Dialectic As Play


The Sis about navigating three-dimensional problem space (like flying), with Hands and Mind.      

                                  The Smart Block.      The Upgrade.

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 

S3 is about pattern, "'voluntarily' reproduced and combined", the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships -- and its simultaneous mental and manual rotation.

The exploration of symmetry, influences among parts and whole, square and circle -- 


S3 is a combinational logic block which contains four tunnels, effectively an array of unconnected 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.

S3 is about awareness of mental and manual environment, modifying self-behavior on the basis of experience, through the optimization of multiple simultaneous paths, in a constant stream of evolving problems, within a topology of paradox.

"[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word." 

                                                                                                  Galileo Galilei, Opere Il Saggiatore, 1623




“One must first of all study geometric forms: the cone, the cube, the cylinder, the sphere.”

                                                                            Paul Cézanne, Letter to Emile Bernard


                                         Hello.  Hands and Mind (30,000 BC)

                                        Mental and Manual Rotation

Mental rotation, the ability to rotate mental representations of two dimensional and three dimensional objects; predictor of math / science achievement. 


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

SHands and Mind activity is a dynamic, hierarchical system of interdependent, nested frames of reference; an interaction of fourteen squares and circles (reorienting connecting chiral, gravity-dependent, tunnels) to be remembered and adjusted. 

A self-generating, self-sustaining, escalating causal loop of cognitive development and intuition


"… 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.”   

                                                                                                   Douglas R. HofstadterGödel, Escher, Bach                              


                                                 Come Fly Me 

                                                              Network Simulation / Modeling

                                                        Like Hand-Flying


"Effective Rotations: Action Effects Determine the Interplay of Mental and Manual Rotations"


                            "Mental and Manual Rotation"

In Cybernetic terms, Sis about regulatory switching, a variety increasing, heuristic routing (ball or vector) system:  seeking best paths (optimization) in an unfriendly (blocking) network / environment of switchable links.


                             Rolling Ball Switch                                                 Binary Processor (0/1)

A distributed processor (S3) programming network.


[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)


                                      Laura E. Berk, Ph.D., Child Development, 8th Edition, ("Sex Differences In Spatial Abilities”  p. 553)                                                                                        

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


S3 is a tangible, lucid demonstration of 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.


                                                      EULER FOR TOTS

In Euler's terms (Topology / Graph Theory -- mathematical study of networks), the S3 demands the manipulator distill a geographic system to its underlying graph, its conceptual abstraction: two fundamental components, simple line and point, abstracted into a set of edges and junctions (paths and switches), a geometry problem independent of distance; a continuously updated network connectivity with rules that control which network objects can properly connect to each other.


                                                       HEGEL FOR TOTS

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


Integration of a set of contradictions, a dialectic, to a higher level to transcend the polarity and form a new proposition.

Each Sis a programmable logic component (In Field Programmable Gate Array terms), a combinational logic block which contains four tunnels, 


                                                                                             Binary Processor (0/1)

effectively an array of unconnected switches to be programmed by the user as the block is rotated in space, which can be connected to other logic blocks to create a hierarchy of reconfigurable interconnects.


(e.g., overlapping subproblems / optimization — straight line:  single path; wavy line:  shortest path between two nodes; bold line:  overall shortest path from start to goal) 

A Cybernetic switch, encompassing the Systems Thinking idea and 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.

A cyclical, dynamic process of analysis (separation into constituent parts) and synthesis (fusion of parts into whole), verification and correction, expressed as programming steps, from formulation to execution.

In summary, the S3 low-tech cognitive tool, with all the benefits of simultaneous high-level reasoning and manual experience:  the reciprocity of Hands and Mind (note embodied cognition, manual and mental rotation studies -- unlike "mouse learning" / computer learning), essential in the very early years.

The Smart Block.   The Upgrade.


                              Thinking Without Boundaries


 S3 is about introducing robust general principles to the confusing prevailing taxonomy of play; cognitive payoff is key.


                    Behavioral Outcomes and Mental Operations,d.cWc

           "Is Rotation of Visual Mental Images a Motor Act?

"Two vs Three-Dimensional Presentation of Mental Rotation Tasks:  Sex differences and effects of training on performance and brain activation."

                  "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)

                      (computer) "Brain Games Are Bogus"


    "Harmful To Children?  The Alliance For Childhood Report."

           "Are Constructivism and Computer-Based Learning       

                            Environments Incompatible?"

