a theoretically sound design, implementation, and evaluation of effective learning environments.

The defining feature of a design problem is that the solution to such problem is the sketch of a new object that can be used to fulfill a public or individual need or motive.

All objects designed and developed by human beings have a certain life cycle.

We distinguish four phases in this life cycle:

a) Design

b) Development

c) Use and maintenance

d) Discarding

The design of a new object is a creative process in which the imagination and originality of the designer are crucial. Trough for all domains the design rules are well known and scientifically founded, the design of a new object is more than less an Art.

The objects to be designed realized and used should meet for different sets of conditions and criteria: 

a) Principles and rules of a “good” design, which means that pertinent laws and principles domain and the regularities of its change are adequately applied.

b) Functional requirements –designs should be functional and objects should fulfill human needs.

c) Aesthetic concerns—the artistic value of a design should satisfy the tastes and preferences of a group of people that often will vary over time and

d) Financial constraints—the financial means that are available should be used efficiently (Dijkstra, 2000).

The solution to an instructional design problem is the design of a communication between and expert and a learner and an environment that supports that communication. The purpose of the communication is for the learner to develop knowledge and skills. The communication can take on many forms, such as providing information, asking questions and giving problems to solve and tasks for training a skill. It can be anticipated in textbooks and electronic learning devices. The designed communications are the means for the “treatment” of the domain and processes of its change, that is, for the “treatment” of cognition and learning. Although the label learning technology might have been more adequate, the label instructional technology was adopted as a standard term because the presentation part of the communication by the expert and the supply of conceptions about the structure and change of the reality that are content of the subjects of the curriculum has been so clearly present in education.

Instructional design is part of a technology, the steps of which should meet the structure of cognition and the processes of knowledge development and learning.

Some confusion is possible due to the use of the label instructional systems design.

This is a “soft” technology or general heuristic for controlling a process of education and training. The use of heuristics instructional (systems) design—including the phases (“steps”)

a) Assessment

b) Design

c) Development

d) Implementation and

e) Evaluation

--is too general. It covers part of the life cycle of an object (e.g., design and development) and parts of the results of its use (e.g., evaluation of what is learned) It does not include detailed rules about how to activate a domain of cognition and how to support the change in knowledge and skills.

A general heuristic of instructional design will be helpful in regulating a process of education and training… In addition, the theory and models of evaluation are different from those of curriculum and instruction. Curriculum, instruction, and evaluation should be nevertheless aligned (see Pellegrino, chap 1, this volume), and the theories should meet this requirement as well.

Chapter I

Complex Learning Environments: Connecting Learning Theory, Instructional design and Technology. James Pellegrino (University of Illinois at Chicago)

The Curriculum-Instruction-Assessment

Triad

Three things are central to the educational enterprise: curriculum, instruction, and assessment.

Curriculum consists of the knowledge and skills in subject matter areas that teachers teach and students are supposed to learn. Generally as code or breadth of content in a given subject area and a sequence for learning.

Instruction refers to methods of teaching and the learning activities used to help students master the content and objectives specified by a curriculum. Instruction encompasses the activities of both teachers and students.

Assessment is the means used to measure the outcomes of education and the achievements of students with regard to important knowledge and competencies. It may include both formal methods such as large-scale state assessments, or less formal classroom-based procedures, such as quizzes, class projects, and teacher questioning.

These three aspects should be aligned and directed towards the same end. Ideally an assessment should measure what students are being taught and this should parallel the curriculum one wants students to master. Each plays a role in making decisions and setting policies. Most current approaches are based on theories and models.

This alignment could be better achieved if the three are derived from scientifically credible and shared knowledge based about cognition and learning in the subject matter domains.[2]

The model of learning would provide the central bonding principle serving as a nucleus around which the three functions would revolve. This approach can result in an undesirable narrowing of the curriculum and a limiting of learning outcomes. Such problems can be ameliorated if, instead, decisions about both instruction and assessment are guided by a model of learning in the domain that represents the best available scientific understanding on how people learn. Then we shall consider what do we actually know about the nature of learning and knowing.

