Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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Investigating Student Understanding of Rate Constants: When is a Constant “Constant”? Kinsey Bain,† Jon-Marc G. Rodriguez,‡ and Marcy H. Towns*,‡ †
Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, United States Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States
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‡
ABSTRACT: The themes discussed in this study relate to how students reason about the information encoded in rate constants, which is important for developing a deep understanding of chemical kinetics at the molecular level. This study is part of a larger project centered more generally on students’ understanding and use of mathematics in chemical kinetics. During semistructured interviews, students36 general chemistry students, five physical chemistry students, and three chemical engineering studentsworked through chemical kinetics problems and were asked to discuss equations used in chemical kinetics, yielding rich data that provided insight regarding students’ understanding of rate constants. Analysis revealed that students often conflated ideas from chemical kinetics and equilibrium, such as rate constants and equilibrium constants. Furthermore, students demonstrated varying levels of sophistication regarding the distinction and relationship between rate and rate constants. Finally, students conveyed different ideas about the mathematical nature of the rate constant quantity. These findings suggest students need more support in order to develop a more sophisticated understanding of rate constants. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Chemistry Education Research, Kinetics, Interdisciplinary/Multidisciplinary FEATURE: Chemical Education Research
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given setting).10 A rate constant would typically be considered a parameter, or a “generalized constant”.10,11 As discussed by colleagues from the physics education research community, the labeling and use of constants, parameters, and variables is very different in scientific communities, such as physics or chemistry, compared to mathematics communities.12,13 Further, scientists also ascribe meaning to these symbols, which can lead to different interpretation of equations and changes how equations are viewed; such differences arise because the goals and purposes for the use of mathematics are so divergent.12,13 These distinctions and differences are often not apparent to students who are concurrently enrolled in mathematics and science courses;12,14 however, little work has investigated how students reason about constants and parameters, especially in the context of the physical sciences. Therefore, this work was guided by the following research question: How do students reason about rate constants in chemical kinetics?
ne of the core ideas in the discipline of chemistry is “change and stability of chemical systems”.1−4 This foundational idea that “energy and entropy changes, the rates of competing processes [emphasis added], and the balance between opposing forces govern the fate of chemical systems” takes many shapes and forms,4 one of which involves understanding the information encoded in the rate constant, which embodies important variables that affect rate, such as temperature dependence and the role of activation energy.2 However, studies of chemistry students at both secondary and tertiary levels demonstrate that students have difficulty navigating these concepts. As illustrated in a topical review of education research situated in chemical kinetics, students often have an incomplete understanding of the relationship between reaction rate and temperature, a relationship that is contained in the rate constant, and students tend to mischaracterize the relationship between temperature and activation energy or the mathematical nature of rate’s time dependence.5 Moreover, conflation between kinetics and equilibrium concepts is well documented in the literature, such as confusion between rate laws and equilibrium expressions or reaction rate and extent.6−8 While these studies give some insight into the nature of student thinking in this area, more work is needed at the undergraduate level to move beyond cataloging alternative conceptions and focus on interdisciplinary contexts.5,9 A robust understanding of rate constants would, among other things, include mathematical resources related to constants, parameters, variables, and functions. Mathematical symbols, like those present in a rate law, encode meaning. These quantities, represented by different symbols, could represent a constant (does not vary ever), a parameter (does not vary within a given setting), or a variable (varies within a © XXXX American Chemical Society and Division of Chemical Education, Inc.
