(3—3:... .., . .. is? r u. 0.5.,” .... ,...“:A1...W.w.fi E... “on: a emf} flu... %1ML4»M ‘ , & - : a. . l u. . «Emma‘s... ., u...a.a..u.lw.th?s... £83.36. \ v .7 3h“- . 1. .15 Z 42:: flaunt-W}- . 5‘}... infi‘a}alstat. 8% a .(n: .mw. 39!. it... at t. .4 1.1:}: c. .3 5.355552%: ‘1: A Fiat}; :93? I. . 1.3.5.1.. :3... tit-i... 21 huh - .f A “1.7.1 1.3 .3. .4331: ,. . . 93.1.5 1.... .. “I. 1 .. .1. I: .i‘iiauteehPI 3, laxOé‘i. eta. .taxblillsk .37. (.9. .t.L..L3I~l¢-y ’33.: .itI-t: u. v96 .4 we. 1: .. . :.. llllllglllllllllllll 02048 8585 THESlS é LIBRARY 7“" Michigan State University This is to certify that the dissertation entitled STUDENTS' UNDERSTANDINGS OF THE BEHAVIOR OF A GASEOUS SUBSTANCE presented by Edward L. Jones II has been accepted towards fulfillment of the requirements for Ph . D . degree in Education Major professor Date December 16, 1999 MSUi: an Affirmatiw Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 11/00 CJCIRCJDaieOuopGS-p‘“ STUDENTS’ UNDERSTANDINGS OF THE BEHAVIOR OF A GASEOUS SUBSTANCE By Edward L. Jones II A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Teacher Education 1999 ABSTRACT STUDENTS' UNDERSTANDINGS OF THE BEHAVIOR OF A GASEOUS SUBSTANCE By Edward L. Jones M One hundred sixteen community college students enrolled in a basic chemistry class who had completed a unit on the behavior of a gaseous substance were given a written instrument that presented several mathematical and conceptual problems describing the behavior of a gas. Nine students representing a range of achievement levels were chosen for more intensive clinical interviews. In the clinical interview, students explained their responses on the written instrument and gave quantitative and qualitative explanations of the behavior of air in the barrel of a real syringe. Interview results revealed that students commonly experience difficulties at three different levels: 1. Mathematical understanding. Most students could manipulate the gas law equations, but few had a real understanding of the equation. There were some unique understanding of proportional relationships. 2. Conceptual understanding. Many students could represent pictorially the notion that gas molecules randomly occupy the entire space of its container. Many, however, had a different conception of this when the air was compressed. The reason for this seemed to be due to a misunderstanding of the kinetic molecular theory. 3. Real-world application. Students’ use of their mathematical understanding to explain the behavior of air in a real syringe revealed some internal consistency found in mathematical explanations of real- world phenomena. Many students used mathematical strategies consistent with their mathematical understanding and satisfactory for producing reasonable estimates of numerical values. All of the 9 students had misconceptions about mathematical proportionality with most of them understanding proportional relationships as being additive in nature. Although some of the students were able to state the relationship between two variables, they could only do so outside of the context of the gas law equation. Only one student was able to propose a reasonable explanation of the proportional relationships between variables in a gas law equation. All 9 students were classified as either transitional or naive in the real- world use of their mathematical understandings with 3 of the 9 clearly having na'ive conceptions of the mathematics of gas behavior. A majority of the 9 students could clearly represent the nature of the submicroscopic level of gas behavior when asked to draw it during the clinical interview. However, only 2 of these students had the Chemist’s understanding of this concept when put to use with a real-world task. Three students were considered transitional in their thinking, having various capacities to understand and use molecular language depending on the context of the problem; while, 4 students were clearly naive in their thinking having various conceptions of the atomic theory which they could not consistently use in describing the behavior of air in real-world situations. The results of this study suggest that students in college basic chemistry classes don’t walk away from instruction on the behavior of a gaseous substance with the understanding teachers intend for them to possess. College chemistry teachers should be aware of the myriad of ways students understand the behavior of a gaseous substance and incorporate better methods of instruction to help students truly understand this behavior. This study analyzes students’ understanding and suggests a possible approach for helping them gain better understanding. This dissertation is lovingly dedicated to my father and high school chemistry teacher, Mr. Edward L. Jones, Sr., who, for 35 years, exposed his high school and college chemistry students to the ideas of conceptual understanding in chemistry. ACKNOWLEDGEMENTS No man is an island entire of itself. Every man is a piece of the continent; apart of the main. If a clod is carried away by the sea, Europe is less. Every man’s death diminishes me, for! am apart of all mankind. 80 never send to know for whom the bell tolls, it tolls for thee. -John Donne I recognize that this work is not the sole result of my efforts alone. In fact, this dissertation and the completion of my graduate experience would never have come to fruition had it not been for the myriad of people who have inspired, counseled, consulted, and nudged me along the way. First of all, I was fortunate enough to have crossed paths with Dr. Charles W. (Andy) Anderson at Michigan State University. He has been an advisor and dissertation director extrordinaire. At many times when my focus seemed to wane, he provided the gentle but firm nudge I needed to get back on course. His gentle nature and persistent counsel has been most needed and appreciated. Much needed guidance has been provided also by my committee members who represent many areas of expertise. I am grateful for Drs. Sharon Feiman-Nemser, Robert Floden, and James Miller for their willingness to serve and eager spirits. Their individual charms and professional expertise has been truly appreciated and valued. I further thank Dr. James Miller who enthusiastically accepted the invitation to step in and fill the void left by Dr. Gordon Galloway, who had to step down as a committee member due to his progressive illness. My thanks and prayers are with him. vi I am grateful to Drs. Maxie Jackson and Dorothy Harper-Jones at Michigan State University, and Drs. Richard Evans, James Thompson, John Vickers, Phillip Redrick, and Margaret Kelley at Alabama A & M University. These individuals set the wheels in motion that provided much needed financial support during my graduate study. I could never adequately thank them enough for what they have accomplished in spite of limited budgets and resources. I was and am continually inspired by the many conversations - professional, academic, and personal - l have shared with my colleagues and office mates at Lansing Community College. I am grateful to Drs. Shannon Briggs, Brian Jordan, Gary Lobel, Laura Markham, and Chris Marschall. Vifith these individuals I shared office space and many wonderful and stimulating conversations. All are excellent science teachers in their own right whose collegiality I will always cherish. The research for this dissertation would not have been possible but for the cooperative nature of other colleagues at Lansing Community College. I am thankful for Gerald Blair, Evelyn Green, Laura Markham, Don Nofzinger, and Mike Waldo. These individuals willingly opened their classrooms and gave me valuable class time to collect the data for this study. Their kindness and generosity have not gone unnoticed. Many of my friends had the patience to listen at times when l lamented about my problems during the graduate experience. I thank them for their support, encouragement, and listening ear. To LaTrese Adkins, Edward Fubara, and Dr. Janice Hilliard, I offer my sincere and profound appreciation for enduring vii through “the struggle” with me and the empathetic ear you were willing to lend. I would also thank Dr. Melvin T. Jones, pastor, and a multitude of friends at the Union Missionary Baptist Church in Lansing who have been paragons of strength for me during my years of graduate study. The individuals in this church are too numerous to name here, but all hold a special place in my heart. Finally, my family has given of themselves the most during this graduate experience. I am grateful to my wife, Rosalyn, for giving me the time, which often belonged to her, to complete this task. I am thankful for Mom and Daddy, Renee, Edwina, and DeJuan for the constant support and encouragement they have provided throughout my life and this experience. To all, I love you and thank God for you. viii TABLE OF CONTENTS LIST OF TABLES ............................................................................................... xiv LIST OF FIGURES .............................................................................................. xv CHAPTER ONE: INTRODUCTION ................................................................... 1 Statement of the Problem ........................................................................... 1 Theoretical Framework ............................................................................... 2 Conceptual change model ............................................................... 2 Three-part model ............................................................................. 4 Chemists’ understanding of gas behavior ............................. 4 Specifics of mathematical understanding ............................. 5 Specifics of conceptual understanding ................................. 7 The connection between conceptual and mathematical understanding ........................... 8 Specifics of real-world understanding ................................... 9 Research Questions ................................................................................. 11 Overview of the Study .............................................................................. 11 Methodological Limitations ....................................................................... 12 CHAPTER TWO: LITERATURE REVIEW ...................................................... 14 Introduction ............................................................................................... 14 Part I: Students’ Mathematical Understandings and Explanations of Gas Behavior ................................................................................. 15 Summary of Part I .................................................................................... 20 Part II: Students’ Conceptual Understandings and Explanations of Gas Behavior ................................................................................. 21 Students’ macroscopic understandings and explanations of gas behavior ....................................................................... 21 Students’ understandings and explanations of the atomic- molecular level of gas behavior ........................................... 27 Summary of Part II ................................................................................... 34 Part III: Real-World Applications ............................................................. 35 Summary of Chapter Two ......................................................................... 39 Mathematical understanding .......................................................... 39 Conceptual understanding ............................................................. 41 Real-world applications .................................................................. 42 CHAPTER THREE: METHODOLOGY ................................................................ 44 Introduction ............................................................................................... 44 Overview of Research Design .................................................................. 44 Subjects and Setting ................................................................................ 47 Data Collection ......................................................................................... 48 The paper-and-pencil instrument ................................................... 48 The clinical interview ...................................................................... 49 Data Analysis ............................................................................................ 50 Stage 1: The case studies of Cameron, Betty, Karen, and Connie ...................................................................... 50 Stage 2: How the remaining five students were analyzed ............ 51 Specific Descriptions of Data Collection and Analysis .............................. 51 Mathematical understanding questions ......................................... 51 Conceptual understanding questions ............................................ 56 Real-world applications questions ................................................. 58 CHAPTER FOUR: RESULTS ........................................................................... 59 Introduction ............................................................................................... 59 Case Study 1: Cameron .......................................................................... 60 Cameron’s mathematical understandings ..................................... 60 Explanation of item 1 on the paper-and-pencil instrument ................................................................ 6O Explanation of item 3 on the paper-and-pencil instrument ................................................................ 61 How Cameron understands and uses the mathematical representations of the gas laws ................................ 64 Direction vs. magnitude with proportional relationships .................................................. 66 Multiplicative vs. additive character of proportional relationships .................................................. 67 Cameron’s conceptual understandings ......................................... 68 Describing the submicroscopic nature of matter ................. 68 Describing the phenomenological behavior of a gaseous substance ................................................................. 