Mathematics in Chemical Kinetics - ACS Publications - American

Chapter 3

Mathematics in Chemical Kinetics: Which Is the Cart and Which Is the Horse? Downloaded via UNIV OF ROCHESTER on May 13, 2019 at 17:14:54 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

Kinsey Bain,1 Jon-Marc G. Rodriguez,2 Alena Moon,3 and Marcy H. Towns*,2 1Department of Chemistry, Michigan State University, East Lansing, Michigan 48824,

United States 2Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States 3Department of Chemistry, University of Nebraska – Lincoln, Lincoln, Nebraska 68588,

United States *E-mail: [email protected]

This chapter is a summary of the results from a large project that is centered on how students understand and use mathematics in chemical kinetics, with an emphasis on the integration or “blending” of chemistry and mathematics during problem solving. Our primary data came from semistructured interviews with 40 general chemistry students, five physical chemistry students, and three chemical engineering students. During the interviews, the students were prompted to reason about chemical kinetics equations and work through chemical kinetics problems. The unique nature of American Chemical Society Symposium Series books has provided a space for us to bring together the different publications that have resulted from this project. Based on this data set, we have previously discussed student engagement in modeling, mathematical reasoning (i.e., symbolic and graphical forms), reasoning regarding the role of mathematics in chemical kinetics, understanding of rate constants, conceptions of zero-order and half-life, and problem-solving approaches. The aim for this chapter is to provide a brief summary of each study, followed by a discussion of the insight gained by looking at all of these studies together, with particular attention to addressing whether chemistry or mathematics drives student reasoning—which is the cart and which is the horse?

Introduction This chapter presents a synthesis of our recent work at the interface of chemistry and mathematics education research (1–6). One of the driving motivations in designing this study was that mathematics is the language we use to describe and model phenomena, and not surprisingly, © 2019 American Chemical Society

Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

mathematical ability is correlated to success in undergraduate chemistry courses (7–17). Given the utility of mathematics in modeling phenomena and its ubiquity across science and engineering, education researchers across science and math disciplines have been exploring how students understand and use mathematics. From a purely mathematical perspective, symbols and symbolic notations are complex, due to the magnitude of information encoded and the variety of ways they can be interpreted and used (18–23). Similarly, mathematical reasoning related to graphs and understanding the information communicated in a graph is challenging (24–26). However, there is a clear distinction between how mathematicians and physical scientists use and conceptualize mathematics (27, 28), which provides insight regarding why students have difficulty applying the mathematics they learn to physical science topics (8). Thus, the ability to reason using equations and graphs is further complicated when considering the additional layer of conceptual reasoning introduced in physical science topics, in which students are expected to model abstract phenomena using abstract mathematical formalisms (29–34). It is within the context of this rich body of literature that spans multiple disciplines that we are interested in the interface between chemistry and mathematics—how students use both mathematics and physical knowledge, how they interpret data and graphical symbolisms, and how competency in one (mathematics) predicts success in another (physics or chemistry). Unifying these research interests is a larger question: How do physical and mathematical knowledge interact as students learn and apply mathematical models in physical contexts? The aim of this chapter, then, is to shed light on this question with results from a project in which we have explored aspects of student problem solving in the context of chemical kinetics (1–6). This context, in which mathematics is used to model the rate of chemical processes, is ripe for exploring the interaction between students’ mathematical knowledge and chemical knowledge. Furthermore, our investigation of how students understand and use mathematics in chemistry contexts has led us to consider the extent to which student reasoning is driven by mathematical or chemistry reasoning: which is the cart and which is the horse?

Chemical Kinetics As discussed in topical reviews related to research situated in chemical kinetics, previous work has primarily focused on elucidating the various alternative conceptions exhibited by students (34, 35). This body of literature provides many claims regarding what students do not know, but as discussed by Cooper and Stowe (36), more work is needed to better understand what students do know and how we can use this knowledge productively: “… there is little to be gained by simply cataloging misconceptions without paying heed to the mechanisms of their emergence, their organization, and their character.” In line with this sentiment, recent work by Becker and colleagues (37, 38) has focused on analyzing student reasoning using a method of initial rates tasks to inform how instruction can support student learning. The project that we describe herein seeks to further build on this work. For more information regarding previous work related to chemical kinetics, we direct the readers to recent reviews (34, 35).

