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Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

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Chemistry and Mathematics: Research and Frameworks To Explore Student Reasoning 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, University of Iowa, Iowa City, Iowa 52242, United States § Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States Downloaded via VOLUNTEER STATE COMMUNITY COLG on August 16, 2019 at 04:12:53 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

ABSTRACT: Chemistry education research (CER) as a field is inherently interdisciplinary and has traditionally borrowed ideas, frameworks, and methodologies from other fields, such as anthropology, psychology, cognitive science, and science education more broadly. In this paper we encourage researchers to continue this tradition by utilizing the rich body of literature that exists outside of CER, specifically looking to communities interested in mathematics education and physics education to expand the research carried out in our field. Previous work situated in university-level mathematics and physics involves considerations and concerns shared by the chemistry education community, such as integrating conceptual and mathematical reasoning during problem solving. We also briefly discuss and draw attention to a recent ACS Symposium Series book, It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics, that illustrates the rich area of inquiry at the interface of chemistry and mathematics. An overview of frameworks that have been productive for investigating students’ mathematical reasoning in a chemistry context is provided, encouraging researchers to engage in work that crosses disciplinary boundaries and promotes discussion between experts from different fields. KEYWORDS: General Public, Interdisciplinary/Multidisciplinary, Mathematics/Symbolic Mathematics, Chemical Education Research, Analogies/Transfer, Problem Solving/Decision Making FEATURE: Chemical Education Research

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s discussed by Cooper and Stowe,1 the use of theoretical frameworks and learning theories to create a foundation for a study has become more common in chemistry education research (CER). Moreover, the importance of theory-based research has been stated, and there is a push from the community to move toward situating work using theoretical or conceptual frameworks, which requires intentionality to draw connections among the research questions, theoretical underpinnings, methods, and framing of the results for the larger education community.2−5 In terms of the direction for future research carried out in CER, more work is needed that moves beyond alternative conceptions and operates across interdisciplinary boundaries, with the interface between chemistry and mathematics being a rich area for inquiry, particularly in upperlevel chemistry courses that tend to be math-heavy and less represented in the education literature.1,5−9 Previous research related to mathematical reasoning indicates mathematical ability is a predictor of success in chemistry across the undergraduate curriculum6,10−20 and that students tend to perform better on algorithmic problems as opposed to conceptual problems.21−29 Upon the basis of the results of our research, we have advocated for instruction and assessment to emphasize not only algorithmic and conceptual © XXXX American Chemical Society and Division of Chemical Education, Inc.

reasoning but also meaningful blending of mathematical and chemical reasoning during problem solving.30−32 It is also important to note that mathematics is framed differently and used differently by mathematicians in comparison with how physical scientists use mathematics.33,34 This distinction creates a challenge in teaching and in research that focuses on how students understand and use mathematics, acknowledging that the role of mathematics in science, technology, engineering, and mathematics (STEM) fields is complex and worth investigating from a variety of perspectives. The purpose of this article is to review and highlight frameworks from other disciplines that may be useful in characterizing mathematical reasoning in chemical contexts. We value and acknowledge the importance of other areas of inquiry that are grounded in sociocultural, organizational, and other systemic perspectives, which may further broaden the research foci in CER. For some time, mathematics education researchers have been using sociocultural theories of learning Received: June 4, 2019 Revised: August 2, 2019

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DOI: 10.1021/acs.jchemed.9b00523 J. Chem. Educ. XXXX, XXX, XXX−XXX

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A Modeling Perspective on Supporting Students’ Reasoning with Mathematics in Chemistry

Mathematics in Chemical Kinetics: Which Is the Cart and Which Is the Horse? Graphs: Working with Models at the Crossroad between Chemistry and Mathematics

Graphs as Objects: Mathematical Resources Used by Undergraduate Biochemistry Students to Reason about Enzyme Kinetics Math Self-Beliefs Relate to Achievement in Introductory Chemistry Courses “But You Didn’t Give Me the Formula!” and Other Math Challenges in the Context of a Chemistry Course Transition of Mathematics Skills into Introductory Chemistry Problem Solving Mathematical Knowledge for Teaching in Chemistry: Identifying Opportunities to Advance Instruction