70 How Cameron understands and uses the concepts ........... 71 Cameron’s real-world applications ................................................. 72 Summary of Cameron’s case study ............................................... 76 Mathematical understanding issues .................................... 77 Conceptual understanding issues ....................................... 79 Case Study 2: Betty ................................................................................ 81 Betty’s mathematical understandings ............................................ 81 Explanation of item 1 on the paper-and-pencil instrument ................................................................ 81 Explanation of item 3 on the paper-and-pencil instrument ................................................................ 83 How Betty understands and uses the mathematical representations of the gas laws ................................ 84 Multiplicative vs. additive character of proportional relationships .................... 85 Betty’s conceptual understandings ................................................ 86 Describing the submicroscopic nature of matter ................. 86 Describing the phenomenological behavior of a gaseous substance ................................................................. 88 How Betty understands and uses the concepts .................. 89 Betty’s real-world applications ....................................................... 90 Summary of Betty’s case study ..................................................... 94 Mathematical understanding issues .................................... 95 Conceptual understanding issues ....................................... 96 Case Study 3: Karen ............................................................................... 97 Karen’s mathematical understandings ........................................... 97 Explanation of item 1 on the paper-and-pencil instrument ................................................................ 97 Explanation of item 3 on the paper-and-pencil instrument ................................................................ 98 Understanding of P, V, and T relationships in gas law equations ................................................................ 101 How Karen understands and uses the mathematical representations of the gas laws .............................. 103 Understanding of ratio-and-proportion relationships ................................................ 1 04 Relating the variables in ratio-and-proportion equafions ..................................................... 104 Karen’s conceptual understandings ............................................. 105 Describing the submicroscopic nature of matter ............... 105 How Karen understands and uses the concepts ............... 108 Karen’s real-world applications .................................................... 109 Summary of Karen’s case study .................................................. 113 Mathematical understanding issues .................................. 1 13 Conceptual understanding issues ..................................... 1 14 Case Study 4: Connie ............................................................................ 115 Connie’s mathematical understandings ....................................... 1 15 Explanation of item 1 on the paper-and-pencil instrument .............................................................. 1 15 Explanation of item 3 on the paper—and-pencil instrument .............................................................. 1 16 xi How Connie understands and uses the mathematical representations of the gas laws .............................. 117 Connie’s conceptual understandings ........................................... 1 19 Describing the submicroscopic nature of matter ............... 119 How Connie understands and uses the concepts ............. 122 Connie’s real-world applications .................................................. 123 Summary of Connie’s case study ................................................ 125 Mathematical understanding issues .................................. 125 Conceptual understanding issues ..................................... 126 Summary of the Nine Students Who Were Clinically Interviewed .......... 128 Analysis of mathematical understanding issues .......................... 128 The understanding of proportional relationships exhibited by all students taking the paper-and-pencil instrument .............................................................. 133 Analysis of conceptual understanding issues .............................. 138 Some understandings of atomic theory exhibited by all students taking the paper-and-pencil instrument .............................................................. 143 CHAPTER FIVE: SUMMARY, CONCLUSIONS, AND IMPLICATIONS ........ 148 Summary of Dissertation ........................................................................ 148 The problem and the theoretical basis ......................................... 148 Issues derived from the literature ................................................ 151 Mathematical understanding ............................................. 151 Conceptual understanding ................................................ 151 Methods ....................................................................................... 151 Summary of findings .................................................................... 152 Issues and key findings supported by the data ................. 152 Comparison of the four case study students ..................... 154 Conceptual and mathematical connections ...................... 157 Implications for Curriculum Development, Classroom Teaching, and Teacher Education ....................................................................... 158 Implications for curriculum development and classroom teaching ............................................................................ 158 A specific strategy based on the results of this study for teaching the behavior of a gaseous substance ...... 159 Implications for teacher education ............................................... 164 Conclusions and Implications for Further Research ............................... 165 Some specific considerations arising out of this study ................. 165 Some general considerations arising out of this study ................. 166 APPENDICES ................................................................................................... 170 Appendix A: Paper-and-Pencil Instrument ............................................ 172 xii Appendix B: Course Outline & Learning Objectives .............................. 178 REFERENCES .................................................................................................. 184 xiii Table 1. Table 2. Table 3. Table 4. Table 5. Table 6. LIST OF TABLES Percentage of Students in Mathematical/Conceptual Categories Based on Results of the Paper-and-Pencil Instrument (N = 116) .......................................................................... 59 Breakdown of Students Who Were Clinically Interviewed Into Various States of Understanding .......................................... 130 Some Values for Item 3 Questions on the Paper-and-Pencil Instrument, the Percentage of Students Reporting Each Value, and the Percentage Change in Students Reporting Each Value from the Pretest to the Posttest (N = 116) .......................................... 134 Some Responses for Item 1 on the Paper-and-Pencil Instrument Posttest (N = 116) ........................................................................ 137 Percentage of Students Responding to Each Choice of Item 2 of the Posttest and the Percentage Change of Responses to Each Item From the Pretest (N = 116) .......................................... 144 Comparison of the Mathematical, Conceptual, and Real-world Understandings Possessed By the Four Case Study Students...155 xiv Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. LIST OF FIGURES Syringe System Used for Quantitative Exercise in de Berg (1995) Study ............................................................................... 18 Syringe System Used for Qualitative Exercise in de Berg (1995) Study ............................................................................... 22 Conceptual and Traditional Questions Used in Nurrenbern & Pickering (1987) Study ........................................................ 25 Sample Task Used to Examine the Atomic-molecular Structure in the Novick & Nussbaum (1981) Study .................................... 32 Flowchart for Data Collection and Analysis .............................. 46 XV CHAPTER ONE INTRODUCTION Statement of the Problem Understanding the nature and behavior of matter is important to the professional scientist and layperson alike. The world in which we live is at one level, a collection of solid, liquid, and gaseous substances. Understanding the nature and behavior of these substances is essential for scientists, as they are charged with developing the characteristics of these substances to benefit society, and laypeople, as they are called to be informed and literate citizens in an increasingly technological environment. Particularly useful in this regard, is an understanding of the nature and behavior of a gaseous substance. Air - which we depend on for our very existence - is a gaseous mixture that is extremely important to society. The measurement of barometric pressure, the lifting of a hot-air balloon, or the proper inflation of a bicycle or automobile tire are common examples of everyday occurrences that can be understood using the ideas of gas behavior. Although everybody comes into contact with gaseous substances - particularly, air - in the course of their daily activities, few people can exhibit a scientific understanding of why gaseous systems behave as they do. For example, few people are able to offer scientific explanations of what a barometer is measuring when it measures air pressure, of what makes a hot-air balloon lift and descend, or understand that the pressure in automobile tires changes with the weather. In school, students learn about the behavior of matter at the primary, secondary, and post-secondary levels. Consequently, this topic is an important one throughout the school curriculum. However at all levels, students have difficulties achieving a scientific understanding of the behavior of matter, particularly matter which exists in the gaseous state. Theoretical Framework Conceptual Change Model This is a study of student understandings of the behavior of matter, particularly that of gaseous substances. As such, it is based on a number of assumptions about how students learn and understand science and what it means to say that someone “understands” the behavior of matter. There is a general understanding in the cognitive science tradition that people develop notions about scientific phenomena before they are introduced to them in science classrooms, through their socialization into our general culture and interactions with their environment. The research literature which characterizes students’ notions, beliefs, and interpretations about scientific phenomena is extensive and has been reviewed by a number of researchers (Driver & Easley, 1978; Driver & Erickson, 1983; Osborne, Bell & Gilbert, 1983; Driver et al., 1985; Osborne and Freyberg, 1985; Eylon and Lynn, 1988; Pfundt and Duit, 1985, 1988, 1991; Wandersee, Mintzes, & Novak, 1994). This body of research advances the claim that before coming into classrooms, students develop some informal and useful ways of making sense of the world around them. These beliefs and understandings by students have been shown to be robust to typical science instruction (Driver, Guesne, & Tiberghien, 1985). Students do not enter science classrooms as empty pitchers waiting to be filled with knowledge. Rather, students enter school and science classrooms with well-established beliefs of how and why everyday things behave as they do (Posner, Strike, Hewson, & Gertzog, 1982; Resnik, 1983; Strike, 1983). These beliefs - variably referred to as misconceptions, alternative conceptions, naive theories, etc. - influence how students learn new scientific knowledge, and have been found to hinder successful acquisition of scientific concepts taught in school (Hewson, 1982; Shuell, 1987). Consequently, many researchers have examined students to try and understand how they change their alternative conceptions into scientific conceptions (Clement, 1982; Roth, Smith & Anderson, 1983; Anderson & Smith, 1983; Minstrell, 1985; Yarroch, 1985; Ben-Zvi, Eylon, & Silberstein, 1986; Nussbaum & Novick, 1982; Smith, 1990; Lee, Eichinger, Anderson, Berkheimer, & Blakeslee, 1993). What is clear is that understanding students’ alternative conceptions is instructionally useful. Again, much research in science education is being devoted to determining how it is students change their current alternative conceptions into scientific conceptions. But while this area of inquiry is important, more must be learned about the understandings students have in specific science areas. The primary goal of this study is to present a more detailed understanding of students’ conceptions of the behavior of a gaseous substance. Thre_e—Part Moge_l In addition to the general assumptions about the nature of scientific understanding embodied in the conceptual change model, this study is based upon a three-part model of chemical understanding which may be useful in understanding the explanations given by students of the nature and behavior of a gaseous substance. This model states that students must acquire three different types of understanding in order to produce an explanation about the behavior of a gaseous substance acceptable to a trained chemist: (1) a mathematical understanding, which includes knowledge of the mathematical representations of the gas laws as well as a knowledge of the proportional relationships contained within these representations, (2) a conceptual understanding of the nature of matter, which includes knowledge of the atomic-molecular and kinetic molecular theories, and (3) a real-world understanding, which allows the student to use both their mathematical and conceptual understandings to explain the behavior of real gaseous systems. Chemists’ Urmlerstandmlof the Behavior of a Gaseous