Theoretical Perspectives Blended processing served as the primary theoretical underpinning when designing this study, influencing all aspects from participant sample selection to interview prompts and analysis. This framework comes from cognitive science and describes human information integration (39, 40). 26 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

From this perspective, interactions of knowledge structures (or mental spaces) are “blended” in a way that an external stimulus can be made sense of in an emergent fashion (39, 41). This framework was chosen because in many instances chemistry and mathematics knowledge are used and required to make sense of phenomena. When these two mental spaces, chemistry and mathematics, are blended together, the understanding that results is greater than the sum of its parts; it is an emergent and synergistic understanding that requires selective blending of information from each space. In our project, we set out to describe how students understood and used mathematics in the context of chemical kinetics, seeking to describe their individual chemistry and mathematics mental spaces, as well as the blended processing that occurred during individual student interviews. To do this, we also utilized the resources framework (42–44), which describes knowledge as being constructed in the mind of the learner (consistent with other constructivist frameworks); however, the resources framework describes this constructed knowledge as composed of finegrained resources that are not necessarily neatly connected or organized, as opposed to unitary stable entities. These resources form an interactive, dynamic network that varies in complexity and can be disconnected or fragmented in some areas. Resources are also described as being activated in specific contexts; such activation can be productive or unproductive depending on the given situation. This framework accounts for the observed inconsistency in student responses. Influenced by themes that emerged from the data, observed patterns in activated student resources led to multiple analyses reported in various papers and summarized in this chapter. In addition to conceptual resources, such as ideas about chemistry, we were interested in mathematical resources. One example of mathematical resources are the symbolic forms described by Sherin (45), which reflect intuitive mathematical ideas about equations. The symbolic forms analytic framework was initially developed to characterize the mathematical reasoning used by physics students as they solved problems (45), and most of the symbolic forms identified by Sherin (45) primarily revolve around algebraic operations (e.g., considering proportional relationships, the influence of a coefficient, dependence of terms on other values, etc.). However, Sherin (45) acknowledged that his initial list was not exhaustive, and the framework has been utilized by researchers across disciplines to describe a range of mathematical ideas, including more advanced topics, such as integrals, differentials, and vectors (45–54). Sherin’s symbolic forms (45) describe mathematical resources that involve assigning ideas (conceptual schema) to a pattern in an equation (symbol template). Reasoning using symbolic forms derives its utility from how it supports reasoning about phenomena (8, 55), in a way that is independent of context, since many equations share the same recognizable pattern (e.g., base ± change characterizes expressions in physics used to describe motion, or in more general terms, the slope–intercept form for the equation of a line). We will continue our discussion of symbolic forms in the results section, since it is particularly relevant for our study that focused specifically on students’ mathematical reasoning (2).

Methods Participants were sampled from a large midwestern university during a single academic year (Table 1). Most participants were recruited from a second-semester general chemistry course intended for engineering majors. Other nonchemistry major participants were recruited from two upper-level courses: a physical chemistry for life sciences course and a chemical reactions engineering course. The pilot interviews were conducted with four students from the fall general chemistry II 27 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

course prior to conducting the full study with the remaining participants. The institutional review board approved the study design and interview protocol prior to recruitment and implementation of the project. All participants were recruited through an in-class announcement and follow-up email, and students were compensated with a $10 iTunes gift card. Participant names reported herein are pseudonyms. Table 1. Number of Interview Participants by Course and Semester Number of Interview Participants Recruited Participant Courses

Fall 2015

Spring 2016

Pilot study (general chemistry II)

4

-

General chemistry II

17

19

Physical chemistry for life sciences

-

5

Chemical reactions engineering

-

3

The primary data for this study were individual interviews that used a think-aloud protocol (8). Student audio and written data were recorded in real time using a Livescribe smartpen (56). The four interview prompts were printed on Livescribe paper, while all probing questions were asked by the interviewer (57). The interview prompts were designed so that each participant engaged with two mathematics prompts and two chemistry prompts. To investigate priming effects, we alternated whether participants saw the mathematical prompt or the chemistry prompt first within the two pairs of questions (second-order and zero-order) (Figure 1).

