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PAGE 1 Effects of Using Graphing Calculators with a Numerical Appr oach on Students Learning of Limits and Derivatives in an Applied Calculus Course at a Community College by Arumugam Muhundan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Secondary Education College of Education University of South Florida Major Professor: Denisse R. Thompson, Ph.D Committee Members: Jeffrey D. Kromrey, Ph.D James A. White, Ph.D Fredric J. Zerla, Ph.D Date of Approval: June 24, 2005 Keywords: Calculus, Graphing Calculator Copyright 2005, Arumugam Muhundan PAGE 2 Dedication I dedicate this dissertation to my loving wife Kalaivani for providing me with the love, patience, understanding, and guidance that was needed to reach this goal. I also dedicate this to my wonderful sons Vishnu and Krishna for th eir love, patience, and understanding. I further dedicate this to my loving parents Aru mugam and Leelavathy. PAGE 3 Acknowledgments I sincerely thank my major professor Dr. Denisse R. Th ompson for her numerous reviews and valuable suggestions as my study evolved into fina l form. Without her support, constant guidance, and encouragement this project could not have been completed. I also thank my committee members Dr. Jeffr ey D. Kromrey, Dr. James A. White, and Dr. Fredric J. Zerla for their guidance, valuable suggestions, and support throughout the study. I thank Dr. James A. Condor and Mr. Thomas Hamersma f or their participation in my study, as well as their support and valuable suggestions, Dr. Dennis C. Runde for his administrative support at the college where the study w as conducted, and the students who participated in the study. I wish to acknowledge all o f my colleagues who provided support and assistance throughout the study. I also thank my wife Kalaivani, my children Vishnu and Kris hna, my parents Arumugam and Leelavathy, and the rest of my family for a lways believing in me and thank all others who contributed in any way. PAGE 4 i Table of Contents List of Tables iv List of Figures vii List of Acronyms ix Abstract x CHAPTER 1. INTRODUCTION 1 Calculus and Technology 4 Graphing Calculators (GCs) 6 Purpose of the Study 10 Significance of the Study 12 Limits of Functions 16 Algebraic Approach to Limit Problems 17 Numerical Approach to Limit Problems 18 Derivatives of Functions 20 Algebraic Approach to a Derivative Problem 21 Numerical Approach to the Derivative Problem 21 Definitions of Terms 24 Delimitations 25 Limitations 25 Summary 26 CHAPTER 2. LITERATURE REVIEW 28 Research on Technology Usage in Mathematics Education 29 Research on Usage of Regular (NonGraphing) Calculators (NGCs) in Mathematics Education 30 Research on Usage of Computers and CASs in Mathematics Education 33 Research on Usage of Graphing Calculators in Mathematics Education 35 Graphing Calculator Usage in Precalculus Topics 40 Students Errors and Misconceptions in Calculator Usage 43 Calculus and Students Knowledge in Limits and Derivativ es 45 PAGE 5 ii The Role of Technology in Calculus 49 Research on Usage of Computers and CASs in Calculus 50 Research on Usage of Graphing Calculators in Calculus 55 Summary 60 CHAPTER 3. METHOD 63 Design of the Study 65 Variables 65 Independent Variable 65 Dependent Variable 66 Instruments 66 Student Initial Survey 66 Pretest 67 Unit Exams 67 Class Observation Protocol 68 Validity and Reliability 69 Content Validity 69 Reliability 69 Procedure 70 Participants 70 Selection of Treatment and Control Groups 72 Facilitators 74 The Studied Course 74 Textbook 75 Graphing Calculator 75 ResearcherDeveloped Instructional Materials 76 Instructor Preparation 78 Treatment Phase 79 Data Analysis 84 Pilot Study 86 Summary 88 CHAPTER 4. RESULTS 89 Research Questions and Hypotheses 89 Data Analysis of the Pretest 91 Data Analysis of Unit 1 Exam on Limits 94 Data Analysis of Unit 2 Exam on Derivatives 104 Summary 116 CHAPTER 5. DISCUSSION 118 Hypotheses Results of the First Research Question 119 Hypotheses Results of the Second Research Question 121 Limitations of the Study 124 Implications for Practice 128 Conclusions and Recommendations for Future Research 130 PAGE 6 iii REFERENCES 133 APPENDICES 141 APPENDIX A: Student Initial Survey 142 APPENDIX B: Pretest 144 APPENDIX C: Unit Exams 147 APPENDIX D: Class Observation Protocol 155 APPENDIX E: Course Syllabus 158 APPENDIX F: ResearcherDeveloped Instructional Lessons: Unit Lessons of TI 83 Graphing Calculator Usage in an Applied Calculus Course 160 APPENDIX G: Pilot Study Report 232 APPENDIX H: Student Questionnaire 246 APPENDIX I: Course Implementation Log 247 APPENDIX J: Student Demographic Information 248 APPENDIX K: Consent to Participate in a Research St udy 249 About the Author End Page PAGE 7 iv List of Tables Table 1 Scoring Rubric 68 Table 2 College Demographic Information 70 Table 3 Number of Students Registered for the Four Sectio ns by Instructors and Groups 72 Table 4 Selection of Treatment and Control Groups 73 Table 5 Instructor Demographic Information 74 Table 6 Brief Contents of the ResearcherDeveloped Instructional Lessons 78 Table 7 Weekly Time Table for the Study 80 Table 8 Means, Standard Deviations, Skewness, and Kurtosis f or the Pretest by Instructors 92 Table 9 OneWay ANOVA for the Pretest 93 Table 10 Means, Standard Deviations, Skewness, and Kurtosis f or the Entire Unit 1 Exam on Limits by Instructors 94 Table 11 Means and Standard Deviations for Unit 1 Exam on Li mits by Skills, Concepts, and Applications 96 Table 12 2 X 2 ANCOVA for the Skill Portion of Unit 1 Exam o n Limits 97 Table 13 2 X 2 ANCOVA for the Concept Portion of Unit 1 Ex am on Limits 98 Table 14 2 X 2 ANCOVA for the Application Portion of Unit 1 Exam on Limits 98 Table 15 Mean Percents for Unit 1 Exam on Limits by Skill s, Concepts, and Applications 100 PAGE 8 v Table 16 Effect Sizes for the Skill, Concept, and Applica tion Portions of the Unit 1 Exam on Limits 102 Table 17 Percent Scores for Each Item Type within Skill, Concept, and Application Portions for Unit 1 Exam on Limits by Gr oups 103 Table 18 Means, Standard Deviations, Skewness, and Kurtosis f or the Entire Unit 2 Exam on Derivatives by Instructors 105 Table 19 Means and Standard Deviations for Unit 2 Exam on De rivatives by Skills, Concepts, and Applications 106 Table 20 2 X 2 ANCOVA for the Skill Portion of Unit 2 Exam on Derivatives 107 Table 21 2 X 2 ANCOVA for the Concept Portion of Unit 2 Ex am on Derivatives 108 Table 22 2 X 2 ANCOVA for the Application Portion of Unit 2 Exam on Derivatives 109 Table 23 Mean Percents for Unit 2 Exam on Derivatives by Skills, Concepts, and Applications 110 Table 24 Effect Sizes for the Skill, Concept, and Applica tion Portions of the Unit 2 Exam on Derivatives 112 Table 25 Percent Scores for Each Item Type within Skill, Concept, and Application Portions for Unit 2 Exam on Derivatives by Groups 113 Table 26 Correlations Between the Pretest and Each Port ion of the Unit 1 Exam on Limits and Between the Portions 115 Table 27 Correlations Between the Pretest and Each Port ion of the Unit 2 Exam on Derivatives and Between the Portions 115 Table 28 Scoring Rubric 237 Table 29 Descriptive Statistics for the Pretest of the Pilot Study 238 Table 30 Oneway ANOVA for the Pretest of the Pilot St udy 239 Table 31 Descriptive Statistics for Unit 1 and Unit 2 Exams of the Pilot Study 240 PAGE 9 vi Table 32 Pearson ProductMoment Correlations for the Pre test and Each Portion of Unit 1 and Unit 2 Exams of the Pilot Study 242 Table 33 ANCOVA for the Skill Portion of Unit 1 Exam of t he Pilot Study 243 Table 34 ANCOVA for the Application Portion of Unit 1 Exa m of the Pilot Study 243 Table 35 ANCOVA for the Skill Portion of Unit 2 Exam of t he Pilot Study 244 Table 36 ANCOVA for the Application Portion of Unit 2 Exa m of the Pilot Study 245 PAGE 10 vii List of Figures Figure 1 Mean Comparison for the Pretest for the Treatm ent and Control Groups by Instructors 92 Figure 2 Boxplot Graph for the Pretest for the Treatmen t and Control Groups by Instructor 93 Figure 3 Mean Comparison for the Entire Unit 1 Exam on Li mits for all Groups by Instructors 95 Figure 4 Mean Comparison for the Skill Portion of the Uni t 1 Exam on Limits for all Groups by Instructors 100 Figure 5 Mean Comparison for the Concept Portion of the Unit 1 Exam on Limits for all Groups by Instructors 101 Figure 6 Mean Comparison for the Application Portion of the Unit 1 Exam on Limits for all Groups by Instructors 101 Figure 7 Mean Comparison for the Entire Unit 2 Exam on De rivatives for all Groups by Instructors 105 Figure 8 Mean Comparison for the Skill Portion of the Uni t 2 Exam on Derivatives for all Groups by Instructors 110 Figure 9 Mean Comparison for the Concept Portion of the Unit 2 Exam on Derivatives for all Groups by Instructors 111 Figure 10 Mean Comparison for the Application Portion of the Unit 2 Exam on Derivatives for all Groups by Instructors 111 Figure 11 Mean Comparison for the Pretest for the Treat ment and Control Groups 238 Figure 12 Boxplot Graph for the Pretest for the Treatmen t and Control Groups 239 PAGE 11 viii Figure 13 Mean Comparison for Unit 1 Exam for the Treatme nt and Control Groups 241 Figure 14 Mean Comparison for Unit 2 Exam for the Treatme nt and Control Groups 241 PAGE 12 ix List of Acronyms AMATYC American Mathematical Association of TwoYe ar Colleges ANOVA Analysis of Variance ANCOVA Analysis of Covariance CAS Computer Algebra System GC Graphing Calculator MAA Mathematical Association of America MWF MondayWednesdayFriday NCTM National Council of Teachers of Mathematics TR TuesdayThursday PAGE 13 x Effects of Using Graphing Calculators with a Numerical Appr oach on Students Learning of Limits and Derivatives in an Applied Calculus Course at a Community College Arumugam Muhundan ABSTRACT This study examined the effects of using graphing calculators wi th a numerical approach designed by the researcher on students learning of limits and derivatives in an Applied Calculus course at a community college. The purpose s of this study were to investigate the following: (1) students achievement in solvi ng limit problems (Skills, Concepts, and Applications) with a numerical approach co mpared to that of students who solved limit problems with a traditional approach (primarily an algebra ic approach); and (2) students achievement in solving derivative problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved derivative problems with a traditional approach (primarily an algebra ic approach). Students (n = 93) in all four daytime sections of an Applie d Calculus course in a community college participated in the study during the spri ng 2005 semester. One of two MWF sections and one of two TR sections served as the treatment groups; the other two sections served as the control groups. Two instructors oth er than the researcher participated in the study. Instructor A taught one treatme nt group (a TR section) and one PAGE 14 xi control group (a MWF section); instructor B taught one t reatment group (a MWF section) and one control group (a TR section). Dependent variables were achievement to solve skill, co ncept, and application limit problems and skill, concept, and application derivative problems, measured by two teachermade tests. A pretest administered on the first day of class determined that no significant difference existed between the groups on prerequi site algebra skills. Separate ANCOVA tests were conducted on the skill, concept, and application portions of each of the limit and derivative exams. Data analyses revealed the following: (1) there was n o significant difference found on the skill portion of the limit topic (unit 1 exa m) due to instruction or to instructor ; (2) there was a significant difference found on the con cept portion of the limit topic due to instruction and to instructor ; (3) there was a significant difference found on the application portion of the limit topic due to instruction but not due to instructor ; (4) the interaction effects between instructor and instruction were not significant on the skill, concept, and application portions of the limit topi c; (5) there was a significant difference found on the skill portion of the derivative topic (unit 2 exam) due to instruction but not due to instructor ; (6) there was a significant difference found on the concept portion of the derivative topic due to instruction and to instructor ; (7) there was a significant difference found on the application portion o f the derivative topic due to instruction but not due to instructor ; and (8) the interaction effects between instructor an d instruction were not significant on the skill, concept, and application portions of the derivative topic. All significant differences were in fav or of the treatment group. PAGE 15 1 Chapter 1 Introduction Concerned mathematics educators search for new and effici ent methods to improve students understanding and performance in mathematic s courses (Douglas, 1986; Heid, 1997; Kaput, 1992; Waits & Demana, 2000). Over the years, there have been many calls for the reform of mathematics education in order to improve students performance and understanding. Among them, the most consi stent recommendation from the mathematics community is that all mathematics co urses take full advantage of the availability of calculators and computers (American Mat hematical Association of TwoYear Colleges [AMATYC], 1995, 1999; Dunham, 1999; Heid, 1997; Kaput, 1992; National Council of Teachers of Mathematics [NCTM], 1974, 1980, 1989, 2000; Waits & Demana, 2000). Educators believe technology is useful in mathematics educ ation and technology has been changing at an increasing rate. The use of com puters in education first appeared in the 1960s. The first electronic calculator, a fourfunc tion model, was manufactured in 1970 by Canon, Inc. The first computer algebra system (MuMath ), a computer program, was first introduced in mathematics in 1979. The first graphi ng calculator (Casiofx7000G) appeared in 1986. The first computer algebra systemadded grap hing calculators (supercalculators, like the TI 92) were manufactured in 1996. In 1998, flash ROM PAGE 16 2 technology was introduced in calculators which allows s oftware applications to be run on the calculators. Hence, in a relatively short period of time technology has made tremendous strides. These calculators, known as handheld calculators, and computers have impacted and will continue to impact mathematics education. The NCTM has been a promoter of the use of technology in mathematics classrooms from the beginning (NCTM, 1974, 1980, 1989, 2000). As ear ly as 1974, the NCTM recommended the use of the earliest fourfunction c alculators in schools. The early research on calculators convinced the NCTM to ma ke further recommendations for using these technologies in mathematics education. In it s 1980 publication, An Agenda for Action: Recommendations for School Mathematics the NCTM stated: Mathematics programs must take full advantage of the power of calcu lators and computers at all levels (p. 8). Later, in its Curriculum and Evaluation Standards the NCTM (1989) emphasized that Computer technology is changing the way s we use mathematics; consequently, the content of mathematics programs and me thods by which mathematics is taught are changing (p. 2). In the same publication, th e NCTM stated: The teaching of mathematics is shifting from primary emphasis on paper andpencil calculations to the full use of calculators and computers (1989, p. 83), and there fore, scientific calculators with graphing capabilities will be available to all students at all times (p. 124). In the NCTMs latest document, Principles and Standards for School Mathematics, the council emphasized the importance of the use of tec hnology in mathematics classrooms by stating, technology is esse ntial in teaching and learning mathematics; it influences the mathematics that is ta ught and enhances students PAGE 17 3 learning (2000, p. 24) and when technology tools are available, students can focus on decision making, reflection, reasoning, and problem solving ( 2000, p. 24). Other national and international mathematical organiz ations, such as the AMATYC and the Mathematical Association of America ( MAA), have also consistently made recommendations for the use of appropriate calculato rs and computers in mathematics courses. The AMATYC, in its Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus (1995), adopted a standard that Mathematics faculty will model the use of appropriate technology in the teaching of mathematics so that students can benefit from the oppo rtunities it presents as a medium of instruction (p.15). Later, in its 1999 Crossroads in Mathematics: Programs Reflecting the Standards, AMATYC again emphasized, Students will use appropriate te chnology to enhance their mathematical thinking and understanding an d to solve mathematical problems and judge the reasonableness of their results (p. 9). Many mathematics educators have also made recommendations to use appropriate technology in mathematics courses. For example, Corbi tt (1985) supported the use of technology in mathematics education by saying that The m ajor influence of technology on mathematics education is its potential to shift the f ocus of instruction from an emphasis on manipulative skills to an emphasis on devel oping concepts, relationships, structures, and problemsolving skills (p. 244). Dossey, Mull is, Lindquist, and Chambers (1988) wrote that the use of technology could improve students performance in mathematics: Improving mathematics performance will require educators best efforts to upgrade the curriculum, modify classroom instruction, and us e new teaching materials, including technological resources. The rapid p ace of technological PAGE 18 4 progress necessitates a revised set of priorities for m athematics instruction. To improve their understanding of mathematics and their abi lity to solve mathematical problems, students need the benefit of instruc tion that emphasizes practical experience in solving problems and opportunity to us e calculators and computers (p. 15). The constant recommendations for the use of technology in mathematics encouraged mathematics educators to initiate projects and st udies in mathematics courses with the use of technology. Evidence shows that many fo rms of technology have been used and tested in various areas in mathematics (Demana & Waits, 1998; Dunham, 1999; Heid, 1988; Hembree & Dessart, 1986, 1992; Kaput, 1992; Penglase & Ar nold, 1996). These studies provided evidence that using technology in mathematics education has a great potential to shift the instructional focus from an e mphasis on manipulative skills to an emphasis on developing concepts, reasoning skills, and problem solving skills. These studies concluded that the use of technology in appropriate ways improves students attitudes towards mathematics, increases students motiv ation level, and improves students mathematics achievement. There are many areas in mathematics where technology ha s played and can play an important role to make teaching and learning mathematics e asier. Calculus is one area in mathematics that can benefit from technology, as c alculus is often difficult for many students (Douglas, 1986; FerriniMundy & Graham, 1991; Smith, 1996; Steen, 1987; Tall, Smith, & Piez, 2004). Calculus and Technology Calculus is considered a rich subject in the modern world with applications in many areas, such as engineering, the physical and biologica l sciences, and business. PAGE 19 5 Therefore, those programs require students to have at lea st one semester of calculus. Unfortunately, calculus is probably the most unsuccessful course offered in higher education (Gordon & HughesHallett, 1991). FerriniMundy an d Graham (1991) reported that approximately 50% of the students enrolled annually in th e first semester calculus course either withdraw or fail the course. Students unsuc cessfulness in calculus filters them out of business, science and engineering fields (Whit e, 1987). For many students, calculus is a major stumbling block on the road to their professional careers. To understand the calculus concepts and how these concepts can be applied to other fields, one needs a strong algebraic background. A we ak mathematical background in terms of lack of function concepts, algebraic manipul ations, and geometric visualizations is one reason why students are not successf ul in calculus classes (Douglas, 1986; FerriniMundy & Graham, 1991; Tall et al., 2004). Many calculus reforms initiated in the 1980s addressed the hig h attrition rate in calculus and the lack of student understanding of the conce pts of calculus (Barnes, 1997; Douglas, 1986; Smith, 1996; Steen, 1987; Tall et al., 2004). In all t he calculus reforms and reports, the primary ideas were to develop an alternat e curriculum that is more conceptual and application oriented than the traditional curriculum, to make use of multiple representations of mathematical concepts, to dev elop a variety of teaching methods for calculus, and to use technology. Many studies i n calculus reforms have focused on using technology to increase the emphasis on co nceptual understanding while decreasing the emphasis on routine skills (Crocker, 1990; H eid, 1984; Judson, 1990). Although many of these studies involved the usage of computers and a variety of computer algebra systems (CAS), this technology is not widely available for students PAGE 20 6 because of its high cost, and the required training to operate these systems. Waits and Demana (1996) agreed: Dependence on only desktop computers and e xpensive software housed in computer laboratories is a major barrier to imp lementing serious technologybased curriculum reform in mathematics (p. 713). The inve ntion of graphing calculators (GCs) became a solution for this problem. Foley (1987) stat ed The advantage of computers over calculators disappeared in early 1986 when Ca sio introduced the fx7000G, a programmable scientific calculator (p. 28). Graphing Calculators (GCs) The GC was first introduced by Casio in 1986. Since then, ma ny manufacturers, including Texas Instruments, Casio, Sharp, and HewlettPa ckard, have produced more sophisticated GCs. GCs are programmable calculators that have standard computer processors, display screens, and builtin software. Due to increasingly sophisticated technology, GCs have begun to assume more and more comput ertype capabilities. Before the discovery of GC technology, computers and C ASs were the only available technology in college levelmathematics cours es. Today's easy access to technology in the form of GCs creates new ways for learning through graphical and numerical representations. Because of their low cost, portability, and capability, a GC is an appropriate technology tool to use in upperlevel mathemat ics courses. Particularly, the speed, accuracy, and capabilities of current graphing calcu lators have led many mathematics educators to believe that more emphasis sh ould be placed on numerical methods in calculus classes and less emphasis on technique s of differentiation and integration (Demana & Waits, 1998; Dunham, 1999; FerriniMundy & Gaudard, 1992; PAGE 21 7 Heid, 1997). They believed that this approach can enhance studen ts understanding in calculus concepts. The initial reactions to this technology in mathematic s education are generally positive (Dunham & Dick, 1994). Demana and Waits (1998) stated t hat students who use GCs experience a rich mathematics curriculum that allo ws them to focus on realistic applications. They also believed that the full use of G C technology could deepen students understanding about mathematics concepts. Vonder Embse (1992) mentioned that The large screen display, graphics capability, exp loratory functions of graphing and multiline display calculators afford students and teacher s opportunities to investigate, compare, and explore concepts and problem situations in bet ter ways than when using standard handheld calculators or no technology at all (p. 65). GCs have become more popular among students and teachers for several reasons. GCs offer relatively large screens, interactive graphics and onscreen programming and other builtin features, such as zoomin, zoomout, trac e, and table. Many of these capabilities were previously available only on a mainframe or a microcomputer. The invention of handheld GCs was, in fact, a major breakthrough in mathematics education. The powerful capabilities, togethe r with the decreasing cost, have made the use of GCs a choice for technology use in math ematics classrooms. The greatest benefits seem to come from this technology that is under student and teacher control, promotes student exploration and enables general ization (Demana & Waits, 1992, p.94). Ruthven (1995) supported the use of GCs and mentioned they offer not simply a mechanism for calculating and drawing but a medium for thinking and learning (p. 232). PAGE 22 8 Major mathematics organizations such as the NCTM, AMAT YC, and MAA and experts in mathematics education support the use of GC tech nology in mathematics courses, including calculus. Several projects and studies h ave been initiated in mathematics courses because of the calculators power, and a strong recommendation from the mathematics community. The usage of GCs among students and teachers has grown rapidly. In fact, today a GC is required for students who enroll in any mathematics courses in college algebra and beyond in many universities and commu nity colleges. Also, GCs are required or allowed on many standard mathematics exams, such as PSAT, SAT, and AP calculus by the College Board (Waits, Leinbach, & Dem ana, 1998). The use of GCs in school mathematics has also increased in many parts of the world, including in Australia, Canada, and many countries in Europe (Waits & Demana, 2000). Because of the potentially strong influence the technology can have on mathematics education, educators have observed its impact on mathematics. Waits, Leinbach, and Demana (1998) warned that, At least onefourth of the material that was typical ly taught in a US high school trig/functions course or college precalculus/college or algebra/trig and even some chapters in textbooks dealing with paperandpencil com putation methods became obsolete and disappeared from the curriculum beca use handheld scientific calculators provided better ways to compute t han paperandpencil methods. The same thing (obsolescence) will soon happen wi th paperandpencil symbolic algebraic manipulations common today because o f student use of inexpensive handheld computer algebra systems that now exist and soon will proliferate (p.1). Although major mathematics organizations and experts in mat hematics education support the use of GC technology, some raise questions and concerns about the impact of PAGE 23 9 using this technology in mathematics courses. Foley (1988) r aised some important questions related to the presence and affordability of this t echnology: 1. How will these calculators affect mathematics educati on at the community college? 2. How should these calculators affect it? 3. How should calculators influence what is taught and how it is taught? 4. How can calculators improve students understanding of ma thematics? (p. 29) Harvey (1992) raised a similar concern. He made the foll owing suggestions: 1. We need to analyze carefully the content that we presently teach and that we would like to teach. 2. Once the content of the mathematics curriculum ha s been examined, we need to determine the ways that particular tools can he lp us teach that content. 3. We must not cling to our present ways of teaching (p. 145). In addition, Dunham (1999) and Heid (1997) emphasized that curricu lum development, assessment, the method of instruction, and required training and instructional materials for instructors need to be addres sed in the use of GCs in mathematics education. These questions and concerns are increasingly important wi th the proliferation of GC technology in mathematics education. The mathemati cs education community has a responsibility to react positively to the available t echnology and careful research studies are needed to answer these questions and validate the con cerns. While mathematics educators are working on evaluating the impact of the technology in mathematics education, the use of many fo rms of technology continues to PAGE 24 10 grow. In particular, the use of GCs in mathematics cour ses by teachers and students cannot be stopped. Using a regular scientific calculator is a norm in almost every mathematics course in any school. Now, using a GC has gr adually become a norm in many mathematics courses in colleges and universities. Wit h GCs now available and affordable, Foley (1988) warned that students are likely to b uy and use them whether or not the mathematics teachers do. The AMATYC (1995) wrote about the availability of this technology in students hands: students will use it [GC] whenever they realize its power, regardless of whether professors allow it or not in their classrooms. Mathematics faculty must adapt to this reality and help students use te chnology appropriately so that they can be competitive in the workforce and adequately pr epared for future study (p. 55). Iseri (2003) agreed: The issue, it seems, is not so much on whether to use technology [graphing calculator] in mathematics teaching, b ut when and how (p. 1). Therefore, the question to the mathematics community i s not whether a GC is allowed in mathematics courses but how it is and should be used in s tudents mathematics learning. Purpose of the Study Among these concerns of the use of GC technology in mat hematics, the following question motivated this study: How effectively can and shou ld the GC technology be used so that the usage will have a positive effect on studen ts understanding of mathematics, particularly in limit and derivative topics in an Applied Calculus course? Research evidence shows that the use of GC technology has a positive impact on students calculus learning (Dunham, 1999; Heid, 1997; Penglase & Ar nold, 1996; Waits & Demana, 2000). However, it is also evident that using the GC technology alone in PAGE 25 11 mathematics courses may not make any differences in st udents conceptual understanding (Cassity, 1997; Dunham, 1999; Heid, 1997; Waits & Demana, 2000). Caref ully designed instructional materials need to accompany the use of GC te chnology in the mathematics course. The main purpose of the proposed study, therefore, was to ex amine the effects of the use of GCs with a numerical approach and the resear cherdeveloped instructional materials on students learning of limits and derivatives in an Applied Calculus course (MAC 2233) at a community college. The course requires studen ts to have a GC (the mathematics department recommends a TI 83 GC) for MAC 2233 course at this college. The instructional materials were developed by the researc her with the use of a TI 83 GC and are divided into four unit lessons. The purpose of these lessons is to guide students to use this powerful calculator in their understanding of se lected calculus topics, limits and derivatives. More details of these lessons will be dis cussed in chapter 3. The general research question that this study sought to answ er is To what degree can the use of GCs with a numerical approach and instruct ional materials developed by the researcher affect community college Applied Calculu s students learning of limits and derivatives? In particular, the study sought to answer th e following research questions: 1. How does the students achievement in solving limit problem s with a numerical approach compare to that of students who solved limit probl ems with a traditional approach (primarily an algebraic approach) in a n Applied Calculus course? 2. How does the students achievement in solving derivative pro blems with a numerical approach compare to that of students who solved de rivative problems PAGE 26 12 with a traditional approach (primarily an algebraic approac h) in an Applied Calculus course? The following research hypotheses are used to seek the answ ers for these research questions: 1. Students (GCGL group) who receive instruction with a numer ical approach will have higher achievement in routine (skill oriented) lim it problems than students (GC group) who receive instruction with a traditional appro ach (primarily an algebraic approach). 2. The students in the GCGL group will have higher achievemen t in conceptual oriented limit problems than the students in the GC group. 3. The students in the GCGL group will have higher achievemen t in related applications of limits than the students in the GC group. 4. The students in the GCGL group will have higher achievemen t in routine (skill oriented) derivative problems than the students in the GC group. 5. The students in the GCGL group will have higher achievemen t in conceptual oriented derivative problems than the students in the GC group. 