Substaflcg Chemists have developed ways of predicting and explaining the response of gases to changes in temperature, pressure, volume, and amount that can be expressed in elegantly simple ways: macroscopic - PV = nRT and molecular - atomic-molecular and kinetic molecular theories. The simple expression, however, conceals conceptual difficulties. In order to use the chemists’ conceptual tools for predicting and explaining gas behavior, learners need three different kinds of understanding. Specifics of Mathematical Understanding Mathematical models of gas behavior add precision to the theories used to describe this behavior, thus increasing the predictive power of the theories. These models are important sources of explanations and hypotheses. In describing the behavior of a gaseous substance, these mathematical models are represented by the gas law equations. For the chemist, the generation and refinement of such models is a dynamic process, and is necessary and important in understanding the theories which explain the behavior of a gaseous substance. For the student, mathematical models of gas behavior are “prepackaged” and “distributed” during the instructional process. This is not to suggest, however, that students don’t use their mathematical understanding in a model- like way. However during instruction these models are generally present as static entities unrelated to any conceptual understanding of gas behavior. But, students have, and develop some mathematical understanding before being exposed to the gas law equations in the classroom. That is, students possess mathematical notions, and they use these notions in explicit or tacit ways to inform their understanding. To explain the behavior of a gaseous substance mathematically, students must have some knowledge about proportional relationships, because any good mathematical model used to describe the behavior of a gaseous substance depends on these relationships. The behavior of a gaseous substance is adequately described by its volume, pressure, temperature, amount, and the changes these quantities undergo. The volume and pressure are two quantities which are inversely proportional to each other, whereas temperature and amount are directly proportional to both volume and pressure. A mathematical model of the behavior of a gas allows for the prediction of pressure, volume, temperature, or amount when any one of these quantities is changed. That is, if the volume of a gas is doubled from its original value, the pressure exerted by the gas will be cut by one-half of its original value if the temperature and amount of gas are not changed. Similarly, if the pressure exerted by a gas is increased by two-thirds of its original value, the volume occupied by the gas is decreased by three-halves of its original value if the temperature and amount of gas are not changed. However, if the pressure, for example, of a gas is cut to one-fourth of its original value, the absolute temperature of the gas will also be cut to one-fourth of its original value if the volume and amount of gas are not changed. Chemists use mathematical models in this way to engage in proportional reasoning. For the students in this study, the mathematical relationships governing the behavior of a gaseous substance were presented during classroom instruction in each of three gas laws: P1V1 = P2V2. P1/T1 = P2/T2. V1/T1 = V2/T2. and summed up in the combined gas law, P1V1/T1 = F’2V2/T 2 In displaying mathematical understanding, students use their knowledge in some interesting ways. Some research that has explored the problem of how students use their mathematical understandings will be reviewed in Chapter 2. Specifics of Concethual Understanmg The chemist uses the atomic-molecular and kinetic molecular theories as the explanatory ideal in describing the behavior of a gaseous substance. These theories describe the phenomena of gas behavior based on the actions of molecules. The student often has problems with such explanations. This is because atoms cannot be seen. Therefore, to base explanations on their existence and movement often becomes a “leap of faith” some students are not willing to take. Chemists have a conceptual understanding of what it means to talk about the volume, pressure, temperature and amount of a gas on both a macroscopic and molecular level. For example, on a macroscopic and molecular level, when chemists talk about changing the amount (i.e., mass or number of moles) of gas in a container by withdrawing some of the gas molecules from the container, they don’t consider that the volume of the gas will change. This because mass is understood to be a measure of the quantity of gas molecules, while volume is a measure of the amount of space occupied by the gas molecules. The student often has conceptual difficulties not only with the molecular idea of small particles and their movement, in this regard, but, also, with distinctions between volume as the space occupied by the molecules as opposed to the amount of the substance. The notions of small discrete particles (atomic-molecular theory) and their inherent movement within matter (kinetic molecular theory) are the theories used by chemists to explain and understand the concepts of volume, pressure, temperature, and amount of a gas. Gases always completely occupy the space of the container in which they are placed because the molecules of the gas are in constant and random motion spreading out from one another as far as possible. Therefore, students must come to understand the chemist’s conception that removing some particles of the gas does not affect the ultimate volume because the remaining particles will spread out to fill the space of the withdrawn particles. The Connection Between Conceptual and Mathematical Understanding Chemists use their mathematical understanding to enhance the predictive power of and add precision to their conceptual understanding. Chemists’ mathematical model describing the behavior above should show a direct proportional relationship between the number of molecules and the pressure of the gas (fewer molecules, fewer collisions with the wall, the lower the pressure) with no change in the volume: V oc k (n/P) where k is a proportionality constant that includes the constant value of the temperature at which the gas must be maintained for this equation to adequately describe the behavior of a gaseous substance under these conditions. The variable n represents the number of molecules of the gas while P represents the pressure exerted by the molecules. In the direct proportion relationship, both would decrease by the same factor so that the ratio never changes. Consequently, the volume, V, would also not change. This mathematical model successfully predicts the submicroscopic and macroscopic behavior of a gaseous substance. To adequately model the conceptual behavior of a gaseous substance, the student must focus on these conceptual connections through mathematical equations. In general, students find it difficult to use mathematical equations as models for conceptual understanding. Instead, they often use equations as devices for computation. Specifics of Real-World Understariflrjg A real-world understanding of gas behavior is evidenced when students, like chemists, can use the mathematical models they develop and link the behavior of a real-world gaseous system with their conceptual understandings. For example, a person who understands the scientific nature of a hot-air balloon system should be able to (1) make a mathematical prediction of the temperature at which the air in a hot air balloon must be maintained in order to establish a constant drift when atmospheric conditions are known, (2) understand why the balloon will change its drift when the temperature of the gas in the balloon is changed, and (3) describe at the atomic level what happens to the air in the balloon when it is heated. As evidenced by the above description of what it means to understand the behavior of a gas, a practical understanding requires that students are able to meander between their conceptual and mathematical understanding of the behavior of a gas and connect both kinds of understanding to cues from real- world situations that have not been labeled as “data” and quantified for the student. It has been pretty well documented (Nurrenbern & Pickering, 1987; Sawrey, 1990; Nakhleh, 1983; Nakhleh & Mitchell, 1993) that students are, in general, better at using mathematical representations than they are conceptual problem solvers. That is, students typically develop the ability to manipulate mathematical representations but are often unsuccessful with more conceptually based problems. What we seem to be less clear about is how students use their mathematical and conceptual understandings when explaining the behavior of a gas. In order to understand this, it is important to examine the models of understanding, both mathematical and conceptual, that students adopt when learning about the behavior of a gas. Because a scientific understanding involves using mathematical and conceptual understandings together, an interesting phenomenon is how students tend to use these understandings in parallel. Therefore, this study seeks to examine the ways students use their personal mental models - both mathematical and conceptual - to understand the behavior of a gas. Research Questions The above discussion suggests a complexity to teaching and learning about chemistry. To examine the nature of students’ learning in chemistry, this study will examine particularly the nature of students’ mathematical and conceptual explanations about the behavior of a gas. What understandings and processes promote or inhibit students’ development in these areas as they come to understand and explain gas behavior? The following questions give focus to the study: 1. What mathematical understandings do introductory-level, basic chemistry college students use when they describe the behavior of a gaseous substance? 2. What conceptual understandings do the students have about the behavior of a gaseous substance? 3. How do students use their mathematical and conceptual understanding when explaining the behavior of a real-world gaseous system? Overview of the Study The heart of this study is an in-depth examination of how four students explained the behavior of gaseous substances. These students were chosen from four groups of students who had been categorized as medium mathematical/high conceptual (MMHC), medium mathematical/medium conceptual (MMMC), medium mathematical/low conceptual (MMLC), and low mathematical/low conceptual (LMLC) based on their performances on a paper- and-pencil instrument. In all, nine students were clinically interviewed out of 116 tested using a written instrument. Some comparisons are drawn between the four students who formed the basis of my study and the remaining five who were also clinically interviewed. All the students in this study had studied chemistry for about six weeks with one week devoted specifically to the topic of the behavior of a gaseous substance. Methodological Limitations This study is based on an epistemological model of conceptual change. As such it examines the cognition or understandings that students possessed about the behavior of a gas and does not attend to other issues, such as social/affective issues, attended to in other models (e.g., Tyson et al., 1997). No attempts are made to trace student learning as it changed during instruction or as a consequence of classroom interactions. Neither is any attempt made to monitor the attendance of the interviewees in class during instruction on the unit describing the behavior of gases. Second, this study examines students from five different sections of a community college, basic chemistry course with different instructors. Although all sections shared a common syllabus and common lecture outline, there are no assumptions made that all instructors taught the same way or strictly followed the lecture outline. Neither is any formal attempt made to monitor student interaction with the objectives in the. syllabus or readings in the textbook. Consequently, this study is strictly an examination of the cognitive understandings community college students have about the behavior of a gaseous substance after instruction. It is assumed that the students’ expressed knowledge during the clinical interview is an indication of how they came to understand the behavior of a gas as a result of instruction in their chemistry course. 