Introduction
Calculus is one of the most important areas of mathematics. Therefore understanding basic ideas and concepts of calculus which are normally introduced in secondary school is extremely important for those wishing to pursue further studies in mathematics at university level. However a common complaint among students from almost all the countries in the world is that learning calculus is very difficult (Robert & Speer, 2001). More importantly, mathematics lecturers criticize that,
Students were not emerging from the courses with a strong understanding of calculus concepts and they were also not choosing to pursue further mathematics courses (Robert & Speer, 2001, p.291).
Therefore identifying the difficulties students face in learning calculus and understanding the related concepts in calculus are essential to the mathematics world.
The purpose of this essay is to review the literature on students’ perception of learning calculus at higher secondary and first year at university level. In this essay I will also try to identify the difficulties students face in learning calculus at higher secondary and first year at university level. The essay also includes a discussion on how such difficulties could be minimized and the teachers’ roles in doing that.
Learning calculus concepts
Function
The concept of function is central to the study of calculus. So the ability to comprehend function is crucial to understanding the concepts in calculus. Poor understanding of the underlying concept of function has been shown not only among mathematics students (Eisenberg, 1991), but also among teacher education students and teachers (Even, 1993). The apparent tendency of calculus courses to promote rote, manipulative learning without much understanding has been a subject of concern for more than a decade (Hassan, Mohamed & Mitchelmore, 2000). It might be entirely possible that the difficulties students face in understanding calculus are highly related to their concept of function, leading to narrow understanding of concepts such as derivative, limit and integral (Ferrini-Mundy & Lauten, 1993). Santos-Trigo’s (1998) findings offer further support that lack of understanding of elementary concepts is a major source of difficulty for first year calculus students. Santos-Trigo analyzed errors among first year students and discovered that many of the errors were due to lack of understanding of basic concepts such as functional notation, evaluation of limits and so on.
Therefore, it is extremely important for students to have a good foundation of the basic concepts of function in order to learn calculus concepts.
The role that functions play in mathematics is so extensive that it is impossible to give any summary that is both brief and adequate. There is no branch of mathematics whose developments since 1800 can be studied in their present form without an understanding of general function concept (Buck sited in Ferrini-Mundy & Lauten, 1993 , p. 156).
There is convincing evidence indicating that function concepts develop over time and even very young children can learn the basic ideas of function (Greeno, Leinhardt, Smith & Thomas cited in Ferrini-Mundy & Lauten, 1993). However research suggest that many students cannot think of a function as it behaves over intervals or in a global way rather they deal with functions pointwise, that is they can plot and read points (Bell & Janvier, 1981; Lovell, 1971; Monk, 1988).This suggests that it is essential to emphasize the development of the fundamental concepts of calculus like functions and ensure thorough understanding of these with different representations.
Limits, Series, Sequence and Continuity.
Limits, Series, Sequence and Continuity are the other important basic concepts that need to be developed in order to successfully progress with learning calculus. However, this also seems to be an area that students consider to problematic. According to Ferrini-Mundy & Lauten (1993), researchers examining limits, series, sequence and continuity indicate that students face conflicts between formal definitions and informal explanations that are used for convenience in discourse. Informal meanings of these terms also contribute to student confusion regarding these concepts (Ferrini-Mundy & Lauten, 1993). Additionally, some scholars like Tall & Schwarzenberger (1978 ) cited in Ferrini-Mundy & Graham (1991) have also observed that difficulties arise due to “ informal translation of very sophisticated ideas” (p. 630). Other research also indicate that students often have only a superficial understanding of some concepts like equating function with evaluation which in turn contribute to misinform students about the nature of these concepts.
Derivative
Derivative is one of the central concepts of calculus and as such it is essential that students have an in-depth understanding of the main concepts in this area. However, literature and research investigating the extent of understanding of concepts of derivatives suggests that this is an area that is riddled with difficulties and misconception for students. A large-scale qualitative study conducted by Orton (1983) cited in Ferrini-Mundy and Graham (1991) provides evidence to this effect. The study revealed that students had a good grasp of procedural aspect of differentiation but their understanding of differentiation of quadratic polynomials was very poor. Furthermore, the results of this study suggest that the majority of the students had limited “intuitive” understanding of derivative. Additionally, it was suggested that areas that students had most difficulty include “tangent as limit of secants” and “ rate of change of a line versus rate of change of a curve and rate of change of a point versus rate of change over an interval” ( p. 631).
