COGNITIVE TOOL 

    

                


       

                                                                              Adventure By Topology

                       Cognitive Payoff Under Uncertainty

                                     With Hands and Mind



We have in effect already turned the world over to the children -- our challenge is to respect that fact and support their efforts to reinvent the world, to rethink it from the ground up.


                                                                         

                                                                        KICK STARTERS 


                                                                                                                                             


It’s all about people.

This is the story of the Sommer Cube (S3), a cubical maze module — leveraging Binary and Analogical reasoning in a hybrid toy:  a cognitive tool (analogical and logical arguments presented visually — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity).


                                           


                                         Scientists in Action - Euler is with us.


 MG 2305-2 crop text



 MG 2306 copy 2 crop rotate

“Those holes on top are not the same in terms of collections of features —  now one exits between my legs, the other exits under the couch …  I wonder if my judgements of similarity of the common and distinctive features, which are dynamic, are dependent on the effects of context on the relations between my judgements of similarity and difference, and asymmetric similarities, which are also dynamic.  I wonder if my clumsy efforts to unscramble all this is considered irrational, therefore unabstract reasoning, by the people in white coats watching me.  Amos Tversky, where are you?"


Conventional cognitive thinking maintains that it is not possible for this five month old baby girl to exercise the necessary level of abstraction to work with S3 — but she did just that:  she chose and formulated interesting problems; experience embraced adaptation.  Which leads us to question the foundation of the scientific reasoning (Is there systematic early education bias because early abstract thinking —use logic or separate, combine or transform ideas — is too hard to discern / measure, and is therefore ignored in the education ladder?):  Can she facilitate her own cognitive development, transfer learning to new stimuli?  Is there sudden acquisition of new abstract concepts, or their decline, independent of age-spectrum?  Will Sfull age-spectrum behavioral system, a single evolving conceptual Hands and Mind framework of nested manipulator-formulated tests of simplest to most complex abstract  >>  analogical reasoning, answer these questions? 

A multidisciplinary benchmark, both diagnostic and learning: a new standard.  Analytic and Holistic thinking:  a competition of Rational (programming desired logic functions) and Nonrational; a fundamental change of basic concepts and experimental practices.  

A question.  (What are the “signature" cognitive messages at play with S3?)

(Hint:  Birds do it too.) Watch..


                                                   


For anyone, across the age spectrum.  

                                         


In other words, S3 is a systematic confusion of whole(s) and part(s), a topology of paradox (different sorts of paradoxes) requiring a back and forth of cognitive and perceptual faculties where the manipulator must rise to successive higher levels of abstraction:  Hegel’s dialectic.

One of the most powerful forces in the universe:  the human mind as Gestalt Engine — mind as machine — a dynamic field where the parts are at war with each other.  

Problem space as evolving construct:  a furious evolutionary  competition of patterns, structure and logic at multiple levels of abstraction and relational complexity, recursive processes of self-observation and self-reference — parts to their contexts, filling in missing information (Pragnanz).  

The foundation of cognitive process and Artificial Intelligence. 

Gestalt thinking, descriptive and explanatory.


"I have always found Gestalt ideas to be the most interesting and, I think, the most fertile in psychology and related disciplines. I also think that Gestalt work has suffered from a poverty of conceptual tools and from the technical inability to do the kind of research required to reveal the power of its ideas.”  

                                                                                                          Ray Pavloski, to MSS June 8, 2017 


The challenge:  understand the dynamic, structural features of navigation in topology of paradox, where the strategies and tactics are evolutionary.  Key is the difference between objective and subjective reality, and factoring that in.

                                          

It’s about survival of the creativist.

Each S3 reorientation requires the construction of a new field map:  blind alleys, paths, routes and environmental relationships must be continuously revisualized.

In other words, a switchiness of learning to adjust processes which were previously, but are no longer, successful.

The question is, Machine-think or Human-think?  Remember, we can switch how we think  re-program ourselves.  We are all programmers  but not limited to programmatic thinking.  

                                                       

Enter S3.

S3 is about exploration of situation awareness, feedback and symmetry, transformation of parts and whole, square and circle -- adaptation to accelerating rates of change and complexity of system and environment:  a full spectrum cognitive development tool (manual / mental rotation studies),  

It's about low-tech in a high-tech world and understanding, and therefore balancing, the benefits of each.

A competition between analogical and binary reasoning.   

Think of that black swan which disproves the contention that all swans are white (S3 is a physical hybrid as well as a hybrid of ideas), like the promise of Spatial Thinking, a long overdue consolidation of important scientific inquiries and educational initiatives.

With S3, the human mind (context), with its evolutionary field effects (parts), confronts an external Gestalt confusion, which demands an evolutionary cycle of cognitive development.  That is, the S3 is both agent and object of change; the mind is both agent and object of change.  In an eminently researchable partnership.


"Steve Jobs:  The Next  Insanely Great Thing." 

"I used to think that technology could help education.  I've probably spearheaded giving away more computer equipment to schools than anybody else on the planet.  But I've had to come to the inevitable conclusion that the problem is not one that technology can hope to solve.  What's wrong with education cannot be fixed with technology.  

"No amount of technology will make a dent."

                                          Steve Jobs (as quoted by Gary Wolf, The Wired Interview, 1996)


Be it Computer Learning or toy, all this relates to long standing formal ideas / implementations concerning Child Development and Artificial Intelligence, particularly cognitive Constructivism (process discovery / task-based learning) which with rare exceptions (e.g., Friedrich Froebel, 1837, with his “sphere and cube linked by an elucidating cylinder”) never reach the child’s Hand and Mind.

Toward a Science of Play (here defined as evolutionary regulatory arena) — cognitive and educational assessment.

Thus, Science Technology Engineering Mathematics in a low-tech / high-yield, cost-effective format, where questions of function, role and position determine meaning.  What seems to be not understood is that science is a question, not an answer — any child will understand that.  

 

                           Peel the conceptual onion to follow Slinks in science.

                                                  What it is.  What it does. 

                                                   

Full-spectrum cognitive tool.

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must simulate the future, conceptually "drill down” a hierarchy of nested coordinate systems.  

Each of these nested frames of reference establishes conditions / opportunities as the focal point of each frame is moved in any of three perpendicular directions.


