It's Just Math: Research on Students' Understanding of Chemistry and Mathematics 2019011824, 9780841234345, 9780841234352, 0841234345

Table of contents :
Contents......Page 8
1How Did We Get Here? Using and Applying Mathematics in Chemistry......Page 10
2A Modeling Perspective on Supporting Students’ Reasoning with Mathematics in Chemistry......Page 18
3Mathematics in Chemical Kinetics: Which Is the Cart and Which Is the Horse?......Page 34
4Graphs: Working with Models at the Crossroad between Chemistry and Mathematics......Page 56
5Graphs as Objects: Mathematical Resources Used by Undergraduate Biochemistry Students To Reason about Enzyme Kinetics......Page 78
6Math Self-Beliefs Relate to Achievement in Introductory Chemistry Courses......Page 90
7But You Didn’t Give Me the Formula!” and Other Math Challenges in the Context of a Chemistry Course......Page 114
8Transition of Mathematics Skills into Introductory Chemistry Problem Solving......Page 128
9Mathematical Knowledge for Teaching in Chemistry: Identifying Opportunities To Advance Instruction......Page 144
10The Logic of Proportional Reasoning and Its Transfer into Chemistry......Page 166
11Making Sense of Mathematical Relationships in Physical Chemistry......Page 182
12What Education Research Related to Calculus Derivatives and Integrals Implies for Chemistry Instruction and Learning......Page 196
13Developing an Active Approach to Chemistry-Based Group Theory......Page 222
14Systems Thinking as a Vehicle To Introduce Additional Computational Thinking Skills in General Chemistry......Page 248
15Video-Based Kinetic Analysis of Period Variations and Oscillation Patterns in the Ce/Fe-Catalyzed Four-Color Belousov–Zhabotinsky Oscillating Reaction......Page 260
Editors’ Biographies......Page 280
Indexes......Page 282
Author Index......Page 284
Subject Index......Page 286

Citation preview

It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics

ACS SYMPOSIUM SERIES 1316

It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics Marcy H. Towns, Editor Purdue University West Lafayette, Indiana

Kinsey Bain, Editor Michigan State University East Lansing, Michigan

Jon-Marc G. Rodriguez, Editor Purdue University West Lafayette, Indiana

Sponsored by the ACS Division of Chemical Education

American Chemical Society, Washington, DC

Library of Congress Cataloging-in-Publication Data Names: Towns, Marcy H., editor. | Bain, Kinsey, editor. | Rodriguez, Jon-Marc G., editor. | American Chemical Society. Division of Chemical Education. Title: It's just math : research on students' understanding of chemistry and mathematics / Marcy H. Towns, editor (Purdue University, West Lafayette, Indiana), Kinsey Bain, editor (Michigan State University, East Lansing, Michigan), Jon-Marc G. Rodriguez, editor (Purdue University, West Lafayette, Indiana) ; sponsored by the ACS Division of Chemical Education. Other titles: It is just math Description: Washington, DC : American Chemical Society, [2019] | Series: ACS symposium series ; 1316 | Includes bibliographical references and index. Identifiers: LCCN 2019004262 (print) | LCCN 2019011824 (ebook) | ISBN 9780841234345 (ebook) | ISBN 9780841234352 (alk. paper) Subjects: LCSH: Chemistry--Mathematics. | Chemistry--Study and teaching. | Mathematics--Study and teaching. Classification: LCC QD39.3.M3 (ebook) | LCC QD39.3.M3 I87 2019 (print) | DDC 540.1/51--dc23 LC record available at https://lccn.loc.gov/2019004262

The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48n1984. Copyright © 2019 American Chemical Society All Rights Reserved. Reprographic copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Act is allowed for internal use only, provided that a per-chapter fee of $40.25 plus $0.75 per page is paid to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. Republication or reproduction for sale of pages in this book is permitted only under license from ACS. Direct these and other permission requests to ACS Copyright Office, Publications Division, 1155 16th Street, N.W., Washington, DC 20036. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. Registered names, trademarks, etc., used in this publication, even without specific indication thereof, are not to be considered unprotected by law. PRINTED IN THE UNITED STATES OF AMERICA

Foreword The purpose of the series is to publish timely, comprehensive books developed from the ACS sponsored symposia based on current scientific research. Occasionally, books are developed from symposia sponsored by other organizations when the topic is of keen interest to the chemistry audience. Before a book proposal is accepted, the proposed table of contents is reviewed for appropriate and comprehensive coverage and for interest to the audience. Some papers may be excluded to better focus the book; others may be added to provide comprehensiveness. When appropriate, overview or introductory chapters are added. Drafts of chapters are peer-reviewed prior to final acceptance or rejection. As a rule, only original research papers and original review papers are included in the volumes. Verbatim reproductions of previous published papers are not accepted. ACS Books Department

Contents 1. How Did We Get Here? Using and Applying Mathematics in Chemistry . . . . . . . . . . . . . . . . . . . . .  Marcy H. Towns, Kinsey Bain, and Jon-Marc G. Rodriguez 2. A Modeling Perspective on Supporting Students’ Reasoning with Mathematics in Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Katherine Lazenby and Nicole M. Becker

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3. Mathematics in Chemical Kinetics: Which Is the Cart and Which Is the Horse? . . . . . . . . .  25 Kinsey Bain, Jon-Marc G. Rodriguez, Alena Moon, and Marcy H. Towns 4. Graphs: Working with Models at the Crossroad between Chemistry and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47 Felix M. Ho, Maja Elmgren, Jon-Marc G. Rodriguez, Kinsey R. Bain, and Marcy H. Towns 5. Graphs as Objects: Mathematical Resources Used by Undergraduate Biochemistry Students To Reason about Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  69 Jon-Marc G. Rodriguez, Kinsey Bain, and Marcy H. Towns 6. Math Self-Beliefs Relate to Achievement in Introductory Chemistry Courses . . . . . . . . . . . . .  81 Michael R. Mack, Cynthia A. Stanich, and Lawrence M. Goldman 7. “But You Didn’t Give Me the Formula!” and Other Math Challenges in the Context of a Chemistry Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  105 Amy J. Phelps 8. Transition of Mathematics Skills into Introductory Chemistry Problem Solving . . . . . . . .  119 Benjamin P. Cooke and Dorian A. Canelas 9. Mathematical Knowledge for Teaching in Chemistry: Identifying Opportunities To Advance Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  135 Lynmarie A. Posey, Kristen N. Bieda, Pamela L. Mosley, Charles J. Fessler, and Valentin A. B. Kuechle 10. The Logic of Proportional Reasoning and Its Transfer into Chemistry . . . . . . . . . . . . . . . . . . . . . . . .  157 Donald J. Wink and Stephanie A. C. Ryan 11. Making Sense of Mathematical Relationships in Physical Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . .  173 Renée Cole and Tricia Shepherd 12. What Education Research Related to Calculus Derivatives and Integrals Implies for Chemistry Instruction and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  187 Steven R. Jones vii

13. Developing an Active Approach to Chemistry-Based Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .  213 Anna Marie Bergman and Timothy A. French 14. Systems Thinking as a Vehicle To Introduce Additional Computational Thinking Skills in General Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  239 Thomas Holme 15. Video-Based Kinetic Analysis of Period Variations and Oscillation Patterns in the Ce/Fe-Catalyzed Four-Color Belousov–Zhabotinsky Oscillating Reaction . . . . . . . . . . . . . . . .  251 Rainer Glaser, Marco Downing, Ethan Zars, Joseph Schell, and Carmen Chicone Editors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  271 Indexes Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  275 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  277

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Chapter 1

How Did We Get Here? Using and Applying Mathematics in Chemistry Marcy H. Towns,1,* Kinsey Bain,2 and Jon-Marc G. Rodriguez1 1Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States 2Department of Chemistry, Michigan State University, East Lansing, Michigan 48824,

United States *E-mail: [email protected]

Concerns about the ability of students to apply mathematics in chemistry courses have been part of the chemistry landscape in the United States since at least the 1920s. Improving the mathematical preparation of undergraduate and graduate students has been a sore point that has led to research documenting deficiencies. This book may help researchers and chemists understand current research on mathematical reasoning including covatiational reasoning, graphical reasoning, and mathematical modeling in the context of chemistry. But to accomplish that lofty goal, we’d like to take readers back in time to the turn of the 20th century to briefly examine the landscape as it pertained to mathematics in chemistry. Taking a historical perspective may help researchers understand where we are now in the 21st century, and it may drive forward the type of novel and innovative consideration of frameworks and literature that can guide important research questions that span both chemistry and mathematics.

Introduction “Inadequate experience in mathematics is the greatest single handicap in the progress of chemistry in America” (1). The ability of students to transfer and meaningfully apply their mathematical knowledge in chemistry has been a concern in undergraduate chemistry for decades, as Dr. Farrington Daniels pointed out in 1929 (1). One might ask, how did we get here and why did Daniels frame the lack of experience and knowledge of mathematics in this way? To begin this symposium series book, we shall look back over the educational landscape in the United States and in Europe to gain a bit of perspective about the field of mathematics and undergraduate and graduate programs in chemistry, then launch forward toward the present day. © 2019 American Chemical Society

The Fields of Mathematics and Chemistry in the United States The transformative rise of American science can be traced back to the decades between 1880–1930 (2). Although different science fields grew and matured at different rates and under different influences, our focus will be on chemistry and mathematics. American chemists during these decades relied heavily on laboratory measurements and experiments and de-emphasized areas that were considered to be theoretical in nature that required an adept understanding of mathematics. Thus, at that time, the United States was a nation with more experimentalists than theorists. As Servos noted for chemists and educators, laboratory instruction was surrounded with a “special mystique” (2). “For many of them the laboratory was, first and foremost, a place to mold character, to inculcate in young men the virtues of honesty, perseverance, and fidelity in the little things, and to instill respect for painstaking manual labor” (2). Thus, there was a privileged position given to experimentalists who made their discoveries by laboring in the laboratory versus those who might discern how nature worked by using mathematical models to probe and reveal its secrets. That norm in American science scraped against the theories that had been developed by the 1880s to describe thermodynamics, electricity, and magnetism, all of which required an understanding of differential equations (3–5). By the early 1900s, kinetic molecular theory and statistical thermodynamics required more mathematical prowess. The early decades of the 20th century saw the development of quantum theory, and with it, a host of mathematical formalisms required to understand, explain, and predict how atoms and molecules behaved. Servos notes the educational systems in particular in Germany and France emphasized engagement in mathematics in what Americans would consider secondary school, undergraduate, and graduate programs (2). To the detriment of the growth of theoretical understandings of chemistry and physics, American school systems at the secondary, undergraduate, and graduate level simply did not provide the same level of rigorous preparation in mathematics. For example, the Harvard University Catalogue in 1900 for the Lawrence Scientific School and Harvard College describes elementary studies in algebra through quadratic equations and plane geometry, and courses at the advanced level included logarithms and trigonometry, solid geometry, analytic geometry, and advanced algebra (6). The requirement for advance study in chemistry was a course of at least 60 experiments performed at the school. There was no coursework in calculus at Harvard at that time, and calculus was not a requirement for chemistry majors at a majority of U.S. institutions in the early 20th century. Two more examples by way of the career trajectories of prominent chemists serve to illustrate the standards in American universities. Nobel laureate Irving Langmuir received his undergraduate degree in metallurgy from the Columbia School of Mines in 1903 and went to Göttingen to study physical chemistry (2). In letters to his family, he described his classroom experiences and frustration at his inability to meet the mathematical expectations of the coursework. He dropped a course in mechanics and theoretical physics at the midway point due to the challenging level of mathematics. He began work in Nernst’s laboratory on physical properties of electrolytes but was removed from the project due to his inadequate mathematical preparation (2). Langmuir later won the Nobel Prize in 1932 for his discoveries and investigations in surface chemistry (7), and he mastered a great deal of mathematics that was applied to his research 2

endeavors. However, Servos notes that Langmuir preferred to work on problems guided by visual analysis and models that were more concrete in nature and not entirely mathematically based (2). Farrington Daniels, the author of the quote at the beginning of the chapter, supplies another example. Daniels was a physical chemist who was a pioneer in kinetics and solar energy research. He was a leading chemical educator and received the 1957 James Flack Norris Award for Outstanding Achievement in Teaching Chemistry for his impact on undergraduate physical chemistry instruction via his publications, presentations, and textbooks (8). He earned his bachelor’s degree in chemistry from the University of Minnesota in 1910 and his doctorate in physical chemistry from Harvard in 1914 (2, 9). He accomplished this without taking any coursework in calculus (2). The global events of World War I scuttled a postdoctoral appointment with Fritz Haber in Germany (9). By 1920, he was an assistant professor in chemistry at University of Wisconsin Madison where he spent the majority of his career. Initially he was tasked with teaching physical chemistry to undergraduate and graduate students and developing and teaching a course in calculus for chemists (10). Daniels’ experiences as an assistant professor teaching these courses led him to publish Mathematical Preparation for Physical Chemistry in 1928 (9, 10). He organized a symposium on The Teaching of Physical Chemistry at the fall 1928 ACS meeting that attracted over 600 audience members (11). During his talk, he reviewed the mathematical requirements for physical chemistry at that time. Some of his remarks about using slide rules and drawing graphs on coordinate paper are reminders of a time long before calculators and Excel were used. However, his blunt remarks emphasized the perspective that calculus is imperative to understand how differential equations such as the van’t Hoff equation and rate equations are used, that partial differentiation is the cornerstone of thermodynamics, and how partial molal quantities can be introduced and used (which perhaps was a sore point relating back to his doctoral research) (9). He concluded that calculus should be a prerequisite for physical chemistry. What changed in the 1920s was the role of mathematics in American schools and universities. Daniels represented part of that wave demanding that chemistry majors take a year of calculus. During the 1920s, many universities began including a year of calculus in their chemistry coursework (2). By the 1930s, America had a contingent of theoreticians in chemistry and physics including Linus Pauling, Robert Oppenheimer, and Harold C. Urey. Mathematicians were not eager to abandon pure mathematics; however, a growing desire emerged after World War I to apply mathematics to physical problems that ultimately supported changes in undergraduate curricula. Driven in part by the development of quantum theory, applied mathematics was so closely associated with mathematical physics that it risked being narrowly defined as such. However, the events of World War II emphasized the utilitarian aspects of mathematics in physics, chemistry, and engineering and thus broadened the perspectives of mathematicians and scientists around the globe.

The Influence of Mathematical Preparation on Success in Chemistry Daniels’ quote about the mathematical preparation and changes in the undergraduate chemistry curriculum that required increasing levels of mathematical fluency has inspired a great deal of research about mathematical preparation and student success in chemistry. Studies carried out by chemistry faculty in the United States at their own institutions have firmly established that mathematical preparation as measured by SAT Math score, ACT Math score, or grade in last high school mathematics classes is related to course performance (as measured by final grade) in first semester or 3

second semester general chemistry (11–19). In a study spanning 12 colleges and universities from across the country, SAT Math score and grade in last high school mathematics class were the two most influential predictors of introductory chemistry course grade (20). Essentially, the higher the Math SAT or ACT score, or the higher the grade in the last high school mathematics class, the stronger the performance in either first semester or second semester general chemistry. There also has been a great deal of research and commentary about problem-solving techniques and specific areas of mathematics in which fluency is required for success, such as algebra, logarithms, and graphing techniques (18, 21–23). Mathematicians and chemists have written about the difficulty of translating English into mathematical inscriptions and mathematical inscriptions into English (24–29). It is clear that the development of meaning of mathematical inscriptions does not occur simply by taking a course or courses in calculus, differential equations, or linear algebra. If that were the case, then students could relate an equation such as to the macroscopic phenomenon it represents, and there is evidence in chemistry and physics education research that it is simply not the case (25–29).

It’s Just Math To answer the call to improve mathematical preparation and to understand how students apply mathematics to problems in chemistry, we have gathered together research from the august group of authors in this symposium series book. The transfer of learning between chemistry and mathematics is a rich area for inquiry. As stated above, research demonstrates that preparation in mathematics is strongly related to success in chemistry; however, until relatively recently, little work has been done to investigate how undergraduate students understand and use mathematics in the context of chemistry (30–38). Growing interest in this area also includes studies at the university level regarding chemistry topics that promote applications of mathematical modeling to understand chemical phenomena (39, 40). This book seeks to provide potential avenues for future research by drawing attention to the myriad of complex issues that exist at the interface between chemistry and mathematics, while simultaneously supporting national efforts expressed in the Next Generation Science Standards science practices regarding the importance of skills such as quantitative reasoning, analyzing and interpreting data, and developing and using models (41). The collection of research presented in this book complements the symposium on which it is based, in which many of the presentations focused on chemical and mathematical reasoning and their integration (or lack thereof). There are chapters by Wink and Ryan; Lazenby and Becker; Bain, Rodriguez, Moon, and Towns; Posey and colleagues; and Ho and colleagues, all of which bring rich theoretical perspectives to readers that may help to further ground research in this area. Further, there are chapters that give in-depth reports of studies carried out on a wide variety of student participants. Phelps carried out her research in the context of electrochemistry with general chemistry students. Holme emphasizes the value of systems thinking in chemistry as a vehicle to improve mathematical reasoning skills in general chemistry. Cooke and Canelas carried out their work in an introductory chemistry course and introduce an instrument that may parse out student performance on domain-general and domain-specific word problems. Mack, Stanich, and Goldman focus on introductory chemistry and general chemistry courses, and their research calls attention to mathematics self-efficacy. Rodriguez and Bain focused on rates of change through the context of kinetics and used upper division students and general chemistry 4

students. Cole and Shepard focus on mathematical reasoning of physical chemistry students. Glaser and colleagues describe a project wherein a group of students studied a particular class of oscillating reactions requiring an understanding of chemical analysis, multiequlibria problems, and nonlinear dynamics. The thread that runs through these chapters is the quest to forge meaningful connections between chemistry and mathematics—to create a yoke, in the words of both Bain and Wink in their chapters, that helps students connect these domains of knowledge. Additionally, there is a repeated call to move away from a deficit model of learning toward one that recognizes (or assesses) where students are and helps them move forward toward more meaningful and normative understandings of chemistry. This book expands the horizons of the symposium because of its broader scope that includes contributions from the field of research on undergraduate mathematics education, known as RUME. In particular, the chapters by mathematicians Steven Jones and Annie Bergman focus on derivatives and integrals and group theory. In both cases, chemists have the opportunity to learn how mathematicians approach these areas of mathematics and to consider where there is overlap between applications in chemistry and mathematics. Further, and perhaps more important, by reading and considering the chapters by Jones and Bergman, we as chemists can be stretched to understand how mathematicians consider helping students learn the mathematics we desperately want them to apply meaningfully in our classes. Ultimately, what we find most exciting to share with the chemistry community is the opportunity to acquaint readers with literature and approaches that may be novel, innovative, and creative. More simply, they may strike the reader as new. Pursuing research questions positioned at the interface between chemistry and mathematics may be enhanced by the consideration of new frameworks and an acquaintance with and synthesis of new literature. New vistas for research may emerge from careful contemplation of the ideas found within this book.

Conclusion We hope that as you read these chapters, you keep in mind a quote by Peter Atkins, who wrote in Advances in Teaching Physical Chemistry (28): “We should be sensitive to the difficulty that large numbers of students have with mathematics, and never fail to interpret the salient features of the equations we derive.” In every chemistry course, there is a need to facilitate student understanding of mathematical equations and representations (e.g., graphs, orbitals). The ability to ascribe physical meaning to equations and representations is about applying mathematical understanding, and seat time in a mathematics course does not automatically confer this ability upon students. In a broad disciplinebased education research sense, we hope that the chapters in this book inspire chemists and perhaps mathematicians and physicists to consider how students apply and understand mathematics in the context of the physical sciences. More research in this area is well-warranted and implications for classroom practices derived from that research have the potential to positively impact, broaden, and deepen student learning. Specifically for chemistry education researchers, we hope that these chapters broaden your perspectives about research in this area and inspire you to synthesize the research found herein with the goal of asking creative and important questions designed to drive the field forward. 5

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Daniels, F. Mathematical Requirements for Physical Chemistry. J. Chem. Educ. 1929, 6, 254–258. Servos, J. W. Mathematics and the Physical Sciences in America, 1880–1930. Isis 1986, 77, 611–629. Partington, J. R. A History of Chemistry; MacMillan: New York, 1970. Segré, E. From Falling Bodies to Radio Waves: Classical Physicists and Their Discoveries; Dover: Mineola, NY, 2007. Segré, E. From X-Rays to Quarks: Modern Physicists and Their Discoveries; Dover: Mineola, NY, 2007. The Harvard University Catalogue, 1899–1900; Harvard University: Cambridge, MA, 1900. See pp 289–295, 303–306. Irving Langmuir – Biographical. NobelPrize.org. Nobel Media AB 2018. https://www. nobelprize.org/prizes/chemistry/1932/langmuir/biographical (accessed Oct. 3, 2018). James Flack Norris Award for Outstanding Achievement in the Teaching of Chemistry Recipients. http://www.nesacs.org/awards/norris/awards_norris_recipients.html (accessed Oct. 3, 2018). National Academy of Sciences. Farrington Daniels. In Biographical Memoirs; The National Academies Press: Washington, DC, 1994; Vol. 65. https://www.nap.edu/read/4548/ chapter/6 (accessed Oct. 3, 2018). Daniels, F. Teaching Physical Chemistry – Forty Years of Change. J. Chem. Educ. 1958, 35, 322–323. Pickering, M. Helping the High Risk Freshman Chemist. J. Chem. Educ. 1975, 52, 512–514. Ozsogomonyan, A.; Loftus, D. Predictors of General Chemistry Grades. J. Chem. Educ. 1979, 56, 173–175. Craney, C.; Armstrong, R. Predictors of Grades in General Chemistry for Allied Health Students. J. Chem. Educ. 1985, 62, 127–129. Carmichael, J. W.; Bauer, S. J.; Sevenair, J. P.; Hunter, J. T.; Gambrell, R. L. Predictors of FirstYear Chemistry Grades for Black Americans. J. Chem. Educ. 1986, 63, 333–336. Spencer, H. Mathematical SAT Test Scores and College Chemistry Grades. J. Chem. Educ. 1996, 73, 1150–1153. Angel, S. A.; LaLonde, D. E. Science Success Strategies: An Interdisciplinary Course for Improving Science and Mathematics Education. J. Chem. Educ. 1998, 75, 1437–1441. Wagner, E.; Sasser, H.; Diabiase, W. Predicting Students at Risk in General Chemistry Using Pre-Semester Assessments and Demographic Information. J. Chem. Educ. 2002, 79, 749–755. Leopold, D.; Edgar, B. Degree of Mathematics Fluency and Success in Second-Semester Introductory Chemistry. J. Chem. Educ. 2008, 85, 724–731. Stone, K.; Shaner, S.; Fendrick, C. Improving the Success of First Term General Chemistry Students at a Liberal Arts Institution. Educ. Sci. 2018, 8, 1–14. Tai, R.; Sadler, P.; Loehr, J. Factors Influencing Success in Introductory College Chemistry. J. Res. Sci. Teach. 2005, 42, 987–1012. Dierks, W.; Weninger, J.; Herron, J. D. The Special Nature of Quantity Equations. J. Chem. Educ. 1985, 62, 839–841. 6

22. Kennedy-Justice, M.; DePierro, E.; Garafalo, F.; Sunny Pai, S.; Torres, C.; Toomey, R.; Cohen, J. Encouraging Meaningful Quantitative Problem Solving. J. Chem. Educ. 2000, 77, 1166–1173. 23. Niaz, M.; Herron, J. D.; Phelps, A. J. The Effect of Context on the Translation of Sentences into Algebraic Equations. J. Chem. Educ. 1991, 68, 306–309. 24. Lamb, H. An Elementary Course of Infinitesimal Calculus; Cambridge University Press: Cambridge, UK, 1898. 25. Hadfield, L. C.; Wieman, C. E. Student Interpretations of Equations Related to the First Law of Thermodynamics. J. Chem. Educ. 2010, 87, 750–755. 26. Loverude, M. E.; Kaurz, C. H.; Heron, P. R. L. Student Understanding of the First Law of Thermodynamics: Relating Work to Adiabatic Compression of an Ideal Gas. Am. J. Phys. 2002, 70, 137–148. 27. Thompson, J. R.; Bucy, B. R.; Mountcastle, D. B. Assessing Student Understanding of Partial Derivatives in Thermodynamics. AIP Conf. Proc. 2006, 818, 77–80. 28. Atkins, P. The Evolution of Physical Chemistry Courses. In Advances in Teaching Physical Chemistry; Ellison, M. D., Schoolcraft, T. A., Eds.; ACS Symposium Series 973; American Chemical Society: Washington, DC, 2007; pp 44−55, DOI: 10.1021/bk-2008-0973.ch005. 29. Becker, N.; Towns, M. Students’ Understanding of Mathematical Expressions in Physical Chemistry Contexts: An Analysis Using Sherin’s Symbolic Forms. Chem. Educ. Res. Pract. 2012, 13, 209–220. 30. Rodriguez, J. G.; Bain, K.; Ho, F. M.; Elmgren, M.; Towns, M. H. Covariational Reasoning and Mathematical Narratives: Investigating Students’ Understanding of Graphs in Chemical Kinetics. Chem. Educ. Res. Pract. 2019, 20, 107–119. 31. Rodriguez, J. G.; Bain, K.; Towns, M. H. Graphical Forms: The Adaptation of Sherin’s Symbolic Forms for the Analysis of Graphical Reasoning Across Disciplines. Int. J. Res. Undergrad. Math. Educ. 2018, submitted. 32. Rodriguez, J. G.; Bain, K.; Santos-Diaz, S.; Towns, M. H. Using Symbolic and Graphical Forms to Analyze Students’ Mathematical Reasoning in Chemical Kinetics. J. Chem. Educ. 2018, 95, 2114–2125. 33. Bain, K.; Rodriguez, J. G.; Towns, M. H. Zero-Order Chemical Kinetics as a Context To Investigate Student Understanding of Catalysts and Half-Life. J. Chem. Educ. 2018, 95, 716–725. 34. Bain, K.; Rodriguez, J. G.; Towns, M. H. The characterization of cognitive processes involved in chemical kinetics using a blended processing framework. Chem. Educ. Res. Pract. 2018, 19, 617–628. 35. Rodriguez, J. G.; Bain, K.; Hux, N. P.; Towns, M. H. Productive Features of Problem Solving in Chemical Kinetics: More Than Just Algorithmic Manipulation of Variables. Chem. Educ. Res. Pract. 2019, 20, 175–186. 36. Bain, K.; Towns, M. H. A Review of the Research on the Teaching and Learning of Chemical Kinetics. Chem. Educ. Res. Pract. 2016, 17, 246–262. 37. Bain, K.; Moon, A.; Mack, M. R.; Towns, M. H. A review of the research on the teaching and learning of thermodynamics at the university level. Chem. Educ. Res. Pract. 2014, 15, 320–335.

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38. Hadfield, L. C.; Wieman, C. E. Student Interpretations of Equations Related to the First Law of Thermodynamics. J. Chem. Educ. 2010, 87, 750–755. 39. Becker, N. M.; Rupp, C. A.; Brandriet, A. Engaging Students in Analyzing and Interpreting Data to Construct Mathematical Models: An Analysis of Students’ Reasoning in a Method of Initial Rates Task. Chem. Educ. Res. Pract. 2017, 18, 798–810. 40. Brandriet, A.; Rupp, C. A.; Lazenby, K.; Becker, N. Evaluating Students’ Abilities to Construct Models from Data Using Latent Analysis. Chem. Educ. Res. Pract. 2018, 19, 375–391. 41. National Science Teachers Association Science and Engineering Practices. https://ngss.nsta. org/PracticesFull.aspx (accessed Oct. 3, 2018).

