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Visualizing Student's Difficulties in Learning Calculus
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Procedia Social and Behavioral Sciences 8 (2010) 377–383
International Conference on Mathematics Education Research 2010 (ICMER 2010)
Visualizing Students’ Difficulties in Learning Calculus Rohani Ahmad Tarmizi* Institute for Mathematical Research & Faculty of Educational Studies, Universiti Putra Malaysia
Abstract Visualization is well recognized as a powerful problem representation process for solving mathematical problems. Van Garderen and Montague (2003) emphasizes that good problem solvers generally construct a representation of the problem to facilitate understanding. The purpose of this study was to investigate students’ performance in solving Calculus problems and further analysed students’ difficulties in solving the problems. It is clear from the protocol analyses that some students require special treatment such as further tutorial session in correcting their misconception. Also students need to monitor their steps in problem solving or deriving problem solution. Much attention should be directed to fostering students’ ability to plan for problem solution. If this is not done, the students are likely to become progressively more confused and in the long run they may not survive in their mathematics problem solving experiences. © 2010 Elsevier Ltd. All rights reserved. Keywords: Mathematics problem solving; Visualization; Learning difficulties; Mathematical thinking; Learning calculus
1.
Introduction
The aim of learning mathematics for university undergraduates is surely success for some whilst meeting the requirements for others. It does seem to be a fact of life that whilst a few prosper in mathematics, a much greater number find mathematics difficult. Hence, however successful a course may appear to be, there are students who begin to struggle and who will need appropriate help to be able to pursue mathematics further. Gray and Tall (1991) gave evidence that the less successful students have greater difficulties with the processes and concentrate on the lesser target of being able to carry them out satisfactorily whilst more successful students develop a more flexible method of handling notation, being able to see it either as a process to calculate a needed result or as an object to be manipulated as part of a more complex piece of symbolism. Tall (2004) explained that mathematical thinking develop through two distinct process. The first builds on our perception of real-world objects and our actions upon them which he referred to conceptual embodiment. The second builds on actions that are symbolised as process or concept, which may remain at procedural level or may be operated at a higher level which he referred to as proceptual symbolism. An example in calculus can be seen as examining the rate at which functions change is
* Corresponding author. E-mail address: [email protected] . 1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.12.053
378
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
analogous to seeing the steepness of the graph changes as a finger is traced along the graph, then the graph of the changing slope can be seen as a new graph of the rate of change, symbolised as the derivative. This study examined undergraduate students’ performance in solving Calculus problems and further analysed students’ difficulties in solving the problems. In the vast literature, many methods in identifying the difficulties encountered by students during solving mathematics problems were cited. Many researchers have considered interview techniques, such as getting the students to think aloud (Newell & Simon, 1972; Bartlett, 1954). Using this method, students were asked to verbalize every thought that comes to their mind in the course of solving the given mathematics problem. Another method was proposed originally by Jean Piaget, which is by conducting clinical interview (Greer, 1987). This differs from the “thinking aloud” method. Here, the tutor may follow the thoughts of his students by asking questions here and there during the course of solving the particular problem. Another method is the used of the Diagnostic test paper consisting of questions whose design was based on expected conceptual difficulties with questions formulated to induce conceptual errors. The questions may elicit several errors such as mental, emotional, motivational, reading or instructional related problems (Bell, 1978). Before proceeding, let us look at the four-step Polya’s mathematical problem solving (Polya, 1957) which is to understand, plan, conduct and reflect on the problem. Van Garderen and Montague (2003) emphasizes that good problem solvers generally construct a representation of the problem to facilitate understanding. Thus, visualization is well recognized as a powerful problem representation process for solving mathematical problems. In this study, we aimed to examine problem solving procedures and into the diagnosis of the difficulties experienced by students in learning Calculus I (MTH 3100). Twenty students participated in this study on a voluntary basis. 2.
