The Logic of Proportional Reasoning and Its Transfer into Chemistry

Chapter 10

The Logic of Proportional Reasoning and Its Transfer into Chemistry Downloaded via AUBURN UNIV on August 20, 2019 at 22:37:17 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

Donald J. Wink*,1 and Stephanie A. C. Ryan2 1Department of Chemistry, University of Illinois at Chicago, Chicago, Illinois 60607,

United States 2Ryan Education Consulting LLC, Carmel, Indiana 46032, United States *E-mail: [email protected]

Although mathematical reasoning is central to chemistry, the specific link of even basic mathematical ideas with chemistry is not always made explicit, potentially undermining student learning. This includes proportional reasoning, a key part of a great many chemistry calculations and based on important concepts such as the persistence of atoms. Whichever is the “cart” and whichever is the “horse,” a strong and reliable link (or “yoke”) between mathematical reasoning and chemical reasoning is needed, both for chemistry educators and for students. To strengthen this linkage, this chapter reviews the structure of mathematical reasoning as it relates to the concept of variables, quantification, expressions, and equations. This review will then be applied to questions of the reasoning in chemistry about quantity—and in particular the relationships among quantities that arise as a result of specific proportional relationships among different units. With this in place, a study of student thinking in proportional reasoning in a general mathematics task and in a domain-specific chemistry task is presented.

The Problem of Mathematical Reasoning in Chemistry Mathematical reasoning is a major factor in student learning across a wide variety of contexts (1), including the science classroom. This chapter presents an analysis of the nature and application of a particular part of mathematical reasoning—proportional reasoning—in the context of chemistry. When one considers the role of mathematics in science and science education, it is common to consider mathematically complex systems covered in algebra, calculus, and reasoning using mathematical models as presented in equations and graphs, and in connection with statistical methods. This is common in the literature when chemists consider mathematical reasoning (2–5). But simpler aspects of mathematics are also important, something suggested by many studies that emphasize the role of precollege mathematics aptitude in college chemistry (6–9). Learning theory, © 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.

