Session M04: Research on Student Reasoning and Understanding

Identifying Student Resources on Integration across Mathematics and Physics Questions

An essential skill for success in an undergraduate curriculum is the utilization of math to solve physics problems. Mathematics is a well-documented barrier to success throughout the physics curriculum, especially for persons within marginalized groups. Further, there is a growing understanding of subtle but impactful differences between the mathematics taught in math class and the mathematics students are expected to use in their physics courses. Investigations in the fields of PER and RUME (Research in Undergraduate Mathematics Education) tend to be performed by only mathematicians (from RUME) or physicists (from PER), which may cause bias in the results. To address this issue, four one-on-one interviews were conducted with students at the end of a calculus-based physics sequence wherein they answered questions on integration written by a researcher from RUME and subsequent questions on integration written by a PER researcher. Analysis using a Resources Theoretical Framework identified several resources; however, across math and physics questions, different resources were often activated, and previously activated resources were not necessarily called upon when the questions switched from those written by a mathematician to those written by a physicist. This provides a signal that there are likely impactful differences when thinking about integration depending on who is authoring the questions.   

Student reasoning with series expansions in introductory mathematical methods

As part of a project to investigate student use of mathematics in upper-division physics courses and develop instructional materials, we have examined how students use series expansions as a means of approximating and simplifying expressions in theory-oriented physics courses. Student responses were collected on pre- and post-instruction written tasks in which students were prompted to use a series to approximate an expression arising from a problem in electricity and magnetism. Students were required to identify an appropriate variable in which to expand and use the expansion to generate an approximation. Despite prior experience with series in calculus and physics, students struggled to determine appropriate quantities in which to expand and most did not attend to the units of the terms or the convergence of their series. While student success rates improved after targeted instruction, many students continued to expand in quantities that were neither dimensionless nor small, and thus unproductive for modeling.   

Students' reasoning with multi-variable expressions in the context of potential difference

Research suggests that some reasoning difficulties persist even after targeted instruction. One such instance is the student’s incorrect reasoning with multi-variable expressions in the context of Electromagnetism in the calculus-based introductory physics course. This talk will focus on student reasoning with Potential Difference, DELTA V=-W_EF/q_test. Specifically, after relevant instruction, students struggle to recognize that if the value of a test charge moving between two points is changed, the potential difference between the two points remains the same.  We designed instructional intervention (a blend of web-based assignments and classroom instruction) to prob students' reasoning with multivariable expressions in the context of (1) math problems and (2) analogous problems in the context of physics. We interpret the results through the lens of the dual-process theory of reasoning. 

    

Understanding how students recognize and connect mathematics ideas in physics contexts: A pilot study

A long standing problem within physics education (and education more widely) is the difficulty of promoting transfer of ideas learned in one context into another one. In session at the APS April 2023 meeting, researchers met and discussed needs for further research aimed at understanding and addressing how students connect mathematics and physics ideas when solving problems. To address this at the University of Edinburgh, first and second year mathematics courses are taught to physics majors by physics faculty rather than mathematics faculty (as is typically done). With the goal of evaluating this pedagogical change, we have devised a new instrument using a problem categorization task to evaluate students’ abilities to recognize and connect ideas presented in a purely mathematical context to similar problems presented in a physics context. 12 items relating to vectors were chosen from the Test of Calculus and Vectors in Mathematics and Physics to pilot the viability of this task. Six students were given this task and interviewed after its completion. We will present the findings of this pilot, discuss the viability of this task, and reflect on future research pathways that can meet the aims of understanding how students recognize and connect mathematics and physics knowledge.   