6. The students in the GCGL group will have higher achievemen t in related applications of derivative problems than the students in th e GC group. Significance of the Study It is often noted by calculus educators that students in calculus show lack of understanding in the concepts of calculus topics (Douglas, 1986; FerriniMundy & Graham 1991; Gordon & HughesHallett, 1991; Tall et al., 2004; Wai ts & Demana, PAGE 27 13 2000). One of the criticisms is that students may learn h ow to answer limit and derivative problems (typically, routine skill oriented problems) but not necessarily understand the concept of the limit and derivative. In fact, getting an answer for a problem without knowing the meaning of it does not help students gain conceptual understanding in mathematics. FerriniMundy and Graham (1991) criticized th at [The] calculus course frequently becomes one of memorization of techniques and procedures with little time focused on the concepts (p. 29). Similarly, FerriniMundy and Gaudard (1992) noted: The old calculus becomes a litany of procedures and temp late problems which too often results only in giving students some rather mindless algebr a practice (p.58). They further mentioned that Many students who can find derivatives mecha nically and solve problems using them nevertheless often have little idea [ about] what a derivative actually means (p. 62). This criticism about lack of students understanding in cal culus still continues. Recently, the AMATYC, in its 2003 report, A Vision: Mathematics for the Emerging Technologies wrote, Knowing how to differentiate and integrate functi ons is worthless unless students know how these operations are used and can i nterpret the results in the context of a reallife situation (p. 8). Calculus, which is about 350 years old, is considered a ric h subject by the mathematics community. Unfortunately, students do not think o f calculus in the same way. Gordon & HughesHallett (1991) mentioned: Students who come out of calculus do not have any feel for the beauty and grandeur of the subject and an appreciation for its power to solve dynamic problems in almost all areas of human endeavor. Instea d, the students have been mired in a series of mindless mechanical manipulations t hat many believe to be the substance and raison d etre of calculus (p. 50). PAGE 28 14 Further they blamed the role of the calculus textbooks a s a part of the reason why students do not value the power of calculus. They noted: Historically, the development of calculus was always problemdriven; it is only the s tandard course and the associated textbooks that are technicallydriven (p. 54). They furth er mentioned that in the calculus curriculum the evaluation of limits becomes a mechanical process to produce an answer to a variety of artificial questions; different iation is a manipulative process to produce an answer, but not to answer a question of any subst ance; and integration is a mechanical process to produce an answer which matches the expression at the end of the book. Tall et al. (2004) pointed out that students do not see the power of calculus because of the way calculus instruction is presented to t hem. He suggested that the derivative of a function should be introduced as a rate of change of a function. He wrote, Students see differentiation as a sequence of some algeb raic manipulations applied for a specific symbolic expression, rather than a conceptual i dea of rate of change. Analyzing these various concerns about students conceptual understanding in calculus topics, this study aimed to promote students con ceptual understanding of limits and derivatives. My teaching experience in the Applied Calcu lus course at a community college over the last 15 years has given me an opportuni ty to understand students difficulties in this course. The experience shows that one of the reasons for students difficulties in this course is their weak mathematical background in terms of lack of function concepts, algebraic manipulations, and geometric visualizations. There are two entry levelcourses in calculus that are offered in colleges and universities. One is a regular CalculusI course (MAC 2311) and the other is an Applied PAGE 29 15 Calculus course (MAC 2233). Students get to choose one of the se calculus courses based on their major and requirements. As mentioned earlier, this study was conducted in an Applied Calculus course at a community college. The Appli ed Calculus students, the accessible population, at this college, like other coll eges in the state, are even weaker in function concepts and algebraic skills than the Calcul usI students. Part of the reason is that, in this college, the prerequisites for MAC 2311 are P recalculus Algebra (MAC 1140) and Trigonometry (MAC 1114); however, the prerequisit e for applied calculus is only Basic College Algebra (MAC 1105). Previo usly, MAC 1140 was the prerequisite for the Applied Calculus course; the state h as lowered the algebra requirements for the Applied Calculus course, perhaps beca use many students were not able to register for the course. Hence, it is unfortunate that less algebra is now required for the Applied Calculus course and students typically ex perience greater difficulty in the class than do students in Calculus I. It is, therefore, a challenge for teachers to teach calculus concepts to students who are weak in function c oncepts and algebraic manipulative skills. The Applied Calculus course, also known as Business Calc ulus in some colleges, is a calculus course designed to fulfill requirements for business and other nonmathematics majors but not for mathematics and science majors. For many students, this course is probably their last mathematics course. The m ain purpose of the course is to study calculus applications in students related majors. R igorous calculus is not a goal of the course. Topics include limits, differentiation, and integration of algebraic, exponential, and logarithmic functions, integration techniques and related applications in the management, business, and social sciences. PAGE 30 16 Limits of Functions Students in the Applied Calculus course study limits of functions as the first topic in their calculus learning. They spend a great deal of time studying limit problems without studying the formal definition (the deltaepsilon definition) of a limit Because, perhaps, the formal definition of the limit of a functi on is too abstract for students to understand at this level, Applied Calculus students are int roduced to the concept of the limit of a function through examples; most Applied Calcu lus textbooks, if not all, do not even mention the formal definition of the limit of a function. Generally, therefore, a limit problem can be approached by three methods: Algebric Numerical and Graphical An algebraic approach to a limit problem depends highly on algebraic techniques and manipulations. Further, it i s also possible that any algebraic techniques or manipulations may not work at all fo r certain limit problems. Students may not do well by this method if they are weak i n their algebraic skills. A numerical approach to a limit problem depends on computational skill s. This method may be time consuming if no calculators are allowed. T he required computations are relatively easy and quick with GCs using special features s uch as a table feature. The third approach, a graphical approach, requires a GC but not all functions give nice graphs. Also, students need to be warned about the misleadi ng behaviors and limitations of graphs. It is possible, of course, that a limit proble m can be approached by a combination of these three methods. For the purpose of this study, an algebraic and a numeric al approach are discussed in the following section. PAGE 31 17 Algebraic Approach to Limit Problems Consider the following examples of limit problems found in a typical Applied Calculus textbook in the chapter on the limits of func tions: Example 1. Given 3 5 2 12 4 3 ) ( 2 2 3 + + + + = x x x x x x f find ), ( lim 3 x f x if it exists. Answer: Even though finding the limit of ) ( x f as x approaches 3 does not mean finding the function value at x = 3, if the function is a continuous function the n the above two values will be the same. If students "substitute x = 3" in the expression, they get "0/0" (indeterminant form). So they simplify the expressi on by factoring each polynomial in the numerator and the denominator before substituti ng x = 3. The steps may look like this: 3 5 2 12 4 3 lim 2 2 3 3 + + + + x x x x x x = )3 )(1 2( )3 ( )4 ( lim 2 3 + + + x x x x x = )1 2( )4 ( lim2 3+x xx = 7 13 Therefore, 3 lim() x fx = 7 13. Example 2. Given 2 4 ) ( = x x x f find 4 lim(), x fx if it exists. Answer: If students "substitute x = 4" in the expression, they again get "0/0" (indeterminant form). This time students simplify t he expression by rationalizing the denominator and then are directed to "substitute x = 4". The steps may look like this: 2 4 lim4x xx = )2 )(2 ( )2 )(4 ( lim 4 + + x x x x x = )4 ( )2 )(4 ( lim4+ x x xx = 2 lim4+xx = 4 Therefore, 4 lim() x fx = 4. PAGE 32 18 In these two examples, students obtain the correct a nswers for the problems, but it is doubtful how much of the limit concept of a function is understood. Students use their algebraic knowledge ( factoring knowledge in example 1 and rationalization knowledge in example 2) to complete the problems. This approach rel ied upon algebraic techniques and manipulations. At the same time, this method of doing t hese limit problems provided students the opportunity to reinforce their algebraic skill s without focusing on the meaning of the concept of the limit of a function. Furthe rmore, this approach may not work if the expressions in the given function cannot be factored or rationalized Instead of using these algebraic techniques to find the limits, the limits can be found with the following numerical method. Numerical Approach to Limit Problems Graphing calculators, such as the TI 83, have a table feature that can be used to find limits numerically as indicated below. To answer example 1, first enter the expression in the calculator as a function 1 y = )3 5 2 /() 12 4 3 3 ^ ( 2 2 + + + + x x x x x and then use the table feature to get the following: PAGE 33 19 The table is actually providing the function values at var ious x values. From the table it is clear that the function approaches 1.857 (correct to 3 decimal places) as x approaches 3 from both sides. Note that the exact answer for this problem was 13 7 approximated to 3 decimal places as 1.857. Also it is worth noting th at the function is undefined at x = 3. Likewise for example 2, first enter the expression as the function 1 y = )2 ) ( /()4 ( x x and then use the table feature to obtain the following: PAGE 34 20 From the table it is clear that the function approaches 4 (correct to 3 decimal places) as x approaches 4 from both sides. Note that the exact answer was 4 as well. Also, the function is undefined at x = 4. When a limit problem is done numerically using a table, i t is easier for students to understand the left hand limit and the right hand limit c oncepts which are crucial in understanding the limit concept of a function. Derivatives of Functions The second topic the Applied Calculus students study is differentiation. Differentiation is a process of finding the derivative of a given function. Because the derivative of a function at a given point is defined through a limit process, understanding the limit of a function is important to understand the derivative of the function. In the derivative problems, first, one has to apply the co rrect derivative rule(s) for the given function; second, one has to perform the correc t algebraic steps in order to get the derivative in a simple form. Finding the derivative of a function can be a tedious procedure in many realworld applications. The first and direct application of finding the derivative of a function at a given value is that the derivative value provides the rate of change of the function at the given value. The class of rate of change applications is one of the important applications that can be solved by the concept of derivative. To solve a rat e of change application problem, the derivative of the particular function needs to be o btained from the derivative rule(s) and then the derivative of that particular function needs to be evaluated for a given x value. The second part is relatively easy because students simply plug in the x value PAGE 35 21 into the derivative they obtain. Of course, if students make mistakes in finding the derivative, then the plug in part gives an incorrect a nswer. Consider the following examples of derivative problems foun d in a typical Applied Calculus textbook in the chapter on the derivati ve of functions. As before, an algebraic approach is first discussed and then a numerical approach for the same problem is discussed. Algebraic Approach to a Derivative Problem Given ) ( x f = x x 02 0 1 ) 7 10( 20 + + find ).5(' f Answer: Students first need to note that the given function is in a quotient form, and use the quotient rule to get ) (' x f and then substitute x with 5 to get the answer. The steps may look like: ) 02.0 1( 136 ) 02.0 1( 8.2 4 8.2 140 ) 02.0 1( ) 02.0 )( 7 10( 20 ))7( 20 )( 02.0 1( ) (' 2 2 2 x x x x x x x x f + = + + = + + + = Then 3966942 112 21.1 136 ))5( 02.0 1( 136 )5(' 2 = = + = f and is rounded to 112.4. Students may get the solution by using the necessar y formula and some algebraic work. Again, the concern is: Do students understand the m eaning of the number 112.4 in the answer they found? At the same time, the derivative of ) ( x f at ,5 = x ),5(' f can be found by using the following numerical method: Numerical Approach to the Derivative Problem Given ) ( x f = x x 02 0 1 ) 7 10( 20 + + find ).5(' f PAGE 36 22 From the definition of the derivative, )5(' f = h f h f h )5( ) 5( lim 0 + Therefore, this derivative problem is a limit probl em; thus, students can solve this as a limit problem with the same numerical method they h ave used earlier. The work is as follows: To use a calculator to find this limit, use x for h (the calculator has a variable key as x ) and let ))5( 02.0 1( ))5(7 10( 20 )5( 1 + + = = f y )) 5( 02.0 1( )) 5(7 10( 20 ) 5( 2 x x x f y + + + + = + = and 1 2 3 x y y y = Therefore, lim )5('3 0y fx = To find the limit of 3 y as x approaches 0, the following x values are entered in the table feature of a TI83 graphing calculator. Students ar e able to get the following: From the table, students can conclude that .4. 112 lim3 0=yx That is, .4. 112 )5(' = f PAGE 37 23 Note that because finding the derivative of a function a t a given x value is a limit problem, the whole process of finding the limit was done n umerically with help from a graphing calculator. Because students have experience in doing limit problems, they can approach this derivative problem as doing a limit problem. A numerical approach using a graphing calculator for finding the limit of a function is a consistent approach that students can use to find the derivative at a given number. While using a graphing calculator for this problem, it is possible for students t o focus on concept development with derivatives. For example, the other benefit of doing the problem this way is that the table gives the average rate of change of that function on various intervals around the given x value. That is, in the previous example, students are able to determine the average rate of change of ) ( x f on various intervals around 5 = x. For example, the average rate of change of ) ( x f between 5 = x and 9.4 = x is 112.6 (see the first number in the previous table). Also, students notice that the average rate of change of ) ( x f on an interval approaches the instantaneous rate of change of ) ( x f as the intervals approach zero. Both finding the limit of a function and the deriva tive of a function have traditionally relied upon a high level of algebraic manipulations. As mentioned earlier, students in an Applied Calculus course are not stro ng in algebraic skills and often get frustrated if they do not get the correct answer or the correct form of the answer that the book provides. This frustration causes students to have difficulty in focusing on calculus concepts. Therefore, teachers in these classes spen d time teaching students to use necessary algebraic skills in order to find the cor rect answer for the limit and derivative PAGE 38 24 problems. Thus, teachers have less time, if any, to spend us ing these limits and derivatives to solve the related real world applications The limit and derivative problems are approached by a numeri cal method with the help of a TI 83GC and researcherdeveloped instructional ma terials in this study. Then the same approach is used to solve the related applicatio ns in limits and derivatives so that students can gain conceptual understanding of limits an d derivatives. Definition of Terms Regular (NonGraphing) Calculator (NGC) is either a basic calculator or a scientific calculator. A basic calculator is a calculator that has the fourfunctions (addition, subtraction, multiplication, and division) with an eig htdigit or more display with floating decimals. A scientific calculator is a calculator tha t has trigonometric and logarithmic functions in addition to the basic features. Graphing Calculator (GC) is a calculator that has a 2.5 by 2.5 inch display scre en with some advanced features such as graphing, table, solving matrices etc. in addition to the features that a NGC has. Computer Algebra System (CAS) is a computer program that is run in a computer or builtin a graphing calculator. It has the capability to do algebra ic operations symbolically. HandHeld Calculator is a calculator that is either a basic, scientific graphing calculator, or CASadded graphing calculator. Numerical Method is an approach that is used to solve limit problems and deriva tive problems using the table feature of a TI 83 graphing calculator as explained in the previous worked examples. PAGE 39 25 Traditional Approach is an approach that is used to solve limit problems and der ivative problems using primarily algebraic techniques and algebraic mani pulations as shown in the previous worked examples. Delimitations The study was conducted in the spring of 2005 at a public communi ty college in southwest Florida that serves two counties. According t o the college 20032004 Factbook, the college service area has a population of approximately 617,000 of which 82.2 % is 18 years or older. In spring 2004, the college enrollm ent was 8393 students, of whom 3357 were fulltime and 5036 were parttime; the gender r atio is 64% female to 36% male. In the same term, the average age of fulltim e students was 23 years and the average age of parttime students was 28 years. The generali zation of this study is limited to the students who enroll in an Applied Calculus course at the community college that is described here. This study was conducted in all of the available daytime s ections of an Applied Calculus course. Therefore the generalization of this study is limited to such sections of the course. Also, the generalization of this study is limited to the courses that have the same contents as this course. Limitations There are only two instructors who participated and both ar e males. Any influence of this selection cannot be controlled because these w ere the two available instructors for this course during the time of the study. PAGE 40 26 Communication between students enrolled in differen t sections of the course cannot be controlled. In fact, due to the availability of resources outside the classroom, such as the math lab and personal relationships that exis ted between students, there is a possibility that students from different sections would have communicated with each other and would have shared instructional materials and meth ods between the treatment and control groups. Summary Evidence shows that technology can have a positive impa ct on students conceptual understanding in mathematics, including calculus. Various organizations and mathematics educators consistently recommend the use of t echnology in mathematics education, including calculus. Calculus is a subject that c an benefit from the use of technology, as calculus is a difficult subject for ma ny students. At the same time, using technology in any mathematics c ourse without giving careful thought and supporting instructional materials will n ot ensure students understanding in mathematics. Carefully designed guided less ons along with appropriate technology can improve students understanding in mathemat ics. Ellington (2000) stated that curriculum designed specifically for instruction w ith [graphing] calculators can enhance student achievement in operational and problem solving skills (p. 177) and recommended that teachers should design lessons which in tegrate calculatorbased explorations of word problems and mathematical concepts with regular instruction (p. 178). PAGE 41 27 Also, it should be noted that there has been a critic al need for these supported instructional materials in mathematics courses. Dunham (1999) also emphasized such a need: There is a critical need for research in instruct ional design to create curricula that use calculators to their best advantage, to find effecti ve materials to combat calculatorinduced errors, and to evaluate programs that incorporate cal culators (p. 23). Understanding these concerns in mathematics education and an attempt to respond to such a critical need, particularly in calculus, this study examines the effects of using GCs with a numerical approach and the researcherde veloped instructional materials on community college students learning of limi ts and derivatives in an Applied Calculus course (MAC 2233). It is the researchers hope t hat teaching calculus topics with this method along with the use of GCs and the supporte d instructional materials can increase students conceptual understanding in limits and deri vatives in the Applied Calculus course. PAGE 42 28 Chapter 2 Literature Review This study was designed to investigate the hypotheses that the effects of using graphing calculators (GCs) with a numerical approach and the researcherdeveloped instructional materials to limit and derivative problems will increase students conceptual understanding in limit and derivative topics in an Applied Calculus course at a community college. An Applied Calculus or Business Calcu lus course is a part of the calculus curriculum and several closely related areas i n the literature were included in this chapter. The chapter is organized into four sections. The first se ction examines technology usage in mathematics education, particularly regular (non graphing) calculator (NGC) usage, computer and computer algebra system (CAS) usage, and gr aphing calculator (GC) usage in mathematics education. The second section discusses students errors and misconceptions in calculator usage. The third section foc uses on the contents of calculus, particularly, limits and derivatives and students knowledge and learning in those areas. The fourth section discusses the role of technology in calculus, particularly computer and CAS usage, and GC usage in limit and derivative topics. PAGE 43 29 Research on Technology Usage in Mathematics Education Technology has taken on an increasingly important role i n mathematics education for quite some time as it gains support from the mathemat ics community (AMATYC, 1995, 2003; Demana & Waits, 1990, 1992, 1998; Dunham, 1999; Heid, 1988, 1997; Kaput, 1992, NCTM, 1989, 2000; Suydam, 1982; Suydam & Brosnan, 1994). Ther e are several forms of technology that are widely used in mat hematics education. These include regular (nongraphing) calculators (NGC), computers, c omputer algebra systems (CASs), microworlds, dynamic geometry tools, calculato rbased laboratory devices (CBLs), microcomputerbased laboratory devices (MBLs), gr aphing calculators (GC), and CASadded GCs (Heid, 1997). Pea (1987) called these technolo gies cognitive technologies because these technologies help transcend the limitations of the mind in thinking, learning, and problemsolving activities (p. 91). Heid (1997) supported the use of these cognitive technologies noting, the use of a co gnitive technology has the potential for affecting subject matter, curriculum, inst ruction, learning styles, and problemsolving activities (p. 7). Because of the appearance of these cognitive technologies, many areas in mathematics education required changes. Some of the cha nges are the way students study, the way the instruction is delivered, the way questio ns are asked on exams, the way assessments are made, and the way curriculum is de veloped. Bitter (1987) stated this as, Traditional mathematics curriculum components are being outdated as new technology expands its capabilities (p. 46). Although ther e were several forms of technology available, mainly NGCs, GCs, computers, and CASs are the technology forms that are widely accepted in schools, colleges, an d in many universities to promote PAGE 44 30 students understanding in mathematics. Over the last twe nty years, several dissertation studies and projects were initiated in mathematics courses focusing on increasing the emphasis on conceptual understanding while decreasing the emp hasis on routine skills with the use of these technologies (Dunham, 1999; Heid, 1988, 1997; Juds on, 1990; Palmiter, 1991; Penglase & Arnold, 1996). The following sectio n focuses on the research in mathematics education with the use of NGCs. Research on Usage of Regular (NonGraphing) Calculators (NGCs ) in Mathematics Education NGCs were probably the first piece of technology that received attention in mathematics education. NGCs have become more popular amo ng students and teachers because they are simple, inexpensive, and friendly to use. The NCTM (1974) was probably the first organization that recommended the use of these calculators in schools and many mathematics educators welcomed the recommendation Bell (1978) supported the use of calculators in school and mentioned that, Calculators can provide a direct alternative to the arithmetic and computational methods that makeup the principal component of the first eight years of mathematics tr aining as schools move away from answer oriented instruction to a concentration on t he more important concepts (p. 405). Kaput (1992) wrote that heavy use of calculators in the early grades as part of instruction and assessment does not harm computational abil ity and frequently enhances problemsolving skill and concept development (p. 534). Troutman & Lichtenberg (1995) also supported this little machi ne: The minicalculator is truly a revolutionary device a calcul ating machine that is accurate, inexpensive, durable, and small. Not only is the use of this m achine in the teaching of PAGE 45 31 mathematics appropriate, but it will have a direct and s ignificant consequence for the school We will be spending more of our allotted time em phasizing mathematical ideas and less time on long and tedious computational procedures ( p. 38). Many educators believed that calculators could be used to aid algorithmic ins truction; facilitate concept development; reduce the demand for memorization; enlarge th e scope of problem solving; provide motivation and encourage discovery, explorat ion, and creativity. One of the earliest technologybased projects was star ted in the Ohio State University (OSU) by Waits and Leitzel in 1974 (Waits & Leit zel, 1976). The project was an effort to reform the college remedial mathematics curriculum that required the use of the very earliest fourfunction calculators (NGCs) by all students. The results of this project motivated the developers to develop two other majo r projects at the OSU: Approaching Algebra Numerically (AAN) and Calculator and Computer Precalculus (C 2 PC) project. Among these two, the C 2 PC project targeted mainly precalculus courses and it was widely conducted in many high schools, colleges, and universities. Computers, CASs, and GCs were heavily used in this project. A discus sion on this project is covered later in this chapter. Much research has occurred on the effects of NGCs on s tudent achievement in basic skills, in problem solving, and on student attitudes, typically showing positive results. Many of these studies were analyzed and reported by several researchers, with three such review reports being examined in this section. Suydam (1976) was perhaps the first to give a report on NGCbased studies. She reviewed 24 studies and reported that most studies favored the use of calculators but most of them were conducted with poor research designs. In 1982, sh e again conducted a PAGE 46 32 research review on the effects of using calculators in m athematics classes (Suydam, 1982). In the research review of 75 studies, Suydam reported that 95 comparisons were drawn between the achievement scores of groups that used or did not use calculators within traditional instruction. In 43 comparisons, the ca lculator group had a significantly higher mean score than the noncalculator group; in 47 com parisons, the difference was not significant; and in 5 comparisons, the noncalculat or group had a significantly higher mean score. She further reported that the use of calcu lators will result in as high or higher achievement as with paperandpencil and recommended that c alculators are good for promoting achievement and that all students can benefit from using calculators (p. 27). Hembree and Dessart (1986) conducted another metaanalysis on the effects of NGCs in precollege mathematics courses (Hembree & Dess art, 1986). The purpose of their study was to integrate the findings of the research on students using calculators in learning mathematics in grades K12. The analysis included 53 d issertations, 12 journal articles, 12 Educational Resources Information Center rep orts, a project report, and an unpublished report. They found advantages for average students in calculator using groups when assessed with noncalculator problemsolving tes ts. Students in calculator groups at all ability levels showed positive effects when calculators were allowed on posttests. Hembree and Dessart concluded: Students who use calculators in concert with traditional instruction maintain their paperandpencil skills without apparent harm. Indeed, a use of calculators can improve the average students basic skills with paperandpe ncil, both in basic operations and in problem solving (Hembree & Dessart, 1986, p 96). PAGE 47 33 They also reported that students using calculators posses s a better attitude towards mathematics and an especially better selfconcept in m athematics than noncalculator students for all grade and ability levels. They also conduc ted a followup metaanalysis of 88 studies in 1992. With the exception of one study, Hembree an d Dessart found that the studies showed positive effects on students problem solving a bilities and attitudes towards mathematics (Hembree & Dessart, 1992). In 1996, Smith conducted a metaanalysis on 24 studies with calculator use from 1984 to 1995 and reported that the calculator had positive eff ects for students in third grade, grades seven through ten, and twelfth grade and had no significant effect on achievement for grades four through six and grade eleven. Furt her, he stated that problem solving, computation, and conceptual understanding were the a reas that provided positive results. From these review reports there was strong evidence that the use of NGCs in fact had positive effects on students understanding in mathemat ics. Because of NGCs limited capabilities, many of those studies reviewed in the metaanalyses were in entrylevel mathematics courses. The research on upperlevel mathematics courses was mainly conducted with the use of computers, various CASs, and GCs. The review of studies with the use of computers and various CASs follows in the nex t section. Research on Usage of Computers and CASs in Mathematics Educ ation Computer technology was first brought to education in vari ous disciplines in the 1960s in the form of mainframe units (Kaput 1992). However, the cost and size of a mainframe computer and needed training to use a mainframe we re main factors that prevented their widespread use in schools and colleges. Only after the invention of PAGE 48 34 microcomputers have many areas in education, including math ematics education, begun to benefit from the computers power (Demana & Wait, 1990, 1992, 1998; Dunham, 1999; Heid, 1988, 1997; Kaput, 1992). The Ohio State Universitys C 2 PC project was one of the first projects that took advantage of the available technology. The main purpose of the C 2 PC project was to improve the mathematics preparation of college bound high school students through the use of computers and GCs (Waits & Demana, 1998). In particul ar, the project provided a computergraphingintensive precalculus curriculum to improve the preparation of calculusintending students. Computers were used exclusively in the project for the first two years. Because of the discovery of GCs, the projec t continued after the first two years with the use of GCs. The project created instructional materials that made effective use of computerandcalculatorbased graphing to improve student understa nding of functions and strengthen student problemsolving skills. As an outgrowth of the C 2 PC project, several high schools, colleges, and universities adopted the project in their precalculus curric ulum. As a result of this project, developers claimed that they observed students learning to va lue mathematics, the participating students became more flexible problem solvers, and the technology that they used gave students a better feeling about mathematics (Waits & Demana, 1998). Computer technology in mathematics education was mainly used as tutoring, computermanaged teaching, simulation, and programming to solve problems (Kulik, Kulik, & Cohen, 1980). CASs, such as Mathematica, Derive, Maple, Mathcad, and MuMath, are computer programs that allow a user to solve algebraic and calculusbased problems. The potential of computers and CASs technology on teaching and learning PAGE 49 35 mathematics was widely initiated in the 1980s. The NCTM (1989) understood the power of this piece of technology and suggested that mathematics programs should use the full power of computer technology at all levels. They made t he point that students can solve a problem with a greater degree of accuracy with the computer and the computer can remove the necessity for long, complicated computations The use of computers also offers the potential of solving real world applications t hat were not previously possible in the mathematics curriculum by paper and pencil techniques (Hei d, 1988, 1997; Kaput, 1992; Waits & Demana, 1996, 2000). In fact, with the use of comput ers as an instructional tool, teachers can interact more with st udents and give students the opportunity to build on their skills, conceptual understa nding, and problem solving abilities in mathematics. Because the purpose of this study is particularly the effe cts of using GCs in a calculus course, the further discussion of the litera ture review first focuses on GCbased studies in mathematics education that are related to this study and later it discusses the literature review on computers, CAS, and GCbased studies i n calculus. Research on Usage of Graphing Calculators in Mathematics Educ ation Until the discovery of GCs, only computers and CASs wer e used as instructional technologies in college level mathematics courses. The cost, teacher training, and other factors prevented the widespread use of these technologie s in schools and colleges. It may not be possible for every school to set up computerb ased instruction in every mathematics classroom. GCs became a solution for this problem. GCs are programmable calculators that have standard computer processors, displ ay screens, and builtin software. Due to increasingly sophisticated technology, GC s have begun to assume even PAGE 50 36 more computer type capabilities. Foley (1987) and Dick (1992) ca lled GCs handheld computers, and Waits and Demana (1996) called GCs pocket com puters for their power. Heid (1997) claimed, Perhaps the single most importan t technological influence on the high school and early college mathematics classr oom has been the graphing calculator (p. 21). Because of their low cost, portabil ity, speed, accuracy, and other advanced capabilities, the use of GCs became a better cho ice for use in upperlevel mathematics courses. Many positive things have been said about GCs by mathemat ics educators (Foley, 1988; Ruthven, 1990; Demana & Waits, 1998; Dunham & Dick, 1994; Dunham, 1999; Heid, 1997). Demana and Waits (1998) have continuously supported t he use of GCs in mathematics, noting, the use of hand held technology can provide more classroom time for the development of better understanding of mathemat ical concepts by eliminating the time spent on mindless paper and manipulations (p. 5). The first GCs appeared in 1986 and research studies began to a ppear in 1990. There were a number of studies related to the use of GC s conducted in a wide range of areas in mathematics such as precalculus, calculus, st atistics, geometry, trigonometry, and algebra on such topics as function concepts, graphing con cepts, modeling, limit concept, and derivative concept. A few studies addressed i ssues on equity, gender differences, students errors and misconceptions; developm ent of spatial visualization skills, and problemsolving skills with the use of GCs. So me other studies examined students and teachers impact in terms of beliefs, at titudes, and perceptions in GC usage. Research on GCs, however, is still relatively new an d the direction for GC use in mathematics education is still unclear (Penglase & Arn old, 1996). Early research on GC PAGE 51 37 use in mathematics had mixed results. Many studies and proj ects reported increases in student achievement, understanding of mathematical concepts, attitudes towards mathematics, and problemsolving ability when comparing GC groups to nonGC groups (Allison, 2000; Carter, 1995; Cassity, 1997; Dunham, 1999; Dunham & Di ck, 1994; Ellington, 2000; Heid, 1997). At the same time, there were som e studies showing no difference between the GCusage group and nonGC groups (Ar my, 1991; Girard, 2002). There were also many studies showing mixed results in ma thematics achievement between GCgroups and nonGC groups (Blozy, 2002; Dimiceli, 1999; Ellison, 1993; Estes, 1990; Ganter, 2001; Oster, 1994; Penglase & Arnold, 1996). Dunham and Dick conducted one of the earliest reviews on st udies in mathematics education that used GC technology (Dunham & Di ck, 1994). Their finding generally supported the use of GCs in mathematics education They reported: The early reports from research indicate that graphing cal culators have the potential dramatically to affect teaching and learning mathe matics, particularly in the fundamental areas of functions and graphs. Graphing cal culators can empower students to be better problem solvers. Graphing calculator s can facilitate changes in students and teachers classroom roles, resulting i n more interactive and exploratory learning environments (p. 444). Penglase and Arnold conducted a critical review on published di ssertation studies during 1990 to 1995 that examined the effects of GCs in high sch ool and college mathematics (Penglase & Arnold, 1996). In their review, t hey sought to answer the following questions: 1) How did the GC benefit student achievement in mathematic s? 2) What kind of learning environment allowed for maximum benefit s to be attained? PAGE 52 38 Penglase and Arnold mentioned that the majority of the studies they reviewed focused on two major areas: a) testing the effects of the use of GCs within specific areas of mathematical study; and b) making judgments regarding t he effectiveness of such use. From their finding, they concluded that the studies using GCs in mathematics education had mixed results and that the GC research failed to provide clear direction