Chaos Theory, Fractals, knowledge and learning – Part 2

Fractals are indeed also embedded in curation, and in subsequent conversation, by the agent (learner) through internal conversation with him/herself, and that with others in a complex learning environment, social networks, and community of networks.

Such fractals are part of the phenomena associated with the Chaos Theory.  It seems nearly impossible to make long-term predictions about online conversation where large number of agents are interacting with each others, as in the case of MOOCs.  Such conversations are highly sensitive to small initial perturbations (Fractals and Chaos Theory).  This also explains the often difficult to predict and control sort of conversation in open spaces, where constraints over what and how conversation is based on moderation by the agents (the professors, educators, and certain participants in the case of MOOCs).

How would fractals and Chaos Theory help in understanding more about the changes and transformation of our education system?

Helpful concepts include co-evolution, disequilibrium, positive feedback, perturbance, transformation, fractals, strange attractors, self-organization, and dynamic complexity. These concepts can help us to understand (a) when a system is ready for transformation, and (b) the system dynamics that are likely to influence individual changes we try to make and the effects of those changes.

Furthermore, chaos theory and the sciences of complexity can help us to understand and improve the transformation process as a complex system that educational systems use to transform themselves. Strange attractors and leverage points are particularly important to help our educational systems to correct the dangerous evolutionary imbalance that currently exists.  (Reigeluth, 2004)

How have strange attractors impacted on MOOCs, in particular on xMOOCs?

The most powerful strange attractors are core ideas and beliefs like those described earlier: ownership and empowerment, customization and differentiation, and shared decision making and collaboration.

How is Chaos Theory used in lesson planning and delivery?

The use of Chaos Theory in lesson planning and delivery is discussed in this paper. The author argues that planning for a lesson needs to take into account any changes in the lesson, building in elements of interests, and responding to the chaos in a dynamic way so as to make order out of chaos, especially when there are always strange attractors changing the stability of the equilibrium of the system.

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#Change11 How would a Shift of Framework help? A New Model for Understanding Time in Pedagogical Contexts

I enjoyed reading this Multidimensional Facets of Time in Online Learning by Pekka Ihanainen and John W. Moravec.

Ihanainen and Moravec provide a typology of Learning:

1. Temponormative Learning

2. Pointillist Learning

3. Cyclical Learning

4. Overlapping Learning

I am particularly interested in how they have elaborated on each category of  learning.  Here are my short notes.

Pointillist learning – Pointillist behavior and learning implies an ability to tolerate the insecure, uninterrupted, un anticipated and obvious absurdity of the “moment,” but at the same time it indicates a capacity to differentiate the essential from the unessential and to perceive the whole from fragments, almost as a fractal construction of personal experiences and understanding.  Such fractal construction of personal experiences and understanding also resonates with what I describe here, here and here.  I would like to expand this fractal construction in future research, where learning as conversation and social interaction could be viewed and conceptualised in a holistic perspective, under an ecology, or an experience in MOOC.

Pontillist pedagogy is the pedagogy of serendipity.  This sounds useful and I would like to relate to my experience here, here and Carmen’s post here where she reflected beautifully: Stepping out of a normal routine, finding novelty, being open to serendipity, enjoying the unexpected, embracing a little risk, and finding

In such a scenario, learning happens in instances and waves, independent of a definable pedagogical time.

In the overlapping Learning – “Pulsating waves of new knowledge generation within the learning group, beyond the learning group, and in the spaces between.”

I have conceived knowledge and learning as waves here, and so I would like to see if the Temponormative Learning, Pointillist Learning, Cyclical Learning, and Overlapping learning be metaphorically conceptualised as different waveforms, based on fractals and chaos patterns, where the different temponormative waves, pulsating waves and cyclical waves meet, causing interferrence and or resonance in the media, under sets, nets, groups, or collectives, and thus exhibiting different patterns under a Chaordic (chaos and order) ecology.  This requires further research and validation 🙂

 Picture: Google images

De-pedagogy means that as facilitators of learning, we have to give up our role as teachers and start working as colearners and peers within our own pointillist environments.  This sounds challenging to those facilitators who are accustomed to the instructivist paradigms – as sage on the stage, with lectures as the primary approach towards knowledge dissemination.

In reflection, I would like to dig deeper into our previous research here to see how the different learning pans out in CCK08, and subsequent MOOCs.

I would surely be excited if  Pekka Ihanainen and John W. Moravec include more empirical and grounded research findings and claims to their model.

It could be interesting to research into this learning typology with the Change11 MOOC.