Important principles about learning and teaching

The first important principle about how people learn is that student come to class with existing knowledge structures and schemas that include preconceptions about how the world works. Sometimes those understandings are accurate, but sometimes they are not. In science, students often have misconceptions of physical properties that cannot be easily observed. In humanities, their preconceptions often include stereotypes or simplifications, as when history is understood as a struggle between good guys and bad guys.

The second important principle is to develop competence in an area of inquiry, students must

a) Have a deep foundation of factual and procedural knowledge

b) Understand facts, procedures, and ideas in the context of a conceptual framework and

c) Organize knowledge in ways that facilitate retrieval and application.

This principle emerges from research that compares the performance of experts and novices, and from research on learning and transfer.

The ability to plan a task, to notice patterns, to generate reasonable arguments and explanations, and to draw analogies to other problems are all more closely intertwined with factual and procedural knowledge that was once believed.

To develop competence in an area of inquiry, students must have opportunities to learn with understanding. Key to expertise is a deep understanding of the domain in which they are working that transforms factual and procedural information into “usable knowledge”. This allows them to see patterns, relationships, or discrepancies that are not apparent to novices.

A key finding in the learning and transfer literature is that organizing information into a conceptual framework allows for greater “transfer”. It allows the student to apply what was learned in new situations and to learn related information more quickly.

A third critical idea about how people learn is that a “metacognitive” approach to instruction can help students to take control of their own learning by defining learning goals and monitoring their progress in achieving them. In research with experts, who were asked to verbalize their thinking as they worked, it has been revealed that they monitor their own understanding carefully. This metacognitive monitoring activities are an important component of what is called adaptive expertise (Hatano, 1990). Research has demonstrated that individuals can be taught these strategies, including the ability to predict outcomes, explain to oneself in order to improve understanding, and note failures to comprehend. They can learn to activate background knowledge, plan ahead, and apportion time and memory.

Three core learning principles

1. Teachers must draw out and work with the pre-existing understandings that their students bring with them, inquiring into students’ thinking, creating classroom tasks and conditions under which student thinking can be revealed.

2. Teachers must teach some subject matter in depth, providing many examples in which the same concept is at work and a firm foundation of factual and procedural knowledge. This will allow the students to grasp defining concepts in specific domains or areas of occupational skill.

3. The teaching of metacognitive skills should be integrated into the curriculum in a variety of content areas. Metacognition often takes the form of an internal dialogue. Students may be unaware of its importance, unless the processes are explicitly emphasized by teachers. Integration of metacognitive instruction with discipline-based learning can enhance student achievement and develop their ability to learn independently.

Implications for Curriculum, instruction and Assessment

Focusing on issues of how people learn with regard the matters of curriculum, instruction and assessment. At the level of curriculum, knowledge of how people learn will help teachers and the educational system move beyond either-or dichotomies that have plagues the educational system.  One such issues is whether curriculum should emphasize “the basics” or teach thinking a problem-solving skill. Both are necessary.  Student´s abilities to acquire organized sets of facts and skills are actually enhanced when they are connected to meaningful problems-solving activities, and when students are helped to understand why, when and how those skills are relevant. And attempts to teach thinking skills without a strong base of factual knowledge do not promote problem-solving ability or support transfer to new situations.

Focusing on how people learn also helps bring order to a seeming chaos of instructional choices. Consider the many choices for teaching strategies: Lecture-based teaching, text-based teaching, inquiry-based teaching, technology-enhanced teaching, teaching organized around individuals versus cooperative groups, and so forth.

Are some of these techniques better than the others? Is lecturing a poor way to teach, as many seem to claim? Is cooperative learning good? Does technology-enhanced teaching help achievement help achievement or hurt it?

Research and theory on how people learn suggests that these are the wrong questions. Asking which teaching technique is best is analogous to asking which tool is best—a hammer, a screwdriver, a plane or pliers. In teaching, as in carpentry, the selection of tools depends on the task at hand and the materials one is working with.

