# Students’ mathematical performance, metacognitive experiences and metacognitive skills in relation to integral-area relationships

Students’ mathematical performance, metacognitive experiences and metacognitive skills in... Abstract Previous studies have explored students’ understanding of the relationship between definite integrals and areas under curves, but not their metacognitive experiences and skills while solving such problems. This paper explores students’ mathematical performance, metacognitive experiences and metacognitive skills when solving integral-area tasks by interviewing nine university and eight Year 13 students using a think-aloud protocol. The findings show that the students could have benefitted from their teachers and lecturers placing greater emphasis on both their conceptual understanding of integral-area relationships and their metacognitive experiences and skills. 1. Introduction Research has reported that students have difficulty with integral calculus concepts (e.g., Jones, 2013; Kouropatov & Dreyfus, 2013; Kiat, 2005; Sealey, 2014; Thomas & Hong, 1996). While the majority of students studying integral calculus can successfully apply basic procedures to find antiderivatives, their understanding of the underlying concepts is limited. For example, Thomas & Hong (1996) report that many students regard integral calculus ‘as a series of processes with associated algorithms and do not develop the grasp of concepts which would give them the necessary versatility of thought (p. 577).’ Studies have also shed light on students’ understanding of the relationship between definite integrals and areas under curves (e.g., Jones, 2013, 2015; Mahir, 2009; Sealey, 2014); however, little attention has been paid to students’ metacognition while solving integral-area problems. Metacognition is knowledge or cognitive activity that individuals have about their cognitive activities (Flavell et al., 1993). It is a ‘meta-level knowledge and mental action used to steer cognitive processes (Jacobse & Harskamp, 2012, p. 133).’ To successfully solve mathematical problems, metacognitive knowledge and activities are necessary (Lester, 1982; Özsoy, 2011; Schoenfeld, 1985; Silver, 1982; Verschaffel, 1999). However, the use of metacognition to improve problem-solving skills is sometimes ignored in teaching (Schoenfeld, 1985), and a lack of development of students’ metacognition has been reported (e.g., Jacobse & Harskamp, 2012). Metacognition has three facets (Efklides, 2006, 2008; Kim et al., 2013; Schneider & Artelt, 2010; Tarricone, 2011): metacognitive knowledge or knowledge of cognition, metacognitive skills or regulation of cognition and metacognitive experiences or concurrent metacognition.1 As these three facets have not been well explored for the integral-area relationship at either the upper secondary or undergraduate level, there is the possibility that students’ outcomes with integral-area questions might be improved by studying students’ metacognition while solving such questions. This study uses the research question: What are students’ metacognitive experiences and skills while solving integral-area problems? to explore that possibility. 2. Literature In this section the relevant integral calculus and metacognition literature is reviewed to frame the study and to help justify the study’s findings. 2.1 Integral calculus Studies have shown that many students who are able to undertake routine procedures that utilize integral techniques to find areas have a limited understanding of why such procedures should be undertaken (e.g., Artigue, 1991, Thomas & Hong, 1996). For example, Kiat (2005) found that if the graph of the integrand is not given, students can incorrectly set up area integrals, suggesting a procedural understanding of definite integrals that does not include the connection between definite integrals and area. Kiat also identified that 55% of students could not set up the correct integrals to find a shaded area in a question where one of the curves was above and the other below the x-axis. Mahir (2009) explored the conceptual and procedural knowledge of undergraduate students in integral calculus. He found that the students had developed a satisfactory level of procedural knowledge in that 92% and 74% correctly answered two questions that could be solved procedurally. However, for the three questions based on conceptual knowledge, only 8%, 16% and 24% were able to do so. In addition, Rasslan & Tall (2002) have presented evidence of students’ difficulties in understanding the definite integral as an area under a curve when working with piecewise functions and improper integrals. Important concepts, such as functions, limits, rate of change and multiplication, are involved in Riemann sums and definite integrals, $${\mathop{\lim }\nolimits _{n\rightarrow \infty } \sum ^{n}_{i=1}{f\left (c_{i}\right )\triangle x } }$$ (Sealey, 2006). However students’ difficulties with understanding the definite integral as the limit of a sum has also been highlighted (Grundmeier et al., 2006). Sealey (2014) sought to gain an insight into how students might develop the concept of the Riemann integral, proposing a framework for characterising students’ understanding of Riemann sums and definite integrals. This framework has a pre-layer that involves symbolicrepresentation $$\big[\frac{1}{c}\,\cdot\ f(x_{i})\big]$$2 and or $$\left [c\cdot\Delta x\right ]$$, and four layers, including product $$\big[\frac{1}{c}\,\cdot\, f(x_{i})\big]\,\cdot\,\left [c\,\cdot\,\Delta x\right ]$$, summation $$\sum \nolimits _{i=1}^{n}{f(x_{i})\Delta x}$$, limit $${\mathop{\lim }\nolimits _{n\to \infty } \sum ^{n}_{i=1}{f(x_{i})\triangle x}\ }$$ and function $$f\!\left (b\right )={\mathop{\lim }\nolimits _{n\to \infty } \sum ^{n}_{i=1}{f(x_{i})\Delta x} }$$. Using this framework, Sealey found that layer 1, the product of f(x) and △x, is the most complex part for students. ‘Difficulties in this layer are not necessarily related to the operation of multiplication and performing calculations, but are typically related to understanding how the product is formed and understanding how to use each factor within the product (p. 238).’ In terms of the symbols and notations of integral calculus, the ‘dx’ in $$\int{f(x)dx}$$ causes conflicts and contradictions for a number of students. For example, in integral calculus dx shows that the integration should be done with respect to x, however, some students are told not to cancel dx in $$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}$$ when solving questions related to the chain rule as it has no separate meaning (Tall, 1992). Students can also have great difficulty learning the symbolic definition of a definite integral (Grundmeier et al., 2006). From a sample of 52 students, only one was able to provide the symbolic definition of a definite integral and only 35% were able to provide a correct verbal definition. However, the lack of such a definition did not affect many students’ ability to answer routine integration problems, as more than 60% of the students were able to evaluate the definite integral of a trigonometric function. Other research supports that a majority of students may not be able to write meaningfully about the definition of a definite integral (e.g., Rasslan & Tall, 2002). One reason given relates to the learning and teaching approach of some teachers and lecturers, who focus on the procedural aspects of calculus (Bezuidenhout, 2001). 2.2 Facets of metacognition That there are three general facets of metacognition is recognized in a number of studies (e.g., Efklides, 2006, 2008; Kim et al., 2013; Schneider & Artelt, 2010). In this section each facet is unpacked in some detail. 2.2.1 Metacognitive knowledge Metacognitive knowledge derives from a person’s long-term memory (Efklides, 2006) and can be defined as ‘knowledge of cognition in general as well as awareness and knowledge of one’s own cognition (Anderson et al., 2001, p. 46).’ It is a declarative knowledge about cognition that refers to both an individuals’ explicit or implicit knowledge about persons, tasks, goals and strategies (Efklides, 2006, 2008). This explicit or implicit knowledge can be ideas, beliefs and theories about how individuals perform and feel about different tasks, the features of different tasks and how they work; the goals individuals follow within different tasks and situations; and finally when, why and how different strategies should be used (Efklides, 2006, 2008). Metacognitive knowledge also includes knowledge about the different cognitive functions, for example thinking, in terms of what they are and how they work (Efklides, 2006, 2008). Moreover, it encompasses knowledge of the criteria of the validity of knowledge, including knowledge about limits of knowing, criteria of knowing and certainty of knowing (Kitchner, 1983). Finally, theories of mind (Bartsch & Wellman, 1995)—beliefs that individuals have about people’s minds, including themselves, can be considered as metacognitive knowledge (Kuhn, 2000). Communicating with others, self-monitoring, monitoring other people’s cognitive activities and being aware of personal metacognitive experiences help develop, revise and update metacognitive knowledge (Efklides, 2006, 2008; Flavell, 1979; Kim et al., 2013). 2.2.2 Metacognitive experience Metacognitive experience, is ‘what the person is aware of and what she or he feels when coming across a task and processing the information related on it (Efklides, 2008, p. 279).’ Unlike metacognitive knowledge and skills, metacognitive experiences take place in working memory (Efklides, 2006). They include feelings of knowing, difficulty, familiarity, confidence and satisfaction, as well as estimations of effort and time needed to be spent on tasks, judgements of learning and estimates of the correctness of solutions (Efklides, 2006, 2008). Online task-specific knowledge is another aspect of metacognitive experiences (Efklides, 2001, 2006,2008; Schneider & Lockl, 2002). This refers to the task information plus the ideas and thoughts an individual have about the task they are working upon, along with the metacognitive knowledge they retrieve from memory and use to complete the task. 2.2.3 Metacognitive skills The last facet of metacognition, metacognitive skills are deliberate activities that help control cognitive activities (Schraw, 1998). They include task orientating, planning, monitoring, regulating and evaluating (Efklides, 2006, 2008). Task orientating is about understanding the task requirements, planning relates to the steps that need to be taken to achieve a goal or complete a task, monitoring refers to activities undertaken while implementing a strategy to assess its progress and effectiveness, while evaluating and regulating relate to checking the product of cognitive processing and adjusting it as required (Efklides, 2006, 2008; Garofalo & Lester, 1985; Schraw, 1998; Veenman & Elshout, 1999). 3. Research Method To develop an understanding of students’ metacognitive experiences and skills while completing integral-area tasks, a multiple case study (Yin, 2014) was used. In the research, a case was considered to be an educational institution in New Zealand (i.e., a university or college) in which integral calculus was taught in 2014. The two selected cases consist of a sample of students who were interviewed. This design was chosen because the topic of integral calculus at Year 13 and first year university have a substantial content overlap in New Zealand, but, due to the different contexts in which they are learning, students at the two levels can have quite different metacognitive experiences and skills. As the research was exploratory, emerging theory was intended to be derived from an analysis of the data collected rather than existing beforehand. As such, grounded theory (Charmaz, 2006) was chosen as an approach to the data analysis because of its usefulness for constructing explanations of complex phenomena. 3.1 Study participants Case 1 is one of the top five universities in New Zealand (QS World University Rankings, 2014). To identify potential interview candidates, because the lecturers of the integral calculus course did not know their students’ calculus backgrounds, convenience sampling was used. All students enrolled were invited to participate by email and nine students (seven males and two females) volunteered. All were interviewed. Case 2 is one of 11 colleges in Wellington city that offer calculus courses to students. As the teachers did know their students’ calculus backgrounds, theoretical sampling was used to select potential candidates, meaning the sample was ‘suitable for illuminating and extending relationships and logic amongst constructs (Eisenhardt & Graebner, 2007, p. 27).’ Students with different calculus backgrounds were sought from two classes, a regular and a scholarship class. Based on their performance in differentiation, the mathematics teacher of the regular class selected six students; two each with low, medium and high achievement. Two scholarship students were selected by the Head of Mathematics to represent students with very high achievement. By choosing students with different calculus backgrounds, it was hoped a broad understanding of students’ metacognitive skills and experiences might be developed. Two students were chosen from each achievement group to avoid a single student representing a group. The students’ calculus backgrounds can be identified through their student code (Table 1) so potential connections between students’ metacognitive experiences and skills and their calculus background can be noted. Table 1 College interview participants Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Note. Identifying codes are based on the student’s calculus background (see column 3) Table 1 College interview participants Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Note. Identifying codes are based on the student’s calculus background (see column 3) During the interviews, different levels of mathematical performance were observed amongst the volunteers from Case 1. To ascertain if the sample also represented a range of achievement, students’ responses to the nine items were later graded, with a possible total of 36 marks3. Subsequent analysis produced a mean score of 23.4 with standard deviation 8.0, indicating the students generated a range of different scores, suggesting that, similar to the college students, a range of achievement levels were represented in the sample. 3.2 Instruments In this section the measures used for collecting data are described, including those used to discover students’ metacognitive experiences and skills (Section 3.2.1), and for exploring students’ solution of integral-area questions (Section 3.2.2). 3.2.1 Instruments and procedures for measuring metacognitive experiences and skills Two main approaches are used to measure metacognition, offline and online measures (e.g., Jacobse & Harskamp, 2012; Schneider & Artelt, 2010). Offline measures evaluate metacognition without concurrent problem solving (Schneider & Artelt, 2010), whereas online measures seek to evaluate metacognition during problem solving. Both interviews (e.g., Kreutzer et al., 1975) and questionnaires (e.g., Schraw & Dennison’s 1994 Metacognitive Awareness Inventory) have been used previously to measure metacognition. Questionnaires are a common form of offline measure as they are usually undertaken after a task has been completed. They are generally comprised of statements about metacognitive monitoring and the regulation of tasks in which participants indicate the degree to which the statements apply to them (McNamara, 2011; Jacobse & Harskamp, 2012; Schneider & Artelt, 2010). Using questionnaires to measure metacognition has the advantage of being easily administered (Jacobse & Harskamp, 2012). However, as they require a participant to recall from memory what they did, or would do, when completing a task the results may be affected by memory distortion (Jacobse & Harskamp, 2012; McNamara, 2011), or the social desirability factor, ‘the basic human tendency to present oneself in the best possible light (Fisher, 1993, p. 303).’ One effective way to measure metacognition online is to conduct an interview using a think-aloud protocol (Ericksson & Simon, 1993). In such a protocol, a participant’s thinking should be verbalized and recorded while they are working on a task. This thinking should then be transcribed and coded based on a scheme (e.g., Kim et al., 2013), or, if transcribing is not undertaken, be coded based on a scheme (e.g., Jacobse & Harskamp, 2012). Using a think-aloud protocol to measure metacognition is time-consuming, however, it provides more reliable information than questionnaires as data are collected in-the-moment while a participant works on a task so those data are less affected by memory distortions or the social desirability factor (Jacobse & Harskamp, 2012; Veenman, 2011). Consequently, a think-aloud protocol was chosen for this study. Before their interview started, each student was asked to verbalize what they were thinking while working on each question; that is, think aloud rather than work silently. Prior to the interview, an information sheet that described the research project and the process for answering the questions had been given to students. Several approaches were considered for analysing students’ metacognitive experiences and skills (Veenman et al., 2000; Jacobse & Harskamp, 2012). The chosen approach combined four items from these two sources (Table 2). The two metacognitive experience items (ME1 & ME2) were adapted from the VisA instrument (Jacobse & Harskamp, 2012). The two metacognitive skills items (MS1 & MS2) were chosen from Veenman et al. (2000). MS1 was selected as making a drawing for a given problem has been found to be an important factor in mathematical problem solving (Jacobse & Harskamp, 2012). MS2 was selected to explore whether or not students monitor their work and progress when answering a mathematical question. Table 2 Items explored in relation to metacognitive experiences and skills Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Note. ME: Metacognitive Experiences; MS: Metacognitive Skills Table 2 Items explored in relation to metacognitive experiences and skills Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Note. ME: Metacognitive Experiences; MS: Metacognitive Skills To use ME1 and ME2 to explore metacognitive experiences, students were initially given access to paper and writing materials. An item was then shown to the student and they were given a chance to read it. Before attempting the task, the student was asked: How well do you think you can solve this problem? They could select one of the following options: I am sure I will solve this problem; I am not sure whether I will solve this problem correctly or incorrectly; or I am sure I cannot solve this problem. After selecting an option, they were asked to provide a reason for their choice. Once they had finished work on the problem, a similar question was asked: rate your confidence for having found the correct answer. Similarly, students had three options and were encouraged to provide a reason for their selection. For MS1 and MS2, interviews were video recorded and copies of students’ written work made. Examining the written work and watching the interview transcripts identified the use of drawings, written or mental, for MS1. For MS2 the first author watched the videos and recorded any checking of calculations and answers that students were seen to undertake while, or subsequent to, answering the questions. 