including the framework of Knowledge-in-Pieces (10), suggests that one reason students may struggle to use or understand a problem may lie in their need to transfer knowledge from one domain to another. This suggests that instruction in chemistry may benefit by having students explicitly consider how the mathematics that they know is applied in chemistry. Such instruction, in turn, requires that instructors themselves reflect upon the basic mathematical logic behind chemistry calculations and, in some cases, consider how to make that logic explicit for students. One particular example of fundamental precollege mathematics that is important in chemistry is proportional reasoning. Put simply, proportional reasoning is applied whenever two variables are in proportion to one another, especially in cases of direct proportion. In that case, a change of one quantity x is accompanied by a proportionately equal change in another quantity y. This is often expressed by the simple mathematical equation y = kx. Proportional reasoning is involved in considering a host of chemical phenomena, including ones that permit chemists to make measurements using one system (such as mass) and to reason with a different system (such as the counting of elements, either at the atomic or the mole scale). Often, these are treated algorithmically in classroom settings, including with unit analysis (sometimes also called dimensional analysis). However, if we are to get students to understand the conceptual basis of such reasoning (11), then it is important that they be aware of how mathematical logic of proportional reasoning can, along with “big ideas” such as the atomic nature of matter, conceptually justify the application of proportional reasoning, as has been described by some previous workers (12–15). In this chapter, we review ideas from the literature in chemistry education, research about how people learn, and studies of how knowledge is transferred across domains, to argue for the importance of thinking about fundamental aspects of mathematical reasoning in chemistry. We focus on the logic behind proportional reasoning to clarify how that logic is the basis of a great many chemical calculations. To do this, we consider what it means to have an expression about a system and how this is then the basis of mathematical notions of quantification and variables. This then allows us to consider how expressions, quantification, and variables are present in proportional reasoning, including the way that proportional reasoning works within chemistry. Finally, we include data that shows how, in at least one system, college students demonstrate smooth facility with proportional reasoning in a nonchemistry task but often fail to transfer that to a chemistry task of related content involving molarity. The Importance of Considering Mathematical Reasoning in Chemistry It is easy to consider that mathematics plays an important role in chemistry as a tool—as a way to accomplish transformations and representations in the service of chemical reasoning. If this is the case, then it may not be important to explicitly teach mathematical reasoning at any level. Instead, instruction may make use of mathematics, but in an algorithmic form. However, there is ample evidence and arguments that simply using algorithmic or heuristic strategies will not contribute to a deep conceptual understanding of chemistry (16–18). It is considered a basic principle of learning theory that deeper learning requires a strong conceptual understanding, something summarized well within the National Research Council’s How People Learn (11), which includes the principle that: To develop competence in an area of inquiry, students must (a) have a deep foundation of factual knowledge, (b) understand facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application. 158 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Further evidence for the importance of mathematical reasoning in chemistry is available from recent standards about what and how science should be learned. The National Research Council’s Framework for K–12 Science Education (19) considered research on how science is done and formulated a set of Science and Engineering Practices, including the practice of “mathematical and computational reasoning.” Thus, in the appendix for the Next Generation Science Studies (which were developed from the Framework), we read that students engaged in mathematics and computational reasoning will (20): • Use mathematical, computational, and/or algorithmic representations of phenomena or design solutions to describe and/or support claims and/or explanations; • Apply techniques of algebra and functions to represent and solve scientific and engineering problems; [and] • Apply ratios, rates, percentages, and unit conversions in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/ mL, kg/m3, acre-feet, etc.). These standards point to how mathematics is also a critical part of the conceptual foundation of the sciences. As one example, the measurement of chemical substances in mole amounts is necessary to reason about and understand chemical formulas, reaction stoichiometry, and important applications such as concentration. Chemical systems react on the basis of individual particles and formula units. However, we rarely count moles in any direct manner. Instead, we rely on the conceptually important relationship of mass and count, which depends on the important and “big” idea that the atoms of the chemical elements are conserved in all chemical and physical transformations. This relationship of mass and count, however, is a proportional relationship—nothing more and nothing less. Therefore, mathematical reasoning using proportions is vital to conceptually understanding how to measure and reason about chemical reactions. However, it is unlikely that students will automatically use mathematics, even mathematics that they understand well, in new contexts. Instead, there is the problem of students transferring relevant knowledge into a new domain, such as chemistry. This follows from the research of workers such as diSessa (10), who considered, through his Knowledge-in-Pieces theory, that students can struggle to apply knowledge in coherent ways in new domains. Again, this argues for making explicit the reasoning of mathematics whenever it is applied in new settings. Proportional Reasoning in Mathematics The mathematics education community has paid careful attention to both research about and teaching associated with proportional reasoning. This chapter cannot review all of that literature, of course, but some highlights do stand out. First, it is important to note that most educational systems, including in the United States, consider ratios and proportions to be topics for upper elementary or middle school grades, which is where the corresponding mathematics education literature focuses (21–23). These are well separated from when students encounter algebra or any sort of chemical reasoning using mathematics (in secondary and postsecondary settings). The resulting instructional offset, along with issues of how mathematics is taught in science, means students who have a good general understanding of proportion and quantity may not be able to apply that properly in specific areas of science. The same instructional offset problem with ratio, proportion, and quantity is also found in examining the relevant standards. The National Council of Teachers of Mathematics in 2000 159 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

recommended, in Principles and Standards for School Mathematics, that students should be able to use fractions, percents, and decimals interchangeably and appropriately in the sixth to eighth grades (24). Specifically: …students should understand not only that 15/100, 3/20, 0.15, and 15% are all representations of the same number but also that these representations may not be equally suitable to use in a particular context. For example, it is typical to represent a sales discount as 15%, the probability of winning a game as 3/20, a fraction of a dollar in writing a check as 15/100, and the amount of the 5% tax added to a purchase of $2.95 as $0.15. It is easy for those with expertise to overlook the complexity of proportions and their composition. Lamon (25) notes that ratio reasoning is “abstract… [I]t is related to many rational number ideas: equivalence, the idea of showing the same relative amount, and the relationship between the four quantities, just to name a few.” Hence, there are multiple mathematical concepts present in proportional reasoning. This includes the issue that some of the quantities related to “amount” vary with the size of the sample, making them extensive quantities. Other quantities, related to relative amounts, may not vary with the size of a sample, making them intensive variables. Lobato and coworkers, examining middle school students, suggest essential understandings with respect to ratios and proportions (26). The first is that “reasoning with ratios involves attending to and coordinating two quantities.” Therefore, for a change in understanding to occur, “students need to make a transition from focusing on only one quantity to realizing that two quantities are important.” The problem is also found in higher education settings. Lobato (27) also reported on an interview study of high school students learning about steepness of a ramp. She found more than half of the students had difficulty isolating steepness (an intensive quantity) from its extensive variables of the height of the ramp and the length of the base. To draw a parallel to solutions chemistry, the intensive quantity would be molarity, and the extensive variables would be the moles of solute and the volume of solution. For students to understand the intensive quantity of molarity, they must understand the effect of changing each variable on the molarity. And this, we suggest, would be much stronger if the mathematical reasoning that is done with these variables—two extensive, one intensive—were clearer. Concentration problems are important in mathematics education, dating back to Piaget. Such an approach was followed by Schwartz and Moore in a study of sixth graders (21). In that case, they considered that students’ mathematical reasoning was not at the level to understand the proportional reasoning required. With junior high school students who did demonstrate mathematical reasoning skills, Eilam (28) showed that many students, observing that it takes fewer drops of water than of soap solution to dispense a constant volume, could not reason that the soap solution drops were smaller, and therefore could not make sense of the molecular basis of the phenomenon. The mathematics education literature often takes a simple approach to concentration with a focus on percentage or some other “part to whole” proportion. In chemistry, though, there is the added complication that students need to attend to different measurement units. For example, it is possible to represent the percent composition of titanium in an ore as 15%, the concentration of glucose as 7.5 g of sucrose/500 mL of water, and the concentration of potassium bromide as 0.15 mol. A 15% solution describes either a mass/volume ratio (e.g., g of A/L of B) or a mass/mass ratio (e.g., g of A/g of B). The choice of units is often context-dependent. Determining the economic value of an ore that is used to obtain a metal is best discussed with a weight percentage (15 tons of iron/100 tons of iron ore), calculating the density of a substance is best done with a mass/volume ratio (15 g 160 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