I would surely like to respond to their challenge: “So in lieu of a conclusion, we leave educators—particularly online educators—with a challenge: Afforded the post-temponormative enabling of online environments, how can we best leverage these opportunities of pedagogical time to facilitate multidimensional learning and meaningful new knowledge production?”  How about you?

Reference:

Mak, Sui, Fai, J., Williams, R. & Mackness, J. (2010). Blogs and Forums as Communication and Learning Tools in a MOOC. In Networked Learning Conference, Aarlborg (pp. 275-284). Retrieved from http://www.lancs.ac.uk/fss/organisations/netlc/past/nlc2010/abstracts/Mak.html

Research, Wave Theory and Curiosity

I would explore and reflect on Research, Wave Theory and Curiosity in this post.

Research

I have changed my way of doing research through my blog posts: quite a bit.  I trust that we could experiment with blog-post followed by peer & community review approach in doing research.  I have been thinking about narrative and case study researches instead of mere surveys.  This would ensure the theory model building is based on mixed research methods, and grounded on application-theory combination – grounded theory.

Wave Theory

I have been thinking knowledge and learning along the lines of wave theory – i.e. learning as waves – resulting from the neuron-connection, that waveform as shown on the fMRI scan denotes the knowledge pattern and wave propagation as a learning both at a micro and macro -level.  Here, the concept of fractals could be useful to denote the propagation of knowledge growth.  Complexity Theory and Chaos Theory – where emergent learning arises could be explained when different waveform meet, causing interference patterns (either constructive or destructive interference – similar to the amplification and dampening action as in self-organizing networks), and different types of waves would be formed upon interaction.

I haven’t shared any of the above ideas to any one yet, as I started to think right now except the one here.  It didn’t resonate with anybody else.  Has anyone explored about these?

Wonders of Nature

Video that may be of interest

My curiosity

My wild questions on assumptions pop up about nature of everything, relating to the interconnected nature of science, where one wave sets up the changes (a ripple stirs up another set of waves, like the butterfly effect).  We speak and hear – through sound waves, we see through light (light waves), we sense through touches – sensory “waves” – muscles contract and extend, we breathe and smell through air (air in the form of wave), we think with brain wave, we send signals through electromagnetic waves.  Even earthquake, air turbulence, typhoon, hurricane, tsunami and black holes are the results of huge waves, some visible, some invisible!

Photos: Wikipedia Butterfly Effect

Would wave be a way of connection?

I wrote these before realizing the video on Curiosity, wonderfully crafted below, with narratives of Feynman.

Postscript: I just found this video about Waves, narrated by Feynman.  What a coincidence of the concepts behind!

I found this video useful in explaining about Waves in Brain

Using Fractals to explain social interaction, synergy and life creation

This is a series of post on how fractals and their development would explain about the development of social interaction, synergy and social capital.

I would like to start with this video on fractals

The basic formula for fractal is Z=z*2 + C

If we were to denote z = (a+bi) where a and b  are “real number” and i is the imaginary number of square root -1, then by definition z is a complex number.

Let’s replace z = a + bi by an idea “created” due to an interaction between two ideas (or two nodes).

a denotes a known idea or information (e.g. known knowledge or information that can be explicitly described, explained) though it is a “variable” that would depend on the scope and complexity.  This is the explicit idea or information (or knowledge) that we could refer to under existing knowledge framework

b denotes a variable that is associated with the tacit idea, knowledge that are yet likely embedded in one’s “mind”, or hidden within thoughts, that needs to be mined out, or could only be manifested through “tinkering” as explained by John Seely Brown.

i is the fundamental imaginary idea or “intuitive” idea that might not be known by the person, or what is in the dream, and could be the power particle that we are looking for.

bi is then the number of such fundamental imaginary ideas that emerged out of the mind (or brain of human)

Then z= a + bi could represent the ideas that are emerged from both explicit knowledge and tacit knowledge

What happens if two ideas of a network interact with each other?  We could denote the result as z*2 = (a+bi)(a+bi) = a*2 + 2abi – b*2

a*2 – b*2 denotes the resultant explicit knowledge and

2abi denotes the resultant implicit tacit knowledge

Now f(z) or z = z*2 + c could then be represented as

z (on the left hand side of the equation) = emergent knowledge (or learning)

z*2 = zxz refers to the interaction between z and z

c= boundary condition (which could be a stimulus, catalyst, but an initial condition which could “spark” the interaction, this could be based on the mediation of technology or social media).  So c could be the tools used, in the case of social networking, or the mediator such as teacher, classmate, in the case of classroom teaching.  These are the extrinsic motivators or factors that would initiate the interaction.