to mathematics education. They stated that most studies tha t favored the use of GCs did not show significant differences between GC and nonGC groups Penglase and Arnolds review found that students understanding of the connection between functions and their graphs, capabilities with spatial visualization skills, an d attitudes toward mathematics were the areas that provided positive results. They, h owever, questioned the GC usage and testing procedures in several studies and suggested a need for new methods to evaluate students who have been exposed to GC technology. Heid (1997) also provided a review on the research studies using GCs. She reported that the review studies provided positive results; in particular, the GC usage in those studies increased conceptual understandings of graphs a nd functions, understanding of connections among a variety of representations, and st udents problemsolving ability. At the same time, she criticized these studies as, I n light of the almost uniformly positive results, it is important to note that most of these studies reported on projects that involved students in different mathematical activities as they worked with graphics calculators. They did not merely place calculators i n students hands in the context of an unchanged curriculum (p. 23). Again in 1999, Dunham provided a thorough research review on studie s that used GCs in various ways. She mentioned that the GCbased s tudies improved students PAGE 53 39 problem solving ability, conceptual understanding, and computatio nal skills. Dunham also stated that the review of some studies supported that the use of GCs in mathematics classes helped some special populations, such as lowabilit y and atrisk students, students with learning disabilities, and nontraditional students, e tc. Ellington (2000) conducted a metaanalysis on the effects of handheld calculators (NGCs and GCs) on precollege students in mathematics cl asses. This analysis contained fiftythree calculusbased studies from 1984 through 2000. Her finding in the GC usage sector included the following: when the GC was a signific ant element in all aspects of high school mathematics classes, the basic operational skills of students improved; and students who used GCs during mathematics instruction had bette r attitudes toward mathematics than their nonGC counterparts. There were numerous studies conducted in mathematics educat ion with the use of GCs in various topics in various courses. The current st udy is interested in the effect of using GCs in an Applied Calculus course. Further, to learn calculus concepts students need to have a strong foundation in precalculus topics incl uding function concepts and problemsolving ability. Also, to learn calculus concepts with the use of GCs, students need to know how a GC can be used in a correct and effect ive way. Therefore, the following review sections concentrate on studies that f ocused on students understanding in precalculus topics, including function concepts and calc ulus concepts with the use of GCs, and the studies that focused on students difficulties errors, reluctance, and misconceptions in using GCs in their mathematics learning. PAGE 54 40 Graphing Calculator Usage in Precalculus Topics There were a number of studies with the use of GCs co nducted at the precalculus level in high school, colleges, and universities to exami ne the effect on students understanding. For example, Oster (1994) examined the effects of instruction using a GC on conceptual and procedural understanding in precalculus. Th ree teachers and 98 students participated in this quasiexperimental study to exa mine aspects of constructivism as applied to the instructional use of a GC i n collegelevel precalculus. The experimental group was taught precalculus graphics strate gies with the use of a GC and the control group was taught using traditional teaching me thods. Pretestposttest and endoftreatment surveys were used as instruments in this study. The results of the study showed a significant increase in students conceptual know ledge but no evidence was found to support significance for students procedural or ov erall achievement in the precalculus topics. Oster, however, claimed that the r esult of her study supported the constructivist view that recommends teachers involve student s in an interactive problemsolving situation. Also, Oster reported that the student and teacher survey responses showed generally positive views on the use of the GCs for learning and teaching in precalculus topics. Doenges (1996) also conducted a study to examine the effect of using a GC in high school precalculus classrooms. She examined several factors in her study, such as gender, spatial skills, achievement, confidence, and attitude s in using GCs. Doenges selected six sections of a precalculus course in four hig h schools for her study, with a total of 134 students. She concluded that confidencewithcal culator scores were significantly higher than confidencewithoutcalculator scores. She also reported that the PAGE 55 41 majority of the students regarded GCs positively, with m ost negative comments coming from females and low spatial ability students. The signif icant differences in the confidencewithcalculator scores favored the males. A significant amount of research has been conducted into the effectiveness of the GC as a tool for instruction and into students learning ab out functions and graphing concepts. For example, Carter (1995) investigated the effect s of GCs on student achievement and understanding of the function concept. The treatment group was introduced to function concepts graphically with the use of a GC and the control group was taught in a traditional manner. Pretestposttest, t wo questionnaires, and an interview were used as instruments in this study. From the data c ollection and analysis, Carter concluded that there was a significant gain between prete st and posttest scores for both groups and the treatment group showed more improvement than the control group. He also reported that the GC instruction produced a favorabl e influence on student achievement; however, the difference in the outcomes fo r the two groups was not statistically significant. Carters study also examined the effects of GCs on studen ts difficulties and misconceptions with the function concept. From the questi onnaires and interviews, the researcher concluded that the students who used GCs understoo d function transformations, understood the connections between the graphical and algebraic representations, were able to make connections between a po int on the graph and the two distinct values that the point encodes, were able to so lve nonroutine problems, and were more active than students not using a GC. PAGE 56 42 Rich (1990) conducted a study to examine the ways in which the use of GC as a teaching tool affects precalculus students learning of func tions and related concepts. Two classes used a Casio fx7000G calculator and three cont rol groups studied the same materials without the use of the GCs. She reported that there was indication that students who used a GC had a better understanding of the relations hip between an algebraic function and its graph; comprehended that problems in algebr a can be solved graphically as well as through algebraic manipulation; and tended to do m ore conjecturing and generalizing. Rich also claimed that the use of the GC i n precalculus provided a new problemsolving technique, and changed classroom dynamics. Problem solving is another area where the use of GCs pro vided some positive results. For example, Allison (2000) sought to determine the impact of the GC on four high school students mathematical thinking while solving problem s. The students were given both contextual nonroutine problems and noncontext ual exploratory problems. From the interviews with the students, Allison was able to observe the following: GCs amplified the speed and accuracy of students problemsolving s trategies; GCs encouraged the students to use graphical approaches to solve probl ems; and GCs enhanced the students ability to focus on reasoning for their answers to the problems. The students, however, commented that the GC added time t o the problemsolving when syntax errors occurred. Cassity (1997) reported that when GCs were used in a colle ge algebra course, spatial visualization and mathematical confidence were i mproved but no gender difference was observed. She, however, pointed out tha t many studies with a GC focused on procedural or algorithmic understanding rather than conc eptual understanding. PAGE 57 43 A large portion of the body of GC research has focused on students achievement in various mathematics courses; only a few studies examine d students errors and misconceptions that are related to GC technology (Dunham, 1999; Gaston 1990; Mitchelmore and Cavanagh 2000; Tuska 1992). Dunham (1999) referred to these errors as calculatorinduced errors. In order to use GCs in t heir mathematics learning effectively, students need to be observed and informed about their GC based errors and misconceptions. The next section discusses some of the se new classes of students errors and misconceptions. Students Errors and Misconceptions in Calculator Usage Evidence shows that a new class of errors and misconc eptions that students experience are introduced by GCs (Dunham, 1999; Gaston 1990; Mitchel more and Cavanagh 2000; Tuska 1992). This is an area that needs carefu l attention but it often receives less attention. Mitchelmore and Cavanagh (2000) pointed out that, Researchers have rarely investigated how individual studen ts actually use a GC. In particular, although there is anecdotal evidence that students occasionally misinterpret the graphic image, we [they] have found no sy stematic research on the types of misconceptions which arise or their causes (p. 254). Tuska (1992) attempted such a study. She studied students general errors in GCbased precalculus classes and identified eight GC assoc iated misconceptions. These misconceptions fell into four categories: misunderstand ing of the domain of a function, misunderstanding of the end behavior and asymptotic behavior o f functions, misconception of the solution of inequalities, apparent mis belief that every number is rational. PAGE 58 44 Mitchelmore and Cavanagh (2000) investigated students difficu lties in operating a GC. This particular study investigated how grades 10 and 11 st udents interpret linear and quadratic graphs on a Casio fx7400G through clinical intervi ews. Their finding showed that students errors in using a GC were attributabl e to four main causes: a tendency to accept the graphic image uncritically without a ttempting to relate it to other symbolic or numerical information; a poor understanding of the concept of scale; an inadequate grasp of accuracy and approximation; and a limited gras p of the processes used by the calculator to display graphs. Another study by Gaston (1990) reported that certain student s in remedial and nonremedial precalculus classes exhibited a reluctance to use calculators and/or had difficulty with calculator use in classes where such use was permitted. The study was conducted with community college students who took Basic Ar ithmetic, Introductory Algebra, Intermediate Algebra, and Trigonometry courses. The data was collected through students responses to the researchermade questionnai re and a series of interviews with certain students. Her finding showed that students reluctance to use calculators was observed in students who had poor attitudes toward the use of calculators in the mathematics classroom; had little perception of the usefulness of calculators; had little experience with calculator use; and/or were un able to achieve the successful integration of appropriate levels of mathematical compete ncy and calculator competency. The next section discusses the contents of calculus that are related to this study and students knowledge in calculus, in particular with l imits and derivatives. The final part of this chapter examines the role of technology in ca lculus instruction. PAGE 59 45 Calculus and Students Knowledge in Limits and Derivatives The first part of this section reviews the contents of calculus, particularly in limits and derivatives. Understanding the nature of the contents of calculus will help to understand how students learn these calculus topics and als o the knowledge they have in these topics. Differentiation and integration are two main components in calculus. The history of calculus shows that the development of calculus by Newton (16421727) and Leibniz (16461716) resulted from the investigation of the following pro blems: 1. Finding the slope of a tangent line to the graph of a gi ven function at a given point. 2. Finding the area of a region bounded by the graph of a function. The first problem led to the creation of differential calculus in which students learn the derivative of functions and related application s. The second problem led to the creation of integral calculus in which students learn the antiderivative of f unctions, definite integrals, and related applications. Finding the derivative of a function is developed from the ide a of a limit of the function. Likewise, the limit concept is a foundation for the development of integration as well. So, understanding the limit concept is crucial in u nderstanding the derivative and integration. Students who enter into a calculus course, therefore, fi rst study the limit topics. However, it is acknowledged by mathematics educators that students have difficulties in PAGE 60 46 understanding the limit concept (Barnes, 1997; Bergthold, 1999; Da vis & Vinner, 1986; Tall et al., 2004; Smith, 1996). Bergthold (1999) wrote that the limit concept is harder for students to understand because the mathematical definiti on of limit differs substantially from intuitive limit ideas. The formal definition of the limit of a function, th e delta epsilon definition, was first given by Cauchy (1789 1857). The definition was given as fo llows: The description of L x f c x = ) ( lim is that for each 0 > e there exists a 0 > g such that if 0g< < c x then ) (e< L x f This formal definition, however, for the limit conc ept of a function is rather complex to understand for students who just enter into a calcu lus class. Barnes (1997) stated, the limit concept is inherently difficult and causes pr oblems no matter how it is taught, partly because many students intuitive ideas are in confl ict with the formal definition (p.1). Tall, Smith, and Piez (2004) agreed: the formal li mit is a grievously difficult concept to use as a foundation for teaching the calculus (p. 6). Teaching the limit of a function through the formal definition of a limit is, in fac t, a difficult task for mathematics teachers. It is also noted by calculus educators that even s tudents who have passed a calculus class could not define the limit of a func tion correctly. Students who have studied calculus are often unable to define limit c orrectly, or to explain why the concept of limit is fundamental to calculus (Davis & Vinne r, 1986). After the limit section is covered, students in a c alculus course study derivatives of functions. The derivative of a function at a giv en point is defined as the slope of the PAGE 61 47 function at that point. Further, the slope of a funct ion at a particular point is a measure of how quickly the output ) ( y changes as the input ) ( x changes at that point. This is called the instantaneous rate of change or simply the rate of change of the function at that point. Thus, the problem of finding the derivative of a function at a point is mathematically equivalent to finding the rate of ch ange of the function at that given point. The process of finding the derivative of a f unction at a point is known as differentiation. Therefore, all applications that a re related to finding the rate of change of functions require finding the derivative of those f unctions. A description of finding the derivative of a function at a point is given as fol lows: Let A = (,()) xfx be the given point on the graph of ). ( x f Choose another point that is close to the given point on the graph of ). ( x f Say, the second point B = (,()). xhfxh ++ Then the slope of the line (called secant line) th at goes through these two points is ()() () fxhfx xhx ++= ()() fxhfx h +Now the argument is that when h approaches 0, the point B approaches the point A. Therefore, when h approaches 0, the secant line AB approaches the tangent line at point A. So, when h approaches 0, the slope of the secant line AB approaches the slope of the tangent line at A. Tha t is, the slope of the tangent line to the graph of ) ( x f at x is ) ( ) ( lim 0 h x f h x f h + Now, because the tangent line goes through the point A on the gr aph of the given function, the slope of ) ( x f at x equals to h x f h x f h ) ( ) ( lim 0 + and is known as the derivative of ) ( x f at x PAGE 62 48 Students, therefore, need to understand that finding the der ivative of a function at a point is, in fact, a limit problem. Like students difficulties with the limit concept, st udents experience difficulties with the concept of derivatives (Smith, 1996; Tall, 1990; Tal l et al., 2004). Some argue that students have difficulties in understanding the deri vative because of the way the derivative is presented. For example, Tall (1990) thought tha t students experience difficulties in understanding the derivative concept becau se the derivative is presented as a limit problem, not because of the students inability t o grasp such an abstract concept. His argument was that the limiting process may be intui tive in a mathematical sense but not in a cognitive sense, therefore students have diffi culties in understanding the derivative concept. To understand the calculus concepts and how these concepts can be applied to other fields, one needs a strong algebraic background. A we ak mathematical background in terms of lack of function concepts, algebraic manipul ations, and geometric visualizations is one reason why students are not successf ul in calculus classes (Douglas, 1986; FerriniMundy & Graham, 1991). As a result of these s tudents difficulties, students study calculus courses by memorizing the techniques and procedures with little time focused on the concepts (FerriniMundy & Graham, 1991; Tall et al., 2004). Educators and researchers are concerned about students di fficulties in calculus topics. They look for new ways to deliver calculus instruct ion so that students will have a better understanding of what they study in calculus. Tech nology has been thought by many to aid calculus instruction in order to improve students conceptual understanding PAGE 63 49 in calculus concepts. The next section examines such s tudies in which technology, particularly GCs, computers and CASs were used in calculus The Role of Technology in Calculus Like other areas in mathematics, calculus is one of t he areas in mathematics that has been and can benefit from technology, as calculus is often difficult for many students (Douglas, 1986; FerriniMundy & Graham, 1991; Tall et al., 2004). Mostly, computers, various CASs, and GCs are widely used in calculus courses to improve students learning in the subject. Waits and Demana (1996) state that Compute r generated numerical, visual, and symbolic mathematics is revolutionizing the teaching and learning of calculus (p. 712). A concerned mathematics faculty group initiated calculus reform in the 1980s to address the high attrition rate in calculus and the lac k of student understanding of the concepts of calculus. In their official report, Toward a Lean and Lively Calculus they wrote about their ideas (Douglas, 1986). Immediately after that report, over six hundred mathematicians, scientists, and educators gathered for th e Calculus for a New Century Colloquium sponsored by the National Academy of Sciences and the N ational Academy of Engineering. A well known report, Calculus for a New Century: A Pump, Not a Filter, came out of the conference and discussed issues in calcul us instruction (Steen, 1987). Similar calculus reform movements initiated in Britain (Tall, 1992) and in Australia (Barnes, 1997) generated the same concerns in calculus ins truction. In all calculus reforms, the primary ideas were to dev elop an alternate curriculum that is more conceptual and application oriented than th e traditional curriculum, to make PAGE 64 50 use of multiple representations of mathematical conce pts, to develop a variety of teaching methods for calculus, and to use technology. The calculus reformers have recommended the widespread use of technology in the form of computers, CASs, and appropriate graphing and symb olic calculators. The increasing availability of computers and highly sophisticated new generations of calculators helped to push the development of calculus ref orms forward. HughesHallet (1991) wrote I believe that in calculus most of the idea s should be presented in three ways: graphically and numerically, as well as in the tra ditional algebraic way. Technology is invaluable here (p. 33). These calculus refo rmers strong belief is that using technology will help students in the following ways: 1. To free students from algebraic manipulations. 2. To reduce the drudgery of calculations. 3. To use visualization to understand abstract ideas. 4. To explore what if situations. Most of these studies were conducted with the use of com puters and various computer software and CASs and a few studies were conducted with t he use of GCs. Research on Usage of Computers and CAS in Calculus Since the beginning of the calculus reform movement, t here have been many calculus studies initiated and conducted with the use of di fferent forms of technology in high schools, colleges, and universities (Crocker, 1990; Fr eese, Lounesto, & Stegenga, 1986; Heid, 1988; HughesHallet, 1991; Judson, 1990). Most of the studies emphasized less time on paper and pencil methods and more time on app lications, problem solving, and concept development in calculus. For example, the Harvard Core Calculus PAGE 65 51 Consortium Project emphasized the importance of multiple representations w ith technology that should be used in calculus classes in o rder to understand calculus concepts (HughesHallet, 1991). The project emphasized that m ost of the ideas in calculus should be presented in three ways: graphically, numerically, and algebraically with the appropriate use of technology, and therefore, t he project was known as the Rule of Three. The project had five major thrusts: 1. Deemphasizing the current stress on manipulative sk ills by achieving a balance among visual interpretations of the concept s, numerical interpretations of the ideas, and the traditional manipulative approaches. 2. Presenting a more intuitive approach to the concepts a nd methods of calculus to improve student understanding. 3. Introducing more modern mathematical ideas. 4. Including a wider variety of more realistic applications to reflect better the modern uses of calculus in the client disciplines. 5. Incorporating the use of appropriate technology (computers, CASs, and sophisticated graphing calculators) to improve student underst anding of the ideas of calculus. This project later expanded to be referred to as the Rule of Four, including a writing or verbal descriptions component (FerriniMundy & Gaudar d, 1992). This component emphasizes the need for students to explain their mathe matical work. The other wellknown calculus project, called Project CALC, was developed and conducted at Duke University between 1989 and 1993. The project, developed by Moore and Smith, created a new calculus course at Duke Univ ersity that differed from PAGE 66 52 the traditional calculus course in several fundamental w ays. The traditional course emphasized acquisition and memorization of pencilandpaper computational skills whereas the content of the project was based on solving real world problems with the use of the computer lab. The project developers claimed that these projects provided evidence that students conceptual understanding can be improved by usi ng technology and suggested placing more emphasis on conceptual development an d less emphasis on computational skills. Ganter (2001) evaluated calculus reform projects that were conducted with the use of computers, or CASs, or GCs during 1988 to 1998 that were mai nly funded by the National Science Foundation (NSF). Ganter observed th at the common belief from the project developers was that the students in the reform c lasses did at least as well as those in the traditional classes. He found positive student achi evement in some projects but not in all. Many individual mathematics researchers conducted studies in various topics in calculus with the use of computers and of CAS technology Heid (1984, 1988, & 1997) was one of the first to investigate the effects of te chnology in calculus classes. For example, for her doctoral dissertation study, Heid (1984) examined the effects of an applied calculus course during which students focused on concept s and applications and used the CAS to execute routine symbolic manipulations. S he participated in her own study and compared a traditional calculus and a treatment group who used a CAS called muMATH. The study lasted for twelve weeks with data that was collected from interview transcripts, conceptual comparison questions, and final exa mination results. Heid claimed that Computers have decreased the time and attention usua lly directed towards mastery PAGE 67 53 of computational skills and have lent flexibility to the analysis of problem situations (p.24). She concluded that the treatment group had a better un derstanding in the calculus concepts than did the control group by saying, students from the experimental classes spoke about the concepts of calculus in more detail, wi th greater clarity, and with more flexibility than did students of the comparison group (p. 21). A similar study was conducted by Judson in 1990 to examine the ef fects of using a CAS (MAPLE) in an introductory calculus class. The r esults of the study indicated that there were no significant differences between the expe rimental group who used the CAS and the control group who was taught with a traditional a pproach in skills and concepts. Palmiter (1991) conducted a study on integration techniques in a calculus course with a CAS (MACSYMA). The experimental group used the CAS as an aid in their homework problems and the control group learned those int egration techniques with a paper and pencil method. The results showed that the experi mental group gained significantly in achievement over the control group. She also claimed that the use of technology increased student attitude and confidence in mathe matics. The author admitted some internal threats to the study because it lasted 5 weeks for the experimental group but 10 weeks for the control group. A study by Cunningham (1991) examined the effects on achievemen t of using computer software to reduce handgenerated symbolic manipulat ion in freshmen calculus. The study used a pretestposttest design to compa re treatment and control group performance on calculus computation and conceptual skill m easures. He concluded that the use of software improved achievement and did not do h arm when access was denied. PAGE 68 54 He further stated that success required instructor use in the classroom in tandem with extensive student use both outside of the classroom and on te sts. Most CAS programs offer multiple representations that allow students to use graphical, numerical, and analytical representations to s olve calculus problems. For example, Porizo (1994) examined the effects of using three instructional approaches in three firstyear calculus classes. One group was treat ed with a CAS program ( Mathematica ), the second group was treated with the use of a GC (TI 81), and the third class received a traditional approach. He found that student s in the CAS group made stronger connections between graphical and symbolic repres entations than did students in the other two groups. A study by Connors (1995) investigated relationships between gende r and achievement as well as gender and attitudes in computer and n oncomputer groups in a first year college calculus. The study contained both quan titative and qualitative components. She collected final exam scores from four s emesters and attitude survey from two semesters. Connors concluded that the students i n the experimental group performed significantly better on the common final exam in fall of 1993 and female students in the experimental group benefited more than the other group. She also concluded from the attitude survey that the students in the experimental group showed a positive result but there was no gender difference in th e attitude results. The benefits of using CAS and computers in calculus instruc tion are being discussed and various projects and studies are being conducte d. Yet, the usage of these forms of technology in calculus instruction is still li mited because of cost and required teacher preparation and training. The GC technology that w as introduced in 1986 offers PAGE 69 55 similar capability to the capability possible with CASs. B ut because the GCs are portable and inexpensive, this technology has become a better cho ice for educators to use in mathematics classrooms. The next section examines ca lculus studies with the use of GCs. Research on Usage of Graphing Calculators in Calculus Researchers took advantage of the GC technology in various mathematics courses, including calculus, to improve students understandi ng because of its affordability, capability, and portability. Research studies in calculus with GC technology, like other areas in mathematics, were initi ated beginning in the early 1990s (Barton, 1995; Bergthold, 1999; Blozy, 2002; Dunham, 1999; Ellison, 1993; Estes, 1990; Heid, 1997; Stiles, 1994). There are two entry levelcourses in calculus that ar e offered in colleges and universities. One is a regular calculus course (CalculusI, MAC 2311) and the other is an Applied Calculus course (MAC 2233). Students get to choose one or the other based on their major and requirements. This study was conducted in Applied Calculus at a community college and examined the effects of a numerica l method along with the use of GCs and the researcherdeveloped guided lessons in limits, derivatives, and related applications in these two topics. Limits and derivatives a re common topics for both calculus courses. Because an Applied Calculus course has a relatively small place in the calculus literature, this section includes a review of research studies in calculus from both of those calculus courses. Stiles (1994) conducted a study in using GC in first semeste r calculus. The study explored the use of GC in the relationship between a fun ction and its tangent line, Newtons method for finding real zeros of polynomials, a nd curve sketching. The Casio PAGE 70 56 fx7000 G graphing calculator was used in the study and Stiles co ncluded that the usage of the GC in the calculus class improved students geomet ric intuition and improved students understanding of calculus. Bergtholds (1999) study provided some useful information about students early understanding of limit concepts with the use of GCs. The purpose of the study was to identify and describe students patterns of analytical thin king and knowledge use in solving limit problems. The research was a qualitative desig n involving four interviews with each of 10 firstsemester calculus students in a unive rsity. Written and oral responses were analyzed relative to the researchermade f ourelement framework: 1. Analyzing functions locally in graphical and numerical settings; 2. Conjecturing limits from representative graphs and tabl es; 3. Understanding advantages and limitations of tables and graphs to conjecture limits; and 4. Producing multiple sources of evidence to justify a limi t conjecture. The researcher observed two factors that influenced studen ts early understanding of the limit concept: a) students knowledge and understandi ng of functions and b) students knowledge and use of GCs. Bergthold pointed out that Graphing calculators allow easy access to numerous intuitive limit ideas ( p. 5). She further listed benefits of using a GC: graphs and tables can be produced quickly and easily; the trace feature permitted a dynamic sense of the limit process; and the zoom feature sometimes helped to show how smaller viewing windows lead to more accurate limi t conjectures. She also reminded students that these features sometimes have drawba cks, such as a) calculatorproduced tables and graphs can be misleading due to computational limitations, b) poor PAGE 71 57 input choices can mislead the limit result, c) the inte resting behavior at a particular point is hiding between two pixels, and d) misleading graphs are pos sible in limit problems that involve vertical asymptotes. Some calculus studies were conducted with the combined use o f computers and GCs. For example, Estes (1990) conducted a study to examine the effect of implementing GC and computer technologies as instructional tools and the impact of the technology on the instructor and student in an Applied Calculus class. Th e treatment group experienced graphics techniques via a microcomputer and a GC and the contro l group was taught by traditional methods and allowed to use a GC. The pretestp osttest design was used to measure the achievement in both conceptual and procedural kn owledge. Estes reported that the experimental group scored significantly higher on conceptual achievement but there was no significant difference on procedural achiev ement. The examination of the impact of technology on the instructor showed four major categories of problems for instructors to implement the technology: instructionalde sign, syllabusschedules, computerperipherals, and environmental difficulties. The examination of the impact of the technology on the students showed that computer demonst rations along with the use of GCs enhanced conceptual understanding. Further, the studen t survey data indicated that students believed that the GC and computer were help ful in their learning, if the student understood how to use the technology. Also, the studen ts in the study indicated a preference for the GC over the computer. Ellison (1993) examined the effect of using computer softwa re and a GC on students ability to construct calculus concepts. She used a qualitativecasestudy research methodology in her study. Ten students were draw n from two technologically PAGE 72 58 enhanced sections of Calculus and Analytical Geometry offered at a university. A series of interviews were conducted for the purpose of the study. The tasks of the interviews were to see whether students were able A. to distinguish the graphs of functions and their derivativ es; B. to sketch the graph of the derivative from its parent fun ction; C. to draw conclusions about characteristics of the deriva tive from the graph of the derivative; and D. to link the formal definition of the derivative with a vi sual image of the limiting slope of secant lines. Ellison reported that all ten students were able to do t ask A; six of them were able to do task B; five students were able to do task D; and only four students were able to do task C. Overall she concluded that the technologicalba sed instruction had a positive effect on students ability to construct a mental concept image of the derivative but a number of students had only a partiallyformed understanding of the connections between derivatives, functions, and graphs. The study, however, reported some serious limitations for the study: One of the two classes from which the te n students were chosen was taught by the researcher; the researcher had a more thorough knowl edge of and proficiency with the computer software and GC than the other instructor, a nd the researcher used GC to a greater extent during class sessions than did the other i nstructor. There have been very few studies conducted in mathemat ics education with the next generation of GCs, known as supercalculators or CAS graphing calculators (Blozy, 2002; Keller & Russell, 1997). These calculators (eg. TI92 and TI89) not only PAGE 73 59 have both numeric and graphic features but also have buil tin computer algebra systems computer programs. Blozy (2002) conducted a study to analyze students performance in calculus between students who used a nonCASGC (TI 83) and students who used a CASGC (TI 89). The study also analyzed students performance in calcu lus when those students were not allowed to use any type of graphing calculator at all. In both cases, the study examined students performance in concepts, algorithmic skil ls, graphing and interpretation, and related applications in calculus. A group of 56 students from two different Advanced Placement Calculus classes from a high school participated in Blozys study. He collected data from two tests (one all owed the use of a GC and the other did not allow the use of any GC) and a clinical inte rview with 10 students (five from each group). The study showed mixed results. On the ca lculatorallowed tests, the CASGC group performed significantly better on the algorit hmic skills but the nonCASGC group performed significantly better on concepts and gra phing and interpretation areas. On the calculatornotallowed tests, overall, students performed equally. He also reported that his clinical interviews showed similar res ults on the two tests but he was able to notice overwhelmingly that the CASGC group approac hed and solved problems using algebraic representations and the nonCASGC group a pproached and solved problems using graphical and numerical representations. A study by Barton (1995) examined classroom instructional prac tices and teachers professed conceptions about teaching and learning col lege calculus in relationship to the implementation of GCs. The study provided information on how the college teacher responded to the call for reform in tea ching calculus through a graphing PAGE 74 60 approach supported by use of GCs. The researcher selected f ive teachers as subjects out of ten teachers who were assigned to teach the calcul us course at a university at that time. She collected data through interviews with the teachers and classroom observations. She concluded that the college teachers conceptions of thei r teaching approach were largely consistent with their instructional practice. Teachers showed skepticism about the usefulness of the use of GCs because of a) inexperience in operating the GCs. b) teachers limited time both within the classroom and in preparation for class c) teachers rigid conceptions of an appropriate teaching approa ch to calculus d) teachers strong conceptions toward a theoretical approac h emphasizing precise wording of definitions and proofs of theorems e) lack of interest from students, and f) the calculator display unit and physical arrangement of the teaching environment. Summary This chapter reviewed the literature on technology usage i n mathematics education, including NGC, GC, computer, and CAS usage in math ematics education. Then it reviewed students knowledge and learning in calculu s, particularly with limits and derivatives. Studies indicated that students experience di fficulties in limits and derivative concepts. Then the last part discussed how tech nology played and can play a role in calculus instruction to improve students conceptua l understanding in limits and derivative concepts. PAGE 75 61 The literature indicates that several national organi zations in mathematics, as well as mathematics educators, have urged the use of technology in mathematics teaching and learning. Several studies have indicated positive results in using technology in mathematics education, including calculus. Many researcher s believe that GC technology can play an important role in improving students conceptual understanding in mathematics courses, including calculus. Students in many fields need to take at least one semest er calculus for their major. Yet, calculus is an unsuccessful course and a difficul t course for many students. These students can benefit from the available sophisticated techn ology we have today. Several studies with various technology indicated positive result s in students mathematics learning. A critical need for instructional materials wit h the use of GCs that would enhance students conceptual understanding in calculus topics was observed. It has been almost 20 years since the first GC appeared in mathematics education. Its usage by students and teachers has been growing at a rapi d rate. A national survey indicates that as of 2000, 80% of high school teachers used GC technology in their classrooms. This figure will likely continue to grow. GCs are powerful and capable of doing things that were not possible before or only possible with computers. The question is whether or n ot we really take advantage of this powerful calculator. At the same time, there are many questions that remain open when it comes to the usage of technology in mathemati cs education. For example, a recent report, Handheld Graphing Technology in Secondary Mathematics: Research Findings and Implications for Classroom Practice (Burrill, 2000), raises the following questions to the GC technology research community: PAGE 76 62 1. What is the nature of the tasks for which the technolog y is used? 2. How do students and teachers choose to use the technology? 