Based on the proposed model (see section 4.2.2 and figure 32), an intervention was designed. The intervention includes arranging the teaching-learning environment according to the proposed criteria and working on sets of activities. The activities aimed to encourage attaining the constructs of conceptual knowledge specified in the proposed model and to lift students‟ knowledge to a higher-level aspect of mathematical thinking which in turn reduces observed difficulties and enhances conceptual knowledge. The term “activity” refers to an open-ended or closed-ended item of a classroom, homework and formative assessment tasks, which the students are asked to work on either on their own or in a group at the end of the teachers‟ conventional introduction of each concept. The activities are compiled together and quoted as an “activity sheet”.
The activities are designed for these concepts- limit of sequences, the limit of functions, continuity, and derivatives. Most of the activities were selected from
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previous study instruments, national exams, and books and some of them were designed by the researcher. Of course, even for those taken from the literature, all of them were modified to fit the intended purpose.
The purpose of the activities was addressing observed difficulties, so that students enhance their conceptual knowledge. The items were collected based on the required constructs of conceptual knowledge and content of grade 12 mathematics syllabuses. With regard to the type of items, the activities consist of both open-ended and closed-ended. But the closed-ended items also ask not only selecting the correct answer, but justification why a certain alternative is selected. The items also include scripts from students‟ work. This is deliberately done so that students exercise how to “analyse errors” and think of their own thinking.
A month before the intervention, three-day training was provided to 21 selected upper secondary school mathematics teachers by the researcher, in collaboration with the researchers‟ employ University and the zone education department. There were 21 participants (18 males and three females). In the training entitled, “error analysis: a tool to enhance students conceptual knowledge”, issues like assessment practice, common student errors, feedback as a pedagogical tool, constructs of conceptual knowledge and mathematical thinking practice, was presented. The experimental group teacher was part of the training. Besides the training, an individual orientation and subsequent discussions were conducted with the teacher, so that the intervention was implemented as intended.
The intervention was administered for eight weeks, 80 minutes per week running parallel to the normal teaching-learning program. In the intervention session, students were arranged in mixed ability groups of five to six. After the first, the sessions were arranged as group work, presentation, reflection on the presentation and stabilization, group discussion and homework for the next class meeting. A week after the intervention was terminated; the post-test that aimed to examine students‟ conceptual knowledge in calculus was administered.
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3.4.1. Expert validation and pilot test of the items in the activity sheet
Initially, 35 activities were selected. Informed by the comments from a panel of experts and pilot tests, the qualities of the items in the activities were improved. First, a pilot test of the items was conducted with incoming first-year mathematics department students at a University. Twenty-eight students (12 males and 14 females) participated. The purpose of the pilot test had been to obtain feedback about the items before they were used in the study. During the pilot, the researcher observed students doing the activities to assess the quality of the items in the activity sheet. The researcher‟s observation was focused on whether the activities are appropriate for the intended method of instruction (individual work, group work, qualitative description, quantitative description), encouraging or not, helped to construct the intended components of conceptual knowledge (interiorization, encapsulation, and coordination), and whether the language of the items and the instructions of the activities are clear and understandable. Based on the experience gained adjustment was made on the time frame, work-load and level of difficulties on each item.
Besides the pilot test, the judgement of experts was also implemented to improve the quality of the activities. A panel of four experts- two grade 12 mathematics teachers, who have extensive experience teaching calculus and two university mathematics lecturers (who have masters in mathematics education) have participated. Based on the feedback collected, some of the activities were modified, some of them were removed and some new activities were added.
Finally, 30 activities were selected for final administration (see appendix H). In the development of the activity sheet, different sources were used. Although the present description of some items may not be the same as to the description in the sources, the beginning sources are the following: Areaya & Sidelil (2012); Bezuidenhout (2001); IER and AAU10 (2013, 2014, 2015, & 2016); Jordaan (2005); Maharaj (2010); Moru (2006); Rabadi (2015); Wangle (2013).
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The purpose of activity one to five is to establish students‟ conceptual knowledge in the limit of sequences. The activities were focused on overgeneralization, conflicting concept image due to linguistic ambiguity, knowledge of the definition of terms, and the relations and conditions among these terms. Activity six to 19 aims to address the difficulties in limits and continuity. Most items in this section will demand object-level concept formation, reconstructive generalization, and reasoning level problem-solving skills. Activity 20 to 30 is intended to address difficulties in the derivative. In this section too, most items demand object-level conception, reconstructive generalization, and multi-step reasoning level problems solving ability.