Let me put all these into a social context.

(a) Social interaction

When a person A interacts with a person B, within a social media (e.g. a blog, Twitter, Facebook, or Quora), which is denoted by c, then the emergent learning (or knowledge) that may result from such interaction is

z = z*2 + c

The development of this fractal into different fractal patterns would depend on (a) the z (ideas, information, knowledge) themselves

(b) the boundary conditions (and the ecology)

(c) how the interaction occurs

This concept could be applied to the interaction in case of communication between two persons, or interaction between an actor (a person) with a non human actor (could be an animal, a machine, a media, an artifact) or a network.

The product of such interaction would be emergent and its development is also based on the initial boundary condition.

So, the synergy emerging out of an interaction of actors (in networks) is greater than the sum of its parts mainly because z*2 +c = (a+bi)(a+bi) +c= a*2 + 2abi – b*2 +c is normally greater than a*2.  and so “network” collaboration or cooperation would likely generate more ideas than those coming from individuals.

However, using the above formula, there may also be noises involved in the interaction, which may be denoted by c being a negative value.  So if the noise – c value is big enough, then the resultant value of z*2 +c could be less than z*2 meaning that there could be a loss of synergy.  This also explains why conflicts (which may be denoted by c) could often hinder or even lessen the resultant synergy out of the interaction.  This explains why some ideas are amplified, resonated, and other ideas being dampened.

Further explanation about how these ideas are resonated, developed are explained here.

The above could also be used to explain the formation, development of networks, Networks, ecology under Connectivism, and that of Actor Network Theory and Community of Practice, all based on the Fractals.

(b) Origins of life.  This is a re-post from my previous blog post on Fractals

I have written an article (never published) back on 22 November 1997 at 11:36 pm.  Here is part of it (though some other parts have been added/revised now):

A basic mathematical theory on the multiplication of cell.
(details not shown)

As life begins in the multiplication of cells:
Imagine a round cell is now sub-divided into two identical cells of the same size as the original one.

The surface area of the multiplied cells could be equal to the surface of another cell of radius r2

By equating 4x pi x r2xr2 = 2 x 4 x pi x r1 x r1

r2 = square root 2 x r1

This suggest that the cells increase by square root 2 the original size of the radius but two-folds in area for any division of cells.

The formula f(Z) = ZxZ +C could be a general formula for explaining this division of cells (I don’t think I have learnt anything about fractals in 1997, and I just made up that formula to prove my point in the division of cells).  So, when I watch the video now and realise that the same formula was used in fractal formation, I was so amazed.  I am just not sure where I got that idea from.  Remember in 1997, the Internet wasn’t popular as yet, and I don’t think I knew the formula as mentioned in the M-SET.

But I really did a lot of original work without referencing to any literature.

Let  Z be a complex number denoted by a+bi, you could derive that

C=square root 2 x a (working not shown here)

If we apply this principle to the multiplication of basic cell, it means that one cell will be divided into two, and the two cells will be divided into four and so onwards.

It seems that the square root of 2 is the mysterious figure that should be investigated.  And I suggest that this is fundamental “number” that could unfold the origins of life – i.e. square root 2.  That is, if we keep squaring root the 2 objects, then it would become 1.

This is similar to the concepts of uniting the sperm and ovary in human birth, where the ovary will begin to subdivide once the sperm unite with ovary.  So the sperm is C and ovary is Z, where the ovary multiplies when the sperm entered into the equation, and the life begins.  Does it make sense?

I will pause at this point.

I will use the above to explain the development of online resonance, Instructional Design, PLENK, Community of Practice, and Social Capital in coming posts.

About Fractals – a revisit and reflection

I have been thinking about fractals for the last few years, and found some renewed interests here in its application in social media.

I believe that there might be some ground breaking ideas behind fractals especially in describing how it could be represented as a pattern in
(a) our brain
(b) our origins of life
(c) our way of communication and networking, especially in a complex open networked environment
(d) our future of education.

Here are my meanderings:

The basis of fractals is routed from (you could watch the video series 1-6 and it was mentioned in one of them on the formula of fractals basis https://suifaijohnmak.wordpress.com/2010/03/30/the-map-is-the-treasure/

f(Z)= Z x Z +C  where Z is a complex number (a+bi) – where a and b are real numbers and i is the square root of -1, an imaginary number.  I include this as a brief background only, and I must admit that I am too new to it. 🙂

This is similar in concept to the Newton’s iteration equation in solving numerical problem, which is also fundamental in computation using iteration.