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THEORETICAL FRAMEWORK We have framed our data analysis and discussion of results in terms of the resources framework, which is a model of cognition that defines knowledge as a network of fine-grained resources, or cognitive units, that are activated and constructed in response to a task or prompting.15,16 The resources perspective builds on diSessa’s17 knowledge-in-pieces conceptualization, which accounts for the observed inconsistency of student responses, since different resources or groups of Received: January 3, 2019 Revised: June 4, 2019
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DOI: 10.1021/acs.jchemed.9b00005 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Table 1. Second-Order and Zero-Order Math and Chemistry Prompts
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METHODS The study that we discuss in this paper is part of larger project interested in investigating how students integrate chemistry and mathematics knowledge when solving chemical kinetics problems. For this project we have previously reported on student engagement in modeling,21 student conceptions regarding zero-order systems,22 productive features of problem solving,23 and student use of symbolic and graphical forms;24 here we focus on student reasoning related to rate constants. Our primary data source for this study is semistructured interviews involving students working through a series of prompts (Table 1), with data collection involving the use of a Livescribe smartpen to digitally synchronize audio and written work.25 The prompts focused on second- and zero-order kinetics contexts because while familiar to the students, they are less commonly encountered compared to first-order systems. Participants were undergraduate chemistry students from a second-semester general chemistry course (n = 40), an upper-level physical chemistry course (n = 5), and an upperlevel reactions engineering course (n = 3). They were recruited in class prior to instruction and interviewed after all instruction and assessment related to kinetics was completed. This project was completed with the approval of the university’s Institutional Review Board. Participants were compensated with a $10 iTunes card after their interview; each was assigned a pseudonym to protect his/her identity. Student interviews were transcribed and open coded using constant comparison.21,26 Data analysis involved two researchers coding in tandem, discussing coding discrepancies and requiring 100% consensus for code assignments.27 Further-
resources may be activated when reasoning about different contexts.16 The resources perspective is in contrast to an alternate model of cognition that presupposes student understanding as composed of unitary, stable conceptions that are applied generally across contexts.15,16 This has implications for understanding the role of instruction in relation to how student ideas change over time; instead of targeting and replacing large entities or conceptions, conceptual change involves adding fine-grained resources and modifying connections between resources, ultimately restructuring students’ local cluster of ideas to create a more coherent network of meaningfully connected resources.18 We are interested in identifying the resources students used to reason about rate constants, acknowledging that certain groups of resources, as well as connections between resources, result in a more coherent and sophisticated understanding. Therefore, our analysis involved characterizing students’ resources (i.e., ideas about rate constants) and establishing a relationship among these ideas, which resulted in different hierarchical categories pertaining to levels of sophistication.6,19 As stated by Becker and colleagues, “In this perspective, students’ responses go beyond being correct or incorrect, and rather, students’ knowledge and skills lie somewhere on a continuum.”6 This approach toward analyzing the data provides insight regarding the range of students’ responseswith less of an emphasis on what students do not know and more emphasis on understanding what students do know and how we can use this to support students in developing a more sophisticated understanding.20 B
DOI: 10.1021/acs.jchemed.9b00005 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Figure 1. Side-by-side comparison of two mathematical equations and their corresponding symbol templates that model various aspects of this generic equilibrium reaction.
(Figure 1), and as discussed in previous work related to symbolic forms, focusing on surface-level features in an equation might limit students’ ability to draw connections to phenomena.24,29−31 The sentiment that symbols and topics in general chemistry are similar and difficult to differentiate is summarized in a statement made by a general chemistry student, Nelly: Nelly: “That’s like equilibrium [constant]. Not rate constant. I don’t know. That’s also another thing that’s hard about chemistry. It just seems that everything is the same almost, and it’s hard to distinguish each equation and each principle.” This discussion stemmed from her reasoning about if and how rate constants change for different reactions. She began reasoning about rate constants as equilibrium constants but realized that she was thinking about the inappropriate constant, correcting herself. The similar nature of the symbols and equation structure caused temporary conflation of the ideas due to the activation of inappropriate resources during her interview. Another general chemistry student, Georgina, demonstrated conflation of equilibrium and rate constants as well, utilizing an equilibrium-like expression to solve for reaction order. • Georgina: “I remember from zero order you didn’t have to do anything to do the concentration of A for it to be a straight line.” • Interviewer: “Why do you think that is?” • Georgina: “I know it has something to do...I kinda remember vaguely that...Say that your equation would be A plus B equals C plus D [writes chemical equation, top of Figure 2]. Concentrations of your products go over your concentration of the reactants [writes variation of
more, during data analysis it was determined that data saturation had been reached based on the observation of no new additional themes emerging from the data and adequate representation of the themes across the data.28 This process yielded a coding scheme for the larger project reflecting multiple themes, from which our previous work is based.21−24 In this paper, we focus on themes that provided insight into students’ conceptualization of rate and rate constants, organizing students’ conceptions based on sophistication, as discussed previously.