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Making Sense of Mathematical Relationships in Physical Chemistry

What Education Research Related to Calculus Derivatives and Integrals Implies for Chemistry Instruction and Learning Developing an Active Approach to Chemistry-Based Group Theory

Systems Thinking as a Vehicle To Introduce Additional Computational Thinking Skills in General Chemistry Video-Based Kinetic Analysis of Period Variations and Oscillation Patterns in the Ce/Fe-Catalyzed Four-Color Belousov−Zhabotinsky Oscillating Reaction

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See ref 51.

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The Logic of Proportional Reasoning and Its Transfer into Chemistry

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How Did We Get Here? Using and Applying Mathematics in Chemistry

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Chapter Number and Title

R. Glaser, M. Downing, E. Zars, J. Schell, C. Chicone

T. Holme

A. M. Bergman, T. A. French

S. R. Jones

R. Cole, T. Shepherd

L. A. Posey, K. N. Bieda, P. L. Mosley, C. J. Fessler, V. A. Kuechle D. J. Wink, S. Ryan

B. P. Cooke, D. A. Canelas

K. Bain, J.-M. G. Rodriguez, A. Moon, M. H. Towns F. M. Ho, M. Elmgren, J.-M. G. Rodriguez, K. Bain, M. H. Towns J.-M. G. Rodriguez, K. Bain, M. H. Towns M. R. Mack, C. A. Stanich, L. M. Goldman A. J. Phelps

M. H. Towns, K. Bain, J.-M. G. Rodriguez K. Lazenby, N. M. Becker

Authors

Brief Description

Qualitative interview pilot study that explored mathematics graduate student understanding of group theory and students’ ability to reinvent a classification scheme for chemically important point groups Proposed example of how a systems thinking approach can improve student application of mathematical reasoning in general chemistry Development of a video-based method for measuring the kinetics of oscillating reactions that mathematically mimics reflection UV−vis spectroscopy

Review of the structure of mathematical reasoning and reasoning about quantities in chemistry, supported by a study of student thinking about proportional reasoning in both a general mathematics task and a domain-specific chemistry task Exploration of the use of mathematics in physical chemistry, as well as areas that students struggle to make connections between the mathematics and the physical chemistry concepts or phenomena they represent Literature review of education research on calculus concepts of derivatives and integrals in relation to chemistry topics

Qualitative study using semistructured interviews to investigate how second-year students enrolled in an introductory biochemistry course interpret graphs in the context of enzyme kinetics Quantitative study exploring the effect of students’ math abilities on achievement in chemistry and the mediating role of math self-beliefs underlying this process Qualitative interview study that investigated general chemistry students’ perspective on the issues they face when translating between chemistry problems and mathematical relationships Eight-year study of the relationship between student performance on a quantitative problem-solving assessment instrument and success in a first-semester general chemistry course Initial investigation of what mathematical knowledge for teaching could support chemistry instructors’ efforts to help students develop robust understanding of the mathematics used in general chemistry

Historical perspective on mathematics in chemistry and this history’s relationship to current research on mathematical reasoning in the context of chemistry Review of research on modeling approaches to undergraduate-level chemistry instruction and discussion of the role of epistemological knowledge, specifically metamodeling knowledge, in students’ reasoning with and about mathematical representations Summary of results from a large project 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 Discussion of the design and use of open-ended problems using various frameworks as a lens grounded in analysis of a chemical kinetics task from a recent study to provide practical examples

Table 1. Content Synopsis of It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematicsa

Journal of Chemical Education Article


Journal of Chemical Education

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the mathematics community) has relevance for the use of calculus in chemistry courses. Other notable chapters involve an investigation centered on group theory,52 a discussion of student understanding of proportional reasoning,53 and research on metamodeling related to mathematical representations.54 The resulting book uses frameworks from a variety of disciplines and perspectives to reveal student reasoning in mathematical and chemical contexts across the undergraduate curriculum. A complete list of chapters, including authors and brief descriptions, is shown in Table 1. Chapters include an introduction and historical perspective from the editors, various studies on student engagement in mathematical modeling in chemistry, explorations of student beliefs and perspectives on mathematics in chemistry contexts, investigations of the mathematical knowledge required for teaching and problem solving in chemistry, perspectives from the Research in Undergraduate Mathematics community, and examples of infusing mathematical science practices into the classroom.