                                         


                                                             Hand-Flying 

                     


                      



                                                        Switchiness

Sis about the tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement — reversal learning":  analogical reasoning, instrumental learning):  switching of directed attention, relative motion and form, and navigational strategies — winning tactics become losing tactics.   (Gestalt Engine (analog)  x  Tilt Switch (binary)  =  "Directed Attention" (flip-flop)  =  "Cognitive Flexibility" — think Gestalt Architecture.)


                                                         

A dialectic.


                                                      


                                                                                                                                                          

                                                     Asynchronous Analog-Binary Processor (0/1)


                                    

                                                                   Rolling Ball “Tilt-Switch

Each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow.  (Think Field Programmable Gate Array.)


From the perspective of information processing, a system of competing parts, demanding simultaneous analogical and binary reasoning, in ascending levels of abstraction, which kick starts a continuous pattern of cognitive development (not Piaget’s development in stages).

In musical (math / pattern) termsSis about polyphony:  simultaneously combining modules, independent in melody (rhythm and contour of ball trajectory) yet interdependent harmonically (periodic variations upon a theme).

Sis simply about variations on a theme of square and circle   -- flip-flop, a counterpoint of logic and intuition.  Symmetry, and feedback, under transformation. 

A generative process.  An “unfolding”.

                                                            

Sis about Spatial Thinking  (space, representation, process:  multifaceted, interconnecting competencies, human and robot), the holistic approach —  manifested by a cubical maze module -- the rational in conflict with the intuitive, and why it is important.

"Recognizing Spatial Intelligence", SCIENTIFIC AMERICAN (2010) 

                        

In particular, Wayfinding:  spatial problem solving critically examined as cognitive tool, in conjunction with an overdue operational definition of Play (an evolutionary regulatory arena), a theoretical framework in Cybernetic regulatory terms:  the hidden architecture of play.

A simple block and ball networking system which leverages full-spectrum cognitive flexibility / perceptual processing:  learning algorithms, and the art of design, with an emphasis on advanced thinking and intuition -- and self-reflection (not "mere facts", but principles); functional relations, particularly goals and feedback (What information is relevant?  What are my assumptions?  Are they justified?)

Thus, Sis about adaptation to uncertainty:  unleashing the instinct to think, with Hands and Mind; play, with freedom of means and ends, and why it is important, in scientific terms.  A challenge to prevailing assumptions, integrating concepts that have been carefully examined in radically different fields.

The perspective:  Brain as evolutionary union (the whole) of analogical and binary processes, biology and experience (the parts).  The Gestalt engine (de Kreyser’s “we are furious pattern-matchers”), resolving dissonance by escalating perception:  bird’s-eye view — like Zen.


"For Hegel a Gestalt is a “formation of consciousness” understood as the dissonant unity of a way of thought …. practical activity and subjective thought all aspects of a single whole or figure, that is Gestalt, but always moving, always with internal contradictions."       

                                                                          Anthony Blunden, forward to “ Hegel’s Logic” 

  

Thus, an intiative to see S3 cognitive flexibility get directly into the hands and minds of the children and the researchers; then the researchers can get to work to make the necessary “sea change”.

Hegel for Tots.

What is S3? 

Combinatory play. 

Exploration in depth of the contrapuntal possibilities inherent in a cubical maze module:  order, structure and wholeness.  

A dialectic.

A continuous interweaving, user created, network of tunnel paths (inner voices -- relating to bass line -- cantus firmus), imitative / developed contrapuntally into an orientation-independent and network-interdependent system of harmony.

While the S3 exterior has six faces (three sets of opposing faces), each "quartered" face with one or two holes (mismatched hole quadrants between modules are dead-ends, and proliferate erratically), the S3 interior has four intertwined chiral tunnels (each tunnel with independent logic gate orientations) with eight entrance / exits.

Thus, each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow. (Think Field Programmable Gate Array.)

In other words, Sis a programmable logic component where the manipulator can program the desired logic functions (physical logic arrays) — actually see and touch and test them.  Algebraic analysis by other means.

From another perspective, S3 is about problems sharing a common medium:  (attention researcher:  regeneration of field map) variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition.  An interplay of objective and subjective reality, concept learning  where manipulator sets up placeholders for more complicated sets and abstractions, and fills in the details over time ("Quinian bootstrapping”?). 


"Further discussion suggests that in a way, the creative possibilities of the mind are generally dormant.  This state of ‘sleep’ is probably the result of a very widespread and deep confusion between the creative and the mechanical.  As a result of this confusion it is very hard even to perceive that one is not creative.

                                                                                       David Bohm, ”On Creativity1968

What does Sdo?

It kick starts intuition, the nonrational. 

S3 is about creativity -- playfulness, imagination and rigor -- in the face of real and abstract variables:  coordinating mental and manual environment, modifying self-behavior on the basis of experience, through the optimization of multiple simultaneous paths (make most paths begin / cross within a single S3), in a constant stream of evolving problems, within a topology of paradox.  (“How complex environments push brain evolution”


                   

                            S3 Day at Pennsylvania Academy of the Fine Arts -- April 9, 2002


Simply put, S(a motile) is a work of art (SPACE AS ARTof bare utility, like mobile and stabile (remember Calder’s play circus), a machine which offers abstraction of form and function, as well as cognitive payoff, for the manipulator.


"Newton also regarded the geometric approach as more intuitive, certain, and direct, as opposed to algebraic techniques which he once characterized as nauseating. (Mathematical Papers, Volume 4, p. 277)"  

                                                            Robert Fox, The Oxford Handbook of the History of Physics

[Hands and Mind -- compass and straightedge, synthetic / axiomatic geometry 


A combinational (technically “permutational", like a combination lock) logic block which contains four tunnels, effectively an array of unconnected, gravity-dependent switches to be programmed by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous, routes by reconfigurable interconnects.

Thus, Sis about simultaneous mental (rational and nonrational) and manual rotation, giving form to binary and analogical information, a coupling between physical objects and binary information where bits are directly manipulable and perceptible. 

To be precise, S3 is a technological mechanism, a dynamic router, an Analog-Binary Processor which employs corrective feedback (non-linear feedback controlled "purpose”) which generates oscillating, intermittent restriction of cognition and perception within a dynamic logic hierarchy, a topology of paradox:  the cognitive mechanics of Gregory Bateson's Double Bind paradox: contradictory communication at different levels, where acknowledgement of that contradiction is forbidden); and Wieners’s oscillatory response, "pathological oscillations”, to "set-theoretical" paradox). 