3.2.2 The integral-area problems Three integral-area problems that were used to probe students’ mathematical performance, metacognitive experiences and skills are described in this section. The questions were chosen as they each had the potential to evoke several cognitive processes. The appropriateness of each question for the sample students was ascertained by observing the integral calculus lessons at both the university and college, and by undertaking a document analysis. Q1. Please calculate the area enclosed between the curve $$x=y^{2}$$ and y = x − 2 in two ways. Which way is better to use? Why? The first part of Q1 (i.e., to calculate the area enclosed between curves) is a typical question in integral calculus that explores whether students know how to use the definite integral to find areas between curves. Similar questions were used in teaching and assessments in both Cases. While the two functions are not sophisticated, the curves cross the x-axis and the lower curve changes if the integral is set up with respect to the x-axis. Therefore, the question could challenge some students. While solving the question two ways and evaluating which way was better was not a standard question format, it had the potential to trigger metacognitive experiences and skills. Q2. ‘The graph of f′(x), the derivative of f(x), is sketched below. The area of the regions A, B and C are 20, 8 and, 5 square units, respectively. Given that $$\ f\!\left (0\right )=-5$$, find the value of $$f\left (6\right )$$’ (Mahir, 2009, p. 203). This question was taken from Mahir (2009) and was designed to explore students’ conceptual knowledge. Students from both Cases might have difficulty with it as $$f\!\left (x\right )$$ is not explicitly stated in the item (Thomas & Hong, 1996). In addition, the question is not typical of those used in the teaching and assessment of integral calculus at either Case. Again the item was chosen as it had the potential to activate several cognitive processes; such as the ability to analyze and distinguish which of the areas A, B and C should be used to find f(6). Other possible cognitive processes include remembering how the area under the graph of f′(x) is linked to f(x) through the Fundamental Theorem of Calculus and executing the remembered process. Q3. Please can you pose a problem about the area enclosed between a curve and a line with any two arbitrary bounds that will give an answer of 1 (i.e., the enclosed area will be equal to one). Christou et al. (2005) designed a taxonomy for problem-posing processes that can be useful when designing problem-posing questions. According to this taxonomy, there are four problem-posing question types: a) editing, b) selecting, c) comprehending and d) translating, quantitative information. Q3 is classified as selecting quantitative information because it requires students to pose a problem that is appropriate to the given answer. This task is more difficult than editing as students need to focus on relationships between the given information (Christou et al., 2005). This question might be challenging for students at both cases as they had not seen problem-posing questions in integral calculus during class. 4. Results In this section, the students’ mathematical performance on the three integral-area items is detailed (Section 4.1) then their metacognitive experiences and skills are described (Section 4.2). 4.1 Students’ mathematical performance in the integral-area problems 4.1.1 Question 1: Using the integral-area relationship to calculate areas under curves For Q1 only four students, three from the university sample (U589) and one from the college sample (Y8), were able to find the enclosed area with respect to the x-axis correctly (Table 3). As drawing an incorrect graph could affect how students set up their integral, the integrals the students created were compared to their drawings to investigate whether a student had the skills to find the area with respect to the x-axis. Four students (U13; Y67) set up integrals that related directly to their drawings. One student (Y2) did not integrate with respect to the x-axis while the remaining eight students (U2467; Y1345) had difficulty attempting this method. Table 3 Summary of students abilities to find the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Table 3 Summary of students abilities to find the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Of the university students who had difficulty, U4 struggled with the fact that $$x=y^{2}$$ is not a function, and set up the integral incorrectly (i.e., $${{{\int ^{4}_{0}}}}{(x-2-\sqrt{x}\ )dx+{{{\int ^{0}_{1}}}}{(x-2-\sqrt{x}\ )dx}}$$); even though he had sketched the graph correctly. U7 did not realize that if the upper/lower limit changes during the integration interval, the area function will change accordingly. He said, ‘I find a general formula that would give me the correct answer with whatever bound I put in. So having integrated it, then I can decide what bound I can use (see Fig. 1).’ Fig. 1. View largeDownload slide Part of U7’s working on Q1. Fig. 1. View largeDownload slide Part of U7’s working on Q1. U2 inappropriately used the disk formula for finding volume ($${{{\int ^{b}_{a}}}}{\pi f^{2}\left (x\right )dx}$$). The last University student, U6, could not find the second intersection point for the two curves. She only found the x = 4 intersection point then set up an incorrect integral for finding the area ($$\int ^{4}{(\sqrt{x}}-x-2)\ dx$$). Of the college students who had difficulty, Y1 considered only $${{{\int ^{2}_{0}}}}{(x-2)\ dx}$$ when finding the enclosed area. Y5 had difficulty with finding the area below the x-axis, commenting ‘I am not too sure whether it [$$x=y^{2}$$] actually stops at x = 0 or would continue around like a parabola in which case there will be some area there that is missed out. I am not sure how to calculate that.’ Two students (Y34) struggled to find the intersection points, which prevented them from setting up the correct integrals. Eleven students (U23456789; Y278) attempted to find the area with respect to the y-axis. Of these, seven students (U45689; Y78) did so correctly, with four students (U569; Y2) using y-axis integration as their first choice method for the task. The four students who answered incorrectly (U267; Y2) made various errors when setting up their integrals. Three students (U67; Y2) mistakenly identified that the curve was the top function. Two students used the x-axis bounds (Fig. 2) of integration rather than calculating the equivalent bounds for the y-axis (U2; Y2). As mentioned earlier, U2 used an incorrect formula for finding the enclosed area while U3 created an incorrect drawing of the graph of $$x=y^{2}$$, leading to an incorrect lower bound for the integral, $$^{-}$$2. The remaining six students (U1; Y13456) did not attempt this method. Fig. 2. View largeDownload slide Part of Y2’s working on Q1. Fig. 2. View largeDownload slide Part of Y2’s working on Q1. When asked, the 10 students (U23456789; Y78) who found the enclosed area with respect to both axes (whether correctly or incorrectly) highlighted that integration with respect to the y-axis was easier for finding the enclosed area in this question, because only one integral is involved and they did not need to work with the square root and negative area: ‘I could do it in one equation and also it is all above, all positive, [so] I don’t have to deal with negative area (U5).’ Two students from Case 2 (Y58) mentioned their first choice was always integration with respect to the x-axis when finding enclosed area. Y8 believed it is easier to conceptualize when you are integrating with respect to the x-axis: ‘By default, I go x because it is easier to conceptualize, because you have positive to negative, but y, sort of is inverted in terms of positive to negative. But sometimes it is just easier to do in terms of y [sic].’ Y5 mentioned, ‘In all questions it is easier to use the integration with respect to x-axis because you don’t need to write the function in terms of x = f(y), at least in questions that we see in our school.’ Two broad inter-related themes emerged from students’ explanations of which method was better to use: 1. choice based on the graph of the curves (U34589; Y1234); and 2. choice based on the algebraic manipulations needed to solve the problem (U123456789; Y567). When making graph-based choices, five students (U3489; Y2) noted they chose the method that involved the lesser change to the lower/upper functions. For example, U8 said ‘when you have to break it up less times like the previous example [Q1].’ U5 mentioned she would make her choice based on which alternative had the lesser negative signed area. Three students from Case 2 (Y134) said that their choice would be based on the graph being enclosed by the x or y axis, indicating they were thinking about the area under one of the curves rather than between the two curves. This error was also found when illustrating A and B in Q2 (Section 4.1.2). When finding the enclosed area between curves and two bounds, the axes are not important; however, most college students had not developed this understanding as they had not developed an understanding of integral-area relationships through Riemann sums (Section 4.1.2). The second theme related to students basing their choice of method on the least amount of algebraic manipulation required, a decision that could take into account both the finding of the lower and upper bounds and possible form changes for the integrand. If the bounds were given in terms of x = a and x = b, they would be more likely to choose integration with respect to x, and if they were given in terms of y = c and y = d, they would choose integration with respect to y. If the integrand was presented as a function of x, they would tend to integrate with respect to the x-axis, and if it was given in terms of y, they would tend to integrate with respect to the y-axis: ‘I would use it [the method] depending on what formula I have for the functions. I am integrating relative to the thing the function is of (U7).’ A lack of proficiency in algebraic manipulation was a major barrier to successfully solving the integral-area problems. For example, in Q1, six students (U136; Y356) had difficulty solving $$x-2=\sqrt{x}$$ to find the intersection points. Five students (U13; Y145) made a mistake when finding y from $$x=y^{2}$$, assuming $$y=\sqrt{x}$$. Y6 squared $$x-\sqrt{x}$$ as $$x^{2}-x$$. Y5 thought these two curves have no intersection points, and Y3 wanted to solve $$x-2=y^{2}$$ to find the intersection points. 4.1.2 Question 2: Using the integral-area relationship for the graph of a derivative function Students’ responses to Q2 were used to explore whether they were able to use the integral-area relationship for f′(x). Seven students from Case 1 (U3456789) and four from Case 2 (Y5678) realized that $${{{\int _{a}^{b}}}}{f^{\prime}(x)dx}$$ is equal to the area under the graph of f′(x) between x = a and x = b. However, only six students from Case 1 (U345689) realized this integral is equal to the signed net area underneath the graph of f′(x). One student, Y7, made the mistake of considering $${{{\int _{a}^{a}}}}{f^{\prime}(x)dx=f^{\prime}(b)-f^{\prime}(a)}$$. 4.1.3 Question 3: Posing an integral-area relationship question For Q3, eight students (U3456789; Y7) posed a correct question about the area enclosed between a curve and a line using the given information. Of the other nine students, seven (U12; Y23458) could not pose a question and two (Y16) posed an incorrect question (Table 4). Table 4 Questions posed by students for the integral-area relationship Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Table 4 Questions posed by students for the integral-area relationship Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Posing a question was not an easy task for many students. For example, Y4 said, ‘I am struggling with only the area is given and you have too many things.’ However, those not able to pose a question did not realize that the task could be simplified by choosing the x-axis as the line. For instance, Y8 chose $$y=x^{2}$$ as the curve and y = x as the line for his first try, then tried $$y=\sqrt{x}$$ as the curve. After he was unable to find an area equal to one, he changed the functions to a more general form by considering the curve $$y=a\sqrt{x}$$ and the line as y = bx. 4.2 Students’ metacognitive experiences and skills while solving the integral-area problems This section describes the metacognitive experiences students had and the metacognitive skills they used while solving the integral-area tasks. As shown in Section 2.2, metacognitive experiences and skills have several aspects. Those explored here relate to feelings of familiarity, knowing, confidence, estimating the correctness of the solution and making a judgement of learning (Efklides, 2006, 2008). The two metacognitive skills investigated involve making a drawing related to the problem, and checking calculations and answers. 4.2.1 Having an accurate pre-judgement of whether they can solve the problem Students had different metacognitive experiences when dealing with Q1. Eight students (U127; Y12567) made their judgement based on their familiarity with how to find an area using integral calculus. An example of these responses is ‘We have recently learnt this [topic] in class and I am practising these questions at the moment (Y1).’ Four students (U568; Y8) based their judgement on their ability to integrate the form of the integrand. They provided such reasons as ‘equations [are] not particularly difficult to integrate (U6). Three students (U3; Y34) based their decision on their familiarity with the shape of the graph, providing such reasons as ‘I can imagine it graphically (U3).’ U9 highlighted the importance of the shape of the enclosed area for making his judgement: ‘What I will do before I would have known if I am sure or not I will draw the graph. Then I decide whether I am sure I can solve it or not. That is hard for me to look at those two functions and say, oh yes, it is easy I can find the area between functions [sic].’ Drawing a graph is an element of metacognitive skills (Jacobse & Harskamp, 2012), as would be imagining a graphical representation of the task. It is also an important part of solving integral-area problems as it helps students decide whether to integrate with respect to the x- or y-axis, and shows whether the curves have any discontinuity. The responses of these four students (U39; Y34) illustrate how the metacognitive experiences students can be related to the metacognitive skills they possess. The judgement of the final student, U4, was affected by the fact that $$x=y^{2}$$ was not a function, which meant he was unsure if he could solve the problem. Overall, when comparing their metacognitive experiences to the results of Q1 (Table 5), students from Case 1 had more accurate predictions than students from Case 2. Table 5 Students’ prediction of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 Table 5 Students’ prediction of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 For Q2, four students (U348; Y4) were sure they could solve the question; however, only two answered correctly (U34). Of the four, three (U348) had that feeling because they thought they understood why the information was given. For instance, U8 said, ‘we know the area of important parts of the f′(x) and area is related to the anti-derivative.’ Y4 thought he had seen similar questions; however when he started solving the problem he said he had not properly understood what the question asked. He said, ‘I have seen question like this before…I think I misinterpret what is asking [sic].’ Eleven students (U12567; Y123568) were not sure if they could solve the question correctly, and only U5 did so correctly. U5 was unsure because she believed she had not ‘encountered any question like this.’ Two other students (U7; Y3) also felt they had not seen a question like this before. Two students (Y68) were unsure because they felt they needed to think more about it to know if they were able to solve it. For instance, Y6 said, ‘I am not sure because I need time to think about it mentally in my head to understand the question better... I think I might be able to do it in time... I know it seems there are some familiar parts in it but all together in one question [I am not sure whether I am able to solve it].’ The remaining students had different reasons for their feeling. Three students (U1; Y12) recognized the question, but were unsure how to solve it. For instance, U1 said, ‘I have seen questions like this before, but cannot remember how to do it.’ U6 was unsure because she thought guessing may be required: ‘not sure, because some guessing may be required.’ U5 misunderstood the given information, thinking the given graph was that of f(x). He said, ‘$$f\left (0\right )=-5$$ is confusing me.’ Y1 was unsure because he did not know ‘where to start.’ Y5’s feeling was specifically related to how the problem needed to be solved. He said, ‘I am not sure because I have to change the area to f′(x). but I am not sure I can calculate it correctly.’ Two students (U9; Y7) were sure they could not solve this question, but U9 did so correctly. He felt he had not seen a similar question before, saying ‘I never seen this before [sic].’ Y7’s feeling was related to how the problem should be solved. He said ‘I do not know how to find out the f(x) because I do not recognize the graph type of f′(x).’ Finally for Q3, nine students (U3456789; Y57) were sure they could pose a problem to meet the given constraints, and of these, all but Y5 posed a problem with the correct area. Most of these students either believed they could find an example (U689; Y57), or thought they could use simple functions for posing a problem (U357). For example, U7 said, ‘sure, I am going to use simple stuff’, and Y7 said, ‘I can think of an example.’ Apart from those reasons, U4 was sure he could pose a problem because he ‘understand[s] the theory behind the task.’ Seven students (U12; Y24568) were unsure if they could pose a problem based on the given information, and either did not pose a problem or posed a problem with an incorrect area. These students were unsure for a variety of reasons, including being unsure: if they could find an example that fitted the given information (U1; Y26); that they could ‘do it backward’ (U2; Y8); because they had not posed a problem before (Y3); and because they ‘may make a mistake’ (Y4). Only one student (Y1) was sure he could not pose a problem based on the given information because he had not posed a problem before. 4.2.2 Having an accurate post-judgement of how effectively the task was completed For Q1, students’ judgements of how well they answered the item were varied (Table 6) and to some extend influenced by whether or not a student was able to solve the task with two methods. Four students (U589; Y8) were sure they had solved it correctly because they could find the same answer using both methods (their answers were correct). Similarly, two students (U27) were sure they had solved the question incorrectly as they found different answers using the two methods (although both answers were incorrect). Table 6 Students’ post-judgement of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Table 6 Students’ post-judgement of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Students who were only able to use one method needed to seek other rationales for their judgement. For example, two students (U3; Y6) made an inaccurate post-judgement, saying they were sure they had found the correct answer because ‘it makes sense graphically’ (U3), and ‘looking at graph it seems right visually [sic]’ (Y6). However, their drawings were incorrect. Two students (Y15) were sure they had solved it incorrectly because their answers were negative. Other students used drawn curves to check their answers and intersection points. For example, Y4 was unsure whether or not he had solved the question correctly because the intersection points he found did not match the curves. Three students (U46; Y7) were unsure if they had solved the item correctly for the following reasons; U6 could not distinguish which function was the top function: ‘not sure which one [in] $$\int{[f\!\left (x\right )-g\!\left (x\right )]\ dx}$$ is f(x)’ (U6). U4 had difficulty with $$x=y^{2}$$ and said: ‘$$x=y^{2}$$ is not a function so that confused me’; Y7 was unsure because ‘[I] forget to account for the other part [the part which is under x-axis] of $$x=y^{2}$$.’ U1 was not confident about his response, saying, ‘usually with math question you are pretty sure when you have got it right. I was pretty hazy when I go through. I was over confident when I started.’ Finally, Y2 was sure he had solved the question correctly but could not explain why; his answer was incorrect. Five students (U58; Y568) thought they had solved Q2 correctly, but only U5 made an accurate post-judgement. She was sure because ‘I have utilized all the information I have in the question.’ Y5 could not provide a reason why he thought he had solved the question correctly. The other three students did not provide any reliable justification for their judgement. For instance, Y6 said, ‘It works logically and reasonably, I am confident with the working’, and Y8 responded: ‘I think I solved it correctly… because it was a lot simple one to do once I was start looking at it in more depth [sic].’ Four students (U349; Y7) were unsure if they had solved the question correctly, but three (U349) did so correctly. Their reasons for their lack of confidence differed. U4 was unsure because he did not use a part of the information given in the problem, the area of region C. U9 was ‘not sure because I do not feel I can justify the method I have used.’ U3 was not confident because: ‘It is a question that I haven’t come across for a long time… It is just being so long since I have to do a non-calculation–based integration and or differentiation [question] that it’s pretty much taken me by surprise. Always expect calculation heavy question. Not prepared.’ Y7 was unsure because he was not confident about being able to integrate both sides of $$f^{\prime}(6)+12=f^{\prime}(0)$$, and in his solution he had incorrectly written $$f(6)+12x+c=f(0)$$ for the intergral of both sides. The remaining eight students (U1267; Y1234) could not come up with an answer to this item so were categorized as being sure they had not solved the question correctly. To sum up, five factors were found in relation to students’ feelings about whether they had solved Q2 correctly, including how much of the given information was being used; how well the method could be justified; how confident the student was with his/her working; how familiar the student was with the question; and how easily the answer was found. For Q3, eight students (U345689; Y67) were sure they had posed the problem correctly; all but Y6 being correct. Five students (U45689) were certain because they had solved their posed problems and had obtained the answer one. Y6 believed that if someone solved his problem, the answer would be one. Y7 was sure because he thought he had set up the integral correctly. U3 thought his solving ‘was thorough and multiple wrong solutions excluded.’ One student who was unsure posed a correct problem (U7), but his reasoning was not related to the content and indicated that he was not confident with his working. He said, ‘still I can see a smile on the researcher’s face’ indicating he attempted to gain feedback about the correctness of the solution from the interviewer. Eight students (U12; Y123458) could not pose a problem so were coded as being sure that they had not correctly solved the task. 4.2.3 Making a drawing related to the task For Q1, all students drew curves while answering the item but only 11 (U2456789; Y2378) did so correctly. For the six other students, there were two common reasons for incorrect curves. Firstly, neglecting the part of $$x\!=\!y^{2}$$ under the x-axis (U13; Y456) (Fig. 3b). Secondly, trying to sketch the graph from memory (Fig. 3a) and not checking their drawing by substituting some values from the domain to confirm the relationship between x and y. In fact, during think-alouds, none of these six students mentioned substituting values into the function/relation to help sketch or check the graph. Had they used this checking procedure they might have been more able to identify errors more effectively. Fig. 3. View largeDownload slide Examples of students’ errors when drawing the curves and a correct drawing. Fig. 3. View largeDownload slide Examples of students’ errors when drawing the curves and a correct drawing. Students’ ability to find the area enclosed between the curves correctly was closely related to their ability to draw the curves correctly, especially when the upper and lower functions did not change in the enclosed area. Seven students (U45689; Y78) sketched correct curves and successfully found the area with respect to the y-axis. Four of these (U589; Y8) correctly drew the graph and integrated with respect to the x-axis successfully (Table 6). The lower success level with respect to the x-axis was due to changes in the lower function at x = 1 from $$y=-\sqrt{x}$$ to y = x − 2. The final piece of evidence that supports the importance of curve sketching when using integrals to find areas is that no student who drew an incorrect curve was successful with the item (Table 7). Table 7 Relationship between a correct drawing and finding the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 Table 7 Relationship between a correct drawing and finding the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 For Q2, students did not need to make a drawing as a sketch was already provided. For Q3, 11 students (U34579; Y235678) tried to sketch a curve and a line to develop a better understanding of what they should consider a curve and a line when creating their problem. Of those, six successfully posed a correct question (Table 8). However, when Q3’s results are compared to Q1’s, making a drawing was not as closely related to success with the item, partly because some students who created drawings were still not able to pose an appropriate question. Table 8 Relationship between making a drawing and being successful with Q3 Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) Table 8 Relationship between making a drawing and being successful with Q3 Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) 4.2.4 Checking calculations and answers For Q1, more than half of the students in the sample did not check their solutions, even though the question asked them to solve the question in two ways so they had the opportunity to compare their final answers. Four students from Case 1 (U1589) and three from Case 2 (Y246) did some form of checking but only checked their calculations and, as identified earlier, only Y2 checked his drawing of the curves. Of these, four (U1; Y246) had errors in their working; and three (U1; Y46) could not find their errors. Y2 was able to amend his drawing for $$x=y^{2}$$ (Fig. 4) on his fourth try. Of the three students who solved the question correctly and did some form of checking (U589)4, two of the three checked to see if they had found the same answer in both integrations, and one checked his calculations to ensure he had not made any mistakes when finding the intersection points. Fig. 4. View largeDownload slide Y2’s attempts in drawing $$x=y^{2}$$. Fig. 4. View largeDownload slide Y2’s attempts in drawing $$x=y^{2}$$. When solving Q2, only U5 did some form of check. After she found the correct answer, she revisited her working to make sure she did not make a mistake. Six students (U45689; Y6) used a form of check in Q3; after posing their problem they solved it to see if they got 1 as the answer. However Y6, who had posed a question with a different answer, could not identify where he had made the mistake. 5. Discussion and conclusion In this study, a multiple case study was used to explore what students had learned about the relationship between integrals and area. The study sample comprised nine first-year university and eight Year 13 students who each participated in an individual semi-structured interview. The results in this paper adds to the literature about students’ understanding of the integral-area relationship in several ways. Firstly, creating a profile of students’ metacognitive experiences and skills in relation to the integral-area relationship has not been attempted previously. Students’ responses show that aspects of their metacognitive experiences and skills could be further developed. Secondly, students’ problem posing ability for the integral-area relationship (Q3) has not been studied previously. The students’ responses show such question types could be useful for probing students’ understanding of integral-area relationships. For Q1, several students based their pre-judgement of whether they were able to solve the question on their knowledge of techniques to find the antiderivative of integrands, or knowledge of how to find enclosed area in general. However, making such a judgement prior to making a sketch of the relations is premature as the shape of the enclosed area affects both the methods that can be used to find that area and the ease of use of different methods. This suggests that teachers and lecturers need to be aware of the potential interplay of students’ mathematical content knowledge with their metacognitive experiences and skills, and how the order in which thinking occurs can affect students’ chances of success with a task. With Q1, all students made a drawing to help them answer the question, which could be an indication of the presence of that metacognitive skill, but did not naturally want to do this as part of decision-making about the solvability of the problem. Emphasizing the value of preliminary exploration, such as identifying the shape of the enclosed area before making any judgement about an integral-area task, reduces the chance of making an incorrect pre-judgement and increases the chance of setting up the integral correctly, especially the top function. Similarly, the interplay between content knowledge and metacognitive experiences and skills is important for students to be able to judge their success with a task. The 10 students who were unsure they had found the correct answer, or were sure they had solved Q1 incorrectly, were not seen to revisit and check their solution. This suggests two things. First, these students had not learned from their teachers and lecturers that checking is a normal part of answering mathematical questions; second, they may not possess strategies to check their answers for integral-area questions. For their pre-judgement of Q2, fewer students were sure they could solve the question correctly (12 for Q1 and 4 for Q2), suggesting that they were not as confident about answering questions that focus on conceptual knowledge. In addition, the reasons given for their post-judgement shows students were not confident with their solutions. This suggests that the students could have benefited from being exposed to more conceptually based tasks as part of their class work. Classroom observations identified that problem posing (Q3) was not part of the work in class or the assessment at either case, yet eight students (seven from Case 1) were able to pose a question to meet the provided constraint. Students from Case 1 may have been more successful because of their greater experience working with integral calculus questions, having likely passed Year 12 and Year 13 calculus before starting their university course. Students from Case 1 also had more accurate pre-judgements of their ability to solve integral questions and better post-judgements of whether they had solved the questions correctly than students at Case 2. However, several students from both Cases made an incorrect pre- or post-judgement. Therefore, as for Q1, highlighting the drawing of the enclosed area and using monitoring strategies more often might help some students understand where they can make mistakes. Overall, more than half the students did not appear to check their calculations and answers to any of the integral questions during the interviews, suggesting that this metacognitive skill should be emphasized in teaching. Ways in which students can check their work should be discussed and if needed suggested to them (e.g., approximating area using geometric shapes, differentiating antiderivatives and checking that area is positive). Furthermore, if lecturers and teachers themselves used monitoring strategies regularly, especially when modelling the answering of questions on the board, and encouraged students to do so as a routine part of their work, students might use this metacognitive skill more often. Such a change in practise may not seem natural to teachers or lecturers as the questions they work on publicly are seldom genuine problems, rather they tend to be familiar tasks that have been seen and solved previously. However, as they are working with novices, it is important to model the actions of mathematicians working on unfamiliar problems, especially as the practise of checking work has the potential to lead to fewer errors and students developing greater confidence in their understanding of the mathematics they are learning. Evidence has also presented that suggests if a student is not introduced to recognized and accepted strategies for checking their work they may create their own proxies, such as whether or not all information provided in a problem has been used. While this is an astute measure that can be used on many text and assessment items, it is not one that can be used on genuine mathematical problems. By using this strategy, rather than learning about working as a mathematician, students may be learning the skills of creating text and assessment items. As already noted for Q1, a typical question about the integral-area relationship, all students made a drawing to help them solve the problem. However, for non-typical questions, like Q3, only eleven of the seventeen students made drawings. This suggests that for some students making a drawing may be part of a memorized procedure for a particular type of problem rather than being a general strategy to help understand the requirements of any question involving functions and relations. This in turn suggests that the importance of making a drawing to help understand mathematical questions could be emphasized more generally. For instance in Q3, if students had made a drawing for their proposed curve and line, there was a greater chance they could find the suitable curve and line that fitted the given condition. As such, lecturers and teachers should be encouraged to make a drawing for each question they solve in classes for students, and encourage students to also do this. While freehand sketches are powerful as they may be the only method available in assessment situations, free online programmes such as https://www.desmos.com/ could also be used. In terms of their mathematical performance on the integral-area relationship, students’ procedural knowledge was better developed than their conceptual knowledge. This is also consistent with previous studies showing students are able to undertake routine procedures to find area using integral techniques; however their knowledge about why such a procedure is used is limited (Artigue, 1991; Grundmeier et al., 2006; Mahir, 2009; Rasslan & Tall, 2002; Thomas & Hong, 1996). A lack of algebraic manipulation skills and prior knowledge were barriers for several students, due to their not being able to find intersection points, sketch the graph correctly or find the equation of lines and curves. These findings, also highlighted by previous studies (e.g., Kiat, 2005), indicate that some students would have benefitted from improving their knowledge of functions and relations, and/or algebraic manipulation and/or graph sketching prior to starting integral calculus (Kiat, 2005). Conceptually, several students seemed to believe the area or net area underneath the graph of f′(x) is equal to the $$\int{f^{\prime}\!