with a focus on equity to better describe classroom interactions and to determine what actions might foster more equitable classrooms.35−38 Given that mathematics is the language of the sciences and that mathematical fluency is required to engage in the science practice of modeling, where the model is mathematical in nature and representation, fostering equitable classrooms in mathematics is of importance to scientists. Additionally, institutional perspectives and institutional change are areas of emerging and expanding research importance in mathematics and other STEM disciplines.39−46 Although many change efforts have been supported, documented, and disseminated, the desired transformation of STEM education has not occurred.47−49 As faculty across STEM disciplines and in chemistry specifically (see, for example, Emory University’s initiative which was highlighted as an ACS Editor’s Choice Article50) engage in changing classroom, departmental, and institutional practices, understanding how change occurs, what actions hinder or help it to flourish, and what sustains classroom innovations after the original innovators move on is of critical importance. Narrowing our scope to one aspect of the complex role occupied by mathematics, more chemistry education research is required to investigate how undergraduate students use and understand mathematics in the sciences. We have looked to other fields for frameworks to aid in characterizing such student reasoning in a way that moves beyond simply cataloging alternative conceptions, affording the opportunity to capture various nuances that are to be expected in university-level contexts. In this paper, we emphasize the utility of chemistry education research that uses frameworks from other education communities to characterize students’ mathematical reasoning in chemistry contexts and investigate constructs that are inherent to the use of mathematics, providing examples of how frameworks have been used and suggestions to promote future work. The overview of frameworks herein is not intended to be exhaustive but rather to initiate a dialogue and encourage others to collaborate across disciplines. We begin this discussion by providing a brief overview of a recently released ACS Symposium Series book that illustrates current work at the interface of chemistry and mathematics.

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FRAMEWORKS FOR INVESTIGATING MATHEMATICAL REASONING

Overview

The frameworks described in this section provide an analytical lens to characterize students’ mathematical reasoning and are not intended to be an exhaustive list. Instead, the intention is to open a dialogue within the chemistry education community and continue to encourage research informed by work that spans different fields of study. This discussion begins with a brief overview of the resource-based model of cognition, which is used here to organize and situate the presentation of the various analytical frameworks. Resources: A Manifold View of Cognitive Structure

The resources framework is founded in constructivist ideas regarding building knowledge from experiences, acknowledging the fragmented and manifold view of cognitive structure, which is consistent with the knowledge-in-pieces perspective.55 Within the resource-based model of cognition, resources are conceptualized as fine-grained cognitive units that form complex networks, where the resources can be characterized on the basis of the nature of the ideas discussed (e.g., conceptual resources, mathematical resources, ontological resources, epistemological resources, etc.).56 As an example, in one of the chapters presented above, Lazenby and Becker57 focused on epistemological resources (i.e., ideas about the nature of knowledge and learning). In their chapter, they discussed the utility of framing epistemological resources as metamodeling knowledge: ideas related to the nature and purpose of models, such as the idea that models are empirically derived and context-dependent. In a recently published paper related to this data set, Lazenby and colleagues58 discussed how students have a variety of productive ideas related to the nature of models, but because the students were less likely to view equations and graphs as models, students might not use metamodeling ideas when considering these mathematical representations. The resources framework posits that knowledge structures are emergent and specific resources or clusters of resources are activated in concert by a context. This fractal and manifold perspective of cognition is in contrast to viewing knowledge as stable unitary entities, a view that suggests simply identifying