Relating again on the video https://suifaijohnmak.wordpress.com/2010/03/30/the-map-is-the-treasure/development of fractals based on just the growth of two lines (could be both equal in length, or unequal, with same or different directions) is fascinating, as this may provide some cues in lots of areas:

(a) Our brain.  First, I suspect that the development of dendrites in the nerve cells and the connections are formed could be simulated using the fractals development pattern.  I don’t know if there are any simulation based on that, but surely the full brain scan provides a pattern that could be studied in greater details, to see how such “fractals” are formed and developed.  I realise patterns of fractals appear when I wash my hairs with hot water, and such images of fractals could be “visualised” naturally.  Have you got similar experience?

(b) Origins of life.  I have written an article (never published) back on 22 November 1997 at 11:36 pm.  Here is part of it (though some other parts have been added/revised now):

A basic mathematical theory on the multiplication of cell.
(details not shown)

As life begins in the multiplication of cells:
Imagine a round cell is now sub-divided into two identical cells of the same size as the original one.

The surface area of the multiplied cells could be equal to the surface of another cell of radius r2

By equating 4x pi x r2xr2 = 2 x 4 x pi x r1 x r1

r2 = square root 2 x r1

This suggest that the cells increase by square root 2 the original size of the radius but two-folds in area for any division of cells.

The formula f(Z) = ZxZ +C could be a general formula for explaining this division of cells (I don’t think I have learnt anything about fractals in 1997, and I just made up that formula to prove my point in the division of cells).  So, when I watch the video now and realise that the same formula was used in fractal formation, I was so amazed.  I am just not sure where I got that idea from.  Remember in 1997, the Internet wasn’t popular as yet, and I don’t think I knew the formula as mentioned in the M-SET.

But I really did a lot of original work without referencing to any literature.

Let  Z be a complex number denoted by a+bi, you could derive that

C=square root 2 x a (working not shown here)

If we apply this principle to the multiplication of basic cell, it means that one cell will be divided into two, and the two cells will be divided into four and so onwards.

It seems that the square root of 2 is the mysterious figure that should be investigated.  And I suggest that this is fundamental “number” that could unfold the origins of life – i.e. square root 2.  That is, if we keep squaring root the 2 objects, then it would become 1.

This is similar to the concepts of uniting the sperm and ovary in human birth, where the ovary will begin to subdivide once the sperm unite with ovary.  So the sperm is C and ovary is Z, where the ovary multiplies when the sperm entered into the equation, and the life begins.  Does it make sense?
(c) way of communication and networking, especially in a complex open networked environment.  I have written a post on this – the map is the treasure https://suifaijohnmak.wordpress.com/2010/03/30/the-map-is-the-treasure/
(d) our future of education.  Given the current trend, could we predict the future using fractals as pattern?  May be.  I am still pondering on this, but I would postulate Z (future education) = Z (present education + imaginary or predicted condition of education) x Z (present education + imaginary or predicted condition of education) + C (paradigm shift: based on networking, social media and technology affordance, and promotion and support from government, institutions, communities, local and global networks, social medias etc.)

The following is just my intuitive thoughts only.
Finally, assume that a is our present life, b is our future life (imaginary) in the Z=a+bi then if our present life is “equated” to our future life, then we will have eternal life (as a Catholic, that is a teaching by Jesus Christ).  The eternal life could also be a reality by applying the principle of multiplication of cells.  Also our faith will multiply as a multiple of two just as the cell division.  This is also how religious beliefs gain its life in its spread of good news, IMHO.

Hope you don’t mind me sharing these really “strange thoughts” with you, but I could assure that you would never find it in any literature, as I made all these up myself.

I am posting it here onto my blog, though such are just half-baked ideas, not yet ready for “cooking”.

Let me know what you think.

Photo: From wikipedia on Fractals

I enjoyed watching this video about chaos and order

Postscript:

I love watching these videos on fractals too.  This is Part 6 of 6.  I have shared it in my previous blog post.

About fractals and some of my ideas

I have been thinking about fractals for the last few years, and found some renewed interests here in its application in social media – here are some sharing http://connectivismeducationlearning.ning.com/forum/topics/future-of-education?commentId=2749904%3AComment%3A9897&xg_source=activity

I believe that there might be some ground breaking ideas behind fractals especially in describing how it could be represented as a pattern in
(a) our brain
(b) our origins of life
(c) our way of communication and networking, especially in a complex open networked environment
(d) our future of education.