Such scientifically verifiable full-spectrum Spatial Thinking (spatial:  ability, reasoning, cognition, concepts, intelligence, environmental cognition, cognitive mapping, mental maps, etc.) tools and learning opportunities are needed, for pilots, astronauts, surgeons, etc., as well as teachers and students, diagnostic as well as development.

S3 is an eminently measurable (think signal-to-noise ratio:  mental opposed to manual rotation / neurophysiological) interface (invites interaction and responds to it); a working definition of that cognitive processing (in a paradoxical environment -- hierarchies of different rules and changing order of operations where principles have to be unlearned) versus logical algorithms, "naive physics”.  The functionality of machine, the learning of organism.

Measurable utility.

                                                    

"Part of the underlying motivation for Cognitive Science is a dissatisfaction with the orthodox methods of studying cognition, and an impetus to change the fashion in which we think about the mind and investigate its operations …. Cognitive science, however, needs theories that both cohere and correspond to the facts.

                                                                       P. Johnson-Laird, “Mental Models in Cognitive Science”, 1980


In John von Neuman’s words, "after much ‘abstract' inbreeding, [the discipline] is in danger of degeneration.… whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source:  the reinjection of more or less empirical ideas.”    

Thus, S3 is about challenging orthodox assumptions, integrating concepts that have been carefully examined in radically different fields.

              

                 S3 supporters speak out at  CONSTRUCTION BLOCKS (Analog-Binary Processor)

S3 is about the need for more learning proficiency than problem solving competence:  ("tin-can”) creativity, in operational terms, and the careful construction of logical arguments, and their limitations.  And situation awareness.

                                                            

Control of complex systems: dynamic decision making.  Uncertainty as tool.  Strategies for learning and control under continuously varying context (among eccentrically rotating local" and expanding “absolute" coordinate systems) — multivariate, celestial mechanics.  

                                                           

“The basic thesis of gestalt theory might be formulated thus:  there are contexts in which what is happening in the whole cannot be deduced from the characteristics of the separate pieces, but conversely; what happens to a part of the whole is, in clear-cut cases, determined by the laws of the inner structure of its whole.”

                                                        Max Wertheimer, "Gestalt Theory”, 1924  ("Social Research", 1944)


Instead of viewing space as fixed, a passive arena (simple games, puzzles and mazes), the S3 network architecture uses space and geometry as active participants (think of space as individual construct) in the problem universe (Animal and Machine) -- balanced, opposing, and apparently random forces.

Thus, S3 is unequivocally transformative, a kick-start, a self-generating, self-sustaining, escalating causal loop:  an exploration of deep causal structure (in "hard science”, beyond behaviorist psychology terms -- think endogeneity); an evolutionary cycle of cognitive development where the manipulator is both agent and object of change.

Whatever the theory, what is important is knowledge generation that manipulator can both express and demonstrably use -- avoiding what Whitehead called “inert ideas” -- propositional knowledge that the student could express but not use. 

S3 is about manual, and its inseparable partner mental rotation, the ability to rotate mental representations of two dimensional and three dimensional objects, predictor of math / science achievement. 


"Nature appears to have built the apparatus of rationality not just on top of the apparatus of biological regulation, but also from it and with it."

                                                            (Daniel Dennett’s ReviewAntonio Domasio, "Descarte’s Error”, 1995


                                                

                                                    Sit on your hands -- then try talking for 15 minutes.



                                    

                      Children formulate a Systems Problem         (checkered paths allow exit)


The S3 physical state embodies the binary state of the system, as well as the analogical. (think Tangible User Interface -- TUI)


                                                                  

                                                    (VIRTUAL REALITY paths confirmed)



                          The Hidden Architecture of Play 

                                     Play Without Boundaries


“I was particularly taken with the possibilities the S3 offers for strengthening spatial, mathematical, and problem-solving skills in children, beginning in the preschool years.  The S3 readily engages and captivates children, sustains their attention, and challenges and stretches them cognitively.  Moreover, it enables children to calibrate their own complexity of play--making the S3 ideal for any age and age-mix of players, child-child or child-adult.  We have much scientific evidence to indicate that children benefit enormously from such collaborative endeavors, both cognitively and socially.” 

                                     Laura E. Berk, Ph.D.  Child Development, 8th Edition (May 13, 2008, cc m.s.)


Think of play as “defragmentation" / optimization (balancing discrete parts of a dynamic development equlibrium -- intelligence whole -- with Hands and Mind).  A self-regulation, load reduction:  "… NFB (neurofeedback) tunes brain oscillations toward a homeostatic set-point which affords an optimal balance between network flexibility and stability”, "Tuning pathological brain oscillations with neurofeedback”.  

In other words, mental processes reach beyond brain.  Play capitalizes on that.

The question:  What invisible forces are driving Play, and intuition?

Swith its nested reference frames creates an interdependence of analogical and binary reasoning, transformation to higher and higher levels of abstraction, much as algebra (“al-jabr”:  reunion of broken parts) converts geometry into algebra (and vice versa) with the Cartesian Plane.   CHALLENGES / GAMES 

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must simulate the future, conceptually "drill down” a hierarchy of nested coordinate systems.  

In other words, navigation of nested levels of abstraction (sets: binary and analog), mental discrimination / integration of different problem models, starkly contrasting conceptual frameworks (each with its own set of rules).

Combinatory play.

The hypothesis:  Play as regulatory arena (survival of the fittest) for neural housecleaning (rearranging / loosening’ rigidly held intellectual content” — “pruningunused synapses), a harmony-seeking oscillatory activity; regulation of neural binary and analogical computation and plasticity.

In other words, it’s about survival of the creativist.

Ongoing bidirectional exchange of analog and binary signals to the brain along reciprocal avenues.  In Edelman’s neural Darwinism terms, reentrant strengthening of mapping.  

Play = Ashby’s Cybernetic / homeostatic "good regulator" of Intelligence (like REM sleep):  a dynamic development equlibrium. 

                                                       

              (Analogical x Binary reasoning  >>  Probabilistic reasoning  >>  Intuition.)  

                                                       

Thus, S(cognitive tool) leverages analogical and binary harmony (coherent states of opposition are preserved while that decoherence is maintained, in an evolutionary fashion).

And intuition — the nonrational — is the cognitive payoff.

Ask Einstein.