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IT’S JUST MATH: RESEARCH ON STUDENTS’ UNDERSTANDING OF CHEMISTRY AND MATHEMATICS The authors of this article recently coedited an ACS Symposium Series book, It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics, which was an effort to bring together interdisciplinary research at the interface of chemistry and mathematics.51 The opportunity and inspiration for this stemmed from a pair of symposia organized by the authors at the 2018 ACS National Meeting in New Orleans, LA, and the 2018 Biennial Conference on Chemical Education in Notre Dame, IN. Many presenters from these symposia were invited to contribute chapters to this book. Additional authors were also recruited to expand the scope of chapter types, with a few chapter authors recruited from the Research on Undergraduate Mathematics Education (RUME) community to provide rich interdisciplinary insight. For example, Steven Jones, an associate professor of mathematics education at Brigham Young University, wrote a chapter about research on student understanding of integrals and derivatives and discussed how this research (carried out in C



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and replacing misconceptions.56 Although resources activated by a prompt may not be productive for a particular problemsolving scenario, those same resources may be productive in a different context. Thus, special consideration is given to understanding the activated resources and providing insight regarding how to activate more productive resources. This view of knowledge has implications for curriculum, instruction, and research, in which the goal is to support students in using resources productively, in order to encourage a more sophisticated understanding of phenomena.59,60 As stated by Cooper and Stowe,1 “...there is little to be gained by simply cataloging misconceptions without paying heed to the mechanisms of their emergence, their organization, and their character.” From a theoretical perspective, the resources framework provides the language and the lens to make the transition from focusing on what students do not know to considering how students can use what they do know productively. Recent work situated using the resources perspective has yielded rich data across topics in chemistry17,31,32,59−63 and more broadly in physics and mathematics.64−70 In the sections that follow, the resources perspective is used to frame the remainder of the discussion, in which each of the analytic frameworks can be viewed as tools to characterize mathematical resources (e.g., symbolic forms) or the interaction between resources or groups of resources (e.g., blended processing). To illustrate this idea, a resource graph is provided below in Figure 1, which illustrates

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CONCEPTUAL INTEGRATION: MEANINGFULLY BLENDING RESOURCES Blended processing or conceptual integration is a framework from cognitive science that characterizes the integration of ideas from different knowledge spaces or domains, resulting in a synergistic and unique product.75,76 Within the context of the resources perspective, engaging in blending can be framed as drawing connections between resources or groups of resources.77 In previous work completed by the authors, blended processing was used to characterize student engagement in modeling, focusing on how students coordinated and combined chemistry and mathematics ideas.17,30 Alternatively, in a different study, Rodriguez and colleagues62 focused on knowledge integration by analyzing the mathematical resources students combined with chemistry resources to describe the story associated with a graph, operationalizing Nemirovsky’s construct of mathematical narratives to characterize students’ particulate-level explanations.78 More generally across the physical sciences, blended processing has been used to investigate how students meaningfully combine conceptual and algorithmic reasoning while solving problems.77,79 Here’s Why It Matters: Affording a More Sophisticated Understanding of Phenomena

Taken together, the body of work discussed above highlights the important role of mathematical reasoning, to which students can anchor their discussions, affording a rich understanding of dynamic phenomena.17,30,62,77,79 For example, consider the discussion provided in Figure 2, with a student quote from a previous project reported by the authors, which involved analyzing students’ engagement in chemical kinetics problem solving.30 In the example provided, Blair, a student enrolled in a physical chemistry course for life science majors, was prompted to reason about the role of a catalyst in a reaction. Blair began by discussing chemistry ideas related to biological catalysts (1), followed by a quantification of this description (2), which resulted in a sophisticated understanding of the observed constant rate for a reaction catalyzed by an enzyme (3). Blair’s conceptual integration of chemistry and mathematics ideas afforded a deeper understanding of the dynamic processes related to enzymes and reaction rate, connections that are not trivial, as illustrated in the chapter discussed above related to enzyme kinetics.80 Moreover, as posited in the literature, special instances of blending, such as the use of metaphors, in which one domain is used to describe another, and metonymy, in which one aspect of a domain is used to describe another aspect of the same domain, are fundamental for understanding contexts ranging from advertisements to calculus.75,81,82 For example, a robust understanding of the derivative involves consideration of related ideas (ratios, limits, and functions) and the various ways these ideas can be represented; however, depending on the context, a particular perspective of the derivative may be more productive and a metonymic shortening of the derivative definition (e.g., derivative as rate) may be sufficient.82 Derivative as rate is an example of metonymy in the way that this abbreviated phrase is a piece of a larger knowledge structure that is used as a stand-in for the entire knowledge structure. For clarity, a more colloquial use of metonymy is reflected in phrases such as “Wall Street is in crisis”, in which a physical location (an actual road located in the Financial District of New York City) can be used to reference U.S. financial markets in general terms.81 As discussed by Zandieh