Methodology
Clinical interview series were conducted with the students who were mostly mathematics majors. Subjects were interviewed independently in undisturbed classroom settings. Each interview began with a presentation of a set of question related to the course. The subjects were asked to derive problem solution on the question set. The subjects were also asked to verbalize their thought processes. For this purposes, the tutor will elicit questions to subjects while they solve the given problem. All interviews were recorded, transcribed, and coded based on the four-steps of Polya problem solving steps. An overall analysis of problem solution along with the protocol analysis was performed. An initial scan of the students’ protocols and solutions revealed a wide array of difficulties with specific knowledge. Thematic analyses were drawn from these transcriptions. The following is the first question given to the students. Question 1: If
( x)
4x 2 2 , x z 1 , find f (3x) f ( ), x z 0 x 1 x
Probable Solution to Question 1:
§2· f (3x) f ¨ ¸ © x¹
§2· 4¨ ¸ 2 4(3x) 2 x © ¹ 3x 1 §2· ¨ ¸ 1 © x¹
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
379
§ 8 2x · 12 x 2 ¨© x ¸¹ 3x 1 § 2 x · ¨ ¸ © x ¹ 12 x 2 § 8 2 x · § x · ¨ ¸¨ ¸ 3x 1 © x ¹ © 2 x ¹ 12 x 2 § 8 2 x · ¨ ¸ 3x 1 © 2 x ¹ 3.
Findings
The findings of this study will be described based on the Polya’s four-step in mathematics problem solving which is understand, plan, conduct and reflect on the answer to the problem. The results are presented as follows: 3.1
Understanding the Problem
Interview result found as many as 16 out 20 subjects really understand that question. Nevertheless, two subjects said that they slightly understand whilst two subjects do not understand this question. Students understanding were supported by protocols in which, they were able to elaborate the question’s goals and givens of the problems. Overall, the pattern on describing their understanding of the question is .... by substituting 3x 2 into function
4x 2 I’ll be able to derive the answer. Generally, the students were able to show their level of x 1 understanding of the question. However, three students get confused with the information x z 1 . They do not know f ( x)
the role of that information. Apart from that, all respondent were successfully identifying important information from the question namely the function which needs to be substituted by f (3x 2) But, there is one subject who had contravened concept in which he/she substituted function 3.2
f ( x)
4x 2 into f (3x 2) . x 1
Devising a Plan
Subjects were asked to verbalize their plans in obtaining the problem solution. All of the subjects mentioned that they actually do not planned on using any law or rule except for one respondent. This particular subject mention about using the substitution method that is replacing the given values into the new function x . 3.3
Carry Out the Plan
All respondents explain their problem solving step which was based on their own prior understandings or conceptions from previous class and tutorial sessions. For respondents that fully understand, they wrote correct solution and explained clearly where they substituted function f (3x 2) into the function given,
f ( x)
4x 2 . Two example of problem solution are as follows: x 1
380
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
On the other hand, there were also a few misconception of the problem situation thus the students’ solution steps resulted in wrong solution. Their solutions are as follow:
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
3.4
381
Looking Back
To assess whether students do evaluate, checked or reflect their answer solution, they were questioned whether they had actually showed the correct solution step. Five out of 20 subjects said that they were not sure. About four subjects did not check their answers. They just wrote their solution without evaluating, checking or reflecting on their answer solution. Further probing on their mental thought processes during the interview revealed the following confusions. When asked whether they know any other solutions form...only two of them said that maybe there is another form of solution but they do not know how. 15 respondents said that they do not know any other way except for they had done. Only three respondents feel that there is another way to solve this question but it is beyond their present knowledge. A respondent said that even he is not sure of his own method, let alone any other method. One of them started with .. I do not understood the question from the start. One of them stated that I do not know how to add fraction to whole number .. this is so confusing. Student 17 ... I am confused whether to substitute the x value in the denominator or not. However, after taking a few minutes to think, the student replaced the value in both numerator and denominator part. Students were asked again whether they understood what they had done or they just memorize the procedures. Three subjects indicated they did not understand clearly their solutions whilst one student responded ... I did not simply memorized the method of solution but understand what I was doing.
382
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
One of them said that I am confused while eliminating the value at the final answer. However, the method used was totally wrong. Below is the work written by this subject:
4.