3. What is the impact of its use on student understanding? 4. Which students benefit from using technology? These unanswered questions, perhaps, lead the future GC tec hnology research in the right direction. This study, however, searched for evidence that students i n an Applied Calculus course can enhance their conceptual understanding in limi ts and derivatives. To examine students improvement in understanding these topics, this study provided a numerical method along with the use of GCs and the researcherdevel oped instructional materials. GCs have great potential that can be used to enhance student s understanding in limit and derivative topics. In fact, a GC is required for this App lied Calculus course. However, it has been noted from the literature and personal experienc e that students have problems using this GC effectively. The researcherdeveloped instruc tional materials in this study have a component that helps students to use their GCs cor rectly and effectively by providing instructional supporting materials. Also, it has be en noted from the literature and personal experience that students experience a greater difficulty in understanding the limit and derivative concepts. Another component of the researcherdeveloped instructional materials in this study demonstrated a way of approaching limit and derivative problems numerically with the help of a TI 83 GC so that students can improve their conceptual understanding in limits and derivatives. PAGE 77 63 Chapter 3 Method This chapter describes how the study was conducted. It begi ns by discussing the purpose of the study, the research questions, and hypotheses that are needed to answer the stated research questions. Next, the overall main study design and procedures are described along with information about facilitators, parti cipants, instruments, and data analysis. Also, changes to the study design based on resul ts from a pilot study conducted in the fall of 2004 are discussed. The purpose of this study was to examine effects of the use of GCs with a numerical approach and the researcherdeveloped instruction al materials on the limit and derivative topics in an Applied Calculus course at a community col lege. The general research question that this study sought to answer is To w hat degree can the use of GCs with a numerical approach and instructional materials dev eloped by the researcher affect community college Applied Calculus students learning of li mits and derivatives? In particular, the study sought to answer the following resea rch questions: 1. How does the students achievement in solving limit problem s with a numerical approach compare to that of students who solved limit probl ems with a traditional approach (primarily an algebraic approach) in an Applied Ca lculus course? PAGE 78 64 2. How does the students achievement in solving derivative pro blems with a numerical approach compare to that of students who solved de rivative problems with a traditional approach (primarily an algebraic approac h) in an Applied Calculus course? The following research hypotheses were used to answer th e research questions. The first three hypotheses are stated to answer the fir st research question and the last three hypotheses are stated to answer the second researc h question. 1. Students (GCGL group) who receive instruction with a numer ical approach will have higher achievement in routine (skill oriente d) limit problems than students (GC group) who receive instruction with a traditional appro ach (primarily an algebraic approach). 2. The students in the GCGL group will have higher achie vement in conceptual oriented limit problems than the students in the G C group. 3. The students in the GCGL group will have higher achie vement in related applications of limits than the students in the GC group. 4. The students in the GCGL group will have higher achie vement in routine (skill oriented) derivative problems than the students in the GC group. 5. The students in the GCGL group will have higher achie vement in conceptual oriented derivative problems than the students in t he GC group. 6. The students in the GCGL group will have higher achie vement in related applications of derivative problems than the students in the GC group. PAGE 79 65 Design of the Study The main design of the study was a 2 X 2 (Type of Inst ruction X Instructor) factorial analysis of covariance (ANCOVA). The prete st was used as a covariate. The two types of instructions were: Instruction with a numer ical method with the use of TI 83 GC; and Instruction with a traditional approach (primarily algebraic approach). Two instructors (Instructor A and Instructor B) participated i n this study. The design of the study was a quasiexperimental treatmentcontrol group. Variables Independent Variables The main independent variable was method of instruction The two levels of instruction on the selected calculus topics were instruc tion with a numerical approach with the use of GCs and instruction with the traditional approach (primarily algebraic approach). The other independent variable for this study was instructor There were two instructors other than the researcher used in the study. The study was designed with two instructors to show that different instructors can repli cate the effectiveness of the instructional treatment. It is, however, possible to argue that any differences in performance between the two groups are due to the instructor and not necessarily the method of instruction. It was expected that the effects due to possible differences in individual teaching style were minimal because the instruc tors had similar characteristics. The instructors are both male, have almost the same nu mber of years of experience in college teaching (one has 16 years of teaching and the other h as 17 years of teaching), and have the same number of years of teaching experience i n an Applied Calculus course PAGE 80 66 and both have positive reputations among students. Further, to help minimize effects due to possible differences in individual teaching style, each instructor taught one MondayWednesdayFriday (MWF) section and one TuesdayThursday ( TR) section and each instructor taught one treatment group and one control group. Also, to investigate potential interaction effects between the instructors and the met hod of instruction, instructor was included as a second independent variable. Dependent Variables The dependent variables were the achievement in solving sk ill oriented limit problems, conceptual oriented limit problems, and related ap plication problems of limits, skill oriented derivative problems, conceptual oriented der ivative problems, and related application problems of derivatives and were measured by re searchermade tests. These dependent variables were measured at two different times dur ing the semester. Instruments A student initial survey, a pretest, two unit exams, and a c lassroom observation protocol were used as instruments for this study. Student Initial Survey All students were given a student initial survey to compl ete on the first day of class. The purpose of the survey was to obtain informatio n about students, such as their major, previous math courses they have taken, their GC o wnership and usage, etc. See Appendix A for the student initial survey. PAGE 81 67 Pretest The researchermade pretest was administered on the fi rst day of the semester. The Basic College Algebra (MAC 1105) course is a prerequisit e for the Applied Calculus (MAC 2233) course. Therefore, a set of 13 questions were cho sen for the pretest from the objectives of the Basic College Algebra course. The purp ose of the pretest was to determine whether students in the four sections were simi lar in their mathematical ability before they received any new instruction in applied calcul us. The pretest was used as a covariate. The question items in the pretest were cons tructed response rather than multiple choice. See Appendix B for the pretest. Unit Exams The unit 1 and 2 exams were used to measure students achievem ent on limit and derivative topics, respectively. In particular, to measur e students achievement in skill oriented problems, conceptual oriented problems, and applicat ion problems, each unit test contained problems for those three areas. The research er developed these two unit exams. See Appendix C for the two unit exams. The two unit exams were scored in two different ways. B ecause these two unit exams were part of the students final grade for the cour se, the instructors who participated in the study graded the tests for their own s ections for the purpose of giving students a final grade for the course. For the purpose of the statistical analyses of the study, the researcher graded the two unit exams separately. T he two unit exams as well as the pretest were graded and scored by the researcher using t he following scoring rubric: PAGE 82 68 Table 1 Scoring Rubric Point(s) Guideline 2 Correct work and Answer 1 Partially Correct work and Answer 0 Incorrect Work or No Response Class Observation Protocol A classroom observation protocol was developed by the rese archer to use every time he visited the treatment and control groups during the period of the study. The researcher visited all experimental groups and control groups once a week to observe that the progress in the classes was in agreement with the course syllabus and the instructional methods were being utilized as planned in the main study. S ee Appendix D for the description of the classroom observation protocol. In addition to the classroom protocol, each of the inst ructors completed a courseimplementation log sheet after every class (see Appen dix I). The categories listed on the log sheet were created to learn how each class was run. A free response section was provided to list details of any problems that occurred. Obser vations made on this were included with the class observation protocol. PAGE 83 69 Validity and Reliability Content Validity To test for content validity of the instruments, the pretest and the two unit exams were given to two faculty members who have extensive e xperience in these two courses. These two faculty members were chosen from the same de partment but were not the two participating instructors or the researcher. The faculty members were asked to evaluate the content, number of questions, and the timing of the pr etest and two unit exams. Also, they were asked to evaluate any ambiguousness, wording, gramm ar, consistency, and reasonability of each item in all tests. Appropriate ch anges and corrections to these exam items were made by the researcher based on the suggestions and corrections that were received from the faculty members who reviewed the test s. Reliability The pretest, unit 1 exam, and unit 2 exam internal consist ency were measured using Cronbachs alpha and found to be .86, .78, and .72, respect ively. These coefficients support the reliability of the tests based on the number of questions on the instruments and the size and variability of the groups (Gall, Borg, & G all, 1996). To ensure the consistency of the grading process of all tests, the researcher and another faculty member who was not participating in the study but who has extensive experience in this course scored 10 anchor papers chosen ra ndomly from the pretest and the two unit exams in the study. Both instructors graded 10 anc hor papers for all three of these tests using the scoring rubric in Table 1. To assess the reliability of the grading, a total score for each student for each test from each gr ader was obtained and correlation coefficients were computed to obtain interrater reliabi lity coefficients between the PAGE 84 70 graders. The interrater reliability coefficients for the pretest, unit exam 1, and unit exam 2 were .93, .98, and .95, respectively. According to Gall et al. (1996), the obtained interrater reliability coefficients validated the reliabilit y of the grading. Procedure Participants The study was conducted in the spring of 2005 at a public communi ty college in southwest Florida that serves a two county area with a population of approximately 617,000. In spring 2004, the college enrollment was 8393 students; 3357 st udents were fulltime and 5036 students were parttime. The gender ratio is 64% female to 36% male. In the same term, the average age of fulltime student s was 23 years and the average age of parttime students was 28 years. Table 2 shows the demogr aphic information of the community college as of the 20032004 school year. Table 2 College Demographic Information Race Percentages (%) Caucasian 78.1 AfricanAmerican 10.0 Hispanic 6.2 American Indian & Asian 2.3 NonU.S. Residents 1.6 Unknown Race 1.8 PAGE 85 71 The target population was community college students enr olled in an Applied calculus course in the United States. The sample for t his study consisted of students who enrolled in four daytime sections of an Applied Calculus course (MAC 2233) scheduled during the 2005 spring semester at the community college where the study was conducted. The class size is limited to 30 students per sec tion for this course at this college. However, due to students work schedules and their schedules of other classes, earlier sections (MWF 8 A.M. and TR 8 A.M.) have smalle r enrollments than later sections (MWF 11 A.M. and TR 12:30 P.M.). The attrition was monitored during the study period. Because the study period was the first four weeks of the semester, attrition w as not expected to be a significant problem for this study. However, the students attendance was recorded during the study period. If a student missed more than half of the class meetings during the study, his or her score on the tests was not included in the statist ical analyses. Although the students were not randomly selected from the target population, i t is reasonable to assume that the students were fairly representative of the population beca use community college students are very similar around the country, according to Watkins (1992). There was a total of 93 students enrolled for the four sections of MAC 2233. The n umber of students who registered for these four sections by instructors and groups are reported in Table 3. These students were not randomly assigned to these four sections That is, students who registered for these four sections knew who their ins tructor would be. PAGE 86 72 Table 3 Number of Students Registered for the Four Sections by I nstructors and Groups Control Treatment Total Instructor A 17 30 47 Instructor B 18 28 46 Total 35 58 93 Selection of Treatment and Control Groups Four daytime sections (between 8:00 AM and 2:00 PM) of the A pplied Calculus course were offered in the spring of 2005 at the college, in cluding two (8 A.M and 11 A.M.) MondayWednesdayFriday (MWF) sections and two (8 A. M. and 12:30 P.M.) TuesdayThursday (TR) sections. Two other sections of this course were also offered at the college in the evening between 5:30 PM and 9:00 PM. Typica lly, the students who take day classes are fulltime students and are younger an d the students who take evening classes are parttime students and are older. To keep the study among a similar population, only the four daytime sections were chosen to participate in the study. Two groups (11 A.M. MWF section and 12:30 P.M. TR section) se rved as treatment groups and the other two groups (8 A.M. MWF secti on and 8 A.M. TR section) served as control groups. Two instructors other than the re searcher participated in the study and each instructor taught a MWF schedule class and a TR schedule class and each instructor taught one treatment and one control group. Due to scheduling concerns between the participating instructo rs, A and B, and the researcher, both participating instructors knew which two sections they were going to PAGE 87 73 teach but they did not know which section was treatment and which section was control until the first day of the classes. The MWF schedule c lasses meet 53 minutes for three days a week and the TR schedule classes meet 80 minutes for two days a week. To avoid the treatment variable being confounded by the number of day s the classes meet each week, one of the two MWF sections served as a treatmen t group and the other MWF section served as a control group; the same arrangement existed for the two TR sections. The two treatment groups received instruction with a nume rical method with the use of GCs and the researcherdeveloped instructional materials; the two control groups received instruction in a traditional manner (primarily a lgebraic approach) with the use of GCs in limit and derivative topics. A flip of a coin between the two TR sections determined that TR at 12:30 P.M. (by Instructor A) was assigned as a treatment group. There fore, instructor As MWF at 8 A.M. section was assigned as a control group. This sele ction determined instructor Bs MWF at 11 A.M. section as a treatment group and TR at 8 A.M. as a control group. Table 4 below summarizes the selection of treatment and co ntrol groups. Table 4 Selection of Treatment and Control Groups Section Time Slot Class Length Instructor Group MWF Class 8:00 8:53 A.M. 53 minutes A Control MWF Class 11:00 11:53 A.M. 53 minutes B Treatment TR Class 8:00 9:20 A.M. 80 minutes B Control TR Class 12:30 1:50 P.M. 80 minutes A Treatment PAGE 88 74 Facilitators Two fulltime instructors (Instructor A and Instructor B) in the mathematics department who were scheduled to teach these four section s participated in the main study. The researcher is also a fulltime faculty mem ber in the same department and has been teaching at least one section of this course for th e last 15 years but did not teach any section of this course in the spring of 2005. The two facilitating instructors have positive reputations among students and instructors and extensive experience teaching the Applied Cal culus course at this college. Table 5 below summarizes the demographic data of the instruct ors. Table 5 Instructor Demographic Information Instructor Gender Race Age Highest Degree Years Teaching A Male Caucasian 45 Ph.D Mathematics Education 16 B Male Caucasian 56 M.S Mathematics Education 17 The Studied Course The study was conducted in the Applied Calculus (MAC 2233) co urse at a community college. This course, also known as Business Ca lculus, is a threecredit course and is designed to fulfill requirements for business and nonmathematics majors. For many students, this is the last mathematics course t hey need to graduate. Topics in this course include limits differentiation and integration of algebraic, rational, PAGE 89 75 exponential, and logarithmic functions, and related applica tions of limits, differentiation, and integration in the management, business, and social s ciences. The main purpose of this course is to study calculus applications in the rela ted majors. Rigorous calculus is not a goal of this course. Textbook A required text, Calculus for Managerial, Life, and Social Sciences, 6 th Edition, by Tan (2003) was used for both groups. Chapters 2 through 6 were c overed during the course. See Appendix E for the course syllabus and a tentati ve academic calendar. Graphing Calculator A graphing calculator was required for this course. Students we re allowed to use any type of graphing calculator but any CASadded graphing calculat or (like TI 92 or TI 89) was not allowed during any exams because of their symbolic manipulative capabilities. The mathematics department strongly recom mends a TI 83 model because the TI83 is a powerful graphing calculator in terms of i ts affordability, portability, and capability. It is also user friendly and a popular graphing cal culator among college students. This study focused on instruction in limit and deri vative topics in the Applied Calculus course with a numerical approach with the use o f the TI 83 graphing calculator. Both groups were allowed to use their GCs in the classroom for homework, and for all exams. PAGE 90 76 ResearcherDeveloped Instructional Materials In addition to the text and a graphing calculator, the r esearcher developed a set of instructional materials in four unit lessons along with th e use of the TI 83 graphing calculator to use in the study. See Appendix F for the ent ire unit lessons. The purpose of developing the instructional materials is given below. Generally, there are three types of representations possible to teach mathematical concepts such as limits and derivatives in calculus. Thes e multiple representations are namely graphical, analytical, and numerical representations. A graphical representation is a way of approaching a particular concept via visual images For example, the graph (picture) of a given function can be used to explain the behavior and properties of the given function. A problem with this approach is that, in general, the functions that are described in real world applications (in the limit and deriv ative topics) may not produce nice graphs. An analytical representation uses mainly algebraic techniques, properties and manipulations to explain a mathematical concept. Because t his approach depends highly on algebraic techniques and manipulations, this approach may not work well for students if they are weak in their algebraic skills. Further, i t is quite possible that any algebraic techniques or manipulations may not work at all for certa in mathematical problems (in the limit and derivative topics). A numerical representation refers to the use of numerical computati ons to explain or solve mathematical problems. Because this approach depend s highly on computational work, this method may be a time consuming process if no c alculators are allowed. The PAGE 91 77 required computations are relatively easy and quick with G Cs using special features such as the table feature. As stated earlier, the purpose of this study was to exami ne students performance in learning limit and derivative topics with a numerical approach using GCs. Therefore, the researcher developed two unit lessons for the topic of limits (Lesson 3) and derivatives (Lesson 4). That is, these two lessons focus ed on a numerical approach to find limits and derivatives of functions and related applicat ions in limits and derivatives. These two lessons were used only in the treatment groups. Ea ch unit lesson included an instructor note section that provided an instructional guide along with a variety of examples for instructors to use and student activity sectio n that consisted of questions for students as practice. Because the GC technology is used in the previous unit less ons, one needs to make sure that students are using this technology correctly and effectively. The researcher constantly observed students lack of confiden ce and familiarity, errors, and misconceptions when using a GC in this course. Similar obs ervations were made by some in the literature (Dunham, 1999; Gaston 1990; Mitchelmore an d Cavanagh 2000; Tuska 1992). Understanding this problem fully and in order to help st udents use a GC correctly and confidently, the researcher developed two unit lessons (L esson 1 and 2) just to help students learn to use the GC effectively. These two lesso ns, therefore, were used in both treatment and control groups. These unit lessons were previo usly used by the researcher in his Applied Calculus classes. The researcher receiv ed positive comments from students for the supplied guided lessons. These four unit lessons we re given to three faculty members (two from the same college where the researc her teaches and one from a nearby PAGE 92 78 university) to evaluate the contents of the unit lessons. The faculty members were encouraged to make any comments, corrections, and/or suggest ions on the unit lesson pages as needed. Those unit lessons were collected from the faculty panel and appropriate changes were made by the researcher based on the suggestio ns received from the faculty members. The brief contents of each lesson are given in Table 6. Table 6 Brief Contents of the ResearcherDeveloped Instructiona l Lessons Unit Lesson Contents of Each Unit Lesson 1 A. Entering arithmetic and algebraic expressions in a graphing calculator. B. Simplifying arithmetic expressions by using a graphing calcula tor. C. Graphing functions (equations) with a graphing calculator D. Student activities. 2 A. Finding function values using a graphing calculator. B. Student activities. 3 A. Finding the limit of a function as x approaches a number. B. Finding the limit of a function as x approaches a number from the right and left. C. Finding the limit of a function as x approaches and D. Student activities. 4 A. Finding the derivative of a function at a given va lue. B. Finding the slope of a function at a given point. C. Finding the rate of change of a function at a give n value. D. Student activities. Instructor Preparation One month prior to the main study, the two particip ating instructors were given a copy of the researcherdeveloped unit lessons, incl uding student handouts. Then the researcher met the participating two instructors as a team four times before the end of the PAGE 93 79 fall of 2004. The purpose of the meeting was to explain the purpose, procedure, and method of the study and to review the details of the treat ments that both the treatment and control groups received during the study. The researcher provi ded instruction on using the researcherdeveloped unit lessons along with the use o f the TI 83 graphing calculator during the meeting. Each meeting lasted about 30 to 45 minutes. When the main study began in the spring of 2005, the resear cher met the instructors together once a week outside the classrooms th roughout the study period. The purpose of these meetings was to provide opportunities for the instructors to share any problems implementing the plans, discuss the progress, s hare successful strategies, and ask questions. In addition to the meetings, the researc her visited all treatment and control groups unannounced once a week during the study to assure that th e methods and the procedures were being employed as planned. Treatment Phase Table 7 provides brief information about the time schedul es of the treatment phase including the first day activity for the study. PAGE 94 80 Table 7 Weekly Time Table for the Study Week Number Descriptions of Activities 12 Sections were randomly assigned to both facilitatin g instructors as one treatment group and one control group for each instructor. The instructors met students and explained the students ro le in the study. Student Demographic Information and Student Initial Survey wer e given to all groups. Pretest was given to all groups. Sections 2.1, 2.2, and 2.3 were covered for all groups. These 3 sections were review of functions and both treatment and control groups were taught these sections along with the use of TI 83 GCs and the researcherdeveloped guided unit lessons 1 and 2. 23 Sections 2.4 and 2.5 were covered for all groups. These t wo sections were the topic of the limit of functions. The two treatment groups were taught this topic with the use of TI 83 GCs and the resear cherdeveloped guided unit lesson 3. The two control groups were taught witho ut the researcherdeveloped guided unit lessons. Unit 1 Exam was given to all groups. 