Figure 1. Overview of frameworks discussed in this paper, visualized as a resource graph, indicating the various analytic frameworks that can be used to characterize mathematical reasoning. The dotted line indicates potential integration of ideas across knowledge spaces or groups of resources (mathematics and chemistry), which can be described using frameworks such as blended processing and engagement in discussing mathematical narratives.

the connections among resources;71,72 for examples of how resource graphs have been used as a data visualization tool, see recent work by the authors.73,74 As shown in Figure 1, symbolic forms, graphical forms, and covariational reasoning can be framed as examples of mathematical resources, and when these resources are combined with resources related to chemistry, conceptual integration (blended processing and engaging in mathematical narratives) occurs between the chemistry and mathematics domains. These frameworks are discussed in more detail in the sections that follow. D



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Figure 2. Example of student engagement in blended processing, which involves reasoning about (1) chemistry ideas related to biochemistry and enzymes, (2) mathematic ideas related to quantifying rate, and (3) combining these ideas to draw a conclusion about the rate of a reaction involving an enzyme. Adapted from ref 30. Copyright 2018 Royal Society of Chemistry.

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and Knapp,82 when characterizing student reasoning, it is important to elucidate whether students’ metonymic abbreviated phrases reflect part of a larger conceptual understanding or a limited perspective of an idea (e.g., an individual is unaware of the larger network of ideas implied by metonymy). That said, although metonymy can be a productive tool, it is important for instructors to make their metonymic reasoning more explicit and be aware of potential instances in which students may be misapplying metonymy, as observed with students’ reasoning about the relationship between reaction rate and the rate constant.61

SYMBOLIC AND GRAPHICAL FORMS: MATHEMATICAL RESOURCES Symbolic and graphical forms can be characterized as types of mathematical resources that reflect similar types of reasoning. According to Sherin,86 reasoning involving symbolic forms occurs when a conceptual schema (mathematical idea) is assigned to a symbol template (the pattern in the equation). For example, consider the symbolic form, base ± change, which has the symbol template □ ± Δ (the box denotes a term or group of terms, and the delta symbol indicates change) and a conceptual schema that involves ideas related to having an initial value that is altered. Thus, when these ideas are mapped onto this equation pattern, it can be characterized as reasoning using symbolic forms. Additional examples of symbolic forms are provided below in Table 2, and in a forthcoming manuscript, we provide a comprehensive list of symbolic forms identified in the literature. As an analytic framework, reasoning described by symbolic forms was first discussed by Sherin86 in a study that involved analyzing students’ mathematical reasoning while they worked through classical mechanics problems in a physics course. On the basis of the nature of the problem-solving scenario, most of the symbolic forms identified by Sherin86 were related to algebraic manipulations (e.g., prop-, the meaning encoded in a reciprocal relationship) or were implicitly more relevant for physics contexts (e.g., balancing, terms or groups of terms on opposite sides of the equation are equivalent and reflect a balance of opposing influences). However, more recently, the symbolic forms framework has been used in many contexts that move beyond classical mechanics, describing a range of reasoning, including advanced mathematical ideas such as the meaning encoded in integrals or vector notation.10,17,64−70,80,87,88 Building on the idea of symbolic forms, graphical forms is a logical extension that provides a lens to characterize ideas associated with patterns in graphs. In his original work and in a subsequent paper, Sherin86,89 suggested the existence of graphical reasoning analogous to symbolic forms; however, this idea was not developed further or explored in the literature. In a forthcoming paper, the authors describe the