There is no universal best teaching practice. If, instead, the point of departure is a core set of learning principles, then the selection of teaching strategies, mediated, of course, by subject matter, age, and grade level, and desired outcome, can be purposeful.

Perhaps no area stands to gain more from knowledge of how people learn than the area of assessment, a persistent concern in the educational process…. The attributes are measured are mental representations processes that are not outwardly visible. Thus, an assessment is a tool to observe students’ behavior and produce data that can be used to draw reasonable inferences about what students know…. These targets of inference should be determined by cognitive models of learning. That describe how people represent knowledge and develop competence in the domain of interest.

The process of reasoning from evidence can be portrayed as a triad of three interconnected elements known as the assessment triangle.

                                                                                 Cognition

                                                                              Model of student

                             Observations                                                                        Interpretations

                             Set of Beliefs                                                         Process of making  sense of the evidence

The three represented as vertices of a triangle because each is connected to and dependent on the other two. A major tenet of Knowing what students know (Pellegrino et al., 2001) is that for an assessment to be effective, the three elements must be in synchrony.

The cognition corner of the triangle refers to a theory of beliefs about how students represent knowledge and develop competence in a subject domain. Is needed to identify a set of knowledge and skills that is important to measure for the task at hand, whether that be characterizing the competencies students have acquired thus far or guiding instruction for further increase learning. A central premise is that cognitive theory should represent the most scientifically credible understanding of typical ways in which learners represent knowledge and develop expertise on domain. These findings should derive from cognitive and educational research about how people learn, as well as the experience of expert teachers. Cognition is not meant to imply that the theory must necessarily come from a single cognitive research perspective. Theories and data on student learning and understanding can take different forms and encompass several levels and types of knowledge representation that include social and contextual components. These inference for a given assessment will be a subset of a larger theory of how people learn the subject matter. Targets for assessment could be expressed in terms of numbers, categories, or some mix; they might be conceived as persisting over long periods of time or apt to change at the next problem step. They might concern tendencies in behavior, conceptions of phenomena, available strategies, or level of development.

Implications for The Design of Learning Environments

How to take knowledge about how people learn, as well as the implications for curriculum, instruction, and assessment, and use it productively to design effective learning environments? What role is there for information and communication technologies in this process?

These questions do not have simple answers. One implication is to achieve higher level thinking and learning outcomes we want for our students, we will need to build learning environments that more carefully and consistently implement design principles that foster an effective integration of curriculum, instruction and assessment…

To address the design challenges, we need to ask what findings from contemporary research on cognitive and social issues in learning and assessment… Four such characteristics have been identified for powerful learning environments:

1.- Effective learning environments are knowledge centered. Attention is given to what is taught (central subject matter concepts), why it is taught (to support “learning with understanding” rather than merely remembering), and what competence or mastery looks like.

2.-  Effective learning environments are learners centered. Educators must pay close attention to the knowledge, skill, and attitudes that learners bring into the classroom. This incorporates preconceptions regarding subject matters, and it also includes a broader understanding of the learner. Teachers in learner-centered environments take careful attention to what students know as well as what they don´t know, and they continually work to build on students’ strengths. 

3.- Effective learning environments are assessment centered. Especially important are efforts to make students thinking visible through the use of frequent formative assessment. This permits the teacher to grasp the students’ preconceptions, understand where students are on the “developmental corridor” from informal to formal thinking, and design instruction accordingly. They help both teachers and students monitor progress.

4.- Effective learning environments are community centered. This includes the development of norms for the classroom and school, as well as elections to the outside world, that support core learning values. Teachers must be enabled and encourages to establish a community of learners among themselves. These communities can build a sense of comfort with questioning rather than knowing the answers and can develop a model of creating new ideas that builds on the contribution of individual members.

Consistent with the ideas about multiple and interacting elements of a powerful learning environment, all driven by concerns about how people learn, are four principles for the design of instruction within such a contextual perspective:

1. To establish knowledge-centered elements of a learning environment, instruction is organized around meaningful problems with appropriate goals.

2.    To support a learner-centered focus, instruction must provide scaffolds for solving meaningful problems and supporting learning with understanding.

