5

Problem Solving, Spatial Thinking,
and the Use of Representations
in Science and Engineering

Chapter 4 explored students’ conceptual understanding in science and engineering, with the goal of helping students advance toward a more expert-like understanding. This chapter addresses how students use those understandings to solve problems, and how scientific representations, such as pictures, diagrams, graphs, maps, models, and simulations facilitate or impede students’ problem solving and understanding of science and engineering. Although we recognize that there are other important dimensions to promoting a deep understanding of science and engineering—including strong mathematical knowledge—we start with these topics because they are vital to acquiring greater expertise in the disciplines, and discipline-based education research (DBER) on them is relatively extensive and robust.

The discussion of each topic in this chapter begins with an introduction of that topic and its importance to undergraduate science and engineering education. Following these introductions, we provide an overview that summarizes the focus of DBER on the topic, the theoretical frames in which DBER is grounded, and the typical methods used. We then discuss the research from each discipline and summarize key findings across disciplines. The discussion of each topic concludes with an identification of directions for future research.

PROBLEM SOLVING

Problem solving may be the quintessential expression of human thinking. It is required whenever there is a goal to reach and attainment of that goal is not possible either by direct action or by retrieving a sequence of



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5 Problem Solving, Spatial Thinking, and the Use of Representations in Science and Engineering Chapter 4 explored students’ conceptual understanding in science and engineering, with the goal of helping students advance toward a more expert-like understanding. This chapter addresses how students use those understandings to solve problems, and how scientific representations, such as pictures, diagrams, graphs, maps, models, and simulations facilitate or impede students’ problem solving and understanding of science and engi- neering. Although we recognize that there are other important dimensions to promoting a deep understanding of science and engineering—including strong mathematical knowledge—we start with these topics because they are vital to acquiring greater expertise in the disciplines, and discipline-based education research (DBER) on them is relatively extensive and robust. The discussion of each topic in this chapter begins with an introduction of that topic and its importance to undergraduate science and engineering education. Following these introductions, we provide an overview that sum- marizes the focus of DBER on the topic, the theoretical frames in which DBER is grounded, and the typical methods used. We then discuss the research from each discipline and summarize key findings across disciplines. The discussion of each topic concludes with an identification of directions for future research. PROBLEM SOLVING Problem solving may be the quintessential expression of human think- ing. It is required whenever there is a goal to reach and attainment of that goal is not possible either by direct action or by retrieving a sequence of 75

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76 DISCIPLINE-BASED EDUCATION RESEARCH previously learned steps from memory (Bassok and Novick, 2012; Martinez, 2010). That is, during problem solving the path to the intended goal is uncertain. This characterization describes much of what people do on a daily basis, from (a) mundane activities like deciding what to cook for dinner given the ingredients at hand or how to get from work to home given cer- tain street closures, to (b) student activities such as interpreting laboratory results, figuring out how to organize a term paper on evidence for specia- tion, or designing a roller coaster for an engineering class, to (c) professional work such as curing illnesses or determining the best way to structure a class so that students will understand a key concept. Clearly, problem solving is central to science and engineering as well to everyday life. Researchers in numerous disciplines have drawn a distinction between well-defined and ill-defined problems (Hsu et al., 2004; Reitman, 1965). Most of the problems students encounter in their science and engineer- ing classes are well-defined, such as a mechanics word problem. In these problems, the initial conditions, the goal, the means for generating and evaluating the solution, and the constraints on the solution are all clearly specified for students. For other types of problems, however, such as a more open-ended laboratory or an authentic design problem in engineering, students have to define one or more of the problem components on their own (Fay et al., 2007; Whitson, Bretz, and Towns, 2008). In a laboratory, the means of generating the solution may be ill-defined. For an engineering problem, the goal may be ill-defined; as a result, it may not be clear how to determine whether the goal has been accomplished. For example, what constitutes a better coffee cup, and how does one decide that a new cup design represents a big enough improvement over the status quo to declare the design finished? Society’s most important problems are usually ill-defined in some way. Consider two examples: (1) How can the rapid regrowth of human skin be promoted so that life-threatening infections in burn patients are prevented? (2) How can affordable, alternative energy to power cars be generated, thereby limiting reliance on fossil fuels? These are the kinds of problems students will have to solve after they graduate. Students who have scant experience with ill-defined problems during their undergraduate education may be poorly prepared to grapple with the most significant problems in their fields. This discussion of problem solving is structured around important find- ings from DBER that are consistent with prominent themes from the cogni- tive science literature, namely problem representation and the nature of the solution process. In the cases for which the findings apply to only a small number of problem domains or disciplines, their broader applicability to problem solving within the disciplines of interest here is an open question. For example, as the following discussion will show, research has shown that

