# The role of context and gender in predicting success in a modified laboratory course

код для вставкиСкачатьTHE ROLE OF CONTEXT AND GENDER IN PREDICTING SUCCESS IN A MODIFIED LABORATORY COURSE BY KERON SUBERO A dissertation submitted to the Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy Major Subject: Physics Minor Subject: Applied Statistics New Mexico State University Las Cruces, New Mexico December 2010 UMI Number: 3448957 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI Dissertation Publishing UMI 3448957 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 "The role of context and gender in predicting success in a modified laboratory course," a dissertation prepared by Keron Subero in partial fulfillment of the requirements for the degree, Doctor of Philosophy, as been approved and accepted by the following: ^**%. HJoijaotQ Linda Lacey Dean of the Graduate School J J^a-r»^ 1 <Cs>- Stephen E. Kanim, Chair Chair of the Examining Committee Date Committee in charge: Dr. Stephen E. Kanim, Chair Dr. Dennis L. Clason Dr. William R. Gibbs Dr. Heinrich Nakotte Dr. Dominic A. Simon Dr. Theodore B. Stanford 11 ACKNOWLEDGEMENTS To my wife Katurah, this is the result of your perseverance and support. To my kids for growing up and supporting each other when I could not be there Especially to Anisa for fathering Zolani and Nya. Dr. Steve Kanim, thank you for going above and beyond always, for putting your care and time into this research, and for shaping my view on everything. Thanks to everyone on my committee for all of your help at various stages in this dissertation. Thanks to my parents for giving me simple mores that guided me through many rough times. iii VITA Nov. 4, 1978. Born in Port of Spain, Trinidad and Tobago. 1997-2001 B.Sc. North Carolina Central University. 2001-2003 Graduate Program , Oregon State University. 2004-2010 Ph.D. New Mexico State University. PROFESSIONAL SOCIETIES American Association of Physics Teachers PUBLICATIONS S. Kanim and K. Subero, "Introductory labs on the vector nature of force and acceleration" Am. J. Phys., 78 (5), 461-466, 2010. PRESENTATIONS K. Subero, "What is proportional reasoning, anyway?" October 19, 2007, APS Four Corners Section Meeting, Flagstaff, AZ. K. Subero, "Promotion of vector use in introductory mechanics laboratories." July 24, 2006, AAPT Summer Meeting, Syracuse, NY. FIELD OF STUDY Physics Education Research IV ABSTRACT THE ROLE OF CONTEXT AND GENDER IN PREDICTING SUCCESS IN A MODIFIED LABORATORY COURSE BY KERON SUBERO Doctor of Philosophy New Mexico State University Las, Cruces, New Mexico, 2010 Dr. Stephen Kanim, Chair We designed and implemented curriculum intended to be used by students in an algebra-based introductory physics laboratory course. Our curricular goal was to foster, through observations in the lab, a coherent framework in students' understanding of general principles presented in the introductory mechanics course, while addressing known student difficulties. The research that guided our curriculum development efforts, however, was previously implemented in an intervention setting which was quite different from ours, and was conducted on students enrolled in calculus-based physics courses who were generally academically better prepared than our students. We describe the development of laboratory materials, designed to fit the specific curricular constraints of a lab course at NMSU. We present some results from post-testing of our labs, which were not as favorable as results obtained by v researchers at other institutions implementing similar curricula in their courses. We attempted to quantify differences in preparation among our introductory physics student populations who use these laboratory materials. We developed a short proportional reasoning pretest, which we found to be a relatively efficient predictor of student success in our courses. We investigated the effect of context variations on performance by various student populations on this pretest, and found that the effect of context variation was not the same for all of our student populations. Results from our calculus-based population showed a small but significant increase in performance when we modified the context of our pretest, while the performance of our algebrabased population showed very little sensitivity to the variation in pretest context. Finally, when considering students' gender, we found in both algebra-based and calculus-based physics courses that female students were significantly affected by context variation, while male students' performance remained relatively unchanged when we varied our pretest context. vi TABLE OF CONTENTS Page LIST OF TABLES xiii LIST OF FIGURES xv CHAPTER 1: OVERVIEW 1 Introduction 1 Research questions 5 Context for investigation 7 Student population at NMSU 10 Introductory physics courses at NMSU and the students enrolled in them 12 Predicting success in these labs 15 Epistemology 16 Summary 17 CHAPTER 2: MODIFICATION OF THE LABORATORIES 18 Background 18 Structure of the labs at NMSU 19 Example of curricular modification: Two-dimensional motion 21 Motivation 21 Tutorial on motion in two dimensions 24 Laboratories using long-exposure digital photography 25 Equipment 26 vii Page Two-dimensional motion lab 29 Procedure for finding acceleration direction 30 Introductory pencil-and paper exercises 31 Analysis of long exposure photographs of moving obj ects 31 Central force motion: Spherical pendulum 32 Parabolic motion: Hovercraft on a ramp 32 General two-dimensional motion: Roller coaster 34 Newton's Second Law lab 35 Introductory pencil-and paper exercises 36 Lab exercises connecting force to acceleration 36 Physical pendulum 37 Constant velocity: Hovercraft on a level surface 37 Linear motion: Block on a ramp 38 Other labs using long-exposure digital photography 40 Energy 40 Rotation 41 Measurements of effectiveness of the labs 45 Results on questions about acceleration in two dimensions 46 Results on questions relating force direction to acceleration direction 47 Discussion 50 viii Page Conclusion 52 CHAPTER 3: PROPORTIONAL REASONING AS A PREDICTOR OF SUCCESS IN PHYSICS 55 Introduction 55 Description of statistics used 57 Test statistics 57 Correlation Coefficients 59 Review of previous studies 62 Predictors of physics grade, mathematics and education reform 62 Mathematics as a predictor 64 Piagetian ability as a predictor 66 Paper-and-pencil tests of Piagetian ability 69 Various other factors used as predictors 70 Summary of previously used predictors 75 Lawson test questions as predictors of success 76 Results of using questions from the Lawson test as a pretest 78 Proportional reasoning pretests 80 Performance on pretests 81 Performance on final exam 83 Pretest version and correlation 85 IX Page Fall 2009 pretest versions 87 Working memory capacity 90 Description of working memory capacity 90 Previous use of working memory as a predictor 93 Working memory capacity as a predictor at NMSU 94 Description of our N-back test 95 Scoring of our N-back test 97 Results of our N-back test 98 Spatial working memory test Results using spatial working memory capacity Summary of working memory results Conclusion 100 102 102 103 CHAPTER 4: CONTEXT DEPENDENCE OF PERFORMANCE ON PROPORTIONAL REASONING TASKS 105 Two perspectives on student difficulties 105 Example of context-dependent performance 109 Background to study at NMSU 113 Method 114 Results for all students 115 Difference in performance by course 117 Discussion of performance by course Gender effects on performance 122 124 x Page Results of NMSU study separated by gender 125 Fall 2009 pretest versions 129 Pretest correlation with final exam 135 Discussion and conclusion 137 CHAPTER 5: CONCLUSION 140 Introduction 140 Summary of findings 141 Implications for instruction 144 APPENDICES A. Motion in two dimensions tutorial at the University of Washington 149 B. Selected labs and associated homework exercises 152 C. Selected final exam questions 200 D. Lawson Test of Scientific Reasoning 203 E. 15 Question subset of Lawson pre-test 208 F. Three versions of proportional reasoning pretest 212 G. Correlation coefficients between Lawson subcomponents, pretest and final H. Various scatterplots of final exam score vs measures of working memory capacity 216 218 I. Various tables showing differences in performance on pretests by two introductory physics populations 221 J. Various tables showing gender separated differences in performance on pretests by two introductory physics populations 225 xi Page K. Fall 2009 pretest versions 229 L. Correlation matrix of TA ratings of students with other measures of student success 234 REFERENCES 235 xii LIST OF TABLES Page Table 2.1 Student performance on matched pre/post questions 49 2.2 Comparison of performance for sections with two instructional emphases 49 Correlations among Lawson test subcomponents and final exam score for Physics 211 students 79 Correlations among Lawson test subcomponents and final exam score for Physics 221 students 79 Descriptive statistics of difference in student performance on two pretest versions 116 T-test summary of difference in student performance on two pretest versions 116 Descriptive statistics of differences in performance on original pretest questions by students in two intro physics courses 221 T-test summary of differences in performance on original pretest questions by students in two intro physics courses 222 Descriptive statistics of differences in performance on modified pretest questions by students in two intro physics courses 223 T-test summary of differences in performance on modified pretest questions by students in two intro physics courses 224 Descriptive statistics of difference in student performance on two pretest versions within each course 120 T-test summary of difference in student performance on two pretest versions within each course 120 Descriptive statistics of gender separated differences in student performance on two pretest versions for Physics 211 225 3.1 3.1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 xiii Page 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 T-test summary of gender separated differences in student performance on two pretest versions for Physics 211 225 Descriptive statistics of gender difference in performance on two pretest versions for Physics 211 students 226 T-test summary of gender difference in performance on two pretest versions for Physics 211 students 226 Descriptive statistics of gender separated differences in student performance on two pretest versions for Physics 215 227 T-test summary of gender separated differences in student performance on two pretest versions for Physics 215 227 Descriptive statistics of gender difference in performance on two pretest versions for Physics 215 students 228 T-test summary of gender difference in performance on two pretest versions for Physics 215 students 228 Gender separated correlation coefficients of pretests score with final exam score 136 Correlations among TA's rating of students with students' final exam scores and pretest scores 234 xiv LIST OF FIGURES Page Figure 1.1 Five bulbs Question 2 1.2 Distribution of ACT math scores for NMSU students vs All high school seniors 11 1.3 Distribution of class and maj or of introductory physics courses 13 2.1 Blinkie Circuit 28 2.2 Equipment for motion in two dimensions lab 29 2.3 Photograph of pendulum moving in an ellipse 33 2.4 Long-exposure photograph for the toy hovercraft 34 2.5 Long-exposure photograph of the roller-coaster 35 2.6 Photograph of a single swing of the physical pendulum 38 2.7 Photograph of the wooden block on an inclined ramp 39 2.8 Roller coaster on steep and shallow tracks 41 2.9 Photograph of plywood disk rotating with constant angular velocity... 42 2.10 Photograph of plywood rotating disk with constant angular acceleration 43 2.11 Plywood disk with variable moment of inertia 44 2.12 Questions on two-dimensional motion 47 3.1 Sample scatterplots showing different correlations 60 3.2 Scatterplots showing R-squared 62 3.3 Stages of Piagetian development 67 xv Page 3.4 Scatterplots of score on final vs pretest score 78 3.5 Distribution of student scores on proportional reasoning pretests 82 3.6 Distribution of final exam scores 84 3.7 Scatterplots of final score vs pretest score for different courses 86 3.8 Model of memory processing cycle 91 3.9 Model of inner workings of working memory 92 3.10 N-back slide - question 3 96 3.11 N-back slide - question 5 97 3.12 Scatterplot of final score vs working memory score 99 3.13 Scatterplot of final score vs time to take working memory test 218 3.14 Scatterplots of final score vs working memory correct response rate... 219 3.15 Spatial working memory test slide 101 3.16 Scatterplot of final score vs spatial working memory score 219 3.17 Scatterplot of final score vs time to take spatial working memory test 220 4.1 Two time of flight questions with different context cuing 108 4.2 Three contexts of question on predicting trajectory Ill 4.3 Distributions of students' scores on two pretest versions 121 4.4 Student performance on each set of paired questions 122 4.5 Distributions of students' scores on two pretest versions separated by gender for the algebra-based physics course 126 xvi Page 4.6 4.7 Distributions of students' scores on two pretest versions separated by gender for the calculus-based physics course 127 Frequency counts of correct responses on students' paired pre-post test scores on two versions of ratio questions 130 4.8 Diagram of two interlocked gears 132 4.9 Student performance on gear question 132 4.10 Student Fall 2009 performance on two pretest versions 134 xvn CHAPTER 1: OVERVIEW Introduction Physics education research (PER) has evolved as a field of research conducted in many physics departments worldwide out of a desire to systematically study issues related to students' learning of physics. Much of the work in the field has consisted of pinpointing and correcting reasoning or conceptual difficulties on specific topics which were repeatedly observed to affect students' learning of physics. The impetus behind many early physics education investigations was a dissatisfaction with the physics knowledge of students after completing traditionally taught physics courses.1' There was evidence that traditional instruction invariably resulted in students using formula-driven problem solving strategies and that students often left physics courses with a largely unmodified, incorrect view of the physical world.5 In response many physics education researchers have modified curricula to include inquiry-based and other interactive-engagement techniques.5 These modified curricula have led to significant gains in students' physics content knowledge. Oftentimes, curriculum development is thought of as the primary purpose of o all physics education research. However, McDermott points out that researchers in PER can be categorized by their perception of the various teaching processes occurring in physics classrooms. She identifies researchers who use the context of 1 students studying physics to focus on understanding human cognition as one group; those researchers focusing on physics subject matter as another; and those who use the context of physics as a means to focus on instructional strategies as a third. Regardless of focus, results from PER influences and often forms the basis of curriculum development in physics. PER has been reasonably successful to date in finding specific topical difficulties that are encountered by students enrolled in undergraduate physics courses, and in designing curricula based on this research that successfully addresses these difficulties.10 An example of a research question that elicits common student difficulties is the 'five bulbs question' shown in Figure l.l. 11 Students are asked to rank the brightness of the five bulbs, assuming all bulbs are identical and the batteries are ideal and identical. B A T c T (j) D E (j) Figure 1. 1. 'Five bulbs Question' asked by McDermott and Shaffer. "Students are asked to rank the brightness of five identical bulbs connected to identical ideal batteries. Only about 15% of students in introductory calculus-based courses correctly answer this question. This result does not depend strongly on whether the question is 2 asked before or after relevant standard instruction in electric circuits. Students' incorrect answers suggest several widely held incorrect beliefs about electric current in circuits, for example that current is 'used up' as it passes through resistive elements in a circuit (consistent with a student ranking of bulb B brighter than bulb C); or that a battery acts as a constant current source (consistent with a student ranking of bulb A brighter than bulb D or E). By uncovering and categorizing these common incorrect models or misconceptions, researchers have formed a basis on which to develop more effective curriculum. Research-based instruction typically refers to curricular design based on investigations into student understanding and incorporating instructional strategies that actively involve the learner in the construction of correct physical models. These researchers have shown that scores can be significantly improved on conceptual questions such as this if students are carefully guided through the reasoning necessary to develop a model that is consistent with physical laws. For example, on the five bulbs question, 15% of students in an introductory calculusbased electricity and magnetism course, who have completed two 50-minute recitation section exercises (instead of more typical recitation section problemsolving instruction) from Tutorials in Introductory Physics, correctly answer the 5bulbs question. For comparison, 70% of graduate students at the University of Washington give a correct ranking on this question. When curricula developed at one institution and with one group of students is adopted at another institution, it has been shown that improvements can be obtained 3 that are similar to those reported from the institution that conducted the research and developed the materials. For example, at the University of Colorado, 15% of students answer the 5-bulbs question described above correctly after traditional instruction, whereas upon completion of the relevant Tutorials, 57% of students correctly answer this question upon post-testing.12 These values are very similar to the ones observed at the University of Washington. Based on these results, it is tempting (especially for physicists) to treat student populations as essentially interchangeable, much in the same way that the results of physics experiments do not depend on which particular batch of electrons are being manipulated. With this assumption, when implementation of curriculum falls short of reported results, it is assumed that the problem lies in the implementation. However, in the past 8 years researchers have established correlations between student performance on various pretests given at the beginning of instruction with the gains in conceptual understanding that these students make during physics instruction. For example, Meltzer55reports that student performance on a mathematics pretest is significantly correlated with normalized gain in student performance on a standard conceptual test on electricity (CSE), while Coletta reports that students' reasoning ability correlates strongly with their improvement in score on a popular test of the concept of a force (FCI). These results have implications for the degree to which we might expect curriculum developed at one institution and with one student population to be as effective at improving student understanding when it is implemented at a different 4 institution or with another student population. Many initial PER investigations were primarily conducted in calculus-based physics courses at Research I institutions with selective admission requirements. These results are often applied at less selective institutions, and with more poorly prepared students. For example, Kanim14 reports that while about three-quarters of all PER research on topics involving introductory physics curricula is done using students enrolled in calculus-based introductory physics courses, this population accounts for only about one-third of the total introductory physics population. At NMSU we have reported using curriculum materials similar to that used elsewhere yet our success rates were less than to be expected. ' Similarly, differences in performance were also noted by our colleagues at Cal State Fullerton. Anecdotally, instructors describe differences in algebra-based and calculus-based student populations or between pre-med and pre-engineering students and we were curious to find out if there were measurable differences between these populations that might help explain differences in performance. This led us to pay close attention to the success of curricular development among various introductory physics populations as is explored in greater detail in chapters three and four. Research Questions Among the guiding questions we asked ourselves throughout the planning and implementation of the research presented in this dissertation were: 5 1) Can we modify our laboratory course to include exercises which significantly improve our introductory physics students' conceptual understanding of physics, while at the same time maintaining faculty and student expectations of laboratory-like activities? 2) Can we help develop an expert-like, coherent view of physics formalism in our students through curricular modifications? 3) How do the anecdotal differences in algebra-based and calculus-based introductory physics populations play out in terms of measurable differences in reasoning abilities and in performance on conceptual physics problems? 4) Are there limits to how successful our curriculum are expected to be, and what factors influence students successfully understanding the material presented? 5) What is the role of student epistemology in determining student performance on typical tests? 6) Do various styles of questions produce differing effects on different student populations? These overarching questions, constrained by the practical issues of integrating our research into the normal conduct of the laboratory courses, led to the following questions that we attempted to answer in the course of our investigation: 1) Through curriculum developed to emphasize a vector treatment of displacement, velocity and acceleration, can we achieve results on post-test questions indicating student understanding of acceleration as vector quantity, comparable to research-based materials used at other institutions? 6 2) Can we further develop materials that lead to measurable improvements in students' understanding of Newton's second law as a vector equation? 3) Can we design a pretest, based on previous research, that would give us some indication of which among our students would be most likely to succeed in our laboratory course, and which among our students would be most likely to have difficulty with the materials presented? 4) Is pretesting of scientific reasoning ability context-dependent? And if so, does variation in the context of questions asked on this pretest influence student performance? 5) Are there differences in the level of context-dependence of student performance, for various subsets of our student populations at NMSU? Context for Investigation At NMSU, unlike at many other universities nationally, there are no recitation sections associated with the introductory physics courses. The lecture meets for about 150 minutes per week and forms the main arena for student learning of physics. A lab that is closely aligned with each lecture course is offered, however, some majors (such as engineering technology, mechanical engineering and mathematics) do not require their students to take these labs. As such, students' physics content knowledge is largely left up to individual instructors' lectures. In comparison, at the University of Washington a typical engineering student taking an equivalent introductory physics 7 course is required (along with lecture) to enroll in a recitation section as well as take the associated laboratory course, for a total of over 300 minutes of physics instruction per week. In light of this discrepancy of time dedicated to physics learning, we decided to modify the structure of our lab course to make more efficient use of the time our students spent with physics. Prior to the year 2000, the introductory physics labs at NMSU were traditional.1 In an effort to improve students' conceptual understanding of physics, the Tutorials in Introductory Physics16 (from henceforth referred to as the Tutorials) were introduced in both first semester (mechanics) and second semester (electricity and magnetism) introductory physics lab courses as the basis of our lab instruction. The Tutorials, while designed to be used in a recitation section (which we do not have at NMSU), is arguably the most widely adopted research-based curriculum in use at many universities today. The Tutorials were developed at the University of Washington through an iterative development sequence of: 1) systematic investigation of student understanding, 2) application of what is learned from this research as a guide towards the development of new curriculum and 3) design, testing and revision of 17 instructional material based on classroom use. For instance, if through exploratory testing of physics content it is observed that students misunderstand some topic, in subsequent semesters the curricular materials used are revised to try to address this 1 A more detailed description of the differences between traditional and inquiry-based labs is given at the beginning of chapter 2. 8 issue. In light of these results, the effectiveness of the curricular modification is ascertained and it is determined which, if any, .further modifications are necessary. This cycle is repeated until there is satisfaction that the curricular materials successfully address student misunderstandings. The structure, focus and instructional strategies used in the labs we describe in chapter two of this dissertation, owe a great deal to the Tutorials16 and to the extent that we could, we also copied this iterative developmental sequence. At NMSU, while these tutorials were somewhat successful in improving students' understanding of some concepts, it was felt that many of the tutorials relied on pencil-and-paper exercises that were less appropriate for a laboratory than for the recitation sections they were designed for. In addition it was found that some of the labs, such as the Motion in Two Dimensions tutorial shown in appendix A, were not successful in improving our students' content knowledge. It is against this backdrop, beginning in Spring 2002 with a few laboratory exercises each semester, that we began to replace the tutorial exercises with labs which we modified for use within our lab sections at NMSU. As part of a collaborative project with Michael Loverude at California State University Fullerton and Luanna Gomez at Buffalo State College, we designed laboratories intended to strengthen students' conceptual understanding of topics in introductory mechanics. Funding from the National Science Foundation's Course, Curriculum, and Laboratory Improvement (CCLI) program allowed us to design, implement, test, and improve a set of inquiry-based labs intended for students in the 9 algebra-based course. In addition, we conducted research into student understanding of some of the topics associated with these labs in concert with the curriculum design. A total of 17 labs were developed. At New Mexico State University, we use about 12 of these labs for the mechanics semester. Our curriculum goal was to develop a set of labs that improved student conceptual understanding, and that allowed students to connect the rules and equations that they were learning in their lecture to physical events in the laboratory. We wanted to tailor the labs to suit the needs of students in the algebra-based course, and a secondary goal of our project was to better understand the differences in needs between students in the algebra-based and calculus-based courses. In practice, at NMSU, we use almost exactly the same sequence of labs for both populations, as we have seen them to be more beneficial than the labs that they have replaced for both student populations. Student Population at NMSU We believe that our descriptions of the curriculum development we have implemented and our efforts towards predicting success in our labs are not complete without at least some understanding of the student population we serve at NMSU. New Mexico State University is the state's land grant university. We are a minoritymajority institution: 39% percent of our students are Hispanic. Three-fourths of our students are from New Mexico, which is one of only four states considered 10 Distribution of ACT Math scores for NMSU freshmen compared to High School seniors nationwide. NMSU incomin freshmen i.o%0.8% All High School Seniors 0.6% 0.4% 4 0.2%- 1L 0 2 4 6 8 10 12 14 Vy*— 16 18 20 22 24 26 28 30 32 34 36 ACT Math Score Figure 1.2. Relative distribution of ACT math scores for All High School Seniors nationally compared to ACT scores for NMSU incoming freshmen. 1R minority-majority in the nation. About 43% of incoming freshmen at NMSU qualify for Pell grants based on income. For comparison, at the University of Washington, a i io% veloper of PER-based curricula used nationally, 16% of all undergraduates i ilify for Pell grants. The distribution of ACT scores for incoming freshmen at ] 1SU is a close match to the distribution for all high school seniors taking the ACT. ] 4SU has an acceptance rate of about 96%, which places it on the high end of acceptance rates of colleges and universities nationwide.19 11 The results we report in this dissertation will primarily be from students enrolled in labs associated with the first semester of introductory physics courses. As mentioned previously, at NMSU, students do not automatically enroll in a lab and some majors require the lecture but do not require the lab. For grading purposes, then, the mechanics labs we describe in this dissertation are stand-alone 1-credit courses. The lab coordinator generally communicates with the lecture instructors to ensure that the lab sequence is aligned with the lecture, in order to best support the lecture courses. About one-third of the students enrolled in our lab courses have taken high school physics. Roughly 90% of students enrolled in the labs are simultaneously enrolled in the corresponding lecture course. There are four such first semester introductory physics courses offered at NMSU - Physics 211, Physics 213, Physics 215 and Physics 221 - only two of which (Physics 211 and Physics 215) are offered every semester. Introductory Physics Courses at NMSU and the Students Enrolled in Them The bulk of data presented in this dissertation is from students enrolled in the lab sections associated with two main courses Physics 211 and Physics 215. Physics 211 is a standard, algebra-based course with a non-calculus treatment of mechanics. Knowledge of simple algebra and trigonometry is required of students enrolled in this 12 Distribution of Class and Major of three introductory physics courses Physics 211 • Freshmen Sophomores Ujniore Engineering Technology Seniors Biology Agriculture Education Mcrobiology Biochemistry Animal Sdercc® Crvil Engineering fwfechanical Engineering Chemical Engineering • • • • • Other Physics 22: 25» 9SW 0% (M Freshmen Soonomores Juniors • Hology Seniors Physics 215 2M4_ Freshmen Soononwes Juiiors Electrical Engineering Seniors Other Engineering Figure 1.3 Spring 05 histograms of the relative distributions of Class (left column) and Major (right column), of students enrolled in three of our introductory physics courses - Physics 211, Physics 221 and Physics 215. 13 course. About two-thirds of Physics 211 students are juniors and seniors (see Figure 1.3). These students major in a variety of fields, however, depending on semester and whether or not Physics 221 is offered, biology majors usually make up the slight majority of Physics 211 students. Physics 221 is an algebra-based physics course where special emphasis is given to applications in the life sciences, and is also recommended for students preparing for the physics part of the MCAT. Most majors requiring their students to take an algebra-based physics course do not distinguish between Physics 221 and Physics 211. The context of the physics material presented within the course is usually biology-oriented, thus this course tends to attract majors with an interest in biology. This course is not offered every semester, and class sizes are relatively small. Students enrolled in the two algebra-based lecture courses (Physics 211 and Physics 221) are enrolled in the same lab sections (called Physics 211 labs). We do not distinguish between these two groups in the results reported from the labs. Physics 215 is a calculus-based introductory mechanics course primarily made up of various engineering majors (see Fig. 1.3). Calculus is a prerequisite for students enrolled in this course. In contrast to the algebra-based courses, about two-thirds of Physics 215 students are freshmen and sophomores. This course is offered every semester and is usually large-enrollment - above 80 students per semester. Physics 213 is a calculus-based course for physics and chemistry majors. Calculus is also a prerequisite for this course, however students can also be concurrently enrolled in calculus and still be allowed to register. Physics 213 is only 14 offered in the Fall semesters (as opposed to Physics 211 and Physics 215 which are offered every semester) and class sizes are relatively small - around 20 students. Because of its small student population, in many cases throughout this dissertation, we omit results from students enrolled in this course. These students usually perform markedly differently from the rest of our introductory physics student populations. Predicting Success in These Labs A second theme of this dissertation, discussed in chapter three, centers on predicting success in these laboratories through the use of pretests. The grading and structure of the laboratory course is similar to many lectures in the sense that weekly homework scores, participation, and scores on a final exam all contribute to students' lab course grade. Aspects such as homework scores and participation are difficult for researchers to control for and influence, especially for lab sections graded and run by multiple instructors. As such, we believe the uniform final exam to be the most unbiased estimate of what students actually know upon leaving our lab course. For the purposes of this dissertation, then, success in the laboratory course is primarily measured by students' scores on the final exam of the lab course. We will explore some of the factors which guided the development of tests used by other researchers seeking to predict success in their courses, with particular emphasis on the role of Piaget's idea of cognitive development, in influencing items on historically used predictive tests. We will briefly describe how predictive some of 15 these tests have been at different institutions. We will also detail a short diagnostic test we have developed, which we have found to be reasonably predictive of student success in our laboratory course. Epistemology In chapter four of this dissertation we explore two views of the source of differences in student performance on physics problems such as our short diagnostic test. The view that has guided the development of many of the assessments used as predictors of success to date, portrays differences in students' incoming ability as the reason for varying degrees of success. Some researchers portray these tests as good measures of scientific reasoning ability. Other researchers21 believe that what students 'know' is highly context dependent and rarely transferred intact from one instance to the next, thus tempering their reliance on such tests in characterizing students. In chapter four we take a look at how student epistemology plays into performance on questions appearing on a commonly used assessment of proportional reasoning ability. Further, we take a look at how different student populations are affected by the contexts in which the questions are presented. A summary of our findings in this dissertation can be found in chapter five. 16 Summary This dissertation was motivated by a desire to improve our students' knowledge of physics within the constraints of our course structure. As explored further in chapter two, our initial results from curriculum changes were not as we would have hoped. This led us to consider if curriculum developed based on difficulties observed and interventions attempted with one student population might not be as successful with a different population as a result of student preparation among other factors. We explored factors that other researchers have identified as pertinent to student success in physics, and highlight our experience at NMSU using these in accounting for different success rates among our students. Finally we observed the interplay between students' epistemologies, the use of these predictive factors, and students' successful acquisition of physics knowledge. 17 CHAPTER 2: MODIFICATION OF THE LABORATORIES" Background Before the year 2000 the labs at NMSU could be described as traditional labs. Traditional labs as described in PER22include labs during which students are asked to complete sequential steps, much like a recipe in a cookbook, performing measurement and error analysis with an eye towards verification of quantitative relationships. At the end of traditional labs students typically complete a lab report following a prescribed format. Researchers have shown however that students develop little conceptual understanding of relevant physical phenomena as a result of these traditional lab experiences. The Tutorials in Introductory Physics (commonly known as the Tutorials) developed at the University of Washington were implemented at NMSU after the year 2000 to improve students' conceptual understanding in introductory physics courses. As measured by analysis of homework and exam questions, this intervention was moderately successful for some of the topics covered in the tutorials, while for others there was little improvement in students' understanding of the materials presented. While today the second semester course in our introductory physics sequence (which covers topics in electricity, magnetism and optics) continues to use the Tutorials, we have been modifying the laboratory materials for our mechanics course since 2002.24 " Substantial portions of this chapter are reprinted with permission from Am. J. Phys. 78(5), 461-466 (2010). Copyright 2010, American Association of Physics Teachers. 18 Structure of the Labs at NMSU The focus of the labs we describe in this chapter is on developing a solid conceptual understanding, on making connections between observations and physics formalism, on promoting reasoning based on observation, and on strengthening scientific reasoning skills. Students download each week's lab worksheet from the course website before coming to the lab room, but we do not expect them to have read the lab. The first page of every lab summarizes the underlying concepts and provides a summary of the lab goals and procedures. Typically, the topics of the labs have previously been discussed in the course lecture. At the beginning of the lab period, students are given a pretest that we use to gauge their understanding of the concepts and procedures related to the lab material and that have been helpful in designing the lab exercises. The pretest takes less than 10 minutes. Some teaching assistants then give students a brief summary of the physics that they will be using in the lab and of any procedural issues that are anticipated; others just let students start immediately after the pretest. The labs follow a guided inquiry format. Students work in groups of 3 or 4 to answer the questions on the lab worksheet. The teaching assistant circulates from group to group, helping with procedures and asking students questions to assess their understanding of the material. Students are often asked to predict what they expect to happen before performing experiments, and they are expected to interpret their observations in light of physics principles. 