3.    To support assessment-centered activities, instruction provides opportunities for practice with feedback, revision, and reflection.

4.    To create community in a learning environment, the social arrangements of instruction must promote collaboration and distributed expertise, as well as independent learning.

…

Problem solving is the core of inquiry- or project-based learning.

Several contemporary educational reform efforts use dilemmas, puzzles, and paradoxes to “hook” or stimulate learners’ interests in the topic of study.

One major challenge for inquiry-based learning environments is developing problems that are rich and complex enough to engage students in the kinds of sustained inquiry that will allow them to deeply understand important new concepts. Bringing complex to the classroom is an important function of technology. Graphics, video and animation can be explored again and again.

These multimedia formats capture children’s interest and provide information in the form of sound and moving images that is not available in text-based problems and stories

The cumulative work on The Adventures of Jasper Woodbury Problem Solving Series (CGTV, 1994, 1997, 2000) is one example of an attempt to develop meaningful problems and use of instructional design principles based on cognitive theory.

Final Comments: Making It All Work.

The revision process is quite important and seems to lead in changes in students’ perspectives of the nature of adult work as well as conceptual growth. The larger model that has emerged is known as SMART model, which stands for Scientific and Mathematical Arenas for refining Thinking they include a Smart lab

a) virtual community

b) Toolbox, visual representations for problem solving.

c) Kids on-line which features students making presentations pointing typical errors as to allow students to engage in critical analysis arguments and see same age peers explaining their work in sophisticated ways.

The SMART model has provided students and teachers with access to information about how people outside their classroom have thought about the same problem that they are facing. Such access can help students be more objective about their own view and realize that even with the same information other people may come to different conclusions or solutions. In addition, discussion of these differences of opinion can support the development of shared standards of reasoning, a truly rich and useful understanding of complexities of connecting learning theory, instructional design and technology.

This puts theory into practice, provide the feedback we need about what works and why and the basis for much richer and evolving theories of learning and instructional design.

[1] SEEL, Norbert M. (Albert-Ludwigs-University of Freiburg, Germany). and DIJKSTRA, Sanne (University of Twente, The Netherlands), Lawrence Erlbaum Associates, Inc. New Jersey 2004.

[2] Typically, subject matter domains are recognizable bodies of knowledge that are the focus of instruction for some significant length of time. A broad subject matter domain like physics must really be understood in terms of subdomains that define the discipline.

Personal Development Plan

PDP

The challenging task of a life-long learner is to write or put ideas together to visualize them in a simple timeline. 

Achievement will always takes more time than one can project and presume, this has been one of the lifetime constructs in the author’s teaching experience. However, this PDP might carry a prediction to take the author to some interesting unexpected ports and will definitively help to clarify some of the many plans that are ahead.

Usually the work that teachers face, ought to ground scientific foundations or at least expose statements that have base on what is provable. It must be planned carefully and experience his labour as a delicate deed, that must be kept in mind, in deep thought, and research as tentative path of resourceful improvisation regarding ‘processes’.

Because, even when the target might not be changing the spirit of the students we shall be handing to them philosophical insight  in many ways. It is also a mission of great responsibility for teachers to mould their students’ thinking, so the task must have plenty of wishing and imagining as to be full of choices, findings, and takeaways; in one-word: learnings.

In the case of the author’s PDP there are two seeming realities moving along;  on one side, the mystic and ancient knowledge, which carry many of the discoveries of humankind; and in the other, the one that moves along with a scientific thirst which englobes the immediate target, in the author’s case the recognition here in Finland, of a pedagogical degree. 

Knowing that this effort will take this paper further into deep thinking and useful reflection, it carries also future projects and the many ways they might become a real contribution to Education, for the demanding changes that are needed in times to  come, hopefully these will be taken into consideration.

When looking to the future most of us will try to imagine, or at least can imagine that we might need a few innovative answers to face the unknown; the certainty of  constant change carries forward us at an amazing speed, and has brought us to feel that these dramatic changes in technology and science, which we all have witnessed in the last three decades, have been so vertiginous that we do not know what the next decade will bring to us, leave alone next year.