"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;view=fulltext



                 Marriage of Constructivism and Computer Learning                 

                               Innovation in Educational Technology 

                  Think Low-tech -- Reciprocity of Hands and Mind



                                 “Problem-Solving Deficit Disorder"

"PSDD describes the condition in which children are no longer active agents of their involvement with the world…. They have trouble becoming deeply engaged in unstructured activities. They lack creativity and imagination and experience difficulty in playing cooperatively with others or resolving conflicts without aggression. They do better when they are told what to do. They prefer structured activities at school, DVDs to watch, or video games to play at home."

                                                         Diane Levin:  "Paradigm Magazine", 2009

        "Recognizing Spatial Intelligence", SCIENTIFIC AMERICAN (2010) 



                  "The Dialectic Incarnate -- Hegel for Tots"

“Froebel developed the concept for the second gift by 1836.  He considered it the most profound of all, claiming that to understand the second gift was to understand kindergarten.  Consisting of a sphere and a cube linked by an elucidating cylinder .... [emphasis added] Alike in their perfection, the sphere and the cube are, in respect to form, pure opposites …. the dialectic incarnate – Hegel for tots – and the clearest expression of Froebel’s law of opposites.”

                                                        Norman Brosterman:  Inventing Kindergarten, 1997                                                                                                                           


      "Yes, Your Toddler Really Is Smarter Than A 5-Year-Old"

"Very small children can reason abstractly, researchers say, and are able to infer the relationships between objects that elude older children who get caught up on the concreteness of things.

'Learning may actually harm these kids' abilities to do abstract reasoning.’ 

Walker is working in the lab of Alison Gopnik, a developmental psychologist who has made a career out of devising experiments that reveal the inner thoughts of children still too young to talk. Her take is that babies are smart, and in many ways smarter than adults."

                                                               Caren M. Walker, Alison Gopnik, Psychological Science, 2013

                                               The "Smart" Playground

                                           Child Designed


                                   (checkered paths allow exit)

                               The Developmental / Cognitive View

"It focuses on the cognitive mastery of reasoning and problem-solving strategies that are enhanced by object play.  A developmental / cognitive perspective sees object play as helping the child to reach forward toward mental or developmental challenges which are not yet a part of the child’s day-to-day repertoire.  It is this perspective which Jean Piaget used in his analysis of the child’s play with objects (Flavell, Miller, & Miller, 2002; Piaget, 1962).  


“Children’s play with parquetry blocks or tangrams exemplifies play with objects for the purpose of cognitive mastery.  The manipulation of individual pieces in order to reproduce design patterns or to create original designs requires reasoning and problem-solving strategies to be employed and enhanced through the play.                        

"Adults’ ability to observe and encourage without intruding on the reasoning aspect of this form of object play is crucial to children’s development of internal problem-solving strategies. …. 


“When children use the computer mouse to “paint” a picture, they are manipulating the representation of a paint brush to master the movements needed to satisfactorily complete a picture.  When they manipulate blocks and small classification toys, they may be using the objects to master either the motoric challenges involved, or the cognitive challenges of space, pattern, or relationships.”

                                                        Shirley K. Morgenthaler, The Meanings in Play with Objects              

             “Children Must Master the Language of Things 

              Before They Master the Language of Words.”

                                                       Friedrich Froebel, Pedagogics of the Kindergarten, 1895”

“In one sentence, Froebel, father of the kindergarten, expressed the essence of early-childhood education. Children are not born knowing the difference between red and green, sweet and sour, rough and smooth, cold and hot, or any number of physical sensations. The natural world is the infant’s and young child’s first curriculum, and it can only be learned by direct interaction with things. There is no way a young child can learn the difference between sweet and sour, rough and smooth, hot and cold without tasting, touching, or feeling something. Learning about the world of things, and their various properties, is a time-consuming and intense process that cannot be hurried.”

“This view of early-childhood education has been echoed by all the giants of early-childhood development -- Froebel, Maria Montessori, Rudolf Steiner, Jean Piaget, and Lev Vygotsky. It is supported by developmental theory, which demonstrates that the logical structure of reading and math requires syllogistic reasoning abilities on the part of the child.  Inasmuch as most young children do not attain this form of reasoning until the age of five or six, it makes little sense to introduce formal instruction in reading and math until then.”

                                                        David Elkind:  “Much Too Early”, 2001

                             Systems Thinking in Schools

                                                                      Systems Blocks:  

           A physical Interface for System Dynamics Learning


Low-tech cognitive tool.  Intelligent toy.

Reciprocity of Hands and mind. 

S3 Play


                  Lest we forget -- Archimedes' tombstone.



© Michael S. Sommer, Ph.D. 2015