of a substance/1 cm3 of water), and determining the concentration of salt in water is best done with a molarity ratio (0.15 mol of potassium bromide/L of water). From Mathematical Logic to Proportional Reasoning in Chemistry In order to more fully consider how proportional reasoning works in chemistry, we next describe some basic notions and terminology in mathematics and, following the logic embedded in them, how they lead to proportional reasoning in chemistry (29). Specifically, mathematics has very careful ideas about expressions, statements, quantities, and variables. An expression is a designation of an object (“A number”); expressions are neither true nor false—they are given. Statements are claims about objects (“The number is 5”); statements may be true or false, depending on the situation. As we will see later, equations are a special kind of statement, but one that provides the opportunity to do important work in delineating useful relationships within a system. As a way of considering these ideas, consider Figure 1. A central photograph of a piece of mineral graphite is presented. The photograph itself is an expression. So, too, are the noun phrases in the boxes surrounding it. Each of the expressions are examples of different variables: a category that can take on different characteristics. We are, of course, accustomed to variables that have a measurement and number associated with them, such as “183.4 g.” But logical variables are not restricted to what we can measure or relate to a unit. Each of the verbal or graphical expressions are variables, also. As noted, statements arise when we assert that two expressions have a relationship. This is done in Figure 1 through linking verbs, including the important “be” verbs, to link the two expressions.

Figure 1. A photograph of graphite and a series of statements about it. In the case of the statements in Figure 1, a valid logical proposition we could write is “there is some piece of graphite (the one in the photograph) that has a mass of 153.4 g.” An important note here is that the quantification we have in this case, if it is to be valid, needs three parts: a number (153.4), a unit of measurement (grams), and a specific substance (the graphite piece). If this was all that mathematics allowed us to do with measurement, we would indeed have little use of mathematics—the measurements of mass would be no different than assertions of identity. However, mathematical reasoning lets us seek out ways to relate variables to one another. Because we want to make statements that apply to other forms of graphite, and that are not just about individual cases, we also use mathematical reasoning to relate variables to one another. The notion of proportional reasoning depends on the property that certain properties of substances, such as mass and the count of atoms, are present in a simple ratio (proportion) that is the same for all samples. These properties are not individual masses, mole amounts, combustion energy, or the like. The proportional properties we are interested in are those that are ratios among the particular mass, mole, and combustion energy amounts, and the like. What we reason here is that the ratio of any two 161 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

extensive variables (for example, mass of copper in a mass of copper [II] oxide) is the same for all samples:

This means that, if we have two samples, we can relate the ratio of the variables in one sample directly to the other. And the “reference” may be a previously determined amount. For example, say we have a sample of 32.5 g of copper (II) oxide and wish to know the amount of copper. If we also know that a property of copper (II) oxide is that it contains 79.8 g of copper per 100 g of copper oxide, we can use proportional reasoning as in Figure 2.