Conclusion
Calculus is seen as a difficult subject among mathematics students and often they misunderstood the notion of function. A function y f ( x) is a process: input x, carry out a procedure, output y. Majority of students may only follow rules ... multiply out brackets, collect together like terms, look for common factors etc. In addition, students may have acquired learned procedures from prior discussion and instruction to use the substitution rule before doing the required expansion. It is clear from this protocol analysis that some students are likely to need special treatment such as further tutorial session in correcting their misconception. Also students need to monitor their steps in problem solving or deriving problem solution. Much attention should be directed to fostering students’ ability to plan for problem solution. If this is not done, the students are likely to become progressively more confused and in the long run they may not survive in their mathematics problem solving experiences. The literature makes some suggestions as to what may be happening. One possible solution is to reemphasize teaching of problem-solving strategies in mathematics courses thus providing students with logical step towards solution formulation. Tall (2004) proposed that revision of procedures to give success and concrete visualization of these procedures is necessary. Reflection on these procedures will provide learners more meaningful learning. References Bartlett F. C., 1958: Thinking, George Allen ~ Unwin, London. Bell F. H., 1978: Teaching and Learning Mathematics (in Secondary schools), W.C.Brown Co. Pub., USA. Gagné R. M., 1970: The Conditions of Learning, Holt Rinehart and Winston Inc., New York, 2nd. edition. Gray E. & Tall D. O., 1991: “Duality, Ambiguity and Flexibility in Successful Mathematical Thinking”, Proceedings of PME14, Assisi, II, 72– 79. Greer B.,1987: “Nonconservation of multiplication and division involving decimals”, Journal for Research in Mathematics Education 18 37-45. Newell A. & Simon H.A.,1972: Human Problem Solving, Prentice Hall, N.J., USA. Tall, D. O. & Bakar, M. N., 1993: “Students’ Mental Prototypes for Functions and Graphs”, International Journal of Mathematics Education in Science and Technology, 23 1, 39–50.
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
383
Tall, D. O. (2004). Thinking through three worlds of mathematics, Proceedings of the 28 th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281-288. Van Garderen, D. & Montague, M. (2003). Visual-spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practices, 18(4), 246-254.
Procedia Social and Behavioral Sciences 8 (2010) 377–383
International Conference on Mathematics Education Research 2010 (ICMER 2010)
Visualizing Students’ Difficulties in Learning Calculus Rohani Ahmad Tarmizi* Institute for Mathematical Research & Faculty of Educational Studies, Universiti Putra Malaysia
Abstract Visualization is well recognized as a powerful problem representation process for solving mathematical problems. Van Garderen and Montague (2003) emphasizes that good problem solvers generally construct a representation of the problem to facilitate understanding. The purpose of this study was to investigate students’ performance in solving Calculus problems and further analysed students’ difficulties in solving the problems. It is clear from the protocol analyses that some students require special treatment such as further tutorial session in correcting their misconception. Also students need to monitor their steps in problem solving or deriving problem solution. Much attention should be directed to fostering students’ ability to plan for problem solution. If this is not done, the students are likely to become progressively more confused and in the long run they may not survive in their mathematics problem solving experiences. © 2010 Elsevier Ltd. All rights reserved. Keywords: Mathematics problem solving; Visualization; Learning difficulties; Mathematical thinking; Learning calculus
1.