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77 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS experts adopt a working forward strategy in certain situations (e.g., solving introductory mechanics problems from physics). That strategy may reflect the nature of the particular problem-solving tasks that have been inves- tigated; other problems, from the same or other disciplines, may require different strategies or approaches for successful and/or efficient solution. Where findings have been replicated in numerous disciplines within and outside the sciences, it is probably safe to presume that those findings generalize to new problem domains or disciplines yet to be investigated. A prime candidate for such a finding is the differential reliance of experts and novices on structural versus superficial features of problems, respectively. A potential complication of generalizing from cognitive science research relates to the research setting and nature of the problems studied. DBER is typically conducted in classroom settings with discipline-specific prob- lems, whereas much of the cognitive science research—especially early in that field’s history—has been conducted in laboratory settings with puzzle problems or brain teasers. However, cognitive science research on problem solving in more ecologically valid settings and in domains such as physics and mathematics has often yielded comparable results to studies of puzzle problems (Bassok and Novick, 2012). What changes from one problem to another in these situations is the specific knowledge students need to bring to bear on their solution attempts, rather than the underlying cognitive processes. This general pattern of consistent results across disparate types of problems lends support to the committee’s view that findings from cognitive science research on problem solving may be applicable in undergraduate science and engineering domains in which they have not yet been investi- gated. After all, humans have a single cognitive system, with specific operat- ing parameters and constraints, that underlies their learning and problem solving regardless of the problem or discipline under investigation (Simon, 1978). At the same time, domain knowledge, which the general processes take as input, is important as well. In the inevitable cases where different patterns of results are found across problems or disciplines, these patterns will point to specific areas of science learning where disciplinary knowledge and perspectives are especially critical. Overview of Discipline-Based Education Research on Problem Solving Problem solving is a significant focus of DBER in physics (see Docktor and Mestre, 2011, for a review), chemistry (for reviews, see Bodner and Herron, 2002; Gabel and Bunce, 1994), and engineering (see Svinicki, 2011, for a review), and an emerging area of study in biology and the geosciences. Because problem solving is not taught frequently enough in astronomy, the committee did not find peer-reviewed astronomy education

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78 DISCIPLINE-BASED EDUCATION RESEARCH research on problem solving. As a result, this discussion does not include astronomy. A significant body of research on problem solving also exists in cogni- tive science, and that research overlaps considerably with DBER. Cognitive science research corroborates some DBER findings, can help to explain or extend others, serves as the theoretical basis for some studies, and provides potential building blocks for future DBER on problem solving (Bassok and Novick, 2012). Because of these linkages, this section interweaves discus- sions of DBER and cognitive science. Research Focus DBER studies on problem solving range from investigations of general problem solving strategies, to behavioral differences between novices and experts, to measurements of the effectiveness of instructional strategies that teach problem solving. Most of these studies investigate how students solve quantitative, well-defined problems. Accordingly, unless otherwise noted, the bulk of the discussion in this chapter refers to well-defined problems. The rich research base on problem solving in physics builds on many studies in cognitive science dating back more than 50 years. Many of these studies are based on the information-processing approach to understand- ing thinking, which comes from cognitive psychology (e.g., Simon, 1978). Key ideas from this framework include a step-by-step approach to problem solving, the importance of both internal knowledge representations and processes for understanding human thinking, the role of prior knowledge (which supports analogical transfer of knowledge gained from previously solved problems to solve new problems), and a limited capacity processing system. In chemistry, the study of problem solving is muddied by disagree- ments over what constitutes a problem (Bodner, 2004). These debates notwithstanding, a large group of studies has examined problem solving strategies in a specific content area of chemistry, such as stoichiometry or equilibrium. Studies on these topics have used several models of problem solving as a framework for inquiry, including the Pólya model (1945)— originally developed in the context of mathematics—of understanding the problem, devising and carrying out a plan, and checking work; Wheatley’s (1984) model of the steps successful problems solvers take to solve novel problems in mathematics; and the expert novice paradigm described in this chapter. More recently, chemistry education research studies have drawn on knowledge space theory, which describes possible states of knowledge (Taagapera and Noori, 2000) and the ACT-R theory for understanding human cognition (Taatgen and Anderson, 2008).

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79 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS Most research on student learning outcomes in engineering focuses on problem solving and engineering design (ABET accrediting criteria C and D; see Chapter 3). Early studies on this topic drew on information processing theory (Simon, 1978). More recent studies are grounded in constructivist (Piaget, 1978) or, less commonly, socioconstructivist (Lave and Wenger, 1991; Resnick, 1991) theories of learning. Research on problem solving in biology and the geosciences is sparse. The six biology studies that the committee reviewed examined individual differences in problem-solving strategies and did not explicitly situate them- selves in broader learning theory. In the geosciences, one emerging line of research draws on the cognitive science field of naturalistic decision making (Klein et al., 1993; Marshall, 1995) to investigate student problem solving in the field setting using global positioning satellites. Methods DBER scholars use a wide variety of qualitative and quantitative meth- ods to study problem solving. Some data are gathered using think-aloud interviews, in which students are asked to solve problems and verbalize their thoughts while being video and/or audio taped.1 Many studies com- paring expert and novice problem solvers have used categorization tasks, which require participants to group problems based on the similarity of the solution method. Other methods of tracking student problem-solving strat- egies include computer systems that use knowledge space theory (Taagepera and Noori, 2000), and artificial neural networks and Hidden Markov Models (Cooper et al., 2008). Some studies use student-generated summaries of their problem-solving approaches, course exam scores, and final grades to measure proficiency with problem solving, rather than examining problem solving processes. In physics, students’ written solutions to problems that have been designed by the researcher(s) and/or adapted from existing problem sources such as textbook end-of-chapter problems are another common data source. These data are typically analyzed by scoring students’ solutions relative to rubrics that characterize expert problem solving. Study populations range from high school students to community col- lege students to graduate students, with the preponderance of studies focus- ing on students enrolled in introductory college courses. Sample sizes in these studies range from fewer than 20 to several hundred students. The research typically is conducted in classroom settings—although in physics, this research also involves students both in research settings. 1 Ericsson and Simon (1980, 1993) have provided important theoretical and practical guid- ance for collecting and interpreting think-aloud protocols.

