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We investigate the interplay between mathematics and physics resources in intermediate mechanics students. In the mechanics course, the selection and application of coordinate systems is a consistent thread. At the University of Maine, students often start the course with a strong preference to use Cartesian coordinates, in accordance with their pr...
Contexts in source publication
Context 1
... both interviews, students are presented with the same problem: given a drawing of a simple pendulum Fig. 2, with polar coordinates shown in a way students did not see, find the position of the pendulum bob as a function of time. So that students do not spend time figuring out the forces on the bob, and to predispose the students into thinking of force- based solutions, the forces on the bob a weight force and a tension force are given ...
Context 2
... solve for the position of the bob as a function of time, a physicist might first write Newton's second law for the system, a vector second-order differential equation. The physicist would then choose a polar coordinate system as shown in Fig. 2. This coordinate system takes advantage of the natural geometry and symmetry of the situation, and it is a calculationally easy choice. With the coordinate system in place, the vector equation of motion can be split into two scalar differential equations and then solved. As the focus of the interviews was on the coordinate system ...
Context 3
... week 10, the task in Fig. 2 is posed to the students again. In the intervening weeks, students have studied damped and driven harmonic motion in class and have been assigned a homework problem on the equation of motion for the pendulum derived using both Lagrangian and Newton- ian methods. Their responses typify their approaches to the class: Wes says that he ...
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Citations
... Understanding readout strategies in coordination classes is crucial for educators and researchers as it sheds light on how students are organizing and utilizing their knowledge. Previous research has shown that knowledge pieces can be used by learners towards such coordination classes (Levrini & diSessa, 2008;Sayre & Wittmann, 2008;Wittmann, 2002), but there are no investigations in the context of the greenhouse effect. ...
The greenhouse effect is a complex scientific phenomenon that plays a crucial role in understanding climate change. Grasping students' understanding of this phenomenon on the content-specific level but also how students' conceptions are organized is vital for effective climate change education. This study addresses both levels and delves into the relationship between students' frameworks and knowledge pieces of the greenhouse effect through the analysis of multiple-choice questions, employing Bayesian correlations and multiple logistic regression. We thereby focus on specific types of conceptualizations of the greenhouse effect that have been identified in previous research and furthermore investigate the coherency of them. To do so, we analyzed answers of N = 604 grade 11 students in Austria and Germany and interpreted them from different theoretical perspectives. The findings showed that students hold various ideas about the greenhouse effect that are only seldom coherent, in particular when it comes to adequate ideas about the greenhouse effect. However, especially for a reflection-based framework of the greenhouse effect, our results demonstrate that students' conceptions show some form of coherency. We argue that our results can inform the development of effective teaching strategies that address students' existing knowledge and alternative conceptions. In terms of practical implications, the findings suggest that teaching strategies should provide opportunities for students to integrate their knowledge pieces into a more coherent understanding of the greenhouse effect. The study highlights the need for further investigation into the relationship between knowledge pieces and frameworks not only for the greenhouse effect, but for science education in general.
... Resource theory seeks to characterize the resources that feed into the construction of new knowledge, namely, their structure, organization, and how they change over time. Resources have been defined as "small reusable pieces of thought that make up concepts and arguments" [6]. Rooted in a computational metaphor for learning, resources can be thought of as chunks of computer code that are incorporated into more complex procedures [3]. ...
... Related work used a framing and transfer lens to examine the activation of resources across contexts [18]. Many PER scholars have used resource theory to characterize knowledge systems across a variety of physics topics, including coordinate systems [6], kinematics [19], force [20], momentum [21], mechanical waves [22], fluid mechanics [23], thermodynamics [24,25], optics [26], electromagnetism [27], inertial forces [28,29], energy [30], and quantum mechanics [31]. Other studies have emphasized the dynamics of resources, including the process of activation [32] and the connections and development of resources [6]. ...
... Many PER scholars have used resource theory to characterize knowledge systems across a variety of physics topics, including coordinate systems [6], kinematics [19], force [20], momentum [21], mechanical waves [22], fluid mechanics [23], thermodynamics [24,25], optics [26], electromagnetism [27], inertial forces [28,29], energy [30], and quantum mechanics [31]. Other studies have emphasized the dynamics of resources, including the process of activation [32] and the connections and development of resources [6]. Some research has explored how physics students conceptualize equations [33] and utilize mathematical skills such as the separation of variables [34]. ...
