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 deep understanding of these concepts. The power of graphical 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” which is a 2D and 3D graphing software that lets you bring mathematical concepts, related to pre calculus, calculus and statistics, to life. It is a very powerful tool that helps students explore mathematical concepts using dynamic objects.
In this paper, I will present examples of mathematical topics explored by students in the classroom. Using these examples, I will also discuss 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 deep understanding of mathematical concepts are also discussed.
Mathematical concepts are closely linked which makes 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 transiting from arithmetic to algebra because this transition was not well “managed” by instructors . This is due to the fact that mathematics was presented as a set of rules, formulas, procedures and facts to be memorized . Mathematics instructors often spend more time teaching procedures and facts, and students practicing and memorizing algorithms. Consequently, students view mathematics as a set of isolated procedures to be memorized only .
In general students with deep understanding of mathematical concepts, objects and procedures are less likely to have major difficulties in learning and understanding new topics in mathematics. In fact they are well prepared to develop and understand new topics. They also enjoy their mathematics classes and are intrinsically motivated . Also students abilities in mathematical problem solving, which is the heart of any math curriculum, depends on a deep understanding of mathematical concepts and the ability to apply them in different situations not necessarily seen before .
Learning and deep understanding of topics in mathematics involve processes in which students connect to and build on knowledge acquired in the past . Making connections between prior knowledge and new information to construct new knowledge is an indication of learning with deep understanding . We therefore need to design and develop classroom activities in which prior knowledge is activated in order to gain deep understanding, learn and develop new mathematical concepts .
One of the most important concepts in mathematics is the concept of functions. Acquiring a deep understanding of functions is an important facet of mathematical thinking in that it leads to better problem solving and understanding other mathematical concepts. Functions may be represented algebraically, numerically and graphically and the linkage between these different representations can provide the learners with a deep understanding of functions .
Students using graphing calculators are better able to relate graphs to their equations, to understand the characteristics of functions and find algebraic representations for graphs . It was also found that the use of graphing calculators change the classroom environment in that students become more active with more group work, investigation and exploration .
We, at the math unit in UGRU, acquired Autograph software version 3 in spring 2009 and have already started using it in our math classes. We also organized several professional development sessions in fall 2009 and intend to continue with these sessions in the future. Autograph has strong capabilities in visualizing and animating math objects and may therefore be used to explore math concepts more deeply. It is a highly interactive software that can be used to create students centered environments. In this paper, I will discuss ideas on how to use the graphing calculator Autograph to design and develop lessons in order to create student centered environments, where students build new knowledge using prior knowledge and more information. In these activities students are actively involved in building their own knowledge. In this paper, I will discuss situations where autograph is used to help students
- visualize a mathematical concept for deep understanding,
- explore new concepts and definitions,
- animate a mathematical objects for more understanding,
- explore mathematical functions graphically,
- explore mathematical problems and solve them graphically,
- model a situation and solve the problem associated with it,
- and explore mathematical concepts and objects for deeper understanding.