The good procedural performance in computing derivate suggests that many students actually mimic examples provided in the text books and their understanding of geometric representations of derivate is limited. Therefore, it is essential to evaluate the kind of understanding students have and find ways to develop a deeper understanding of the main concepts in derivative in order to be successful in calculus at higher secondary and university.
Integrals
As with other areas of calculus discussed previously, the main difficulty that students come across in integration also relate the primitive understanding of the underlying concepts of integrals. For instance, integration of the limit of sum as well as the procedure of computing area or volume using the limit process does not seem to be central to students’ understanding of integrals. Additionally, many students are unable to reason out why methods to do with these areas actually work (Ferrini-Mundy 1991).
Based on his study, Orton (1983) cited in Ferrinii-Mundy (1991), suggests that students also posses misunderstandings regarding the underlying concepts of integration.
Thus, the main hindrance to success in calculus evidently relates to the lack of in-depth understanding of the concepts central to certain important areas of calculus such as function, integrals, derivative and Limits, Series, Sequence and Continuity. Students also lack the necessary “intuition” required to fully master the main concepts of these areas. As a result, calculus at higher secondary and university becomes exceptionally difficult for students.
Difficulties students face in learning calculus
Research indicates that calculus is an area in mathematics that students find problematic. For instance, a survey by mathematical Association of America Conference Board on mathematical sciences (Anderson & Loftsgarden 1987 cited in Ferrini-Mundy & Gaudard 1992) revealed that failure rates in the 1st year calculus courses were exceptionally high and only less than 50% completed the course with a D grade or better. It is possible to rectify weaknesses in other areas of mathematics like computation of algebraic problems which are important to calculus through remediation. Nonetheless, it makes one wonder why college students still find calculus difficult, what could be contributing to these difficulties and why such high rates of failure in calculus at university level happen. One possible explanation could be that calculus involves understanding of many concepts that are interrelated. For example, the extent to which students understand limits and functions will influence their subsequent understanding of derivatives and other fundamental concepts of calculus (Robert & Speer 2001). This means that if students do not have adequate understanding of one area of calculus, it is very likely that they will face additional difficulties in doing calculus. Additionally, some areas in calculus like functions and their associated sub-concepts can be very abstract to varying degrees. Students are unable to conceive functions and related notions visually and this in turn impedes understanding of these concepts (Eisenberg 1991).
The other reason related to difficulty in learning calculus at university level has to do with inadequate preparation of the students leaving high school. According to White & Mitchelmore (1996), the main concern lies in the fact that rote and manipulative learning takes place at high school. For instance, students might be good in calculating derivates and anti-derivatives as a result of continuous practice of similar procedures, but their understanding of concepts like rate of change is limited. Thus, students leaving high school have only a narrow understanding of concepts and as a result, they face more difficulties when it comes to application of these concepts at a higher level calculus courses.
Minimizing difficulties
The Mathematics teachers could play a central and an essential role in minimizing the difficulties that students face in learning calculus. For this to be operational, it is necessary that teachers have a good through understanding of the central concepts like limits, sequence, series and functions. However, a research done by Even (1993) indicates that knowledge of function of secondary mathematics teachers is limited. Teachers’ conceptions of function were similar to students and they did not hold modern conception of functions. Thus, in order to minimize the difficulties students face in learning, it is very important to train qualified teachers with good understanding of concepts in calculus like functions. Developing pedagogical knowledge is equally as important as developing mathematics content knowledge. Many of us tend to believe teachers having good content knowledge are good teachers. It does not necessarily have to be so. However, it is becoming widely recognized that teachers content knowledge has a strong influence on their pedagogical content knowledge, and hence on the learning of students (Thomas, 2003). Hence, teachers who have strong content knowledge and specific pedagogical preparation can create a learning environment that foster the development of students’ mathematical power (Even, 1993).
