Chapter 2
A Modeling Perspective on Supporting Students’ Reasoning with Mathematics in Chemistry Downloaded via UNIV OF ROCHESTER on May 13, 2019 at 17:15:00 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
Katherine Lazenby and Nicole M. Becker* Department of Chemistry, University of Iowa, 305 Chemistry Building, Iowa City, Iowa 52242, United States *E-mail:
[email protected] Undergraduate general chemistry courses typically feature a substantial amount of mathematical problem solving. Research shows that many students approach mathematical problem solving algorithmically and without recognition of either the assumptions and limitations of such models or their relationship to the particulate level. The question becomes: How can instructors support students in more meaningful engagement with mathematical models in chemistry contexts? In this chapter, we present research on modeling approaches to undergraduatelevel chemistry instruction and discuss the role of epistemological knowledge, specifically metamodeling knowledge, in students’ reasoning with and about mathematical representations.
Mathematical models (and related models such as computational models) are central to contemporary research in chemistry and are important tools for predicting and explaining chemical behavior. For students of chemistry, mathematical models are important, not only because they can serve as conceptual bridges between macroscopic and particulate-level scales (1), but also because they represent part of an important scientific process—the practice of developing and using models. This practice is one of eight highlighted in the National Research Council’s Framework for Science Education (2) as critical elements of contemporary scientific inquiry. Other practices include analyzing and interpreting data, using mathematical and computational thinking, and constructing explanations (2). The Framework’s argument for engaging students in scientific practices is twofold; by giving students opportunities to use disciplinary core ideas and interdisciplinary (“cross-cutting”) ideas to engage in scaffolded forms of science practices such as constructing and using models, students will develop both deeper knowledge of core disciplinary concepts and a better understanding of the nature of scientific practice. That is, they will develop more robust epistemological knowledge about the nature of scientific inquiry (2).
© 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.
While the Framework forms the basis of K–12 level STEM education reforms, researchers and educators have argued that curricula centered on having students engage in science practices as they learn core and cross-cutting ideas would also support student learning at the undergraduate level (3, 4). Several “three-dimensional” (3-D) approaches to undergraduate curricular reform have been described in the literature (3, 5), and researchers have developed tools for helping faculty evaluate the extent to which their assessments address 3-D learning and examine whether classroom activities reflect 3-D learning (4, 6). Early assessments of the impact of these instructional approaches on student learning suggest greater content learning gains (5, 7). To date, however, there is limited information available on the impact of 3-D instruction or other modeling-focused approaches on students’ ideas about the nature and purpose of models, such as mathematical models. In the remainder of this chapter, we set the stage for future examinations of the impact of 3-D and modeling-focused instructional approaches on students’ epistemological ideas about models and modeling. First, we provide an overview of the practice of constructing and using models. We then discuss elements of metamodeling knowledge and their development and conclude with a review of curricular models that show promise for supporting students’ ideas about models and modeling.
Scientific Models and the Practice of Developing and Using Models Scientific models often take the form of representations that embody ideas about how and why a phenomenon occurs or about relationships between components within systems being studied. Models can take a range of representational forms (e.g., equations, graphs, diagrams, words) and are defined more by how they are used—that is, to predict and explain phenomena—than by the form they take (8). That is, they are sense-making tools that help predict and explain the natural world. The practice of modeling can be understood as a bridge between wondering why something works or happens the way it does and arriving at an explanation for the phenomenon that is grounded in theory and empirical evidence. When generating explanations about natural phenomena, people often implicitly use a set of ideas and assumptions they have about the system or problem. The process of modeling makes these ideas explicit (8). For students, we can think of two ways of engaging in the practice of developing and using models: thinking “with” models and thinking “about” models (8). Thinking “with” models has also been referred to as “model-based reasoning” in the research literature (9) and involves the application of models as tools for predicting or explaining macroscopic phenomena. For example, one might use the ideal gas law, PV = nRT, to predict how the volume of a balloon might change when moving from indoors to outdoors on a cold winter day or use a simulation of particulate-level motion and spacing to help construct a particulate-level account of how and why volume changes. Thinking with models is essential for helping students see how modeling enables new ways of thinking about a system under study (8). Thinking “about” models, in contrast, involves constructing, testing, and refining models rather than the straightforward application of existing models. Thinking “about” models may involve iterative cycles of model development, testing, and model refinement. These cycles are guided by the broader goals of sense-making, a term we use to refer to the process of resolving gaps or conflicts in comprehension in the service of understanding how or why a system works the way it does (10). Thinking “about” models is especially important for helping students see how theoretical ideas (e.g., about the particulate level) connect to the models under study. Additionally, thinking “about” models is important for supporting students’ understanding of what goes into a model and why as 10 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.
well as for their knowledge of the relationship between the model and the target system (that which is being modeled). The nature and importance of epistemological ideas such as these are discussed further in the next section.