19 As in the Tutorials, students are occasionally presented with student dialogues and are asked to critique the reasoning presented in these dialogues. In some places, students work on developing the procedural skills (for example, drawing free-body diagrams or subtracting vectors) that are associated with the lab topic. Often the physics principle that is under study is only presented after students have done experiments related to that principle; in these cases they are asked to verify that the observations they have made are consistent with this principle. The lab worksheets are typically 6-10 pages long. Selected relevant labs and their associated homework are included in appendix B. Although the lab period is 3 hours long we have tried to design the labs so that most students are done in 2 - 2.5 hours. Each lab section has one teaching assistant and there are usually around 20 students in a section. There are no lab writeups. Each lab has an associated homework assignment that is intended to reinforce the underlying concepts, to require students to extend these concepts to other contexts, and to give students practice with the associated procedural skills. These homework assignments are turned in at the beginning of the lab at the following meeting. For research purposes, success in our lab course is primarily measured through their scores on our lab final exam. The lab final is given during the last lab meeting and has roughly 36 questions (3 per lab), and counts for 20% of the students' grade. Most of the questions are qualitative and focus on concepts that have been explored during the lab meetings. Students may take up to the full 3-hour lab period to complete the final: In practice, most are done in half an hour. 20 In the next section, we give a detailed example in the context of 2-dimensional acceleration, of the modifications we have made to the introductory mechanics labs at NMSU, and of our attempts to measure the effectiveness of these changes. Example of Curricular Modification: Two-Dimensional Motion Motivation Physics instructors would like to have students leave an introductory physics course with an appreciation of the laws of physics as powerful unifying principles. A more typical outcome, unfortunately, is that students come to the conclusion that physics is a collection of specialized rules that apply to specific cases, and that there is a lot to remember. The Motion in Two Dimensions and Newton's Second Law labs we describe in this chapter (also in appendix B) were motivated by a desire to improve our students' conceptual understanding of Newton's second law as an equation relating two vector quantities. In the past few years, physics education researchers have documented25'26'27 introductory mechanics students' difficulties with vector addition and subtraction and with acceleration and force as vector quantities. In particular, at NMSU, we had found that students in our introductory algebra- and calculus-based courses typically could not (1) reason qualitatively about the magnitude and direction of vector sums and differences; (2) find the direction of an acceleration vector based on a change in velocity; or (3) reason about the relative magnitudes of individual forces in a free-body diagram based on knowledge about the 21 direction of the acceleration. Without an understanding of how the direction of acceleration is obtained, and without recognizing how the acceleration direction constrains the relative magnitudes of the forces acting on a body, it is hard to imagine that students form a sense of Newton's laws as unifying principles. Instead, they are left with rules that apply in individual circumstances but that do not connect to one another. For example, motion in a line and motion in a circle become separate situations with different equations that govern them. In a study of concept interpretation in physics, Reif and Allen outline five steps necessary to apply the definition of acceleration to the motion of a particle: (1) Identify the velocity v of the particle at the time of interest; (2) Identify the velocity v' of the particle at a slightly later time t' ; (3) Find the small velocity change Av = v' — v of the particle during the short time interval At — t' — t ; (4) Find the ratio Av/At and (5) Repeat the preceding calculation with a time t' chosen progressively closer to the original time /. They note that understanding and applying the definition of acceleration has many complexities that are hidden by the seemingly simple definition a = Av/At - f o r example the procedure for subtracting vectors. Reif and Allen suggest that the procedural knowledge necessary to interpret the definition of acceleration is usually not adequately taught. They recommend that the presentation of operational definitions of concepts such as acceleration be followed by the opportunity for practice in a variety of special cases. This emphasis on operational definitions is reiterated by Arons and by Shaffer and McDermott. ' 22 As part of their study, Reif and Allen asked students in an introductory physics course for scientists and engineers qualitative questions about the directions and relative magnitudes of accelerations in 13 different situations. Even though these students had been using acceleration as part of the course for at least two months, none of them were able to answer more than 3 of the 13 questions correctly. In answering the same questions, physics faculty members rarely used the definition of acceleration, and often used information about forces to answer questions that were purely kinematical. Both experts and novices had difficulty applying the definition of acceleration to objects moving in a curved path. They tended to use case-specific knowledge about acceleration rather than working from a more general definition, and both groups often used this case-specific knowledge incorrectly. Shaffer found similar difficulties: Asked to indicate the direction of the acceleration of a swing at five points along its path, none of 124 students in a calculus-based introductory physics course answered correctly for all points. Of 22 physics teaching assistants asked the same question, only 3 drew the acceleration correctly for all 5 points. Similar results were obtained on physics comprehensive exams for graduate students. Our content goals for these two labs, which form the main basis for this chapter were for students to (1) gain practice at using the definition of acceleration to describe the direction and to compare the magnitudes of accelerations for objects in two-dimensional motion; (2) recognize that Newton's second law requires that the direction of acceleration and of net force be the same for objects in two dimensional motion; and (3) be able to use kinematical knowledge to make inferences about the 23 relative magnitudes of forces acting on an object in two-dimensional motion. Our epistemological goal was to have the labs contribute to students' sense of the coherence of the kinematics and dynamics they were learning. Tutorial on Motion in Two Dimensions In order to address the conceptual and procedural difficulties described, the implementation of the Tutorials in the labs at NMSU in 2000 included a tutorial on two-dimensional motion called Motion in Two Dimensions that is included in appendix B. This tutorial was based on extensive research conducted by the Physics •in Education Group at the University of Washington. McDermott and Shaffer report that, after relevant lecture instruction, only about 20% of students could consistently find the direction of acceleration for an object moving at constant speed in two dimensions. After completion of the Motion in Two Dimensions tutorial, about 80% recognized that the acceleration was perpendicular to the trajectory at all points. For an object moving with increasing speed on a two-dimensional trajectory, about 60% could sketch the direction of acceleration after tutorial instruction. However, for questions involving vertical motion (for example, questions about the acceleration of a pendulum bob) only about 15% could answer correctly after tutorial instruction. While there is clearly room for improvement, these post-test results are better than the pretest results for physics graduate students working as teaching assistants in the introductory course. 24 We wanted to replace the Motion in Two Dimensions tutorial in our lab sequence for three reasons. First, the tutorial consisted entirely of pencil-and-paper exercises, and so did not fit well with student or instructor expectations, and lacked the opportunity for students to reason based on observation that we would like as part of all of our labs. Second, we wanted to be able to use the labs with students in the algebra-based course, and a significant portion of the tutorial is devoted to exploring what happens to the direction of the acceleration vector in the limit of small time intervals, a topic more appropriate to the calculus-based course. Finally, our posttesting revealed that our students were not performing as well after the tutorial as students at the University of Washington. Laboratories Using Long-exposure Digital Photography We wanted to create a set of laboratory exercises that went beyond the penciland paper exercises of the tutorial, allowing students to start with observations of actual motion. At the same time, we wanted to emphasize the procedural steps necessary to find the acceleration of an object as described previously, and to allow students to practice these steps as part of the laboratory. In addition, we hoped to use the same lab procedures in subsequent labs to allow students to connect the acceleration vector for an object moving in two dimensions to the net force acting on that object at an instant in time. As an overall objective, we hoped to offer students 25 lab experiences that would allow them to think conceptually about Newton's second law as a vector equation. We decided to incorporate long-exposure digital photography as a means of connecting actual motions with representations of that motion that were amenable to graphical analysis. In the Motion in Two Dimensions lab described in this section, and in the other labs described in this chapter, students use inexpensive digital cameras to record the motion of objects with attached blinking light emitting diodes ('blinkies'). These photographs can then be used to make inferences about the velocity and acceleration of the objects at different points along their paths. Here we give details about the equipment we use and describe the sequence of exercises that we developed for a two dimensional motion lab. In the next section we discuss the lab on Newton's second law. We will also briefly describe a more recent introduction of the same digital photography techniques into labs on conservation of energy and on rotational motion. Finally, we present some pre- and post-test data from questions related to the concepts underlying these labs. Equipment The labs we describe here require a digital camera that has a manual setting or a 'shutter priority' setting.31 We use older cameras that we buy used online for about $60. Students photograph a moving flashing light emitting diode (LED) using exposure times of between one and four seconds. A similar photographic technique has been described by Terzella et al. for measuring the acceleration of gravity. 26 Because we intend the lab to be used in the algebra-based course, we restrict the lab to finding average accelerations over short time intervals. By using flashing LEDs with only short 'off periods we have students create photographs with light streaks that look very much like motion diagrams. Although we now have one camera for each lab group, we have run the labs with as few as three cameras for 20-24 students without causing significant 'wait times' since the photographs do not take very long to make and download. We have instructions for how to use the cameras at each lab station: Generally students are not familiar with using a shutter priority setting, but have no difficulty downloading the pictures and sending them to the black and white laser printer. The cameras use AA batteries, which often need to be recharged during a one-week lab cycle. Software at each lab station (currently Adobe Photoshop Elements) allows students to invert the photographs before they print them, so that the dark background is printed as a light grey and the light streaks are printed as dark lines. This saves toner and makes it easier to write on the prints. Additional equipment includes cables for downloading the pictures from the camera, an inexpensive 'gooseneck' lamp at each lab station (so that students can read and write while other groups are taking photographs), and several tripods. We tried a variety of flashing light sources34 before deciding to build our own so that we could choose a flash rate and control the fraction of each period that the LED was on (the duty cycle). Each blinkie has an on/off switch, a second switch to change the flash rate, and a 'trimpot' variable resistor to control the duty cycle. 27 In the future we will probably replace the variable resistor with two fixed resistors that set the duty cycle at about 90%. The circuit is powered by two CR2032 3-volt disc batteries installed in a holder on the back of the board. We have not yet had to replace a battery except when the circuits were accidentally left on. The circuits are about 2.4 cm wide and 4.8 cm long. A +6V 555 Timer - ^ I.C. 1 3 C1TJC2]_ VLHD Y S2 4r0V Figure2.1. Blinkie circuit. Switch SI turns the circuit on; S2 increases the time constant. The potentiometer controls the fraction of time in each cycle that the LED is on. For the Motion in Two Dimensions lab, the circuit is mounted on a toy hovercraft, on a Pasco roller coaster cart, and in a PVC plumbing pipe cap used as a spherical pendulum bob, as shown in Figure 2.2.35 28 1& ^W-^\ f «fe4 the change in velocity for a short time interval. Finally, we want them to recognize that this direction is the direction of average acceleration. Procedure for Finding Acceleration Direction As described by Van Heuvelen, the procedure for using motion maps begins with a drawing of velocity vectors along the object's path. The change-in-velocity vectors for points of interest are then found by subtracting the vector just before that point from the vector just after that point, and this allows a qualitative determination of the direction of acceleration for that point. We have modified this procedure for our labs. A long-exposure photograph of an object with a blinkie attached shows the path of the object. The photograph is printed, and students draw vectors with tails at the beginning of each light stripe and heads at the end. Each vector they have drawn, then, represents the displacement during the time that the blinkie was on. Since each of these vectors represents a displacement for the same duration, their lengths are proportional to the average speed of the object, and they can also be used as average velocity vectors for the time intervals that the blinkie is on. Two adjacent vectors can be used to determine the change in (average) velocity from one light stripe to the next: we assign this change-in-velocity vector (and the associated average acceleration) to the space between adjacent vectors. The lengths of these vectors allow a qualitative comparison of the magnitudes of the acceleration for various points along a path. 30 Introductory Pencil-and-paper Exercises In the first section of the lab the operation of the blinkie circuit is described and an example photograph is given. A drawing of what a blinkie trail might look like for a hovercraft that is kicked so that it changes direction and speed is shown, and students are asked to make inferences about the speed of the hovercraft based on measurements taken from the drawing. Students draw scaled vectors to represent the hovercraft's velocity before and after the kick. They are guided through the procedure for graphical vector subtraction and then determine the change-in-velocity vector for the hovercraft. The direction of the acceleration of the hovercraft is determined for the kick. Students then find the change-in-velocity vector for a portion of a drawing based on a blinkie photograph, and are asked some qualitative questions about this vector. Analysis of Long-exposure Photographs of Moving Objects Following this extended introduction, students take long-exposure digital photographs of three different objects in two-dimensional motion and determine the acceleration direction for each motion following the procedure described in section i. Because we have only one station for each experiment and there are up to 8 lab groups, we let groups take these photographs in any order. The three motions - of a spherical pendulum, a hovercraft on a ramp, and a toy roller coaster - are intended to provide practice with the procedure for determining an acceleration, and to illustrate that the same procedure and definition are applicable across a range of motions. For 31 each of these motions students are asked to print out an inversion of the photograph they obtain, and to find the acceleration vector for various points along the path. We briefly describe details of each exercise. Central force motion: Spherical Pendulum. A spherical pendulum approximately 2 meters long is suspended from the ceiling. The LED protrudes through a hole in the center of the bottom surface of the pendulum bob. The camera is placed on the floor (about 1.5 meters below the pendulum bob) pointing up towards the suspension point of the pendulum. Students push the pendulum bob so that it moves in an ellipse. Figure 2.3 is a photograph of this motion, with an example of the vector construction we expect our students to perform in order to find the direction of the change in velocity. The point of suspension can be seen in the photographs, and when asked to give a description for the acceleration direction that is true for all points on the path, students generally state that it is toward the center of the ellipse. Parabolic Motion: Hovercraft on a Ramp. Students push a toy hovercraft so that it travels in a parabola along a piece of plywood with one edge raised so that it is at about a 4° angle. The camera is mounted on a tripod that is placed at the bottom of the plywood ramp. The board has a grid painted onto it so that the pictures that are taken have horizontal and vertical reference lines. A long-exposure photograph taken of this motion is shown in Figure 2.4. For this exercise, the camera is further from the top of the plywood ramp than from the bottom, so a light stripe in the photograph will be smaller for a hovercraft in 32 motion near the top of the ramp than it will be for a hovercraft moving at the same speed near the bottom. When the vector generated from a light stripe at one location is subtracted from one generated at another location this introduces an error in magnitude and in direction for the change-in-velocity vector. In general, these errors W .' r\tMr.+ is*rk*-if Aw 4/ <••?- «*'-" Figure 2.3. Photograph of spherical pendulum moving in an ellipse. The blinkie flash rate is about 10 Hz, and the exposure is 1.5 seconds, less than the time of a complete orbit to avoid overlap of stripes. Superimposed on the photograph are examples of vector differences based on the length and direction of the light streaks just before andjust after the point of interest. 33 Figure 2,4. Long-exposure photograph for the toy hovercraft moving in a parabolic path. The blinkie flash rate is about 3 Hz, and the exposure is 3 seconds. are small enough that they do not prevent the students from concluding that the acceleration direction is down the ramp, parallel to the grid lines in the photograph. We intend to modify this lab so that the camera is mounted above the plywood and is pointed directly down to minimize this distortion. General two-dimensional motion: Roller coaster. In this exercise, a toy roller coaster cart travels down a flexible track and is launched into a container filled with Styrofoam peanuts. The cart has a blinkie circuit mounted on the top with the LED placed through a hole in the side of the cart close to the center of mass. 34 mmmm»:mmmmM Figure 2.5. Long-exposure photograph of a roller coaster cart traveling down the track and then being launched into a catch basin. The blinkie flash rate is about 20 Hz, and the exposure is 2.5 seconds. Students are asked to find the acceleration direction for points A—Eas a homework exercise. The resulting photograph (see Figure 2.5) Is the basis for the initial homework exercises associated with this lab. Since the lab generally takes less than 2.5 hours9 many student groups choose to do this part of the homework in the lab room together. Subsequent homework exercises give students practice with graphical vector subtraction with finding a ehange-in-velocity vector from two velocity vectors, and with determining the direction of an acceleration vector for objects moving along various paths. N©wtom's §®<s©ondl Law Lalb The lab that follows the Motion in Two Dimensions lab, called Forces, introduces weight, normal forces, tension, and friction. The directions of these forces 3 and the factors affecting these forces are explored, and students are given some practice drawing free-body diagrams. The fifth lab, Addition of Forces, provides practice with vector addition, with reasoning about the relative magnitudes of forces for situations where the net force is zero, and culminates in exercises using the force table. The lab we describe in this section, Newton's Second Law, is the sixth lab in the sequence, and requires students to reason about relative force magnitudes for situations where the acceleration is not zero. Introductory Pencil-and-Paper Exercises. The initial exercises for the lab are again pencil-and-paper. Students use a photograph of the two light stripes adjacent to the bottom of the roller coaster to determine that there is a large change in velocity and therefore a large acceleration upward at that point. They draw a freebody diagram of the coaster at the bottom of the track, and are guided through the reasoning required to compare the magnitudes of the normal force on the cart and the weight at that point. Lab Exercises Connecting Force to Acceleration. In addition to the equipment mentioned earlier, for the Newton's Second Law lab, blinkie circuits are also mounted onto a block of wood cut to fit a 1.2-meter Pasco track, and onto a thin board that is used as a physical pendulum. The Pasco track is mounted on a hinge so that we can adjust its angle to the horizontal. In this lab, students take photographs of four different motions: circular motion with changing speed, linear motion with acceleration both in and opposite to 36 the direction of motion, and motion with no acceleration. For each motion, they determine the direction of acceleration and draw a free-body diagram for the object at one or more points, and use the direction of the acceleration to reason about the relative magnitudes of the forces in the free-body diagrams. Our intention is to emphasize that Newton's second law is a unifying principle that applies to all motions, and serves as a common theme for much of what they have studied up to this point. Physical Pendulum. The physical pendulum is a board approximately 2 meters long with a blinkie circuit at one end. From a photograph of the swing (see Figure 2.6) students determine the direction of the acceleration at two points. They draw a free-body diagram of the blinkie circuit for these points, and are asked to determine the relative lengths of the tension and the weight vectors that results in a net force direction consistent with the acceleration. Constant Velocity: Hovercraft on a Level Surface. The toy hovercraft is pushed across a plywood board that is level on the floor. Students take a 2-second photograph of the hovercraft, and determine that there is no change in velocity and therefore no net force. From a free-body diagram of the hovercraft, they reason about the relative magnitudes of the forces on the hovercraft. 37 Figure 2.6. Photograph of a single swing of the physical pendulum. Students determine that the acceleration is not toward the center of the circle for points where the speed of the blinkie is changing. Lnnnear M©ttt©ms Bltaxek ©m a Manrnpo In the last two exercises In this lab, students photograph a blinkie attached to a block that slides down a 1.2-rneter long Pasco ramp. The angle of the ramp is such that the block speeds up when it slides with the wood surface In contact with the rarnp but slows down when it Is turned over so that a cork surface on the other side of the block is In contact with the ramp (see Figure 2.7). For these two cases, students determine the relative magnitudes of the three forces in the free-body diagram consistent with a net force in the direction of the 3 eieration. They then compare the forces for 1 T>* * Igure 2.7. The # « * » * » " * f t , , .* ^ s - * » - > «*fH«-»Ci(^fr~*«» *." if,'*"''- • fl^NO -'VJSWt.f' J ^ * » k ! ~ -*|j9»5&»* "^ Other Labs Using Long-exposure Digital Photography In this section we briefly describe two additional labs that we have developed in the past couple of years that use the blinkie circuits and long-exposure photographs. These are the seventh and ninth labs in the sequence, Energy, and Rotational Motion (also see appendix B). Energy. The Energy lab is intended to introduce kinetic and potential energies and the conservation of mechanical energy. Again starting with a photograph of the roller coaster, students observe that the length of the light streaks is about the same for locations that are at the same height. They measure the lengths of light streaks at different heights, and from this find that the speed of the coaster doubles when the height below the release point quadruples. Expressions for kinetic and potential energy are introduced, and the measured results are compared to what would be obtained if the potential energy lost from the release point to a point below it on the track was all converted into kinetic energy. Students then look at factors that influence the speed of the roller coaster as it travels down a ramp. Two tracks are set up with different slopes and shapes but with the same starting and ending heights, and students are asked to predict whether the steeper track will result in a higher terminal speed. In addition, they are asked whether adding mass to the cart would result in a higher terminal speed. Students then photograph the coaster as it travels down the shallow track (see Figure 2.8), 40 down the steep track, and down the steep track again with added mass, and the lengths of the light streaks at the bottom of the tracks are compared. Figure 2.8. Long-exposure photograph for the roller coaster cart traveling down the lesser steep of two tracks. At this point the idea of energy conservation is introduced and the results obtained up to this point are discussed in light of energy conservation. We make use of energy bar charts to emphasize the idea of a conserved quantity,37 and students construct these bar charts with different reference heights. Rotation. The Rotation lab is an introduction to the kinematics of rotation and includes a qualitative introduction to moment of inertia. The lab uses two plywood disks, each about 25 centimeters in diameter, that rotate on bearings. One disk has 4 LEDs, one at the center and the other three at different radii, all driven by 41 the same blinkie circuit. Students measure the flash rate of the blinkie circuit, and then spin this disk and photograph it using a one-half second exposure. Figure 2.9. Long-exposure photograph for a plywood disk rotating at approximately constant angular velocity. In addition to the LED in the center, there are 3 LEDs at different radii, all controlled by the same blinkie circuit. From this photograph they verify that the arc length of each light streak is proportional to its distance from the LED at the center of the disk, and that the angular displacement during one blinkie period is the same for the three rotating LEDs. (See Figure 2.9) They then calculate the angular velocity of the disc. The second disk has a blinkie at the center of the disk and one near the outside of the disk. A groove in the edge of the disk allows students to wind a string attached 42 to a weight around the disk. When the weight is released, the disk rotates with Increasing angular velocity. Students photograph the disk and measure the chan m — AB/M from one blinkie cycle to the next (See Figure 2.10). Figure 2.10. Long-exposure photograph for a plywood disk rotating approximately constant angular acceleration. The torque is created by a falling weight attached to the string visible at lower right. !##%*« • ' W • v • <0fo&®&- Figure 2.11. Plywood disk with four removable lOOg cylindrical masses inserted into four holes near the outer edge of the disk. Students compare the angular acceleration of the rotating disk for this configuration of masses, to a configuration where the masses are inserted into the inner four holes (vacant in this picture). From this measurement they calculate the (roughly constant) angular acceleration of the disk. This disk has four holes a small distance from the center and four more holes near the outside. For the first photograph, brass cylinders are inserted in the Inner holes. The experiment is repeated with the same falling mass but with the brass cylinders moved to the outer holes (see Figure 2.11). Students again calculate the angular acceleration of the disk and compare their result to what they obtained previously. These results are used to motivate a qualitative introduction to Measurements of Effectiveness of the Labs In this section, we report results from the pretesting and limited post-testing that we have conducted in order to measure the labs' effectiveness. As mentioned in chapter one, free-response pretests are given at the beginning of each lab that give us some sense of students' initial understanding of that lab topic. We use questions on the multiple-choice lab final as a crude measure of their understanding at the end of the semester. As part of the lab development, we have also used student responses to homework questions to gauge lab effectiveness. We encourage students to work together on the homework with the caution that what they turn in should be their own work. In practice, it is often hard to tell whether a correct homework response reflects individual or collective understanding of the topic. There are only a few questions on nationally administered assessments that ask for information about force based on knowledge of acceleration. We report below, results from two of these questions asked on one of our final exams. Most questions about force and acceleration on widely-used assessments (such as the Force Concept Inventory) are for one-dimensional motion, or are for one component of parabolic motion, and do not test for student understanding of force and acceleration as vector quantities. Because the focus of the acceleration labs we have described is on the vector nature of force and acceleration, most of the assessment questions we have asked are questions that we have written, or are questions that have 45 been asked as part of other research projects into student understanding of twodimensional kinematics and dynamics. ' Results on Questions About Acceleration in Two Dimensions In Fall 2009, when questions about the direction of acceleration for an object in two-dimensional motion were asked on a pretest, about 4% of our students answered correctly, which is consistent with pretest data reported by Shaffer and McDermott.27 On the Mechanics Baseline Test, the question shown in Figure 2.12a asks for the direction of the acceleration of the block when it is at the position shown. Reported results39 range between 12% and 18% correct after standard university instruction. About 39% of 135 students in our algebra-based course and 51% of 88 students in our calculus-based course answered this correctly on the lab final. Nagpure38 gives results for a question about a dog that is speeding up as it moves along a curved path (Figure 2.12b). After tutorial instruction using a modified version of the Motion in Two Dimensions tutorial at the University of Maine, 44% of the students in their algebra-based course and 69% of the students in the calculusbased course answered correctly. We asked a multiple-choice version of this question on our lab final: About 20% answered correctly in both algebra- and calculus-based sections. For a similar question about an object changing both direction and speed, Shaffer and McDermott report27 about 60%) of students in a calculus-based course answering correctly after tutorial instruction. 46 11a lib. lid. r"7 11c. Steel cable ,200 m/'s FJevator going up at constant speed 350 m/s Figure 2.12a. Question from Mechanics Baseline Test about acceleration for twodimensional motion. Figure 2.12b. Question about acceleration direction for a dog speeding up. Figure 2.12c. Question about direction of net force for an object changing direction and speed. Figure 2.12d. Question from Force Concept Inventory requiring reasoning about forces based on information about kinematics. Results on Questions Relating Force Direction to Acceleration Direction. One question on the Force Concept Inventory40 asks for a comparison of forces based on information about acceleration: For an elevator moving upward at a constant speed (Figure 2.12d), 61% of students recognized that the tension in the cable would be equal to the weight of the elevator after standard university-level instruction. We asked this question on our lab final exam: About one-third of the 223 students in both courses answered correctly. For an object that is speeding up as it moves in a curved path (Figure 2.12c), about one-quarter of the students in our lab courses correctly chose the direction of the net force (about one-third gave a direction toward the center of the curve). We have obtained similar results after extensive 47 practice in the lecture portion of a course at NMSU. We do not have data from other institutions to compare this to. For Spring 2009, there were 6 questions that were asked both as free response questions as part of the pretests and as multiple-choice questions on the lab final. (Five of these questions were about the rotation and energy labs, because these labs were introduced in the 2008/2009 academic year for the first time and we wanted to see whether they were working.) The distractors for the multiple-choice questions are based on answers given on the pretest. Students were not given answers to the pretests, nor were the pretests discussed after they were administered. In Table 2.1 we give results from the four questions from the Spring 2009 exam (one question based on each of the four labs that have been described in this chapter). Questions 16 in appendix C are the multiple-choice final exam versions of these questions. No data are shown for students enrolled in the physics course for majors because it is not offered in spring semester. In addition, we show some data that compares results from students enrolled in different lecture sections of the calculus-based course in Fall 2008, one taught by one of the lab designers (SK) with an instructional focus that was well-matched to the labs, and with homework assignments that included conceptual free-response questions as well as quantitative problems. The other section was taught with less emphasis on graphical techniques or on qualitative analysis, and used online homework that was primarily quantitative. Questions 7, 8, and 9 in appendix C are 48 Table 2.1. Student performance on matched pre/post questions for spring 2009. Algebra-based sections Calculus-based sections Pretest Lab Final Pretest Lab Final Net Force direction (Question 1): 22% (N = 50) 44%(N = 71) 19%(N = 79) 45% (N = 85) Energy (Question 2): 26% (N = 65) 61%(N = 71) 47% (N = 83) 75% (N = 85) Energy (Question 3): 52% (N = 65) 82%(N = 71) 58% (N = 83) 87% (N = 85) Energy (Question 4): 14%(N = 65) 39%(N = 71) 22% (N = 83) 47% (N = 85) Rotation (Question 5): 41% (N = 63) 65%(N = 71) 43% (N = 60) 71%(N = 85) Rotation (Question 6): 41%(N = 63) 62%(N = 71) 40% (N = 60) 69% (N = 85) Table 2.2. Comparison of student performance on Newton's Second Law questions for lecture sections with different instructional emphases. Lecture not aligned with lab (N = 46) Lecture aligned with lab (N = 40) Newton's Second Law (Question 7): 41% 65% Newton's Second Law (Question 8): 15% 38% Newton's Second Law (Question 9): 63% 55% the questions related to the Newton's second law lab taken from the lab final exam for Fall 2008. In Table 2.2 we compare results for these three questions for students in the section taught by the designer ("aligned") with students in the other section ("not aligned"). 49 Discussion Student performance in general on the lab final exam is disappointing. In Spring 2009, the average score for students in the algebra-based lab sections was 40% and for students in the calculus-based lab sections it was 47%. (We grade on a curve!) In contrast, for the 10 students in our calculus-based course for majors, the average score was 78%. While we believe that the labs are improving our students' conceptual understanding of the material, there are still a large number of students who are not benefiting as much as we would like. We do not know whether there are significant improvements that can be made to student performance by improving the labs, or whether the constraints inherent in a 1-credit course with only one meeting per week means that only incremental gains can be achieved. We will continue to assess student performance and to modify the labs where appropriate. Generally, however, there is improvement in student performance from pretest to final, as can be seen in Table 2.1. It is almost certainly true that not all of this improvement is due to the lab, as students are also learning about and applying the related physics concepts in their lecture course. Nonetheless, we believe that for our students these labs have been at least as successful in promoting student conceptual understanding as the more traditional labs and tutorial that they replaced. When the goals and instructional focus of the lecture portion of the course is aligned with the goals and instructional focus of the lab, students generally do somewhat better on the lab questions. (There are, however, exceptions, such as the results from question 9 shown in Table 2.2) Final exam scores averaged 60% for 50 students enrolled in one of the lab developers' lecture section; the average score was 50% for students from the other section. We believe that this reinforcement is important in both directions, and that the lab experiences contribute strongly to student understanding of the material in the lecture course. One of the test sites for this lab development project, Chicago State University, has chosen to intersperse portions of the labs with their lecture instruction, and this seems to be a very promising model for promoting student understanding. When looking at student understanding of force and acceleration as vector quantities, our pretest results seem to be consistent with results reported elsewhere and suggest that without intervention only a small fraction of students will be able to determine the direction of the acceleration for general two-dimensional motion. There is generally improvement in student performance from pre- to post-test. However, most students do not seem to be applying the procedural knowledge that we are attempting to encourage with these labs. Results are also poor for questions requiring reasoning about forces based on knowledge about acceleration direction. Our post-testing indicates that most students in our lab do not develop a functional understanding of Newton's second law as an equation relating two vector quantities. 