\left (x\right )dx}$$, rather than its signed net area. A lack of conceptual knowledge about the definite integral is also shown in the literature (e.g., Mahir, 2009; Thomas & Hong, 1996). The fact that six students did not use the integral-area relationship to solve Q2 suggests that lecturers and teachers should also use the graph of the derivative in integral-area questions they solve in their classes to help students to understand that $${{{\int _{a}^{b}}}}f^{\prime}(x)dx=f(b)-f(a)$$ is another representation of the first part of the FTC and equivalent to $${{{\int _{a}^{b}}}}f(x)dx=F(b)-F(a)$$ where $$F^{\prime}\!=f$$ . This practise might provide a better opportunity for students to realise that the relationship between the graph of the function and the area under the curve also exists for the derivative function. In general, students from Case 2 had less developed conceptual knowledge of the definite integral and the integral area relationship than students from Case 1. One simple explanation is that the university students had more experience with this topic as they have been exposed to it for longer. However, another possible explanation relates to the teaching of the Riemann sums5. In Case 1, Riemann sums were the focus of teaching definite integrals, and examples were solved in this regard. In addition, during the teaching of other topics, such as finding volumes by slicing and cylindrical shells, the proofs for volume formulas were taught to Case 1 students. The ideas used in those proofs are related to Riemann sums, therefore help students to develop a better understanding of Riemann sums and the Riemann integral. At Case 2, the teacher did not introduce Riemann sums until end of the integral calculus topic and no example was solved in the classroom. This teacher started with definite integrals, declaring his procedural approach towards teaching the topic by saying: ‘I am going to take the expedient route... I am going to give you the application…saying without proving….’ Such an approach towards teaching mathematics is related to the development of instrumental understanding (Skemp, 1976), which can have negative consequences for students’ learning. For example, it influences students’ attitudes towards mathematics and their understanding of the structure of mathematics. One reason why this teacher did not focus on Riemann sums is likely related to The New Zealand Curriculum (Ministry of Education, 2007) and the assessment of learning in Year 13 through the National Certificate of Educational Achievement (NCEA) level 3 mathematics achievement standards (New Zealand Qualifications Authority, 2013). At this level, this curriculum and these assessments are used in most schools in New Zealand. Being able to use numerical methods of integration is prescribed by these documents; however Riemann sums are not highlighted. Similarly, the textbook used in the college, Delta Mathematics (Barton & Laird, 2002) does not focus on the Riemann integral and only provides it as an appendix. Rather, the trapezium method and Simpson’s rule are provided in the main body of the textbook and were emphasized during teaching. However, it seems the proofs behind these methods are more complicated than the Riemann sums approach as their formulas have more elements. Another possible reason for this teaching approach is that numerical methods are presented at the end of integral calculus topic in this text so were also taught at the end of the integral calculus topic, rather than being an introduction to finding areas. However students at Case 1 first learnt about Riemann sums, then were exposed to the Fundamental Theorem of Calculus and integral techniques. This may be because numerical methods are more conceptual and need more time to understand, and while students from Case 1 had more time to understand them as they were introduced to different topics in integral calculus, this opportunity was not available to students from Case 2 due to the curriculum and assessment regime being followed. The differences between students’ performance in Case 1 and 2 in relation to definite integrals suggests changes in the New Zealand schools’ mathematics curriculum document to focus more on teaching Riemann sums might be beneficial. This could help to increase students’ conceptual and procedural knowledge. By knowing the rationale behind the relationship between a definite integral and the area under a curve through Riemann sums, students are likely to develop their conceptual knowledge as to why a definite integral can be used for finding area. Teaching Riemann sums would also help students develop their procedural knowledge by knowing what to integrate and how to set up the bounds of an integral (Sealey, 2006, 2014). Finally, it needs to be reinforced that in this exploration of student learning of the integral-area relationship, the responses of only 17 students have been analyzed, so it is unlikely that the results obtained represent all students’ problem-solving behaviours in this topic. The Cases were chosen from an area that was geographically accessible to the authors. The school decile of the College, a measure of the Socio-Economic Status (SES) in New Zealand, was 10; 10 being high. As a consequence, the findings might not be applicable to students who attend colleges with lower decile ratings. Possible gender differences were also not explored because of the small sample size, although in New Zealand, gender differences in mathematics achievement are generally not large. Nor was it possible to explore whether there may a difference between students of different ethnic groups. Such questions are important to address in future studies, ones with larger sample sizes set in different contexts. Acknowledgements The first author wishes to acknowledge his gratitude to Associate Professor Robin Averill for her support, guidance and many hours of valuable discussions in relation to this study. Farzad Radmehr received his first PhD in Mathematics Education at the Ferdowsi University of Mashhad in 2014. He recently received his second PhD at Victoria University of Wellington in 2016. He is an assistant professor in the Department of Applied Mathematics of Ferdowsi University of Mashhad. Farzad is interested in the teaching and learning of mathematics at upper secondary and tertiary level, in particular, introductory level undergraduate courses. Michael Drake received his PhD in Mathematics Education at Victoria University of Wellington in 2010. He is a senior lecturer in the School of Education of Victoria University of Wellington. Michael is interested in the teaching and learning of mathematics, in particular, number and numeracy, algebra, measurement, probability and statistics during the middle years of schooling. Footnotes 1  These facets are described in the literature section. 2  c is a constant and included to describe situations that quantities do not directly correspond to the $$f\left (x_{i}\right )$$ and $$\Delta x$$ (Sealey, 2014). 3  This study only reports part of the students’ interviews. 4  Students who compared their answers when responding to ME2 are not included in this analysis as solving the task was completed before responding to ME2. 5  All teaching of integral calculus was recorded and observed by the first author. References Anderson , L. W. (ed) , Krathwohl , D. R. (ed) , Airasian , P. W. , Cruikshank , K. A. , Mayer , R. E. , Pintrich , P. R. , Raths , J. & Wittrock , M. C. ( 2001 ) A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (Complete edition) . New York, N.Y : Longman . Artigue , M. ( 1991 ) Analysis. In Advanced mathematical thinking (D. Tall ed) . Boston, M.A : Kluwer , pp. 167 -- 198 . Google Scholar CrossRef Search ADS Anton , H. , Bivens , I. & Davis , S. ( 2012 ) Calculus: Early Transcendentals . New Jersey, N.Y : Wiley Global Education . Barton , D. & Laird , S. ( 2002 ) Delta Mathematics . Auckland, New Zealand : Pearson Education . Bartsch , K. & Wellman , H. M. ( 1995 ) Children talk about the mind . New York, United State of America : Oxford University Press . Baxter , P. , & Jack , S. ( 2008 ) Qualitative case study methodology: Study design and implementation for novice researchers . Qual. Res , 13 , 544 -- 559 . Bezuidenhout , J . ( 2001 ) Limits and continuity: some conceptions of first-year students . Int. J. Math. Educ. Sci. Technol. , 32 , 487 -- 500 . Google Scholar CrossRef Search ADS Charmaz , K. ( 2006 ) Constructing grounded theory: A practical guide through qualitative analysis . London, England : Sage Publication Ltd. Christou , C. , Mousoulides , N. , Pittalis , M. , Pitta-Pantazi , D. & Sriraman , B. ( 2005 ) An empirical taxonomy of problem posing processes . ZDM , 37 , 149 -- 158 . Efklides , A. ( 2001 ) Metacognitive experiences in problem solving: Metacognition, motivation, and self-regulation . In Trends and prospects in motivation research (A. Efklides, J. Kuhl & R.M. Sorrentino eds) . Dordrecht, the Netherlands : Kluwer , pp. 297 -- 323 . Google Scholar CrossRef Search ADS Efklides , A . ( 2006 ) Metacognition and affect: What can metacognitive experiences tell us about the learning process? Educ. Res. Rev. , 1 , 3 -- 14 . Google Scholar CrossRef Search ADS Efklides , A . ( 2008 ) Metacognition: Defining its facets and levels of functioning in relation to self-regulation and co-regulation . Eur. Psychol. , 13 , 277 . Google Scholar CrossRef Search ADS Eisenhardt , K. M. & Graebner , M. E. ( 2007 ) Theory building from cases: opportunities and challenges . Acad. Manag. J. , 50 , 25 -- 32 . Google Scholar CrossRef Search ADS Ericsson , K. A. & Simon , H. A. ( 1993 ) Protocol analysis: Verbal reports as data (Revised Edition) . Cambridge, Massachusetts : The MIT Press . Fisher , R. J. ( 1993 ) Social desirability bias and the validity of indirect questioning . J. Consum. Res. , 20 , 303 -- 315 . Google Scholar CrossRef Search ADS Flavell , J. H. ( 1979 ) Metacognition and cognitive monitoring: A new area of cognitive developmental inquiry . Am. Psychol. , 34 , 906 -- 911 . Google Scholar CrossRef Search ADS Flavell , J. H. , Miller , P. H. & Miller , S. A. ( 1993 ) Cognitive development . New Jersey, United State of America : Prentice-Hall . Garofalo , J. & Lester Jr , F. K. ( 1985 ) Metacognition, cognitive monitoring, and mathematical performance . J. Res. Math. Educ. , 16 , 163 -- 176 . Google Scholar CrossRef Search ADS Grundmeier , T. A. , Hansen , J. & Sousa , E. ( 2006 ) An exploration of definition and procedural fluency in integral calculus . PRIMUS , 16 , 178 -- 191 . Google Scholar CrossRef Search ADS Hong , Y. Y. & Thomas , M. O. ( 2015 ) Graphical construction of a local perspective on differentiation and integration . Math. Educ. Res. J. , 27 , 183 -- 200 . Google Scholar CrossRef Search ADS Jacobse , A. E. & Harskamp , E. G. ( 2012 ) Towards efficient measurement of metacognition in mathematical problem solving . Metacognition Learn. , 7 , 133 -- 149 . Google Scholar CrossRef Search ADS Jones , S. R. ( 2013 ) Understanding the integral: Students’ symbolic forms . J. Math. Behav. , 32 , 122 -- 141 . Google Scholar CrossRef Search ADS Jones , S. R. ( 2015 ) Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts . J. Math. Behav. , 38 , 9 -- 28 . Google Scholar CrossRef Search ADS Kiat , S. E. ( 2005 ) Analysis of students’ difficulties in solving integration problems . Math. Educ. , 9 , 39 -- 59 . Kim , Y. R. , Park , M. S. , Moore , T. J. & Varma , S. ( 2013 ) Multiple levels of metacognition and their elicitation through complex problem-solving tasks . J. Math. Behav. , 32 , 377 -- 396 . Google Scholar CrossRef Search ADS Kitchner , K. S. ( 1983 ) Cognition, metacognition, and epistemic cognition . Hum. Dev. , 26 , 222 -- 232 . Google Scholar CrossRef Search ADS Kouropatov , A. & Dreyfus , T. ( 2013 ) Constructing the integral concept on the basis of the idea of accumulation: Suggestion for a high school curriculum . Int. J. Math. Educ. Sci. Technol. , 44 , 641 -- 651 . Google Scholar CrossRef Search ADS Kreutzer , M. A. , Leonard , C. , Flavell , J. H. & Hagen , J. W. ( 1975 ) An interview study of children’s knowledge about memory . Monogr. Soc. Res. Child. Dev. , 40 , 1 -- 60 . Google Scholar CrossRef Search ADS PubMed Kuhn , D. ( 2000 ) Theory of mind, metacognition, and reasoning: A life-span perspective . Children’s reasoning and the mind (Mitchell, P. & Riggs, K. J. eds) . Hove, UK : Psychology Press , pp. 301 -- 326 . Lester , F. K. ( 1982 ) Building bridges between psychological and mathematics education research on problem solving . Mathematical problem solving (Lester, F. K. & Garofalo, J. eds) . Philadelphia, Pennsylvania : The Franklin Institute Press , pp. 55 -- 85 . Mahir , N . ( 2009 ) Conceptual and procedural performance of undergraduate students in integration . Int. J. Math. Educ. Sci. Technol. , 40 , 201 -- 211 . Google Scholar CrossRef Search ADS McNamara , D. S. ( 2011 ) Measuring deep, reflective comprehension and learning strategies: challenges and successes . Metacognition Learn. , 6 , 195 -- 203 . Google Scholar CrossRef Search ADS Ministry of Education . ( 2007 ) Achievement objectives by Learning Area, The New Zealand Curriculum . Wellington, New Zealand : Learning Media . New Zealand Qualifications Authority (NZQA , 2013 ) NCEA level 3 mathematics achievement standard . Wellington, New Zealand : Learning Media Limited . Özsoy , G . ( 2011 ) An investigation of the relationship between metacognition and mathematics achievement . Asia Paci .Educ. Rev. , 12 , 227 -- 235 . Google Scholar CrossRef Search ADS QS World University Rankings . ( 2014 ) Study in New Zealand . Retrieved from http://www.topuniversities.com/where-to-study/oceania/new-zealand/guide Rasslan , S. & Tall , D. ( 2002 ) Definitions and images for the definite integral concept . Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, Norwich (Cockburn, A. D. & Nardi, E. eds) . UK . Rösken , B. & Rolka , K. ( 2007 ) Integrating intuition: The role of concept image and concept definition for students’ learning of integral calculus . Montana Math. Enthusiast , 3 , 181 -- 204 . Schneider , W. & Lockl , K. ( 2002 ) The development of metacognitive knowledge in children and adolescents . Applied metacognition (Perfect, T. J. & Schwartz, B. L. eds) . Cambridge, England : Cambridge University Press . Google Scholar CrossRef Search ADS Schneider , W. & Artelt , C. ( 2010 ) Metacognition and mathematics education . ZDM , 42 , 149 -- 161 . Google Scholar CrossRef Search ADS Schoenfeld , A. H. ( 1985 ) Math. Probl. Solving . New York, United State of America : Academic Press . Schraw , G. ( 1998 ) Promoting general metacognitive awareness . Instr. Sci. , 26 , 113 -- 125 . Google Scholar CrossRef Search ADS Sealey , V. ( 2006 ) Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient? Proceedings of the 28th Annual Meeting of the North American Chapter of the international Group for the Psychology of Mathematics Education (Alatorre, S., Cortina, J. L., Saiz, M. & Mendez, A. eds) . Mexico : Merida , pp. 46 -- 53 . Sealey , V . ( 2014 ) A framework for characterizing student understanding of Riemann sums and definite integrals . J. Math. Behav. , 33 , 230 -- 245 . Google Scholar CrossRef Search ADS Silver , E. A. ( 1982 ) Knowledge organization and mathematical problem solving . Mathematical problem solving (Lester, F. K., & Garofalo, J. eds) . Philadelphia, Pennsylvania : The Franklin Institute Press , pp. 55 -- 85 . Skemp , R. ( 1976 ) Relational understanding and instrumental understanding . Mathematics Teaching , 77 , 20 -- 26 . Tall , D. ( 1992, August ) Students’ difficulties in calculus. In Proceedings of Working Group 3 on Students difficulties in calculus, ICME-7 . Québec, Canada , pp. 13 -- 28 . Tarricone , P. ( 2011 ) The taxonomy of metacognition . East Sussex, England : Psychology Press . Thomas , M. O. J. & Hong , Y. Y. ( 1996, June ) The Riemann integral in calculus: Students’ processes and concepts . In Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA) , pp. 572 -- 579 . Veenman , M. & Elshout , J. J. ( 1999 ) Changes in the relation between cognitive and metacognitive skills during the acquisition of expertise . European Journal of Psychology of Education , 14 , 509 -- 523 . Google Scholar CrossRef Search ADS Veenman , M. V. , Kerseboom , L. & Imthorn , C. ( 2000 ) Test anxiety and metacognitive skillfulness: Availability versus production deficiencies . Anxiety, Stress and Coping , 13 , 391 -- 412 . Google Scholar CrossRef Search ADS Veenman , M. V. J. ( 2011 ) Learning to self-monitor and self-regulate . Handbook of research on learning and instruction (Mayer, R. E., & Alexander, P. A. eds) . New York, N.Y : Routledge , pp. 197 -- 218 . Verschaffel , L. ( 1999 ) Realistic mathematical modelling and problem solving in the upper elementary school: Analysis and improvement . Teaching and learning thinking skills: Contexts of learning (Hamers, J. H. M., Van Luit, J. E. H. & Csapo, B. eds) . Lisse, the Netherlands : Swets & Zeitlinger , pp. 215 -- 240 . Yin , R. K. ( 2014 ) Case study research: Design and methods . United State of America : Sage publications Inc. © The Author(s) 2018. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Teaching Mathematics and Its Applications: International Journal of the IMA Oxford University Press

# Students’ mathematical performance, metacognitive experiences and metacognitive skills in relation to integral-area relationships

, Volume Advance Article – Mar 21, 2018
22 pages

/lp/ou_press/students-mathematical-performance-metacognitive-experiences-and-odcn2VmyVn
Publisher
Oxford University Press
© The Author(s) 2018. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com
ISSN
0268-3679
eISSN
1471-6976
D.O.I.