Metamodeling Ideas: Epistemological Knowledge Associated with the Practice of Developing and Using Models Scientists aim to develop models that can provide insights into how or why a phenomenon occurs, for example, by providing a mechanism that drives the phenomenon. Disciplinary norms and ideas about what counts as a “good” model or a robust modeling process play key roles in framing model development and evaluation. For example, it is important that models be general enough to be applicable to other phenomena and useful for the modelers (11). It is also essential that models be accurate with respect to predicting and explaining phenomena within certain levels of tolerance (8, 11). These ideas and others represent elements of knowledge that, together with content knowledge and knowledge of mathematics, support students’ engagement in developing and using models. Epistemological knowledge specific to the practice of developing and using models has been termed metamodeling knowledge. Metamodeling ideas include the idea that models are developed and refined on the basis of empirical data, that there may be multiple models for a phenomena, and that models highlight some features of a system while simplifying others (12). Such ideas previously have been considered part of students’ understanding of the nature of science but have more recently been considered through the lens of science practices (13, 14). Studies examining students’ metamodeling knowledge commonly address five key topics: the nature, purpose, and testing of models; the role of multiple models; and the way in which models are revised (15, 16). These five aspects of metamodeling knowledge are summarized in Table 1. In the next section, we briefly define each of these aspects and highlight research evidence on students’ thinking about each one.
Aspects of Metamodeling Knowledge and Their Development Grünkorn, Upmeier zu Belzen, and Krüger developed a “model of model competence” (Table 1), a five-dimensional hierarchical progression of knowledge about models and modeling, based on the open-ended survey responses of 7th–10th grade students. The five dimensions of metamodeling knowledge in their model are the nature, purpose, testing, changing, and multiplicity of models. These dimensions and their developmental progression according to Grünkorn and colleagues’ data are summarized in Table 1 (15, 16). The nature of models refers to the relationship between the model and the phenomenon it represents (14, 17). Experts’ understanding of this aspect of metamodeling knowledge includes the idea that models may be abstract, theoretical representations of phenomena and that models may simplify some aspects of a system while highlighting others (16, 18). Research on students’ ideas about models and modeling suggests that students view models through a naïve realist perspective and see models as exact replicas of the phenomenon (19, 20). Grünkorn et al. found that students often consider the extent to which the model “looks like” or “matches” the target instead of considering that assumptions and simplifications are necessary in the representation of phenomena. 11 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. Grünkorn, Upmeier zu Belzen, and Krüger’s Model of Model Competence. Reproduced with permission from reference (15). Copyright 2014 Taylor and Francis. Aspect
Complexity Level 1
Level 2
Level 3
Nature of models
Replication of the target phenomenon
Multiple models
Due to differences between Complex phenomena allow the target phenomena for multiple models
Due to different hypotheses about the target phenomenon
Describing or showing phenomena
Explaining mechanisms of or variable relationships within phenomena
Predicting phenomena
Testing models Testing the model itself
Comparing the model to the target phenomena
Testing hypotheses about the target phenomena with the model
Purpose of models
Changing models
Idealized representation of the Theoretical reconstruction of target phenomenon the target phenomenon
Would occur in order to Would occur due to new correct errors in the model findings regarding the target phenomena
Would occur due to new hypotheses about the target phenomena
The purpose of models relates to their function as tools used to predict and explain a phenomenon (12, 21, 22). Experts may focus on models’ use as a way to illustrate the underlying mechanism of a phenomenon, which enables the development of testable hypotheses about the target system. Students, in contrast, may believe that the purpose of modeling is to describe or show the original as precisely as possible (22). The idea of multiple models relates to the idea that different models may make different theoretical assumptions and hence have unique limitations and affordances that enable them to be useful in addressing different hypotheses about the target system (19, 20, 23). Students, however, may see multiple models as presenting the same information through different representational conventions (21), see different models as related to differences between phenomena, or see different models as focusing on different aspects of complex phenomena (16, 18). The testing aspect of metamodeling knowledge addresses the idea that models are developed, tested, and revised on the basis of data. Although experts may think about testing hypotheses about the target phenomena with the model by collecting empirical data, students may think about comparing the target phenomena directly with the model and evaluating their similarity (16, 18). Finally, changing models relates to the way in which models are revised based on the outcome of testing. Although experts recognize that models can be revised based on the results of experiments or advances in theory, research suggests that students may not recognize that models can be revised (15). There is evidence that students’ ideas about these five dimensions of metamodeling knowledge develop independently and at different rates (24). Additionally, students’ knowledge about metamodeling ideas may be highly context-specific and tied to their understandings of models within specific STEM disciplines (22). For example, Krell and Krüger administered an open-ended survey to elicit undergraduate and graduate students’ ideas about both models in general and specific models in order to investigate the interaction between students’ understanding of models and 12 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.
situational factors such as the respondent’s self-identified discipline and item features, in this case, the models that students specified in survey prompts. With regard to students’ discipline, the authors reported that students from STEM disciplines expressed advanced understandings more often than their peers in the social sciences or linguistics; however, even STEM students expressed “prospective” or expertlike understanding of models and modeling in only 20% of responses (compared with 12% overall). Regarding specific versus general metamodeling knowledge, university students expressed more expertlike ideas about models generally than they did about specific models from their discipline, suggesting that metamodeling knowledge may be situated and contextualized. Those students who did express expertlike views with regard to specific models had specified abstract models such as mathematical equations, supporting the view that students’ ideas about models and modeling are highly context-dependent (13).
