Abstract
The interactive online platforms Desmos and GeoGebra can be used to support students' development of conceptual understanding in mathematics. They provide opportunities to support algebraic manipulations with visual representations, facilitate the derivation of formulas and procedures, and can help students verify and make sense of their answers. These platforms can also be used to increase student interest and engagement while promoting a student-centered classroom environment. This chapter presents several practical ideas for using these free platforms in the context of an Algebra 2 unit on trigonometry and a Precalculus unit on lines and planes in space. Selection, creation, modification, and implementation of resources are discussed.
TopIntroduction
Mathematics is a difficult subject for many students and is made even more difficult when they are asked to simply memorize definitions and procedures without understanding, or when they are asked to perform algebraic manipulations without any alternative representations which could be used to make sense of their work. It does not have to be this way. Free and interactive technological platforms exist which can empower students to make sense of algebraic work with visual, graphical representations. These graphical platforms can assist with derivations of mathematical procedures, can help students make sense of what they are doing and the answers they are getting, and can promote student engagement and discovery.
The use of technology in teaching mathematics has been called for by various organizations. The PISA 2018 mathematical framework included references to students’ use of technology to portray mathematical relationships and to approximate solutions and called for the use of dynamic geometry software to manipulate and interpret shapes (OECD, 2019). One of the Common Core Mathematics Practice standards affirms that mathematically proficient students can use technology to visualize the results of varying assumptions and to explore and deepen their conceptual understanding (National Governors Association, 2010).
In the context of secondary mathematics courses at the Algebra 2 and Precalculus levels, these goals can be addressed through intentional use of the free online platforms Desmos (https://www.geogebra.org/). As a high school mathematics teacher, the author has used these platforms extensively with his students and can provide numerous detailed examples of how to: a) select, modify, and create applets using these platforms; b) use these platforms in a classroom setting to enhance student learning and engagement; and c) ensure that students use these resources in ways that develop their conceptual understanding and promote meaning-making.
The objectives of this chapter are to provide specific and varied examples of how Desmos and GeoGebra can be used to teach mathematics topics at the Algebra 2 and Precalculus levels, with advice on how to create, select, modify, and use these resources with students. Examples from two specific units of instruction will be highlighted: a unit on trigonometry and a unit on lines and planes in space.
Key Terms in this Chapter
Student-Centered Instruction: Instruction that involves student engagement and exploration, with the teacher as facilitator.
Preservice Teachers: Students in a teacher preparation program who are working towards becoming certified teachers.
Teacher-Centered Instruction: Instruction that is involves teacher dissemination of material, often in a lecture format. The students are recipients of knowledge who may write down notes but are not otherwise involved in knowledge production.
Procedural Competency: Basic knowledge about how to perform a mathematical algorithm and arrive at an answer, without necessarily understanding what the answer represents or knowing if it is correct.
Parametric Functions: Functions written as x in terms of a parameter such as t and y in terms of the same parameter. The dependent variables are x and y and t is the independent variable.
Conceptual Understanding: Deep knowledge about a topic that involves knowing why and how it works; ability to work with novel but related topics due to knowledge of underlying structures and relationships.
Applet: A webpage that is usually interactive which can be used for demonstrations. In this chapter, webpages created using Desmos and GeoGebra are referred to as applets.