Math Class Languages

Features of the Languages of the Math Classroom

• If two people don't speak the same "language," communication cannot occur unless an interpreter or another "language" is used.
• People have a fondness, even a loyalty, for their primary language. REPRESSION of another language is often strong.
• People have a hesitance, even a reluctance, to use a language if they do not feel they have sufficient proficiency in the language to express themselves clearly, perhaps even expertly.

It is often the case that:

• Fluency in one language does not imply fluency in other languages.
• Fluency in one language of a family facilitates mastery of a new language in the same family.
• Introduction of a new language often introduces a new dimension of thought.

The Levels of Proficiency include:

REPRESSION

 

REPRESENTATION

 

OPERATION

 

CREATION

 

INTERPRETATION

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."

• DTWYP
• Hands to Head to Hands
• Not the format in which you learned it.
• 19+ Languages
  Pictorial Communication:: sine-cosine
  More Concrete Communication:: sine-cosine
• Always the 3-Day Plan:
  Foreshadow, Teach, Review
• Offer A Rerun before the Lesson
• A Mother-Tongue Sound Bite Goes Straight to the Head
        - Fraction -- Numberer/Namer
        - A Number -- Not Just Any Old Number
        - Slope -- Rise/Run
        - Limit -- Approach
        - Derivative -- Tangent
        - Derivative -- Rate of Change
        - Second Derivative -- The Change in the Change

  Customary ruler

-- lab, "hands to head"

  Group & Coral Recitation

-- "sing a song" vs. "apply a formula"

  Math Exercises

Term Tiles

-- for expressions & word problems

  Distance in 1-, 2-, 3-Dimensions

  Model for Distance

  Sine & Cosine on Unit Circle

  Area Formulas by Paper Folding

  Describe Slope In Words,

Derivative and 2nd Derivative

    w/Words, Graph, & Pointer

      The Game for Two Players