10.1093/teamat/hry006
Publisher site
See Article on Publisher Site

### Abstract

Abstract Previous studies have explored students’ understanding of the relationship between definite integrals and areas under curves, but not their metacognitive experiences and skills while solving such problems. This paper explores students’ mathematical performance, metacognitive experiences and metacognitive skills when solving integral-area tasks by interviewing nine university and eight Year 13 students using a think-aloud protocol. The findings show that the students could have benefitted from their teachers and lecturers placing greater emphasis on both their conceptual understanding of integral-area relationships and their metacognitive experiences and skills. 1. Introduction Research has reported that students have difficulty with integral calculus concepts (e.g., Jones, 2013; Kouropatov & Dreyfus, 2013; Kiat, 2005; Sealey, 2014; Thomas & Hong, 1996). While the majority of students studying integral calculus can successfully apply basic procedures to find antiderivatives, their understanding of the underlying concepts is limited. For example, Thomas & Hong (1996) report that many students regard integral calculus ‘as a series of processes with associated algorithms and do not develop the grasp of concepts which would give them the necessary versatility of thought (p. 577).’ Studies have also shed light on students’ understanding of the relationship between definite integrals and areas under curves (e.g., Jones, 2013, 2015; Mahir, 2009; Sealey, 2014); however, little attention has been paid to students’ metacognition while solving integral-area problems. Metacognition is knowledge or cognitive activity that individuals have about their cognitive activities (Flavell et al., 1993). It is a ‘meta-level knowledge and mental action used to steer cognitive processes (Jacobse & Harskamp, 2012, p. 133).’ To successfully solve mathematical problems, metacognitive knowledge and activities are necessary (Lester, 1982; Özsoy, 2011; Schoenfeld, 1985; Silver, 1982; Verschaffel, 1999). However, the use of metacognition to improve problem-solving skills is sometimes ignored in teaching (Schoenfeld, 1985), and a lack of development of students’ metacognition has been reported (e.g., Jacobse & Harskamp, 2012). Metacognition has three facets (Efklides, 2006, 2008; Kim et al., 2013; Schneider & Artelt, 2010; Tarricone, 2011): metacognitive knowledge or knowledge of cognition, metacognitive skills or regulation of cognition and metacognitive experiences or concurrent metacognition.1 As these three facets have not been well explored for the integral-area relationship at either the upper secondary or undergraduate level, there is the possibility that students’ outcomes with integral-area questions might be improved by studying students’ metacognition while solving such questions. This study uses the research question: What are students’ metacognitive experiences and skills while solving integral-area problems? to explore that possibility. 2. Literature In this section the relevant integral calculus and metacognition literature is reviewed to frame the study and to help justify the study’s findings. 2.1 Integral calculus Studies have shown that many students who are able to undertake routine procedures that utilize integral techniques to find areas have a limited understanding of why such procedures should be undertaken (e.g., Artigue, 1991, Thomas & Hong, 1996). For example, Kiat (2005) found that if the graph of the integrand is not given, students can incorrectly set up area integrals, suggesting a procedural understanding of definite integrals that does not include the connection between definite integrals and area. Kiat also identified that 55% of students could not set up the correct integrals to find a shaded area in a question where one of the curves was above and the other below the x-axis. Mahir (2009) explored the conceptual and procedural knowledge of undergraduate students in integral calculus. He found that the students had developed a satisfactory level of procedural knowledge in that 92% and 74% correctly answered two questions that could be solved procedurally. However, for the three questions based on conceptual knowledge, only 8%, 16% and 24% were able to do so. In addition, Rasslan & Tall (2002) have presented evidence of students’ difficulties in understanding the definite integral as an area under a curve when working with piecewise functions and improper integrals. Important concepts, such as functions, limits, rate of change and multiplication, are involved in Riemann sums and definite integrals, $${\mathop{\lim }\nolimits _{n\rightarrow \infty } \sum ^{n}_{i=1}{f\left (c_{i}\right )\triangle x } }$$ (Sealey, 2006). However students’ difficulties with understanding the definite integral as the limit of a sum has also been highlighted (Grundmeier et al., 2006). Sealey (2014) sought to gain an insight into how students might develop the concept of the Riemann integral, proposing a framework for characterising students’ understanding of Riemann sums and definite integrals. This framework has a pre-layer that involves symbolicrepresentation $$\big[\frac{1}{c}\,\cdot\ f(x_{i})\big]$$2 and or $$\left [c\cdot\Delta x\right ]$$, and four layers, including product $$\big[\frac{1}{c}\,\cdot\, f(x_{i})\big]\,\cdot\,\left [c\,\cdot\,\Delta x\right ]$$, summation $$\sum \nolimits _{i=1}^{n}{f(x_{i})\Delta x}$$, limit $${\mathop{\lim }\nolimits _{n\to \infty } \sum ^{n}_{i=1}{f(x_{i})\triangle x}\ }$$ and function $$f\!\left (b\right )={\mathop{\lim }\nolimits _{n\to \infty } \sum ^{n}_{i=1}{f(x_{i})\Delta x} }$$. Using this framework, Sealey found that layer 1, the product of f(x) and △x, is the most complex part for students. ‘Difficulties in this layer are not necessarily related to the operation of multiplication and performing calculations, but are typically related to understanding how the product is formed and understanding how to use each factor within the product (p. 238).’ In terms of the symbols and notations of integral calculus, the ‘dx’ in $$\int{f(x)dx}$$ causes conflicts and contradictions for a number of students. For example, in integral calculus dx shows that the integration should be done with respect to x, however, some students are told not to cancel dx in $$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}$$ when solving questions related to the chain rule as it has no separate meaning (Tall, 1992). Students can also have great difficulty learning the symbolic definition of a definite integral (Grundmeier et al., 2006). From a sample of 52 students, only one was able to provide the symbolic definition of a definite integral and only 35% were able to provide a correct verbal definition. However, the lack of such a definition did not affect many students’ ability to answer routine integration problems, as more than 60% of the students were able to evaluate the definite integral of a trigonometric function. Other research supports that a majority of students may not be able to write meaningfully about the definition of a definite integral (e.g., Rasslan & Tall, 2002). One reason given relates to the learning and teaching approach of some teachers and lecturers, who focus on the procedural aspects of calculus (Bezuidenhout, 2001). 2.2 Facets of metacognition That there are three general facets of metacognition is recognized in a number of studies (e.g., Efklides, 2006, 2008; Kim et al., 2013; Schneider & Artelt, 2010). In this section each facet is unpacked in some detail. 2.2.1 Metacognitive knowledge Metacognitive knowledge derives from a person’s long-term memory (Efklides, 2006) and can be defined as ‘knowledge of cognition in general as well as awareness and knowledge of one’s own cognition (Anderson et al., 2001, p. 46).’ It is a declarative knowledge about cognition that refers to both an individuals’ explicit or implicit knowledge about persons, tasks, goals and strategies (Efklides, 2006, 2008). This explicit or implicit knowledge can be ideas, beliefs and theories about how individuals perform and feel about different tasks, the features of different tasks and how they work; the goals individuals follow within different tasks and situations; and finally when, why and how different strategies should be used (Efklides, 2006, 2008). Metacognitive knowledge also includes knowledge about the different cognitive functions, for example thinking, in terms of what they are and how they work (Efklides, 2006, 2008). Moreover, it encompasses knowledge of the criteria of the validity of knowledge, including knowledge about limits of knowing, criteria of knowing and certainty of knowing (Kitchner, 1983). Finally, theories of mind (Bartsch & Wellman, 1995)—beliefs that individuals have about people’s minds, including themselves, can be considered as metacognitive knowledge (Kuhn, 2000). Communicating with others, self-monitoring, monitoring other people’s cognitive activities and being aware of personal metacognitive experiences help develop, revise and update metacognitive knowledge (Efklides, 2006, 2008; Flavell, 1979; Kim et al., 2013). 2.2.2 Metacognitive experience Metacognitive experience, is ‘what the person is aware of and what she or he feels when coming across a task and processing the information related on it (Efklides, 2008, p. 279).’ Unlike metacognitive knowledge and skills, metacognitive experiences take place in working memory (Efklides, 2006). They include feelings of knowing, difficulty, familiarity, confidence and satisfaction, as well as estimations of effort and time needed to be spent on tasks, judgements of learning and estimates of the correctness of solutions (Efklides, 2006, 2008). Online task-specific knowledge is another aspect of metacognitive experiences (Efklides, 2001, 2006,2008; Schneider & Lockl, 2002). This refers to the task information plus the ideas and thoughts an individual have about the task they are working upon, along with the metacognitive knowledge they retrieve from memory and use to complete the task. 2.2.3 Metacognitive skills The last facet of metacognition, metacognitive skills are deliberate activities that help control cognitive activities (Schraw, 1998). They include task orientating, planning, monitoring, regulating and evaluating (Efklides, 2006, 2008). Task orientating is about understanding the task requirements, planning relates to the steps that need to be taken to achieve a goal or complete a task, monitoring refers to activities undertaken while implementing a strategy to assess its progress and effectiveness, while evaluating and regulating relate to checking the product of cognitive processing and adjusting it as required (Efklides, 2006, 2008; Garofalo & Lester, 1985; Schraw, 1998; Veenman & Elshout, 1999). 3. Research Method To develop an understanding of students’ metacognitive experiences and skills while completing integral-area tasks, a multiple case study (Yin, 2014) was used. In the research, a case was considered to be an educational institution in New Zealand (i.e., a university or college) in which integral calculus was taught in 2014. The two selected cases consist of a sample of students who were interviewed. This design was chosen because the topic of integral calculus at Year 13 and first year university have a substantial content overlap in New Zealand, but, due to the different contexts in which they are learning, students at the two levels can have quite different metacognitive experiences and skills. As the research was exploratory, emerging theory was intended to be derived from an analysis of the data collected rather than existing beforehand. As such, grounded theory (Charmaz, 2006) was chosen as an approach to the data analysis because of its usefulness for constructing explanations of complex phenomena. 3.1 Study participants Case 1 is one of the top five universities in New Zealand (QS World University Rankings, 2014). To identify potential interview candidates, because the lecturers of the integral calculus course did not know their students’ calculus backgrounds, convenience sampling was used. All students enrolled were invited to participate by email and nine students (seven males and two females) volunteered. All were interviewed. Case 2 is one of 11 colleges in Wellington city that offer calculus courses to students. As the teachers did know their students’ calculus backgrounds, theoretical sampling was used to select potential candidates, meaning the sample was ‘suitable for illuminating and extending relationships and logic amongst constructs (Eisenhardt & Graebner, 2007, p. 27).’ Students with different calculus backgrounds were sought from two classes, a regular and a scholarship class. Based on their performance in differentiation, the mathematics teacher of the regular class selected six students; two each with low, medium and high achievement. Two scholarship students were selected by the Head of Mathematics to represent students with very high achievement. By choosing students with different calculus backgrounds, it was hoped a broad understanding of students’ metacognitive skills and experiences might be developed. Two students were chosen from each achievement group to avoid a single student representing a group. The students’ calculus backgrounds can be identified through their student code (Table 1) so potential connections between students’ metacognitive experiences and skills and their calculus background can be noted. Table 1 College interview participants Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Note. Identifying codes are based on the student’s calculus background (see column 3) Table 1 College interview participants Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Identifying code Gender Calculus background Y1 Male Low Y2 Male Low Y3 Male Medium Y4 Male Medium Y5 Male High Y6 Male High Y7 Male Scholarship student Y8 Male Scholarship student Note. Identifying codes are based on the student’s calculus background (see column 3) During the interviews, different levels of mathematical performance were observed amongst the volunteers from Case 1. To ascertain if the sample also represented a range of achievement, students’ responses to the nine items were later graded, with a possible total of 36 marks3. Subsequent analysis produced a mean score of 23.4 with standard deviation 8.0, indicating the students generated a range of different scores, suggesting that, similar to the college students, a range of achievement levels were represented in the sample. 3.2 Instruments In this section the measures used for collecting data are described, including those used to discover students’ metacognitive experiences and skills (Section 3.2.1), and for exploring students’ solution of integral-area questions (Section 3.2.2). 3.2.1 Instruments and procedures for measuring metacognitive experiences and skills Two main approaches are used to measure metacognition, offline and online measures (e.g., Jacobse & Harskamp, 2012; Schneider & Artelt, 2010). Offline measures evaluate metacognition without concurrent problem solving (Schneider & Artelt, 2010), whereas online measures seek to evaluate metacognition during problem solving. Both interviews (e.g., Kreutzer et al., 1975) and questionnaires (e.g., Schraw & Dennison’s 1994 Metacognitive Awareness Inventory) have been used previously to measure metacognition. Questionnaires are a common form of offline measure as they are usually undertaken after a task has been completed. They are generally comprised of statements about metacognitive monitoring and the regulation of tasks in which participants indicate the degree to which the statements apply to them (McNamara, 2011; Jacobse & Harskamp, 2012; Schneider & Artelt, 2010). Using questionnaires to measure metacognition has the advantage of being easily administered (Jacobse & Harskamp, 2012). However, as they require a participant to recall from memory what they did, or would do, when completing a task the results may be affected by memory distortion (Jacobse & Harskamp, 2012; McNamara, 2011), or the social desirability factor, ‘the basic human tendency to present oneself in the best possible light (Fisher, 1993, p. 303).’ One effective way to measure metacognition online is to conduct an interview using a think-aloud protocol (Ericksson & Simon, 1993). In such a protocol, a participant’s thinking should be verbalized and recorded while they are working on a task. This thinking should then be transcribed and coded based on a scheme (e.g., Kim et al., 2013), or, if transcribing is not undertaken, be coded based on a scheme (e.g., Jacobse & Harskamp, 2012). Using a think-aloud protocol to measure metacognition is time-consuming, however, it provides more reliable information than questionnaires as data are collected in-the-moment while a participant works on a task so those data are less affected by memory distortions or the social desirability factor (Jacobse & Harskamp, 2012; Veenman, 2011). Consequently, a think-aloud protocol was chosen for this study. Before their interview started, each student was asked to verbalize what they were thinking while working on each question; that is, think aloud rather than work silently. Prior to the interview, an information sheet that described the research project and the process for answering the questions had been given to students. Several approaches were considered for analysing students’ metacognitive experiences and skills (Veenman et al., 2000; Jacobse & Harskamp, 2012). The chosen approach combined four items from these two sources (Table 2). The two metacognitive experience items (ME1 & ME2) were adapted from the VisA instrument (Jacobse & Harskamp, 2012). The two metacognitive skills items (MS1 & MS2) were chosen from Veenman et al. (2000). MS1 was selected as making a drawing for a given problem has been found to be an important factor in mathematical problem solving (Jacobse & Harskamp, 2012). MS2 was selected to explore whether or not students monitor their work and progress when answering a mathematical question. Table 2 Items explored in relation to metacognitive experiences and skills Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Note. ME: Metacognitive Experiences; MS: Metacognitive Skills Table 2 Items explored in relation to metacognitive experiences and skills Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Themes of analysis in relation to metacognitive experiences and skills ME1 Student has an accurate pre-judgement of whether they are able to solve the problem MS1 Making a drawing related to the problem MS2 Checking calculations and answer ME2 Student has an accurate post-judgement of how they solved the problem Note. ME: Metacognitive Experiences; MS: Metacognitive Skills To use ME1 and ME2 to explore metacognitive experiences, students were initially given access to paper and writing materials. An item was then shown to the student and they were given a chance to read it. Before attempting the task, the student was asked: How well do you think you can solve this problem? They could select one of the following options: I am sure I will solve this problem; I am not sure whether I will solve this problem correctly or incorrectly; or I am sure I cannot solve this problem. After selecting an option, they were asked to provide a reason for their choice. Once they had finished work on the problem, a similar question was asked: rate your confidence for having found the correct answer. Similarly, students had three options and were encouraged to provide a reason for their selection. For MS1 and MS2, interviews were video recorded and copies of students’ written work made. Examining the written work and watching the interview transcripts identified the use of drawings, written or mental, for MS1. For MS2 the first author watched the videos and recorded any checking of calculations and answers that students were seen to undertake while, or subsequent to, answering the questions. 