Graphing calculators can do so much more than the simple arithmetic of calculators of old. These amazing technological inventions enable teachers to enhance, develop, and enrich learning experiences in mathematics. Research has shown that using the graphing calculator can change the environment in which students learn (Berger, 1998). Some ways graphing calculators have changed the face of mathematics today are: as tools for greater student motivation, as tools for greater student empowerment, as tools for expediency, as amplifiers for conceptual understanding, as vehicles for multiple representations, and as catalysts for critical thinking.
The use of graphing calculators has revolutionized the way students learn mathematics and the way teachers teach mathematics. The graphing calculator allows teachers the opportunity to explore mathematical concepts in a manner that facilitates an intricate understanding of mathematical concepts. For example, an algebra class studying graphs can explore many different graphs quickly, simultaneously in fact, with the use of the graphing calculator. These multiple representations seen with such expediency could not be achieved prior to the use of the graphing calculator. Before the graphing calculator, some graphs may have required an entire class period to examine. During these explorations, concepts such as slope and the y-intercept can be explored and create a greater understanding of how these two concepts affect the graphical appearance of an equation in slope-intercept form. The graphing calculator expedites this exploration and many others, and in many cases, allows the student to make connections quickly that might otherwise have been lost.
Again the use of multiple representations can be used to study functions, a mathematical topic usually quite difficult for a student to understand conceptually and visually. Students can solve problems and explore functions that might not be understood so fully when manipulated by traditional pencil and paper techniques alone. Graphs of functions can be explored. The maximums, minimums, zeros, points of intersections, and other mathematics can be calculated quickly when working with functions and then represented numerically as well as analytically. Each of these representations emphasizes and suppresses various aspects of the concept so that students can gain a more thorough understanding of a function (Piez & Voxman 1997). Hollar and Norwood (1999) studied the effects of a graphing approach in the study of functions and found that the graphing-approach class demonstrated significantly better understanding of functions than did students in the traditional-approach class. This research is fundamental to mathematics education since the function concept is considered to be one of the most central concepts in all mathematics. Traditionally, it is also one in which students rarely develop adequate understanding. This can now be changed through the use and integration of the graphing calculator in the mathematics classroom.
The many varied functions of the graphing calculator can free a student from the burdens of complex computations. The graphing calculator has spreadsheet and table capabilities. The graphing calculator can perform recursions. Programs can be written to perform tasks on a calculator. Regression analysis can be used for mathematical modeling using complex statistical methods to allow the students to better understand the connections between real-life statistical generation and the formulation of equations or functions to define and explore these real-life statistics. All of these same functions can be done on a computer but who can argue the ease, accessibility and portability the smaller graphing calculator gives versus a computer. While a computer may be easier for a student to navigate at first, the graphing calculator has the advantage of being portable and affordable. Graphing calculators can be purchased for less than $100. This makes them ideal for student and school systems with limited funds, and teachers with limited access to computer laboratories.
The graphing calculator allows teachers to explore topics in more detail and allows students the ability to attempt more difficult problems. For example, non-integer zeros of a polynomial can be approximated with a graphing calculator using a graph or using a table. The table feature can be used to solve problems using an iterative type process. The students can approach a concept using different methods. In a study of 143 teachers in a large northeaster city, Milou (1999) found that 81.2% of the teachers surveyed agreed that the calculator allowed the students to solve more difficult problems and understand concepts better. Brown (1999) used the TI-92 graphing calculator with dynamic software in a geometry project and stated that the technology moved student understanding from the concrete experience to abstract mathematical ideas. Graphing calculators when used effectively will enhance student’s abilities in mathematical visualization, problem solving, exploring, formulating and testing of conjectures, and assessing and verify these conjectures. The graphing calculator affords teachers and students an opportunity to generate data and then formulate and test conjectures, giving students the chance to do mathematics as mathematicians do. As an example, the Fibonacci sequence can be used along with the table and list features of the graphing calculator to discover the various numerical patterns found within this sequence (Dion 1995).
The most compelling implication for instruction for the graphing calculator is the power of the calculator to give students an active problem-solving tool. Students can be engaged in doing mathematics as opposed to passive recipients of mathematical concepts and procedures that must be regurgitated on tests and quizzes every few weeks. The use of the graphing calculator allows teachers to respond to student’s questions that inspire further exploration. The interactivity of the calculator in giving immediate feedback along with the greater control experienced within the learning environment by both the teacher and the student enable the calculator to be called a problem-solving tool (Kaput 1994).
The calculator is powerful in promoting divergent thinking where a problem may have several correct answers using various approaches. This is found in the many problem-based learning units centered on using the graphing calculator. An almost incidental benefit of these graphing calculator explorations of the problem is found in the many false starts that occur when investigating a problem. These false starts often lead the student back to basic mathematical principles, which are then reinforced. Many of the students will sometimes conduct further investigations into topics included in the curriculum during their explorations. This investigation into multiple what-if scenarios allows the students to discover which properties of mathematics vary and which are invariant. Students perform extensive student-centered exploration with hands-on interactive technology giving immediate visual feedback. These explorations reveal the power of graphing calculator technology. Students are experiencing critical thinking and analysis while being reinforced in basic mathematical concepts. Real life problems and investigations that yield multiple solutions lend a sense of reality and ownership to the results students eventually propose. This innovative use of the graphing calculator has shown greater motivation in the student and given the student a sense of empowerment in their learning (Alper, Fendel, Fraser, & Resek 1996)
In conclusion this technology puts the necessary tools in the hands of the students to discover basic concepts, rules and patterns for themselves, to explore open-ended problems, and to make real world applications accessible in the classroom. The applications and implications of hand-held graphing calculators are reshaping methods for teaching mathematics. Graphing calculators now provide students with the opportunity to interact visually with mathematics in ways never experienced before in their education. Using the graphing calculator diversifies the typical approach by supplementing teaching in a way to make the classroom more exciting. With the instructor using the overhead panel projection system in the classroom, what was once a passive atmosphere is now a more active one.
Using the graphing calculator, mathematics is made more exciting to the students by getting them personally involved in experimentation and discovery. With the power of the graphing calculator students are allowed to actually see what they could never before even imagine. With the help of the graphing calculator, students are able to focus on achieving an understanding of the concept of functions, to develop competency in problem solving, to acquire a more confident understanding of graphs and other quantitative materials that they will encounter in their daily lives. Using the graphing calculator as a tool for mathematics exploration, allows students to strengthen their understanding of the relationship between graphs and symbolic forms. The visualization aspects of the graphing calculator enables students to fit graphs of functions to pictures and real-world situations.
Technology has frequently been viewed as a widely useful asset to education. It is only in recent years that concerted efforts have been undertaken to mandate the inclusion of technology in educational settings. Proponents of technology generally contend that technology should not replace the learning of the basic concepts but supplement the curriculum to encourage deeper and more substantial explorations into the mathematics concepts. The research on technology in education presents evidence that the infusion of technology can prepare students for the entrance into an increasing technological workforce. Research on graphing calculators presents opportunities to involve more students learning through technology and for students to identify with mathematics. Each year there has been a growing recognition among educators of the important role technology can play to enhance many aspects of the teaching and learning of mathematics.
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