"The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined... this combinatory play [emphasis mine] seems to be the essential feature in productive thought before there is any connection with logical construction in words or other kinds of signs which can be communicated to others."

                                                            Albert Einstein:  letter to Jacques Hadamard,

                                                                             The Psychology of Invention in the Mathematical Field, 1945   


It’s important to repeat, in Minsky's Connectionist and Symbolic terms, Sis predicated on Spatial Thinking   (space, representation, process:  multifaceted, interconnecting competencies, human and robot), as manifested by a cubical maze module -- the rational in conflict with the intuitive. 

In other words, Sis about seeing Play in operational intelligence terms (not silliness or behaviorist truism), as Adaptability (neurocognitive plasticity”).  


                                                          INTELLIGENCE

                         Conscious Awareness (Self)           User Interface           

                                   Unconscious                      Machine Language       

                                   

                                               Defragmentation       Play       Defragmentation

                                         Intuition                       Hard Drive Optimization


Toward a science of Play (here defined as evolutionary regulatory arena) — cognitive and educational assessment.

To repeat, Sis predicated on the tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement— "reversal learning"):  switching of directed attentionrelative motion and formand navigational strategies. 


Logical-mathematical intelligence                 S3              benchmark

Verbal-linguistic intelligence

Spatial-visual intelligence                              S3              benchmark

Naturalist intelligence

Musical intelligence                                       

Interpersonal intelligence                               S3

Intrapersonal intelligence                               S3

Bodily-kinesthetic intelligence                        S3  

(Howard Gardner)


Think of play as “defragmentation" / optimization (balancing discrete parts of a dynamic development equlibrium -- intelligence whole -- with Hands and Mind).  A self-regulation, load reduction:  "… NFB (neurofeedback) tunes brain oscillations toward a homeostatic set-point which affords an optimal balance between network flexibility and stability”, "Tuning pathological brain oscillations with neurofeedback”.  

In other words, mental processes reach beyond brain.  Play capitalizes on that.


"Nature appears to have built the apparatus of rationality not just on top of the apparatus of biological regulation, but also from it and with it .… viewing the brain (and hence the mind) as one organ among many, a relatively recent usurper of control, whose functions cannot properly be understood until we see it not as the boss …"

                                                            (Daniel Dennett’s ReviewAntonio Domasio, "Descarte’s Error”, 1995

 

As simply as possible:  play with freedom of means and ends, “free" play learning.

                                              


"Children play naturally, and it is through natural play that children practice reasoning. Children who are manipulated into play by teachers who think that this will improve their reasoning will soon learn to resist the manipulations.  Play, in the long run, is only play if it is self-chosen and self-directed. Children practice reasoning in their own ways, through their own self-chosen play; we can't do it for them and shouldn't try.

                                                Peter Gray, "The Value of Play II: How Play Promotes Reasoning”


Another question:  What is the functional significance of the difference states of play?  Does state make a significant difference?  Why?  

In a sense, the snowstorm of uses of the term Play, in behavioral psychology descriptive terms, is the problem  (think endogeneity) with inevitable confusion across the aging spectrum and cost of lost opportunity.  


"It might seem that one could make practically any assertion one likes ... and then ensure its truth simply by adjusting the definitions. 

                                                       Conant / Ashby, "Every Good Regulator of a System Must Be a Model of That System” 1970 


Experts are missing the point, in the sense of Steven Weinberg’s "anything can be explained, and no explanation can be verified” [the supernatural]:  “Intelligence" and “Play” are analytically isolated, made into equivocal benchmarks, instead of synthesized as dynamic equilibrium) PLAY AS THINKING — “DEFRAGMENTATION" / OPTIMIZATION

The question is:  Early Spatial Thinking:  "gateway or barrier”

To repeat, in operational terms, Snested reference frames create an interdependence of analogical and binary reasoning, transformation to a higher level of abstraction, as algebra (“al-jabr”:  reunion of broken parts) facilitates conversion of geometry into algebra (and vice versa) with Descartes' Cartesian Plane).   CHALLENGES / GAMES 

 Manifested as Full-spectrum COGNITIVE TOOL

An interplay of objective and subjective reality, concept learning where manipulator sets up placeholders for more complicated sets and abstractions, and fills in the details over time ("Quinian bootstrapping”?). 

                                  


                            


This is the story of a "Distributed Adaptive Message Block Network", in Paul Baran’s Internet terms, in a low-tech format. (Think of the ball as “ping”; think graph theory.)

                                                          

To put it another way, Sconstruction is about seeing Play in operational intelligence terms (not silliness or behaviorist truism), as adaptability (“neurocognitive plasticity”):  cognitive payoff.

S3 offers Hands and Mind in conjunction with juxtaposition of Analogical and Binary reasoning, the Rational in conflict with the Intuitive:  Apex reasoning, patterns of thought exceeding the parameters normally experienced in logical operations -- where the strategies and tactics are evolutionary; discovering order in what is seemingly random, in starkly contrasting conceptual frameworks, different problems sharing a common medium.  

And, the resulting opportunity to "self-program” (neuroplasticity), to shape and direct the organization of the brain, over its lifetime.


                                           

                                                       Scientists in Action


Thus demanding Cognitive flexibility (across the age spectrum) -- switch in thinking between two different concepts, and multiple concepts simultaneously

   

(Set Switch:  between task rules; Response Switch:  different resource mapping; Simple Switch:  between a circle and square).  (note photos of individuals manipulating S3 

Look for specific examples of sets of cognitive flexibility at CHALLENGES / GAMES).

S3 is about the tension of switchiness (compound switching -- cognitive, perceptual, mechanical flip-flop):  counterpoint, an architecture of continuous interweaving, user created, network of tunnel paths, imitative / developed contrapuntally into an orientation-independent and network-interdependent system of harmony — polyphony.

Simply put, Albert Einstein’s "musicality in the sphere of thought.

Zen koans employ the same flip-flop mechanics:  enlightenment by contradiction. 

    


                          Complex and Real World Stimuli


                                        *

"I'd like to see the S3 in every preschool and elementary school classroom, and in the home of every family with young children.”  

                                                      Laura E. Berk, Ph.D.  Child Development, 8th Edition (April 10, 2008, cc m.s.)


Formal early childhood education needs a powerful learning / research tool (research on spatial cognition and documentation on the transfer of skills and logic acquisition from Sto STEM -- complex and real world stimuli) -- discussed in depth later.