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RESULTS Our analysis revealed three primary themes: (1) conflation of rate constants with equilibrium constants, (2) levels of sophistication in differentiating the concepts of rate and rate constants, and (3) various types of understanding regarding the mathematical nature of rate constants. Conflation of Rate Constant (k) with Equilibrium Constant (K)
As reported in prior research, students often confuse kinetics and equilibrium concepts.5−8 In the case of Cakmakci7 and Tastan et al.,8 students expressed a general conflation of reaction rate and equilibrium, using Le Chatelier’s principle to describe reaction rate, whereas in Becker et al.6 students confused the mathematical expressions used in kinetics (rate laws) and equilibrium (equilibrium expressions). In light of this, it was unsurprising to see that almost a quarter of our participants demonstrated rate constant (k) and equilibrium constant (K) conflation, a finding similar to the results of Becker et al.6 In their method of initial rates task, Becker et al.6 suggested students’ conflation of rate laws and equilibrium expressions could be the result of the need for more support in developing metamodeling knowledge (i.e., epistemological resources related to developing and using models), advocating for providing scaffolding to help students move beyond recalling algorithms toward understanding the empirical basis of models such as equations. Based on the reasoning exhibited by students in this data set, additional explanation is provided regarding students’ conflation of equilibrium and kinetics, which appears to be 2-fold. First, the symbols for each constant are represented by the letter “K”, which are only distinguishable by capitalization (or lack thereof). Second, from the perspective of Sherin’s29 symbolic forms, the pattern of terms in the equations (symbol templates) is somewhat similar
Figure 2. Chemical and mathematical equations written by Georgina (general chemistry student). The mathematical equation is structured like that of an equilibrium expression. C
DOI: 10.1021/acs.jchemed.9b00005 J. Chem. Educ. XXXX, XXX, XXX−XXX
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constant, describing the quantities as constant (k) or variable (rate being proportional to concentration). Jenny: “So your k for this [first] reaction and your k for this [second] reaction [where the initial concentration is doubled] should be the same because of this rate law. ... So what would happen is as you raise the concentration, the rate would be different because of the rate law. The rate constant always stays the same. Like in this example, as you were doubling this [initial concentration], this [rate] also changes, which means the rate constant is staying the same the whole time ... ” The quote above demonstrates how Jenny clearly distinguished between rate and rate constant, primarily drawing on the rate law relation. This is distinct from participants that activated even more resources, explicitly noting factors that affect changes in a rate constant (level 3). Some provided a qualitative description of factors that the affect rate constant term (e.g., changing temperature or addition of catalysts), whereas other participants utilized mathematical expressions to describe these relationships (most often through the Arrhenius equation). For example, Skyler, a general chemistry student, provided the Arrhenius equation (Figure 4) and discussed the variables that can affect k (level 3).
equilibrium expression, bottom of Figure 2]. I know it has something to do with whatever exponents you ended up with here.” In this passage, Georgina was using the inappropriate equation to solve for order; she should have been using rate law equation, which contains a rate constant term, rather than the equilibrium constant equation. As shown in Figure 1, the symbol templates of the two equations are similar in structure. In general, each equation contained a variable related to the product of bracketed quantities, each raised to a power. It is this similarity that caused some participants to activate the inappropriate resource for this context. Levels of Sophistication in Understanding the Distinction between Rate and Rate Constants
There were a wide variety of resources characterized regarding student understanding of rate constants. One theme that emerged revealed varying participant understanding of the relationship between rate and rate constant. Some students expressed conflation of these ideas, while others conveyed distinctive understanding of these two concepts with differing degrees of sophistication (Figure 3). Among those who did
Figure 4. Arrhenius equation written by Skyler (general chemistry student).
Skyler: “This equation [Figure 4] ... might tell you like if you know that [two reactions] are at the same temperature, if you knew two things were at the same temperature, but they had different k’s, you could say, okay, well maybe this frequency constant is different or maybe they are different activation energies, so you could probably extrapolate some information from that.” As the data and Figure 3 show, the resources activated were increasingly more sophisticated when comparing level 0 through level 3. Relatively speaking, levels 1 and 3 were the most prominent across the data set.
Figure 3. Levels of sophistication in participant understanding of the distinction between rate and rate constants.