34 Section 2.6 was covered for all groups. This section was the topic of the derivative of functions. The two treatment groups were taught this to pic with the use of GCs and the researcherdeveloped guided unit lesson 4. The two control groups were taught without the researcher developed guided unit lessons. Unit 2 Exam was given to all groups. On the first day of class, instructors briefly explain ed the purpose of the research study to students. Students were informed that participation i n the study was optional, but that they would receive extra credit for scores obtai ned on all studentactivity homework assignments. All students enrolled for these sections a greed to participate in the study and PAGE 95 81 students were asked to sign a consent form (see Appendix K). The students, however, were not told whether their group would be a treatment group or a control group. Next, students were asked to complete a demographic information she et (see Appendix J) and a student initial survey (see Appendix A). The survey focuse d on the students major, previous math courses, and graphing calculator ownership and usa ge. Also, the pretest (see Appendix B) was administered the first day of class The treatment phase began in the second day of class w ith the review of functions for all groups. Both treatment and control groups followed the same curriculum with the same textbook, guidelines, and time schedules. All subjects too k the same pretest and unit exams at the same scheduled times and were graded based on the same score rubric previously discussed. As mentioned earlier, for the Applied Calculus course, a graphing calculator (GC) is required. TI 83 model GCs were used by all students and th e instructors. Although a GC is required for this course, prior experience showed t hat many students come to this course with less experience and lack of familiarity wit h a GC. As a result, students constantly use this machine incorrectly and interpret its outcome incorrectly. Therefore, the first two unit lessons out of the four unit lessons deve loped by the researcher were designed to meet students needs in order to use the TI 83 GC correctly and efficiently. Therefore, these lessons were utilized in both treatme nt and control groups. All students in the treatment and control groups were given a copy of these two lessons as handouts. In these handouts, the instruction of operating the machine s, possible mistakes and misconceptions by students, a number of worked out examples, and a homework activity section were discussed. Also, according to the syllabu s, all students need to spend one PAGE 96 82 week reviewing some algebra topics that they previously lea rned, including functions. Therefore, all students in both groups reviewed these sect ions along with the researcherdeveloped unit lessons 1 and 2. After the review sections, both groups of students studied the limit of functions. The two treatment groups of students learned the limit to pic with the researcherdeveloped unit lesson 3 that focused on solving limit problems with a numerical method with the use of TI 83 GCs. The students in the treatmen t groups were given a copy of this lesson as a handout. In this handout, the instruction of solving a variety of limit problems with worked out examples and a homework activity sectio n were discussed. That is, the two treatment groups of students used the table feature of a TI 83 GC along with the researcherdeveloped unit lesson 3 to solve limit problems including applications. The two control groups of students, however, learned the same topic in a traditional manner that includes primarily algebraic techniques to solve limit pr oblems. The students in the control groups were not given the researcherdeveloped unit le sson 3. At the end of the first treatment phase, all students took the same unit 1 e xam which they were given 53 minutes to complete. This test consisted of limit problems from three areas: skill oriented, concept oriented, and related applications. Each portion was graded and scored separately. Immediately after the unit 1 exam, all students learned derivatives of functions with the limit definition. The two treatment groups were taught to find the derivative of a function at the given x value as a limit problem with the help of the research erdeveloped unit lesson 4. That is, the students in the trea tment groups were introduced to derivative problems as rate of change problems; thus, th ey approached the derivative PAGE 97 83 problems the same way they previously solved the limit pro blems. All students in the treatment groups were given a copy of this lesson as a han dout. In this handout, the instruction of solving a variety of derivative problems wit h worked out examples, and a homework activity section were discussed. The control gro ups were introduced to the derivative with the same limit definition but they wer e taught to find the derivative in a traditional manner that includes primarily algebraic techni ques and then substituted the given x value at the end. Both groups learned to solve the related a pplications of derivatives by whatever method they learned to find the der ivative for the given functions. At the end of the second and the last treat ment phase, all students took the same unit 2 exam which they were given 53 minutes to comple te. This test consisted of derivative problems from three areas: skill oriented, c oncept oriented, and related applications. Each portion was graded and scored separately. The treatment phase ended with the unit 2 exam and the entire treatment phase last ed for four weeks. Even though it was expected that students would solve the exam problems by the method they were taught in the classroom, they could hav e solved a problem by any method but they were required to show the work of their m ethod. It is possible that students from both the treatment and control groups could have interacted outside the classrooms. Also, the college has a math lab where students can visit and receive help on their math courses. To find out the interaction betwe en students in the treatment and control groups and how students prepared (studying with the sam e class mates, studying with other class mates, or receiving any tutoring help) for both unit exams, a student questionnaire was given to students to complete after ever y unit test was completed. PAGE 98 84 A discussion on the results of the student questionnaire s is given in chapter 5. See Appendix H for the student questionnaire. Data Analysis The data analysis was divided into three parts. The fir st part examined the initial group comparisons with the pretest test scores. Descriptiv e statistics, including means and standard deviations for the pretest for all four groups w ere reported. A mean comparison graph and a boxplot graph for the pretest data by instructors were also shown. Then the pretest test scores were tested with a simple oneway ANOVA. An alpha level of .05 was used for the test. According to Ste vens (1986), the following are the ANOVA assumptions: (1) the observations are normal ly distributed on the dependent variable in each group; (2) the population variances for t he groups are equal; and (3) the observations are independent. Assumption (1) was examined by plotting pretest scores and using histograms. Inspection showed that the scores we re approximately normally distributed. Further, as reported in Table 8 (chapter 4), the values of skewness and kurtosis for the treatment and control groups were within 1 to +2, indicating that there were no major departures from normality. Assumption (2) was examined using Levene statistic. The pretest data was found to have homogeneous va riances. It was assumed that the students were independent (Assumption (3)) of one ano ther since the pretest was adminsisrated in the classrooms on the first day of cl ass. The second part examined the test scores of the unit 1 exam on limits for all four groups. Mean comparison graphs were shown for each portio ns of skill, concept, and application of the unit 1 exam for the treatment and cont rol groups. Then three separate ANCOVA tests were conducted on the skill, concept, and application portions of the PAGE 99 85 unit 1 exam. The third part is similar to the second part b ut examined the test scores of the unit 2 exam on derivatives. An alpha level of .05 was used in all ANCOVA tests. According to Stevens (1986), the following are the ANCOVA assumptions: (1) the observations are normally distributed on the dependent var iable in each group; (2) the population variances for the groups are equal; (3) the observ ations are independent; (4) a linear relationship exists between the dependent variable a nd the covariate; and (5) the slope of the regression line is the same in each group. Assumption (1) was examined by plotting the scores of the unit 1 and 2 exams and using histograms. Inspection showed that the scores we re approximately normally distributed. Further, as reported in Tables 10 and 18 (chapte r 4), the values of skewness and kurtosis for the treatment and control groups were wit hin 1.61 to +1, indicating that there were no major departures from normality. Assumpt ion (2) was examined using Levene statistic. The unit 1 and 2 exam data were found to have h omogeneous variances. Even though it was assumed that the students were indepen dent (Assumption (3)) of one another, there is some concern. The lecture settings for instruction for all groups provide independent observations in the classrooms but it was alm ost impossible to control the students independence outside the classroom. The examinati ons for assumptions 4 and 5 showed that the data used in the ANCOVA failed to meet th ese assumptions. Part of the reason is, as discussed in chapter 4, that the correlati ons between the pretest and unit 1 and 2 exams were found to be very low. Therefore, the pre test score might not be a good measurement for testing the groups equivalence. Students pre vious math scores or grades could have been used as a covariate in all ANCOVA t ests instead. PAGE 100 86 Finally, the next section provides a summary of the pil ot study that was conducted prior to the main study. A full report of the pilot study, including the data analysis, is given in Appendix G. The Pilot Study A pilot study was conducted in the fall of 2004 by the resea rcher prior to the main study. The purpose of the main study was to examine the e ffects of using GCs with a numerical approach and the researcherdeveloped instruction al materials in limits and derivatives in an Applied Calculus course at a community college. It is the researchers belief that a numerical method with the use of a TI 83 GC and supported instructional materials would improve students understanding of limit an d derivative topics. The purpose of the pilot study was to evaluate the characteris tics of the used numerical method with the use of a TI 83 GC and researcherdeveloped instructional materials in the limit and derivative topics of the Applied Calculus c ourse and to determine critical situations that might be encountered during the main study. It was expected that the pilot study would help to explore whether the instruction with the numerical method along with the use of TI 83 GC and instructional materials could lead to better understanding on limits and derivatives among community college students. Furthermore, the pilot study was used to access th e validity and reliability of all exams that were used in the main study. The researcher used parallel forms of the pretest, and two unit exams in his pilot study during the fall of 2004. Th e following changes were made to the main study as a result of the pilot study: PAGE 101 87 1. In the pilot study, there were two parts on each unit exam. The first part of each unit exam dealt with a number of skilloriented probl ems for measuring students ability to solve these routine (skillorient ed) problems. The second part dealt with a number of applications for measuring st udents conceptual understanding for a particular topic. Some of these problem s were based on graphs. It was noted in the pilot study that some students were able to solve the application problems but did not solve the graphicalbas ed questions and vice versa. Therefore, to learn how students understand a particular component of the topic, each unit exam was changed to hav e three parts: a number of skill oriented problems; a number of conceptual oriented problems based on graphs; and a number of application problems. 2. In the pilot study, the researcherdeveloped materia ls (four unit lessons) were used only in the treatment group. Because the first two unit lessons were only about how to use a TI 83 GC effectively, it was decided th at these two unit lessons would be given to both treatment and control stude nts. 3. A number of items that would be helpful to students were added to the student activity sections in all instructional guided lesso ns. 4. A number of items that would be helpful to students were added to the instructor notes sections in all instructional guide d lessons. 5. Some question items were edited for purposes of clar ity. 6. Both unit exams were shortened slightly because of timing issues. PAGE 102 88 Summary This study was conducted to examine effects of the use of GCs with a numerical approach and the researcherdeveloped instructional materials on the limit and derivative topics in an Applied Calculus course at a community colle ge. The pilot study supported the validity and the reliability of all instruments. The main study involved four daytime sections with 87 part icipants and 2 instructors as facilitators. Two sections served as trea tment groups and received instruction with the numerical approach along with the re searcherdeveloped unit lessons and TI 83 graphing calculators on limit and derivative topics; the other two sections served as control groups and received instruction on those two topics with the traditional manner (primarily algebraic approach). Hypotheses were st ated to answer stated research questions. Instruments used to gather data included the pretes t and two unit exams to test hypotheses. To analyze the data gathered with these instr uments, a simple oneway ANOVA (pretest) and a set of twofactorial ANCOVA stat istical tests (unit 1 and 2 exams) were used. The next chapter discusses the results of the data analysis. PAGE 103 89 Chapter 4 Results This chapter begins by restating the research questions and hypotheses that were tested in order to answer the research questions for th e study. Next, the data analysis is discussed in three parts. The first part discusses the de scriptive statistics and an ANOVA test for the pretest. The purpose of this part is to ex amine the initial equality of the study because full random assignment of students to the study groups was not possible. The second part discusses the descriptive statistics of the dependent measures of the entire unit 1 exam on limits and each subset (Skills, Concepts, a nd Applications) of the unit 1 exam and then necessary ANCOVA tests that are needed to answer the studys first three research hypotheses. The third discussion is similar to the second part but on the entire unit 2 exam on derivatives and each subset (Skills, Conce pts, and Applications) of the unit 2 exam and then necessary ANCOVA tests that are ne eded to answer the studys last three research hypotheses. Research Questions and Hypotheses The general research question that this study sought to ans wer is To what degree can the use of GCs with a numerical approach and instruct ional materials developed by the researcher affect community college Applied Calculu s students learning of limits and derivatives? In particular, the study sought to answer th e following research questions: PAGE 104 90 1. How does the students achievement in solving limit problem s with a numerical approach compare to that of students who solved limit probl ems with a traditional approach (primarily an algebraic approach) in an Applied Ca lculus course? 2. How does the students achievement in solving derivative pro blems with a numerical approach compare to that of students who solved de rivative problems with a traditional approach (primarily an algebraic approac h) in an Applied Calculus course? The following research hypotheses were used to answer the research questions. The first three hypotheses are stated to answer the fir st research question and the last three hypotheses are stated to answer the second researc h question. 1. Students (GCGL group) who receive instruction with a numer ical approach will have higher achievement in routine (skill oriented) lim it problems than students (GC group) who receive instruction with a traditional appro ach (primarily an algebraic approach). 2. The students in the GCGL group will have higher achie vement in conceptual oriented limit problems than the students in the G C group. 3. The students in the GCGL group will have higher achie vement in related applications of limits than the students in the GC group. 4. The students in the GCGL group will have higher achie vement in routine (skill oriented) derivative problems than the students in the GC group. 5. The students in the GCGL group will have higher achie vement in conceptual oriented derivative problems than the students in t he GC group. PAGE 105 91 6. The students in the GCGL group will have higher achie vement in related applications of derivative problems than the students in the GC group. Data Analysis of the Pretest The pretest data was used to determine whether students in the treatment and control groups were similar in their prerequisite mathem atical ability before they received any instruction for the calculus topics. The pretest item s were selected from College Algebra (MAC 1105), the prerequisite course to Applied Calculus (MAC 2233). A total of 93 students showed up in the selected four sectio ns of the Applied Calculus course on the first day of class and all of t hem took the pretest. The first week is also the dropadd period at the college. Six students dropped t he course after the first day for various reasons that were not related to the study. Also, the other 87 students stayed in the course and took the unit 1 exam but only 82 students score s were considered for the statistical analysis because 5 students were absent 50% o r more during the first experiment phase. Descriptive statistics for the pretest (n = 82) for all four groups are reported in Table 8. Small effect sizes of 0.24 and 0.08 were o btained for the classes of instructors A and B, respectively, and favored the treatm ent group for both instructors. A mean comparison graph for the pretest data by instructors is shown in Figure 1 and a boxplot graph to illustrate the spread of the data is sho wn in Figure 2. A visual comparison from Figure 1 also indicates that the mean s core of the control group was slightly lower than the mean score of the treatment group for both instructors A and B. PAGE 106 92 Table 8 Means, Standard Deviations, Skewness, and Kurtosis for the Pretest by Instructors Group n M SD Skewness Kurtosis Min Max Effect Size Control A 16 23.00 6.98 0.03 0.60 11 35 0.24 Treatment A 25 24.48 5.60 0.24 1.84 9 37 Control B 15 23.40 8.54 0.50 0.15 6 37 0.08 Treatment B 26 23.96 7.32 0.13 0.44 8 35 Note : Maximum possible score on pretest was 40 points. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. 15 20 25 30 ControlTreatment PretestScore Instructor A Instructor B Figure 1: Mean Comparison for the Pretest for the Treat ment and Control Groups by Instructors PAGE 107 93 Figure 2: Boxplot Graph for the Pretest for the Treatme nt and Control Groups by Instructor To determine whether the control groups and treatments groups were statisticallysignificantly different on the pretest, the means of t hese groups were compared using a oneway ANOVA and the results are reported in Table 9. Table 9 OneWay ANOVA for the Pretest Source Sum of Squares Df Mean Square F p Between Groups 24.81 3 8.27 0.17 .918 Within Groups 3844.80 78 49.29 Total 3869.61 81 Control A Treatment A Control BTreatment B Group 0 10 20 30 40 Pretest PAGE 108 94 The ANOVA table confirmed that the groups were not statist icallysignificantly different, F (3, 78) = 0.17, p = .92. That is, there was no statistically significant difference between the groups in their mathematical ability prior t o their study of Applied Calculus. Data Analysis of Unit 1 Exam on Limits The unit 1 exam was used to measure students achievement on limit topics of Applied Calculus. The exam contained problems from three areas: Skills, Concepts, and Applications. The means, standard deviations and other descr iptive statistics for the entire unit 1 exam for all groups by instructors are reported in Table 10. Also, Figure 3 shows a visual comparison of the mean scores of the control gro ups and treatment groups for the entire unit 1 exam for each instructor. The graph shows tha t the mean score of the treatment group is higher than the mean score of the c ontrol group for both instructors A and B. Table 10 Means, Standard Deviations, Skewness, and Kurtosis for the Entire Unit 1 Exam on Limits by Instructors Group n M SD Skewness Kurtosis Min Max Control A 16 33.69 7.53 0.01 0.83 18 48 Treatment A 25 41.88 10.22 0.27 0.56 20 57 Control B 15 41.73 11.25 0.34 0.89 21 60 Treatment B 26 45.38 10.62 1.05 0.83 20 59 Note : Maximum possible score on unit 1 exam was 60 points. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. PAGE 109 95 30 35 40 45 50 ControlTreatment Unit 1 Exam Score Instructor A Instructor B Figure 3: Mean Comparison for the Entire Unit 1 Exam on Limits for all Groups by Instructors As mentioned earlier, the unit 1 exam has three portions: Skills, Concepts, and Applications. Each portion is graded and scored separately for testing any statistically significant differences between the groups on each of the se portions. First, the means and standard deviations of the skill, concept, and application portions of the unit 1 exam were obtained (see Table 11). PAGE 110 96 Table 11 Means and Standard Deviations for Unit 1 Exam on Limits by Skills, Concepts, and Applications Skill Concept Application Group n M SD M SD M SD Control A 16 14.37 2.83 10.81 7.09 8.50 1.27 Treatment A 25 15.48 3.28 15.88 6.87 10.52 2.42 Control B 15 15.00 4.15 15.73 6.77 9.93 1.98 Treatment B 26 16.07 4.55 19.77 5.95 10.62 2.35 Note : Maximum possible score on unit 1 exam was 60 points. Maxim um possible scores for skill, concept, and application portions were 20, 26, and 14, respectively. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. The next section discusses the statistical analysis of the skill, concept, and application portions of the unit 1 exam for any statistic ally significant differences between the control and treatment groups with three sepa rate ANCOVA tests with the pretest as covariate. First, Table 12 reports the result s of the ANCOVA test for the skill portion of the unit 1 exam. PAGE 111 97 Table 12 2 X 2 ANCOVA for the Skill Portion of Unit 1 Exam on Limits Source Sum of Squares Df Mean Square F p Pretest 5.92 1 5.92 0.40 .527 Instructor (Ir) 7.13 1 7.13 0.49 .488 Instruction (In) 0.01 1 0.01 0.01 .981 Ir X In 21.94 1 21.94 1.49 .225 Error 1131.01 77 14.69 As reported in Table 12, there were no statistically signi ficant differences on instruction and instructor effects between the groups found for the skill portion of the unit 1 exam, F(1, 77) = 0.01, p = .98 and F(1, 77) = 0.49, p = .49, re spectively. Also, there was no interaction effect between instruction and instructor found in the data, F(1, 77) = 1.49, p = .23. Table 13 reports the result of the ANCOVA test for the c oncept portion of the unit 1 exam. PAGE 112 98 Table 13 2 X 2 ANCOVA for the Concept Portion of Unit 1 Exam on Limits Source Sum of Squares Df Mean Square F p Pretest 35.76 1 35.76 0.82 .369 Instructor (Ir) 374.79 1 374.79 8.55 .005 Instruction (In) 380.07 1 380.07 8.67 .004 Ir X In 4.28 1 4.28 0.10 .756 Error 3376.86 77 43.86 The table confirmed that the effect of instruction was significant, F(1, 77) = 8.67, p = .004, and the effect of instructor was also significant, F(1, 77) = 8.55, p = .005 on the concept portion of the unit 1 exam. But the interaction effect between instruction and instructor was not significant, F(1, 77) = 0.10, p = .756. Table 14 reports the result of the ANCOVA test for the a pplication portion of the unit 1 exam. Table 14 2 X 2 ANCOVA for the Application Portion of Unit 1 Exam on Limits Source Sum of Squares Df Mean Square F p Pretest 1.57 1 1.57 0.34 .561 Instructor (Ir) 11.29 1 11.29 2.44 .122 Instruction (In) 33.92 1 33.92 7.34 .008 Ir X In 8.38 1 8.38 1.81 .182 Error 355.75 77 4.62 PAGE 113 99 The table confirmed that the effect of instruction was significant, F(1, 77) = 7.34, p = .008, and the effect of instructor was not significant, F(1, 77) = 2.44, p = .122 on the application portion of the unit 1 exam. Also, the interaction effect between instruction and instructor was not significant, F(1, 77) = 1.81, p = .182. In addition to the ANCOVA tests on the skill, concept, and application portions of the unit 1 exam on limits for any statistically signific ant differences, some further analysis was conducted to help understand how students in e ach group scored on each of those sections. First, it should be noted that the maxim um possible points for each portion of the unit 1 exam vary. The maximum possible score for t he entire unit 1 exam was 60 points and the maximum possible scores for the skill port ion, concept portion, and application portion are 20, 26, and 14, respectively. Therefor e, the mean score percent of the skill, concept, and application portions of the unit 1 exam was calculated and is reported in Table 15. PAGE 114 100 Table 15 Mean Percents for Unit 1 Exam on Limits by Skills, Con cepts, and Applications Skill Concept Application Group n M % Earned M % Earned M % Earned Control A 16 14.37 71.9 10.81 41.6 8.50 60.7 Treatment A 25 15.48 77.4 15.88 61.1 10.52 75.1 Control B 15 15.00 75.0 15.73 60.5 9.93 70.9 Treatment B 26 16.07 80.1 19.77 76.0 10.62 75.9 Note : Maximum possible score on unit 1 exam was 60 points. Maxim um possible scores for skill, concept, and application portions were 20, 26, and 14, respectively. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. Also, visual comparisons of the percent mean scores of the control and treatment groups by instructors on the Skill portion (Figure 4), Concept portion (Figure 5), and Application portion (Figure 6) are presented. 40 50 60 70 80 90 100 ControlTreatment Unit 1 Exam (Skill Portion)Mean Percent Instructor A Instructor B Figure 4: Mean Comparison for the Skill Portion of the Unit 1 Exam on Limits for all Groups by Instructors PAGE 115 101 40 50 60 70 80 90 100 ControlTreatment Unit 1 Exam (Concept Portion)Mean Percent Instructor A Instructor B Figure 5: Mean Comparison for the Concept Portion of the Unit 1 Exam on Limits for all Groups by Instructors 40 50 60 70 80 90 100 ControlTreatment Unit 1 Exam (Application Portion)Mean Percent Instructor A Instructor B Figure 6: Mean Comparison for the Application Portion of the Unit 1 Exam on Limits for all Groups by Instructors. Several interesting results can be noted in the percen t mean scores of the unit 1 exam. The most notable observation is that each treat ment group outperformed their respective control group in the skill, concept, and applic ation portions of the limit topic (unit 1 exam). In general, groups scored highest in the skill portion, with scores on the application portion next highest, and the scores on the concepts the lowest. That is, all four groups scored higher (between 71.9% and 80.1%) on the skil l portion than the application or concept portion. Comparing the percent me an scores, Treatment B group PAGE 116 102 scored the highest and Control A group scored the lowest in every portion of the unit 1 exam. These observations made on the skill, concept, and appl ication portions of the unit 1 exam were confirmed by the calculated effect sizes and are reported in Table 16 below. That is, both instructors had small to large (ac cording to Cohen, 1969) effect sizes but positive, favoring the treatment groups. Table 16 Effect Sizes for the Skill, Concept, and Application P ortions of the Unit 1 Exam on Limits Skill Concept Application Instructor A 0.36 0.73 1.09 Instructor B 0.25 0.64 0.32 Furthermore, one needs to note that the question items in each portion of the unit 1 exam on limits differ in terms of the number of ques tions, content, difficulty level, etc. Therefore, to help understand how students scored on each item of the skill, concept, and application portions of the unit 1 exam on limits, the pe rcent score of each group by item type was calculated and is reported in Table 17. PAGE 117 103 Table 17 Percent Scores for Each Item Type within Skill, Conce pt, and Application Portions for Unit 1 Exam on Limits by Groups Control A Treatment A Control B Treatment B Skill Portion Question #s 1 and 2: Finding the limit of a function as x approaches a number. 75 77 78 82 Question #s 3, 4, and 5: Finding the limit of a function as x approaches a number from the left and/or from the right. 72 78 77 80 Question #6: Finding the limit of a function as x approaches and/or 68 75 70 78 Concept Portion Question #7: Explaining the meaning of the limit of a function as x approaches a number. 36 57 48 66 Question #s 8, 9, and 10: Finding the limit of a function as x approaches a number from the left and the right from the graph of the function. 46 64 65 80 Question #11: Finding the limit of a function as x approaches a number from the given table. 43 62 65 83 Application Portion Question #12(a) and 12(b): Finding the function value and interpreting the answer. 82 87 84 90 Question #12(c): Finding the limit of a function as x approaches a number from the left and interpreting the answer. 48 68 60 67 Question #13: Finding the limit of a function as x approaches and interpreting the answer. 52 70 68 70 PAGE 118 104 Some interesting results can also be noted in the pe rcent mean scores of each question type for the skill, concept, and application port ions of the unit 1 exam for all four groups. Among the skill oriented limit questions, all groups scored lower on the question of finding the limit of a function as x approaches and/or than any other questions. This difficulty might be associated with studen ts understanding of the meaning of and On the concept portion of the limit topic, the questio n of explaining the meaning of the limit of a function as x approaches a number was the hardest question for all groups. On the application portion of the limit topic, the question of finding the limit of a function as x approaches a number from the left and interpreting the answer was the hardest question for all groups. A possible explanation for students poor performance on this type of question is th at students, in general, dislike an interpret your answer part of a question. Experience teaching Applied Calculus course has shown that even good students tend to avoid answe ring such questions. Data Analysis of Unit 2 Exam on Derivatives The unit 2 exam was used to measure students achie vement on derivative topics of Applied Calculus. This exam also contained probl ems from three areas: Skills, Concepts, and Applications. Table 18 reports the me an scores and other descriptive statistics for the entire unit 2 exam for all group s by instructors. Figure 7 shows a visual comparison of the mean scores of the control groups and treatment groups for the entire unit 2 exam for each instructor. The graph shows th at the mean score of the treatment group is higher than the mean score of the control group for both instructors A and B. PAGE 119 105 Table 18 Means, Standard Deviations, Skewness, and Kurtosis for the Entire Unit 2 Exam on Derivatives by Instructors Group N M SD Skewness Kurtosis Min Max Control A 14 27.57 4.14 0.45 0.24 21 36 Treatment A 23 34.09 6.08 0.08 0.31 20 46 Control B 14 29.00 8.89 0.36 0.27 13 45 Treatment B 25 35.96 7.39 0.04 1.61 25 46 Note : Maximum possible score on unit 2 exam was 48 points. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. 20 25 30 35 40 ControlTreatment Unit 2 ExamScore Instructor A Instructor B Figure 7: Mean Comparison for the Entire Unit 2 Exam on Derivatives for all Groups by Instructors As mentioned earlier, the unit 2 exam was also divided into three portions as skills, concepts, and applications. These three portions of the unit 2 exam were graded and scored separately for all groups and the means and standa rd deviations of these portions are reported in Table 19. PAGE 120 106 Table 19 Means and Standard Deviations for Unit 2 Exam on Derivativ es by Skills, Concepts, and Applications Skill Concept Application Group n M SD M SD M SD Control A 14 12.86 2.69 5.00 3.55 9.71 1.59 Treatment A 23 14.78 2.35 8.22 4.27 11.13 1.87 Control B 14 12.14 3.66 8.07 4.45 8.79 2.72 Treatment B 25 14.76 2.83 10.36 5.23 10.84 3.04 Note : Maximum possible score on unit 2 exam was 48 points. Maxim um possible scores for skill, concept, and application portions were 18, 16, and 14, respectively. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. The next section discusses the statistical analysis of the skill, concept, and application portions of the unit 2 exam for any statistic allly significant differences between the control and treatment groups. Three separate ANCOVAs were conducted with the pretest as covariate on these three portions. An alpha level of .05 was used in each case and Table 20 reports the results of the ANCOVA test for the skill portion of the unit 2 exam. PAGE 121 107 Table 20 2 X 2 ANCOVA for the Skill Portion of Unit 2 Exam on Deri vatives Source Sum of Squares Df Mean Square F p Pretest 2.67 1 2.67 0.33 .569 Instructor (Ir) 2.42 1 2.42 0.30 .588 Instruction (In) 93.81 1 93.81 11.50 .001 Ir X In 2.38 1 2.38 0.29 .591 Error 579.23 71 8.16 The table confirmed that the effect of instruction was significant, F(1, 71) = 11.50, p = .001, and the effect of instructor was not significant, F(1, 71) = 0.30, p = .588 on the skill portion of the unit 2 ex am. Also, the table found that the interaction effect between instruction and instructor was not significant, F(1, 71) = 0.29, p = .591. Table 21 reports the results of the ANCOVA test for the concept portion of the unit 2 exam. PAGE 122 108 Table 21 2 X 2 ANCOVA for the Concept Portion of Unit 2 Exam on Derivatives Source Sum of Squares Df Mean Square F p Pretest 32.01 1 32.01 1.57 .214 Instructor (Ir) 120.51 1 120.51 5.92 .018 Instruction (In) 106.20 1 106.20 5.21 .025 Ir X In 5.14 1 5.14 0.25 .617 Error 1446.59 71 20.37 The table confirmed that the effect of instruction was significant, F(1, 71) = 5.21, p = .025, and the effect of instructor was also significant, F(1, 71) = 5.92, p = .018 on the concept portion of the unit 2 exam. But, th e table found that the interaction effect between instruction and instructor was not significant, F(1, 71) = 0.25, p = .617. Table 22 reports the result of the ANCOVA test for the a pplication portion of the unit 2 exam. PAGE 123 109 Table 22 2 X 2 ANCOVA for the Application Portion of Unit 2 Exam on Derivatives Source Sum of Squares Df Mean Square F p Pretest 27.46 1 27.46 4.88 .030 Instructor (Ir) 6.48 1 6.48 1.15 .287 Instruction (In) 38.21 1 38.21 6.79 .011 Ir X In 1.09 1 1.09 0.19 .661 Error 399.72 71 5.63 The table confirmed that the effect of instruction was significant, F(1, 71) = 6.79, p = .011, and the effect of instructor was not significant, F(1, 71) = 1.15, p = .287 on the application portion of the unit 2 exam. Also, the table found that the interaction effect between instruction and instructor was not significant, F(1, 71) = 0.19, p = .661. As before, some further analysis was conducted to help unde rstand how students in each group scored on each of those portions. It shou ld be again noted that the maximum possible points for each portion of the unit 2 exa m also vary. The maximum possible score for the entire unit 2 exam was 48 points and t he maximum possible scores for the skill portion, concept portion, and application portion are 18, 16, and 14, respectively. Therefore, the mean score percent of ski ll, concept, and application portions of the unit 2 exam was calculated and is reported in Table 23. PAGE 124 110 Table 23 Mean Percents for Unit 2 Exam on Derivatives by Skills Concepts, and Applications Skill Concept Application Group n M % Earned M % Earned M % Earned Control A 14 12.86 71.4 5.00 31.3 9.71 69.4 Treatment A 23 14.78 82.1 8.22 51.4 11.13 79.5 Control B 14 12.14 67.4 8.07 50.4 8.79 62.8 Treatment B 25 14.76 82.0 10.36 64.8 10.84 77.4 Note : Maximum possible score on unit 2 exam was 48 points. Maxim um possible scores for skill, concept, and application portions were 18, 16, and 14, respectively. Control A = Control group for Instructor A, Treatment A = Treatment group for Instructor A, Control B = Control group for Instructor B, Treatment B = Treatment group for Instructor B. Further, visual comparisons of the percent mean scores o f the control and treatment groups by instructors on the Skill portion (Figure 8), Concept portion (Figure 9), and Application portion (Figure 10) are presented. 30 40 50 60 70 80 90 100 ControlTreatment Unit 2 Exam (Skill Portion)Mean Percent Instructor A Instructor B Figure 8: Mean Comparison for the Skill Portion of the Unit 2 Exam on Derivatives for all Groups by Instructors PAGE 125 111 30 40 50 60 70 80 90 100 ControlTreatment Unit 2 Exam (Concept Portion)Mean Percent Instrucor A Instructor B Figure 9: Mean Comparison for the Concept Portion of the Unit 2 Exam on Derivatives for all Groups by Instructors 30 40 50 60 70 80 90 100 ControlTreatment Unit 2 Exam (Application Portion)Mean Percent Instructor A Instructor B Figure 10: Mean Comparison for the Application Portion of the Unit 2 Exam on Derivatives for all Groups by Instructors Several interesting results can be noted in the percen t mean scores of the unit 2 exam. As in the unit 1 exam, the most notable observatio n was that each treatment group of both instructors outperformed their control groups on the skill, concept, and application portions of the derivative topic (unit 2 exam). Every group in both treatment and control groups scored highest (between 67.4% and 82.1%) on the skill portion, second highest on the application portion (between 62.8% and 79. 5%), and lowest (between 31.3% and 64.8%) on the concept portion. That is, again it was found that the PAGE 126 112 concept portion was the hardest for all groups. Comparing th e percent mean scores, the Treatment B group scored the highest on the skill and con cept portions of the topic of derivative, but Treatment A scored the highest on the a pplication portion; Control B group scored the lowest in the skill and application porti ons and Control A group scored the lowest on the concept portion. These observations made on skill, concept, and applicati on portions of the unit 2 exam were confirmed by the calculated effect sizes and a re reported in Table 24 below. That is, both instructors had medium to large (according to Cohen, 1969) effect sizes but positive, favoring the treatment groups. Table 24 Effect Sizes for the Skill, Concept, and Application P ortions of the Unit 2 Exam on Derivatives Skill Concept Application Instructor A 0.76 0.82 0.82 Instructor B 0.81 0.47 0.70 As noticed earlier on the unit 1 exam, the question item s in each portion of the unit 2 exam also differ in terms of the number of questions content, difficulty level, etc. Therefore, to help understand how students scored on each item of the skill, concept, and application portions of the unit 2 exam on derivatives the percent score of each group by item type was calculated and is reported in Table 25. PAGE 127 113 Table 25 Percent Scores for Each Item Type within Skill, Conce pt, and Application Portions for Unit 2 Exam on Derivatives by Groups Control A Treatment A Control B Treatment B Skill Portion Question #1: Finding the derivative of a function at a given x value. 75 85 71 84 Question #2: Finding the slope of a function at a given point. 68 79 64 80 Concept Portion Question #4: Explaining the meaning of the derivative of a function at a given x value. 25 46 42 59 Question #s 5, 6, and 7: Determining the derivative of the given graph of a function at given points as positive, negative, and zero. 36 55 56 68 Question #8: Determining the point of the given graph of a function at which the derivative is the greatest. 33 53 53 66 Application Portion Question #s 3(a) and 9(a): Finding the average rate of change of a function on a given interval. 84 89 80 87 Question #s 3(b) and 9(b): Finding the instantaneous rate of change of a function at a given value. 64 77 56 75 Question #10: Finding the derivative of a function at a given value and interpreting the answer. 