Involving Technology in the Teaching and Learning Mathematics

This study proposes a way to involve technology in the area of teaching mathematics. The technology tool is Geogebra4, which is computer software in the teaching and learning mathematics. This technology emphasizes on the use of multiple representations of mathematical concepts by computer software. The objective is to make students consider the representation of mathematical concepts and help them to enjoy studying mathematics. From that thought, hopefully student understanding will improve and their mathematical achievement will increase. The result of this study is five lesson plans for teaching and learning integral using Geogebra


PREFACE
Most teachers teach mathematics without representation. As a result, students have difficulties visualizing many concepts. The teachers just teach our students with the formula and symbol-letter, and then the students try to solve problems with the formula, without having knowledge of the visualization of the function or solution look likes. It is a little bit weird because students just remembering the formula without knowing what are the curves or representation look like.
For example, when teachers teach integral concept about area between two curves; teachers just give students the formula that is , where is above . For instance, teacher gives question abouthow to calculate the area between two functions, and , from 0 to 1. To solve this problem, the students in my country take minutes to draw picture of these two functions. For example, the picture is like this: After drawing the picture, students begin to calculate the area, by solving the integral equation like this: Of course, for some students it is easy to draw the picture and to solve the integral equation. However, for the other students, it is hard to draw picture and to solve the integral, moreover when the functions are not simple like on the above example.
Technology (GeoGebra 4 ) can help students get more understanding about this problem; that is how to make representation to calculate integral: area between two curves. My opinion is based on Bruner's insights on the role of representation have greatly influenced mathematics educator. Most of mathematics educators state that mathematical idea can be represented in three ways: enactively (concrete representation), iconically (pictorial representation), and symbolically (written symbols) (Bruner, 1960). In this context, technology (GeoGebra 4 ) plays role as iconically or pictorial representation.
The purpose of this project is to involve technology in the mathematics classroom. I do believe that mathematics must be taught in a joyful learning environment with multiple representations, including representation from technology. With this project, I want to help students to enjoy studying mathematics with technology, because nowadays, technology is the focus of their attention. I also want to make mathematics become less abstract with the representation from the technology (Geogebra 4 ). The result of this project is five lesson plans for teaching integral with Geogebra 4 .

LITERATURE REVIEW
Most of the mathematics teachers just teach mathematics in the level of theory or concept. Students then have perception that mathematics is an abstract subject matter. The teachers rarely use representation or make connection between mathematics and the real life. From that historical background, I have a dream to change "the theoretical teaching style" in my country. In my teaching philosophy, I do believe that mathematics must be taught with realistic representation, involving technology, and trying to make students enjoy when they learn mathematics My paper is based on Bruner's insights on the role of representation have greatly influenced mathematics educator. Most of mathematics educators state that mathematical idea can be represented in three ways: enactively (concrete representation), iconically (pictorial representation), and symbolically (written symbols) (Bruner, 1960). In this article, technology (GeoGebra 4 ) plays role as iconically or pictorial representation.
In the Prepare and Inspire (President's  Council  of  Advisors on Science and Technology,  2010), there is a belief that technology has the potential to transform K-12 education, just as it has many other sectors of the US and global economy and our society. It can enable real-time and meaningful data gathering that allow learning and innovation in the education system. It can power innovative learning tools that prepare and inspire students. Furthermore, the report states thatone of the most powerful tools to propel innovation in education is computation and information technology.
The report also explains that technology supports innovation in three fundamental ways: (1) continuous evaluation and improvement based on data, (2) rapid and inexpensive dissemination of successful solutions, and (3) mass customization. The report also notes some important points: (1) educational technology has been advancing rapidly in recent years and is likely to create major strides in the near future; (2) there will be a growing need for new instructional materials, new professional development materials, and new kinds of assessments that are aligned with higher standards and provide much richer learning experiences and more vibrant sources of information; (3) the ''collection and use of data'' is one of the U.S. Department of Education's four assurances; (4) technology is becoming increasingly affordable, accessible, and versatile-a trend that will continue over the next decades, and will encompass personal and mobile devices; and (5) today's students are increasingly digital natives.
Many research findings conclude that technology is a great instrument to enhance mathematics teaching and learning process. Hatfield and Kieran (1972) explain that that was believe that "the activity of writing, processing, and studying the output of computer algorithms should promote the development of mathematical concepts and principles, computational skills, and problem-solving abilities of the students". Ellington (2003) also states that when calculators were included in instruction, the ability to select the appropriate problem solving strategies improved for the participating student. Furthermore, she states that students who used calculators while learning mathematics reported more positive attitudes towards mathematics than their non-calculator using counterparts on survey taken at the end of calculator treatment.
From another research finding using computer-intensive algebra (CIA), O'Callaghan (1998) found that the CIA students achieved a better overall understanding of function and were better at the component of modeling, interpreting, and translating. Moreover, CIA students showed significant improvements in their attitudes toward mathematics, were less anxious about mathematics, and rate their classes as more interesting.
From the other research findings, Kaput, Hegedus, and Lesh (2007) state that technology become infrastructure in mathematics education.
They explain that technology is a fundamental yet invisible role similar to the electricity in our homes. Moreover, technology will lead to emphasize to new level and types of ideas and abilities, as well as new ways to think about traditional concepts and skills. In the school, technology will facilitate new type of social interaction and thinking, and new way to make mathematics less abstract and more accessible to a wider population of students. The authors said that to realize the potential of technology, new type of pedagogical diversification will be needed, and of course teacher development must be done. Kaput et al. (2007) also show their result of classroom connectivity (CC) (i.e., classroom that involves technology); there are significant improvements in low-achieving students' abilities to solve standardize and applied problems. They also state that there are significant shifts in participation structures from non-CC to CC context. They also explain that the use of representationally rich software in mathematics education calls for a reconceptualization of both traditional and applied mathematics concepts. They also see distinct differences in fundamental process such as posture and gesture as well as discourse, teachers using CC more positive and effectual in the classroom. They state that connectivity support pedagogical manipulation of students' focus of attention.

