An Intro to Math Classroom Languages


A Walk Down The Hall To
Introduce The Language Families


  • Most Abstract
  • May Not Be Spoken in the Mother Tongue(s) at All


  • Most Abstract
  • Most Linear
  • Easiest to Lose Thread of Discourse
  • Possibly Listner Passive


  • Less Abstract
  • Symbols & Words
  • Chalk & Talk
  • Two Paths of Input to Listner


  • More Concrete
  • "Worth A Thousand Words"
  • Precurser to Written?
  • The Better the Picture, The Better the Picture


  • Not in Listner's Hand
  • Movement by Another
  • Model - Teacher Moves
  • Tokens


  • Learner Centered
  • Manipulative - Listner Moves & is
    No Longer A Listner
  • Most Basic
  • Active

    Mother Tongue(s)

  • Most Concrete & Abstract
  • Easiest Way to
    Make Things Concrete
  • Abstractly Make Things Concrete

Math Class Languages

The Mother Tongue, Other tongue(s)
formal spoken mathematics
informal spoken math
spoken symbol
symbol speak
web speak
written word
written symbol
calculator symbol
nonverbal body language

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:


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
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."

Key Points
  • 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

Things to Do: Make Math More Concrete & More Abstract
  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

Game for Two Players

      The Game for Two Players

Just Make It More Concrete
  • A good story is worth retelling.
    Tell it in as many languages a possible.
    Start with the least sophisticated version.
  • Introduce in the concrete.
  • Debrief in the abstract.
  • Discuss This With Your Partner
  • Make It More Concrete.
  • You Can Do More Than Chalk & Talk

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© October 4, 2003, Agnes Azzolino,
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The Languages of the Math Classroom

    Verbal, Written, Pictorial, and Concrete (the Hundreds Board, for example) are the four broad mathematics language families discussed in this electronic monograph found at and additional pages (ISBN: 1-929-870-01-9 © 1998, Agnes Azzolino).

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