The study examined classroom instructional practices and teacher's professed conceptions about teaching and learning college calculus in relationship to the implementation of scientific-programmable-graphics (SPG) calculators. The study occurred at a university not affiliated with any reform project. The participants were not the catalysts seeking to implement calculus reform, but expressed a willingness to teach the first quarter calculus course with the SPG calculator. The research design was based on qualitative methods using comparative case studies of five teachers. Primary data were collected through pre-school interviews and weekly classroom observations with subsequent interviews. Teachers' profiles were established describing general conceptions of teaching calculus, instructional practices, congruence between conceptions and practice, conceptions about teaching using SPG calculators, instructional practice with SPG calculators, and the relationship of conceptions and practice with SPG calculators. Initially, all the teachers without prior experience using SPG calculators indicated concern and skepticism about the usefulness of the technology in teaching calculus and were uncertain how to utilize the calculator in teaching the calculus concepts. During the study the teachers became less skeptical about the calculator's usefulness and found it effective for illustrating graphs. Some of the teachers' exams included more conceptual and graphically-oriented questions, but were not significantly different from traditional exams. Findings indicated the college teachers' conceptions of teaching calculus were generally consistent with their instructional practice when not constrained by time. The teachers did not perceive a dramatic change in their instructional practices. Rather, the new graphing approach curriculum and technology were assimilated into the teachers' normal teaching practices. No major shifts in the role of the teachers were detected. Two teachers demonstrated slight differences in their roles when the SPG calculators were used in class. One was a consultant to the students as they used the SPG calculators; the other became a fellow learner as the students presented different features on the calculator. Use of the calculator was influenced by several factors: inexperience with the calculator, time constraints, setting up the classroom display calculator, preferred teaching styles and emphasis, and a willingness to risk experimenting with established teaching practices and habits. / Graduation date: 1996

Not available / text

<p>The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective.</p> <p><br /> Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for f(x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate f(0) as the limit value. This method is appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually.</p>

<p>The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective.</p> <p><br /> Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for f(x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate f(0) as the limit value. This method is appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually.</p>

This study investigates the effects of implementing mathematical problem solving instruction in a regular calculus course taught at the college level. Principles associated with this research are: i) mathematics is developed as a response to finding solutions to mathematical problems, ii) attention to the processes involved in solving mathematical problems helps students understand and develop mathematics, and iii) mathematics is learned in an active environment which involves the use of guesses, conjectures, examples, counterexamples, and cognitive and metacognitive strategies. Classroom activities included use of nonroutine problems, small group discussions, and cognitive and metacognitive strategies during instruction. Prior to the main study, in an extensive pilot study the means for gathering data were developed, including a student questionnaire, several assignments, two written tests, student task-based interviews, an interview with the instructor, and class observations. The analysis in the study utilized ideas from Schoenfeld (1985) in which categories, such as mathematical resources, cognitive and metacognitive strategies, and belief systems, are considered useful in analyzing the students' processes for solving problems. A model proposed by Perkins and Simmons (1988) involving four frames of knowledge (content, problem solving, epistemic, and inquiry) is used to analyze students' difficulties in learning mathematics. Results show that the students recognized the importance of reflecting on the processes involved while solving mathematical problems. There are indications suggesting that the students showed a disposition to participate in discussions that involve nonroutine mathematical problems. The students' work in the assignments reflected increasing awareness of the use of problem solving strategies as the course developed. Analysis of the students' task-based interviews suggests that the students' first attempts to solve a problem involved identifying familiar terms in the problem and making some calculations often without having a clear understanding of the problem. The lack of success led the students to reexamine the statement of the problem more carefully and seek more organized approaches. The students often spent much time exploring only one strategy and experienced difficulties in using alternatives. However, hints from the interviewer (including metacognitive questions) helped the students to consider other possibilities. Although the students recognized that it was important to check the solution of a problem, they mainly focused on whether there was an error in their calculations rather than reflecting on the sense of the solution. These results lead to the conclusion that it takes time for students to conceptualize problem solving strategies and use them on their own when asked to solve mathematical problems. The instructor planned to implement various learning activities in which the content could be introduced via problem solving. These activities required the students to participate and to spend significant time working on problems. Some students were initially reluctant to spend extra time reflecting on the problems and were more interested in receiving rules that they could use in examinations. Furthermore, student expectations, evaluation policies, and curriculum rigidity limited the implementation. Therefore, it is necessary to overcome some of the students' conceptualizations of what learning mathematics entails and to propose alternatives for the evaluation of their work that are more consistent with problem solving instruction. It is recommended that problem solving instruction include the participation or coordinated involvement of all course instructors, as the selection of problems for class discussions and for assignments is a task requiring time and discussion with colleagues. Periodic discussions of course directions are necessary to make and evaluate decisions that best fit the development of the course. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

