19+ Languages of the Math Classroom

© 11/15/01, A. Azzolino


      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 Pre-Test (Mentally if you wish.)

  1. Determine the inverse of .
  2. Explain why this is the inverse.
  3. Consider the sequence of statements:
    Solve: x2 + 2x - 3 =0
    (x + 3)(x - 1)= 0
    x + 3 =0     x - 1 =0
    x = -3, x = 1
    Explain why this is true.
  4. Explain the term "math wars."
  5. 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.
  6. What is the standard US railroad gauge?

Features of the Languages of the Math Classroom

It is often the case that:

The Levels of Proficiency include:


Levels of Proficiency

REPRESSION -  "Ignore it. It's substandard communication."
REPRESENTATION - "How do you say/represent this or that?"
OPERATION - "What can you do with this?" or "How do you do this?"
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."
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.

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

Strategies for Each Language Family

- mother tongue, other tongue(s)
- formal/informal spoken mathematics
- spoken symbol, symbol speak
- calculatoreze/computereze
- web speak
- written word, written symbol
- semisymbolic
- calculator symbol
- picture, numeral, graph, nonverbal body language
- object, model, manipulative/token

A good story is worth retelling.

      The Romans had two-horse 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, 4-feet-8.5-inch or 1,435-millimetres, 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