Language aspects of algebra – from a semiotic/linguistic perspective

Language aspects of algebra

The symbolic language of algebra has evolved over time to become an

efficient describe medium to

powerful generate

Mathematical ideas

Aspects of Symbolic Language

• Language relies on– Economy of symbol use– Limited set of linguistic patterns

• These features make it possible to express powerful mathematical ideas squeezed into manageable “chunks” of symbol strings

Power of Symbolic Language

Precision

Economy Efficiency

Ambiguity in Language

• Power and efficiency – ambiguity• Problem for learners

• Instead of removing the ambiguity, for example, by introducing new symbols, remove the ambiguity by focusing instruction on the linguistic form of a given symbol string

Linguistic Aspects

There are a limited number of modes of algebraic expression, which, once mastered enable an individual to– read, write, speak mathematics– undertake transformations– develop conceptual understanding

Linguistic Aspects

• Notion of representation – Different forms of variables

• Abstraction of methods – formulas, identities, theorems

• Truth of sentences• Patterns of mathematical

reasoning

• Teachers and students underestimate the importance of the explicit study of these linguistic elements

• What is opaque to the student may be transparent to the teacher

2x + 3 double the number and add 3

opaque transparent

• Focus on physical signs and placement order

• Look through the symbols to underlying mathematical meaning

Multiple Aspects of Symbolic Competence

Understanding of

linguistic categories Form

Practical How you say

(symbolic) Process

Capability How you do

Understanding Concept

concepts what you do

Example

a) 2(x + 1) = 3xb) 2(x + 1) = 2x + 2

What is each equation about?

a) numbers, unknown, equality of number

b) Operations, equivalence of operations

As students grasp the nature of symbolic algebra’s linguistic aspects, they gain power to use mathematical contexts to remove ambiguity from any given symbol string.

Explicit instruction within this focus

helps students gain autonomy and fully master algebraic thinking.