TECHNOLOGY OVERVIEW
Geogebra 4 is an application for exploring and demonstrating Geometry and Algebra. It is an open source application and is freely available for non-commercial use. There are currently versions available for Windows, Mac OS X, Linux and other java-enabled platforms. To start Geogebra 4 go to http://www.geogebra.org where we will see links to Web start or Download. The Web start option downloads the necessary java files to our computer and starts the application immediately. The advantage of choosing this option is that the application is always up to date.
The Download option downloads files to our computer and we must then install. The big advantage here is that we can continue to work offline.
The installation process is very straightforward. After we have downloaded on a Windows machine just double-click the downloaded file. An Install Wizard will guide us through every step. It is strongly advised that we select the typical configuration when given the choice. Full instructions are given on the Geogebra4 site.
Double click the Geogebra 4 icon on the desktop to start the application. We will be presented with a launch screen as shown in Figure 1. The button menu along the top (see Figure 2) contains a submenu of actions. By clicking on the downpointing arrow at the bottom right corner of any of these buttons the submenu is displayed.

LESSON PLANS OVERVIEW
On the next pages, five lesson plans about teaching integral using Geogebra 4 will be explained. The first lesson plan is on teaching lower sum concept with representation from Geogebra 4 . In this lesson, students will investigate the properties of lower sum as a basic concept to understand Riemann integral. Students also will construct a conjecture and then they will try to analyze their conjecture by Geogebra 4 .
The second lesson plan is teaching upper sum concept with representation from Geogebra 4 . In this lesson, students will investigate the properties of upper sum as a basic concept to understand Riemann integral. Students also will construct a conjecture about the relation between the number of rectangles and the value of the upper sum. Additionally, students will construct a conjecture and then they will try to analyze their conjecture by Geogebra 4 .
The third lesson plan is an investigation of the phenomenonwhen the number of rectangles goes to infinity. In this lesson, students will prove their conjecture in the previous lesson (the second lesson). They will construct the lower sum and the upper sum with a large enough number of rectangles, and then they will analyze whether their previous conjecture in the second lesson is true or not.
The fourth lesson plan is on teaching definite integral (Riemann Integral). In this lesson, students will investigate the properties of definite integral. Students also will construct a conjecture about the relation between the value of the definite integral (positive or negative) and the position of the area under the curves. Furthermore, students will construct a conjecture and then they will try to analyze their conjecture by Geogebra 4 .
The fifth lesson plan is on teaching area between two curves with representation from Geogebra 4 . In this lesson, students will investigate the properties of area between two curves. Students also will construct a conjecture about the relation between the value of area between two curves and the position of the function f and g. Moreover, students will construct a conjecture and then they will try to analyze their conjecture by Geogebra 4 .

LESSON PLANS
Lesson Plan 1

Investigation: Lower Sum
Lower Sum is ∑ ∆ , where is x-value at which f(x) attains a minimum on interval , . We can make Lower Sum representation in the Geogebra 4 software.

Sketch
Teacher have to makes sure that students have a representation as shown in the student worksheet. Teacher should moves to each group to see whether they have an intended representation.