60 72 52 70 PAGE 128 114 Some interesting results can also be noted in the perc ent mean scores of each question type for the skill, concept, and application port ions for the unit 2 exam for all four groups. Among the skill oriented derivative questions, a ll groups scored lower on the question of finding the slope of a function at a given poin t than the other type question, finding the derivative of a function at a given x value. Even though these two questions are exactly the same, students often do not connect the fact that the derivative of a function at a given x value gives the slope of the function at that x value. On the concept portion of the derivative topic, the question of explaining the meaning of the derivative of a function at a given x value was the hardest question for all groups. This finding was similar to a finding on the limit topic wher e explaining the meaning of the limit of a function as x approaches a number was the hardest question for all groups On the application portion of the derivative, the question o f finding the derivative of a function at a given value and interpreting the answer w as the hardest question for all groups. Here also, students failed to connect the fact that the derivative of a function at a given value in an application gives the rate of change o f the function at that given value. Further, students again dislike the interpret your answe r part of the question. As this study compared the achievements in solving skill, concept, and application portions on the limit (unit 1 exam) and deriva tive (unit 2 exam) topics, the researcher also looked at Pearson productmoment correla tions of the following. The correlations between the pretest and each portion of the unit 1 exam and between the portions of the unit 1 exam are reported in Table 26. The correlations between the pretest and each portion of the unit 2 exam and between the portions of unit 2 exam are reported in Table 27. PAGE 129 115 Table 26 Correlations Between the Pretest and Each Portion of the Unit 1 Exam on Limits and Between the Portions Skill Concept Application Pretest Skill 1 Concept .38* 1 Application .44* .48* 1 Pretest .08 .11 .09 1 Note: Total Number of Students Used in Correlation Computat ions was 82. *Correlation is significant at the .01 level. Table 27 Correlations Between the Pretest and Each Portion of the Unit 2 Exam on Derivatives and Between the Portions Skill Concept Application Pretest Skill 1 Concept .28* 1 Application .48* .10 1 Pretest .01 .19 .30* 1 Note: Total Number of Students Used in Correlation Computat ions was 76. *Correlation is significant at the .01 level. Tables 26 and 27 indicated that the correlations between the pretest and each portion of both unit 1 and 2 exams were very low except th e correlation between the application portion of the unit 2 exam and pretest (the cor relation was .30, see Table 27) and that was significant. The pretest was administrated on the first day of class and PAGE 130 116 students typically did not expect to take an exam on the first day of class. Even good students can do poorly on the pretest if they had a longer gap between mathematics courses. It was noted in the initial student survey that 14% of students took the prerequisite course 2 years ago, another 14% took it more th an 2 years ago, 6 students (about 5%) took it more than 5 years ago, and 2 students (ab out 2%) took the course 10 years ago. It was quite possible that these students did no t remember anything from their last math course. However, it was noted by the research er that about 5% who scored low on the pretest made perfect scores on the unit 1 and 2 exam s. One other possible explanation for the low correlations was that, because the pretest score was not a part of students final grade, some students did not do their best o n such a test. Summary The analysis of the data does show some positive effe cts of using the GCs with the proposed numerical approach in the study to understan d better concepts and application problems in both limit and derivative topics. S tatistically significant differences between the control and treatment groups w ere found on the concept and application portions of the limit topic (unit 1 exam) and on the skill, concept and application portions of the derivative topic (unit 2 exam), but not on the skill portion of the limit topic. Also, there were no statisticallysi gnificant differences found on the interaction effect between the instruction and instruc tor on any portions of the limit and derivative topics. It is, however, important to note from the data that the treatment groups of both instructors outperformed their control groups in each porti on of both limit and derivative PAGE 131 117 topics. Thus, apparently, one can at least conclude that the used numerical approach along with the use of GCs and instructional materials in this study helped the students learn the limit and derivative topics better. The next c hapter (Chapter 5) discusses the results of each of the stated hypotheses along with the implications and limitations of this study. PAGE 132 118 Chapter 5 Discussion This study examined the effects of using graphing calculators (G C) with a numerical approach and the researcherdeveloped instruction al materials on limits and derivatives in an Applied calculus course at a community college. Th e general research question that this study sought to answer is To what degree can the use of GCs with a numerical approach and instructional materials developed by t he researcher affect community college Applied Calculus students learning of li mits and derivatives? In particular, the study sought to answer the following resea rch questions: 1. How does the students achievement in solving limit problem s with a numerical approach compare to that of students who solved limit probl ems with a traditional approach (primarily an algebraic approach) in an Applied Ca lculus course? 2. How does the students achievement in solving derivative pro blems with a numerical approach compare to that of students who solved de rivative problems with a traditional approach (primarily an algebraic approac h) in an Applied Calculus course? When limit and derivative topics were taught with two instructional approaches (numerical versus algebraic), students achievement was measured on those topics in three different areas: Skills, Concepts, and Application s. This study was interested in determining the treatment and control groups performance o n these areas due to the PAGE 133 119 treatment effect. In the preceding chapter the details o f the procedure of the study were described and the results of the statistical tests for the collected data were presented in order to answer the stated research questions. On the first day of class, all students took the pretest and an ANOVA test on the pretest confirmed that there was no statistically signi ficant difference on prior mathematical ability (pretest) between the control an d treatment groups. The first treatment phase began with the study of the limit topic The students in the treatment group solved the limit problems (skill, concept, and applicat ion types) with the use of GCs and a numerical approach and the students in the cont rol group solved the limit problems in a traditional manner (primarily an algebraic approach). Both groups used GCs but the treatment groups were provided with the resea rcherdeveloped instructional materials (the groups used unit lesson 3 on limits). Immedi ately after the end of the first treatment phase, all students took a test on the limit topic (unit 1 exam) and the problems were graded separately as skill, concept, and application portions. Hypotheses Results of the First Research Question The following research hypotheses (Numbers 1, 2, and 3) were used to answer the first research question on the limit topic. 1. Students (GCGL group) who receive instruction with a numer ical approach will have higher achievement in routine (skill oriente d) limit problems than students (GC group) who receive instruction with a traditio nal approach (primarily an algebraic approach). PAGE 134 120 2. The students in the GCGL group will have higher achievemen t in conceptual oriented limit problems than the students in the G C group. 3. The students in the GCGL group will have higher achievemen t in related applications of limits than the students in the GC group. Three separate ANCOVA tests on the skill, concept, a nd application portions of the limit topic with the pretest as covariate were con ducted and the results are briefly given below: There was no statistically significant difference foun d on the skill portion of the limit topic (unit 1 exam) due to instruction or to instructor There was a statistically significant difference foun d on the concept portion of the limit topic (unit 1 exam) due to instruction and to instructor There was a statistically significant difference foun d on the application portion of the limit topic (unit 1 exam) due to instruction but not due to instructor The interaction effects between instructor and instruction were not statistically significant on the skill, concept, and application portio ns of the limit topic (unit 1 exam). Even though there was no statistically significant di fference found between the treatment and control groups on the skill portion of the limit topic, the study supported that students (treatment group) who solved the skill orient ed limit problems with the numerical method were better able to solve the concept and application problems in the PAGE 135 121 limit topic than the students (control group) who solved t he skill oriented limit problems algebraically. That is, the numerical approach in the limit topic helps students to do better in the concepts and applications of limit problems. There fore, this finding supports the second and third research hypotheses but did not support the fi rst research hypothesis. Hypotheses Results of the Second Research Question As mentioned earlier, immediately after the first uni t 1 exam on limits the second treatment phase began with the study of the deriva tive topic. Again, the students in the treatment group solved the derivative problems (sk ill, concept, and application types) with the use of GCs and a numerical approach and t he students in the control group solved the derivative problems in a traditional manner (pr imarily algebraic approach). Again, both groups used GCs but the treatment groups were provided with the researcherdeveloped instructional materials (the groups use d the unit lesson 4 on derivatives). At the end of the second treatment phase, all students took a test on the derivative topic (unit 2 exam) and the test problems were gr aded again separately as skill, concept, and application portions. The following three research hypotheses (Numbers 4, 5, and 6) were used to answer the second research question on the derivative topi c. 4. The students in the GCGL group will have higher achievemen t in routine (skill oriented) derivative problems than the students in the GC group. 5. The students in the GCGL group will have higher achievemen t in conceptual oriented derivative problems than the students in t he GC group. PAGE 136 122 6. The students in the GCGL group will have higher achievemen t in related applications of derivative problems than the students in the GC group. Three separate ANCOVA tests on the skill, concept, a nd application portions of the derivative topic with the pretest as covariate wer e conducted and the results are briefly given below: There was a statistically significant difference foun d on the skill portion of the derivative topic (unit 2 exam) due to instruction but not due to instructor There was a statistically significant difference foun d on the concept portion of the derivative topic (unit 2 exam) due to instruction and to instructor There was a statistically significant difference foun d on the application portion of the derivative topic (unit 2 exam) due to instruction but not due to instructor The interaction effects between instructor and instruct ion were not statistically significant on the skill, concept, and application portio ns of the derivative topic (unit 2 exam). The ANCOVA tests supported the fact that the students ( treatment group) who used the numerical method were better able to solve not only the skill oriented derivative problems but also the concept and application problems on derivatives than the students (control group) who solved the derivative problems algebraicall y. Therefore, this finding supports the fourth, fifth, and sixth research hypotheses. PAGE 137 123 One might wonder why the treatment group students did better than the control group students on the skill portion of the derivative topic but not on the skill portion of the limit topic. One explanation was that treatment gr oup students first solved the skill oriented limit problems with the numerical method and th en solved the skill oriented derivative problems; these derivative problems were presen ted as limit problems. So the experience the treatment students gained in the early pa rt of the study (limit topic) may have helped them to see a direct relation to the second pa rt (derivative topic); therefore, they may have done better than their control group count erparts. Also, at the beginning of the treatment period, students were slow in entering the expressions and settingup the correct table in their GCs in order to find the require d limits. Students in the control group solved the skill portion of the limit and derivative problems algebraically but the types of algebraic work they had to do in both sections w ere different. That is, the algebraic experience the control students gained from the first topic (limit) may not have helped in the second topic (derivative). As noted earlier, students in all groups were able to scor e higher on the skill portion than the concept and application portions in the limit and derivative topics. It has been mentioned by many in the literature that students ar e able to solve routine (skill) limit and derivative problems whether they understand the concept of limit and derivative or not (FerriniMundy & Graham, 1991; Tall et al., 2004). The finding of this study also confirmed those claims. PAGE 138 124 Limitations of the Study Several limitations of the research method of this st udy must be noted when interpreting conclusions drawn from the results of this s tudy. Group Selection: Full random assignment of students to the study groups was n ot possible. All available four daytime sections of Applied C alculus during spring 2005 at the college were chosen to participate in this study. T wo other sections of this course offered in the evening during spring 2005 at the same college were not chosen because of the student differences in the population. Two of the four daytime sections were offered on TR at 8 a.m. and 12:30 p.m.; the other two were offered on MWF at 8 a.m. and 11:00 a.m. Due to students work schedules and their schedules of other classes, morning sections (8 a.m.) have smaller enrollments than late r time sections (11 a.m. and 12:30 p.m.). As a result of a flip of a coin, TR at 12:30 (taught by Instructor A) was assigned as a treatment group; therefore, MWF at 11:00 a.m. (taught by I nstructor B) was assigned as the other treatment group. That is, the two larger (a tot al of 58 students) and later time sections (11 a.m. and 12:30 p.m.) ended up serving the treatment groups and the two smaller (a total of 32 students) and earlier time section s (8:00 a.m.) ended up serving the control groups. Sample Size: A total of 93 students (17 students were in Control A; 18 st udents were in Control B; 30 students were in Treatment A; and 28 students were in Treatment B) enrolled for the course and took the pretest administ rated on the first day of class. The first week is also the dropadd week for the college. Stude nts who were misplaced or did not meet the prerequisite needed to drop the class. When the treatment phase began on the second day of class, 6 students had dropped out (Control A class lost one student, PAGE 139 125 Control B lost 2 students, Treatment A class lost 2 st udents and Treatment B class lost 1 student). Further, students attendance was monitored; if a student missed 50% or more of the classes during the treatment period, his or her t est score was not included for the statistical analysis. A total of 87 students took the unit 1 e xam; because of absences, only 82 students had test scores considered for the unit 1 exam; 16 students in Control A, 15 students in Control B, 25 students in Treatment A, and 26 st udents in Treatment B. Therefore the ANOVA test on pretest and three ANCOVA tests on unit 1 exam were conducted with n = 82. After the second treatment phase, a total of 79 students took the unit 2 exam and 3 students scores were not included because they missed 50% or more of the classes; therefore, only 76 students had test scor es considered for the unit 2 exam: 14 students in Control A, 14 students in Control B, 23 students in Treatment A, and 25 students in Treatment B. Therefore, the three ANCOVA tests on unit 2 exam were conducted with n = 76. Facilitators: Two instructors A and B participated and both were white an d male. As mentioned earlier, both have some similarities but they certainly have differences such as teaching style, the belief about using technology ( GC), and the amount of experience they have in using GC technology. The participa ting two instructors in this study were volunteers. Applying an instructional method by ot hers under volunteer conditions might be subject to bias in terms of motivati on, belief, and responsibility. However, the participating instructors showed positive at titudes and worked with the researcher to promote the goals of this study. Also, it is a policy in the department that the names of the instructors are published in the semeste r schedules so students know who is teaching what section. One other factor was that one of the instructors teaches the PAGE 140 126 prerequisite course (MAC 1105) for the Applied Calculus cours e and the other one never teaches MAC 1105. If students took and passed MAC 1105 with a parti cular instructor then they tend to stay with the same instructor for the next mathematics course if they have a choice. It was not clear whether students who ha d the instructor previously would do better than the students who did not. Participants: As mentioned earlier, the students knew who their instr uctor would be. But they were not told whether they were in the tr eatment or the control group. There was no statistical difference between the groups on th e pretest scores. From the students initial survey, it was noted that students were differe nt in terms of their mathematical experiences and background and the familiarity with their calculator ownership. About 92% of the students took MAC 1105 as the prerequisite course wit h about 27% of them completing the course in the previous semester, 37% in th e last year, 14% 2 years ago, and the other 14% more than 2 years ago. About 3% are permi tted into this class by their placement scores and about 5% had Precalculus (MAC 1140) or higher. About 86% of students had used a GC in their previous math course and about 99% used the TI 83 model (the model that was used in the study). Further, it was not clear how many students had repeated this course. Measurement Error: The dependent variables were the achievement scores of the skill, concept, and application portions of the unit 1 and 2 e xams that were measured using paper and pencil instruments. Although the scores from these measures were moderately reliable, they were limited by the testing con ditions (e.g., the time and the day of the testing). PAGE 141 127 Instrument Items: Although the content validity of the instruments item s was measured, the items selected for skill, concept, and appl ication portions for the limit and derivative topics were subjective. Further, there are dif ferent types of problems that may be chosen for the same category of skill, concept, an d application. Therefore, just because a student answered a type of problem correctly in a categ ory does not necessarily mean the student can answer another type of problem in the sa me category. It is also possible that solving an application problem in a topic is often as sociated with the concept of that topic. That is, a question or a part of a question in the application portion could be a question for the concept portion. Also, the number of que stions for each portion was limited because of the available testing time. Selection of the Course: The study was carried out in an Applied Calculus course (MAC 2233) and the topics under investigation were limits an d derivatives at a community college. The population of a community college reflects the surrounding community, and therefore, the results of this study mig ht not be generalized to all students taking Applied Calculus. Further, the limit and deri vative topics are taught in other calculus courses (Calculus I, MAC 2311) too. The differ ence is that if students take MAC 2233, then that is the only calculus course they have to take; if students take MAC 2311 then most probably students have to take additional calc ulus courses (Calculus II and Calculus III). So the populations of MAC 2233 and MAC 2311 ar e different; therefore, the results of this study may not be genera lized to any calculus course. Students Preparation before the Exams: Even though students in the treatment groups and control groups received different treatments, th e interaction between students and the type of help students received from the available math lab and private tutoring PAGE 142 128 cannot be controlled. Students were required to fill out t wo student questionnaires (Appendix H) right after each unit exam to learn about h ow they prepared for these exams. The majority of them (about 80%) studied by themsel ves and the other students (about 15%) studied either with students from the same clas s or the other classes. There are a few students (about 5%) who received help from privat e tutors before they took the exam. These situations might have influenced some students performance. ANCOVA Assumptions: The assumptions of ANCOVA tests were discussed earlier but any violations of these assumptions might b e a threat to the finding of this study. Implications for Practice It was repeatedly mentioned in the literature that st udents are able to do better on skill oriented problems than the concept and application problems on the limit and derivative topics whether they understand the topic or not (FerriniMundy & Graham, 1991; Tall et al., 2004). In this study also, all groups except o ne group (Treatment B) on the limit topic and one group (Control A) on the derivat ive topic scored better on the skill portion than the concept and application portions of the limit and derivative topics. However, the results are encouraging in that both trea tment groups did better than their control groups on the concept and application portions of t he limit and derivative topics and the skill portion of the derivative topic when the pr oposed numerical approach with the use of GCs was presented on these topics. As reported earlier, about 86% of the students had used a GC in their previous mathematics course but it was not clear at what level the calculator was used. Because of PAGE 143 129 its capabilities, the GC can be used as a regular fourfu nction calculator to do just basic operations. This was suspected because when students were asked on the initial survey about their comfort level of using a GC, about 25% felt v ery comfortable, another 25% felt moderately comfortable, and 50% felt not comfortable about using their calculator even though they had previously used it for at least one s emester. It seemed like some students used a GC just as a regular fourfunction calculato r, not as a GC. Therefore, when a GC is required for a course instead of a regular calculator, special features of that GC must be utilized by both instructors and students. Otherw ise, a part of the available time in the next course has to be spent on learning how t o use a GC. Also, when students are taught certain topics or certai n types of problems by a specific method (like the numerical approach), the approac h must be consistent throughout their mathematics learning whenever possible. F or example, finding the rate of change of functions is a type of application Applied Calculus students need to solve throughout the semester at different times as they lea rn different functions. If students have to solve a rate of change problem algebraically, th ey need to apply different types of algebraic techniques, manipulations, and/or formulas to solv e the problem based on what type of function is given in that problem. If students lea rned how to solve a rate of change problem on a function by a numerical approach with the h elp of a GC, students will be ready to solve a rate of change problem on any type of f unction because the procedures and the amount of work needed to do the problem are the sa me. Instructors beliefs, knowledge, and willingness are impo rtant factors to implement new teaching methods successfully in classroo ms. Cross (1990) noted that classroom research should be planned and used with teache r involvement to try better PAGE 144 130 teaching methods. This would be successful if instructors ar e provided the needed training for the technology that is intended to be used for a course. They also need to be provided with detailed instructional materials that focus di fferent ways of teaching various mathematics concepts. Also, instructors should be aware not only of what the GC is capable of doing but also the things a GC cant do, its limitations, and misleading behaviors along with other pitfalls. Students beliefs, knowledge, and willingness are equally important factors to implement new teaching methods successfully in their le arning. Schoenfeld (1985) recognized that a students belief system is an importan t factor in his/her ability to learn concepts. Also, there is a perception in students minds t hat if their class is under a treatment, they are being used; they do not necessari ly consider the treatment as a way of trying to improve their understanding. Students att itudes about the new method are not known nor any changes in attitudes towards mathema tics learning. For many students (about 87%) the Applied Calculus course is the la st mathematics course they must pass to graduate; therefore, they do not necessarily think that this new approach will help them much in the long run. Conclusions and Recommendations for Future Research As stated earlier, the purpose of this study was to identif y a method with the available graphing technology that helps improve students conceptual understanding in the limit and derivative topics. Otherwise, these topic s are hard for many students (FerriniMundy & Graham, 1991; Tall et al., 2004). Several studies and projects reported positive results about using appropriate pieces of technolog y along with the instruction to PAGE 145 131 improve students conceptual understanding in those areas (C rocker, 1990; Heid, 1988, 1997; Gordon & HughesHallet, 1991). The experience with the use of GC technology in mathema tics courses and other studies showing positive results in this area motivated th e researcher to try a numerical approach with the GC technology in limit and derivative topics in Applied Calculus at a community college. The researcher proposed a numerical method with the use of GC that can be used to solve limit and derivative topics. This metho d with various examples was given as a set of supplemental handouts to all treatment students (unit lesson 3 and 4). But the researcher realized that an important initial a nd basic step needed is knowing how to operate the piece of technology comfortably. The resea rcher constantly observed students lack of confidence and familiarity, errors, and m isconceptions when using a GC in this course. For example, students had trouble noticing the difference in syntax, even in simple cases like 3/2 ^8 and )3/2(^8 or the difference between 3 / + x x and ).3 /( + x x Understanding these concerns and in order to help s tudents to use a GC correctly and confidently, the researcher prepared two unit l essons just to help students learn to use the GC effectively. These two unit lessons (unit le sson 1 and 2) were given as a set of supplemental handouts to all students. Then the oth er two unit lessons (unit lesson 3 on limits and unit lesson 4 on derivatives) were used in the treatment groups. The use of these unit lessons and numerical methods in the tre atment groups in this study yielded some positive results in learning limit and derivat ive topics in calculus. With todays technology sophistication, a friendly usable and affordable graphing technology like the TI 83 has a lot to offer studen ts and instructors to promote students PAGE 146 132 conceptual understanding in difficult topics such as the li mit and derivative. Only a fraction of its capability is being used by instructors and s tudents, if it is used at all. It is the mathematics communitys responsibility to promote ne w ways with available technology that would help students understanding in mathem atical topics and to conduct research studies on those new methods to validat e its usefulness. This study mainly focused on one type of application (ra te of change) with certain types of functions (Algebraic Functions) in the deriva tive topic with the proposed numerical method along with the use of a GC. Further re search studies are needed to inquire about this numerical approach on the same type of application (rate of change) but with other types of functions (e.g., Transcendental Functions). Also, it would be worthwhile for studies to investigate the effects of this method on other types of applications in derivative topics (finding the maximum and m inimum, etc). Also, research studies are needed to inquire into the e ffect of this approach on the integration topic (this topic also develops from the idea of limit concept) that calculus students study immediately after they study the derivative topic. This study was a quantitative study. The conclusions of this study were made based on the data collected from the test scores. A por tion of a qualitative study (for example, student interviews) would have given a better pic ture of students understanding in the topics studied. 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Proceedings of the conference on the teaching of mathematics, Village of Pythagorion, Samos, Greece. Waits, B. K., & Leitzel, J. (1976). Handheld calculators in the freshman mathematics classroom. American Mathematical Monthly, 83(9), 731 733. Watkins, A. (1992). Remedial statistics?: The implication s for colleges of the changing secondary school curriculum. Mathematical Association of America, Notes #26, Washington, D.C: Mathematical Association of America. White, R. M. (1987). Calculus of reality. Mathematical Association of America, Notes #9, Washington, DC: Mathematical Association of Ame rica. PAGE 155 141 Appendices PAGE 156 142 Appendix A: Student Initial Survey 1. What is your major? If undecided, please state so. 2. Did you take the course MAC 1105 College Algebra? Ye s: _____ No: ______ If not, what course did you take as a prerequisite for the applied calculus course? 3. When did you take this course? Last semester: _________ Last year: __________ Other (specify): _________ 4. Do you plan to take any other mathematics course after this course? If yes, please indicate which course. 5. Do you plan to transfer to a university? Yes: _____ No: _____ Maybe: ______ 6. Before this course, have you ever used a graphing calculat or in a mathematics course? Yes: _____ No: _____ If the answer is Yes, go to question #7; if the answer i s No, skip question #7 and go to question #8. 7. a) For which mathematics courses have you used a graphi ng calculator? _____________________________________________________ b) Specify the type of the graphing calculator: ___________________________________ c) Was it required for the course? Yes: _____ No: _____ PAGE 157 143 Appendix A: (Continued) d) How often did you use it in this class? A. Daily B. 2 to 3 times a week C. 2 to 3 times a month D. once a month E. Never e) How useful was it in your mathematics course? A. Very useful B. Useful C. Moderately useful D. Not useful at all 8. What type of graphing calculator will you use in this Appl ied Calculus course? _______________________________________________________ 9. What are the reason(s) for choosing that type of gra phing calculator? Circle all that apply. A. Cost B. Recommended to you C. Given to you by somebody D. Capabilities of the calculator E. Other reason (specify): 10. How comfortable, in terms of proficiency, are you us ing your graphing calculator? A. Very comfortable B. Comfortable C. Moderately Comfortable D. Not comfortable 11. How helpful do you expect the graphing calculator to be to your understanding and learning of calculus in this course? A. Very helpful B. Somewhat helpful C. Not helpful D. Not sure 12. What features (if any) do you like the best about your gr aphing calculator? ___________________________________________________ 13. Is there anything you do not like about your graphing calculat or? ___________________________________________________ PAGE 158 144 Appendix B: Pretest MAC 2233 Pretest Name: Spring 2005 Show all work. Section: MWF or TR 1. Given 6 ) ( 2 x x x g + = find the following: a) (4) g b) 3) ( g 2. Given ()310, fxx =find the following: a) () fa b) () fah + 3. Given + < + = 4 ,1 4 4 8 3 x ) ( 2 x x x x x f find a) )2( f b) )4( f 4. Find the slope of the line given by the following e quation: .8 4 3 = y x PAGE 159 145 Appendix B: (Continued) 5. Solve the following equation: 20 3 2 2 = x x 6. Find the domain of the following function: 3 2 ) ( = x x f 7. Sketch a line that has a positive slope. 8. Find the slope and the y intercept of .8 3 5 = y x 9. Factor the following polynomial: t t t 15 2 2 3 + PAGE 160 146 Appendix C: (Continued) 10. Find the domain of = ) ( x f x x x + 2 2 11. Find the slope of the line that passes through the points (4,3) and (6,8). 12. Consider the following graph for a function ) ( x f and the following : 13. A population of town is given by x x x P 15 28 340 52 ) ( 2/3 + + = where ) ( x P denotes the population x months from now. a) Find )0( P and interpret your answer b) Find )9( P and interpret your answer a) )2( f b) the x value such that = ) ( x f 0 PAGE 161 147 Appendix C: Unit Exams MAC 2233 Unit 1 Exam: Limits Name: Spring 2005 Show all work. Section: MWF or TR Find the limit of each of the following functions, if it exists, by any method. If the limit doesnt exist, then state so and explain why the limit doesnt exist. If the answer seems to be or state so. Show your work! 1. = ) ( x f ; 4 4 7 2 2 x x x ) ( lim 4 x f x = 2. = ) ( x f ; 25 5 x x ) ( lim 25 x f x = 3. = ) ( x f ; 2 2 x x a) ) ( lim 2 x f x = b) ) ( lim 2 x f x+ = c) ) ( lim 2 x f x = PAGE 162 148 Appendix C: (Continued) 4. = ) ( x f ; 3 7 2 + x x ) ( lim 3 x f x+ = 5. < + = 0 ,1 0 ,1 2 ) ( 2 x x x x x f a) ) ( lim 0 x f x = b) ) ( lim 0 x f x+ = c) ) ( lim 0 x f x = 6. = ) ( x f ; 3 7 4 3 10 2 2 x x x + ) ( lim x f x  = 7. Suppose that for a function ), ( x f it is given that .3 ) ( lim 5 = x f x Explain what this means. PAGE 163 149 Appendix C: (Continued) 8. Consider the graph of ) ( x f and find the following. If the answer seems to be to state so. 9. Consider the graph of ) ( x g and find the following. If the answer seems to be to state so. 10. Consider the graph of ) ( x f and find the following. If the answer seems to be to state so. a) )2 ( f b) ) ( lim 2 x f x c) ) ( lim 2 x f x+ d) ) ( lim 2 x f x a) )2( g b) ) ( lim 2 x g x c) ) ( lim 2 x g x+ d) ) ( lim 2 x g x a) )2( f b) ) ( lim 2 x f x c) ) ( lim 2 x f x+ d) ) ( lim 2 x f x PAGE 164 150 Appendix C: (Continued) 11. Consider the following table. The function values of ) ( x f are computed for different x values. Is it reasonable to estimate the limit, ?) ( lim 4 x f x If yes, give the answer for the limit and if not possible, tell why tha t is not possible. = ) ( lim 4 x f x x 3.9 3.99 3.999 4 4.001 4.01 4.1 ) ( x f 46.81 47.88 47.988 Error 48.012 48.12 49.21 12. The cost ), ( x C in thousands of dollars, of removing % xof a citys pollutants discharged into a river is given by 100 93 ) ( x x x C = a. Find ) 25( C and interpret your answer. b. Can you find ?) 