51 Conclusion The labs described in this chapter were designed to try to improve our introductory physics students' understanding of the vector nature of acceleration and force as well as to improve their conceptual understanding of conservation of energy and the relationships among quantities characterizing rotational motion. After taking the traditional labs and even after TUP we found our students' understanding in these areas in particular to still be deficient and sought to use digital photography of various moving objects fitted with blinking LED's to deepen student understanding in some of these content areas. We hoped that students practice with vector manipulation in real world examples of objects moving along curved paths would help develop the necessary tools for them to appreciate the utility of Newton's Second Law in many situations and thus seek to use it in understanding a broad range of circumstances. Previous research at other universities indicated that most students cannot readily determine the direction of an acceleration after standard instruction. Our labs seem to also have been marginally successful in improving student performance in this regard. Students do not seem to have significant difficulties with the lab procedures or with the equipment, and they generally like making the photographs and are able to perform the required analysis correctly in the lab. However, post-testing of student understanding of the underlying concepts has yielded disappointing results. Some students, for example our physics majors, do quite well with the post-test questions. On the other hand, many students in the algebra-based course — our intended 52 audience — do not. There are also different success rates among the various topics presented, with students improving performance on concepts related to energy and rotations more than they improved on concepts related to Newton's Second law. The differences in performance among our different introductory physics populations, as well as differences in the successful application of curricular modifications at the University of Maine and the University of Washington heightened our interest in attempting to predict which among our students were more likely to grasp the concepts presented in lab. We sought through the use of pretests to identify potential factors that may contribute to student understanding of the topics presented in the labs. Our attempt to identify some of these factors is detailed in chapter four of this dissertation. Before these labs were developed we attempted to address the same instructional goals through extensive use of motion maps as part of the lecture instruction.24 This also met with limited success. As a stand-alone intervention, the labs seem to be only marginally successful, at least as measured by our multiplechoice lab final. There is some encouragement however in the fact that there is some improvement in student performance when the labs are closely aligned with lecture. We hope to teach a lecture section that emphasizes the vector nature of kinematic and dynamic quantities in the near future with these labs in place as reinforcement of this approach to further improve student understanding. We believe that the goals that have motivated the design of these labs are important ones and we will continue to 53 attempt to explore factors which may assist in helping us achieve these goals, as well as potential barriers to students' understanding. 54 CHAPTER 3: PROPORTIONAL REASONING AS A PREDICTOR OF S U C C E S S IN P H Y S I C S Introduction The lab curriculum development project described in chapter 2 was an attempt to improve our students' understanding of physics concepts through the application of physics education research that was primarily done at other universities with different student populations. We had less success using the research-based materials developed at the University of Washington than the developers of those materials had. One obvious possibility was that the implementation of these materials was deficient in some way. However, our collaborators at California State University Fullerton and Arizona State University had also seen poorer student performance when implementing the Tutorials than they had observed at the University of Washington. A second possibility was that the difference in student populations made a significant difference in what we might expect. The Tutorials were developed through research with students who were in the calculus-based introductory physics course at the University of Washington. The majority of these students have taken high school physics, and admission to the UW is reasonably selective. These students are pre-engineering majors, and must pass these physics courses to be admitted into the engineering program. The engineering program is competitive: for aeronautics engineering, "admitted students generally have grade point averages well above 3.0;" for electrical engineering "the average admission prerequisite GPA varies between 55 3.4 and 3.6." In contrast, NMSU is less selective, and fewer than half of our students have had high school physics. Students are admitted directly into the engineering program, and often do not take physics until after they have taken some engineering courses. In general, our students are not as well prepared for physics as the students who participated in the research at the University of Washington. One of the motivations that we had for focusing on students in the algebrabased introductory mechanics course was that we had observed that the students in this course seemed to struggle with the ideas presented in the lab more than students in the calculus-based course. While this might not be surprising, this difference in performance was apparent even on questions that required very little math. Anecdotally, physics instructors often comment on the differences between these two student populations in terms of both ability and attitude toward physics. For these reasons, we wanted to have some measure of the level of our student's preparation for physics courses. Are there predictors for success in physics that would allow us to compare how well one student population is prepared compared to some other student population? Is it possible to characterize what makes students in the algebra-based course different than students in the calculus-based course? In this chapter we describe previous attempts to measure preparation for physics, and our own experiments with using proportional reasoning pretests as a predictor for success. We do not believe that one can have a complete understanding of how a factor, such as proportional reasoning, influences success in a physics course without also considering other closely linked factors. To this end, we also 56 attempt to give a brief perspective on related influential cognitive theories that have guided previous attempts to identify factors predictive of success in physics courses. Description of Statistics Used Throughout this dissertation we will use standard (frequentist) statistical hypothesis testing when measuring the significance of differences in 'treatment' and 'control' groups. Central to this technique is the idea of the null hypothesis. Under the null hypothesis, it is assumed that the two groups being compared are not inherently different. In general, data is collected and a test statistic is calculated that describes how far the data fall from the null hypothesis value.42 Test Statistics One of the oldest test statistics still in use today is the chi-squared statistic, x2It establishes the contingency between two variables by comparing the observed frequency of data values (f0) against the frequency one would expect to observe for these data values under the null hypothesis (i.e. no association between the two variables), fe. Its value is calculated from x2 = E f 'e where the summation is taken over all cells in a contingency table. Larger values of this statistic indicate that the null hypothesis of independence of variables may be false. Another test statistic used extensively in chapter four of this dissertation is the t-statistic. The t-value measures the number of standard errors (dy) that the sample 57 mean (Y) falls from the null hypothesis value (uo) - t = -—•. For a random sample of size n, the standard error in 7, is related to the population standard deviation a by: dy = -j=. Again, using a t-test, larger t-values indicate increasing un-likeliness of the null hypothesis. Associated with each test statistic is the P-value. This value tells us the probability of exceeding the value of a test statistic under the null hypothesis condition49 (that there was no inherent difference between experimental and control groups to begin with). We will keep with what has become convention since Fisher's 1925 work43 in considering p<0.05 as being statistically significant for test statistics - i.e. there is sufficient evidence to reject the null hypothesis. It should be kept in mind that non-significantly different sample means do not imply equal population means, but may imply that there is not a large enough sample size to detect any difference.44 Often throughout chapter four, our sample sizes from the two populations will be of different sizes and will possibly have different variances. The WelchSatterthwaite t-test is used when the two population variances are assumed to be different (the two sample sizes may or may not be equal)45 and hence must be estimated, and therefore in this dissertation it is often also quoted separately. The effect size46 of a treatment, also called Cohen's d, is a measure of the difference between the treatment and control means (ma - nib) as a fraction of their common standard deviation (a) - d = ™a mb 58 . Researchers have kept with Cohen's convention in referring to effect sizes of about 0.2 as small, effect sizes around 0.5 as medium, and effect sizes above 0.8 as large. Correlation Coefficients The correlation coefficient is a well defined numerical index which describes the degree to which a relationship between two random variables can be modeled linearly. It is commonly represented by the letter 'r' or the greek letter 'p' and provides a number to associate with the correspondence between variables observed on a scatterplot diagram. It ranges in value from negative 1.00 to positive 1.00. The magnitude of the correlation coefficient describes the degree of association of the two variables, while the sign tells us the nature of the relationship: a positive correlation coefficient tells us that a large positive value of one variable coincides with large positive values of the correlated variable, while a negative coefficient tells us that large positive values of one variable coincide with large negative values of its correlated variable (see Fig. 3.1). Researchers are usually interested in predicting some measure of student success such as students' scores on a final exam or students' final grades. In a scatterplot diagram, these variables will usually appear on the y-axis. Factors which may possibly be predictive of our chosen success measure will usually appear on the x-axis. In general factors which are good predictors will be well correlated with our response (success) measure. 59 r=-0.9 # m« • * v y r=-0.8 •* • a • •• • » » «» • •• • • • •• * » •^ x e » . . r=0.0 ' * . • • » « • • •.» *« » r=0.8 *« r=0.3 y «» r=1.0 > • ••» • • » • ** * x Figure 3.1. Sample scatterplot diagrams for two variables x andy and their associated correlation coefficients. Adapted from Kachigan. 47 Cohen46 has popularized a rough, qualitative scale applicable to correlation coefficients found in the social sciences. He referred to correlations of magnitude above 0.5 as large, correlations between 0.3 and 0.5 moderate, correlations 0.3 to 0.1 small, and correlations less than 0.1 insignificant. It is often instructive to researchers using correlation coefficients to focus on their relationship to the coefficient of determination - commonly called R . As its name implies the value of R is equivalent to the square of the value of the correlation coefficient, r. Where multiple predictors are used the coefficient of multiple 60 correlation R, and the coefficient of multiple determination still called R , are the multivariate analogs and satisfy the same relationship. Usually the variable that we would like to predict, e.g. student scores on a final exam, will have some distribution (it is unlikely that all students will have the same score on a final exam). Nearly all reasonable distributions will have an associated mean and (sample) variance. The variance of a distribution is a measure of the average spread of the distribution's values from its mean. It is found by: s2 = — Xi=i(yi — y)2- Whe n two variables are correlated, the average spread of values from the 'line of best fit' is less than the average spread of values from the mean. Using SStotaias a measure of the spread of values from the mean, and SSerroras a measure of spread from the line of best fit, we can see that the ratio of ^ ^ plays an ss Total important role in describing how well the line of best fit (linear regression model48) 'describes' some of the variance in the observed y-values (Fig. 3.2). Statisticians talk about 'partitioning' the variance in y-values into that which is attributable to regression (and by extension to the predictor or x-variable) and that which is attributable to error.49 The value of/? 2 is calculated from these measures by: R2 = 1 - f ^ = 55f/re"i0" Taking, for example, students' final exams as our chosen measure of physics success, R2 tells us what fraction of total variation in students' final exam scores (measured by SSjotai) is accounted for by our predictor variable(s). Many researchers quote the correlation coefficient, with the knowledge 61 3.2(b) r=0.3 SS Totai = d x 2 + d 2 SSrotai = d i 2 + d 2 2 + d 3 2 + ••• + d 1 2 2 SSurror = ^ 2 + e 2 2 + e 3 2 + - + e 1 2 2 = 0 2 SSError K = l-f^=l--p— = 1 scrotal i? 2 = S-'Total = l- e l + e2 : Scrotal + d32 + + e3 + ' • + d27 • + e 27 2 = (0-3)2 Figure 3.2. Scatterplots ofy vs x with mean y-value (dashed line) and line of best fit (red solid line) superimposed. Circles below show enlarged view for various data points' (counting from left for point # n) distances from the mean - d„, and distances from line of best fit - en. that squaring this number gives us an estimate of the fraction of total variation that is accounted for. Review of Previous Studies Predictors of Physics Grade, Mathematics and Education Reform In an early study of factors correlated to success in a physics course, Blumenthal50 reported in 1961 on the correlations between grades on a mathematics 62 pretest and a final exam of an introductory physics course. His study involved 159 students over three semesters at City College of New York. The final exam for the course consisted of two exams: Final exam I was a twenty-five question, multiple choice exam 'concerned with physics knowledge' and final exam II was an 8question 'physics problem' exam. In his own words: "The term 'physics problemsolving abilities' as used herein refers to success in solving problems in physics involving the use of mathematics." Blumenthal found no significant correlation between students' scores on the mathematics pretest with their scores on final exam I. However, there was a 0.4 correlation between final exam II and mathematics pretest. Both exams are undoubtedly measures of success in a physics course, however, the difference in correlation serves as a useful reminder of an inherent problem associated with predicting success in physics - that various styles of exams used as measures of success in physics courses may produce different correlations with predictors. This phenomenon of different correlations based on qualitative vs quantitative physics problems has been noticed by others.77In this dissertation, we will refer to exams that have a focus on numerical solutions or on symbolic manipulation of equations as a 'traditional exam'. Exams that have significant focus on physics concepts and on qualitative reasoning we will refer to as 'conceptual exams'. In recent years, this distinction has been of particular importance to physics education researchers since Hake54 has shown that courses structured where there is interaction among students and between students and instructors, referred to as 'interactive engagement' courses, 63 tend to produce better gains in students conceptual understanding than traditional or passive courses. Mathematics as a Predictor Mathematics is a natural first candidate as a predictor of success in physics. The results obtained using mathematics as a predictor have been varied across institutions and also seems to depend on the type of exam (whether traditional or conceptual) used as a measure of success. Adams and Garrett51 in their 1954 publication of their study conducted at Louisianna State University, reported a correlation coefficient of 0.435 between grades in a physics course and grades in a mathematics course. Only students who successfully completed the sequence: mathematics course, first physics course, second physics course, in consecutive semesters were included in the study. As described later in this section, Rottmann et al. claim that this selection may have had some effect on the measured correlations. The study was conducted over 3 years from 1947-1949, and included 877 students in total. Physics grades in the first physics course (taken in the Fall semesters) were used as the measure of success in physics. In contrast, Hudson and Mclntire used an eighteen-question mathematics pretest with topics including parametric equations, linear equations, graphical analysis, and trigonometry as a potential predictor of success. They conducted their 64 study on 200 algebra-based physics students at the University of Houston and found only weak correlations1" between math pretest score and physics final score. Hudson and Rottmann53 over a 3 year time-span, reported in a study involving 1403 students in an introductory physics course, 913 of whom completed the course and 490 of whom subsequently dropped the course. They found a 0.42 correlation with final grade and a mathematics diagnostic pretest for students that completed the course and a 0.23 correlation coefficient between math pretest score and 'projected final grade' for students who subsequently dropped the course. The studies reported in this section, have found mathematics to be a useful predictor of success in physics courses. Most of the studies cited also sought to use other predictors but have found mathematics to be most useful for their course. Even today, with emphasis on non-traditional modes of instruction, and with significant gains being reported by instructors using these methods54'55 it still holds true that 'mathematical skill' is a good predictor of success in physics courses regardless of instruction method.55 It is important to note however, that none of these researchers found correlations above 0.5 in any of their studies even given the traditional nature of some of their courses. 111 They did not calculate a value, however using their tabulated results, using their binning of 8 pretest bins and 5 final grade bins, a correlation coefficient of 0.3 can be found. This coefficient is not terribly different from results reported by other researchers. 65 Piagetian Ability as a Predictor As a part of his study mentioned in the first section of this chapter, Blumenthal50 also included 'ability to reason' and 'general scientific knowledge' based on scores on college entrance examslv as predictors of final exam scores. The only significant predictor for exam I was general scientific knowledge. For exam II, however, he found coefficient of multiple correlation (essentially the square root of R-squared where more than one predictor variable is used) of r=0.55 for prediction of this final exam score. Final exam II correlation with general scientific knowledge (r=0.2) and mathematics ability (r=0.4) were the two primary variables contributing to this multiple correlation coefficient. In the decades that were to follow, formal operational capacity (sometimes referred to as Piagetian ability) became a second popular variable used by many researchers as a predictor of success in physics courses. Jean Piaget was a professor of experimental psychology and genetic epistemology at the University of Geneva who for many years served as the director of the International Bureau of Education and of the Institut J.J. Rousseau. He sought to form a theoretical framework to explain how the human mind advances from a lesser state of knowledge to a state of 'higher knowledge'. He posited that all humans progress through the logical formalization of scientific knowledge by a process of 1V This entire entrance exam was given across 4 municipal colleges in the City of New York. The author breaks up scores on the exam into three sections: 1) Ability to reason 2) general scientific knowledge 3) mathematics ability. 66 equilibration of thought structures, and in some cases transformation from one cognitive level to another.56 Piaget described human intellectual development as a progression through four stages: sensory-motor, pre-operational, concrete operational, and formal operational stages (see Fig 3.3). Piagetian Stages of Intellectual Development (1958) Formal Operational 11-15 yrs old / Concrete Operational 7-11 yrs old ~7 Pre-Operational 2-7 yrs old ~1 r Sensory-Motor 0-2 yrs old ~1 Figure 3.3. Representation of human intellectual development (from left to right) according to Piaget's 1958 model. He called the first stage the sensory-motor stage, characterized by an infant's developing muscles used in performing reflex actions, and on framing the 67 permanence of an object's existence. He referred to the second stage of development (observed from about two to seven years of age) as the pre-operational stage of development. A major feature of this stage is the development of a child's framing of the world in a less 'ego centric' fashion and the development of moral characteristics such as empathy. He believed that in this stage that children observe how objects as well as people 'behave'. The third stage of intellectual development occurs between the ages of seven and eleven years old, and is called the concrete operational stage. During this stage children are able to organize, seriate, and match objects, and can manipulate data. However at this stage, while they are able to determine how their own action influences the world around them, they experience difficulty in generating hypotheses. The fourth and final stage of intellectual development, called the formal operational stage, is marked by the ability to perform abstract manipulation on classes of objects, with 'concrete reality' taking a less significant role, and with 'possibility' playing a more prominent role. At around the age of fifteen, humans undergo the last transition of intellectual development from the concrete operational to the formal operational stage. In the late 1960's and throughout the 1970's, researchers in the US began to take note of the work done by Piaget.57 They began testing their students for the presence of certain elements of mathematical and logical structure called 'operational capacities' that signal the last transition in human intellectual growth from concrete operational to 'formal' reasoning, even though Piaget's model assumes that a collegeaged population would be largely at the formal operational stage.57 They argued that 68 the main logical operations which characterize formal reasoners can be summarized by: 1) The ability to draw (verbal) analogies; 2) The ability to identify correlations; 3) The ability to deduce the generating principle of, and sequentially follow, verbal and numerical chains; 4) The ability to isolate a variable; 5) The ability to use proportional reasoning. Piaget himself proposed that age was overwhelmingly the dominant factor in determining one's transition through these levels of development. The evidence is strong that it does indeed have a large role to play in performance on 'Piagetian tasks'.58,59'60 It is not, however, the only pertinent factor: many researchers have reported that a large fraction of college students were not at the formal level as measured by Piagetian guidelines.61'62'63'64'65'66 Piaget's is not the only theory on intellectual development. Other very influential cognitive psychologists like Bruner, Sternberg or Gardner might argue that Piaget only measured one aspect of intelligence (for instance Gardner might call the intelligence measured by Piaget 'logico-scientific intelligence' as opposed to six other 'types of intelligences'). Because Piaget used experiments closely related to physics and he focused specifically on scientific reasoning, Piaget's theory of development has held particular attraction for physics education. Paper-and-Pencil Tests of Piagetian Ability Physicists, and especially physics education researchers, have been at the forefront of investigating how Piaget's theory of genetic epistemology applies to 69 physics classrooms. ' ' Two researchers in particular - Robert Karplus and Anton Lawson - devoted significant effort towards trying to develop pencil and paper tests that measure the same abilities as the original Piagetian interview tasks. The Lawson Test of Scientific Reasoning Ability (see appendix D) resulted from some of this 7ft research. Lawson was also very instrumental in summarizing the research demonstrating the unity of 'formal thought'71 and in showing its relationship to a variety of other measures of cognitive ability. In the past researchers used other pencil-and-paper tests of scientific reasoning ability such as the Tomlinson-Keasey Campbell test73 and the Science Logic test.74 The Lawson Test, however, appears to be the most widely used of these tests currently. Various Other Factors Used as Predictors With the rise in acceptance of the Piagetian model of development, many physicists sought to modify the focus of their courses to promote the development of 'reasoning ability'. As a result, when reporting on prediction studies, researchers tended to be a bit more descriptive of the pedagogical content of their courses. Piagetian ability, mathematics ability as well as other factors were explored as possible predictors of success in physics courses. Cohen et al. used a battery of tests at the beginning of their physics courses, as well as SAT verbal and math scores as predictors of physics success for a group of 195 students enrolled at the University of Vermont. Included in each student's battery of tests was at least one Piagetian task. Students in this study (reported inl978) were 70 randomly chosen from 4 different intro physics courses (with grades assigned by different instructors). Choosing a combined subset of 53 students drawn from two of the courses - one course with the highest average math SAT score and the other course with the lowest average SAT math score - they found that verbal SAT score had no correlation with student's final course grade. They also found that Piagetian level only just significantly correlated with final course grade and that SAT math score was the best predictor of course grade with a correlation coefficient of 0.43. 77 Griffith, working with small class sizes of between 20-30 students over multiple semesters at Pacific University, used the Science Logic Test (a test of Piagetian ability developed by Griffith and Weiner) as a predictor of success. He found correlations of between 0.2 and 0.5 (from one semester to another) with students' scores on a conceptual final. Liberman used the Tomlinson-Keasey / Campbell test as a measure of Piagetian ability and found a correlation of 0.49 with scores on this test and final exam score for a group of 67 undergrads enrolled in a physics course for non-majors. 7R In 1981, Halloun and Hestenes as part of an effort to develop separate tests to measure incoming physics knowledge and mathematics ability, reported on the 70 sn predictive ability of these tests. ' The mechanics test was designed to highlight differences between common-sense and Newtonian concepts. The questions on this test went on to form the basis for two of the most commonly used assessments of o i conceptual understanding in mechanics - The Mechanics Baseline Test and the R7 Force Concept Inventory. The math test used as a predictor consisted of 33 71 questions containing (a) ten algebra and arithmetic items (b) six reasoning items (c) four items on graphs (d) six reasoning items (e) five calculus items. This math test was obtained by choosing from a long list of math questions that had the highest correlation with the physics test. In a study of more than 1000 students over 3 years enrolled in both calculus-based and algebra-based introductory physics course at Arizona State U, as well as 50 high school students, Halloun et al. found that factors such as gender, age, major and high school mathematics showed no correlation with physics performance. However, they found a 0.48 correlation coefficient between the mathematics pretest and students' final score for the calculus-based physics students. They also found a 0.43 correlation coefficient between mathematics pretest scores and final scores for algebra-based physics students. They further reported a 0.56 correlation between students' pretest scores and post test scores on the same mechanics test described above (asked at the beginning and ending of the course respectively). In 1979, in a study involving 60 undergraduate physics students in a course for non- physics majors, Liberman and Hudson83 used the Tomlinson-Keasey / Campbell test battery (a composite of pre and post tests), as a pretest measure of formal reasoning ability. The final course score, a composite of 4 exams and homework scores, was used as the predicted variable. They found a 0.49 correlation coefficient between pretest and final score. It is interesting to note that because students scored very well on the proportional reasoning aspects of the TKC test, the scores from this section were left out of the analysis. 72 Whether or not students took high school physics is another variable used to attempt to predict success in introductory physics courses. In the Fall semesters of 1990, 1991 and 1992, Hart and Cottle84 collected a total of 508 questionnaires at Florida State University with students reporting on their: 1) grade in last math course; 2) major; 3) high school physics background; 4) whether or not they completed a degree at a community college or not. Using the Mann-Whitney test, they found a statistically significant difference in physics course grades of students who: 1) Took a physics in high school compared to those who did not (Z=5.4, p<10"4); or who 2) had a math grade above B- in their last math course compared to those who did not (Z=5.0, p<10"V 5 This study was later replicated by Alters86 with over 200 introductory algebra-based physics students at the University of Southern California during the academic year 1993 through 1994. An identical questionnaire was given on the first day of class. He reported a similar significant difference in course grade between students who took high school physics and those who did not. In one of the landmark studies that introduced gender as a contributing factor into predicting success in introductory physics courses, McCammon87 studied 206 students in various introductory physics courses at East Carolina University. They purposefully elected not to use Piagetian ability as a predictor variable, citing the low correlations observed by Griffith in his 1985 study. The final exam in McCammon's study would most likely be considered traditional, as she describes questions appearing on the exams as "problems which require translation between english and mathematics." The predictor variables used in this study were 1) critical thinking 11 appraisal using the Watson & Glaser Thinking Appraisal Test, which tests students on assumption making, deduction, and the interpretation and evaluation of arguments; 2) primary mental ability, an assessment of reasoning, perceptual speed, spatial relations and tests of verbal meaning; 3) a measure of math anxiety; 4) Arithmetic Skills Test - a test developed by the college entrance exam board which typically contains algebra problems including problems involving inequalities, solving equations etc. and 5) Elementary Algebra Skills Test. No clear indication was given on how variation in assessment instruments used by different instructors was controlled for in her study. In this study, these multiple predictor variables held a multiple correlation coefficient with grade of 0.4. Of the individual factors investigated, the most significant predictors of final grade were the algebra test with a correlation coefficient of 0.32 and the arithmetic test with a correlation coefficient of 0.3. The other predictors had lower correlation coefficients and were therefore considered below the commonly accepted level of significance for this number of students. In a separate analysis of the 91 women in the study, however, they found a correlation coefficient of 0.56 between algebra test score and final grade. This result suggests that gender might be an important factor in attempting to account for differences in student performance in physics. We will explore this in further detail in Chapter 4 of this dissertation. 74 Summary of Previously Used Predictors In the large majority of studies conducted so far, attempting to find predictors of success in physics courses, there appears to be a pattern that the most predictive of variables tend to be centered around mathematics, with some sensitivity to which mathematics questions are included on the mathematics pretest. There also appears to be some sensitivity to whether the measure of success on the course was traditional or conceptual. There have also been some potential problems in the studies conducted. Some of the previous studies may have placed insufficient emphasis on specifying when their 'predictor' assessment was administered (in many cases predictors and achievement tests were all used in the same battery of tests). In some studies, course grade, itself composed of multiple midterm (and sometimes even homework) scores, was used as a measure of success in physics. This complicates our understanding of which aspects of success the predictors are tied to. In some studies, results from only subsections of researchers' physics populations were included, which may not have been representative of the entire introductory physics population. In general the correlations found, even those using many factors, have not consistently yielded correlations above r =0.6. A large fraction of the variance in predicting student success in physics remains unexplained, with even the most generous estimates leaving more than 50% of the variance unaccounted for. This estimate agrees with the findings of Sadler and Tai (in their analysis placing this figure at 64%) as well as Kost et al.90 The large fraction of unexplained variance 75 points to the need for either improving our prediction instruments, or for the identification of yet to be discovered predictive factors. In the subsequent sections of this chapter, we will detail some of our attempts to use pretests as a predictor of score on the final exam in our laboratory course (described in chapter two). Our goal was to have a pretest that would help us to reasonably identify underprepared students, so that in the future we would be able to suggest an appropriate intervention for these students. A further criterion influencing the design of our pretests was that they should be short, so as not to significantly infringe on the time allocated to covering physics material in the lab course. Lawson Test Questions as Predictors of Success The Lawson Test was found by Coletta to be a good predictor of FCI gain.91 In our earliest attempts to use diagnostic pretests as a predictor of final performance at NMSU, we used a subset of fourteen questions taken from the Lawson Test of Scientific Reasoning (see appendix D) which we deemed contextually applicable to a physics course. We added one additional question (question 15) which asked students to compare the densities of pieces of a broken block. The final form of the fifteen questions given to the students is shown in appendix E. Lawson suggested various subcomponent ability measures associated with each question appearing on the Lawson test.70 The questions appearing on the fifteen question pretest included questions testing proportional reasoning ability, 76 conservation of mass, conservation of volume, control of variables and probabilistic reasoning. These pretests were given during the lecture portion of two introductory algebra-based physics courses in the Fall of 2005. Seventy-seven students from the Physics 211 lecture course turned in answers to the pretest, while from the Physics 221 lecture course, 48 responses were turned in. For the Physics 211 lecture course 65 students' pretest scores were compared to the final exam of the lab course for which they were simultaneously enrolled. Of the 48 students taking the 15 question scientific reasoning diagnostic in Physics 221 however, few were simultaneously enrolled in the lab course, so instead their pretest scores were compared to their lecture course final exam. In Physics 221, 33 students completed the course and 15 students dropped the course. The average pretest score of the students who completed the course was 10.7 out of 15, while for those students who ended up dropping the course, the average score on the pretest was 8.3. Correlation of students' pretest scores with scores on the final grade shows that about 30% of the variation in students' final scores is accounted for by their performance on the 15-question diagnostic. These results are on the order of the best predictive pretests used in physics courses that we have seen. Our students generally had no problem with the conservation of mass problems. However, they seemed to struggle the most with the proportional reasoning problems. Since the 15-question pretest was still a bit time consuming, we decided to construct a shorter pretest consisting only of proportional reasoning items. 77 Results of Using Questions From the Lawson Test as a Pretest Physics 211 — 30 ra • • c i*Z 25 Si . 5 20 • °1S §io t i$ t* •*• \ • • c • **•• •ft • • 1 • : » • • • 1 1 1 10 15 20 Score on pretest Physics 221 E re 120 • ^ • x 100 cu 80 £ 60 o 40 o u 1% A1 • _< t • • i! • ? • 20 1 1 1 1 10 15 20 Score on pretest Figure. 3.4. Scatterplots of 'Score on final' vs 'pretest score 'for two algebra based physics courses. Top Graph: For students completing the Physics 211 course (N=65) the correlation coefficient was 0.54. Bottom Graph: For the Physics 221 course (N=33) the correlation coefficient 0.58. 