[This paper is part of the Focused Collection on Qualitative Methods in PER: A Critical Examination.] Within physics education research (PER), resource theory has proven to be a useful framework for investigating knowledge and learning and informing instructional design. To analyze learning over longer timescales and across cases, PER scholars must first identify and describe the resources activated within and across physics contexts and domains. Despite its importance, a reliable method for identifying resources has not been clearly outlined. This paper presents guidelines for the design of research aimed at identifying knowledge resources. We begin by describing the origin, assumptions, and utility of resource theory. We then introduce methods of data collection and analysis. We end with a discussion of validity and reliability, drawing connections with general principles of qualitative research. With this work, we hope to promote coordination among the many PER scholars who utilize resource theory and to invite new scholars to join in its application and development.
... We want to emphasize that we are not claiming that a certain distribution of activity types (e.g., an equal percentage of create and interpret) is ideal. Studies have shown that students have difficulties in creating or interpreting coordinate systems (e.g., Earnest, 2015;Herscovics, 1989;Mevarech & Kramarsky, 1997; Sarama et al., 2003), and selecting appropriate coordinate systems for a given problem (Sayre & Wittmann, 2008). A lack of opportunities to independently create coordinate systems may contribute to students' difficulties with establishing coordinate systems. ...
... Third, and more generally, the pieces perspective should be applied to additional investigations of student thinking in the life sciences. Only a handful of pieces' studies exist in biology education research (Nehm and Ha, 2011;Gouvea and Simon, 2018;Lira and Gardner, 2020;Slominski et al., 2020), even though the perspective has been productively used in physics (e.g., Hammer, 2000;Sayre and Wittmann, 2008;Scherr and Hammer, 2009;Weliweriya et al., 2019) and chemistry (e.g., Heisterkamp and Talanquer, 2015;Becker and Towns, 2012;Becker et al., 2017;Rodriguez et al., 2018Rodriguez et al., , 2020Rodriguez and Towns, 2019). The pieces perspective facilitated our discovery of many knowledge elements students used when solving metabolism problems, and the biology and biochemistry education communities are now better equipped to create instructional materials that support student learning. ...
Research on student thinking facilitates the design of instructional materials that build on student ideas. The pieces framework views student knowledge as consisting of independent pieces that students assemble in fluctuating ways based on the context at hand. This perspective affords important insights about the reasons students think the way they do. We used the pieces framework to investigate student thinking about the concept transformations of energy and matter with a specific focus on metabolism. We conducted think-aloud interviews with undergraduate introductory biology and biochemistry students as they solved a metabolism problem set. Through knowledge analysis, we identified two categories of knowledge elements cued during metabolism problem solving: 1) those about the visual representation of negative feedback inhibition; and 2) those pertaining to student focus on different metabolic compounds in a pathway. Through resource graph analysis, we found that participants tend to use knowledge elements independently and in a fluctuating way. Participants generally showed low representational competence. We recommend further research using the pieces perspective, including research on improving representational competence. We suggest that metabolism instructors teach metabolism as a concept, not a collection of example pathways, and explicitly instruct students about the meaning of visual representations associated with metabolism.
... Many theories have been developed pertaining to the structure and function of knowledge (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). One theory, Resource Theory, considers knowledge to be made up of smaller, broadly-applicable units known as resources (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). ...
... Many theories have been developed pertaining to the structure and function of knowledge (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). One theory, Resource Theory, considers knowledge to be made up of smaller, broadly-applicable units known as resources (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). Resources are like tools in that they are basic and versatile enough to be applied to a variety of problems and scenarios (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). ...
... One theory, Resource Theory, considers knowledge to be made up of smaller, broadly-applicable units known as resources (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). Resources are like tools in that they are basic and versatile enough to be applied to a variety of problems and scenarios (Hammer, 2000;Sayre & Wittmann, 2008;Wittmann, 2006). They often take the form of abstract generalizations; for example, the idea that phenomena (such as the ringing of a bell or a bouncing ball) tend to die away over time is an example of a resource (diSessa, 1993). ...