distinguish between rate and rate constant ideas, some simply noted they were different with little elaboration (even after probing questions), while others described the relationship between these two terms. The nature of the described relationships also varied, where some participants described proportional relationships (e.g., as rate constant increases, so does the rate) and others provided specific factors that affect rate and rate constant (e.g., defined mathematical relationships). Physical chemistry student Jack conflated rate and rate constant ideas during his interview (i.e., level 0). When discussing the second-order integrated rate law, he provided the definition below for rate constant and had difficulty delineating the two ideas throughout the interview. Jack: “I think the rate constant tells you ... the consistent rate, so at any given concentration the rate that this reaction is going to proceed forward, that number is the rate constant.” Alternatively, the quote below from a general chemistry student, Jenny, illustrates reasoning characterized as level 2, based on how she distinguished between rate and rate
Levels of Sophistication in Understanding the Mathematical Nature of Rate Constants
When analyzing participant understanding of rate constants among the students who did conceive of rate and rate constants as distinct, there were three levels of understanding that emerged with respect to what type of mathematical quantity rate constants were (Figure 5). First, participants sometimes conveyed the idea that rate constants were like universal constants, that is, the quantity was the same at all times (level 0). A general chemistry student, Penelope, demonstrated this sentiment during her interview. Penelope: “Rate constant, I think it’s always the same number no matter what rate we’re using ... Yeah, it’s just a constant that you plug in. I think it has something to do with how they calculate it, like they use hydrogen or something?”
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DOI: 10.1021/acs.jchemed.9b00005 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Figure 5. Levels of sophistication in participant understanding of the mathematical nature of rate constants.
making suggestions regarding the developmental levels that should characterize instruction related to rate constantsis beyond the scope of this work. Furthermore, we do not make claims regarding whether students move stepwise through the levels discussed (e.g., students may not necessarily have to “go through” level 2 to get to level 3).
This is distinct from other participants, such as general chemistry student Andrew, who stated that rate constants were only constant for a given reaction (even after probing questions), demonstrating a more parameter-like understanding (level 1). Andrew:“k is the ... rate constant. ... What I’m thinking, it’s the probability of the reaction happening. It’s usually a standard value...for that reaction. It can vary between reactions, but it’s just basically the probability of a reaction happening in creating a product.” Finally, some participants were able to elaborate further to describe on what rate constants depend. These participants cited specific variables, such as temperature, or provided the Arrhenius equation, demonstrating an even more sophisticated parameter-like understanding (level 2) through the activation of additional, connected resources. Similar to Jenny and Skyler (Figure 4) in the section above, general chemistry student Andrea expressed this type of level 2 understanding. Andrea: “k is specific for each experiment. ... The way you can change k would be by ... increasing temperature, or adding a catalyst, and that’s just about it. Even if I add in more concentration, in this case would be A, that wouldn’t change k.” Comparing levels 0, 1, and 2, the resources activated were increasingly more sophisticated, being more complex and nuanced (Figure 5). Level 0 was the least common among the participants, while level 1 was the most.
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CONCLUSIONS AND IMPLICATIONS The work described herein reveals themes that emerged from our analysis regarding student understanding of rate constants. Among our sample, participants sometimes conflated rate constants with equilibrium constants, had a range of ability in distinguishing between rate and rate constant concepts, and varied in their understanding of the mathematical nature of rate constants. Each of these findings has implications for both research and practice. We elaborate on these ideas below. While differences in symbols and equations may be obvious to experts, they may not be to novices. Symbols such as “k” and “K” may not appear distinct; further, rate law and equilibrium constant equations, while different, may appear similar (Figure 1). Instructors should explicitly call attention to symbols that look alike (e.g., rate constant, k; equilibrium constant, K; and potentially others, such as Boltzmann’s k (kB); t, time; T, temperature), as well as equations that may appear similar. Further, care should be given to explain and assess how and why they are different and what each represents at the particulate or macroscopic level. This may provide students with more sophisticated resources that can be activated in appropriate contexts. As referenced earlier, Becker et al.6 suggested that the conflations, like those observed in both their and our studies, reveal a need for developing student facility with metamodeling, resources related to developing and using models. Students need scaffolding to move them beyond simply recalling equations to understand the components of mathematical models (equations). To do this, further work is needed to characterize student understanding of mathematical models, as well as their metamodeling knowledge, in chemistry contexts so that instruction and assessment can be shaped to better support students. Participants revealed various levels of sophistication in distinguishing between rate and rate constant ideas. This may suggest that not enough attention is given to rate constants in traditional introductory-chemistry curricula, or at the very least
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LIMITATIONS The small number of the upper-level students limits any claims or comparisons among the participant groups. For this reason, we did not attempt to draw any conclusions from differences among the introductory- and upper-level students; rather, we used the diversity in the sample to capture a variety of possible student resources. Additionally, as these findings emerged from the data and were not a part of the originally conceived research agenda, more targeted research is needed to further investigate student understanding of the mathematical nature of chemistry concepts, such as rate constants. Additionally, for clarification, our analysis involved establishing a hierarchy between students’ conceptions, but our intention was to characterize reasoning in a way that is descriptive (not perspective); that is, developing a learning progressione.g., E
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ACKNOWLEDGMENTS The National Science Foundation under grant DUE-1504371 supported this work. Any opinions, conclusions, or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the National Science Foundation. We wish to thank Tom Holme, Ryan Bain, and the Towns research group for their support and helpful comments on the manuscript.