Here are my meanderings:

The basis of fractals is routed from (you could watch the video series 1-6 and it was mentioned in one of them on the formula of fractals basis https://suifaijohnmak.wordpress.com/2010/03/30/the-map-is-the-treasure/

f(Z)= Z x Z +C  where Z is a complex number (a+bi) – where a and b are real numbers and i is the square root of -1, an imaginary number.  I include this as a brief background only, and I must admit that I am too new to it. 🙂  Sorry Jenny I don’t want to fill you with jargon of mathematics, but this is important to understand.  And Roy and Matthias, please point out if I am wrong in any of the interpretation.  I love to learn more on this.

This is similar in concept to the Newton’s iteration equation in solving numerical problem, which is also fundamental in computation using iteration.

Relating again on the video https://suifaijohnmak.wordpress.com/2010/03/30/the-map-is-the-treasure/ development of fractals based on just the growth of two lines (could be both equal in length, or unequal, with same or different directions) is fascinating, as this may provide some cues in lots of areas:

(a) Our brain.  First, I suspect that the development of dendrites in the nerve cells and the connections are formed could be simulated using the fractals development pattern.  I don’t know if there are any simulation based on that, but surely the full brain scan provides a pattern that could be studied in greater details, to see how such “fractals” are formed and developed.  I realise patterns of fractals appear when I wash my hairs with hot water, and such images of fractals could be “visualised” naturally.  Have you got similar experience?

(b) Origins of life.  I have written an article (never published) back on 22 November 1997 at 11:36 pm.  Here is part of it (though some other parts have been added/revised now):

A basic mathematical theory on the multiplication of cell.
(details not shown)

As life begins in the multiplication of cells:
Imagine a round cell is now sub-divided into two identical cells of the same size as the original one.

The surface area of the multiplied cells could be equal to the surface of another cell of radius r2

By equating 4x pi x r2xr2 = 2 x 4 x pi x r1 x r1

r2 = square root 2 x r1

This suggest that the cells increase by square root 2 the original size of the radius but two-folds in area for any division of cells.

The formula f(Z) = ZxZ +C could be a general formula for explaining this division of cells (I don’t think I have learnt anything about fractals in 1997, and I just made up that formula to prove my point in the division of cells).  So, when I watch the video now and realise that the same formula was used in fractal formation, I was so amazed.  I am just not sure where I got that idea from.  Remember in 1997, the Internet wasn’t popular as yet, and I don’t think I knew the formula as mentioned in the M-SET.

But I really did a lot of original work without referencing to any literature.

Let  Z be a complex number denoted by a+bi, you could derive that

C=square root 2 x a (working not shown here)

If we apply this principle to the multiplication of basic cell, it means that one cell will be divided into two, and the two cells will be divided into four and so onwards.

It seems that the square root of 2 is the mysterious figure that should be investigated.  And I suggest that this is fundamental “number” that could unfold the origins of life – i.e. square root 2.  That is, if we keep squaring root the 2 objects, then it would become 1.

This is similar to the concepts of uniting the sperm and ovary in human birth, where the ovary will begin to subdivide once the sperm unite with ovary.  So the sperm is C and ovary is Z, where the ovary multiplies when the sperm entered into the equation, and the life begins.  Does it make sense?
(c) way of communication and networking, especially in a complex open networked environment.  I have written a post on this – the map is the treasure
(d) our future of education.  Given the current trend, could we predict the future using fractals as pattern?  May be.  I am still pondering on this, but I would postulate Z (future education) = Z (present education + imaginary or predicted condition of education) x Z (present education + imaginary or predicted condition of education) + C (paradigm shift: based on networking, social media and technology affordance, and promotion and support from government, institutions, communities, local and global networks, social medias etc.)

The following is just my intuitive thoughts only.
Finally, assume that a is our present life, b is our future life (imaginary) in the Z=a+bi then if our present life is “equated” to our future life, then we will have eternal life (as a Catholic, that is a teaching by Jesus Christ).  The eternal life could also be a reality by applying the principle of multiplication of cells.  Also our faith will multiply as a multiple of two just as the cell division.  This is also how religious beliefs gain its life in its spread of good news, IMHO.

Hope you don’t mind me sharing these really “strange thoughts” with you, but I could assure that you would never find it in any literature, as I made all these up myself.

I am posting it here onto my blog, though such are just half-baked ideas, not yet ready for “cooking”.

Let me know what you think, and Roy please, as I know you would also like something new and novel, as a discourse scholar, and Matthias as a brilliant Mathematician, and Jenny, as a critical thinker and scholar.

Please fill free to share in the Ning or your blog or your wikispace (Roy) too.

Wishing you all an enjoyable Easter Holiday.

John