Facing this uncertainty, then, is a tremendous responsibility. We, as educators, shall bring forward not only thoughtful processes, but also some emotional answers for a global society that is falling into the abyss of individualism and irrational consumerism.

Upon graduation the author’s thesis on “The relation between psychophysical training of theatre and its influence in the learning of mathematics” was regarded, at least at the very beginning, as not only a too bold project for a thesis, but as something that would prove unconnected in terms of scientific grounds. Nevertheless, the author overcame scepticism and under the assistance of a philologist, a philosopher and a thesis specialist, and also a calculating device (FLASHMASTER), achieved to prove the correlation between maths and dramatic arts, and how understanding the connection between these two apparently separate worlds was not such. For example, an actor has the power to shock an audience calculating the tone and volume of his voice and the way to communicate a thought using pause and or gesture. This semantic features allow him to touch the audience’s emotions. This might be seen as a mathematical equation. In this same way, the connection between these two ways of approaching reality: the art and the sciences, shall clarify the relationship between design and mathematics. 

There is a complementary vision in Andean culture, called Yanantin and Masintin, which is regarded as dualism, and not as two disconnected matters or opposite dialectic explanation. This world view could be understood and compared to the Chinese Taoism. While the Western world sees dialectic concepts as antagonistic and contrasting, Andean concepts refer to differences as a way to understand the whole, a complementary cosmovision, a wholesome universe that provides the counterpart with what it is missing, so that in this sense there is nothing opposed but a holistic and wholesome live. Andean philosophy also sees the opposites of existence (such as male/female, dark/light, inner/outer) as interdependent and essential parts of a harmonious whole. 

This relation of unity and complementarity appears in all their cultural manifestations: specially the designs appear to show what can be learnt from nature, and they were observant enough to reflect it in temples’ architecture, in iconographic adorn, in textiles, in utilitarian pottery, in metallurgic handcrafts, rituals, music and even in their mathematical calculations, when they were looking into nature or to outer space,  to the constellations in the heavens.

Surely this philosophy might help us to face future, taking not only a glance to the past, but a deeper look into ancient cultures which lived in harmony with their environment, and that stated the importance of learning from nature to give response to needs. And for this PDP, the author found a comforting way to put together holistic ideas to work by matching artistic design and mathematical calculations.

Tokapu Kelka is one example of this ancient knowledge based in the Andean dualism. It englobes not only the iconographic representations of the southern star, which inspired pre-Inca and Inca cultures to perform the most diverse designs, but also a complementary way of understanding the world. The designs are based on a symmetrical progression from which scholars presume writings and sacred icons are derived.  The discovery of these two, apparently, different languages, one signed by formal calculations and semiotic rules and the other by the daily things that signify culture seem to be unified in each doing and handcraft. The whole culture was harmonizing ritualistically with their environment, looking to nature and considering themselves just as another part of it.

This amazing legacy has endured the vassalage that 500 years left over American cultures. The survival of this knowledge and the tradition of its legacy has surpassed the barriers of time and now returns to us in the form of humble wisdom.

When the author´s approach to the Tokapu puzzles commenced, there was a sceptical awe in the wonders of what he was witnessing. The marvellous designs appeared to be just an entertaining game and that alone. But in deepening in the study, the discoveries and the bibliography showed that there was more to be discovered. So then the use of the Yupana came along; this is a mathematical device that was used to calculate and was being used again, nowadays, studied as a concrete material to clarify mathematical notions to children. Other artefacts like Quipus, an ancient way of accountancy, and the use of the Kelka concept used in fabric designs as a writing ideogram by weavers happened to be only a few of “the skeins of this long hair”.

Deepening into these many subjects is not the matter of my PDP, since there is still a long way to deal with research, and many scholars are actually doing this studies; but this might be a proper way to introduce these two interesting notions and to point out its importance, and how to match them for education would be the purpose of this paper.