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80 DISCIPLINE-BASED EDUCATION RESEARCH Many DBER studies of problem solving have compared undergradu- ate students (novices) to more expert problem solvers such as graduate students, faculty, or professionals outside academia (e.g., Larkin et al., 1980; Petcovic, Libarkin, and Baker, 2009). Similarly, a major focus of cognitive science research on problem solving has been to compare the per- formance of novices (usually, although not always, college students) with that of experts. The definition of expert varies across studies, ranging from graduate students in an academic discipline such as physics, to grandmaster chess players or practicing physicians with 20 years of experience in their field. In cognitive science research, the typical study has used an extreme groups design, comparing a group of novices to a single group of (relative) experts. Fewer studies have compared problem solving across multiple lev- els of expertise. Regardless of the number of groups included, these studies provide important information for discipline-based education researchers because they give insight into the nature of the transition that needs to occur and the goal toward which students should strive (Lajoie, 2003). As discussed in Chapter 4, the committee recognizes that students are not expected to become experts within a single class, or even across the four years of their undergraduate education. They are, however, expected to progress along the path of increasing expertise. Thus, our frame of reference for this discussion is focused on helping students move toward the more expert end of the continuum. The Nature of the Solution Process A problem representation is an internal (i.e., existing in memory) or external (e.g., drawn on paper) model of the problem that is constructed by the solver to summarize his or her understanding of the problem. Ide- ally, this model includes information about the objects or elements in the problem, their interrelations, the goal, the types of operations that can be performed on the elements (e.g., algebraic operations for certain types of problems), and any constraints on the solution process. A student’s rep- resentation of the problem at hand is critical because the representation constructed affects the types of operations that can be applied (i.e., the steps that can be taken) to solve the problem (see “The Role of Visualization and Representation in Conceptual Understanding and Problem Solving” in this chapter). According to a review of cognitive science research (Bassok and Novick, 2012), for some problems, getting the right representation is the key to solving the problem, or at least to solving it in a straightforward manner. Although problem representation is especially important for ill-defined problems, it can also be critical for solving well-defined problems. For other problems, determining the best representation is a relatively straightforward process,

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81 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS and the primary work is to discover a (or the best) path connecting the situ- ation as presented to the goal state. Representation and step-by-step solution are interactive processes, however, and both are important in most cases of problem solving. As noted, the solver’s representation of the problem guides the process of generating a possible solution. The step-by-step solu- tion process, in turn, may change the solver’s representation of the problem, leading to corresponding changes in the solution method attempted. This iterative process of representation and step-by-step solution continues until the problem is solved or the solver abandons the goal. One difference between relative experts and novices concerns how they allocate their problem solving time between creating a representation and working to find a solution. In some disciplines, experts spend relatively more of their time on understanding the problem, that is, on analyzing the structure of the problem, developing a coherent representation of the prob- lem, and enriching that representation with relevant information retrieved from long-term memory (Simon and Simon, 1978; Voss, et al., 1983). Because experts construct better developed representations and have stored in memory effective procedures for responding in the face of familiar pat- terns (Gobet and Simon, 1996), they have been found to adopt a working- forward strategy for solving certain problems. Thus, in certain cases experts proceed from the information given, to inferences based on that informa- tion, to further inferences, and so on until the goal is reached. Novices, in contrast, often proceed backward from the goal to an equation to calculate that goal, to a second equation to calculate an unknown quantity in the first equation, etc., until an equation is found for which all the quantities needed are part of the given information of the problem. Such a difference between experts and novices has been observed repeatedly in physics, and one small-scale study involving genetics problems in biology adds to the support for this emerging consensus (Smith and Good, 1984). Students’ working backward strategy (referred to as a means-ends anal- ysis in the cognitive science literature), although often effective for solving problems, places a heavy load on working memory, leaving little capacity for learning more general information from the solution attempt, such as a general schema for solving such problems. Working memory refers to the information processing resource that allows a person to (a) hold informa- tion in mind temporarily while completing a task or solving a problem and (b) do the work of problem solving (reasoning, language comprehension, etc.). This information burden, known as the working memory load, can be taxing because the working memory system is limited in its capacity to store information and engage in cognitive work (Baddeley, 2007). Thus, it is easy to forget one or more crucial elements of a problem. As noted by Sweller (1988), when using means-ends analysis, “a prob- lem solver must simultaneously consider the current problem state, the goal

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82 DISCIPLINE-BASED EDUCATION RESEARCH state, the relation between the current problem state and the goal state, the relations between problem-solving operators and lastly, if subgoals have been used, a goal stack must be maintained” (p. 261). Solving a problem with a nonspecific goal (e.g., to calculate the value of as many variables as possible) obviates the need to keep several of the aforementioned items in working memory. Indeed, computer simulation work by Sweller (1988) in kinematics, geometry, and trigonometry has demonstrated that problems with nonspecific goals (i.e., open-ended problems) reduce working memory load. Physics Research in physics provides support for the working forward/working backward finding. This research has shown that expert problem solvers typically begin by describing problem information qualitatively and using that information to decide on a solution strategy before writing down equations (Bagno and Eylon, 1997; Chi, Glaser, and Rees, 1982; Eylon and Reif, 1984; Larkin, 1979, 1981a, 1981b; Larkin et al., 1980). A successful solver’s strategy includes the appropriate physics concept or principle and, usually, a plan for applying the principle to the particular conditions in the stated problem (Finegold and Mass, 1985; Larkin et al., 1980). This plan leads experts to work forward from the given information to the desired solution. Experts also monitor their progress while solving problems and evaluate the reasonableness of the answer (Chi, 2006; Chi et al. 1989; Larkin, 1981b; Reif and Heller, 1982; Singh, Granville, and Dika, 2002; see Chapter 6 for a more detailed discussion of metacognition). In contrast, beginning physics students typically start by writing down equations that match given or desired quantities in the problem statement and then work backward, somewhat less efficiently, to find an equation for which the unknowns are given directly in the problem (Larkin et al., 1980). When beginning students get stuck using this approach, they lack strategies to go further (Reif and Heller, 1982).2 To illustrate the working forward/working backward contrast, consider a problem for which the goal is to determine the final velocity of a block when it reaches the bottom of an inclined plane. As discussed by Larkin, (1981b), expert physicists begin by noting that the motion of the block on the inclined plane depends on gravitational and frictional forces. This approach leads them to retrieve from memory the equation F = ma (force = mass × acceleration). That equation, in turn, leads to retrieval of an equa- tion relating final velocity, the goal of the problem, to acceleration. Novices, 2 Thissection draws heavily on a review of physics education research that the committee commissioned for this study (Docktor and Mestre, 2011).

