"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 (emphasis mine) that are slowly (if ever) refined and updated.”  

                                                   Marvin Minsky,  A Framework for Representing Knowledge, 1975

                                UNCERTAINTY AS TOOL

"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

The Sommer Cube (S3) is all about people, choosing and formulating interesting problems — enlightenment:  concocting and evaluating new hypotheses; experience embraceing adaptation, to create explanatory knowledge  "change the fashion in which we think about the mind and investigate its operations.  

Inquiry in the face of uncertainty.  

S3 is about hidden architecture, predicated on the manipulation of objective and subjective reality.  But, it is also about challenging orthodox assumptions, integrating concepts that have been carefully examined in radically different fields.

These issues are embedded in conflicting ideas, the foundation of rational thought and inquiry, and span numerous fields / constituencies / special interest groups which unfortunately, do not necessarily talk to each other.

The S3 dialectical premise, uncertainty as tool, was first employed as a new strategic planning / forensic market research tool for a multinational 


facing precipitous loss of core market share for no understandable reason, using the best research money could buy:  billions being spent on advertising, with poisonous results.

In other words, a situation where, as John Wanamaker put it:  “Half the money I spend on advertising is wasted;  The trouble is I don’t know which half.


Something needed to be created, quickly; It was called the Listening Post:  a two-stage formal manipulation of feedback and symmetry; a true partnership between researcher / producer and subject / consumer (the CEO and Executive Staff were face-to-face with the consumers in stage-two).

The "bad half” of  the advertising was isolated:  Focus Group built-in “False Positives” turned out to be the key problem.  (The counterproductive advertising was immediately revamped — clear brand idea: share was regained, and increased.)

In other words, its all about people, but not the preconceived notions of the researchers (if the researchers can’t see the forrest, all they will pick up is trees):  the hidden architecture of the research was systematically biasing that research.  The consumers (with their view of objective reality) were offended and took their business elsewhere;  The consumers had to choose and formulate problem (just like Smanipulator) in stage-one).

To be described in detail later.

What follows is the story of the Sommer Cube (S3), a cubical maze module:  classic toy (Tate Museums) and cognitive tool (developmental and research).  A work of art

It’s about the same thing.  Inquiry.

Explanatory knowledge — enlightenment, concocting and evaluating new hypotheses

It’s at once fundamental and obscure, leveraging both analogical and binary reasoning as the exoskeleton circumscribes a system of linkable, but asymmetric, tunnels which depend on the law of gravity — effectively switches.  A dynamic 3D maze.  A labyrinth.


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


                                                  We Are All Programmers


To put it succinctly, S3 architecture 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.

Thus, aptation 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.

The question is:  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. 


                                     2002                                                1988       

                                       S3                                                   Listening Post

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.

"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”                                              

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

It’s about survival of the creativistst.

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

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

Thus, S3 Play offers the opportunity to regulate analogical and binary competition. 

And intuition — the nonrational — is the cognitive payoff.

What is S3?

A dialectic.

Exploration in depth of the contrapuntal possibilities inherent in a cubical maze module.


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.

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”.


Thus, Sis about the genesis of thinking, navigating three-dimensional problem space; analogical reasoning between superficially distinct models, which share common underlying structure.  Transformation of parts and whole, square and circle, adaptation to accelerating rates of change and complexity of system and environment.  

With Hands and Mind. 


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?)

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.

As well as relative motion (flip-flop of manipulator frame of reference:  egocentric  >>  allocentric) and patterns of thought (higher levels of abstraction) in a paradoxical environment, and why and how to get the most out of them:  partnership of feedback and symmetry.  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)


                                                                                         Rolling Ball "Tilt Switch

Thus, Sis about continuous synchronous and asynchronous switching:  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. 

In other words, S3 is about  giving form to binary and analogical information, a coupling between physical objects and binary information where bits are directly manipulable and perceptible.  In other words, the S3 physical state embodies the binary state of the system, as well as the analogical. (think Tangible User Interface — TUI) 

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

Thus, S3 is unequivocally 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 variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition (Immanuel Kant's thoughts on the subject, Critique of Pure Reason), levels of abstraction, pattern, "'voluntarily' reproduced and combined”, the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships.

