Several procedures were followed to accomplish this research.
3.3.1 Diagnostic Test
A diagnostic test (Appendix A) was administered to both groups to assess their mathematical background and record their weaknesses and gaps. The test includes nine questions that cover the following topics: quadratic equations, domain of definition of functions, rate of change, trigonometry, lines, parabola, limits, and derivative. The test took place during the lab time of the course (once per week), and students were given 65 minutes to complete it. The researcher was present to make sure that everything was clear and to answer any technical questions not related to the solutions.
3.3.2 Two calculus books (Book 1 and Book 2)
Book 1, which emphasizes the symbolic approach of the derivative concept, was used in the control group (fall 2013). In the experimental group (spring 2014), Book 2 was used which, emphasizes the multiple- representation of the concept. A general comparison of the development of the derivative concept in the two books (Book 1 and
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Book 2) is provided. Then, the content and the structure of each of the four selected sections on derivative, in the two books, are discussed. Finally, the modes of
representation (symbolic, graphical, and numerical) used in Book1 are discussed and compared to those used in Book 2. Section 4.1 provides detailed analysis of the books.
3.3.3 Derivative Questionnaire
A questionnaire on “Students’ perceptions of the notion of derivatives”
(Appendix B) was administered to all students before the implementation of the unit on derivatives. The questionnaire was developed by the researcher, and it consists of three parts (I, II & III). The first part (I) is an open-ended question that asks students to freely write what they know about derivative. Parts (II) and (III) are a multiple – choice items that ask students to choose the correct statements. Students took 15 minutes to complete the questionnaire.
The questionnaire aims to record students’ dominant image of the concept of derivative. In addition, it aims to examine students’ conceptions related to derivative. The questionnaire was administered again, immediately after the implementation of the unit on derivative to check the development and the progress of students’ conceptions of derivative. Data are analyzed both qualitatively, and then quantitatively ana lyzed using the test. The null hypothesis cla ims that there is no significant difference in the means between the two groups.
3.3.4 Implementation of the derivative unit
In the control group (fall 2013), four sections on derivative are selected from Book 1, which address the concept derivative using different modes of representation.
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The three modes of representation include symbolic (formal definition of derivative using the limit of quotient of differences and denoted by,
– ), geometrical (as the slope of a curve or the slope of a tangent line), and numerical (as the instantaneous rate of change). The emphasis, however, is on the symbolic approach (using algebraic expressions, equations and the formal definition of derivative).The implementation of the unit was carried out over five sessions of work (50 minutes each). Homework was usually assigned at the end of each teaching session. The homework exercises and problems are selected from Book 1. It is worth noting that there is no group work nor is there any use of technology. The lecture method was frequently used; the teacher explains the lessons, students take notes and answer questions when needed. Section 4.2.1 describes the flow and the content of sessions addressed by instructor X during the teaching of derivative.
On the other side, four sections on derivative are selected from Book 2, which also address the derivative concept using different modes of representations. The three modes of representations are equitably used, with more emphasis on the graphical representation. The four selected sections in Book 2 cover almost the same objectives as those sections selected in Book 1. The implementation of the unit was carried out over 6 sessions and a half of work (50 minutes each) because group work and technology were used, which required more time for technical support. The researcher with the
cooperation of instructor Y designed activities (Appendix F) that encourage the
exploration of the derivative through different representations (tables, graphs, formulas, word problems). Moreover, the activities were designed to guide students to promote the formation of the mental structures (Action- Process- Object- Schema) as described in the
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genetic decomposition of the derivative concept (see section 3.3.7). It is worth noting that the teaching sessions were interactive where students felt free to ask questions, listen to others and express their ideas. During the activity part, the teacher encouraged discussions among group members and sharing results before having class discussion and presentation. The activities were then followed by homework exercises that were similar to the activities. Section 4.2.1 describes the flow and content of sessions addressed by instructor Y during the teaching of derivative. It is worth mentioning that the pedagogy used in the experimental group (cooperative learning, technology and activities) was only done in the derivative chapter. The approach was traditiona l in the earlier chapters (functions, limits, continuity).
It is important to note that the lack of activities in the control group is due to several factors. First, Book 1 seems to follow a traditional approach where the
definitions of the concepts are directly stated at the beginning of the Text without giving students the opportunities to discover and be active learners. Moreover, in Book 1, the dominant mode used in the given part of the exercises is the symbolic representation (using algebraic equations and formulas) which are used as tools to find the equation of the derivative function, slope of a curve at a given point or slope of a tangent line at some point and others. However, the graphical and numerical representations are used mostly as an end and not as teaching tools.
3.3.5 Observations
During the implementation of the unit on derivatives, the researcher is present in both groups (control and experimental) to observe the instructional method/s and
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strategies ( lecture, group work, technology, individual work) used in the classroom as well as students’ participation and work, using an observation work log (Appendix C). In addition, a major attention is directed toward the different representations of the concept of derivative that are emphasized or de-emphasized in the two groups. The researcher took detailed notes about how each lesson was conducted. Moreover, at the end of each teaching session, notes were taken about students' opinions and attitudes toward the teaching approach used.
3.3.6 De rivative Test
After the implementation of the unit on derivatives, a common written test (Appendix D) was adminstered to all students. It includes five conceptual –
understanding based problems on derivatives , constructed keeping in mind the APOS levels. Four questions out of five are developed by the researcher, and one question (question I) was adopted from a study by Asiala et al. (1997). In this test, students were asked to justify their answers and reasoning and were given 65 minutes to complete it. The main characteristic of the test is that only graphs and table of values of the functions are given without their algebraic expressions. The purpose of the test is to identify the types of difficulties that students face in relation to the notion of derivative. In addition, it aims to assess students’ ability to translate among the different representations of derivatives. The researcher was present to make sure that everything was clear and to answer any technical questions not related to the solutions of the questions. Students' responses on the test are qualitatively analyzed based the apriori analysis provided in section 4.4.1. Moreover, a quantitative analysis using descriptive and t-test is applied to check whether the difference in the mean scores between the two groups is significant.
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The null hypothesis claims that there is no significant difference in the performance between the two groups.
3.3.7 Interviews
Interviews are conducted with 12 students while solving the derivative test. The purpose of the interview is to obtain a clear, explicit and better picture of students’ conceptual understanding of derivative. Students were asked to think aloud while solving the test. Moreover, the interviewees were asked about their educational
background, their opinions toward math, their opinions toward the teaching approaches used in their classes, and finally about the meaning of the derivative. The interviewed students include five males and seven females who have different levels of
achievements. Even though the researcher explained to the interviewees the purpose of the interview, and assured them both confidentiality and anonymity, she could not audiotape nor videotape the interviews upon their request.