Algebraic Reasoning
Algebraic reasoning is the ability to think logically about unknown quantities and the relationships between them. It allows students to reason about expressions, functions, graphs, equations and so forth, and makes it possible for them to draw conclusions and make conjectures about the relationships between them. In elementary school, students are often presented with problems such as, "6 + = 11. What number goes in the box?" Most students will be able to fill in the box correctly, remembering from addition tables that 6 + 5 = 11. In elementary school, students work consistently with arithmetic concepts in developing number sense. Students in a traditional high school algebra class would solve that same problem by translating the problem to the form 6 + x = 11 and using the algorithm which mandates that 6 must be subtracted from both sides of the equation to solve for x. In high school, students work with symbolic manipulation and the advanced use of functions. A student who has developed algebraic reasoning skills, however, would contend that the answer is 5 since "the difference between 11 and 6 is 5." In middle school, a student's intellectual development bridges the two areas between understanding how numbers function and the formal study of number systems. That conceptual development is the realm of algebraic reasoning.
There are three prerequisite skills necessary prior to beginning a formal study of algebra: understanding the language of algebra; understanding the concept of variable; and understanding the concepts of relation and function. These developments occur in the middle school classroom. To develop these skills, a student should be exposed to the following activities: Expression and Equation Manipulation Understanding the intimate relationships between parts of a mathematical statement is the very foundation of algebra. In middle school, this understanding can be developed especially through the use of formulas. Pattern Recognition As algebra is often termed the "science of patterns" and can be defined as the "generalization of arithmetic", the ability to identify patterns from charts, graphs and sets of data is an inherently necessary and valuable skill. Conversely, a student should be skilled at using tables and graphs to represent data. Formal and Informal Problem Solving Algebra problems can be solved with symbolic manipulation, and traditionally this was the only method available to students. The ability to manipulate symbols is a valuable skill, but the logic behind symbolic manipulation is fundamental to success in mathematics. Solving problems by various problem-solving methods (such as those described on pages 2637 in this handbook) is key to reasoning algebraically. Real-World Applications Students need to use the ideas of algebra as they apply to the world around them. An irrelevant classroom is not a place where student understanding is fostered. By seeing the relevance of algebra to the real world, students develop algebraic reasoning skills in a meaningful way.
Parallels can be drawn between algebraic reasoning and number sense. Number sense allows a student to understand the patterns between numbers; likewise, algebraic reasoning allows a student to recognize the interrelationships between unknown quantities. A student with well-developed number sense understands when it is appropriate to use a calculator and when it is appropriate to rely on mental mathematics; similarly, a student who reasons algebraically knows when symbolic manipulation would be necessary or advantageous and when another method would be appropriate.
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