Tokapu and Yupana

The Tokapu puzzle is a game designed and developed into its actual form by some scholars of the Peruvian tradition in Perú and Finland. César Aguilar, a Peruvian artist settled in Finland, introduced the author to the Tokapu puzzles in 2016 and they immediately dragged him into its beautiful forms and designs to start studying them. Since the author’s thesis corresponded to the holistic connection between the arts and the science, the discovery of integrating the Yupana, the Inka mathematical device, to the Tokapu Puzzles did not take long. Again, the complementary Andean philosophy guided us towards an enlightening path on firm grounds.

The multiple answers allowed by the puzzle suggested also many possible variations. The playful colours, the way of accommodating the pieces in layers according to different sizes that kept a Fibonacci progression relationship, and the use of more pieces and colours in each layer, can guide our interest further on.

Playing freely with this concept, we can assign a value to each colour and then we would have created problems spontaneously; calculating certain quadrants of the puzzle to justify the answers comes up naturally, and players are conducted to playing with designing concepts, like symmetrical or asymmetrical compositions, and later on to instructional design an iterative mathematical practice. 

But then some other concepts came up, such as the wavelength of the colour spectrum, and even some other considerations, like angle studies of geometry and volumes that could help students for the understanding of more scientific knowledge.

Moving even further into accidental discoveries for the game, there was a fire in César Aguilar’s house and some of the pieces that were saved from the fire, because of the water, swelled and from that accident the idea of a Braille version for the puzzles was developed. The puzzles are now more than a possibility, they are a hope for learning and understanding other scientific concepts, and a chance to be introduced to the young, to people with special needs and even the elderly may take profit from it.

The immense potential that lays on the Tokapu and Yupana match inspires a whole new way of approaching maths and science for teachers. Measuring the world has always been a way to understand reality, to grasp the universe, and through these games we may do so in a friendlier way.

When addressing somebody about their talent in maths, usually the reaction is a defensive one -I am not very good at math- would be a very frequent answer, but this responds to a conditioned behaviour.no one is good enough in Math, and this is because this is the ground of uncertainty, where only the wise move or those with deep capacity of thought and reflection. Troubled by the lack of understanding we move away from it.

But if someone can make you see the way to solve a problem usually you look at this teacher as an angel descended from heaven to enlighten you, and you will never forget the teacher and the feeling of the discovery, probably the mathematical notion may disappear, but the feeling of achieving the right answer never will. And many times, the trick to solve the problem is to be able to enunciate it clearly.

“Von Neuman was a fantastic artisan of the theory…He could write a problem from the first time he heard it, with an excellent sense to express it…he would put great care in what was said and that what was written was justly what he meant to say”

The wonderful thing about our Mathematical Tokapu Puzzles and the Yupana Device is that they explain the exercises themselves.

Since the designs are created by the players, they decide which is the problem to solve and they are the ones to decide where is the logic of their findings. In a way you are encouraging the student to expose these findings and to look at with critical eyes at  his own decisions and at those of others. So somehow interaction is also guaranteed.

The procedure has proved to be fun and rewarding, of course there are a lot of questions to be answer in this process of development:

What is the best material to produce the Tokapu Puzzles?

How does the Yupana integrate the Tokapu?

What are the best ages to introduce this game?

How training children would benefit math teachers best?

What kind of training shall art and math teachers go through to be assertive and helpful to each other?

How easily will the heuristics of the Tokapu Yupana spread using appropriately its mechanisms?

Would it encourage elderly people to play making new synaptic connections and favouring neuroplasticity?

Will people with special need will take real profit from the use of the game and mathematical device?

These are just some of the many questions regarding the academic research that has to be undertaken.

On the other hand, answering to these many questions we might be able to give answer to some other referred to the entrepreneurship side.

How much will this cost?

How many can be produced and who may be interested in buying the games? 

Which channels of distribution will be used to make it more effective? 

Many other rises immediately from these basic ones.

