Many researchers (Zimmermann 1991; Arcavi, 2003; Zarzycki, 2004, Rolka & Rosken, 2006; Presmeg, 2006; Ubuz 2007; & Natsheh &Karsenty; 2014) have
investigated the role of visualization in mathematics as well as the strengths and difficulties associated
with it.
Arcavi (2003) defined visualization as follows:Visualization is the ability, the process and the product of creation,
interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about, developing previously unknown ideas, and advancing understandings. (p. 217).
This definition emphasizes the powerful role that visualization has in teaching and learning mathematics. Visualization is not just seeing pictures, graphs or any other visual tool, but it gives meaning, depth understanding, helps in problem solving and encourages discoveries. Giaquinto (2007) argues that visualization's role goes beyond just being a supportive role (demonstrating cases or providing examples for a definition etc), but it provides a mean for discovery, understanding and can be seen as a proof itself.
21 2.6.1 Difficulties around visualization
It has been said that "a picture worth of thousands of words." This proverb is true only if one is able to use it effectively. A student may be able to plot a graph or a draw a diagram, but he may not be able to extract information or use it to solve the problem. Interpreting graphs, tables, diagrams and other visual tools involves analytic and synthetic skills, intuition and sense of judgment. The difficulties around visualization and students' reluctance to use visual images have been widely investigated (Alcock and Simpson 2004; Arcavi, 2003; Guzman, 2002). Reasons behind this may be because many instructional and curriculum materials emphasize the symbolic approach over the visual one, and visual thinking demands higher cognitive thinking than the algorithmic one (Dreyfus and Eisenberg, 1991). In addition, many students and even teachers believe that a visual proof is not accepted or not considered a "real" proof. Another reason is that many students have weak visualization skills, or are not visualizers as defined by Presmeg (1986), "individuals who use visual methods in solving mathematical problems when there is a choice "(p. 298).
Since visualizing a mathematica l concept is not an easy task, teachers should provide students with enough preparation, training and time. Students should be taught how to read, interpret any visual tool (graphs, tables, etc), extract information and translate it into other forms of representation (verbal using words, symbolic using equations etc). A student should have a repertoire of images , pictures, properties,
relations, and different forms of representation associated with the concept as a musician who has a repertoire of melodies.
22 2.6.2 Technology and dynamic software
The calculus reform movement since 1980's has encouraged the integration of technology in the teaching and learning mathematics. Graphic calculators, computer programs and dynamic mathematical software (GeoGebra, Maple, Cabri, Geometer's
Sketchpad, Autograph and others) have been used for teaching and learning many math
concepts such as: polygons, triangles, quadratic equations, functions, limits, derivatives, integrals and others. The dynamic math programs allow users to create objects (points, lines, graphs etc), make measurements (angles, areas, slopes of lines etc), do
transformations (rotation, reflections, symmetry, enlargement etc) and perform other manipulation of the selected or constructed object. In addition, the 'dragging' and
'animation' features of most dynamic programs provide students with an environment for discovery, experimentation, seeing patterns, generating and testing conjectures and visualizing mathematical objects (Gonzalez & Rodriguez, 2011; and Herceg, 2010). Moreover, most of these dynamic programs allow the user to visualize the concept in many representations (algebraic, numeric, and graphical). For example, a student can create a table of values, enter an equation and draw the graph of any function. Dynamic representations of mathematical objects allow learners to visualize mathematical
problems or processes in ways that are not possible using paper and pencil (Sacristán et al., 2010). For example, the use of both the 'slope function' and 'slow plot' buttons, in
Autograph, allow the user to plot gradually the derivative of the selected graph or
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2.6.3 Studies on the effects of technology on students' learning
Several studies have been conducted on the impact of technology (internet, simulation ion of the games, graphic calculators and dynamic programs etc) on students' math achievement, understanding and attitudes.
In their study, Simonsen and Dick (1997) found that graphing calculators enhanced students’ conceptual understanding, and allowed them to be active learners and autonomous who construct their knowledge. Students were able to visualize the problems using multiple representations. In contrast, Porzio (1999) has shown that students who used graphic calculators did not show a better understanding of the concept. Heid (1984) conducted a study on two groups of college calculus students to investigate their understanding of functions, limits, integrals and derivatives. One group used dynamic computer software in the course while the other group did not. Heid collected her data based on the interviews she conducted with students in addition to their class work, assignments, quizzes and tests. Heid asked students to explain the meaning of derivative and interpret real world problems that include derivatives, and others. In general, the results showed that students in the experimental section held rich conceptual understanding, while students in the traditional section showed litt le and superficial understanding.
Habre (2006) investigated university students’ conceptual understanding of a function and its derivative in an experimental calculus course. Students in the course were given the opportunities to use both the graphic calculator and Autograph, a dynamic mathematical software. At the end of the study, Habre noticed that some students struggled and faced difficulties in the course, while many enjoyed the course
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and showed a good understanding of derivative particularly the ideas of derivative as the slope of the tangent line and as the rate of change. However, Habre noted that even with the instructor emphasis on the graphical/visual approach of derivative, several students preferred the symbolic approach (equations, formulas).
Further studies have been conducted on the effects of dynamic software on students' achievement. Naidoo (2007) developed an interactive and dynamic module for teaching derivative. A group of 33 engineering students was taught using an interactive software, while 30 students were taught using the traditional lecture method. Students were tested on the ideas of average rate of change, instantaneous rate of change, limit of sequence and some rules of differentiation. Some students were clinically interviewed while solving the math tasks. Students' scores on the test were significantly different in favor of the experimental group. The findings showed that students in the experimental exhibited deep understanding of the concepts while the control group had superficial understanding and exhibited more structural errors compared to the experimental group. In their study, Zulnaid and Zakaria (2012) examined the effects of using dynamic software, GeoGebra, on the procedural and conceptual knowledge of functions. A total of 124 high school students participated in the study where 60 students were in the experimental group and 64 were in the control group. The difference in the mean scores between the two groups was significantly different at p < 0.05 in favor of the
experimental group. Students who took lessons on functions using GeoGebra, showed better conceptual and procedural knowledge of the function concept, compared to the control group.
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