8

Chapter 2

A Modeling Perspective on Supporting Students’ Reasoning with Mathematics in Chemistry Katherine Lazenby and Nicole M. Becker* Department of Chemistry, University of Iowa, 305 Chemistry Building, Iowa City, Iowa 52242, United States *E-mail: [email protected]

Undergraduate general chemistry courses typically feature a substantial amount of mathematical problem solving. Research shows that many students approach mathematical problem solving algorithmically and without recognition of either the assumptions and limitations of such models or their relationship to the particulate level. The question becomes: How can instructors support students in more meaningful engagement with mathematical models in chemistry contexts? In this chapter, we present research on modeling approaches to undergraduatelevel chemistry instruction and discuss the role of epistemological knowledge, specifically metamodeling knowledge, in students’ reasoning with and about mathematical representations.

Mathematical models (and related models such as computational models) are central to contemporary research in chemistry and are important tools for predicting and explaining chemical behavior. For students of chemistry, mathematical models are important, not only because they can serve as conceptual bridges between macroscopic and particulate-level scales (1), but also because they represent part of an important scientific process—the practice of developing and using models. This practice is one of eight highlighted in the National Research Council’s Framework for Science Education (2) as critical elements of contemporary scientific inquiry. Other practices include analyzing and interpreting data, using mathematical and computational thinking, and constructing explanations (2). The Framework’s argument for engaging students in scientific practices is twofold; by giving students opportunities to use disciplinary core ideas and interdisciplinary (“cross-cutting”) ideas to engage in scaffolded forms of science practices such as constructing and using models, students will develop both deeper knowledge of core disciplinary concepts and a better understanding of the nature of scientific practice. That is, they will develop more robust epistemological knowledge about the nature of scientific inquiry (2).

© 2019 American Chemical Society

While the Framework forms the basis of K–12 level STEM education reforms, researchers and educators have argued that curricula centered on having students engage in science practices as they learn core and cross-cutting ideas would also support student learning at the undergraduate level (3, 4). Several “three-dimensional” (3-D) approaches to undergraduate curricular reform have been described in the literature (3, 5), and researchers have developed tools for helping faculty evaluate the extent to which their assessments address 3-D learning and examine whether classroom activities reflect 3-D learning (4, 6). Early assessments of the impact of these instructional approaches on student learning suggest greater content learning gains (5, 7). To date, however, there is limited information available on the impact of 3-D instruction or other modeling-focused approaches on students’ ideas about the nature and purpose of models, such as mathematical models. In the remainder of this chapter, we set the stage for future examinations of the impact of 3-D and modeling-focused instructional approaches on students’ epistemological ideas about models and modeling. First, we provide an overview of the practice of constructing and using models. We then discuss elements of metamodeling knowledge and their development and conclude with a review of curricular models that show promise for supporting students’ ideas about models and modeling.

Scientific Models and the Practice of Developing and Using Models Scientific models often take the form of representations that embody ideas about how and why a phenomenon occurs or about relationships between components within systems being studied. Models can take a range of representational forms (e.g., equations, graphs, diagrams, words) and are defined more by how they are used—that is, to predict and explain phenomena—than by the form they take (8). That is, they are sense-making tools that help predict and explain the natural world. The practice of modeling can be understood as a bridge between wondering why something works or happens the way it does and arriving at an explanation for the phenomenon that is grounded in theory and empirical evidence. When generating explanations about natural phenomena, people often implicitly use a set of ideas and assumptions they have about the system or problem. The process of modeling makes these ideas explicit (8). For students, we can think of two ways of engaging in the practice of developing and using models: thinking “with” models and thinking “about” models (8). Thinking “with” models has also been referred to as “model-based reasoning” in the research literature (9) and involves the application of models as tools for predicting or explaining macroscopic phenomena. For example, one might use the ideal gas law, PV = nRT, to predict how the volume of a balloon might change when moving from indoors to outdoors on a cold winter day or use a simulation of particulate-level motion and spacing to help construct a particulate-level account of how and why volume changes. Thinking with models is essential for helping students see how modeling enables new ways of thinking about a system under study (8). Thinking “about” models, in contrast, involves constructing, testing, and refining models rather than the straightforward application of existing models. Thinking “about” models may involve iterative cycles of model development, testing, and model refinement. These cycles are guided by the broader goals of sense-making, a term we use to refer to the process of resolving gaps or conflicts in comprehension in the service of understanding how or why a system works the way it does (10). Thinking “about” models is especially important for helping students see how theoretical ideas (e.g., about the particulate level) connect to the models under study. Additionally, thinking “about” models is important for supporting students’ understanding of what goes into a model and why as 10

well as for their knowledge of the relationship between the model and the target system (that which is being modeled). The nature and importance of epistemological ideas such as these are discussed further in the next section.

Metamodeling Ideas: Epistemological Knowledge Associated with the Practice of Developing and Using Models Scientists aim to develop models that can provide insights into how or why a phenomenon occurs, for example, by providing a mechanism that drives the phenomenon. Disciplinary norms and ideas about what counts as a “good” model or a robust modeling process play key roles in framing model development and evaluation. For example, it is important that models be general enough to be applicable to other phenomena and useful for the modelers (11). It is also essential that models be accurate with respect to predicting and explaining phenomena within certain levels of tolerance (8, 11). These ideas and others represent elements of knowledge that, together with content knowledge and knowledge of mathematics, support students’ engagement in developing and using models. Epistemological knowledge specific to the practice of developing and using models has been termed metamodeling knowledge. Metamodeling ideas include the idea that models are developed and refined on the basis of empirical data, that there may be multiple models for a phenomena, and that models highlight some features of a system while simplifying others (12). Such ideas previously have been considered part of students’ understanding of the nature of science but have more recently been considered through the lens of science practices (13, 14). Studies examining students’ metamodeling knowledge commonly address five key topics: the nature, purpose, and testing of models; the role of multiple models; and the way in which models are revised (15, 16). These five aspects of metamodeling knowledge are summarized in Table 1. In the next section, we briefly define each of these aspects and highlight research evidence on students’ thinking about each one.

Aspects of Metamodeling Knowledge and Their Development Grünkorn, Upmeier zu Belzen, and Krüger developed a “model of model competence” (Table 1), a five-dimensional hierarchical progression of knowledge about models and modeling, based on the open-ended survey responses of 7th–10th grade students. The five dimensions of metamodeling knowledge in their model are the nature, purpose, testing, changing, and multiplicity of models. These dimensions and their developmental progression according to Grünkorn and colleagues’ data are summarized in Table 1 (15, 16). The nature of models refers to the relationship between the model and the phenomenon it represents (14, 17). Experts’ understanding of this aspect of metamodeling knowledge includes the idea that models may be abstract, theoretical representations of phenomena and that models may simplify some aspects of a system while highlighting others (16, 18). Research on students’ ideas about models and modeling suggests that students view models through a naïve realist perspective and see models as exact replicas of the phenomenon (19, 20). Grünkorn et al. found that students often consider the extent to which the model “looks like” or “matches” the target instead of considering that assumptions and simplifications are necessary in the representation of phenomena. 11

Table 1. Grünkorn, Upmeier zu Belzen, and Krüger’s Model of Model Competence. Reproduced with permission from reference (15). Copyright 2014 Taylor and Francis. Aspect

Complexity Level 1

Level 2

Level 3

Nature of models

Replication of the target phenomenon

Multiple models

Due to differences between Complex phenomena allow the target phenomena for multiple models

Due to different hypotheses about the target phenomenon

Describing or showing phenomena

Explaining mechanisms of or variable relationships within phenomena

Predicting phenomena

Testing models Testing the model itself

Comparing the model to the target phenomena

Testing hypotheses about the target phenomena with the model

Purpose of models

Changing models

Idealized representation of the Theoretical reconstruction of target phenomenon the target phenomenon

Would occur in order to Would occur due to new correct errors in the model findings regarding the target phenomena

Would occur due to new hypotheses about the target phenomena

The purpose of models relates to their function as tools used to predict and explain a phenomenon (12, 21, 22). Experts may focus on models’ use as a way to illustrate the underlying mechanism of a phenomenon, which enables the development of testable hypotheses about the target system. Students, in contrast, may believe that the purpose of modeling is to describe or show the original as precisely as possible (22). The idea of multiple models relates to the idea that different models may make different theoretical assumptions and hence have unique limitations and affordances that enable them to be useful in addressing different hypotheses about the target system (19, 20, 23). Students, however, may see multiple models as presenting the same information through different representational conventions (21), see different models as related to differences between phenomena, or see different models as focusing on different aspects of complex phenomena (16, 18). The testing aspect of metamodeling knowledge addresses the idea that models are developed, tested, and revised on the basis of data. Although experts may think about testing hypotheses about the target phenomena with the model by collecting empirical data, students may think about comparing the target phenomena directly with the model and evaluating their similarity (16, 18). Finally, changing models relates to the way in which models are revised based on the outcome of testing. Although experts recognize that models can be revised based on the results of experiments or advances in theory, research suggests that students may not recognize that models can be revised (15). There is evidence that students’ ideas about these five dimensions of metamodeling knowledge develop independently and at different rates (24). Additionally, students’ knowledge about metamodeling ideas may be highly context-specific and tied to their understandings of models within specific STEM disciplines (22). For example, Krell and Krüger administered an open-ended survey to elicit undergraduate and graduate students’ ideas about both models in general and specific models in order to investigate the interaction between students’ understanding of models and 12

situational factors such as the respondent’s self-identified discipline and item features, in this case, the models that students specified in survey prompts. With regard to students’ discipline, the authors reported that students from STEM disciplines expressed advanced understandings more often than their peers in the social sciences or linguistics; however, even STEM students expressed “prospective” or expertlike understanding of models and modeling in only 20% of responses (compared with 12% overall). Regarding specific versus general metamodeling knowledge, university students expressed more expertlike ideas about models generally than they did about specific models from their discipline, suggesting that metamodeling knowledge may be situated and contextualized. Those students who did express expertlike views with regard to specific models had specified abstract models such as mathematical equations, supporting the view that students’ ideas about models and modeling are highly context-dependent (13).

Research on Students’ Metamodeling Knowledge in Chemistry Contexts To date, there is little research on how students think about these metamodeling ideas, specifically with respect to mathematical models. The progression in Table 1, for example, was developed through a characterization of students’ ideas about various types of biological models such as theoretical reconstructions of a Neanderthal man or of tyrannosaurus rex or maps of the taste regions on the human tongue (15). Other work on metamodeling knowledge has focused on diagrammatic models (e.g., diagrams of evaporation and condensation) (12). In chemistry contexts, some research has been conducted to develop and use assessments of students’ ideas about models and modeling from a domain-general perspective. One notable and widely used assessment of students’ metamodeling knowledge is the Students’ Understanding of Models in Science (SUMS) instrument, a Likert-scale instrument intended for use in assessing students’ knowledge about scientific models in general (20). Items on the SUMS instrument were based on Grosslight et al.’s (19) analysis of students’ ideas about models. The SUMS includes the following five subscales: multiple representations (MR), models as exact replicas (ER), models as explanatory tools (ET), uses of scientific models (USM), and the changing nature of models (CNM) (20). The SUMS has been used to characterize the metamodeling knowledge of various student populations (20, 21, 25–28) and as an assessment of the efficacy of modeling-focused curricula (25–28). For example, Gobert et al. administered SUMS to students in three different high school science classes (biology, physics, and chemistry) and found higher SUMS scores (based on mean differences on three subscales) in physics classes than in biology classes (25). This result may suggest that different courses impact students’ metamodeling knowledge differently. Park et al. used the SUMS to examine the impact of a modeling-focused instructional intervention in which students used computer-based models over the course of one year. Simulations depicted both macroscopic and particulate-level phenomena and were used in each classroom between 4 and 20 times, depending on the instructor. A class that did not use simulations served as the control group. Park and colleagues found nonsignificant differences between the control and treatment groups on four of the five SUMS subscales. On the fifth subscale (CNM), there were statistically significant differences that favored the control group (28). Although it is possible that the intervention did not affect students’ ideas about models, it may also be the case that the domain general assessment (SUMS) was not able to detect changes in students’ ideas about the specific type of models addressed by the intervention. 13

Context Specificity of Students’ Metamodeling Ideas There is growing evidence that domain-general instruments may not be robust indicators of students’ ideas about specific types of models (22). As we have noted, Krell and colleagues found that students expressed more expertlike ideas about models when discussing models in general than when thinking about models specific to a STEM discipline (13). Our own work on students’ ideas about models suggests that students may have very different ideas about even specific models within the chemistry curriculum (29). We administered an openended survey about models in chemistry contexts to 773 undergraduate general chemistry students. We asked students to indicate whether they thought six representations common to the chemistry curriculum would be considered scientific models and to explain their reasoning. We found significant differences in the ways that students categorized and discussed mathematical and graphical models (energy diagram [ED], the ideal gas law [IG], and equilibrium constant expression [EQ]) compared with models of particulate-level entities (a representation of the motion and spacing of gas particles [RP], physical model of a molecule [PM], and a Lewis structure [LS]) (Figure 1).

Figure 1. Proportion of students who classified six representations from general chemistry as “a scientific model” or “not a scientific model”; N = 773. Pairwise t-tests indicate significant differences between proportions of students who classified particulate-level entities (RP, PM, and LS) as scientific models and the proportions who classified mathematical models (ED, IG, EQ) as scientific models; all pairwise pvalues < 0.001, with one exception—the pairwise p-value = 0.02 for the difference between ED and LS. We also identified qualitative differences in students’ reasoning about their classifications of different representations. Students who did not categorize the mathematical and graphical models (ED, IG, and EQ) as scientific models commonly discussed the representational form of the item, for example, noting that equations and graphs can never be considered models because models must be visual in nature. When discussing the ideal gas law specifically, students also discussed the ontological status of the model. For example, one student noted: “It [the ideal gas law] has law in the name... It goes something like model, theory, law.” To us, this suggests confusion about what counts as a model compared with a scientific theory or law. Students who did classify the mathematical and graphical representations as scientific models discussed their utility for predicting or explaining phenomena much less frequently (|t|)

(Intercept)

72.945

2.772

26.316

0 implies increasing entropy towards the maximum. In most cases, what is important is the symbolic representation that allows one to predict the direction of changes. However, students often struggle with this and tend 177

to prefer substituting in values for the variables to do calculations rather than engaging in symbolic reasoning. In cases where variables are replaced with equations, such as in Schrodinger’s Equation, students can write the initial equation but often struggle past that point. This also involves the concept of operators, which can be problematic for many students, who often want to group like terms. This results in expressions such as rather than . As most chemical phenomena require multivariate models, partial derivatives are required to represent how one variable changes in response to changes in another variable (while holding the others constant). While students can write the total differential for functions after a brief review, they often struggle to understand the representation. It is common for students to conflate holding a variable constant as implying that “x” is 0 rather than recognizing that it is d“x” that is zero (“x” representing any variable of interest). The reverse is sometimes seen in quantum mechanics, where students want to set the derivative to zero rather than the function to zero when finding nodes. Relationships between partial derivatives of state functions are also key to solving for changes that cannot be easily calculated or measured in terms of other changes. The most common example of this in physical chemistry is the use of Maxwell relations. In his tribute to the legacy of James Clerk Maxwell, Reid (29) writes that “Maxwell’s thermodynamic relations were among his less popular output with students of thermodynamics because lecturers considered them so important that they were worth memorizing.” While it is still true that many instructors require students to memorize Maxwell equations (or teach mnemonics to help students remember relationships), a recent online conversation among a group of experienced physical chemistry instructors indicates that some instructors are moving away from this. The question was raised regarding the amount of attention given to Maxwell’s equations in a typical coverage of thermodynamics. There was a diversity of responses with many instructors opting for minimal coverage if they were addressed at all. The following exchange illustrates some of the challenges students encounter in addressing these concepts. The form of the total derivative does not depend on which gas law equation is being used to model the system, but students have difficulty separating the general form from a specific application. Student 2: Number three…“Through the following series of questions we will demonstrate that one can define (dT/dP)V [referring to in the text] in terms of a different slope. Write the total derivative for P(V,T)” Student 4: Are we talking about the Van der Waals gas equation or the ideal gas equation? Student 2: That isn’t the question. Student 3: Would it be the partial derivatives added together? Student 5: Yes. Student 2: The total derivative, yes, but we’re… Student 4: …probably the Van der Waals. Student 2: That’s funny because I was going to say the ideal Student 1: Yeah. Student 3: I mean it’s a partial derivative Student 5: Do we have to add them together? Student 4: I think so. Student 2: for the derivative? Student 5: Yes. Student 1: This doesn’t say whether… Student 2: …Yeah well, it doesn’t actually say which one. 178

Student 3: I think you just write it Student 2: I mean the slope is technically different. Excerpt from small group discussion (23) of activity U.4 Slope Relationships (24) It is also important for instructors to discuss why a derivation is important or why the skill is necessary for understanding physical chemistry. Too often derivations appear to be done for the sake of completing a derivation rather than in service to explaining the significance of the resulting equation. This approach is being adopted in the POGIL physical chemistry materials where an activity that involves derivations concludes with questions that ask students to reflect on the process or the significance of the resulting expression. Activities have also been developed in physics that provide scaffolding for students to think about thermodynamic derivatives in terms of experimental measurements (30).

Integrals Integration is another area of mathematics that is critical for mastery of physical chemistry. While students can generally do calculations involving integrals, they often do not understand what is represented. Additional discussion of education research on the concept of integrals in relation to chemistry topics is provided in this volume (28). As was noted by Christensen and Thompson (27), the findings from multiple studies indicate that students do not have sufficient conceptual understanding of the relationship between definite integrals and the areas under the curve to allow them to successfully complete problems requiring this understanding. The topic of pressure-volume work is the ideal illustration in review of these concepts. Students can be guided to consider a single step expansion or compression process and plot a line representing the value of constant external pressure, Pex, over a range of volumes on a P versus V graph. The area under the curve is the magnitude of work equivalent to , which corresponds to the maximum compression work or the minimum expansion work. As more steps are added, the sum of each individual step can be be assessed and compared to a single step process until the limit of an infinite number of infinitesimal steps is considered. This is defined as a reversible process where Pex is replaced by a gas equation like . Students can be led to draw a PV isotherm and solve the integral that corresponds to an area that is the same for both expansion and compression work. One might be hopeful that this logic could be generalized to other PV curves, for example a plot where is a constant. Instead, students typically report the total amount of work as PidV + PfdV. In addition, Hadfield and Wieman (13) found in a study of student understanding of the First Law of Thermodynamics that many students interpreted the equation for work as representing a transfer of energy from pressure and volume rather than as the area under the curve. Similar errors were observed for student interpretations of the equation for heat. This fundamental mathematical reasoning of integrating functions that can either be constant or changing over an interval is extremely important in order for students to appreciate the assumptions inherent in a relationship. Instructors often emphasize the importance of either recognizing or deriving changes in entropy or enthalpy over a temperature range while either assuming that Cp is a constant or that it is function of temperature. But when given Cp = a + bT, many students will evaluate Cp at one (or both) values of temperature defined by the integration range, and then treat 179

Cp as constant when evaluating the integral. As later relationships build on prior assumptions, these same integration errors accumulate. One example is the evaluation of changes in the equilibrium constant K with temperature, which can assume either that the reaction enthalpy change, ΔrHrxn, is a constant or that it depends on temperature, assuming Cp is independent of temperature. It is about this point in the semester when a student will wonder out loud, “but isn’t K a constant?” Similar challenges are in seen in physics, where students also have difficulty applying mathematical concepts and skills in relevant situations (31). Bajracharya and Thompson note that it is important for instructors to “emphasize the appropriate framing of a problem, helping students decide what strategies (e-games), and thus which resources, to bring to bear” (31). Student challenges with integration are illustrated in a whole class discussion of a question where students were asked to provide an expression for the calculation of entropy of H2O(s) at 125 K relative to that at 0 K. Student G: OK so we were asked to provide an expression for the change in entropy from ice from the initial temperature of zero degrees to the final temperature of 125. So we took this general expression for delta S is equal to this integral and when you do the integration you get specific heat times your difference in logs while the log of zero is undefined so you can’t really evaluate that but you do know by definition that at zero degrees kelvin entropy is zero so you can just ignore that term and so this provides you with delta S at ... equals Cp at constant pressure time the log of the final temperature which is 125. Instructor: I have a question. What does delta S T represent (delta S sub T)? Student G: um well ... delta S for delta T for the change in temperature but since our initial temperature was zero, I don’t think... Instructor: I don’t think you answered the question. Student G: ok well delta S, this would be the change in entropy going from initial temp to final temp the change in entropy depending on ... Instructor: I don’t want a general thing, I want a specific in this example what does it represent Student G: a specific thing, it represents the absolute.... for the ice. Student M: doesn’t that notation usually mean for keeping the temperature constant. Student G: I don’t know that just the notation that they used S sub T..... Student R: Can we pull Cp out for sure through the integration, because then the Cp ... for a lot of our problems they state specific heat not varying with temperature but for this case specific heat could vary with temperature, so it’s as a function of temperature that it could vary with temperature in that sense, and you couldn’t just pull it out of the integral either. Student G: right so then you are going to have to do the product rule. Instructor: So you could have left it with the integral until you know what Cp is, so you made an assumption by pulling it out. Student G: that it’s constant with temperature. Excerpt from whole class discussion (32) of activity T8 (33) In addition to the single integral examples provided above, quantum mechanics also regularly requires the use of multiple integrals. Work in mathematics education has illustrated that moving to an understanding of multiple integrals from an understanding of single integrals is not an easy task (34, 35). The understanding students have of multiple integrals is strongly influenced by their interpretation of integrals as representing the “area under a curve” (34, 36). Jones indicated that 180

students have productive knowledge of the intergral and that difficulties may be due to which interpretation of the integral is most appropriate to use in the given context (37). He also suggested that instructors can better support student learning by helping students choose which interpretation to use when applying their knowledge to solve problems.

Differential Equations When you look through any quantum textbook and consider that many students viewing the same equations may have only had two semesters of calculus, it is understandable the dread most students have of the subject of Quantum Mechanics and its use of differential equations. Although undergraduates are generally not expected to solve a differential equation, it is assumed they can demonstrate whether a function is a solution, how boundary conditions effect the solution, and whether a solution function is normalized or orthogonal. However, the idea that the “solution” to a differential equation can be problematic for students (38). It is easy for students to quickly lose sight of the chemical insight these functions convey. This is where guided inquiry activities, such as POGIL activities (39), and collaborative learning strategies (40) are particularly useful to guide students through exploration of the details. Because of the simplicity of the general solution, the one-dimensional particle-on-a-line model is the perfect example for students to gain experience with unfamiliar mathematical content related to the nature of differential equations while simultaneously constructing conceptual knowledge of a simple quantum system. Students can be prompted to predict the types of mathematical functions that would or would not work as a solution. They can be guided to recognize constraints on the solutions that arise from the application of boundary conditions leading to the origination of quantum numbers. Finally, they can be directed to propose the impact these integers have on the properties of the system. However, care must be taken such that students do not view verifying a solution to be the same as obtaining it by solving a differential equation (41, 42).

Factorials (Combinatorics) While a factorial is a relatively simple idea, its application to represent permutations and combinations is another area of mathematics that is typically less familiar to students (and that may also be less emphasized by instructors). The direct connections that can be made to a particulate representation of chemical systems, however, are an opportunity for students to construct a more meaningful understanding of entropy, equilibrium, and other aspects of statistical thermodynamics. Even the application of mathematical reasoning in terms of simple logarithm rules offer insight when used in the context of entropy defined as S = k ln W. Students discover the connection between the way multiplicities combine, k ln WAWB = k ln WA + k ln WB, and the extensive nature of entropy. When students fail to understand differences in mathematical models, they can struggle to determine which particular equation is appropriate for a given situation. This is evident in the following dialogue from a group of students completing an activity on probability and combinatorics. In the activity, students first complete a section on determining the multiplicity in terms of the probability of a unique sequence of events occurring using the following equation: W = ninjnt (equation 5 in the text). This is followed by a section on permutations where the goal is to determine a particular composition of events. Students are provided the equation W = N! (equation 6 in the text) for a sequence of N different objects for which each object can only be chosen once and (equation 7 in the text). After the different equations for determining multiplicity are 181

introduced, students are given a series of scenarios and are prompted to identify the appropriate equation (5–7) and then to use it to calculate the probability of the event occurring. In the following dialogue, the extent to which students struggled to understand the difference in these equations is evident. Student 3: I guess it depends on how you’re, whether you’re saying using equation five demands that they be independent or if you can still look at them independent and then like… But then you’d just be like… You’d be writing out equation six. Student 1: No. I think the point of having two equations is that they describe two different things. But… … Student 1: I still don’t see why we would define them as different equations when we were not going to assume they were different. Excerpt from small group discussion (23) of activity U.1 Probability and Combinatorics (24) This highlights the importance of the use of informal assessments in the classroom that have students reflect on “how do you know” so that more effective approaches to problem solving and the nature of mathematical models can be supported.

Determinants and Matrices In order to represent a wavefunction or variational approximation of the Schrodinger equation for multielectron systems, matrices and determinant solutions are essential. Using matrix equations to express symmetry operations is also useful. Although linear algebra is typically not a prerequisite course for quantum mechanics, most students will have had some prior exposure to matrix representations and solving a determinant. The transition from solutions expressed explicitly as integrals to notation that masks integrals as “matrix elements,” however, is jarring to students. Students often struggle to construct and interpret appropriate matrices (43), and the beauty of solving n simultaneous linear equations can fall a bit flat. Nonetheless, if students spend time actively engaged in translating the functional form of the wavefunction into the commonly used abbreviated symbols, they can be guided to appreciate the utility of a Slater Determinant and gain insight regarding Pauli’s Exclusion Principle. Using the ground state of helium as an example, students can differentiate the spatial and spin components of the wavefunction and propose all possible combinations of these functions. Students can be guided to make a distinction between those that are indistinguishable and those that are antisymmetric and then notice that a determinant gives the same result.

Student Reasoning While facility with mathematical techniques is important for success in chemistry, logical thinking skills are also critical (17, 44). These skills are related to a student’s ability to analyze and synthesize relevant information to form an argument or reach a conclusion supported with evidence. Analysis of students’ chemical reasoning as evidenced in scientific argumentation indicates that students primarily engage in relational reasoning when discussing topics related to mathematical expressions, rarely providing any sort of causal justification (17). Students tended to take definitions and equations at face value and did not engage in significant interpretation or sense-making unless 182

explicitly prompted to by the materials or the instructor. Complex causal arguments that incorporated mathematical reasoning and explicit connections to the phenomena being described were observed infrequently. Most mathematical reasoning that went beyond describing relationships was characterized as linear reasoning and focused on the sequential reasoning (or mathematical steps) employed to arrive at a solution. While most of the relationships used in physical chemistry are inherently multivariable, multicomponent arguments were the least frequently observed. Moon et al. interpreted this as evidence of the difficulty students had in considering how multiple variables contribute to an outcome (17).

Conclusions and Implications for Practice When teaching physical chemistry, there is a constant tension between focusing on the practical, experimentally relevant variable spaces that are “easier” to control (pressure and temperature) and introducing the wide range of ensemble spaces that can be modeled. There is also the tension between ensuring students can “do the math” and evaluating the extent to which they have developed a conceptual understanding of the material. This is highlighted in a quote from a student in Tsaparlis’ study of the structure of physical chemistry and graduate students’ opinions on the difficulties of the subject (2). Surely the concepts and the mathematical formulas [of quantum chemistry] are difficult. They do not make the relevant concepts understandable. That is, they are based more on mathematical reasoning than [non-mathematical, physical] arguments (2). While it is important for students to be able to solve mathematical problems in physical chemistry contexts, it is also important for instructors to ask students to explain their reasoning and provide opportunities to discuss approaches to problem solving or interpreting complex information. Instructors must clarify their goals for student learning and provide opportunities for students to practice as well as receive feedback on desired learning outcomes (45–47). Connecting visualizations (including diagrams and graphical representations) to mathematical models has been shown to result in more meaningful understanding of both the process being modeled and the mathematical techniques (48–50). Cooperative learning activities have been demonstrated to support students in more meaningful learning that integrates concepts rather than on rote learning strategies and algorithmic problem solving (15, 40). This also allows instructors to “attend to the meanings that students are actually constructing and adjust instruction appropriately” (47). The importance of engaging students in dialogue regarding their understanding is also important given evidence that students’ interpretations of the same mathematics lecture differed based on their perceptions of mathematics and the purpose of lecture (51). Strategies suggested to help students develop more appropriate reasoning skills include instructor modeling of complex causal reasoning in their lectures and coconstructing arguments with students, particularly contributing variables not considered by students, during whole class discussion to encourage more multicomponent reasoning (17). Scientists model physical systems by mapping measurements of interest into “mathematical symbols and expressing the physical-causal relations between the measured quantities in terms of mathematical operations between the symbols” (52). If students are to appreciate the ways in which mathematics is used to model chemical processes, instructors must do more to facilitate those connections and engage students in 183

meaningful dialogue as to what is being modeled and why specific mathematical techniques are useful.

