OVERVIEW




                                                                            


                                                A Subtle Form of Tunnel Vision.

                                          Eliminating It.


"A too restricted view of human nature … even though only briefly ascendant, can significantly alter the expectations and, hence, the behavior of men and societies and may thus provide its own bogus validation.”

                                                                                                Geoffrey Vickers, The Art of Judgment, 1965


"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)


The salient issue in education today is low-tech in a high-tech world and understanding, and therefore balancing, the benefits of each.  

The new reasoning is that if it is not electronic, it can't be significant.  Which is too bad  and dumb.  A subtle form of tunnel vision.

In other words, today's Machine-Think / computer-video learning precepts have become a problem in their own right:  the fix  the silver bullet"  the big “silo”.

The world of cognitive studies / learning development (not to mention the toy companies) is often too comfortable making new inquiry into old developmental precepts.  (Think endogeneity:  the solution is married to the error.) 

Unfortunately, early abstract / spatial thinking cannot be discerned / measured precisely enough by the conventional cognitive development models and associated research methods, and is therefore ignored up the education ladder (think learning trajectories).  The foundation is missing.  The benchmark is not there.  

Thus, the cost of the lost opportunity, developmental and research, is hidden.  

Because there are so many issues involved, across so many disciplines, and so many constituencies with competing agendas, we need to integrate Early Learning / Development within a hands-on, bottom-up study of abstract reasoning.  (Technology — use it well.)

The Sommer Cube (S3), described here at length, is both classic toy and cognitive tool (developmental and research) dedicated to the special genius of each individual and the opportunity to exercise it without constraint.

A coupling between physical objects and binary information, where bits are directly manipulable and perceptible.  

The premise is that S3 is both agent and object of change; the mind is both agent and object of change.  In an eminently researchable partnership. 

It’s about revealing the cost of that lost developmental and research opportunity.  Something new, but very simple — a Hands and Mind approach with "intellectual teeth”.   

Thus, the SInitiative:  introducing SPlay (within an operational definition of Play as evolutionary regulatory arena) to a wide audience; proposing Spatial Thinking / neuroscience Learning Research / Children’s Museum creative partnership (Sis seeking) as well as S3 availability in homes and schools.  Evidence-based practice for teaching and learning.

The early years of cognitive development are an untapped gold mine.


                          

                                                       Let’s raise the bar.


"The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea."

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

"… the main achievement in the solution of a problem is to conceive the idea of a plan.” 

                                                                                                                G. Polya, How to Solve It (1945)


It’s about examining Mind Waste; in particular, systematic Early Education test bias / failure, all within the context of today's dominant educational catalyst, STEM / STEAM.

Which leads us to the first problem:  The computer, and related / dependent technologies are not the solution.


"Calls for the widespread use of computer-based educational technology often justify themselves by the potential to support some version of constructivism, seen to be a major improvement in education over more behaviorist or information transfer notions of teaching. The strong claim that computer-based educational technologies are inherently constructivist cannot be sustained because current technology largely fails to provide four key components of a constructivist environment: an engaged learner, hands-on interaction with the materials of the task, an authentic problem-solving context, and human interaction during the learning process….”    

     Stephen Gance:  "Are Constructivism and Computer-Based Learning Environments Incompatible?", 2002


Which leads us to the second problem:  The efforts to solve the first problem by agressive conflict of educational-interest constituencies (STEAM vs. STEAM) are counterproductive, and there is an inherent bias in the methods of inquiry employed by both.  Equality paradox.  All STEM fields are not created equal.

In particular, there is a need to eliminate STEM / STEAM bias against "girls and minorities” by illuminating a side of the manipulator often not otherwise seen.  STEM / STEAM development should be about power  first, the power of abstract reasoning and then its practical application  right from the beginning (bottom-up, in addition to top-down).

What is needed is foundation, a multi-disciplinary framework to document claims for the already jammed instructional time in K-12 curriculua; greater coordinated (participation as well as theory) long-term involvement of Brain and Psychology Sciences; innovative Children's Museum research collaboration (such as    S3) on spatial cognition, transfer of skills, logic acquisition, etc.  

(S3 learning / development research is a first step toward rigorous, scientific demonstration that STEAM is as effecitve as STEM.)

Thus, new methodology, producing a unified picture of the spatial manipulator in a single session:  structured, with specific spatial prompts to respond to; unstructured, constructing whatever comes into one’s head; and circumventing our conscious processes, to graphically reveal our “programming", our unconscious decision process architecture.

It’s about 3D "Rorschach Test” for Spatial Thinking.