As discussed earlier in this essay, the major difficulty lies in understanding the main concepts of calculus such as functions, derivatives, integrals, series and sequence. Therefore teachers need to put more emphasis on developing concepts rather than spending more time with the procedural knowledge with repeated algorithms in the classrooms. One possible mean of achieving this is suggested by Ferrini-Mundy & Graham (1991). They suggest that activity based teaching could contribute to make lessons more engaging for the students thereby ensuring a deeper conceptual understanding of various areas in mathematics such as calculus. Questions that we present must engage the students through classroom, laboratory or project activities. Evidence supporting the effectiveness of activity based learning is presented in a study done by Heid (1988). The study compared two groups of students in an introductory calculus course. The control group spent more time with activity based learning. On the last three weeks were spent on skill development. On the other hand the other group spent entire 15 weeks on skill development. Class transcripts, student’s interviews and test results were analyzed for patterns of understanding. The result showed that students in the control group have better understanding of course concepts and even performed almost as well on the final examination as the comparison group. Their performance was remarkably suggestive that compressed and minimal attention to skill development was not necessarily harmful, even on a skills test.
Conclusion
Calculus is one of the most important areas of mathematics. As Young (1986) points out “Calculus is our most important course. The future of our subject depends upon improving it…” (Cited in Ferrini-Mundy & Graham 1991: 627). Despite the relative importance of calculus, research indicates that students leaving high school wishing to pursue mathematics at university do not have the necessary strong foundation in this area. Most of these students may be able do calculus problems in an automatic rote learned fashion. However, they lack the in-depth understanding of some of the central concepts and as a result have a major weakness when applications of these concepts are required at higher level courses. Thus, it is essential that relevant concepts in calculus are introduced to the students in some fashion at high school in order to develop a stronger base for higher level calculus courses. The first step in doing this is to examine if there exists a gap between the high school and the university curriculums.
Teachers can also play a crucial role in making calculus less difficult and more appealing for students.
References
Bell, A. (1981). The interpretation of graphs representing situations. For the learning of Mathematics. 2(1), 34-42.
Einsenberg, T. (1991). Functions and associated learning difficulties. In Tall, D.O. (Ed.), Advanced Mathematical thinking (pp. 140-152). Dordrecht, The Netherlands: Kluwer.
Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics education, 24, 94-116
Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and Learning Calculus. In Wilson, P.S.(Ed.), Research Ideas for the classroom: High school mathematics (pp.155-176). New York: Macmillan Publishing Company.
Ferrini-Mundy, J., & Graham, K. (1991). An over view of the calculus curriculum reform effort: Issues for learning, teaching and curriculum development. American Mathematical Monthly, 98(7), 627-635.
Hassan, I., Mohamed, K., & Mitchelmore, M. (2000). Understanding of Functions Among Maldivian Teacher Education Students. In Bana, J., & Chapman, A. (EDS.), Mathematics Education Beyond 2000. Proceedings of the Twenty –third Annual Conference of the Mathematics Education Research Group of Australasia Incorporated.1.291-298. Perth: Mathematics Education Research Group of Australasia Incorporated.
Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19(1), 3–25.
Lovell, K. (1971). Some aspects of the growth of the concept of a function. In Myron, F., Rosskopf, L. P., Steffe, & Stanley, T. (Eds.). Piagetian cognitive development research and mathematical education (pp. 12-33). Washington, D.C.:NCTM
Monk, G.S.(1988). Students’ understanding of functions in calculus courses, Humanistic Mathematics Network Newsletter, 2.
Ortan, A. Students Understanding of Differentiation. Educational Studies in Mathematics. 15 (1983a) 235-250.
Robert, A., & Speer, N. (2001). Research on the teaching and learning of calculus /elementary analysis. In Holton, D. (Ed.), The teaching and learning of mathematics at university level. 283-299. Dordrecht: Kluwer Academic Publiher.
Santos-Trigo, M. (1998). Patterns of mathematical misunderstanding exhibited by calculus students in a problem solving course. In Oliver, A. & Newstead, K. (Eds.). Proceedings of the 22nd annual conference of the international Group for the psychology of Mathematics Education. 4.89-96.
Thomas, M. O. J. (2003). The role of representation in teacher understanding of function, Proceedings of 27th Conference of International Group for the Psychology of Mathematics Education, Honolulu, Hawai.