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17. Moon, A.; Stanford, C.; Cole, R.; Towns, M. The Nature of Students’ Chemical Reasoning Employed in Scientific Argumentation in Physical Chemistry. Chem. Educ. Res. Pract. 2016, 17, 353–364. 18. Stanford, C.; Moon, A.; Towns, M.; Cole, R. Analysis of Instructor Facilitation Strategies and their Influences on Student Argumentation: A Case Study of a Process Oriented Guided Inquiry Learning Physical Chemistry Classroom. J. Chem. Educ. 2016, 93, 1501–1513. 19. Stanford, C.; Moon, A.; Towns, M.; Cole, R. The Impact of Guided Inquiry Materials on Student Representational Level Understanding of Thermodynamics. In Engaging Students in Physical Chemistry; American Chemical Society: Washington, DC, 2018; Vol. 1279, pp 141–168. 20. Yeatts, F. R.; Hundhausen, J. R. Calculus and Physics: Challenges at the Interface. Am J. Phys. 1992, 60, 716–721. 21. Roundy, D.; Weber, E.; Dray, T.; Bajracharya, R. R.; Dorko, A.; Smith, E. M.; Manogue, C. A. Experts’ Understanding of Partial Derivatives Using the Partial Derivative Machine. Phys. Rev. Spec. Top. - Phys. Educ. Res. 2015, 11, 020126. 22. Dray, T.; Gire, E.; Kustusch, M. B.; Manogue, C. A.; Roundy, D. Interpreting Derivatives. PRIMUS 2018, 1–32. 23. From classroom discourse collected in a Physical Chemistry course in Fall 2014. See references 16–19 for further details. 24. Shepherd, T.; Grushow, A. Thermodynamics & Statistical Mechanics: A Guided Inquiry. Unpublished Work, 2014. 25. Muniz, M. N.; Crickmore, C.; Kirsch, J.; Beck, J. P. Upper-Division Chemistry Students’ Navigation and Use of Quantum Chemical Models. Chem. Educ. Res. Pract. 2018, 19, 767–782. 26. Mack, M. R.; Towns, M. H. Faculty Beliefs about the Purposes for Teaching Undergraduate Physical Chemistry Courses. Chem. Educ. Res. Pract. 2016, 17, 80–99. 27. Christensen, W. M.; Meltzer, D. E.; Ogilvie, C. A. Student Ideas Regarding Entropy and the Second Law of Thermodynamics in an Introductory Physics Course. Am J. Phys. 2009, 77, 907–917. 28. Jones, S. R. What Education Research Related to Calculus Derivatives and Integrals Implies for Chemistry Instruction and Learning. In It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics; Towns, M., Bain, K., Rodriguez, J.-M. G., Ed.; American Chemical Society: Washington, DC, 2019; Vol. 1316, pp 187−212. 29. Reid, J. S. Maxwell’s Thermodynamic Relations. https://homepages.abdn.ac.uk/j.s.reid/ pages/Maxwell/Legacy/MaxRelations.html (accessed Oct. 15, 2018). 30. Roundy, D.; Bridget Kustusch, M.; Manogue, C. Name the Experiment! Interpreting Thermodynamic Derivatives as Thought Experiments. Am J. Phys. 2013, 82, 39–46. 31. Bajracharya, R. R.; Thompson, J. R. Analytical Derivation: An Epistemic Game for Solving Mathematically Based Physics Problems. Phys. Rev. Phys. Educ. Res. 2016, 12, 010124. 32. From classroom discourse collected in a Physical Chemistry course in Fall 2010. See references 14–19 for further details. 33. Spencer, J. N.; Moog, R. S.; Farrell, J. J. Physical Chemistry: Guided Inquiry Thermodynamics; Houghton Mifflin Company: Boston, 2004. 185

34. Jones, S. R.; Dorko, A. Students’ Understandings of Multivariate Integrals and How They May Be Generalized from Single Integral Conceptions. J. Math. Behav. 2015, 40, 154–170. 35. McGee, D. L.; Martinez-Planell, R. A Study of Semiotic Registers in the Development of the Definite Integral of Functions of Two and Three Variables. Int. J. Sci. Math. Educ. 2014, 12, 883–916. 36. Jones, S. R. The Prevalence of Area-Under-a-Curve and Anti-Derivative Conceptions over Riemann Sum-Based Conceptions in Students’ Explanations of Definite Integrals. Int. J. Math. Educ. Sci. Technol. 2015, 46, 721–736. 37. Jones, S. R. Understanding the Integral: Students’ Symbolic Forms. J. Math. Behav. 2013, 32, 122–141. 38. Rasmussen, C. L. New Directions in Differential Equations: A Framework for Interpreting Students’ Understandings and Difficulties. J. Math. Behav. 2001, 20, 55–87. 39. Moog, R. S.; Spencer, J. N. Process Oriented Guided Inquiry Learning (POGIL); ACS Publications: Washington, DC, 2008. 40. Towns, M.; Grant, E. R. ‘I Believe I Will Go Out of This Class Actually Knowing Something,’ Cooperative Learning Activities in Physical Chemistry. J. Res. Sci. Teach. 1997, 34, 819–835. 41. Czocher, J. A. How Can Emphasizing Mathematical Modeling Principles Benefit Students in a Traditionally Taught Differential Equations Course? J. Math. Behav. 2017, 45, 78–94. 42. Raychaudhuri, D. Dynamics of a Definition: A Framework to Analyse Student Construction of the Concept of Solution to a Differential Equation. Int. J. Math. Educ. Sci. Technol. 2008, 39, 161–177. 43. Zhu, G.; Singh, C. Improving Student Understanding of Addition of Angular Momentum in Quantum Mechanics. Phys. Rev. Spec. Top. - Phys. Educ. Res. 2013, 9, 010101. 44. Nicoll, G.; Francisco, J. S. An Investigation of the Factors Influencing Student Performance in Physical Chemistry. J. Chem. Educ. 2001, 78, 99. 45. Biggs, J. Aligning Teaching and Assessing to Course Objectives. Presented at Teaching and Learning in Higher Education: New Trends and Innovations Aveiro, Portugal, April 13–17, 2003. 46. Biggs, J. Constructive Alignment in University Teaching. HERDSA Rev. Higher Educ. 2014, 1, 5–22. 47. Thompson, P. W. In the Absence of Meaning… In Vital Directions for Mathematics Education Research; Leatham, K. R., Ed.; Springer: New York, 2013; pp 57–93. 48. Lang, P. L.; Towns, M. H. Visualization of Wavefunctions Using Mathematica. J. Chem. Educ. 1998, 75, 506. 49. Asiala, M.; Cottrill, J.; Dubinsky, E.; Schwingendorf, K. E. The Development of Students’ Graphical Understanding of the Derivative. J. Math. Behav. 1997, 16, 399–431. 50. Breidenbach, D.; Dubinsky, E.; Hawks, J.; Nichols, D. Development of the Process Conception of Function. Educ. Stud. Math. 1992, 23, 247–285. 51. Krupnik, V.; Fukawa-Connelly, T.; Weber, K. Students’ Epistemological Frames and Their Interpretation of Lectures in Advanced Mathematics. J. Math. Behav. 2018, 49, 174–183. 52. Redish, E. F.; Kuo, E. Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology. Sci. Educ. 2015, 24, 561–590.

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Chapter 12

What Education Research Related to Calculus Derivatives and Integrals Implies for Chemistry Instruction and Learning Steven R. Jones* Department of Mathematics Education, Brigham Young University, 167 TMCB, Provo, Utah 84602, United States *E-mail: [email protected]

This chapter discusses education research on the calculus concepts of derivatives and integrals in relation to chemistry topics. Using the topics of chemical kinetics and thermodynamics as examples, themes regarding derivatives and integrals that are explicitly or implicitly contained in these topics are identified. The themes discussed in this chapter are: (1) rate and covariation, (2) representations and layers, (3) differentials, (4) integrals, and (5) multivariation. Brief reviews of research concerning each of these five themes are provided with discussions of practical implications for chemistry instruction (or other science, technology, engineering, and mathematics [STEM] education fields) making use of derivatives or integrals.

Encouragingly, there has been a concerted effort to bring people and knowledge together from education circles across science, technology, engineering, and mathematics (STEM) disciplines. In this spirit of collaboration and knowledge-sharing, I present this chapter on how education research on understanding and using calculus derivatives and integrals provides insight into using them in the context of chemistry. The derivative is a major concept across STEM fields as a rate of change, and the integral is a major concept as the amount (or net change in amount) of a quantity present in a system. There is a rather large body of research related to derivatives and integrals that have implications for teaching courses. This chapter has the format of a research review focusing on a few interrelated topics, organized as follows. The reader is first acquainted with my background, and discussion topics to be used as examples for discussion are introduced. For these topics, themes are identified that are pertinent to derivatives and integrals with corresponding research bases. The education research literature is then reviewed including some practical implications for chemistry instructors (or other STEM educators) based on that research.

© 2019 American Chemical Society

Background To begin this chapter, let me be clear that I am a mathematics education researcher, not an expert in chemistry. My line of research has focused on undergraduate calculus education with an emphasis on how students apply their calculus knowledge to physics, engineering, and other scientific areas. While that has made me somewhat familiar with science topics, I was initially less informed about how undergraduate chemistry students would encounter the derivative and integral in their course of study. Guided by the knowledgeable editors of this book, my first objective was to identify topics within chemistry that could serve as examples of how these mathematical concepts are used. For the purposes of this chapter, I have decided to focus on two content areas: chemical kinetics and thermodynamics. These content areas were chosen because they contain the main themes of derivatives and integrals which are discussed in this chapter. However, these themes would relate to most derivatives and integrals throughout chemistry, such as quantum mechanics or other areas. In this first section, I use a short recap of chemical kinetics and thermodynamics to extract themes regarding derivatives and integrals that are explicitly or implicitly contained in them. This section is meant only to explain the themes. The remainder of the chapter then takes each theme, one by one, and discusses the research literature relevant to it. Derivative Themes that Show Up in the Topic of Chemical Kinetics Silberberg and Amateis define chemical kinetics as how fast reactants change into products (1). It involves the rate at which a concentration, [A], changes as time, t, changes. This rate can either be an average rate over a finite time interval Δt, as described by

or an instantaneous rate at a single point in time, expressed as the derivative

While chemical kinetics involves both derivatives and integrals (e.g., differentiated and integrated rate laws), at the level of undergraduate general chemistry discussion of this topic deals mostly with derivatives (outside of the brief mention that taking anti-derivatives can lead to the standard integrated rate law equations) (1, 2). Because of this, the focus is on derivatives within chemical kinetics, and integrals are discussed later in connection with thermodynamics. The first derivative theme in chemical kinetics is rate, which by definition is a central part of this topic. Of course, one can talk about rates without necessarily invoking derivatives, but instantaneous rates are all intrinsically derivatives. In fact, in many places in chemical kinetics where the word “rate” is used, a derivative could be inserted instead, such as rate = k[A] being reinterpreted as

Thinking of rates as derivatives allows for the formulation of the integrated rate laws. Rate is viewed as the quantification of covariation, which is how two quantities change with respect to each other. 188

For this theme, I draw on work done by Pat Thompson, Marilyn Carlson, and others to review what is known about covariational reasoning and developing strong images of rate. Note that I intend to use the term “image of rate” in the way Thompson used it when he stated, “By ‘image’ I mean much more than a mental picture” (4). He explained that in mathematical thinking, “image” can include internalizations of actions, primitive thought experiments, and relationships (4). The second theme discussed is what elements even make up the derivative concept. For example, chemical kinetics distinguishes between average rate of change over a time interval, the instantaneous rate of change at a point, and the different rates of change at different points. While I have claimed that instantaneous rates are inherently derivatives, average rates of change and changing rates of change are also connected to derivative understanding. Michelle Zandieh called these aspects different “layers” of the derivative concept, and her conceptual framework is used to discuss them (5). Within this theme, Zandieh’s framework also describes how these layers of the concept are manifested inside different representational contexts (5). Such representations of the derivative can be seen: (a) symbolically as a difference quotient, (b) graphically as a slope, or (c) “in words” as a ratio of changes (Figure 1).

Figure 1. Representations of rate including (a) symbolic, (b) graphical, and (c) in words.

Derivative and Integral Themes that Show Up in the Topic of Thermodynamics The second topic used as an example is thermodynamics, which is replete with derivatives and integrals (2, 3, 6, 7). Thermodynamics is the study of energy and related quantities. Thermodynamics can be treated algebraically without calculus, as in Silberberg and Amateis, but deeper explorations require derivatives and integrals (1, 6). The same themes about derivatives from chemical kinetics also hold true for thermodynamics, such as rates and representations. However, there are additional derivative and integral themes in thermodynamics, which is discussed in this chapter. The third theme of the chapter involves a key idea related to both derivatives and integrals in thermodynamics: differentials, or infinitesimally small amounts of quantities. For example, the equation

includes differential quantities for pressure (dP), temperature (dT), and volume (dV). To discuss this theme, historical context is shared about how differentials are often viewed within mathematics and contrasted with research about understanding and using them. The fourth theme is integration, a key part of working with and defining relationships between quantities in thermodynamics, as seen in 189

McQuarrie and Simon (2). Some integrals can be used to define a quantity, like

which defines work in terms of pressure and volume. Other integrals can give relationships between quantities, like

which gives a relationship between energy, heat capacity, and temperature for a certain context. I have been involved in a growing body of research on students’ understanding and usage of integrals, and I share results from that research. The fifth, and final, theme in this chapter is that many of the equations in chemical kinetics and thermodynamics involve more than two quantities, such as

which requires additional cognitive work. I discuss my work on multivariational reasoning and research on understanding partial derivatives.

Reviewing Research on Each of the Themes Having identified five themes discussed in this chapter, I now provide brief reviews of the research on each of the themes that are relevant to chemistry education (or other STEM education fields). Research implications for instruction are included for each theme. Theme 1: Covariation and Rate Experts can easily take ideas of covariation (two quantities changing in relation to each other) and rate (the quantification of covariation) for granted as fairly straightforward concepts. However, a large body of research demonstrates how much cognitive work goes into developing mature images of covariation and rate and reasoning about them (4, 8–13). Even undergraduate students can have difficulties with aspects of covariation and rate (13, 14). To start, covariation is generally thought of as how two quantities change simultaneously and interdependently (15). Here, a “quantity” refers to any measurable quality of an object or system (8), which can be “extensive” or “intensive.” Extensive quantities depend on the amount of substance (such as mass or volume), while intensive quantities do not depend on the amount of substance (such as density). Schwartz gave a rough (though not perfect) delineation explaining that extensive quantities are directly measurable, while intensive ones are not (16). As such, rates are essentially always intensive quantities because one cannot measure rates like speed or reaction rates directly.

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One must measure two other quantities (such as distance and time or concentration and time) and use mathematical operations to create the rate quantity. What goes into reasoning about simultaneous changes in two quantities? Thompson and Carlson claim that students go through a progression of reasoning levels that increase in sophistication (12). At the lowest level, students do not coordinate the two quantities but focus on only one quantity at a time. While it seems like secondary and tertiary students would be above such a level, Thompson described upper-level undergraduate and graduate college students who interpreted a rate as how fast a function changes “without interpreting the details of the expression as an amount of change in one quantity in relation to a change in another (4).” Thompson called this type of thinking “accruals as solitary objects” because the attention is only on changes in one quantity. In chemical kinetics, this could be akin to imagining a reaction rate as Δ[A] with time having a more subdued and implicit role. Or you canimagine time in isolation with the reaction rate essentially being just amount of reaction time (17). In thermodynamics, this could be akin to envisioning ΔP by itself, without thinking explicitly about ΔV. Students at times are less attentive to the “per [quantity]” part of rates. Covariational reasoning, in Thompson and Carlson’s framework, develops as one imagines simultaneous changes between the two quantities and coordinates between the quantities’ simultaneous values. In the highest type of covariational reasoning (continuous covariation) a person envisions these simultaneous values as continuously persistent at all points. Thompson and Carlson further distinguish between two types of continuous covariation, based on Castillo-Garsow’s work on “chunky” versus “smooth” covariation (18, 19). In one of Castillo-Garsow’s studies, a high school student was asked what it means to go 65 mph. The student responded, “That in one hour you’ve gone, you should have gone sixty-five miles” (18). The interviewer then asked if one can travel for 1 s at 65 mph, and the student answered, “No. You have to do… You would have to do, um… Well, yeah, you could” (18). Castillo-Garsow interpreted this answer to mean that the student conceptualized the rate only in terms of completed “chunks.” An hour must be completed for “65 mph” to make sense. Of course, she changed her answer, but Castillo-Garsow claimed that, “She then reconceptualize[d] that hour as being composed of smaller ‘one second’ chunks” (18). Thompson and Carlson allowed that in chunky continuous covariation, in-between values can “come along” by being part of the chunk, though “the person does not envision that the quantity has these values in the same way” as the endpoints of the completed chunks (12). Smooth continuous covariation, on the other hand, includes the explicit awareness and anticipation of the quantities’ values existing continuously as they change. The difference between chunky and smooth reasoning has ramifications for how students create and interpret graphs, an important part of representing scientific quantities. For example, Castillo-Garsow gave high school students a scenario in which a person starts with $320 in a savings account (with no interest) and contributes $55 to the account each month (18). The students drew graphs of the amount in account versus time. The two graphs in Figure 2 show two of the student responses. While the image on the left in Figure 2 shows a continuous graph, it suggests the student is thinking “chunkily” in terms of completed month chunks. The graph simply connects the dots between the completed chunks, which is incorrect for this scenario. The image on the right in Figure 2 shows a stronger image of “smooth” continuous thinking, as it suggests anticipation of the amounts in the savings account at all points as time increases.

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Figure 2. On the left, reasoning in completed chunks with less attention to in-between values, and, on the right, reasoning continuously with attention to in-between values. Reproduced with permission from reference (18). Copyright 2015 Castillo-Garsow. The idea of simultaneity was problematized by Johnson, who observed students attending to both quantities simultaneously but in a way that each quantity was considered independently (10, 11, 20). She gave secondary students a picture of four different bottles and, for each bottle, asked them to draw a graph that related height and volume of a liquid being poured into the bottle. One student, while creating a graph for a thin bottle, stated that the liquid would be “filling up the height faster but it’s not putting in as much volume” (20). Another student, while creating a graph for a wider bottle, stated “it’s getting more volume in it but not that high” (20). Johnson interpreted these statements as suggesting that students were reasoning about volume and height as independent quantities that both happened to be related to the filling of the bottle. While these two separate quantities changed simultaneously, “the simultaneous variation did not require that one quantity change with respect to changes in [the other] quantity. In contrast, it was as if each quantity was changing independently of the other quantity with respect to time” (20). In this interpretation, the reasoning even suggests an image of dual solitary accruals. In fact, some students can interpret rates as always in respect to time (even for non-time rates), like ∂P/∂V, because many rates include time (e.g., speed, acceleration, power, payrate). Unfortunately, this can lead “rate” and “with respect to time” to become synonymous, which was seen in a study of mine on non-time derivatives in real-world contexts (21). I observed that some students had a tendency to force derivatives to be with respect to time, even though they were with respect to a different variable. For example, one student was explaining the derivative dV/dr in the context of the volume of a cylinder (V) with radius (r) and constant height. The student stated, “Like say [r] is changing at a rate of one meter per second… [the volume] changes smaller at first, but then bigger” (21). In this case, the inclusion that the quantities r and V change over time was not problematic, except that the context did not stipulate whether r would be changing quickly or slowly over time. However, consider another student in that same study describing the meaning of the same derivative for the same context. Student: I don’t know [pause]. Alright, [dV/dr] means change in volume over change in radius, so [long pause]. The rate at which – there’s no time involved!... So, as we’re changing the radius, imagine the radius is time, because… you can’t do it without time… If we negate the middle-man and negate the change in radius, then we’d just have the change in volume as the time changes (21).

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In this case, forcing rates to include time was certainly problematic for this student, and seemed to lead to a “solitary accrual” of volume that was “simultaneous yet independent” to theradius. It may be that some students in thermodynamics are trying to interpret derivatives like ∂V/∂P, ∂V/∂T, or ∂S/∂P in this way. This thinking could mistake ∂S/∂P as how fast entropy changes as time goes on. However, this derivative deals with changes in entropy as pressure changes, regardless of whether pressure changes quickly or slowly over time, or whether it even changes at a constant rate over time. Overall, we can see that rate, as the quantification of covariation, is a challenging construct. Thompson even noted disagreement among experts about what constitutes a “rate” (8). He described how, within the same research book on students’ understanding of middle school mathematics, authors for different chapters used different meanings for rate, including: (a) a comparison of quantities of unlike nature (22), (b) a ratio between a quantity and a period of time (23), and (c) a relationship between one quantity and one unit of another quantity (24). If experts vary in their images and meanings, it is no wonder that students may struggle! Thompson suggested that to have a mature image of rate, rate must be reasoned about multiplicatively, or in terms of a constant ratio (4). In other words, rate involves proportional correspondence, which is the idea that “a/b-ths of one accumulation corresponds to a/b-ths of the other accumulation” (4). As an example, suppose a reaction rate is 0.001 mol/L·s. A mature image of rate would include imagining that as time increases continuously, concentration simultaneously increases by exactly 0.001 times the change in time. It is more than simply being able to solve for a change in concentration given, say, a 3-s time change. Rather, it is the idea that changes in time and concentration are multiplicatively linked by a constant factor of 0.001. One should imagine time as continuously evolving and that each instantaneous increase in time brings with it an associated increase in concentration;each time value has an associated concentration value. Time should be an explicit quantity in this multiplicative relationship, rather than simply being what rates are always “with respect to.” While some experts might think that rates should not be an issue for their students, research suggests that even college students do not have a fully mature image of rate. Byerley and Thompson conducted a study where they gave college graduates with STEM training the following question: “Every second, Julie travels j meters on her bike and Stewart travels s meters by walking, where j > s. In any given amount of time, how will the distance covered by Julie compare with the distance covered by Stewart (25)?” Options provided included j – s, j·s, and j/s. Only 30 out of 100 respondents chose the correct j/s option. Interestingly, 47 out of 100 respondents chose j – s, which is indicative of an “additive” accrual for rates, instead of a “multiplicative” accrual. College students may also struggle with more basic concepts. Byerley et al. found that some STEM college students held unproductive meanings for division that could impede their image of rate (14). In their study, they noted that some calculus students, while certainly able to compute division, tended to reason about division from a difference (i.e. “additive”) perspective, rather than a proportional (i.e. “multiplicative”) perspective. This view of division led some students to think of rate in terms of adding on a constant amount, rather than as a multiplicative factor or proportion between two quantities. For example, one student, when discussing the growth rate of an animal, could only talk about that rate over unit-sized time intervals by describing the amount of added height over each unit time. When time intervals were not uniform, he struggled to show that it still corresponded to a constant rate. Another student, who also thought of division additively, had difficulties explaining slope, saying, “I don’t really see [slope] as division… I see that there is division but when I think of it in terms of slope I don’t, I don’t see that” (14). These results suggest that we cannot take for granted the level of understanding our students have, or the level of reasoning they can employ, about rate. 193

What Research on Rate Implies for Instruction In helping our students develop mature images of rates, it would beneficial to focus on helping students articulate rates more carefully. For example, instructors and students alike may inadvertently refer to rates as solitary accruals, even if the person’s own image of rate includes time explicitly. In chemical kinetics, it could be easy to refer to a reaction rate as “how fast reactants change into products” without explicit attention to time, because the “action” happens with the change in concentration. If students use such language, instructors could help the students better conceptualize the idea by asking them to consider what is missing in such a description. The language could be clarified to “how fast reactants change into products as time changes.” Of course, time is implicitly contained in “how fast,” but because time is so experiential, students might not always think about it directly (8). Requiring explicitness can help students attend to both quantities in their images. It can also help prevent derivatives such as ∂P/∂V from being viewed as solitary accruals with P and V changing simultaneously but independently. Similarly, because “fast” and “slow” are closely connected to our intuitive experience with changes in time, it pays to be careful how or when such adverbs are used. They might make students think of all rates, even non-time rates (like ∂P/∂V), as being time-dependent. If students did so for this derivative, they would try to separately and independently coordinate pressure and volume with respect to time, without coordinating them in direct relation to each other. Johnson argued that such reasoning inhibits students from thinking of derivatives as true rates (11). Within classroom activities on these topics, tasks should require students to directly coordinate how, for example, pressure and volume change in direct relation to each other without time as a required mediator. Finally, we should evaluate whether we, and our students, tend to talk only in “chunks,” or if we are attending to smooth continuous change. This could take the form of asking students to imagine what is happening with concentration as time continuously increases from one point to another. Students would have to deal with the in-between values and not simply the endpoint of certain intervals. Similarly, in thermodynamics, rather than give only problem situations where volume has changed by chunks, it is conceptually useful to ask students to describe what is happening with pressure as volume continuously increases from one size to another. Rodriguez et al. suggested imagining a graph as representing the “unfolding story” of covariation between two variables (26). Doing so can help students to see the persistent and simultaneous existence of both quantities where each point on the graph represents an event within that overall story. Theme 2: Layers and Representations that Make Up the Derivative Concept This subsection discusses the elemental parts of the derivative concept using Zandieh’s conceptual framework of the derivative (5). To give some background, Zandieh’s framework came from a need to articulate what the derivative concept even consists of. Student understanding of the “derivative” had been discussed in calculus education research, but different possible interpretations of what counts as “understanding the derivative” are based on what a person believes constitutes the derivative in the first place. The framework had its genesis in expert conceptual analysis about the parts that make up the “derivative,” and Zandieh used student data to show how her framework could be used to portray a particular student’s understanding or to contrast between students. The framework is helpful because it makes explicit some aspects of the concept that might otherwise be overlooked.

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Zandieh’s framework consists of two main parts. The first describes the structure of the derivative by breaking the concept into three layers: the ratio layer, the limit layer, and the function layer (5). Each layer consists of a process (i.e. an imagined mathematical action) that is then reified (27) into a conceptual object. This newly created conceptual object is then used in the process of the next layer. Zandieh defined the ratio layer as a process that “takes two objects… and acts by division,” which is reified into a ratio object (5). As an example, the average reaction rate in chemical kinetics matches the ratio layer, where a change in concentration is divided by a change in time producing a ratio of changes. The limit layer is defined as the “limiting process” of this ratio over smaller and smaller intervals. This process is reified into the limit value at that point, which is also the derivative value at a point. In the chemical kinetics example, this layer would be equivalent to the instantaneous reaction rate. The function layer is defined as imagining the limiting process occurring at every point in the domain to create a limit/derivative value at each point. This process is reified into the derivative function. In chemical kinetics, this would be imagining that there is an instantaneous reaction rate at every point in time. The second part of Zandieh’s framework describes how this common structure can be seen in different representational contexts including graphical, symbolic, verbal/rate, and physical contexts. In the graphical context, the ratio layer consists of the “rise over run” of a secant line (i.e. a line crossing through two points of the graph) close to the point of interest, which is reified into the “slope.” The limit layer consists of imagining the two points of the secant line coming closer together and converging to a single point of interest, which is reified into a tangent line whose slope is the derivative at that point (an example tangent line is shown in Figure 1b). The function layer consists of imagining tangent lines at every point on the graph, each with its own slope. In the symbolic context, the ratio layer is the difference quotient,

the limit layer is the limit of this symbolic expression as x → a, and the function layer is the repeatability of this limit for any a-value in the domain. In the verbal/rate context, the ratio layer is the description of a change in one quantity divided by a change in the other, the limit layer is the description of the instantaneous rate of change at a point, and the function layer is the description of an instantaneous rate being applicable at every point in the domain. Examples of symbolic, graphical, and verbal/rate contexts are shown in Figure 1. For the physical context, Zandieh used the paradigmatic velocity example, but Roundy et al. further developed this context through the idea of scientific measurement (28). The ratio layer consists of measuring two quantities at different points to determine the ratio of their differences, the limit layer consists of doing this measurement with two points very close to each other, and the function layer consists of the “tedious repetition” of this measurement process to measure the ratio of differences at different points. Taken altogether, the layers and contexts form a matrix (Figure 3, left). If a student is interviewed or assessed in terms of their understanding, a filled-in dot is placed in any cell they show evidence of having within their understanding (Figure 3, right). Of course, it is possible for students to use an object for a layer where they do not understand the process that led to that object from the previous layer. For example, a student might say the derivative is the “slope of the tangent line” without knowing where that tangent line comes from. In this case, such an object is referred to as a pseudoobject (27) and is denoted with a hollow dot. 195

Figure 3. The derivative matrix with an example of it being filled in by a student.