This study examines a conversation between a group of three in-service teachers during a professional development workshop. During the workshop, the teachers collaborated to answer a set of survey questions pertaining to energy concepts in physical science. The teachers began the survey expressing low self-efficacy in their knowledge of energy. Through this study, we were able to observe teachers as they learned new concepts. This study used a narrative analysis of their conversation. The analysis centers on changes in the teachers’ common content knowledge of conservation of energy and their expressed self-efficacy regarding that knowledge. Their knowledge was evaluated from a resources perspective, as many components of conservation of energy appeared in their conversation. We found that the teachers, in the course of one hour, developed an understanding of conservation of energy, and that they expressed greater self-efficacy regarding this knowledge. Further study may expand our understanding of this process.
... resource is influenced by the lived experience of the learner, the social setting in which the question is asked, and the question at hand [2,6,17]. Thus, resources theory does not view student thinking as stable, coherent, and consistently applied; instead it assumes that specifics of the context in which a question is asked will influence the resources activated in a student's mind [1,2,6,16,19]. At the same time, resources theory leaves open the possibility of predictable patterns of resource use, because patterns in the context may lead to patterns in the resources activated. ...
Resources theory assumes that resource activation is context sensitive, and that an important dimension of context is the question students are answering. The context sensitivity of resource activation has been demonstrated empirically by case studies that show students using different resources to answer questions that are similar in focus. In this paper, we further substantiate and add specificity to the field’s understanding of the context sensitivity of resource use by demonstrating a pattern in resource activation for questions about mechanical pulse propagation, superposition, and reflection. In particular, our analysis shows a pattern in the kinds of resources students use to answer different styles of questions about the same physics topic. Questions that ask for a prediction tend to elicit rules or procedures, while questions that ask students to explain an observation elicit these plus ideas about force, energy, and motion. Our results call both researchers’ and instructors’ attention to the style of question that they ask and the impact it has on the resources commonly cued for students.
... Therefore, researchers in physics education expanded the scope of their studies to explore how students use mathematics tools to learn the physics content and to solve the physics problems. There are some studies in specific physics content areas including classical mechanics [39], electrodynamics [40][41][42], thermodynamics [43], and statistical mechanics [44][45][46]. These studies have identified how students' difficulties with a general mathematics tool interact with their understanding of specific physics content. ...
Upper-division physics students solve partial differential equations in various contexts in quantum mechanics courses. Separation of variables is a standard technique to solve these equations. We investigated students’ solutions to midterm exam questions and utilized think-aloud interviews. We also applied a framework that organizes students’ problem-solving process into four stages: activate, construct, execute, and reflect. Here we focused on students’ problem-solving process for two typical problems in the context of quantum mechanics: an energy eigenfunction problem in two spatial dimensions and a time evolution problem in one spatial dimension. We found that the students encountered various difficulties when they used the separation of variables technique to solve these partial differential equations. Common difficulties included recognizing when separation of variables is the appropriate method, deriving the correct separated equations from the original equation, deciding the signs of the separation constants, justifying when using the summation form of the wave function, and using an effective reflecting tool for their final solutions. In addition, we observed qualitatively and quantitatively different errors in students’ solutions to the two problems. Finally, we discussed the possible implications of our findings for instruction.
... These connections may all have differing strengths and correlations. DiSessa (1993) refers to these connections as structured priorities, and Wittmann (2006) and Sayre and Wittmann (2008) build on this work with their explanations of resource graphs and cuing priorities. The map of the connections between resources can often be visualized as ball and stick style graphs (Sayre and Wittmann 2008;Wittmann 2006), and the relative arrangements of and connections between the individual elements are described by the cuing priorities (DiSessa 1993;Sayre and Wittmann 2008). ...
