The First and Greatest Commandment

Deuterenomy 6:4-9

Israel, remember this! The LORD and the LORD alone - is our God. Love the LORD your God with all your heart, with all your soul, and with all your strength. Never forget these commands that I am giving you today.Teach them to your children. Repeat them when you are away, when you are resting and when you are working. Tie them on your arms and wear them, on your foreheads as a reminder. Write them on the doorposts of your houses and on your gates.

As believers this bible verse ought to be a vital part of our daily lives, instead of just 

simply reading and memorizing it we ought to meditate and actually feel each and every 

word wholeheartedly. From experience, I am adamant that LOVE encompasses all. The 

most important of all is to love GOD.  There is an unseen power bestowed upon us when 

we love GOD. Most importantly, when we love GOD in spirit and in truth, then we will be 

able to love others  no matter their characteristics.I feel that this bible verse, carries the 

most weight compared to all verses. It represents the central being of the bible. As 

clarified by 1 Corinthians 13:13 " Meanwhile these three remains, faith, hope and love 

and the greatest of these is Love.

REALISTIC MATHEMATICS EDUCATION

All structures that exist mathematically also exist physically.

Tegmarks Max (en.wikipedia.org)

I would like to relate the statement above in the explanation and description of Realistic Mathematics Education. Realistic comes from the word real, of which according to the Dictionary.com “is based on what is real or practical”. Thus, in my opinion, Realistic Mathematics Education is realistically mathematizing daily problems for solutions solely for the benefit of each individual’s daily lives. After numerous readings and internet search, RME originated from the Netherlands by Hans Freudenthal a Dutch mathematician, who proposed that since students’ cannot be viewed as passive receivers of ready-made mathematics, mathematics education should then be directed to the use of a variety of situations and opportunities to enable students to reinvent mathematics (Hadi, 2002). In this he infers that Mathematics should be a human activity (Hadi, 2002). Specifically, the maths problem begins with a daily real world issue, which is then deduced, structured and simplified into a real model incorporating concepts, relations, conditions and assumptions for the final formal mathematical model.

Realistic Mathematics Education Model.

I would like to use various models to specifically explain and describe Realistic Mathematical Education.

According to the diagram above taken from powermathematics.blogspot.com there are 4 levels in the process of RME which is similar to Blum & Niss (1989), 4 steps of applied problem solving.

1. Mathematical world orientation or a real problem situation.

2. Model material or a real model of the original situation

3. Building stones, number relations or mathematising

4. Formal mathematics or a mathematical model.

The first step mathematical world orientation or a real world problem situation refers to daily lives problems, for instance, in my case, how can I travel to Fiji, using the shortest and cheapest flight route? That is an illustration of a real world problem, that individual mostly travellers are faced with often. The second step is to  create a real model of the situation from cognition unto paper through simplification, structuralism and deductivism. In this example, I will then have to identify the various flight routes to Fiji, their costs and the duration of each flight.There are quite a number flights from Yorgyakarta to Fiji, for the sake of this short paper, I will use two, which is flying from Bali (Denpasar) or from Jakarta. If I have to depart from Bali, then I can transit through Sydney, Melbourne or Brisbane. However, these 3 transit cities have different costs as well as duration of flight. But, if I have to depart from Jakarta, then I will have to transit through Hong Kong. The third step, which corresponds with Building Stones, number relations or mathematizing is the tagging of prices (numbers) on each of the routes in relation to their flight duration.  Then, the computation of the ready made mathematical model for evaluation, in solving the problem of the shortest and cheapest flight route to Fiji which depicts the final step of the process of RME.