As for the process, it is irreversible. The author sees it as something very useful nowadays, an ancient knowledge that forces to look back to its wisdom. Looking with respect at it, and seeing this iterative practice as a cornerstone for further research. So we see the urge of this knowledge to be shared. It shows us an awesome opportunity to give mankind, yet, another way to look at mathematical knowledge and to practice it in a friendlier, more interesting and as a natural matter. Maths shall be a part of daily life and knowledge, easy to everyone and never referred to as a troublesome discipline, but as an enjoyable experience.

Regarding this PDP entrepreneurship project, the author made acquaintance with the Start-up programme in Haaga-Helia in the person of Ms. Irma Mäkäräinen-Sumi whith whom three appointments were made. During the first one it was a short exposure among other students of their own PDP. Soon enough the idea captured her attention and we had an invitation for the Autumn Start-up in Helsinki city where we met Tommo Kuisuvalo and other people regarding the Start-up initiative in Helsinki city. After that we had had two more meetings and in the last one we agreed that Ms. Mäkäräinen-Sumi will be coaching this PDP.

She referred to it as a very solid Product Development already done where: Physical product(s), educational  grounds and that the  IT development project have a fine projection.

The entrepreneurship experience will be under construction on the Business-wise target. This will include: manufacturing, team-work (formation and responsibilities), finance (grants to apply to startup the business), customers (potential) and customers coming from a pilot experience. Seeking for partners, and the projection to global or international partners shall build into customers interested in supporting the development of the product in other places around the world.

Since this a product brings with it so many sinergic possibilities, being an  educational tool, the ways to develop it may include digital, format, videos, manuals, books, apps, and those referred to training purposes as tutorials, workshops and others. As these can be considered for the Value Proposition that Ms, Mäkäräinen-Sumi exposed and suggested to think of during our last meeting.

The author hopes that this starting point takes us to develop and share with students from all around the world, the fascinating many possibilities of the wonderful knowledge that Tokapu & Yupana Mathematical Puzzles enclose.

Here some links from our late teaching experience:

https://www.youtube.com/watch?v=reDfmnHpQvg&t=12s

https://www.youtube.com/watch?v=B_XNg0X0H_g&t=14s 

Tokapu and Yupana project

1. Product development/idea development

   1. physical product/service, mathematical puzzle, cardboard box

Tokapu and Yupana project is an idea that comes from Andean tradition of design and mathematics working together.

The strength of the project is based on the opportunity and advantage of an educational concrete tool for children of all ages and the possibility to play it with their elders having a way to play and compete in designing following certain rules of symmetry in design and space.

A plus would take us to students with Special Needs when we can take design according to textures and widths of the pieces of the puzzles which will give a kind of versatile approach into a sort of Braille design.

• 1. additional material, products /services

The puzzles are not only a static matter because applying its rules, the creativity may also take part because it will not stay into the pieces that we make or provide, but by changing the designs through the colours, the sizes and the cuts for the puzzles they will give the opportunity to develop personal characteristics and will offer quality time to spend with children for parents and a clever tool for teachers on the learning of mathematical calculations as they involve measures of sizes proportions, angles, volumes and percentages.

• 1. digital apps, games? etc.

The digitalization of the product may include games, calculating devices (Yupana variations) and the usage of recycling elements.

1. Business side

   1. Marketing, market research

Educational official systems, libraries, schools and Special Needs Institutes and programmes

• 1. Finland/Peru education system

According to the research on the use of the puzzles and the origin of the Tokapu design we may offer it as an ancient knowledge that can be rediscovered by applying it in different schools and stages in both Finland and Perú to start with.

• 1. Business Model Canvas, how this idea could be a business?

  2. Name, trademark, logo etc.

  3. IPR intellectual property rights (trademark, copyright etc.), legal

  4. team? what you have as a knowledge, what are you missing as a skill, which kind team do you need

  5. Pricing

  6. Question of funding?

                                                   i.     friends and fools

                                                  ii.     own capital

                                                 iii.     crowdfunding

                                                 iv.     Tekes, grants

1. Viranomaisluvat, official permits

2. HOW WILL YOU DO THIS?

   1. to get the feedback simultaneously as you are creating this

To do:

1. Prioritize, what are the most urgent, most important

2. Plan of action for next year.