Levels of Proficiency
 REPRESSION  "Ignore it. It's substandard communication."
 "It's not another language, it's slang. It's not math the way I've always spoken it."
 In the late 1970s, college professors refused to use calculators and used slide rules instead.
 In the 1990s, students refused to use pencil & paper and used calculators instead.
 In the early 1990s, teachers refused to use calculators and used pencil & paper instead.
 Still some high school teachers refuse to use manipulatives and use symbol manipulation instead.
 REPRESENTATION  "How do you say/represent this or that?"
 Speak of things/nouns: an equation, an expression, an exponent, an exponential.
 positional and additive
 Cognition is enhanced by Reflection, summary/debriefing,visualization
 Once representation is begun, one might "go overboard" it using a new language.
 OPERATION  "What can you do with this?" or "How do you do this?"
 Speak of doings/verbs: solve an equation, simplify an expression.
 addition, subtraction, multiplication on the 100s board
 additions, subtractions, dilations on the coordinate plane w/GWM

 Upon REPRESENTATION or OPERATION, there exists a desire to
 restate familiar ideas in the new language.

 CREATION  "Watch/Listen to what I can do," and
 "See how well I speak & understand."
 Using the identity as input to compare or analyze the square root
 Students' coordinate plane art work with equations

 REPRESENTATION and OPERATION initiate CREATION 
 the stage in which a new idea or clarity of an old theme occurs because of the
restatement or translation.

 INTERPRETATION  "Which language suits my purpose best?" or
 "Should I say the same thing in both/all languages?" and
 "It doesn't mean exactly that: it's means this."