Figure 1. Two possible sequences of interview prompts, “chemistry first” (top) or “math first” (bottom). The interview prompts were designed in a manner similar to Kuo et al. (55), where the mathematics prompts provided students with a familiar equation (second-order integrated rate law or zero-order integrated rate law) and the chemistry prompts were reminiscent of chemistry problems they had done in course-related activities. The chemistry prompts were written so that they could be solved in a number of ways, such as using conceptual understanding about mathematical relationships in chemical kinetics or solving for values using chemical kinetics equations and the provided data (Figure 2).

28 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Figure 2. Second-order (top) and zero-order (bottom) chemistry interview prompts. Audio data from the interviews were transcribed verbatim. Images of written student work were then incorporated into each transcript. Interpreted narratives were then created for each interview, organizing the data into “steps” of problem solving as a means of preliminary analysis that allowed us to manage our multimodal data sources (58). Data analysis continued through open coding keeping in mind the frameworks underpinning the study. The resulting codes characterized problem-solving approaches, various activated resources, and blended processing (Figure 3). All interviews were coded by two researchers, and any disagreement was reconciliated, requiring 100% intercoder agreement (59). Constant comparison was used throughout the coding process to refine and modify the coding scheme and its application (60). For each of the studies summarized in this chapter, analysis of data from different participants and prompts was conducted and is detailed in the sections below. Furthermore, in order to help illustrate the themes in this chapter, we created resource graphs, which visually indicate links between different resources activated in a specific context (43). Examples of resource graphs and further discussion about resource graphs are provided in subsequent sections of this chapter.

Results and Discussion In the sections that follow, we detail the main analyses conducted as a part of this large-scale project. The first study we report on is the characterization of blended processing, the original intent of this work. We then discuss an analysis of the mathematical resources activated in a purposefully sampled subset of the project participants, followed by a study that focused on student reasoning about the use of mathematics in chemical kinetics. Studies regarding student understanding of the mathematical nature of rate constants and zero-order contexts are then discussed. Next, we review our analysis of productive features of problem solving among the general chemistry II participants. Finally, we further illustrate the findings of these studies by using an exemplar from the data in order to afford deeper insight.


Figure 3. Outline of the open coding scheme that emerged from the data analysis, broadly characterizing problem-solving approaches, activated resources, and blended processing.

Characterizing Cognitive Processing Using a Blended Processing Framework The primary aim of our large-scale project was to investigate how students understood and used mathematics in chemical kinetics contexts through individual, semistructured interviews. As described above, our prompt and probing questions were designed to elicit chemistry and mathematics resources and the opportunity to engage in blended processing involving these mental spaces. Our open coding revealed many chemistry and mathematics resources activated, as well as instances where blended processing occurred. We further characterized blending themes by describing common contexts when blending occurred, perceived directionality of blending, varying quality and complexity of blending, and trends in problem solving (1). This analysis encompassed all interview data (every participant and both prompts). Blended processing typically occurred when participants were discussing three contexts: molecularity and order, catalysts, and concentration. When discussing molecularity and order, students often discussed the interaction of particles (molecularity) and its relationship to order. This was typically tied to formalizing order mathematically, as a number and in at least one equation (e.g., rate law). Catalysts were also often at the center of blended processing instances, where rate dependence was often considered. Particulate-level discussions of catalyst mechanisms (e.g., reaction at a metal catalyst surface) often tied to implications for the mathematical rate. These students often recognized that a first- or second-order rate law model would not suffice, as the rate depended on other factors besides reactant concentration, such as surface area. The most common context for blended processing was the discussion of how reactant concentration changes throughout the duration of a reaction. While most students could identify that reactant concentration decreases during a reaction, those that exhibited blending tied this mathematical behavior to the physical 30 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