A dynamic visuospatial (analogical mapping and processing) cycle of concrete and abstract reasoning:  nested coordinate systems (think Rene Descartes’ analytic geometry, "Omnia apud me mathematica fiuntMathematics  -- the science of patterns:  nothing but patterns;  Albert Einstein’s “Combinatory Play”.  


The system viewpoint has penetrated, and has indeed proved indispensable, in a vast variety of scientific fields …. the parallelism of general cognitive fields."  …. the name 'general systems theory' is here used broadly ….  It is the introduction of a new paradigm that matters." 

                                           Von Bertalanfy, Ludwig, General Systems Theory: Foundations,    

                                                       Development, Applications, George Braziller, 1968

Sis about thinking within a topology of paradox, Watzlawick’s "contradiction that follows correct deduction from consistent premises”;  reasoning as play:  invitation to double-loop, learning about learning;  messy, intrinsically destabilizing, Open Systems Thinking (“An open system is a system whose behavior is jointly determined by its internal structure, and by the input it receives from the environment.

Unfortunately, the Systems Thinking axiom / mantra, The Whole Is “Greater" Than The Sum Of its Parts, is a misquote, a misunderstanding, and represents thinking not robust enough for an environment of paradox or chaos (think Bayesian inference -- epistemic tool).  (Gestalt psychologist Kurt Koffka’s original statement was "the whole is other than the sum of its parts” (the whole may be different from its parts studied in isolation); “this is not a principle of addition” .) 

Thus, S3 gestalt (like the Listening Post below) is messy:  it's about situation awareness as well as naive commonsense reasoning, which, according to Artificial Intelligence precepts makes us smarter, more resourceful, than computers; reasoning which relates to the sense of touch, which overrides the human interaction with complex cognitive and perceptual content in paradoxical ways.  Its about Spatial Thinking.

"While in the past, science tried to explain observable phenomena by reducing them to an interplay of elementary units investigable independently of each other, conceptions appear in contemporary science that are concerned with what is somewhat vaguely termed 'wholeness', i.e. problems of organization, phenomena not resolvable into local events, dynamic interactions manifest in difference of behaviour of parts when isolated or in a higher configuration, etc.; in short, 'systems' of various order not understandable by investigation of their respective parts in isolation. Conceptions and problems of this nature have appeared in all branches of science, irrespective of whether inanimate things, living organisms, or social phenomena are the object of study.”

                                                      von Bertalanfy, Ludwig, General Systems Theory: Foundations, 


                               "Big Brother Is Watching

To repeat, Sis about challenging assumptionsintegrating concepts that have been carefully examined in radically different fields.  

“Normal science does not aim at novelties of fact or theory and when successful, finds none." … "puzzle solving" …  [instead of] paradigm shift”.

                                                       Thomas Kuhn, The Structure of Scientific Revolutions, 1970


The S3 dialectical premise, uncertainty as tool, was first employed in a new research / strategic planning methodology for group (a social sciences paradigm shift), the Listening Post flip-flop of consumer and producer frame of reference:  egocentric  >>  allocentric.  The switch of environments from stage one >> stage 2:  consumer switches to superordinate, producer switches to subordinate (Executive Staff dutifully, respectfully listens (face-to-face) to the consumer:  the ultimate “researcher.  In other words, The consumers had to choose and formulate problem (just like Smanipulator) in stage-one).

Thus, both S3 and LP are about Spatial Thinking, patterns of thought (higher levels of abstraction) in a paradoxical environment, and how to get the most out of them:  the hidden architectures of inquiry and play. 

In other words, in solar system terms, the consumer does not revolve around the producer, but the producer revolves around the consumer:  the producer must be able see the marketing issues from the perspective of the consumer’s world view (subjective reality — personal perspective) not the psychologists' world view (objective reality). 

The Listening Post was developed to understand and regain Anheuser-Busch's (Budweiser) plummeting share of market (losing share to Millers, among 18 - 25 year olds -- the future of AB brands).  Big Brother was watching (behind the researchers window: and the consumer skewed their responses accordingly), to the detriment of consumer marketing research.  

A reappraisal was needed.

Reality-grounded science and tools (#1 marketing research: LP; #2 cognitive, S3).


Spanning Open Systems learning (Ashby’s double feedback loop) to Closed Systems negative learning:  from dialectic to self-fulfilling prophecy, learning proficiency to problem solving competence.