Introduction
The aim of learning mathematics for university undergraduates is surely success for some whilst meeting the requirements for others. It does seem to be a fact of life that whilst a few prosper in mathematics, a much greater number find mathematics difficult. Hence, however successful a course may appear to be, there are students who begin to struggle and who will need appropriate help to be able to pursue mathematics further. Gray and Tall (1991) gave evidence that the less successful students have greater difficulties with the processes and concentrate on the lesser target of being able to carry them out satisfactorily whilst more successful students develop a more flexible method of handling notation, being able to see it either as a process to calculate a needed result or as an object to be manipulated as part of a more complex piece of symbolism. Tall (2004) explained that mathematical thinking develop through two distinct process. The first builds on our perception of real-world objects and our actions upon them which he referred to conceptual embodiment. The second builds on actions that are symbolised as process or concept, which may remain at procedural level or may be operated at a higher level which he referred to as proceptual symbolism. An example in calculus can be seen as examining the rate at which functions change is
* Corresponding author. E-mail address: [email protected] . 1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.12.053
378
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
analogous to seeing the steepness of the graph changes as a finger is traced along the graph, then the graph of the changing slope can be seen as a new graph of the rate of change, symbolised as the derivative. This study examined undergraduate students’ performance in solving Calculus problems and further analysed students’ difficulties in solving the problems. In the vast literature, many methods in identifying the difficulties encountered by students during solving mathematics problems were cited. Many researchers have considered interview techniques, such as getting the students to think aloud (Newell & Simon, 1972; Bartlett, 1954). Using this method, students were asked to verbalize every thought that comes to their mind in the course of solving the given mathematics problem. Another method was proposed originally by Jean Piaget, which is by conducting clinical interview (Greer, 1987). This differs from the “thinking aloud” method. Here, the tutor may follow the thoughts of his students by asking questions here and there during the course of solving the particular problem. Another method is the used of the Diagnostic test paper consisting of questions whose design was based on expected conceptual difficulties with questions formulated to induce conceptual errors. The questions may elicit several errors such as mental, emotional, motivational, reading or instructional related problems (Bell, 1978). Before proceeding, let us look at the four-step Polya’s mathematical problem solving (Polya, 1957) which is to understand, plan, conduct and reflect on the problem. Van Garderen and Montague (2003) emphasizes that good problem solvers generally construct a representation of the problem to facilitate understanding. Thus, visualization is well recognized as a powerful problem representation process for solving mathematical problems. In this study, we aimed to examine problem solving procedures and into the diagnosis of the difficulties experienced by students in learning Calculus I (MTH 3100). Twenty students participated in this study on a voluntary basis. 2.
Methodology
Clinical interview series were conducted with the students who were mostly mathematics majors. Subjects were interviewed independently in undisturbed classroom settings. Each interview began with a presentation of a set of question related to the course. The subjects were asked to derive problem solution on the question set. The subjects were also asked to verbalize their thought processes. For this purposes, the tutor will elicit questions to subjects while they solve the given problem. All interviews were recorded, transcribed, and coded based on the four-steps of Polya problem solving steps. An overall analysis of problem solution along with the protocol analysis was performed. An initial scan of the students’ protocols and solutions revealed a wide array of difficulties with specific knowledge. Thematic analyses were drawn from these transcriptions. The following is the first question given to the students. Question 1: If
( x)
4x 2 2 , x z 1 , find f (3x) f ( ), x z 0 x 1 x
Probable Solution to Question 1:
§2· f (3x) f ¨ ¸ © x¹
§2· 4¨ ¸ 2 4(3x) 2 x © ¹ 3x 1 §2· ¨ ¸ 1 © x¹
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
379
§ 8 2x · 12 x 2 ¨© x ¸¹ 3x 1 § 2 x · ¨ ¸ © x ¹ 12 x 2 § 8 2 x · § x · ¨ ¸¨ ¸ 3x 1 © x ¹ © 2 x ¹ 12 x 2 § 8 2 x · ¨ ¸ 3x 1 © 2 x ¹ 3.
