# The Languages of the Math Classroom

## 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.
• This paper outlines, and briefly elaborates upon, specific Levels of Proficiency in the use of a language. The Levels of Proficiencey 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.
• Reflection/summary/debriefing/visualization enhances cognition.
• 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.
• Reflection/summary/debriefing/visualization enhances cognition.
• CREATION - "Watch/Listen to what I can do," and "See how well I speak & understand."
• Using the identity as input to compare or analyse 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."
• When deciding which language(s) to speak -
• Introduce in the concrete.
• Debrief in the abstract.
• Don't force anyone to speak a "less sophisticated language." Use it yourself. Speak it to them. Do not force them to reply using that language.

The next page introduces the Math Class Language Families.

## 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 www.mathnstuff.com/papers/langu/page0.htm and additional pages (ISBN: 1-929-870-01-9 © 1998, Agnes Azzolino).