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83 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS in contrast, begin by focusing on the goal of determining the final velocity. This focus leads them to first find an equation that involves that unknown quantity, in this case the equation relating final velocity to acceleration. Acceleration is an unknown in that equation, so novices then look for a new equation that relates acceleration to information given in the problem, in this case F = ma. Chemistry Some scholars in chemistry focus not on expert-novice comparisons, but on general problem-solving approaches and on identifying the charac- teristics of successful problem solvers. According to Herron and Greenbowe (1986), successful problem solvers have a strong command of basic facts and principles; construct appropriate representations; have general reason- ing strategies that permit logical connections among the different elements of the problem, and apply verification strategies at multiple points during the problem-solving process. However, similar to findings from physics and cognitive science, research on problem solving in stoichiometry and equilibrium indicates that students are sometimes able to solve a problem using algorithmic/algebraic strategies or analogous problems, with only a superficial understanding of the underlying concept (Camacho and Good, 1989; Chandrasegaran et al., 2009; Gabel and Bunce, 1994; Tingle and Good, 1990). Similarly, a limited amount of research on how students approach organic chemistry problems that involve the use of representa- tions but no calculations suggests that many students memorize the relevant reaction and apply it to a novel task, rather than applying more general skills they have been taught to solve a novel problem (Bhattacharyya and Bodner, 2005). Some research in chemistry has explored the dichotomy between algo- rithmic problem solving and problem solving that involves only concep- tual understanding (e.g., a multiple-choice question measuring conceptual understanding of gases using no mathematics) (Nurrenbern and Pickering, 1987). A spate of papers in the 1990s probed this subject (e.g., Nakhleh and Mitchell, 1993; Sawrey, 1990), but much of that research used questions that centered on visualizations of the particulate nature of matter (PNOM) as alternatives to algorithmic problem solving. At that time, most textbooks did not contain PNOM problems; because such problems were unfamiliar to students (and many faculty), inferences cannot be readily drawn from those studies. Although PNOM problems are now more common, there is no clear evidence demonstrating whether the use of particulate representa- tions leads to improvements in conceptual problem solving or in problem solving using algorithmic calculations.

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84 DISCIPLINE-BASED EDUCATION RESEARCH Engineering Similar to the cognitive science findings presented above, a limited amount of engineering education research shows that translating the prob- lem into a visual representation and then into a mathematical representa- tion is an important step in solving problems (Eastman, 2001). Even so, students often go straight to a mathematical formula without creating a visual representation of the problem. This approach usually results in failure or misapplication of a formula leading to a dead-end rather than a deeper understanding of the phenomenon under study. One obstacle is students’ lack of understanding of concepts that serve as gatekeepers to more sophisticated conceptions of a field (Baillie, Goodhew, and Skryabina, 2006; Meyer and Land, 2005). These concepts are difficult for most stu- dents, often abstract, and not recognized by students as keys to new ways of thinking about the discipline. For engineering students, the difficulty also can manifest itself as dependence on ritualistic algorithmic problem solving rather than true understanding, sometimes resulting in the inability of the student to even recognize the problem.3 Summary Taken together, these findings suggest that it is important for science and engineering instructors to help students understand that both a good representation of the problem at hand and a good solution method are needed for successful problem solving. Moreover, when students encounter difficulty in solving problems, they need to learn to consider alternative procedures for figuring out the answer and alternative representations of the problem itself (or at least refinements to their current representation). Perhaps one component of an effective instructional strategy would be to provide a compelling example of how much difference a good representa- tion can make for the ease of solution (Posner, 1973, provides one such example). Problem Representation Another consistent finding from DBER and cognitive science is that superficial characteristics of problems have an undue influence on novices’ problem solving. One source of cognitive science evidence for this claim is that isomorphic problems (problems that have the same underlying struc- ture) may lead people to construct very different representations of their 3 This section draws heavily from a review of the literature that the committee commissioned for this study (Svinicki, 2011).

