According to Hiebert and Lefevre, the main concepts of integral theory are that the definite integral of a function is the limit of Riemann sums, the integral-area relation and the fundamental theorem of calculus. However, research has found learners’ limited understanding of underlying concepts related to integration. For example, while students could perform routine procedures for finding the area under a curve, the learners rarely could explain their procedures. Also, many learners were found to be able to perform procedures and basic techniques of integration, but they failed to understand what behind the procedures. Thus, regarding knowledge of integration, it indicates learners’ conceptual knowledge, i.e. knowledge of concepts, which are abstract and general principles, is not as good as learners’ procedural knowledge, i.e. knowledge of procedures, i.e. a series of steps, or actions done to accomplish a goal. Thus, the aim of this study is to investigate the extent to which prospective teachers who completed calculus course involve their procedural or conceptual knowledge in relation to the concept of definite integral through a problem-solving task.

 

This is a descriptive-explorative research which involved 30 prospective teachers who were taking ‘problem-solving’ course in a mathematics teacher education program at a university in Surabaya, Indonesia. Data were collected through the prospective teachers’ work on a problem-solving task related to Integral. The problem-solving task asks the prospective teachers to find out the possible integrand of f(x) if both the value of definite integral associated with such f(x) and the interval of f(x) is given. The prospective teachers were then provided a week after solving such task to reflect on whether they need to revise their work and to what extent they carried out any changes. Data were analyzed by categorizing the prospective teachers’ work into whether they considered f(x) meeting the property of uniqueness or not. Also, in what Polya’s problem-solving stages the prospective teachers find the most difficult.

 

Results point out that more than a half of prospective teachers (n= 17) answered that the integrand of f(x) is unique, which means there is only one f(x) which satisfy the equation int (f(x), x=2..5)=-6 , while the remaining 13 prospective teachers answered  f(x) is not unique (more than one possible form). The reasons of those who answered f(x) is unique were found around procedural errors related to the technique of integration, missundertanding of using fundamental theorem of calculus, and misconception regarding the concept of definite integral. For those who answered that f(x) is not unique, none of their answers were indicated to show possible various algebraic form of f(x)  which represent the integrand. Instead, students were found to let  as constant function, linear function, and quadratic function, instead polynomial functions which has degree more than 2 or other types of form of functions.  More specifically, there were no prospective teachers who related their performance with the concept of definite integral as area-relation. Thus, it indicates prospective teachers’ lack of conceptual understanding of definite integral. However, once the prospective teachers revised their work, most of them (28), primarily for those who answered is unique, indicated that f(x) is not unique, and only 2 prospective teachers who kept their answers, i.e. f(x) is unique. Regarding problem-solving stages, prospective teachers experienced the stages of devising a plan as the most difficult steps to determine. This is because the prospective teachers found difficulties in determining the possible form of f(x) which can be included in the integral calculation. Also, at ‘understanding the problem’ stages, prospective teachers were also found to have a misunderstanding of the meaning of ‘unique’ for f(x). The results of this study encourage future empirical research to further study about the impact of conceptual understanding on procedural fluency, particularly about integral concept, and vice versa.