78 Table 3.1. For Physics 211 students, this matrix shows the correlation between score on the final exam and various subcomponent ability measures of the Lawson test (according to Lawson10) including proportional reasoning ability, conservation of mass, conservation of volume, control of variables and probabilistic reasoning. Correlation Matrix - Physics 211 Final exam Final exam 15 0 pre Gonsvmass Propreas Cons vol Control var Probaility 1 0.54 1 -0.01 0.18 1 Propreas 0.44 0.78 -0.02 1 Cons'vol 0.16 0.46 -0.05 0.20 1 Gontrolvar 0.22 0.42 0.28 0.03 0.19 1 Probability 0.42 0.69 -0.02 0.34 0.03 0.16 i5Qpre Consvmass 1 Table 3.2 Similar matrix to Table 3.1. for Physics 221 course. Correlation Matrix - Physics 221 Final exam Final exam iSQpre Consvmass Propreas Cons vol Controlvar Probaility 1 ijOpre 0.58 1 Consvmass 0.15 0.03 1 Propreas 0.37 0.80 0.01 1 Cons'vol 0.07 0.36 -0.15 0.14 1 Gontrolvar 0.59 0.53 -0.10 0.24 -0.08 1 Probability 0.40 0.76 -0.09 0.37 0.04 0.47 79 1 Proportional Reasoning Pretests Each semester since Fall 2007 we have used two of three different, short proportional reasoning pretests, assigned to students within the first few weeks of the lab course. Each pretest was composed of five multiple-choice questions. Version one of the pretest consisted of the four proportional reasoning items on the Lawson test, plus one additional density question described in the section above. We refer to this version as the original pretest version (see appendix F). In this chapter, we are primarily concerned with the use of proportional reasoning ability as a predictor of success in our physics course. An additional goal involving use of our pretest, however, was to determine the effect of context variations on performance on our proportional reasoning pretests and in our course. As a consequence of this goal (explained further in chapter four), we also present data for two pretests which were context-variations of our original proportional reasoning pretest. These two pretest versions are referred to as the modified and discrete modified pretest versions respectively. The modified and discrete modified pretest versions were interleaved with the original pretest version during different semesters (the discrete modified pretest version was only used during the Fall 2007 semester). For the Spring semester of 2008, the pretests were only asked of Physics 211 students, and were administered in the latter half of the course. We present data on student performance on these pretest versions in the next section. 80 Performance on Pretests There is a difference in student performance on the three versions of the pretests given to our algebra-based physics population, with Physics 211 students having a mean score of 2.31 out of five on the original pretest version (Std. Error 0.11); of 2.55 (Std. Error 0.11) on the modified pretest version; and of 2.77 (Std. Error 0.20) on the discrete modified pretest version. For the calculus-based course, Physics 215, there was a significant difference in performance on the three versions of the pretest (explored in further detail in the next chapter), with students achieving mean scores out of five of 3.07 (Std. Error 0.13) on the original version; mean 3.47 (Std. Error 0.13) on the modified version; and 3.18 (Std. Error 0.25) on the discrete modified version of the pretest. For reference we also include the results from a course comprised of a small number of students who are mostly physics majors, Physics 213: The mean score on the original pretest version was 3.87 (Std. Error 0.36); on the modified pretest version the mean was 4.17 (Std. Error 0.48); and on the discrete modified pretest version the mean was 3.43 (Std. Error 0.43) for this group of students. 81 Physics 215 Physics 211 £ 8 „ r mean=2.31 2S CD N=254 20 I" Original Version § 10 T3 55 N=161 CO CO £ mean=3.07 §25 5 1 2 3 4 £ 20 3 CO 15 -4-< § 10 "O 3 CO 5 5 Pretest score N-118 N=191 £ mean=2.55 o 25 o mean=3.47 §25 CO CO .c 20 Modified S 10 T3 3 55 £ 20 > CO 15 <D 10 T3 3 5 CO 5 0 1 N=44. 2J mean=2.77 o 25 o CO a N=62 CO o 25 o CO 20 20 Discrete Modified Version 15 £ 10 •D 3 55 mean=3.18 5 0 3 CO c T3 3 55 15 10 5 1 Figure 3.5. The left column graphs show the distribution of scores on the 5 question pretest for Physics 211 students. The right column corresponds to Physics 215 students. Each row represents one of the 3 pretest versions. Superimposed in the background of each graph is the mean score for each pretest population (a single vertical line) and the corresponding standard error of the mean (a narrow band in red on either side of the mean). 82 Performance on Final Exam As described in Chapter one of this dissertation, students enrolled in the lab course at NMSU are given a lab final that is composed of about 36 multiple-choice conceptual questions (about three questions per topic covered on each meeting during the semester). The test is administered on the last day of the lab course and accounts for about 20% of their lab grade. The labs covered in the lab course for our algebrabased, calculus-based and physics majors courses are exactly the same except for one lab each semester (the algebra-based students are generally not required to do the Torque lab and in its stead often do the Buoyancy lab). For ease of comparison among the introductory physics courses, the lab final results presented from here onwards are of the 33 questions that were common for all lab sections. The labs covered from semester to semester were the same and the conceptual questions asked on the lab finals each semester were very similar in nature. Class averages and distributions are virtually the same every semester for each course. We show the distribution of class scores over the period Fall 2007 to Spring 2010 in Figure 3.6 below. For the Physics 211 course, the mean score on final was 15.60, with a standard error of 0.26, out of a possible 33 points. For the calculus-based physics group, Physics 215, the mean score was 17.02 with as standard error of 0.33. For reference the Physics 213 course had a mean score of 21.77 with a standard error of 1.02. There is much overlap in performance on the final exam between the algebra83 10% 8% U) (ft $ '3 6% Phys211 ^ •Ln —f 4% 2% "TTftf B ~ ™. 10 12 N=513 14 _.._...:if 16 18 20 22 24 26 i 28 i r.-M—, 30 32 34 Score on Final Exam 10% •i-i I£ 8% 3 u3 6% - Phys215 ^ N=398 V) # i ™" 4% 2% m _ _rT t—*ra B 10 12 14 16 18 20 22 24 26 2B 30 32 34 Score on Final Exam Phys213 N=43 3 8 6% ^ i 4 % 2% 10 12 14 16 1B 20 22 24 26 28 30 32 34 Score on Final Exam Figure 3.6. Shows the distribution of final exam scores for the three introductory mechanics courses. The red hollow bar in the background is centered on the mean score for each course and its width is indicative of the standard error of the mean. based and calculus-based introductory physics populations. This speaks favorably to our goal of designing labs that were accessible to the algebra-based students, however, we are somewhat disappointed that our calculus-based students in particular, do not perform better. 84 Pretest Version and Correlation The overall correlation of the original pretest version with the final exam for N=211 students over four semesters was 0.38 (significant at the p<10"4 level) for our algebra-based physics students, and 0.54 for N=140 calculus-based physics students (significant at the same level). For the modified pretest version there was generally a lower correlation with N=168 algebra-based physics students, having a 0.29 correlation (significant at the p<10"3 level) with score on final, while for N=109 calculus-based physics students, there was a correlation coefficient of 0.34 (same pvalue). Due to the low numbers of students taking the discrete modified pretest version, we only include these results for the sake of completeness. There was a correlation of 0.49 between student scores on this pretest with the final for 50 Physics 211 students and a 0.42 correlation coefficient with 39 Physics 215 students. These results show that our proportional reasoning pretests account for about 1/5 the total variation in students' final grades. This is slightly better than the mathematics pretests used in the past, although it is not as predictive as some of the other studies that have included multiple factors. Given that the pretests were only 5 questions long and asked at the beginning of the semester, however, they exhibit a high efficiency in predictive ability, indicating that proportional reasoning ability may be a central skill required for student success in conceptually driven introductory physics courses such as ours. It is interesting to note that each pretest version had different predictive ability for our two main introductory mechanics populations, with 85 Physics 211 Original Pretest Version Physics 2 1 1 Modified Pretest Version 30 El a u fe 1 i> » a "« ->0 <S 1 A 15' I •a 1 a : 1 d 101 1 • "ill Fin d c • a» q 1 1 J d .3 ^ (5 8 B 1 , l<! '• C3 • • .3 8 B cq B •a • D •a ii d I 2 0 3 I 2 Physics 215 Original Pretest Version in • 30 D 1 ES] i" « KJ D j D 13 1 {) I 2 3 D 4 I , 0 ! ! •! | 11 • • i 0 EJ 1 ! 1 q d C «is| I Physics 2 1 5 Modified Pretest Version ; r J Pretest Score Pretest Score A A A a • ; , i • 4 1 | 0 q ! a' B. ; d q Q d d 0 i i p ."> Pretest Score Pretest Score F/g 3.7 Scatterplots of student score on final vs score on pretest for two pretest versions for students enrolled in two introductory physics course: Physics 211 and Physics 215. For each group of students with a particular pretest score (e.g. 1) there is a corresponding distribution of scores received by these students on the final exam. 86 a generally higher correlation between final score and pretest score for our calculusbased populations. The fact that there were different correlations for different types of proportional reasoning questions will be explored further in chapter four. This result seems to show that 'proportional reasoning ability' may not be the only factor at work when using our pretests as predictors. Fall 2009 Pretest Versions Fall 2009 was a unique semester in our efforts to optimize our use of proportional reasoning pretests as predictors of final grade. In this semester, we used a pair of pretest versions (described in further detail in chapter 4) that were different from the questions we used on our proportional reasoning diagnostics in previous semesters. On the two (Fall 09) versions, we retained the corresponding versions of questions two and three from our 'original' and 'modified' pretests. The other questions used this semester were new or significantly modified from previous versions. We also had an opportunity to use the entire Lawson Test as one of our weekly administered pretests on the last lab meeting of the semester - one week prior to the final exam. The temporal proximity of these two tests may have had some effect on the correlations observed. For comparison we have included in appendix G the correlations with various components of the Lawson test with the final exam to 87 give a further picture of how the correlations compare with various subsections of the Lawson test. For 128 algebra-based physics students there was a 0.47 correlation coefficient between the proportional reasoning questions on the Lawson test and the final exam score. This was the highest correlation among all of the subsections of the Lawson test. The Total Lawson test score had a 0.65 correlation with the final. Considering the fact that the Lawson test has no questions that are based on physics content presented in the course, we were surprised to learn that it accounted for nearly 40% of the variance in students' final exam scores. Our proportional reasoning pretest, asked at the beginning of this semester, had a 0.42 correlation with the final for these students. For 63 Physics 215 students taking pretests, final exam and Lawson test, there was 0.64 correlation with the final exam score (a value similar to Physics 211). Again the proportional reasoning items on the Lawson test correlated best with the final with a coefficient of 0.63 (a value about as high as the total score on the Lawson test). Interestingly also for this group, the pretest at the beginning of the semester had a correlation of 0.62 with the final score. These findings suggest that proportional reasoning may stand on its own as a valid predictor of success in a physics course such as our own that emphasizes conceptual understanding. It is also interesting to note that the correlations found with proportional reasoning items were about the same whether these questions were asked at the beginning or at the end of the semester. 88 It should be noted that our labs do not attempt to directly improve our students' proportional reasoning abilities. In a lab course at Rutgers University for engineering students, the same pretest used in Fall 2009 at NMSU (unmodified version) was given to students near the end of Fall 09 in the lab section of their course. (Since this was given some time after relevant instruction it might be more accurately considered a proportional reasoning post-test). In that course, students' lab and lecture grades are combined into an overall physics grade and there is greater synchronization of the material presented. Students' ratio reasoning is explicitly elicited as part of invention tasks designed to improve students' conceptual understanding of kinematics. Those students performed markedly better on this pretest than our mechanics students, averaging 3.6 out of 5. Also, the correlation between students' final exam score and pretest score for this group of students was 0.22 (N=101; p-0.023). While this correlation was significant at the 0.05 level, it was not nearly as strong as the correlation that we have observed at NMSU with our of Phys 211 and Physics 215 populations. In an effort to improve the predictive ability of our pretests, we have experimented with adding measures of students' working memory capacity into the prediction of students' final exam score at NMSU. We were led to this construct by anecdotal instructors' characterization of algebra-based physics students as 'memorizers', while this characterization was not made of calculus-based students. As described in subsequent sections, some psychologists have found significant correlations between measures of working memory capacity and higher level 89 cognitive functions. We hoped that measuring our students' working memory would highlight any differences in the efficient use of memory between our introductory physics students, and how these differences affected student success in our physics course. A description of our efforts in this regard, along with some preliminary results, is presented in the next section. Working Memory Capacity In an attempt to explore potential measurable factors that would help us characterize some of the differences between the algebra-based and calculus-based introductory physics populations, we conducted preliminary experiments to measure our students' working memory capacities. Further, we determined how predictive these measures were of students' final exam scores. As far as we are aware, there have been no attempts to use the construct of working memory capacity as a predictor of success in physics, although as described in subsequent sections, a few researchers have used similar constructs in predicting physics grades. Description of Working Memory Capacity Working memory as a construct falls within the domain of informationprocessing theories, which generally view all cognitive processes in terms of a relatively small number of mental sub-processes and representations which interact with stimuli to produce definite responses.93 Lautrey94 notes the shift in dominance 90 from Piaget's theory to information-processing models of cognition towards the end of the 80's, with the new emphasis on exploring the limits imposed by processing capacities to explain individual differences. Early neo-Piagetian researchers have used measures of 'processing capacities' as a means of explaining the persistent problem of Piagetian horizontal decalage the inconsistent solution of problems supposedly requiring the same cognitive processes according to Piaget's stage theory.95'96 Among the capacities used by these early researchers are M-Capacity (central computing space) and field dependence / independence - a measure of cognitive style closely linked to the ability to extract useful information hidden among useless information (one common task to measure this is the popular hidden figures task). Other researchers abandoned the Piagetian model altogether. ENVIRONMENT (INPUT) MtRCEPTUAL PROCESSING CYCLE '1 PROCEDURAL MEMORY COGNmVE PROCESS!!*! CYCLE womcwa MEMORY COGKTT1V6 PROCESSING CYCLE DECLARATIVE MEMORY MOTOR SCESSWO ro< CYCLE ' RESPONSE (OUTPUT) Figure 3.8. Information-processing model of memories and processing cycles. Individual differences occur in either the capacity of the memories or the speed of the processing cycles. 91 Figure 3.9. Baddeley and Hitchl974 model of the inner components of Working Memory Baddeley and Hitch describe Working memory (see Figures 3.8 and 3.9) as a highly dynamic form of memory that operates over periods of seconds and temporarily stores selected information for detailed analysis." Working memory span tasks typically involve some complex activity (such as reading sentences aloud) performed concurrently with an item retention task (such as remembering the last word of each sentence read).100 The span or capacity of a span task is usually quoted as the maximum number of items one can remember while efficiently performing the 'complex activity' associated with the task. Working memory has been linked to cognitive roles such as: memory, focus, attention101, and inhibiting irrelevant information.102'103 Researchers also find the working memory construct useful in explaining why processing speed varies with age (with adults typically exhibiting greater processing speed than children),93 and in offing insights into reduced performance when experiencing math anxiety (Ashcraft and Kirk10 ) and general 'choking under pressure' (Beilock and Carr 105 ). We believed WMC would make a 92 good candidate to account for variance in individual students' lab final exam scores independent of that which is accounted for by the coarser, Piagetian four-stage measure. Previous Use of Working Memory as a Predictor In one of the most cited experimental results in support of the success of executive function (shown in Fig 3.9 as a component of WMC) variables as predictors of cognitive success, Bull and Scerif,106 tested 93 third grade students from six Scottish elementary schools on a variety of executive function measures. The mathematics test consisted of single and multi-digit addition and subtraction as well as the Group Mathematics Test. It was found that a number of executive function measures correlated significantly with achievement on their mathematics test, the highest of which were 0.44 (p<.01) with counting span (a working memory span task that involves counting aloud while remembering specified numbers) and -0.46 (p<.01) with a measure of field dependence/ independence . More recently, Lepine et al.100used various span measures as potential predictors of scores on nationally administered tests of reading and mathematics for 93 French 6 graders. They found correlations of about 0.35 with test scores when compared to scores on span tasks as described by Barrouillet et al (2004). Aside from the results of studies involving children's reading and mathematics,108 Hambrick and Engle109 noted the lack of research on the role of working memory in problem solving or performance on real 93 complex cognitive tasks (as opposed to 'toy model' cognitive tasks), echoing the sentiments of Hutchins (1995). In one of the few studies to use elements of the working memory capacity construct as a predictor of success in a physics course, Cilliers et al110 measured 2D and 3D spatial rotation ability, memory tests of meaningless words and symbols buried within a paragraph, as well as visual perception speed on a group of 75 students at the University of South Africa at Unisa. They did not find significant correlation with any of these psychometric measures with exam or final score. Whereas these processing measures have been shown to be highly correlated with one another, their relation to achievement in physics has been disappointing in general.95'111 Working Memory Capacity as a predictor at NMSU As a preliminary investigation into whether or not working memory capacity bore any relationship with student success in our introductory physics labs, we solicited volunteers from among students enrolled in our physics labs between Spring 2008 and Fall 2009. The study was strictly voluntary and students were not offered any extra-credit or other incentive to participate in our study - as a result of this, the number of students participating in this study was relatively low. We used the Blackboard online learning management system's quiz tool to host and time our span tasks. In order to prevent interviewees from anticipating which 94 spans they were to be tested on, the ordering of the individual screens was such that the spans tested were random. Engle et al.112 described a similar random ordering of set sizes in their experiment. Our span tasks were also self paced working memory span tasks. The time savings automated span tasks affords over individualized, experimenter-run task measures have been also noted by Engle et al.112 The ease and time savings of computer run span tasks have been found to translate into larger potential experimental populations, a problem with many of the studies involving working memory thus far. Our span tasks were available to students from the lab course's online course management web page throughout the duration of the course this meant that students could participate whenever they found some free time. In spite of this only about 120 students participated in the study. Description of Our N-Back Test The n-back test is a simple span task where the student is shown a series of cards (or slides) after each of which the student is asked to recall the card that was shown to her "n" cards previously. In our version of the task, in order to minimize mental processing power used in choosing students' responses (leaving nearly all of their mental processing power to be devoted to the memory task), we narrowed our available responses to two choices: same if students thought that the slide in the current question item was the same as the slide in the question item 'n' slides back; or different if they thought they were different. It is precisely because of the relatively low processing component (complex activity usually associated with working 95 memory span tasks) associated with the n-back task that some researchers have questioned the validity of using this task as a measure of working memory 1 1 1 capacity. We will however accept the n-back task as a valid measure of working memory capacity. After an introduction into the rules of the 'game,' including information on what is meant by a slide (see Figures 3.10 and 3.11.) and which slide we are asking them to compare the present slide to, students are led into a practice-run quiz, where they are allowed to play the game with feedback given at the end of a short 'practice run' quiz. The release of the scored version of the quiz is conditional upon volunteers successfully completing at least two items correctly on the 5-question practice run, including the final question which asks whether or not they understand the rules of the game. 3 . (Points: 5,0) n-back=2 O a. same G b. different 1 Save and View NextJ jj^ext Question J Finish j ' Help • Figure 3.10. Question 3. The large number one with the box around it is referred to as a slide. In this question students are asked to compare this slide (number 1) with the slide encountered two slides previously (n-back=2). 96 5 . (Points: 5.0) n-back=4 ',; a, same ..' b. different Save and View Next | Next Question F i n i s h ]>•.«• ••! H e l p j . Figure 3.11. Question 5 (shown to students two screens later). Asks students to compare this slide (number 2) to the slide shown 4 slides ago. Scoring of Our N-back Test There are differences in the scoring criteria used by researchers in determining memory spans. For instance, Bull106 considered the span to be one less than the value where there was a double failure on recall of both cards used to determine the span length (i.e. a success rate greater than or equal to 50%), while Leather114 considered success rates of more than two out of three trials to be indicative of successful span capacity at that level. Further, some of the original researchers into the construct such as Daneman and Carpenter115 define (word) span as the maximum number of sentences the subject could recall while maintaining perfect recall of final words (i.e. a perfect success rate). Conway et al.116 have shown, however, that there still exists high construct validity even when different scoring schemes are used. The results quoted in the remainder of this chapter are for a scoring schema where spans are not 97 calculated directly, instead the percentage of correct responses is recorded for each student. We wish to emphasize, however, that we have also calculated these results using scoring schema similar to the researchers cited above without much difference in the correlations quoted. Results of Our N-back test In our preliminary study to use the working memory capacity construct as a predictor of success in our laboratory physics course, we used an n-back test as a measure of our students' working memory capacity. Over four semesters at NMSU about 60 Physics 211 and about the same number of Physics 215 students volunteered to take our standard working memory test (WMT). The mean final exam scores of these students were slightly higher than the average score of the general introductory physics population reported earlier in this chapter. There was a correlation slightly above 0.2 for students enrolled in both courses, when comparing their score on the working memory test ('Grade' on the scatterplot) with student's score on the final exam (see Fig 3.12). When brought in as a second predictive variable of final exam score into a regression equation already including our proportional reasoning pretest scores, however, the combination of low correlation and low student numbers leads to the result that as a predictor variable, score on the WMT may not be a significant factor (p>0.4). 98 Scatterplot of Final Score vs WMT score Final33 40 i i i i I i i 30 ' • • i > i 40 i i i I i 5 0 i i i i i i i i I i i i i i i i i i I i i i i i i • i i I i i i i i i i i i I i i i i i i i i i I 6 0 Grade Course 000phys211 + + Phys213 + + + Phys215 Figure 3.12. Scatterplot of students' final exam score vs percentage of working memory capacity item correctly answered. The time students took to take the test was also not a good predictor of success on the lab final exam for our students (see Fig 3.13 in appendix H). Neither was the efficiency of working memory score, which was calculated by dividing the score on the working memory test by the time taken to make this score (see Fig 3.14 in appendix H). In summary, even though our experiment was preliminary, we feel that the results from our n-back test were not encouraging. It is possible that relying on results from a subset of students who volunteer to take a test biases our results to show weak to no correlation with final exams. It is also possible that some students may have 99 played unfairly to inflate their scores. While the aforementioned are both possible causes of the low correlations observed between n-back scores and final exam scores, we believe the lack of correlation to be indicative of a weak connection between student ability to perform well on the n-back task, and student understanding of physics as measured by our lab final exam. Further testing may be necessary to show otherwise. Spatial Working Memory Test 117 Daneman and Carpenter suspected that the reason that traditional memory span tasks failed to correlate with comprehension ability was because the source of individual differences in memory capacity rested with the brain's functional capacity - the amount of capacity left over for storage after the requirements for processing had been met. They presumed that less skilled cognitive performers process information less efficiently, resulting in reduced storage capacity. At NMSU, we also attempted to measure students' working memory capacity through use of a spatial working memory task, also commonly referred to as counting span task.106These tasks involved a higher level of processing power as students had to count the number of squares, located at different screen positions, while remembering the counts from previous slides. The release of this test was conditional upon successful completion of the practice test and the n-back task. This was done because we expected low participation rates and wanted to have the highest number of students volunteering for the same task (the n-back task) as possible. The number of students completing the 100 spatial working memory task was 25, lower than the number of students taking the nback task. The task we used was identical to the one described by Towse et al.,118except that the time for which each slide was displayed was not computer driven, but was instead user defined (the user clicked 'next question' when she was ready to move on to the next question). 3 . (Points: 5.0) • • • • • • • • • • • • n-back=l C a. 6 blue squares O b. 7 blue squares O c. 8 blue squares O d. 9 blue squares Save and View Next ! Next Question [Finish J,..:": >;;•.:•:] H e l p i Figure 3.15. Spatial working memory test slide. 101 Results Using Spatial Working Memory Capacity. Neither the spatial working memory capacity score (r= 0.03) nor the time taken to complete this task (r=0.14) correlated well with students' final exam grade for the 25 students who volunteered to take this test. It also failed to significantly improve the overall pretest predictive ability when added as a second variable when trying to predict final grade. The low numbers of students taking this test makes it difficult to say whether or not this test was more or less predictive of physics grade than the n-back test (see Figures 3.16 and 3.17 in appendix H). Summary of Working Memory Results The results of our working memory capacity experiments did not lead us to finding a secondary predictor of success in physics as we had hoped. In our estimation, neither n-back nor spatial tasks, upon preliminary investigation, held out much hope of being more predictive of success in physics than the predictors already identified by other PER researchers. In addition, we believe the working memory construct may not hold out much promise of adding to the predictive ability of other pretests. We do wish to emphasize, however, that our investigation into this construct was preliminary, and the number of students participating was low, leaving open the possibility that other researchers may find this construct predictive of success in physics. 102 Conclusion One common theme in many of the studies conducted so far is that there are many problems associated with the variables used in attempting to predict success in physics courses. Using tests in a battery may introduce an unwanted time factor119 as does using final grade which is composed of a collection of many midterm grades. Equating success measures across courses taught by different instructors using different exams is also problematic. Tests of scientific reasoning ability such as the Lawson test seemed to be at the heart of many of the most predictive pretests used. Our results using at first the 15 item subset of these questions, then with the smaller subset of proportional reasoning items, resulted in pretests which were shorter than many of the traditionally used tests of Piagetian ability, without the loss of much predictive power. In chapter four, we explore factors which affect students' performance on these proportional reasoning pretests. We try to determine what effect surface-features such as question context have on student performance, and whether any differences in performance have deeper implications for some student populations over others. Finally, whereas other researchers have found the working memory capacity construct to be a useful predictor of higher cognitive ability97so far we have not found this to be the case for our physics courses. A study with larger student numbers may be necessary in order to observe if working memory span accounts for unique variance that is unaccounted for by proportional reasoning ability. Such a test in 103 conjunction with our proportional reasoning test, however, will lose the feature of brevity and may not be a better predictor than the Lawson test. 104 C H A P T E R 4: C O N T E X T D E P E N D E N C E OF P E R F O R M A N C E ON PROPORTIONAL REASONING TASKS Two Perspectives on Student Difficulties As described in the previous chapter, Piaget saw learning as a process of acquisition of abilities, resulting in progression through various stages of development. In an example of a description of such a process, he states:120 "Once the child possesses the operation of seriation he has opened himself to a whole range of new behaviors.. .he becomes able to construct, understand and cope with new relationships among objects not possible before. Indeed the acquisition of seriation is a Copernican revolution for the child."In Piaget's view of intellectual development (as described in the previous chapter), learning is achieved through the equilibration of thought structures in a process of adaptation through assimilation and accommodation. Many physics education researchers viewing the repeated and replicable student difficulties encountered by students when asked certain physics problems122 believe that students come into their physics classrooms with certain relatively strongly held beliefs which are usually referred to as misconceptions. Researchers framing student difficulties in this manner tend to adopt a model for the development of physics knowledge similar to Piaget's adaptation and often employ strategies such as elicit, confront and resolve, in attempting to correct these student difficulties. 105 McCloskey et al. for instance, asked students to consider the horizontal motion of a ball tied to a string whirled in a circle above a person's head. He asked students to draw the horizontal path of the ball if the string were suddenly cut. He found that about one-third of the students interviewed drew curved paths upon release. He went on to use apparatus similar to that shown in Figure 4.2c to ask students to predict the path of a ball bearing initially constrained to move in a circular path. He found that most students were surprised that upon leaving the circular segment, the ball bearing did not continue to move in a circular trajectory. McCloskey attributed these findings to students possessing a strongly held impetus theory of motion, where any object set in motion was given an embedded 'force' called impetus, which would keep it moving in this way until this impetus died out. This theory of motion was prevalent in the 14th-16th Centuries, and as such is sometimes referred to as a 'medieval' impetus misconception when used to explain students' wrong answers on similar questions. An alternate view on repeatedly observed student difficulties is held by researchers holding to the knowledge in pieces model espoused by DiSessa126 and others. In this view of student thinking, often called fine-grained constructivism, DiSessa states that in answering physics problems students are heavily guided by their sense of 'intuition'. He claims that it is only through the reorganization of the elements of intuitive knowledge that scientific knowledge can be attained. He refers to the smallest elements of intuitive knowledge: things that 'are' because "that's the way things are," as phenomenological primitives (p-prims). He believes that 106 students' cognition is a manifestation of the activation of a certain set of p-prims which are called and ranked based on their reliability (i.e., past experiences in 'similar' situations determine which p-prims are called and which among these are called first). One example of a p-prim is the Ohm's p-prim, where an agent acts against a resistance to produce some result. This p-prim often leads students into framing many physics situations as whether or not an agent or the resistance is 'winning' or 'overcoming'. Many fine-grained constructivists share DiSessa's view that much of students' intuitive knowledge is loosely connected and inarticulate. Researchers such as Hammer and Elby refer to these pieces as 'resources' while others such as Minstrell129 call these units of thought 'facets'. These researchers believe that when students answer questions, they activate and assemble these elements 'on the fly', and their answers are strongly sensitive to context.130'131 Experiments showing that students can be cued into answering virtually the same problem in markedly different ways support a knowledge-in-pieces view. The instructional implications of viewing knowledge in such a fine-grained manner are immediate. Rather than viewing student learning as a process of replacing misconceptions with more productive beliefs, these researchers view teaching as a process of utilizing student 'schema', or activating student 'resources'. In attempting to help students construct physics knowledge from their own sense of intuitions, programs such as the Maryland Tutorials in Physics, 107 encourage students to activate the correct set of resources when solving physics problems. They often refer to this activation of correct resources as sense-making. In one example where the cuing of different resources elicited different student responses, Frank et al.132asked students to consider the time of flight for three balls rolled off of a table (see Figure 4.1). Half of the students were asked to compare the time of flight for the three balls given that each ball had a different initial (horizontal) velocity when sent rolling off of a table (they referred to this as a speedcue). The other half of the students were asked to compare the time of flight for the three balls given that each of the three balls travelled different horizontal distances before they hit the floor (they referred to this as a distance-cue). SO. (b) Experiment 1: Ejperiitifint2: _2*_ E3£)eriment3: I _^J -3*- Projectile question: distance-cuing. Figure 4.1. Students are asked to rank the time it takes for the ball to travel from the table to the floor in three different situations: a) speed-cuing: students are given that the three balls have different initial speeds b) distance-cuing: students are given that the balls travel different horizontal distances before hitting the floor. McCloskey asked similar trajectory-based questions in his earlier mentioned study, and attributed students' incorrect responses to students misconceptions of each 108 ball containing different 'impetuses' which would 'resist' the pull of gravity differently. Viewed from this misconceptions standpoint, one would expect both speed-cuing and distance-cuing situations to equally lead students to the incorrect conclusion that the fastest (or furthest-travelling) ball would have the most impetus, enabling it to resist the pull of gravity longest, and therefore having the longest time of flight. Frank et al. found however, that for the speed-cued situation, students were more likely to answer that the fastest-travelling ball would travel for the shortest time, than for similarly distance-cued students to answer that the furthest-travelling ball would travel for the shortest time. Students choosing this answer on the speed-cued questions based their answers on the fact that faster travelling objects take a shorter amount of time to travel, a reasoning pattern inconsistent with the 'medieval impetus misconception' mentioned earlier. This result led researchers such as Frank to conclude that depending on the context of questions asked different resources (more distance equals more time etc.) were activated. Example of Context-Dependent Performance Figure 4.2c shows a question appearing on the FCI40 in which students are asked to predict the trajectory of a ball after it exits a circular channel. Over half of all students predict that the ball continues in a circular path (answer A). McCloskey,134 who introduced this question, was an early proponent of the misconceptions interpretation of student difficulties on physics problems. As mentioned earlier, he 109 saw this result as part of students' naive impetus theory of motion developed through their every-day experience with moving objects. Kaiser et al. however, in a study conducted at the University of Michigan, sought to challenge whether or not students answered this question incorrectly because of a strongly-held 'circular impetus' misconception of motion or whether students' responses were more context dependent, and therefore more likely to be explained by a resources model of student thinking. In Kaiser et al.'s study 80 students recruited from a hallway were asked two versions of this circular channel question (see Figure 4.2). Half of the students had taken physics either at the high school or college level. There were equal numbers of males and females in the sample. The researchers asked half of the students to predict the path of a ball that rolls through a coiled tube (shown in Fig 4.2a and described in their paper). These students were then asked to predict the path that water coming out of a hose would take if the hose was coiled as in Figure 4.2a. These two questions were posed in reverse order to the other half of students. The researchers chose the water problem since "water shooting from a curved garden hose is a closely related event that is familiar to most people." Students performed significantly better on the water question than on the ball question, x2 (1) = 12.13, p<.005. They also found a smaller gender effect on performance for the water problem than there was for the ball problem (x2 (1) = 4.59, p<.05 compared to x2 (1) = 9.45, p<.005). Notably, they found no evidence of transfer, as students answered at about the same rate on both problems whether asked 110 Figure 4.2. Students are asked to predict the trajectory of: a) water exiting a curved hose b) a bullet exiting the barrel of a curved gun and c) a marble as is leaves a circular tube (appears on FCI). first or last x = 2.26, p>.10. In a modification of this experiment involving 81 female students a second intuitive context of the same problem was added (Figure 4.2b). Here students were asked to draw the trajectory of a bullet as it leaves the curved barrel of a gun. In a similar experimental format, the two intuitive examples were asked before the ball problem for half the students and after for the other half. Again there was no evidence of transfer of correct solutions from the familiar problems to the less familiar problem, x 2 (1) = 1-29, P<.005; and there was also similarly better performance on the intuitive problems. Ill The findings of Kaiser et al.'s study brought to light two important results that are pertinent to physics education research. That students could solve this problem in the familiar contexts and not in less familiar contexts leads one to think that students do not hold rigid beliefs that all objects constrained to move in circular paths inherit a property of 'circleness' of their motion (a 'misconceptions' view posited by McCloskey). Researchers such as Elby would argue, on the contrary, that this is an example of different contexts activating different resources, and that the significantly different response rates support a resources view. Another important finding highlighted in this study was that the difference in performance between males and females was less marked for the contexts that were 'familiar'. We wanted to explore further whether the context of our proportional reasoning pretest would have some effect on student performance. In consultation with Andy Elby we discussed the possibility that familiarity of contexts used in physics problems may matter to some students and have some effect on their performance. Elby suggested that familiar contexts may activate more appropriate resources, which would enable students to solve our proportional reasoning problems. It is against this backdrop that we decided to modify the versions of the pretest used to test proportional reasoning ability. 