Teachiing Adaptive and Strategic Reasoning through Formula Derivation: beyond Formal Semiotics

Conventional logic in teaching American secondary level students (secondary level meaning students comprised of grades 7-12) tells us that mathematics instruction should be both efficient and informative. Logic also suggests that efficiency and information should not be mutually exclusive, and yet American students continue to struggle with adapting their learning beyond the rudiments of their textbooks. This is possibly because the most common instructional method used in secondary mathematics classrooms is highly structured and is based primarily on the semiotic processes so commonly illustrated in mathematics textbooks (Watanabe, 2007). Semiotics in mathematics is simply defined as the use of symbols that are helpful in understanding the processes of thinking, symbolizing, and communicating (Radford, Schubring, & Seeger, 2008). Semiotics in mathematics instruction typically refers to the teaching of mathematics as a process of symbol manipulation through structured algorithms and rigorously defined theorems. The outcome for students in this strategy of teaching and learning is referred to as procedural fluency. Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately (National Research Council, 2001). Procedurally fluent students ostensibly develop the ability to evaluate and simplify various expressions, solve simple equalities, and represent mathematical relationships in graphical form. However, as the mathematics performance of American secondary students continues to remain low relative to students in other countries, the value of teaching for procedural fluency without identifying the relevance of the underlying mathematical logic remains a matter of debate. The manuscript at hand describes how pedagogy including Strategic Competence and Adaptive Reasoning can supplement procedural fluency to create a more comprehensive learning experience for secondary mathematics students.