I find it determinant.

I am always amazed of how surprising research can be. We must do our work, keep thinking, keep looking, and of course sometimes there are creative moves but I feel that often our attention is guided towards those hidden precious facts, whether through a mistake, an accident, a search for something else, or simply a small and unpredictable event that seems to underline what needed to be underlined.

We play a part in qualifying these "events". It is up to us to realize (or not) and give sequence (or not) to what came right before our eyes.

I believe that no matter how planned and controlled a research project is, there is a lot to gain from keeping flexible arranjements and an open mind towards whatever surprises may come.

You presented the psychoanalytic interpretation.

I also find room in here for a religious interpretation. For a religious mind these accidents might fit the category of "god's interaction with the world", according to the work of John Polkinghorne (famous physicist and priest) on the links between science and religion. But this would lead us somewhere else.

I find this topic very interesting because it emphasizes the unpredictable side of science discovery. We already know that we have to work. What we need to be reminded is that there are surprises on the way.

Jose Saramago (late portuguese writer, Nobel prize winner) wrote the following words:

"If you can look, see. If you can see, notice."

Good luck.

Dear Fernando,

Thanks for taking time to share from your unique perspective.

Best Regards.

We often forget that the most creative time of our lives occurs when we are young.

The incredibly vast amount of learning that we do when we are infants is all achieved without rational thought to guide us and occurs, I believe, as a consequence of a fundamental thermodynamic 'economy' achieved within the context of a dynamical system - the brain.

My own analysis suggests that cognitive states equate to catalytic waves (solitons) that mediate transitions. This is achieved by mediating transitions via points of invariance (fixed points) implicit in the stimulus (external or internal). Creativity equates to the initial discovery and the dynamic establishment of these  points of mediation. Additionally, intuitive creativity may involve the re-arrangement, substitution or the adding/subtracting of points of invariance. For example, the great leap of the imagination that led to relativity involved Einstein dropping what was previously thought to be a fundamental axiom of physics.

I believe that knowledge that is derived from analysis alone amounts to the dotting of i's and the crossing of t's within the context of an established paradigm.

Observation, meditation and analysis are all tools that serve only to prepare the mind for inspiration and insight!. 

Dear Christopher,

Very interesting arguments on the brain-centric view of the discovery process!

Best Regards

First, I'd like to say that this is a treasure-trove question...the opposite of a Pandora's box, if you will.

Though I focus more on classroom environments than psychoanalytic or counseling domains, I am inclined to believe that many of the learning processes are similar.  That said, I can recall only a few times when students "stumbled" onto an answer for a given problem.  In those instances, the "problem solving" process was entirely encapsulated within the moment of "serendipity."  That is, with the quick discover, the problem was instantly solved.

As the solving of a problem is one of the goals of helping students become better problem solvers, this obviously can't be viewed as a bad thing.  However, from the pedagogic perspective of building a problem solving tool-kit, I would suggest that serendipitous failure probably is more important to the problem solving process.  Working through a proposed solution to discover that it is not correct (i.e., will not solve the problem at hand) is a powerful way to learn and discover more about the problem.  Furthermore, being able to convince yourself that the solution indeed does not work is a useful skill for both critical thinking and problem solving.

I will be very curious to read others' responses to this query.

Dear Allen,

I am inclined to echo your views- somehow the missteps along the way towards a solution leads to a greater sense of ownership and personalization for the student. Serendipitous failure has in hindsight been taken for serendipitious success, and has been the source of many innovative products (e.g. as a consequence of the quest for the philospher's stone); and much newfound opportunities (e.g.the discovery of the New World).

Best Regards

Dear Shian loong,

I think I understand why you suggest that my response indicates a 'brain-centric' view of intuitive insight - and you are right. However, the Fractal Catalytic model argues that the brain is not what might be described as an 'epistemological engine' that somehow 'operates' upon stimulus to achieve knowledge or answers - rather, the utility of the brain is that rather, the utility of the brain is that it provides a unique thermodynamic environment (an excitable medium) that is structured in such a way as to force the environment (via stimulus) to interact or intefere, with ITSEF! - just like a lens.

From this perspective it becomes rather ambiguous as to whether this is a brain based approach or an environmental based approach. Actually, it is probably a bit of both. The implicit environmental structures that are more likely to be 'registered' and so play a significant role in the 'unseen' dynamic process would be dependent upon the individual experiences of the problem solver (translated into structural memories) and so would be brain based. However, immediate random experiences that may trigger insight would be more(although not entirely) environmental. 

The song that the world sings is not the same for everyone!