112 Background to study at NMSU As part of our efforts to explore whether or not some part, or all of what is claimed to be measures of proportional reasoning ability by our pretests is really a manifestation of a conglomeration of non-ability factors such as student familiarity with scientific contexts and vocabulary, for 6 semesters we have been using two versions of a pretest designed to measure proportional reasoning ability. One version of the pretest (referred to as the original version throughout this chapter), has been administered every semester (except Fall 2009) in our lab course since Fall 2007. This version consists of four proportional reasoning questions drawn from the Lawson Test of Scientific Reasoning136 in addition to one density item, as explained in Chapter 3. Each semester since Fall 2007 we have used an alternate pretest within our lab course, interleaved among pretests of the original version. Our goal was to see if there was: 1) A difference in student performance on the two versions of the pretests by controlling for other factors on the two pretests except for the contexts of these pretests and 2) a difference in predictive ability for these two pretests. There does seem to be some predictive power in questions about 'scientific reasoning' in terms of student performance in physics classes, as has been described in the previous chapter of this dissertation. We wanted to see, however, if a pretest based on a more familiar situation might trigger students to use 'sense-making' in their solution of physics problems, as described by researchers such as Elby and 113 Redish, ' and whether this might have an effect on their performance on such questions or on the predictive ability of these pretests. Method In the Fall of 2007, the second version used as an alternate to the original pretest version was a pretest designed to highlight common-sense reasoning in proportional reasoning items. In this example, referred to as the 'discrete modified' version of the pretest in the previous chapter [see appendix F], the Lawson questions were re-written to include the same numeric quantities, but with a context that we thought would be more familiar. The mathematical steps required to solve the ratio problems were exactly the same. The context however, was changed to one based on serving rice in a soup kitchen, a more everyday context in our estimation. The two versions of the tests were alternated, so that every consecutive pretest was of a different version. Students arriving at the lab were then given a pretest from the top of the stack, thus randomizing the version of the test each student received. As shown in Figure 3.5 of the previous chapter, there was a significant difference in performance on the alternate version of the pretest for our algebra-based physics population. This version (discrete modified) was used as an alternate version for only one semester however, because of its failure to account for an additional difference between the two sets of problems: The discrete modified pretest version was composed of proportional reasoning problems involving materials that were discrete 114 (a countable number of scoops of rice) while the original Lawson questions involved materials that were continuous (water poured from one container to another). The difference in student performance on discrete versus continuous proportional reasoning questions has been shown by Lawton139 and cited by Pulos and Tournaire as a context variable that affects student performance. From Spring 2008 to Spring 2010 (except Fall 2009), the second version that we used as a more commonsense alternative to the original pretest version was in the context of a mother using two different sized dosage cylinders to pour medicine for her children (see appendix F). For the alternate version of the broken block question (question one on the 5 item pretest), we used 'cheesiness' as a commonsense alternative to mass density, except in Spring 2010 where we asked about the density of a broken brownie (appendix F). Results for all students Because the discrete modified version of the pretest was only asked for one semester (Fall 2007) and the total numbers of students taking this pretest was low, we will ignore the results obtained in the data we present here. Of the over 700 students taking either the original pretest version or the modified pretest version during the period Fall 2007 to Spring 2010, there was a 0.27 difference in score out of 5 (by students taking the different versions of the pretests, with students scoring marginally (10%) better on the modified version of the test version. This difference meets the 115 commonly-used threshold level of significance of 5% (see Table 4.1 and Table 4.2) using the appropriate Satterwhite t-test for unequal variances (as described in chapter 3). The effect size for this difference is, however, small (d~0.2). Table 4.1 and Table 4.12 show summary statistics of student performance on the pretests. Table 4.1. Descriptive statistics for 2 pretest versions Fall 2007 to Spring 2010 Pretest Version N Mean Score Std. Std. (out of 5) Dev Error Original 431 2.654 1.760 0.085 2.924 Modified 315 1.588 0.090 1.690 0.125 Difference (2-1) 0.270 Table 4.2 Summary oft-test analysis for difference in Table 4.1 Method Variances df t-Value Pr > |t| Pooled Satterthwaite Equal Unequal 744 711.8 2AS 2.19 0.034 0.031 This result implies that for a significant portion of our introductory mechanics population, the original pretest version is more difficult to answer than the modified pretest version, while to an expert, these pretests require the same proportional reasoning steps and abilities. We can then attribute this small effect to the difference in contexts, controlling for proportional reasoning ability (this difference is significant at the p<0.05 level which is commonly considered significant140). 116 Difference in Student Performance by Course A primary motivating force behind the development of lab exercises described in chapter two of this dissertation was to address student difficulties faced by our algebra-based physics population. Many instructors recognize a difference in the algebra-based and calculus-based physics students beyond that which can be attributed to a difference in mathematical ability. Instructors often complain that the engineers are obsessed with 'asking for the right formula', while the algebra-based students are sometimes known as 'grade grubbers' with little to no interest in actually understanding the material. Despite these apparent differences, the overwhelming majority of research in PER has been done with calculus-based students141 with the (perhaps incorrect) expectation that difficulties revealed by these students are also present in the algebra-based population. As a first attempt to explore whether or not the pretest versions were 'more different' for one group of students over another, we looked for differences in the results for our calculus-based physics students compared to those for our algebrabased physics students. A look at differences in overall score on each pretest version can be gleaned from Figures 4.3 and 3.5. In this chapter we will look at the differences in performance of the two introductory physics populations on individual questions as well as differences in performance on the two pretest versions within each course. 117 The calculus-based physics students outperformed the algebra-based physics students on every question on both versions of the pretests. On the original pretest version, Physics 215 students scored about 0.8 points higher on average than Physics 211 students, and on the modified version of the pretest, the difference in the mean Physics 215 score was about one full point higher than the score for their Physics 211 counterparts. The effect size was medium for both pretests, with course having a slightly larger effect on the modified version (see Tables 4.3 and 4.4 in appendix I). On every question on the standard pretest version, the average Physics 215 student performed significantly better than the average Physics 211 student. This difference in performance between Physics 211 students and Physics 215 students was at the (p<0.05) level for question number five and (p<0.01) for question number four. For all other questions, the probability the observed difference in performance was due to random sampling was less than a 0.1% as can be seen in Tables 4.3 and 4.4 in appendix I. The effect sizes of these differences were all small-medium. In total, based on performance on this test, these two introductory physics populations are significantly different. On the modified version of the pretest however, there was not a significant difference in performance for students enrolled in the two mechanics courses on question number five. There was however a statistically significant difference in performance between the two groups on all of the other questions of the pretest. The difference in average total score on this pretest between the Physics 211 and Physics 215 students was also statistically significant [see Tables 4.5 and 4.6 in appendix I]. 118 The effect sizes of these differences were medium (larger than those for the corresponding questions on the original pretest), for all questions except number five - which virtually showed no effective difference in the two populations' scores. Using this pretest version as a guide (except probably for question five), there remains a significant difference in performance between these two introductory physics populations. By looking into achievement on each of the pretests within the two main courses, we were able to explore each population's sensitivity to version of the pretest taken. Students enrolled in the algebra-based course performed about the same on the two versions of these pretests. For our Physics 215 (calculus-based mechanics) population however, students who answered the modified pretest version scored about 0.4 points out of 5 better on average than students who answered the original version. This difference was significant at the 0.05 level [see Figure 4.3 and Tables 4.7 and 4.8 below], however the effect size was small. When comparing question-byquestion performance within each course, for Physics 211 there was a significant difference in performance on the two versions of question number two and question number three (p<0.05 andp<0.01 respectively), but there was not a significant difference in performance on the other questions of the pretest. For the Physics 215 students however, there was a statistically significant difference in performance on the two versions of the pretest only for question three (p<0.01). 119 Table 4.7. Descriptive statistics for difference in performance on 2 pretest versions within each course. Course (Physics) Pretest Version N Mean Score (out of 5) Std. Dev Std. Error 211 I 2 Diff(2-1) 254 191 2.307 2.550 0.243 1.749 1.572 1.675 0.110 0.114 0.160 215 1 2 Diff(2-1) 161 118 3.075 3.466 0.392 1.664 1.448 1.577 0.131 0.133 0.191 Cohen's d: Physics 211 d=0.15, Physics 215 d=0.25. Table 4.8. Summary oft-test analysis for difference described in Table 4.7. Course Method Variances df t-Value (Physics) Pooled Equal 443 1.51 211 429.2 1.54 Satterthwaite Unequal Pr>|t| 0.131 0.125 Pooled Equal 277 2.05 0.041 Satterthwaite Unequal 268.9 2.09 0.037 215 120 Algebra-based physics Pretest: 'Standard' version 40% 30% 20% 10% Algebra-based physics Pretest: Modified version 40% 30% 20% 10% 0 1 2 3 4 N = 245 5 0 40% 30% 20% 10% 0 1 2 3 4 5 Calculus-based physics Pretest Modified version Calculus-based physics Pretest 'Standard'version 40% 30% 20% 10% 1 2 3 4 N=191 - l_l 0 1 2 3 4 N=118 5 N=161 5 Figure 4.3. Distribution of student scores on two versions of the 5-question pretest. Top graphs shows results for algebra-based physics course - Physics 211, while the bottom graphs shows Physics 215 results. 121 Physics 21 1 80 o <D 6 0 O 40 O sP 20 I Q1 PH Q2 Q3 I Q4 Q5 Original Version Modified Version Physics 21 5 80 id 60 40 o u 20 iI i [—1 Q1 Q2 Q3 Q4 I Q5 Figure 4.4 Comparisons of student performance (including error bars) on each test question of the original version and the modified version of our proportional reasoning pretests. Separate results are given for Physics 211 and Physics 215 students. Discussion of Performance by Course The result of differing overall performance between the algebra-based and calculus-based introductory mechanics populations on both proportional reasoning pretests may not be surprising to many researchers. Barnes,69 in one of the earliest reports of Piagetian-type tests used as a predictor of success in introductory physics courses, also observed that mean scores on his pretest varied as he considered introductory physics courses with different prerequisites and consisting of different introductory physics populations (e.g. biology majors etc). Variation in performance 122 on these tasks across institutions, has also been noted by Loverude et al. Viewed from a Piagetian standpoint one might attribute the difference in performance by our two introductory physics populations (and as in Loverude's case: by students at different institutions) on these pretests as indicative of these students' differing proportional reasoning abilities. Generally speaking, when glancing at student performance on the two pretest versions within each of our introductory physics courses, the results on these two versions appear similar. This result lends credence to the view that regardless of context, both tests are largely measures of proportional reasoning ability and that this ability is an issue that is separate from context in which the question is posed. One may say that both of these proportional reasoning tests show that calculus-based physics students are better proportional reasoners than their algebra-based physics counterparts. However, context does play a significant role in student performance on these tests, especially within our calculus-based physics population. For the algebrabased students, even though we do not observe statistically significant differences in the mean performance on the two pretests, we do observe subtle shifts in the distribution of scores on the two versions (see Fig 4.2). We believe these differences in our students' overall performance indicate that familiar contexts might be more successful at eliciting and activating the right set of resources, necessary for some students to tap into their proportional reasoning abilities. The implications of this result to our curriculum development goals are then immediate, as familiar contexts can be used as a bridge to tap into students' hidden abilities, while building up to 123 broader understanding of many physics concepts. As part of our efforts to further investigate which groups among our student were more sensitive to context dependence of our proportional reasoning pretest, we looked for possible gender effects as explained in the next section. Gender Effects on Performance We further wished to investigate whether males and females performed differently on our proportional reasoning pretests, within each of our courses. In a study very similar to our own, McCollough143 constructed a 'stereotypically female' version of the FCI which she called the Revised FCI (RFCI). The contexts appearing on the RFCI were "shopping, cooking, jewelry and stuffed animals," as opposed to (what she saw as) the male-dominated contexts appearing on the regular FCI of cannonballs and hockey, as well as the all-male illustrations. She interleaved the versions of the FCI and RFCI given to students. She asked the questions to over 300 students enrolled in general education classes (such as English) because at her university she saw that the physics classes were male-dominated. She found that whereas there was no difference in performance in the female students' scores, the male students scored significantly worse on the RFCI. We looked at whether or not student performance on the two versions of the proportional reasoning diagnostic here at NMSU exhibited any gender-related effects. 124 Results of NMSU Study Separated by Gender Within the algebra-based physics course two very important differences were observed when data collected from the students enrolled in this course were separated by gender. For the male Physics 211 students, there was almost no difference in performance on the two versions of the proportional reasoning pretests. However, there was a significant difference in performance on the two versions of the pretests for female students, who scored on average about 0.6 points higher out of five (d~0.3) on the version of the pretest which contained the more familiar context (see Tables 4.9 and 4.10 in appendix J and Figure 4.5 below). Another very important feature of the gender difference in performance on the two versions of this pretest taken by the Physics 211 students is that the difference between mean male students' scores versus mean female students' scores on the original (standard) version of the pretest (mean 2.53 vs 2.09 respectively - d~0.3) is significant, with a p-value less than 0.05 (see Table 4.11 and 4.12 in appendix J); whereas the difference in the two genders' mean scores on the modified version of the pretest is not significantly different (p>0.55). This result carries with it the implication that while the context of the original pretest version, carries with it the commonly observed gender difference in performance on tests like the FCI and in physics courses,144 there is no evidence that the context of the modified pretest version carries with it a bias in either direction. There was a similar pattern for our Physics 215 students. For the male students in this course, there was no significant difference in performance on the two 125 versions of the pretests. Even though there is a relatively low number of female calculus-based physics students, the difference in performance on the two versions of the pretest was statistically significant at the (p<) 0.05 level as was the case for Physics 211 female students [see Tables 4.13 and 4.14 in appendix J and Figure 4.6 below]. The effect size for this difference was also medium (d~0.5) Algebra-based physics Pretest: 'Standard' version Algebra-based physics Pretest: Modified version 40% 30% 40% 30% 20% 10% 20% 10% 0 1 2 3 4 Males: N = 122 5 0 Algebra-based physics Pretest: 'Standard' version 1 2 3 4 Males: N = 93 5 Algebra-based physics Pretest: Modified version 40% 30% 20% 10% 40% 30% 20% 10% 0 1 2 3 4 5 Females: N = 88 0 1 2 3 4 5 Females: N = 128 Figure 4.5. Distribution of Physics 211 students' total scores on two versions of 5item pretest. Results for male students in top graphs andfemale students in bottom graphs. 126 Again, in our calculus-based physics course, for the original pretest version there was a significant male bias in terms of performance: Males scored on average about 0.7 points higher out of 5 than their female counterparts on this test version (d~0.4). In contrast, on the modified pretest version the two genders' scores were not significantly different [see Tables 4.15 and 4.16]. Calculus-based physics Pretest: 'Standard'version Calculus-based physics Pretest: Modified version 40% 30% 20% 10%. 40% 30% °0% 10% 0 1 1 1 2 3 4 5 Klale s:N [ = 1 22 0 Calculus-based physics Pretest: 'Standard' version 11 1 2 3 4 Males: N = 87 Calculus-based physics Pretest: Modified version 40% 30% — 20% — 10% — 40% 30% 20% 10% J L 0 1 2 3 4 5 Females: N = 37 0 1 2 3 4 Females: N = 28 5 Figure 4.6. Distribution of Physics 215 students' total scores on two versions of 5item pretest. Results for male students in top graphs andfemale students in bottom graphs. 127 In short, altering the context of our proportional reasoning test served to improve the proportional reasoning performance of our female introductory physics students. We did not observe the same significant decrease in male performance that McCullough did. Our findings are consistent with observations made by Trimmer145 in reviewing the literature on gender differences pertaining to physics: that females tend to perform better than males on questions that are contextualized and on questions that involve a human presence. We posit that these contextual characteristics are primarily responsible of our modified pretest version having significantly improved female performance, while leaving male performance virtually unaffected. It is instructive to consider some further observations made by Trimmer on contextual characteristics of questions where females outperform males. She observed that questions that contain open instructions or that require extended writing, with words such as 'explain', 'evaluate', 'transform' and 'analyze' and questions that require 'routine use of mathematical knowledge and manipulation', generally led to females outperforming males. In contrast she found that males outperformed females in questions requiring very short answers or where only one correct answer was sought (such as multiple choice questions); questions involving calculations; questions beginning with the words 'identify', determine', 'locate', or 'calculate'; questions that involve the use of addenda such as diagrams, tables and graphs; complex information-dense questions; and questions framed in the negative. 128 Our study provides further evidence that by carefully choosing the contexts and wording of our physics problems, we can reduce the male gender bias prevalent in our classrooms today. Steele145 has showed that, by cuing people in genderdiscriminatory and race-discriminatory situations in math and science testing, one can cause women and minority populations to do more poorly than otherwise. We believe that context factors play a larger role in standardized test scores than it is currently given credit for, and that highlighting these factors is a potentially valuable contribution to be made by physics education researchers. Fall 2009 Pretest Versions As explained in chapter three, the pretests used in Fall of 2009 were slightly different. In addition we also had an opportunity to ask the entire Lawson Test during the last lab of this semester. For students who were present for both pretests, we had matching pre-course and (virtually) post-course testing data for two questions which appeared on the pretests given during the first lab as well as on the Lawson test. For half of these students the pre-test and post-test questions were the same, while (depending on which pretest they took at the beginning of the semester) the other half of these students had matching 'modified pre-' and 'original post-' testing results. We thought these results would be useful in showing how individual students responded to the two versions of questions - Whether or not there would be many 129 frequency count of algebra-based physics students'paired preand post-test scores Score on original pretest Score on original post-test Score on original post-test 1 1 17 1 7 0 2 2 3 3 26 Score on modified pretest 10 2 2 2 2 5 2 2 33 N=60 N=si Jbrequency count ofcalculus -based physics students'paired preand post-test scores Score on original pretest Score on original post-test Score on original post-test 1 1 5 0 A 1 1 1 1 1 13 Score on modified pretest N=27 6 1 2 0 1 0 1 2 16 N=29 Figure 4.7 Frequency counts of students' corresponding pre- and post- test scores (out of two) on two questions - Questions two and three on 'original' and 'modified' pretests shown in appendix F. For instance, in the algebra-based course, of the 61 students taking the original pretest and original post-test, seven of them scored zero on the pretest but scored two on the post test on these two questions. Also, of the 29 students taking the modified pretest version in the calculus-based course, 16 students got both these questions correct (had a score of two) on both the pre-test and posttest. 130 students who could answer the modified question versions at the beginning of the semester yet fail to answer the original question versions at the end of the semester. Due to the low numbers of students taking both pretests however, we cannot make conclusive statements based on our results about whether or not more students who took the modified version of questions two and three at the beginning of the semester, performed worse later on in the semester when taking the original version of these questions. Theoretically one could use the paired results of those students who took the original pretest version before and after as a baseline measure of proportional reasoning improvement through the semester, to assist in making this case. The large number of students receiving the same pre- and post-test scores on these two proportional reasoning questions, as can be seen by the frequency counts along the diagonals on Figure 4.7 above, implies that there was little change in students' proportional reasoning throughout the semester. A sixth question, placed at the end of both pretest versions used at the beginning of the Fall 2009 semester, was a question on the revolution of two interlocked gears [see appendix K]. This question was suggested to us during an earlier (Spring 2004) student interview as a question that might be more recognizable as a 'proportional reasoning problem.' As mentioned in chapter three, this pretest was also given towards the end of the same semester to a group of underprepared engineering students at Rutgers University. These students, as part of a special course on introductory mechanics that spans two semesters, worked explicitly throughout the 131 Fall of 2009 (the first semester of their 2-semester course), on invention tasks designed to build scientific reasoning ability and to help students decide "when to use Figure 4.8. Two interlocked gears. Students were asked to determine the number of revolutions made by the smaller gear, given revolutions of the larger gear. Performance^ on Gear Qyestion Phys211 Phys215 Rutg Eng Figure 4.9. Student Performance on gear question with error bars. For NMSU students performance on the two pretest versions is shown. 132 math and what math to use." This pretest was administered to these students near the end of their first semester in this course. As predicted by our interviewee, students performed very well on the gear question. Over 75 percent of students in all groups answered this question correctly, which cannot be said of any other proportional reasoning question we have asked. This finding brings home the fact that students' poor performance on the proportional reasoning questions appearing on the Lawson test cannot be taken on its own as evidence that students don't 'have proportional reasoning ability'. It also brings up the important question of why many students recognize that ratio reasoning is required in problems such as this one, but do not recognize that it is to be used in other situations. The fact that this same question was the last question appearing on two different versions of proportional reasoning pretests allowed us to test for the effect of context cuing on performance on this question. For 40 calculus-based physics students, about 98% of students correctly answered this gear question appearing on the modified version of the pretest, whereas only 78% of 41 students enrolled in the same course answered this question correctly on the original version of the pretest. This was a very significant difference (t=2.75, P<10"4). The difference in performance on this question appearing at the end of the two randomly assigned pretest versions was not significant for 145 Physics 211 students (t=0.58, P=0.56). About 80% of students got this question correct regardless of pretest version chosen. 133 On the entire 5-question pretests used in Fall 09, students performed comparably to previous semesters. During this semester, however, there was a statistically significant difference in performance on the two versions by Physics 211 students (df 143, t=2.17, P=0.03) while this difference was not observed for the Physics 215 group (df 79, t=0.19, 0.85). We believe this to be an artifact of the relatively low numbers of students taking each pretest and place much higher significance on the results collected over 6 semesters (mentioned in previous sections). Mean score on 5 Question Fall09Pretest 3 _.__ _ m w. 1 0 Fall 09 Original _h. 2 W>. Phys211 W V//A Phys215 Fall 09 Modified Rutg Eng Figure 4.10. Student Performance five-question pretests (excluding gear question). For NMSU students, performance on the two pretest versions is shown. 134 Pretest Correlation with Final Exam As stated in chapter three, the modified pretest version had a generally lower correlation with success in our physics lab course as measured by our lab final exam. For the original pretest version the correlation with the final was between 0.4 and 0.5, whereas for the modified pretest version the correlation was about 0.3. Here we describe the correlations with final when the different populations are divided by gender. Results from the discrete pretest version as well as from our female Physics 215 students' are included for completeness of the presentation However, as these numbers are low, these correlations are not as reliable as the other coefficients presented. There is some small gender difference between the two course populations in terms of how predictive the pretests are [see Table 4.19]. It should also be noted that again it appears as though the original pretest version is more predictive of success on our final exam than the modified pretest version regardless of gender. This holds true in both the algebra-based and calculus-based courses. In the Fall semester of 2008 we used one additional measure of student success in our lab course. During the last week of lab we asked the Teaching Assistants (TAs) to rate their students based on factors such as how well students understood what was going on in lab and the quality of the questions they asked in lab. 135 Table 4.17. Correlation coefficients of score on different pretest versions with final exam score. pretest Version Physics 211 Males Physics 215 Females Males Females Original 0.29(72)* 0.46(82)*** 0.50 (91 )*** 0.40 (25)* Modified 0.28(61)* 0.39(64)** 0.36(63)** 0.26(19) Discrete 0.68(18)** 0.27(14) 0.37(33)* 0.20(6) Modified * p < 0.5 * * P< ICr2 *** P< 10"4 Table 4.20 in appendix L shows the correlations among TAs' ratings of students, the students' scores on the proportional reasoning pretest (taken at the beginning of the semester), and the students' scores on the final exam. The TAs' ratings of students were fairly well correlated with students' scores on the final exam (correlation coefficients generally about 0.6). The TAs' ratings of students were also generally better correlated with the original pretest version than the modified pretest version. Again, if one were to choose between these two pretests based solely on their predictive ability of success in physics as defined by TAs' ratings one would be inclined to choose pretest version one. In designing our final exams, we generally try to choose 'real life' contexts in which to frame our questions. This fact does not seem to translate into higher correlations between our students' scores on our modified version (everyday context) proportional reasoning pretest with their final exam scores (compared to correlation between the original proportional reasoning pretest with final). In Karplus' 136 interpretation of Piaget's framework of accounting for different student thinking patterns,148 it can be said that the original pretest, being more abstract, probes a higher stage of development (the formal stage) than the modified pretest which probes the concrete stage. In Karplus' view, abstract reasoning is preferable to concrete and as a consequence the more abstract of the two pretests is preferable. The fact that an independent measure of accounting for student differences - rankings by TA's correlated better with the more abstract of the two pretests, lends credence to the view that the abstract measure may be a better predictor of success in physics regardless of measure of success used. It may be argued however, that this lower correlation is an indication that despite our best efforts, the labs we have designed were unsuccessful in promoting our students' reconciliation between mathematical and everyday reasoning.137'149 We have not done expectations surveys150 within our labs to test for changes in student views on what the purpose of the labs were. One way to test this would be to use both pretest versions in a course designed around sense-making ideas and see if we get a different result in the pretests' correlations with the success measures used in such a course. Discussion and Conclusion We have observed that our different introductory physics populations perform differently on our proportional reasoning pretests. We saw that our calculus-based 137 students performed better than our algebra-based students on these pretests regardless of context. Using a strict Piagetian interpretation of these results one would conclude that there are more formal reasoners among the calculus-based physics population than among the algebra-based physics population. That contexts affects performance on our proportional reasoning tests (at least among our calculus-based physics population), leads us to lean towards a less strict reasoning ability interpretation of these results towards including more of a resource-based model of student thinking. We believe that tests such as the original proportional reasoning diagnostic incorrectly lump students who have the ability to reason proportionally in some contexts along with students who cannot reason proportionally at all - both of these groups receiving the wrong answer on these tests. In the past a few other researchers have observed that context plays some role in student performance on Piagetian tasks. Tschopp and Kurdek151 for example concluded that Piagetian tasks often differ in levels of difficulty oftentimes based on the students' level of familiarity with the experimental materials. Tournaire,95 in studying elementary school students in grades 3, 4 and 5 on various proportional reasoning problems across various contexts, concluded that "proportional reasoning begins as an essentially fragmented ability. It is likely that solving proportional problems in different contexts, or with different number structures, involves different abilities." We observed that when we shifted contexts in proportional reasoning tasks that our female students tended to perform better, while our male students' 138 performance remained the same. This finding was surprising to us, and alerted us to the fact that we may have paid insufficient attention to the contexts used in our curriculum development materials and in our courses. One application of these findings may be in the promotion of 'human' context-based problems and text-books within our courses, and (of particular importance to us) in the curriculum we develop, as a means of tapping the abilities of female students. One potential problem of using this approach, however, is observed where we noted that the abstract problems may be more closely related to success in physics. This would necessitate that we design curriculum that initially probes students' everyday thinking, but eventually providing a clear path from everyday thinking to full scientific reasoning. 