Paper presented by Dr. A. Dendane at the 11th Annual Research Conference - UAE University, April 2010
It is well established that graphing calculators and computer algebra systems may be used to create student-centered environments where students learn by exploring mathematical concepts and hence gain a deep understanding of these concepts. The power of graphing calculators lies in their ability to allow different representations of the same mathematical concept. For example, an algebraic function may be defined by an algebraic expression, a graph, or a set of numerical values.
In spring 2009, the mathematics program in UGRU acquired “Autograph,” 2D and 3D graphing software that brings mathematical concepts related to precalculus, calculus, and statistics to life. It is a powerful tool that helps students explore mathematical concepts using dynamic objects.
In this paper, examples of mathematical topics explored by students in the classroom are presented. Using these examples, the paper discusses the ability to use this software to explore complex mathematical concepts and mathematical problem solving. The advantages of using “Autograph” and the conditions under which students may gain a deep understanding of mathematical concepts are also explored.
Mathematical concepts are closely linked, making mathematics a hierarchical subject where conceptual understanding of new ideas depends on mastering earlier ideas. One encounters difficulties in understanding topics in calculus, for example, if one does not have a deep understanding of algebra concepts and procedures. Earlier, learners may have encountered even greater difficulties when transitioning from arithmetic to algebra because this transition was not well managed by instructors [1][2]. This is due to the fact that mathematics was often presented as a set of rules, formulas, procedures, and facts to be memorized [3]. Instructors frequently spend more time teaching procedures and facts while students practice and memorize algorithms. Consequently, students view mathematics as a set of isolated procedures to be memorized [4][5].
In general, students with a deep understanding of mathematical concepts, objects, and procedures are less likely to have major difficulties in learning and understanding new topics. In fact, they are well-prepared to develop and understand new topics, enjoy their mathematics classes, and are intrinsically motivated [6]. Furthermore, student abilities in mathematical problem solving—the heart of any math curriculum—depend on a deep understanding of mathematical concepts and the ability to apply them in unfamiliar situations [7][8][9][10].
Learning and deep understanding of topics in mathematics involve processes in which students connect to and build on knowledge acquired in the past [11]. Making connections between prior knowledge and new information to construct new knowledge is an indication of learning with deep understanding [12]. We therefore need to design and develop classroom activities in which prior knowledge is activated in order to gain deep understanding and develop new mathematical concepts [6].
One of the most important concepts in mathematics is the concept of functions. Acquiring a deep understanding of functions is an essential facet of mathematical thinking, leading to better problem solving and comprehension of other mathematical concepts. Functions may be represented algebraically, numerically, and graphically, and the linkage between these different representations provides learners with deep insight [14].
Students using graphing calculators are better able to relate graphs to their equations, understand the characteristics of functions, and find algebraic representations for graphs [15][16]. Research also shows that the use of graphing calculators changes the classroom environment, making students more active with increased group work, investigation, and exploration [17].
At the math unit in UGRU, Autograph software version 3 was acquired in spring 2009 and integrated into math classes, accompanied by professional development sessions. Autograph possesses strong capabilities for visualizing and animating math objects, making it ideal for exploring math concepts deeply and creating interactive, student-centered environments. This paper discusses strategies for using Autograph to design lessons where students actively build new knowledge using prior knowledge. Specifically, situations are examined where Autograph helps students: