Teaching concept of calculus

Calculus may be described as the mathematical study of change.It provides us with the tools to study and analyse change of specific phenomena in the real world. The change may be with respect to time or with respect to another quantity

The symbolic world of mathematics

In understanding upper secondary school students’ interpretations of information in symbolic representations of a distance-time-relation, little attention has been paid to the analysis of the condition of the conceptual development related to utterances. Understanding this better can help improve the teaching of attribute and information in symbolic representations of different phenomena. Two theoretical perspectives have been used to conduct the analysis: Tall and Vinner's theoretical perspectives on learning and Gray’s & Talls’s theory of three mathematical worlds. The findings provide evidence that a detailed analyse of student’s utterances show difference in quality related to student’s interpretations of a distance-time relation. The qualities were related to student’s concept images of functions and derivatives.

Upper secondary school students’ interpretations of analytical and graphical representation of linear motion–From event to process

The ability to interpret mathematical concepts is a prime concern in science in general and physics in particular. The aim of this licentiate thesis is to investigate and analyze upper secondary school students’ interpretation of graphical and analytical representation of linear motion. The report presents a study focused on 17 students from two upper secondary schools, located in two different areas in Gothenburg. Video recording was arranged for documentation of students’ activities. The observer was not present during the recording. The collected material consists of about 10 hours video recording divided into 20 sequences about 15 – 50 minutes each. The task used to approach students’ interpretations of graphical representation of linear motion where related to a distance-time graph and symbolic representation of linear motion where related to a distance-time function. The theoretical framework is based on constructivism, including the theory of concept image and concept definition of Tall and Vinner and models of conceptual change developed by diSessa and Chi. The result shows that iconic interpretations were most prevalent among students who used only everyday concepts to explain the phenomenon and students who used relevant concepts to the tasks, but did not manage to apply them correctly. The outcome further indicates those students’ alternative conceptions about graphical patterns and distance-time graph can be explained by ontological categorization of existing concept. There is evidence to support that iconic interpretations could be stimulated and generated as a result of student categorization of distance-time graph as an event, when in fact graph such as distance – time graph or velocity – time graph are used to describe and communicate phenomenon or processes. In this case conceptual development could be specified when a concept has to be re-assigned to an ontological lateral category. Most students are comfortable in using distance function to calculate difference quotient and average velocity in different intervals. Less than half of student used the derivative to determine instantaneous velocity of the object. Students’ concept image of the derivative of distance function is related to instantaneous rate of change. Most students in this study have acquired knowledge to link symbols and operations. There were cases where students’ concept image was found to be incoherent to establish a link between symbols and concepts in the proceptual - symbolic learning of mathematical concepts

The symbolic world of mathematics

In understanding upper secondary school students’ interpretations of information in symbolic representations of a distance-time-relation, little attention has been paid to the analysis of the condition of the conceptual development related to utterances. Understanding this better can help improve the teaching of attribute and information in symbolic representations of different phenomena. Two theoretical perspectives have been used to conduct the analysis: Tall and Vinner's theoretical perspectives on learning and Gray’s & Talls’s theory of three mathematical worlds. The findings provide evidence that a detailed analyse of student’s utterances show difference in quality related to student’s interpretations of a distance-time relation. The qualities were related to student’s concept images of functions and derivatives.