Making the transition from calculus to advanced calculus/real analysis can be challenging for undergraduate students. Part of this challenge lies in the shift in the focus of student activity, from a focus on algorithms and computational techniques to activities focused around definitions, theorems, and proofs. The goal of Realistic Mathematics Education (RME) is to support students in making this transition by building on and formalizing their informal knowledge. There are a growing number of projects in this vein at the undergraduate level, in the areas of abstract algebra (TAAFU: Larsen, 2013; Larsen & Lockwood, 2013), differential equations (IO-DE: Rasmussen & Kwon, 2007), geometry (Zandieh & Rasmussen, 2010), and linear algebra (IOLA: Wawro, et al., 2012). This project represents the first steps in a similar RME-based, inquiry-oriented instructional design project aimed at advanced calculus. The results of this project are presented as three journal articles. In the first article I describe the development of a local instructional theory (LIT) for supporting the reinvention of formal conceptions of sequence convergence, the completeness property of the real numbers, and continuity of real functions. This LIT was inspired by Cauchy's proof of the Intermediate Value Theorem, and has been developed and refined using the instructional design heuristics of RME through the course of two teaching experiments. I found that a proof of the Intermediate Value Theorem was a powerful context for supporting the reinvention of a number of the core concepts of advanced calculus. The second article reports on two students' reinventions of formal conceptions of sequence convergence and the completeness property of the real numbers in the context of developing a proof of the Intermediate Value Theorem (IVT). Over the course of ten, hour-long sessions I worked with two students in a clinical setting, as these students collaborated on a sequence of tasks designed to support them in producing a proof of the IVT. Along the way, these students conjectured and developed a proof of the Monotone Convergence Theorem. Through this development I found that student conceptions of completeness were based on the geometric representation of the real numbers as a number line, and that the development of formal conceptions of sequence convergence and completeness were inextricably intertwined and supported one another in powerful ways. The third and final article takes the findings from the two aforementioned papers and translates them for use in an advanced calculus classroom. Specifically, Cauchy's proof of the Intermediate Value Theorem is used as an inspiration and touchstone for developing some of the core concepts of advanced calculus/real analysis: namely, sequence convergence, the completeness property of the real numbers, and continuous functions. These are presented as a succession of student investigations, within the context of students developing their own formal proof of the Intermediate Value Theorem.

This thesis investigates the mathematical cognitive errors made in elementary calculus concepts by first-year University of Technology students. A sample of 34 first year students, the experimental group, from the Durban University of Technology Faculty of Engineering were invited to participate in project in elementary calculus using computer technology (CT). A second group, the control group, also consisted of 34 first year engineering students from the same University were given a conventional test in elementary calculus concepts. The experimental group was then given the same conventional test as the control group on completion of the project in elementary calculus using computer technology (CT). The purpose of the analysis was to study the effect of technology on the understanding of key concepts in elementary calculus. The major finding was that technology helps students to make connections, analyse ideas and develop conceptual frameworks for thinking and problem solving. The implications include: • Improvement of curriculum in mathematics at tertiary level; • New strategies for lecturers of elementary calculus; • An improved understanding by students taking the course in elementary calculus. • Redesign of software to improve understanding in elementary calculus. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2007.

Covariational reasoning is a cognitive activity that attends to two or more varying quantities and how their changes are related to each other. Previous studies indicate that covariational reasoning seems to have levels. Content analysis was used to examine the pedagogy and development of covariational reasoning levels in the sections that conceptually introduce derivatives in four calculus textbooks. One widely used calculus textbook was selected for the study in each of the four categories: U.S. college, U.S. high school, China college, and China high school. Two qualified investigators and I conducted the study. We used a framework of five developmental levels for covariational reasoning. The conceptual analysis of four calculus textbooks found that the U.S. college and the U.S. high school textbooks emphasize the average and instantaneous rate of change. However, both lack development of the direction and magnitude of change. On the other hand, this study's Chinese high school calculus textbook has a greater degree of development in the direction and magnitude of change while having a deficit in the average rate of change. This study's Chinese college calculus textbook does not have any meaningful development regarding covariational reasoning pedagogy. The relational analysis of the concepts previously identified in the conceptual analysis phase revealed that this study's U.S. college calculus textbooks provide abundant examples and exercises to transition between the average and instantaneous rate of change. On the other hand, all other calculus textbooks in this study lack any significant transition among passages that stimulate covariational reasoning. The textbook analysis in this study provides insights into the current focus of calculus textbooks in both the U.S. and China. In addition, the study has implications for learning and teaching calculus at both high school and college, as well as future editions of calculus textbooks. Finally, limitations and recommendations are discussed.