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85 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS (identical) underlying structure because of the different situations they pres- ent (e.g., discs of different sizes stacked on a peg versus acrobats of differ- ent sizes standing on one another’s shoulders), with clear consequences for solution time, accuracy, and method of solving (Bassok and Novick, 2012). These findings mean that although experts in a domain may easily see that two apparently different problems are really the same kind of problem “deep down,” students are likely to assume these problems are of distinctly different types. This assumption impairs students’ ability to apply what they learned on one problem to new problems that have similar underly- ing structures despite superficial differences. Although the initial research in this area involved brain-teaser-type puzzle problems (Hayes and Simon, 1977; Kotovksy, Hayes, and Simon, 1985), similar results have been found more recently for problems from academic domains such as mathematical word problems (Bassok and Olseth, 1995; Bassok, Chase, and Martin, 1998; Martin and Bassok, 2005). A second source of evidence comes from a large number of studies, in cognitive science and DBER, showing that (relative) experts and nov- ices in a domain (or even good and poor students) differ with respect to the problem features they highlight in their representations. In particular, novices often focus on superficial features of problems, such as the specific objects and terms mentioned and the particular way the question happens to be phrased. Experts, in contrast, typically focus on underlying structural features concerning the relations among the elements in the problem. The structural features are critical for solving the problem and the surface fea- tures are not. The following sections on physics, chemistry, and biology education research present results from problem-solving studies that are consistent with this finding. Additional supporting evidence exists based on the results of memory tasks using stimuli from engineering (electric circuit diagrams: Egan and Schwartz, 1979) and biology (clinical cases in medi- cine: Coughlin and Patel, 1987). Indeed, using a variety of experimental tasks, this finding of expertise differences in problem representations has been replicated in numerous domains, including chess (Gobet and Simon, 1996), computer programming (McKeithen et al., 1981), mathematics (Schoenfeld and Herrmann, 1982), and sports such as basketball and field hockey (Allard and Starkes, 1991). Clearly, this difference is a fundamental aspect of human cognition, relevant to science and nonscience disciplines alike. Part of acquiring expertise in a domain involves learning to identify the important structural features of that domain, and part of being an expert means seeing problems through the lens of the domain’s principles (i.e., its deep structure). Representational differences between experts and novices have implica- tions for problem solving accuracy, solution time, and the ability to transfer analogous solutions across superficially dissimilar problems (Hardiman,

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108 DISCIPLINE-BASED EDUCATION RESEARCH BOX 5-6 Visualization/Representation in Biology Cladograms, or diagrams that depict evolutionary relationships among a set of taxa, are typically depicted in either a tree or ladder format. Trees are much more common in the evolutionary biology lit- erature, while ladders are slightly more common in high school and college biology textbooks (Catley and Novick, 2008; Novick and Catley, 2007). Novick and Catley (2007) asked college students who had taken at least the first semester of the introductory biology class for majors to translate hierarchical relationships from the nested circles format to the tree and ladder formats and from the tree format to the ladder format and vice versa. The figure in this box shows an example cladogram of each type (the experiment used Latin names of real taxa rather than letters). The figure also shows that students’ translation accuracy was reduced whenever the ladder format was involved. Common errors when trans- lating between the tree and ladder formats are consistent with students’ interpretation of the long, slanted “backbone” line of ladders as a single entity (i.e., hierarchical level), consistent with the Gestalt principle of good continuation (Kellman, 2000). Figure for Box 5-6 Bitmapped--FPO

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109 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS Some geoscience education research has focused on production tasks, which are posed in the context of the real world and require students to make or modify a representation. Studies in which participants produce a geological map by direct observation of rocks and structures in nature exemplify this category (Petcovic, Libarkin, and Baker, 2009; Riggs, Balliet, and Lieder, 2009; Riggs, Lieder, and Balliet, 2009). Those studies have revealed varia- tion in the levels of sophistication of advanced geology students’ maps, and suggest that more effective traverses of the territory lead to better representations. Representational correspondence tasks provide information in one rep- resentation and require the student to transfer or translate that informa- tion to another form of representation. A pioneering example is Kali and Orion’s (1996) study in which students were shown a block diagram of a three-dimensional geological structure and asked to sketch a vertical profile down through the center of the block. Most of Titus and Horsman’s (2009) tasks also fall in this category (see Box 5-7 for a more detailed discussion of this research). A very limited amount of geoscience education research has been con- ducted on metarepresentational tasks, which require students to explain how a representation works. The work of Liben, Kastens, and Christensen (2011) in which students explain dip and strike (i.e. how to measure the orientation of a sloping planar surface) falls in this category (see “The Role of Spatial Ability in Visualization and Mental Model Formation”). No research yet exists on comprehension tasks, which are posed in the context of the representation and require students to respond by performing an action in the real world. An example of this task from the geosciences would be to present students with an interpretive model from the published literature, and ask them to find evidence in an outcrop to support or refute this model (Mogk and Goodwin, 2012). The Role of Spatial Ability in Visualization and Mental Model Formation In many science and engineering disciplines, spatial thinking is a vital component of expertise in the discipline. Some evidence suggests that spa- tial ability is correlated with learning and reasoning in specific science disciplines, although the strength and importance of the relationship may vary across disciplines (see Hegarty, 2011, for a review). Although these studies have limitations, most importantly concerning causation and causal direction, they do provide evidence for the importance of further investi- gating the role of spatial abilities in understanding science concepts across disciplines. They also raise issues about how spatial thinking should be promoted, a topic that is considered in some depth by Hegarty (2011).

