What is Algebra?

Algebra as a subject seeks to establish and use generalizations about objects and operations in symbol systems to describe quantitative relationships. We define general structures and their properties by noticing how their patterns of use are related. Then properties identified for one representative structure can be interpreted about another representative structure.

In high school algebra, we have number objects, and addition and multiplication operations, and a concept of set. We extend these operations to sets of number objects and variable objects, and use these new expressions to establish sets of ordered number pairs, which can describe relationships between paired measurements. This typifies elementary study of functions.

In abstract algebra at the college level, we have number objects (integers, or pick your favorite), and addition and multiplication operations. We note that each operation is associative and also commutative. Each operation has an identity number object. We define these properties. Then when we explore different operations on our number objects, we can test whether these properties are present or not. When we explore sets of different objects with their own operations, we can test whether these properties are present or not. This typifies study of groups and rings.

Algebra seeks to establish higher levels of generalization by introduction of abstract concepts to classify object structures and their properties.

Algebraic Thinking in Middle School

Introduction of algebraic exercises at middle school is consistent with what we recognize of intellectual development from the basic observations of Piaget. This age group for children is at the cusp of transition from the “concrete operational” to the “formal operational” stage.

Concrete operational stage thinking (prevalent from age 7 to up to 12) requires actual physical objects, and can comprehend few abstract reasoning concepts, though can begin to understand reversibility. The concrete operational stage is also characterized by the ability to coordinate two dimensions of an object simultaneously, arrange structures in sequence, and transpose differences between items in a series. Thus sets of ordered pairs and Cartesian graphs are natural objects to learn about at this stage.

The formal operational stage begins at approximately 11 to 12 years of age, and continues throughout adulthood. The level of abstraction individuals are comfortable with will vary. In terms of mathematics, formal operational thinking allows understanding of the form or structure of a problem,

It is characterized by the ability to formulate hypotheses and systematically test them to arrive at an answer to a problem, and ability to reason contrary to fact. A purely formal hypothesis, even a false one, can be used as the basis of an argument.