phenomena. Many went further to provide implications for other reacting species, other features of the mathematical model (typically the integrated rate law), or both. Another theme that emerged from our analysis was that there seemed to be directionality associated with the blended processing. Students often appeared to begin reasoning in one mental space (e.g., mathematics), move or make connections to another mental space (e.g., chemistry), and then blend these spaces together, demonstrating an integrated understanding, an emergent claim, or both. A general example of “chemistry to mathematics” blending is depicted in Figure 4. In these instances, blended processing was clearly anchored in one of the domains, most often mathematics. However, this was not always explicitly clear, as interview participants do not always voice all of their thoughts aloud. Further, in some cases, students weaved back and forth between mental spaces numerous times, making it hard to determine if blending was anchored in one primary mental space. Future studies are needed to further investigate and characterize blending with special attention to anchoring or directionality of blending.

Figure 4. An example of one type of blending observed and characterized, where this type originated in the chemical domain (enzyme–substrate interaction) and moved to incorporate the mathematical domain (rate implications).

The quality of blending also varied, as participants revealed a spectrum. While future studies could expand on this characterization, we classified blending into low and high categories within each common blending context. For example, when blended processing occurred during discussion of concentration, the number of connections students were making between chemistry and mathematics differed—a student simply tying the mathematical decrease in reactant concentration to the chemical processes is vastly different than others who tied these same ideas to other features of the relevant integrated rate law (e.g., signs, variables, constants) or graphical representations. The final blending theme related to trends in problem solving. We compared students who exhibited five or more instances of blending, high-frequency blenders (5 students—4 general chemistry, 1 physical chemistry), with those who did not blend during their interview, nonblenders (12 students—8 general chemistry, 2 physical chemistry, 2 chemical engineering). High-frequency blenders often began problem solving by predicting or answering conceptually, then supporting their claim through mathematical calculations. Many also suggested alternative approaches to solve the problem. These students were also more likely to consider the empirical nature of the data provided and its implications for their calculations. These problem-solving themes are in contrast to those of the nonblenders. Nonblenders exhibited greater variation in their approaches. They were much less likely to approach the problem conceptually, often used numerous unproductive problem-solving routes, and were more likely to reach an incorrect conclusion based on their calculation (even when the calculation was correct). Nonblenders also encountered “dead starts” and “dead ends,” not 31 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

knowing how to solve the problem and not providing any answer at all (61). Interestingly, when considering the priming effect of prompt ordering in each question pair, all high-frequency blenders had the math prompt first, followed by the chemistry prompt, while the nonblenders were evenly distributed between math first and chemistry first (Figure 1). Analyzing Mathematical Reasoning Using Symbolic and Graphical Forms Building off the results from the initial paper that characterized student engagement in blending (1), we purposefully sampled four student interviews from the larger sample to analyze using the symbolic forms framework, with the intention of comparing the mathematical reasoning of highfrequency blenders and nonblenders (2). This analysis utilized student responses to all four prompts, and we selected interviews for additional analysis using the following criteria: general chemistry students interviewed during the spring of 2016 (i.e., the students had the same instructor), “mathfirst” students (to account for any potential priming effects related to prompt order), and students that had the correct final answer to both chemistry prompts in their interview. For clarification, we selected students that had the correct answer to the chemistry prompts because we wanted to illustrate that our goal for instruction should move beyond simply getting the correct answer, and as illustrated by our analysis, simply getting the correct final answer is not an indication of a deeper and more meaningful understanding of chemical phenomena. Based on our selection criteria, we analyzed two high-frequency blender interviews (Steven and Howie) and two nonblender interviews (Louis and Isabel) using the symbolic forms framework (45). During the process of analyzing the data for our project, we noted that, in addition to reasoning that could be characterized using Sherin’s symbolic forms (45), students also illustrated analogous reasoning about graphs, which led us to consider how we can expand the symbolic forms framework to analyze graphical reasoning. Thus, based on this data set we developed a new construct, “graphical forms,” which characterizes students’ intuitive ideas about graphs and involves associating mathematical ideas to regions in a graph (or an entire graphical shape). For example, consider the graphical form steepness as rate, which describes reasoning that involves associating rate with the relative steepness of a curve. Due to limited space, only a brief summary of graphical forms is provided here, but for more information we direct readers to a forthcoming paper that provides a review of previously identified symbolic forms, describes our conceptualization of graphical forms, and illustrates its utility for analyzing graphical reasoning across disciplines (62). Moreover, in a recently published paper, we utilized the graphical forms framework to analyze students’ graphical reasoning, focusing on student engagement in covariational reasoning (note that this involved a data set distinct from the work described in this chapter) (63). Analysis revealed that in comparison to the nonblenders (Louis and Isabel), the high-frequency blenders (Steven and Howie) utilized a larger variety of symbolic and graphical forms, which were used to describe more varied contexts. Thus, we posit that students that have access to more symbolic and graphical forms are able to better reason about a phenomena from different perspectives, providing a more complete understanding of a chemical system. As discussed in the broader chemistry education literature, mathematical ability is correlated with success in undergraduate chemistry (7–17). We contribute to this body of literature by asserting that mathematical reasoning (engaging in symbolic and graphical forms) has the potential to afford students a deeper understanding of concepts across the physical sciences by supporting engagement in blended processing, a sentiment shared by Kuo et al. (55).