In my view, the ratio layer provides the basic meaning for derivatives in science as the change in one quantity per change in the other quantity. The ratio and limit layers represent the average rate of change and instantaneous rate of change, respectively. The limit and function layers then distinguish between the rate at a point and the changing rate function. It could seem like once a student understands the limit layer at a point, they should be able to easily extend it to the idea of a derivative value existing at every point in the domain. However, research has suggested that the derivative at a point and the derivative function are separate concepts each needing development (29, 30). Further, the derivative function involves changing rates of change, which require higher-order reasoning (13). Park observed that while students had ideas about both a derivative at a point and the derivative function, they did not have a complete grasp about the connections between them (29). She suggested that this may be because mathematics instructors often begin by defining the derivative at a point (30). This approach uses secant lines converging to a tangent line at one point (x = a) to create the derivative expression

at that point (note that [f(a + h) − f(a)]/h is equivalent to [f(x) − f(a)]/(x − a) by defining h = x − a). Calculus instructors then tend to simply change a to be a generic x-value,

to go from a derivative at a point to the derivative function. Park stated that the instructors assumed that “the transition between the point-specific and interval views of the derivative are clear to their students,” even though they were not (30). Equivalent jumps can be seen in presentations of chemical kinetics or thermodynamics. For example, Silberberg and Amateis introduce chemical kinetics by first defining average rate, instantaneous rate, and initial rate, each at a point (1). From there, they immediately show graphs of concentrations and discuss how different rates compare with each other across time. This discussion requires thinking about how rates change over time. Thus, the rate (i.e. derivative) is defined at a point, and an assumption is made that students can easily transition to having a well-understood notion of rates at all points (i.e. derivative function). Further, Seethaler et al. noted that chemistry textbooks often do not explicitly depict rate versus time graphs, instead opting for concentration versus time graphs, making changing rates implicit to the concentration graphs (31). In fact, changing rates of change is an incredibly complicated idea that needs time to develop. In an earlier framework on covariational reasoning, Carlson et al. put reasoning about changing rates of change at the highest level (13). In their study, they interviewed a group of 20 second-semester 196

calculus students about their understanding of function and how function values change. In one task, the students were shown an image of a bottle with a large spherical bottom and a thinner tube at the top. The students were told that it was being filled with water and were asked to “sketch a graph of the height as a function of the amount of water that’s in the bottle” (13). They found that only 2 of the 20 students produced an acceptable graph for this bottle problem, with many problems involving the difficulty conceptualizing the changing rate of volume increase. In another task, students were shown a graph that represented the rate of change of temperature over an 8-h period of time. The graph included increasing portions, decreasing portions, positive portions, and negative portions. The students were then asked: “Given the graph of the rate of change of the temperature over an 8-h period, construct a rough sketch of the graph of the temperature over the 8-h time period. Assume the temperature at time t = 0 is 0 °C” (13). Only 4 of the 20 second-semester calculus students produced an acceptable graph for this problem. Similarly, the issues often involved difficulty in thinking about how different rates corresponded to different types of increases or decreases in the temperature. These results show that students can struggle to reason about changing rates of change, and that assumptions should not be made that understanding rate at a point guarantees fluency in reasoning about changing rates. In one of my studies, I also observed students at the end of first-semester calculus struggle to use changing rates of change in real-world contexts (32). Several students confused changing rates of change with the actual function values at those points. That is, increasing rates were, at times, confused with increasing function values, and inflection points were sometimes confused with the maxima or minima in the function values. Students also, at times, relied on graphs-as-pictures (33) representations without knowing how to connect them to changing quantities. The results of Carlson et al. (13) and Jones (32) suggest that even mid-level undergraduate students can struggle quite a bit with the idea of changing rates of change, and that their mathematics courses may not have prepared them to deal with them in science courses. This subsection on representations and layers concludes by singling out the representation of the derivative as a slope, because it is so commonly used across mathematics and science (1, 34–36). Experts often have strong conceptual connections between slope and rate of change. However, research has shown that many undergraduate students have more naïve notions of slope (37), and can have separate conceptions for slope and rate of change that are more loosely connected than that of an expert (38). Specifically, Nagle et al. observed differences between how students and instructors conceptualized “slope” (37). The two most common interpretations by students were (a) that slope indicated behavior (i.e., a line was increasing, decreasing, or flat) and (b) that slope was simply the “m” in the equation y = mx + b. By contrast, the most common instructor interpretations included (a) slope as a geometric ratio, or the vertical versus horizontal displacement on a graph and (b) slope as the rate of change of a function. While it is true that about half of the students in that study also showed evidence of thinking of slope as a geometric ratio, these results indicate an overall disparity between the invoked understanding of “slope” for students and instructors. Bingolbali and colleagues demonstrated that students with slanted-line-focused meanings were not always able to answer corresponding questions about rate of change (38, 39). Christensen and Thompson also observed that a higher percentage of their physics students answered pure “slope” questions correctly than corresponding “derivative value” questions (40). These studies suggest that it cannot be taken for granted whether a student who understands slope also understands the corresponding rate of change idea, or vice versa.

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What Research on Representations and Layers Implies for Instruction Zandieh’s framework applies readily to derivatives encountered in chemical kinetics and thermodynamics. Graphical, symbolic, rate, and physical contexts are used in these topics, which also distinguish between average and instantaneous rates of change. This seems to require that a student would need to have filled-in dots across the entire first and second rows of the derivative matrix (Figure 3) to fully understand these presentations. However, research that has used Zandieh’s matrix show that many students’ understanding does not tend to be even close to that advanced (5, 41–43). A related issue is the fact that expert understanding is typically highly “packed” (44, 45), or bundled in such a way that one part of knowledge immediately connects with other parts of knowledge. As such, an expert might view the slope of the tangent line, the instantaneous rate of change, and the expression

as equivalent. Experts might also understand that the limit is “obviously” the limit of a ratio or that if a derivative can be found at one point, it can “clearly” be found at any point. Breaking the derivative down into its layers and contexts can help an expert unpack their knowledge, allowing them to see more closely a student’s perspective of the elemental parts that come together to build the overall concept. Unpacking knowledge can give instructors insight into mental processes students need to undergo to construct their own stronger understandings. Unpacking knowledge can also aid in making sure instructors and students are on the same conceptual page. For example, Silberberg and Amateis introduce chemical kinetics squarely through the verbal/rate-ratio cell in the matrix by focusing on changes in quantities over finite periods of time, Δ[A]/Δt (1). However, when they come to define rate, they state, “the average rate is the slope of the line joining two points along the curve” (1). This interpretation is squarely in the graphical-ratio cell of the table. An expert easily sees verbal/rate and slope as equivalent, but novice students might not. The studies by Bingolbali and colleagues (38, 39) demonstrate that having an understanding of slope does not guarantee an equivalent understanding of rate of change. Because of the major emphasis on graphical representations in typical calculus courses (34, 46), undergraduate chemistry students likely come into classrooms without fully developed connections between slope and rate. To be clear, this certainly does not mean that instructors should be constrained to only one representational context to simplify the cognitive load because we know that multiple representations are, in fact, beneficial to learning (47). Doing so would reduce the conceptual gains possible for students. However, it does mean possible inconsistency, from a student’s perspective, in terms of what concepts mean. It becomes critical for instructors to keep in mind where students are in Zandieh’s matrix, because this can give insight into the students’ background, help identify how students are talking about rates and derivatives, assist in task creation that target areas of weakness within Zandieh’s matrix, and support the design of activities that foster connections. Instructors should also be explicit about movement between representational contexts, or connections between them, to help students develop the expertise that comes with seeing the various representations as equivalent. For example, if chemical kinetics is introduced through the verbal/rate context, an instructor could ask students to consider how change in concentration per change in time can be graphically represented as the slope, and how rate and slope definitions are equivalent. Doing so can bolster connections across the representations, which helps to develop that expertise. 198

Finally, instructors should be aware that there is a difference between the ratio, limit, and function aspects of the derivative. For example, while Silberberg and Amateis focus mostly on rate as a division between finite changes in concentration and time,

McQuarrie and Simon immediately discuss rates using instantaneous derivatives,

The difference between these foci is whether the rate is seen at the ratio level (as in Silberberg and Amateis) or at the limit level (as in McQuarrie and Simon). There is certainly nothing wrong with either level, but it is important for instructors to keep in mind where we are at and where we want our students to be. For a complete understanding, it is important to make sure we include ratio, limit, and function layers in learning. An approach by Weber et al. provides an idea for how such a transition can be done (48). The authors suggest focusing on rate of change and graphical slope simultaneously by drawing not just a secant or tangent line whose slope is the derivative, but imagining a small right triangle, like in the image on the left of Figure 4, which they call a “calculus triangle.” The triangle’s hypotenuse coincides with the secant or tangent line and its legs represent the changes in x and y. With this triangle, students can attend to both the graphical slope interpretation and the rate of change interpretation at the same time. The figure also displays changes in both x and y simultaneously, perhaps helping with the “solitary accrual” or “simultaneous yet independent” issues. This lone triangle is only the rate at a point, but Weber et al. then discuss imagining “that a calculus triangle exists at every point on a function’s graph and that these triangles can be as small as one desires” (48). The image on the right in Figure 4 shows such triangles “at every point,” which can begin to convey the fact that rates exist at all points. The differences in “heights” of the various triangles can create discussions about how rates change. This in turn can scaffold the notion of a rate function, which is necessary to understand the key idea of changing rates of change.

Figure 4. An example of a calculus triangle (left) and calculus triangles existing at every point (right). As changing rates of change come to the forefront of class discussions, additional activities can help bolster students’ reasoning. For example, Seethaler et al. recommend asking students to predict how reaction kinetics graphs will look by having them consider whether the concentration versus time graph should decrease linearly, decrease rapidly at first and then level off, or decrease slowly at first and then drop off quickly (31). This makes rates explicit and makes changing rates a focal point. Seethaler et al. also suggest asking the students to describe what these graph “shapes” mean to the progress of the reaction. Such questions can even lead to students creating rate-versus-time 199

graphs or rate-versus-concentration graphs. These activities drive changing rates to the forefront of class discussion, allowing students to develop their reasoning about this challenging idea. Theme 3: Differentials The next theme describes the tension in the mathematical community that exists in using differentials (e.g., dt, dP, dT) as infinitesimal quantities. It is important for STEM educators to know what messages their students receive about differentials from their mathematics classes. Being aware can help instructors transition their students to using differentials, as needed, in their science classes. Historically, as Fermat, Cavalieri, Leibniz, and Newton developed the body of mathematics now known as calculus, there was great debate and disagreement about how to handle this idea of “infinitesimally small” quantities. While these four mathematicians used these quantities to good effect, critics abounded, such as the well-known mathematician George Berkeley who asked: “And what are these same evanescent [i.e. infinitesimal] Increments? They are neither finite Quantities, nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?” (as quoted by Kleiner (49)) The next 100 years in mathematics saw great minds attempting to put calculus on axiomatic, rather than conceptual, footing, resulting in the (unfortunate, in my opinion) disposal of differentials as legitimate entities. Mathematics as a university discipline was also beginning to separate from its natural science counterparts, meaning that mathematicians focused calculus instruction on the axiomatic underpinnings and discarded differentials from the curriculum (50). Despite an attempt by mathematician Abraham Robinson in the mid-20th century to bring infinitesimals back through its own rigorous development (49), the movement went largely unheeded by the general mathematics community (50). To this day, many mathematics instructors are reluctant to use the idea of infinitesimals, even if they “privately” think of them (51). Instead, mathematicians often insist that notations such as “dx” or “dt” are meaningless in their own right. In fact, the commonly used calculus text by Stewart explicitly states, “The symbol dy/dx, which was introduced by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for f′(x) (34).” Unfortunately, this tradition runs in direct opposition to how many scientists use infinitesimals in their work (52). Infinitesimals show up regularly in discussions in thermodynamics (2, 6, 7). Because of this mismatch, students may think of differentials purely as algebraic objects (53). Moreover, because of their undeveloped intuitions about differentials, some students perceive them as simply meaning a calculated derivative of some function, such as dt meaning the derivative of some function “t,” or dV meaning the derivative of some function “V” (54). These interpretations cause difficulties for students because making sense of differentials as infinitesimal quantities is key for understanding integral equations (55–57), setting up integral equations (58), and interpreting equations governing scientific contexts (53). A few calls have been made for infinitesimals to be included back into typical calculus discussions (52), but traditional curriculum certainly can take significant time and energy to change. Thus, STEM instructors should expect their students to continue to receive such counterproductive messages from their mathematics courses about differentials not representing infinitesimal quantities. What Research on Differentials Implies for Instruction While some readersmight be aware that many mathematicians avoid discussing differentials as infinitesimal quantities, it may be surprising news to others. I hope that being aware of such 200

issues helps STEM instructors to transition their students from pure mathematics courses to science courses. It is useful to know that students leave their calculus courses with superficial meanings for symbols like dt or dV. For example, consider the total differential for the Gibbs function,

relating Gibbs free energy, entropy, temperature, volume, and pressure. A chemistry expert might interpret this equation in terms of the relationships between small changes in temperature, pressure, and Gibbs free energy. However, students may think of this as computing separate derivatives for algebraic functions G, T, and P with the algebraic derivative functions being equal. Or, some students may simply think something is wrong with this equation because these are isolated differential symbols and differentials should only exist as attachments to an integral symbol such as ∫f(G)dG. If one’s class makes use of differentials, it is critical in this case to explicitly give meaning to the differential by explaining that it is a tiny amount of that quantity. I like to tell my own students that we (roughly) use “delta,” as in ΔT, when we are talking about relatively larger-scale changes, and “d”, as in dT, when we are talking about very small changes. Of course, I do not mean to imply that ΔT cannot be used for very small changes in temperature, only that ΔT and dT have essentially the same meaning and we use them to distinguish the scale we are talking about. It may be required to explicitly state that a differential such as dP does not necessarily mean a derivative of some algebraic P function. It is true that there are important relationships between differentials and derivatives, but I have at times seen my own students in second- and third-semester calculus classes so hung up on the derivative interpretation of differentials that they have difficulty viewing it as anything else. Theme 4: Integrals This subsection discusses what research on integrals implies about integrals as used in chemistry. My own research has focused on how students understand integrals and how they use them in both pure mathematics and applied science contexts. While nuances certainly exist from one student’s conception to another, my research has documented three main conceptualizations students have of integrals. The first conception, perimeter and area, aligns with the common “area under the curve” interpretation of integrals (56). The word perimeter here means “physical boundary” and not a “measure of length.” In this conception, the integral expression refers to a shape in the x-y plane bounded on top by the graph of f, on the two sides by vertical lines a and b, and on the bottom by the x-axis. The integral symbol refers to the area contained within this shape. The second conception, function matching, aligns with the “anti-derivative” meaning for integrals (56). In this conception, f(x) is the derivative of some other “original” function, and the point of the integral is to identify this original “pre-derivative” function by undoing the derivative operations. The differential simply refers to the variable with respect to which the derivative/anti-derivative is taken. Students sometimes even refer to dx as a “the derivative of x,” and the integral “is like the reverse version of the derivative” (54). This interpretation is in line with students’ anemic notion of what a differential element is. The limits of integration are simply inputs into the anti-derivative function. The third conception, adding up pieces, is similar in structure to the Riemann sum process (56). Here, the integral is viewed as the (infinite) summation over (infinitesimally-sized) pieces of the domain. In other words, [a,b] is broken into infinitesimal pieces, dx, and the product f(x)·dx is found within each piece. The integral then sums up the results of the product over all of the pieces between 201

[a,b]. While the mathematical definition requires a sequence of finite summations ( for every n, whose limit as n→∞ is taken), the adding up pieces conception imagines those pieces as already being infinitesimal in size. Thus, the limit is implicit, but the structure contained within this conception matches the structure of the Riemann sum. In other places, I have also named this conception the multiplicatively-based summation (MBS) conception to highlight the multiplication between f(x) and dx (55). However, others have shown the broader use of thinking of adding up pieces that might not have products inherent in them, such as

or

Consequently, I have gone back to using the original adding up pieces name. Several studies have pointed to the importance of the adding up pieces conception compared to the other two conceptions, especially in scientific contexts (55, 58, 59, 61–63). It is needed to both interpret written integral expressions as encountered by students (55, 59) and to set up integral expressions (57, 58). Because of its importance, one might hope that calculus instruction would develop this understanding of integrals in students. Unfortunately, the exact opposite is true. In a survey of a large sample of students, a large majority showed evidence of understanding integrals only through areas under curves and anti-derivatives (54). Fewer students invoked any kind of summation conception for integrals, even for science-based integrals like the integral of pressure over surface area,

Why do calculus instructors not develop this type of conception? For two reasons: (1) the adding up pieces conception contains the idea of infinitesimally-sized differentials (which some mathematicians dislike), as discussed in the previous subsection; and (2) well-meaning instructors sometimes undermine Riemann-sum-based conceptions of integrals by explicitly stating that Riemann sums become obsolete in the face of easier anti-derivative techniques for calculating integrals (64). Of course, anti-derivatives are an easier method for computing derivatives, but the adding up pieces conception is often crucial for seeing how integrals apply to a given situation, or how integral expressions are created. Having stated the importance of the adding up pieces conception, it is certainly true that many sciences make use of the “area-under-the-curve” interpretation. For example, physicists use it to represent the size of an impulse (35), economists use it to represent a consumer surplus (36), engineers use it to represent the total force from a distribution of weight (65), and chemists use it to represent the work done in compressing a gas from an external pressure (2). Thus, it might be tempting to view the area-under-the-curve conception as the important conception for integration. However, my research has indicated that unless a student has a well-developed understanding of the adding up pieces interpretation, they are left wondering why the area under the curve would yield such a result (55). For example, one interview question I gave students involved engineers testing a 202

motor that was being run at varying levels of rpms (denoted “R” rev/min) over a 10-h period. The students were asked the meaning of the integral

The students tended to draw an arbitrary graph of R, shaded in the region underneath it between t = 0 and t = 600, and stated that the integral was the shaded-in area. This response clearly shows use of the perimeter and area conception, but, when pushed to explain what that area represented, the students were often at a loss. For example, one student working on this task pointed to the area he had drawn and stated, “What would that signify? Like, what does this area represent?” (55). Progress was not made until the students in my studyinvoked the adding up pieces conception (55). For example, on the rpm task, one pair of students was stuck after drawing the area under the curve. One of them, though, began to think of tiny time segments and what happened over each segment: Student 1: [Draws a thin rectangle under the graph] So this would be just dt [writes “dt” along the bottom of the rectangle], it would be the number of revolutions for this dt [writes “# revs” next to the rectangle]. And when you integrate this, it adds up each component. Interviewer: So why does that little rectangle, why does that end up counting up the number of revolutions for that piece? Student 2: Yeah, so like [to Student 1] good call. We have R, which is revolutions per minute. So that means this side is revolutions per minute [traces finger down the height of the rectangle]. So we’re going to multiply the length times width [writes “L·w” under the graph]. Length would be revolutions per minute [writes “rev/min”]. And then our width is dt and that’s just saying dt is essentially just going to be minutes, the time value [writes “rev/min·min”]. So the area of that rectangle would just be revolutions [cancels out both “min”] (54). Such explanations helped the students make sense of several integrals, including:

and

In fact, many thermodynamics integrals require summation perspectives, such as

where “1” and “2” correspond to initial and final states (2). As written, what could this integral possibly mean in an area conception? This interpretation would view the integrand function as a constant f(U) = 1, with the integral being the area under the horizontal line U = 1, in between the limits of integration (ostensibly U = 1 and U = 2). On a related note, Thompson et al. noted a student 203

thatremarked about the integral of velocity, “I don’t understand how a distance can be an area” (66). Alternatively, what would this integral mean in the anti-derivative conception? The absence of an “integrand function” would mean that f(U) = 1, making its only possible anti-derivative function f(U) = U. The apparent limits suggest this integral always has the value of . The adding up pieces conception offers the only sensible interpretation of this integral expression as the summation of many small amounts of energy that accumulate into the total change in energy, ΔU. What Research on Integrals Implies for Instruction What does this research mean for instructors using integration in their classes? First, if the integrals require no more than simple anti-derivatives, as in the presentation of chemical kinetics in Silberberg and Amateis (1), then the students will not face any major difficulty as students already see integrals as anti-derivatives. Second, if the area under the curve is discussed, as in the familiar Figure 5 from thermodynamics, students would see a connection between area and integrals. However, if we want the students to understand why integration can be used in these places, like in Figure 5, it is critical for them to see how infinitesimal quantities are being added. Explaining why the area in Figure 5 represents work requires an additional understanding of tiny changes in volume together with the pressure for that tiny change. In another thermodynamics example, Silberberg and Amateis describe quantifying entropy using changes in heat (1). They describe making infinitesimal changes in pressure through the analogy of removing single grains of sand at a time. These lead to infinitesimal changes in heat. They state, “If we continue this nearly reversible expansion to 2L and use calculus to integrate the tiny increments of heat together, qrev is 1718 J” (1). If students only have the antiderivative or area conceptions of integration, this key explanation of how calculus is used would be incomprehensible. Where is the anti-derivative or the area in that statement? The justification of this process lies squarely in the interpretation of integration as adding up pieces. Of course, this means that the instructor would likely have to take time to help their students think of integrals as fundamentally being about adding up infinitesimal quantities because few students would naturally think of them that way (54).

Figure 5. Work represented as the “area under the pressure versus volume curve” in thermodynamics. As instructors, we should not be satisfied with our students having only an area conception of integrals. Instead, students should be acquainted with the idea of accumulations of tiny amounts over “infinitesimal” pieces. I wish I could say we do an adequate job of this within standard mathematics courses, but that is simply not the case. A typical calculus class discusses Riemann sums, but as I observed in one of my studies, calculus instructors may directly undermine the Riemann sum by casting it as obsolete and unnecessary (64). In my study, one student described Riemann sums as, “I’m sure, back in the day, before they had [anti-derivatives], that Riemann sums were probably big. 204

And that would just take a lot of time” (64). It therefore falls to our STEM education colleagues to further develop the idea of integrals as summations . Doing so would consist of helping students see integrals through a lens of partitioning, evaluating within each piece, and adding/accumulating across all pieces. This three-part understanding can help students make sense of integrals in a wide range of contexts (55, 58–60). Theme 5: Multivariation and Partial Derivatives The final theme focuses on how many equations within chemical kinetics and thermodynamics involve more than two quantities and that reasoning with multiple quantities can entail greater cognitive work. The equation

involves two concentrations [A] and [B] and the intensive quantity of reaction rate. The equation

involves work, mass, and distance, and the equation

involves four quantities: pressure, volume, number of moles, and temperature. Because the term covariation has a history of being applied to contexts involving only two quantities, I have used the term multivariation for situations where more than two quantities could be changing (67). While I have identified four distinct multivariation structures (67), only three are described here because they are the most pertinent to the chemistry topics I am focusing on. The first, independent multivariation, is essentially when all but two quantities can be held constant. The rate law equation for initial quantities satisfies this structure because (for example) the initial concentration [B] could be held constant while the initial concentration [A] is changed. If initial [B] is constant, then initial [A] and the initial reaction rate can covary independent of initial [B]. In the second type of multivariation, dependent multivariation, it is not possible to hold all but two quantities constant. A change in one quantity induces simultaneous changes in other quantities. For example, the thermodynamic quantities of energy, temperature, entropy, pressure, and volume do not have enough degrees of freedom in a system to keep all but two of these quantities constant (68). The third type of multivariation, nested multivariation, deals with the function composition structure, z(y(x)), where a change in x induces a change in y, which in turn induces a change in z. This type of multivariation also shows up in reaction rates where there is a rate inside of a rate, such as mol/L·s. In this rate example, if one adds a certain amount of a chemical to a solution, they increase the number of moles. This increase in moles then increases the concentration, mol/L, which in turn can increase the reaction rate, mol/L·s. Having multiple quantities in equations also complicates the derivatives. Basic derivatives are applied to functions of only one variable, like f(x), P(V) or [A](t). Equations with multiple quantities inherently require partial derivatives, or derivatives applied to changes in two specific quantities while holding other quantities constant, like (∂S/∂U)V,N. However, students may struggle to use 205

the idea of “holding constant” appropriately (7, 69, 70). Bucy et al. observed that some students may think that if a quantity is held constant for a particular partial derivative, then that quantity should always be constant throughout the entire context (69). Additionally, for contexts involving dependent multivariation, like thermodynamics, Roundy et al. stated that, “students are taught that when taking partial derivatives, all the independent variables are held fixed (68). Nevertheless, we have found that most students come into our course with a firm belief that when taking a partial derivative everything else is held fixed.” This tendency is problematic for thermodynamics because, in certain cases, one can make choices to keep certain (but not all) variables constant. These studies imply that, while students may be able to algebraically compute partial derivatives, they do not always have a developed sense of how to think about partial derivatives. What Research on Multivariation and Partial Derivatives Implies for Instruction Realizing that not all multivariable equations have the same relationship structures is important because we need to know that different contexts require different types of reasoning in order to understand these equations. What can be done to help students become aware of the specific structures in a given context? One activity that appears to be successful with students is called “name the experiment” (7, 70). In this activity, partial derivatives, like (∂V/∂P)T or (∂U/∂T)V, are given to students, and the students are asked to outline a possible experiment that could calculate (or approximate) the value of that derivative. Roundy et al. state that this activity helps students to develop an appreciation for “the meaning and importance of the quantity that is held fixed” (70). As a student tries to grapple with how to keep, say, temperature constant while changing the pressure by a small amount and measuring the corresponding change in volume, they develop a stronger sense of the relationships between them. Such activities can also explore what it means for one quantity to be held constant for a particular measurement, even when it is not required to always remain constant. Even for simpler derivatives, like d[A]/dt, it could be beneficial to ask students how such a derivative could be empirically measured in a laboratory. Such questions can cement the covariational nature between t and [A], and help them think about the concentration of A as an intensive quantity (mol/ L) rather than thinking only of the amount of A. In another study, Thompson et al. claimed that using multiple representations can help students make sense of the relationships between quantities (7). For example, in trying to figure out how multiple quantities relate to each other, the authors suggest creating “derivative tree” diagrams (as shown in Figure 6). In this diagram, the student would draw connections between each quantity that directly affects another quantity, thereby seeing how a change in one would relate to changes in others. The authors also suggest using graphical representations of partial derivatives to let students see how making changes in one quantity while holding others constant plays out. The image on the left in Figure 7 shows the how changing T or P, while holding the other constant, can each affect volume. The middle image in Figure 7 shows how that rate of change changes at different T or P values. The image on the right in Figure 7 portrays mixed partial derivatives by displaying how the tangent vector pointing along one dimension (pressure or temperature) changes as it slides in the direction of the other dimension. Another useful activity in helping students understand relationships between quantities in thermodynamics is the partial derivative machine (68, 71). While this device is too complicated to adequately describe in this short subsection (refer to Roundy and colleague’s work), the point of the machine is to confront students with situations where all variables cannot be held constant. The machine is an exact mechanical analogy to thermodynamic systems, and students have to work with 206

the constraints of choosing which variables to hold constant, which to manipulate, and which to expect to change along with those manipulations. This activity provides students with insight into how partial derivatives work in contexts involving dependent multivariation.

