## The development of curricular modules using learning theories and research results to enhance student learning of completing the square

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Geometry and Algebra are central to the study of mathematics and have been called its two formal pillars. Although they are seemingly separate branches, the two overlap in the curriculum, as in the Texas standards for high school mathematics, or Texas Essential Knowledge and Skills (TEKS). These state standards dictate what students should know after completion of a given school course. The strategic overlap of Geometry and Algebra in the high school curriculum emphasizes their importance in leading students to a higher level of conceptual understanding of mathematics.

The purpose of this study is to review effective, research-based techniques in Mathematics Education and to develop two well-organized lesson plans for teaching Completing the Square, an important concept in the high school curriculum that combines both algebraic and geometric reasoning and skills. This topic is a well-known technique for solving quadratic equations and is the process of converting a quadratic equation into a perfect square trinomial by adding or subtracting terms on both sides. Completing the Square is praised for its ability to be applied to quadratic equations regardless of the nature of the roots or factorability of the quadratic over the rational numbers and the real numbers. The concept of completing the square is a good example of incorporating both vertical and horizontal articulation in the curriculum and is able to bridge the ideas learned in geometry and algebra, and pave the way for understanding more difficult and complex mathematical topics.

Two alternative methods for teaching completing the square are developed, using an algebraic approach for one lesson plan and a geometric approach for the other. These lessons cater to different types of learners and present the material in such a way that incorporates significant learning theories and acknowledges the overlap in the concepts of Algebra and Geometry in the state curriculum.

The Algebraic Approach is influenced by a cognitive information processing theory because of its influence on the retention of knowledge. Additionally, utilizing schema structures makes it easier to retrieve the knowledge when similar problems reappear in assessment and proceeding classes. The Geometric Approach utilizes more of a constructivist learning theory by allowing the students to become active learners who construct knowledge and understanding from building upon their prior knowledge. This approach makes use of visual representations to enhance retention of knowledge and accessibility in a student's working memory, which will be beneficial to the student in the problem solving process. The Geometric Approach also provides students an opportunity to make a connection between what they see, what they know, and from what is to be learned, while developing the formulas and schematic structures provided to the students in the Algebraic Approach. A thought process such as this can influence a deeper level of conceptual understanding and prepare the student to exploit these learning techniques in problem solving needed for real world challenges.