Mathematical Knowledge for Teaching in Chemistry: Identifying

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Mathematical Knowledge for Teaching in Chemistry: Identifying Opportunities To Advance Instruction Lynmarie A. Posey,1,* Kristen N. Bieda,2 Pamela L. Mosley,1 Charles J. Fessler,3 and Valentin A. B. Kuechle3 1Department of Chemistry, Michigan State University, 578 S. Shaw Lane, East Lansing,

Michigan 48824-1322, United States 2Department of Teacher Education, Michigan State University, 620 Farm Lane, East Lansing,

Michigan 48824-1600, United States 3Program in Mathematics Education, Michigan State University, 354 Farm Lane,

East Lansing, Michigan 48824-5602, United States *E-mail: [email protected]

The Mathematical Knowledge for Teaching (MKT) theoretical framework posits that effective mathematics teaching relies on both the instructor’s subject matter knowledge (SMK) and pedagogical content knowledge (PCK). In further unpacking the MKT construct, Ball, Thames, and Phelps (Ball, D. L.; Thames, M. H.; Phelps, G. J. Teach. Educ. 2008, 59, 389-407) proposed that SMK is comprised of common content knowledge (CCK), specialized content knowledge (SCK), and horizon content knowledge (HCK); PCK consists of knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of content and curriculum (KCC). This work represents an initial effort to understand MKT in the context of chemistry instruction, namely can MKT support chemistry instructors’ efforts to help students develop a robust understanding of the mathematics used in general chemistry. In the domain of SMK, CCK, an understanding of the mathematics learned in school and used in everyday life, has been characterized using the Common Core State Standards in Mathematics (CCSS-M). The PCK that has been identified to date falls into the categories of known difficulty, pedagogical opportunity, anticipated gaps in prior knowledge, and areas of difference between chemistry applications and mathematics instruction. Examples of CCK and PCK coding are provided for three important problem types in general chemistry: balancing chemical equations, reaction stoichiometry, and functional relationships involving covariation, such as the ideal gas law.

© 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.

Introduction Students’ challenges with the mathematics used in general chemistry are long-standing (1) and persistent (2). As general chemistry is often the first college-level science course taken by students pursuing a STEM degree, addressing these mathematical challenges so that students can succeed in general chemistry is an important first step in retaining these students in these majors. Much of the mathematics found in general chemistry falls in the domain of algebra (3, 4), including ratios and proportional reasoning, covariance, and linear rates of change as ratios. Unfortunately, recent data for 2017 high school graduates show that only 48% of students interested in STEM met the ACT college readiness benchmark in mathematics, corresponding to a 50% probability of earning a B or higher and a 75% chance of earning a C or higher in a college algebra course (5). Prior efforts to address the deficiencies in mathematics preparation that impact outcomes in general chemistry have largely targeted students by providing more opportunities to practice wellestablished procedures for solving chemistry problems. Such work has been carried out in a variety of settings, including online (6) and face-to-face (7, 8) preparatory courses and supplemental sessions for general chemistry courses, commonly with additional peer support (9–13). Unfortunately, simply giving students more practice without building an understanding of the underlying mathematics is unlikely to have long-term benefits and will not prepare students to address novel problems. Another approach in which algebra and chemistry instruction were coordinated showed some improvement in student outcomes in the first semester of general chemistry, but not in the second (3). The recent Chem-Math project (4, 14) developed instructional materials for recitations serving at-risk students to explicitly connect the mathematics used in general chemistry to the “formal math” that students learn in mathematics courses; general chemistry failure rates were reduced for the students placed in these recitations. This prior work has relied primarily on faculty with chemistry expertise teaching problem-solving procedures. These efforts are often not grounded in knowledge of how students learn and make meaning of mathematics and, thus, fail to address student understanding of the underlying mathematics. Suppose that the focus was shifted to building chemistry instructors’ capacity to anticipate, identify, adapt, and respond to their students’ difficulties with mathematics in the context of chemistry. Chemistry instructors generally have a strong operational understanding of mathematics in terms of applying mathematics to model phenomena and solve problems specific to the discipline, but they do not necessarily know research-based strategies for teaching mathematics. Furthermore, they are often far removed from their experiences in first learning the mathematics used in general chemistry. We propose that building chemistry instructors’ mathematical knowledge for teaching (MKT), combining subject matter knowledge (SMK) with pedagogical content knowledge (PCK), would better prepare them to both address the mathematical challenges faced by their students and support students in building a deeper understanding of the mathematics used in general chemistry in anticipation of new situations and future learning. First, it is necessary to identify and articulate what MKT looks like in the context of chemistry. In other words, what is mathematical knowledge for teaching in the context of chemistry (MKT-C)?

