The Languages of the Math Classroom

© '98, Agnes Azzolino

The Written Language Family

written word

written symbol


calculator symbol

[Top] 9. written word 10. written symbol

The written word codes the mother tongue, other tongues, formal mathematics, and informal mathematics. The written symbol also codes these. Alone each is a powerful tool. Together their power is magnified.

In the math classroom situation the written word is found most often in the text in paragraph or word problem form. The traditional uses of the written mathematics languages have been:

In the math classroom situation the written symbol is found most often on the board in line-by-line coding of the solution of a problem. Traditional uses of the written mathematics languages have been:

Fortunately, that is changing. Math teachers have opted for "chalk & talk" over "just talk" - "written & verbal" over "written." They have begun to employ strategies championed by the critical thinking and writing-to-learn movements. Math teachers have begun to expreiment with longer written execises and the use of technology in place of more symbol-manipulating exercises.

These efforts are to be applauded. The simultaneous, bilingual if you will, communication of mathematics in more than one language and more than one language family is even more powerful and useful than communication in only one language.

Example: Solving Quadratic and Higher Order Equations by Factoring

Traditionally, write on the board:
"Solve: AB= 0, then below it A=0 or B=0"
Now, also write on the board:
"Solve: AB= 0, then below it A=0 or B= 0"
"If the product of two numbers is zero, one or both numbers must be zero."

Example: Reviewing the 'Laws of Exponents'

Write the Laws in Words!
Traditionally, write on the board:

Now, also write on the board:
"When multiplying two numbers with same base, copy the base and add the exponents."

Teaching Strategy: Have students write their own written, non symbolic, version of the 'Laws of Exponents'

See the Fall '98 Precalc QB, Questionbook

[Top] 11. semisymbolic

This is a term coined by the author to represent a transitional stage from the written word to the written symbol. Please examine it through the page entitled The Semi/Pseudo Expression/Equation", the page which follows this one.

[Top] 12. calculator symbol

Example: Solving Quadratic Equations with a Calculator

There can be no doubt that calculator symbols are another language. The above graphic illustrates 3 different dialects of this language.

Teaching Strategy: Use as many language families as possible.

Think of all the ways one might go about solving a quadratic equation!
You might solve:
  1. The square of a number is twenty-five. Find the number.
  2. One less than the square of a number is twenty-four. Find the number.
  • mentally, or
  • verbally (with the mother tongue), or
  • through a written symbolic language (with symbolic manipulation), or
  • through keystroking a written symbolic language (with a graphing calculator), or
  • pictorially (with a graphing calculator), or
  • more concretely with Graphing With Manipulatives.
Please, consider solving the equation with more than one language.

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[Verbal] [Written] [Pictorial] [Concrete]
[written word] [written symbol] [semisymbolic] [calculator symbol]

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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