100 ( C Why or why not? c. Find ) ( lim 100 x C x and interpret your answer. 13. The local game commission decided to stock a lake with trout. To do this, 200 trout were introduced into the lake. The population of th e trout can be approximated by 02 0 1 ) 7 10( 20 ) ( t t t P + + = ,0 t where t is time in months since the lake was stocked. Find ) ( lim x P x and interpret your answer. PAGE 165 151 Appendix C: (Continued) MAC 2233 Unit 2 Exam: Derivatives Name: Spring 2005 Show all work. Section: MWF or TR Use the definition of the derivative for all derivative probl ems. Show work. 1. 25 10 2 ) ( 2 + = x x x f find ).3(' f 2. Given 3 ) ( x x f = find the slope of ) ( x f at the point ). 2 3 ,2( PAGE 166 152 Appendix C: (Continued) 3. Let 4 ) ( 2 x x x f = a) Find the average rate of change of ) ( x f with respect to x in the interval from 5 = x to 5.5 = x and 5 = x to .1.5 = x b) Find the (instantaneous) rate of change o f ) ( x f at .5 = x 4. Consider the graph of .) ( x f The tangent line to the graph of ) ( x f at (1, 2) is drawn and the equation of the tangent line is given as 2 x y = What is the value of )1 (' f ? Explain your answer. PAGE 167 153 Appendix C: (Continued) Consider the points A, B, C, D, E, and F on the graph of the function ) ( x f and answer the questions (5) (8). 5. At what point(s) the derivative of ) ( x f is positive? 6. At what point(s) the derivative of ) ( x f is negative? 7. At what point(s) the derivative of ) ( x f is zero? 8. At what point the value of the derivative of ) ( x f is the greatest? PAGE 168 154 Appendix C: (Continued) 9. Under a set of controlled laboratory conditions, t he size of the population, ), ( t P of a certain bacteria culture at time t (in minutes) is described by the function .1 2 3 ) ( 2 + + = t t t P a. Find the average rate of the population of the bacteria between 5 = t minutes and 10 = t minutes; 5 = t minutes and 7 = t minutes b. Find the (instantaneous) rate of change of the population of the bacteria at 5 = t minutes. 10. The quarterly profit (in thousands of dollars ) of Cunningham Realty is given by 30 7 ) ( 2 + + = x x x P ) 50 0( £ £ x where x (in thousands of dollars) is the amount of money C unningham spends on advertising per quarter. a. Find ) 10( P and interpret the result. b. Find ) 10(' P and interpret the result. PAGE 169 155 Appendix D: Class Observation Protocol The researcher visited all experimental groups and control groups once a week. The purpose of the visit was to make sure the groups are fo llowing the lessons as planned in the study. All groups used GCs throughout the semester. The experimental groups used TI 83 GCs and the researcherdeveloped instructional guided l essons to learn limit and derivative problems with numerical approach. The control groups solved the problems in those topics in a traditional approach (algebraic approac h). Instructors A and B and all students in the treatment groups were given a copy of the entire four units as handouts. The researcher visited and observed all sections to insur e that the progress in the classes are in agreement with the course syllabus and the instructional methods are being utilized as planned in the main study. The researcher made class observations in the following manner: For the Treatment Groups: 1. Observed that students in this group studied the review secti ons with the use of TI 83 GCs and the researcherdeveloped unit lessons #1 and #2. The unit lessons have examples that helped students to use the ir GCs effectively and that reinforced the function concepts. 2. Observed that the students in this group learned the limit problems with the table feature of a TI 83 GC and the researcherdevel oped unit lesson #3. The unit lesson has numerous examples to make students understand the nature of the limit problems, how to solve those pr oblems numerically with the table feature of a TI 83 GC, and interpret the a nswers to those problems. After that students learned to solve related app lications of PAGE 170 156 Appendix D: (Continued) limits. The researcher observed that the instruction w as carried out as planned. 3. Observed that students in this group learned the derivative problems as limit problems and solved them with the table feature of a TI 83 GC (the same way they have solved the limit problems previously) and the researcherdeveloped unit lesson #4. The unit lesson has num erous examples to make students understand the meaning of the deri vative problems, how to solve those problems numerically with th e table feature of the GC, and interpret the answers to those problems. After that students learned to solve related applications of derivatives with the help of the unit lesson and a TI 83 GC. The researcher observed that th e instruction was carried out as planned. For the Control Groups: 1. Observed that students in this group studied the review secti ons (Lessons 1 and 2) with the use of TI 83 GCs only. It was noted that t he instruction was done as planned without any treatment effect influence s. 2. Observed that students in this group studied the limit problem s the way that was done in the textbook with the use of TI 83 GCs only. It was noted that the instruction was done as planned without any treat ment effect influences. No unexpected events noted by the researcher. 3. Observed that students in this group studied the derivative pro blems the way that was done in the textbook with the use of TI 83 only. It was noted PAGE 171 157 Appendix D: (Continued) that the instruction was done as planned without any treat ment effect influences. The instructor observed in all sections that the instr uction for the particular session was done as planned in terms of time management amount of material covered for the session, and the type of instruction deli vered to the students as planned in the main study. The researcher took notes of ea ch session visited. PAGE 172 158 Appendix E: Course Syllabus Tentative Academic Calendar Spring 2005 MAC 2233 APPLIED C ALCULUS Week Sections Topics Covered Suggested Homework Assignments 1 R 1/6 F 1/7 2.1 2.2 2.3 Functions and Their Graphs The Algebra of Functions Functions & Mathematical Models 2.1p.58:333(eoo),3747(o),51,53,55,62 2.2p.75: 141(eoo),47,51,55,57,6770 2.3p.87:117(eoo),19,36,39,45,55, 65 2 M 1/10 F 1/14 2.4 2.5 2.6 Limits OneSided Limits The Derivative 2.4p.111:177(eoo),59,79 2.5p.127:119(o),21107(eoo) 2.6p.148:157(eoo) 3 M 1/17 F 1/21 M 1/17 Holiday Review Exam 1 4 M 1/24 F 1/28 3.1 3.2 Basic Rules of Differentiation The Product and Quotient Rules 3.1p.168: 169(eoo) 3.2p.182: 161(eoo) 5 M 1/31 F 2/4 3.3 3.4 3.5 The Chain Rule Marginal Functions in Economics HigherOrder Derivatives 3.3p.195:185(eoo) 3.4p.212:1,5,9,13,23,27,31,33 3.5p.219:145(eoo) 6 M 2/7 F 2/11 3.6 3.7 Implicit Differentiation & Related Rates Differentials 3.6p.231:351(eoo),53,55,57,59 3.7p.241:145(eoo) 7 M 2/14 F 2/18 F 2/18 Holiday Review Exam 2 4.1 Applications of the 1 st Derivative 4.1p.262:9,1957(eoo),63,69,73,77,81 8 M 2/21 F 2/25 4.2 4.3 Applications of the 2 nd Derivative Curve Sketching 4.2p.280:141(eoo),49,61,63,65,69,75,81 4.3p.296:165(eoo) 9 M 2/28 F 3/4 4.4, 4.5 Optimization Problems 4.4p.311:169(eoo) 4.5p.325: 125(eoo) 10 M 3/7 F 3/11 Exam 3 5.1, 5.2 Exponential and logarithmic Functions 5.1p.337:341(eoo),4346; 5.2p.346:155 11 M 3/14 F 3/18 5.3 5.4 5.5 Compound Interest Differentiation of Exponential Fns Differentiation of Logarithmic Fns 5.3p.359:137(eoo) 5.4p.368:173(eoo) 5.5p.379:565(eoo),64 12 M 3/21 F 3/25 5.6 Review Exam 4 Exponential Fns as Math Models 5.6p.388:3,5,13,15,17,23,27 13 M 3/28 F 4/1 Spring Break College Closed 14 M 4/4 F 4/8 6.1 6.2 Antiderivatives and Integration Integration by Substitution 6.1p.407:185(eoo) 6.2p.419:165(eoo) 15 M 4/11 F 4/15 6.3 6.4 6.5 Area and the Definite Integral Fundamental Theorem of Calculus Evaluating Definite Integrals 6.3p.430:3,7,15 6.4p.439:149(eoo) 6.5p.449:149(eoo) 16 M 4/18 F 4/22 6.6 6.7 Area between Two Curves Applications of Definite Integral 6.6p.461:151(eoo) 6.7p.478:117(eoo) 17 M 4/25 F 4/29 Review Exam 5 18 M 5/2 F 5/6 FINAL EXAMS PAGE 173 159 Appendix E: (Continued) MAC 2233 APPLIED CALCULUS Spring 2005 Instructor: Math Office: Instructors Office: Math Office phone: Instructors Phone: Math Lab: Instructors email: Math Lab Hrs Office hours: Adjuncts available prior to or afte r class or specific times available by appointment PREREQUISITES: MAC 1105 with a grade of C or better or equivalent. Students already with credit for MAC 2311 cannot subsequently get credit for this course.. S tudent Enrollment in any mathematics course is contingent upon approval of the mathe matics department. This means that students who have been misplaced may have their s chedule changed. COURSE DESCRIPTION: Topics in this course include limits, differentiation and i ntegration of algebraic, exponential and logarithmic functions, integration technique s and related applications in the management, business, and social sciences. This course is not designed for science majors. Course performance standards are available at www.mccfl.edu/academ/math/math.htm and in the Math Labs. TEXT: Calculus for Managerial, Life and Social Sciences by Tan, 6 th edition MATERIALS: A graphing calculator is required. It is allowed during e xams. The TI83 model is strongly recommended. Calculators with symbolic manipulation capabilities (e.g. T I 89, TI92) will not be allowed for use during exams. ADDITIONAL MATERIALS: Student Solutions manual is available in the bookstore. EXAMINATIONS: There will be 5 exams and a required comprehensive final exa mination. NO MAKEUP EXAMS WILL BE GIVEN. GRADING: Your grade in the course is determined by the percentage of points earned during the semester. A grade of 60% or better must be earned on the final e xam in order to pass the course. POINTS SCALE 5 Exams 500 90 100%=A Quizzes/Participation/Projects/Homework* 100200 80 89%=B Final Exam (cumulative) 200 70 79%=C 60 69%=D 0 59%=F Instructor will choose composition of these points. GORDON RULE: This course meets the Florida State Board of Education Rul e Number 6A10.30. For the purpose of this rule, a grade of C or better shall be co nsidered successful completion. ATTENDANCE: All late arrivals, early departures and absences must be discussed and cleared with the instructor. More than 3 hours of unexcused absences or exces sive tardiness may result in your withdrawal from the course. WITHDRAWAL: March 16, 2005 is the last day to withdraw with a W. If you remain in the course after that date, you must receive a grade. Withdrawal af ter the midpoint may be granted only by the Associate Dean of Instruction and shall be based on major extenuating circumstances. As of Fall 1997, a student is allowed a maxi mum of three attempts per course. On the third attempt the student cannot be withdrawn and must receive a grade. More information regarding this policy is available in the current Manatee Community College Catalog or the class schedule. PAGE 174 160 Appendix F: ResearcherDeveloped Instructional Lessons: Unit Lessons of TI 83 Graphing Calcu lator Usage in an Applied Calculus Course Unit Lesson 1: A Guide that Helps to Use a TI 83 Graphing Calculator A. Entering Arithmetic and Algebraic Expressions in a G raphing Calculator B. Simplifying Arithmetic Expressions by Using a Graphing Cal culator C. Graphing Functions (Equations) with a Graphing Calcul ator D. Student Activities for Lesson 1 Unit Lesson 2: Function Values and the Table Feature of a TI 83 Graphing Calculator A. Finding Function Values by Using a Graphing Calculator B. Student Activities for Lesson 2 Unit Lesson 3: Limits of Functions A. Finding the Limit of a Function as x Approaches a Number B. Finding the Limit of a Function as x Approaches a Number from the Right and Left C. Finding the Limit of a Function as x Approaches and D. Student Activities for Lesson 3 Unit Lesson 4: Derivatives of Functions A. Finding the Derivative of a Function at a Given Value B. Application: Finding the Rate of Change of a F unction C. Student Activities for Lesson 4 PAGE 175 161 Appendix F: (Continued) Lesson 1: A Guide that Helps to Use a TI 83 Graphing Calculator Instructor Notes: Introduction A graphing calculator can be a good learning tool in studying m athematics concepts, if it is used correctly and appropriately. Othe rwise, its use could seriously harm students mathematics learning. If students are going to use a graphing calculator as a tool, they need to learn to communicate with their cal culators. For example, if students work with 3 1 + = x y and enter it in a calculator as 3 /1 + = x y the calculator understands this as ,3 1 + = x y and therefore, gives a different result than expec ted. Suppose students have an exponential function, such as 2 x e y = If it is entered in a calculator as ) 2(^ x e y = the calculator understands the syntax as the corr ect exponential function; however, if it is entered as x e y )2(^ = the calculator understands the syntax as a linear function, 2 x e y = Likewise, when students perform an operation such as 5 00014 .0 in a graphing calculator, the answer appears as 7E 4. The calculator gives the answer in scientific notation and students need to know that means 0.0007, not just 7 or something else. Therefore, it is important that stu dents understand what they are typing in their calculator and also what they are reading fro m their calculator. PAGE 176 162 Appendix F: (Continued) A. Entering Arithmetic and Algebraic Expressions in a G raphing Calculator When using graphing calculators in a mathematics course, the first thing students need to know is how an arithmetic or algebraic expressi on needs to be entered in the calculator. If students fail to do this properly, the cal culator gives a false result or no result at all. This section addresses how mathematical expressions need to be entered in a graphing calculator. a) The minus key and the negative key in a graphing calculator A minus sign, indicating the operation, and a negative si gn, indicating the opposite of a number or an expression, are represented by distinct symbols on a graphing calculator. The symbol may not be a problem when it is used in a paperpencil mode, but it is often a problem when it is entered in a graphing calc ulator. Students need to know the locations of these two keys. The minus key, is located in the right column, third key from the bottom; the negative key, ()" is lo cated in the bottom of the second right column. For example, 10 minus 7 gives 3 as an ans wer and 10 negative 7 gives an error message. b) The role of a parenthesis key The parenthesis key is an important key for students to unde rstand. In some cases a parenthesis key needs to be placed, in some cases a pa renthesis key should not be placed, and in other cases placing a parenthesis key may n ot make any difference. Consider the following examples: 1. Negative of five squared means 2 5so the answer is 25. In this case, if it is enter ed as 2 5) ( the answer is 25. Therefore a parenthesis key sho uld not be placed. PAGE 177 163 Appendix F: (Continued) 2. a) Negative five squared means 2 5) ( so the answer is 25. In this case, if it is entered as 2 5the answer is 25. Therefore a parenthesis key sh ould be placed. b) The expression 3 5 1 + equals 8 1 or 0.125. But, if it is entered as 3 5/1 + the answer is 3.2 because the calculator understand s the entered expression as .3 5 1 + However, if it is entered as ),3 5 /(1 + the calculator gives the correct answer. Therefore a parenthesis key should be placed. 3. The equation 1 2 + = x x y is entered as either )1 /( 2 + = x x y or )1 /() 2( + = x x y in a graphing calculator and produces the same graph. Therefore a parenthesis key for the numerator does not make any differenc e in this case. The following table gives additional examples to e mphasize the importance of using the minus key, negative key, and parenthesis key. Furthermore, the table identifies a list of common mistakes that are experienced by m any students when they use graphing calculators. These mistakes are not necessarily com mon mistakes for students when they work with paperpencil. The purpose of this lesson is to address the mistakes students make and provide students with a guide that helps t hem to use their calculators properly. PAGE 178 164 Appendix F: (Continued) Desired Expression Correct way to input Incorrect way to in put 5 + x 5 + x (negative of x ) + 5 5 + x (minus x ) + 5 2 5 2 5 (5) 2 (7) 2 (7) 2 7 2 2 5 2 3 7 2 5 4 3 2 + + )2 5 2 3 /()7 2 5 4 3( 2 + + 2 5 2 3/7 2 5 4 3 2 + + 2 + x )2 ( + x 2 ) ( + x 3 1 + x )3 /(1 + x 3 /1 + x 3 2x )3/2(^ x 3/2 ^ x (x )4 (4 )^ x 4 ^ x 3ln x )3^ ln( x 3 )^ (ln x 5 5 4 32+ x x x )5 /()5 4 3(2+ x x x 5 /5 4 32+ x x x x e2 ) 2(^ x e x e )2(^ 3 ln + x 3 ) ln( + x )3 ln( + x PAGE 179 165 Appendix F: (Continued) B. Simplifying Arithmetic Expressions by Using a Graphing Calculator In this section, students learn how to use a graphing calcul ator to simplify a given arithmetic expression. Students can simplify any arithmet ic expression by correctly entering the expression in a graphing calculator. Example 1: Simplify the expression by using a graphing calculator. 2 2 3 2 4 3 5 37 4 6 4 3 + + Enter the expression as ) 3 2 4 3 5 /()37 4 6 4 3( 2 2 + + and press PAGE 180 166 Appendix F: (Continued) Because the result is a decimal representation of the exact answer, students can change the result to a fraction which is the exact answer to the expression. Press $and\; select\; option\; 1:\; Frac.\; Ans\; Frac\; appears\; on\; the\; screen.\; Pressto\; get\; the\; exact\; answer,\; 19\; 72\; on\; the\; screen.$ PAGE 181 167 Appendix F: (Continued) Example 3: Simplify the expression by using a graphing calculator. 6 1 5 07.0 0025 .0 Enter the expression as 6.1/5 07.0 0025 .0 and press PAGE 182 168 Appendix F: (Continued) Because the answer is a decimal form, students may chan ge it to a fraction form. Press $and\; select\; option\; 1:\; Frac.\; Ans\; Frac\; appears\; on\; the\; screen.\; Pressthen\; the\; exact\; answer,\; 9\; 19\; appears\; on\; the\; screen.\; b)\; Enter\; the\; expression\; as\; )\; 3\; 16\; /()3\; 2\; 11\; )\; 15\; 3\; (23\; (2\; +\; +\; and\; press.\; The\; answer\; 2.023593056\; appears\; on\; the\; screen.$ PAGE 183 169 Appendix F: (Continued) Because the answer is a decimal form, students may chan ge it to a fraction form. Press $and\; select\; option\; 1:\; Frac.\; Ans\; Frac\; appears\; on\; the\; screen.\; PressThis\; time\; the\; same\; decimal\; number\; appears\; on\; the\; screen.$ PAGE 184 170 Appendix F: (Continued) Students could be asked why this happens, providing an opportunity for the instructor to explain that the answer to the expression is an irrational number and therefore a fraction form cannot be obtained for the a nswer. PAGE 185 171 Appendix F: (Continued) C. Graphing Functions (Equations) with a Graphing Calculat or In this section, students learn to use a graphing calculato r to graph a given function or equation. Example 1: Graph 4 = x y with a graphing calculator. Press PAGE 186 172 Appendix F: (Continued) Press PAGE 187 173 Appendix F: (Continued) Example 3: Graph 13 )2 ( ) ( 2 + = x x g with a graphing calculator. Press PAGE 188 174 Appendix F: (Continued) (2, 13). Students need to change the viewing window. Press PAGE 189 175 Appendix F: (Continued) Press PAGE 190 176 Appendix F: (Continued) Press PAGE 191 177 Appendix F: (Continued) D. Student Activities For Lesson 1 Worksheet #1A: Entering Expressions in a Graphing Calculator For each of the following expressions, write the way yo u would enter it in your graphing calculator. Show the written expressions in the gi ven table. 1. 7 4 5 x 2. 6 5 3 x x 3. 10 2 3 3 2 + x x 4. 13 2 x 5. x x 3 7 5 + 6. 4 5 3 x Recording Table Expression Write the expression the way you would enter it in the calculator 7 4 5 x 6 5 3 x x 10 2 3 3 2 + x x 13 2 x x x 3 7 5 + 4 5 3 x PAGE 192 178 Appendix F: (Continued) Worksheet #1B: Simplifying Arithmetic Expressions with a G raphing Calculator 1. Enter the following arithmetic expressions in your calcul ator and then simplify. If the answer is a decimal, convert the decimal to a fr action if possible; if the answer is in scientific notation, convert it to a st andard number. Record the work on the given sheet. a) 2 4 b) 2 4 7 2 4 3 2 2 + c) 3 28 d) 24 3 6 5 4 3 + e) 3 2125f) 2)3( 2 5 3 6 4 2 + g) 2 5 2 3 h) 2 00045 .0 i) 5 2 0034 .0 002 .0 Recording Table Expression Copy the expression the way that is entered in the calculator Write the answer from the calculator 2 4 2 4 7 2 4 3 2 2 + 3 28 24 3 6 5 4 3 + 3 2125 2 )3( 2 5 3 6 4 2 + 2 5 2 3 2 00045 .0 5 2 0034 .0 002 .0 PAGE 193 179 Appendix F: (Continued) Worksheet #1C: Graphing Functions (Equations) with a Graphi ng Calculator Graph the following equations with your graphing calcula tor. Record the work on the given sheet. Adjust the window when necessary. Co py the graphs from the calculator on the other side. 1) 3 = x y 2) 2 4 ) ( + = x x f 3) 2 3 ) ( + = x x x g 4) 3 ) ( + = x x f 5) 3 7 4 3 ) ( = x x g 6) 3 7 4 3 = x y 7) 14 3 4 = y x 8) 50 ) ( = x f 9) + < = 2,3 2,4 3 ) ( 2 x x x x x f Recording Table Problem Write the way the function that it is entered in the calculator Copy the graph from the calculator 3 = x y 2 4 ) ( + = x x f 2 3 ) ( + = x x x g PAGE 194 180 Appendix F: (Continued) Problem Write the way the function that it is entered in the calculator Copy the graph from the calculator 3 ) ( + = x x f 3 7 4 3 ) ( = x x g 3 7 4 3 = x y 14 3 4 = y x 50 ) ( = x f + < = 2,3 2,4 3 ) ( 2 x x x x x f PAGE 195 181 Appendix F: (Continued) Lesson 2: Function Values and the Table Feature of a TI 83 Graphing Calc ulator Instructor Notes: The table feature in a graphing calculator is a useful feat ure to determine the values of a function for various x values. Consider the following examples. Example 1: Given 4 3 ) ( 2 + = x x x f find the following: a) )4( f b) )1( f c) )0( f d) )3( f e) ) 10( f f) ) 100 ( f Instead of substituting the given x value in each case, students can enter the function as 1 y in a graphing calculator and then use the table fe ature to find the required function values. Before entering the function in the calcula tor, students need to set up the table as follows. This is required for the first time only. Setting up the Table: Press the <2nd> and PAGE 196 182 Appendix F: (Continued) Set up the table by using the arrow keys to bring the cursor down to highlight the option Ask at the Indpnt prompt; press PAGE 197 183 Appendix F: (Continued) Press the <2nd> and PAGE 198 184 Appendix F: (Continued) Example 2: Given < + = 1, 14 3 1,2 5 ) ( 2 x x x x x x f find the following: a) )3( f b) )1( f c) )0( f d) )4( f e) )6.8( f f) ) 53( f Press PAGE 199 185 Appendix F: (Continued) d) )4( f = 2 e) )6.8( f = 11.8 f) ) 53( f = 145 Example 3: The number of commercial FM radio stations in the U nited States can be modeled by x x f ) 036 .1(6. 2149 ) ( = for 26 1 £ £ x where x represents the number of years since 1969 and ) ( x f represents the number of commercial FM radio stati ons (Source: U.S. Census Bureau, www.census.gov). A. Find the following and interpret your answers. a) )1( f b) )3( f c) ) 10( f d) ) 25( f B. (i) Is it possible to compute )4( f and ) 28( f ? (ii) Is it meaningful to compute )4( f and ) 28( f ? Why or why not? Answer: A. Compute the function values as before. Press PAGE 200 186 Appendix F: (Continued) Then press the <2nd> and PAGE 201 187 Appendix F: (Continued) Student Activities For Lesson 2 Work Sheet #2: Evaluating Functions with a Graphing Calculator Determine the function values for the given x values in each case. Enter the given function in your calculator as 1 y and then use PAGE 202 188 Appendix F: (Continued) 3. Given 12 ) ( = x f find the following: a) )5( f b) )0( f c) ) 4 13 ( f Answer: Write the way the function is entered in the calcul ator: Write the answers from the table: a) )5( f = b) )0( f = c) ) 4 13 ( f = 4. Given < + = 3, 10 2 1 3,5 3 ) ( 2 x x x x x x f find the following: a) )4 ( f b) )3( f c) )4.2( f Answer Write the way the function is entered in the calcul ator: Write the answers from the table: a) )4 ( f = b) )3( f = c) )4.2( f = PAGE 203 189 Appendix F: (Continued) 5. The number of milligrams of cholesterol consumed each day per person in the United States can be modeled by 02. 445 04.4 11.0 ) ( 2 + = x x x C 23 1 £ £ x where x represents the number of years since 1974 and ) ( x C represents the number of milligrams of cholesterol consumed each day per person. (Source: U.S. Department of Agriculture). A. Compute the following function values with you r calculator (using the PAGE 204 190 Appendix F: (Continued) Lesson 3: Limits of Functions Instructor Notes: Introduction The notion of a limit of a function is fundamental to the study of calculus. The investigation of the following two problems led to the crea tion of calculus: 1. Finding the slope of a tangent line to the graph of a given f unction at a given point. 2. Finding the area of a region bounded by the graph of a funct ion. The first problem led to the creation of differential c alculus and the second led to the creation of integral calculus. The idea of a limit is used in the process of solving these two problems. Therefore understanding the limit concept is important in calculus. Generally, limit problems can be approached graphically, an alytically, or numerically. The graphical method depends on the graph of the function; if the graph of the function is not given or not easy to get, this metho d may not be a good choice. The analytical method depends on how well the expression in the function can be simplified; if the expression in the function is hard to simplify or not possible to simplify, this method does not help. The numerical method can be used fo r any limit problem as long as the function rule is given. These numerical computat ions are easy to perform with the table feature of a graphing calculator. This method also h elps to understand the meaning of the limit of a function. PAGE 205 191 Appendix F: (Continued) In this lesson students learn to solve limit problems nu merically by using the table feature of a graphing calculator and develop understanding of th e limit concept. This lesson is divided into three sections. The first sectio n deals with the problems of finding the limit of a function as x approaches a number; the second section discusses the problems of finding the limit of a function as x approaches a number from the right or left; and the last section examines problems of finding the limit of a function as x approaches and A. Finding the limit of a function as x approaches a number The question is to find the limit of a function ) ( x f as x approaches a number a or in notation, find ). ( lim x f a x The meaning of this question is: Does ) ( x f approach a number as x approaches a from the right and the left? Students enter the fu nction in the calculator as 1 y and use the table to find the corresponding functi on values as x approaches a from the left and the right. If the function appro aches a particular value, say l, as x approaches a from the left and the right, then the function has a limit of l as x approaches a this is written as l x f ) ( as a x or in notation, ) ( lim l x f a x = Otherwise the limit of ) ( x f does not exist as x approaches a Consider the following examples. Example 1: Given 9 12 4 ) ( 2 = x x x f find ), ( lim 3 x f x if it exists. Press PAGE 206 192 Appendix F: (Continued) Press the <2nd> and PAGE 207 193 Appendix F: (Continued) The values in the table suggest that the limit of the fu nction as 3 x from both sides is 0.667, that is, 667 .0 9 12 4 lim 2 3 = x x x Also, note that the function is undefined (error message in the calculator) when .3 = x Example 2: Find 25 5 lim 25 x x x if it exists. Press PAGE 208 194 Appendix F: (Continued) The above table can be rewritten as follows. Note the function values as 25 x from both the left and right. x 24.9 24.99 24.999 25 25.001 25.01 25.1 ) ( x f 0.1001 0.10001 0.1 Error 0.1 0.09999 0.0999 The values in the table suggest that 1.0 25 5 lim 25 = x x x Also, note that the function is undefined (error message in the calculator) when 25 = x Example 3: Find 2 2 lim 2 + + x x x if it exists. Press PAGE 209 195 Appendix F: (Continued) The above table can be rewritten as follows. Note the function values as 2 x from both the left and right. x 2.1 2.01 2.001 2 1.999 1.99 1.9 ) ( x f 1 1 1 Error 1 1 1 From the values in the table, it appears that 1) ( x f as 2x and 1 ) ( x f as 2+ x Because ) ( x f approaches two different values as x approaches 2 from the left and right, ) ( x f does not have a limit as 2 x and therefore, 2 2 lim 2 + + x x x does not exist. Also, note that the function is undefined (e rror message in the calculator) when 2. = x Example 4: Find 9 5 2 lim 2 3 x x x if it exists. Press PAGE 210 196 Appendix F: (Continued) Press the <2nd> and PAGE 211 197 Appendix F: (Continued) The values in the table suggest that ) ( x f as 3 x and ) ( x f as + 3 x so 9 5 2 lim 2 3 x x x does not exist. Also, note that the function is und efined (error message in the calculator) when .3 = x Example 5: Given < + = 3 ,1 3 ,8 2 ) ( x x x x x f find ), ( lim 3 x f x if it exists. Press PAGE 212 198 Appendix F: (Continued) The above table can be rewritten as follows. x 2.9 2.99 2.999 3 3.001 3.01 3.1 ) ( x f 2.2 2.02 2.002 2 2.001 2.01 2.1 Thus, 2 ) ( x f as 3 x from both sides and therefore, .2 ) ( lim 3 = x f x PAGE 213 199 Appendix F: (Continued) B. Finding the limit of a function as x approaches a number from the right and left Suppose the question is to find the limit of a function ) ( x f as x approaches a number a from the right, denoted ), ( lim x f a x+ or to find the limit of a function ) ( x f as x approaches a number a from the left, denoted ). ( lim x f a x Consider the following examples. Example 1: Find 4 5 lim 4 + x x x if it exists. Press PAGE 214 200 Appendix F: (Continued) The above table is the same as the following. Note the function values as + 4 x x 4 4.0001 4.001 4.01 4.1 4.5 ) ( x f Error 200005 20005 2005 205 45 From the table, it appears that ) ( x f as 4 + x Example 2: Find x x x 2 1 5lim if it exists. Press PAGE 215 201 Appendix F: (Continued) The above table can be rewritten as follows: x 0.5 0.9 0.99 0.999 0.9999 1 ) ( x f 20 55.556 505.05 5005 50005 Error Thus, ) ( x f as 1 x and therefore, x x x 2 1 5lim = Example 3: Find x xx + 0lim, if it exists. Press PAGE 216 202 Appendix F: (Continued) The above table can be rewritten as follows. x 0 0.001 0.01 0.1 0.5 1 ) ( x f Error 1 1 1 1 1 From the values in the table, it appears that 1 ) ( x f as + 0 x so x xx + 0lim= 1. PAGE 217 203 Appendix F: (Continued) C. Finding the limit of a function as x approaches and Examples in this section are about examining the l imit of a function as x approaches and Consider the following examples. Example 1: Find 9 2 3 4 lim 2 2 + x x x if it exists. Press PAGE 218 204 Appendix F: (Continued) x 10 100 200 500 1000 5000 ) ( x f 1.5916 1.5009 1.5002 1.5 1.5 1.5 Thus, it appears that .5.1 9 2 3 4 lim 2 2 = + x x x Example 2: Find 4 5 12 lim 2 +  x x if it exists. Press PAGE 219 205 Appendix F: (Continued) x 10 100 500 1000 2000 5000 ) ( x f 0.024 0.0002 0.0000 0.0000 0.0000 0.0000 Thus, it appears that .0 4 5 12 lim 2 = +  x x Example 3: Find 3 2 5 4 lim 2 + x x x x if it exists. Press PAGE 220 206 Appendix F: (Continued) x 10 100 200 500 1000 5000 ) ( x f 3.8235 48.756 98.753 248.75 498.75 2498.8 Thus, it appears that 3 2 5 4 lim 2 + x x x x = Applications for limits Example 1: The concentration (in mg/cubic cm) of a certain dru g in a patients bloodstream t hours after injection is given by 1 2.0 ) ( 2 + = t t t C a) Compute ),5.0( C ),1( C ),2( C and )3( C and interpret the answers. b) Evaluate ) ( lim t C t and interpret the result. Press PAGE 221 207 Appendix F: (Continued) a) That is, 08.0 )5.0( = C ,1.0 )1( = C 08.0 )2( = C and 06.0 )3( = C This means that the concentrations (in mg/cubic cm) of the dru g in the patients bloodstream are 0.08, 0.1, 0.08 and 0.06 after 0.5 hour, 1 hour 2 hours, and 3 hours of injection, respectively. b) To answer ) ( lim t C t enter x values of 10, 20, 50, 100 and 150 and determine th e limit of the function as x x 10 20 50 100 150 200 ) ( x C 0.0198 0.00998 0.004 0.002 .00133 0.001 Thus, it appears that ) ( lim t C t = 0. This means that the concentration of the drug in the patients bloodstream is vanishing as time increas es. PAGE 222 208 Appendix F: (Continued) Example 2: The average cost/disc in dollars incurred by Herald Records in pressing x videodiscs is given by the average cost function 2500 2.2 ) ( x x C + = a) Compute ), 50( C ), 100 ( C ), 500 ( C and ) 1000 ( C and interpret the answers. b) Evaluate ) ( lim x C x and interpret the result. Press PAGE 223 209 Appendix F: (Continued) a) That is, 20. 52 ) 50( = C 20. 