Figure 6. Derivative tree for relating multiple quantities. Reproduced with permission from reference (7). Copyright 2012 AIP Publishing.

Figure 7. Graphical representations of partial derivatives. Reproduced with permission from reference (7). Copyright 2012 AIP Publishing.

Conclusion Chemistry topics involve the important derivative and integral concepts by making use of the ideas of rates, covariation, representations, differentials, summations, and relationships between quantities. These two concepts are rich, are complex, and involve a great number of ideas, reasoning abilities, and conceptualizations. Thus, if we wish for our students to productively and effectively use derivatives and integrals, work and effort will be required on the part of STEM instructors to help their students do so. Students may need help developing their images of rate, their connections between representational contexts, their notion of differentials, and their conception of integration. It would be wonderful if such things were all taken care of in the students’ pure mathematics courses, but this is unfortunately rarely the case. If such ideas are not developed for students, then derivatives and integrals can become meaningless or confusing. Or, as can happens, these powerful calculus concepts might simply be relegated to marginal comments without the intention of students actually making sense of them. However, I believe that the practical implications suggested in this chapter could reasonably be incorporated into chemistry (or other STEM) classrooms in ways that might not take too much time away from the core chemistry concepts. Attending to students’ and our own language can help develop direct discussions about explicit attention to both quantities in a rate. Asking students to think about non-time-based rates can develop an understanding of “rate” as not always being “per time.” Giving students tasks that require connections across representation contexts can develop expertise. Having a brief conversation about differentials can help students see how they are used. Calling attention to different meanings for integrals can open the door for a discussion on the key “adding up pieces” conception. Rather than discouraging instructors by discussing the many areas of difficulties students might face, this chapter instead gives instructors 207

some ideas about how to better make use of derivatives as rates and integrals as summations in their classrooms.

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Chapter 13

Developing an Active Approach to Chemistry-Based Group Theory Anna Marie Bergman*,1 and Timothy A. French2 1Fariborz Maseeh Department of Mathematics and Statistics, 1825 SW Broadway, Portland

State University, Portland, Oregon 97207, United States 2Department of Chemistry and Biochemistry, DePaul University, 1110 W Belden Avenue,

Chicago, Illinois 60614, United States *E-mail: [email protected]

Group theory, particularly the concept of symmetry, has applications in many different scientific fields and is an important part of the undergraduate curriculum in mathematics. In upper-level mathematics courses, group theory is only discussed abstractly, and students are rarely given an opportunity to apply these ideas to real-world problems. To better appreciate the applicability of group theory and symmetry, a local instructional theory is being developed, where students reinvent a classification scheme for chemically important point groups. In a pilot study, two mathematics education graduate students with limited knowledge of chemistry were given ball and stick models of water, ammonia, and ethane and were asked to develop and describe a procedure for efficiently and comprehensively finding all the symmetries of any given molecule. Video recordings of the students successfully completing this task and the corresponding inscriptions they made were interpreted using the emergent model heuristic in order to understand the evolution of the students’ model from model-of to modelfor. Implications of these results on the development of the local instructional theory and for future experiments are also discussed.

“To be sure, mathematics is a precious treasure-chest of tools, precious that is for those who can put them to good use.” —Hans Freudenthal, mathematician and mathematics educator

Introduction The use of group theory, in particular the study of symmetry groups, has become an essential tool for chemists when interpreting experimental data and predicting the macroscopic properties of materials based on molecular structure. An understanding of molecular symmetry is necessary © 2019 American Chemical Society

from a quantum mechanical standpoint as well, because molecular wave functions must conform to the symmetry of the equilibrium nuclear framework of the molecule (1). The mathematical consequences of symmetry have profound implications for many important chemical applications, such as electron configurations, molecular orbital theory, vibrational and rotational motion, and optical and nuclear magnetic resonance spectroscopy. Therefore, as Cotton and Wilkerson explain, “from a knowledge of symmetry alone it is often possible to reach useful qualitative conclusions about molecular structure and to draw inferences from spectra about molecular structures” (2). Nowadays, the use of symmetry to describe molecular structure is widely accepted in the chemistry community, so much so that it is commonly introduced and invoked throughout undergraduate courses in organic, inorganic, and physical chemistry. The associated concepts and conventional notation provide evidence toward the creation of a precise description of the underlying structure of symmetry when applied to chemical contexts. For example, the symbol C2v conveys precise structural information to a trained recipient about a given molecule that could otherwise require long verbal descriptions (2). Because of this utility, the use of symmetry notation in chemistry has become commonplace in the research literature. Therefore, knowledge of the basic concepts, conventions, and symbols is necessary in order to read and appreciate the findings reported in many contemporary research papers. Symmetry is also one of the most powerful and pervasive concepts in mathematics and has intrigued mathematicians for centuries. The mathematical study of symmetry has been systemized and formalized into what is called group theory. Historically, the group concept emerged out of a variety of mathematical lines of inquiry, including those in algebra, geometry, number theory, and analysis. Eventually the theory of groups came to be seen as a unifying thread for much of mathematics (3). In general, to a mathematician, a symmetry is an intrinsic property of an object that causes the object to remain invariant under certain classes of transformations. Practically, symmetries offer insight into regularities and are widely applicable in many fields of study, not only in pure and applied mathematics but also in the natural and physical sciences, like the radial symmetry of a sea star or the screw axis of a crystal (4). In general, group theory exploits symmetry when it exists, and the systematic collection of symmetries for a given object can offer powerful insights to the overall structure of the object, like its shape. As Mackey suggests, “many mathematical systems, including those which model the physical world, also have symmetries and symmetry groups and the study of the structural and other properties of these symmetry groups provides profound insights into the more immediately interesting properties of these systems and the key to the solution of many important problems” (5). Mathematically, a group, , can be defined as the set of elements in a set G together with an associative binary operation * on G. The set G must contain an identity element, contain inverses for each element in the set, and be closed under the operation *. Chemists use a particular set of groups, known as point groups. The set of elements for a point group is a set of symmetry operations where each included operation leaves a specific point of a molecule unchanged when applied to a three-dimensional molecular structure. This set of symmetries is then paired with the binary operation of composition of successive symmetry operations (i.e., perform one operation from the set, immediately followed by another—or the same—operation from the set) to form the point group. A symmetry operation applied to a geometrical figure can be thought of as performing some motion to the figure that returns the figure to itself, thus appearing unchanged. For example, a square has a rotational group consisting of only four elements: the symmetry operations corresponding to 90°, 180°, 270°, and 360° rotations about the center of a square. The rotational 214

group for a circle, on the other hand, consists of an infinite number of elements because a rotation through any angle about the center of a circle will return the figure back to itself. Although a square and a circle are simple, geometric shapes, this example shows an important implication of group theory when applied to molecular structures—each shape has a different symmetry group that consists of different symmetry operations. In mathematics, the overall structure of a group is often described in terms of a particular subset of the group called generators. Generators are group elements such that repeated application of these elements on themselves and each other are capable of producing all the elements in the group (6). For example, the square has a rotational group with four elements, each of which can be produced by composing a 90° rotation with itself a different number of times. Therefore, the rotational group of a square would be described completely as the elements generated by a 90° rotation. Given the restrictions of three-dimensional space and chemical bonding, there are only a relatively small number of combinations of symmetry elements that can occur. Despite the nearly infinite number of molecules that can exist, the total number of chemically important symmetry groups is bound to 32 (7). Therefore, group theory is especially powerful for identifying and differentiating molecules based on their valid symmetry operations associated with their shape. The findings presented in this chapter come from a larger body of work where the overarching goal is to develop a generalized path for student-guided reinvention of a classification algorithm for molecular structures. The results of the pilot study describe the mathematical activity of a pair of students as they engaged in classifying symmetry groups, starting with ball-and-stick models of particular molecules.

Group Theory in the Undergraduate Curriculum Due to its powerful ability to simplify problems and guide intuition, symmetry is invoked in undergraduate courses in chemistry. Group theory specifically is often formally introduced in undergraduate inorganic chemistry courses, where the basic formalism of point groups, symmetry operations, and character tables are discussed. These symmetry operations are applied to relatively simple molecules in order to categorize them into their respective point groups. Character tables are also sometimes discussed in physical chemistry courses during units on vibrational spectroscopy. The American Chemical Society (ACS) requires certified graduates in chemistry to have the equivalent of at least one semester of both inorganic and physical chemistry (8). The Committee on Professional Training lists geometries and symmetry point groups as one of the topics in the inorganic chemistry curriculum but does not require that it be taught (9). Abstract algebra, the larger field of mathematics containing group theory, is an essential part of the undergraduate mathematics curriculum as well (10–12). According to the most recent College Bureau of Mathematics Sciences report, 80% of departments offering undergraduate degrees in mathematics offered some kind of modern algebra course between 2009–2011, which is an increase from 61% in departments offering such courses between 2004–2006 (13). Modern algebra is a requirement for a bachelor’s degree in mathematics in 88% of mathematics departments. Unfortunately, while many students are taking abstract algebra in general, much of the current abstract algebra research highlights student difficulties in learning fundamental concepts in group theory in particular (14, 15). As noted in Dubinsky, Dautermann, Leron, and Zazkis, “Mathematics faculty and students generally consider it to be one of the most troublesome undergraduate subjects” (16).

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Roughly every 10 years, the Mathematics Association of America charges the Committee on the Undergraduate Program in Mathematics (CUPM) with making recommendations to guide mathematics departments in designing curricula for undergraduates. In 2015, the CUPM shared their most recent recommendations, including those specifically for undergraduate curricula in abstract algebra. The CUPM recognized that courses in abstract algebra are valuable for a wide variety of students, including mathematics majors and majors in science, technology, engineering, and mathematics (STEM) disciplines, such as chemistry (17). One of the cognitive learning goals highlighted by the CUPM is the integration and application of course concepts, more specifically “students should be able to describe connections between abstract algebra and other mathematics courses they have taken and they should be able to apply algebra to solve problems in other areas of mathematics and in other disciplines” (17). While applications of abstract algebra and connections to other disciplines are being encouraged and promoted by the CUPM, they remain mostly absent in undergraduate group theory courses offered in mathematics departments. A recent study aimed at establishing a meaningful consensus on the valued and important topics of undergraduate group theory among experts in the teaching and learning of group theory found that applications and/or uses were never mentioned at all, which might be quite shocking to most chemistry faculty (18). Research focusing on symmetry groups within the chemistry education community also highlights student difficulties with learning symmetry theory in chemical contexts. Students often struggle with visualizing molecules in three dimensions and determining the relevant symmetry operations. According to Mackey, “Significant logical-visual spatial skills, including visualization and rotation (i.e. spatial relations), are required to identify symmetry elements and place molecules in point groups. These skills are then partnered with conceptual knowledge to predict vibrational spectra and chirality” (19). However, it has been shown that even with the use of three-dimensional modeling software and the pervasive use of modeling kits, students still have a difficult time visualizing three-dimensional molecular representation of molecular structures (19–21). Students also struggle with interpreting complex representations of molecular structures and then connecting the microscopic structure of a substance to predictions about its macroscopic behavior (20). From Cooper, Corley, and Underwood, “Without a robust understanding of the underlying ideas that allow the structure-property connection, there is no organizing framework for most of chemistry and students, out of necessity, resort to memorization, and generation of heuristics” (22). In traditional instruction, both formal mathematical definitions and rich molecular representations are often presented from the perspective of an expert. These artifacts are representative of complex concepts that are meaningful to the expert given in a relevant representation, from which the novice—the student—is meant to extract particular meaning (23). Renowned mathematician and mathematics educator Hans Freudenthal once said, “to be sure, mathematics is a precious treasure-chest of tools, precious that is for those who can put them to good use” (24). Complex and abstract concepts are typically presented in a distilled, ready-made form intended to be easily digestible for students, but in reality these ideas are more often the result of years, decades, or possibly centuries worth of thoughtful pursuits riddled with preliminary conjectures and false attempts. This creates a problem known as “the learning paradox”: “How is it possible to learn the symbolizations, you need to come to grips with new mathematics, if you have to have mastered this new mathematics to be able to understand those very symbolizations?” (25). Too often in traditional instruction, the results of the mathematical (and/or chemical) insights of experts are taken as the starting point for student activity, rather than from the viewpoint and knowledge base of the student. 216

Activating Student Learning Entering mathematics education from a career as a mathematician, Hans Freudenthal saw little use in the ready-made mathematics presented to students and sensed that “things were upside down if one started by teaching the result of an activity rather than by teaching the activity itself” (26). He characterized this approach to instruction as an anti-didactical inversion. Freudenthal’s answer to this inversion was embodied by the idea that mathematics should be taught instead as an activity in which students are expected to participate in what he called mathematizing. Mathematizing can be described as, “the process by which reality is trimmed to the mathematician’s need and preferences” (4). Students who are provided an opportunity to reinvent mathematics by mathematizing are given a high level of autonomy, and thus the mathematics learned is on one’s own and through one’s own mental activities. Consequently, this provides students with a sense of ownership over the mathematics they reinvent, which seems common sense to the student since it is a product born of their own activities and the mathematics learned has the characteristics of cognitive growth, not of stacking pieces of knowledge (23). Freudenthal argued that this approach to instruction and learning mathematics more honestly represented the activity of actual mathematicians. University undergraduate classroom instruction can mostly be described as a one-way transfer of information, with homework problems being the primary means of trying to engage students in active learning (27). Within the last half-century, various attempts have been made within both chemistry and mathematics education to increase the level of student engagement. Many of the encouraging findings of these studies have recently been amplified in a meta-study conducted by Freeman et al., who analyzed 225 studies (28). The studies in the meta-analysis included data on examination scores or failure rates when comparing student performance in undergraduate (STEM) courses (28). Freeman and collaborators found that on average, student performance on examinations and concept inventories increased by about 6% in active learning sections, and that students in classes with traditional lecturing were 1.5 times more likely to fail than those in classes with active learning. Freeman’s findings raise serious questions about the continued use of traditional lecture methods and strongly supports active learning as the “preferred, empirically validated teaching practice in regular classrooms” (28). A similar meta-analytic study recently analyzed quantitative studies that examine the effects of cooperative learning (CL) on achievement outcomes in chemistry (29). Cooperative learning was conceptualized as “structured small group activities with five essential components: positive interdependence, face-to-face promotive interactions, individual accountability, interpersonal and small group skills, and group processing” (29). In total, 25 chemical education studies published from 2001–2015 were analyzed, involving 3985 participants. The report found that CL increased student achievement outcomes by 0.68 standard deviations, which implies that a student learning within a CL group setting would perform 25 percentile points better than a student in a traditional group performing at the 50th percentile. The findings of this meta-analysis suggest that cooperative learning is highly recommended as an important pedagogical tool for teaching chemistry at all educational levels. As the paradigm of STEM education gradually shifts toward cooperative, inquiry-based, active learning, the call for an increase in student engagement has been answered by educational researchers in both chemistry and mathematics. Process-Oriented Guided Inquiry Learning is one of the most well-known evidence-based curricula in chemistry and is used at both the secondary and postsecondary levels (30). Another pedagogical resource specifically geared toward inorganic chemistry is the Virtual Inorganic Pedagogical Electronic Resource that is curated by the Interactive 217

Online Network of Inorganic Chemists (31). One important example of research-based attempts at enhancing student engagement in abstract algebra is the Inquiry-Oriented Abstract Algebra (IOAA) curriculum, which contains materials designed for an introductory group theory course (32). These materials enable students to learn new mathematics through engagement in genuine exploration and argumentation and guide them through the reinvention of important mathematical ideas (33), specifically groups, subgroups, isomorphisms, and quotient groups. In the IOAA curriculum, students begin by reinventing the group concept via a local instructional theory, best described as a sequence of steps in terms of students’ progressive mathematical activity (32). Guided reinvention of the group concept begins in the context of the symmetries of an equilateral triangle as students identify, describe, and symbolize the set of symmetries. As the students begin to analyze the symmetries of the triangle, the group structure begins to emerge as a model of the students’ mathematical activity. As the students’ activity transitions from analyzing combinations of symmetries geometrically to calculating combinations algebraically, the rules they develop include axioms featured in the definition of group. Students work together to reduce their list to a minimal set of rules needed to completely determine an operation table for combining pairs of symmetries, at which point the students have transitioned from mathematizing the geometric context to mathematizing their own activity. The reinvention process concludes with students analyzing other groups and defining the group concept in terms of the properties shared by these systems (34).

Our Efforts in Activating Student Learning In an effort to engage students in the richness of group theory and its applicability, the first author (AMB) is currently conducting a design research study aimed at developing a local instructional theory (LIT) for student reinvention of the classification of chemically important symmetry groups. A local instructional theory describes a generalized roadmap for student reinvention of a particular mathematical concept (35). This path is a generalized sequence of steps described in terms of student strategies and ways of thinking that have been identified as important milestones in the development of the fundamental ideas of a particular mathematical concept (36). Both the curriculum and the theory are developed together during the design research process through a series of teaching experiments, each based on the constructivist teaching experiment (37, 38). A teaching experiment methodology allows researchers to experience firsthand students’ mathematical learning and reasoning through an interview, like setting where the researcher plays a duel role of teacher/ researcher. Instructional activities are tested along with the microtheories that describe how the instructional activities provoke the mental activities of the students, and how these mental activities contribute to the presumed growth in mathematical ability and understanding (35). Eventually, these microtheories serve as the rationale that the curriculum being developed links up with the informal situated knowledge of the students, and they also describe how the curriculum enables the students to develop more sophisticated, abstract, formal knowledge, all the while complying with the basic principle of intellectual autonomy. As students mathematize the given context, the activity they participate in can be described as a kind of mathematical modeling. In the Realistic Mathematics Education (RME) approach, the models are not prederived from the intended mathematics. Instead, the models are student generated and initially grounded in the contextual problems that the students are meant to solve. Such models consist of student strategies, inscriptions, and symbols that together address the contextualized 218

problems; therefore the term model should be understood in a holistic sense. These models are built out of a kind of modeling activity in which students are engaged in progressive mathematization. In the process of progressive mathematization students mathematize in two distinct ways to construct new mathematics. First, students participate in horizontal mathematization, where the contextual problem at hand is described in mathematical terms in order to solve it with mathematical means. Students can then also mathematize their own mathematical activity to reach a higher level of mathematics through vertical mathematization. It is important to distinguish the type of modeling the students participate in during this type of study from the more common notion of mathematical modeling. Traditionally, mathematical modeling can be described as a “translation activity,” where students have to translate the problem situations into mathematical expressions that can then function as models. In the more traditional kind of modeling, it is important that students are aware of the distinction between the model and the situation so that they can learn to assess whether the model is more or less adequate, given the particular goals of the modeler. Alternatively, a model like those found in an RME approach that are the result of an organizing activity emerge from the process of structuring the problem situation. In this kind of modeling, the model and the situation modeled co-evolve and are mutually rooted in the organizing activity (39). Because this type of model is a result of organizing the situation and structuring it in terms of the mathematical relationships, eventually the distinction between the model and the situation modeled dissolves.

Methods The overarching goal of a design experiment is to develop a preliminary LIT through a series of teaching experiments with students in order to produce an initial model of successful innovation (33, 36, 40). The experimental method of a design experiment is particularly well suited for developing, testing, and refining a preliminary hypothesis, such as a preliminary local instructional theory, because a design experiment has a built-in method of refinement through the implementation of multiple iterations. The design experiment methodology distinguishes three stages of research: the first stage is the preparation, the second is conducting a series of teaching experiments, and the third is retrospective analysis (41). Additionally, ongoing analysis is conducted within and between each iteration of the experiment. These complimentary ongoing and retrospective analyses are meant to inform not only the refinement of the protocol for the next experiment but more importantly the development of the overall local LIT. The ultimate goal of this project is to develop an LIT that can be used in an undergraduate course, so the pilot study was conducted with a pair of graduate students. As per the reinvention principal of RME, before the design research cycle can begin, a learning route has to be mapped out along which the students can reinvent the mathematical content for themselves (4). To do so, the curriculum developer often starts with a thought experiment imaging how they may have reinvented the concept (42). During my thought experiment, I imagined that it might not be obvious to students that a classification algorithm for molecular structures was a result of group theory, and so I was particularly interested in knowing the extent to which students would use group theory to solve the problem. Therefore, I decided to conduct the pilot study with students who had a background in group theory. Furthermore, the pilot study reported here is only the first part in a larger design research study with multiple teaching experiments planned with pairs of undergraduate mathematics students. 219

The mathematical activity and guided reinvention of the graduate students were meant to serve as both a sort of existence proof and an initial model of success. Graduate students were used in the pilot study instead of undergraduates to better ensure that the students would be able to successfully complete the task of classifying chemically important point groups, especially in the amount of time available (four meetings). As per the design research methodology, the approach of these pilot students becomes a hypothetical roadmap for the instructional sequences that will be tested with undergraduates in subsequent iterations of teaching experiments. In this report, I will be discussing the findings of the pilot study by sharing the activity of the graduate students, which serves to describe a way in which students might successfully classify chemically important symmetry groups. These findings also raise important questions about the probable/possible approaches of undergraduate mathematics students, which will also be discussed. The pilot study was conducted with a pair of mathematics education graduate students at a large public university on the West Coast, under the approval of an Internal Review Board. The students, referred to by the pseudonyms Emmy and Felix, had both completed a yearlong graduate sequence in abstract algebra, including a term in which they classified various groups of finite order. In addition to strong group theory backgrounds, the students had good rapport with one another. Emmy and Felix had worked as partners in a previous mathematics course and were extremely supportive work partners, especially in difficult situations. The pilot consisted of four 60- to 90-minute episodes, with time between each episode for ongoing analysis and subsequent construction of appropriate instructional activities based on the ongoing analysis. Data consisted of video recordings of each episode along with all accompanying written work produced by the students. The participants were compensated monetarily for their time. To ensure student engagement and ownership of the knowledge created, the entire study utilizes the instructional design theory of RME. The underlying theoretical perspective of RME aligns with Freudenthal’s belief that mathematics is first and foremost a human activity (26). Accompanying this perspective is a theoretical framework that includes three design heuristics, the reinvention principal, didactical phenomenology, and emergent models (35). The first heuristic informed the development of the tasks used in the teaching experiment in that a context was chosen that offered an opportunity for the students to begin by using their own intuitions and experiences to develop informal highly context-specific strategies (23). These context-specific strategies can then be used in a more general mathematical reality. Didactical phenomenology concerns the relationship between a mathematical content and the “phenomenon” it describes and analyses, or, in short, organizes (43). In this sense, the heuristic helped at a global level to inform a good starting point, the investigation of particular molecular models, which begged to be organized by the very same mathematical activity intended in the reinvention process. Didactical phenomenology was also used at a more local level during the teaching experiment to drive the study by helping to identify ways in which I as the researcher could support the students to transform their informal approaches to specific molecules into more powerful arguments about molecules in general (44). Much of the retrospective analysis and data from the pilot are framed using the emergent model heuristic. These models refer to the evolving process a student undertakes while constructing formal, abstract mathematical knowledge from an initial informal, context-dependent understanding. Gravemeijer highlights three interrelated mechanisms of emergent models: Firstly, there is the overarching model, which first emerges as a model of informal activity, and then gradually develops into a model for more formal mathematical reasoning. 220

Secondly, the model-of/model-for transition involves the constitution of some new mathematical reality—which can be called formal in relation to the original starting points of the students. Thirdly, in the concrete elaboration of the instructional, there is not one model, but the model is actually shaped as a series of symbolizations” (39). This heuristic can also be used to describe the qualities and features of the process, not just the process itself. By observing students as their mathematical activity progresses from contextually situated to more formal in a new mathematical reality, the overarching emergent model can be extracted (42). Transcripts of video excerpts from the pilot highlighting the progressive levels of mathematical activity the students participated in and the accompanying symbolizations and inscriptions they constructed through the reinvention process are provided to show how these observations explicate the overarching emergent model in this work of developing a classification system for chemically important point groups. It should be mentioned that this global model took on various manifestations and a much broader definition for the term model. “Model” here should be understood in a holistic sense. The model is not simply the inscriptions the students create, but also all of the meaning surrounding these inscriptions, as well as the students’ strategies for creating the inscriptions and of using them. Therefore, the various manifestations of the model are represented by the cascade of inscriptions and symbolizations that were an integral part of the students’ organizing activity as they investigated the symmetry groups of molecules.

Results Task Setting In a typical undergraduate inorganic chemistry curriculum, students are usually introduced to symmetry theory in a traditional way, beginning with definitions. Symmetry elements are defined to be geometric objects that a molecule may contain, such as mirror planes, rotational axes, and inversion centers, whereas symmetry operations are defined to be the reflections, rotations, and inversions preformed on or about these objects. Once students have a sense of the various symmetries found in three-dimensional space, they are often given a flowchart, similar to that found in Figure 1, to aid in identifying the specific symmetry group for any given molecule. Much attention is then spent developing students’ proficiency in symmetry group identification through memorization of the flowchart and its use along with significant practice applying it to various objects, often starting with simple geometric shapes before moving to molecular shapes of increasing complexity. This drill-based approach reflects a long-held belief that these flowcharts are central to understanding molecular symmetry “since a flow chart serves as a mnemonic device, the beginner very quickly acquires a feeling for molecular symmetry classification” (45). An important overall objective of this study is to avoid the use of mature, conventional symbolizations as mathematical starting points for instruction. Toward this end, the students had no a priori experience with the conventional classification flowchart; instead, they were given a set of three ball-and-stick model representations of water, ammonia, and ethane (as seen in Figure 2) and asked “to develop and describe a procedure for efficiently and comprehensively finding all the symmetries of any given molecule.” The molecules chosen for the initial task are canonical examples used to introduce symmetry groups, as they contain many, but not all, of the symmetry elements present in three-space. (Notably, they are lacking an inversion center.) 221

Figure 1. A typical flowchart given to undergraduates in inorganic chemistry to aid in determining the symmetry groups of molecular shapes.

Figure 2. Initial task prompt and accompanying manipulatives. Photos courtesy of Anna Marie Bergman.

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The resulting activity of the students can be described in two phases: • Phase 1: activity around finding the symmetry group of a particular molecule • Phase 2: activity around classifying symmetry groups in general Phase 1 In this first phase, the students’ mathematical activity was characterized as situational because their interpretations and solutions were dependent on the symmetry relationships observed within specific molecules and their particular symmetry groups. The pilot students began the experiment by determining and describing the symmetry groups of specific molecules. Both of the students used the same approach: (1) identify all symmetry operations to be considered as elements of the symmetry group, (2) distinguish which symmetry operations could and should be considered as generators, (3) determine the relations between each pair of generators, and lastly (4) decide to which “familiar group” the new- found group was isomorphic. Neither of the students wavered from this approach at any time during the pilot, and it proved to be very powerful for them as they were able to successfully identify the unique symmetry group for each molecule. While participating in the situated mathematical activity, questions arose that initiated important conversations that were crucial to the students’ subsequent success. One of the first realizations the students had was that the structure of a molecule can be considered somewhat fixed due to some underlying set of chemical and physical laws, thus implying that each molecule has a unique shape. Because neither of the students in the pilot had much experience with molecular representations in general, or ball-and-stick models in particular, they initially wondered whether the atoms could move around the molecule or if the bonds could be broken or if both were possible simultaneously. Almost immediately after grabbing the molecular model of water, Felix asks (while holding the hydrogen atoms), “Can we move these parts? Do these parts not move?” Only after Emmy and Felix realized that a unique shape would be necessary to have a well-defined classification system were they able to move forward in the activity of identifying symmetry operations. Determining that the molecules were a fixed shape also allowed them to use their pre-existing understanding of triangles while considering the symmetry operations allowed on a water molecule. They quickly determined that the set of symmetry operations of a water molecule would be more similar to that of an isosceles triangle, which is fewer than an equilateral triangle; ultimately this creation of bounds on the problem proved quite fruitful in constructing their procedure. Another important realization the students made early on during the situational activity occurred when they had to wrestle with the more mathematical question of if the definition of a symmetry is in the context of molecular shapes and what these symmetries might look like. Emmy was quick to offer her idea of symmetry: Emmy: So in this case, since these are like 3-d symmetry means if I had a shape, if I had it oriented like this [see Figure 3], I wanna do something (she rotates the model 180 degrees) so that it’s in the same orientation? Felix: I think so yeah. Emmy: Ok.