Figure 2. Solution of a chemical ratio problem using explicit proportional reasoning. It is important to note the wide application of even the simplest relationship of direct variation. For example, we might write: 1. All lengths of 1 decimeter are also 1/10 of a meter. 2. All masses of 1 mole of fluorine atoms are 18.998 grams. 3. All 100-gram samples of copper (II) oxide have 79.8 grams of copper. These relationships, as stated, are all for a very special case of samples—one decimeter, one mole of fluorine, or 100 grams of copper oxide. The power of proportional reasoning is found when we claim that these are also true for all samples on a ratio basis: 1. All samples contain 1/10 of a meter per 1 decimeter. 2. All samples of fluorine atoms have 18.998 grams per one mole. 3. All samples of copper (II) oxide have 79.8 grams of copper per 100 grams of copper. We can then write all three of these as ratios as standard amounts to be used in other calculations:

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

Constants of Proportionality The reasoning sequence in Figure 2 describes an important chemistry problem in terms of the underlying mathematical terminology and logic. In addition to this, mathematics has the ability to relate variables in the form of mathematical equations. In the case of a simple proportion, the basic equation relating two variables y and x is:

The term k is, mathematically, a constant: a special variable that, in the context of a mathematical system, does not change. In this case, the value of k is:

This is valid for all pairs of x and y. This means:

Of course, the discussion above, for copper oxide, is a special case of this. And in this case, as shown in Figure 3, we can lay out the calculation as a process: multiplying the “given” amount (32.5 g of copper oxide) by the constant of proportionality yields the “target” amount of grams of copper.

Figure 3. Reasoning through a problem using a known ratio as a constant of proportionality.

It is easy to overlook the importance of the y = kx relationship. This applies for any situation in which two variables are in a direct proportional relationship. These relationships arise in many different situations in chemistry. Table 1 contains a partial listing of different proportions as they appear in general chemistry. The different constants of proportionality in Table 1 arise from three sources. The first source is associated with defined ratios, most commonly found with the conversions among metric units of the same type. Another source of ratios is also fixed: the ratios of the amounts of atoms and substances. But the constant ratio of composition or of stoichiometric ratios reveal an important fact: that atoms are persistent in their identity. The third source of ratios has one or more experimental components to the ratio. The numbers in this case are therefore subject to some imprecision.


Table 1. Common Constant of Proportionality in Chemistry

are chemical formulas for certain condensed materials (such as the Berthollet minerals) that have variable composition, and therefore, their composition is subject to less than absolute precision. * There

Explicitly writing the equation of an unknown and a known ratio is rarely done, although it provides a clear link to the underlying mathematical reasoning. Instead, common problem solving makes use of a different strategy, where mathematical operations transform an expression. For example, this is what happens when we simplify an expression, as in reducing a fraction to its simplest terms:

In this case, we divide the original expression (16/4) by the number 1, in the form of 4/4. As a result, we can now express the original fraction in a much simpler term—as an integer.


The same logic we used for 16/4 is what happens when we convert one unit into another by multiplication with a constant of proportionality (formally, remember that all constants of proportionality are equal to 1). This is almost always denoted by an equal sign because the resulting statement does have the form of an equation. In the general case, we are taking the unit x and carrying out the procedure of multiplying it by a ratio (the constant of proportionality) between x and y:

This logic, applied to the problem in Figures 2 and 3, would then give:

This is the familiar form of unit conversion problem found throughout chemistry. It can also be applied to several units in sequence, covering many different unit types. Another set of related problems operates on both the numerator and the denominator of a ratio (30).

Student Proportional Reasoning on Mathematical and Chemical Tasks This review of the basic logic and the chemical application of proportional reasoning demonstrates how a relatively simple form of mathematics can be applied in many different chemistry settings. But, as discussed at the start, it is well known that problems with mathematical knowledge cause many students to struggle in chemistry. This raises an important question: Is this difficulty because of problems with their knowledge of proportional reasoning or with their understanding of how to apply this in chemistry? To investigate this, a qualitative study was conducted with first-year students entering college who had taken a chemistry placement test that signaled their potential interest in a major that required at least one semester of general chemistry. Full details of this study have been published and can be found elsewhere (31). The work includes an important finding that is summarized here for its relevance to the problem of how students’ knowledge of proportional reasoning does, or does not, relate to their ability in chemistry settings. The work was done with 24 students entering college, recruited prior to their first classes. By comparing results on a chemistry placement test that included both quantitative and conceptual questions, a sample of students with high and low quantitative knowledge and high and low conceptual knowledge was recruited. Students were given an interview that included several parts, including an assessment, not pertinent here, of basic reasoning skills in math and chemistry. One part of the interview asked students to reason on four tasks designed to get insight into students’ use of proportional reasoning on domain-general (DG) problems and chemistry domain-specific (DS) problems. In this manner, we used the method of contrasting cases to examine how individual students reasoned in similar but different situations. This enabled direct examination of their ability to reason with proportions in both familiar and a not-as-familiar settings (21, 28). All interviews were conducted before students had encountered college-level chemistry, and interview data (transcripts and drawings) were open-coded. As mentioned, deeper analysis of the codes permitted further