Theoretical Framework: MKT Shulman (15) introduced the concept of PCK, arguing that effective teaching requires more than sound content or SMK. He proposed that instructors who possess PCK bring an understanding of learning and learners as well as knowledge of curriculum and effective pedagogy (both general and discipline-specific) to the work of teaching disciplinary content. Evaluation of how instructors use 136 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

content knowledge and PCK and assessment of the impact of these two components of teaching knowledge on instruction require having defined constructs to measure. In the realm of mathematics, Ball and coworkers (16) incorporated Shulman’s ideas about the knowledge required for teaching (17) to develop the MKT theoretical framework. This theoretical framework articulates the components of content knowledge and PCK necessary to support effective teaching of mathematics. As illustrated in Figure 1, MKT consists of two domains, SMK, which corresponds to content knowledge, and PCK. Each domain is broken down into subdomains to characterize the knowledge that comprises each domain.

Figure 1. Domains of MKT. Adapted with permission from reference (16). Copyright 2008 SAGE Publications. Within SMK, there exist three subdomains: specialized content knowledge (SCK), common content knowledge (CCK), and horizon content knowledge (HCK). SCK (16) is the knowledge of mathematics related specifically to the teaching of mathematics that permits instructors to build appropriately on student thinking. It includes unpacking student work to assess the validity and generality of alternate approaches and to identify patterns in student errors. In contrast, CCK refers to an operational understanding of how to do or apply mathematics that may extend to other contexts outside of school mathematics. The third component, HCK, represents an understanding of the discipline of mathematics as a whole, including the central ideas, that allows an instructor to see how current instruction fits into the larger picture. It is arguable that the normative SMK expertise of most chemistry instructors falls in the subdomain of CCK, the material learned in mathematics classes. Furthermore, mathematics CCK is the most relevant subdomain to chemistry instruction, followed by SCK. HCK is the least important as chemistry instructors will not be teaching mathematics in order to support future mathematics learning. Our efforts to develop MKT in the context of chemistry (MKT-C) will focus on the CCK component of SMK. The PCK domain in MKT (16, 18) is comprised of three components: knowledge of content and students (KCS), knowledge of content and teaching (KCT), and knowledge of content and curriculum (KCC). KCS reflects an understanding of the ideas held by learners and the developmental progression in learning specific mathematics content. An instructor’s KCS could include knowledge of when students are ready to learn specific mathematical ideas and eventually master them, ideas that are easily grasped by students, concepts that are more difficult to comprehend and the reasons why, how students use mathematical ideas, and common errors made by learners. KCT connects knowledge of mathematics with pedagogy, which prepares instructors to effectively design instruction and respond to student ideas, both correct and incorrect, with appropriate 137 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

instructional moves. The third component, KCC, provides a context for current mathematics instruction based on anticipated prior knowledge held by students. An understanding of the mathematical ideas that students bring to introductory chemistry courses for STEM majors as well as the nature of their prior mathematics instruction is essential in helping students construct new knowledge (19–21) when they are asked to draw upon these ideas in learning chemistry. When chemistry instructors understand the K–12 mathematics curriculum, they can identify gaps between students’ prior mathematics experience and the mathematics used in general chemistry. Furthermore, an awareness of the vocabulary used in mathematics courses as well as a willingness to use this vocabulary in chemistry or at least point out where descriptions of mathematical ideas commonly used in chemistry differ from those in mathematics instruction is essential to helping students make connections across the disciplines (as noted in the Chem-Math project) (4, 14). While the initial work on MKT focused on grades K–8, subsequent work has extended MKT to secondary and post-secondary levels of instruction (22–24).

Research Question The work reported here represents our initial efforts to address the research question: What elements of MKT, particularly PCK, are needed to enact the curriculum in general chemistry to successfully support students in using mathematics to build understanding in chemistry? The long-term goal of this work is to map the terrain of MKT-C to provide a foundation for examining the current state of mathematics-related instruction in chemistry and to support professional development.