Research on Students’ Metamodeling Knowledge in Chemistry Contexts To date, there is little research on how students think about these metamodeling ideas, specifically with respect to mathematical models. The progression in Table 1, for example, was developed through a characterization of students’ ideas about various types of biological models such as theoretical reconstructions of a Neanderthal man or of tyrannosaurus rex or maps of the taste regions on the human tongue (15). Other work on metamodeling knowledge has focused on diagrammatic models (e.g., diagrams of evaporation and condensation) (12). In chemistry contexts, some research has been conducted to develop and use assessments of students’ ideas about models and modeling from a domain-general perspective. One notable and widely used assessment of students’ metamodeling knowledge is the Students’ Understanding of Models in Science (SUMS) instrument, a Likert-scale instrument intended for use in assessing students’ knowledge about scientific models in general (20). Items on the SUMS instrument were based on Grosslight et al.’s (19) analysis of students’ ideas about models. The SUMS includes the following five subscales: multiple representations (MR), models as exact replicas (ER), models as explanatory tools (ET), uses of scientific models (USM), and the changing nature of models (CNM) (20). The SUMS has been used to characterize the metamodeling knowledge of various student populations (20, 21, 25–28) and as an assessment of the efficacy of modeling-focused curricula (25–28). For example, Gobert et al. administered SUMS to students in three different high school science classes (biology, physics, and chemistry) and found higher SUMS scores (based on mean differences on three subscales) in physics classes than in biology classes (25). This result may suggest that different courses impact students’ metamodeling knowledge differently. Park et al. used the SUMS to examine the impact of a modeling-focused instructional intervention in which students used computer-based models over the course of one year. Simulations depicted both macroscopic and particulate-level phenomena and were used in each classroom between 4 and 20 times, depending on the instructor. A class that did not use simulations served as the control group. Park and colleagues found nonsignificant differences between the control and treatment groups on four of the five SUMS subscales. On the fifth subscale (CNM), there were statistically significant differences that favored the control group (28). Although it is possible that the intervention did not affect students’ ideas about models, it may also be the case that the domain general assessment (SUMS) was not able to detect changes in students’ ideas about the specific type of models addressed by the intervention. 13 Towns et al.; It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics ACS Symposium Series; American Chemical Society: Washington, DC, 2019.
Context Specificity of Students’ Metamodeling Ideas There is growing evidence that domain-general instruments may not be robust indicators of students’ ideas about specific types of models (22). As we have noted, Krell and colleagues found that students expressed more expertlike ideas about models when discussing models in general than when thinking about models specific to a STEM discipline (13). Our own work on students’ ideas about models suggests that students may have very different ideas about even specific models within the chemistry curriculum (29). We administered an openended survey about models in chemistry contexts to 773 undergraduate general chemistry students. We asked students to indicate whether they thought six representations common to the chemistry curriculum would be considered scientific models and to explain their reasoning. We found significant differences in the ways that students categorized and discussed mathematical and graphical models (energy diagram [ED], the ideal gas law [IG], and equilibrium constant expression [EQ]) compared with models of particulate-level entities (a representation of the motion and spacing of gas particles [RP], physical model of a molecule [PM], and a Lewis structure [LS]) (Figure 1).
Figure 1. Proportion of students who classified six representations from general chemistry as “a scientific model” or “not a scientific model”; N = 773. Pairwise t-tests indicate significant differences between proportions of students who classified particulate-level entities (RP, PM, and LS) as scientific models and the proportions who classified mathematical models (ED, IG, EQ) as scientific models; all pairwise pvalues < 0.001, with one exception—the pairwise p-value = 0.02 for the difference between ED and LS. We also identified qualitative differences in students’ reasoning about their classifications of different representations. Students who did not categorize the mathematical and graphical models (ED, IG, and EQ) as scientific models commonly discussed the representational form of the item, for example, noting that equations and graphs can never be considered models because models must be visual in nature. When discussing the ideal gas law specifically, students also discussed the ontological status of the model. For example, one student noted: “It [the ideal gas law] has law in the name... It goes something like model, theory, law.” To us, this suggests confusion about what counts as a model compared with a scientific theory or law. Students who did classify the mathematical and graphical representations as scientific models discussed their utility for predicting or explaining phenomena much less frequently (