139 CHAPTER 5: CONCLUSION Introduction In this dissertation we have described curricular modifications that we have made to our introductory mechanics laboratories in an attempt to improve students' conceptual understanding. Although these modifications were made with a research basis, our post-testing suggests that the conceptual gains that our students made were modest. For the four laboratories that we described in Chapter 2, we remain satisfied with the laboratory procedures we have developed in terms of their efficacy in connecting the motion of actual physical objects to a representation of this motion that is amenable to analysis. Students like the procedure, and seem to benefit from the procedural practice that is offered in the labs. Moreover, for our Physics 213 students (physics majors), there was reasonable success in students' understanding of Newton's second law as a vector equation as measured by our final exam questions. For our Physics 211 students (the intended audience for our lab development efforts) and our Physics 215 students, however, we did not observe the conceptual gains that we had hoped for. The labs appeared to be least successful with the population that we had hoped to have the most impact on - the students in the algebra-based course. As part of attempting to tailor our laboratories to the students in the algebrabased course, we tried to characterize the differences in preparation between these students and students in the calculus-based course. A brief summary of the results of 140 our efforts to characterize differences in student preparation between these two students groups is presented in the next section. Summary of Findings In a review of literature relevant to predicting success in introductory physics courses highlighted in Chapter 3, we saw that many of the most successful predictors in physics were based upon tasks used to determine Piagetian stages of intellectual development. In our efforts to characterize differences that were relevant to the successful implementation of the conceptual labs that we designed in our NMSU introductory physics populations, we also found that questions designed to measure students' scientific reasoning ability were quite useful predictors of success. Questions excerpted from the widely used Lawson Test of Scientific Reasoning plus one additional item accounted for about one-third of the total variance in students' final exam scores. This fraction of total variance is cited by other researchers as among the largest accounted for in studies in physics, even using multiple predictors.89'90 Upon further investigation of results from the Lawson test, we found that our students struggled most on proportional reasoning items, and in fact, pretests based on these items alone also bore good predictive power of student success in our physics lab courses: We found the predictive ability of a short 5-question pretest on proportional reasoning ability - four proportional reasoning items from the Lawson 141 test, plus one additional density item - to be of comparable predictive power to other longer pretests used as predictors in physics in the past. That so many of our students do so poorly on this pretest shows that a significant portion of our incoming students lack this fundamental skill. In a later study, we attempted to use our students' working memory capacities to improve on the predictive ability of our pretests, as well as to potentially provide a further measurable characterization of differences between our introductory physics populations. Our results from this investigation were largely inconclusive, with measures of our students' working memory capacities using the n-back test only weakly correlating with their final exam scores. In addition, there was virtually no correlation between final exam scores and: (1) A spatial working memory test; (2) the time taken do the working memory task; or (3) a measure of student efficiency in answering our working memory questions. Results from our introduction of contextual modifications to the proportional reasoning pretests used as predictors were guided by a desire to explore whether contextual changes influenced students' epistemic dispositions as indicated by their success rates on proportional reasoning problems. Initially, we introduced an alternate proportional reasoning diagnostic which was set in the context of serving rice in a soup kitchen. This pretest, however, failed to control for the difference in reasoning required to solve for proportional reasoning questions involving discrete items, as has been noted to influence success on proportional reasoning problems by other researchers.96 Later, as an alternate to the original proportional reasoning pretest, we 142 used a proportional reasoning pretest that (similar to the original) involved the use of a liquid (modified pretest). We found that these contextual modifications had significant consequences on our students' success rates on proportional reasoning questions. In particular, we observed that on a pretest version modified to include a more every-day context (in our estimation) - one pouring medicine into two different sized dosage containers that students were generally more successful at answering our proportional reasoning problems. When broken up by course, this difference in performance was statistically significant for our calculus-based students, but fell short of achieving statistical significance for our algebra-based population. We concluded from this that not all of what we measured using the original proportional reasoning pretest can be attributable to proportional reasoning ability, and that some of our students' poor performance on these questions can probably be attributed to a lack of familiarity with the context with which the problems are presented . In what was perhaps the most surprising result of our study, we found that for students in both algebra-based and calculus-based introductory physics courses, the effect of context modification was significant for female students, whereas for male students there was not a noticeable difference in performance on the two pretest versions. Furthermore, within each course the original proportional reasoning pretest version showed males significantly outperforming females, whereas this was not the case on the modified pretest version. 143 Finally, in measuring the correlations between students' scores on both versions of the pretest with their final exams scores, we generally found lower correlations between the modified pretest version with final exams than the original pretest version with final exam. This was true for male and female students alike, and was also true for calculus-based and algebra-based introductory physics courses. Implications for instruction At NMSU we have seen that whereas there is general improvement from pretest to final exam for many of our students across different introductory physics course populations, our majors seem to benefit the most from our reform attempts while our algebra-based population seems to benefit least. In spite of this disappointment, some hope is presented in the fact that students in lectures closely aligned with the labs' content, generally exhibit above average performance, as was highlighted in chapter two. We believe that reinforcement may be key towards further improving results from our underperforming student populations. It is possible that through increasing awareness of the methods utilized in the laboratory to our lecture instructors, that the coherence between physics presented in lecture and lab might lead to improved performance. Results reported by the University of Maine seem to suggest that their students have benefited from breaking the acceleration along a curved path into tangential and radial components in addition to the method of vector subtraction we 144 have presented in this dissertation. Our past experience at NMSU leaves us hesitant to attempt such an approach, as our algebra-based students (the intended audience for this lab development) have traditionally struggled with understanding vector components. This fact, however, does not preclude the possibility of attempting a dual vector-component approach with our calculus-based population. We believe that different success rates of the same curricula at various institutions implies a need for careful consideration of the target population when designing curricula. Hake,54 in popularizing the use of <g>, made a very important first step in accounting for different incoming states of students. It is our opinion that a further measure needs to be introduced that gives some indication of the maximum possible improvement in performance that can be reasonably expected for different populations, in order to be of further guidance to institutions, instructors etc. We think that such a measure will be useful for instructors who hope to implement curriculum in populations that are significantly different from the populations in which the research was conducted. Meltzer supposes the existence of a 'hidden variable'55 related to identifying successful physics students. Factors such as selfefficacy show promise in predicting success, and are among factors that most educators across various institutions would agree should contribute towards overall student success. That our short proportional reasoning pretests play such a prominent role, on their own, in accounting for which among our students will be successful in our courses at NMSU, highlights the strong connection between this skill and 145 understanding physics. This result gives some merit to Piaget's view - that proportional reasoning ability is one central factor (among others), necessary for students' to reason scientifically. It can be surmised that in any physics course, whether conceptual or mathematical, students need to represent and manipulate very abstract ideas (such as momentum, energy, and torque) and mathematics is an indispensable tool in doing the book-keeping of all of these abstractions. That some mathematics pretests used in the past have not been highly successful as predictors, may indicate that raw mathematical ability is insufficient to guarantee success in physics. In this regard, it may be enlightening for instructors to pay close attention to the abilities identified by Piaget as having particular bearing on scientific reasoning in general. It is very important to note that the context in which questions are asked has a strong effect on which subsets of students enrolled in our courses do better and which do worse. We are encouraged by the finding that in choosing contexts that are familiar to students, we can potentially help some students overcome what at first seems to be deficit in ability. We hope that by guiding students to work with the concepts we are trying to convey in a more familiar context, and then linking these skills directly to formalism necessary for physics, that we will be able to better utilize students' incoming abilities toward physics understanding. The influence of context on our female students' performance suggests a need to pay particular attention to the contexts in which the material that we develop, is presented. Careful selection of contexts may play a significant role in reducing 146 potential gender bias and encouraging more gender-equitable participation in physics. In order to design our physics courses in ways that promote diversity,154 it is important that we pay particular attention to the contexts in which our questions are asked. These contexts influence which among our students will go on and be considered 'successful' and which will not.155 Rennie and Parker156 in research supporting gender-equitable teaching and assessment in physics found that teachers can create gender-neutral or gender-inclusive assessment tools by careful portrayal of stereotypes in particular contexts. They advocate concrete problems over abstract problems and find that appropriate contexts make problems easier to visualize and more interesting. We hope to be guided by these principles in any curriculum development we may undertake in the future. 147 APPENDICES 148 APPENDIX A MOTION IN TWO DIMENSIONS TUTORIAL AT THE UNIVERSITY OF WASHINGTON Mt h MOTION m TWO DIMENSIONS * L Vdtwily Anwtojeci i* rowing around an oval track. Sfceleh itie tr»j««;K»cy <sf rte *>bjeet tin a large sheet of paper. (Make your diagram lai%e.% A. Choose a point to serve as an origin for your «*irdina,!e system. Label Shal point O (for origin). Select twt» locations »F the object ihw are about nnc-icighfh of the oval apart and label them A and B. f. Draw the paoacion vector* f«r isiwh *:*!' „ Co y y l w r f IWP sd , , > », , „. . «'*«'m'« * i * *P«*« * * r df«w»«w- «l»e t w o JctCatKNMt <4 and # and draw tlis vector that represents the displLncemcinl fawn A la-B, 2, \ IJCK+IH* how U» us* the *t:is|rfiK«ncni vector lo determine She «liwcti«i of i!t« av'Miige velocity s»f the object between A and i<. Dww it veet«r t» rejwcNeiM itu; iivwige velocity, 3. Chewsc m ptsmx %m the ovalfeelween ptsittte 4 *«d W. ami label that jwini fl*. A* poiw H * i* ehf**en 1*> tic th}<*r aiul elcwer la j»in« A, ckte* the direction of (tic awrrage velocity «v«r the interval ^ H ' change? i f so, haw? 4, ftesMihe she direction ttf the i«*.ttriia»w*»«s> velocity €»i'th« mbjwt at pniwl A. How w©tth( yuti t'tiaisw-ieriw IIK? diwsettan «f the wsMi»rrt«ie«i:* vcluciiv m tm$ %mM <« ills* waicostff ? P«e* y«t« anwer defwnd mt wteiltw tte «Jtoje£t is speeding rtp. stewing itown, m moving; wish «M*isl«it \jxsert"? Hxphun. B, M ym wefts u> dwwse a different origin (tor tji<? ettwrtltotite syMerit, wbk'h of tin* «**tow that yam have (!*!»»» in pan A. w«dd dw*»g$ tuwl wbk*h wtitihl mit cluing? ? fttft.vittf*ill Meftermaif, Staffer,« M i * i , t ; , ' W i » l i . fr»iri«,tef/-v>"Fkyik'i 149 CClraski? Halt, IIK. FSIM lalidmi-JtJOI 16 m^-—————^mmiBe=SBBB 11, Amterathxn for motion -wltb (.cinyfunt spet-d Suppfls? that the ufojset in s*elioti I is moving around (he track at amstam spend. ITratw vectors te» represent the velocity at two points on she track t t a an? relatively dose together, fDra* ywii vectors tor^*',) Latiel (fee two points Carad J3>. A, On a $tjmm» pm of your paper, copy tte velocity *«et«f* pc and #^ Pnwn these vectors. tietetmifie the cfaiti$i in tneheity m*<w-f M*. I,. Is the i«igte formed by the '"head*" of i% and the "tail" ef Ar greater than, less than, m cqmimW'f As point D is chosen to lie etasc? aiKl closer to point C, dots* Ae above angle incfmvtt deertase,« remain the Mime? E*pljMli h»* yen «M tell, D»e«< (he shove anjgfc upproiKh a Umtimg value"? If so, what is te limiting value? 2.= Describe how tw H « the change in velochy vcdo* ?»<*ete«wii«c the mmrgs mmtemkm irf: the ©fe^t het%«M C and /X Draw SI v%ttMf w *epf*»*nt the avwap aaceknttion Between point* C and 0 , What hippo!* Mi the; rm»piiSiMfc «f Ai» sw» point P isrfwwettSo lie cloxcr aiwl etasesr tit pmituCr1 &«iittesicwl«itJiwt«tanpttMhc:W)*w wty? Ikptaifi, Cansidef the dirertkift «1 the iie«teraii«:i»t «i jMsiiii C, lb HK aitfte kstwcew dhe »t«|ffljlici« m l o r swl the welaelty w&mgtmm than, km $hm, wmjmt t» W f (l¥«ir.;; CMvetrtitwtiHjvtte « $ k tetweett » **«*» Is tfcftned «i (te Mikgle liHimi*t when tltey ««frfwrf '*iill»t»-ttil,"l fuimmk mfaMMte-terj'Ptopics MriDwrooll, Staffer, *>.E«J„ V. Wwh. 0f"wnik» HUf I,, to£> I%si tetiita, W 6 150 Mpt-ifm w IWG dlfttfttmioni mmmm B. StiRpcse she object started/horn resl at pottst £ and mewed towards point F whit iitcrcastiif speed. H»w WCKIM yew find Mie aewteratioii at paint E? Demwibe-ttmdirccliosi of (he m^tarcwtofJ of (lit abjeti m paint E. €• AS j*everat point*on each*rfUse sHaft'siMi*bchw,straw it v*cu*thai rejWEsaHs the acceleration sf she object. AeeeltfEatmn vectors for speeding up (TOHI rest iil piiim 4 '\ ; > Tpjt view (iiafrrajti TV:f- vtew diagram CttafartefKg the direct i<Kt «f Hw irecderMion i t « d » pntat DO site trajectory for « « h >e*se. Is the KJceteratrort directed toward the ''center*' «f the «w»! »t (Wety paint «t» ibc trajectory fyt i*jtrwr of these cases? Sketch arrows to show ihc disreeiiwrii of the ii««lersHofi fur the following tra|cc?txwte<<: Cwtstaitt *|tg#i| • $ p « t l i l l j t Uft- CO ffcSiMMMfrMpNl • eiwwiiir Tutorials in huradmrmri,- IRISH'S © I t e r t i w I W I , fee. Hist Edition, 2002 151 APPENDIX B SELECTED LABS AND ASSOCIATED HOMEWORK EXERCISES Motion in 2 Dimensions Lab Introduction In this laboratory, you will examine the acceleration of objects that are moving in two dimensions, or 'in a plane'. You can imagine this motion as motion that is confined to lie on a giant, flat sheet of paper. The orientation of this 'flat sheet of paper' will change in different instances, but the motion of our objects can always be traced upon this virtual sheet of paper. - Av By definition, (average) acceleration is a = — . This is a vector equation, so the direction of A? the acceleration must be the same as the direction of the change in velocity (since the change in time is a scalar). We will use this fact to help us find the direction of acceleration for motion in two dimensions. Lab Objectives After completing this lab and the associated homework, you should be able to: 1. Subtract two vectors, and identify (Av) as the difference of 2 vectors. 2. Use a change in velocity vector to determine the direction of the acceleration of an object for a small time interval. 3. Describe the direction of acceleration for objects in parabolic, oval and other types of two-dimensional motion. Outline of Laboratory Approximate sequence of the lab and homework: 1. Practice interpreting time-exposure photographs. 2. Learn how to graphically subtract two vectors. 3. Relate displacement to velocity. 4. Deduce the acceleration of an object from velocity vectors for straight-line motion. 5. Find the acceleration vector for an object experiencing parabolic motion. 6. Find the acceleration vectors for an object moving in an oval path. 7. Derive the acceleration vectors for complex paths of motion. 152 8. Draw acceleration vectors corresponding to oval-shaped motion. Recording motion and interpretation of pictures In this laboratory we will be using images similar to the strobe photographs that were used in the Descriptions of Motion lab. We will attach blinking lights ('blinkies') to objects in twodimensional motion and take long-exposure digital photographs of these objects, which will help us to analyze the motion of our objects. These photographs will be similar to the one shown at right. Our blinkie emits light for a period of time (we will refer to this condition as being 'on') and immediately following this period undergoes a period where it does not emit light ('off). The blinkie goes through this sequence of being 'on' then 'off then 'on' continuously. If the room is dark enough, the resulting picture only shows the light that entered the shutter, while the blinkie was on, at the position from which the light was emitted. If the shutter was open long enough to record many cycles of light and darkness (from the blinkie), the resulting image will show lines of light (and spaces of darkness) emanating from the position of the blinkie. We can then analyze this image (of lines of light amid a backdrop of darkness) to determine the direction of the velocity and acceleration of the moving object to which the blinkie was attached. 2.1: The drawing at right represents a photograph taken of the motion of a toy hovercraft initially travelling along a straight line, then suddenly being kicked. It is known that our attached blinkie emits light for 0.1 seconds (one-tenth of a second), and that the floor tiles that can be seen in the background are 10 cm on a side. Based on this drawing: A. Is the speed of the hovercraft at point A: greater than, less than, or equal to the speed of the hovercraft at point B? Explain B. Use a ruler to determine the ratio of the speed of the hovercraft at point A to the speed of the hovercraft at point B. C. Estimate the speed of the hovercraft at point A. D. Estimate the speed of the hovercraft at point B. 153 E. Before the hovercraft was kicked, was it speeding up, slowing down, or moving at constant speed? Explain how you can tell. 2.2: In physics, vectors are often represented by lines whose length is proportional to the magnitude of the vector, and whose direction is depicted by an arrow-head (at the tip of our line) pointing in the direction of our vector. For example, a 200 km per hr wind from the north may be represented by the vector shown at right. 200km/hr A. Measure the length of the vector shown at right above. What speed does 1 cm represent? The relationship between the length of a drawn vector and the magnitude of the quantity that vector represents is known as the scale of the drawing. i B. For the motion of the toy hovercraft shown in section 2.1, draw a vector that represents the velocity at point A. Use a protractor to ensure that you have not changed the directions of the original vectors. Use a scale of 1 mm = 2 cm per second. C. At the same scale, draw | another vector for the velocity of the hovercraft at point B. D. How does the length of the velocity vector at point B (which represents the magnitude of the velocity at B) compare to the length at point A? Does the ratio of the lengths of the vectors you have drawn accurately reflect the ratio between the lengths of lines of light at points A and B in your diagram? 154 => Check your answers above with your lab instructor before continuing. Subtracting two vectors Given a vector^, the vector -A is the vector having the same magnitude as A, but pointing in the opposite direction. _ _ A 3.1: Given the vectors A, B and C shown at right, draw the vectors -A, -B and —C in the space below. ^ ^ Vectors are translation invariant, which means that you can slide the vector A across or down or wherever, as long as it points in the same direction and has the same magnitude as the original vector, then it is the same vector. All of these vectors are equivalent 3.2: Two vectors can be added graphically by placing the tail of one vector against the tip of the second vector. The result of this vector addition, called the resultant vector (R) is the vector that has its tail at the tail of the first vector and has its tip at the tip of the second vector. In the space below, add vectors B and C shown in 3.1 and at right. 155 3.3: Vector subtraction is just a special case of vector addition. One can view A -B as The procedure we have used for adding two vectors can therefore be used for vector subtraction, where in this case the two vectors to be added are A and -B. _ A. In the space below, subtract vector C from vector A. B. Now subtract vector A from vector C. How does A -Ccompare to C-A? Finding change in velocity vectors and acceleration direction. The average acceleration of an object for a time interval defined as the ratio of the change in the velocity of that object divided by the length of the time interval: a = ^At • This is a vector equation - the quantities on either side of the equal sign must have the same direction as well as the same magnitude. Therefore, in order to find the direction of the acceleration we need to be able to determine the direction of the change in velocity vector. (Since time is not a vector quantity, the change in time Af does not influence the direction.) The direction of the acceleration is the same as the direction of the change in velocity. 4.1: To find the change in any quantity, we need to subtract its initial value from its final value. To find the change in velocity, we subtract the initial velocity vector from the final velocity vector. A. Consider the velocity vectors A and B that you drew in section 2.2. In the space at right, redraw these vectors to scale, labeling the velocity at point A asvjmtiaiand the velocity at point B as fmal. The direction of these vectors must be the same as in the original diagram in section 2.1. 156 The change in velocity vector Av is equal to the final velocity final minus the initial velocity v initial • Av - v final ~ v initial = v'final + (~v initial) B. In the space to the right above, find the change in velocity vector for the hovercraft as it moves from point A to point B. C. Draw a small arrow to indicate the direction of the average acceleration for the toy hovercraft between point A and point B. 4.2: It is probably a good idea at this point to comment on why we are doing all this. One of the most important fundamental ideas in introductory mechanics is Newton's second law. This law is a vector equation (again, the direction as well as the magnitude must be the same on either side of the equal sign) that relates the net force acting on the object to the acceleration of that object: I.F -ma . Since we found the direction of the acceleration the hovercraft above, we can now use this to learn something about the direction of the net force. In this case that net force direction is just the direction of the kick. In what direction was the hovercraft kicked? 4.3: The drawing at right was created by drawing the blinkie lines from a photograph similar to the one at the beginning of section 2. We will assume that the object was moving clockwise (That is, the object reached point A before it reached point B.) To find the direction of the acceleration for a portion of the motion, we can determine the change in velocity vector for that region. For example, to find the change in velocity vector for a small region around point A, we use the light stripe just before point A to determine the velocity at the beginning of a small time interval, and we use the stripe just after points to determine the velocity at the end of this small time interval. Often, the change in velocity vectors are small, so it is useful to draw the vectors larger than the light stripes as shown below. In this case we have drawn the vectors five times as large as the light stripes. 157 A/ — Vfiml - Vinitial 'final A. In the space at left above, perform a similar vector subtraction to find the direction of the change in velocity vector for point B. B. For each of the following cases determine whether it is possible for the change in velocity vector to be zero. For each case, include a graphical vector subtraction to support your contention. Case 1: Neither the speed nor the direction of the velocity vector changes in the interval between two adjacent stripes. Case 2: The magnitude of the velocity vector remains the same, but the direction of the velocity vector changes in the interval between two adjacent stripes. 158 Case 3: The magnitude of the velocity vector changes, but the direction of the velocity vector remains the same in the interval between two adjacent stripes. For the remainder of this lab you will be using 'blinkies' attached to objects to create longexposure photographs, and then you will analyze these photographs to study acceleration in two dimensions. You will complete three exercises: (1) motion of a toy hovercraft moving along a ramp, (2) motion of a pendulum moving in an oval, and (3) motion of a toy roller coaster cart moving along a track. You do not need to perform these exercises in any sequence, and you should ask your lab instructor at this point which experiment to perform first. You can do either the hovercraft or the pendulum experiments first. You will use data from the roller coaster experiment as a basis for some of your homework. General instructions for camera use You will be using a digital camera to take these pictures. The camera is set up in a 'shutter priority' setting that allows the user to control the amount of time that the camera shutter is open. In this setting, the ambient light level is measured and the camera adjusts the aperture to control the amount of light. These digital cameras take about one-half of a second for the shutter to open after the shutter control button is pushed. It may take a few tries before you get a photograph that captures the motion you want to analyze. Once you have a picture your group is happy with, use the USB cable to download the picture to the computer at your lab table. Follow the instructions on the handout on the table to invert the picture. (This is similar to creating a negative.) Then print a copy of the picture for each member of your group, as well as a picture for the whole group. As with the exercise on the previous page, we will need to draw fairly large vectors to represent the initial and final speeds compared to the size of the stripes. As long as we are careful to preserve the relative sizes of the vectors as well as their angles, we will be able to determine the direction of the change in velocity vector. 159 Exercise I: Parabolic Motion The diagram at right shows our hovercraft about to move along an inclined plane. ^ — ^ ^ ^ ^^JS"""""""'"^ ^ ^ ^ ^ ^^T^^ """" *4 ^ ^ ^ - ^ ^^*^^ /' ^ ^ \ ^ 5.1: In a darkened room, take a ^S*S%. /' ^s^S^fj picture of the motion of the hovercraft ^'*5!!!!S!^ ' ^a^^^^^^^ as it travels in a parabolic arc as ^ s *S ftsS as* iiS ^^ shown. To get the best possible picture, we should aim for our hovercraft to begin its motion in one of the lower corners, and end its motion on the opposite lower corner of the inclined plane. You may want to repeat this procedure a few times, and only use the 'best run' for your analysis. You will find that the parallel stripes on the ramp do not look parallel in the photograph. But you can use the grid painted on the ramp to tell which direction is 'down the ramp' and which direction is 'across the ramp' at each point in the photograph. 5.2: After printing this photograph: A. Find the change in velocity vector for the hovercraft for a portion of the path when it is on its way up the ramp (somewhere near point A in the diagram). What is the direction of the acceleration vector for this portion of the path? B. Find the change in velocity vector for the hovercraft for a portion of the path when it is at the turnaround point for this path (somewhere near point B in the diagram). What is the direction of the acceleration vector for this portion of the path? Find the change in velocity vector for the hovercraft for a portion of the path when it is on its way down the ramp (somewhere near point C in the diagram). What is the direction of the acceleration vector for this portion of the path? D. For this parabolic motion, summarize your answers about the direction of the acceleration for the hovercraft. That is, how can you describe the direction of acceleration for all points of the parabolic motion? Check your answers above with your lab instructor before continuing. 160 Exercise II: Motion in an oval The diagram at right shows a pendulum swinging in an oval. 6.1: In a darkened room, Place the camera on the floor pointing up at the pendulum bob. Start the pendulum swinging in an oval as shown, and take a picture of the motion of the pendulum. 6.2: After printing this photograph: A-i' A. Find the change in velocity vector for ~~ ~ _ •*--"' B the pendulum bob when it is at its maximum distance from the center of the oval (somewhere near point A in the diagram). What is the direction of the acceleration vector for this portion of the path? B. Find the change in velocity vector for the pendulum bob for a portion of the path when it is at a point closest to the center of the oval (somewhere near point B in the diagram). What is the direction of the acceleration vector for this portion of the path? C. Find the change in velocity vector for the pendulum bob for a portion of the path when it is at an intermediate distance from the center of the oval (somewhere near point C in the diagram). What is the direction of the acceleration vector for this portion of the path? D. For this motion, summarize your answers about the direction of the acceleration for the pendulum bob. That is, how can you describe the direction of acceleration for all points of the oval? Check your answers above with your lab instructor before continuing. 161 Exercise III Motion along a track and parabolic motion The diagram above shows the track that the toy roller coaster will travel along. Unlike a real roller coaster, our toy will be launched so that it travels in a parabolic arc. 7.1: In a darkened room, take a picture of the motion of the coaster as it travels along the path shown. Be sure that your picture includes a clear blinkie signal from the cart for all of the labeled points above, because you will be using the photograph you make as the basis for answering some homework questions. You may want to repeat this procedure a few times, and only use 'the best run' in your final analysis. 7.2: Print one copy of the photograph for each group member so that you can use it to answer homework questions. (If you have time, you may want to complete question 1 of the homework while you are in the lab room.) 162 Motion in 2 Dimensions Homework 1. For each pair of vectors shown below, use a ruler (and protractor if necessary) to perform the indicated subtraction as shown in the example. Example C- C=B-A oD=E-F 11. H c>G=H-I in. J=K-L K 2. Suppose that two vectors A and B have the same magnitude (length). For each part below show an example of possible directions of A and B if subtracting one from the other gives: I. zero. ii. a third vector that has twice the magnitude of one of the original vectors. iii. a third vector that has the same magnitude as one of the original vectors. 163 In each case i. through v. below, find the change in velocity vector Av from the initial point to the final point for an object moving along the path shown. IndicateAv with a dashed line vector, and graphically determine its magnitude. Use a scale 1 centimeter = 1 meter per second. In the space to the right of each case, draw an arrow that shows the direction of the average acceleration for the time that the object travels from the initial to the final point. {Recall that the average acceleration is defined by the vector equation a* = —- Since At is a scalar, the direction of a must be the same as the direction of A v.} Example Initial: / v = 4 m/s H =0 Final: 4 m/s Acceleration direction: None. \4 Acceleration direction: \M- Acceleration direction: \M Acceleration direction: \M Acceleration direction: \M- Acceleration direction: Final: v"= 5 m/s Final: v"= 3 m/s • Initial: v = 4 m/s Final: v*= 4 m/s • Initial: v"=4m/s Finab v = 3 m/s • Initial: 7 = 4 m/s 164 4. A toy car with a blinkie attached moves in a clockwise direction around a racetrack. A drawing of the trail made by the blinkie is shown. The car starts at rest from points. By the time it reaches point D it is traveling at a constant speed, and continues at this speed until it reaches point G. It then slows down to ,B a stop. y^ s' / ^ i. On the diagram at right, draw velocity vectors for each of the points A-G Be sure that the relative magnitudes of your vectors are consistent. / / ii. On the same diagram at right, draw the acceleration vectors for each of the points A-G. If the acceleration is zero at any point(s) indicate that explicitly. iii. How does the magnitude of the acceleration at point E compare to that at G? Explain. 5. A car is slowing down (but not turning left or right) as it passes over the crest of a hill as shown. Indicate the approximate direction of the acceleration of the car for the time that it travels between points A and B. Show how you determined your answer. 165 sF Newton's Second Law Lab Introduction In this laboratory we look at Newton's second law. In the addition of forces lab, we looked at situations where the net force acting on an object was zero. Since Newton's second law states that the net force acting on a body is equal to its mass times its acceleration: S F = ma, as long as there is no acceleration, there is no net force. That is, when an object is not moving, or is moving with constant speed and in a constant direction, then the acceleration is zero and the vector sum of all the forces acting on the body is also zero. In this lab, we look at Newton's second law more generally, including cases where the net force is not zero. In a sense, the purpose of this lab is to connect what you did in the last lab (drawing free-body diagrams to find the net force) to what you did in the lab before that (finding change-invelocity vectors based on photographs of blinkies attached to objects). This connection is the essence of Newton's second law: The net force acting on an object, based on a vector sum of these forces, is proportional to the acceleration of that object, based on the vector difference between final and initial velocities for that object for a small time interval. Moreover, the direction of the net force must be the same as the direction of the acceleration, because Newton's second law is a vector equation. Sometimes we can use what we know about the forces acting on an object to make inferences about the change in velocity that we expect for that object. In other cases, we can use what we know about the change in velocity for an object to make inferences about the forces acting on the object. Lab Objectives After completing this lab and the associated homework, you should be able to: 1. 2. 3. 4. Find the direction of the acceleration by vector subtraction. Determine the direction of the net force based on the vector sum of the forces involved in a free-body diagram. Compare the direction of the net force with the direction of the acceleration. Make inferences about the magnitudes and directions of forces based on Newton's Second Law. Outline of Laboratory Approximate sequence of the lab and homework: 5. 6. Practice subtracting vectors to find the direction of the acceleration. Practice adding the forces from a free-body diagram to find the direction of the net force. 166 7. Based on photographs taken of objects with blinkies, determine acceleration direction, draw free-body diagram, and construct a vector sum diagram that is consistent with Newton's second law. 2. Roller coaster again In the Motion in Two Dimensions lab, you took a photograph of a roller coaster and studied the acceleration of the coaster at various points. An example of a photograph of a coaster from that lab is reproduced here, and is cropped to show the path of the coaster as it moves from left to right at the bottom of the ramp. For this example, the streaks of light are curved because the coaster changed direction during the time interval At that the light was on. But a vector drawn from the beginning of a streak to the end of a streak represents the displacement Ax of the cart during the time interval, and the average velocity during this interval can be found from the definition of average velocity: ave _ Ax ~~At T7' 2.1 On the diagram above, draw vectors to represent the velocity of the cart for light streaks 1 and 2. 2.2 We will use these velocity vectors to give us an indication of the change in velocity for the time interval At' that the blinkie is off between these streaks. Find the change in velocity vector for this time interval. 2.3 The (average) acceleration for the coaster for the time interval between blinks can be Av found from the definition of acceleration aave - — . What is the direction of the acceleration At for the time that the cart is at the bottom of the ramp? Explain. 2.4 Because it is a vector equation, Newton's second law, ~LF = ma, requires that the direction of the net force be the same as the direction of the acceleration. What is the direction of the net force acting on the coaster in the time interval At' ? 2.5 What forces are acting on the cart during the time interval At' ? (Ignore air resistance and friction, because these forces are very small for the coasters.) 167 2.6 Draw a free body diagram for the coaster at the bottom of its path, using the forces that you listed. Be sure that the directions of the forces you drew are consistent with what you learned about these forces in lecture and in the Forces lab. 2.7 Construct a vector sum diagram by adding the forces from your free-body diagram. The resultant is the net force. When adding these vectors, be sure not to change the direction of the vectors! However, you will need to adjust the lengths of these vectors so that the direction of the net force is the same as the direction of the acceleration. •=!> Do not continue past this point before checking your vector sum diagram with your instructor. 2.8 Based on your vector sum diagram, which of the forces acting on the coaster while it is at the bottom of the track is the largest? 3. Pendulum For the blinkie attached to a pendulum bob: 3.1 Hold the pendulum to the side, and release it. Take a picture of the motion. A reasonable time setting for this photograph is 1.3 seconds. It might take a few tries before you have a picture that has most of one swing from left to right, but not part of a return swing. Print out a copy of your inverted photograph for your group to analyze. 