Methods The four problems presented in this chapter are representative examples of several important problem types seen in a larger general chemistry textbook coding project, but they are not the copyrighted content from textbooks coded in the larger project. The problems in this chapter were coded independently by three of the authors (two chemistry education researchers and one mathematics education researcher) to identify the CCK required to understand and answer the questions posed. In this initial coding, each coder also identified PCK that they thought would support instruction on the associated content. The results were compiled in Excel. Coding for CCK The process of coding for CCK described in this chapter was developed for a project analyzing general chemistry textbooks for STEM majors that is currently underway. An example of such a textbook is Chemistry: The Central Science (25). The same approach was used in coding the examples of general chemistry problems presented in this chapter, which represent a subset of the types of items being coded in the textbook analysis. This textbook analysis includes coding of paragraphs of expository text, worked example problems, student exercises within chapters, and supplemental text found in boxes separate from the main text. To identify whether or not CCK is required to understand general chemistry content and solve associated problems, coders queried the units of analysis (the individual example questions provided 138 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

in this chapter) in the following way: Is there recognizable mathematics once the chemistry context is stripped away? For example, while questions such as “If a neutral atom has 15 protons, how many electrons does it have?” involve mathematics, they are unsolvable without chemistry knowledge. This item would be coded as not an example of CCK (no CCK). The coders also queried each unit with the questions: Am I being required to actively engage in a mathematical process to work out the example or understand the concept? Am I being required to follow along with a mathematical process that the textbook is elaborating on (in other words, is the textbook doing the math)? These questions guided the coder to describe mathematical actions that were required for students to understand the statement or solve a problem. We found many cases in textbooks where students were presented with symbolic formulas. However, if the text discussing the formula did not highlight the mathematical relationships represented by the formula, this unit of the text was coded as no CCK. For the four examples provided in this chapter, all coders agreed that CCK was present. Once a unit was identified as relevant for mathematical CCK, coders consulted the Common Core State Standards in Mathematics (CCSS-M) (26) to characterize the mathematics embedded in the coding unit. The standards in the CCSS-M are organized by grade level and mathematics domain, such as Ratios and Proportional Relationships (RP) and Expressions and Equations (EE). The format used for these standards is G.DD.S, where G represents the grade level (K–8 is used for the elementary and middle school grades and HS is used for all high school grade levels), DD is the two-letter abbreviation for the mathematics domain, and S is the number of the standard within a particular grade and domain. While the CCSS-M articulate content standards for grades K–12, they were chosen because of the lack of comparable standards for college-level algebra. Furthermore, in many cases the mathematics in general chemistry aligns with standards for grades 6–8 in the CCSS-M. For the examples presented, disagreement between coders on domains and standards were discussed to reach consensus. Identifying Opportunities To Use PCK We also identified units in the texts that offered opportunities to further develop students’ mathematical understanding to support building their understanding of chemistry and ability to successfully solve problems in chemistry. These are places that require PCK, knowledge of teaching the subject matter (15). As Figure 1 shows, PCK in mathematics has been articulated into three different components (KCS, KCT, and KCC). We operationalized these constructs as the following questions coders used to query whether there was mathematics PCK associated with the unit of analysis that could support chemistry instruction: Does the unit involve mathematics that students typically find challenging or mathematics that will be important to understand future work in chemistry? (KCS and KCC) Does the unit introduce mathematics that could be connected to prior learning or a realworld application to support students’ understanding of the mathematical relationships involved? (KCC and KCT) Is there an opportunity to use multiple approaches or to leverage findings from mathematics education research to support student learning? (KCT and KCS) Four categories of empirical PCK codes emerged as a result of our pilot attempts to code for PCK: (1) Difficulty: Unit contains content that is a known mathematical challenge (e.g. researchbased) or misunderstanding, or unit requires higher-level mathematical reasoning. Recognition of problems or content that pose high cognitive demand is included in this category. 139 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