27 ) 100 ( = C 20.7 ) 500 ( = C and 70.4 ) 1000 ( = C This means that the average costs are $52.20/ disc, $27.20/disc, $7.20/disc, and $4.70/disc when 50, 100, 500, and 1000 videod iscs are produced, respectively. b) To answer ) ( lim x C x enter x values of 10000, 40000, 80000, 100000, 200000, 400000, 1000000 and determine the limit of th e function as x x 10000 40000 80000 100000 200000 400000 1000000 ) ( x C 2.45 2.2625 2.2313 2.225 2.2125 2.2063 2.2025 Thus, it appears that ) ( lim x C t = 2.20. This means that the average cost of producing x videodiscs will approach $2.20/disc in the long ru n. That is, the price does not go below $2.20 per disc. PAGE 224 210 Appendix F: (Continued) D. Student Activities For Lesson 3 Work Sheet #3: Finding the limits of Functions with a Graph ing Calculator Find the limit of each of the following functions, if it exists. In each case, complete the table by using the table feature of your graphing calculator. Then determine the limit, if it exists. If the limit doesnt exist, then state so and e xplain why the limit doesnt exist. If the answer seems to be or state so. 1. Given 3 5 2 ) ( 2 2 x x x x x f + = find ), ( lim 0 x f x if it exists. x 0.1 0.01 0.001 0 0.001 0.01 0.1 ) ( x f Write the answer from the table: ) ( lim 0 x f x = 2. Given 6 5 2 ) ( 2 + + + = x x x x f find ), ( lim 2 x f x if it exists. x 2.1 2.01 2.001 2 1.999 1.99 1.9 ) ( x f Write the answer from the table: ) ( lim 2 x f x = PAGE 225 211 Appendix F: (Continued) 3. Given 9 3 ) ( = x x x f find ), ( lim 9 x f x if it exists. x 8.9 8.99 8.999 9 9.001 9.01 9.1 ) ( x f Write the answer from the table: ) ( lim 9 x f x = 4. Given 2 4 ) (2+ = x x x f find ), ( lim4x fx if it exists. x 3.9 3.99 3.999 4 4.001 4.01 4.1 ) ( x f Write the answer from the table: ) ( lim4x fx = PAGE 226 212 Appendix F: (Continued) 5. Given x x x f = ) (, find ), ( lim 0 x f x if it exists. x 0.1 0.01 0.001 0 0.001 0.01 0.1 ) ( x f Write the answer from the table: ) ( lim 0 x f x = 6. Given 5 5 ) ( = x x x g find ), ( lim 5 x g x if it exists. x 4.9 4.99 4.999 5 5.001 5.01 5.1 ) ( x g Write the answer from the table: ) ( lim 5 x g x = PAGE 227 213 Appendix F: (Continued) 7. Given 3 3 ) ( + + = x x x g find ), ( lim 3 x g x+ if it exists. x 3 2.999 2.99 2.9 2.8 2.7 2.6 ) ( x g Write the answer from the table: ) ( lim 3 x g x+ = 8. Given 10 10 ) ( = x x x f find ), ( lim10x fx if it exists. x 9.6 9.7 9.8 9.9 9.99 9.999 10 ) ( x f Write the answer from the table: ) ( lim10x fx = PAGE 228 214 Appendix F: (Continued) 9. Given x x x x f = 2 5 3 ) ( find ), ( lim 0 x f x if it exists. x 0.4 0.3 0.2 0.1 0.01 0.001 0 ) ( x f Write the answer from the table: ) ( lim 0 x f x = 10. Given 1 2 5 6 ) ( + = x x x f find ), ( lim 2 1 x f x+ if it exists. x 0.5 0.501 0.51 0.52 0.53 0.54 0.55 ) ( x f Write the answer from the table: ) ( lim 2 1 x f x+ = PAGE 229 215 Appendix F: (Continued) 11. Given 5 2 8 ) ( 2 = x x h find ), ( lim x h x if it exists. x 10 100 1000 2000 5000 8000 10000 ) ( x h Write the answer from the table: ) ( lim x h x = 12. Given 2 2 2 3 5 3 ) ( x x x g + = find ), ( lim x g x  if it exists. x 10000 8000 50000 2000 1000 100 10 ) ( x g Write the answer from the table: ) ( lim x g x  = PAGE 230 216 Appendix F: (Continued) 13. Given 5 2 3 4 ) ( 2 = x x x x f find ), ( lim x f x  if it exists. x 10000 8000 50000 2000 1000 100 10 ) ( x f Write the answer from the table: ) ( lim x f x  = 14. Given 5 2 8 ) ( 2 = x x g find ), ( lim x g x if it exists. x 10 100 1000 2000 5000 8000 10000 ) ( x g Write the answer from the table: ) ( lim x g x = PAGE 231 217 Appendix F: (Continued) 15. Given < + = 2 ,1 4 2 10 2 ) ( 2 x x x x x f find ), ( lim 2 x f x if it exists. x 1.9 1.99 1.999 2 2.001 2.01 2.1 ) ( x f Write the answer from the table : ) ( lim 2 x f x = 16. The total worldwide boxoffice receipts for a longrunning blockbuster movie are approximated by the function 4 120 ) ( 2 2 + = x x x T where ) ( x T is measured in millions of dollars and x is the number of months since the movies release. a) What are the total boxoffice receipts after the f irst month? The second month? The fifth month? b) What will the movie gross in the long run? (Hint: Find ).) ( lim x T x PAGE 232 218 Appendix F: (Continued) 17. A major corporation is building a 4325acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Co ve. As a result of this development, the planners have estimated that Glen Cov es population (in thousands) t years from now will be given by 40 5 200 125 25 ) ( 2 2 + + + + = t t t t t P a) What is the current population of Glen Cove? A fter 2 years? After 10 years? b) What will be the population in the long run? PAGE 233 219 Appendix F: (Continued) Lesson 4: Derivatives of Functions Instructor Notes: Calculus is a study of change. It is a study of how quic kly one quantity changes as the other quantity changes. The slope of the quantity (fun ction) is used to measure this change. The slope of a function at a particular point i s a measure of how quickly the output ) ( y changes as the input ) ( x changes at that point. This is called the instantan eous rate of change or simply the rate of change of the function at that point. The mathematical term for the rate of change of a function at a poin t is the derivative of the function at that point. The process of finding a derivative is calle d differentiation. A. Finding the Derivative of a Function at a Given Value Example: Finding the derivative of ) ( x f at c x = Or simply finding ). (' c f Procedure: )) ( ( c f c is the given point on the graph of ) ( x f where the derivative needs to be found. Choose another point that is close to the given poi nt on the graph of ). ( x f Say, the second point is )), ( ( h c f h c + + where h is the difference of the x values of these two points. The slope of the secant line that goes thro ugh these two points is c h c c f h c f + + ) ( ) ( ) ( = h c f h c f ) ( ) ( + and this quantity gives the average rate of change of ) ( x f on the interval ]. [ h c c + Then the slope of the tangent line to the graph of ) ( x f at c x = will be the same as the slope of the secant line w hen h approaches 0. PAGE 234 220 Appendix F: (Continued) That is, the slope of the tangent line to the graph of ) ( x f at c x = is ) ( ) ( lim 0 h c f h c f h + This is also the slope of the function ) ( x f at c x = This quantity is known as the derivative of ) ( x f at c x = which gives the instantaneous rate of change at c x = Therefore, finding the derivative of a function is a li mit problem so students can find this limit in the same way they have found limits in their earlier study of this topic. Consider the following examples. Example 1: Given ) ( x f = 3 3 2 + x x find the rate of change of ) ( x f at .4 = x That is, the question is to find ).4(' f Therefore, from the definition of the derivative, the question is to find )4(' f = )4( ) 4( lim 0 h f h f h + To use a calculator to find this limit, use x for h and let ,3 4 )4(3 )4( 2 1 + = = f y ,3 ) 4( ) 4(3 ) 4( 2 2 + + + = + = x x x f y and 1 2 3 x y y y = Therefore, lim )4('3 0y fx = Press PAGE 235 221 Appendix F: (Continued) Because the table is only needed for 3 y deactivate the equations for 1 y and 2 y by moving the cursor to the = sign of both equations and pres sing PAGE 236 222 Appendix F: (Continued) The values in the table suggest that 23 lim3 0=yx That is, 23 )4(' = f The benefit of doing the problem this way is that the tabl e gives the average rate of change of that function on various intervals around 4 = x Students are able to determine the average rate of change of ) ( x f on various intervals around 4 = x. Also, students notice that the average rate of change of ) ( x f on an interval approaches the instantaneous rate of change of ) ( x f as the intervals approach zero. The meaning of the values of 3 y in the table is as follows: The Average Rate of Change of ) ( x f Between Two Points The Values of 3 y from Table The Average rate of change of ) ( x f between 4 = x and 1.4 = x 3. 23 4 1 4 )4( )1.4( = f f The Average rate of change of ) ( x f between 4 = x and 01.4 = x 03. 23 4 01 4 )4( ) 01.4( = f f The Average rate of change of ) ( x f between 4 = x and 001 .4 = x 003 23 4 001 4 )4( ) 001 .4( = f f The Average rate of change of ) ( x f between 9.3 = x and 4 = x 7. 22 4 9 3 )4( )9.3( = f f The Average rate of change of ) ( x f between 99.3 = x and 4 = x 97. 22 4 99 3 )4( ) 99.3( = f f The Average rate of change of ) ( x f between 999 .3 = x and 4 = x 997 22 4 999 3 )4( ) 999 .3( = f f PAGE 237 223 Appendix F: (Continued) Example 2: Given ) ( x f = 1 2 2 + x x find the rate of change of ) ( x f at .3 = x From the definition of the derivative, the question is to find )3(' f = )3( ) 3( lim 0 h f h f h + To use a calculator to find this limit, use x for h and let ),1 3 /()3(2 )3( 2 1 + = = f y ),1 ) 3 /(( ) 3(2 ) 3( 2 2 + + + = + = x x x f y and /) ( 1 2 3 x y y y = Therefore, lim )3('3 0y fx = Press PAGE 238 224 Appendix F: (Continued) The above table can be rewritten as follows. Note the function values as 0 x from both the left and right. x 0.1 0.01 0.001 0 0.001 0.01 0.1 3 y .1637 .1604 .16 Error .16 .1596 .1565 The values in the table suggest that 16.0lim3 0=yx That is, 0.16 )3(' = f Using the table, students can determine the average rate of change of ) ( x f on various intervals around 3 = x Also, students note that the average rate of chan ge of ) ( x f on an interval approaches the instantaneous rate of change of ) ( x f as the intervals approach zero. The corresponding average rate of ch ange of ) ( x f can be summarized as follows: PAGE 239 225 Appendix F: (Continued) The Average Rate of Change of ) ( x f Between Two Points The Values of 3 y from Table The Average rate of change of ) ( x f between 3 = x and 1.3 = x 0.1565 3 1 3 )3( )1.3( = f f The Average rate of change of ) ( x f between 3 = x and 01.3 = x 0.1596 3 01 3 )3( ) 01.3( = f f The Average rate of change of ) ( x f between 3 = x and 001 .3 = x 0.16 3 001 3 )3( ) 001 .3( = f f The Average rate of change of ) ( x f between 9.2 = x and 3 = x 0.1637 3 9 2 )3( )9.2( = f f The Average rate of change of ) ( x f between 99.2 = x and 3 = x 0.1604 3 99 2 )3( ) 99.2( = f f The Average rate of change of ) ( x f between 999 .2 = x and 3 = x 0.16 3 999 2 )3( ) 999 .2( = f f PAGE 240 226 Appendix F: (Continued) B. Application: Finding the Rate of Change of a Function Example: According to the U.S. Labor Department, the number of temporary workers (in millions) is estimated to be 505 .1 255 .0 025 .0 ) ( 2 + + = t t t N )5 0( £ £ t where t is measured in years, with t = 0 corresponding to 1991. (Source: Labor Department) a) Find ),0( N ),1( N and )3( N and interpret the answers. b) Find the average rate of change of ) ( t N (i) between 1 = t and 5 = t (ii) between 1 = t and 4 = t (iii) between 1 = t and 3 = t (iv) between 1 = t and 2 = t and interpret the answers. c) Find the instantaneous rate of change of )( t N at 1 = t and interpret the answer. Answers: a) To find the function values at different t values, Press PAGE 241 227 Appendix F: (Continued) Then press the <2nd> and PAGE 242 228 Appendix F: (Continued) 505 .1 )1( 255 .0 )1( 025 .0 2 1 + + = y 505 .1 ) 1( 255 .0 ) 1( 025 .0 2 2 + + + + = x x y x y y y /) ( 1 2 3 = Make sure that only equation 3 y is activated. Press the <2nd> and PAGE 243 229 Appendix F: (Continued) Interpretation of the results: The average rate of change of the number of temporary w orkers during the years 1992 and 1996 is 0.405 million per year. The average rate of change of the number of temporary workers during the years 1992 and 1995 is 0.38 million per year. The average rate of change of the number of temporary workers during t he years 1992 and 1994 is 0.355 million per year. The average rate of change of the number of temporary workers during the years 1992 and 1993 is 0.33 million per year. c) Finding the instantaneous rate of change of )( t N at 1 = t is the same as finding ).1(' N From the definition of the derivative, = )1(' N )1( ) 1( lim 0 h N h N h + Thus, = )1(' N lim3 0yx Using the same table, note the function values of 3 y by letting x approach 0. Thus, the table suggests that 3 0lim yx = 0.305 so that 305 .0 )1(' = N The meaning of 305 .0 )1(' = N is that the rate of change of the number of tempor ary workers in 1992 is 0.305 million/year. PAGE 244 230 Appendix F: (Continued) C. Student Activities For Lesson 4 Work Sheet #4: Finding the Derivatives of Functions with a Grap hing Calculator Find the derivative of each of the following functions at the given value. Use the definition of the derivative of a function at x ) ( ) ( lim ) (' 0 h x f h x f x f h + = 1. 25 10 2 ) ( 2 3 + = x x x x f find a) )2 (' f b) ).3(' f 2. 11 8 ) ( = x x f find a) )3 (' f b) ).4(' f 3. 2 1 3 ) ( + = x x x f find a) )0(' f b) ).4.2(' f 4. ,4 ) ( + = x x f find a) )2 (' f b) ). 10(' f 5. )4 3( ) ( 2 = x x f find a) )5.2 (' f b) ). 3 2 (' f 6. Find the slope of the tangent line to the grap h of the given function at the given point. a) ,7 5 2 ) ( 2 + = x x x f (2, 5) b) 2 ) ( + = x x x f (1, 1) c) ,3 ) ( = x x f (4, 1) 7. Let .5 2 3 ) ( 2 + = x x x f a) Find the average rate of change of ) ( x f with respect to x in the interval from 2 = x to ,3 = x from 2 = x to ,5.2 = x and from 2 = x to .1.2 = x b) Find the (instantaneous) rate of change of ) ( x f at .2 = x PAGE 245 231 Appendix F: (Continued) 8. A hotair balloon rises vertically from the ground so that its height after t sec is t t t h 2 1 2 1 ) ( 2 + = feet ). 60 0( £ £ t a) What is the height of the balloon at the end of 40 sec? b) What is the average velocity of the balloon between t = 0 and t = 40? c) What is the velocity of the balloon at the end of 4 0 sec? 9. Lynbrook West, an apartment company, has 100 tw obedroom units. The monthly profit (in dollars) realized from renting x apartments is given by 000 50 1760 10 ) ( 2 + = x x x P a) Find ) 50( P and interpret the result. b) Find ) 50(' P and interpret the result. PAGE 246 232 Appendix G: Pilot Study Report A pilot study was conducted for four and a half weeks in t he fall of 2004 by the researcher to investigate the effects of using TI 83 graphing c alculators (GCs) along with the researcherdeveloped instructional guided unit lessons on limits derivatives and related applications of limits and derivatives in an Applied Calculus course at a community college. The purpose of this pilot study was to evaluate the charac teristics of the use of GCs along with the guided unit lessons and determine critica l situations that might be encountered during the study. It was expected that the pilot study would help to explore whether the instruction with GCs along with the guided u nit lessons could lead to better understanding on limits derivatives and related applications of limits and derivatives among community college applied calculus students. Furthermor e, the pilot study was used to access the validity and reliability of all test instruments, the pretest and the two unit exams. Procedure To conduct this study, two intact sections of an Applied Calculus course (MAC 2233) at a community college were selected. The student s were not randomly assigned. Thus, this was a quasiexperimental study to comp are two instructional methods for solving problems in the following four areas: ro utine (skilloriented) limit problems, related applications of limits routine (skilloriented) derivative problems, and related applications of derivatives PAGE 247 233 Appendix G: (Continued) Design A nonequivalent treatmentcontrol group design was used i n this pilot study. Dependent variables were ability to solve routine (skil l oriented) limit problems, related application problems of limits, routine (skill oriente d) derivative problems, and related application problems of derivatives. Variables were meas ured by researchermade tests. The dependent variables were measured at two different tim es with a unit 1 exam and a unit 2 exam. It was expected that the treatment group would do b etter than the control group on all measures. The main independent variable was method of instruction Participants Two researchertaught intact applied calculus sections f rom a community college were used in the pilot study during the fall of 2004. Because the main study is planned for the spring of 2005, the researcher decided to use these two s ections for his pilot study. One section, the TuesdayThursday (TR) section, met twi ce a week with each class meeting for 80 minutes from 9:30 10:50 a.m.; the other section, the MondayWednesdayFriday (MWF) section, met three times a week with each class meeting for 53 minutes from 9:00 9:53 a.m. Students were not randomly assigne d to these two sections. That is, students who registered for these two sections knew who their instructor would be. Due to students work schedules and their schedule s of other classes, certain sections at certain time periods have larger enrollmen ts than other sections. At the beginning of the semester, 20 students registered for the TR section and 30 students registered for the MWF section. The TR section was c hosen as the treatment group by the outcome of a flip of a coin; thus, the MWF section serv ed as the control group. PAGE 248 234 Appendix G: (Continued) Instruments A pretest and two unit exams were used as instruments fo r this study. The researcher developed all three tests. Pretest A pretest was administered during the first week of the se mester. Because the Basic College Algebra course is a prerequisite for the a pplied calculus course, a set of 20 questions on the content of this course was written. The purpose of the pretest was to determine whether students in the two classes were simil ar in their mathematical ability before they received any new instruction for the applied c alculus course. The pretest is given in Appendix B. Unit Exams Two unit exams were used to measure students achievement o n limits and derivatives The first part of the unit 1 exam was on routine (skil l oriented) limit problems and the second part was on related application pro blems of limits. The first part of the unit 2 exam was on routine (skill oriented) deriv ative problems and the second part was on related application problems of derivatives. Both uni t exams are given in Appendix C. To address the validity of the pretest and two unit exams two experienced mathematics instructors from the same college inspected the tests and found the content of the pretest and two unit exams to be valid measures. PAGE 249 235 Appendix G: (Continued) Treatment Phase For the studied course, Applied Calculus, a textbook and a graphing calculator (GC) are required. The treatment group (GCGL group) recei ved instruction with the use of TI 83 GCs and the researcherdeveloped instructional guide d lessons; the control group (GC group) received instruction with GCs but without th e researcherdeveloped instructional guided lessons on limits and derivative secti ons. There were four unit lessons developed by the researcher to use in the pilot st udy (the entire lessons are given in Appendix F). Although a GC is required for this course prior experience showed that many students come to this course with less experience and lack of familiarity with a GC. As a result, students constantly use this machine incorr ectly and also interpret the calculators outcome incorrectly. The first two unit lessons were developed to meet students needs in order to use GCs correctly and efficien tly. The students in the Applied Calculus course spent the fir st week reviewing some algebra topics that they previously learned, including fun ctions. The GCGL group reviewed these sections along with instructional guided les sons 1 and 2 and the GC group reviewed the sections in a traditional manner. After t he first week, both groups of students studied the limit topics for about one and a half weeks. The GCGL student s used the table feature of a TI 83 GC along with the instructional guided l esson 3 to solve limit problems, including applications; the GC students le arned the same topic in a traditional way with the textbook. The control group did use GCs for this topic but were not supported by the guided lessons. At the end of the first treatment phase, both groups took the same unit 1 exam. This test had two parts and studen ts were given 53 minutes to PAGE 250 236 Appendix G: (Continued) complete the test. The first part of the test consis ted of a set of routine (skill oriented) limit problems and the second part of the test consiste d of a set of related applications of limits. Each part of the test had a maximum of 50 points. Each part was graded and scored separately. Immediately after the unit 1 exam, both students learne d derivatives of functions with the limit definition. The GCGL group was taught to f ind the derivative of a function at the given x value as a limit problem with the help of the instructi onal guided lesson 4. That is, the students in the GCGL group were introduced to the derivative problems as rate of change problems; thus, they approached the derivat ive problems the same way they previously solved the limit problems. The GC group was introduced to the derivative with the same limit definition but they found the deriv ative algebraically in a traditional way and then substituted the given x value at the end. Both groups learned the related applications of derivatives by whatever method they lear ned to find the derivative for the given functions. At the end of this treatment phase, bot h groups took the same unit 2 exam. This test also had two parts and students were given 53 minutes to complete the test. The first part of the test consisted of a set o f routine derivative problems and the second part of the test consisted of a set of related a pplications of derivatives. Each part of the test had a maximum of 50 points. Each part was grade d and scored separately. The treatment phase ended with the unit 2 exam and it took about four and a half weeks. Exam Scoring Multiplechoice questions were not used on the pretest o r either unit exam. Therefore, students had to work the problems to answer the que stions and students were PAGE 251 237 Appendix G: (Continued) required to show their work. Because most questions had mul tiple steps in order to obtain the final answer, the researcher graded the pretest and th e two unit exams using the following scoring rubric: Table 28 Scoring Rubric Point(s) Guideline 2 Correct work and Answer 1 Partially Correct work and Answer 0 Incorrect Work or No Response To address the validity of the scoring rubric the same instructors who inspected the content validity of the tests reviewed the rubric and found that the content of the rubric was a valid measure. Also, a sample of 10 anchor papers chosen randomly from each of the three tests was used to test for interrat er reliability. The selected papers were graded by the researcher and other faculty from the mathe matics department. The faculty member was different from the participating two instructo rs in the main study but has extensive experience in this course. A total score for ea ch student for each test from each grader was obtained and correlation coefficients were co mputed to obtain interrater reliability coefficients between the graders. The inte rrater reliability coefficients for the pretest, unit 1 exam, and unit 2 exam were .93, .98, and .95, re spectively. According to Gall et al. (1996), the obtained interrater reliability coefficients supported the reliability of the scoring. PAGE 252 238 Appendix G: (Continued) Data analyses and Results Pretest Descriptive statistics for the pretest are reported in Table 29. The mean comparison graph for the pretest is given in Figure 11. Al so, a boxplot graph was obtained (see Figure 12) to see how the data of each group ar e spread out. Although the means of both groups were closer to each other, the contr ol group had more variability than the treatment group. Also, a small effect size of 0.16 was obtained for the pretest. Table 29 Descriptive Statistics for the Pretest of the Pilot Study Group n M SD Skewness Kurtosis Effect Size Treatment Control All Students 20 30 50 31.65 30.67 31.06 5.82 6.10 5.95 1.06 .27 .22 2.32 .41 .55 0.16 Note : Maximum possible score on pretest was 60 points. 20 25 30 35 40 Control Treatment PretestScore Mean Figure 11: Mean Comparison for the Pretest for the Treat ment and Control Groups PAGE 253 239 Appendix G: (Continued) ControlTreatment Group 15.00 20.00 25.00 30.00 35.00 40.00Pretest 31 Figure 12: Boxplot Graph for the Pretest for the Treatme nt and Control Groups Then the means of these two groups were compared using a o neway ANOVA. An alpha level of .05 was used and the results reported in Table 30. Table 30 Oneway ANOVA for the Pretest of the Pilot Study Source Sum of Squares Df Mean Square F p Between Groups Within Groups Total 11.603 1723.217 1734.820 1 48 49 11.603 35.900 .323 .572 The ANOVA table confirmed that the groups were not statist ically different, F (1, 48) = 0.323, p = .572. That is, there was no statistic ally significant difference between the groups in their mathematical ability prior t o this course. PAGE 254 240 Appendix G: (Continued) Unit 1 and Unit 2 Exams Unit 1 and unit 2 exams were used to measure students achieve ments in limits and derivatives, respectively. The first part of each exam (maximum of 50 po ints) focused on students ability to solve skilloriented problems and the second part (maximum of 50 points) focused on students ability to solv e related application problems. Table 31 reports the descriptive statistics for ea ch portion of both unit exams. Table 31 Descriptive Statistics for Unit 1 and Unit 2 Exams of the Pilot Study Unit 1 Exam Unit 2 Exam Control Treatment Control Treatment Skill Portion n 29 20 26 17 Mean 35.93 38.70 33.96 38.11 Standard Deviation 5.17 6.90 4.21 4.14 Kurtosis .11 .31 .50 .47 Skewness .69 .39 .01 .16 Application Portion n 29 20 26 17 Mean 31.17 36.50 30.15 35.06 Standard Deviation 5.66 6.91 4.86 4.98 Kurtosis .15 .59 1.18 1.39 Skewness .19 .72 .09 .17 Note: Maximum possible score on each portion of both unit exa ms was 50 points. PAGE 255 241 Appendix G: (Continued) The mean comparisons of each part of both unit exams for both groups were noted. The graphs of mean comparisons are given in Figure 13 and Figure 14. 20 25 30 35 40Skill Application Unit 1 Exam Score Treatment Control Figure 13: Mean Comparison for Unit 1 Exam for the Treatme nt and Control Groups 20 25 30 35 40 Skill Application Unit 2 ExamScore Treatment Control Figure 14: Mean Comparison for Unit 2 Exam for the Treatme nt and Control Groups As this pilot study compared the achievements in solving skil loriented problems and application problems in limit and derivative topics, th e researcher looked at Pearson PAGE 256 242 Appendix G: (Continued) productmoment correlations between the pretest and each part of the unit 1 and unit 2 exams for the treatment and control groups. The results are reported in Table 32. Table 32 Pearson ProductMoment Correlations for the Pretest a nd Each Portion of the Unit 1 and Unit 2 Exams of the Pilot Study Exams Pretest of the Treatment Group Pretest of the Control Group Pretest Skill Portion of Unit 1 Exam Application Portion of Unit 1 Exam Skill Portion of Unit 2 Exam Application Portion of Unit 2 Exam 1.00 .86* .91* .24 .18 1.00 .01 .08 .14 .24 Note: p < .01 Then a number of ANCOVAs were used to compare the means o f each part of both unit exams with the pretest as a covariate. An al pha level of .05 was used for all tests. First, the mean difference of the skill porti on of the limit topic (unit 1 exam) between the control and treatment groups was examined. Th e results are reported in Table 33. PAGE 257 243 Table 33 ANCOVA for the Skill Portion of Unit 1 Exam of the Pil ot Study Source Sum of Squares Df Mean Square F p Pretest Group Error Total 33.540 99.286 1618.522 69046.000 1 1 46 49 33.540 99.286 35.185 0.953 2.822 .334 .100 Although the mean score of the treatment group ( T M = 38.70) is slightly greater than the mean score of the control group ( C M = 35.93), the ANCOVA table found no significant difference between the groups on the skill portion of t he limit topic, F = 2.822, p = .100. Next, the mean difference of the app lication portion of the limit topic (unit 1 exam) between the control and treatment gro ups was examined. The results are reported in Table 34. Table 34 ANCOVA for the Application Portion of Unit 1 Exam of t he Pilot Study Source Sum of Squares Df Mean Square F p Pretest Group Error Total 29.205 349.943 1775.933 56630.000 1 1 46 49 29.205 349.943 38.607 0.953 9.064 .334 .004 PAGE 258 244 Appendix G: (Continued) The ANCOVA table found a statisticallysignificant diffe rence between the achievement of the groups on the application portion of the limit topic, favoring the treatment group, F = 9.064, p = .004. Next, the mean difference of the skill portion of the derivative topic (unit 2 exam) between the control and treatment groups was examined wit h an ANCOVA. The results are reported in Table 35. Table 35 ANCOVA for the Skill Portion of Unit 2 Exam of the Pil ot Study Source Sum of Squares Df Mean Square F p Pretest Group Error Total 9.093 165.369 707.633 55405.000 1 1 40 43 9.093 165.369 17.691 0.514 9.348 .478 .004 The ANCOVA table found a statisticallysignificant diffe rence between the achievement of the groups on the skill portion of the derivative topic, favoring the treatment group, F = 9.348, p = .004. Finally, the mean difference of the application portio n of the derivative topic (unit 2 exam) between the control and treatment groups was exam ined with an ANCOVA. The results are reported in Table 36. PAGE 259 245 Appendix G: (Continued) Table 36 ANCOVA for the Application Portion of Unit 2 Exam of t he Pilot Study Source Sum of Squares Df Mean Square F p Pretest Group Error Total 9.965 231.817 976.361 45522.000 1 1 40 43 9.965 231.817 24.409 0.408 9.497 .526 .004 The ANCOVA table found a statisticallysignificant diffe rence between the achievement of the groups on the application portion of the derivative topic, favoring the treatment group, F = 9.497, p = .004. Further Research To study the effect of using GCs along with the instructio nal guided lessons on students learning of limits and derivatives, a followup st udy is planned with a similar structure to that conducted in the pilot study. Two instruct ors other than the researcher will participate in the main study to reduce the chance of producing preferential instruction for the treatment group. PAGE 260 246 Appendix H: Student Questionnaire Name (Optional): ____________________ Section (Circle One): MWF 8 8:53 MWF 11 11:53 TR 8 9:20 TR 12:30 1:50 Please answer the following questions to the best of your kn owledge: 1. How did you study for the test? A. I studied by myself. B. I studied with other student(s ) who is (are) in this class. C. I studied with other student(s) who is (are) in a diff erent class. D. I studied with a private tutor E. Other (P lease specify.): ______ 2. Have you visited the math lab to get any help for this class ? Yes: ___ No: ___ 3. If you visited the math lab for help, how many times and how long did you spend in the lab? Total number of times visited: ____ Total number of hours spent (approximately): ______ 4. If you visited the math lab for help, was it helpful? Yes: ____ No: ____ Somewhat helpful: _____ 5. If you received any help from the math lab or a private t utor, choose one of the following that fits the nature of the help you received: a. Math lab person(s)/Tutor helped me with the same methods that were discussed in my class. b. Math lab person(s)/Tutor helped me with the methods tha t were different from the class methods. PAGE 261 247 Appendix I: Course Implementation Log Instructor: __________________ Textbook Section: ____________ Date: ________________ Control Group (MWF/TR) 1. In general, were proposed procedures for this section properly implemented? _______ 2. If a problem occurred, circle the following that bes t categorizes the problem(s): A. Graphing calculator B. Not enough time C. Student apathy D. Student comprehension E. Other (Please specify): __________________________ 3. What was done in attempt to remedy the situation? ________________________________________________________________________________________________________________________________________________ Treatment Group (MWF/TR) 1. In general, were proposed procedures for this section properly implemented? _______ 2. If a problem occurred, circle the following that bes t categorizes the problem(s): A. Graphing calculator B. Not enough time C. Student apathy D. Student comprehension E. Other (Please specify): __________________________ 3. What was done in attempt to remedy the situation? ________________________________________________________________________________________________________________________________________________ PAGE 262 248 Appendix J: Student Demographic Information Please fill out the information on this sheet to the bes t of your ability. All information obtained here and elsewhere in this study will remain s trictly confidential. 1. Your Full Name: __________________________________________ Last First Middle 2. Your Instructors Last Name: _________________________________ 3. Time of Class Meeting: _____________________________________ 4. Your Gender: Male: _________ Female: ________ 5. Your Student ID Number (GOO Number): ____________________ 6. Your Home Address: __________________________________________ Street ___________________________________________ City, State Zip Code 7. Your Telephone Number: _______________________________________ 8. Your Date of Birth: _____________________________________ Month Day Year 9. Your Race (Optional): (Circle one that best descr ibes you.) A. African American B. Asian American C. Caucasian D. Hispanic American E. Native American F. Non US Resident G. Other Group (Please indicate): _________________________ PAGE 263 249 Appendix K: Consent to Participate in a Research Study Date ________________ Dear Student Enrolled in MAC 2233: Applied Calculus, I am very interested in students conceptual understanding in limit and derivative topics in Applied Calculus course, and I am conducting some resea rch on this topic. The purpose of the research will be to examine the effect o f using graphing calculators with numerical approach in limits and derivatives. To do this, I am asking you to do a pretest and two unit exams. The two unit exams are part of your norm al course requirements. I will collect the scores from these three tests to an alyze the performance. Only your instructor and I will have access to the test scores with names attached, which will be kept in my office in the department of mathematics. Additionally, authorized research personnel, employees of the Department of He alth and Human Services and the USF Institutional Review Board may inspect the records fr om this research project. There are no anticipated risks involved with your participat ion. Your participation is voluntary and will not have any effect (positive or n egative) on your course grade. You will not be paid for your participation in this study. Resu lts from the study may be shared with other teachers at professional meetings or in publ ished resources for teachers, such as journal articles or books However, no actual names or any other information that would in any way personally identify you will ever be used. I f needed, pseudonyms will be used. There are no direct benefits to you for participation in this research project, but the study may increase research knowledge related to students conc eptual understanding in calculus. If you have any questions regarding the research study, plea se contact me at *** or the Office of Research Compliance (8139745638). If you are willing to participate in this study, please sig n below. You may keep a copy of this letter for your records. Sincerely, Arumugam Muhundan PAGE 264 250 Appendix K: (Continued) University of South Florida Consent to Participate in a Research Study I understand that I am being asked to participate in a resear ch study entitled, Effects of Using Graphing Calculators: A Numerical Approach to Limits, Derivatives, and Related Applications in an Applied Calculus Course at a Communit y College By signing this form, I give permission for Arumugam Muhundan to use resul ts from the test scores. I understand that my name will not be given. I understand t hat my participation is voluntary and will not have any effect (positive or n egative) on my course grade. By signing this form I agree that: (a) I have fully read this informed consent form. (b) I have had the opportunity to ask questions about this researc h project. (c) I understand the risks and benefits, and I freely give my consent to participate in this research project. (d) I have been given a signed copy of this informed consent fo rm, which is mine to keep. Print Name: Social Security Number: Home Address: Signature: Date: __________ Investigator Statement I have carefully explained to the subject the nature of th e above protocol. I hereby certify that to the best of my knowledge the subject signing this consent form understands the nature, demands, risks and benefits involved in participating in this study. Arumugam Muhundan Signature of Investigator Printed Name of Investigator Date Institutional Approval of Study and Informed Consent This research study and informed consent form were review ed and approved by the University of South Florida Institutional Review Board fo r the protection of human subjects. This approval is valid until the date indicated b elow. If you have any questions about your rights as a person taking part in a research s tudy, you may contact a member of the Division of Research Compliance at (813) 9745638. PAGE 265 About the Author Arumugam Muhundan received his Bachelor of Science in Ma thematics from Eastern University in Batticaloa, Sri Lanka and his Mast er of Science in Mathematics from Florida Atlantic University in Boca Raton, Florida He entered the Ph.D program at the University of South Florida in 1993. Since 1990, Mr. Muhundan has been a member of the faculty at Manatee Community College in Bradenton, Florida and has made nume rous contributions to his department and college. Mr. Muhundan currently is a member of American Mathematical Association of TwoYear Colleges and the Mathematical Association of America. xml version 1.0 encoding UTF8 standalone no record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam Ka controlfield tag 001 001670326 003 fts 005 20051216093249.0 006 med 007 cr mnuuuuuu 008 051110s2005 flu sbm s000 0 eng d datafield ind1 8 ind2 024 subfield code a E14SFE0001186 035 (OCoLC)62256830 SFE0001186 040 FHM c FHM 049 FHMM 090 LB1601 (Online) 1 100 Muhundan, Arumugam. 0 245 Effects of using graphing calculators with a numerical approach on students' learning of limits and derivatives in an applied calculus course at a community college h [electronic resource] / by Arumugam Muhundan. 260 [Tampa, Fla.] : b University of South Florida, 2005. 502 Thesis (Ph.D.)University of South Florida, 2005. 504 Includes bibliographical references. 500 Includes vita. 516 Text (Electronic thesis) in PDF format. 538 System requirements: World Wide Web browser and PDF reader. Mode of access: World Wide Web. Title from PDF of title page. Document formatted into pages; contains 265 pages. 3 520 ABSTRACT: This study examined the effects of using graphing calculators with a numerical approach designed by the researcher on students learning of limits and derivatives in an Applied Calculus course at a community college. The purposes of this study were to investigate the following: (1) students achievement in solving limit problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved limit problems with a traditional approach (primarily an algebraic approach); and (2) students achievement in solving derivative problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved derivative problems with a traditional approach (primarily an algebraic approach). 590 Adviser: Dr. D. Thompson. 653 Calculus. Graphing calculator. 690 Dissertations, Academic z USF x Secondary Education Doctoral. 773 t USF Electronic Theses and Dissertations. 4 856 u http://digital.lib.usf.edu/?e14.1186 