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The TetractysTM Template

Learn about...
  • How the template helps the student analyze the puzzle
  • The importance of visualizing the rule of the puzzle
  • Encoding and decoding the rule using digits
Benefits to the student:
  • Analyzing the rule using a template promotes thinking abstractly
  • The template provides benefits connected to writing
  • The template provides a step-by-step procedure for solving the puzzle

Interfacing with the Puzzle

Of all the ways of interacting with someone, nothing beats meeting that person face-to-face. The template of the Tetractys™ Number Puzzle is where the student comes face-to-face with the rule of the puzzle.

The Tetractys™ Puzzle Analysis Template is a design feature of the Tetractys™ Number Puzzle that helps the student solve the puzzle in a step-by-step fashion. It is the ideal interface between the student and the rule of the puzzle. Firstly, the puzzle grid is replicated in the template. Next, the student decodes the grid by translating the four digits of the puzzle into four mathematical equations representing the rule of the puzzle. Finally, the student solves these equations and translates the solutions back into the puzzle grid. This whole process allows the student to visualize the rule in a well-known form and makes it easier to decode the puzzle before encoding the solution. And by doing all this while writing digits into the template, the student gains the benefits of both visually and tactilely interacting with the puzzle.

The Puzzle Encoded

    The Tetractys™ Puzzle Analysis Template starts with the puzzle grid containing the four digits governed by the Tetractys puzzle's rule. This grid compresses four equations into four digits, or put another way, these four digits 'encode' four equations.

  • Four digits encoding four equations shows the multifunctional nature of digits
  • The rule becomes an encoding algorithm or procedure
  • The compressed form of the grid is fit to be analyzed
  • The first half of the template replicates the puzzle grid

Decode the Puzzle

    Once the puzzle grid has been transfered into the template, it is ready to be decoded. A step-by-step guide is given for deriving four equations from the four digits of the puzzle, giving the student the opportunity to write out the four equations in visual form.

  • Analyzing the rule requires decoding the puzzle grid
  • The template's colored boxes provide visual cues for decoding the digits into equations
  • The erasable laminated surface of the template makes it easy to solve for all unknown variables in the derived equations.

Encode the Solution

    After the four equations in the lower half of the template have been solved, the solutions can be entered back into the puzzle grid in compressed form, that is, the solution digits are then encoded back into the puzzle grid before the puzzle's solution is confirmed.

  • The puzzle grid contains not only the four digits of the rule but also the solution to the puzzle.
  • Encoding and decoding information develops rule-following skills
  • Encoding and decoding information using digits facilitates abstract thinking

Visualizing the Rule in Detail

Being able to think abstractly does not exclude the ability to visualize something simultaneously. Rather, like any good metaphor, visualization creates a bridge over which something concrete can be understood abstractly.

Language and mathematics have a lot in common. Both can be said to be systems of representation, but that alone isn't saying all that much. The analogy between them becomes more informative, however, if we were to look at a key feature of these systems of representation, namely, how they extend meaning from within each system. In language metaphors are used to extend the meaning of words, that is, from literal meaning to figurative or metaphorical meaning. In mathematics symbols are used to extend meaning by taking the symbol with a primary meaning and understanding it abstractly by giving it a secondary meaning. In either system, one forges a new meaning from an existing meaning.

The Tetractys™ Puzzle Analysis Template does something very similar in the Tetracty™ Number Puzzle. The meaning of the four digits of the puzzle comes from the four equations which can be derived from the puzzle's rule, and the template allows the student to visualize these equations explicitly, so that the meaning of the four digits becomes fuller, clearer and immediate. Connecting these two different expressions of the puzzle's rule is fundamental to the ability to think abstractly about the rule. In this case, the details really do matter.

Once the student is able to visualize internally (that is, in the mind) what is explicitly shown in the template, then the leap to abstract thinking is nearly complete.

Tetractys Number Puzzle: A Mathematical Learning System for Students

Product Details

  • Publisher: POIS Research; First edition
  • Language: English
  • ISBN-13: 978-0-9959504-0-5
  • Product Dimensions: 8.5" x 5.5" x 0.2"
  • Paperback: 64 pages in full color (Saddle Stitch)

  • Note: Dry Erase Marker and Dry Eraser not included.
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