Returning to the sentence, given above, RME as a thought can be a bridge from the abstract world to the formal world of mathematics.  Eventually, the models give the students access to more formal mathematical knowledge. In order to fulfil the bridging function between the informal and the formal level, models have to shift from a model of to a model for (Streefland, 1985). Apparently the shift from the model of the real world to a model for mathematics, has  often been a challenge to most student’s likewise to mathematics educators. However, according to the Contemporary school of thought of Mathematics Realism such Platonism, Empiricism and Mathematical monism, have provided various thoughts that can be encouraging and stimulating into actually, devising ways of contributing into the simplification of Realistic Mathematics Education. Significantly, as stated by Tegmarks, “all structures that exist mathematically also exist physically”. That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world". In which in relation to Realistic Mathematics Education, modifying the former to become “all structures that exist physical, also exist mathematically” this corresponds to his mathematics universe hypothesis (MUH) which is: Our External Physical Reality is a Mathematical Structure. Thus, in my opinion, realistically, real life problems already do exist mathematically, in which, there exist a solution which needs to be discovered by Man.

Reference

En.Wikipedia.org

powermathematics.blogspot.com

THE BRIDGE

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Reflection # 6 with Professor Marsigit

A unique experience owned by individuals depicting their personal relationship with God. Our 

spirituality differs as it reflects the way we live our lives discovering the essence of our beings. 

There cannot be any other source which this epistemology is obtained from but from our own 

individual beliefs facilitated by the different religions that we belong to, thus it is adamant that we will have different perspectives on the matter.  It is vital for a being to own an experience as such because without it, we are similar to a tree without its roots. Like a tree, in which the roots is the main connection between itself and its source of life our spirituality is the bridge that transcends our phenomenon and noumena towards our source of life.

With the above in mind, we begin to have the perception and feel the sensation of two beings within us, reflected in reality by our physical being with our methods of living.  Professor has often emphasized the importance of praying when studying philosophy. I do understand and support this notion because Philosophy as I have just realized is maybe another way in which God had wanted us to learn about the abstract world that we live in through the minds of outstanding Philosophers such as Immanuel Kant. From experience during this semester, reading a few chapters in the Critique of Pure Reason, definitely has been mind blowing directing us into another dimension of time and space. Thus, without spirituality that will keep you rooted to reality, the experience keeps you afloat endlessly.

 Philosophers, reflects deeply on the contradictory nature of human life, on God’s mercy and justice, on the seasonality of life, the nature of time and space and on many other topics that engage people in this world. Life is spent on working, labouring and what do we have to show for it? Generations come and generations go, but the world stays just the same. The sun still rises and it still goes down, going wearily back to where it must start all over again. The wind blows south, the wind blows north –round and round and back again. Every river flows into the sea, but the river is not yet full. The water returns to where the rivers began, and starts all over again. With such occurences in this world without a spiritual life it is like chasing the wind.

Reference: The Holy Bible (Good News translation)

Teachers are Researchers

Reflection # 5  with Professor Marsigit.

Would the statement given above as my topic be classified analytic or synthetic? If it’s either of the two, can it be “a priori” or “a posteriori” proposition.

As teachers, aspects of our lives have to be within the domains of our profession, theoretically and practically.  Our mission is the same throughout our teaching career, educating the child according to the various school missions that we have in correspondence with the education mission as a state. Consequently, within individual teacher’s intuition, there are contradictions in their own contribution towards the mission and vision with their own selves as teachers of mathematics. Apparently sharing the same sentiments as Earnest,

Individuals’ conceptualisations of their lives and the world about them are through a highly quantified framework. The requirement for efficient workers and employees to regulate material production profitably necessitated the structuring and control of space and time (Taylor 1911) and for workers’ self-identities to be constructed and constituted through this structured space-time-economics frame (Foucault 1976). We understand our lives through the conceptual meshes of the clock, calendar, working timetables, travel planning and timetables, finances and currencies, ie concept of insurance, pensions, tax, measurements of weight, length, area and volume, graphical and geometric representations, etc. This positions individuals as regulated subjects and workers in an information controlling society/state, as consumers in post-modern consumerist society, and as beings in a quantified universe.