(2) Pedagogical opportunity: Unit contains content where discussing multiple approaches to build understanding could benefit students’ learning, or research advances suggest new approaches to pedagogy (e.g. use of multiple representations including tables, graphs (27, 28), or technology) (3) Anticipated gaps in prior knowledge: Unit contains content that students may not have had much experience learning or where research indicates students have difficulties understanding. (4) Areas of difference: Situations in which the application of a mathematical concept in chemistry may differ on the surface from its use in mathematics. An example is the difference between the balanced chemical equation and mathematical equations elaborated upon in the Results and Discussion. Once identified, the specific instances of PCK from these four empirical categories can be mapped to the three components of PCK in the MKT theoretical framework (KCS, KCT, and KCC) as shown for the examples that follow. One aspect of coding for PCK is important to note: Many decisions are based on the coder’s understanding of the chemistry education research base and/or chemistry teaching experience, as well as their understanding of the mathematics education research base and/or mathematics teaching experience. It is difficult at this time to find coders who are experts in the research literature and instruction in both of these fields. Thus, inter-rater reliability is unsurprisingly low for coding for PCK. We chose to have raters review all items coded as PCK based on their respective backgrounds and scholarly expertise and then negotiated to reach a consensus for each item.

Results and Discussion The results of CCK and PCK coding for four representative general chemistry problems are presented and discussed below. The problems coded cover the chemistry topics of balancing chemical equations, reaction stoichiometry calculations, and application of the ideal gas law. Examples illustrating how PCK could be applied in instruction associated with the CCK identified in the reaction stoichiometry and ideal gas law problems are also provided. Balanced Chemical Equations The balanced chemical equation plays a central role in chemistry, representing conservation of matter in the transformation of reactants to products in chemical reactions. On the surface, the mathematics used in balancing the chemical equation for the reaction provided in Table 1 appear to be fairly basic, falling in the CCSS-M domains of Counting and Cardinality (CC), Operations and Algebraic Thinking (OA), and Number and Operations-Fractions (NF) from the elementary grades. However, the balanced chemical equation is problematic when viewed through the lens of mathematics without a chemical interpretation. The equals sign in a mathematical equation indicates that the two quantities are equivalent; however, the reactants and products in a chemical equation are chemically distinct species. Consequently, four aluminum atoms and three oxygen molecules on the reactant side of the balanced equation are not the same as the two formula units of aluminum oxide found on the product side (expressed mathematically as 4x + 3y ≠ 2z, where x, y, and z represent distinct objects). Nevertheless, when given an equation that is already balanced, some students will ask why the sum of the coefficients for the reactants is not equal to the sum of the coefficients for the products. 140 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Table 1. Balancing an Equation Representing a Chemical Reaction

In order to balance a chemical equation, students must drop down to the level of atoms by using their chemistry understanding to first unpack the information provided in molecular formulas and then find the smallest whole number coefficients that will satisfy the equation ?x + ??(2y) = ???(2x + 3y) where x and y represent Al atoms and O atoms, respectively. Expressing the process of balancing a chemical equation in terms of a mathematical equation involves mathematics corresponding to the middle-school level Equations and Expressions (EE) domain in the CCSS-M as 141 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

students must apply the distributive property (6.EE.3) and simultaneously solve two linear equations in two variables (8.EE.8c). In fact, chemical equations could be balanced using matrix algebra (29). The use of matrices to solve systems of linear equations falls in the CCSS-M high school domain of Algebra - Reasoning with Equations and Inequalities (A-REI). This is not to suggest that students should be expected to balance chemical equations using matrix algebra but rather to point out that balancing chemical equations is more complicated mathematically than it appears on the surface. The procedural approach to balancing chemical equations that is commonly taught and found in textbooks obscures this underlying mathematics. The mathematical operations necessary to balance chemical equations following the procedural approach that begins by fixing the coefficient for one substance in the equation and then iteratively determining coefficients for the remaining substances are covered by CCSS-M standards for the elementary grades (Table 1). Awareness of this “area of difference” in the usage of the term “equation” in chemistry and mathematics is important PCK for chemistry instructors to possess because it represents a potential source of confusion for students. There is an opportunity to remind students that the balanced chemical equation represents a process emphasized by the arrow between reactants and products. The equation in the mathematical sense of representing equivalent quantities on opposite sides of the equals sign occurs at the level of the atoms within the substances represented by molecular formulas. One way to reinforce the idea that there are equal numbers of each type of atom before and after reaction in instruction is by providing students with opportunities to use particulate representations of chemical reactions (30–32).