... DiSessa (1993) refers to these connections as structured priorities, and Wittmann (2006) and Sayre and Wittmann (2008) build on this work with their explanations of resource graphs and cuing priorities. The map of the connections between resources can often be visualized as ball and stick style graphs (Sayre and Wittmann 2008;Wittmann 2006), and the relative arrangements of and connections between the individual elements are described by the cuing priorities (DiSessa 1993;Sayre and Wittmann 2008). Whenever a student activates a given resource, the likelihood that he/she will activate a connected resource is given by the cuing priority: high cuing priority indicates a strong correlation between the two elements, while a negative cuing priority can indicate suppression of a resource in a given situation. ...
... DiSessa (1993) refers to these connections as structured priorities, and Wittmann (2006) and Sayre and Wittmann (2008) build on this work with their explanations of resource graphs and cuing priorities. The map of the connections between resources can often be visualized as ball and stick style graphs (Sayre and Wittmann 2008;Wittmann 2006), and the relative arrangements of and connections between the individual elements are described by the cuing priorities (DiSessa 1993;Sayre and Wittmann 2008). Whenever a student activates a given resource, the likelihood that he/she will activate a connected resource is given by the cuing priority: high cuing priority indicates a strong correlation between the two elements, while a negative cuing priority can indicate suppression of a resource in a given situation. ...
We use the framework of cognitive resources to investigate how students construct understanding of a complex physics topic, namely, a photovoltaic cell. By observing students as they learn about how a solar cell functions, we identified over 60 distinct resources that learners may activate while thinking about photovoltaic cells. We classify these resources into three main types: phenomenological primitives, conceptual resources, and epistemological resources. Furthermore, we found a pattern that suggests that when students make conceptual breakthroughs they may be more likely to activate combinations of resources of different types in concert, especially if a resource from each of the three categories is used. This pattern suggests that physics instructors should encourage students to activate multiple types of prior knowledge during the learning process. This can result from instructors deliberately and explicitly connecting new knowledge to students’ prior experience both in and outside the formal physics classroom, as well as allowing students to reflect metacognitively on how the new knowledge fits into their existing understanding of the natural world.
... For example, a search of Institute of Physics (IOP) journals yields over a hundred thousand papers with the phrase "coordinate systems" in the title. Yet, in undergraduate physics, coordinate systems are known to present a number of learning challenges for students-see for example Sayre & Wittmann (2008), and Vega, Christensen, Farlow, Passante, & Loverude (2017). ...
In this paper, we show that introductory physics students may initially conceptualise Cartesian coordinate systems as being fixed in a standard orientation. Giving consideration to the role that experiences of variation play in learning, we also present an example of how this learning challenge can be effectively addressed. Using a fine-grained analytical description, we show how students can quickly come to appreciate coordinate system movability. This was done by engaging students in a conceptual learning task that involved them working with a movable magnetometer with a printed-on set of coordinate axes to determine the direction of a constant field (Earth's magnetic field).
... Regarding the use of mathematics in school physics contexts, a number of studies have shown that students are rather proficient at mathematical operations; however, they have difficulties developing mathematical models based on physical situations and dealing with a variety of mathematical representations, such as vectors [23][24][25], the coordinate system [26], and graphs [27]. ...
In order to support physics students in their future careers, there is a need to understand the relationship between undergraduate education and professional practice in physics-related fields. This study investigated high-level goal driven mathematical problem-solving activities that are found within two disciplinary cultures: physical science research labs in academia and photonics workplaces in industry. We conducted semistructured interviews with 10 Ph.D. students and 22 engineers and technicians. Math use in professional workplaces was characterized through an adaptation of epistemic games framework, which revealed six common epistemic games in these workplaces: conceptual math modeling, analytical-numerical math modeling, design-oriented math modeling, fabrication, improving processes, and making meaning out of data games. The workplace-specific epistemic games capture the goals, starting and ending conditions, constraints and contextual features, moves, tools, and representations. The games involve a broad spectrum of math that ranges from arithmetic to computational modeling. The games reveal how goals and particular contextual features impact approaches to mathematical problem solving. The findings extend prior work on mathematical problem solving in physics to a new population of professional researchers, engineers, and technicians in their workplaces. The research may guide new approaches for developing problems and explicitly teaching problem solving in diverse physics contexts, which may additionally benefit undergraduate students’ preparation for their future careers.