Accommodating all of the above, where do were stand as teachers of mathematics? Why do we teach mathematics? What are the purposes, goals, justifications and reasons for teaching mathematics? Realistically, probably our concepts of the two key words teacher and mathematics have been at the tip of an iceberg. As teachers we ought to continuously research our profession subjectively and objectively. Developing ourselves professionally is essential in the context of both as a mathematics teacher and for mathematics. Considering Earnest (1991), five interest groups and their aims for teaching mathematics, which are the: Industrial Trainers {Acquiring basic mathematical skills and numeracy and social training in obedience (authoritarian, basic skills centred)}, Technological Pragmatists {Learning basic skills and learning to solve practical problems with mathematics and information technology (industry and work centred)}, Old Humanist Mathematician {Understanding and capability in advanced mathematics, with some appreciation of mathematics (pure mathematics centred)}, Progressive Educators {Gaining confidence, creativity and self expression through maths (child-centred progressivism)} and Public Educators {Empowerment of learners as critical and mathematically literate citizens in society (empowerment and social justice concerns)}. Which of those do we individually belong to or do we aspire to belong to?

We consider the concept of accountability and sustainability. As teachers we are accountable for our actions and responsibilities towards stakeholders of the school. We ought to reflect upon every results feedback after an exam. It is very important to reflect within our own epoche, how we have performed after every academic year. Have we been honestly and righteously doing our duties to God, let alone our own selves? It reflects our selves, who we are really and how have we honestly been doing within this academic year. A show of contradiction between our noumena and phenomena, the unseen real vs the unreal seen. Sustaining, our students’ in all dimensions of life within the bounds of the school is an important aspect of our responsibility as a member of a noble profession. We will often be respected throughout our lives as provider of knowledge and wisdom within the lives of our students. We are accountable and sustainer for their knowledge regarding our subject content. Thus, as bapak mentions, we are researchers, we will continue to research for better ways to teach, for simpler notes to provide, and for our students’family and personal backgrounds, our stakeholders and our own selves as teachers.

Thus, to answer the question above, “Teachers are Researchers” are a posteriori analytic proposition. In my experience, we as teachers are definitely researchers.

Reference

Ernest, P (1991). The Philosophy of Mathematics Education, Studies in Education; Taylor & Francis; ROUTLEDGE FALMER.

Know Thy Self

Reflection # 4 with Professor Marsigit.

An eighty year old man or woman, who has already experienced the methods of life in its various scope and dimension probably would have wondered how they did in this universe. If only we can each have an autobiography of our lives a few years before we pass on, many people will have the opportunity to reflect and better their lives. However, if it’s in terms of materialism or formalism it would rather be too late, only exception would be in terms of spiritualism. Methods of our lives are different due to differences in religion, culture, ethics and morals and other aspects of life. But whatever differences are there, it would be rather amazing to know and feel that we all as human beings share commonalities especially in terms of morals and ethics despite our differences in religious views.

Methods of our lives reflect our upbringing, depending on whether we intend to carry on our parents teachings or not. It is very important to always do away with negative attitude. Once, we become negative, it becomes a barrier to all formal positive opportunities of life. Thus, it is important to always ask God for guidance because he enlightens the darkness, strengthen the weakness, mend the brokenness, bind what’s being bruised, heal sicknesses and revive all that has died within us.

I quote what was inscribed on the Temple of Apollo “Know thy self”. Are we consciously aware of our physical and concious being? Are we apparently concious of true reality (noumena) under the apparent reality (phenomenon) an unseen real beneath an unreal seen? What are the dimensions of our life around this materialistic world? It’s really all important questions because it is our very own selves that we are questioning. What impresses us most and who inspires us most determines our thoughts and actions, whether they are positive or negative factors. We ought to make a distinction between what we consider really exist from those that do not exist.  In my opinion definitely it is a material world that we live in, we can actually see things for real, but there is an aspect of true reality that we cannot see. That unseen true reality is our true self incorporating the methods, the dimensions, formality and materiality of our lives that none other but only God can see and know. In my opinion, it is the heart of our thoughts that we feel and sense but we cannot see. It is thus a connection between the real and the unreal because what is unseen real is actually what controls and drives the unreal seen.

Question for Mr Marsigit.

Sir, I do believe that what we actually cannot see are real and permanent and all that a human eye see are not real. They are made for our of living in this earth, consistently trialling us for the true nature of self. What do you think?