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110 DISCIPLINE-BASED EDUCATION RESEARCH BOX 5-7 Visualization/Representation in the Geosciences In a study of visual penetrative ability, high school students were asked to sketch a specific plan view or profile slice through a three- dimensional volume, using information about the outside of the volume in the form of a block diagram (Kali and Orion, 1996; see the figures in this box). In general, students found this task difficult. Careful examination of incorrect responses and follow-up interviews revealed that partially successful students used specific strategies to “penetrate” the volume, such as continuing a pattern vertically or horizontally (such as depicted in panel 5.a of the figure in this box). Other students, however, gave com- pletely “nonpenetrative” answers, in which they merely copied patterns from the outside of the volume (such as in panel 5.b of the figure in this box) (Kali and Orion, 1996). Titus and Horsman (2009) extended this line of inquiry to college students, using more spatially challenging structures and tasks, such as drawing structural contours. Titus and Horsman (2009) and Murphy et al. Figure for Box 5-7 (2011) show that college students’ performance on tasks that require vi- Bitmapped--FPO sual penetrative ability can be improved by instruction and practice, and that such improvements can be accomplished within the time constraints of standard laboratory instruction.

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111 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS Physics The role of spatial ability in problem solving in mechanics has been an active area of research in physics education. These studies typically focus on the mental processes students use when imagining how the components of a mechanical system (e.g., a spring, gear, or pulley system) move and interact when the system is moving (e.g., Clement, 2009; Hegarty, 1992; Hegarty and Sims, 1994; Schwartz and Black, 1996). This task is referred to as mental ani- mation. The findings from this research suggest that students use both mental images and mathematical analysis to understand how the system is moving and answer the question asked. Other studies on relating force and motion events, interpreting graphs of force and acceleration, extrapolating motion, and inferring the motion of machine components using mental animation have found positive correlations between spatial visualization ability and mechanics problem solving (Hegarty and Sims, 1994; Isaak and Just, 1995; Kozhevnikov, Hegarty, and Mayer, 2002; Kozhevnikov and Thornton, 2006). Chemistry In contrast to physics, research on the relationship between spatial abil- ity and success in chemistry is mixed. Some researchers find spatial ability is a predictor of success on spatial organic chemistry tasks such as assign- ing configurations and understanding the mechanisms of some reactions (Bodner and McMillian, 1986; Pribyl and Bodner, 1987). Others do not find such correlations, and provide evidence that students of lower spatial ability can successfully use heuristics that do not rely on spatial ability (Abraham, Varghese, and Tang, 2010; Stieff, 2011). Engineering Despite the importance of spatial thinking to engineering, a relatively limited amount of engineering education research exists on this topic. This research reveals some differences related to gender: A 1993 study of 535 first-year Michigan Technological University engineering students showed that females were three times more likely than males to fail the Purdue Spatial Visu- alization Test on Rotations (PSVT:R) (Sorby, 2009). In addition, engineering education research suggests that spatial skills are correlated with the ability to use computer interfaces to perform database manipulations (Sorby, 2009). The Geosciences The relationship between spatial ability and understanding of geosci- ence concepts are of great interest to geoscience education researchers

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112 DISCIPLINE-BASED EDUCATION RESEARCH (Liben, Kastens, and Christensen, 2011; Orion, Ben-Chaim, and Kali, 1997). Across many fields of the geosciences, the ability to envision a three-dimensional volume from information that is available in one or two dimensions is essential. This task involves mentally extending the available information into the dimension that is unrepresented, using a combina- tion of spatial thinking ability and knowledge of plausible earth processes. Geologists engage in this activity when they envision what lies behind or between rock outcrops; geophysicists when they infer structures from images of seismic reflections; and oceanographers when they interpolate between stations where seawater temperature and salinity were measured. Kali and Orion (1996) called this ability “visual penetrative ability,” and developed a test for it in the context of working with geological structures. As described in Box 5-7, this research provides evidence of students’ dif- ficulties with tasks involving visual penetrative ability, reveals variation in high-school students’ visual penetrative ability, and shows that the visual penetrative ability of college students can be improved with instruction. Across the disciplines, very little DBER on spatial ability and represen- tations attends to individual differences (e.g., ability levels) or group differ- ences (e.g., race/ethnicity). A notable exception is a study of dip and strike in the geosciences (Liben, Kastens, and Christensen, 2011). In that study, the researchers purposefully selected participants so as to populate six bins of 20 students each: high-, medium, and low-spatial ability, by male and female. Low-spatial ability (as assessed on the water-level task) and being female were associated with worse performance on the strike and dip tasks (Liben, Kastens, and Christensen, 2011). Astronomy Considering that astronomy requires learners to imagine a three- dimensional dynamic universe of galaxies and orbiting planets by looking up at a flat sky, it would be reasonable to assume that spatial thinking is an active area of inquiry in astronomy education research. However, sys- tematic research on spatial thinking in astronomy is very limited. Studies are under way to examine the relationships between spatial thinking and astronomy knowledge, but their results have not yet been published in peer-reviewed journals. Instructional Strategies and Tools to Improve Students’ Spatial Ability and Use of Representations Given the difficulties students have with spatial thinking and the use of representations, improving these skills is an important part of moving