Describing Student Reasoning about Mathematics in Chemical Kinetics Whereas in the previous paper we focused on student mathematical reasoning (using symbolic and graphical forms) (2), in this work emphasis was placed on the general chemistry student (n = 40) reasoning about mathematics and its use in chemical kinetics (3). This chapter is distinct from the other papers discussed in that it was written with the intention of disseminating the results to researchers and practitioners interested in undergraduate mathematics education. In this work, we illustrated how chemistry is a rich context to evaluate student understanding of mathematics, encouraging instructors to use chemistry contexts in their mathematics courses and suggesting researchers expand their interests to include the application of mathematics that occurs outside of mathematics courses. Findings in this work indicate students often expressed a preference for reasoning algebraically (rather than conceptually) to solve problems. This was reflected in student conceptions of chemical kinetics equations, which tended to revolve around their use in solving for a value, demonstrating a limited understanding of the purpose and utility of equations in modeling phenomena. This algorithmic approach to chemistry is problematic due to the deeper understanding afforded when students blend chemistry and mathematics (as discussed above). Further illustrating this theme, students were often unable to describe how the integrated rate law was related to the rate law, suggesting students were not using the resources learned in mathematics courses (e.g., the process of integration) in this chemistry context. Taken together, these themes reflect compartmentalized reasoning—students are not integrating chemistry and mathematics ideas—indicating students need more support in order to productively apply and combine the knowledge and skills learned in one course in new contexts. Examining Student Understanding of Rate Constants This study emerged from our blending analysis as well, as there was much variation in student understanding of rate constants. We analyzed all participant interview data, revealing a few key themes (4). First, students often conflated rate constants with equilibrium constants, a finding corroborated in the literature (34, 37, 64, 65). We also found that students often had difficulty differentiating the concepts of rate and rate constant. While some explicitly conflated these two concepts, others displayed varying degrees of complexity in how and why rate and rate constant were distinct. Perhaps more interestingly, we found that there was a spectrum of sophistication in understanding the mathematical nature of rate constants. While some students conceived of rate constants as universal constants that never change, others were able to articulate how, why, and when rate constants are different (describing the conditions necessary for a rate constant to be constant). An example of a more sophisticated set of activated resources relating to rate constant is depicted in the resource graph provided in Figure 5. This study highlights an important need to explicitly address the distinction between types of constants used (e.g., rate constant, equilibrium constant, universal constants) and related concepts (e.g., rate and rate constant). Further, there is a clear need for instruction to address the mathematical nature of terms. Building on findings from other studies, such as the blended processing work (1), this study reveals a need to emphasize the differences between constants, parameters, and variables. Rate constants are parameters that are constant given that certain terms are unchanged. Such nuances are not obvious to many students and limit their understanding.