To repeat, P. Johnson-Laird put it best (“Mental Models in Cognitive Science”, 1980)

"Cognitive science, however, needs theories that both cohere and correspond to the facts.

                                                                  

Toward understanding cognitive development by capturing patterns of changing analogical and binary abilities:  Spatial Thinking learning / research tools with measurable utility.      

      

For example: four paths begin / cross within a single S3.

QUAD / FRAME / ROBOTICS PROBLEM 

Utility = 1.00 for the top front S3 (one path S3 = .25)

QED

Let’s not underestimate the mind of the scientist ... the child.


                 

                            Princeton’s finest, attacking Quad Problem with 3D map / truth table of problem space:  

                             Group Theory (symmetry); analogical and binary reasoning; Hands and Mind.


To repeat, the S3 physical state embodies the binary state of the system, as well as the analogical (think Tangible User Interface, TUI; switchiness in action at CHALLENGES / GAMES).


                                                                   


Be it low-tech or high-tech formmat, S3, a cubical maze module, leverages Binary and Analogical reasoning as a cognitive tool:  analogical and logical arguments presented visually — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity).

                                                     

                                                                              


(Think rotation of geometric variables:  Dimensional Change Card Sort Test -- think rules), Raven’s Progressive Matrices Test, Necker Cube, Miller Analogy Test, Merrill-Palmer Block Building Test, etc.)  

Spatial awareness — understanding complex plans / patterns.                                        


                                                      

                            

Square and circle -- flip-flop; the rest is variation upon theme -- until the irritating reality of a ball which cooperates / does not cooperate (teleological issues:  ENGINEERING — CYBERNETICS / SYSTEMS THINKING, asynchronous binary (0/1), which is different, not consistent.

To repeat, ongoing bidirectional exchange of analog and binary signals to the brain along reciprocal avenues.  In Edelman’s neural Darwinism terms, reentrant strengthening of mapping.

                                                                 


                                                                                    

                                                                          

 

"I cannot state strongly enough my conviction that the preoccupation with Consistency, so valuable for Mathematical Logic, has been incredibly destructive to those working on models of mind .…This obsession has kept us from seeing that thinking begins with defective networks that are slowly (if ever) refined and updated.”  

                                                   Marvin Minsky,  A Framework for Representing Knowledge, 1975


Not consistent.

A three axis rotation problem space (physical and mental) of nested uncertainty (uncertainty of environment, and cognitive flip-flop and perceptual flip-flop -- "unknown unknowns"), of constant and unexpected contradiction, recursive abstractions and relationships; opposing systems, domains of expectation / reference frames, which exhibit completely different properties and changing order of operations (not to mention the unique powers of particular Smodule clusters -- like chess / go; effective teamwork). 

Thus, Sis about dynamic cycle of nested abstraction, recalibration and reflection:  symmetry under changes  of coordinate systems:  analogical mapping, consistently “structured representations made up of objects and their properties, relations between objects, and higher-order relations between relations.” (Structure-Mapping Theory, Dedre Gentner, 1983, 1988)


                                            

Each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow.  (Think Field Programmable Gate Array.)

(Note that Snonlinearity -- change in one variable which does not produce a directly proportional change in the result -- even in the single S3, is effectively a nonlinear expression / experience which allows one to intuitively graph the output as a curve -- very exciting stuff for the manipulator with a questioning mind.   And that’s just the first S3.)


Sis about Binary and analogical reasoning, heuristic framework — ”What Is A Heuristic?” — in the service of research:  Alzheimers, autism, mental / manual rotation, wayfinding, spatial disorientation, etc.

In other words, S3 is about simultaneous mental (rational and nonrational) and manual rotation, giving form to binary and analogical information, a coupling between physical objects and binary information where bits are directly manipulable and perceptible.  

That is, S3 physical state, the map, embodies the binary state of the system, as well as the analogical.  "Thus, a structure-mapping analogy asserts that identical operations and relationships hold among nonidentical things. The relational structure is preserved, but not the objects.” (Flowing Waters 
or Teeming Crowds: Mental Models
 of Electricity) Dedre Gentner, 1983)

(Think Tangible User Interface -- TUI) S3 is about uncertainty as tool.

Generally, our understanding of spatial issues comes from mono-disciplinary Psychology's diagnostic measures — useful for diagnostic testing and measurement of 2D skills, rotation tasks, etc., but not necessarilly for the development of 3D skills, factors which have yet to be isolated and understood.  

Not to mention research on spatial cognition and documentation on the transfer of skills and logic acquisition from S3 to STEM:  knowledge transfer, very important to document claims for already jammed instructional time in K-12 curricula (Note Steve Jobs criticism ... OVERVIEW). 

What about Sindependent isssues of gravity / time / mass / motion / adaptive networks, etc., not to mention hands-on analogical and binary reasoning, and flexibility of perspective (decision space, an individual construct, from within and without), for example, robotics construction / PROGRAMMING tasks?   ARCHITECTURE / GEOMETRY 


"There is not a lot of work on learning to develop one’s analogical abilities; though exposure to different types of analogical comparison (which seems to be what you are proposing) could help; this however would need to be proven empirically.  In Polya’s primer on improving creativity in maths he proposed doing something similar.  For instance, one of his heuristics was, if you have a problem in x dimensions that you can’t solve, then reduce it to a problem in x-1 dimensions and see if you can solve it there; then go back to X-dimensions with that solution.”   

            Mark Keane,  ”Constraints on Analogical Mapping: A Comparison of Three Models"  (May 19, 2016 m.s.)


Networks of forces and interrelationships.

It’s complicated:  Douglas Hofstadter might see this as illustrating "ball shadows” — projection of nested relational abstractions, a higher, more dynamic level of abstraction -- analogy making (not using).  Hofstadter sees analogy as living distributed network of recursion and mental digestion; cognitive scientists, on the other hand, see analogy as instrument of thinking, a tool of mental digestion.  S3, a partnership of feedback and symmetry, an Hegelian dialectical symphony, encompasses both views.

In Polya’s  formal analogical problem solving heuristics (matrix:  diagrammatic / symbolic  x  inductive / deductive) in a dynamic 3D environment / map, Sis about capturing patterns (action steps) of executive function / innovation -- problem solving, fluid reasoning, diverse perspective-taking (calibratable by manipulator, quantifiable outcomes).  


"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements."