12 Finally, this study involves only nine students from five classrooms taught by different teachers. The limited number of subjects allowed me to achieve the stated objective of providing a rich analysis of the understanding selective, students possess about the behavior of a gas. However, the small sample size precludes the use of statistical methods of analyses. CHAPTER TWO LITERATURE REVIEW Introduction This chapter reviews a body of research that forms the theoretical underpinnings of this investigation. This chapter will show how the study draws from and goes beyond previous attempts to explain students” understanding of the behavior of a gaseous substance. Part I of this chapter examines how students have used mathematics in understanding the behavior of a gaseous substance. This section examines an existing body of research that focuses upon how students understand and use the mathematics of gas law equations. Part II of this chapter examines the conceptual understandings students have of gas behavior. In particular, it examines student's notions of the macroscopic and atomic-molecular descriptions of gas behavior. Part III examines notions of real-world understandings of scientific phenomena. In particular, this section discusses what it means to have an understanding of scientific phenomena that is useful. Part I: Students’ Mathematical Understandings and Explanations of Gas Behavior A growing body of research indicates that there are often discrepancies between performance on mathematical tasks used to describe the behavior of a gaseous substance and student understanding of the ideas which underpin the 14 task. Many studies (Nurrenbern & Pickering, 1987; Sawrey, 1990; Nakhleh, 1993; Nakhleh & Mitchell, 1993) have examined college students as they have responded to tasks in which they use the mathematical gas law equations and a conceptual understanding of a gaseous substance. While more than a majority of students in all of the studies could perform well on the mathematical problem, a minority of the students in each study could perform well on the conceptual problem. Students’ conceptual understanding of the behavior of a gaseous substance will be explored in the next section. In this section, my goal is to examine the literature as it relates to students’ mathematical understanding of the gas laws. This understanding incorporates (1) knowledge of the mathematical representations of the gas laws, and (2) knowledge of the proportional relationships described by the mathematical representations. Gabel & Sherwood (1983) studied the use of proportional reasoning strategies by high school chemistry students. They showed that these high school students performed better in instructional situations when they were taught the gas laws using proportional reasoning strategies. That is, when students in the Gabel and Shenlvood study were taught the gas laws using a relationship such as N3 = C/X and asked to find the value of X, these students typically outperformed students who were taught using more visual methods, such as the use of analogies and diagrams. The more visual methods could be thought of as more conceptual in nature because they required the student to think more deeply about the situation at hand. For example, the analogy method compared the molar volume of a gas to a shipping carton of fruit. No matter the size of the fruit, the volume of a dozen pieces of the fruit was always 3 pints. After considering this problem and being asked to determine the number of fruit in 54 pints, the students were then presented with a problem to determine the number of moles in 89.6 liters of oxygen gas. Gabel & Sherwood (1983) found that even for students who did not prefer this approach to problem solving (i.e., the low visual students in their study), this method of instruction was the best in getting them to understand the problem. These researchers suggest, One possible explanation for this is that even though these students did not prefer this approach, it required them to pay greater attention to the material at hand. Because students prefer a certain approach by which to learn does not necessarily mean that they learn better using this approach (p. 175). In a think-aloud interview of high school students solving gas law, molarity, mole concept and stoichiometry problems, Gabel, Sherwood, and Enochs (1984) examined the preferred strategies used by high school chemistry students to solve the problems. They found that the students who were not successful on the written tests given before the interviews tended to use a nonsystematic approach on the problems given to them during the interviews. That is, the unsuccessful students did not organize their information before attempting to solve the problem. However, for problems dealing with the gas laws, even those who used a nonsystematic approach were no more unsuccessful in solving these problems than those students who used a systematic approach. The authors note that this is probably due to the fact that gas law problems can be solved using a mathematical formula. A systematic approach can be avoided by dependence on the mathematical formula. Similarly, these researchers found that, in general, students who used high proportional reasoning outperformed students who were low proportional in reasoning except for the gas laws where dependence on the mathematical formula did not necessarily require proportional reasoning abilities. The studies by Gabel and her colleagues have shown that high school chemistry students prefer and are often taught the use of proportional reasoning to solve gas law problems. The ability to use proportional reasoning, however, is not always necessary in solving gas law problems because the problems can often be solved by simply manipulating a gas law equation. But proportional reasoning strategies are necessary for understanding the mathematics of gas behavior. Beyond the manipulation of the mathematical representations of the gas laws, a true understanding of the mathematics is lost without the ability to understand the relationships between the variables of the equation. A recent study by de Berg (1995) examined the understanding of the inverse proportional relationships between pressure and volume of air compressed in a syringe. The researcher gave 101 college students from England a written exam showing the picture in Figure 1. Based on the pressure- volume relationships shown for each of the situations presented, students were asked to predict either the pressure or volume of gas in the syringe given the value of the other. Using proportional reasoning, 65% of the students stated l7 50 pressure units 7 ‘ 100 pressure units 200 pressure units . J. , l l T 7 ‘ ' l g .. . 1 40 l I ‘ ‘ volume 7’ T] ‘ , ‘ units , 20 volume l ' I0 volume ' units units _ Figure 1. Syringe System Used for Quantitative Exercise in de Berg (1995) Study correctly that if 25 units of pressure were exerted on the syringe, the volume would be 80 volume units. Likewise, 64% of the students stated correctly that if the gas in the syringe was compressed to 5 volume units, 400 units of pressure must be applied to the plunger. Both of these tasks required proportional reasoning using 2:1 inverse ratios. However, when the students were asked to make judgments using inverse ratios which were not whole number multiples of each other, the performance on this task decreased significantly. When asked to use the picture in Figure 1 to determine either the volume or pressure of a given amount of gas when either the volume or pressure of the gas changed, only 3% of the students stated correctly that if 150 units of pressure is exerted on the plunger the gas volume would be 13.3 volume units; and, if the gas is reduced to 30 volume units, the pressure exerted on the plunger would be 66.6 pressure units. The de Berg study begins to show that there are certainly some differential understandings students have of proportional relationships as it relates to describing the behavior of a gaseous substance. The present study seeks to further add to what’s known about students’ understandings. It seems that although in some instances students are able to use the mathematical representations to solve gas law problems, in other instances these representations don’t seem to be as useful. That is, although students in general, tend to solve gas law problems best when using equations which depict proportional relationships, such as AlB = C/X, the use of the equation doesn’t imply an understanding of the relationships presented. As a matter of fact, in the de Berg (1995) study, the roughly 55% of the students who answered the first two problems correctly using proportional reasoning, chose to use the mathematical averaging principle for the latter two problems. That is, those students who decided correctly the inverse ratio in the first two problems reasoned that because 150 pressure units is the mathematical average of 200 pressure units and 100 pressure units, the volume occupied by the gas at this pressure would be the mathematical average of 10 volume units and 20 volume units, or 15 volume units. Likewise, because 30 volume units is the mathematical average of 40 volume units and 20 volume units, the pressure exerted on the plunger to produce this volume of gas must be 75 pressure units, the mathematical average of 50 pressure units and 100 pressure units. Although the mathematical averaging principle would work for a system operating as a direct proportion, it does not work for a system operating as an inverse proportion. There is often some problems in understanding the relationship the formula is intended to show. The de Berg (1995) study, as with the study preceding it (de Berg, 1992), seems to show that students use mathematical relationships describing the behavior of a gas in interesting ways. Students do 19 not tend to use proportional reasoning with numbers that are inverse, non-whole number multiples of each other. Also, students’ uses of mathematical relationships here seem to be explained by the observation that students tend to treat the nature of a chemical or physical system as purely mathematical. That is, students typically did not attend to the conceptual significance of the pressure- volume mathematical relationship. Consequently, it may be difficult to conceive of how the system should act, and thus, use the mathematical relationship beyond its pure mathematical usage. The present study posits the claim that chemists understand how mathematical equations predict the behavior of a system because they treat the equation as a model for the system’s behavior. Students in general, find it difficult to construct mathematical models using their mathematical knowledge. The present study seeks to go beyond what’s presently known by examining not only what mathematical knowledge students have, but how they use their knowledge to form mathematical models. Summary of Part I The studies by Gabel and her colleagues and de Berg pinpoint various ways students use their mathematical understandings of the gas laws. The students in the Gabel et al. (1984) studies preferred the use of proportional reasoning strategies when solving gas law problems. But proportional reasoning was not always necessary because the gas laws could be solved by manipulating mathematical equations. Also, students low in proportion reasoning 20 ability were no more unsuccessful at solving gas law problems than students who had high proportion reasoning ability. De Berg examined college students as they explored the inverse proportional relationships between P and V for air compressed in a sealed syringe. The students in this study exhibited some misconceptions about proportional relationships. For example, although they used proportional reasoning for 2:1 inverse proportional relationships, they chose a mathematical averaging strategy when dealing with 3:1 inverse proportional relationships. Such a strategy would work if density data were used instead of volume data because then the system would operate as a direct proportion. These studies suggest that students have various understandings when explaining the mathematical behavior of a gaseous substance. In this study, I explore this claim within a wider framework. Particularly, how do students use the knowledge they have to develop mathematical models for the behavior of a gaseous substance? Part II: Students’ Conceptual Understandings and Explanations of Gas Behavior Merits Macroscopic Understandings and Explanations of Gas Behavior From the standpoint of a chemist, the atomic-molecular and kinetic molecular theories are the key explanatory ideals for explaining conceptually the phenomenological behavior of a gaseous substance. Several studies, however, indicate that students often don’t get as far as explanations at the molecular level. Many students are trying to understand the phenomenon of mass, volume, 21 pressure, and temperature and often have difficulty connecting their understanding with the explanatory ideal of the chemist. De Berg (1995) studied 101 17- to 18-year old high school students. In responding to a paper-and-pencil instrument, which presented the diagram pictured in Figure 2, describing air in a closed system before and after compression, 66% of these students correctly answered that the enclosed volume of air is greater in situation A than in situation B. However on the average, 34% of these students did not have an adequate understanding of the push down h IIIIIIIQQI' ——‘_ plunger ‘— barrel .— barrel enclosed air «II-— enclosed air A 8 Figure 2. Syringe System Used for Qualitative Exercise in de Berg (1995) Study volume concept. Of those who had alternative conceptions of the concept, 25% said that the volume in situation A is the same as the volume in situation B. De Berg notes that Sere (1985) in a study with 11- and 12-year olds concludes that this alternative conception of the volume of a gas could be because students relate volume with amount of gas. There is some further support for this idea in my study. When asked what would happen to the mass of air from situation A to situation B, 62% of the 116 students stated correctly that the mass would not 22 change, and 38% had alternative conceptions about the concept of mass. Of those who had alternative conceptions about the concept of mass, only 19% said that the mass of gas in situation A was less than the mass of gas in situation B. These students seemed to have reasoned that the air would have a greater mass (weight) when squeezed into a smaller volume. The other 81% of alternative conceptions also suggested a confusion between mass, density, and weight. Such rationalization suggests that students tend to relate the volume of a gas to its mass, weight, and density. De Berg also noted that Stavy (1990) found that students aged 9-15 possessed these same alternative conceptions of weight and density in a floating experiment. Stavy & Rager (1990) found similar results with 66 ninth- and tenth-grade Israeli students. In an interview task that asked them to determine the equality or inequality of masses of different volumes of different substances (solids, liquids, and gases), 83% correctly determined the inequality of masses of substances with different volumes. However 17% of students possessed alternative conceptions about these concepts. A common explanation was “equal volume means equal quantity.” But these alternative conceptions about the variables mass and volume seem to be one-sided. That is, when Stavy & Rager (1990) asked the same students to determine the equality or inequality of the volumes of equal masses of different substances, a smaller number than before (66%) correctly answered that the volumes of equal masses of different substances would not necessarily be equal. More students (about 34%) found this a more difficult task than the reverse task. 23 Students often relate the volume of a gas with amount or quantity. But students also have trouble with the very idea of volume. Although students are taught the definition of volume, there is no evidence that the rote memorization of this definition gives them a sound conceptual understanding. Nurrenbern & Pickering (1987) examined over 300 college students enrolled in a first-year