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113 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS students along the path toward greater expertise, and, in turn, an important focus of DBER. Related research in cognitive science has yielded insights into the reasons for students’ difficulties, which also have implications for instruction. Here we discuss these bodies of research as they relate to improving the use of diagrammatic displays, enhancing students’ spatial ability, and identifying the role of animations in these tasks. Improving the Comprehension of Diagrammatic Displays In addition to being important tools of discipline, diagrammatic d isplays—e.g., general-purpose abstract diagrams such as hierarchies and matrices, domain-specific diagrams such as free body diagrams in physics or a drawing of the components of a cell in biology, and graphs of data— are important tools in instruction. These displays can enhance reasoning and problem solving (Larkin and Simon, 1987; Lynch, 1990; Winn, 1989), whether in the sciences or in other disciplines. By storing information externally, diagrams free up working memory that can be used for other cognitive processes. Moreover, the spatial organization of information helps viewers to integrate related information (Hegarty, 2011). These displays can also enable viewers to offload more complex cognitive processes onto simpler perceptual processes. For example, a linear relationship between variables is immediately apparent in a graph but must be laboriously com- puted from a table of numbers. Similarly, when a display is interactive, people can manipulate the display itself instead of conducting their own computations of the values. These benefits of graphical displays do not mean that their use is easy or transparent, however, as the previous sections clearly illustrate. One rea- son diagrams do not automatically facilitate reasoning and problem solving is that their successful use requires knowledge of the conventions underly- ing their construction (Hegarty et al., 1991). College science is taught by experts who are very familiar with the representations used in their field and may not realize how difficult these representations are for students to master. Instructors may need to provide students with more detailed intro- ductions to the various graphical displays used in science. Such instruction may include the following (Hegarty, 2011): • explicitly highlighting the relationships between alternative dis- plays of the same or related information, • explaining how different representations at different levels of abstraction are optimized for various tasks, and • providing opportunities for extensive practice interpreting and pro- ducing appropriate types of representations.

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114 DISCIPLINE-BASED EDUCATION RESEARCH Instruction and practice in making representations might include making “first inscriptions,” or representations that are made directly from the rep- resented world or from the raw material of nature (Goodwin, 1994, Mogk and Goodwin, 2012). A second reason diagrams are not always beneficial is because the relationship between perception and cognition affects how students process diagrams to draw inferences about the represented world. For example, circles and lines seem naturally suited to imply physical objects or locations rather than relationships or motion, respectively (Tversky et al., 2000). When diagrammatic representations are constructed in accordance with these conventions, college students are faster and more accurate at drawing appropriate inferences (Hurley and Novick, 2010). In addition, students often import their highly practiced left-to-right processing strategy from reading printed text to nonlinguistic visual tasks (Fuhrman and Boroditsky, 2010; Nachshon, 1985). In situations where left-to-right processing makes it difficult to interpret relationships in a scientific diagram, simply reflecting the diagram 180 degrees above the vertical axis can improve comprehen- sion (Novick et al., in press). These results suggest that when instructors construct diagrammatic representations to illustrate scientific concepts or principles, they should attend to how people naturally interpret the com- ponents from which the representations are constructed. The Gestalt psychologists identified many other perceptual features that have psychological importance, including two that have particular implica- tions for DBER. The first is good continuation, which states that a continu- ous line is interpreted as a single entity (discussed in the biology education research section on the role of visualization and spatial ability). The second is spatial proximity, which states that items that are closer together tend to be processed together. This principle has been shown to play an important role in students’ understanding of cladograms (Novick and Catley, in press) and mathematical equations (Landy and Goldstone, 2007). Whether spatial proximity facilitates or hinders students’ performance depends on whether the conclusion based on proximity supports or conflicts with the correct response. In some situations, visual representations may be constructed in such a way that spatial proximity is a useful clue rather than a conflict- ing signal, as when a flow chart or network diagram is constructed with functionally related subsystem elements clustered together within a larger depicted system. In other cases, however, instructors probably need to tackle the issue directly, such as through the use of refutational instructional strategies (Hynd et al., 1994; Kowalski and Taylor, 2009), explaining why spatial proximity is not relevant and what structural features of the diagram instead provide the critical information needed for responding (Novick, Catley, and Schreiber, 2010).

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115 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS Improving Spatial Ability Research from engineering and the geosciences suggests that spatial ability can be improved through instruction. One review article (Sorby, 2009) summarizes more than a decade of applied research—including several longitudinal studies—aimed at identifying practical methods for improving three-dimensional spatial skills, especially for female engineer- ing students. That article identifies strategies that appear to be effective in developing spatial skills. In the longitudinal studies, first-year engineer- ing majors of low spatial ability (as measured by the PSVT:R) who took a specially designed, multimedia training course improved their spatial skills, earned higher grades, and persisted in the university at greater rates than students of similar spatial ability who did not take the course. The multimedia software and workbook that were part of the course materials were shown to improve students’ spatial skills. In addition, sketching was consistently identified as an important component of spatial skill develop- ment (Sorby, 2009). Some research indicates that geoscience education involving visually rich materials can improve students’ spatial visualization skills on domain- specific tasks and general spatial tasks in ways that are practical to incor- porate into instructional practices (Ozdemir et al., 2004; Piburn et al., 2005; Sawada et al., 2002; Titus and Horsman, 2009) (see Box 5-7 for an example). In some of those studies, including a quasi-experimental study (Piburn et al., 2005), an initial gap between males and females in spatial visualization ability (as measured by the surface development task) dimin- ished after visualization-rich instruction in physical geology (Piburn et al., 2005; Sawada et al., 2002). The Influence of Animated and Static Visualizations on Conceptual Understanding In addition to using specialized representations, expert scientists and engineers use technologies such as animations, interactive computer visu- alizations and virtual models to aid their work. These technologies have enormous potential for promoting representational competence and spatial thinking in science (Hegarty, 2011). Indeed, several quasi-experimental studies comparing the efficacy of static visuals with two- and three- dimensional animations of molecular structures and processes appear to improve student learning of stereochemistry, which concerns the spatial arrangement of atoms within molecules (Abraham, Varghese, and Tang, 2010; Aldahmash and Abraham, 2009; Sanger and Badger, 2001). These studies suggest that three-dimensional animations improve learning more than two-dimensional animations, which, in turn, are better than static