                                                                                                            G. Polya, How to Solve It (1945)

                                                                            

In other words, using Sto pick up / capture patterns of abstraction, from action steps / heuristics to neuroimaging, thought to be limited only to humans with a linguistic foundation?  To put it another way:  could a caveman, whom we assume has none of the skills the linguists cherish, manipulate S3 successfully?  Stay tuned.



"And I cherish more than anything else the Analogies, my most trustworthy masters.  They know all the secrets of Nature, and they ought to be least neglected in Geometry.  

                                                                                        J. Kepler (Polya, Induction and Anology in Mathematics)


In Dedre Gentner's construction activity terms, “The essence of an analogical comparison is finding a structural alignment -- a set of correspondences between relations and their arguments.” 


                                 

                             Rolling Ball Switch                                            Analogical Comparison


Thus, relations and their arguments are incorporated within S3:  structural alignment, analogical and binary (0/1) (don’t forget gravity), with general learning principles, within a compound schedule of reinforcement.  Evolving analogical comparison conferring demonstrable “insight”:  in Gentner’s view, “even a single brief analogical comparison can confer insight”.  


     

                                                                                                                                                        

Interesting challenge:  show how Polya’s formal analogical problem solving heuristics neatly fall into a matrix format:  diagrammatic / symbolic inductive / deductive.  Find a common problem to illustrate his heuristics; or the fewest problems with common elements.

Hint:  reduce efforts to “Pre-K math”.


And it’s fun.  (CHALLENGES / GAMES)

                                                                       


"The Sreadily engages and captivates children, sustains their attention, and challenges and stretches them cognitively.  Moreover, it enables children to calibrate their own complexity of play …

                                                                        Laura E. Berk, Child Development, 8th Edition (May 13, 2008, cc mss)


                                                     


                                                   

                                                                            (checkered paths allow exit)


"Consider all the different things that children do, when they play with their blocks. To build a little house one has to mix and match many different kinds of knowledge: about shapes and colors, space and time, support and balance, stress and strain, speed, cost, and keeping track. An expert sometimes can get by with deep but narrow bodies of knowledge - but common sense is, technically, a lot more complicated."

                                                                   M. Minsky, "Why People Think Computers Can't", AI (1982)

  

               

Picture an eleven-month old boy, holding his mother's hand, carefully watching me arrange S3s on the floor for bubble-wrapping at a Kinkos / FedEx.

I reached out (I was on the floor) and offered him the ball.  

He stared at me, then slipped out of his mothers hand and crawled over, took the ball in his hand, and stared at the S3 and the numerous options.  And he stared.  Then with great precision / confidence he placed the ball into a hole, and sure enough the ball exited (a self-checking process; it does not always exit).  

Then he looked at me as if to say, "I knew I could do that."  

By then his mother had noticed what he was doing and apologized for his disturbing us.  

Does that child [every child] have that challenge of abstract reasoning in his daily life?

Why Not?



                 Binary and Analogical Abilities -- Hands and Mind


"From the perspective of a user, the crucial factor is not whether the problem is intrinsically complex (suitable software can make even complex problems seem simple), but rather the cognitive resources that the user is prepared to devote to solving the problem."

                                                     Alan F. Blackwell, ˆWhat is Programming?” (2002)


Thus, to repeat, S3 is transformative, a kick-start, a self-generating, self-sustaining, escalating causal loop:  an exploration of deep causal structure; an evolutionary cycle of cognitive development where the manipulator is both agent and object of change.

S3 is about thinking within a topology of paradox, Watzlawick’s "contradiction that follows correct deduction from consistent premises”.  (Watzlawick, Paul et al., Pragmatics of Human Communication:  A Study of Interactional Patterns, Pathologies, and Paradoxes, 1967.) 


 A “strange loop”:  nested, dynamic, switching of cognitive and perceptual and mathematical dichotomies, where the manipulator must constantly re-examine reigning assumptions, transcend and control nested and evolving recursion and contradiction, by inventing new propositions.


                                            

                                                        I Am A Strange Loop  /  Gödel, Escher, Bach 


In behaviorist operant conditioning terms, the S3 (think "Sommer Box”) manipulator is both experimenter (double-loop learning) and experimental subject (rewarded with a falling ball, or successful “ping (logical / signal topology), we can switch how we think -- re-program ourselves, while the pigeon manipulator in Skinner’s Box (a topology of axioms -- single-loop learning) is simply experimental subject (rewarded with a falling ball of food).  Note S3 compound schedule of reinforcement (a seminal issue ... even for a world-class mathematician).    

Thus, the Smanipulator has the additional satisfaction / cognitive development / fun of the working scientist creating, continuously reformulating / recalibrating dynamic systems problems with changing ground rules, and then solving them -- no small task.


                                            

 

S3 is about levels of abstraction, pattern,

  

the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships; patterns of thought exceeding the parameters normally experienced in logical operations:  Apex reasoning (think Bayesian inference -- subjective probabiity as epistemic tool).  Where the strategies and tactics are evolutionary.

S3 is about uncertainty of analogical / binary interplay.

In the beginning, heuristic or best approximation, will do (small child), but higher levels of abstraction soon become necessary (order of operations changes, principles have to be unlearned).  The Aha moments (shift of frame of reference), "But, why can’t I do this?", are fun.  (note S3 Quad Problem).


                                                                      

                                                       

                                     

                                                                                          Asynchronous Analog-Binary Processor (0/1)

Each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow. (Think Field Programmable Gate Array.)

(To repeat, Snonlinearity -- change in one variable which does not produce a directly proportional change in the result -- even in the single S3, is effectively a nonlinear expression / experience which allows one to intuitively graph the output as a curve -- very exciting stuff for the manipulator with a questioning mind.   And that’s just the first S3.)


Thus, S3 leverages cognitive / perceptual processing to a higher level of abstraction through a robust integration of Hands and Mind (mental and manual rotation).

In Zen (cognitive flip-flop) terms:  the manipulator must embrace the contradiction (instead of accepting the "Law of Non-Contradiction") creating the necessary tension to escape from the underpinning concepts -- to see them as dynamic patterns, part of a process of transformation.

This S"combinatory play" simultaneously demands, in C.S. Peirce’s Pragmatism terms, A) flip-flops of abduction (intuition) / deduction / induction, a dynamic cycle (or spectrum) of inquiry only understandable as a whole, B) systematically impeded by gestalt figure / ground illusion flip-flop (active resistance, error).