From concrete to abstract, binary and analogical reasoning.  A switching of “true" and “false" logical arguments presented visually and tactilely — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity.  Causal interaction within dynamic core.


"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


Illuminating a side of the manipulator often not otherwise seen (or enabled).

Which is S3 turf. 

 

                            

                                                    dialectic -- open your mind


"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” 

 

Hegel for tots.

                              


Genesis of thinking, navigating three-dimensional problem spacecrucial to the development / exercise of visuospatial reasoning (particularly in the formative years) and full-spectrum cognitive engagement and discovery.  (STEM WITHOUT BOUNDARIES:  STEAM)  And enlightenment, concocting and evaluating new hypotheses.

It’s about early geometry.

Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (Studies in Mathematical Thinking and Learning Series).


"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 …

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


It’s about self-regulation.

More learning proficiency than problem solving competence.  Creativity, in operational terms, and the careful construction of logical arguments, and their limitations.  

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”.

Binary and analogical reasoning.  

Try to follow binary   process in the video below.

Switch to analogical        process, continuously — back and forth.

Or answer the QUAD PROBLEM   

                  

  

                                                               

What does Sdo?

In operational terms, S3 with 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 by other means (Cartesian Plane). 


"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


It kick starts intuition, the nonrational. 

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”

It’s about survival of the creativist.

Thus, it’s about "Come play with me”, and think.   

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

Parenthetically, as a systems scientist I have crept into turf — the nature of Play (be it Euler, Bach, or baby) — not my own because I couldn’t figure out how to explain S3 issues without some kind of formal framework for the “natural act of abstraction” of human life, which takes us to Artificial Intelligence, our future:  competition between analogical and binary reasoning.

None of the following excursions is meant to be the final word, but instead the opening for new ideas and discussion.  The picture which should emerge is one of balance:  baby / Man / Woman hunger for abstract reasoning, and its regulation.  Its environment.  

It’s about topology, remember.  

Thinking within a topology of paradox.  Reasoning as Play:  directly visualizing S3 (as geometrical objects and the relations between them) becomes secondary to thinking at higher and higher levels of abstraction (think Descartes algebraic representation of Euclidian spatial relations).

                                                           


                                                           

 Asynchronous Analog-Binary Processor (0/1) /  three axis rotation problem space      


                                                           

                  Rolling Ball “Tilt-Switch             four bit (0/1) — nibble


                                                                                          


In geography terms, to optimize (CHALLENGES / GAMES) 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.  (Try piloting aircraft or submarine, or brain surgery without spatial proficiency.)


                                               

                                                        allocentric / egocentric flip-flop spatial processing


Here, we have to get a little technical.

Because S3 leverages cognitive / perceptual processing to a higher level of abstraction in a clearly intended paradoxical problem-solving environment, it's a useful tool (developmental and research) for illuminating its opposite, the unintended paradoxical environment.

In aircraft terms, “aircraft upset” (exceeding the parameters normally experienced in logical operations) and “upset recovery”. 

Here, Smanipulator (with twenty-four sanctioned attitude options, unlike aircraft) has a cognitive / sensory workload of planning and controlling each individual 0/1, drilling down the logic hierarchy within multiple, simultaneous, nested reference frames (attitudes): roll / pitch / yaw, within another frame of roll / pitch / yaw, within yet another frame of roll / pitch / yaw.    

In oybernetic terms, the evolving errors of interdependent feedback loop information serve as teacher, revising behavior and blueprints for thinking.

It’s all about control under continuously varying context, contradiction of action and environment:  an illogical universe.  Sort of like flying.


                                                  Come Fly Me 

                                         Like Hand-Flying ("Use it or lose it.”)


                                                                     



                                     


                                         


                                            


Let’s look at it another way.

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.

It’s about Gaudet’s classical architectural ideas of axial relationships, symmetry and balanced asymmetry.

Science as the Arts by other means.  "Logic, power, grace" and recursion.


                                  

                                                                                       Rembrandt 1632


Combinatory play.  

Or, in musical terms, exploration in depth of the contrapuntal possibilities inherent in a cubical maze module:  order, structure and wholeness.  

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.

Think like J.S. Bach:  see the “music”.  (Scubedom and fuguedom)

Hands and Mind together create rhythmic stimuli to alter and invert desired hypothetical subject lines:  passage / trajectory of ball through space — simultaneously.


                                                  


                                                                     

Listen.  Multiple simultaneous paths.


