Current conceptions of expert problem solving depict physical/conceptual reasoning and formal mathematical reasoning as separate steps: a good problem solver first translates a physical Current conceptions of quantitative problem-solving expertise in physics incorporate conceptual reasoning in two ways: for selecting relevant equations (before manipulating them), and for checking whether a given quantitative solution is reasonable (after manipulating the equations). We make the case that problem-solving expertise should include opportunistically blending conceptual and formal mathematical reasoning even while manipulating equations. We present analysis of interviews with two students, Alex and Pat. Interviewed students were asked to explain a particular equation and solve a problem using that equation. Alex used and described the equation as a computational tool. By contrast, Pat found a shortcut to solve the problem. His shortcut blended mathematical operations with conceptual reasoning about physical processes, reflecting a view - expressed earlier in his explanation of the equation - that equations can express an overarching conceptual meaning. Using case studies of Alex and Pat, we argue that this opportunistic blending of conceptual and formal mathematical reasoning (i) is a part of problem-solving expertise, (ii) can be described in terms of cognitive elements called symbolic forms (Sherin, 2001), and (iii) is a feasible instructional target.
Mathematical reasoning flexibility across physics contexts is a desirable learning outcome of introductory physics, where the math world and physical world meet. Physics Quantitative Literacy (PQL) is a set of interconnected skills and habits of mind that support quantitative reasoning about the physical world. The Physics Inventory of Quantitative Literacy (PIQL), which we are currently refining and validating, assesses students proportional reasoning, covariational reasoning, and reasoning with signed quantities in physics contexts. In this paper, we apply a Conceptual Blending Theory analys…
Mathematics is the language of science. Fluent and productive use of mathematics requires one to understand the meaning embodied in mathematical symbols, operators, syntax, etc., which can be a difficult task. For instance, in algebraic symbolization, the negative and positive signs carry multiple meanings depending on contexts. In the context of electromagnetism, we use conceptual blending theory to demonstrate that different physical meanings, such as directionality and location, could associate to the positive and negative signs. With these blends, we analyze the struggles of upper-division…
Expressing physics problems in the form of a mathematical model is one of the most important stages in the problem-solving process. Particularly in algebraic symbolization, understanding the meanings of signs and being able to manipulate them becomes a challenging task for students, especially when more than two elements in the mathematical expression could carry a negative sign. We use Conceptual Blending theory to investigate how students attribute emergent meaning to the signs and how they articulate different signs in their algebraic symbolization. The data for this research is drawn from …
Equations are about more than computing physical quantities or constructing formal models; they are also about understanding. The conceptual systems physicists use to think about nature are made from many different resources, formal and not, working together and inextricably linked. By blending mathematical forms and physical intuition, physicists breathe meaning into the equations they use. In contrast, in physics class, novice students often treat mathematics as only a calculational tool, isolating it from their rich knowledge of the physical world. We are interested in cases where students …
A Framework for Understanding the Impact of Integrating Conceptual and Quantitative Reasoning in a Quantum Optics Tutorial on Students' Conceptual Understanding
University of Cincinnati（辛辛那提大学）；Community College of Allegheny County（阿勒格尼县社区学院）；University of Pittsburgh（匹兹堡大学）
There is some evidence that conceptual inquiry-based introductory physics lab curricula, such as RealTime Physics, may improve students' understanding of physics concepts. Thus, these curricula may be attractive for instructors who seek to transform their physics labs to improve student learning. However, the impact of conceptual inquiry-based lab instruction on students' attitudes and beliefs about experimental physics, as measured by the E-CLASS survey, is not yet fully understood. We present data from three curricular approaches over four semesters ($n=701$). We saw no change in E-CLASS sco…
Conceptual inventory surveys are routinely used in education research to identify student learning needs and assess instructional practices. Students might not fully engage with these instruments because of the low stakes attached to them. This paper explores tests that can be used to estimate the percentage of students in a population who might not have taken such surveys seriously. These three seriousness tests are the pattern recognition test, the easy questions test, and the uncommon answers test. These three tests are applied to sets of students who were assessed either by the Force Conce…
Student learning in upper division thermal physics has not been studied to the same extent as in other courses like electromagnetism and quantum mechanics. Studies addressing reasoning and learning at the graduate level are even more limited. In this study, we conducted think-aloud interviews with eight graduate students involving questions centered around a set of entropy related conceptual tasks, two of which are similar to tasks presented to undergraduates in other studies. We discuss patterns in student reasoning on each question then discuss themes that appeared across questions. We ident…
We discuss the development and validation of a conceptual multiple-choice survey instrument called the Survey of Thermodynamic Processes and First and Second Laws (STPFaSL) suitable for introductory physics courses. The survey instrument uses common student difficulties with these concepts as resources in that the incorrect answers to the multiple-choice questions were guided by them. After the development and validation of the survey instrument, the final version was administered at six different institutions. It was administered to introductory physics students in various traditionally taugh…