general chemistry course. The written exam included items testing the students conceptual and “traditional” understanding of the gas laws and stoichiometry. Traditional gas law problems were defined as the mathematical problems generally included on general chemistry exams to test for understanding, while the conceptual problem did not require the use of a mathematical formula or algorithm for its solution, as represented in Figure 3. While about 67% of the students were able to correctly solve the mathematical equation dealing with the gas laws, only about 36% were successful at solving the corresponding conceptual problem. About two-thirds of these students didn’t depict the gas occupying the entire volume of the container. This is in spite of the fact that many students were able to recite the learned definition that gases occupy the entire volume of their containers. These findings have been mirrored in other studies (Sawrey, 1990; Nakhleh, 1993; Nakhleh & Mitchell, 1993). The typical understanding exhibited by students tends to be one that does not consider the notion that molecules are in constant and random motion. In the Sawrey (1990) study, the most chosen alternative conception was choice (d) in Figure 3. This is indicative of the differential understanding students often possess. Knowing that the molecules of a gas should spread out from each other to fill the container, Conce tual uestion The following diagram represents a cross-sectional area of a steel tank filled with hydrogen gas at 20 ° C and 3 atm pressure. (The dots represent the distribution of H2 molecules.) Which of the following diagrams illustrate the distribution of H2 molecules in the steel tank if the temperature is lowered to -20 ° C? (A) (B) (0) (0) Traditional uestions Charles’ Law A certain sample of methane (CH4) gas occupies 4.5 L at 5 ° C and 1 atm. What volume would the gas occupy at 25 ° C and 1 atm? (a) 0.9 L (b) 4.2 L (c) 4.8 L (d) 22.5 L Combined Gas Law A given mass of gas occupies 5 L at a pressure of 0.5 atm and 5 ° C. What pressure must be maintained to store the gas at 3 L and 25 ° C? (a) 0.32 atm (b) 0.89 atm (c) 1.5 atm (d) 4.2 atm Figure 3. Conceptual and Traditional Questions Used in Nurrenbern & Pickering (1987) Study students often adopt such a representation shown in Figure 3 (d). This representation, however, does not accommodate the notion that the particles of a gas are in constant and random motion. Hwang (1995) also studied students’ conceptions about the idea of gas volume. On a written exam that asked 395 Taiwanese students (102 junior high, 176 senior high, and 117 university students) to give the volume of hydrogen gas in a container with a volume of 1-Liter, Hwang reported that 30% of the junior high, 70% of the senior high, and 100% of the university students had the goal conception that the volume of the gas would be the same as the volume of the container. However, when asked to draw the volume of the gas in the container at the atomic-molecular level, about the same percentage of junior high students (30%) could correctly represent the volume of the gas; whereas, 56% of senior high students and 87% of university students could adequately describe the volume of the gas at the atomic-molecular level. Some researchers have suggested that the problems many students have with the concept of volume are due to (1) the multiple meanings attached to the concept which students often cannot distinguish between (e.g., the student’s ability to distinguish between “1-Liter” as the volume of the container, or the volume occupied by the glass which makes up the container); then, having chosen a meaning, (2) the problem with its application out of context, and (3) the confusion between the terms volume and density (Klopfer, Champagne, and Chaiklin, 1992). The Hwang (1995) study shows in addition that students’ understanding of the atomic-molecular level of matter seems to further influence their conceptual understanding of gas behavior. 26 Students’ Understanding and Explanations of the Atomic-Molecular Level of Gas Behavior Students often explain their conceptual understandings with little regard for the atomic-molecular theory. Yet, this is the explanatory ideal the student is expected to grasp to propose good scientific explanations. Ben-Zvi, Eylon, and Silberstein (1982) have suggested that the problems students have with using this explanatory relationship to explain their conceptual understanding are in their ability to coordinate three levels of description, which chemists seem to do effortlessly. That is, students must learn to describe simultaneously (1) what’s happening at the phenomenological level (e.g., the observation that a gas fills any container it is in); (2) the atomic-molecular level (i.e., the notion that a gas is made up of many particles, the most basic of which is like the others), and (3) the multiatomic-molecular level (i.e., the notion that the observed properties of matter is a consequence of the action of all of the particles which compose the matter.) The Phenomenological Level To explain practical systems, students must make observations. Observations of chemical and physical systems are a result of the properties of these systems. These properties present themselves as phenomena. The task of the student is to explain the phenomena observed. Students often use macroscopic language in explaining observed phenomena. That is, they simply describe what they see or feel. The phenomena of interest in the present study is the behavior of a gaseous substance. Therefore, in explaining the phenomena of the temperature of a gas, for example, students will often explain that steam 27 (gaseous water) is hotter than liquid water without any indication of what “hotter” means from a molecular point of view. This analysis is often simply based on what they may feel. Or, they may explain that air in a closed container does not exert a pressure because they can’t see it. Such explanations are not problematic in class when the manipulation of a mathematical formula is all that is required. However when these ideas must be applied to practical situations, this level of description falls far short of a true understanding. The Atomic-molecular Level Understanding the nature of chemical and physical systems requires a conception of the atomic-molecular theory. This theory postulates that all substances are made up of tiny particles, and it is at the very heart of chemistry. The problem is that the atoms and molecules to which this theory applies can never be seen. From the very beginning of chemistry class, students are often asked to think in terms of atoms and molecules. Some research has suggested that unless students are able to function at the Piagetian formal-operational level, understanding the atomic-molecular theory is problematic (Herron, 1975). However understanding the nature of single atoms and molecules is essential for success in chemistry. In describing the molecular makeup of a molecule of water, for example, the student must understand that one water molecule is made up of two hydrogen atoms and one oxygen atom; and, that this entity represents the simplest nature of water. According to Ben-Zvi, et al. (1982), students’ difficulty in providing more significant explanations for what they 28 observe phenomenologically is often due to their lack of understanding of the simplest nature of the phenomenon. The elusive nature of the atom has been a great source of difficulty as students try to “invent” it for themselves. Classroom discourse may not be helpful in this regard as students try to relate what is said in the classroom to their mental model of the atom. In explaining the ability of a gas to occupy its container, for example, the commonly taught definition is that a gas will expand to fill any container in which it is placed. The students’ atomic-molecular description of this phenomenon often becomes one of expanding the atom or molecule itself to fill the container. Without any actual experience of trying to reconcile the observed phenomenon with the theoretical description of the nature of matter, the student is often at a loss when an examination of their knowledge requires more than mere fact presentation. The Multiatomic-molecgar Level As if the nature of an individual atom or molecule is not elusive enough, students are asked very soon in chemistry class to begin thinking of a collection of such units and the behavior of this collection. Explanations of chemical and physical systems require a conception of the action of many atoms and molecules together. This action is best explained using the kinetic molecular theory. The Chemist’s rationalization of phenomena is accomplished through this theory. Conceiving a large collection of atoms and molecules is often a difficult prospect for the student. The sheer number of such units that a mole, for example, represents is astounding. For students to begin to think on this level is a challenging task, especially when many have not convinced themselves that matter is made up of individual units. Explanations that involve a conception of the multiatomic—molecular level are accepted as reasonable explanations of chemical and physical phenomena. Yet, many students find it difficult to conceive of this level (Ben-Zvi, et al., 1982). In the explanation of air pressure, for example, an explanation describing this pressure as the bombardment of many molecules against a given area is an acceptable definition. However, students often offer the explanation that air which shows no sign of movement is not creating a pressure. Other studies have shown that the coordination of these three levels of description is a difficult prospect for the student, and the ability to operate at one or two levels generally suffers due to a lack of ability to operate at the other Ievel(s). For example, Hwang ( 1995) studied the conceptions students at the junior high school, senior high school and university level had of the idea of gas volume. In all cases, except for junior high school where the percentages stayed the same, a greater percentage of students at each level were able to correctly judge that the volume of a gas placed in a 1-Liter container would be 1 Liter (30% junior high, 70% senior high, and 100% university) than could adequately represent that volume at the atomic-molecular level of description (30% junior high, 56% senior high, and 87% university). 30 Novick & Nussbaum (1981) also found that students’ alternative conceptions about what is happening at the atomic-molecular level greatly influences their ability to explain correctly the phenomena of gas behavior. In a study of 576 American students (83 elementary, 339 junior high school, 88 high school, and 66 university students) asked to represent the particle distribution of a gas in a closed container, these researchers found differentiation in the abilities of students to operate at different levels of description. The percentage of students who correctly represented the uniform distribution of particles rose from the lower to the higher levels (60% elementary, 80% junior high, 90% senior high and university). However when the students were given the item represented in Figure 4, and asked to give the best representation of the air in the flask after the balloon becomes inflated, those students choosing a uniform distribution of particles dropped significantly (30% elementary, 40% junior and senior high, and 30% university). Here, more of the university students (40%) reasoned that there would be more particles in the balloon than in the flask.1 There is evidence that students have some apparent difficulties in coordinating their atomic-molecular descriptions to explain the phenomena of gas behavior. ' This could be due to the representation these researchers gave of the system (see Figure 4) which shows the attached balloon, apparently open to the flask, as containing no particles of the gas. The observation is that, uninflated, the balloon contains no air; therefore, the obvious assumption may be that when the balloon inflates, more particles leave the flask to occupy the balloon. 31 A flask containing air was connected to a rubber balloon. Then the air in the flask was heated with a flame and the balloon inflated. TASK NO. 8 Place an X in the square next to the drawing which you think is the best description of the air after the balloon becomes inflated. A B TASK NO. 9 Explain briefly how the heat of the flame affected particles in the flask. Figure 4. Sample Task Used to Examine the Atomic-molecular Structure in the Novick & Nussbaum (1981) Study. Lu IJ Benson, Vthrock, & Baur (1993) further examined student conceptions of the atomic-molecular nature of gases. They found that roughly 27% of the 191, 10-12 grade students and 64% of the 607 university students in their study (compared to 56% 9-12 grade students and 87% university students (Hwang,1995) and 90% 10-12 grade students and 90% university students (Novick & Nussbaum, 1981) were able to correctly represent the atomic- molecular description of a gas. However, these researchers further categorized the particulate representations of their students and found some interesting conceptions even among otherwise correct representations. With university students, they found that about 25% of the students represented their uniform distribution of particles as being highly packed in the container with little room between them. This could be a result of students’ general tendency at all levels not to conceive of empty space between the particles (Novick & Nussbaum, 1981; Lee, Eichinger, Anderson, Berkheimer, and Blakslee, 1993). They also found that roughly 2% of these university students arranged their particles in very ordered ways when depicting the uniform distribution. Students also exhibit differential abilities internalizing certain aspects of the kinetic molecular theory of matter, which postulates that the particles which compose matter are in constant motion. This affects their understanding of certain conceptual aspects of gas behavior. Novick & Nussbaum (1981) found that although most students in their study represented the uniform distribution of the particles of a gas in a container (60% elementary, 80% junior high, and 90% senior high and universitY). significantly fewer attributed this uniform distribution 33 to inherent particle motion (15% elementary, 23% junior high, 40% senior high, and 48% university). These researchers concluded that because students cannot immediately perceive particle motion, they have difficulty with this concept. Therefore, although they are able to accept other statements of the kinetic molecular theory as plausible (e.g., the uniform distribution of gas particles), they tend to least internalize this concept of particle movement because of the cognitive difficulty it presents (Novick & Nussbaum, 1978). This finding seems to explain the conceptual difficulties researchers have discovered students have with the phenomena of gas pressure (Sere, 1985; de Berg, 1992, 1995; and Jones & Anderson, 1998) and gas temperature (Novick & Nussbaum, 1981) Summary of Part II There are several points of interest for this study that can be derived from the existing work on conceptual understandings of gas behavior. First, the concepts of mass, volume, and density are often confused with each other. This seems to be the case with students at all grade levels, including college students. This could have implications for how students understand the inverse proportional relationships between the pressure and volume of a gas, and the direct proportional relationships between the pressure and density of a gas. Second, students often operate well at a phenomenological level without an understanding of the molecular level behavior of a gaseous substance. 