                                    

Contradictory Nature of Human Life

Reflection on meeting #2 with Dr Marsigit.
PMAT Kelas C - Semester 1

It’s getting much clearer now, that the rain is gone. 

Reflecting deeply on the contradictory nature of human life, on God’s mercy and justice, on the seasonality of life, the nature of time and many other  aspects of life that continue to engage people in this world, Philosophising helps us to recollect cognitively with reflections and predictions on our experiences and towards our future respectively.

With such categories of experiences at a mature age we become aware of our multipersonality in the physical and spiritual sense so we strive to unite them as one, further specified by Hegel (1873),

The impressions from feeling and perception are, if we look to their contents, a multiplicity or miscellany of elements; and the multiplicity is equally conspicuous in their form. For sense is marked by a mutual exclusion of members; and that under two aspects, namely space and time, which, being the forms, that is to say, the universal type of perception, are themselves a priori. This congeries, afforded by sensation and perception, must however be reduced to an identity or primary synthesi. To accomplish this the ‘I’ brings it in relation to itself and unites it there in one consciousness which Kant calls pure ‘apperception’.

As we understand, there are two aspects of these categories. First by our instrumental perceptions of sense which rises to objectivity and experiences, secondly all these notions are unities in our conciousness. It is clear that our physical nature is somewhat different from our spiritual nature. In which all is governed, monitored or directed by our cognitive abilities. However, there is often clashes between the two categories, because there are laws which govern the two separately.  

Thus, I do understand that the above alone is a small category of Philosophy of life, which will manuever us into the basics of Philosophy of Science. There still is the theory of Gods existence which incorporates his love and mercy. In addition the seasonality of life and the nature of time, which all come under various philosophical theories.

Question for Dr. Marsigit.

Do you think that every adult in this world are aware of this two categories of life that we have?

Reference

Hegel's, G.W.F, 1873, " The Critical Philosophy: from The Shorter Logic"' translated by William Wallace, Clarendon Press

A Reflection of Self.

A reflection of self

How often do we take time to reflect on ourselves? How often do we take time to reflect on our inner and outerselves?  Why are we here on this earth? What are we doing here? What is our purpose in this life?  I can say that most of us are too occupied with what this world offers then to take quality time in reflecting upon them. It is most important to use these questions to ponder on our lives because if we are ignorant of such then we are not fulfilling our purpose in this life as God wills for us individualy.

But however, one thing is for certain, that God chose us to be borned into this world. Moreso, according to the Bible, he created us in his image (Genesis 1: 26-27). Thus, we are the image of God. So if we have two hands, two legs, two ears and so on, then so does God which makes us the most superior being in this world.

So what discrimates us? Obviously as we are all aware of, our colour, culture, traditions and customs, differences in values and ethics, differences in religious views, political views and so on as mentioned above things that this world offers. What if all of the latter do not exist? Despite our thoughts, God chose us to be what we are such as Indonesians, Fijians, Indians, Europeans and so on. Sometimes I ask myself why amI a Fijian, why can’t I be someone I want to be. Then it dawned on me…Psalms 139: 13 it says, “You created every part of me; you put me together in my mother’s womb”. Powerful words indeed, it tells us so much and how unique we are, individually. If he did all this while we were still in our mother’s womb, then definitely he has already decided who and what we should become in life. Then we ought to respect ourselves first before we can respect others.  We ought to continue to LOVE despite our differences, because without love I believe we are nothing.

Philosophy of science indeed is the philosophy of life. Without life, there is no science. Without science if I may say there is no Math.Thus, without Math, would we be here studying Mathematics Education?  Indeed most of us are still searching their purpose in life. As Bapak mentions, experiences reconstructs lives, but however, with the realisation of self can one reconstruct from experiences. Without realising, they carry on in life without the realisation that their experience is supposed to be the reconstruction of their lives. Thus, for me it is important to understand and practice this philosophy of science, because it is the foundation of the philosophy of Mathematics Education.