Reaction Stoichiometry Reaction stoichiometry calculations are typically introduced early in the first semester of general chemistry and, consequently, are often one of the first places where students encounter difficulties in using mathematics to solve chemistry problems. Examples of two common types of reaction stoichiometry problems are found in Table 2. The example in part a. is a one-step problem that involves finding the number of moles of a reactant that will react with a specified number of moles of a second reactant. The example in part b. is representative of problems that ask students to determine the mass of a product that can be generated given a specified mass of reactant and requires multiple steps to reach a solution. The underlying mathematics for these two problems requires CCK in the domain of Ratios and Proportional Relationships (RP) from grades 6 and 7 (Table 2). However, the centrality of ratios and proportional relationships in these problems can be obscured by the factor-label heuristic found in most general chemistry textbooks, which relies on canceling units when setting up solutions. This heuristic, while efficient, does not promote understanding of the critical concept of an invariant ratio (i.e., the ratio of stoichiometric coefficients or molar mass) corresponding to an intensive property that is independent of quantity. Furthermore, an additional challenge is added when students are expected to use units throughout the calculation, as this has not necessarily been the case in their prior mathematics experience. In fact, correct execution of the factor-label method relies on consistent use of units to guide the solution; the corresponding CCK is found in the high-school Number and Quantity (N-Q) domain.

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

Table 2. Reaction Stoichiometry Problems Relating Molar Quantities of Two Reactants and the Mass of a Reactant to the Mass of Product Formed: CCK

Chemistry instructors know from experience that stoichiometry problems can present a formidable obstacle for some students, which is KCS in chemistry. How could mathematics PCK be used to support students in developing an understanding of the underlying mathematics, namely ratios and proportional relationships? PCK that could be beneficial to chemistry instructors is summarized in Table 3. The orange juice problem from the Connected Mathematics Project (CMP) middle school curriculum (33) addresses mathematics analogous to the problem presented in part a. of Table 2. The design of this CMP activity encourages flexibility in the approaches taken by students as they construct their understanding of ratios and proportions and then use it to solve problems. There is no reason that students should be limited to a single way of reasoning about proportional relationships or solving problems using ratios in chemistry.

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

Table 3. Reaction Stoichiometry Problems Relating Molar Quantities of Two Reactants and the Mass of a Reactant to the Mass of Product Formed: PCK

The potential shortcoming of relying solely on symbolic representations when teaching students to solve chemistry problems involving proportional reasoning is that the underlying quantitative relationships that are important to chemistry can be lost as students focus on learning a procedure. Introduction of additional representations, such as tables of data and graphs in initial instruction on solving reaction stoichiometry problems can reinforce students’ understandings of the proportional relationships at play. Table 4 shows selected prompts from an activity developed for a chemistry bridge course that engages students in using a table of data and a graph to reason about the relationships between quantities of reactants and products using stoichiometric ratios derived from a balanced chemical equation. After students complete the table, they make a prediction about the shape of the graph representing moles of O2(g) as a function of moles of Al(s) and then construct the graph (questions 1–3). They are then asked to explain how they could use the graph to determine the number of moles of O2(g) that would react with 3.5 moles of Al(s) before doing any calculations (question 4). 144 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Table 4. Selected Prompts from an Intervention to Build Understanding of Ratios and Proportional Relationships in the Context of Reaction Stoichiometry Calculations

Questions 5 and 6 are intended to draw students’ attention to the fact that the slope of the graph is the same as the conversion factor used to find the moles of oxygen that will react with a specified molar quantity of aluminum; this ratio is independent of the quantity of aluminum specified. It is important for chemistry instructors to understand that while slope, or rate of change, is a concept that is second nature to scientists, it can be challenging for students to grasp (35, 36). Furthermore, successful calculation of a numerical value for slope does not necessarily correspond to an understanding of what the slope represents. Students can reason about slope and proportional relationships in one of two ways: in terms of multiple batches (variable numbers of fixed quantities) or variable parts (fixed numbers of variable parts) (35, 36). In the problem given as part a. in Table 2, the slope or ratio of stoichiometric coefficients would be interpreted as three-fourths of a mole of 145 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

O2(g) per mole of Al(s) from the multiple-batches perspective and as three moles of O2(g) for every four moles of Al(s) from the variable-parts perspective. The variable-parts perspective connects more directly to the functional relationship

which aligns with the factor-label method. Beckmann and Izsák (35, 36) have noted that the variableparts perspective has not been widely studied by mathematics education researchers. Question 7 prompts students to first explain how they could use the slope (ratio) to determine the number of moles of O2(g) that would react completely with 30.0 moles of Al(s) and then carry out the calculation. Finally, students are asked in question 8 to write a mathematical expression for the line that describes the points on the graph and to compare this equation to their calculation from question 7. Figure 2 provides a visual summary of the mathematical work that students are asked to do in this activity. The goal is to demystify the factor-label method and add meaning to “multiplying by a conversion factor.”

