Tuesday, May 10, 2011
Define algebraic thinking.
According to some, algebraic thinking is:
Kaput (NCTM, 1993): Algebraic thinking involves the construction and representation of patterns and regularities, deliberate generalization, and most important, active exploration and conjecture.
Greenes and Findell (1998): The big ideas of algebraic thinking involves representation, proportional reasoning, balance, meaning of variable, patterns and functions, inductive reasoning, and deductive reasoning.
Herbert and Brown (1997): Algebraic thinking is using mathematical symbols and tools to analyze different situations by (1) extracting information from the situation...(2) representing that information mathematically in words, diagrams, tables, graphs, and equations; and (3) interpreting and applying mathematical findings, such as solving for unknowns, testing conjectures, and identifying functional relationships.
LUMR Project (Driscoll, 1997): The facility with algebraic thinking includes the ability to think about functions and how they work and to think about the impact on calculations a system’s structure has.
NCTM Standards (5-8) - Algebra (NCTM, 1989): Understand the concept of variable, expression, and equation; represent situations and number pattern with tables, graphs, verbal rules, and equations, and explore the interrelationships of these representations; analyze tables and graphs to identify properties and relationships; develop confidence in solving linear equations using concrete, informal, and formal methods; investigate inequalities and nonlinear equations informally; apply algebraic methods to solve a variety of real-world problems and mathematical problems.
NCTM Standards (5-8) - Patterns and Functions (NCTM, 1989): Describe, extend, analyze, and create a wide variety of patterns; describe and represent relationships with tables, graphs, rules; analyze functional relationships to explain how a change in one quantity results in a change in another; use patterns and functions to represent and solve problems
Usiskin (1997): Algebra is a language. This language has five major aspects: (1) unknowns, (2) formulas, (3) generalized patterns, (4) placeholders, (5) relationships. At any time that these ideas are discussed from kindergarten upward, there is opportunity to introduce the language of algebra.
Vance (1998): Algebra is sometimes defined as generalized arithmetic or as a language for generalizing arithmetic. However algebraic more than a set of rules for manipulating symbols: it is a way of thinking.
As a math teacher and thinker, I am pretty ok with most of these definitions, but I do not agree with reducing algebra to a simple language, as Usiskin tries to do. It isn't the language or the symbols that matter, but rather, the abstract way our minds work with ideas of generalized arithmetic and pattern description.