Algebraic thinking is an approach to thinking about quantitative situations in general and relational manner. This kind of thinking is optimized by a considerable understanding of the objects of algebra, a disposition to think in generality, and engagement in high-level tasks which provide contexts for applying and investigating mathematics and the real-world.
Objects of Algebra
The objects are the content of algebra which I classify into three overlapping categories. The first category and the most basic are those for representing changing and unchanging quantities and relationships. These include the idea of variables, numbers, graphs, equations, matrices, etc. The second category are ideas for working with unknown quantities which involve solving equations and inequalities under which are linear equations and inequalities in one variable, systems of linear equations and inequalities, exponential equations, quadratic, trigonometric equations, etc. The third and last category involves the ideas for investigating relationships between changing quantities which include directly and inversely proportional relationships; relationships with constant rate of change; relationships with changing rate of change; relationships involving exponential growth and decay; periodic relationships, etc.
Knowledge of algebraic content do not necessarily translate in algebraic thinking. Computational fluency in simplifying, transforming, and generating expression for example, while important, do not necessarily involve a person in algebraic thinking if one is doing it for its own sake. Thinking processes that contribute to the development of algebraic thinking are those that require purposeful representations of quantities and relationships, multiple interpretations of representations, finding structures, and generalization of patterns, operations and procedures. These should become part of students’ thinking disposition.
High level tasks
The higher-order tasks in mathematics include problem solving, mathematical investigations (sometimes referred to also as open-ended problem solving tasks), and modeling.