Introduction
Focus and purpose of the study
Proficiency in using the computer software was not the primary concern of this study. This also causes point S to move, creating a trace of the gradient function in the process.
Literature review
The computer in mathematics education
Borwein (2005, p.2), in his description of experimental mathematics, provides an example of where the computer can be used to support both forms of mathematical activities in the classroom. The role of the computer in the classroom is also considered by Salomon et al.
Physical versus cognitive mathematical tools
Both de Villiers (1999) and Mudaly (1998) agree that such quasi-empirical investigations can provide a basis for teaching evidence as explanation. The student must often make an effort to obtain feedback from interactions with the physical tool.
Virtual manipulatives
They will respond to the student's actions, and the immediate feedback gives the student an opportunity to assess the importance of each action. The student can use a compass to open holes while giving instructions, so with physical means the degree to which meaning flows is at the student's discretion (Zbiek et al., 2007).
What is Geogebra and what does it offer?
Furthermore, the open source nature allows teachers to create interactive online learning environments and to share them with other teachers worldwide (Hohenwarter et al., 2008). In the method of mathematical experiments, the teacher can provide incomplete interactive sketches, and students then use them to explore and rediscover mathematical concepts (Hohenwarter et al., 2008).
Understanding student involvement in computer learning environments
In accordance with the Zbiek et al. 2007) characterization, the students were engaged in an exploratory activity while completing the tasks. The purpose of a drag test therefore involves a search, not for results, but for confirmation (or disapproval) of some result (Zbiek et al., 2007).
Understanding teacher involvement in computer learning environments
For example, students in the experimental group showed a deeper understanding of the concept of derivative. When using a white box computer, students are actively aware of the operations performed by the computer program.
A brief history of the derivative
He argued that everything in the world changes as time passes, and this was crucial in his formulation of the laws of motion (Schechter, 2006). Furthermore, Newton's laws of motion and Hooke's law of elasticity were practical examples of the effectiveness of the derivative.
Contemporary issues in the teaching and learning of the derivative
In addition, Park argues that the colloquial use of the derivative exacerbates the students' problems when they try to think of it as a function. It is therefore appropriate that the concept of the derivative (gradient function) is introduced with the help of visual mediators such as graphs.
The role of inductive (plausible) reasoning in mathematics
The condition-property scheme involves understanding each analytic condition in relation to a graph property of the function and matching these conditions. The history of the derivative discussed earlier is a classic example of inductive reasoning.
Teaching conjecturing in the classroom
As Tinashe worked through the straight lines, he noticed that the coefficient of x becomes the equation of the gradient function. Is there a rule for finding the equation of the gradient function for a function of the form c.
Theoretical framework
Introduction
Constructivism is one such learning theory that has been at the forefront of educational reform, and has rich and significant implications for mathematics education (Lerman, 1989). Therefore, this chapter will discuss constructivism with a view to positioning this research within the current literature. It will culminate in a discussion about the Experiential Learning Theory (ETL), the theory that underlies this research.
Constructivism
According to Lerman (1989, p.211), constructivism is defined by a generally accepted hypothesis that knowledge is actively constructed by the knowing subject, and not passively received from the environment. This approach to teaching provides the teacher with the opportunity to look at learning from the child's perspective, an opportunity for the teacher to walk in the child's shoes (Olivier, 1989). One of the teacher's most important responsibilities is to facilitate deep cognitive restructuring and conceptual reorganization.”
An overview of Experiential Learning theory
It can be argued that tools such as Geogebra can be used to facilitate such cognitive restructuring.
Characteristics of Experiential Learning
When learning is viewed as a holistic process, it provides conceptual bridges between life situations such as school and work, portraying learning as a continuous, lifelong process. Experiential learning theory states that the quality of the learning experience depends on the characteristics of the individual and the learning environment. Social knowledge is more refined and is the result of the accumulation of previous human cultural experiences.
The Experiential Learning cycle
Experiential Learning theory in this study
In addition, asking the students to sketch a graph of the gradient function against the original required them to apply the knowledge they had acquired in the concrete experimentation stage. This is important because Kolb (1984) argues that for learning to be useful, the learner must be able to place learning in context.
Another voice in experiential learning
A supportive environment is important to ensure that learners have the time and space to explore individual cases towards an articulation of underlying similarity. In such an environment, students are free to discuss their guesses without fear of being judged and to modify their guesses as a result of subsequent discussions with other students (James, 1992). In addition, he claims that in the supportive environment students should be comfortable to ask for some equipment to model any particular case.
Conclusion
In the quadratic functions, the coefficient of the x² term was no longer the equation of the gradient function. Researcher: Try to relate your sketch to the gradient of the curve at different x values. Using the applet also helped students derive the shape of the gradient graph.
Research design and methodology
Introduction
The interaction between the research questions, the chosen learning theory and the research paradigm prescribes the overall design of a research study, especially the method(s) of data collection. Therefore, the focus of this chapter is to reveal the study's research design and methodology. This is followed by an overview of the research participants, including the research setting and the sampling technique used.
Qualitative research methodology
It starts with a general discussion of the qualitative research methodology, the interpretive paradigm and the action research approach as used in this study. As a result, the researcher in this study drew conclusions about how students used applets to arrive at the differentiation rule, to sketch graphs of functions, and to convince themselves that their rule will work every time. In essence, the interactive methods allow the researcher, through the use of interviews, conversations, field notes, recordings and photographs, to observe, interpret or make sense of the participants' engagement or response to a phenomenon under consideration in a natural setting, such as a typical mathematics. the classroom (Denzin & Lincoln, 2005).
