Hands and Mind

                                      Classic Toy.  Full-spectrum Cognitive tool

Exactly why did Einstein and Frank Lloyd Wright love blocks as children?

Because it's Cutting Edge Play.  PLAY AS THINKING  

The challenge of uncertainty, and the power of Constructivism.

This is the story of the Sommer Cube (S3) and manual and mental rotation, spatial thinking, the ability to rotate mental representations of two dimensional and three dimensional objects, strong predictor of math / science achievement. 

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 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 satisfying our hunger for abstract reasoning, and Spatial proficiency

Solving and displaying everyday, complex functional relationships.  .  Hello, algebra.  



Eeach logic block has equal power to switch and route; tunnels switch differentially.

Step By Step.


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

                                          Mathematician too:  QUAD / FRAME / ROBOTICS PROBLEM

From child to super-mathematician, the Smanipulator has only one path:  reason / intuit up the original "evolution of mathematics” ladderfrom things, to abstractions of things, to binaryness, to geometryness, to algebraness, to setness -- create more sophisticated problem models, know what the laws are.)

Keep it simple  "Pre-K math" . 

Or not.

In Programming / Cybernetic terms.

[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)


                                                     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.

(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.) 


That opportunity should be made available to all.  

Enter S3 — a cubical maze module.  A kickstarter.


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”

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).

It's about leveraging Binary and Analogical reasoning as both classic toy (Tate Museums) and Cognitive Toolconcrete to abstract (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).  

Thinking beyond structured ideas.


A combinational (technically “permutational”, like a combination lock) logic block which contains four tunnels, effectively an array of unconnected, gravity-dependent switches / routers to be programmed (with or without a ball) by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous routes (network theory), depicted physically or logically.  An architecture of nested dichotomies.An architecture of complexity.  (Remember, link multiple simultaneous paths.) 

In other words, simple as it is, 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.

Thus, S3 is 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. 


                                                                            (checkered paths allow exit)

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, giving form to binary and analogical information, a coupling between physical objects and binary information where bits are directly manipulable and perceptible.  

Child Development speaks.


                                      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.”  (April 10, 2008, cc m.s.)

"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 …”  (May 13, 2008, (cc mss)


On the other hand … 

Mathematics speaks:

math forum copy

“Description:  The faces of the modular Sommer Cube (S3) feed into twisted conduits that comprise a three-dimensional, tunnelling maze. The website provides example arrangements and construction questions such as "Given that a ball must exit at the lowest S3 level and obey gravity, what is the minimum number of cubes necessary to make four quadruple path cubes, or S3s which utilize all four conduits as passageways?" Available with transparent or opaque walls.”

“Levels:  High School (9-12), College, Research

Languages:  English

Resource Types:  Games, Manipulatives

Math Topics:  Higher-Dimensional Geometry, Topology”

Thus, S3 is about following the vision of Leonardo da Vinci:  Arts and Science as one (STEM / STEAM).  A hunger for experimentation, abstraction, a manipulation of concepts.  Enlightenment

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


Einstein kept it simple.

"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 


Cézanne kept it simple.

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

                                                                                  Paul Cézanne, Letter to Emile Bernard


Galileo kept it simple.

"[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




Isaac Newton kept it simple.

“Newton’s mathematics in the Principia is not very different from what Archimedes had used in calculating the areas of circles, or what Kepler had used in calculating the volumes of wine casks.” 

                                                                                  Steven Weinberg, To Explain The World

"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)


In Hegel’s terms, not so simple.  

A dialectic, a flip-flop, of pattern and process, a mixing of different levels of abstraction, self-reference.

                                                       HEGEL FOR TOTS



"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     

Thus, S3 is 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.


Euler kept it very simple.

                                                      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.    

A system of logic blocks:  S3 "Distributed Adaptive Logic Block Network” , in Paul Baran’s Internet terms, in a low-tech format.  A fractal command and control, strategic and tactical planning challenge (for child and mathematician).   


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



                                          HYBRID TOY

                                    Challenges / Games 



    (S3s in every pre-K “block corner” / elementary school: explore abstract concepts)

PROGRAMMABLE LOGIC BLOCKS (Analog-Binary Processor:  binary gate arrays / switches)

    (“Levels:  High School (9-12), College, Research …  

     Math Topics:  Higher-Dimensional Geometry, Topology”)




THEME PARK / PLAYGROUND (children change design at will)


                                                            Come Fly Me 

                                                            Like Hand-Flying


    Marriage of cognitive  Constructivism and Computer Learning 

                          Full-Spectrum Cognitive Partners    



                     The Smart Block

                                                        Binary and Analogical Reasoning

                                                        (switch from fixed to changing values).     

                                                        "May The Force Be With You.”

                                                        The Upgrade.

                                              The Smart Playground

                                            Child Designed


                               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

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

                            "Mental and Manual Rotation"

                    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"


           "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

        "Recognizing Spatial Intelligence", SCIENTIFIC AMERICAN (2010) 



      "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 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, 2018