Figure 5. Resource graph of high-frequency blender, Steven (general chemistry student), showing a sophisticated network of activated resources relating to rate constants. Investigating Student Understanding of Catalysts and Half-Life in Zero-Order Contexts During our analysis, we also noticed rich discussion of the chemistry being modeled in zeroorder systems. We further characterized all participant data from the chemistry and mathematics zero-order prompts (5). Independent of prompt order, students demonstrated great facility with zero-order systems with respect to their mathematical definition, associated equations and graphs, as well as the constant nature of rate and its independence of reactant concentration. One of the more sophisticated resource graphs regarding zero-order reactions is shown in Figure 6. However, in most cases students did not understand the particulate-level chemistry modeled by zero-order kinetics. Most participants stated that the order of the system was not affected by the presence of a catalyst, and less than half knew that a catalyst physically interacts with reaction species, influences the reaction mechanism, or both. Among other findings, we also found that students often used first-order kinetics reasoning inappropriately when considering zero-order systems. The zero-order chemistry prompt (bottom of Figure 2) asked students to consider how the half-life of a zero-order reaction might change if the initial reactant concentration were to double. While most students did answer correctly that the half-life would double, they often defined half-life using language that indicated students were thinking about radioactive decay concepts, such as radioactivity or carbon dating. Such reasoning also influenced the observed problem-solving trends. Many students engaged in conceptual reasoning, predicting that the half-lives should be the same, which is only true for the half-life of a first-order system. Continuing to work through the problem by solving mathematically, some were unable to reconcile the difference between their conceptual idea of a constant half-life and the conflicting result of their calculation. These findings indicate that instruction needs to guide and help students activate and use the appropriate resources for a given context. Additionally, it is likely that students would benefit from varied examples of reaction order and half-life, promoting the development of a more robust understanding.


Figure 6. Resource graph of high-frequency blender, Steven (general chemistry student), showing a sophisticated network of activated resources relating to zero-order reactions. Exploring Productive Features of Problem Solving in Chemical Kinetics In this chapter we focused on the different approaches used by the general chemistry students (n = 40) to solve the two chemistry prompts, framing each of the problem-solving strategies as student responses to the activation of resources (6). Thus, problem-solving routes reflected how students made sense of and made use of the activation of resources. Based on the context in which a student used a particular problem-solving route, it was characterized as productive or unproductive in terms of moving students closer to the final correct answer, as depicted in the problem-solving map (Figure 7). Across our data set, students utilized a variety of problem-solving approaches that varied in terms of how productive they were for reasoning through the problems. In the case of the second-order chemistry problem, which asks students to compare the rate constant between two reactions, students had difficulty reasoning about the data table provided, with some students utilizing problem-solving routes that indicated they were not considering the nature of the values provided in the table. For example, some plugged the provided concentration and time values directly into the rate law, set up a ratio of the values in a way that is similar to the method of initial rates, or attempted to determine patterns and trends in the data by qualitatively or quantitatively comparing the values in the table. Furthermore, these problem-solving routes were often paired together, in which students tried multiple unproductive problem-solving routes, resulting in an incorrect or undecided final answer. For the zero-order chemistry problem, students were prompted to reason about the half-life of a zero-order reaction, and as discussed previously (5), students often overgeneralized first-order half-life reasoning to describe zero-order reactions. Consistent with the results from the other studies related to this data set, the importance of conceptual reasoning was one of the key themes in this paper. For example, for the second-order prompt, students that initially responded to the prompt using conceptual reasoning were more likely to get the answer correct (as opposed to quickly plugging values into an algorithm). Along the same lines, our results emphasized the importance of students utilizing conceptual reasoning, not just to provide an answer, but to


evaluate their work and consider the feasibility of a problem-solving approach (engaging in reflection and self-regulating behavior).