Figure 2. Using multiple representations to support students’ understanding of the relationship between moles of reactants and products from the balanced chemical equation.

This activity is also designed to introduce multiple approaches to solving stoichiometry problems by using equivalent ratios (question 7) and by also having students arrive at the equation used in the factor-label method by writing a mathematical equation modeling the relationship shown in the graph of moles of O2(g) plotted as a function of moles of Al(s) (question 8). The follow-up classroom discussion demonstrates that the two approaches are mathematically equivalent. While experts recognize that these two approaches are equivalent, students need to be given opportunities 146 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

to see the relationship between the two approaches. The intent of having students work with the factor-label and equivalent-ratios methods is two-fold: to reinforce the idea that there can be more than one mathematically valid approach to solving a problem and to offer students two options for solving stoichiometry problems from which they can select the approach that makes the most sense. The factor-label method for solving stoichiometry problems is efficient because it permits multiple ratio problems to be solved at once. However, if students do not understand the underlying mathematics and cannot execute this approach reproducibly, then it loses its value. An alternate approach based on the mathematics already presented uses a series of equivalent ratios to solve multistep problems. Setting up equations with equivalent ratios and solving for the unknown quantity emphasizes an intensive relationship between quantities represented by a ratio. Conversions between units of measure, such as grams to milligrams, also involve ratios and proportional relationships. Multiple representations (34) and approaches are equally applicable to the associated instruction. A noteworthy distinction in stoichiometry problems involving masses of reactants and/or products is that there are two types of ratios involved: ratios of stoichiometric coefficients from the balanced chemical equation and the unit factor relating grams and moles (molar mass). The presence of two types of ratios is a potential source of confusion that instructors should be aware of and address explicitly when helping students make sense of stoichiometry problems.

Functions and Covariation (Modeling) A crosscutting concept common to all scientific disciplines is cause and effect (37), which can be represented quantitatively by mathematical functions that model the phenomenon of interest. A guiding principle in the design of experimental investigations to discern the underlying cause of an observed phenomenon and quantify its dependence on a particular variable is to change the variable of interest while keeping all other conditions constant. In using mathematical functions that model phenomena by providing the relationships between covarying quantities, it is essential for students to be able to first identify the dependent variable corresponding to the effect or phenomenon of interest and then identify which of the independent variables are changing and which are constant in the situation specified (38). A potential source of confusion for students when interpreting mathematical functions that model scientific phenomena is the use of symbols to represent both quantities that can vary and physical constants that are irrational numbers. In general chemistry, the ideal gas law equation PV = nRT provides the relationship between the covarying quantities of pressure (P), volume (V), temperature (T), and moles (n) to model the behavior of ideal gases. This equation is well-suited to having students consider which variables are independent (the causes) and which variable is dependent (the effect) as all of the variables can play either role depending on the situation. Table 5 provides an example of such a problem. The CCK required to work through this task falls in the domain of EE at the grade 6 level (6.EE). There are several standards within this domain that are relevant (Table 5). Ideally, students would rearrange the expression to highlight the variable corresponding to the phenomenon of interest, which is mathematical knowledge that falls in the high school domain of Algebra-Creating Equations (ACED); however, this might require prompting from the instructor (PCK).