That premise was introduced in Michael Sommer's, “Open Systems Listening:  Conversation As a Research Tool” Ph.D. Diss.,1988, as the foundation for an alternative to the Focus Group, Robert Merton and Pat Kendall’s FocusGroup (according to Sommer, a topology of paradox, with one-way mirror and "Chilling Effects of Surveillance":  supported by Philip Zimbardo / Gregory White’s “aversive surveillance on opinion inhibition”, etc.).  

It described the discovery and avoidance of the Focus Group paradoxical environment, its inherent research "confirmatory bias" (the system of rules, the blocking environment of paradoxical verbal / nonverbal messages).  The research subjects had made a negative adaptation to uncertainty:  what the psychologists heard was not what was going on (think signal-to-noise ratio, desired information versus misinformation).

Thus the Listening Post was born.  

For example, in the recordings of the the sample of U.S. Focus Group interviews (compared with comparable Listening Post interview data) one subject interrupted (in the middle of the moderator’s “feel-good / we value you” disclaimer about the executives watching througn the one-way mirror — “Just ignore them".)  "Right, anything we say can and will be" [… held against us] — the Miranda criminal warning:  "You have the right to remain silent …"  The moderator smiled, laughed along with the young adults and continued, clueless, with his disclaimer.  Like the criminals given the Miranda warning in the television police procedurals, the young adult subjects were suspicious, and shared their thoughts accordingly;  Focus Group research was clearly biased, contaminated.  But the research continued.  The paradox went unnoticed by the researchers.  


                                                                        "Big Brother Is Watching”

The situation was dire:  the consumers knew more than the experts, but the conventional (self-perpetuating and very expensive) processes of inquiry guaranteed that they would not be heard; AB was driving away its market foundation.


Picture a multi-national corporation that had to become a better strategic planning / intelligence, “learning system”, forthwith.  To act intelligently in the marketplace, the world" of the actor, the consumer, had to be understood before the motives of that consumer.  (The key is, of course, the Executive Staff is only human, hostage to the experts who serve them, not to the consumers.)  Hence, literally and figuratively switching consumer from one side of the looking glass to the other.)

To put it another way (flip-flop of context), 

Instead of viewing space as fixed, a passive arena (Focus Group), the LP architecture uses space as active participant in the problem universe:  after a private (individual researcher / subject) Stage One, 


the consumer was empowered to sit down face to face with the Executive Staff as equals (if not superiors), a public Stage Two, 


and ridicule the corporate research, the “everybody knows that ...” (provided by Focus Groups: information, not intelligence), and suggest alternatives, before the consumer deserted the brand.

Formal manipulation of feedback and symmetry.  

"If we allow that which is fed through the feedback loop to be a pattern or even a concept then we have arrived at a realm where feedback and symmetry exist hand-in-hand.  Such an arena does indeed occur in cognition .... a domain where re-entry, self-similarity, and feedback are all different expressions or exfoliations of the structure of awareness.  Primary awareness opens the possibility of the self.

                                                                 ”Louis H. Kaufman, "Self-reference and Recursive Forms”, 1987

Research subject now empowerd as research partner.

A synthesis of Cybernetics and Systems Thinking:  a learning system of communication, control and feedback loops optimizing Communication Network / Signal operating in an unfriendly (blocking) environment (contradictory cognitive and perceptual signals / messages at different levels, with a negative injunction forbidding acknowledgement of that contradiction:  Double Bind).

Thus, both S3 and LP are about information, thinking within a topology of paradox, Watzlawick’s "contradiction that follows correct deduction from consistent premises”:  invitation to Ashby’s double feedback loop learning (intrinsically destabilizing, Open System).

The subtlety here is that the whole (the Gestalt) has a reality of its own: subjective contours are real, according to the brain, and therefore must be taken into account (illusion of whole affects interpretation of each part); illusion is operational fact.  Actions are taken, or not taken, accordingly.

Whether LP two-stage process, (flip-flop of consumer  >>  Anheuser-Busch's Executive Staff frame of reference: “figure-ground”, projective vs. objective process of inquiry)


or S3it’s all about flip-flop of manipulator frame of reference:  egocentric  >>  allocentric, and patterns of thought (higher levels of abstraction, in a paradoxical environment, and how to get the most out of them:  partnership of feedback and symmetry.  (Systems Thinking in action.)