Action research
- Limitations of action research
Action research is a small-scale intervention in the functioning of the real world and a careful examination of the effects of such intervention. The interpretive perspective is based on the premise that each person's way of making sense of the world is worthy of respect (Patton, 2002). This characterizes the interpretive paradigm: the researcher tries to make sense of the world from the participants' point of view.
The sample
Patton (2002) argues that a paradigm is a worldview, a way to break through the complexity of the world and that paradigms are deeply embedded in the socialization of adherents and practitioners. At the time of the study, all were pursuing Advanced Subsidiary (AS) level. Consequently, if the students in this study were unable to determine the equation of the trace, they might have found it impossible to discover the differentiation rule.
Research procedure
They were the school's top performing Year 10 students in the Cambridge International Examinations (CIE), written in the November/December 2013 session. In South African schools, Calculus topics are generally taught as an extension of the Dear students. It allowed me to easily schedule interviews with the boys and provided easy access to the school computer lab.
Data collection
- The task-based interview
- The electronic environment (Geogebra Applet)
- Observation
The process of pattern recognition began with the realization that the equation of the gradient function in slanted straight line graphs was equal to the coefficient of x. Tinashe: As I continue with the parabola, I've noticed so far that the power of x gives you the coefficient of x in the gradient function. Tinashe: When drawing the gradient graph with x³ I notice that it results in a parabola.
Data analysis and findings
Introduction
This is followed by an analysis of the findings for research questions 1 and 2 in section 5.2. To answer these questions, the students' answers to questions 1 to 4 (see Appendix 1) and the interview protocol (see Appendix) were used. 2). The answers to questions 5 and 6 (see Appendix 1) were the main sources of data in answering the above question.
Considerations in data analysis
In contrast, this study uses a priori coding, whereby the researcher comes up with the codes before examining the collected data (Nieuwenhuis, 2007, p. Testing the conjecture with numerous cases and plotting the results on Geogebra can convince the students that. interviews the students were pressed to articulate how they arrived at their rule and why they thought it would work every time.
Using Geogebra to empirically arrive at a conjecture
- Findings based on Section 5.2: Using Geogebra to empirically arrive at a conjecture
- Levels of conviction and need for an explanation
- Findings based on section 5.2.2: Levels of conviction and need for an explanation
As they went through the quadratic functions, all six boys were quick to realize that the exponent of the function graph became the coefficient of the x-term of the gradient function. A consideration of the other cases, exponents of two and three, resulted in all six boys generalizing that the exponent becomes the coefficient of the gradient function. The results of the reconsideration led them to observe that the exponent of the function graph becomes the new coefficient of the gradient function.
Does the use of dynamic graphing software such as Geogebra enhance conceptual
- Results and findings for question 5 (a)
- Results and findings for question 5 (b)
- Results and findings for question 6
Apparently he didn't seem to realize that the turning point meant a change in the sign of the gradient function, i.e. where the graph intersects the x-axis. He was able to correlate: (i) his differentiation rule which he derived to state that the resulting graph should be that of a quadratic function, (ii) the shape of the graph (concave upwards) in relation to the values of the gradient along the function graph, (iii) the x-intercepts and their relation to the turning points on the function graph and (iv) the position of the turning point. Researcher: Where would point S be positioned now, as it calculated the value of the gradient.
Findings based on Section 5.3: Does the use of dynamic graphing software such as
Limitations of the research study
The study also found that using the software could help students solve some non-routine graphing problems in calculus. The teacher can perform the drag exercise while the students calculate the equation of the trail. Action research as a research methodology for the study of science teaching and learning.
Discussion of findings
Introduction
Second, the presence of the tangent line attached to point A on the applet drew the students' attention to the fact that although the derivative is a function, it is also a point-specific object. The above proof relies on students' understanding of the binomial theorem and the limit concept. Below is the graph of the derivative (gradient function) ƒ'(x) of a function f(x) . Which choice a) to e) could be a graph of the function f(x).
Conclusion, recommendations and directions
Concluding remarks
This action research study aimed to contribute to the ongoing discourse on how to integrate computer technology into middle school secondary education. Specifically, he aimed to find out whether experimental use of the Geogebra computer software could enable students to discover the rule for differentiating elementary polynomials. Based on the findings in Chapter 5, this study tentatively draws a similar conclusion in that Geogebra can help students' conceptual understanding of the derivative and their ability to solve non-routine graph problems.
Recommendations for the teacher
An elementary proof, adapted from Finney and Thomas (1990, p.142), which makes use of the binomial theorem, is presented herein. The order in which these topics are taught is essential to ensure that the pupils understand the proof.
Directions for further research
Plenary address, in the Proceedings Program of the Eighteenth Annual Meeting of the Southern African Association for Research in Mathematics, Science and Technology Education. Journal for Research in Mathematics Education Combining dynamic geometry, algebra and calculus in the GeoGebra software system. Using the provided Geogebra applets, determine the equation of the path traced by point S in each function.
Task-based interview questions
Interview protocol for levels of conviction
The electronic environment (Geogebra applet)