Figure 7. A problem-solving map of the second-order chemistry prompt that was ultimately solved correctly through a series of productive and unproductive steps by high-frequency blender, Steven (general chemistry student). Illustrating the Broader Themes: The Case of Steven In the sections above, we provided a general overview of each of the studies without diving into the data, due to the limited space available. In this section, we utilize a case study approach to illustrate the themes across each of the studies. For the presentation discussed herein, we selected Steven based on his more sophisticated reasoning and problem solving in comparison to the larger sample (recall that Steven was a general chemistry student characterized as a high-frequency blender) (1, 2). Nevertheless, it should be noted that Steven reflects a special case and not all students were able to productively blend chemistry and mathematics ideas, with most students in the sample reflecting a preference for algorithmic approaches, as opposed to solving conceptually. The rationale for selecting a more effective problem solver was to illustrate the insight gained and the understanding afforded when students more frequently integrate chemistry and mathematics ideas. In Table 2 we have provided two examples in which Steven (high-frequency blender) engaged in blended processing, juxtaposed with Isabel (nonblender) discussing the same contexts. For example, when discussing order, Steven mentioned the concentration dependence of rate and related it to molecularity, whereas Isabel’s discussion focused on the algebraic expression of order (i.e., exponents in the rate law). Similarly, in contrast to Isabel, Steven was able to provide an explanation for what could account for the constant rate of a zero-order reaction. In addition to exhibiting blended processing in multiple contexts, this high-frequency blender, Steven, demonstrated activation of many productive resources throughout his interview. When considered from a symbolic and graphical forms perspective, Steven displayed a wide variety of productive mathematical ideas, which were used to describe many contexts (2, 3). This ability afforded him a deep, connected understanding of chemistry compared to many of the other participants. As seen in Figures 5 and 6, Steven also demonstrated sophisticated and nuanced activation of resources relating to rate constants and zero-order reactions (4, 5). He was able to clearly distinguish between related concepts (e.g., rate and rate constant) and between different reaction orders (e.g., half-life for first- and zero-order). The productive activation of these highly interconnected resources contributed to his holistic understanding and successful problem solving.


Table 2. Student Discussions that Highlight the Differences Between Student Responses that Involve Blending (Steven, High-Frequency Blender) and Those That Do Not (Isabel, Nonblender)


To convey how many of these aspects contributed to his successful problem solving (6), we have compiled two large figures (Figures 8 and 9). When solving the second-order chemistry problem, Steven began conceptually reasoning and avoided a common pitfall of getting caught in the data (possibly doing a rate calculation)—largely as a result of engaging in reflection about his problem solving. Similarly, during the zero-order chemistry problem solving, he initially used conceptual reasoning and avoided the common urge to inappropriately utilize first-order half-life thinking. Figures 8 and 9 further depict how the successful problem-solving routes related to his activation of sophisticated and interconnected resources and blended processing, in which the examples of blended processing involve the use of graphical and symbolic forms. While reasoning of this nature was not pervasive among the participant sample, these excerpts are meant to show the type of understanding that is possible among chemistry students, even at a general chemistry level, such as Steven. This is an example of interconnected, contextualized, and useful knowledge that most instructors desire for each of their students, here in the context of chemistry and mathematics. If instruction were designed to explicitly develop and foster blended processing and understanding such as this, we would have more students like Steven continuing on in science, technology, engineering, and mathematics fields.

Figure 8. An overview of Steven’s problem solving of the second-order chemistry prompt, including his problem-solving map, associated resources about rate constants, and blended processing in the context of the rate law.


Figure 9. An overview of Steven’s problem solving of the zero-order chemistry prompt, including his problem-solving map, associated resources about zero-order reactions, and blended processing in the context of catalysts.

Discussion and Implications Reflecting on the results from each of our studies, a common theme is that the students were more comfortable utilizing algorithmic approaches in comparison to reasoning conceptionally. This is perhaps not surprising, given the large body of literature that has reported similar results regarding student ability to solve conceptual problems in comparison to algorithmic problems across various chemistry contexts (66–76); however, whereas this literature base is framed in terms of “conceptual understanding versus problem solving” (which may perpetuate compartmentalized reasoning), our work investigated the importance of utilizing conceptual understanding during problem solving. In our data set, this emerged as instances in which students blended chemistry and mathematics ideas, affording a more sophisticated understanding of chemical phenomena. In addition, students in our sample utilized conceptual reasoning to reflect and evaluate their thinking process, exhibiting selfregulating behavior. Integrating chemistry and mathematics and engaging in reflection serves to move the problem solver beyond the dichotomy of conceptual and algorithmic problems toward reasoning that is effective and sophisticated by being rooted in qualitative reasoning, rather than immediate algorithmic processing (37, 38, 76–79). Additionally, student ability to blend knowledge from various domains is critical, as it is the finegrained processes that comprise engagement in three-dimensional learning (3DL)—a goal recently put forth in the Framework for K–12 Science Education (80) and adapted for use at the university level (36, 81–86). The Framework describes science learning as the development and use of 39 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