Findings
The findings of this study will be described based on the Polya’s four-step in mathematics problem solving which is understand, plan, conduct and reflect on the answer to the problem. The results are presented as follows: 3.1
Understanding the Problem
Interview result found as many as 16 out 20 subjects really understand that question. Nevertheless, two subjects said that they slightly understand whilst two subjects do not understand this question. Students understanding were supported by protocols in which, they were able to elaborate the question’s goals and givens of the problems. Overall, the pattern on describing their understanding of the question is .... by substituting 3x 2 into function
4x 2 I’ll be able to derive the answer. Generally, the students were able to show their level of x 1 understanding of the question. However, three students get confused with the information x z 1 . They do not know f ( x)
the role of that information. Apart from that, all respondent were successfully identifying important information from the question namely the function which needs to be substituted by f (3x 2) But, there is one subject who had contravened concept in which he/she substituted function 3.2
f ( x)
4x 2 into f (3x 2) . x 1
Devising a Plan
Subjects were asked to verbalize their plans in obtaining the problem solution. All of the subjects mentioned that they actually do not planned on using any law or rule except for one respondent. This particular subject mention about using the substitution method that is replacing the given values into the new function x . 3.3
Carry Out the Plan
All respondents explain their problem solving step which was based on their own prior understandings or conceptions from previous class and tutorial sessions. For respondents that fully understand, they wrote correct solution and explained clearly where they substituted function f (3x 2) into the function given,
f ( x)
4x 2 . Two example of problem solution are as follows: x 1
380
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
On the other hand, there were also a few misconception of the problem situation thus the students’ solution steps resulted in wrong solution. Their solutions are as follow:
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
3.4
381
Looking Back
To assess whether students do evaluate, checked or reflect their answer solution, they were questioned whether they had actually showed the correct solution step. Five out of 20 subjects said that they were not sure. About four subjects did not check their answers. They just wrote their solution without evaluating, checking or reflecting on their answer solution. Further probing on their mental thought processes during the interview revealed the following confusions. When asked whether they know any other solutions form...only two of them said that maybe there is another form of solution but they do not know how. 15 respondents said that they do not know any other way except for they had done. Only three respondents feel that there is another way to solve this question but it is beyond their present knowledge. A respondent said that even he is not sure of his own method, let alone any other method. One of them started with .. I do not understood the question from the start. One of them stated that I do not know how to add fraction to whole number .. this is so confusing. Student 17 ... I am confused whether to substitute the x value in the denominator or not. However, after taking a few minutes to think, the student replaced the value in both numerator and denominator part. Students were asked again whether they understood what they had done or they just memorize the procedures. Three subjects indicated they did not understand clearly their solutions whilst one student responded ... I did not simply memorized the method of solution but understand what I was doing.
382
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
One of them said that I am confused while eliminating the value at the final answer. However, the method used was totally wrong. Below is the work written by this subject:
4.
Conclusion
Calculus is seen as a difficult subject among mathematics students and often they misunderstood the notion of function. A function y f ( x) is a process: input x, carry out a procedure, output y. Majority of students may only follow rules ... multiply out brackets, collect together like terms, look for common factors etc. In addition, students may have acquired learned procedures from prior discussion and instruction to use the substitution rule before doing the required expansion. It is clear from this protocol analysis that some students are likely to need special treatment such as further tutorial session in correcting their misconception. Also students need to monitor their steps in problem solving or deriving problem solution. Much attention should be directed to fostering students’ ability to plan for problem solution. If this is not done, the students are likely to become progressively more confused and in the long run they may not survive in their mathematics problem solving experiences. The literature makes some suggestions as to what may be happening. One possible solution is to reemphasize teaching of problem-solving strategies in mathematics courses thus providing students with logical step towards solution formulation. Tall (2004) proposed that revision of procedures to give success and concrete visualization of these procedures is necessary. Reflection on these procedures will provide learners more meaningful learning. References Bartlett F. C., 1958: Thinking, George Allen ~ Unwin, London. Bell F. H., 1978: Teaching and Learning Mathematics (in Secondary schools), W.C.Brown Co. Pub., USA. Gagné R. M., 1970: The Conditions of Learning, Holt Rinehart and Winston Inc., New York, 2nd. edition. Gray E. & Tall D. O., 1991: “Duality, Ambiguity and Flexibility in Successful Mathematical Thinking”, Proceedings of PME14, Assisi, II, 72– 79. Greer B.,1987: “Nonconservation of multiplication and division involving decimals”, Journal for Research in Mathematics Education 18 37-45. Newell A. & Simon H.A.,1972: Human Problem Solving, Prentice Hall, N.J., USA. Tall, D. O. & Bakar, M. N., 1993: “Students’ Mental Prototypes for Functions and Graphs”, International Journal of Mathematics Education in Science and Technology, 23 1, 39–50.
Rohani Ahmad Tarmizi / Procedia Social and Behavioral Sciences 8 (2010) 377–383
383
Tall, D. O. (2004). Thinking through three worlds of mathematics, Proceedings of the 28 th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281-288. Van Garderen, D. & Montague, M. (2003). Visual-spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practices, 18(4), 246-254.