34 Hwang’s work showed that students at all grade levels could well articulate what was meant by the volume of a gas. However, there was a significant decrease of students who could adequately represent this volume at the molecular level. In addition, the work of Benson et al. (1993) showed that even college students who seemed to have an understanding of the particulate nature of a gaseous substance, had some interesting understandings when further pursued. Many of them saw the particles as being highly packed and uniform in their distribution. The work of Novick & Nussbaum also showed the phenomenological understandings of students influenced by molecular understandings in interesting ways. Many of the students understood that the balloon fitted to a flask would inflate when the flask was heated. However, many attributed this behavior to more molecules of air moving out of the flask and into the balloon. Third, students often have difficulty moving across levels of understanding. The works of Hwang and Novick & Nussbaum show the difficulties students have in explaining phenomenological behavior based on atomic-molecular descriptions. Ben-Zvi suggests that students must coordinate three levels of description for explaining the behavior of matter. The chemist tends to cross these levels with ease. Part III: Real-World Application Parts I and II of this chapter indicated that the problems students have in adequately explaining the behavior of a gaseous substance can be attributed to the problems they have in understanding the mathematics and conceptual nature 35 of a gaseous substance. I believe, however, that mathematical and conceptual understanding are not sufficient in and of themselves for examining how students truly understand the behavior of a gaseous substance. Students must be able to use their understanding in real-world situations. As noted in Parts I and II of this chapter, students have various explanations about the mathematical and conceptual nature of a gaseous substance. These explanations have often been examined outside of a meaningful context; that is, a context in which students have a reason to apply their understanding. More can be learned about student understanding as they display their understanding while performing meaningful tasks. This study is, in part, based on a model of conceptual change that examines how students come to change their alternative conceptions into scientific conceptions. It is only through the display of knowledge that students come to reveal their true understandings. In their model of conceptual change, Posner et al. (1982) focus on the conditions which they view as necessary for conceptual change to take place. These conditions seem more favorable and find salience as students try to use their understanding to do something. These researchers see the four conditions necessary for conceptual change as follows: (1) There must be dissatisfaction with current conceptions. That is, the student must no longer have confidence that their way of thinking is sufficient; (2) A new conception must be intelligible, or able to be understood by the student; (3) A new conception must appear initially plausible, or have a capacity for explaining the phenomena; and, (4) A new conception should be fruitful, or able to be 36 extended to explain other relevant systems. The condition of fruitfulness is the most relevant in this study as students try to make real-world applications. Smith (1990) used the model of Posner et al. (1982) in his work with preservice elementary teachers. In the classroom, these prospective teachers were presented with a real task to explain: for a book resting on a table, does the table push on the book? By explaining and ratifying their understandings as a class, in a socially meaningful environment the author suggests that the students were able to gain a better understanding. The students’ experience in the demonstration lesson was unusual or unique for them not only because they felt that they understood, but also because of what they were and were not doing. Rather than simply receiving and remembering information, they engaged in a process in which they drew on their own knowledge, reasoned and argued, inferred and concluded. During this process they became convinced of the plausibility and value of thinking about phenomena in a new and, not only different, but initially counterintuitive way. Such a process is frequently required for learning science with understanding. (p. 52) By so stating, the implication by Smith is also that students found knowledge acquired in this socially meaningful environment as useful in explaining discrepant events. In other words, the knowledge they acquired became fruitful to them as they explained real-world systems. Anderson & Roth (1989) built upon the conditions of conceptual change proposed by Posner et al. by proposing two broad aspects of how students come to achieve conceptual change. The first they refer to as “conceptual integration.” That is, students are considered to have achieved conceptual change and understood a scientific principle or theory to the extent they have integrated an accurate formulation of that principle or theory with their current ways of 37 understanding. The second, they refer to as “usefulness.” That is, students are considered to understand a principle or theory if they can use it to make sense of the world around them. It is this use of value that seems particularly salient here. Anderson & Roth (1989) identify four general categories that group the activities of scientifically literate people. These are description, explanation, prediction, and control. Description, as one activity of a scientifically literate adult, is the ability to provide precise and accurate names, descriptions, or measurements of natural systems or phenomena. Explanation is the process of using scientific knowledge and theories to explain natural phenomena. Prediction involves the ability to generate accurate predictions about future observations or events. And, finally, a scientifically literate adult should be able to use scientific knowledge to control natural systems and phenomena. Examining how students use their mathematical and conceptual knowledge to explain the behavior of a gaseous substance when performing real tasks should be helpful in analyzing how students truly understand the behavior of a gaseous substance. From the discussion above, it seems evident that a complete understanding of how students develop in their understanding of the behavior of a gaseous substance is not yet available. A deeper understanding of science seems evident when students can effectively put their knowledge to use in order to describe, explain, predict, and control their environment. An understanding of how students do this when explaining the behavior of a gaseous substance is useful. The literature - mainly, concentrated in the misconceptions literature - is 38 replete with accounts of what students understand about the behavior of a gaseous substance. It is silent on how students use their understanding to explain the behavior of a gas, and, thus, how students truly understand this behavior. This study addresses this issue. Summary of Chapter Two The purpose of Chapter 2 is to present the reader with a theoretical basis for understanding the major premises of this study. It has been documented that students have many difficulties understanding and explaining the behavior of a gaseous substance in a way that a chemist would understand and explain this behavior. The reasons for these difficulties are complex. The source of these difficulties are assumed to lie in student problems in acquiring three kinds of understanding: (1) mathematical understanding, (2) conceptual understanding, and (3) real-world application. This chapter has reviewed some of the available literature that has addressed each area. Each of these has raised some issues of interest for this dissertation and for chemistry education. These issues are identified and summarized below. Mathematical Understangi_ng Chapter 2 reviewed a few studies broken down along two lines: (1) how students understand proportional relationships relating to the gas law equations, (2) how students use proportional relationships relating to the gas law equations. 39 All studies used paper-and-pencil instruments as a means to assess this understanding. An important finding of the studies by Gabel and her colleagues was that although high school chemistry students preferred the use of proportional reasoning while being taught the gas laws, many of them could solve gas law problems without using proportional reasoning strategies. This suggests that even students who are considered to really understand the mathematics of the gas laws, don’t truly understand the mathematics the gas law equations are meant to convey. The de Berg study, which had students explain mathematically the compression of air within a syringe, found that when students use their understanding of proportional relationships, they do so in some differential ways. De Berg advances the claim that the students are being forced to apply context- specific knowledge out of context. This is a useful theoretical framework from which to examine how students use their mathematical knowledge. There are three issues pertaining to students’ acquisition of mathematical knowledge that are raised by these studies. ISSUE 1: The Understanding of Proportional Relationships. How do college students understand proportional relationships? ISSUE 2: The Creation of a Cohesive Mathematical Model. In what ways do college students use their mathematical knowledge? ISSUE 3: The Use of Gas Law Equations. How useful do college students find the gas law equations? 40 Concegtpal Understandi_ng This chapter has reviewed some studies which examined the understandings students across all levels have about the variables used to describe gas behavior. Particularly, the studies examined how students have understood the concept of volume. These studies have examined both the phenomenological and atomic-molecular understandings of volume possessed by students. De Berg, Stavy and Rager showed that students are often confused by the concept of volume. Many times they confused volume with mass and density. An interesting theoretical consideration is how students’ confusion about these three quantities play into their mathematical notion of the pressure/volume inverse proportional relationship and the pressure/density direct proportional relationship. Novick & Nussbaum (1981), Hwang (1995), and Nurrenbern & Pickering (1987) showed that students phenomenological understanding of the behavior of a gas does not have to be in synchrony with their notions of the particulate nature of matter. Students are often well able to explain what they see apart from an adequate understanding at the submicroscopic level. Ben-Zvi proposes a theoretical framework to examine why students often don’t connect the molecular and the phenomenological. She and her colleagues suggest that students need to connect their knowledge across three levels of understanding: the phenomenological, atomic-molecular, and multiatomic levels. She would suggest that a great deal of specific knowledge is required at each of 4] these three levels before the student will have the knowledge to explain the conceptual nature of gas behavior. I will use the theoretical framework of Ben- Zvi as a basis of analysis for the conceptual knowledge used by students as they explain the behavior of a gaseous substance. There are two issues pertaining to students’ acquisition of conceptual knowledge that are raised by these studies. ISSUE 4: The Understanding of the Atomic-molecular and Kinetic Molecular Theories. How do college students understand these theories? ISSUE 5: Atomic-molecular vs. Macroscopic Descriptions of Phenomena. What are the explanations used by college students when describing the behavior of a gaseous substance? Real—World Applications Ideas about the application of scientific knowledge in real-world contexts was derived from the work of Posner et al. (1982), Smith (1990), and Anderson & Roth (1989). Particularly, scientific knowledge was examined for its use value. The model for conceptual change proposed by Posner et al. (1982) implies that students come to understand scientific ideas after becoming dissatisfied with their current notions when using them, and then adopting a view which has a capacity to make sense to them (intelligibility), is capable of being understood by them (plausibility), and is able to explain other discrepant events (fruitfulness). It is my contention that these four conditions for conceptual change are best achieved as students’ knowledge is tested in use. Smith (1990) demonstrated 42 such a test of knowledge by presenting a use task to a group of prospective teachers. As they worked to solve the problem, many of their misconceptions were made apparent and they were eventually able to achieve conceptual change. Anderson & Roth (1989) suggested that the use value of scientific knowledge of scientifically literate adults is grouped according to four categories of activities: description, explanation, prediction, and control. As scientifically literate adults are able to perform these tasks, they are thought to have achieved scientific understanding. The theoretical framework as presented in these studies is a useful one in which to examine how the students in this study perform the real-world task of compressing air in a syringe and explain the conceptual problems on the paper-and-pencil instrument. Real-world understanding will be examined in the context of the students’ mathematical and conceptual understandings, and, therefore, examined with the five issues listed above. 