interconnected, contextualized knowledge; that is, knowledge is tied to fundamental core ideas, used through engagement in scientific practices, and connected to other domains through crosscutting concepts (the three dimensions in 3DL) (36, 80). Further work is needed to understand the use of blended processing by undergraduate students, as well as how instruction can promote it to support 3DL. As illustrated with the case of Steven, sophisticated reasoning that exhibits meaningful connections between ideas is a reasonable instructional target for students; however, this type of reasoning was less common in our sample. Findings from our studies, particularly the blended processing analysis, revealed that students are more comfortable with and commonly use mathematics to reason and solve chemistry problems. Students indicated preference for algorithmic problem solving over utilizing conceptual reasoning, and even when exhibiting blending, it was typically anchored in mathematical reasoning and then incorporated chemistry (1, 3). This general sentiment was further reflected in student problem-solving approaches, which often involved an emphasis on equation recall, without considering if the algorithm was appropriate for the context, a trend that was observed for both the second-order and zero-order chemistry prompts (6). For example, students often generalized the first-order half-life equation to use for other reaction orders (5) or focused on surface-level features of mathematical expressions (4)—reasoning that would benefit from the incorporation of chemistry ideas. This type of understanding of and engagement with chemistry concepts is antithetical to the vision of 3DL. Based on secondary data sources collected during our study, such as classroom observations and course materials, it seems possible that this type of chemistry understanding and engagement is promoted by instruction and assessment that highly values rote learning and skillsbased assessments. Student learning is driven by what instructors both implicitly and explicitly convey to be important, chiefly through what is assessed and graded (87). Instruction and assessments should reflect the values of the 3DL framework, infusing core ideas, science practices, and crosscutting concepts into the classroom and assessments. While rules- and skills-based questions are needed, so too are questions that target fundamental concepts in chemistry (core ideas), the processes of doing science, and connections across disciplines (36, 81–86). The work presented in this chapter affords a few implications for instruction and assessment: utilize contexts where mathematics and chemistry modeling connections can be emphasized (e.g., reaction concentration, order, molecularity, catalysts), use laboratory coursework as an opportunity for students to make connections between mathematics and chemistry and to reason about models (e.g., limitations, assumptions, empirical basis), and design assessment questions that require the integration of chemistry and mathematics (the Three-Dimensional Learning Assessment Protocol (82, 84) is a great tool for writing and modifying assessment items). Among the participants in our project, it was clear that mathematics was driving student understanding and problem solving in chemistry—math as the “horse,” with chemistry coming along for the ride. Though this is a limited data set, the large midwestern university from which our data was collected seems to be representative of many schools with large class sizes and traditional chemistry curricula. Given that the primary goal for learning in a chemistry course should be learning chemistry, instruction and assessment should more closely align with the 3DL framework, moving chemistry core ideas to the center. In this light, mathematics becomes one of the many ways scientists use their knowledge through engagement in scientific practices, such as developing and using models; thus, mathematics becomes a tool that can be used to support an understanding of chemistry. Although our work reveals that mathematics seems to be the proverbial horse pulling the cart, along with other authors in this collection (88), we believe that chemistry instruction and 40 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

assessment should transform to develop a robust connection between chemistry and mathematics, creating a yoke that facilitates “deep, transferable knowledge” (81).

Acknowledgements This work was supported by the National Science Foundation under grant DUE-1504371. Any opinions, conclusions, or recommendations expressed in this chapter are those of the authors and do not necessarily reflect the views of the National Science Foundation. We thank Tom Holme, Ryan Bain, and the Towns Research Group for their support and feedback on this project.

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