Algebraic Thinking

What is Algebra?

"Arithmetic is the basis for algebra, and algebra is the language of calculus - the gateway to higher mathematics and science"

In our study of math, we have studied some aspects of arithmetic & geometry. Arithmetic is the study of numbers & geometry the study of shapes. We will study now another area of math which is called Algebra.

The term Algebra came from the word "al-jbr" which basically hinted at computation. As the study of math developed, the idea of "algebraic thinking" evolved. Algebraic Thinking was thinking about the relations between numbers, shapes and operations on them in a broader abstract sense.

The Essence of Algebra is Abstraction

We say Math is full of abstract concepts and that it why it is a difficult subject to learn & teach. Abstraction is a generalization of multiple ideas into a single concept. In many cases the multiple ideas could also belong to vastly different domains.

It is almost like looking at a city from the air. From the air, many characteristics of the city become easily observable. We are also able to see connections of various parts of the city with one another. These connections may not be easily visible to a person on the street.

A concept like "noun" in language is a generalization of a million different kinds of ideas about objects, events and ideas. The concept like "what is the moral of the story?" is a generalization drawn from many different stories & incidents from widely varying cultures.

Arithmetic itself contains many abstract ideas.

Algebra takes the abstraction of Arithmetic to another higher level.

Algebra has the capability to show how mathematics exponentially increases the human capacity for analogical thinking in symbolic and maneuverable ways that have dramatically impacted our thinking & hence our civilization.

Abstract ideas in Arithmetic

In math a statement like "2 + 3 = 5" is a generalization of an infinite number of real life stories involving objects, incidents, ideas and actions. The individual objects, events etc have been removed leaving only the "essence" of the meaning of a situation.

Such an "essence" allows us to manipulate the abstractions in the relations and explore many properties of these relationships without the burden of having to consider their real life contexts. We focus on the relation between 3 & 2 and not worry about if the story is about apples, oranges, sheep or something else!

For example, 3 + 2 = 5 also implies that 5 - 3 = 2 & 5 - 2 = 3.

7 - 3 = 4 implies that 7 - 4 = 3 & 7 = 3 + 4

Similarly, 3 X 4 =12 implies that 12 ÷ 4 = 3, 12 ÷ 4 = 3, 3 + 3 + 3 + 3 =12 & 4 + 4 + 4 = 12!

Such multiple relations also exist in a number of arithmetic operations, involving infinity of numbers. They also exist between arithmetic & geometry. Ratios between numbers are related to the idea of similarity of geometric figures. The relation between surface area and volumes of objects can be studied using arithmetic. How do we capture such abstractions which exist in all kinds of arithmetic & geometric relations?

This process of abstracting or generalizing relations in arithmetic & geometry is called algebraic thinking. It is an important step in achieving the objectives of learning mathematics; Mathematical Thinking, Logical Thinking, Critical Thinking, Problem Solving etc etc.

The primary school curriculum in arithmetic and geometry provide a lot of opportunities to develop algebraic thinking. But in most cases such opportunities are not used. Arithmetic should also be taught in terms of understanding the concepts and the relation between the concepts. But it is mostly taught as a list of facts to be memorized.

Algebra

The discipline of Algebra was an outcome of algebraic thinking. It emerged as a language to "express patterns & relations" in a very general sense. Algebra starts with numbers & shapes but goes beyond the realm of arithmetic & geometry. One immediate example is the algebra of logic (Boolean Algebra) which is the base of logic circuits and computers.

Whereas arithmetic operated on numbers, algebra gradually evolved to be a form of calculation that operates on classes of numbers. This is where variables come in. A variable represents a class of numbers in a general way and draws conclusions about that class.

Continuation of Patterns

As we transition from arithmetic to algebra, we can see that patterns connecting numbers remains same. It would help children to learn easily, if attention of children is drawn to this fact.

2 apples + 3 apples = 5 apples

2 hundreds + 3 hundreds = 5 hundreds

2 tenths + 3 tenths = 5 tenths

2x + 3x = 5x

2x^2 + 3x^2 = 5x^2