Conceptual Understanding of Quadratic Expressions

Date of Completion


Document Type

Open Access Capstone

Degree Name

Master of Arts (MA)

First Advisor

Patricia S. Davidson


The learning theories of Skemp (1987) and various conceptual change theorists are synthesized to delineate four levels of understanding: instrumental understanding which is equivalent to unconnected or weakly connected surface knowledge; relational understanding which is the beginning of deeper, more connected knowledge; symbolic understanding which provides a sound connection between surface and deep structures; and conceptual understanding which demands a level of commitment to and trust in a framework of constant and connected knowledge. A teaching model is presented to assist the educator in teaching for the goal of conceptual understanding in students. The model illustrates the premise that teaching for conceptual understanding within a specific content area must take into account content skills, student dispositions, creative and critical thinking skills, and teaching/learning strategies, all within the realm of critical thought. Perkin's (1986) Knowledge as Design questions of purpose, structure, model and argument are offered as a strategy to harness and focus the student's content skills, dispositions and thinking skills to form a viable means of constructing conceptual understanding in a content area. The topic of quadratic expressions is used as an illustration of the learning theory and the model of teaching. The four levels of understanding are outlined for the structure of a quadratic expressions, the evaluation of quadratic expressions, binomial multiplication and factoring. A geometric model is used to build relational, symbolic and conceptual understandings. The teaching model is applied to the teaching of quadratic expressions with the possible source of knowledge, prior misconceptions and required content skills being delineated. The Knowledge as Design strategy, with accompanying creative and critical thinking skills, is analyzed for its effectiveness as a tool in enabling students to construct conceptual understanding. Although it may not always be practical to guide students all the way to conceptual understanding for every concept, it should be an educator's ideal goal. If teachers would begin to teach for the minimum of relational understanding, students would exhibit fewer misconceptions in their understanding and have more faith in their own knowledge.


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