White, P. & Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus. Journal for Research in Mathematics education. 27(1),79-95
Calculus is one of the most important areas of mathematics. Therefore understanding basic ideas and concepts of calculus which are normally introduced in secondary school is extremely important for those wishing to pursue further studies in mathematics at university level. However a common complaint among students from almost all the countries in the world is that learning calculus is very difficult (Robert & Speer, 2001). More importantly, mathematics lecturers criticize that,
Students were not emerging from the courses with a strong understanding of calculus concepts and they were also not choosing to pursue further mathematics courses (Robert & Speer, 2001, p.291).
Therefore identifying the difficulties students face in learning calculus and understanding the related concepts in calculus are essential to the mathematics world.
The purpose of this essay is to review the literature on students’ perception of learning calculus at higher secondary and first year at university level. In this essay I will also try to identify the difficulties students face in learning calculus at higher secondary and first year at university level. The essay also includes a discussion on how such difficulties could be minimized and the teachers’ roles in doing that.
Learning calculus concepts
Function
The concept of function is central to the study of calculus. So the ability to comprehend function is crucial to understanding the concepts in calculus. Poor understanding of the underlying concept of function has been shown not only among mathematics students (Eisenberg, 1991), but also among teacher education students and teachers (Even, 1993). The apparent tendency of calculus courses to promote rote, manipulative learning without much understanding has been a subject of concern for more than a decade (Hassan, Mohamed & Mitchelmore, 2000). It might be entirely possible that the difficulties students face in understanding calculus are highly related to their concept of function, leading to narrow understanding of concepts such as derivative, limit and integral (Ferrini-Mundy & Lauten, 1993). Santos-Trigo’s (1998) findings offer further support that lack of understanding of elementary concepts is a major source of difficulty for first year calculus students. Santos-Trigo analyzed errors among first year students and discovered that many of the errors were due to lack of understanding of basic concepts such as functional notation, evaluation of limits and so on.
Therefore, it is extremely important for students to have a good foundation of the basic concepts of function in order to learn calculus concepts.
The role that functions play in mathematics is so extensive that it is impossible to give any summary that is both brief and adequate. There is no branch of mathematics whose developments since 1800 can be studied in their present form without an understanding of general function concept (Buck sited in Ferrini-Mundy & Lauten, 1993 , p. 156).
There is convincing evidence indicating that function concepts develop over time and even very young children can learn the basic ideas of function (Greeno, Leinhardt, Smith & Thomas cited in Ferrini-Mundy & Lauten, 1993). However research suggest that many students cannot think of a function as it behaves over intervals or in a global way rather they deal with functions pointwise, that is they can plot and read points (Bell & Janvier, 1981; Lovell, 1971; Monk, 1988).This suggests that it is essential to emphasize the development of the fundamental concepts of calculus like functions and ensure thorough understanding of these with different representations.
Limits, Series, Sequence and Continuity.
Limits, Series, Sequence and Continuity are the other important basic concepts that need to be developed in order to successfully progress with learning calculus. However, this also seems to be an area that students consider to problematic. According to Ferrini-Mundy & Lauten (1993), researchers examining limits, series, sequence and continuity indicate that students face conflicts between formal definitions and informal explanations that are used for convenience in discourse. Informal meanings of these terms also contribute to student confusion regarding these concepts (Ferrini-Mundy & Lauten, 1993). Additionally, some scholars like Tall & Schwarzenberger (1978 ) cited in Ferrini-Mundy & Graham (1991) have also observed that difficulties arise due to “ informal translation of very sophisticated ideas” (p. 630). Other research also indicate that students often have only a superficial understanding of some concepts like equating function with evaluation which in turn contribute to misinform students about the nature of these concepts.
Derivative
Derivative is one of the central concepts of calculus and as such it is essential that students have an in-depth understanding of the main concepts in this area. However, literature and research investigating the extent of understanding of concepts of derivatives suggests that this is an area that is riddled with difficulties and misconception for students. A large-scale qualitative study conducted by Orton (1983) cited in Ferrini-Mundy and Graham (1991) provides evidence to this effect. The study revealed that students had a good grasp of procedural aspect of differentiation but their understanding of differentiation of quadratic polynomials was very poor. Furthermore, the results of this study suggest that the majority of the students had limited “intuitive” understanding of derivative. Additionally, it was suggested that areas that students had most difficulty include “tangent as limit of secants” and “ rate of change of a line versus rate of change of a curve and rate of change of a point versus rate of change over an interval” ( p. 631).