                                            


"Try to acquire the weird practice of savoring your mistakes, delighting in uncovering the strange quirks that led you astray. Then, once you have sucked out all the goodness to be gained from having made them, you can cheerfully set them behind you, and go on to the next big opportunity. But that is not enough: you should actively seek out opportunities to make grand mistakes, just so you can then recover from them.”

                                                               Daniel C. DennettIntuition Pumps and Other Tools for Thinking


In this way S3 is a cognitive tool offering the benefits of simultaneous high-level reasoning and manual experience (note manual / mental rotation studies) through a powerful exercise of recursion and contradiction, requiring self-examination of processes and management:  reconciling feedback loops to accomplish a goal.  

For example, in spatial processing switching (flip-flop) terms, when you brush your teeth do you watch the mirror image for reference?  Or, do you avoid watching the reflection and use egocentric reference (location with respect to somatic perspective of perceiver)?  Or do you ever brush, switching frames of reference back and forth, and experience visuospatial disorientation?  


But in the common way of taking the view of any opake object [S3], that part of its surface, which fronts the eye, is apt to occupy the mind alone, and the opposite, nay even every other part of it whatever, is left unthought of at that time:  and the least motion we make to reconnoitre any other side of the object, confounds our first idea, for want of the connexion of the two ideas, which the complete knowledge of the whole world would naturally have given us, if we had considered it the other way before."

                                                          W. Hogarth, 1753, The Analysis of Beauty, (Oxford, 1955) 


                  


That's how Sworks, switching from one mode to the other -- confusing the mind with alternative frames of reference, crossing levels of abstractionover and over again.


"Whenever one encounters a new situation (or makes a substantial change in one's viewpoint [S3 rotation] he selects from memory a structure called a frame:  a remembered framework to be adapted to fit reality by changing details as necessary....  Attached to each frame are several kinds of information… how to use the frame… what one can expect to happen next… what to do if these expectations are not confirmed …. a simplified frame-system to represent the perspective appearances of a cube." 

                                                                   Minsky, Marvin, The Psychology of Computer Vision

                                                                   McGrawHill, 1975


That is, in Gentner’s terms, S3 physical state, the map, embodies the binary state of the system, as well as the analogical.  "Thus, a structure-mapping analogy asserts that identical operations and relationships hold among nonidentical things. The relational structure is preserved, but not the objects.” 

AHegelian logic hierarchy (holistic -- Thesis / Antithesis / Synthesis). 

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must simulate the future, conceptually "drill down” a hierarchy of nested coordinate systems.  

In Michio Kaku’s, The Future of the Mind terms, “self awareness; the ability to put one’s self inside a model of the environment.”  Think like the ball.  (Visualize the necessary systemwide adjustments from the ball’s moving frame of reference  (among eccentrically rotating local" and expanding “absolute" coordinate systems).

To discover order in what is seemingly random.

In other words, navigation of nested levels of abstraction (sets: binary and analog), mental discrimination / integration of different problem models, starkly contrasting conceptual frameworks (each with its own set of rules):

CUBE exoskeleton (six faces)  >>  

HOLES quadrant (eight holes)  >>  

TUNNELS / HOLES (four couplingsgravity gate array; (four binary switches)  >> 

TUNNEL / HOLES (one binary switch). 


                        

                                                      Asynchronous Analog-Binary Processor (0/1)


To repeat, while the S3 exterior has six faces (three sets of opposing faces), each "quartered" face with one or two holes (mismatched hole quadrants between modules are dead-ends, and proliferate erratically), the S3 interior has four intertwined chiral tunnels (each tunnel with independent  “logic gate” orientations) with eight entrance / exits.

Thus, each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow.    

Each of these nested frames of reference establishes conditions / opportunities as the focal point of each frame is moved in any of three perpendicular directions.

A dynamic cycle of nested abstraction, recalibration and reflection:  symmetry under changes  of coordinate systems:  analogical mapping, consistently “structured representations made up of objects and their properties, relations between objects, and higher-order relations between relations.” (Structure-Mapping Theory, Dedre Gentner, 1983, 1988)

With each rotation (precession of the rotational axis of each tunnel), and each additional S3, the manipulator must repeatedly focus on the binary (0/1) issues of a given tunnel (part), proceed up the logic hierarchy to the S3 (whole), then to the expanding universe of S3 (overarching whole / system of systems), while coping with contradiction of action and environment -- dissonance of conflicting cognitions / actions

Thus, S3 is about exploration of situation awareness, feedback and symmetry, transformation of parts and whole, square and circle -- adaptation to accelerating rates of change and complexity of system and environment:  a full spectrum cognitive development tool (manual / mental rotation studies) —   


                                                       


unlike Rubik’s Cube (Machine-think:  about problem solving competence) cognitive “exercise” (narrow spectrum) without paradox).  

A robust exploration of abstract concepts in visuospatial terms:  transformative play through nested dichotomies, logic and intuition, recursion of reason and perception.

Dialectic. 

                               

   Play with this:  study the square and circle patterns below and find the next shape.  

                                                          

Inductive (& abductive) reasoning allows us to make a generalization based on pattern and relationship (arguably, we have been exercising this reasoning from birth).

                                                          

The Sarchitecture, its patterns and relationships of square and circle, capitalizes on this inductive (& abductive) reasoning and its opposite, deductive reasoning, as well as our dependency on the perceptual polarity of Gestalt figure / ground Illusion,  


                                                     

            Raven’s Progressive Matrices Test                                                                   Necker Cube


to demand seemingly paradoxical chains of reasoning:  

1)  Illusion; 

2)  Reality; 

3)  Meta-level of the two.  

A mixing of different levels of abstraction, self-reference.

Just like the viewer of Escher’s paradoxical drawing must rise to another level and reflect.

"… a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level.”    

                                                                                                       I Am A Strange Loop  /  Gödel, Escher, Bach 


                                            

                                                                 

In operational terms, the S3 architecture creates nested, dynamic, cognitive and perceptual loops / dichotomies / switches:

a)  Flip-flop between inductive (& abductive) reasoning (discovery driven / bottom up); 

and deductive reasoning (rule driven / top down); 

b) Flip-flop of figure / ground (distinctive-feature information);

which dictate the manipulator employ analogical mapping and perceptual processing --visuospatial strengths -- "combinatory play", to retrace and reinforce associations, to substitute and manipulate Piaget's "pure proposition" for possible action, within a three-dimensional information (manual / mental rotation) framework.  