Uncertainty as tool.

In other words, both S3 and Listening Post (each a hybrid, and each a paradigm shift, in different fields) are about switchiness (compound switching -- cognitive, perceptual, mechanical / environmental -- and compound schedule of reinforcement) and hypothesis testing, seminal to "control and communication in the Animal and the Machine”.  Both address Human adaptation to paradox.  Both, in Ashby’s terms, adaptation generated by double feedback loops.


                                                                         Problem Universe

                                       "Control and Communication in the Animal and the Machine


The Listening Post (wide spectrum, epistemic, Open System) was created to overcome the “chilling effects’ of the research community’s cherished Focus Group -- (Merton’s narrow spectrum, single feedback loop, Closed System), which, in Vickers terms above, “provide its own bogus validation”.  Contradiction of action and environment -- dissonance of conflicting cognitions / actions

" … request for conversation is … the cornerstone of our misunderstanding …. It implies a simple (means / ends) as opposed to a joint causality [S3] .... Not just a crude, inept version of written propositional statements, e.g., "what they were trying to say is … rather multi-layered messages in which a communication about the nature of the exchange is negotiated on many levels."  

                   M. S. Sommer, "Open Systems Listening: Conversation As A Research Tool", Ph.D. Diss.,1988 

Thus, both of these tools, Sand Listening Post, are dialectical engines, capitalizing on contradiction / paradox; assume the existence of, and address a "naive linguistics" (independent of formal language and symbols, to counter Wittgenstein: “The limits of my language [does not] mean the limits of my world”), recursive processes of self-observation and self-reference which can be put to work, for example in research and cognitive play; and are rooted in the idea and operation of “wholeness", 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, Animal and Machine. 

Again, the subtlety here is that the whole (the Gestalt) has a reality of its own:  subjective contours are real, according to the brain, and therefore must be taken into account (illusion of whole affects interpretation of each part); illusion is operational fact.  Actions are taken, or not taken, accordingly.

Using that wide-spectrum (variety increasing, epistemic tool) "Listening Post” group research methodology, AB's Budweiser brand quickly regained share of market: marketing research efforts were gutted and rebuilt; now the "mind of the consumer" (not the "mind of the researcher”) was in the market planning / strategic intelligence driver's seat, delivering perceptive insights into marketing strengths and weaknesses.  


To sell beer to those "citizens of Wonderland” (from the adult perspective), the young adults (several of the young adult research subjects were not beer drinkers), the client corporation had to hear and understand the web of coherent relationships from within their “irrational" environment, the paradoxes of their logic. 


                                                                          Sommer office truck

Thus, as with the Listening Post, S(communication / signal operating in an unfriendly (blocking) environment) embodies the “New Cyberneticsthe functionality of machine, the learning of organism: the necessary conceptual fusion of Systems Thinking and Cybernetics. 

Navigating topology of paradox.

Regulation of things with “purpose”, joint causality; a multi-layered "conversation" in which the nature of the exchange is negotiated on many levels. 

topology (properties and relations unaffected by continuous change -- symmetry under transformation) 

of paradox (contradictory cognitive and perceptual signals / messages at different levels, where acknowledgement of that contradiction is forbidden).  

Thus, the S3 is about regulatory switching (synchronizing concurrent communication / interference among rotating switching stations), a heuristic routing (ball or vector) system, in Paul Baran’s Distributed CommunicationAdaptive Message Block Network terms:  seeking best paths (optimization) in an unfriendly (blocking) network of switchable links; where dynamic programming is required to optimize incompletely understood and changing systems (using locally implemented full switching at all nodes, not a hierarchy of routing control centers). 


(Think of the ball as “ping”:  a network test for reachability.  Part of the methodological process of logical reasoning, the logical argument.)  


        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.

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

               S”Parallelism of General Cognitive Fields”

“A too restricted view of human nature … even though only briefly ascendent, 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 Judgement, 1965

The Srepresents a long overdue paradigm shift away from that compression of reality: Marcuse's "Happy Consciousness”  (“the belief that the real is rational and that the system delivers the goods -- reflects the new conformism …")

"Logic is a poor model of cause and effect.