"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 Creativity, 1968


A good foundation for what follows is Norbert Wiener’s The Human Use of Human Beings, and Bohm and Peat's Science, Order, and Creativity)  


                                                                  

It’s about thinking skills and evolution.

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).  

                                                              

Be it Computer Learning or toy, the Sstory 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.  

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


                                    

         Children formulate a Spatial Systems Problem  (checkered paths allow exit)


Thus, it’s about a question:  

What invisible forces are driving Play, and intuition?  

Exegesis —  cubical maze module:  classic toy  and cognitive tool (developmental and research).                                    

                      Drill down the conceptual onion to follow Slinks in science.

                                                  What it is.  What it does. 

                                               


"The native and unspoiled attitude of childhood, marked by ardent curiosity, fertile imagination, and love of experimental inquiry, is near, very near to the attitude of the scientific mind …"

                                                                                                                   John Dewey, How We Think, 1910 


_MG_6368 copy crop


Full-spectrum COGNITIVE TOOLconcrete to abstract, (manual / mental rotation studies)

An exploration of situation awareness, transformation of parts and whole, square and circle, feedback and symmetry -- adaptation to accelerating rates of change and complexity of system and environment.  

Spatial Thinking  (multifaceted, interconnecting competencies, human and robot), the holistic approach — manifested by a cubical maze module (S3) -- the rational in conflict with the intuitive, and why it is important.

In other words, S3 is about exercising cognitive resources, the essence of programming; it demands, gently at first, the manipulator simultaneously plan and direct multiple lines of thought, "factors of variation" (orientation, clutter, etc.), to mentally and physically rotate, differentiate and process dynamic parts and wholes, separate evolving tasks into manageable, recursive subsequences, all within evolving layers of abstraction.


[Programming] "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) 


                                                       

                                                                                     FPGA

It’s about a programmable logic component (in Field Programmable Gate Array terms) where the “customer", the manipulator can program the desired logic functions (physical logic arrays) — actually see and touch and test them.

Instead of viewing space as fixed, a passive arena (typical games, puzzles and mazes), the S3 network architecture uses space and geometry as active participants in the problem universe.In other words, Sis a programmable logic component (in Field Programmable Gate Array terms) where the manipulator can program the desired logic functions (physical logic arrays) — actually see and touch and test them.  Algebraic analysis by other means.

In particular, Wayfinding:  spatial problem solving critically examined as cognitive tool, in conjunction with an overdue definition of Play, in operational intelligence terms, as Adaptability:  cognitive payoff.

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?)

It's about simultaneous mental (rational and nonrational) and manual rotation. 


                                                                          

                                                        allocentric / egocentric flip-flop spatial processing


                                                     

 

                                                                            


                                        Geography is dynamic, never static.”  

                                                                                                              Zoe A. Thralls,1949


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.  In depth. 

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”.

To repeat, scientifically verifiable Spatial Thinking   (spatial:  ability, reasoning, cognition, concepts, intelligence, environmental cognition, cognitive mapping, mental maps, etc.); a cognitive tool for pilots, astronauts, surgeons, etc., as well as teachers and students.

Thus, Sconstruction is about seeing Play in operational intelligence terms , as adaptability (“neurocognitive plasticity”):  cognitive payoff.  

Toward a Science of Play — cognitive and educational assessment — STEM:  in a low-tech, high- yield, cost-effective format which requires fluid intelligence, and nurtures it.

The hypothesis:  Play as evolutionary 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. 

Thus, 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 paradox of science education is that its goal is to impart new schemata to replace the student's extant ideas, which differ from the scientific theories being taught.

                                                                      Susan Carey, Cognitive Science and Science Education”, 1986


While Rubik’s Cube (Machine-think:  about problem solving competence)without that paradox, exercises logic (according to MIT / Rubik’s Cube"Solving the cube becomes almost trivial once a certain core set of algorithms, called macros, are learned.” [mind as pocket calculator])  (note "Brain Games are Bogus")S(about human learning proficiency) demands logic and intuition.  

To my astonishment (most just don’t see this), one little boy (note images at PAFA) exclaimed, “Look, It’s an inside out Rubik’s Cube!”.  

To repeat, 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 in the problem universe (Animal and Machine) -- balanced, opposing, and apparently random forces.

Thus, S3 is both agent and object of change, and the mind is both agent and object of change.  An eminently powerful, researchable partnership.

A dynamic structure of ideas and their logical expression.


                                   

               Rolling Ball “Tilt-Switch                                                  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.


A tangible, lucid demonstration of basic behaviors (consciousness) of the brain through mechanical concepts (switches).  (It’s all about “permutations” of dead-ends and thruways:  particular order matters, like a combination lock.)  Recursive processes of self-observation and self-reference. 