in the settings from which these participants came. Increasing the focus on the Arrhenius equation could be one way for students to develop a deeper understanding of rate constants, as well as distinguish between rate and rate constant. Participants characterized at the highest level of sophistication often leveraged the Arrhenius equation to define rate constants and discuss factors that affected the rate constant quantity. Additionally, utilizing this equation brings the chemistry being modeled to the forefront of the kinetics unit via mathematical terms such as the pre-exponential factor, temperature and activation energy. Further work is needed to explore student knowledge related to rate and rate constants in a chemistry context. One such avenue of study could be informed by research in undergraduate mathematics education. Such work suggests that metonymy, the substitution of a word or phrase for another that is related to it, is common in mathematical contexts. Zandieh and Knapp32 studied this in the context of derivatives and found that students can use metonymy in both productive and unproductive ways. More exploration into chemistry student understanding of the distinction and relationship between rate and rate constant may reveal modifications to the levels of sophistication reported herein, perhaps due to metonymy-related findings. Though we did not specifically characterize our data in order to study this, the interviewers often probed students regarding their use of the terms “rate” and “rate constant”; this sometimes revealed that students used one as a “stand in” for the other, while other times students were stating that they were equivalent. Finally, findings from this project indicate that an important instructional target for undergraduate chemistry (and likely other science and mathematics) courses is a nuanced understanding of the distinction between constants, parameters, and variables. Because names like “rate constant” and “equilibrium constant” may be misleading for students, explicit discussion of the mathematical nature of equation terms is important in developing a deep understanding of the chemistry being mathematically modeled. Additional research is needed to study how students reason about the mathematical nature of equations and terms in chemistry contexts. Not only do constants, parameters, and variablesnot to mention equationslook different in the physical sciences when compared to mathematics contexts, but they are also used for different purposes.12−14 Scientists load meaning onto symbols, which necessarily changes how equations are interpreted and used. How students understand the relationship between their mathematics and chemistry knowledge needs much more exploration if we are to support student engagement in scientific practices, such as developing and using models or using mathematical and computational thinking.4,20,33
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REFERENCES
(1) Cooper, M. M.; Posey, L. A.; Underwood, S. M. Core Ideas and Topics: Building Up or Drilling Down? J. Chem. Educ. 2017, 94, 541− 548. (2) Holme, T.; Luxford, C.; Murphy, K. Updating the General Chemistry Anchoring Concepts Content Map. J. Chem. Educ. 2015, 92, 1115−1116. (3) Holme, T.; Murphy, K. The ACS Exams Institute Undergraduate Chemistry Anchoring Concepts Content Map I: General Chemistry. J. Chem. Educ. 2012, 89, 721−723. (4) Laverty, J. T.; Underwood, S. M.; Matz, R. L.; Posey, L. A.; Jardeleza, E.; Cooper, M. M. Characterizing College Science Assessments: The Three-Dimensional Learning Assessment Protocol. PLoS One 2016, 11, 1−21. (5) Bain, K.; Towns, M. H. A Review of Research on the Teaching and Learning of Chemical Kinetics. Chem. Educ. Res. Pract. 