It has been held that heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. Many educational researchers have proposed problem-based curricula to improve students' views of mathematical thinking. The present study reports findings regarding effects of a problem-based calculus course, using historical problems, to foster Taiwanese college students' views of mathematical thinking. The present study consisted of three stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by a six-item, open-ended questionnaire and nine randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week problem-based calculus course in which mathematical concepts were problematized in order to challenge their personally expressed empirical beliefs in doing mathematics. Several tasks and instructional approaches served to reach the goal. Near the end of the semester, all participants answered the same questionnaire and the same students were interviewed to pinpoint their shift in views on mathematical thinking. It was found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students leaned toward a conservative attitude in the certainty of mathematical knowledge. Participants focus seemingly shifted from mathematics as a product to mathematics as a process. / Graduation date: 2003

It is common for students to make mistakes while solving mathematical problems. Some of these mistakes might be caused by the false ideas, or misconceptions, that students developed during their learning or from their practice. Calculus courses at the undergraduate level are mandatory for several majors. The introductory course of calculus—Calculus I—requires fundamental skills. Such skills can prepare a student for higher-level calculus courses, additional higher-division mathematics courses, and/or related disciplines that require comprehensive understanding of calculus concepts. Nevertheless, conceptual misunderstandings of undergraduate students exist universally in learning calculus. Understanding the nature of and reasons for how and why students developed their conceptual misunderstandings—misconceptions—can assist a calculus educator in implementing effective strategies to help students recognize or correct their misconceptions. For this purpose, the current study was designed to examine students’ misconceptions in order to explore the nature of and reasons for how and why they developed their misconceptions through their thought process. The study instrument—Calculus Problem-Solving Tasks (CPSTs)—was originally created for understanding the issues that students had in learning calculus concepts; it features a set of 17 open-ended, non-routine calculus problem-solving tasks that check students’ conceptual understanding. The content focus of these tasks was pertinent to the issues undergraduate students encounter in learning the function concept and the concepts of limit, tangent, and differentiation that scholars have subsequently addressed. Semi-structured interviews with 13 mathematics college faculty were conducted to verify content validity of CPSTs and to identify misconceptions a student might exhibit when solving these tasks. The interview results were analyzed using a standard qualitative coding methodology. The instrument was finalized and developed based on faculty’s perspectives about misconceptions for each problem presented in the CPSTs. The researcher used a qualitative methodology to design the research and a purposive sampling technique to select participants for the study. The qualitative means were helpful in collecting three sets of data: one from the semi-structured college faculty interviews; one from students’ explanations to their solutions; and the other one from semi-structured student interviews. In addition, the researcher administered two surveys (Faculty Demographic Survey for college faculty participants and Student Demographic Survey for student participants) to learn about participants’ background information and used that as evidence of the qualitative data’s reliability. The semantic analysis techniques allowed the researcher to analyze descriptions of faculty’s and students’ explanations for their solutions. Bar graphs and frequency distribution tables were presented to identify students who incorrectly solved each problem in the CPSTs. Seventeen undergraduate students from one northeastern university who had taken the first course of calculus at the undergraduate level solved the CPSTs. Students’ solutions were labeled according to three categories: CA (correct answer), ICA (incorrect answer), and NA (no answer); the researcher organized these categories using bar graphs and frequency distribution tables. The explanations students provided in their solutions were analyzed to isolate misconceptions from mistakes; then the analysis results were used to develop student interview questions and to justify selection of students for interviews. All participants exhibited some misconceptions and substantial mistakes other than misconceptions in their solutions and were invited to be interviewed. Five out of the 17 participants who majored in mathematics participated in individual semi-structured interviews. The analysis of the interview data served to confirm their misconceptions and identify their thought process in problem solving. Coding analysis was used to develop theories associated with the results from both college faculty and student interviews as well as the explanations students gave in solving problems. The coding was done in three stages: the first, or initial coding, identified the mistakes; the second, or focused coding, separated misconceptions from mistakes; and the third elucidated students’ thought processes to trace their cognitive obstacles in problem solving. Regarding analysis of student interviews, common patterns from students’ cognitive conflicts in problem solving were derived semantically from their thought process to explain how and why students developed the misconceptions that underlay their mistakes. The nature of how students solved problems and the reasons for their misconceptions were self-directed and controlled by their memories of concept images and algorithmic procedures. Students seemed to lack conceptual understanding of the calculus concepts discussed in the current study in that they solved conceptual problems as they would solve procedural problems by relying on fallacious memorization and familiarity. Meanwhile, students have not mastered the basic capacity to generalize and abstract; a majority of them failed to translate the semantics and transliterate mathematical notations within the problem context and were unable to synthesize the information appropriately to solve problems.