19+ Languages of the Math Classroom
© 11/15/01, A. Azzolino
Abstract
 Refine your interpretative skills in using the languages of the math class.

 Consider regularly selecting from the verbal, written, pictorial, and concrete language familiies.

 Consider the employment of more than one family to achieve and enhance communication.
Please Complete This PreTest (Mentally if you wish.)
 Determine the inverse of .
 Explain why this is the inverse.
 Consider the sequence of statements:
 Solve: x^{2} + 2x  3 =0
 (x + 3)(x  1)= 0
 x + 3 =0 x  1 =0
 x = 3, x = 1
 Explain why this is true.
 Explain the term "math wars."
 Answer showing work. Find the integers.

 Three consecutive integers are involved. The sum of the first,
triple the middle, and four times the largest is 131.
 What is the standard US railroad gauge?
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."
 A good story is worth retelling.
 Tell it in as many languages a possible.
 Start with the least sophisticated version.
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 Mother Tongue, Other tongue(s) 
 VERBAL
 formal spoken mathematics
 informal spoken math
 spoken symbol
 symbol speak
 calculatoreze/computereze
 web speak

 WRITTEN
 written word
 written symbol
 semisymbolic
 calculator symbol

 PICTORIAL
 picture
 numeral
 graph
 nonverbal body language

 CONCRETE
 object
 model
 manipulative/token

Strategies for Each Language Family
 VERBAL
  mother tongue, other tongue(s)
 DTWYP, Discuss This With Your Partner
 CREATE time in class for math discussion
  formal/informal spoken mathematics
 Speak Both Formal and Informal Mathematics
  spoken symbol, symbol speak
 Speak each language. State in one language, then restate in the other.
 Don't symbol speak sound bytes, use formal spoken math.
"Laws of Exponents" ...
  calculatoreze/computereze
 Alternate between spoken symbols and calculatoreze  say both: the
"first function" and "why won" for "Y1"
  web speak
 Create a listserv for your students and for colleagues.
 Post notes and animation
 WRITTEN
  written word, written symbol
 Have students write their own written, non symbolic, version of the 'Laws of Exponents'
  semisymbolic
 Here are the rules:
 Use words (including nicknames) and arithmetic to express an idea.
 Perhaps encircle the words with parentheses for clarity.
 Use no variable numbers. Use only constant numbers.
 Use order of operations.
  calculator symbol
 Use as many languages as possible.
 Think of all the ways one might go about solving a quadratic equation!
 PICTORIAL
  picture, numeral, graph, nonverbal body language
 Use as many languages as possible.
 CONCRETE
  object, model, manipulative/token
 Speak both "Mother Tongue" and "Formal Mathematics" when in the concrete.
 Permit free play before structured work.

A good story is worth retelling.
The Romans had twohorse chariots and built roads all over
to accommodate these vehicles. The distance between chariot wheels was determined by the
width of the horses that pulled the chariots.
The roads of the Roman Empire, including those in what is now
England, received ruts created by the chariot's wheels.
All vehicles on the roads, even nonchariots, had to deal with
the ruts and, since safety and durability were desired, vehicles other than chariots were
built to accommodate this width.
The English train was a vehicle built with the same body as
other vehicles of the time and therefore used the same wheelbase.
Men who built English trains knew this wheel width and used it
in building trains.
Men who built English trains came to America and built trains
to the same specs as in their homeland.
In order for trains to move from track to track in America,
even those made by different companies conformed to the standard gauge, 4feet8.5inch or
1,435millimetres, the width determined by the width of a pair of Roman horses.
We can say lightly that the standard US railroad gauge was
determined by a pair of horses' ass's. We can also learn from a history lesson  about
doing thing the way they've always been done, but also about conformity for safety and
durability  a lesson to remember in the "math wars."
Please debrief, then, DTWYP. My comments follow.
Most Important Remarks
 Discuss This With Your Partner! is the most valuable strategy!
 Introduce in the Most Concrete
 Debrief in the Most Abstract
 Regularly Require Translation from One Language to Another
 Good Stories Are Worth Retelling. Retell you mathematical stories
in as many languages as possible.
Determine the inverse of
I. Verbally  mother tongue, mathematics, math
 The function:
 1st: Take a number.
 2nd: Add 3.
 3rd: Take the reciprocal of this.
 4th: Take the opposite of this.
 5th: Subtract 4 from this.
  The inverse must:
 1st: Add 4.
 2nd: Take the opposite of this.
 3rd: Take the reciprocal of this.
 4th: Subtract 3.
 5th: Get a number.

x becomes x + 4 becomes (x+4) becomes 1/(x+4) becomes 1/(x+4)  3 
II. Pictorially, Concretely  graphically, manipulatively
III. Symbolically, w/pencil and paper
Closing Remarks
Discuss This With Your Partner is the most valuable strategy!
 Introduce in the Most Concrete
 Debrief in the Most Abstract
 Regularly Require Translation from One Language to Another
 Good Stories Are Worth Retelling.
 Retell your mathematical stories in as many languages as possible.
© A^{2}, 12/29/2001 /www.mathnstuff.com/papers/nov1601.htm