3.2 For the pendulum near position A: Graphically determine the change in velocity vector. 168 3.3 What is the direction of the acceleration of the pendulum? What is the direction of the net force? 3.4 A free body diagram for the pendulum bob is shown at right for a time when it is near point A. The force along the wooden pendulum is a tension force on the bob by the wooden stick. / T BW XBV W BE Note that a free-body diagram tells us about the directions of the forces acting on an object, but it does not tell us about the magnitudes (sizes) of these forces. Y However, the net force and the acceleration must point in the same direction. When we add the vectors together (without changing their directions!) we can adjust the relative lengths of the vectors in order to ensure that the net force points in the same direction as the acceleration. 3.5 Construct a vector sum diagram based on this free body diagram. Again, you will need to adjust the lengths of the vectors without changing their directions so that the net force points in a direction that is consistent with Newton's second law. 3.6 Repeat the steps of exercises 3.2 to 3.5 for the pendulum blinkie when it is near point B, near the bottom of its swing. (This time you will have to draw your own free-body diagram.) 3.7 For which of the two positions shown above does the acceleration (and the net force) point toward the center of the circle that the pendulum blinkie is moving in? What must be true about the speed of an object moving in a circle in order for the acceleration of that object to point toward the center of the circle? 169 3.8 Compare the vector sum diagram for the pendulum bob near the bottom of its swing with the vector sum diagram you obtained for the coaster near the bottom of its path. What similarities and differences do you observe? 4. Hovercraft on a level surface For the blinkie mounted on the hovercraft: 4.1 Push the hovercraft across the board so that its motion is roughly parallel to the stripes and so the hovercraft motion is horizontal in the photograph. Take a picture of the motion. A reasonable time setting for this photograph is 2 seconds. Print out a copy of your inverted photograph for your group to analyze. 4.2 Choose a position somewhere close to the center of your photograph, and graphically determine the change in velocity vector. 4.3 What is the acceleration of the hovercraft? What is the net force? 4.4 Draw a free body diagram for the hovercraft on the board. 4.5 Construct a vector sum diagram based on your free body diagram. Again, you will need to adjust the lengths of the vectors without changing their directions so that the net force is consistent with Newton's second law. 170 4.6 In your own words, describe the special case of Newton's second law that is illustrated by the hovercraft's motion. 5. Block on track - Speeding up For the blinkie attached to the wood block: 5.1 Place the block at the top of the track with the wood side against the track (cork side up). Make sure that the blinkie is facing the camera. Release the block from rest, and take a picture of its motion as it slides to the bottom. A reasonable time setting for this photograph is 2 seconds. Print out your inverted photograph for your group to analyze. 5.2 For the block on the ramp: Graphically determine the change in velocity vector. 5.3 What is the direction of the acceleration of the block? What is the direction of the net force? 5.4 Draw a free body diagram for the block when it is on the ramp. 5.5 Construct a vector sum diagram based on your free body diagram. Again, you will need to adjust the lengths of the vectors without changing their directions so that the net force points in a direction that is consistent with Newton's second law. 171 •%> Check your vector sum diagram with your instructor.. 6. Block on track - Slowing down For the blinkie attached to the wood block: 6.1 Place the block at the top of the track with the cork side against the track (wood side up). Make sure that the blinkie is facing the camera. Give the block a firm shove toward the bottom of the ramp, and take a picture of its motion as it slides downward. You do not need the block to reach the bottom of the track -just be sure that you have a photograph that includes the block slowing down. A reasonable time setting for this photograph is 2 seconds. Print out a copy of your inverted photograph for your group to analyze. 6.2 For the block on the ramp: Graphically determine the change in velocity vector. 6.3 What is the direction of the acceleration of the block? What is the direction of the net force? 6.4 Draw a free body diagram for the block when it is on the ramp. 6.5 Construct a vector sum diagram based on your free body diagram. Again, you will need to adjust the lengths of the vectors without changing their directions so that the net force points in a direction that is consistent with Newton's second law. 172 6.6 Compare your free-body diagram in this case with your free-body diagram for ifHS. jX) '•kslKft 5l ^,fc-'?¥#,^'*«fc' 1-& / * L A roller coaster is set up as shown above. An Inversion of the bliiriMe „^- "=v X st after ii. What is the direction of the acceleration at point A? Of the net force? Explain. iii. Draw a free-body diagram for the coaster at point A, and use this to make a vector sum diagram. Be sure to label the net force on the vector sum diagram. iv. Of the forces you have drawn on your free-body diagram, which is the largest? Explain how you can tell. 2. Repeat parts i., ii., and iii. of question 1 for point B shown in the coaster photographic inversion 3. A skateboarder falls as she travels down the right side of the bump. At the instant shown, she is slowing down. Use velocity vectors to find the direction of the acceleration of the skater. Then draw a free-body diagram of the skater, and show that the net force on the skater is in the same direction as the acceleration. 4. In each case shown on the next page: 174 a. Draw and label velocity vectors for times just before and just after the instant shown. b. Use these velocity vectors to find the direction of the acceleration of the object at the instant shown c. Draw a free-body diagram for the instant shown. d. Add force vectors to show that the net force is in the same direction as the acceleration. Example: A skier speeding up on a ski slope. •&1 A skier slowing down on a ski slope. Free-body diagram Acceleration direction Force addition A car coasting (i.e., no friction) uphill. 175 iii. A football at the highest point of its trajectory. (Ignore air resistance.) Free-body diagram Acceleration direction ^>-<•>•--^^ Force addition iv. A bouncing ball for the time interval during which it is in contact with the floor. lSi,fcsaB-^i:.^iM- ^iitafc-jiaiMMiBB^BMBi v. A roller coaster slowing down as it travels upwards on a loop. • 176 Changes in Energy Lab Introduction With this lab, we introduce a different way of analyzing motion, by considering the changes in the energy for moving objects. In this lab we will introduce two kinds of energy - kinetic energy and potential energy - and observe how these quantities change along an object's path. Kinetic energy is the energy associated with the motion of an object. When the center of mass of an object moves, we say that the object has translational kinetic energy, and when an object rotates about its center of mass, we say that it has rotational kinetic energy. Potential energy is energy associated with the position of interacting objects. For example, when an object moves further from the center of the earth, we say that the gravitational potential energy (PEG) has increased. (Strictly speaking, it is the gravitational potential energy of the system of earth + object that increases.) When a mass attached to a spring stretches that spring, we say that the elastic or spring potential energy of the spring-mass system has increased. In this lab we will only consider translational kinetic energy and gravitational potential energy.) In some situations we can use either Newton's laws or energy ideas to solve a problem, and deciding which approach to use requires practice. A few general rules of thumb: First, in cases where forces change as a function of position (for example, the normal force for an object moving on a curved surface), Newton's laws become difficult to use. Energy is not a vector quantity, and it may be more easily used in these cases. Second, when we apply rules about energy, we do not learn anything about how much time a motion takes. So if a problem does not require or contain information about time, considering the energy of the system under study may be useful. Lab Objectives After completing this lab and the associated homework, you should be able to: 177 5. Relate the gravitational potential energy (PEG) of an object to its position, and determine whether the PEQ increases, decreases, or stays the same in a given motion. 6. Relate changes in potential energy of a simple system to changes in the object's speed. 7. Calculate gravitational potential and translational kinetic energies for simple situations. Outline ofLaboratory Approximate sequence of the lab and homework: 8. Analyze a blinkie photograph of a moving object in order to relate the position of the object to its speed. 9. Make qualitative predictions about the speed of a roller coaster as its position changes. 10. Evaluate whether the shape of a track makes a difference in changes in potential energy. 11. Evaluate the extent to which differences in mass affect the resulting motion of a body. Roller Coaster and Moving Coaster In this lab you will be using the roller coaster and track. The roller coaster has a blinkie on it that flashes at a constant rate. You will take a digital photograph of the coaster as it moves along the track, and the streaks in the photograph created by the blinkie will allow you to make inferences about the speed of the coaster at each position. A long-exposure photograph of the type you will be taking is shown below. Some points along the coaster's path are labeled A through E. (Your group will also be given a larger (inverted) copy of a portion of this photograph for analysis.) 178 \ nz^w^%r-yt .^<m&ifi*r'T-'&.' s? (At this poJEt, you do mc of the §peeds so omly qualitative W-M-^ : to dletennffiui© the auisenca!! what point on the track would you expect the speed to be greatest? Explain. Expl Examine the enlarged photograph given to your group, in which the coaster moves down a ramp, starting from rest. On the diagram, the release point is labeled with the letter A, and the release height is marked by a line across the photograph. Choose a point that is near the top of the ramp where the light stripe is several millimeters long in the photograph. On the picture, label this location as Point B. Prediction: Let's call the speed of the coaster at point B: vB. and the vertical distance between A and B: AyAB. Imagine finding the point where the coaster achieves a speed that is double its speed at point B, or a speed of 2vB. Call that point C. Will the vertical distance between point C and point B be greater than, less than, or equal to the distance between point B and the release point? Explain why you think so. Now examine your diagram. Look for a streak on the photograph that is about twice as long as the streak at point B. Call this location Point C. Is the vertical distance between point C and point A greater than, less than, or equal to double the vertical distance between point B and the point A? (It doesn't matter to which point on the streak you measure the height, as long as you are consistent. The start or end of the streaks are probably easiest.) You should have found that the vertical distance between point C and point A is greater than two times the vertical distance between point B and the release point. Approximately how many times greater is this distance? Now examine your diagram again. Look for a streak on the photograph that is about 3 times as long as the streak at point B. Call this location Point D. Compared to point B, approximately how many times below the release height is point D? In the absence of friction and air resistance (as well as any distortions due to the photograph), the spacing of points B, C, and D should be such that C is four times as far below the release point as B, and D is nine times as far. In practice, your ratios will probably be a bit smaller than that. We find in general that the increase in speed is proportional to the square root of the change in vertical height. Alternatively, the change in height is proportional to the square of the speed. This observation leads to 180 the following definitions of gravitational potential energy (PEG) and kinetic energy (KE):. A PEQ = mg A h KE = lA mv2. In SI units, with mass in kg, height in m, and velocity in m/s, both PEQ and ICE will be measured in Joules, the SI unit for energy. For most of this lab, we talk about the change in PEG between two positions. There are two important points to make about this definition. First, the PEG is, strictly speaking, defined for the system of earth and object together, and depends on the distance between them. Secondly, for most problems, the change in PEG is more relevant than the value of PEQ itself, so PEG is typically defined to be zero at a fixed reference height that is convenient. This height can be chosen at will, but it is important to stick with a single reference height. For the measurements you have just taken, we measured all of the heights from the height of the release point, and we might have chosen it as zero height. For that choice of reference, the potential energies at points is zero, and the potential energy of the coaster at points B, C, and D are negative. Reexamine your results from above. Compared to point B, the coaster at point C has twice the speed. How much greater is the kinetic energy of the coaster at point C compared to point B? Explain. Is your result consistent with the difference in height that you observed in the two cases? Compared to point B, the coaster at point D has three times the speed. How much greater is the kinetic energy of the coaster at point D compared to point B? Is your result consistent with the difference in height that you observed in the two cases? Based on your results, at approximately what height would you expect a speed 2.45 times the speed of point B? You should check your prediction using your diagram, though you will probably find that the distance doesn't quite match your prediction. (Why not?) 181 Factors influencing the speed of the coaster In this section we will examine how the shape of the track and the mass of the cart affect the speed of the coaster. We start by making some predictions. Prediction: A. Imagine that two identical coasters are released from rest from the same height on tracks attached to the same board. One track is steeper, but both reach the same lower level. Will the speed of the coaster at the bottom be greater on the jCoaster Shallower track Steeper track steeper track, greater on the shallower track, or the same on both tracks? B. Now imagine that we add some mass to the coaster, and again release it from rest on the steep track. Will the speed at the bottom of the track be greater with the added mass, greater without the added mass, or the same in both cases? Explain the basis for your prediction. Your group will now take three photographs to test your predictions. Since you will be comparing these photographs, you should take the photographs one after another without moving the camera. In addition, it is important to remember the sequence in which you took the photographs. First, release the coaster from rest on the steep track without added mass. Second, release the coaster from rest on the shallow track without added mass. Finally, release the coaster from rest on the steep track with 60 grams of mass added to the coaster. For the first two photographs, compare points at the same height about halfway down the two tracks. What do you notice (approximately) about the speeds of the two coasters? 182 Compare the speeds of the coasters at the bottom of their respective tracks? What do you notice (approximately) about the speeds of the two coasters? Based on your diagrams, what can you conclude about the change in potential energy for the two coasters? Did the change in potential energy depend on the shape of the track? Note that the change in gravitational potential energy between two points depends only on the difference in vertical height between the points. The shape of the track does not matter, only the change in height. 4.3 Now compare the photographs showing the coaster on the steep track both with and without the added mass: At the bottom of the track, is the speed of the coaster with greater mass greater than, less than, or equal to that of the coaster with the smaller mass? It may seem counterintuitive that the speed of the coaster does not depend on the mass. Note that the formulas for both potential energy and kinetic energy are proportional to the mass of the object. Imagine that the heavier coaster had twice the mass. Its change in potential energy would be twice as much, and it would therefore have twice the kinetic energy at the bottom of the track, but the change in speed would still be the same! Indeed, the change in speed over this interval depends only on the height difference between top and bottom. Conservation of energy In the examples you have examined in this lab, as the coaster travels down the track its gravitational potential energy decreases, and its kinetic energy increases. At any point along the path of the coaster, the potential energy plus the kinetic energy add up to the same value. In the absence of friction and air resistance, we find that the total energy (kinetic plus potential) stays the same. For this reason, physicists call the total energy a conserved quantity, We can choose a reference height and define the potential energy of the earth-coaster system at that point to be zero. We can then define the potential energy at every point 183 at height h as PEG = mgh, where h is the height of the coaster above our refere .1 The coaster picture from section 2 is reproduced below. nergy as zero at the release height. — • • " #,, ~~ }ms«o <* tS. nfcU tin o ^ J*' For the coaster at the instant of release at points, the potential energy is zero (since stgy: sr w i l l K It is helpful to represent the two kinds of energies as bars on a bar graph. For units, and we can re Points K^ram PE Grav Point £ ^rans PE Qav Point C K^T™* PE &av Point D ^ans ^Oav Point £ ^ ^ PE ( 1 A. Draw the bar to represent the kinetic energy of the coaster at point C. Explain how you knew to draw it as you did. B. Fill in the bar chart diagram for points B, D, and E. C. Is it possible to have a negative potential energy? Is it possible to have a negative kinetic energy? Explain. D. Compare the dash in the photograph that is just above point C to the one just above point F. Based on your comparison, has mechanical energy (i.e., potential energy plus kinetic energy) been conserved from C to F? Explain how you can tell. 5.3 Suppose instead that we had chosen point E as our reference potential. Redo the bar chart for this case. 185 Points KETram P E ^ Point B KE^ PE^ Point C KE^ PE^ Please check your results above with your instructor. 186 Point D KE^ PE^ Point £ KETrans P E ^ Changes in Energy Homework 1. Three toy roller coasters are set up on different tracks as shown. Roller coaster A has a mass of 30 grams, roller coaster B has a mass of 50 grams, and roller coaster C has a mass of 70 grams. The coasters are released from rest at the top dashed line, and there is no friction or air resistance. Coaster^ Coaster B Coaster C Rank, from greatest to least, the speeds of the carts when they reach the dashed line. Explain how you determined your ranking. 2. A toy hovercraft is given a quick shove at the bottom of a ramp. The ramp makes an angle of 30° with the horizontal. There is no friction between the hovercraft and the ramp. The hovercraft has a mass of 0.4 kg. At the 187 instant shown, the hovercraft is at point A and has a speed of 2 m/s. The hovercraft travels up the ramp, comes to rest at point B, and then travels down the ramp. i. What is the kinetic energy of the hovercraft at point A? Show your work, and express your answer in Joules. ii. What is the potential energy of the hovercraft at point B? Explain how you can tell. iii. How high above point A is point B? (Give the vertical distance, not the distance along the ramp.) Show how you obtained your answer. iv. What is the speed of the hovercraft when it reaches point A again? Explain your reasoning. 188 3. A roller coaster is set up as shown above. The coaster is released from rest at the location of the coaster on the track in the photograph. There is no friction or air resistance. The white horizontal lines that are superimposed on the photograph are evenly spaced. i. If we define the gravitational potential energy to be zero at the release point A, what is the total energy (potential plus kinetic) of the cart when it reaches point C? Explain. ii. If the speed of the cart is 12 cm/s at point B, what is the speed at point D? Explain how you determined your answer. iii. Fill in the bar chart graph below, showing the potential and kinetic energies at the points labeled A - E. The kinetic energy of the coaster at point C is shown. Points Points Point C l 189 Point D Point £ ^Trans ^Grav iv. The bar chart graph below is for the same five points, but with a different height chosen as zero potential energy. The scale of the chart is the same as for part ii. Fill in the rest of this chart showing the kinetic and potential energies at points A - E. Point/i Kenans PEQav PointB ^^Trans ™Om Point C ^^Trans 190 ^Gnv Point D ^^Tans ^Grav Point £ ^Trans ^ G H V Rotational Motion Lab Introduction All of the previous labs have dealt with systems whose centers of mass were moving from one place to another - systems undergoing translational motion. In this lab, we take a first look at systems in rotation - systems where the object is rotating about an axis. When we studied systems in translation we first focused on the kinematics of translation, i.e., the study of the motion of the systems without worrying about the causes and influences of that rotation. Later we looked at dynamics, or the study of how the forces acting on a system and the mass of the system affected the motion. Similarly, for the study of rotation we will first investigate rotational kinematics, paying attention to the motion, and then we will look at some of the factors that influence rotation. For rotating objects, we can define an angular displacement, the change in angle that a line connecting a point on the object to the center of the object makes in some time interval; an angular velocity, the rate of rotation; and an angular acceleration, the time rate of change of the angular velocity. Mathematically, the relationship between these quantities is similar to the relationships between their translational analogues of displacement, velocity, and acceleration. In addition, we can connect the translational and rotational variables for rigid bodies. In this lab, we explore the relationships between these kinematic quantities. In addition, we introduce the idea of moment of inertia, which is a rotational analogue to mass. Lab Objectives. After completing this lab and the associated homework, you should be able to: 8. Determine angular displacement, angular velocity and angular acceleration based on measurements of a rotating object. 9. Relate rotational kinematic quantities to translational kinematic quantities for points on a rotating object. 10. Reason qualitatively about the effect of moment of inertia on angular acceleration. Outline of Laboratory Approximate sequence of the lab and homework: 191 12. Analyze a blinkie photograph of a disc moving with roughly constant angular velocity. 13. Calculate angular velocity and angular displacement. 14. Analyze a blinkie photograph of a disc moving with roughly constant angular acceleration. 15. Calculate angular acceleration. 16. Change the moment of inertia of a rotating disc, and again calculate angular acceleration. Angular displacement A plywood disc is set up with a set of 4 blinkies attached to it. The blinkies are all attached to the same circuit, and they all turn on and off at the same time. 2.1: The distance from the center of the disc to each blinkie is shown here. Measure the flash rate of the blinkies Count the number of times one of the leds flashes in thirty seconds. (It is probably best to do this as a class and write the count on the blackboard.) Blinkie A B C D Radius 0 4 cm 8 cm 11 cm How many times does it flash per minute? How many times does it flash per second? 2.2: Set the camera set for a one-half second exposure. Turn on both switches on the blinkie circuit, start the disc spinning, and take a picture of the disc as it rotates. Make sure that each blinkie makes several light streaks, but that they don't go around the disc completely. Print out an inversion of the picture for study. 2.3: Does blinkie B travel the same distance as blinkie D during each time interval? Explain how you can tell. 2.4: Do the four blinkies rotate through the same angle during each flash? On your printed picture, measure the angle that each blinkie travels through during a single on/off cycle, and fill in the table. (You can measure the angle from when a blinkie turns on until it again turns on.) 192 Blinkie A B C D Angle When an object is translating (when its center of mass is moving), the displacement of that object in some time interval is the distance Ax from the initial position to the final position. For rotation, we can define the angular displacement as the change in angle from the final position to the initial position A9 during some time interval. It is useful to measure this angle in radians. A6 Blinkie In one complete revolution, the angular displacement of an object is 360° or 2K radians. Convert the angular displacements that you measured above to radians, and record your result here. A B C D Arc length Radius r Blinkie A6 2.5: Using your data above, find the A 0 distance (arc length) traveled by each B 4 cm blinkie during one on/off cycle. You can C 8 cm do this by determining what fraction or D 11 cm percent of a circle the blinkie travels in one on/off cycle, and then calculating what fraction this is of the distance traveled in a complete revolution (i.e., one circumference). 2.6: For a point on a rotating object, write down an algebraic relationship between the radius r, the angular displacement A0, and the arc length As. Angular velocity 3.1: From the number of flashes per second found in section 2.1, find the time At for one complete on/off cycle. 3.2 Use the information from parts 2.5 and 3.1 to find the speed v of each blinkie, measured in cm/sec. Blinkie A B C D Speed When an object is translating in one dimension, the velocity of that object in some time interval is the distance Ax from the initial position to the final position divided Ax by the time interval At:v- — . At For rotation in a plane, we can define the angular velocity oo as the angular displacement A6 (again measured in radians) from the initial position to the final position 9f - 9i divided by the time interval At: Radius r CD Speed v Blinkie co =A0_ A 0 At ' 193 B C D 4 cm 8 cm 11 cm 3.3: Find the angular velocity for each blinkie on the disc, and fill in the table at right. 3.4: For a point on a rotating object, write down an algebraic relationship between the radius r, the angular velocity co, and the speed v. For rotating objects, the use of radian measure for angles simplifies the relationships between translation and rotation. A radian is defined as the ratio of the arc length to the radius, and since this is a length divided by a length, a radian is dimensionless. The angular displacement A0 is therefore also dimensionless, and the angular velocity co has units of 1/seconds. 4. Angular acceleration Since the bearing on the disc has very little friction, the motion you studied in sections 2 and 3 was roughly constant. That is, the angular velocity of the disc did not change very much from one blinkie flash to another. As with the translational kinematics that you studied at the beginning of the semester, a more general treatment of rotation includes cases where the angular velocity of an object is changing with time. Just as acceleration in translation is defined as the change in velocity per unit time, we can define angular acceleration (a) as the change in angular velocity per unit time. The disc you will be using for this experiment has a blinkie to mark the center of the disc, and another blinkie mounted near the edge of the disc. The flash rate for this outer blinkie is approximately 3 flashes per second, so we will use this value in the calculations we make. A third blinkie circuit with (but with no led!) is mounted across the disc and is only used to balance the disc. A small weight attached to a string wrapped around the edge of the disc will be used to change the angular velocity of the disc. 4.1: Place the four 100-gram brass cylinders into the holes that are closest to the center of the disc as shown. Turn on both blinkies. Wind the string around the edge of the disc until the small hanging weight is close to the disc. Release the disc, and take a 1-second exposure photograph of the disc as it rotates. Print out an inverted photograph of your picture for analysis. 4.2: Is the angular velocity w of the disc constant? Explain how you can tell from your photograph. 194 4.3: Measure the angular displacement of each on/off cycle in your photograph. (You can do this by measuring the angle from when the blinkie turns on to the next time it turns on.) You should find that the change in A0 between adjacent cycles is approximately constant - the angle made by each flash increases from the previous one by about the same amount. What is this change in A0 measured in degrees? In radians? 4.4: Pick two adjacent on/off cycles, and use them to calculate the change in angular velocity Aco. Show how you calculated this value below. 4.4: Pick two different on/off cycles that are also adjacent and again use them to calculate the change in angular velocity Aco. You should find that that you get about the same value. 4.5: The angular acceleration a is defined as the change in angular velocity Ac; divided by the change in time (i.e., the time interval over which this change takes place): Aco a = At Since each on/off cycle for the blinkie at the edge of the disc takes about one-third of a second, we can use this time interval to calculate angular acceleration. What is the angular acceleration of the disc? What are the units of angular acceleration? 195 The acceleration of a point on the disc in the direction of its motion is called the tangential acceleration. For a point that is rotating and changing speed such as the blinkie that we have measured here, the tangential acceleration is related to the angular acceleration by atan = ocr. (Note that this point also has an acceleration that points toward the center of the circle - the radial acceleration - that is equal to the velocity squared divided by the radius.) 4.7 If the distance from the center of the disc to the blinkie is 11 cm, what is the tangential acceleration of the blinkie? 5. Moment of inertia So far in this lab we have focused on the kinematics of rotation, paying attention to the relationships between angular displacement, angular velocity, and angular acceleration, and connecting these quantities to their translational counterparts. We now begin to pay attention to the dynamics of rotation. - a study of the factors that affect rotation. In translational motion, the acceleration of an object is related to the net force acting on that object through Newton's second law, HF = ma. In rotational motion, the moment of inertia I is analogous to mass m in translation. If we think of the mass of an object as a measure of how hard it is to change the velocity of that object, then the moment of inertia is a measure of how hard it is to change the angular velocity of a rotating object. 5.1: Take the four brass cylinders from the inner holes and place them in the holes closest to the edge of the disc as shown. Note that you have not changed the mass of the rotating system, and you will use the same hanging weight. Predict what will happen to the angular acceleration of the disc when you repeat the exercise from section 4.1. 6 5.2: Repeat the exercise from section 4.1, again taking a 1 -second photograph. Print out an inverted photograph for analysis. 5.3: Repeat the procedure from sections 4.3 - 4.5 to find the angular acceleration of the disc over two different intervals. Is the angular acceleration greater than, less 196 than, or equal to the angular acceleration of the rotating system with the brass cylinders closer to the axis of rotation? 5.4: If we think of the moment of inertia as a measure of how hard it is to change the angular velocity of a rotating object, does the rotating system have a greater moment of inertia with the cylinders closer to the axis of rotation or further from the axis of rotation? As we have seen, the moment of inertia of a rotating system depends on the position of the mass of that system - the mass distribution. When mass is further from the axis of rotation, the moment of inertia increases, as it becomes more difficult to change the angular velocity of the system. In addition, the moment of inertia depends on the amount of mass in the system - if we had put a brass cylinder in all eight holes, we would have found that the angular acceleration was smaller than with only four cylinders. Rotational Motion Homework 1. Three children, Aricelia, Bao, and Chuck, are playing on a merry-go-round. Their positions on the merry-go-round are shown in the top-view picture at right. The merry-go-round is rotating clockwise and is neither speeding up nor slowing down. A. Rank the speeds of the children. If allI of the children are moving at the same speed, state that explicitly. Explain your reasoning. 197 B. Rank the angular velocities of the children. If all of the children have the same angular velocity, state that explicitly. Explain your reasoning. C. Rank the times that it takes each child to complete one revolution. If all of the times are the same, state that explicitly. Explain your reasoning. D. With Aricelia, Bao, and Chuck on the merry-go-round, a fourth child, David, wants to bring the merry-go-round from rest to a rotation rate of 1 revolution every 4 seconds in as little time as possible. That is, he wants to be able to speed the merry-go-round up from rest as quickly as possible. Does it matter where the other three children sit on the merry-go-round? If you were David, where would you ask the other three children to sit? Explain based on the idea of moment of inertia. 2. In lab you released a mass tied to a string that was wrapped around a disc, and took a picture of the blinkie as the mass fell and the disc rotated. Suppose that instead you had unwound the string until the mass was just above the floor, and then someone in your group had pulled quickly down on the edge of the disc as shown. If you took a picture of the blinkie after the quick pull but while the mass rose was still rising, what would the picture look like? Explain. 198 3. What is the angular velocity of the earth as it spins on its axis? Express your answer in radians per second. Show how you determined your answer. 4. A blinkie photograph made using the same procedure as in section 4 of the lab is shown at right. The blinkie is flashing five times per second. i. What is the duration of a single on/off cycle for the blinkie? ii. What is the approximate angular velocity (in radians per second) of the blinkie at point P? Show how you determined your answer. iii. If the speed of the blinkie at point P was 32 cm/sec, what is the distance from the center of the disc to the blinkie? Show how you determined your answer. iv. What is the approximate angular acceleration of the blinkie (in radians per second squared) of the blinkie at point P? Show how you determined your answer? 199 APPENDIX C SELECTED FINAL EXAM QUESTIONS A football is thrown so that it follows the path indicated by the dashed line (from left to right). Air resistance can be neglected. 1. When the ball is at point A on this path, what is the direction of the net force on the ball? A A heavy steel ball is attached to a light rod to make a pendulum. The pendulum begins at rest at instant^, as shown. The ball is then pulled to one side and held at rest. At instant B, the pendulum is released from rest. It swings down and passes its original height at instant C, as shown. Ignore air resistance. 2. The sum of gravitational potential Ball energy (PEG) + kinetic energy (KE) of the A pendulum is: a) Greater atB. b) Greater at C. c) Equal at both instants. d) It is impossible to determine without more information. A roller coaster starts at rest at the top of a track. The coaster is released and rolls down the shallower track as shown in the diagram. At the ~"~£-£™1'5tpr bottom of the track, the speed of the .Shallower track coaster is recorded. • » • • Steeper track ^ ^ The experiment is repeated with the 1.... ,. coaster rolling down the steeper track shown in the diagram. 6 6 6' 3. Compared to the original experiment down the shallower track, the speed of the coaster at the bottom of the steeper track is: a) The same as at the bottom of the shallower track. b) Greater than at the bottom of the shallower track. 200 c) Less than at the bottom of the shallower track. d) Impossible to tell without more information. A new roller coaster is designed that has twice the mass as the original coaster. The experiment was again repeated with the heavier coaster placed on the top of the shallower track and released from rest. At the bottom of the track, the speed of this coaster was recorded. 4. Compared to the original coaster rolling down the shallower track, the speed of the heavier coaster at the bottom of the track is: a) The speed of the heavier coaster will be 2x the original speed. b) The speed of the heavier coaster will be greater than the original speed;, but without more information, it is impossible to know by how much. c) The speed of the heavier coaster will be 0.5x the original speed. d) The speed of the heavier coaster will be less than the original speed, but without more information, it is impossible to know by how much. e) The speed of the heavier coaster will be the same as the original speed. Three children, Aricelia, Bao, and Chuck, are playing on a merry-go-round. Their positions on the merry-go-round are shown in the top-view picture at right. The merry-go-round is rotating clockwise and is neither speeding up nor slowing down. Each of the children will move in a circle as the merry-goround rotates: The radius of the circle for Aricelia is one meter; for Bao it is two meters; and for Chuck it is 3 meters. 5. A correct ranking of the speeds of the children would be: a) Aricelia > Bao > Chuck. b) Chuck > Bao > Aricelia. c) They are all the same. d) It is impossible to tell. 6. a) b) c) d) A correct ranking of the angular velocities of the children would be: Aricelia > Bao > Chuck. Chuck > Bao > Aricelia. They are all the same. It is impossible to tell. 7. A skateboarder goes over a circular bump. Velocity vectors are shown for the skateboarder just before and just after the time he is at the top of the bump. 201 ^ Which of the following statements is true about the skateboarder at the instant he is at the top of the bump as shown in the diagram? a) His acceleration is zero, and the normal force on him is equal to his weight. b) His acceleration is straight down, and the normal force on him is equal to his weight. c) His acceleration is straight down, and the normal force on him is less than his weight. d) His acceleration is straight down, and the normal force on him is greater than his weight. e) His acceleration is straight up, and the normal force is equal to his weight. f) None of the above statements is correct. 