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

Table 5. Exploring the Relationship Between the Temperature of a Gas and its Volume Using the Ideal Gas Law

Typically, the type of problem represented by the question in Table 5 would ask students to complete a calculation, in this case find the final volume after the temperature is decreased to 300 K when given an initial volume of the gas at 350 K. However, experts would argue that understanding the relationship between the volume of a gas and its temperature is more important to developing an 148 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

understanding of the behavior of gases than calculating a numerical value for volume at a given set of conditions with no further consideration. Often students can substitute and solve for numerical values, but difficulty arises when instead of calculating a numerical value, students are asked to reason about covariance. The initial volume of the gas was intentionally omitted in this problem to force students to reason about how changing temperature impacted volume instead of simply calculating a numerical answer. PCK that would support instructors in helping students to reason about relationships between variables begins with recognizing students’ potential difficulties with the mathematics (Table 5). This includes using context first to identify quantities that vary and those that remain constant and then to distinguish the independent and dependent variables found in the symbolic representation. The practice of using symbols to represent irrational constants in science can cause students to confound constant quantities with variables. Some symbols represent quantities that are always constant, whereas other symbols represent quantities that are variable in some contexts but constant in others. Chemistry instructors can help students begin to make sense of covarying relationships represented by equations (such as the ideal gas equation) by explicitly asking students to identify quantities that are constant and those that are varying and to connect the independent and dependent variables to cause and effect, respectively. Introduction of additional representations, such as data tables and graphs, can also support students in making meaning of the relationships between variables represented symbolically by a mathematical equation. For example, students can be asked to complete a table in which values are provided for either the independent or dependent variable by solving the mathematical equation for the unknown quantity. Then using their understanding of which variable is independent and which is dependent combined with their knowledge of graphing conventions, students can prepare graphs to represent the relationship. An alternate approach involves providing students with several graphs from which they are asked to select the one that best represents the relationship between covarying quantities and then support their claim. A series of scaffolded questions used to explore relationships between covarying quantities and address some of the difficulties commonly encountered by students is provided in Table 6. These questions were developed collaboratively by chemistry and mathematics education researchers as part of a mathematics intervention for a chemistry bridge course; they illustrate application of MKT in the context of chemistry instruction. In the activity, students dissect the function by identifying quantities that are constant and those that vary in the context provided. They are then asked to relate the graph of the temperature change as a function of the heat added for aluminum and liquid water to the quantities represented by symbols in the mathematical equation q = mcΔT. This helps to build understanding of the relationship between the independent and dependent variables and to relate symbols in the equation to slope. When asked which substance has the greater specific heat, some students incorrectly claim that aluminum has a greater specific heat by associating the larger slope with the greater specific heat, either because they have rearranged the equation incorrectly (i.e., ΔT = mcq instead of the correct rearrangement to ΔT = q/mc) or they have used “larger is larger” reasoning. When asked to explain the basis for this incorrect claim, we have observed that students often realize their error when reasoning about what the graph is showing, namely that more heat must be added to liquid water than to aluminum to get the same temperature change resulting in a greater specific heat. Coupling a symbolic representation with a graph and a chemistry context can help students to understand the mathematics and discover errors in symbolic manipulation. It also supports what should be the primary objective in a chemistry course: developing a deeper understanding of chemistry. 149 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Table 6. Exploring the Relationship Between Specific Heat and the Temperature Change that Accompanies the Addition of Heat to a System

Figure 3. Graph showing the relationship between the temperature change and heat added for 10 g of aluminum and 10 g of liquid water for the activity on covariation provided in Table 6.

Implications for Instruction The cases discussed in the previous sections represent initial work to identify and incorporate opportunities to improve students’ success in chemistry by targeting students’ understanding of mathematics needed to solve chemistry problems. For a successful implementation of these approaches, the instructors’ awareness, among other things, of the underlying mathematics of the tasks and students’ thinking about the mathematics in the tasks is critical. Since chemistry instructors typically have a strong foundation in mathematics CCK, this work offers an opportunity to improve general chemistry instruction by expanding chemistry instructors’ mathematics PCK to support student learning, which includes developing an understanding of students’ difficulties with the 150 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