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Figure 3. Emmy considers a 180-degree rotation of a water molecule while describing her definition of symmetry. Photo courtesy of Anna Marie Bergman.

This description of symmetry aligns closely with the common mathematical definition of a symmetry as “a rigid motion that takes a figure to itself” (32). Although the students possessed a productive idea of symmetry in general and extensive experience with both rotations and reflections in two dimensions, the specific rigid motions allowed in the case of molecular structures were less obvious to them. The set of possible “somethings” that Emmy could “do” to the molecule evolved throughout Phase 1. Initially, both students seemed to gravitate toward rotational symmetries, presumably because these were tangible motions that could be performed on the models, and subsequently agreed that a water molecule would have at least two rotational symmetries, including a trivial 360-degree rotation. When asked if there were any other distance-preserving motions other than rotations, Emmy shared that she had an urge to “slice things in half.” From this statement, it is reasonable to assume she is considering the existence of mirror planes and whether these are valid symmetries for molecules. Unlike rotations, which can be physically performed on the provided molecular models, reflections and inversions must be strictly mental operations due to the rigidity of the ball-and-stick models; this added an additional concern the students needed to address. Additionally, the result of a two-dimensional reflection can also be accomplished with a threedimensional rotation (out of the plane of the molecule). Though they had extensive experience with two-dimensional symmetries, three-dimensional symmetries provided a new realm to eventually explore. Both Emmy and Felix seemed less comfortable with the reflection symmetries, and they continued to refine their idea of reflection throughout Phase 1 by carefully testing their ideas about the effect of a reflection symmetry. Initially their idea of a reflection included the need for the plane of reflection to include an atom from the given molecule. While this constraint worked for both water and ammonia, it failed to describe all the possible reflections observed in ethane. The students wondered about the behavior of specific atoms while considering possible vertical and horizontal reflections. For instance, in the case of ethane in the eclipsed configuration, as seen in Figure 4, Emmy considers a reflection through the plane that is orthogonal to the main (carbon-carbon) bond at its midpoint asking, “are they [the hydrogen atoms] jumping back and forth? Can they do that?”

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Figure 4. Emmy investigating the effect of “slices,” often referred to as horizontal reflections, on ethane. Photos courtesy of Anna Marie Bergman. Distinguishing vertical planes (those coincident with the principal axis) from horizontal planes (those orthogonal to the principal axis) was important for the students in the situated context, and they spent quite a bit of time discussing how each type of reflection had distinct effects on the peripheral atoms: Emmy: What we were thinking Annie, was that before when we were slicing stuff, like when we were slicing this guy [grabs the water molecule], we were thinking that the molecules (atom) here [motions to top side of atom; see Figure 5a] are going down to here [motions to bottom side of the same atom; see Figure 5b]. But if you slice like that [makes a chopping motion through the center bond of the ethane molecule; see Figure 5c], they don’t, that’s not what’s happening. That’s a different kind of slice.

Figure 5. Emmy considers the effects of vertical and horizontal reflections on particular atoms. Photos courtesy of Anna Marie Bergman. Eventually, the pilot students distinguished each of the reflections by giving them different names: reflections through vertical planes they called planes, and reflections through horizontal planes they called slices. The students felt that this differentiation between reflection planes was critical to their progress, and it allowed them to accurately describe the symmetry group for each of the given molecules. The distinction between these two types of reflections, coupled with the understanding of how each reflection combines with other symmetry operations, also served as a key mathematical insight into the overall classification algorithm as the various orientations of reflection planes lead to distinct group structures. More specifically, the existence of at least two vertical planes yields a group structure with a semidirect product, which is noncommutative, whereas horizontal reflections commute with all other symmetry elements, yielding a group structure containing a direct product.

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Lastly, the students determined that certain symmetries, such as inversions or 180-degree rotations orthogonal to the principal axis, could be described using combinations of their previously identified symmetries (principal rotations, reflections coincident with the principal rotations, reflections orthogonal to the principal rotations). For example, when investigating the effect various symmetries had on ethane in both the eclipsed and staggered configurations, the students created a kind of “dot diagram,” where the hydrogen atoms were enumerated 1 to 6 to differentiate them from one another in order to record and keep track of information about the molecule. The starting orientation (as seen in Figure 4) corresponded to projecting the molecule straight down onto the paper, where the numbers for the atoms pointing toward the observer are drawn closer toward the center. As seen in Figure 6, the students used the dot diagram to determine whether a rotation of 180 degrees orthogonal to the principal axis was equivalent to a “slice” (a horizontal reflection). Once they agreed the rotation was not equivalent to simply a slice, they quickly recognized that it was instead equivalent to a slice combined with a “plane” (a vertical reflection). Realizing that this “new” three-dimensional rotational symmetry was achievable by combining two of their known symmetries reinforced the students’ belief in their approach of identifying which symmetries could be considered generators. It was this approach that they ultimately reflected on and continued to use throughout their mathematical activity in Phase 2.

Figure 6. Examples of the students’ “dot diagram” used to determine equivalent symmetries found in the eclipsed configuration of ethane.

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Phase 2 After the students had correctly described the symmetry groups of water, ammonia, and eclipsed ethane, the students’ attention was redirected to the original task of classifying symmetry groups in general by redirecting their attention to the original prompt: “develop and describe a procedure for efficiently and comprehensively finding all the symmetries of any given molecule.” The students began to reflect on their own experiences with the molecules at hand, signaling a transition of their mathematical activity from situational to a referential activity. During this new mathematical activity, their focus shifted to generating models of their own activity in the situational context presented in Phase 1. These models were tested and refined throughout Phase 2 and eventually led to a productive algorithm for identifying a substantial subset of possible symmetry groups. (Due to time restrictions, the students were never asked to consider molecules with more than one rotational axis of order greater than two; therefore, they never attended to groups with very high order, such as cubic groups and icosahedral groups.) The first algorithm they created was a model-of their situational activity and mirrored the reasoning they used to categorize water, ammonia, and eclipsed ethane. Similar to the approach for identifying point groups suggested by the traditional flowchart method, the pilot students started by identifying the highest-order rotational axis, often referred to as the principal axis, of each molecule. After identifying the principal axis, the students attended to both vertical and horizontal reflections. This initial model included both a preliminary flowchart and a kind of “user’s manual” that defined the terms they used in their flowchart. It included both their definitions of symmetries and particular assumptions that could not be generalized beyond their observations about the three given molecules because this initial model-of originated through reflecting on highly situated activities (see Figure 7). For example, the students first describe the principal axis as “rotations about non-hydrogen guys.” While this description accurately captures the principal axis in water, ammonia, and ethane, it would fail when considering a molecule with either no hydrogen atoms, such as chlorine pentaflouride ClF5, or a molecule with only hydrogen atoms, such as H2. Once the students had created an initial model-of their activity in the situated context, they began a process of refinement as they continued to reflect on their own activity. The students refined their model in two different ways to accomplish two different goals. First, the students began adjusting their model to better reflect their prior activity with the ball-and-stick models by testing conjectures about the efficiency of their model. Felix observed, “we need to count rotations for all branches of our flowchart, so perhaps we do that first.” This led them to make a change to their flowchart (Figure 8). By thinking through these hypothetical (at least, to them) limiting cases, the students further refined their categorization scheme and produced a more robust version of their flowchart (Figure 9). This refinement activity led to an increasingly accurate and usable algorithm for identifying symmetry groups. The lines of questioning centered around the efficiency of the model and the possibility of different molecular structures than those already encountered are examples of “vertical mathematizing,” or activities where the reinvented mathematics itself is reorganized, generalized, or formalized (46).

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Figure 7. Initial model for finding symmetry groups. (Left) Inscriptions associated with the pilot students’ initial model-of their approach. (Right) The researcher’s (AMB) model-of the student’s model using more conventional terminology. While constructing their flowchart, the students refined many of their descriptions of mathematical objects, including which symmetries are valid and how they should be identified. Their initial descriptions were informal and highly contextualized, such as “slicing this guy,” “different kind of slice,” “identify rotation about non-hydrogen guys.” The symmetries at this stage were dependent on the existence and placement of hydrogen atoms in each molecule. Throughout Phase 2, though, the students’ descriptions of rotations, planes, and slices became more rigorous through the refining process. The descriptions were less rooted in the context of the initial task and more so on the various geometric entities and symmetry elements found within the molecule under investigation. They also became more formal, general, and mathematically accurate. The evolutions of each of these descriptions can be found in Table 1. The students continued to adapt their model via iterations of this “observation-refinementreflection” process and ultimately created an algorithm for correctly identifying and classifying the symmetry groups of a large subset of possible molecular shapes (Figure 10). The comprehensiveness and scope of the students’ reinvented solution compare well to the traditional flowchart provided to chemistry undergraduates (Figure 11). Although the symmetry groups they identified for various molecular shapes were mathematically correct, their level of differentiation between groups was much coarser than that seen in the standard approach found in chemistry. There were four possible distinct groups according to the students’ flowchart, whereas the comparable subset of the traditional flowchart shows 10 possible distinct groups. The students valued efficiency in their model, and this (along with a background in mathematics rather than chemistry) most likely led to an algorithm that classified chemically important symmetry groups by their mathematically isomorphic counterparts, rather than the standard chemical point groups.

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Figure 8. The evolution of the student’s initial model through the refining process. The student’s observation upon reflection and its corresponding effect on the model are shown for three iterations. Note that the phrase “cyclic group” here refers to a mathematical context, rather than a chemical one.

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Figure 9. Intermediate model for finding symmetry groups. (Left) Inscriptions associated with the pilot students’ intermediate model-of their approach. (Right) The researcher’s (AMB) model-of the student’s model using more conventional terminology. Table 1. Evolution of the students’ descriptions of symmetries. Students’ own words are italicized. Students’ Terminology

Rotation

Plane

Slice

Accepted Terminology

Initial Models v.0-v.1

Intermediate Model v.2

Final Model v.3

Identify rotation about non-hydrogen guys

A rotation is a symmetry about the axis through the center of the central atoms that is perpendicular to the plane incident with noncentral atoms

Choose the most symmetric axis, i.e., most number of rotational symmetries

Vertical Reflection

Identify planes/ reflection incident with non-hydrogen guys

A reflection symmetry through a plane incident with the nonhydrogen (central atom)

A reflection symmetry through a plane incident with the rotation axis of symmetry

Horizontal Reflection

Identify “slices,” i.e., planes reflections not incident with nonhydrogen guys

A reflection symmetry through a plane not incident with the central atom

A reflection symmetry through a plane orthogonal to the rotation axis of symmetry

Principal Axis

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Figure 10. Final model for finding symmetry groups. (Left) Inscriptions associated with the pilot students’ intermediate model-of their approach. (Right) The researcher’s (AMB) model-of the student’s model using more conventional terminology.

This focus on efficiency was evident when comparing the two flowcharts. The students determined that eclipsed ethane had an equivalent symmetry group as staggered ethane, both of which were isomorphic to the group D2n × ℤ2 according to their final model. Chemists, on the other hand, categorize ethane as D3h and D3d for the eclipsed and staggered conformations, respectively. Another difference is that chemists consider the point groups Cs, Ci, and C2 to be different from one another even though, mathematically, they are isomorphic to the group ℤ2 because they have the same structure—an order 2 element and the identity element. Chemists differentiate these isomorphic groups by the identity of the order 2 symmetry element: reflection for Cs, inversion for Ci, rotation for C2.

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Figure 11. The traditional flowchart, as seen in Figure 1, where the shaded portion corresponds to the symmetry group structures that are identifiable by the algorithm “reinvented” by the students in the pilot study. Implications for Further Study The disconnect between the mathematically correct and the chemically relevant seen throughout this modeling process presents an interesting source of future work. One important goal of future work would be to learn how to motivate and support mathematics students into developing a scheme that aligns more closely with that of chemists, rather than one that aligns with what algebraists would probably prefer. What is needed is a reason for a chemist to create such a scheme to provide relevance for the chemically important point groups. The discrepancies that arose during the pilot study could provide starting points for this search. For example, the conformations of ethane have different energies in addition to different point groups. Also, the very fact that the order 2 symmetry element in each of these groups is different (i.e., reflection for Cs, inversion for Ci, rotation for C2) is a necessary distinction in chemical applications of group theory, such as determining orbital symmetries, calculating reducible representations, and predicting infrared and Raman active vibrational modes. The design heuristic of didactical phenomenology will be particularly useful in discovering a specific problem or context that begs to be organized in a way meaningful to chemists and where the most appropriate solution is the exact organizational scheme the students are meant to learn. 232

It is important to note that the students during the pilot study were not asked to consider special classes of molecular shapes including linear molecules and groups very high order, such as cubic groups and icosahedral groups. To exclude linear groups with infinite rotational subgroups, the pilot students were told that a typical rotational axis in a molecule does not have order higher than 8, as it would lead to an unstable molecule. The decision to limit the task to rotations of finite order helped the students because, from this constraint, they immediately recognized that all the symmetry groups they were meant to discover would be finite. This seemed to reassure the students and offer them a sense of relief that the problem was, in fact, solvable. Groups with very high order were removed from consideration due to time concerns, and students were never asked to consider molecules with more than one rotational axis of order greater than 2. Because the students never attended to these types of groups, linear molecules and the various classes of highly symmetric molecules will need to be addressed in future work. Working with graduate students in the pilot study offered both advantages and disadvantages. The approach of the graduate students offers an existence proof of sorts for successfully completing the exercise, which can inform the creation of a preliminary local instructional theory ready to be tested with undergraduates. The pilot students were able to create an algorithm to successfully identify the symmetry groups of not only the three molecules they had manipulatives for (i.e., water, ammonia, ethane) but also many different hypothetical molecular shapes in a reasonable amount of time. Although the mathematical activity of the graduate students offers an approach for creating an algorithm for symmetry group classification, it is important to remember they have had formal instruction on group theory and are entering this task with extensive, structured background knowledge. Their approach was mathematically sophisticated from the start: identifying symmetries, determining which could be considered generating elements, and determining group structure by identifying the relations between various generators. This approach revealed a sophisticated understanding of group structures and the ways in which they can be constructed. Therefore, the path outlined here may not be one that students with less experience with group theory—for whom the LIT and reinvention task is ultimately intended—might find productive or consider at all. In follow-up teaching experiments, it will be important to see how students with a less sophisticated understanding of group theory tackle this classification process. Seeing the variation in multiple student-produced approaches by undergraduates will contribute to the overall robustness of the LIT, as well as provide evidence and rationale for including this LIT as a worthwhile student activity in future courses.

Conclusion The overall goal of the work presented here was to create a preliminary local instructional theory that can eventually be developed into a revised LIT for the guided reinvention of an algorithm for the classification of chemically important symmetry groups. This revised LIT will be designed for any undergraduate students, in mathematics or chemistry, who have completed an introductory group theory course. An important aspect of the LIT is its foundation in a real-world application, in this case, the identification of symmetry groups for molecules. To determine the tractability of this task and better understand possible student activity during the reinvention process, a pilot study was carried out with two mathematics education graduate students who had limited previous knowledge of chemistry and a rich understanding of group theory. These students were given ball-and-stick molecular models for water, ammonia, and ethane and ultimately created a classification system for accurately identifying symmetry group structures. Even though they had experience with only these 233

three molecular structures, their algorithm was applicable for many other molecular shapes with which they did not have personal familiarity. The evolution of the students’ mathematical activity from informal and context-dependent to more mathematically formal and generalized was observed over time in the video transcripts and corresponding inscriptions produced by the students. Their activity can be broadly described in terms of two distinct phases: classifying specific molecules and generalizing the algorithm for unfamiliar molecules. Interpreting the entirety of these students’ activity provides a set of possible productive (and unproductive) lines of inquiry undergraduates may pursue, a better understanding of the construction of knowledge related to group theory, and new research questions to investigate in future, follow-up experiments. The sheer fact that the graduate students successfully accomplished the particular task at hand should not be overlooked, as it shows that a solution is, in fact, possible—an obvious necessity for reinvention that was not known or assumed a priori. Through further revisions informed by follow-up studies with both mathematics and chemistry students, the revised LIT could eventually be developed to serve both populations. This LIT, possibly coupled with the group concept LIT described earlier, could provide an interesting opportunity for chemistry students to gain a deeper understanding of molecular structure (34). Although this specific study immersed mathematics students in a chemical context, this is not a necessary requirement of the method. It is reasonable that chemistry students without formal mathematical knowledge beyond minimal introductory group theory could successfully accomplish this task. It is also reasonable that mathematics students could extrapolate their results and see patterns in chemical properties and behavior, given some basic knowledge in chemistry. This LIT could provide chemistry students a new theoretical way of considering the implications of molecular shape on chemical and physical problems and mathematics students an opportunity to see how mathematical theories are applied in various fields. Under the heuristic of didactical phenomenology, educators and education researchers can brainstorm new contexts in the subject matter of interest or new concepts relevant to a given context to use in future coursework and experiments. This breadth of applicability is a powerful facet of this method for both instruction and research in mathematics and chemistry. Knowledge of symmetry theory and its applications is important for practicing chemists and mathematicians to have; therefore, a better understanding of the process by which these concepts are constructed and organized benefits educators and discipline-based educational researchers alike. Typically, during the study of symmetry groups, students are introduced to abstract concepts and rich representations that are too often constructed from the perspective of an expert. This manner of presenting material during traditional instruction, where the results and mathematical and chemical insights of others are taken as the starting point for student activity, contributes to what researchers have referred to as “the learning paradox”: “How is it possible to learn the symbolizations necessary for new mathematics if the very mathematics you are attempting to learn is presented in a manner that assumes its previous mastery?” (25). The results discussed here suggest that students can learn the target content by engaging them in the reinvention of the mathematics they are meant to learn. This method leverages the characteristics of cognitive growth, rather than a simple stacking of seemingly independent and possibly unrelated pieces of knowledge, and provides students with a sense of ownership of the knowledge they have created. This work offers a basis toward the creation of a pedagogically viable alternative to the antididactical inversion that students traditionally experience in both undergraduate mathematics and chemistry courses related to group theory.

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Chapter 14

Systems Thinking as a Vehicle To Introduce Additional Computational Thinking Skills in General Chemistry Thomas Holme* Department of Chemistry, Iowa State University, 2415 Osborn St., Ames, Iowa 50011, United States *E-mail: [email protected]

Water chemistry is proposed as an example where incorporating systems thinking approach can improve mathematical reasoning for students in general chemistry. These advantages are gained with relatively modest mathematical manipulations, such as how does one compute the volume of a pollutant entering an ocean/gulf from a river. Students are accustomed to calculating volume using the familiar lenght x width x height formula; changing the demands on this idea, even modestly to account for river flow rates rather than a length, provides important opportunities for enhancing computational thinking.

Introduction General chemistry courses have long included important quantitative skills as key learning objectives. Carrying out quantitative tasks in chemistry is sufficiently important that a number of studies have established connections between math preparation and course performance in general chemistry (1–9). This level of importance for mathematical skills has a strong influence on the perceptions of both teachers and students in general chemistry. For example, instructors have noted concerns that context-based approaches to chemistry curricula may not provide students with the level of mathematical rigor needed for continued chemistry studies (10). Additional studies have found significant differences between male and female students in terms of liking mathematics and physical sciences (11), with secondary school-aged girls notably less interested in these subjects. These types of studies suggest that (a) connections between math and chemistry are important when developing a curriculum, and (b) there are factors that influence student impressions of the desirability of studying chemistry that may be connected to mathematics. In addition to these overall concerns, specific math-heavy topics within chemistry, such as kinetics, have been the subject of studies on the role of mathematics in student understanding (12). One feature of these studies is that they are largely focused on the role of math in the fundamental aspects of the quantitative description of chemistry. Researchers have identified numerous concerns with regard to student understanding © 2019 American Chemical Society

of traditional mathematically related chemistry topics and the ability of students to utilize their math skills within the chemistry context (13, 14). With prior work suggesting that students struggle with math applications in the traditional topic curriculum of chemistry, how can chemistry educators respond to new demands that arise with curricular intervention? For example, the Next Generation Science Standards (NGSS) incorporate mathematics with skills that go beyond calculations and include computational thinking (15). While still important, it is not clear if the prior emphasis on skills such as algebraic manipulation of common expressions in chemistry (13) would be able to address this new push for computational thinking. In addition, the NGSS also notes the importance of systems models as a cross-cutting component of education across all of the sciences, including chemistry (16). Recent work in defining skills as represented in the NGSS identifies two concepts: computational thinking and systems thinking, which both present challenges to define (17–20), let alone incorporate in existing general chemistry curricula. For the purposes of discussions in this chapter, we will utilize the initial set of computational thinking skills identified by Weintrop et al. (20), to designate characteristics associated with computational thinking: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Ability to deal with open-ended problems Persistence in working through challenging problems Confidence in dealing with complexity Representing ideas in computationally meaningful ways Breaking down large problems into smaller problems Creating abstractions for aspects of a problem at hand Reframing a problem into a recognizable problem Assessing the strengths/weaknesses of a representation of data or representational system Generating algorithmic solutions Recognizing and addressing ambiguity in algorithms

It is clear that some aspects of the problem solving traditionally encountered in general chemistry classes touch on some of these skills. At the same time, there are aspects of this listing that tend to go beyond the “end-of-chapter” style exercises commonly found in the chemistry curriculum. Systems thinking also presents a challenge when identifying which definition is to be considered in a given context. Assaraf & Orion (21) have categorized many of the skills and abilities needed to understand the hydro-cycle system into eight categories. Because the specific example described in this chapter also touches on the biogeochemistry of water, the following set of categories represents a useful way to organize the definition of systems thinking for the present discussion: 1. 2. 3. 4.

The ability to identify the components of a system and the processes within the system The ability to identify dynamic relationships among the system components The ability to identify dynamics relationships within the system The ability to organize the systems’ components and processes within a framework of relationships 5. The ability to understand the cyclic nature of many systems 6. The ability to make generalizations 7. Understanding the hidden dimensions of the system 240

8. Thinking temporally, including retrospection and prediction To incorporate this set of attributes into a curriculum represents a relatively high level of instructional dedication to the importance of systems thinking. This chapter does not suggest that this high level is required in general chemistry, but rather that finding ways to connect more traditional chemistry topics to this type of reasoning is helpful not only for introducing the idea of systems thinking to students, but also in honing their knowledge of fundamental chemistry content and how it fits in with the bigger picture of science knowledge. In particular, we will argue that students are accustomed to thinking of chemistry content at what amounts to the laboratory scale. By asking questions about whether that is the correct boundary to consider, we can provide enhancements in terms of how well students learn the chemistry, how well they can use computational thinking, and how they can begin to approach systems thinking attributes. Ultimately, the longstanding challenges students have traditionally faced with mathematical reasoning in general chemistry might now include new challenges for computational and systems thinking. Therefore, the process by which traditional chemistry content coverage can be expanded now incorporates key components of systems thinking merits consideration. Another component of the NGSS that is relatable to system challenges lies in the cross-cutting concept of scale (22). In chemistry, the most commonly considered challenge related to scale has been how students understand the atomic/molecular scale (23). Nonetheless, the ability to apply fundamental chemical concepts to understand regional or global scale issues is also an important skill related to this crosscutting concept. Context and Systems Thinking The idea that presenting chemical ideas within real world contexts has been an active area of curriculum development for decades (10, 23–26). This prior work represents an important resource for enhancing systems thinking components in chemistry classes. For example, in biology the connections between contexts and more complete systems thinking have been studied (27). The idea of expanding previously considered concepts of context-based curricular ideas in chemistry with approaches that explicitly include the previously enumerated components of systems thinking appears to be worth considering. One important concept for establishing a rich context basis lies in establishing connections between chemistry and the needs of the earth and human society. One useful noteworthy context for the curricular concepts presented here is that of “planetary boundaries” (28, 29). This is the concept that there are aspects of the human/planet system that have, according to best scientific estimates, progressed past tipping points that would assure long-term sustainability. While there are several categories to explore in the description of planetary boundaries, the two with the most direct connection to the system thinking ideas presented here are “land system changes” and “biochemical flows”. It should be noted that prior work (21) on systems thinking has identified the latter as “biogeochemical” in nature, but because of the connection to the planetary boundary framework, this chapter will use the term biochemical flow for this concept. Land system change research currently tends to focus on rain forest changes in locations such as Amazonia (30), but the land use changes made centuries ago in North America can still provide important context and systems thinking perspectives today (31). Historical land use change has served as a valuable context for enhancing ideas around both sustainability and systems thinking in the first semester general chemistry course at Iowa State 241

University for several years. Like most of the Midwestern United States, a large fraction of the land in Iowa is planted with row crops each year. In 2017, over 23 million acres of corn and soybeans were planted in Iowa out of a total area of roughly 36 million acres. Prior to the large-scale settlement of the area by European immigrants to North America, this area was mostly tall-grass prairie. These land use changes have a direct impact on biochemical flows, and the combination of these factors provide an accessible way for students to apply water chemistry knowledge commonly taught in general chemistry to the understanding of larger scale challenges for earth and societal systems.

Water Chemistry Content Water chemistry is of sufficient importance for the understanding of chemistry that it commonly appears in several subjects of current general chemistry textbooks. Because the textbook is arguably the most influential artifact for content coverage in this (or almost any undergraduate chemistry) course, one way to assess the content coverage is to catalogue textbook coverage. We considered the coverage in nine relatively recent textbook editions (32–40) and found that the water chapter routinely follows the initial chapter on stoichiometry in traditional general chemistry textbooks. The information in Table 1 shows a remarkable level of similarity in coverage of fundamental concepts. It also shows relatively modest application coverage; this judgement is based on the observation that at least a section of the narrative has been dedicated to the application. Water comes up again in many of these textbooks, and while applications are possible there, the key point for the current discussion is that there is only a modest level of context-based motivation presented in most general chemistry textbooks when water chemistry is introduced. There are also four areas where calculations of some kind are included (as indicated by the shading of the rows with such mathematical content). Table 1. “Early” Water Chemistry Chapter Content

a Equilibrium ideas for water.

from seawater.

b Hard water.

c Ion exchange for hard water treatment.

d Obtaining metals

e Spectrophotometry to determine concentration.

The Opportunity for Systems Thinking Based on Water Chemistry The event that brought a sense of immediacy to the issue of water chemistry and water quality in central Iowa (where Iowa State is located) was a lawsuit filed by the Des Moines Water Works (DMWW) against upstream rural Iowa counties in response to the cost the DMWW incurred for 242

treating for nitrates (41). Because water treatment to create potable drinking water includes all of the reaction types commonly covered (see Table 1) in acid/base chemistry, precipitation, and ultimately redox chemistry, this topic seemed ideal to add context components related to systems thinking. Our method for incorporating systems thinking in this course is to ask the following questions: “When we consider this chemistry, is the boundary of what’s going on in a beaker in the laboratory too limiting? What if we expand the boundary of what we are considering?” These questions cause students to think about precipitation as a reaction that can do more than produce a (sometimes colorful) solid at the bottom of a test tube in the teaching laboratory. Asking these questions near the beginning of the coverage of water chemistry allows us to introduce the fundamental steps of water treatment: (1) screening; (2) particle removal including flocculation; (3) chlorination; (4) ozonation; and (5) fluoridation. We note concentration levels of nitrates in the Des Moines and Raccoon Rivers and their fluctuation levels throughout the year by graphically presenting the levels as reported by the Department of Natural Resources monitoring stations. Because the nitrate levels vary with weather conditions, they allow for emphasis on both dynamic components (systems thinking attribute [STA] #2) and temporal components (STA #8) because of the seasonality of the nitrate loading in rivers. This information can then be used as a topic for further discussion as new concepts are introduced about water chemistry. For example, once solubility rules have been introduced, we refer back to the conversation on flocculation in water treatment and how it uses precipitation. Students are then asked, without further direct instruction, “Why would nitrates be difficult to remove from drinking water?” as a clicker question. Looking over several semesters, the results are quite consistent with an example from a fall 2018 class shown in Figure 1. This conceptual connection provides a valuable basis upon which to build the treatment of nitrates in water sources, and incorporate numerical aspects of the problem as well.