43 CHAPTER THREE METHODOLOGY Introduction The purpose of this chapter is to identify how the students were selected, and the methods of data collection and analysis. In this chapter, I will show how the questions on the paper-and-pencil instrument allowed me to collect information on students’ mathematical, conceptual, and practical understandings. I will show how the questions asked of students in the clinical interviews allowed me to gather the necessary data to address the research questions posed in the previous chapter. This chapter contains an overview of the study; a description of the subjects and setting; a description of how the data were collected, including an explanation of the paper-and-pencil instrument and the clinical interview technique; and an explanation of how the data were analyzed. Overview of Research Design A flowchart for the data collection and analysis in this study is shown in Figure 5. The heart of the study is an in-depth examination of how four students came to understand and explain gas behavior. In all, 9 students were clinically Interviewed out of approximately 116 who took the posttest paper-and-pencil instrument. The instrument (Appendix A) was designed to measure the mathematical, conceptual, and practical understandings of the students. 44 The instrument was used to categorize each of the 116 students into one of nine groups: high mathematical/high conceptual (HMHC), high mathematical/medium conceptual (HMMC), high mathematical/low conceptual (HMLC), medium mathematical/high conceptual (MMHC), medium mathematical/medium conceptual (MMMC), medium mathematical/low conceptual (MMLC), low mathematical/high conceptual (LMHC), low mathematical/medium conceptual (LMMC), and low mathematical/low conceptual (LMLC). Three students were chosen from each of the four groups above that contained the highest percentage of students (see Table 1): LMLC (28.4%), MMLC (24.1%), MMMC (14.7%), and MMHC (11.2%). In all, twelve students were slated to participate in the clinical interviews. However, I was not able to get more than two students in the LMLC and MMLC categories who would agree to talk about their understanding. In addition, one of the audio tapes produced from the clinical interview of one MMMC student was inaudible due to a faulty microphone. Consequently, this study ultimately involves nine students who were clinically interviewed. Four of these students form the crux of this study: Cameron (MMHC), Betty (MMMC), Karen (MMLC), and Connie (LMLC). General claims are made about the other five students based on case study analyses of these four students. The five students are Nina (MMHC), Janice (MMHC), Donna (MMMC), Sherry (MMLC), and Hilda (LMLC). 45 Paper-and-pencil instrument administered to 5 sections (116 students) in Basic Chemistry after instruction on the gas laws. l Students grouped into categories based on their performance. Three representative students chosen from highest populated categories for clinical interviews. \l/ PART I Students redo paper-and-pencil instrument, or respond to the instrument they previously wrote talking aloud while explaining their understanding. \L PART II Students manipulate a syringe and explain the behavior (quantitatively and qualitatively) of air as it is compressed at various volumes. \ Establish protocols for 9 students from their clinical interviews. l Develop four case studies of students representative of the 9 which were clinically interviewed (Cameron, Betty, Karen, and Connie) l Group the remaining five students (Nina, Janice, Donna, Sherry, and Hilda) based on the 4 case studies. Figure 5. Flowchart for Data Collection and Analysis 46 Subjects and Setting This study involved five of the six introductory, basic-level chemistry classes in a Midwestern community college. Each section of the course was taught by different instructors, except for two sections which were taught by the same instructor. All sections used a common syllabus and very similar exams. In general, each instructor followed a lecture presentation which presented the objectives outlined in Appendix B. Although there were some differences based on the style of the instructor, for the most part, due to stringent guidelines of exam dates, instructors for each section taught the same material in reasonably consistent ways. All students were grouped into one of nine categories according to their responses on the paper-and-pencil instrument. Four of the nine groups that contained the highest percentage of students were chosen for study, and three students from each of these four groups were chosen to participate in the clinical interview. Because this is not a quantitative study for which I seek to make any statistical claims, a random sampling of representative students from each category was not attempted. Rather, I chose students from each of the four groups based on their placement in that group by their scores on the paper-and- pencil instrument, and their willingness to talk about their understanding. Data Collection The Paper-and-Pencil Instrument This instrument was designed to uncover students’ mathematical, conceptual, and practical knowledge of gas behavior. The problems are 47 presented as mathematical/conceptual pairs. That is, one problem in a pair is a question which requires the use of an equation or algorithm for its solution. The other problem in the pair describes a similar situation as the first problem, but requires a conceptual understanding of the volume, pressure, and mass of a gas as explained by the atomic-molecular and kinetic molecular theories. The instrument contains five items, four of which (items 1-4) have been taken from the literature used in studies which have examined students’ mathematical and conceptual understandings (Nurrenbern & Pickering, 1987; Sawrey, 1990; Nakhleh, 1993; Nakhleh & Mitchell, 1993; de Berg, 1995; and, Noh & Scharmann, 1997). The second mathematical/conceptual pair of problems actually contains one mathematical problem and two conceptual problems. The second conceptual problem (item 5) was added because of the results of two pilot studies and other reports in the literature (de Berg, 1995) which suggest that students may be showing a different conception about pressure than what the first conceptual problem was intended to measure. Because the conceptual problems modeled real-world tasks, they were also used to examine students’ practical understanding of the behavior of a gaseous substance. The Clinical Interview In all, 10 students were clinically interviewed after they completed the paper-and-pencil instrument. The interviews were scheduled at a time convenient for the students and conducted in a conference room away from their classroom setting. The interviews lasted between 30-45 minutes. 48 The clinical interview consisted of two parts. In the first part, students were asked to exhibit their understanding in one of two ways. Some of the students were given the paper-and-pencil instrument they originally completed as a posttest and asked to talk aloud as they explained their thinking on selected items. Other students were given a blank copy of the paper-and-pencil instrument and asked to resolve selected problems while explaining their understanding aloud. It was found in some instances that students reworked items differently than they had worked them before. These differences are noted and analyzed for their significance. In the second part of the interview, students were given a syringe and examined on how they used their knowledge of gas behavior to answer questions pertaining to the behavior of air on the inside of the syringe. Questions asked of the students during this part of the interview were used to analyze how students used their mathematical and conceptual knowledge while performing a real-world task. Data Analysis The data analysis focused on the 9 students who were interviewed after instruction. There are two stages to the data analysis process. During the first stage, detailed case studies of four students were prepared. These students were given the pseudonyms Cameron, Betty, Karen, and Connie. During the second stage, the analytical framework developed for the case studies of Cameron, Betty, Karen, and Connie was extended to the other five students. All 49 nine students are then classified according to their mathematical and conceptual understanding as having goal conceptions, having naive conceptions, or being in a transitional state between goal and naive conceptions. Stage 1: The Case Studies of CameroLBetty. Karen, and Connie The literature review in Chapter 2 produced five issues which seemed relevant to this study. All of the nine students who were clinically interviewed addressed a majority of these issues in a satisfactory manner. Cameron, Betty, Karen, and Connie were chosen for in-depth analysis because they were articulate in explaining their views and possessed a great ability to talk about what they understood. In the development of the case studies, emphasis was placed upon development of a coherent framework that would provide a sensible and consistent explanation of Cameron, Betty, Karen, and Connie’s responses to the paper-and-pencil instrument and in the clinical interviews. The guidelines used to develop this framework were the categories of mathematical understanding, conceptual understanding, and real-world application, and the five issues which emerged from the literature review in Chapter 2. The central problem of these case studies lies in trying to determine where Cameron, Betty, Karen, and Connie stand on the five issues. 50 Stage 2: How the RemainingFive S_tpgents Were Ana_lyz;ed The four students chosen as the case studies were representative of the other students in the sample. While all students were clinically interviewed, detailed case studies were not prepared as part of this dissertation. Rather, the comparisons between the four and the remaining five were done by focusing upon the similarities in responses to the relevant issues identified in the four case studies. All of the nine students were then classified according to their mathematical and conceptual understanding as possessing the goal conception, naive conceptions, or being in transition. Specific Descriptions of Data Collection and Analysis Mathematical Understanding Questions The literature review in Chapter 2 indicated that while the manipulation of mathematical representations is generally the way students use math to solve gas law problems, mathematical understanding is often elusive. Students use some differential knowledge when applying their mathematical understanding of the gas laws. I argue that mathematical knowledge is a prerequisite for the development of a student’s ability to use the mathematical representations in a meaningful way; that is, as models to describe the behavior of a gaseous substance. From the literature review in Chapter 2, three issues emerged from the discussion of mathematical knowledge. These issues seem relevant in understanding how students use their understanding of mathematics to form a 5l mathematical model of gas behavior. Each issue will be listed and the questions on the paper-and-pencil instrument that address these issues will be reviewed with some commentary on the expected response. ISSUE 1: The Understanding of Proportional Relationships. The understanding students have of how the gas laws represent proportional relationships is used as one measure of how they use gas laws as mathematical models. This issue is addressed on the paper-and-pencil instrument and again during the clinical interview. Item 1 on the paper-and- pencil instrument asks students to solve a typical gas law problem as presented during instruction. A student who understands this item would use a gas law equation to solve it. The student would either use the algebraic representation of the law or a ratio method in which the initial pressure is multiplied by a ratio of absolute temperatures. The ratio of absolute temperatures would be written with the smaller absolute temperature in the numerator and the larger absolute temperature in the denominator because such a ratio, when multiplied by the initial pressure, would give a decrease in the value of the initial pressure in accord with the direct proportional relationship between the pressure and temperature of a gaseous substance. During the first part of the clinical interview, students are asked to discuss their understanding of the gas law equation they used to solve this problem. Responses mainly to the mechanics of the equation’s setup is considered as one indication of a mechanistic understanding of the equation. With a mechanistic understanding, the student attends only to the mechanics of the equation (i.e., plugging in appropriate numbers, solving for given variables, canceling units, etc.) without giving attention to the nature of the relationships between the variables (i.e., proportionality) contained within the equation. Consequently, responses attending only to the mechanics of the equation are considered indicative of the students’ lack of mathematical model development. Responses which give some indication of a relationship between the variables of the equation are considered as an indication of the students’ mathematical model development. For example, the student will explicitly state or imply some relationship between variables in the equation (e.g., use of ratio method). Item 3 on the paper-and-pencil instrument is also used to explore students’ understanding of proportional relationships. On