exploration of their reasoning (31), but for this chapter we report summary observations concerning proportional reasoning in particular. Figure 4 summarizes the two sets of tasks the students did, following the research methodology of contrasting cases. Both cases, as will be shown later, are logically equivalent. But one is a familiar setting of different shades of paint on wood blocks of different sizes; this is considered a DG task. The other case is a different setting of different concentrations of CaCl2 in solutions of different volumes; this is considered a DS task.

Figure 4. DG and DS tasks for proportional reasoning protocol. Important differences in students’ proportional reasoning were apparent between the two tasks. In the case of the DG tasks, 21 of the 24 students represented the concentration of the paint as an intensive quantity for both tasks. Students specifically referred to an intensive view of concentration, as reflected in their discussion of the color and height of the blocks. This enabled them, for example, to reason correctly on the Different Color Same Height task that the blocks with a deeper red color needed less white color and more red color but the same total amount of paint. This indicated that they understood how a ratio could vary in its composition even as the total amount was constant. 166 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 addition, most students recognized that, on the Different Height Same Color task, the same ratios of colors were needed, but different amounts of paint would be needed. They were therefore able to describe that different amounts of both white and red paints would be needed. Three students were unable to represent concentration as an intensive quantity in one of the tasks. In all three cases, the student focused on the salient feature of color in one of the two tasks instead of the intensive nature of concentration. The success on the DG task was found for students at all ability levels. Hence, we found that, for this task at least, students entering college did have proportional reasoning ability that they could use to describe a novel situation. Things were quite different on the DS chemistry task. In this case, only three of the 24 students used intensive quantities and proportional reasoning about the tasks. In those cases, they used proportional reasoning correctly, with recognition that there was more calcium chloride in a jar with greater volume (at the same molarity) and with higher molarity (at the same volume). Most students, though, saw the value presented for molarity as an extensive amount and reasoned accordingly. Importantly, on the Same Volume Different Molarity task, this still enabled them to reason correctly, and they described that the jars with higher molarity had more calcium chloride. This used a “more is more” schema (10, 32). But because they saw molarity as an extensive amount, these students were unable to reason that more solute was present in the Different Volume Same Molarity task. Almost all of these students had used proportional reasoning correctly on the blocks task, highlighting that the issue was how they understood proportion in the context of chemistry. The origin of the misunderstanding for so many students in chemistry has several possible explanations. Some students explicitly indicated that they thought “M” on the label represented moles, not molarity or any other concentration unit. Hence, they may never have considered the problem in proportional terms at all. Others did not simply declare that “M is moles” but still reasoned in extensive terms. This is highly reminiscent of the finding of Lobato (27) with high school students considering steepness of ramps. The implications of this portion of our larger study are important for thinking about the role of proportional reasoning in chemistry settings. First, this indicates that simple reasoning using proportions is something that students can do, at least on a DG task. Second, even in the case when students are able to reason proportionately, they are using a relatively simple set of explanations. When most students came to the chemistry application, this lack of mathematical rigor in their thinking meant that they went with a simple and accessible extensive reasoning scheme. This scheme was successful in the case of the Same Volume Different Molarity task, however, not because the students applied proportional reasoning correctly but, rather, because the proportional unit of molarity is itself directly proportional to the number of moles. The scheme was unsuccessful in the case of the Same Molarity Different Volume task because the number of moles total is also directly proportional to the total volume (not just to the molarity)—something most students failed to recognize because they were not reasoning in intensive terms. The mathematical description presented of proportional reasoning in chemistry provides a useful way of analyzing what was effective and what was not effective in student reasoning. In the case of the “blocks” tasks, we can consider that the Different Color Same Size task represented a change in the proportion of red paint that was needed to paint a block of a standard size:


The blocks all required the same total amount of paint, but the red paint increased as the paint color went from white to red. The Different Height Same Color task is an example of a case where the proportion of red paint to total paint is constant but the total amount of paint changes with the units of the block. As a result, larger blocks, with the same color of pink paint, require proportionately more red paint:

Most students did have the proportional reasoning ability to carry this out. However, in the Same Molarity Different Volume task, students who could reason about the different amounts of paint needed were often incorrect in their reasoning. They were able to recognize that, as the value of M changed, there would be different amounts of solute:

This is the “correct” reasoning: the ratio solute/solution volume changes, and therefore the amount of solute changes. But students could also reason to the right answer in this case if they totally neglected the proportional reasoning; thinking, as some did, that “M” represents moles, they could reason that a solution with “0.15 mol CaCl2” must have more CaCl2 than a solution with “0.050 mol CaCl2.” The problem with some students’ application of proportional reasoning does not appear until they are asked about the Same Volume Different Molarity task. In this case, students should have reasoned that there was a change in the amount of solute, given the change in the solution volume while the value of solute/solution volume was kept constant. However, they did not engage in the proportional reasoning at all and kept to the idea that the amount of solute would be the same in all solutions. Using the theory of transfer of domain knowledge, we can interpret these results as evidence that students who clearly had domain general ability in proportional reasoning did not spontaneously transfer that to a task involving reasoning about solutions.

Implications This chapter has presented a mathematically logical description of how proportional reasoning, including the valid use of ratios and constants of proportionality, should be specifically considered in the context of general chemistry. The explication of this logic would potentially present students with a clear basis for their reasoning—perhaps (though we have not tested this hypothesis) overcoming heuristic and algorithm-driven schemes. Our empirical work shows that these proportional schemes are available to students for their reasoning in a DG sense. This suggests, finally, that a key step in helping students use proportional reasoning correctly is not simply to teach the relevant chemical setting; it should explicitly show students how the logic of proportional reasoning is structured, is available to them in DG settings, and should be transferred into chemistry. In other words, considering the question used to organize this symposium, it may not matter whether mathematics is the “cart” or the “horse.” What is important is that mathematics is linked to the chemistry by a strong and easily visible yoke of mathematical reasoning properly and explicitly used.


The work presented here, both in the logical analysis of proportional reasoning and the specific findings with respect to DS and DG reasoning, suggest several specific steps for the chemistry classroom: • When proportional reasoning is first introduced, for example with metric unit conversions or with density, it should be explicitly linked to the mathematics of proportional reasoning. This can be as simple as the demonstration that unit conversion problems all fit within the simple y = kx formulation. • The importance of units within problems is a key part of proportional reasoning also, which is why reasoning about metric units is often a starting point. Making units a consistent focus, not just algebraic factors, should be emphasized. For example, to convert from grams to moles of a substance, the process can be described as “multiplying by the reciprocal of the molar mass, with moles in the numerator and grams in the denominator” instead of simply saying “divide by the molar mass.” • Tracking units very carefully, and noting how proportional reasoning allows us to express the same quantity using different measurement systems, should be a consistent part of problem solving, with unit analysis a key part of “showing the work” in solving a problem. • The importance of proportional reasoning for applying central concepts of chemistry—such as the big idea that atoms, with fixed masses, are the persistent building blocks for matter—should be brought into the discussion of mathematical conversions. • The use of mnemonics for problem solving can add speed to the work done by students but at the expense of conceptual understanding. It should only be done (if ever) after a full mathematical explanation. • We could tie in the importance of units within problems to help them keep track of the relationship between variables. Note that taking a closer look at their problem-solving steps allows us to find simple mistakes in proportion use instead of just marking items wrong. • As indicated in our research, particulate-level drawings helped make the different conceptions of molarity evident. These should be used widely in chemistry instruction, of course, and also where possible in illustrating the link of chemistry to the mathematics. • Molarity and similar concentration tasks should be taught, at least initially, using structurally similar tasks as a precursor to teaching molarity, as in the DG examples such as blocks and paints discussed here. • Assessment of concepts such as molarity should include a student’s understanding of molarity as an intensive quantity and pay careful attention that the numbers in the problems do not include a unit value or multiplier. For research, the work presented here brings out the way in which mathematical reasoning can be based on assumptions and heuristics about the way variables behave—specifically the “more is more” phenomenological primitive. Examining this in other areas, including ones with nonlinear mathematics (such as equations involving logarithmic and exponential relationships) is needed. Similarly, the way in which algorithmic and mnemonic systems may support and confuse more accurate reasoning can be included whenever students are asked about mathematical applications.

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The Logic of Proportional Reasoning and Its Transfer into Chemistry

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