8. Velocity vectors are shown for two positions, A and B, for a child on a swing. The average acceleration between points A and B is a) zero. b) upward. c) downward, in the direction of the gravitational force. d) to the left, in the direction of motion. e) to the right, to balance the forces acting on the child. A skateboarder slows down as she coasts up a ramp. Velocity vectors are shown for the skateboarder just before and just after the time she is at the position shown in the diagram. 9. Which of the following statements about the skateboarder at the instant shown in the diagram is true? a) Her acceleration is down the ramp, and the net force acting on her is toward the center of the earth in the direction of gravity. b) Her acceleration is up the ramp in the direction she is moving, and the net force acting on her is toward the center of the earth in the direction of gravity. c) Her acceleration is down the ramp, and the net force acting on her is up the ramp in the direction she is moving d) Her acceleration is down the ramp, and the net force acting on her is also down the ramp. e) None of the above statements is correct. 202 APPENDIX D LAWSON TEST OF SCIENTIFIC REASONING Version Directions: This is a test of your ability to apply aspects of scientific and mathematical reasoning to analyze a situation to make a prediction or solve a problem. Make a dark mark on the answer sheet for the best answer for each item. If you do not fully understan what is being asked in an item, please ask the test administrator for clarification. 1. Suppose you are given two clay balls of equal size and shape. The two clay balls alsc weigh the same. One ball is flattened into a pancake-shaped piece. Which of these statements is correct? a. The pancake-shaped piece weighs more than the ball b. The two pieces still weigh the same c. The ball weighs more than the pancake-shaped piece 2. because a. the flattened piece covers a larger area. b. the ball pushes down more on one spot. c. when something is flattened it loses weight. d. clay has not been added or taken away. e. when something is flattened it gains weight. To the right are drawings of two cylinders filled to the same level with water. The cylinders are identical in size and shape. Also shown at the right are two marbles, one glass and one steel. The marbles are the same size but the steel one is much heavier than the glass one. When the glass marble is put into Cylinder 1 it sinks to the bottom and the water level rises to the 6th mark. If we put the steel marble into Cylinder 2, the water will rise a. to the same level as it did in Cylinder 1 b. to a higher level than it did in Cylinder 1 c. to a lower level than it did in Cylinder 1 because a. the steel marble will sink faster. b. the marbles are made of different materials. c. the steel marble is heavier than the glass marble. d. the glass marble creates less pressure. e. the marbles are the same size. 203 OtASaHMUKE o frl STEB.MAR8UE CYUNOBU 5. To the right are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4th mark (see A). This water rises to the 6th mark when poured into the narrow cylinder (see B). Both cylinders are emptied (not shown) and water is poured into the wide cylinder up to the 6th mark. How high would this water rise if it were poured into the empty narrow cylinder? a. to about 8 b. to about 9 c. to about 10 d. to about 12 e. none of these answers is correct 6. because a. the answer can not be determined with the information given, b. it went up 2 more before, so it will go up 2 more again. c. it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. one must actually pour the water and observe to find out, 7. Water is now poured into the narrow cylinder (described in Item 5 above) up to the 11 th mark. How high would this water rise if it were poured into the empty wide cylinder? a. to about 7 1/2 b. to about 9 c. to about 8 d. to about 7 1/3 e. none of these answers is correct 8. because a. the ratios must stay the same. b. one must actually pour the water and observe to find out. c. the answer can not be determined with the information given. d. it was 2 less before so it will be 2 less again. e. you subtract 2 from the wide for every 3 from the narrow. 9. At the right are drawings of three strings hanging from a bar. The three strings have metal weights attached to their ends. String 1 and String 3 are the same length. String 2 is shorter. A 10 unit weight is attached to the end of String 1. A 10 unit weight is also 1 ?L * „A attached to the end of String 2. A 5 unit M ! weight is attached to the end of String 3. I 1 I The strings (and attached weights) can be swung back and forth and the time it takes to make a swing can be timed. Suppose you want to find out whether the length of the string has an effect on the time it takes to swing back and forth. Which strings would you use to find out? © a. only one string b. all three strings c. 2 and 3 d. 1 and 3 (*) e. 1 and 2 204 10. because a. you must use the longest strings. b. you must compare strings with both light and heavy weights. c. only the lengths differ. d. to make all possible comparisons. e. the weights differ. 11. Twenty fruit flies are placed in each of four glass tubes. The tubes are sealed. Tubes 1 and 11 are partially covered with black paper; Cubes 111 and IV are not covered. The tubes arc placed as shown. Then they arc exposed to red light for five minutes. The number of flics in the uncovered part of each tube is shown in the drawing. REDUOHT 1 T J i i in 9 f IV • t t 0 10 .: 10 u * t 1 t ) t REDUGHT IJiis experiment shows that flies respond to (respond means move to or away from) a. red light but not gravity b. gravity but not red light e. both red tight and gravity d. neither red light nor gravity 12. because a. most flies are in the upper end of Tube 111 but spread about evenly in Tube n . b. most flies did not gotothe bottom of Tubes I and III. c. the flies need light to sec and must fly against gravity. d. the majority of flies are in the upper ends and in the lighted ends of the tubes. c. some flics arc in both ends of each tube. 13. In a second experiment, a different kind of fly and blue light was used. The results are shown in the drawing. BLUE UGKT I \ I r 1 1 i ! * 1 t e § t I i (J w t t t t • io t ) t These data show that these flies respond to (respond means move to or away from): a. blue light but not gravity b. gravity but not blue light c. both blue light and gravity d. neither blue light nor gravity 14. because a. some flies are in both ends of each tube. b. the flies need light to see and must fly against gravity. c. the flies are spread about evenly in Tube IV and in the upper end of Tube III. d. most flies are in the lighted end of Tube II but do not go down in Tubes I ar III. e. most flies are in the upper end of Tube 1 and the lighted end of Tube II. 205 000 15. Six square pieces of wood are put into a cloth bag and mixed about. The six pieces arc identical in size and shape, however, three pieces are red and three are yellow. Suppose someone reaches into the bag Iv I I I I 1 (without looking) and pulls out one piece. What are \ Y | | Y | I YI the chances that the piece is red? a. 1 chance out of 6 b. I chance out of 3 c. 1 chance out of 2 d. 1 chance out of I c. can not be determined 16. because a. 3 out of 6 pieces are red. h. there is no way to tell which piece will be picked. c. only 1 piece of the 6 in the bag is picked. d. all 6 pieces arc identical in size and shape. e. only 1 red piece can be picked out of the 3 red pieces. 17. Three red square pieces of wood, four yellow square pieces, and five blue square pieces are put into a cloth hag. Four red round pieces, two yellow round pieces, and three blue round pieces are also put into the bag. All the pieces are then mixed about. Suppose someone reaches into the bag (without looking and without feeling for a particular shape piece) and pulls out one piece. 000 0000 00 0 0 0 000® 0© 000 What are the chances thai the piece is a red round or blue roundpiece? a. can not be determined b. 1 chance out of 3 c. 1 chance out of 21 d. IS chances out of 21 e. 1 chance out of 2 18. because a. 1 of the 2 shapes is round. b. 15ufthe21 pieces are red or blue. c. there is no way to tell which piece will be picked, d- only 1 of the 21 pieces is picked out of the bag. e. I of every 3 pieces is a red or blue round piece. 19. Parmer Brown was observing the mice that live in his field. He discovered that all of them were either fat or thin. Also, all of Ihem had either black tails or white tails. This made him wonder if there might be a link between the size of the mice and the color of their tails. So he captured all of the mice in one part of bis field and observed them. Below arc the mice that he captured. Do you tltirtk there is a link between the size of the mice and the color of their tails? a. appears to be a link b. appears not to be a link c. can not make a reasonable guess 20. because a. there are some of each kind of mouse. b. there may be a genetic link between mouse size and tail color. c. there were not enough mice captured. d. most of the tat mice have black tails while most of the thin mice have white tails. e. as the mice grew falter, their tails became darker. 206 21. The figure below at the left shows a drinking glass and a burning birthday candle stuck in a small piece of clay standing in a pan of water. When the glass is turned upside down, put over trie candle, and placed in the water, the candle quickly goes out and water rushes up into the glass (as shown at the right). This observation raises an interesting question: Why does the water rush up into the glass? Here is a possible explanation. The flame converts oxygen into carbon dioxide. Because oxygen does not dissolverapidlyinto water hut carbon dioxide does, the newly-formed carbon dioxide dissolvesrapidlyinto the water, lowering the air pressure inside the glass. Suppose you have the materials mentioned above plus some matches and some dry ice (dry ice is frozen carbon dioxide). Using some or all of the materials, how could you test this possible explanation? a. Saturate the water with carbon dioxide and redo the experiment noting the amount of water rise. b. The water rises because oxygen is consumed, so redo the experiment in exactly the same way to show water rise due to oxygen loss. c. Conduct a controlled experiment varying only the number of candles to see if that makes a difference. d. Suction is responsible for the water rise, so put a balloon over the top of aji open-ended cylinder and place the cylinder over the burning candle. e. Redo the experiment, but make sure it is controlled by holding all independent variables constant; then measure the amount of water rise. 22. What result of your test (mentioned in #21 above) would show that your explanation is probably wrong? a. The water rises the same as it did before. b. The water rises less than it did before. c. The balloon expands out. d. The balloon is sucked in. 23. A student put a drop of blond on a microscope slide and then looked at the blood under a microscope. As you can see in the diagram below, the magnified red blood cells look like little round balls. After adding a few drops of salt water to the drop of blood, the student noticed that the cells appeared to become smaller. Magnified Red Blood Cells Afusr Adding Salt Water This observation raises an interesting question: Why do the red blood cells appear smaller? Here are two possible explanations: I. Salt ions (Na+ and C1-) push on the cell membranes and make the cells appear smaller. II. Water molecules are attracted to the salt ions so the water molecules move out of the cells and leave the cells smaller. To test these explanations, the student used some salt water, a very accurate weighing device, and some water-filled plastic bags, and assumed the plastic behaves just like red-blood-cell membranes. The experiment involved carefully weighing a water-tilled bag in a salt solution for ten minutes and then reweighing the bag. What result of the experiment would best show that explanation I is probably wrong? a. the bag loses weight h. the bag weighs the same c. the bag appears smaller 24. What result of the experiment would best show that explanation II is probably wrong? a. the bag loses weight b. the bag weighs the same c. the bug appears smaller 207 APPENDIX E 15 QUESTION SUBSET OF LAWSON PRE-TEST 1. Suppose you are given two clay balls of equal size and shape. The two clay balls also weigh the same. One ball is flattened into a pancake-shaped piece. Which of these statements is correct? a. The pancake-shaped piece weighs more than the ball b. The two pieces still weigh the same c. The ball weighs more than the pancake-shaped piece 2. because a. the flattened piece covers a larger area. b. the ball pushes down more on one spot. c. when something is flattened it loses weight. d. clay has not been added or taken away. e. when something is flattened it gains weight. GUSSUARBUE . 3. To the right are drawings of two cylinders filled to the same level with water. The cylinders are identical in size and shape. o Also shown at the right are two marbles, one glass and one steel. The marbles are the same size but the steel one is much heavier than the glass one. When the glass marble is put into Cylinder 1 it sinks to the bottom and the water level rises to the 6th mark. If we put the steel marble into Cylinder 2, the water will rise a. to the same level as it did in Cylinder 1 b. to a higher level than it did in Cylinder 1 c. to a lower level than it did in Cylinder 1 4. because a. the steel marble will sink faster. b. the marbles are made of different materials. c. the steel marble is heavier than the glass marble. d. the glass marble creates less pressure. e. the marbles are the same size. 208 STEEL U*MLE Kx-& ^ iilfll 5.) To the right are drawings of a wide and a narrow •^ cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4th mark (see A). This water rises to the 6th mark when poured into the narrow cylinder (see B). Both cylinders are emptied (not shown) and water is poured into the wide cylinder up to the 6th mark. How high would this water rise if it were poured into the empty narrow cylinder? a. to about 8 b. to about 9 c. to about 10 d. to about 12 e. none of these answers is correct 6. because a. the answer can not be determined with the information given. b. it went up 2 more before, so it will go up 2 more again. c. it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. one must actually pour the water and observe to find out. 7. Water is now poured into the narrow cylinder (described in Item 5 above) up to the 11th mark. How high would this water rise if it were poured into the empty wide cylinder? a. to about 7 1/2 b. to about 9 c. to about 8 d. to about 7 1/3 e. none of these answers is correct 8. because a. the ratios must stay the same. b. one must actually pour the water and observe to find out. c. the answer can not be determined with the information given. d. it was 2 less before so it will be 2 less again. e. you subtract 2 from the wide for every 3 from the narrow. 2 209 At the right are drawings of three strings hanging from a bar. The three strings have metal weights attached to their ends. String 1 and String 3 are the same length. String 2 is shorter. A 10-unit weight is attached to the end of String 1. A 10-unit weight is also attached to the end of String 2. A 5-unit weight is attached to the end of String 3. The strings (and attached weights) can be swung back and forth and the time it takes to make a swing can be timed. d ® Suppose you want to find out whether the lengdi of the string has an effect on the time it takes to swing back and forth. Which strings would you use tofindout? a. only one string b. all three strings c. 2 and 3 d. 1 and 3 e. 1 and 2 ® <J1Q! because a. you must use the longest strings. b. you must compare strings with both light and heavy weights. c. only the lengths differ. d. to make all possible comparisons. e. the weights differ. (ly Six square pieces of wood are put into a clodi bag and mixed about. The six pieces are identical in size and shape, however, three pieces are red and three are yellow. Suppose someone reaches into the bag (without looking) and pulls out one piece. What are the chances that the piece is red? a. 1 chance out of 6 b. 1 chance out of 3 . c. 1 chance out of 2 d. 1 chance out of 1 e. can not be determined //uj because a. 3 out of 6 pieces are red. b. there is no way to tell which piece will be picked. c. only 1 piece of the 6 in the bag is picked. d. all 6 pieces are identical in size and shape. e. only 1 red piece can be picked out of the 3 red pieces. 210 0 00 0 00 13. Three red square pieces of wood, four yellow square pieces, and five blue square pieces are put into a cloth bag. Four red round pieces, two yellow round pieces, and three blue round pieces are also put into the bag. All the pieces are then mixed about. Suppose someone reaches into the bag (without looking and without feeling for a particular shape piece) and pulls out one piece. 000 0000 00000 ®©0® 00 0.0 0 Wliat are the chances that the piece is a red round or blue round piece ? a. can not be determined b. 1 chance out of 3 c. 1 chance out of 21 d. 15 chances out of 21 e. 1 chance out of 2 14. because a. 1 of the 2 shapes is round. b. 15 of the 21 pieces are red or blue. c. there is no way to tell which piece will be picked. d. only 1 of the 21 pieces is picked out of the bag. e. 1 of every 3 pieces is a red or blue round piece. 15yA uniform block of cheese is cut into two unequal pieces, labeled A and B. The mass density of an object is defined as the mass of that object divided by its volume . A correct ranking of the mass densities (from largest to smallest) of the original block, the largest piece (A), and the smallest piece (B) is: a. Original block, largest piece, smallest piece. b. Smallest piece, largest piece, original block. c. All mass densities are the same. d. Not possible to determine without additional information. 211 y x Original block • >^ / A/VA ,*y APPENDIX F THREE VERSIONS OF PROPORTIONAL REASONING PRETEST Original Pretest Version 1. A uniform block of trinitramine is cut into two unequal pieces, labeled piece A (larger piece) and piece B (smaller piece). The mass density of an object is defined as the mass of that object divided by its volume. A correct ranking of the mass densities (from largest to smallest) of the original block, piece A, and piece B is: a. b. c. d. Original block sT/ / Original block, piece A, piece B. Piece B, piece A, original block. All mass densities are the same. Not possible to determine without additional information. To the right are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4th mark (see A). This water rises to the 6th mark when poured into the narrow cylinder (see B). / / / A / B / 4- 2. Both cylinders are emptied (not shown) and water is poured into the WIDE cylinder up to the 6th mark. How high would this water rise if it were poured into the empty narrow cylinder? a. to about 8 b. to about 9 c. to about 10 d. to about 12 e. none of these answers is correct B' 3. because a. the answer can not be determined with the information given. b. it went up 2 more before, so it will go up 2 more again. c. it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. one must actually pour the water and observe to find out. 4. Water is now poured into the narrow cylinder (described above) up to the 11th mark. How high would this water rise if it were poured into the empty wide cylinder? a. to about 7 1/2 b. to about 9 c. to about 8 d. to about 7 1/3 e. none of these answers is correct 212 5. because a. the ratios must stay the same. b. one must actually pour the water and observe to find out. c. the answer can not be determined with the information given. d. it was 2 less before so it will be 2 less again. e. you subtract 2 from the wide for every 3 from the narrow. Modified Pretest Version 1. A brownie is cut into two unequal pieces, labeled Annie's piece (larger piece) and Russell's piece (smaller piece). The mass density of an object is defined as the mass of that object divided by its volume. A correct ranking of the mass densities (from largest to smallest) of the original brownie, Annie's piece, and Russell's piece is: a. b. c. d. Original brownie, Annie's piece, Russell's piece. Russell's piece, Annie's piece, original brownie. All mass densities are the same. Not possible to determine without more information. The cold medicine you bought came with two cylindrical containers: a wider one marked off in adult doses and a narrower one marked off in children's doses. When you fill the adult (wide) cylinder to the fourth mark (see A), and then pour it into the children's (narrow) cylinder, it rises to the 6th mark (see B). By mistake you pour your child's medicine up to the 6th mark in the ADULT dosage cylinder. How high would this dose be if you pour it into the children's cylinder? a. To about 8 b. To about 9 c. To about 10 d. To about 12 e. None of these answers is correct = : £3@l 6—1= B<gi because a. the answer can not be determined with the information given. b. it went up 2 more before, so it will go up 2 more again, c it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. one must actually pour the water and observe to find out. 4. On another occasion you poured an adult dose into the children's (narrow) cylinder (described above) up to the 11th mark. How high would this medicine rise if it were 213 poured into the empty adult (wide) cylinder? a. To about 7 1/2 b. To about 9 c. To about 8 d. To about 7 1/3 e. none of these answers is correct 5. because a. the ratios must stay the same. b. one must actually pour the medicine and observe to find out. c. the answer can not be determined with the information given. d. it was 2 less before so it will be 2 less again. e. you subtract 2 from the adult (wide) cylinder for every 3 from the children's (narrow) cylinder. Discrete Modified Pretest Version 1. A wedge is cut out of a wheel of cheese. Let's define the "cheesiness" of any piece of cheese as the weight of that piece divided by its volume. Suppose you were to rank the "cheesiness" of the original wheel, the wedge, and the remaining wheel. From largest to smallestthis ranking would be: a. Original wheel, remaining wheel, wedge. b. Wedge, remaining wheel, original wheel. c. The cheesinesses are all the same. d. We can't tell without additional information. Suppose you've decided to volunteer at a homeless shelter, and you are asked to serve rice. There are two scoops for the rice, one for an adult portion and one for a child's portion. You notice that a bowl with enough rice for 4 adult portions is also enough for 6 child portions. 2. A second bowl holds enough rice for 6 adult portions. This would be enough for: a. about 8 child portions. b. about 9 child portions. c. about 10 child portions. d. about 12 child portions. e. none of these answers is correct 2. because a. the answer can not be determined with the information given. b. there were 2 more child portions before, so there will be 2 more again. c. there are 3 child portions for every 2 adult portions. d. the child portions are smaller. e. you'd need to actually serve the rice and see what happens to find out. 214 4. A third serving bowl is now filled with enough rice for 11 child portions (described above). How many adult portions could you serve out of this serving bowl? a. about 7 1/2 b. about 9 c. about 8 d. about 7 1/3 e. none of these answers is correct 5. because a. the ratios must stay the same. b. you'd need to actually serve the rice and see what happens to find out. c. the answer can not be determined with the information given. d. there were 2 less adult portions before so there will be 2 less again. e. you subtract 2 from the adult portions for every 3 from the children's portions. 215 APPENDIX G CORRELATION COEFFICIENTS AMONG VARIOUS COMPONENTS OF LAWSON TEST (A-G), PRETEST (H) AND FINAL EXAM (I). Phys 211 - Table of correlation coefficients among various reasoning ability measures A B C D E F G H I 1.00 0.07 -0.01 0.04 0.06 -0.07 0.11 -0.10 0.20 1.00 0.17 0.27 0.06 0.18 0.44 0.18 0.37 1.00 0.26 0.32 0.18 0.63 0.65 0.47 1.00 0.30 0.40 0.73 0.27 0.41 1.00 0.30 0.68 0.35 0.41 1.00 0.63 0.23 0.38 1.00 0.55 0.65 1.00 0.43 A: Conservation of M a s s (L1-L2). B: Conservation of Volume (L3-L4). C: Proportional R e a s o n i n g - L a w s on (L5-L8). D : Control of Variables (L9-L14). iL: rroDaDil lty r\easoning {LiLo-LilzU}. F: Hypo-deductive Reasoning (L21-L24). G: Total Law son Score. H: Proportional Reasoning -pretest. I : Final Exam. * in parentheses are the question numbers corresponding to reasoning abilities measured by Lawson test (appendix D). 216 1.00 Phys 215 - Table of correlation coefficients among various reasoning ability measures A B C D E F G H I 1.00 0.14 0.26 0.19 0.03 0.13 0.32 0.07 0.19 1.00 0.37 0.07 0.34 0.11 0.48 0.40 0.38 1.00 0.53 0.58 0.34 0.89 0.69 0.63 1.00 0.08 0.24 0.65 0.44 0.51 D 1.00 0.20 0.66 0.49 0.39 E 1.00 0.56 0.24 0.13 A: Conservation of Mass (L1-L2). 1.00 0.69 B: Conservation of Volume (L3-L4). C: Proportional Reasoning-Lawson (L5-L8). 1.00 D: Control of Variables (L9-L14). E: Probability Reasoning (L15-L20). F: Hypo-deductive Reasoning (L21-L24). G:Total Lawson Score. H: Proportional Reasoning -pretest. I : Final Exam. * In parentheses are the question numbers corresponding to reasoning abilities measured by Lawson test (appendix D). 217 B 0.64 G 0.63 H 1.00 APPENDIX H VARIOUS SCATTERPLOTS OF FINAL EXAM SCORE VS MEASURES OF WORKING MEMORY CAPACITY Scatterplot of Final Score vs WMT time Final33 40 0 0 + + 0 + + + + + + -* <0> -fl- o 0 I 1 o o + + o + + + + 0 OD + 0 + 0 0 + o 0 + o 0 0 -to o O 0 o ++ o a-i + + + o o 4f • 0 O o o +t + o + o o o o + I 2 0 o I I I I I I I 3 i i i—i i i i i i i i—f-i i I i i i 4 5 6 7 8 TimeWMT Course * * * °OOphys211 + + + Phys213 +-!- + Phys215 Figure 3.13. Scatterplot of students 'final exam score vs time spent to complete working memory test. 218 Scatterplot of Final Score vs WMT ratio Final33 40 oo 0+ «- o + o 0 + 0 ++ ++ o+ o +o + o * o o o + t 0- 0 + o + 0 ++ 0 + o o oo o e> o o + o + o o o o 20 30 SpanScrRatio Course * * * o o o Phys211 • Phys213 + + + Phys215 Fig 3.14. Scatterplot of students 'final exam score vs time-based correct response rate. Scatterplot of final score vs Spacial WMT score scoreSPWMT Course + + +Cross-Li + + +Ptiys211 + + + Phys213 + + + Phys215 Fig 3.16. Scatterplot of students 'final exam score vs percentage of spatial working memory capacity item correctly answered. 219 Scatterplot of final score vs Time of Spacial WMT TimeMinsSPWMT Course + + + + + + Crass-LI + + +Phys211 + + + Pbys 213 + + + Phys215 Fig 3.17. Scatterplot of students 'final exam score vs time spent to complete spatial working memory test. 220 APPENDIX I VARIOUS TABLES SHOWING DIFFERENCES IN PERFORMANCE ON TWO PRETEST VERSIONS FOR TWO INTRODUCTORY PHYSICS POPULATIONS Table 4.3. Summary of difference in performance on each question on original pretest version between Physics 211 students and Physics 215 students. Question Course N Mean Std. Std. Score Dev. Error (out of 1) Phys211 254 0.461 0.500 0.031 Phys215 161 0.627 0.485 0.038 Qi Diff(2-1) 0.494 0.167 0.050 Q2 Phys211 Phys215 Diff(2-1) 254 161 0.484 0.646 0.162 0.501 0.480 0.493 0.031 0.038 0.050 Q3 Phys211 Phys215 Diff(2-1) 254 161 0.453 0.640 0.187 0.499 0.482 0.492 0.031 0.038 0.050 Q4 Phys211 Phys215 Diff(2-1) 254 161 0.295 0.441 0.146 0.457 0.498 0.473 0.029 0.039 0.048 Q5 Phys211 Phys215 Diff(2-1) 254 161 0.614 0.721 0.106 0.488 0.450 0.474 0.031 0.036 0.048 Phys211 Phys215 254 161 2.318 3.086 1.754 1.666 0.110 0.131 0.769 1.720 0.173 Total Score Diff(2-1) * Cohen's d: Ql d=0.34; Q2 d=0.33; Q3 d=0.38; Q4 d=0.31; Q5 d=0.23; Total Score d=0.45. 221 Table 4.4. Summary oft-test analysis for difference described in Table 4.3. Question Method Variances df t-Value Pr> Pooled Equal 413 3.35 0.001 Unequal 347.9 3.37 0.001 Equal 413 3.26 0.001 Unequal 351.3 3.29 0.001 Equal 413 3.77 0.000 Unequal 349.4 3.80 0.000 Equal 413 3.06 0.002** Unequal 318.9 3.00 0.003** Equal 413 2.23 0.026* Unequal 360.5 2.27 0.024* Equal 413 4.44 0.000 Satterthwaite Unequal 353.1 4.49 * Significant at the P < 0.05 level; **significant at the PO.01 level 0.000 Ql Satterthwaite Pooled Q2 Satterthwaite Pooled Q3 Satterthwaite Pooled Q4 Satterthwaite Pooled Q5 Satterthwaite Pooled Total Score 222 Table 4.5. Summary of difference in performance on each question on the modified pretest version between Physics 211 students and Physics 215 students. Course N Mean Std. Std. Dev. Error Score (out of 1) Phys211 191 0.036 0.429 0.496 118 0.627 0.486 0.045 Phys-215 Ql 0.492 Diff(2-1) 0.198 0.058 Q2 Phys211 Phys215 Diff(2-1) 190 118 0.592 0.754 0.163 0.493 0.432 0.471 0.036 0.040 0.055 Q3 Phys211 Phys215 Diff(2-1) 191 118 0.581 0.788 0.207 0.495 0.410 0.464 0.036 0.038 0.054 Q4 Phys211 Phys215 Diff(2-1) 191 118 0.304 0.576 0.273 0.461 0.496 0.475 0.033 0.046 0.056 Q5 Phys211 Phys215 Diff(2-1) 191 118 0.644 0.720 0.076 0.480 0.451 0.469 0.035 0.042 0.055 Phys211 Phys215 191 118 2.550 3.466 1.572 1.448 0.114 0.133 Total Score Diff(2-1) 0.916 1.526 0.179 * Cohen's d: Ql d=0.40; Q2 d=0.35; Q3 d=0.45; Q4 d=0.57; Q5 d-0.16; Total Score d=0.60. 223 Table 4.6. Summary oft-test analysis for difference described in Table 4.5. Question Method Variances df t-Value Pr> Pooled Equal 307 3.43 0.001 252.1- 3.45 0.001 Equal 307 2.95 0.003 Unequal 272.2 3.04 0.003 Equal 307 3.81 0.000 Unequal 281.6 3.98 0.000 Equal 307 4.90 0.000 Unequal 234.1 4.82 0.000 Equal 307 1.39 0.165* Unequal 259.9 1.41 0.159* Equal 307 5.13 0.000 Unequal 263.4 5.23 0.000 Ql — Satterthwaite Pooled Unequal Q2 Satterthwaite Pooled Q3 Satterthwaite Pooled Q4 Satterthwaite Pooled Q5 Satterthwaite Pooled Total Score Satterthwaite * NOT significant at the 0.05 level. 224 APPENDIX J VARIOUS TABLES SHOWING GENDER SEPARATED DIFFERENCES IN PERFORMANCE ON TWO PRETEST VERSIONS FOR TWO INTRODUCTORY PHYSICS POPULATIONS Table 4.09. Descriptive statistics for difference in performance between Physics 211 students on 2 pretest versions for male andfemale students, — Pretest Version N Male Original Modified Diff(2-1) 122 93 Female Original Modified Diff(2-1) 128 88 Gender Mean Score (out of 5) 2.525 2.495 -0.030 Std. Dev. Std. Error 1.726 1.592 1.670 0.156 0.165 0.230 2.086 2.636 0.550 1.748 1.548 1.669 0.155 0.165 0.231 * Cohen's d: males d=0.02; females d=0.33 Table 4.10. Summary oft-test analysis for difference described in Table 4.09. Gender Method Variances df t-Value Pr > |t| Pooled Equal 213 0.13 0.896 Unequal 205.4 0.13 0.895 Equal 214 2.38 0.018* Unequal 200.8 2.44 0.016* Male Satterthwaite Pooled Female Satterthwaite * significant at the 0.05 level. 225 Table 4.11. Descriptive statistics for gender difference in performance by Physics 211 students on 2 pretest versions. Pretest version Original Gender N Male Female Diff (MaleFemale) 122 128 Mean Score (out of 5) 2.525 2.086 0.439 Male 93 2.495 Female 88 2.636 Diff -0.142 (MaleFemale) * Cohen's d: original pretest d=0.25; modified d=0.09. Modified Std. Dev. Std. Error 1.726 1.748 1.737 0.156 0.155 0.220 1.592 1.548 1.570 0.165 0.165 0.234 Table 4.12. Summary oft-test analysis for difference described in Table 4.11. Pretest Method Variances df t-Value Pr > |t| Version Original Pooled Equal 248 2.00 0.047* Unequal 247.7 2.00 0.047* Satterthwaite Pooled Equal 179 0.61 0.545 Unequal 178.9 0.61 0.544 Modified Satterthwaite * significant at the 0.05 level. 226 Table 4.13. Descriptive statistics for difference in performance between Physics 215 students on 2 pretest versions for male andfemale students. Gender Pretest Version N Male Original Modified Diff(2-1) 121 87 Female Original Modified Diff(2-1) 37 28 Mean Score (out of 5) 3.223 3.575 0.352 Std. Dev. Std. Error 1.615 1.467 1.555 0,147 0.157 0.219 2.541 3.357 0.817 1.709 1.254 1.531 0.281 0.237 0.383 * Cohen's d: males d=0.23; females d=0.54. Table 4.14. Summary oft-test analysis for difference described in Table 4.13. Gender Variances t-Value Pr > |t| Method df Pooled Equal 206 1.61 0.109 Unequal 195 1.63 0.104 Equal 63 2.13 0.037* Unequal 62.96 2.22 0.030* Male Satterthwaite Pooled Female Satterthwaite * significant at the 0.05 level. 227 Table 4.15. Descriptive statistics for gender difference in performance by Physics 215 students on 2 pretest versions. Pretest version Original Gender N Male Female Diff (MaleFemale) 121 37 Mean Score (out of 5) 3.223 2.541 0.683 Male 87 3.575 Modified Female 28 3.357 Diff 0.218 (MaleFemale) * Cohen's d: original pretest d=0.42; modified d=0.15. Std. Dev. Std. Error 1.615 1.709 1.637 0,147 0.281 0.308 1.468 1.254 1.420 0.157 0.237 0.308 Table 4.16. Summary of t-test analysis for difference described in Table 4.15. Pretest t-Value Pr>|t| Method Variances df Version 1 Pooled Equal 2.22 0.028* 156 Satterthwaite Unequal 57.1 2.15 0.036* Pooled Equal 113 0.71 0.482 Unequal 52.9 0.77 0.448 2 Satterthwaite * significant at the 0.05 level. 228 APPENDIX K FALL 2009 PRETEST VERSIONS Fall 2009 Original Pretest Version 1. A uniform block of trinitramine is cut into two unequal pieces, labeled piece A (larger piece) and piece B (smaller piece). The mass density of an object is defined as the mass of that object divided by its volume. A correct ranking of the mass densities (from largest to smallest) of the original block, piece A, and piece B is: a. b. c. d. / Original Wocfc Original block, piece A, piece B. Piece B, piece A, original block. All mass densities are the same. Not possible to determine without additional information. To the right are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4th mark (see A). This water rises to the 6th mark when poured into the narrow cylinder (see B). /T/ / A / B / *- 2. Both cylinders are emptied (not shown) and water is poured into the WIDE cylinder up to the 6th mark. How high would this water rise if it were poured into the empty narrow cylinder? a. to about 8 b. to about 9 c. to about 10 d. to about 12 e. none of these answers is correct 3. because a. the answer can not be determined with the information given. b. it went up 2 more before, so it will go up 2 more again. c. it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. one must actually pour the water and observe to find out. A ruler is suspended at its center from a frictionless pivot shown at right. The ruler balances when the two masses: wy and m2 are 120 mm 3fc s ^ 90 mm CZJ m2 18 kg 229 suspended from the positions shown. The mass of mi is 18 kg while the mass of m2 is unknown. 4. A student moves the unknown mass (rri2) to a position that is 60 mm from the pivot. To what position should the 18 kg mass (mi) be moved for the ruler to balance again? a. 80 mm from the pivot b. 90 mm from the pivot c. 45 mm from the pivot d. 120 mm from the pivot e. Not possible to determine without additional information. Three points: A, B and C on a rotating disk are shown in the top-view picture at right. The disk rotates clockwise about a pivot at its center, and is neither speeding up nor slowing down. Each point moves in a circle as the disk rotates: The radius of the circle for point A is one meter; for point B it is two meters; and for point C it is 3 meters. In one second, point B travels 3 meters along the circle it is moving in. What distance does Point C travel through in one second? a. 1.5m b. 2m c. 4m d. 4.5m e. none of these answers is correct 5. Three points: A, B and C on a rotating disk are shown in the top-view picture at right. The disk rotates clockwise about a pivot at its center, and is neither speeding up nor slowing down. Each point moves in a circle as the disk rotates: The radius of the circle for point A is one meter; for point B it is two meters; and for point C it is 3 meters. In one second, point B travels 3 meters along the circle it is moving in. What distance does Point C travel through in one second? a. 1.5m b. 2m c. 4m d. 4.5m e. none of these answers is correct 230 6. Two interlocked gears are shown at right. For every 40 revolutions made by the smaller gear, it is noticed that the larger gear only completes 16 revolutions. How many revolutions does the larger gear complete, given the smaller gear completes 30 revolutions. a. 40 revolutions b. 12 revolutions c. 6 revolutions d. 20 2/3 revolutions e. none of these answers is correct Fall 2009 Modified Pretest Version 1. A brownie is cut into two unequal pieces, labeled Annie's piece (larger piece) and Russell's piece (smaller piece). The mass density of an object is defined as the mass of that object divided by its volume. A correct ranking of the mass densities (from largest to smallest) of the original brownie, Annie's piece, and Russell's piece is: a. Original brownie, Annie's piece, Russell's piece. b. Russell's piece, Annie's piece, original brownie. c. All mass densities are the same. d. Not possible to determine without more information. The cold medicine you bought came with two cylindrical containers: a wider one marked off in adult doses and a narrower one marked off in children's doses. When you fill the adult (wide) cylinder to the fourth mark (see A), and then pour it into the children's (narrow) cylinder, it rises to the 6th mark (see B). 2. By mistake you pour your child's medicine up to the 6* mark in the ADULT dosage cylinder. How high 231 would this dose be if you pour it into the children's cylinder? a. To about 8 b. To about 9 c. To about 10 d. To about 12 e. None of these answers is correct 3. because a. the answer can not be determined with the information given. b. it went up 2 more before, so it will go up 2 more again, c it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. one must actually pour the water and observe to find out. A see-saw pivoted at its center is shown in the diagram at right. The see-saw balances when two children: Ai and Bao, are seated in the positions shown. Ai's mass is known to be 18 kg but Bao's mass is unknown. 240 cm- 180 cm Ai 18 kg Bao X 4. Bao decides to slide himself to a position 60 cm from the pivot. To what distance from the pivot must Ai position herself so they balance again? a. 80 cm from the pivot b. 90 cm from the pivot c. 45 cm from the pivot d. 120 cm from the pivot e. none of these answers is correct 5. Three children, Aricelia, Bao, and Chuck, are playing on a merry-go-round. Their positions on the merry-go-round are shown in the top-view picture at right. The merry-go-round is rotating clockwise and is neither speeding up nor slowing down. Each of the children will move in a circle as the merry-go-round rotates: The radius of the circle for Aricelia is one meter; for Bao it is two meters; and for Chuck it is 3 meters. In one second, Bao travels 3 meters along the circle he is moving in. What distance does Chuck travel through in one second? a. 1.5m b. 2m c. 4m d. 4.5m 232 e. none of these answers is correct 6. Two interlocked gears are shown at right. For every 40 revolutions made by the smaller gear, it is noticed that the larger gear only completes 16 revolutions. How many revolutions does the larger gear complete, given the smaller gear completes 30 revolutions. a. 40 revolutions b. 12 revolutions c. 6 revolutions d. 20 2/3 revolutions e. none of these answers is correct 233 APPENDIX L CORRELATION MATRIX OF TA RATINGS OF STUDENTS AND OTHER MEASURES OF STUDENT SUCCESSS. Table 4.18. Table of correlation coefficients between TAs' ratings of students with these students 'pretest scores (sorted by version of pretest taken) as well as with their scores on the final exam. Fall 2008 Correlation of TA ratings of students N Version Total Prop Scr Final Scr 26 25 1 2 0.26 0.05 0.64 0.61 24 25 1 2 0.66 0.46 0.68 0.68 18 21 1 2 0.54 -0.17 0.63 0.49 19 17 1 2 0.25 0.15 0.60 0.24 Version 1 = original pretest version; version2 = modified pretest version. 234 REFERENCES 1 L. McDermott and E. Redish, "Resource Letter: PER-1: Physics Education Research," Am. J. Phys. 67(9), 755-767 (1999). 2 L. 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