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
It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics Downloaded from pubs.acs.org by 185.89.101.124 on 04/29/19. For personal use only.
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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