Student understanding of vector products: The effects of context and experience

As part of an ongoing investigation of student understanding of vectors, vector operations, and their physical interpretations, we present findings from responses to survey questions about the magnitude and direction of vector products in different contexts, asked to students at different class levels. The contexts include mathematics with no physics applied, mechanical work, flux of a uniform magnetic or electric field, and magnetic force on a charged particle. This pseudo-longitudinal study included students enrolled in introductory calculus-based physics, a sophomore-level mathematical methods in physics course, and two junior-level electricity and magnetism courses. Between 12% and 30% of introductory students correctly answered that the result of a dot product does not have a direction across all the different contexts. The most common incorrect responses were that the dot product points in a direction between the two vectors in the mathematics context and that the direction of the work is in the same direction as the displacement. Performance improves as the class level increases.   

Analysis of Introductory Physics Computational Thinking Assessment with Classical Test Theory

Computational thinking (CT) in physics is important both because physicists recognize computation as a third pillar of the discipline along with theory and experimentation, and because physicists believe that CT will serve as an important skill for students' future careers. To contribute to the definition of CT in physics and help instructors evaluate their computationally integrated introductory physics courses, we have been developing a CT assessment to add to PhysPort. In this talk we present an analysis of an assessment of CT for introductory physics using classical test theory. Our assessment was administered in the Fall of 2023 by four different instructors as a pre- and post- multiple-choice assessment. In total there were 1,134 responses to the pre- and 964 responses to the post-. We will characterize the factor structure of the assessment, and report on the mean score, item difficulty and discrimination, Cronbach's alpha, and learning gains from the preliminary administration of the assessment.   

Modern Physics: Understanding The Content Taught in the US

The Modern Physics course is a crucial gateway for physics majors, introducing new concepts beyond K-12 experiences. Despite its significance, content varies widely among institutions. This study analyzes 167 Modern Physics syllabi from 127 R1 and R2 US institutions, employing emergent coding on data from public sources (51.5%) and private correspondence (48.5%). Public course catalogs were consulted to identify pre- and co-requisites, with 37.1% of students having completed Calculus II. Foundational topics like Newtonian Mechanics (94%), Electricity and Magnetism (84.4%), and Waves or Optics (77.2%) were frequently required. Quantum Physics (94%), Atomic Physics (83%), and Relativity (70%) were most commonly taught. The study highlights the lack of uniformity in Modern Physics curricula, emphasizing the importance of a consistent and comprehensive education for physics majors across universities. This insight contributes to the ongoing discourse on optimizing physics education in higher education.   

Do students think that objects have a true value?

The idea of a true value is central to the definitions of point and set paradigms, which is a model for thinking about how students view measurements and uncertainty. Several studies have investigated how students' responses to questions about measurement and uncertainty reflect point- and set-like thinking, but none have asked whether a true value exists. In this work, we focus on the idea of a deterministic true value and whether students (and expert physicists) think it exists. We asked over 700 participants from lower- and upper-division physics courses a survey question that directly asks whether objects have a true, definite position. Results show differences between students at the lower level and upper level in their answer choice, and their reasoning. These findings give more context into students' thinking about measurements and uncertainty, allowing us to make better decisions when developing new instructional materials.   

Improving Student Construction of a Quantum Mechanical Position Eigenvalue Equation: Preliminary Findings

Spins-first quantum mechanics (QM) courses are increasingly common due to multiple perceived affordances. Prior research in both physics and mathematics education has characterized student understanding of and reasoning about QM eigenvalue equations, particularly in spins-first courses. As part of an effort to examine and support students’ mathematical sensemaking in the transition from discrete to continuous systems in spins-first QM, students were asked to construct an eigenvalue equation for a one-dimensional, continuous position operator after approximately six weeks of instruction in the contexts of spins, but prior to any instruction in the context of positions. Few students constructed an equation of the correct form, and only a subset of those appropriately interpreted their equation. To address this, we are developing an instructional sequence (clicker question series and tutorial activity) to scaffold continuous position eigenvalue equation construction with explicit references to discrete spin eigenvalue equations. We discuss some aspects of the instructional materials development and implementation. While the sequence generally guided students to a correct equation and appropriate interpretation, there is evidence of room for improvement.