Suggestions for Future Inquiry on Conceptual Integration

As discussed by Holme et al.,83 the general consensus is that chemistry instructors want students to develop a deep conceptual understanding, which includes reasoning that spans different representations and moves beyond rote memorization of ideas. Blended processing provides the language to frame the analysis of student engagement in tasks that require students to productively use concepts from different domains and coordinate different types of knowledge. One area of inquiry that readily lends itself to analysis of this type is systems thinking, which involves the use of rich contexts with global implications to prompt students to make use of interdisciplinary ideas and explain interactions that occur within a larger complex network.84 For example, in the chapter described above, Holme85 proposed the utility of using a systems-thinking approach to provide the opportunity for computational reasoning, using water contamination as a context for problem solving. Investigations involving student engagement in blended processing during tasks framed using systems thinking would be a notable contribution to the field, providing insight regarding how we can promote a more holistic understanding of chemistry principles, transcend a reductionist view of chemistry’s application, and connect chemistry to students’ lives. E



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Table 2. Examples of Symbolic Formsa Symbolic Form

Symbol Template

Base ± change Coefficient

□±Δ [x□]

Dependence

[...x...]

No dependence

[...]

PropScaling

[...] x □ ...

É ÄÅ ÅÅ ... ÑÑÑ Ñ ÅÅ ÅÅ ... x ... ÑÑÑ ÑÖ ÅÇ [n□]

have productive ideas for reasoning about the equation even if they are unfamiliar with the context. By recognizing a specific pattern, an individual brings in useful ideas about how a value changes or perhaps how to model this change mathematically, which can then be used to support reasoning about a system or process. To further illustrate this idea, consider the graph provided in Figure 3. Although there are no labels for the axes

Conceptual Schema Change increases or decreases an initial value Value that multiplies a group of factors, scaling and controlling the size of an effect Whole depends on a quantity associated with an individual symbol Whole does not depend on a quantity associated with an individual symbol Indirectly proportional to a quantity, x, which appears as an individual symbol in the denominator Similar to a coefficient, but the coefficient is unitless; a scaling coefficient is seen as operating on the rest of the factors to produce an entity of the same sort that is larger or smaller than the original Template Key

Expression in brackets corresponds to an entity in the schema Individual symbols in an expression Term or group of terms Omitted portions of an expression that are inconsequential or continue a pattern

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Adapted with permission from ref 86. Copyright 2001 Taylor & Francis.

Figure 3. Example graph without context. Even without providing context (e.g., axis labels), individuals may associate productive ideas (e.g., the graph is periodic, it could be modeled using sine and cosine functions, etc.) with specific graphical patterns, which are useful for reasoning about unfamiliar contexts.

adaptation of symbolic forms to characterize graphical reasoning and provide a compilation of the various symbolic forms discussed in the literature (at the time of writing there were 45 unique symbolic forms). When characterizing graphical forms, the general term registration90 was used to describe the graph or the region in a graph attended to; thus, reasoning involving graphical forms occurs when an individual registers a mathematical idea to a graphical pattern. For example, the graphical form steepness as rate characterizes the idea that the relative steepness of a curve provides information about rate. For additional examples of graphical forms, see Table 3.

to indicate the context of interest, it is likely that readers may associate productive ideas to the graph on the basis of its shape, such as ideas about frequency, conclusions that the graph is periodic, or that it could be modeled using sine and cosine functions. Thus, graphical forms afford the ability to draw conclusions from mathematical representations, which can then be connected to the phenomena (e.g., wavefunction, motion of a pendulum, height of the tide, etc.). Moreover, as discussed in the literature, student access to symbolic and graphical forms supports students in blending, resulting in a more sophisticated understanding, a claim that has been supported in chemistry and physics contexts.17,30,62,79,90

Table 3. Examples of Graphical Formsa Graphical Form

Registration and Conceptual Schema

Steepness as rate

Varying levels of steepness in a graph correspond to different rates Straight line indicates a lack of change or constant rate

Suggestions for Future Inquiry on Symbolic and Graphical Forms

Straight means constant Curve means Curve indicates continuous change or changing rate change Trend from shape General shape of the graph suggests information directionality regarding the graph’s tendency to increase or decrease