Problem solving is a part of the process of (1). planning (2). observations (3). problemsolvings (4). results. The emphesis of these steps could be varies based on being a matter of the hard-science, social science, or Arts & Designs, to reach a desireable result. Moreover, the outcomes and resuts could be different among various cultures and believes, even if the focal point would be the same, because of being the product of a creative deep thinking in more than one perspective, to tackle the problem.  

http://www.sciencedirect.com/science/article/pii/S1877042815010290

Article Theory of Problem Solving

Dear Christoper,

I am intrigued by your idea of the brain as a sort of lens and "attractor"- perhaps more of the self-organizing sort? Also brings to mind the idea of the observer effect that quantum mechanics alludes to, where the observer, far from being independent of the phenomenon, actually influences the phenomenon. Don't know whether it is too much of a quantum leap to suggests that parapraxes are less the result of serendipity than it is causality. We often hear of the phrase, "where there is a will there is a way" and the fact that the subconscious has use especially in the incubation phase of discovery/problem solving. Somehow an amalgamation of ideas comes about almost "accidentally" when there are perhaps some latent pattern recognition facilitated by the subconscious and its interaction with the environment.

https://www.researchgate.net/profile/Michael_Helms/publication/228802786_Compound_Analogical_Design_Or_How_to_Make_a_Surfboard_Disappear/links/0912f50b63aa8eae99000000.pdf

Cheers

Article Compound Analogical Design, Or How to Make a Surfboard Disappear.

Dear Shian-Loong,

My work is most certainly closely associated with the dynamical systems group. However, it was/is my frustration with the idea that dynamical systems (via emergence) may give rise to 'order for free' that distinguishes my work from much of the rest. My work implicates a type of non-linear wave found at all scales in biological processes as being the principle agent of catalysis - from the scale of the enzyme upwards.

The soliton is an interesting wave in many respects. It is extremely robust and requires symmetry/invariance in the boundary conditions for its' emergence. The soliton is a discreet wave that behaves like a particle. The soliton embodies information relating to its' boundary conditions as an aspect of its' dynamic structure and effectively represents a dynamic solution to those boundary conditions. 

The Fractal catalytic model implicates structured coherent solitons as the neural correlate of  cognitive states (and indeed - consciousness). The solitons represent dynamical solutions to the structured boundary conditions imposed upon the brain by the body and the senses. These dynamic solitonic dynamics pre-empt where energy (in the form of an excitable medium) is to be located in the brain consequent upon stimulus and  effectively catalyze its release as a consequence of its ongoing dynamic structure.

A surprising insight that emerges is that cognition does not follow stimulus - it pre-empts stimulus!

From the scale of the enzyme to the brain living systems facilitate transitions in the environment to more favorable thermodynamic states by creating the necessary dynamic  conditions for the emergence of quantum states that embody information relating to those boundary conditions - catalytic solitons.

Chapter Life, Catalysis and Excitable Media: A Dynamic Systems Appro...

Data On creativity

Dear Christopher,

Thanks for sharing your research, it is indeed a novel approach in dynamical systems that I have not come across. Especially in the sense that it emphasizes inter-scale dynamics.

Best Regards,

Bernard

Dear Bernard,

The question you raised is one of the foci of my research in the field of math education. In the instructional environment where the students learn mathematics through inquiry projects  serendipitous solutions occurred.  In our search for analytical tools that can be helpful in analysing a long-term exploration of the students towards their AHA moment, we turn to the theory of shifts of attention proposed by Mason. The approach is complementary to the "representation change" and "prepared mind"  theories. The chain of events seems to be serendipitous, however there is some logic in what objects were in focus of students attention, how they were attended to (structures of attention) and why the student shift from one object to another. 

The emotional component of AHA moment is very strong and contributes to students sense of ownership and sense of creativity. 

Best regards

More here: 

Conference Paper RECONSTRUCTION OF ONE MATHEMATICAL INVENTION: FOCUS ON STRUC...

Conference Paper WHAT COUNTS FOR BEING CREATIVE? A MATHEMATICALLY GIFTED STUD...

Dear Alik,

Thanks for sharing your views from the realm of mathematical education. The AHA moment seems to be conditioned upon some degree of preparation and awareness. Perhaps our subconscious once set on a course, tends like a gyroscope, to lead us to "answers".Sometimes not the intended ones but answers nonetheless.And under a right frame mind (some would say "attitude"), to chance discovery?

Would you agree that a certain level of problem curation, like a shortlisting of key problems tend to cultivate chance discovery more than a brute-force, take-on-the-unknown approach?

Cheers

https://www.researchgate.net/post/What_problem-scoping_skills_can_we_learn_from_the_famous_Hilberts_problems

My own experience suggests that the problem solver must form a 'personal' relationship with the problem - a 'visceral' or intuitive 'feel' for the problem developed through observation and meditation. It is often  the case that the first step (or insight) is the identification of what the 'REAL' problem is!

Dear Christopher,

Reminds me of sensitivity to initial conditions and path dependence.Far from being an abstract framework, Chaos and complexity finds applications even in the psychology of problem-solving. The paradox is that "right footing" is critical and yet problem solving requires a "leap of faith". Reminds me of bifurcations and perhaps the 'problem space' as the 'phase portrait'?