3.2.2 The integral-area problems Three integral-area problems that were used to probe students’ mathematical performance, metacognitive experiences and skills are described in this section. The questions were chosen as they each had the potential to evoke several cognitive processes. The appropriateness of each question for the sample students was ascertained by observing the integral calculus lessons at both the university and college, and by undertaking a document analysis. Q1. Please calculate the area enclosed between the curve $$x=y^{2}$$ and y = x − 2 in two ways. Which way is better to use? Why? The first part of Q1 (i.e., to calculate the area enclosed between curves) is a typical question in integral calculus that explores whether students know how to use the definite integral to find areas between curves. Similar questions were used in teaching and assessments in both Cases. While the two functions are not sophisticated, the curves cross the x-axis and the lower curve changes if the integral is set up with respect to the x-axis. Therefore, the question could challenge some students. While solving the question two ways and evaluating which way was better was not a standard question format, it had the potential to trigger metacognitive experiences and skills. Q2. ‘The graph of f′(x), the derivative of f(x), is sketched below. The area of the regions A, B and C are 20, 8 and, 5 square units, respectively. Given that $$\ f\!\left (0\right )=-5$$, find the value of $$f\left (6\right )$$’ (Mahir, 2009, p. 203). This question was taken from Mahir (2009) and was designed to explore students’ conceptual knowledge. Students from both Cases might have difficulty with it as $$f\!\left (x\right )$$ is not explicitly stated in the item (Thomas & Hong, 1996). In addition, the question is not typical of those used in the teaching and assessment of integral calculus at either Case. Again the item was chosen as it had the potential to activate several cognitive processes; such as the ability to analyze and distinguish which of the areas A, B and C should be used to find f(6). Other possible cognitive processes include remembering how the area under the graph of f′(x) is linked to f(x) through the Fundamental Theorem of Calculus and executing the remembered process. Q3. Please can you pose a problem about the area enclosed between a curve and a line with any two arbitrary bounds that will give an answer of 1 (i.e., the enclosed area will be equal to one). Christou et al. (2005) designed a taxonomy for problem-posing processes that can be useful when designing problem-posing questions. According to this taxonomy, there are four problem-posing question types: a) editing, b) selecting, c) comprehending and d) translating, quantitative information. Q3 is classified as selecting quantitative information because it requires students to pose a problem that is appropriate to the given answer. This task is more difficult than editing as students need to focus on relationships between the given information (Christou et al., 2005). This question might be challenging for students at both cases as they had not seen problem-posing questions in integral calculus during class. 4. Results In this section, the students’ mathematical performance on the three integral-area items is detailed (Section 4.1) then their metacognitive experiences and skills are described (Section 4.2). 4.1 Students’ mathematical performance in the integral-area problems 4.1.1 Question 1: Using the integral-area relationship to calculate areas under curves For Q1 only four students, three from the university sample (U589) and one from the college sample (Y8), were able to find the enclosed area with respect to the x-axis correctly (Table 3). As drawing an incorrect graph could affect how students set up their integral, the integrals the students created were compared to their drawings to investigate whether a student had the skills to find the area with respect to the x-axis. Four students (U13; Y67) set up integrals that related directly to their drawings. One student (Y2) did not integrate with respect to the x-axis while the remaining eight students (U2467; Y1345) had difficulty attempting this method. Table 3 Summary of students abilities to find the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Table 3 Summary of students abilities to find the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Difficulty with the method Didn’t use the method Correct Difficulty with the method Didn’t use the method Case 1 (total = 9) 5 (U13589) 4 (U2467) 0 5 (U45689) 4 (U2367) 1 (U1) Case 2 (total = 8) 3 (Y678) 4 (Y1345) 1 (Y2) 2 (Y78) 1 (Y2) 5 (Y13456) Of the university students who had difficulty, U4 struggled with the fact that $$x=y^{2}$$ is not a function, and set up the integral incorrectly (i.e., $${{{\int ^{4}_{0}}}}{(x-2-\sqrt{x}\ )dx+{{{\int ^{0}_{1}}}}{(x-2-\sqrt{x}\ )dx}}$$); even though he had sketched the graph correctly. U7 did not realize that if the upper/lower limit changes during the integration interval, the area function will change accordingly. He said, ‘I find a general formula that would give me the correct answer with whatever bound I put in. So having integrated it, then I can decide what bound I can use (see Fig. 1).’ Fig. 1. View largeDownload slide Part of U7’s working on Q1. Fig. 1. View largeDownload slide Part of U7’s working on Q1. U2 inappropriately used the disk formula for finding volume ($${{{\int ^{b}_{a}}}}{\pi f^{2}\left (x\right )dx}$$). The last University student, U6, could not find the second intersection point for the two curves. She only found the x = 4 intersection point then set up an incorrect integral for finding the area ($$\int ^{4}{(\sqrt{x}}-x-2)\ dx$$). Of the college students who had difficulty, Y1 considered only $${{{\int ^{2}_{0}}}}{(x-2)\ dx}$$ when finding the enclosed area. Y5 had difficulty with finding the area below the x-axis, commenting ‘I am not too sure whether it [$$x=y^{2}$$] actually stops at x = 0 or would continue around like a parabola in which case there will be some area there that is missed out. I am not sure how to calculate that.’ Two students (Y34) struggled to find the intersection points, which prevented them from setting up the correct integrals. Eleven students (U23456789; Y278) attempted to find the area with respect to the y-axis. Of these, seven students (U45689; Y78) did so correctly, with four students (U569; Y2) using y-axis integration as their first choice method for the task. The four students who answered incorrectly (U267; Y2) made various errors when setting up their integrals. Three students (U67; Y2) mistakenly identified that the curve was the top function. Two students used the x-axis bounds (Fig. 2) of integration rather than calculating the equivalent bounds for the y-axis (U2; Y2). As mentioned earlier, U2 used an incorrect formula for finding the enclosed area while U3 created an incorrect drawing of the graph of $$x=y^{2}$$, leading to an incorrect lower bound for the integral, $$^{-}$$2. The remaining six students (U1; Y13456) did not attempt this method. Fig. 2. View largeDownload slide Part of Y2’s working on Q1. Fig. 2. View largeDownload slide Part of Y2’s working on Q1. When asked, the 10 students (U23456789; Y78) who found the enclosed area with respect to both axes (whether correctly or incorrectly) highlighted that integration with respect to the y-axis was easier for finding the enclosed area in this question, because only one integral is involved and they did not need to work with the square root and negative area: ‘I could do it in one equation and also it is all above, all positive, [so] I don’t have to deal with negative area (U5).’ Two students from Case 2 (Y58) mentioned their first choice was always integration with respect to the x-axis when finding enclosed area. Y8 believed it is easier to conceptualize when you are integrating with respect to the x-axis: ‘By default, I go x because it is easier to conceptualize, because you have positive to negative, but y, sort of is inverted in terms of positive to negative. But sometimes it is just easier to do in terms of y [sic].’ Y5 mentioned, ‘In all questions it is easier to use the integration with respect to x-axis because you don’t need to write the function in terms of x = f(y), at least in questions that we see in our school.’ Two broad inter-related themes emerged from students’ explanations of which method was better to use: 1. choice based on the graph of the curves (U34589; Y1234); and 2. choice based on the algebraic manipulations needed to solve the problem (U123456789; Y567). When making graph-based choices, five students (U3489; Y2) noted they chose the method that involved the lesser change to the lower/upper functions. For example, U8 said ‘when you have to break it up less times like the previous example [Q1].’ U5 mentioned she would make her choice based on which alternative had the lesser negative signed area. Three students from Case 2 (Y134) said that their choice would be based on the graph being enclosed by the x or y axis, indicating they were thinking about the area under one of the curves rather than between the two curves. This error was also found when illustrating A and B in Q2 (Section 4.1.2). When finding the enclosed area between curves and two bounds, the axes are not important; however, most college students had not developed this understanding as they had not developed an understanding of integral-area relationships through Riemann sums (Section 4.1.2). The second theme related to students basing their choice of method on the least amount of algebraic manipulation required, a decision that could take into account both the finding of the lower and upper bounds and possible form changes for the integrand. If the bounds were given in terms of x = a and x = b, they would be more likely to choose integration with respect to x, and if they were given in terms of y = c and y = d, they would choose integration with respect to y. If the integrand was presented as a function of x, they would tend to integrate with respect to the x-axis, and if it was given in terms of y, they would tend to integrate with respect to the y-axis: ‘I would use it [the method] depending on what formula I have for the functions. I am integrating relative to the thing the function is of (U7).’ A lack of proficiency in algebraic manipulation was a major barrier to successfully solving the integral-area problems. For example, in Q1, six students (U136; Y356) had difficulty solving $$x-2=\sqrt{x}$$ to find the intersection points. Five students (U13; Y145) made a mistake when finding y from $$x=y^{2}$$, assuming $$y=\sqrt{x}$$. Y6 squared $$x-\sqrt{x}$$ as $$x^{2}-x$$. Y5 thought these two curves have no intersection points, and Y3 wanted to solve $$x-2=y^{2}$$ to find the intersection points. 4.1.2 Question 2: Using the integral-area relationship for the graph of a derivative function Students’ responses to Q2 were used to explore whether they were able to use the integral-area relationship for f′(x). Seven students from Case 1 (U3456789) and four from Case 2 (Y5678) realized that $${{{\int _{a}^{b}}}}{f^{\prime}(x)dx}$$ is equal to the area under the graph of f′(x) between x = a and x = b. However, only six students from Case 1 (U345689) realized this integral is equal to the signed net area underneath the graph of f′(x). One student, Y7, made the mistake of considering $${{{\int _{a}^{a}}}}{f^{\prime}(x)dx=f^{\prime}(b)-f^{\prime}(a)}$$. 4.1.3 Question 3: Posing an integral-area relationship question For Q3, eight students (U3456789; Y7) posed a correct question about the area enclosed between a curve and a line using the given information. Of the other nine students, seven (U12; Y23458) could not pose a question and two (Y16) posed an incorrect question (Table 4). Table 4 Questions posed by students for the integral-area relationship Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Table 4 Questions posed by students for the integral-area relationship Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Find the area enclosed between Curve Line Bounds N Lower Upper Suitable problems $$y=x^{2}$$ y = 0 a = 0 $$b=\sqrt [3]{3}$$ 5 (U3456; Y7) a = 1 $$b=\sqrt [3]{4}$$ 1 (U9) $$y=3x^{2}$$ y = 0 a = 0 a = 1 1 (U7) $$y={\sin x\ }$$ y = 0 a = 0 $$b=\frac{p}{2}$$ 1 (U8) Unsuitable problems $$y=x^{2}+0.$$5 y = 2x a = 0 a = 1 1 (Y1) $$y=x^{2}$$ y = 1.31 – – 1 (Y6) Posing a question was not an easy task for many students. For example, Y4 said, ‘I am struggling with only the area is given and you have too many things.’ However, those not able to pose a question did not realize that the task could be simplified by choosing the x-axis as the line. For instance, Y8 chose $$y=x^{2}$$ as the curve and y = x as the line for his first try, then tried $$y=\sqrt{x}$$ as the curve. After he was unable to find an area equal to one, he changed the functions to a more general form by considering the curve $$y=a\sqrt{x}$$ and the line as y = bx. 4.2 Students’ metacognitive experiences and skills while solving the integral-area problems This section describes the metacognitive experiences students had and the metacognitive skills they used while solving the integral-area tasks. As shown in Section 2.2, metacognitive experiences and skills have several aspects. Those explored here relate to feelings of familiarity, knowing, confidence, estimating the correctness of the solution and making a judgement of learning (Efklides, 2006, 2008). The two metacognitive skills investigated involve making a drawing related to the problem, and checking calculations and answers. 4.2.1 Having an accurate pre-judgement of whether they can solve the problem Students had different metacognitive experiences when dealing with Q1. Eight students (U127; Y12567) made their judgement based on their familiarity with how to find an area using integral calculus. An example of these responses is ‘We have recently learnt this [topic] in class and I am practising these questions at the moment (Y1).’ Four students (U568; Y8) based their judgement on their ability to integrate the form of the integrand. They provided such reasons as ‘equations [are] not particularly difficult to integrate (U6). Three students (U3; Y34) based their decision on their familiarity with the shape of the graph, providing such reasons as ‘I can imagine it graphically (U3).’ U9 highlighted the importance of the shape of the enclosed area for making his judgement: ‘What I will do before I would have known if I am sure or not I will draw the graph. Then I decide whether I am sure I can solve it or not. That is hard for me to look at those two functions and say, oh yes, it is easy I can find the area between functions [sic].’ Drawing a graph is an element of metacognitive skills (Jacobse & Harskamp, 2012), as would be imagining a graphical representation of the task. It is also an important part of solving integral-area problems as it helps students decide whether to integrate with respect to the x- or y-axis, and shows whether the curves have any discontinuity. The responses of these four students (U39; Y34) illustrate how the metacognitive experiences students can be related to the metacognitive skills they possess. The judgement of the final student, U4, was affected by the fact that $$x=y^{2}$$ was not a function, which meant he was unsure if he could solve the problem. Overall, when comparing their metacognitive experiences to the results of Q1 (Table 5), students from Case 1 had more accurate predictions than students from Case 2. Table 5 Students’ prediction of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 Table 5 Students’ prediction of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method I am sure I will solve this question (N = 12) Case 1 (N = 6) 3 3 0 4 2 0 Case 2 (N = 6) 1 5 0 1 1 4 I am not sure whether I will solve this question correctly or incorrectly. (N = 5) Case 1 (N = 3) 0 3 0 1 2 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I cannot solve this question. (N = 0) Case 1 (N = 0) 0 0 0 0 0 0 Case 2 (N = 0) 0 0 0 0 0 0 For Q2, four students (U348; Y4) were sure they could solve the question; however, only two answered correctly (U34). Of the four, three (U348) had that feeling because they thought they understood why the information was given. For instance, U8 said, ‘we know the area of important parts of the f′(x) and area is related to the anti-derivative.’ Y4 thought he had seen similar questions; however when he started solving the problem he said he had not properly understood what the question asked. He said, ‘I have seen question like this before…I think I misinterpret what is asking [sic].’ Eleven students (U12567; Y123568) were not sure if they could solve the question correctly, and only U5 did so correctly. U5 was unsure because she believed she had not ‘encountered any question like this.’ Two other students (U7; Y3) also felt they had not seen a question like this before. Two students (Y68) were unsure because they felt they needed to think more about it to know if they were able to solve it. For instance, Y6 said, ‘I am not sure because I need time to think about it mentally in my head to understand the question better... I think I might be able to do it in time... I know it seems there are some familiar parts in it but all together in one question [I am not sure whether I am able to solve it].’ The remaining students had different reasons for their feeling. Three students (U1; Y12) recognized the question, but were unsure how to solve it. For instance, U1 said, ‘I have seen questions like this before, but cannot remember how to do it.’ U6 was unsure because she thought guessing may be required: ‘not sure, because some guessing may be required.’ U5 misunderstood the given information, thinking the given graph was that of f(x). He said, ‘$$f\left (0\right )=-5$$ is confusing me.’ Y1 was unsure because he did not know ‘where to start.’ Y5’s feeling was specifically related to how the problem needed to be solved. He said, ‘I am not sure because I have to change the area to f′(x). but I am not sure I can calculate it correctly.’ Two students (U9; Y7) were sure they could not solve this question, but U9 did so correctly. He felt he had not seen a similar question before, saying ‘I never seen this before [sic].’ Y7’s feeling was related to how the problem should be solved. He said ‘I do not know how to find out the f(x) because I do not recognize the graph type of f′(x).’ Finally for Q3, nine students (U3456789; Y57) were sure they could pose a problem to meet the given constraints, and of these, all but Y5 posed a problem with the correct area. Most of these students either believed they could find an example (U689; Y57), or thought they could use simple functions for posing a problem (U357). For example, U7 said, ‘sure, I am going to use simple stuff’, and Y7 said, ‘I can think of an example.’ Apart from those reasons, U4 was sure he could pose a problem because he ‘understand[s] the theory behind the task.’ Seven students (U12; Y24568) were unsure if they could pose a problem based on the given information, and either did not pose a problem or posed a problem with an incorrect area. These students were unsure for a variety of reasons, including being unsure: if they could find an example that fitted the given information (U1; Y26); that they could ‘do it backward’ (U2; Y8); because they had not posed a problem before (Y3); and because they ‘may make a mistake’ (Y4). Only one student (Y1) was sure he could not pose a problem based on the given information because he had not posed a problem before. 4.2.2 Having an accurate post-judgement of how effectively the task was completed For Q1, students’ judgements of how well they answered the item were varied (Table 6) and to some extend influenced by whether or not a student was able to solve the task with two methods. Four students (U589; Y8) were sure they had solved it correctly because they could find the same answer using both methods (their answers were correct). Similarly, two students (U27) were sure they had solved the question incorrectly as they found different answers using the two methods (although both answers were incorrect). Table 6 Students’ post-judgement of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Table 6 Students’ post-judgement of their ability to solve Q1 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Find area with respect to x-axis Find area with respect to y-axis Correct Incorrect Didn’t use the method Correct Incorrect Didn’t use the method I am sure I solved this question correctly (N = 7) Case 1 (N = 4) 3 1 0 3 1 0 Case 2 (N = 3) 1 1 1 1 1 1 I am not sure whether I solved this question correctly or incorrectly (N = 5) Case 1 (N = 3) 0 3 0 2 1 0 Case 2 (N = 2) 0 2 0 1 0 1 I am sure I solved this question incorrectly (N = 5) Case 1 (N = 2) 0 2 0 0 2 0 Case 2 (N = 3) 0 3 0 0 0 3 Students who were only able to use one method needed to seek other rationales for their judgement. For example, two students (U3; Y6) made an inaccurate post-judgement, saying they were sure they had found the correct answer because ‘it makes sense graphically’ (U3), and ‘looking at graph it seems right visually [sic]’ (Y6). However, their drawings were incorrect. Two students (Y15) were sure they had solved it incorrectly because their answers were negative. Other students used drawn curves to check their answers and intersection points. For example, Y4 was unsure whether or not he had solved the question correctly because the intersection points he found did not match the curves. Three students (U46; Y7) were unsure if they had solved the item correctly for the following reasons; U6 could not distinguish which function was the top function: ‘not sure which one [in] $$\int{[f\!\left (x\right )-g\!\left (x\right )]\ dx}$$ is f(x)’ (U6). U4 had difficulty with $$x=y^{2}$$ and said: ‘$$x=y^{2}$$ is not a function so that confused me’; Y7 was unsure because ‘[I] forget to account for the other part [the part which is under x-axis] of $$x=y^{2}$$.’ U1 was not confident about his response, saying, ‘usually with math question you are pretty sure when you have got it right. I was pretty hazy when I go through. I was over confident when I started.’ Finally, Y2 was sure he had solved the question correctly but could not explain why; his answer was incorrect. Five students (U58; Y568) thought they had solved Q2 correctly, but only U5 made an accurate post-judgement. She was sure because ‘I have utilized all the information I have in the question.’ Y5 could not provide a reason why he thought he had solved the question correctly. The other three students did not provide any reliable justification for their judgement. For instance, Y6 said, ‘It works logically and reasonably, I am confident with the working’, and Y8 responded: ‘I think I solved it correctly… because it was a lot simple one to do once I was start looking at it in more depth [sic].’ Four students (U349; Y7) were unsure if they had solved the question correctly, but three (U349) did so correctly. Their reasons for their lack of confidence differed. U4 was unsure because he did not use a part of the information given in the problem, the area of region C. U9 was ‘not sure because I do not feel I can justify the method I have used.’ U3 was not confident because: ‘It is a question that I haven’t come across for a long time… It is just being so long since I have to do a non-calculation–based integration and or differentiation [question] that it’s pretty much taken me by surprise. Always expect calculation heavy question. Not prepared.’ Y7 was unsure because he was not confident about being able to integrate both sides of $$f^{\prime}(6)+12=f^{\prime}(0)$$, and in his solution he had incorrectly written $$f(6)+12x+c=f(0)$$ for the intergral of both sides. The remaining eight students (U1267; Y1234) could not come up with an answer to this item so were categorized as being sure they had not solved the question correctly. To sum up, five factors were found in relation to students’ feelings about whether they had solved Q2 correctly, including how much of the given information was being used; how well the method could be justified; how confident the student was with his/her working; how familiar the student was with the question; and how easily the answer was found. For Q3, eight students (U345689; Y67) were sure they had posed the problem correctly; all but Y6 being correct. Five students (U45689) were certain because they had solved their posed problems and had obtained the answer one. Y6 believed that if someone solved his problem, the answer would be one. Y7 was sure because he thought he had set up the integral correctly. U3 thought his solving ‘was thorough and multiple wrong solutions excluded.’ One student who was unsure posed a correct problem (U7), but his reasoning was not related to the content and indicated that he was not confident with his working. He said, ‘still I can see a smile on the researcher’s face’ indicating he attempted to gain feedback about the correctness of the solution from the interviewer. Eight students (U12; Y123458) could not pose a problem so were coded as being sure that they had not correctly solved the task. 4.2.3 Making a drawing related to the task For Q1, all students drew curves while answering the item but only 11 (U2456789; Y2378) did so correctly. For the six other students, there were two common reasons for incorrect curves. Firstly, neglecting the part of $$x\!=\!y^{2}$$ under the x-axis (U13; Y456) (Fig. 3b). Secondly, trying to sketch the graph from memory (Fig. 3a) and not checking their drawing by substituting some values from the domain to confirm the relationship between x and y. In fact, during think-alouds, none of these six students mentioned substituting values into the function/relation to help sketch or check the graph. Had they used this checking procedure they might have been more able to identify errors more effectively. Fig. 3. View largeDownload slide Examples of students’ errors when drawing the curves and a correct drawing. Fig. 3. View largeDownload slide Examples of students’ errors when drawing the curves and a correct drawing. Students’ ability to find the area enclosed between the curves correctly was closely related to their ability to draw the curves correctly, especially when the upper and lower functions did not change in the enclosed area. Seven students (U45689; Y78) sketched correct curves and successfully found the area with respect to the y-axis. Four of these (U589; Y8) correctly drew the graph and integrated with respect to the x-axis successfully (Table 6). The lower success level with respect to the x-axis was due to changes in the lower function at x = 1 from $$y=-\sqrt{x}$$ to y = x − 2. The final piece of evidence that supports the importance of curve sketching when using integrals to find areas is that no student who drew an incorrect curve was successful with the item (Table 7). Table 7 Relationship between a correct drawing and finding the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 Table 7 Relationship between a correct drawing and finding the area with respect to the x- and y-axes Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 Finding the area with respect to the x-axis Finding the area with respect to the y-axis Correct Incorrect Did not use the method Correct Incorrect Did not use the method Correct sketch (N = 11) Case 1 (N = 7) 3 4 0 5 2 0 Case 2 (N = 4) 1 2 1 2 1 1 Incorrect sketch (N = 6) Case 1 (N = 2) 0 2 0 0 1 1 Case 2 (N = 4) 0 4 0 0 0 4 For Q2, students did not need to make a drawing as a sketch was already provided. For Q3, 11 students (U34579; Y235678) tried to sketch a curve and a line to develop a better understanding of what they should consider a curve and a line when creating their problem. Of those, six successfully posed a correct question (Table 8). However, when Q3’s results are compared to Q1’s, making a drawing was not as closely related to success with the item, partly because some students who created drawings were still not able to pose an appropriate question. Table 8 Relationship between making a drawing and being successful with Q3 Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) Table 8 Relationship between making a drawing and being successful with Q3 Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) Posed correct problem Posed incorrect problem Did not pose a problem Made a drawing 6 (U34579; Y7) 1 (Y6) 4 (Y2358) Did not make a drawing 2 (U68) 1 (Y1) 3 (U12; Y4) 4.2.4 Checking calculations and answers For Q1, more than half of the students in the sample did not check their solutions, even though the question asked them to solve the question in two ways so they had the opportunity to compare their final answers. Four students from Case 1 (U1589) and three from Case 2 (Y246) did some form of checking but only checked their calculations and, as identified earlier, only Y2 checked his drawing of the curves. Of these, four (U1; Y246) had errors in their working; and three (U1; Y46) could not find their errors. Y2 was able to amend his drawing for $$x=y^{2}$$ (Fig. 4) on his fourth try. Of the three students who solved the question correctly and did some form of checking (U589)4, two of the three checked to see if they had found the same answer in both integrations, and one checked his calculations to ensure he had not made any mistakes when finding the intersection points. Fig. 4. View largeDownload slide Y2’s attempts in drawing $$x=y^{2}$$. Fig. 4. View largeDownload slide Y2’s attempts in drawing $$x=y^{2}$$. When solving Q2, only U5 did some form of check. After she found the correct answer, she revisited her working to make sure she did not make a mistake. Six students (U45689; Y6) used a form of check in Q3; after posing their problem they solved it to see if they got 1 as the answer. However Y6, who had posed a question with a different answer, could not identify where he had made the mistake. 5. Discussion and conclusion In this study, a multiple case study was used to explore what students had learned about the relationship between integrals and area. The study sample comprised nine first-year university and eight Year 13 students who each participated in an individual semi-structured interview. The results in this paper adds to the literature about students’ understanding of the integral-area relationship in several ways. Firstly, creating a profile of students’ metacognitive experiences and skills in relation to the integral-area relationship has not been attempted previously. Students’ responses show that aspects of their metacognitive experiences and skills could be further developed. Secondly, students’ problem posing ability for the integral-area relationship (Q3) has not been studied previously. The students’ responses show such question types could be useful for probing students’ understanding of integral-area relationships. For Q1, several students based their pre-judgement of whether they were able to solve the question on their knowledge of techniques to find the antiderivative of integrands, or knowledge of how to find enclosed area in general. However, making such a judgement prior to making a sketch of the relations is premature as the shape of the enclosed area affects both the methods that can be used to find that area and the ease of use of different methods. This suggests that teachers and lecturers need to be aware of the potential interplay of students’ mathematical content knowledge with their metacognitive experiences and skills, and how the order in which thinking occurs can affect students’ chances of success with a task. With Q1, all students made a drawing to help them answer the question, which could be an indication of the presence of that metacognitive skill, but did not naturally want to do this as part of decision-making about the solvability of the problem. Emphasizing the value of preliminary exploration, such as identifying the shape of the enclosed area before making any judgement about an integral-area task, reduces the chance of making an incorrect pre-judgement and increases the chance of setting up the integral correctly, especially the top function. Similarly, the interplay between content knowledge and metacognitive experiences and skills is important for students to be able to judge their success with a task. The 10 students who were unsure they had found the correct answer, or were sure they had solved Q1 incorrectly, were not seen to revisit and check their solution. This suggests two things. First, these students had not learned from their teachers and lecturers that checking is a normal part of answering mathematical questions; second, they may not possess strategies to check their answers for integral-area questions. For their pre-judgement of Q2, fewer students were sure they could solve the question correctly (12 for Q1 and 4 for Q2), suggesting that they were not as confident about answering questions that focus on conceptual knowledge. In addition, the reasons given for their post-judgement shows students were not confident with their solutions. This suggests that the students could have benefited from being exposed to more conceptually based tasks as part of their class work. Classroom observations identified that problem posing (Q3) was not part of the work in class or the assessment at either case, yet eight students (seven from Case 1) were able to pose a question to meet the provided constraint. Students from Case 1 may have been more successful because of their greater experience working with integral calculus questions, having likely passed Year 12 and Year 13 calculus before starting their university course. Students from Case 1 also had more accurate pre-judgements of their ability to solve integral questions and better post-judgements of whether they had solved the questions correctly than students at Case 2. However, several students from both Cases made an incorrect pre- or post-judgement. Therefore, as for Q1, highlighting the drawing of the enclosed area and using monitoring strategies more often might help some students understand where they can make mistakes. Overall, more than half the students did not appear to check their calculations and answers to any of the integral questions during the interviews, suggesting that this metacognitive skill should be emphasized in teaching. Ways in which students can check their work should be discussed and if needed suggested to them (e.g., approximating area using geometric shapes, differentiating antiderivatives and checking that area is positive). Furthermore, if lecturers and teachers themselves used monitoring strategies regularly, especially when modelling the answering of questions on the board, and encouraged students to do so as a routine part of their work, students might use this metacognitive skill more often. Such a change in practise may not seem natural to teachers or lecturers as the questions they work on publicly are seldom genuine problems, rather they tend to be familiar tasks that have been seen and solved previously. However, as they are working with novices, it is important to model the actions of mathematicians working on unfamiliar problems, especially as the practise of checking work has the potential to lead to fewer errors and students developing greater confidence in their understanding of the mathematics they are learning. Evidence has also presented that suggests if a student is not introduced to recognized and accepted strategies for checking their work they may create their own proxies, such as whether or not all information provided in a problem has been used. While this is an astute measure that can be used on many text and assessment items, it is not one that can be used on genuine mathematical problems. By using this strategy, rather than learning about working as a mathematician, students may be learning the skills of creating text and assessment items. As already noted for Q1, a typical question about the integral-area relationship, all students made a drawing to help them solve the problem. However, for non-typical questions, like Q3, only eleven of the seventeen students made drawings. This suggests that for some students making a drawing may be part of a memorized procedure for a particular type of problem rather than being a general strategy to help understand the requirements of any question involving functions and relations. This in turn suggests that the importance of making a drawing to help understand mathematical questions could be emphasized more generally. For instance in Q3, if students had made a drawing for their proposed curve and line, there was a greater chance they could find the suitable curve and line that fitted the given condition. As such, lecturers and teachers should be encouraged to make a drawing for each question they solve in classes for students, and encourage students to also do this. While freehand sketches are powerful as they may be the only method available in assessment situations, free online programmes such as https://www.desmos.com/ could also be used. In terms of their mathematical performance on the integral-area relationship, students’ procedural knowledge was better developed than their conceptual knowledge. This is also consistent with previous studies showing students are able to undertake routine procedures to find area using integral techniques; however their knowledge about why such a procedure is used is limited (Artigue, 1991; Grundmeier et al., 2006; Mahir, 2009; Rasslan & Tall, 2002; Thomas & Hong, 1996). A lack of algebraic manipulation skills and prior knowledge were barriers for several students, due to their not being able to find intersection points, sketch the graph correctly or find the equation of lines and curves. These findings, also highlighted by previous studies (e.g., Kiat, 2005), indicate that some students would have benefitted from improving their knowledge of functions and relations, and/or algebraic manipulation and/or graph sketching prior to starting integral calculus (Kiat, 2005). Conceptually, several students seemed to believe the area or net area underneath the graph of f′(x) is equal to the $$\int{f^{\prime}\!