Symbolic and graphical forms provide a unique way to characterize students’ mathematical reasoning. Many quantitative topics, particularly at the upper-division level, are wellsuited for these analytic frameworks. Within the chemistry community, Becker and Towns10 used symbolic forms to analyze student reasoning related to derivatives and partial derivatives using the Maxwell relations for thermodynamics as a context. In addition, in the chapter discussed above, using a biochemical context, the symbolic forms and graphical forms frameworks were used to characterize students’ understanding of equations and graphs relevant for enzyme kinetics.80 In terms of moving this body of literature forward, the graphical forms framework is still in its early stages, and most of the graphical forms identified revolve around ideas related to rate, which is largely a function of the kinetics contexts utilized thus far. Therefore, more work is needed to focus on how students utilize mathematical resources to support their understanding of processes across chemistry topics. For example, wavefunction graphs, heat of vaporization curves, and the variety of plots students generate in laboratory courses (e.g., titration

a Adapted with permission from ref 62. Copyright 2019 Royal Society of Chemistry.

Here’s Why It Matters: Providing Support for Drawing Connections

The symbolic and graphical forms frameworks provide a way to analyze and characterize mathematical reasoning, which is important because this reasoning serves as an anchor to allow individuals to draw conclusions and make connections to phenomena. For example, going back to the symbolic form base ± change, there are a variety of equations across contexts that share this same basic pattern, including equations used in physics to describe motion, integrated rate laws in chemistry, or, in more general terms, the slope−intercept form for the equation of a line. The key point here is that individuals may F



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each x value), whereas the object view frames the function as a fixed entity or concept with associated descriptors, with the former more explicitly involving ideas related to covariation. Moore and Thompson95 discuss related reasoning called shape thinking, characterizing students’ reasoning about graphs as emergent (involving a mapping of x and y values and focusing on the relationship between varying quantities) and static (involving an emphasis on the shape of the curve and associating ideas to the graphical pattern); moreover, David and colleagues101 emphasized student conceptions of single points on the graph, categorizing reasoning as value-thinking and location-thinking, depending on whether students attended to the numerical quantity of an ordered pair or the relative spatial position of the point, respectively. However, as mentioned in Moore and Thompson’s discussion of shape thinking95 and in David et al.’s101 discussion of value-thinking and location-thinking or process and object perspectives,98−100,102,103 the interplay between different types of reasoning is complex, and a holistic understanding of representations involves utilizing different views depending on the context.

curves, calibration curves, etc.) are rich areas for inquiry that can be framed using graphical forms.

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COVARIATIONAL REASONING: COORDINATING RELATIONSHIPS BETWEEN VALUES As discussed by Saldanha and Thompson,91 covariational reasoning involves “holding in mind a sustained image of two quantities (magnitudes) simultaneously.” Stated differently, covariational reasoning involves attending to the relationship between values, with an emphasis on how both values change.92 The foundational nature of covariational reasoning and its role in reasoning about functions, graphs, and other topics across mathematics is well-documented in the literature.91−97 Moreover, reasoning involving ideas related to covariation is complex and has been described as more of spectrum with levels of increasingly sophisticated reasoning reflecting developmental stages.92 As described in the covariational reasoning framework presented by Carlson et al.92 (illustrated in Figure 4), a lower level of covariational reasoning involves

Here’s Why It Matters: Understanding Relationships in Mathematical Models

Mathematical representations such as graphs and equations are used with ubiquity across science, technology, engineering, and mathematics.10,104 Although students may have difficulty viewing mathematical representations as models,57,58 in the physical sciences graphs and equations are empirically based with explanatory and predicative power, tersely summarizing the relationships among relevant constants, parameters, and variables. Thus, in order for students to draw connections to dynamic phenomena, they must be able to understand these relationships and coordinate how values change. This emphasizes the productive role of viewing a graph as a process. Although the object perspective may be productive in some contexts, solely viewing a graph as an object limits students’ understanding, because reasoning dominated by the shape of a graph does not attend to relevant features such as the axes.62,80 A recent study by the authors illustrates the sophisticated graphical reasoning that is possible when students attend to changes in both variables, such as considering concentration as a function of time (rather than simply focusing on the general shape of the graph).62 In this study, the authors investigated students’ covariational reasoning in chemical kinetics, noting that some students interpreted a concentration versus time graph as representing an acid−base reaction because of the sigmoidal shape of the curve, neglecting the meaning encoded in the axes (concentration vs time, not pH vs volume).62 Moreover, in the enzyme kinetics chapter discussed above, similar object-centric reasoning was observed, in which the students associated reaction order with graphical patterns without discussion of the axes (e.g., “zero-order is straight”, “second-order is curved”, etc.), which limited the connections made by the students to the particulate-level phenomena.80

