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.
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.