The good procedural performance in computing derivate suggests that many students actually mimic examples provided in the text books and their understanding of geometric representations of derivate is limited. Therefore, it is essential to evaluate the kind of understanding students have and find ways to develop a deeper understanding of the main concepts in derivative in order to be successful in calculus at higher secondary and university.
Integrals
As with other areas of calculus discussed previously, the main difficulty that students come across in integration also relate the primitive understanding of the underlying concepts of integrals. For instance, integration of the limit of sum as well as the procedure of computing area or volume using the limit process does not seem to be central to students’ understanding of integrals. Additionally, many students are unable to reason out why methods to do with these areas actually work (Ferrini-Mundy 1991).
Based on his study, Orton (1983) cited in Ferrinii-Mundy (1991), suggests that students also posses misunderstandings regarding the underlying concepts of integration.
Thus, the main hindrance to success in calculus evidently relates to the lack of in-depth understanding of the concepts central to certain important areas of calculus such as function, integrals, derivative and Limits, Series, Sequence and Continuity. Students also lack the necessary “intuition” required to fully master the main concepts of these areas. As a result, calculus at higher secondary and university becomes exceptionally difficult for students.
Difficulties students face in learning calculus
Research indicates that calculus is an area in mathematics that students find problematic. For instance, a survey by mathematical Association of America Conference Board on mathematical sciences (Anderson & Loftsgarden 1987 cited in Ferrini-Mundy & Gaudard 1992) revealed that failure rates in the 1st year calculus courses were exceptionally high and only less than 50% completed the course with a D grade or better. It is possible to rectify weaknesses in other areas of mathematics like computation of algebraic problems which are important to calculus through remediation. Nonetheless, it makes one wonder why college students still find calculus difficult, what could be contributing to these difficulties and why such high rates of failure in calculus at university level happen. One possible explanation could be that calculus involves understanding of many concepts that are interrelated. For example, the extent to which students understand limits and functions will influence their subsequent understanding of derivatives and other fundamental concepts of calculus (Robert & Speer 2001). This means that if students do not have adequate understanding of one area of calculus, it is very likely that they will face additional difficulties in doing calculus. Additionally, some areas in calculus like functions and their associated sub-concepts can be very abstract to varying degrees. Students are unable to conceive functions and related notions visually and this in turn impedes understanding of these concepts (Eisenberg 1991).
The other reason related to difficulty in learning calculus at university level has to do with inadequate preparation of the students leaving high school. According to White & Mitchelmore (1996), the main concern lies in the fact that rote and manipulative learning takes place at high school. For instance, students might be good in calculating derivates and anti-derivatives as a result of continuous practice of similar procedures, but their understanding of concepts like rate of change is limited. Thus, students leaving high school have only a narrow understanding of concepts and as a result, they face more difficulties when it comes to application of these concepts at a higher level calculus courses.
Minimizing difficulties
The Mathematics teachers could play a central and an essential role in minimizing the difficulties that students face in learning calculus. For this to be operational, it is necessary that teachers have a good through understanding of the central concepts like limits, sequence, series and functions. However, a research done by Even (1993) indicates that knowledge of function of secondary mathematics teachers is limited. Teachers’ conceptions of function were similar to students and they did not hold modern conception of functions. Thus, in order to minimize the difficulties students face in learning, it is very important to train qualified teachers with good understanding of concepts in calculus like functions. Developing pedagogical knowledge is equally as important as developing mathematics content knowledge. Many of us tend to believe teachers having good content knowledge are good teachers. It does not necessarily have to be so. However, it is becoming widely recognized that teachers content knowledge has a strong influence on their pedagogical content knowledge, and hence on the learning of students (Thomas, 2003). Hence, teachers who have strong content knowledge and specific pedagogical preparation can create a learning environment that foster the development of students’ mathematical power (Even, 1993).