                                                                 Gregory Bateson, Mind and Nature: A Necessary Unity, 1979

According to the S3 reciprocity of Hands and Mind epistemological premise, not to mention common sense, the current focus of educational technology, all things computer -- The “Silver Bullet”, “The Fix -- and the assumptions supporting it, are too constricted:  Marcuse’s  pattern of “One-dimensional Thought”.

"Conventional education in physics, biology, or the social sciences treats them as separate domains … each specialty becomes a triflingly small field, unconnected with the rest.

"In contrast, the educational demands of training “Scientific Generalists” and of developing interdisciplinary "basic principles"  [STEM] are precisely those General Systems Theory tries to fill.” … interdisciplinary synthesis and integrated education." 

                                                                 von Bertalanfy, Ludwig, General Systems Theory: Foundations, 

                                                                  Development, Applications, George Braziller, 1968


From this platform, the S3, with its unique "dynamic interactions" (Norbert Wiener,  Cybernetics:  Or Control And Communication In The Animal And The Machine, 1948)), leverages cognitive / perceptual processing to a higher level of abstraction through robust integration of Hands and Mind (mental and manual rotation), 

Thus, Sis a tangible demonstration of of those processes at work, basic behaviors (consciousness) of the brain through mechanical concepts (switches); 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.

In Cybernetic terms, Sis a Leibniz  germinis "machinea rationcinatrix.”

A dynamic domain (Wiener Animal-Machine Control and Communication) of variety increasing (informal as well as formal reasoning), nested and recursive structures / functions / processes; a tangible, lucid demonstration of Cybernetics / Systems Thinking -- goal systems of circular, regulatory chains influencing each other; a learning system of communication, control and feedback loops.

In Fine Arts terms, Sis

"… 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.”    Gödel, Escher, Bach


      Sas Marriage of Constructivism and Computer Learning

                          Full-Spectrum Cognitive Partners

The core process of abstract reasoning, both motor and visuospatial skills, should work together, independent of disciplines, as the learner moves up the education spiral.

“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.”  (May 13, 2008, cc m.s.)

                                                                    Laura E. Berk, Ph.D. [Child Development, 8th Edition] 

The S3 eliminates barriers by  demanding the manipulator visualize and resolve evolving spatial sets and hidden patterns, generating a succession of new mental models.


"The child must not only apply operations to objects -- in other words, mentally execute possible actions on them -- he must also 'reflect' those operations in the absence of the objects which are replaced by pure propositions.  This 'reflection' is thought raised to the second power.  Concrete thinking is the representation of a possible action, and Formal thinking is the representation [abstract thought] of a representation of possible action [with consequences]."

                                                                         Jean Piaget, "The Mental Development of the Child", 1968

These evolving transformations are self-initiated and self-regulated (Ratchet).  By allowing the manipulator to calibrate problem complexity (unintended consequences) the S3 generates abstract, discretionary and flexible thought across a wide range of intuitive-to-logical conceptual understanding.  

In other words, the manipulator must constantly re-examine and adjust chains of contradictory reference frames (boundaries which shape our thought), those of the observer as well as the observed system itself. 

"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."   

                                                                                        Marvin Minsky, "Frames Systems Theory", 1975

By continuously tackling fundamental, increasingly complex and dynamic hypotheticals, the S3 manipulator is “re-inventing” mathematics, manipulating higher levels of abstraction without symbolic / numeric tools (programming evolving goal-directed patterns and relationships of square and circle):

Propositions, not just objects [simple blocks];

Ideas within a hierarchical structure; and 

Alternative possibilities within a combinational array.

In short, the S3 manipulator must develop / intuit sets of rudimentary patterns and operations, inductive (& abductive) and deductive, then revisit and reconstruct them, a spiraling step by step reconceptualization, from operational rigor to comprehensive abstraction.

Back to basics.

Reciprocity of Hands and mind.

                 "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)                                            

As Steve Jobs says above (Is anyone listening?), Hi-Tech by itself is necessary, but not sufficient; computer learning and the S3 learning / play process are interdependent, natural partners:  both cognitive tools are needed, for different reasons, and should be seen as complementary, not competing. 

            "A Silicon Valley School That Doesn’t Compute"


"The Underlying Assumptions of Educational Technology Implementation" 



                             Removing Arbitrary Barriers

Pre-K and up, arbitrary barriers between school subjects restrict the development of abstract  reasoning. 







© Michael S. Sommer, Ph.D, 2018