Calculus is a challenging course in terms of both teaching and learning since it covers many concepts difficult to develop a conceptual understanding for students (Özdemir, 2017; Parrot & Eu, 2018). Integral is one of the concepts of calculus that students from different levels such as secondary school and undergraduate have difficulty (Attorps, Björk, & Radic, 2013; Bezuidenhout & Olivier, 2000; Orton, 1983; Rasslan & Tall, 2002). For example, understanding the formal definitions given as related to integral is a challenging task for some students (Özdemir, 2017). Moreover, it was presented that many students struggle with developing the necessary conceptual understanding regarding Riemann sum, the relation of integral and area, and the Fundamental Theorem of Calculus (Hall, 2010;  Jones, 2015; Mahir, 2009; Rasslan & Tall, 2002).

The literature review stated that not only students but also the instructors in the calculus course offered in various levels focused on the routine procedures rather than developing conceptual understanding (Aspinwell & Miller, 1997; Grundmeier, Hansen, & Sousa, 2006). However, the objectives of the calculus course cover the development of students’ conceptual understanding as well as their procedural understanding (Bezuidenhout, 2001). In addition, being competent in representing the concepts of calculus algebraically, numerically, and graphically and also moving among different representations are among the objectives of the calculus course (Berry & Nyman, 2003; Goerdt, 2007). Biza, Diakoumopoulos and Souyoul (2007) emphasized the importance of designing the learning environments which present the interconnected and multiple representations of the concepts. In particular, Biza et al. (2007) gave the example related to the definite integral by referring to moving between different representations and connecting inferences and meanings related to the definite integral such as the symbolic expressions, the limit of sums, and calculation of the area.

In spite of the fact that visualization is considered as an entailment in terms of understanding mathematics, some studies also pointed out that students’ understanding is generally algebraic rather than visual (Aspinwall et al., 1997). Moreover, symbolic representation is a required component for students while studying in advanced mathematics (Artigue, Batanero, & Kent, 2007). Thus, it can be stated that to be able to carry out both graphical and symbolic representation related to integral properly is an important issue. In this respect, this study focused on the interpretation of the graphs presented as related to definite integral symbolically.

To investigate how prospective middle school mathematics teachers expressed the area given in the graphs symbolically might present detailed data about their thinking process related to the definite integral. For example, the meaning of the subtraction sign in the integrand might give evidence, especially when supported with the interviews, whether they are considering the integral as the area between the graphs (Jones, 2013). Moreover, such a study might present the data about how prospective teachers are considering the functions of the x-axis and y-axis in a given graph in terms of integral. Moreover, since to examine the points that students have difficulty is a main step for framing further teaching (Mahir, 2009), this study might contribute to teaching methods in the calculus course. To summarize, the purpose of this study is to investigate how freshman prospective middle school mathematics teachers interpret the graphs of definite integral symbolically in detail. Considering this purpose, research questions of the study were stated as follows:

1. To what extent are freshman prospective middle school mathematics teachers successful in the interpretation of the graphs of definite integral symbolically?

2. What are the difficulties that freshman prospective middle school mathematics teachers encountered in the interpretation of the graphs of definite integral symbolically?

To be able to answer these research questions, case study was utilized as the research design. While arranging the participants, purposive sampling was used since this method is based on the idea that a sample from which the extensive data can be gathered should be selected due to the need to examine and gain insight related to the concept in detail (Merriam, 2009). In this study, the criteria for selecting the participants were reorganized after the pilot study. The pilot study was conducted 130 freshman and 89 sophomore prospective middle school mathematics teachers from four universities in Turkey at the end of the spring semester of the 2017-2018 academic year. Since freshmen did not take a course involving integral in the first year of the program whereas sophomores took two calculus courses during the second year of the program, the aim was to compare the graphical interpretations of two year levels. However, a change was conducted in the courses of the Elementary Mathematics Teacher Education program and the calculus courses were taken to the first year of the program from the second year. This situation affected both the criteria for the participation selection and the purpose of the main study. In this respect, the participants of the main study were determined as freshman prospective teachers in the Elementary Mathematics Teachers Education in a state university in Ankara, Turkey. Since the selected university involves 75 freshmen in the program in the fall semester, it was aimed to involve a near number of participants in the main study. Moreover, it was decided that the instrument will be applied at the end of the calculus courses which also corresponds to the end of the spring semester of the 2018-2019 academic year. After the pilot study, the questions in the instrument were revised and the follow-up interviews were decided to conduct in the main study. The instrument involves 12 questions, each of which presents a graph with a shaded region and asks to write the area of the given region in terms of integral symbolically. Data sources of this study were documents and video recordings of the interviews. During the data analysis, as the first step, the answers of prospective teachers will be examined to answer the first research question. Then, to answer the second research question, the difficulties they have will be investigated by means of both their answers and the follow-up interviews.

Based on the analysis of the pilot study, it was expected that some freshman prospective middle school mathematics teachers might not be successful in stating area symbolically in the questions that the shaded area is piecewise and involves parts under the x-axis. It was stated that people have a tendency to focus on the prototype images to facilitate their thinking process (Jones, 2017). Since some calculus textbooks (e.g., Adams & Essex, 2010; Stewart, 2001; Thomas, Weir, Hass, & Giordano, 2010) generally focus on the first quadrant of the coordinate plane while introducing the area under a curve and integral relation and presenting how it can be represented symbolically, it was expected that freshmen would be more successful in such questions due to the mentioned familiarity. Moreover, Jones (2013) investigated the symbolic forms of integral and offered three main forms which are “the adding up pieces symbolic form” which was related to the concept of Riemann sum, “the perimeter and area symbolic form” which was deduced via accepting integral as an area, and “the function matching symbolic form” which was associated with the anti-derivation process. By means of follow-up interviews, symbolic forms presented by the participants might be examined and compared to the ones in the study of Jones (2013). The converse version of the questions in the instrument of this study which refers to giving a symbolic representation of the integral and asking to sketch a graph proper to the given can be considered as a suggestion for further studies. Thus, the prototype images regarding the definite integral might also be examined in such a study.

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