Figure 1. Clicker question for students to make connections between solubility rules and water treatment processes prior to any direct instruction. The next new mathematical concept that this issue allows us to add to the topic of water chemistry is the concept of drinking water standards and the motivations behind them in terms of atrisk subpopulations (in this case, infants). A key reason that nitrates are regulated in drinking water lies in the potential for inducing methemoglobinemia, or blue-baby disease, which occurs when infants are exposed to levels of nitrates that are too high. With infants, careful consideration of toxicity 243

and the units for this important public health measure are required. The standard for nitrate is given as 10 mg/L or ppm Nitrate-nitrogen (42), which means the measure is being considered as a unit of nitrogen, the diatomic molecule. As soon as this standard is presented, there are two mathematical reasoning questions that present themselves. First, can students establish that mg/L is a unit that corresponds to ppm? Second, what does the nitrate-nitrogen concept imply, and how would this compare with other logical units? This type of learning exercise is connected to the computational thinking concepts noted earlier, in part because they require students to consider how a new idea (in this case, a mathematical unit) is related to an already familiar unit (in this case, computational thinking attribute [CTA] #7). The first task, allowing students to determine that mg/L is equivalent to ppm, requires the consideration of the density of water. Once the students are reminded that water is so close to 1 g/mL in density, they areable to work out the equivalence. The second component requires the students to reinforce their understanding of chemical formulas and the concept of mass percentage based on the formula (a task that has been covered prior to this chapter). With some prompting from an active learning assignment, students are able to determine the percentage of nitrogen in nitrate. We have noted that the charge on the ion does occasionally cause student concerns, and we have to remind them of the relative mass of electrons so they can feel comfortable leaving the additional mass of the “extra electron” out of the calculation. Once the students have determined the percentage of nitrogen, we note the similar total nitrate standard often used in Europe is 40 mg/L of nitrate. Using the percentage of nitrogen in nitrate results in this standard yields 22%, which seems to be off by a factor of two; this is addressed by the diatomic nature of nitrogen (N2). This set of activities introduces students to the importance of carefully considering concentration units. The vast majority of concentration-related problems introduced in general chemistry are based on the unit of molarity. This is, of course, quite practical from the perspective of connecting lecture and laboratory, but it becomes less all-encompassing when we ask the question that promotes greater awareness of systems thinking: “Are we considering the right boundary for the chemistry we are studying?” Incorporating concentration units within context-based practical applications serves the purpose of pushing the knowledge students gain outside of the comfortable box of information that is only used in the chemistry classroom or laboratory, where students tend to perceive molarity to be used. The extension of mathematical skills required is modest and generally within the coverage of ideas discussed in the course, but the tendency of students to compartmentalize their mathematical understanding can be addressed with even small changes to the math expectations for topics that are commonly taught in the curriculum. The potential for long-term gains associated with incorporating these conceptual and mathematical activities related to nitrates in the environment has been estimated to some degree as well. The course in which this idea has been introduced uses ACS Exams (43) for its final examination. This fact prevents us from providing a specific discussion of any items on the final exam. Nonetheless, in one semester where this context was employed for instruction, an ACS Exam that included an item where knowledge of nitrates would be helpful to correctly answer the question was used. Course-wide performance on this item was 21% better than national statistics for the same question, and this constituted the third highest positive differential in the entire test. This was accomplished with only about 20 additional minutes of instruction time. It is not possible to attribute this strong performance wholly to the contextualization of the concept of nitrates in the environment, but the strong positive effect is nonetheless tantalizing as an indicator of ways that gains from introducting mathematical extensions that promote awareness of systems thinking could be measured in future studies. 244

Stretching Mathematical and Estimation Skills with Systems Thinking There are constraints inherent in the delivery of chemistry in large lecture formats. When teaching over 700 students, most of whom need fundamental chemistry education for coursework in other disciplines in addition to subsequent advanced chemistry classes, the amount of time available for providing mathematical context through systems thinking is somewhat modest. Even within this learning environment at Iowa State, additional opportunities for mathematical education arise. For example, students who wish to take the general chemistry course as an honors course volunteer to sign up for extra activities, and in these small group sessions, it is possible to add content including systems thinking-related activities for water chemistry. The context used for these extra activities remains the same—the biochemical flow of nitrogen. The difference lies, once again, in expanding the boundary of what is considered in the context. In particular, rather than focusing on the need to treat water in a municipal water system close to home, we consider the creation of the “dead zone” 850 miles away in the waters of the Gulf of Mexico. The loading of both nitrogen and phosphorous into the waterways of the Midwest is largely responsible for this concern, which provides a large-scale example of how fundamental chemistry content can be connected to planetary boundaries. Estimates show that roughly 40% of the nutrient loading that is responsible for the dead zone originates in Iowa. We ask the students, “what would we need to do to investigate the accuracy of this statement?” This setup is, once again, capable of engaging both enhanced systems thinking and computational thinking. Initial brainstorming sessions from the students (captured on whiteboards) show how hard it is for them to confront the need to expand their fundamental knowledge to larger, context-based systems. Figure 2 shows the topics noted in these initial sessions discussing the question, “What things have you learned in chemistry that would help you address the question of whether or not this estimate is reasonable?”

Figure 2. Initial group brainstorm responses for what chemistry ideas might be needed to start to estimate the percentage of gulf nutrient loading that originates in Iowa. 245

While some strategies are mentioned by students immediately upon confronting this question, other strategies show the challenges faced in framing a problem as ill-defined as this one. Once the students had this initial reflection time, they were prompted with a more specific question from the discussion leader: “What would you want me to Google?” At that point, the groups began to coalesce around what was perceived to be the most productive approaches, most of which are associated with determining the volume of water. Thus, students begin with essentially a laundry list of water-related chemistry topics, but when prompted anew, they go beyond identifying components. They start to organize those components within a larger framework (STA #4), and because of the incorporation of inherently dynamic river flow, they adopt dynamic features as well (STA #2, STA #3). There was significant interest in trying to gauge water entering and exiting the area where Iowa rivers are. The students were essentially honing in on the idea that in order to determine the percentage, they needed to know the whole sample entering the gulf and also the amount originating from Iowa itself. Thus, they began to engage in the process of breaking down the larger problem into smaller ones (CTA #5). It was at this point that the groups hit an interesting snag. By selecting Memphis as a location to investigate, it took little time to determine the depth and width of the Mississippi River downstream from the confluence with the Missouri River. (In this case, the convenience of having a familiar, wellestablished location for model building became important.) Even so, the students struggled with how to determine the volume in this case. They found themselves stuck with the idea that volume equals a (length x width x depth), and they could not determine what “length” would work. While they did determine that using the flow rate x time could obtain the third dimension in this problem, it was apparent that there was discomfort associated with having to seemingly abandon a trusted mathematical formula. In terms of the computational thinking traits, this suggests moving toward an ability to deal with open-ended problems (CTA #1) and a persistence in working through challenging problems (CTA #2) where familiar algorithms no longer work. The next eye-opening aspect of their problem-solving arose from the magnitude of the water flow in a river as large as the Mississippi. When they ultimately found an estimate of the water volume leaving the river and entering the Gulf, they were hesitant to keep moving forward because the number seemed so large. This reaction suggests that the propensity to provide laboratory-scale exercises in general chemistry problem-solving examples carries a cost with it in terms of student comfort level (at least initially) with global-scale issues. It is ultimately the connection to global-scale issues that provides the clearest incorporation of systems thinking attributes within this topic. Students identify individual components (STA #1) in terms of concentration concepts (and even components of concentration), such as the idea that concentration includes “per unit of volume”. They note that in order to determine volume, they must abandon a comfortable static formula for a dynamic one that includes river flow rates (STA #2). The fact that students need information about the watershed both before and after it flows through Iowa to estimate the percentage of nitrates that originate there helps to establish a framework where they can repeatedly apply mathematical relationships (STA #4). In the time frame over which this problem was discussed, the students did not include factors such as interaction of nitrates with soil, but they recognized the importance of such issues. This type of reasoning represents a form of understanding hidden dimensions (STA #7), or at least the need to consider other dimensions. Finally, the scale of the problem, in this case the vast amount of water that flows within the Mississippi watershed, induces students to think about time frames (i.e., translating flow rates per min into annual nitrate loading). Because of the relative unfamiliarity of calculations at such a large scale, students also showed signs of retrospection and questioned their ability to predict information based on the models they were creating (STA #8). 246

In the end, these fairly strong students grew more comfortable with building models for this problem and applying computational reasoning to the issue at hand. This is reflected in comments provided in the end-of-exercise statements they submitted, which are summarized in Figure 3.

Figure 3. A sample of individual student responses about what they might need to include in a mathematical model that estimates the percentage of Gulf nutrient loading attributable to Iowa. These statements also reveal enhanced aspects of computational thinking. Attributes such as assessing strengths/weaknesses (CTA #8) is notable in the response from Student 1; representing ideas in computationally meaningful ways (CTA #4) is apparent in the way Student 2 phrased their response; and confidence in dealing with complexity (CTA #3) is shown in the response from Student 3 who includes a large number of factors that would need to be considered. These observations begin to exemplify how helping students connect fundamental chemistry concepts to large-scale systems-oriented global issues can enhance both the systems thinking awareness and computational thinking attributes. Despite initial challenges, these students began building models that extended their basic mathematical skills in important ways. The large, systems-based consideration quickly pushed them toward accepting ways to build approximate models to start understanding the problem. Several students noted that improvements would be needed subsequent to these initial efforts, but all of them noted the value of getting a sensible estimate from which to begin considering more global scale implications of chemistry. At the same time, they were able to identify specific skills, such as the concept of concentration multiplied by volume to obtain an amount of nitrates involved in this issue. This occurred, even as they struggled with how to obtain a volume when the “length” dimension was causing confusion.

Conclusions and Extrapolations While it is certainly true that this chapter represents a single observation of the possible utility of incorporating systems thinking approaches within general chemistry, the initial results are largely promising. Modest extrapolations of concepts (such as solubility) that are commonly taught in any general chemistry course show evidence that students can make connections to real-world applications, and such connections appear to enhance long-term retention of conceptual 247

understandings. Mathematical reasoning aspects can also be expanded, particularly in terms of introducing a wider variety of concentration units and using them in practical situations (in this case, exposure limits of potentially harmful compounds). With a smaller group (or more available time), the introduction of biochemical flow issues as a component of water chemistry presents several potentially useful expansions of the topic. Students can be encouraged to use mathematical reasoning to build models, starting with more approximate treatments, and identify ways to build in greater accuracy. The topic, particularly the role of transportation of chemicals in river ways, lends itself to pushing students to expand their numerical reasoning beyond resorting to familiar and comfortable equations. For example, determining the volume of water that flows from a river into a larger body (the Gulf of Mexico) presents students with the challenge of determining one of the three customary variables used for volume calculations. We have carried out similar exercises within other contexts, but not to the same extent as the ideas reported here. Thus, climate change and anthropogenic global warming have been used in the description of gases. We began this chapter discussing how to determine information about the composition of the atmosphere and global temperatures in “deep time” through ice-core data. This approach, starting with ice cores, leading naturally to the idea of beginning the treatment of gases with kinetic gas theory. This idea introduces gases in a sufficiently new way, such that a large fraction of the students in the class did not react with a perception that the topic of gases will be the same ideas they learned in secondary school. Starting with this realization has proven to be an effective motivational tool for many students. The key message lies in the idea that it is possible to consider mechanisms to include systems thinking and models in general chemistry in ways that enhance student engagement and learning. Surprising challenges emerge for students when approaching the mathematical component of this strategy. Because students may compartmentalize their mathematical skills as applied to chemistry problem-solving, identifying contexts where they can logically extend those ideas and enhance their computational thinking can be highly useful, given the context of expectations that have emerged from the NGSS. Water chemistry represents one example of these advantages, but a number of others seem reasonable to propose and will be the topic of future work.

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Chapter 15

Video-Based Kinetic Analysis of Period Variations and Oscillation Patterns in the Ce/Fe-Catalyzed Four-Color Belousov–Zhabotinsky Oscillating Reaction Rainer Glaser,1,2,* Marco Downing,2,3 Ethan Zars,1 Joseph Schell,1,2 and Carmen Chicone3 1Department of Chemistry, Missouri University of Science and Technology, Rolla,

Missouri 65409, United States 2Department of Chemistry, University of Missouri, Columbia, Missouri 65211, United States 3Department of Mathematics, University of Missouri, Columbia, Missouri 65211,

United States *E-mail: [email protected]; [email protected]

The development of a video-based method for the measurement of the kinetics of oscillating reactions is described in this chapter. The method mimics reflection ultraviolet–visible spectroscopy and allows for simultaneous measurement at any target color (RC, GC, BC) with a temporal resolution of 0.01 s. The image analysis results in discrete time traces [DCC(t, RC, GC, BC)], and numerical methods are described for the analysis of the quasiperiodic oscillation patterns over the course of the reactions to determine period lengths PT(t) and other parameters [oxidation times OT(t), reduction times RT(t), etc.] that characterize the redox chemistry of the catalysts within each period. The methods were demonstrated with studies of the Ce/Fe-catalyzed four-color Belousov–Zhabotinsky reactions. The temporal evolution of period and pattern characteristics are described with polynomials, and values measured for reaction times of approximately 500 and 2000 s are discussed to highlight the temporal changes. The measurements provide a wealth of constraints to determine chemically reasonable reaction models. The discovery of the Bactrian-type oscillation pattern of ferriin in the four-color Belousov–Zhabotinsky reactions is a key result of the present study: the concentration of Fe3+ goes through two maxima in every period.

Introduction The original Belousov–Zhabotinsky (BZ) oscillating reaction referred to the cerium-catalyzed bromate oxidation of citric acid (1–9). Today, the term describes the entire class of bromate © 2019 American Chemical Society

oxidations of dicarboxylic acids (malonic acid, malic acid, etc.) catalyzed by a variety of metals including Fe (10). BZ oscillating reactions are the prototypical nonlinear chemical systems (11–16). To fully understand these reactions requires the ability to simulate their kinetics without any simplifying assumptions while considering consumption of substrates and the formation of a myriad of products—a daunting challenge. To approach such complexity requires experimental studies of the reaction kinetics, mechanistic studies to explore plausible reaction mechanisms, and computational simulations of the complete multiequilibria systems. Occasional collaboration of chemists with mathematicians will not suffice; the approach requires faculty and students with a thorough understanding of both chemistry and mathematics. A few years ago, a group of faculty (including the corresponding author) won a National Science Foundation Proactive Recruitment in Introductory Science and Mathematics (PRISM) grant in mathematics and life sciences to promote the integration of mathematics education with science education. The program involved various components, including freshman interest groups, new research-oriented courses and interdisciplinary seminars, and, importantly, early undergraduate research experiences. In this context, two of the authors (Rainer Glaser and Carmen Chicone) created a nonlinear dynamics group to study the mechanism of BZ oscillating reactions using a combination of experimental, computational, and mathematical methods. The group studied aspects of bromine chemistry (17, 18), and attention quickly turned to questions about pH dependence (17, 19, 20). In parallel, the group learned how to solve multiequilibria problems (21), learned how to include effects of ionic strength (22, 23), and explored spectral decomposition (24). In 2018, these pieces came together in a study of the Fe-catalyzed BZ reaction, which, for the first time, achieved agreement between experiments and simulations of the pH dependence over a wide range (25). In this chapter, we present the results of a video-based kinetic analysis of the four-color BZ oscillating reaction (FC-BZR) and an analysis of the temporal evolution of the oscillation pattern. The new method for video analysis of chemical reactions mimics reflection ultraviolet–visible (UVvis) spectroscopy and allows for the study of fast reactions at any wavelength at the same time. This video-based kinetic analysis was applied to the Ce/Fe-catalyzed FC-BZR. An oscillation pattern analysis is described for the characterization of dromedary- and Bactrian-type time traces, and the effects of color-base selection are analyzed in detail. The result of the analyses is a wealth of discrete experimental data for period lengths, oxidation times, and reduction times for every reaction as a function of reaction time, as well as regression functions that describe the temporal evolution of the characteristic parameters. A chemical reaction mechanism is discussed to explain the Bactrian-type oscillation pattern of the Fe catalysts. A full and quantitative understanding of the details of FC-BZR will require simulations of the entire reaction system, and the measured data provide a wealth of experimental constraints to determine chemically reasonable parameters.

Ce/Fe-Catalyzed FC-BZR The Ce/Fe-catalyzed BZ oscillating reaction has attracted much interest, and the chemistry greatly depends on the Ce/Fe concentration ratio (26–30). Lefelhocz (26) described the original Ce/Fe-catalyzed BZ reaction by adding an iron catalyst to Bruice and Kasperek’s Ce-BZR (27); this system with [Ce]/[Fe] ≈ 1 results in oscillations between purple and blue. Citing unpublished work by Rossman, Lefelhocz mentions that a higher Ce/Fe concentration ratio of approximately 10 results in red–green oscillations, the so-called traffic light reaction. In this spirit, the purple–blue system was described as the “alternative traffic light.” Ruoff et al. (28) explored similar systems with even higher Ce/Fe concentration ratios and observed red–blue or red–green oscillations, and D’Alba and 252

Di Lorenzo (29, 30) explored various purple–blue systems with low Ce/Fe concentration ratios. The FC-BZR shows a spectacular oscillation in that the solution periodically cycles through four colors in the sequence red, blue, green, blue, and purple. The reaction has been described by Shakhashiri (31) and has become a popular demonstration reaction. The colors in Shakhashiri’s FC-BZR are caused by the different oxidation states of the iron and cerium ions (Table 1). The solution starts out red because both metals are present in their reduced states, and it turns blue for the first time because Fe(II) is oxidized quickly to Fe(III). Soon thereafter, the cerium is oxidized and the solution turns green because both iron and cerium are present mostly in their oxidized states. Cerium does not stay oxidized for long, and the solution goes back to blue because the iron remains mostly oxidized while Ce(IV) is undergoing reduction to Ce(III). The solution gradually moves toward red as the number of Fe(II) ions increases. This changing proportion of Fe(III) ions to Fe(II) ions is visible as a purple solution gradually becoming red. After all the iron is reduced, the cycle begins anew. Table 1. Observed Colors of the FC-BZR, Characteristic Values (RC, GC, BC), and Dominant Metal Ions at the Various Color Phases

The sequence of the color changes can be explained by eqs 1–5. Equations 1 and 2 describe the oxidations of Fe2+ and Ce3+ by bromate, respectively. Bromine generated in eqs 1 and 2 will brominate malonic acid (MA) to bromomalonic acid [BMA, BrCH(COOH)2]. The systems studied here all start with large amounts of BMA present because the combination of bromate and bromide leads to an initial Br2 burst and the conversion of MA to BMA. Equations 3 and 4a describe the 253

reductions of Fe3+ and Ce4+ by BMA, respectively. While ferriin does not oxidize MA, Ce4+ can (eq 4b). The reduction standard potentials for the redox couples Fe(III)/Fe(II) and Ce(IV)/Ce(III) in acidic media are commonly listed as +0.771 V and +1.76 V, respectively (32, 33). For the ferroin/ ferriin redox couple in 1 M H2SO4, the standard reduction potential of +1.04 V was measured (34). For the Ce4+/Ce3+ couple of ammonium hexanitratocerate(IV) in 1 M H2SO4, the standard reduction potential of +1.44 V was reported (35). Hence, one must consider the oxidation of Fe2+ by Ce4+ (eq 5) (36, 37).

Experimental and Mathematical Methods Experimental Section We followed the protocol reported by Shakhashiri (31) for the Ce/Fe-catalyzed bromate oxidation of MA. Three solutions were prepared. The first beaker contained 250 mL of deionized (DI) water and 9.5 g (57 mmol) of potassium bromate (KBrO3, 99.5%). The second beaker contained 250 mL of DI water, 1.75 g (15 mmol) of potassium bromide (KBr, 99+%), and 8.0 g (77 mmol) of MA [H2C(COOH)2, 99%]. The third beaker contained 75 mL of 18.38 M sulfuric acid, 175 mL of DI water, and 2.65 g of cerium(IV) ammonium nitrate [Ce(NH4)2(NO3)6, >98%]. All three beakers contained 250 mL of solution. A fresh ferroin solution was prepared by mixing 100 mL DI water, 0.23 g (1.51 mmol) of iron(II) sulfate heptahydrate [FeSO4·(H2O)7, 99.999%], and 0.46 g (1.96 mmol) of 1,10-phenanthroline (99+%). In a 1 L beaker on a stir plate, the first two solutions were mixed and the third solution was added, followed by 15 mL of the ferroin solution. The initial concentrations of this parent system, the OCR(1,1) reaction, were [KBrO3] = 74.36 mM, [KBr] = 19.22 mM, [MA] = 100.5 mM, [H2SO4] = 0.882 M, [Ce] = 6.319 mM, [Fe] = 0.1622 mM, [Ce]/[Fe] = 38.96, and [cat.total] = 6.481 mM. The reaction was recorded for 40 min with a Panasonic HC-V110 video camera mounted on a tripod and set approximately 1.5 m away from the reaction beaker. A white poster board was placed behind the reaction beaker, and the zoom on the camera was set to five times. HD Writer 2.0 software was used to edit the iFrame-formatted video and produce an mp4 file. Mathematical Section: Video Analysis and Numerical Aspects Image Selection and Conversion to Red–Green–Blue Matrices The video-editing software Aoao Video to Picture Converter was used to convert the captured video to a series of JPEG images (38, 39). The program Mathematica (40, 41) was employed for image analysis. The images were imported and cropped to isolate the desired data. The cropped region needed to be in a location where the sample was not obstructed or changed. For example, the portion of the image in Figure 1 was selected to avoid the stirring rod in the beaker, the water distortion at the top, and the beaker label on the left side. In some cases, the middle of the beaker was also avoided if a water vortex formed because of high stirring speeds. The cropped images were converted to matrices of red–green–blue (RGB) values using the ImageData command (41).

254

Figure 1. Sample window selection. Distance from Characteristic Color Approach The next step of the analysis involved the choice of a metric to compare the color of the cropped image to a predefined color target. Each image contained RGB data corresponding to the intensities of red, green, and blue on a scale from 0 to 255. The sum of the RGB values was used as in Beer’s law to calculate species concentrations (42–47). A Fourier transform was applied to the data (48), and the digital-image-based method was employed to determine the end point of titrations (49, 50). The square of a difference from a standard or reference color has also been used as a way of extracting concentrations from RGB data (51), and we took a similar approach. For a given image with frame number FN corresponding to reaction time t = FN/FPS (where FPS is frames per second), we chose the average relative distance from the characteristic color target values using the formula

The m × n × 3 matrix id contains the R, G, and B values at the m × n points of the cropped image. The constant target values RC, GC, and BC are the R, G, and B values predetermined for the characteristic target color. Equation 6 evaluates the deviation from the characteristic color, where DCC = 1 indicates no deviation. The characteristic colors “red,” “blue1,” and “yellow” are the RGB values of pure solution of the ions (Table 1). The characteristic colors “green” and “purple” were chosen based on those frames of OCR(1,1) that showed the cleanest and most intense green or purple coloring, respectively. The characteristic values for “blue2” are essentially half of the respective values for “blue1.” For a given characteristic color (RC, GC, BC), the time traces values DCC(t, RC, GC, BC) were recorded over time. This approach recovers important timing information pertaining to the oscillating reactions because the time traces DCC(t, RC, GC, BC) are related to species concentration. The overall approach parallels the process of reflection UV-vis spectroscopy. The camera is the detector, and the computation of DCC(t, RC, GC, BC) for each image functions as a monochromator for the color (RC, GC, BC). It is the main advantage of our approach to allow for the study of fast reactions at any target color (RC, GC, BC) every 0.01 s. Time-resolved studies by UV-vis spectroscopy are limited by the time required to scan the entire visible range (ca. 3 s). 255

The Mathematica notebook for the determination of DCC(t, RC, GC, BC) values through the analysis of sets of images extracted from a video is shown in Figure 2. The code allows for the simultaneous analysis of as many color channels as desired. Interested parties may contact the corresponding author to obtain the notebook along with sample input and output files.

Figure 2. Mathematica notebook to determine DCC(t, RC, GC, BC). Effects of Frame Number The choice of the number of FPS to use in the data analysis requires a balance between accuracy and computer time. In an mp4 format, the upper limit is given by the technical limit of 29 FPS. The selection of 29 FPS for the analysis provides the clearest data but at a very high processing time. Conversely, the selection of only 1 FPS limits the accuracy of the data analysis but allows quick 256

scanning of huge data sets. To determine which FPS rate is the best, finding the optimal balance between both processing time and accuracy really is the key issue. Using a normalized root-mean-square (NRMS) deviation, we analyzed the percentage difference data at one FPS varied from the “ideal” 29 FPS data (Table 2). We chose 10 FPS as the most appropriate compromise between time and accuracy. The 10 FPS runs required only 70 min to analyze, versus 260 min for the 29 FPS runs, while only having an approximate 1% NRMS error. Conveniently, the maximum NRMS error was only approximately 4% at the 1 FPS runs, and so for quick analysis the method is still quite accurate. Table 2. Numerical Aspects of Video Image Analysis: Processing Time for DCC(t, blue1) in Reaction OCR(1,1) with Frame Number and Window Size Dependence Entry

FPSa

Window Sizeb

NRMSDc (%)

Processing Timed (min)

FPS Dependence 1

29

200 × 200

0.000

260.9

2

25

200 × 200

0.374

223.9

3

20

200 × 200

0.674

142.4

4

15

200 × 200

0.712

108.4

5

10

200 × 200

1.283

71.30

6

5

200 × 200

2.070

34.63

Window Size Dependence 7

10

200 × 200

0.000

71.30

8

10

200 × 100

1.356

60.68

9

10

100 × 100

1.777

56.80

10

10

100 × 50

2.204

54.56

11

10

50 × 50

2.182

50.56

12

10

50 × 25

3.025

49.20

13

10

25 × 25

2.970

46.96

14

10

10 × 10

3.139

47.04

15

10

5×5

5.617

45.77

16

10

1×1

5.023

46.34

= Frames per second. b Window size in pixels. deviation. d Time required for the data integration. a FPS

c NRMSD

= Normalized root-mean-square

Effects of Window Selection (Size and Location) With the FPS set, standardizing the window selection is possible. While location is dependent on the application, the key criterion for the selection of the location is the requirement to avoid color distortions. In our applications, this meant avoiding the center where distortion occurred due to the vortex. The choice of the size of the window is constrained by the size of the distortion-free space and by practical limitations to the analysis time. The amount of time saved by selecting a smaller window 257

size is proportional to the number of FPS used. Higher FPS values mean more time gained or lost for increasing or decreasing window size. Using 10 FPS, we found that the processing time needed when going from a 200 × 200 pixel window to just one pixel only changed from 70 to 45 min, a 35% decrease. However, this also introduced a 5% NRMS error. The data indicate that even reducing the frame from 200 × 200 pixels to 200 × 100 pixels introduced a 1.4% NRMS error while reducing the time by approximately 15%. We found using a 100 × 100 pixel window kept the NRMS error under 2% while reducing analysis time by 21%. Using both these methods, we reduced the calculation time by 80% while only introducing an NRMS error of approximately 3%.