Investigate/Conjecture
To increase the number of rectangles, create the slider n to go from 8 to 100 in steps of 0.1 by clicking the . Then type this command: Lower Sum [f, -3, -1, n] or select it from the drop down list in the Input Bar, and press enter.
To make the difference between the value of Lower Sum clear, create text block a: Lower Sum with n = 8 rectangles by clicking and create text block b: Lower Sum with n rectangles by clicking .
The illustration is below: Leading students to conjecture:

When the number of rectangles increases, then the value of Lower Sum also increases
Introduce the term infinity after students have made the conjecture. It is an important concept for understanding Riemann integral in the next lesson.

Explore More
Lead students to a number of rectangles that makes the value of Lower Sum not change significantly. For example, n = 1000, this will illustrate the next lesson about the definition of the definite integral (Riemann Integral), which is the value of Lower Sum and the value of Upper Sum are equal when n goes to the infinity.

Lesson Plan 2 Investigation: Upper Sum
Upper Sum is ∑ ∆ , where is x-value at which f(x) attains a maximum on interval , . We can make Upper Sum representation in the Geogebra 4 software.

Sketch
Teacher have to makes sure that students have a representation as shown in the student worksheet. Teacher should moves to each group to see whether they have an intended representation, and help them if they have difficulties.

Investigate/Conjecture
As students construct their Upper Sum representation, make sure that they do these procedures: The illustration is below: Lead the students to make final conclusion about their conjecture, that is the definition of definite integral (Riemann Integral), which is the value of Lower Sum and the value of Upper Sum are equal when n and m (the number of rectangles) go to the infinity.

Lesson Plan 4 Investigation: Definite Integral
A definite integral is an integral with upper and lower limits. If x is restricted to lie on the real line, the definite integral is known as a Riemann Integral. We can make definite integral representation in the Geogebra 4 software.

Type the equation
in the Input Bar and press enter.

Present Your Findings
Discuss your results with your partner or group. To present your findings you could: 1. Show some set of data about the differences between definite integral that has area above x-axis and definite integral that has area below x-axis. For example, when the upper and lower limits are 1 and 3 (above x-axis), the definite integral is________, and when the upper and lower limits are -3 and -1 (above x-axis), the definite integral is _________.
2. From that data, then you explain your conjecture about the definite integral that has an area above xaxis and the definite integral that has an area below x-axis to your group members.

Explore More
See if

Sketch
Teacher have to makes sure that students have a representation as shown in the student worksheet. Teacher should moves to each group to see whether they have an intended representation, and help them if they have difficulties.

Investigate/Conjecture
To make a representation of the definite integral that has upper and lower limit -3 and -1, you can follow this procedure: 1. Type the equation n the Input Bar and press enter.
2. Type this command: Integral [f, -3, -1] or select it from the drop down list in the Input Bar, and press enter.
The illustration is below: Leading students to the conjecture:

Definite integral that has area above x-axis is positive and definite integral that has area below x-axisis negative
Teacher should use Geogebra 4 to show some example of representations to make sense this conjecture.

Explore More
Leading students to the conclusion that: the definite integral that has area below x-axis is negative because in the whole interval [-3,-1]. Remember the definition of the Riemann Sum: ∑ ∆ as the basic concept of the definite integral. Therefore, if for all ∈ , then Riemann Sum is a negative number.

Lesson Plan 5 Investigation: Area Between Two Curves
The method for determining the area between two curves is an important application of integral calculus. It lets us determine the area of nonstandard shapes by evaluating the definite integral. You will learn and investigate that in this lesson.

Present Your Findings
Discuss your results with your partner or group. To present your findings you could: 1. Show some set of data about the relation between the area of function f and g . For example, when you type f first, and then you type g, the area is________, and when you type g first, and then you type f, the area is _________.

Sketch
Teacher have to makes sure that students have a representation as shown in the student worksheet. Teacher should moves to each group to see whether they have an intended representation, and help them if they have difficulties.

Investigate/Conjecture
As students manipulate their integral: area between two curves representation, you should note that if we type: Integral [f(x), g(x), x(A), x(B)], the area must be a negative number. The reason is the area under f function is less than the area under g function on that interval. So, if you type: Integral [f(x), g(x), x(A) , x(B)], it means you subtract the area under function gfrom the area under function f, and it must be a negative number.
To make this representation clear, you can do these procedures and show it to the students: This representation shows that the area under g, which is b = 8.49, and the area under f, which is a = 4.71. Therefore, if you subtract the area under function gfrom the area under function f, the result must be -3.78.

Leading students to the conjecture:
When the position of g is above f in the coordinate plane, we must type g first, and then type f in the Input Bar (it means we subtract the area under function ffrom the area under function g), in order to get a positive number for the area between two curves.