Figure 4. Covariational reasoning levels, which reflect development stages, such that classification at a level encompasses the mental actions of all previous levels. For example, an individual’s reasoning ability being characterized as Level 5 implies they can reason using ideas related to instantaneous rate and the actions described in Levels 1−4.92

coordinating values (e.g., in a concentration vs time graph, describing how concentration changes with time), whereas a higher level of covariational reasoning involves consideration of concepts related to instantaneous rate (e.g., in a concentration vs time graph, describing the instantaneous rates at multiple points along the curve). In addition, levels of covariational reasoning reflect the developmental stages individuals move through, and classification at a level indicates an individual is able to reason using mental actions described by the level of interest and the preceding levels. For example, a classification of Level 5 indicates the ability to reason using ideas related to instantaneous rate (Level 5), average rate (Level 4), quantitative coordination (Level 3), direction (Level 2), and coordination (Level 1), each of which may be productive and sufficient for solving a problem, depending on the context. It is also worth noting that the levels of the covariation framework reflect movement toward a sophisticated understanding of ideas related to the derivative. Within the mathematics community, the role of covariation in reasoning about equations and graphs can be framed as a distinction between different perspectives of a function, namely, whether the function is viewed as a process or an object.98−100 As discussed by Sfard,100 the process view of a function emphasizes the relationship between the dependent and independent variables (i.e., a corresponding y value for

Suggestions for Future Inquiry on Covariational Reasoning

Because of the foundational role of covariation in reasoning about mathematical representations, this area reflects a rich context for inquiry, especially in physical chemistry contexts. Chemical kinetics, in particular, readily lends itself to ideas related to covariation, such as rate and understanding the role of the derivative in describing rate. Current gaps in the G



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literature related to chemical kinetics involve understanding how students reason about different types of rate (initial rate, average rate, and instantaneous rate), how these types of rate are related to one another, and how they are represented graphically, ideas that could be framed using the covariational reasoning framework described by Carlson et al.92 On a closely related note, future work could involve efforts to support students in making connections among different representations of the same set of chemical kinetics data, (e.g., concentration vs time, rate vs concentration, rate vs time, etc.). Additionally, a myriad of contexts outside of chemical kinetics would be equally fruitful, for example, focusing on gas laws and the role of covariational reasoning in students’ conceptions of the relationships expressed in the equations and corresponding graphs.

CONCLUSION Research that seeks to support students in understanding and using mathematics in combination with chemistry reflects an understudied area that would benefit from future work, supported by the use of frameworks from other communities. Here, we have provided an overview of various frameworks along with specific areas of inquiry that readily lend themselves to be investigated using these frameworks. We encourage the chemistry education community to add to the literature base by utilizing these frameworks. By situating work in the suggested contexts, insight can be provided regarding how instructors can support students in making connections between mathematics and chemistry. Optimistically, we hope these results might transform and improve learning. Furthermore, one way to drive chemistry education research forward is to collaborate across disciplines. The implementation of frameworks and ideas that both span the disciplines in areas of inquiry and delve into crosscutting concepts, such as stability and change or scientific practices such as developing and using models, can bring about unique insights. It is innovative work such as this that has great potential for transforming undergraduate student learning, not only in chemistry but also in other science and mathematics disciplines as well. 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

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. The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under grant DUE-1504371. We wish to thank Tom Holme, Ryan Bain, and the Towns Research Group for their support and feedback on this project. H



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