Abstract
Integral calculus is one of the major topics involved in mathematical courses both at secondary
and tertiary level with several applications in different disciplines. It is part of gateway
mathematical courses at universities for many majors and important for the development of the
science. The purpose of this study was to explore the errors and misconceptions in integral calculus
at one of the primary teachers’ training colleges in Zimbabwe. Lecturers’ awareness of their
student teachers’ errors and misconceptions on a mathematics topic is critical in developing
appropriate pedagogical content knowledge. Thus, college lecturers’ awareness of student
teachers’ level of pre-requisite knowledge is vital in developing appropriate teaching and learning
intervention strategies that support effective learning outcomes. For effective teaching of
mathematics the educators need to study and understand student teachers’ alternative conceptions,
and how they come to have them.
The researcher claims that research on student teachers’ errors and the misconceptions they display
in integral calculus would provide crucial information on the transition from one level of
mathematics conception to the next. During the process of assimilation student teachers grasp new
information and accommodate concepts in cognitive structures that are adjusted when new
mathematical concepts are introduced. Thus, during this process of prior and new knowledge
acquisition, misconception and errors associated with the concepts are displayed. If the newly
attained mathematical knowledge is not properly accommodated, unstable behaviours occur.
During this process, student teachers may harbor concept images that are competing with
established mathematical knowledge. The rationale is therefore that 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 integral calculus.
Apart from the error analysis framework, the researcher assumed that misconceptions held by
student teachers in mathematics, specifically integral calculus, may be explained within the
frameworks of that of Tall and Vinner (1981) and constructivism. The frameworks stress that
students come to understand mathematical concepts based on their prior knowledge. However,
some of the knowledge they construct may appear to be correct to them but conceptually untrue.
This may occur through overgeneralisation of prior knowledge, mis-learning and inappropriate
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definitions of mathematical conceptions during new mathematical knowledge acquisition
situations.
Integral calculus is first introduced to the student teachers in their first year. Student teachers and
lecturers have encountered challenges in the teaching and learning of the various constructs
associated with the topic. So, the researcher found it compelling to conduct this study. The study
focused on errors and misconceptions in integral calculus: A case of primary teachers’ college
students in Zimbabwe. 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 student teachers as data collecting
methods. Content analysis techniques were used to analyse the data on the basis of a conceptual
framework of mathematics and calculus errors obtained from literature. A purposeful sample of
40 student teachers from diverse social backgrounds was selected to write the pre-and post-test.
The interviews were used as a follow up from the test to reinforce 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.
Findings in the pre-test were fluctuating. About 78% of students performed lowly in terms of
calculus concepts related to basic algebra concepts. There were errors that were common across
all the student teachers 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
terminology accepted and used by the mathematics community. Students had difficulties
conceptualising critical integral calculus terms such as: function, surds, limit, trigonometric
functions, logarithmic functions, integration by parts and exponential functions, among others.
Students displayed conflicting definitions of these terms. Further analysis of the student teachers’
pre-test work revealed that most of them could not integrate trigonometric functions correctly. I
used Tall’s (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 student teachers’ conceptual and procedural knowledges of
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integral calculus were weak. To mitigate their weak knowledge acquisition skills that related to
the identified feeble conceptual and procedural knowledge of integral calculus an intervention
programme was developed. Using research based instructional approaches a group of 30 students
were involved in the intervention programme. 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.