2016, 17, 246−262. (6) 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. (7) Cakmakci, G. Identifying Alternative Conceptions of Chemical Kinetics among Secondary School and Undergraduate Students in Turkey. J. Chem. Educ. 2010, 87, 449−455. (8) Tastan, Ö . T. A. S̨ .; Yalçinkaya, E.; Boz, Y. Pre-Service Chemistry Teachers’ Ideas about Reaction Mechanism. J. Turkish Sci. Educ. 2010, 7, 47−60. (9) Singer, S. R.; Nielson, N. R.; Schweingruber, H. A. DisciplineBased Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering; National Academies Press: Washington, DC, 2012. (10) Thompson, P. W.; Carlson, M. P. Variation, Covariation, and Functions: Foundational Ways of Thinking Mathematically. In Compendium for Research in Mathematics Education;Cai, J., Ed., National Council of Teachers of Mathematics: Reston, VA, 2017; pp 421−456. (11) Philipp, R. A. The Many Uses of Algebraic Variables. Math. Teach. 1992, 85, 557−561. (12) Redish, E. F. Problem Solving and the Use of Math in Physics Courses. Proceedings of the World View on Physics Education in 2005: Focusing on Change Conference, New Delhi, India, August 21−26, 2005. (13) Redish, E. F.; Gupta, A. Making Meaning With Math in Physics: A Semantic Analysis. Proceedings of the GIREP International Conference, Leicester, UK, August 17−21, 2009. (14) Tuminaro, J.; Redish, E. F. Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games. Phys. Rev. ST Phys. Educ. Res. 2007, 3, 020101. (15) Hammer, D.; Elby, A. Tapping Epistemological Resources for Learning Physics. J. Learn. Sci. 2003, 12, 53−90. (16) Hammer, D.; Elby, A.; Scherr, R. E.; Redish, E. F. Resources, Framing, and Transfer. In Transfer of Learning from a Modern Multidisciplinary Perspective; Mestre, J. P., Ed.; IAP: Greenwich, CT, 2005; pp 89−119. (17) di Sessa, A. A. Toward an Epistemology of Physics. Cogn. Instr. 1993, 10, 105−225. (18) Wittmann, M. C. Using Resource Graphs to Represent Conceptual Change. Phys. Rev. Spec. Top.-Phys. Educ. Res. 2006, 2, 020105.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Kinsey Bain: 0000-0003-0898-1862 Jon-Marc G. Rodriguez: 0000-0001-6949-6823 Marcy H. Towns: 0000-0002-8422-4874 Notes
The authors declare no competing financial interest. F
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(19) Wilson, M.; Sloane, K. From Principles to Practice: An Embedded Assessment System. Appl. Meas. in Educ. 2000, 13, 181− 208 2000 . (20) Cooper, M. M.; Stowe, R. L. Chemistry Education Research From Personal Empiricism to Evidence, Theory, and Informed Practice. Chem. Rev. 2018, 118, 6053−6087. (21) Bain, K.; Rodriguez, J. G.; Moon, A.; 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. (22) 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. (23) 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. (24) Rodriguez, J. G.; Santos-Diaz, S.; Bain, K.; Towns, M. H. Using Symbolic and Graphical Forms To Analyze Students’ Mathematical Reasoning in Chemical Kinetics. J. Chem. Educ. 2018, 95, 2114−2125. (25) Linenberger, K. J.; Lowery Bretz, S. A Novel Technology to Investigate Students’ Understandings. J. Coll. Sci. Teac. 2012, 42, 45− 49. (26) Strauss, A.; Corbin, J. Basics of Qualitative Research: Grounded Theory Procedures and Techniques; SAGE Publications, Ltd.: Newbury Park, CA, 1990. (27) Campbell, J. L.; Quincy, C.; Osserman, J.; Pedersen, O. K. Coding In-depth Semistructured Interviews: Problems of Unitization and Intercoder Reliability and Agreement. Sociol. Methods Res. 2013, 42, 294−320. (28) Saunders, B.; Sim, J.; Kingstone, T.; Bakjer, S.; Waterfield, J.; Bartlam, B.; Burroughs, H.; Jinks, C. Saturation in Qualitative Research: Exploring Its Conceptualization and Operationalization. Qual. Quant. 2018, 52, 1893−1907. (29) Sherin, B. L. How Students Understand Physics Equations. Cogn. Instr. 2001, 19, 479−541. (30) 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. (31) Rodriguez, J. G.; Bain, K.; Towns, M. H. Graphs as Objects: Mathematical Resources Used by Undergraduate Biochemistry Students to Reason About Enzyme Kinetics. It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics; ACS Symposium Series; American Chemical Society: Washington, DC, 2019; Vol. 1316; pp 69−80. (32) Zandieh, M. J.; Knapp, J. Exploring the Role of Metonymy in Mathematical Understanding and Reasoning: The Concept of Derivative as an Example. J. Math. Behav. 2006, 25, 1−17. (33) Cooper, M.; Caballero, M.; Ebert-May, D.; Fata-Hartley, C.; Jardeleza, S.; Krajcik, S.; Laverty, J.; Matz, R.; Posey, L.; Underwood, S. Challenge Faculty to Transform STEM Learning. Science 2015, 350, 281−282.
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