*"Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. ... However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction." *

* G. Polya, Lecture on Teaching Mathematics*

The Sommer Cube (S^{3}), like algebra, is about solving geometry (spatial) problems by analytic means* *(Cartesian Plane): illumination of diverse problem sets (classes) across a broad range of disciplines. CHALLENGES / GAMES

It's about variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition.

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

*“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”*

http://mathforum.org/library/more_info.html?id=59253

* *

Thus, S^{3 }is about the need for more *learning proficiency* than *problem solving competence*: situation awareness, creativity, and the careful construction of logical arguments, and their limitations.

*The aim of heuristic [*S^{3}*] **is to study the methods and rules of discovery and invention .... Heuristic, as an adjective, means 'serving to discover'. ... its purpose is to discover the solution of the present problem. ... What is good education? Systematically giving opportunity to the student to discover things by himself.** *

* G. Polya**, **How to Solve It** (1945)*

S^{3} is 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 exploration of deep causal spatial structure.

Scientifically verifiable Spatial Thinking (space, representation, process: multifaceted, interconnecting competencies, human and robot) tools and learning opportunities are needed, for teachers as well as students, diagnostic as well as development.

In musical (math / pattern) terms, S^{3 }is about polyphony: simultaneously combining modules, *independent* in melody (rhythm and contour of ball trajectory) yet *interdependent *harmonically (periodic variations upon a theme).

From child to super-mathematician, the S^{3 }manipulator has only one path: reason / intuit up the original *"evolution of mathematics**” , *from things, to abstractions of things, to binaryness, to geometryness, to algebraness, to setness -- create more sophisticated problem models, know what the laws are.

S^{3} is about levels of abstraction, pattern, the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships; patterns of thought (and their systematic breakdown -- learning requires negative knowledge), intuition, under simultaneous mental (rational and nonrational) and manual rotation. Apex reasoning, patterns of thought *exceeding the parameters normally experienced in logical operations*: where the strategies and tactics are evolutionary*.*

S^{3} are "pattern blocks", which create porus polyhedron "chains" formed by distinct collections of figurative elements: circles and squares.

A paradigm shift, a hybrid, an orchestration of ideas: combination 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.

*"Teachers try to convince their students that equations and formulas are more expressive than ordinary words. But it takes years to become proficient at using the language of mathematics, and until then, formulas and equations are in most respects even less trustworthy than commonsense reasoning. *

* **Marvin Minsky, Society of Mind*

Consequently, S^{3} demands recognition of sequence / pattern of repeating events formed in accordance with a definite rule(s) -- but different individuals can perceive the same pattern differently, reach different generalizations.

*As Galileo put it.*

*"[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*

* *

“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*

Thus, S^{3} is about uncertainty as tool.

It’s mysterious.

Thus, the fun of S^{3} and mathematics:

*Do it Yourself.*

Distributed Programming Network

Discovering and Representing A System Of Rules

The exercise of critical thought is fun for scientists and for artists and for musicians.

School is too often not fun.

School is too often about learning -- not thinking.

It's a big secret: *Thinking is good for you*.

S^{3}^{ }is a programmable logic component (In Field Programmable Gate Array terms), a combinational (technically “permutational", like a combination lock) 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.

S^{3}^{ }is a dynamic router, an Analog-Binary Processor: a system of programmable binary (0/1) switches, a “flying junction” (an over-and-under junction of tunnels which weave past each other) housed within a cubical exoskeleton with multiple axes of rotation and alternative, manipulator-controlled ball trajectories, offering exit in all orientations.

Thus, S^{3}^{ }is about continuous switching (synchronous and asynchronous): 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 *(Boolean algebra -- a way of seeing new structures -- fundamental to the design of binary computer circuits and programming language)*. **Note that individual **S ^{3}*

*logic elements do not necessarily have a discrete true / false state at any given time; simple Boolean logic is inadequate for this, thus extensions are required.*

The Dialectic

S^{3 }(about human learning proficiency) is an exploration of symmetry, transformation of parts and whole, square and circle -- adaptation to accelerating rates of change and complexity of system and environment. While Rubik’s Cube (Machine-think: about problem solving competence), without that paradox, exercises logic, S^{3 }topology of paradox demands logic and intuition, naive commonsense reasoning.

The developing S^{3}^{ }network requires constant revision of operational sets and simultaneous consideration of many possibilities from many perspectives: the order of steps is not fixed, there is no algorithm. (learning requires negative knowledge)

In Cybernetic terms, the S^{3} is a tangible demonstration of basic behaviors (consciousness) of the brain through mechanical concepts (switches / logic gates); 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.

Rolling Ball “Tilt-Switch” *Asynchronous *Binary Processor (0/1)

*Each S ^{3}^{ }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 S*^{3 }*nonlinearity -- change in one variable which does not produce a directly proportional change in the result -- even in the single S*^{3}*, 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 S*^{3}*.)*

It demands the manipulator understand and assimilate evolving information about patterns and functional relationships and analyze change in both concrete and hypothetical contexts: a meeting of abstract ideas with the spirit of mathematical rigor.

In logic / mathematical terms, the S^{3}^{ }manipulator must personally (without symbolic / numerical tools) visualize, intuit, reason incommensurate concepts, like Gödel’s Paradox (“this statement is false” requires a broader logical perspective), like the S^{3}, self-referential), like the Pythagorean triangle, 1,1, √2; 1,1,1, √3, and reinvent accordingly (Oops! all numbers are not integers, rational numbers -- let’s look at the bigger picture — think of it as irrational number along a line of possible numbers).