A paradigm shift, a hybrid, in Marvin Minsky’s terms, simultaneouslly Connectionist and Symbolic, an orchestration of ideas:  a system of simple puzzle, labyrinth / maze and rolling ball (apparently unrelated, but in equilibrium, exploiting the symmetry of geometry and algebra, to higher and higher levels of abstraction)  where intuition, naive commonsense reasoning, are prerequisites -- logic is necessary but not sufficient.

 

                                                


In other words, S3 is about the tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement — “probabilistic eversal 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.)


                                             


What is S3

A dialectic.

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). 

A Cybernetic switch, a dynamic router, encompassing the Systems Thinking idea, the operation of wholeness, and the relation of whole and parts, structure of systems and their behavior, laws of regulation, and recursive processes of self-observation and self-reference in the service of reasoning. 

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

Asymmetry as equilibrium (comfortably understood withinin the framework of music and mathematics).

A do-it-yourself quadruple fugue.   

A generative process.  An “unfolding”.  A switching.  


                                                     


In other words, 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 probability as epistemic tool).  Where the strategies and tactics are evolutionary.

A cubical maze module (four tunnels = four binary (0/1) switches = gate array) offering a development of choices (control flow) to create linearly independent / dependent paths, using a ball, or symmetry in mathematics.

A continuous switching of patterns:  square and circle.

                                                           


It’s about patterns. 

Dynamic visuospatial (analogical mapping and processing) cycle of concrete and abstract reasoning:  nested coordinate systems (think Rene Descartes' analytic geometry); Albert Einstein’s “Combinatory Play”;  Mathematics  -- the science of patterns. 

Nothing but patterns  … Descartes’ "apud me omnia fiunt mathematica.” 


                                                            

                                                                            (checkered paths allow exit)


Let’s get back to basics. 

STEM development should be about power -- first, the power of abstract reasoning and then its practical application -- right from the beginning (bottom-up, in addition to top-down). 

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

Spatial Thinking and Learning ResearchChildren’s Museum Collaboration.  


              


2012



    Marriage of Cognitive Constructivism and Computer Learning

                                                    (CHALLENGES / GAMES)


                               


From manual and mental rotation for fun to cognitive research across the age spectrum (Autism, Alzheimer's, visuospatial, etc.) S3 offers a true "parallelism of general cognitive fields”. 

Sis about play and programming a network:  formal (play by the rules) and intuition, and the sense of touch, which overrides the human interaction with complex cognitive and perceptual content in paradoxical ways.  

What does this process and S3 have to do with our new world?  It develops intelligence, the cognitive resources necessary to program a computer (0/1) -- but without the computer; a mechanical user interface (invites interaction and responds to it)

This robust integration of haptics, mental and manual rotation, illusion as operational fact, is missing, from early childhood computer-focused learning (as opposed to child-focused learning) to joint pilot / computer management of aircraft (increasing absence of hand-flying).  

Dynamic interaction of the brain and its environment trumps brain alone.  Be it child, astronaut, surgeon, student:  across the age spectrum.


                                              

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


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 C. Dennett’s ReviewAntonio Domasio, "Descarte’s Error”, 1995


Thus, S3 is about switching between the two:  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.

Generating knowledge that manipulator can both express and demonstrably useavoiding what Whitehead’s called “inert ideas” -- propositional knowledge that the student could express but not use”.

Naive Commonsense reasoning, according to Artificial Intelligence precepts, is what makes us smarter, more resourceful, than computers; it's about making assumptions, making assumptions about assumptions, engaging rule-based as well as analogical based reasoning. 

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

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

S3 is combinatory play.

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   

 


                                          HYBRID TOY 

                                                       Complexity for Tots


                                                                          


                                                


                 The Smart Block.

                                                      Binary and Analogical Reasoning  

                                                      (switch from fixed to changing values).

                                                      The Upgrade.     


                                                            



S3 discovery through paradox, uncertainty as tool.  

Freedom to Think.   Mix things up — reassociate them.  Mystical equation.

                

And it's fun. 


 

Zen koan (cognitive flip-flop) mechanics of enlightenment (contradiction).


The Very Special Children


 

Cybernetics equivalent of model railroad network switches.


                                                                   

                                                                        KICK STARTERS 


                                                                         


    


                                                                                                                

                                         star light KIC7671081Bhttp://www.ilovetoteachkids.

 



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                                                           May The Force Be With You!

Jedi


© Michael S. Sommer, Ph.D, 2018