As discussed earlier in this essay, the major difficulty lies in understanding the main concepts of calculus such as functions, derivatives, integrals, series and sequence. Therefore teachers need to put more emphasis on developing concepts rather than spending more time with the procedural knowledge with repeated algorithms in the classrooms. One possible mean of achieving this is suggested by Ferrini-Mundy & Graham (1991). They suggest that activity based teaching could contribute to make lessons more engaging for the students thereby ensuring a deeper conceptual understanding of various areas in mathematics such as calculus. Questions that we present must engage the students through classroom, laboratory or project activities. Evidence supporting the effectiveness of activity based learning is presented in a study done by Heid (1988). The study compared two groups of students in an introductory calculus course. The control group spent more time with activity based learning. On the last three weeks were spent on skill development. On the other hand the other group spent entire 15 weeks on skill development. Class transcripts, student’s interviews and test results were analyzed for patterns of understanding. The result showed that students in the control group have better understanding of course concepts and even performed almost as well on the final examination as the comparison group. Their performance was remarkably suggestive that compressed and minimal attention to skill development was not necessarily harmful, even on a skills test.
Conclusion
Calculus is one of the most important areas of mathematics. As Young (1986) points out “Calculus is our most important course. The future of our subject depends upon improving it…” (Cited in Ferrini-Mundy & Graham 1991: 627). Despite the relative importance of calculus, research indicates that students leaving high school wishing to pursue mathematics at university do not have the necessary strong foundation in this area. Most of these students may be able do calculus problems in an automatic rote learned fashion. However, they lack the in-depth understanding of some of the central concepts and as a result have a major weakness when applications of these concepts are required at higher level courses. Thus, it is essential that relevant concepts in calculus are introduced to the students in some fashion at high school in order to develop a stronger base for higher level calculus courses. The first step in doing this is to examine if there exists a gap between the high school and the university curriculums.
Teachers can also play a crucial role in making calculus less difficult and more appealing for students.
References
Bell, A. (1981). The interpretation of graphs representing situations. For the learning of Mathematics. 2(1), 34-42.
Einsenberg, T. (1991). Functions and associated learning difficulties. In Tall, D.O. (Ed.), Advanced Mathematical thinking (pp. 140-152). Dordrecht, The Netherlands: Kluwer.
Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics education, 24, 94-116
Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and Learning Calculus. In Wilson, P.S.(Ed.), Research Ideas for the classroom: High school mathematics (pp.155-176). New York: Macmillan Publishing Company.
Ferrini-Mundy, J., & Graham, K. (1991). An over view of the calculus curriculum reform effort: Issues for learning, teaching and curriculum development. American Mathematical Monthly, 98(7), 627-635.
Hassan, I., Mohamed, K., & Mitchelmore, M. (2000). Understanding of Functions Among Maldivian Teacher Education Students. In Bana, J., & Chapman, A. (EDS.), Mathematics Education Beyond 2000. Proceedings of the Twenty –third Annual Conference of the Mathematics Education Research Group of Australasia Incorporated.1.291-298. Perth: Mathematics Education Research Group of Australasia Incorporated.
Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19(1), 3–25.
Lovell, K. (1971). Some aspects of the growth of the concept of a function. In Myron, F., Rosskopf, L. P., Steffe, & Stanley, T. (Eds.). Piagetian cognitive development research and mathematical education (pp. 12-33). Washington, D.C.:NCTM
Monk, G.S.(1988). Students’ understanding of functions in calculus courses, Humanistic Mathematics Network Newsletter, 2.
Ortan, A. Students Understanding of Differentiation. Educational Studies in Mathematics. 15 (1983a) 235-250.
Robert, A., & Speer, N. (2001). Research on the teaching and learning of calculus /elementary analysis. In Holton, D. (Ed.), The teaching and learning of mathematics at university level. 283-299. Dordrecht: Kluwer Academic Publiher.
Santos-Trigo, M. (1998). Patterns of mathematical misunderstanding exhibited by calculus students in a problem solving course. In Oliver, A. & Newstead, K. (Eds.). Proceedings of the 22nd annual conference of the international Group for the psychology of Mathematics Education. 4.89-96.
Thomas, M. O. J. (2003). The role of representation in teacher understanding of function, Proceedings of 27th Conference of International Group for the Psychology of Mathematics Education, Honolulu, Hawai.
White, P. & Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus. Journal for Research in Mathematics education. 27(1),79-95