\left (x\right )dx}$$, rather than its signed net area. A lack of conceptual knowledge about the definite integral is also shown in the literature (e.g., Mahir, 2009; Thomas & Hong, 1996). The fact that six students did not use the integral-area relationship to solve Q2 suggests that lecturers and teachers should also use the graph of the derivative in integral-area questions they solve in their classes to help students to understand that $${{{\int _{a}^{b}}}}f^{\prime}(x)dx=f(b)-f(a)$$ is another representation of the first part of the FTC and equivalent to $${{{\int _{a}^{b}}}}f(x)dx=F(b)-F(a)$$ where $$F^{\prime}\!=f$$ . This practise might provide a better opportunity for students to realise that the relationship between the graph of the function and the area under the curve also exists for the derivative function. In general, students from Case 2 had less developed conceptual knowledge of the definite integral and the integral area relationship than students from Case 1. One simple explanation is that the university students had more experience with this topic as they have been exposed to it for longer. However, another possible explanation relates to the teaching of the Riemann sums5. In Case 1, Riemann sums were the focus of teaching definite integrals, and examples were solved in this regard. In addition, during the teaching of other topics, such as finding volumes by slicing and cylindrical shells, the proofs for volume formulas were taught to Case 1 students. The ideas used in those proofs are related to Riemann sums, therefore help students to develop a better understanding of Riemann sums and the Riemann integral. At Case 2, the teacher did not introduce Riemann sums until end of the integral calculus topic and no example was solved in the classroom. This teacher started with definite integrals, declaring his procedural approach towards teaching the topic by saying: ‘I am going to take the expedient route... I am going to give you the application…saying without proving….’ Such an approach towards teaching mathematics is related to the development of instrumental understanding (Skemp, 1976), which can have negative consequences for students’ learning. For example, it influences students’ attitudes towards mathematics and their understanding of the structure of mathematics. One reason why this teacher did not focus on Riemann sums is likely related to The New Zealand Curriculum (Ministry of Education, 2007) and the assessment of learning in Year 13 through the National Certificate of Educational Achievement (NCEA) level 3 mathematics achievement standards (New Zealand Qualifications Authority, 2013). At this level, this curriculum and these assessments are used in most schools in New Zealand. Being able to use numerical methods of integration is prescribed by these documents; however Riemann sums are not highlighted. Similarly, the textbook used in the college, Delta Mathematics (Barton & Laird, 2002) does not focus on the Riemann integral and only provides it as an appendix. Rather, the trapezium method and Simpson’s rule are provided in the main body of the textbook and were emphasized during teaching. However, it seems the proofs behind these methods are more complicated than the Riemann sums approach as their formulas have more elements. Another possible reason for this teaching approach is that numerical methods are presented at the end of integral calculus topic in this text so were also taught at the end of the integral calculus topic, rather than being an introduction to finding areas. However students at Case 1 first learnt about Riemann sums, then were exposed to the Fundamental Theorem of Calculus and integral techniques. This may be because numerical methods are more conceptual and need more time to understand, and while students from Case 1 had more time to understand them as they were introduced to different topics in integral calculus, this opportunity was not available to students from Case 2 due to the curriculum and assessment regime being followed. The differences between students’ performance in Case 1 and 2 in relation to definite integrals suggests changes in the New Zealand schools’ mathematics curriculum document to focus more on teaching Riemann sums might be beneficial. This could help to increase students’ conceptual and procedural knowledge. By knowing the rationale behind the relationship between a definite integral and the area under a curve through Riemann sums, students are likely to develop their conceptual knowledge as to why a definite integral can be used for finding area. Teaching Riemann sums would also help students develop their procedural knowledge by knowing what to integrate and how to set up the bounds of an integral (Sealey, 2006, 2014). Finally, it needs to be reinforced that in this exploration of student learning of the integral-area relationship, the responses of only 17 students have been analyzed, so it is unlikely that the results obtained represent all students’ problem-solving behaviours in this topic. The Cases were chosen from an area that was geographically accessible to the authors. The school decile of the College, a measure of the Socio-Economic Status (SES) in New Zealand, was 10; 10 being high. As a consequence, the findings might not be applicable to students who attend colleges with lower decile ratings. Possible gender differences were also not explored because of the small sample size, although in New Zealand, gender differences in mathematics achievement are generally not large. Nor was it possible to explore whether there may a difference between students of different ethnic groups. Such questions are important to address in future studies, ones with larger sample sizes set in different contexts. Acknowledgements The first author wishes to acknowledge his gratitude to Associate Professor Robin Averill for her support, guidance and many hours of valuable discussions in relation to this study. Farzad Radmehr received his first PhD in Mathematics Education at the Ferdowsi University of Mashhad in 2014. He recently received his second PhD at Victoria University of Wellington in 2016. He is an assistant professor in the Department of Applied Mathematics of Ferdowsi University of Mashhad. Farzad is interested in the teaching and learning of mathematics at upper secondary and tertiary level, in particular, introductory level undergraduate courses. Michael Drake received his PhD in Mathematics Education at Victoria University of Wellington in 2010. He is a senior lecturer in the School of Education of Victoria University of Wellington. Michael is interested in the teaching and learning of mathematics, in particular, number and numeracy, algebra, measurement, probability and statistics during the middle years of schooling. Footnotes 1  These facets are described in the literature section. 2  c is a constant and included to describe situations that quantities do not directly correspond to the $$f\left (x_{i}\right )$$ and $$\Delta x$$ (Sealey, 2014). 3  This study only reports part of the students’ interviews. 4  Students who compared their answers when responding to ME2 are not included in this analysis as solving the task was completed before responding to ME2. 5  All teaching of integral calculus was recorded and observed by the first author. References Anderson , L. W. (ed) , Krathwohl , D. R. (ed) , Airasian , P. W. , Cruikshank , K. A. , Mayer , R. E. , Pintrich , P. R. , Raths , J. & Wittrock , M. C. ( 2001 ) A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (Complete edition) . New York, N.Y : Longman . Artigue , M. ( 1991 ) Analysis. In Advanced mathematical thinking (D. Tall ed) . Boston, M.A : Kluwer , pp. 167 -- 198 . Google Scholar CrossRef Search ADS Anton , H. , Bivens , I. & Davis , S. ( 2012 ) Calculus: Early Transcendentals . New Jersey, N.Y : Wiley Global Education . Barton , D. & Laird , S. ( 2002 ) Delta Mathematics . Auckland, New Zealand : Pearson Education . Bartsch , K. & Wellman , H. M. ( 1995 ) Children talk about the mind . New York, United State of America : Oxford University Press . Baxter , P. , & Jack , S. ( 2008 ) Qualitative case study methodology: Study design and implementation for novice researchers . Qual. Res , 13 , 544 -- 559 . Bezuidenhout , J . ( 2001 ) Limits and continuity: some conceptions of first-year students . Int. J. Math. Educ. Sci. Technol. , 32 , 487 -- 500 . Google Scholar CrossRef Search ADS Charmaz , K. ( 2006 ) Constructing grounded theory: A practical guide through qualitative analysis . London, England : Sage Publication Ltd. Christou , C. , Mousoulides , N. , Pittalis , M. , Pitta-Pantazi , D. & Sriraman , B. ( 2005 ) An empirical taxonomy of problem posing processes . ZDM , 37 , 149 -- 158 . Efklides , A. ( 2001 ) Metacognitive experiences in problem solving: Metacognition, motivation, and self-regulation . In Trends and prospects in motivation research (A. Efklides, J. Kuhl & R.M. Sorrentino eds) . Dordrecht, the Netherlands : Kluwer , pp. 297 -- 323 . Google Scholar CrossRef Search ADS Efklides , A . ( 2006 ) Metacognition and affect: What can metacognitive experiences tell us about the learning process? Educ. Res. Rev. , 1 , 3 -- 14 . Google Scholar CrossRef Search ADS Efklides , A . ( 2008 ) Metacognition: Defining its facets and levels of functioning in relation to self-regulation and co-regulation . Eur. Psychol. , 13 , 277 . Google Scholar CrossRef Search ADS Eisenhardt , K. M. & Graebner , M. E. ( 2007 ) Theory building from cases: opportunities and challenges . Acad. Manag. J. , 50 , 25 -- 32 . Google Scholar CrossRef Search ADS Ericsson , K. A. & Simon , H. A. ( 1993 ) Protocol analysis: Verbal reports as data (Revised Edition) . Cambridge, Massachusetts : The MIT Press . Fisher , R. J. ( 1993 ) Social desirability bias and the validity of indirect questioning . J. Consum. Res. , 20 , 303 -- 315 . Google Scholar CrossRef Search ADS Flavell , J. H. ( 1979 ) Metacognition and cognitive monitoring: A new area of cognitive developmental inquiry . Am. Psychol. , 34 , 906 -- 911 . Google Scholar CrossRef Search ADS Flavell , J. H. , Miller , P. H. & Miller , S. A. ( 1993 ) Cognitive development . New Jersey, United State of America : Prentice-Hall . Garofalo , J. & Lester Jr , F. K. ( 1985 ) Metacognition, cognitive monitoring, and mathematical performance . J. Res. Math. Educ. , 16 , 163 -- 176 . Google Scholar CrossRef Search ADS Grundmeier , T. A. , Hansen , J. & Sousa , E. ( 2006 ) An exploration of definition and procedural fluency in integral calculus . PRIMUS , 16 , 178 -- 191 . Google Scholar CrossRef Search ADS Hong , Y. Y. & Thomas , M. O. ( 2015 ) Graphical construction of a local perspective on differentiation and integration . Math. Educ. Res. J. , 27 , 183 -- 200 . Google Scholar CrossRef Search ADS Jacobse , A. E. & Harskamp , E. G. ( 2012 ) Towards efficient measurement of metacognition in mathematical problem solving . Metacognition Learn. , 7 , 133 -- 149 . Google Scholar CrossRef Search ADS Jones , S. R. ( 2013 ) Understanding the integral: Students’ symbolic forms . J. Math. Behav. , 32 , 122 -- 141 . Google Scholar CrossRef Search ADS Jones , S. R. ( 2015 ) Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts . J. Math. Behav. , 38 , 9 -- 28 . Google Scholar CrossRef Search ADS Kiat , S. E. ( 2005 ) Analysis of students’ difficulties in solving integration problems . Math. Educ. , 9 , 39 -- 59 . Kim , Y. R. , Park , M. S. , Moore , T. J. & Varma , S. ( 2013 ) Multiple levels of metacognition and their elicitation through complex problem-solving tasks . J. Math. Behav. , 32 , 377 -- 396 . Google Scholar CrossRef Search ADS Kitchner , K. S. ( 1983 ) Cognition, metacognition, and epistemic cognition . Hum. Dev. , 26 , 222 -- 232 . Google Scholar CrossRef Search ADS Kouropatov , A. & Dreyfus , T. ( 2013 ) Constructing the integral concept on the basis of the idea of accumulation: Suggestion for a high school curriculum . Int. J. Math. Educ. Sci. Technol. , 44 , 641 -- 651 . Google Scholar CrossRef Search ADS Kreutzer , M. A. , Leonard , C. , Flavell , J. H. & Hagen , J. W. ( 1975 ) An interview study of children’s knowledge about memory . Monogr. Soc. Res. Child. Dev. , 40 , 1 -- 60 . Google Scholar CrossRef Search ADS PubMed Kuhn , D. ( 2000 ) Theory of mind, metacognition, and reasoning: A life-span perspective . Children’s reasoning and the mind (Mitchell, P. & Riggs, K. J. eds) . Hove, UK : Psychology Press , pp. 301 -- 326 . Lester , F. K. ( 1982 ) Building bridges between psychological and mathematics education research on problem solving . Mathematical problem solving (Lester, F. K. & Garofalo, J. eds) . Philadelphia, Pennsylvania : The Franklin Institute Press , pp. 55 -- 85 . Mahir , N . ( 2009 ) Conceptual and procedural performance of undergraduate students in integration . Int. J. Math. Educ. Sci. Technol. , 40 , 201 -- 211 . Google Scholar CrossRef Search ADS McNamara , D. S. ( 2011 ) Measuring deep, reflective comprehension and learning strategies: challenges and successes . Metacognition Learn. , 6 , 195 -- 203 . Google Scholar CrossRef Search ADS Ministry of Education . ( 2007 ) Achievement objectives by Learning Area, The New Zealand Curriculum . Wellington, New Zealand : Learning Media . New Zealand Qualifications Authority (NZQA , 2013 ) NCEA level 3 mathematics achievement standard . Wellington, New Zealand : Learning Media Limited . Özsoy , G . ( 2011 ) An investigation of the relationship between metacognition and mathematics achievement . Asia Paci .Educ. Rev. , 12 , 227 -- 235 . Google Scholar CrossRef Search ADS QS World University Rankings . ( 2014 ) Study in New Zealand . Retrieved from http://www.topuniversities.com/where-to-study/oceania/new-zealand/guide Rasslan , S. & Tall , D. ( 2002 ) Definitions and images for the definite integral concept . Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, Norwich (Cockburn, A. D. & Nardi, E. eds) . UK . Rösken , B. & Rolka , K. ( 2007 ) Integrating intuition: The role of concept image and concept definition for students’ learning of integral calculus . Montana Math. Enthusiast , 3 , 181 -- 204 . Schneider , W. & Lockl , K. ( 2002 ) The development of metacognitive knowledge in children and adolescents . Applied metacognition (Perfect, T. J. & Schwartz, B. L. eds) . Cambridge, England : Cambridge University Press . Google Scholar CrossRef Search ADS Schneider , W. & Artelt , C. ( 2010 ) Metacognition and mathematics education . ZDM , 42 , 149 -- 161 . Google Scholar CrossRef Search ADS Schoenfeld , A. H. ( 1985 ) Math. Probl. Solving . New York, United State of America : Academic Press . Schraw , G. ( 1998 ) Promoting general metacognitive awareness . Instr. Sci. , 26 , 113 -- 125 . Google Scholar CrossRef Search ADS Sealey , V. ( 2006 ) Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient? Proceedings of the 28th Annual Meeting of the North American Chapter of the international Group for the Psychology of Mathematics Education (Alatorre, S., Cortina, J. L., Saiz, M. & Mendez, A. eds) . Mexico : Merida , pp. 46 -- 53 . Sealey , V . ( 2014 ) A framework for characterizing student understanding of Riemann sums and definite integrals . J. Math. Behav. , 33 , 230 -- 245 . Google Scholar CrossRef Search ADS Silver , E. A. ( 1982 ) Knowledge organization and mathematical problem solving . Mathematical problem solving (Lester, F. K., & Garofalo, J. eds) . Philadelphia, Pennsylvania : The Franklin Institute Press , pp. 55 -- 85 . Skemp , R. ( 1976 ) Relational understanding and instrumental understanding . Mathematics Teaching , 77 , 20 -- 26 . Tall , D. ( 1992, August ) Students’ difficulties in calculus. In Proceedings of Working Group 3 on Students difficulties in calculus, ICME-7 . Québec, Canada , pp. 13 -- 28 . Tarricone , P. ( 2011 ) The taxonomy of metacognition . East Sussex, England : Psychology Press . Thomas , M. O. J. & Hong , Y. Y. ( 1996, June ) The Riemann integral in calculus: Students’ processes and concepts . In Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA) , pp. 572 -- 579 . Veenman , M. & Elshout , J. J. ( 1999 ) Changes in the relation between cognitive and metacognitive skills during the acquisition of expertise . European Journal of Psychology of Education , 14 , 509 -- 523 . Google Scholar CrossRef Search ADS Veenman , M. V. , Kerseboom , L. & Imthorn , C. ( 2000 ) Test anxiety and metacognitive skillfulness: Availability versus production deficiencies . Anxiety, Stress and Coping , 13 , 391 -- 412 . Google Scholar CrossRef Search ADS Veenman , M. V. J. ( 2011 ) Learning to self-monitor and self-regulate . Handbook of research on learning and instruction (Mayer, R. E., & Alexander, P. A. eds) . New York, N.Y : Routledge , pp. 197 -- 218 . Verschaffel , L. ( 1999 ) Realistic mathematical modelling and problem solving in the upper elementary school: Analysis and improvement . Teaching and learning thinking skills: Contexts of learning (Hamers, J. H. M., Van Luit, J. E. H. & Csapo, B. eds) . Lisse, the Netherlands : Swets & Zeitlinger , pp. 215 -- 240 . Yin , R. K. ( 2014 ) Case study research: Design and methods . United State of America : Sage publications Inc. © The Author(s) 2018. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

### Journal

Teaching Mathematics and Its Applications: International Journal of the IMAOxford University Press

Published: Mar 21, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations