Abstract
The research interrogated the learning of introductory calculus at a particular Technical Vocational and Education Training Education (TVET). The major area of interest was the errors and misconceptions students displayed in basic differential calculus. It was apparent from the onset that the TVET college lecturers were in most cases aware of the errors and the misconceptions National Certificate (Vocational) (NC(V)) level 4 students displayed in introductory differential calculus. The crucial point was that the causes of the students’ errors and the misconceptions were not established. There was lack of error analysis methods and intervening strategies that would effectively address students’ deficiencies in the learning of introductory differential calculus. TVET college lecturers’ awareness of students’ level of pre-requisite knowledge is crucial in developing appropriate teaching and learning intervention strategies that supported effective learning outcomes. The researcher argues that research on TVET students’ errors and the misconceptions they exhibit in introductory calculus would provide critical information on the transition from one level of mathematics conception to the next. During the process of assimilation students take in new information and accommodate concepts in cognitive structures that are modified when new mathematical concepts are introduced. During the process of prior and new knowledge acquisition, misconception and errors associated with the concepts are exhibited. If the newly acquired mathematical knowledge is not properly accommodated, interference/ unstable behaviours occur. During this process, students may harbour concept images that are competing with established mathematical knowledge. The rationale is therefore that TVET college mathematics lecturers’ ability to identify and analyse these alternative conceptions and how students came to acquire them is a critical enabler for effective instruction and optimal learning outcomes in differential calculus. Apart from the error analysis framework, the researcher assumed that misconceptions held by students in mathematics, specifically differential calculus may be explained within the frameworks of procept (Tall, 1995), Action Process Object Schema (APOS) (Dubinsky, 1991), and Sfard’s (1991) structural and operational descriptions of mathematical notions. The frameworks assert that students come to understand mathematical concepts based on the prior knowledge. However, some of the knowledge they construct may appear truthful to them but conceptually untrue. This may occur through overgeneralization of prior knowledge, mislearning and inappropriate definitions of mathematical conceptions during new mathematical knowledge acquisition situations. Differential calculus is first introduced at NC(V) level 3 at TVET colleges. Despite students having learnt it at high school (Grades 10 - 12), students and lecturers have registered challenges in the learning and teaching of the various construct associated with the topic. It was therefore found compelling to conduct this study. The study focused on identifying, discussing and categorizing errors and the misconceptions responsible for the errors on differential calculus as displayed by NC(V) level 4 students in their pre- and post-tests. The research study was qualitative in nature and employed a case study design with the pre- and post-tests as well as interviews with the learners as data collecting methods. Content and interpretive analysis as well as Stein et al conceptual frameworks on mathematics that hinged on differential calculus and the errors the students displayed. A purposeful sample of 350 NC(V) level 4 students from diverse social backgrounds were selected to write the pre-and post-test. The interviews were used as a follow -up from the test to cement the students’ responses in the pre-test and establish the source of the misconceptions. The unit of analysis was therefore students’ written pre and post-test responses as well as the spoken responses they provided during the interviews. Reliability and validity of data collection instruments were validated using the Rasch Analysis framework. The Rasch framework was critical for this study because it was used to: authenticate the test items that were used in the pre and post intervention tests; provide customized analysis of differential calculus items against the students whom the test was administered to and provide and analysis how the selected students performed against the set of the test items at different levels of difficulty as located by the Rasch. Findings in the pre-test were varying. About 75% of students performed lowly in terms of differentiating calculus concepts related to basic algebra concepts. There were errors that were common across all the students and those that were specific to a particular group of students. Students experienced difficulties with algebra and procedures. Most students experienced difficulties with the function concept. Their inability to operationalise the function concept affected their understanding of calculus and the application thereof. In addition, students grappled with calculus terminologies accepted and used by the mathematics fraternity. Students had difficulties conceptualising critical differential calculus terms such as: function, surds, limit, power-rule, trigonometric functions, chain rule, maxima-minima concept, logarithmic functions, and quotient rule among others. Students displayed conflicting definitions of these terms. Further analysis of the students’ pre-test work revealed that most of them could not substitute correctly where required which revealed lack of basic differential calculus operations. I used Tall (1981) knowledge acquisition conceptions of concept definition and concept image to interpret students’ conceptual understanding and how that related to their concept definitions. The analysis, hence, established that students conceptual and procedural knowledges of basic differential calculus were weak. To mitigate their weak knowledge acquisition skills that related to the identified feeble conceptual and procedural knowledge of basic differential calculus an intervention programme was developed. Using research based instructional approaches a group of 30 students were involved in the intervention program. The finding were that when students’ prior knowledge of differential calculus is established, using appropriate intervention approaches it is possible to reduce misconception and errors and facilitate concrete learning outcomes.
Ph.D. (Mathematics Education)