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116 DISCIPLINE-BASED EDUCATION RESEARCH representations. The use of animations in biology classes also has been shown to increase retention of content knowledge in the short and long- term (Harris et al., 2009; McLean et al., 2005; O’Day, 2007). The anima- tions in those studies included a three-dimensional animation of protein synthesis (McLean et al., 2005), a narrated animation related to a complex signal transduction pathway (O’Day, 2007), and a combination of a molec- ular imaging program and handheld physical models related to molecular structure and function (Harris et al., 2009). Although this research shows that students can learn more from animations than from static images, McLean et al. (2005) caution that “learning is best achieved when an animation is coupled with a lecture, because this combination provides a reference from which students can appreciate the knowledge presented in the animation” (p. 170). The recommendation of McLean et al. (2005) might reflect the fact that visual technologies are typically more complex than static visualiza- tions. Thus, these technologies may require greater spatial ability and other aspects of representational competence for their successful use. It is not a foregone conclusion, therefore, that all animations will necessarily be beneficial for learning. In this regard, some cognitive science research com- paring static and animated displays of dynamic biological and mechanical processes has found no benefit in performance when animations were used (Tversky, Morrison, and Betrancourt, 2002). Sometimes, animations present information too quickly for students to accurately perceive and comprehend. Although the pace of animations can easily be slowed, other problems are not so readily fixed. For example, animations of a mechanical system can give students the illusion that they understand the information being presented and may even cause them to “see” what they believe is true rather than what is actually presented (Kriz and Hegarty, 2007). Some research has found that animations showing how a mechanical system works were more effective in promoting learn- ing when students had to predict how they thought the machine works before viewing the animation (Hegarty, Kriz, and Cate, 2003). The authors hypothesize that the prediction task improves the effectiveness of the ani- mation by making students aware of what they do not understand, thus cueing them to the type of information to extract from the animation. This finding is consistent with those discussed in Chapter 6 about the benefits of interactive lecture demonstrations. One conclusion to draw is that although animated, three-dimensional, and interactive visualizations might seem to rely less on visualization skills, research to date has suggested that they actually require visualization skills for their use (Hegarty, 2011).

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117 PROBLEM SOLVING, SPATIAL THINKING, AND REPRESENTATIONS Summary of Key Findings on Spatial Thinking and the Use of Representations • How students create, use, interpret, and translate between graphi- cal and mathematical representations provides insight into their understanding of important concepts in a discipline. DBER high- lights discipline-specific challenges that students face when using such representations. • Although equations, graphical displays, and other representations may seem easy to understand for undergraduate faculty who are domain experts, college students have difficulty extracting infor- mation from these representations, and constructing appropriate representations from existing information. College students also have difficulty relating and translating among different representa- tions of the same entity or phenomenon. • There is contradictory evidence about the relationship between spatial ability and performance in science. Consistent with findings from cognitive science that students with low-spatial ability espe- cially have difficulty relating two- and three-dimensional represen- tations, some DBER studies show a relationship between measures of spatial ability and success on specific science or engineering tasks. Other studies do not provide evidence of that relationship. • The evidence on the effectiveness of animations is mixed: The use of animations has been shown to enhance learning in some circumstances, and to be ineffective or even detrimental to student learning in other situations. Directions for Future Research on Spatial Thinking and the Use of Representations DBER and cognitive science have yielded many useful insights into how students use mathematical and graphical representations, but important gaps remain. For example, the research community, instructors, and those who develop representations would benefit from a deeper understanding of students’ use of representations as tools to enhance their learning, and studies along these lines should leverage what is already known about the basic cognitive and perceptual processes that students use to comprehend graphical representations. The role of spatial ability also needs clarification. Spatial ability may be measured in many different ways, any one of which may be more or less rel- evant to any specific science or engineering task. Although several authors have proposed that many tasks (e.g., rotation tasks of three-dimensional models) require mental imagistic models, others have shown that many

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118 DISCIPLINE-BASED EDUCATION RESEARCH students use heuristics and other strategies that do not employ visualization skills and are able to move flexibly between such strategies as needed. In addition to clarifying the overall role of spatial ability, it would be useful to evaluate the contributions of large-scale and small-scale spatial ability to learning in physics, chemistry, engineering, biology, the geosciences, and astronomy. DBER has not yet examined these different spatial abilities. The research base on promoting students’ understanding of and facility with domain-specific representations is less robust. DBER does not provide conclusive evidence about how instructors, illustrators, and authors should design representations for maximum effect, or what the optimal represen- tations are for a given situation. Moreover, additional research is needed to identify the range of instructional approaches that help students use mathematical and graphical representations to enhance their knowledge and understanding. For example, does designing and constructing represen- tations affect students’ understanding differently than merely interpreting existing representations, and if so, how? Given the increasing use of tech- nology, more research is needed on the educational efficacy of computer animations, simulations, and other technology-enhanced techniques that aid with visualization and representations, and the conditions under which those techniques are effective. Representations vary within and across disciplines. As one example, the nature of the representations used in geoscience education varies enor- mously on multiple important dimensions, including the use of spatial representations to represent nonspatial data (Dutrow, 2007; Kastens, 2009, 2010; Libarkin and Brick, 2002). This variation presents a challenge to developing a research agenda for the use of visualizations and repre- sentations in undergraduate science and engineering education, because research using any specific representation may not be generalizable to other representations.