Calls to transform introductory college physics courses to include scientific practices require assessments that can measure the extent to which these transformations are effective. Such assessments should be able to measure students' abilities to blend conceptually important concepts with practices scientists engage in. To design assessment tasks that can measure these abilities, we leveraged Evidence-Centered Design and the Three-Dimensional Learning Assessment Protocol with the focal scientific practice of "Using Mathematics." We conducted video recorded one-on-one think-aloud interviews to…
We discuss research involving the Conceptual Survey of Electricity and Magnetism (CSEM) to evaluate one aspect of the pedagogical content knowledge of teaching assistants (TAs): the knowledge of introductory students' alternate conceptions in electricity and magnetism as revealed by the CSEM. For each item on the CSEM, the TAs were asked to (1) identify the most common incorrect answer choice of introductory physics students and (2) predict the percentage of introductory students who would answer the question correctly in a post-test. Then, we used the CSEM post-test data from approximately 40…
Assessing Student Expertise in Introductory Physics with Isomorphic Problems, Part II: Effect of Some Potential Factors on Problem Solving and Transfer
Measuring student understanding in quantum computing: Development and validation of the Quantum Computing Conceptual Survey
California State University San Marcos（圣马科斯加州州立大学）；George Mason University（乔治梅森大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；California State University Fullerton（富尔顿加州州立大学）
Ishimoto, Davenport, and Wittmann have previously reported analyses of data from student responses to the Force and Motion Conceptual Evaluation (FMCE), in which they used item response curves (IRCs) to make claims about American and Japanese students' relative likelihood to choose certain incorrect responses to some questions. We have used an independent data set of over 6,500 American students' responses to the FMCE to generate IRCs to test their claims. Converting the IRCs to vectors, we used dot product analysis to compare each response item quantitatively. For most questions, our analyses…
Exploring Generative AI assisted feedback writing for students' written responses to a physics conceptual question with prompt engineering and few-shot learning
The cyclic format of the undergraduate physics curriculum depends on students' ability to recall and utilize material covered in prior courses in order to reliably build on that knowledge in later courses. However, there is evidence to suggest that people often do not retain all, or even most, of what they learned previously. How much information is retained appears to be dependent both on the individuals' approach to learning as well as the style of instruction. In particular, there is evidence to suggest that active engagement techniques in the classroom can improve students' retention of th…
Learning to use math in physics involves combining (blending) our everyday experiences and the conceptual ideas of physics with symbolic mathematical representations. Graphs are one of the best ways to learn to build the blend. They are a mathematical representation that builds on visual recognition to create a bridge between words and equations. But students in introductory physics classes often see a graph as an endpoint, a task the teacher asks them to complete, rather than as a tool to help them make sense of a physical system. And most of the graph problems in traditional introductory phy…
Instruction is unlikely to be effective if instructors do not know the common alternate conceptions of introductory physics students and explicitly take into account common student difficulties in their instructional design. Here, we discuss research involving the Conceptual Survey of Electricity and Magnetism (CSEM) to evaluate one aspect of the pedagogical content knowledge of teaching assistants (TAs): knowledge of introductory students' alternate conceptions in electricity and magnetism as revealed by the CSEM. For each item on the CSEM, the TAs were asked to identify the most common incor…
Historically, the implementation of research-based assessments (RBAs) has been a driver of education change within physics and helped motivate adoption of interactive engagement pedagogies. Until recently, RBAs were given to students exclusively on paper and in-class; however, this approach has important drawbacks including decentralized data collection and the need to sacrifice class time. Recently, some RBAs have been moved to online platforms to address these limitations. Yet, online RBAs present new concerns such as student participation rates, test security, and students' use of outside r…
Newton's third law is one of the most important concepts learned early in introductory mechanics courses; however, ample studies have documented a wide range of students' misconceptions and fragmented understandings of this concept that are difficult to change through traditional instruction. This research develops a conceptual framework model to investigate students' understanding of Newton's third law through the knowledge integration perspective. The conceptual framework is established with a central idea emphasizing forces as the outcomes of an interaction instead of using the common actio…
Historically, the implementation of research-based assessments (RBAs) has been a driver of educational change within physics and helped motivate adoption of interactive engagement pedagogies. Until recently, RBAs were given to students exclusively on paper and in-class; however, this approach has important drawbacks including decentralized data collection and the need to sacrifice class time. Recently, some RBAs have been moved to online platforms to address these limitations. Yet, online RBAs present new concerns such as student participation rates, test security, and students' use of outside…