underlying mathematics, identifying opportunities to use pedagogical strategies developed in mathematics education, and building an awareness of gaps in prior instruction. The use of mathematics in general chemistry tends to focus on symbolic representations and solving for numerical values, namely applications of the mathematics “that students should already know.” However, many students arrive in general chemistry without a solid understanding of the prerequisite mathematics. Existing approaches to this lack of understanding have primarily focused on improving learning opportunities for students in their mathematics courses; however, as our analysis shows, there are key differences between mathematics in the context of chemistry and pure mathematical contexts. Moreover, students may transfer some aspects of mathematical understanding in potentially problematic ways when solving chemistry problems (e.g. balanced equations). Chemistry instructors have an opportunity to help students improve their understanding of the mathematics used in their courses by going beyond symbolic representations and using other supporting representations (including tables of data and graphs), while building important sciencerelevant skills, such as an ability to construct and interpret graphs. This work raises questions about best practices for chemistry instructors to develop aspects of MKT for teaching chemistry. While one approach could have chemistry instructors participate in professional development that would unpack the mathematical understanding needed to comprehend key concepts and operations needed for solving chemistry problems, another approach could be to form professional learning communities (PLCs) of chemistry instructors focused on finding approaches to surface students’ understanding of mathematics for key topics in chemistry. PLCs could collectively interpret aspects of students’ work and explore options for further developing students’ understanding of mathematics throughout the chemistry course. This work argues that while CCK can be developed by taking a course, the multi-faceted and contextual nature of MKT demands that chemistry instructors have opportunities to investigate and explore their understanding of students’ mathematical thinking and how to leverage students’ understandings to build more robust mathematical thinking while engaging in instruction – not beside it.

Limitations This chapter reports pilot work on characterizing the terrain of MKT-C. An important goal of the larger project is characterization of the mathematics used in general chemistry through analysis of the mathematics content of general chemistry textbooks. We have chosen to use the standards from the CCSS-M to characterize CCK because no comparable standards exist for college algebra, a common prerequisite for general chemistry. In our pilot work, we found that the mathematics content of general chemistry is largely covered by standards from the middle school grades in the CCSS-M. As evidenced by the examples presented in this chapter, the applications of mathematics in chemistry often involve mathematics from more than one CCSS-M standard. Coders are left to decide which standard is most central to the mathematical work of the problem. Furthermore, the boundaries between different standards within a grade level, or even between adjacent grade levels, are not always clear. The CCK coding for the examples reported here is the result of independent coding by three coders followed by discussion to achieve consensus when there was disagreement. Going forward in coding of textbook content, we are identifying common general chemistry problem types and establishing rules for coding such problems. This work requires the expertise of an interdisciplinary team of chemistry education and mathematics education researchers as most individuals do not possess both knowledge of the relevant literature base from the two disciplines and instructional experience and expertise in the two disciplines. Consequently, the PCK identified for particular types of mathematics encountered 151 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

in general chemistry is a collection of contributions from the members of the research team. What different people from different disciplines define as PCK can vary. It is unrealistic to expect that interrater reliability can be achieved by raters with different disciplinary perspectives and different levels of experience in the classroom. It is valuable to have different perspectives because it should lead to a richer characterization of PCK. For example, chemistry instructors can identify mathematics that pose a challenge within the chemistry context, whereas members of the team with disciplinary expertise in mathematics can speak to whether the same mathematics outside a science context will present difficulties for students. Having both perspectives is important for understanding whether student difficulties arise from the underlying mathematics or the mathematics embedded in the chemistry context. The mathematics educators on the team can connect the chemistry applications of mathematics to the research base in mathematics learning, which is especially important for identifying productive approaches to teaching the mathematics that promote student learning. The process of coding using the CCSS-M can shed light on potential gaps between students’ prior knowledge and chemistry instructors’ assumptions about prior knowledge. Understanding the areas of difference in the use of mathematics also requires collaboration across chemistry and mathematics.

Conclusions This chapter reports our initial efforts to characterize MKT applied to the context of chemistry instruction (MKT-C). This work has focused on two areas that are both important in general chemistry and challenging for students: ratios and proportional relationships and covariation. The approach to coding CCK and identifying PCK provides the foundation for a more comprehensive study that is currently underway in which the CCK in general chemistry textbooks is being systematically coded using the CCSS-M and relevant PCK is being identified by the interdisciplinary research team. This chapter illustrates the nature of the PCK that could support instruction within general chemistry for several important types of problems that involve ratios and proportional relationships and covariation. It also highlights important differences in the use of the concept of equations in chemistry and mathematics.

Acknowledgements The authors would like to acknowledge the Lappan-Phillips-Fitzgerald Endowment and the CREATE for STEM Institute at Michigan State University as well as The Herbert H. and Grace A. Dow Foundation for supporting this work. Jennifer Nimtz and Benjamin Brandicourt contributed to the early development of mathematics interventions for a chemistry bridge course that use multiple representations to support student understanding of the underlying mathematics.

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