                                 Continuous Synchronous Switching


“Conflict is the gadfly of thought. It stirs us to observation and memory. It instigates invention. It shocks us out of sheep-like passivity, and sets us at noting and contriving…conflict is a sine qua non of reflection and ingenuity.”

                           Dewey, John, Human Nature and Conduct: An Introduction to Social Psychology, 1930


Sis a topology of paradox -- an environment of contradictory cognitive and perceptual signals / messages at different levels, with a negative injunction forbidding acknowledgement of that contradiction.  (Paradoxical intervention in behavior modification is a classic tactic which appears to contradict the desired purpose of the intervention.)


                                   


  



                                 



                                                        

                                                                “trompe l’oeil / trompe l’esprit"



That is, Sis an engine of paradox which generates continuous ambiguous, bi-stable stimuli, cognitive and perceptual, such that the brain cannot commit to just one interpretation, but must switch back and forth, providing best guesses, unconscious inference ("unbewusster Schlussin Helmholtz's 1867 terms. 

To be precise:  a pair of contradictory signals (cognitive) which contradict another pair of contradictory signals (perceptual), where those higher level pairs themselves contradict each other, with a negative injunction forbidding acknowledgement of that contradiction. (Note, the brain does not broadcast the operation of its own processes.)  

The only way out of the paradox is for the manipulator to shift point of view to third person perspective (just like the viewer of Escher’s paradoxical drawing must rise to another level and reflect on the paradox).  That is, escape the recursive, nested, back and forth of cognitive and perceptual cycles which impede / allow cognition and perception, simultaneous covert contradictions.

in effect, a mandatory focus on process / frame -- not just outcome.  

http://projects.ict.usc.edu/itgs/papers/Constructivism_KirschnerEtAl_EP_06.pdf


Thus, the S3 demands, gently at first, the manipulator simultaneously plan and direct multiple lines of thought, rotate mentally and physically, differentiate and process visual parts and wholes, separate evolving tasks into manageable, self-referential subsequences (each step is specified in terms of previous steps that the same function has already calculated -- like the Tower of Hanoi classic recursive programming problem).

                                                                 


And escalates from there.

A cognitive development tool which demands the manipulator integrate opposites, by linear and non-linear reasoning with hand and mind; 

Transformative, a kick-start, where voluntary behavior is modified and maintained by its consequences (compound schedule of reinforcement) -- Ratcheting. 

A cycle of concrete to abstract which preserves within it each of the ideas or stages which it has subsumed; a totality, a fractal architecture.  



                                                         Synergy

S3 is about synergy:  “operationalizing” the collapse of our expectation of a logical universe, and how we adapt by advanced thinking and intuition -- and reflect upon those processes -- in an eminently measurable (think signal-to-noise ratio: orientations to solution, etc.) and simple form, an orthogonal manipulative. 

A demonstration of Hegelian dialectics of pattern and of process, a mixing of different levels of abstraction, self-reference:


                            Thesis,                  Antithesis,                     Synthesis; 

                              


In Hegel’s terms (best seen / experienced in practice)

Synthesis: Integration of a set of contradictions to a higher level, transcending the polarity and forming a new proposition.


In C.S. Peirce’s Pragmatism terms, abduction (intuition) / deduction / induction:

A dynamic cycle (or spectrum) of inquiry only understandable as a whole.


In Piaget’s terms:

Assimilation:  a new network scheme into the old network scheme, to accommodate the new scheme  (re-examines and adapts to the new information) to produce a synthesis of the two: an equilibrium, a higher level network scheme. 


In Bateson's terms: 

”Double Bind” paradox:  contradictory signals / messages at different levels, where acknowledgement of that contradiction is forbidden. 

(In Zen Koan Master tells student: "If you tell me this stick is real, I will beat you. If you tell me  this stick is not real, I will beat you. If you say nothing, I will beat you." Student escapes the paradox by changing the "level of communication”.  She takes the stick from the teacher, and breaks it.  Thus,  Zen mechanics of enlightenment, with an implicit injunction forbidding discussion of the contradiction -- the sanity / wisdom of the Master.)


In Russell’s / Cantor's (logic / set theory) terms: 

Reconcilement (manual and mental) of the paradox of sets / groups, symmetry under transformation (transformation geometry -- rotation [turns], reflection [flips], translation [slides], etc.) as learning medium:  evolving reflection symmetry > group theory > analytic geometry > linear algebra, etc. (higher dimensional geometry, topology).

Thus, a topology of reciprocal influence of Hands and Mind. 


In Spenser’s terms: 

the S3  demands the manipulator program: 

Propositions, not just objects [simple blocks];

Ideas within a hierarchical structure; and 

Alternative possibilities within a combinational array. 


Or, from Bruner’s perspective:

the S3 is a manual cognitive development tool which spirals learning:

Enactive:  how to do things -- actions which achieve results;

Iconic:  images as surogates for concepts; and 

Symbolic:  reflect on propositions rather than objects. 


Or in plain english, the S3 demands a conscious, self-reflective process of: 

Reorganizing, again and again, orientations and evolving networks;

Analyzing and intuiting evolving, dynamic patterns and connections;

Manipulating according to rules (deductive reasoning) or inferring new rules from particular instances (inductive / abductive reasoning);

Switching / jumping back and forth from internal to external reference frames, survey perspective (bird’s eye view) and route perspective (environment tour); 

Coping with tricky Gestalt figure / ground illusion and structure (relationship among the parts and whole) and function (step-by-step operation), etc.    


                                         

                                                                (checkered tunnels show paths)


In a nutshell, S3 demands the manipulator constantly re-examine reigning assumptions and embrace a new paradigm: a spiral of self-reinforcing change of cognitive structure; a balance of evolving recursion, dichotomies, pattern matching and creation; transcend and control nested and evolving recursion and contradictions by inventing new propositions.


                                        /span>

                                                    (checkered paths allow exit)


For infant, child or world-class mathematician, from manipulation to study, from intuition to reasoning, from object to proposition, from developing intellect to powerful mind, each at his / her own pace — testing hypotheses, cycling between the concrete and the abstract.

Low-tech cognitive tool.  Intelligent toy.

Reciprocity of Hands and mind. 

S3 Play.


 




© Michael S. Sommer, Ph.D, 2017