It’s the same with S^{3}: connect the abstract with reality, ideas as things, deal *systematically* with changing patterns of:

Structure and transformation (transformation geometry / symmetry; rotation [turns], reflection [flips], translation [slides], etc.);

Change and motion (Calculus);

Visual or tactile (Geometry);

Operations (Algebra); and

Connections (Graph Theory).

Systematically *reinvent* the fundamentals of mathematics; create basic strategies, reasoned / intuitive conceptual structures, mathematical abstractions:

"Just WHAT IS ALGEBRAIC THINKING?”

“While it is unclear what the goal "algebra for all" really means, the trickle-down effect of this goal is clear: elementary and middle school mathematics instruction must focus greater attention on preparing all students for challenging middle and high school mathematics programs (Steen, 1992; Chambers, 1994; Silver, 1997). Thus, "algebraic thinking" has become a catch-all phrase for the mathematics teaching and learning that will prepare students for successful experiences in algebra and beyond.

“ALGEBRAIC THINKING - ACCORDING TO SOME EXPERTS”

“Battista and Brown (1998): For students to meaningfully utilize algebra, it is essential that instruction focus on sense making, not symbol manipulation. Throughout their mathematical careers, students should have opportunities to reflect on and talk about general procedures performed on numbers and quantities. ...Thinking about numerical procedures starts in elementary grades and continues...until students can eventually express and reflect on the procedures using algebraic symbolism.

Greenes and Findell (1998): The big ideas of algebraic thinking involves representation, proportional reasoning, balance, meaning of variable, patterns and functions, inductive reasoning, and deductive reasoning.

Herbert and Brown (1997): Algebraic thinking is using mathematical symbols and tools to analyze different situations by (1) extracting information from the situation...(2) representing that information mathematically in words, diagrams, tables, graphs, and equations; and (3) interpreting and applying mathematical findings, such as solving for unknowns, testing conjectures, and identifying functional relationships.

Kaput (NCTM, 1993): Algebraic thinking involves the construction and representation of patterns and regularities, deliberate generalization, and most important, active exploration and conjecture.

Kieran and Chalouh (1993): Algebraic thinking involves the development of mathematical reasoning within an algebraic frame of mind by building meaning for the symbols and operations of algebra in terms of arithmetic

LUMR Project (Driscoll, 1997): The facility with algebraic thinking includes the ability to think about functions and how they work and to think about the impact on calculations a system’s structure has.

NCTM Standards (5-8) - Algebra (NCTM, 1989): Understand the concept of variable, expression, and equation; represent situations and number pattern with tables, graphs, verbal rules, and equations, and explore the interrelationships of these representations; analyze tables and graphs to identify properties and relationships; develop confidence in solving linear equations using concrete, informal, and formal methods; investigate inequalities and nonlinear equations informally; apply algebraic methods to solve a variety of real-world problems and mathematical problems.

NCTM Standards (5-8) - Patterns and Functions (NCTM, 1989): Describe, extend, analyze, and create a wide variety of patterns; describe and represent relationships with tables, graphs, rules; analyze functional relationships to explain how a change in one quantity results in a change in another; use patterns and functions to represent and solve problems.

Usiskin (1997): Algebra is a language. This language has five major aspects: (1) unknowns, (2) formulas, (3) generalized patterns, (4) placeholders, (5) relationships. At any time that these ideas are discussed from kindergarten upward, there is opportunity to introduce the language of algebra.

Vance (1998): Algebra is sometimes defined as generalized arithmetic or as a language for generalizing arithmetic. However algebraic more than a set of rules for manipulating symbols: it is a way of thinking."

From Concepts in the Middle School: Kregler, Shelley. A special edition of Mathematics Teaching in the Middle School.

*http://blog.coreknowledge.org/2010/10/06/singapore-math-is-our-dirty-little-secret/*

Algebraic Play

"Why Learn Algebra?"

Early Childhood Mathematics: Promoting Good Beginnings.”

** **“Children become intensely engaged in play. Pursuing their own purposes, they tend to tackle problems that are challenging enough to be engrossing yet not totally beyond their capacities. Sticking with a problem—puzzling over it and approaching it in various ways—can lead to powerful learning. In addition, when several children grapple with the same problem, they often come up with different approaches, discuss, and learn from one another [74, 75]. These aspects of play tend to prompt and promote thinking and learning in mathematics and in other areas.

… “Block building offers one example of play’s value for mathematical learning. As children build with blocks, they constantly accumulate experiences with the ways in which objects can be related, and these experiences become the foundation for a multitude of mathematical concepts—far beyond simply sorting and seriating. Classic unit blocks and other construction materials such as connecting blocks give children entry into a world where objects have predictable similarities and relationships [66, 76].

From a joint position statement (section eight) of the National Association for the Education of Young Children (NAEYC) and the National Council for Teachers of Mathematics (NCTM) Adopted in 2002 .

"Supporting [all] Students in Mathematics Through the Use of Manipulatives"

"Is it possible for all children to become mathematically literate? To achieve this goal of mathematics literacy and to meet the needs of all children requires a change in our thinking about the framework of mathematics curricula and how children learn mathematics.

"The National Council of Teachers of Mathematics’ (NCTM) Curriculum and Evaluation Standards for School Mathematics in 1989 and its revised framework of Principles and Standards for School Mathematics in 2000 provides a vision for all students to think mathematically and highlights learning by all students….

"Mathematical tools—whether concrete manipulatives or virtual manipulatives—are supportive tools for learning. The use of mathematical tools shapes the way students think and build mathematical relationships and connections toward conceptual understanding (Fuson et al.1992). Selecting and accessing the appropriate tools and processes for students with disabilities is critical to their understanding mathematics."

From Supporting Students in Mathematics Through the Use of Manipulatives: Terry Anstrom, American Institutes for Research, 2006.