Results and Discussion Measurements of the Time Traces DCC(t) Discrete plots were generated of the time traces DCC(t) for the OCR(1,1) reaction. The DCC(t) values were plotted for approximately 2900 s (48.3 min) for five characteristic colors (using blue1 but not blue2). The discrete plot of the time traces for reaction OCR(1,1) is shown in Figure 3, together with a close-up in the range 500 ≤ t ≤ 700 s. The time traces DCC(t) determined in every color channel show oscillations with various shapes. We needed to develop parameters to characterize the period variations as a function of reaction time, examine their color-selection dependence, and extract mechanistic insights from this information. Oscillation Pattern Analysis and Color-Base Selection We analyzed the shapes of the time traces DCC(t, color) and found that two characteristic types occurred (Scheme 1). We refer to these shapes as the dromedary (one-humped camel) and Bactrian types (two-humped camel). We began with the analysis of time traces with dromedary-shaped periods and specifically determined the times when DCC(t, color) goes through minima and maxima (Scheme 1, top). The time between successive maxima is the period. We also characterized the ascent times (oxidation times) and the descent times (reduction times) to and from the maxima, respectively. While a strictly periodic process is characterized by one period time, chemical oscillations feature period times that vary with reaction time. The data show that the timing characteristics for each oscillation cycle vary with the reaction progress, and we therefore report the timing characteristics for every oscillation over the course of the reaction. In most cases, the variations of the timing characteristics were steady and well-described with simple polynomials (Table 3). Parameters characterizing the oscillation patterns close to the reaction times of 500 and 2000 s are summarized in Table 4. Method of Finding Extrema in DCC(t) and the Start of the Oxidation Phase The local minima and maxima for the time traces DCC(t) were determined using Mathematica and the following process. First, the time trace DCC(t) data were smoothed using a moving-average filter computed by averaging over k frames. The averaging reduced the random noise and produced the time trace DCCA(t). Next, DCCA(t) was analyzed with an extrema-point finder function, which searches for local minima and maxima and generates the discrete functions EPmax(ti) (extrema points max) and EPmin(tj) (extrema points min). The period times PT(ti) were determined as the difference PT(ti) = ti+1 – ti. 258

Figure 3. Discrete plots of DCC(t) values determined by image analysis of the video recording of OCR(1,1) for various characteristic colors.

The determination of the oxidation time OT(ti) leading up to peak EPmax(ti) and of the reduction time RT(ti) following that same peak are trivial for smooth time traces (Scheme 1, top). With the minima EPmin(tj) and EPmin(tj+1) before and after the maximum EPmax(ti), respectively, the oxidation time is OT(ti) = ti − tj, and the reduction time is RT(ti) = tj+1 − ti. However, the times traces are not always sufficiently smooth, and EPmin(tj) does not correspond to the actual onset of oxidation at time tjox, with tj < tjox. Hence, we also determined tjox and defined the oxidation time as OT(ti) = ti – tjox and the reduction time as RT(ti) = tj+1ox − ti. With these definitions of OT(ti) and RT(ti), one can identify the maxima-based period time PT(ti) = ti+1 – ti as the sum of RT(ti) = tj+1ox − ti and OT(ti+1) = ti+1 – tj+1ox. Alternatively, one can use the knowledge of the tiox values and define the period as PTox(tj) = tj+1ox – tjox—that is, the time between successive onsets of metal oxidation. Note that as PTox(tj) equals the sum of OT(ti) = ti – tjox and RT(ti) = tj+1ox − ti, we will use this definition in the following discussion.

259

Scheme 1. Parameters in the oscillation pattern analysis of time traces with one maximum per period (dromedary-type, top) or two maxima per period (Bactrian-type, bottom).

The determination of tjox employs an interpolation function DCCI(t) of the DCCA(t) data and involves a backward search starting at the maximum EPmax(ti). This method works well for time traces that feature steady increases along the entire ascent {d[DCCI(t)]/dt ≥ 0}, which usually occurs for time traces with oscillation periods of >20 s. To determine the ascent time to the ith maximum EPmax(ti), we examined d[DCCI(t)]/dt going backward from ti and found the first point with d[DCCI(t)]/dt < 0. The time of this point is defined as tjox. Period Variation, Oxidation Times, and Reduction Times in the Yellow Channel Figure 4 shows the period times PTox(ti), the oxidation times OT(ti), and the reduction times RT(ti) as a function of time for the OCR(1,1) reaction. These data are the result of analysis of the time traces DCC(t, yellow). Table 3 shows the polynomial coefficients and the associated regression coefficients that best fit the measured data. In almost all cases, the measurements show a steady variation of the timing characteristic, and all data are included in the regression.

260

Table 3. Polynomial Coefficients and Regression Coefficients for Time Variations of Oscillation Parameters for Reaction OCR(1,1)a Parameterb

Channel

a

b

c

d

R2

Primary Parameters PTox(ti)

Yellow

−8E−10

9E−06

4.5E−03

44.997

0.9997

RT(ti)

Yellow

−9E−10

8E−06

4E−03

38.817

0.9998

OT(ti)

Yellow

−4E−11

9E−07

1.4E−03

6.592

0.9663

0.9991

0.0

0.9496

Oxidation Timesc OT(ti)

Greend Primary Parameters

PTox(ti)

Blue

−1E−09

1E−05

−0.0064

45.990

0.9996

RT(ti)

Blue

−1E−09

9E−06

−0.0051

43.378

0.9995

OT(ti)

Blue

−2E−10

9E−07

−0.0014

2.623

0.8383

Secondary Parameters DT1(ti)

Blue

−2E−10

1E−06

−0.0019

3.731

0.9837

IR(ti)

Blue

−1E−10

3E−06

−0.0039

12.175

0.9864

DT2(ti)

Blue

−7E−10

5E−06

0.0007

27.499

0.9943

Primary Parameters PTox(ti)

Purple

−1E−09

1E−05

−0.0063

45.960

0.9995

RT(ti)

Purple

−9E−10

9E−06

−0.0043

40.433

0.9998

OT(ti)

Purple

−2E−10

1E−06

−0.0020

5.527

0.7608

Secondary Parameters DT1(ti)

Purple

−2E−10

1E−06

−0.0020

3.853

0.9834

IR(ti)

Purple

−2E−09

1E−05

−0.0164

14.747

0.9897

DT2(ti)

Purple

1E−09

−4E−06

0.0141

21.833

0.9991

a Trend line function: y(t) = a·t3 + b·t2 + c·t + d.

b PT = period time, RT = reduction time, DT1 = first drop

time, IR = intermediate rise time, and DT2 = second drop time. c·DCC(t, yellow) + d. d Trend line-based range: 0 < t < 1755 s.

261

c Trend

line function: DCC(t, green) =

Table 4. Variations of Period Lengths and Pattern Shape Parameters with Reaction Timea Reaction

Channel

PT

RT

DT1

IR

DT2

IR+DT2

28

40

42

59

t = 500 s OCR(1,1)

Yellow

45

39

OCR(1,1)

Blue

45

42

3

12 t = 2000 s

OCR(1,1)

Yellow

67

57

OCR(1,1)

Blue

65

63

4

17

a Period time (PT), reduction time (RT), first drop time (DT1), intermediate rise time (IR), and second drop

time (DT2) in seconds.

Figure 4. Oscillation pattern analysis for the OCR(1,1) reaction in the yellow channel. Variations over time are shown for period times PTox(ti) (diamonds), oxidation times OT(ti) (triangles), and reduction times RT(ti) (circles). Figure 4 clearly illustrates that the period times PTox(ti) increase during the course of the reactions. PT(ti) and RT(ti) values for the cycles close to the reaction times of 500 and 2000 s are listed in Table 4. The reduction times RT(ti) are always much longer than the oxidation times. The variations of oxidation times over the course of the reaction are much more moderate than those of the reduction times. Period Variation, Oxidation Times, and Reduction Times in the Green Channel We analyzed the DCC(t, green) time traces the same way the DCC(t, yellow) traces were analyzed. Plots of DCC(t, green) versus DCC(t, yellow) show perfect linear correlations with unity slopes for the period times PTox(ti) and the reduction times RT(ti) (Figure 5).

262

Figure 5. Oscillation pattern analysis for OCR(1,1): Comparison of the time traces DCC(t, yellow) (xaxis) and DCC(t, green) (y-axis). Variations over time are shown for period times PTox(ti) (left) and reduction times RT(ti) (right).

Period Variation, Oxidation Times, and Reduction Times in the Blue and Purple Channels The analysis of the time traces in the blue and purple channels is made slightly more complicated by the fact that two maxima can appear within one period. Similar to the primary extrema EPmax(ti) and EPmin(tj), we defined EPsmax(tk) and EPsmin(tl) for the second feature (Scheme 1, bottom), and the additional parameters DT1(ti), IR(ti), and DT2(ti) were needed to describe the shapes. The first drop time DT1(ti) specifies the time between the first maximum EPmax(ti) and the second minimum EPsmin(tl) and is determined by DT1(ti) = tl − ti. The intermediate rise time, which can be calculated as IR(ti) = tk − tl, is the time between the second minimum EPsmin(tl) and the second maximum EPsmax(tk). The second drop time can be determined by DT2(ti) = tj+1 − tk and is the time between the second maximum EPsmax(tk) and the following minimum EPmin(tj+1). For the same reason discussed earlier, we used tj+1ox instead of tj+1 so that DT2(ti) = tj+1ox − tk. The results of the analysis of the DCC(ti, blue) and DCC(ti, purple) time traces of the OCR(1,1) reaction are illustrated in Figure 6. As with the analysis of the yellow channel, solid trend lines are shown for PTox(ti), OT(ti), and RT(ti), and dashed trend lines are shown for DT1(ti), IR(ti), and DT2(ti). The polynomial coefficients are listed in Table 3, and Table 4 lists representative values for ti = 500 s and ti = 2000 s. The first drop times DT1(ti) are always very short—less than 5 s.

Color-Base Dependency of the Oscillation Pattern Analysis Plots of the respective trend lines created with the DCC(ti, yellow), DCC(ti, blue), and DCC(ti, purple) data illustrate in a compelling fashion that the PTox(ti) time traces overlay perfectly for t < 1500 s and in many cases for much longer (Figure 7). The oxidation times OT(ti) for DCC(ti, blue) are always shorter than the oxidation times for DCC(ti, yellow).

263

Figure 6. Oscillation pattern analysis for reaction OCR(1,1) in the blue channel (left) and purple channel (right). The top panels show the variation over time of the primary parameters: period times PTox(ti) (blue diamonds), oxidation times OT(ti) (red squares), and reduction times RT(ti) (green triangles). The panels in the middle show the primary parameters again but also include the variation over time of the secondary parameters: first drop times DT1(ti) (purple circles), intermediate rise times IR(ti) (teal squares), and second drop times DT2(ti) (orange circles). The bottom panels are zoomed-in versions of the center panels and show only the secondary parameters.

Kinetic Origin of the Bactrian-Type Oscillation Pattern of Fe(III) In the Fe-only reactions, the time traces are of the dromedary type as expected; they show a short, single maximum in every cycle (25). In sharp contrast, however, every FC-BZR features a second maximum in the blue trace (Figure 3), and they are of the Bactrian-type. Scheme 2 is helpful in discussing the mechanism responsible for the variation of the Fe3+ concentration within one period of the FC-BZR, and we distinguish four phases. Phase 1 involves the oxidation of the metal catalysts (eqs 1 and 2) by the active oxidant BrO2· with oxidation rates k1[Fe2+][BrO2·] and k2[Ce3+][BrO2·], respectively. The reaction rate constants for the oxidations of Fe(II) and Ce(III) are k1 = 1.66·107 L/(mol·s) (52) and k2 = 6·104 L/(mol·s) (53, 54), respectively. The reaction rates for iron oxidation will always exceed the reaction rates for cerium oxidation, even though [Fe2+]0 is a magnitude lower than [Ce3+]0. Phase 1 ends once all BrO2· is exhausted by the oxidation of metals or organic compounds. 264

Figure 7. Color-base dependency of the oscillation pattern analysis of reaction OCR(1,1). Comparisons of the trend lines are fitted to the variations over time of the primary parameters from the time traces DCC(t, yellow) (brown), DCC(t, blue) (blue), and DCC(t, purple) (purple). Solid trend lines represent period times PTox(ti), dashed lines represent reduction times RT(ti), and dotted lines represent oxidation times OT(ti).

Scheme 2. Phases in the Fe3+ concentration within one period of the FC-BZR. Phase 2 is the first stage of the reduction of Fe3+ and Ce4+ by BMA (eqs 3 and 4a) with reaction rates k3[Fe3+][BMA] and k4[Ce4+][BMA], respectively, and with k3 = 11.7 L/(mol·s) and k4 = 0.09 L/(mol·s), respectively (55–57). The reaction rate constant k4(MA) = 0.23 L/(mol·s) (56, 58) for the reduction of Ce4+ by MA is approximately 2.5 times higher than k4(BMA). The Fe3+ reduction (eq 3) is much faster than the Ce4+ reduction (eq 4). The high reaction rates for Fe3+ reduction ensure narrow peaks in the blue trace of Fe-only reactions, and this rapid Fe3+ decay is indicated by the dashed line in Scheme 2. In contrast, the much slower reduction of Ce4+ by BMA or MA proceeds during the entire duration of each period of the Ce-only reactions and never reaches completion. Phase 3 begins once the rate of oxidation of Fe2+ by Ce4+ (eq 5) exceeds the rate of Fe3+ reduction by BMA. The rates for the formation of Fe3+ and Ce4+ are described by eqs 7 and 8, respectively. The occurrence of the Bactrian shape is the direct consequence of replacing the Ce4+ reduction by BMA (eq 4) with its reduction by Fe2+ (eq 5).

265

The reaction rate constant for the redox reaction of eq 5 is k5 = 1.61·105 L/(mol·s)8 in 1 M H2SO4 and six orders of magnitude higher than k4. The concentration of BMA must be less than the initial concentration of MA ([MA]0 = 100 mM), and it can be at most two to three magnitudes larger than the maximal concentration of the iron catalyst ([Fe]0 = 0.16 mM). The concentration terms will never make up for the difference between k4 and k5, and hence the reaction of Ce4+ with Fe2+ becomes the dominant mechanism of Ce4+ reduction. Bromide recycling becomes highly effective because Fe3+ reduction always produces bromide (while Ce4+ reduction may not). Both Ce4+ and Fe2+ must be present in significant amounts for the reaction in eq 5 to become productive. Equation 5 is not important at the very beginning of the period for lack of Ce4+. In the course of phase 1 (eqs 1 and 2), the concentration of Ce4+ builds up and eq 5 can operate, but it can only do so for a very brief time because eq 1 depletes the concentration of Fe2+ quickly (k1 » k2). The onset of phase 3 occurs when both Ce4+ and Fe2+ become available in significant concentrations and the k5[Ce4+][Fe2+] term drives the Fe3+ concentration up to its second maximum. Phase 4 starts with the decline of the Ce4+ concentration, and the oxidation of Fe2+ by Ce4+ (eq 5) cannot keep up with Fe3+ reduction by BMA (eq 3).

Conclusions We have developed a new video-based approach for the analysis of the kinetics of oscillating reactions that mathematically mimics reflection UV-vis spectroscopy. The main advantage of our approach is that it allows the study of fast reactions at any target color (RC, GC, BC) at the same time and with a temporal resolution of 0.01 s. The image analysis results in discrete time traces DCC(t, RC, GC, BC), and we described numerical methods for the analysis of the quasiperiodic oscillation patterns over the course of the reactions. We have demonstrated that this new approach yields accurate period lengths PT(t) that are independent of the choice of the color base. This approach also recovers important timing information about the evolution of the redox chemistry of the catalysts within each period because the time traces DCC(t, RC, GC, BC) are related to species concentration. The most important mechanistic result of the present study is the discovery of the Bactrian-type oscillation pattern of Fe(III) in the FC-BZR: Fe(III) goes through two maxima in every period. A chemical mechanism was proposed to explain this Bactrian-type oscillation pattern. The outstanding feature of the Ce/Fe-catalyzed FC-BZR in BMA-rich environments is the effective redox-cycling of the metal catalysts ([M]rc ≈ [Ce]0 + [Fe]0) and the effective bromide recycling (BMA oxidation » MA oxidation). An analysis of the Bactrian-type oscillation pattern of the Fe catalyst as a function of the Ce/ Fe concentration ratio is now underway to further probe the kinetics. Yet there are limitations to one’s capacity to rationalize complex systems of reactions based on experimentation alone. Clearly, simulations of the entire reaction system will be needed to fully understand the details of the FCBZR over time, and the measured data (Tables 3 and 4) provide a wealth of constraints to determine chemically reasonable parameters. In particular, simulations should aim to reproduce the temporal evolution of those characteristic parameters, which show the least experimental noise.

266

Acknowledgments This research was supported by the National Science Foundation (PRISM 0928053; CHE 0051007), and we acknowledge the donors of the American Chemical Society Petroleum Research Fund for partial support of this research (PRF-53415-ND4).

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Editors’ Biographies Marcy H. Towns Dr. Marcy H. Towns is a Professor of Chemistry at Purdue University. In 2017 she received both the ACS Award for Achievement in Research for the Teaching and Learning of Chemistry and the James Flack Norris Award for Outstanding Achievement in the Teaching of Chemistry. She is a Fellow of the American Association for the Advancement of Science and a Fellow of the American Chemical Society. She has over 80 publications, over 2300 citations, and over 100 international and national presentations. She is an Associate Editor for the Journal of Chemical Education, focusing on manuscripts pertaining to chemistry education research.

Kinsey Bain Dr. Kinsey Bain earned a B.S. in Chemistry from Bethel University in 2012 and a Ph.D. in Chemistry from Purdue University in 2017. Her doctoral work focused on student understanding of energy in the context of chemical reactions/processes. Concurrently, Kinsey also designed and led a National Science Foundation project that investigated student understanding and use of mathematics in the context of chemical kinetics. As a postdoctoral researcher at Michigan State University, she presently leads a research team focused on transforming STEM education through faculty development. One effort of this project focuses on facilitating and supporting faculty by teaching about three-dimensional learning (an adaptation of the National Research Council’s document, A Framework for K-12 Science Education), developing and validating assessment tools, and implementing support structures (at MSU and other partner institutions). Another aspect of this project investigates the adoption and implementation of instructional innovation across different disciplines and department cultures, as well as the effect on student outcomes.

Jon-Marc G. Rodriguez Jon-Marc G. Rodriguez earned a B.S. in Pharmacological Chemistry in 2014 and an M.S. in Chemistry at UC San Diego in 2016. He is currently a Ph.D. candidate in the Department of Chemistry at Purdue University. His research interests are situated at the interface between chemistry and mathematics, with the goal of learning more about students’ reasoning to provide insight into how we can improve chemistry instruction and curriculum development. His research interests have led him to take an interdisciplinary approach toward research, focusing on the learning challenges shared across disciplines. He is currently investigating students’ mathematical reasoning in chemical kinetics, and his dissertation focuses on students’ graphical reasoning in enzyme kinetics.

© 2019 American Chemical Society

Indexes

Author Index Bain, K., 1, 25, 69 Bain, K., 47 Becker, N., 9 Bergman, A., 213 Bieda, K., 135 Canelas, D., 119 Chicone, C., 251 Cole, R., 173 Cooke, B., 119 Downing, M., 251 Elmgren, M., 47 Fessler, C., 135 French, T., 213 Glaser, R., 251 Goldman, L., 81 Ho, F., 47 Holme, T., 239

Jones, S., 187 Kuechle, V., 135 Lazenby, K., 9 Mack, M., 81 Moon, A., 25 Mosley, P., 135 Phelps, A., 105 Posey, L., 135 Rodriguez, J., 1, 25, 47, 69 Ryan, S., 157 Schell, J., 251 Shepherd, T., 173 Stanich, C., 81 Towns, M., 1, 25, 69 Towns, M., 47 Wink, D., 157 Zars, E., 251

275

Subject Index A Additional computational thinking skills, systems thinking conclusions and extrapolations, 247 biochemical flow, introduction, 248 introduction, 239 context and systems thinking, 241 water chemistry content, 242 chemistry ideas, initial group brainstorm responses, 245f early water chemistry chapter content, 242t individual student responses, sample, 247f nitrogen, percentage, 244 solubility rules and water treatment processes, clicker question for students, 243f water flow, magnitude, 246

C Ce/Fe-catalyzed four-color Belousov–Zhabotinsky oscillating reaction, video-based kinetic analysis Ce/Fe-catalyzed FC-BZR, 252 FC-BZR, observed colors, 253t conclusions, 266 experimental and mathematical methods, 254 determine DCC, Mathematica notebook, 256 sample window selection, 255f video image analysis, numerical aspects, 257t introduction, 251 results and discussion, 258 DCC(t) values, discrete plots, 259f FC-BZR, phases in the Fe3+ concentration, 265 OCR(1,1), oscillation pattern analysis, 263f OCR(1,1) reaction in the yellow channel, oscillation pattern analysis, 262f oscillation parameters for reaction OCR, time variations, 261t period lengths and pattern shape parameters with reaction time, variations, 262t

reaction OCR(1,1), color-base dependency of the oscillation pattern analysis, 265f reaction OCR(1,1) in the blue channel, oscillation pattern analysis, 264f time traces, parameters in the oscillation pattern analysis, 260 Chemistry-based group theory, developing an active approach activating student learning, 217 activating student learning, efforts, 218 conclusion, 233 introduction, 213 methods, 219 results, 221 dot diagram, examples of the students, 226f ethane, horizontal reflections, 225f finding symmetry groups, final model, 231f finding symmetry groups, intermediate model, 230f initial task prompt and accompanying manipulatives, 222f inorganic chemistry, typical flowchart, 222f particular atoms, vertical and horizontal reflections, 225f phase 1, 223 phase 2, 227 refining process, evolution of the student’s initial model, 229f symmetries, evolution of the students’ descriptions, 230t symmetry groups, initial model, 228f symmetry group structures, traditional flowchart, 232f water molecule, 180-degree rotation, 224f undergraduate curriculum, group theory, 215 Chemistry course, math challenges, 105 conceptual chemistry, 109 electrochemistry, lessons, 111 identifying better-prepared students, 106 math raw score, frequency of success or failure compared, 108f percentile score, frequency of success or failure, 108f Toledo Exam scores by final grades, distribution, 107f

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Toledo percentile score versus course grade, 107f translating chemistry into math, 114 Chemistry instruction and learning, education research, 187 background, 188 rate, representations, 189f conclusion, 207 reviewing research, 190 calculus triangle, example, 199f completed chunks, reasoning, 192f covariation, quantification, 193 derivative concept, layers and representations, 194 derivative matrix with an example, 196f differentials, 200 instruction, research on representations and layers, 198 integrals, 201 multiple quantities, derivative tree, 207f multivariation and partial derivatives, 205 partial derivatives, graphical representations, 207f representations and layers, subsection, 197 thermodynamics, work represented, 204f

E Enzyme kinetics, mathematical resources used by undergraduate biochemistry students conclusion and implications, 77 introduction, 69 methods, 71 interview prompt, Michaelis-Menten plot provided, 72f results and discussion, 72 enzyme kinetics, kinetics graphs, 76f graphical forms and symbolic forms, Lex’s reasoning, 77t measurement symbolic form, symbol template, 74f rate laws, student reasoning, 74t reaction order, student reasoning, 75t themes discussed in this chapter, description of relevant codes, 73t science practices, 70 theoretical perspectives, 70

generic rate law, examples of symbolic forms, 71f

G Graphs, crossroad between chemistry and mathematics chemical kinetics, analysis of a problem, 49 chemical kinetics, assessment task, 50f chemistry resources, activation, 55 chemistry resources activated and applied, 56t emergent versus static reasoning, 54t Johnstone’s triangle, 52f proposed by Johnstone, different types of problems, 51t student responses, analysis, 53 titration curve, 57 introduction, 47 modeling and problem solving, interaction between mathematics and chemistry, 57 mathematical modeling cycle, 59f mathematical modeling cycles, utility, 60 possibilities and challenges, 48 processes to graphs, many-to-one mapping, 49f practice, implications, 61

I Introductory chemistry courses, math selfbeliefs implications, 99 introduction, 81 limitations, 98 literature review, 82 mathematics and chemistry achievement, self-beliefs, 84 quantitative studies, summaries, 83t methods, 85 achievement in chemistry, 87 ATMI subscale, distribution of responses, 88f causal relations, path diagrams depicting the variables, 90f computational details, 91 continuous variables, descriptive statistics, 89t math ability, 86 278

results and discussion, 92 model A disaggregated by course, regression data, 93t model B′ disaggregated by course, regression data, 95t model B disaggregated by course, regression coefficients, 94t simple mediation model, regression data, 97t WLSMV method, data-model fit indices for models B-D, 96t summary, 98 Introductory chemistry problem solving, transition of mathematics skills, 119 conclusions and future directions, 129 limitations, 123 methods, 121 matrix of problems, examples of items, 122f results and discussion, 123 algebraic problems, proximal development, 124 alluvial diagram tracing pathways through problem type, 125f answer accuracy, final exam score as a function, 126f hierarchical cluster analysis, dendrogram, 128f quantitative introductory chemistry, conceptual understanding, 127 regression analysis, results, 126f theoretical framework, 120

M Mathematics in chemical kinetics chemical kinetics, 25 discussion and implications, 39 student learning, understanding of and engagement with chemistry concepts, 40 introduction, 25 methods, 27 course and semester, number of interview participants, 28t interview prompts, two possible sequences, 28f second-order and zero-order chemistry interview prompts, 29f results and discussion, 29

blending observed and characterized, example of one type, 31f differences between student responses, student discussions, 37t high-frequency blender, resource graph, 34f high-frequency blender, resource graph relating to zero-order reactions, 35f open coding scheme, outline, 30f rate constants, examining student understanding, 33 second-order chemistry prompt, problemsolving map, 36f Steven’s problem solving, overview, 38f symbolic and graphical forms, analyzing mathematical reasoning, 32 zero-order chemistry prompt, overview of Steven’s problem solving, 39f theoretical perspectives, 26 Mathematics in chemistry, supporting students’ reasoning, 9 chemistry contexts, research on students’ metamodeling knowledge, 13 chemistry-focused modeling resources, 17 developing and using models, epistemological knowledge associated with the practice, 11 developing and using models, scientific models and the practice, 10 mathematical models, students’ understanding, 15 metamodeling knowledge and their development, aspects, 11 model competence, Grunkorn, Upmeier zu Belzen, and Krüger’s model, 12t modeling-focused curricula in physics, student outcomes, 16 modeling in chemistry, supporting students’ ideas, 17 activity development, 18 activity structure, 19 ideal gas law activity, particle attraction and temperature, 19f students’ metamodeling ideas, context specificity, 14 students who classified six representations from general chemistry, proportion, 14f summary and conclusions, 20 Mathematics in chemistry, using and applying conclusion, 5 introduction, 1

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it’s just math, 4 mathematical preparation on success in chemistry, influence, 3 mathematics and chemistry in the United States, fields, 2

proportionality in chemistry, common constant, 164t

T

P Physical chemistry, mathematical relationships derivatives, 177 multivariate models, chemical phenomena, 178 determinants and matrices, 182 differential equations, 181 equations, 176 factorials (combinatorics), 181 graphs, generation and interpretation, 177 illustrate state functions, example P versus T diagram, 177f integrals, 179 introduction, 173 notation and variables, 174 practice, conclusions and implications, 183 student reasoning, 182 Proportional reasoning, logic implications, 168 mathematical and chemical tasks, student proportional reasoning, 165 proportional reasoning in chemistry, mathematical description, 167 proportional reasoning protocol, DG and DS tasks, 166f mathematical reasoning in chemistry, problem, 157 constant of proportionality, reasoning through a problem using a known ratio, 163f explicit proportional reasoning, solution of a chemical ratio problem, 162f graphite, photograph, 161f

Teaching in chemistry, mathematical knowledge, 135 conclusions, 151 instruction, implications, 150 introduction, 136 limitations, 151 methods, 138 identifying opportunities, 139 MKT, theoretical framework, 136 MKT, domains, 137f research question, 138 results and discussion, 140 chemical reaction, balancing an equation, 141t functions and covariation, 147 ideal gas law, exploring the relationship between the temperature of a gas and its volume, 148t molar quantities, reaction stoichiometry problems, 143t reaction stoichiometry, 142 reaction stoichiometry calculations, selected prompts, 145t relationship between the temperature change and heat added, graph, 150f specific heat and the temperature change, exploring the relationship, 150t supporting students’ understanding, using multiple representations, 146f symbolic manipulation, discovering errors, 149 two reactants, reaction stoichiometry problems relating molar quantities, 144t

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