the one hand, a student who understands this item would consider the syringes presented and recognize the pressure and volume relationships there. They would then use proportional reasoning to answer the items. For item 3(i), since 25 pressure units is half of the 50 pressure units exerted on the plunger pictured in the first syringe, then the volume at 25 pressure units will be doubled to 80 volume units. Likewise, for item 3(ii) since 5 volume units is one-half the volume pictured in the third syringe, the resulting pressure at 5 volume units would be twice the pressure at 10 volume units, or 400 pressure units. In item 3(iii), since 150 pressure units is three times the pressure exerted on the plunger pictured in the first syringe, the volume occupied by air at this pressure would be one-third of 40 volume units, or 13.3 volume units. Likewise, in item 3(iv) since 30 volume units 53 is three times the volume represented in the third syringe, the pressure units at 30 volume units would be one-third of the pressure at 10 volume units, or 66.7 pressure units. On the other hand, a student who understands item 3 might use a gas law equation or the ratio method similar to the one used with item 1. With a gas law equation, a student would identify initial and final pressures and volumes then solve the equation for an unknown value. With the ratio method, students would identify the values, compose the appropriate ratio for the inverse relationship, and multiply the initial volume or pressure by this ratio. During the first part of the clinical interview, students are asked to discuss their understanding of item 3. Students’ use of the proportional reasoning strategy mentioned above is considered indicative of their proportional reasoning ability. Using the gas law equation or ratio method does not give a direct indication of the students’ proportional reasoning ability unless specifically indicated by the student. ISSUE 2: The Creation of a Cohesive Mathematical Model. Students’ mathematical understandings allow them to create mathematical models and make predictions based on their models. These predictions at many times take the form of estimations because students often make their predictions based on other values which have in some way resulted from their mathematical model. Particularly, the consistency between how students articulated their knowledge in Part I of the clinical interview when explaining items 1 and 3 and 54 how they used this knowledge for predicting values on the real-world task in Part II of the interview, is taken as indicative of the students’ use of their own mathematical model. I have referred to these models as personal models. Such models are empirical claims about patterns as seen in the interview data. ISSUE 3: The Use of Gas Law Equations. Students’ use of gas law equations will also be considered indicative of their mathematical modeling. The studies reviewed in Chapter 2 showed that students can use gas law equations without understanding them. Student understanding of the gas law equations was explored in item 1 and item 3 on the paper-and-pencil instrument. During the clinical interviews, students were asked to discuss their understanding. The knowledge possessed by a student who understands these items is discussed with Issue 1. The use of a gas law was expected of item 1. If students did not use a gas law here, they were considered to have an extremely limited understanding of the gas laws. If a student used a gas law to solve item 3, they were considered to have a wider appreciation for the value of the gas law and its usefulness as a mathematical model to describe the behavior of a gaseous substance if they could simultaneously talk about proportional relationships between variables in the gas law equation. 55 Conceptual Understanding Questions During the clinical interviews students were asked to discuss their understanding of the atomic-molecular and kinetic molecular theories. Students were given a diagram of a syringe and asked to draw what they thought air in the barrel of the syringe would look like at the submicroscopic level. ISSUE 4: The Understanding of Atomic-molecular and Kinetic Molecular Theories. Students’ drawings and explanations of these drawings during the clinical interview were examined. Although many of the students could not remember these theories by name when mentioned in the clinical interviews, this was not considered as indicative of their lack of understanding. Instead, their notions were pursued simply by having them talk about what they understood of the particulate nature of matter. ISSUE 5: Atomic-molecular vs. Macroscopic Descriptions of Gas Behavior. Items 2, 4, and 5 on the paper-and-pencil instrument allows the analysis of students’ explanations about various phenomena of gas behavior. During the clinical interviews, students were asked to explain their understanding of these items. A student who understands item 2 would say the distribution of molecules in the tank after the temperature drops would be similar to the representation depicted in choice (A). This is because hydrogen would still be a gas at the lowered temperature as it is still above its boiling point. Consequently, according 56 to kinetic molecular theory, the molecules would spread out and randomly fill the entire volume of the tank. A student who understands item 4 would say the volume of air in the syringe would decrease upon compression because the space occupied by the air will decrease. In addition, the student would understand that the mass of the air in the syringe will not change upon compression because no air has leaked in or out of the barrel of the syringe. Finally, the student who understands this item would say the pressure exerted by the air in the barrel of the syringe would increase upon compression because the molecules of air have been squeezed into a smaller space. Consequently, the molecules will hit the walls of the syringe barrel with greater frequency. A student who understands item 5 would say the pressure of enclosed air in the syringe barrel is the same as standard atmospheric pressure if the plunger is not moving. This is because, if the plunger is not moving, the pressure exerted on the plunger in one direction (atmospheric pressure) must be the same as the pressure exerted on the plunger in the opposite direction (air pressure inside the syringe barrel). Students who used explanations that were more visual and phenomenological in nature were classified as using macroscopic explanations. Such explanations are expected for items 4(i), 4(ii), and 5. However, items 2 and 4(iii) are best explained with explanations at the submicroscopic level. Students were classified as attending to submicroscopic explanations when they explained these items based on considerations of molecules and their movement. For 57 example, a response like, "The pressure created in this syringe is greater because the molecules are closer together and bounce off the walls more.” Real-World Application Questions Students’ real-world use of their mathematical understanding was analyzed as they used the real syringe to answer quantitative questions. Students’ real-world use of their conceptual understanding was analyzed as they used their understanding of the atomic-molecular and kinetic molecular theories when responding to items 2, 4, and 5 on the paper-and-pencil instrument. In this study, I have suggested that the meaningful use of mathematical and conceptual knowledge when performing real-world tasks is a measure of how students truly understand gas behavior. Therefore, in this study students’ real-world use of their knowledge is examined as a part of the above five issues and discussed in those sections. 58 CHAPTER FOUR RESULTS Introduction The students presented in the following case studies were picked from the groups containing the higher percentage of students as noted in Table 1. The Table 1. Percentage of Students in Mathematical/Conceptual Categories Based on Results of the Paper-and-Pencil Instrument (N = 116) High Math Medium Math Low Math Totals Achievement Achievement Achievement High Conceptual 0.0 11.2 6.0 17.2 Achievement Medium Conceptual Achievement 2.6 14.7 8.6 25.9 Low Conceptual Achievement 4.3 24.1 28.4 56.8 Totals 6.9 50.0 43.0 99.9 Boldface percentages represent categories of students chosen for interviews. students are grouped initially into categories based on their correct or incorrect responses to the items on the posttest. Most of the students fall into the medium mathematical and low conceptual categories. In the present study, about 50.0% of students are initially categorized as being between high and low in their mathematical achievement, and 56.8% are initially categorized as low in their 59 conceptual achievement. The case studies presented in the next section examine the understanding of a typical student in the MMHC (Cameron), MMMC (Betty), MMLC (Karen), and LMLC (Connie) categories. The results suggest that the initial categorization of these students is not indicative of their true achievement. That is to say, those students achieving medium mathematical and low mathematical proficiency often share some common misconceptions in spite of the categories in which they’re initially placed. The same can be said of students in the high, medium, and low conceptual categories. Case Study 1: Cameron Cameron’s Mathematical Understanding Explanation of Item 1 on the Paper-and-Pencil Instgipient On the paper-and-pencil instrument, Cameron was asked to solve a typical gas law problem presented in his basic chemistry class. He uses the relationship: P2 = P, x T2ff1 During the clinical interview, Cameron is not shown his original problem, but is given a blank copy of the posttest and asked to rework and explain his solution for item 1. During the interview, Cameron sets up and solves the relationship: P1/T1 = P2lT2 6O He is asked about his understanding. Interviewer: Do you remember how you were taught to work the problem? Cameron: ...First you want to set it up with what’s given, which is the volume, and the pressure, and temperature. Calculate the pressure in atmospheres if the temperature is changed. So, the volume’s going to stay the same cause it doesn’t say anything about it. When explaining how to work this problem, Cameron gives some indication that he is attending to the proportional relationships between P and T. Cameron: The pressure’s going to change because the temperature has been changed...Pressure goes down, temperature goes down. . .They’re proportional, directly proportional. Cameron gives an indication that he is aware of the proportional relationships between variables in the equation. QpIa_nation of Item 3 on the Paper-ang-Pencil Instrpment On the paper-and-pencil instrument, Cameron further demonstrates his understanding of proportional relationships. When he initially completed item 3 on the posttest, Cameron began by using a gas law equation to solve the problem. For item 3(i) he sets up the relationship: V2 = 40 x 25/50 6I However, he abandons this relationship for another strategy. This is obvious because he finally reports “10” volume units as an answer instead of the 20 volume units the relationship above would produce. During the clinical interview, Cameron was not shown his initial solution to this problem, but was given a blank copy of the posttest and asked to rework item 3(i) and explain his solution. Cameron does not explicitly use a gas law equation this time, but he does use a proportion strategy. Cameron: Goes from 50 (pressure units) to 25 (pressure units)...and it had 40 volume units when it was full. So you take it to half the pressure. So it's, um, 20 volume units then. Even though Cameron incorrectly uses 2:1 direct ratio reasoning in solving item 3(i) instead of 2:1 inverse ratio reasoning, he does use ratio reasoning. However, for item 3(ii), which is also a 2:1 inverse ratio problem, there is an addition component in his proportion reasoning. Cameron: I think it’s 250...From this one (refem'ng to the diagram), there’s 200 pressure units on 10 volume units...So when it goes 5 volume units, um, I said it was 250 because that’s kind of the descent they all took. Cameron initially believes that as the volume is cut by a factor of one-half, the resulting pressure is increased in increments of 50 units. Although he 62 conceives of the pressures as additive of each other, he conceives of the volumes as some multiple of each other. Cameron: ...So that’s (the volume) twice as much as on this one. Interviewer: ...Twice as much volume? Cameron: Um, or half the volume. When initially working item 3(ii) during the clinical interview, Cameron uses a strategy in which he performs an addition to predict the pressure that a fourth diagram at 5 volume units might have. When asked to comment further on his strategy, Cameron refers to the idea of proportional relationships and experiences dissatisfaction with his initial response. Interviewer: ...So you said this was 250 because...? Cameron: Just the way the proportionate was. From 100 (pressure units), there’s 20 volume units. Then 200 (pressure units), it went to 10 volume units. But, um, for 5 volume units, um, I guess that it was 300 (pressure units). ..Or it doubles. ..maybe it’s 400, because they’re proportionate. ..So that will be, urn, 400 pressure units for 5 volume units. Towards the end of his explanation above, Cameron is once again pursuing proportional reasoning. However, Cameron’s strategy changes again when he considers the 3:1 inverse ratios in items 3(iii) and 3(iv). 63 On the paper-and-pencil instrument he took as a posttest, Cameron responded the same way to these problems as he responds during the clinical interview. During the clinical interview, he explains how he works the problems. For these problems, he uses an averaging strategy. Cameron: When you put 150 pressure units on it, I said it was 15 volume units cause it’s in between these (referring to the second and third syringes at 100 and 200 pressure units, respectively). . .And half of that (referring to the volume of air in the second and third syringes at 20 and 10 volume units, respectively) would be 15. Likewise, Cameron explains how he thought about item 3(iv). Cameron: ...I said it was 75. It’s between 50 and 100 pressure units. How Cmemn Understands and Uses the Mathematical Repre_sentati