arXiv subjects

Eric Burkholder

Publications and source records attributed to Eric Burkholder.

13 recordsLinked to original sources

arXiv2026

Active Learning vs Traditional Lecturing in Introductory Mechanics: A Pooled Pass-Rate Benchmark Under Common Departmental Assessments from a Latin American Institutional Change Initiative

Improving student success in introductory physics remains a persistent challenge despite substantial progress from research-based instructional practices. Evidence from the Latin American context remains limited, where resources for instructional change are often constrained. This study reports a transparent benchmark of student passing outcomes in \textit{Elementary Mechanics I} at a large public university in M\'exico, comparing sections using Active Learning (AL) with those using Traditional Lecturing (TL). The labels AL and TL are operational, referring to section-level implementations by individual instructors rather than standardized protocols. Using aggregated counts from coordinator reports and common departmental assessments -- written by a committee independent of instructional modality -- we estimated pooled student-level pass probabilities for the first and second midterm exams, the global exam, and the final mark. Modality differences are summarized primarily by the risk difference, $RD_a=p_{\mathrm{AL},a}-p_{\mathrm{TL},a}$ (percentage points), with uncertainty quantified using Wilson confidence intervals and a Bayesian reference analysis with Jeffreys priors for binomial proportions. Across assessments, pooled pass rates were higher under AL than under TL, with the strongest separation observed for the global exam and the final mark. For these outcomes, the $95\%$ confidence intervals excluded zero, including under a random-intercept Bayesian model. We emphasize a constrained interpretation: the results provide a student-weighted benchmark of ``AL as implemented'' versus ``TL as implemented'' in this setting, without isolating the causal effect of individual instructional techniques. Implications are discussed for departmental decision-making and feasible next steps in evaluation, including improved student data collection and more robust qualitative analysis.

physics.ed-ph↗

arXiv2026

Living in the tensions: Investigations of gender performativity in STEM

In this work, we present the results of semi-structured interviews with four women to explore how they perceive themselves with respect to three gender constructs (femininity, masculinity, androgyny), and how they believe others perceive them. All the women highlighted the performative nature of gender in science, technology, engineering, and mathematics (STEM), citing (1) stereotypes that women are not analytical thinkers, or femininity being associated with "being stupid"; (2) the pressure to conform to the masculine norms of STEM, and (3) a pressure to perform to prove that they belong in STEM. Some of these women aligned their own perceptions of their gender with these norms, while others expressed frustration with the tension between their gender and how that is perceived by peers in STEM. This work suggests that conceptualizing gender as performance is a useful lens for understanding the oppression and underrepresentation of women and gender minorities in STEM.

physics.ed-ph↗

arXiv2025

Relating Different Definitions of Linear Series on Tropical Curves

We investigate relationships among several recent notions of linear series on tropical curves, including tropical linear series, strongly recursive tropical linear series of Farkas, Jensen, and Payne, and combinatorial limit linear series of Amini and Gierczak. We introduce structured and locally weakly recursive tropical linear series, showing that every locally weakly recursive tropical linear series is a combinatorial limit linear series. Consequently, every strongly recursive tropical linear series is a combinatorial limit linear series. We also obtain an equivalent characterization of combinatorial limit linear series. We establish additional extensions of these results and construct counterexamples showing that the converse implications fail for strongly recursive tropical linear series. Finally, we investigate the extent to which permutation arrays can arise as local combinatorial data of tropical linear series.

math.AG↗

arXiv2025

Refined Brill-Noether Theory for Complete Graphs

The divisor theory of the complete graph $K_n$ is in many ways similar to that of a plane curve of degree $n$. We compute the splitting types of all divisors on the complete graph $K_n$. We see that the possible splitting types of divisors on $K_n$ exactly match the possible splitting types of line bundles on a smooth plane curve of degree $n$. This generalizes the earlier result of Cori and Le Borgne computing the ranks of all divisors on $K_n$, and the earlier work of Cools and Panizzut analyzing the possible ranks of divisors of fixed degree on $K_n$.

math.CO↗

arXiv2024

Real-World Problem-Solving Class is Correlated with Higher Student Persistence in Engineering

Student persistence in science, technology, engineering, and mathematics (STEM) has long been a focus of educational research, with both quantitative and qualitative methods being used to investigate patterns and mechanisms of attrition. Some studies have used machine learning to predict a student's likelihood to persist given measurable classroom factors and institutional data, while others have framed persistence as a function of a student's social integration in the classroom. While these methods have provided insight into broader underlying patterns of attrition in STEM, they have not investigated class structures or teaching methods that promote persistence. In this study we explore how a research-based instructional format for an introductory calculus-based physics class using real world problem-solving (RPS) was correlated with higher persistence for students at a large research-intensive university. We found that the one-year persistence rates for the RPS course were 74% (fall semester) and 90% (spring semester), while the lecture-based class had a persistence rate of 64% and 78%, respectively. In spring, the RPS persistence rate was significantly higher (p=0.037). The RPS also had higher final grades and larger learning gains than the lecture-based class despite lower scores on a physics diagnostic test. We also note that the higher rates of persistence were not completely explained by higher final grades. This study motivates future work to understand the structural mechanisms that promote student persistence in introductory physics courses.

physics.ed-ph↗

arXiv2023

Examining the Potential and Pitfalls of ChatGPT in Science and Engineering Problem-Solving

The study explores the capabilities of OpenAI's ChatGPT in solving different types of physics problems. ChatGPT (with GPT-4) was queried to solve a total of 40 problems from a college-level engineering physics course. These problems ranged from well-specified problems, where all data required for solving the problem was provided, to under-specified, real-world problems where not all necessary data were given. Our findings show that ChatGPT could successfully solve 62.5% of the well-specified problems, but its accuracy drops to 8.3% for under-specified problems. Analysis of the model's incorrect solutions revealed three distinct failure modes: 1) failure to construct accurate models of the physical world, 2) failure to make reasonable assumptions about missing data, and 3) calculation errors. The study offers implications for how to leverage LLM-augmented instructional materials to enhance STEM education. The insights also contribute to the broader discourse on AI's strengths and limitations, serving both educators aiming to leverage the technology and researchers investigating human-AI collaboration frameworks for problem-solving and decision-making.

cs.AI↗

arXiv2023

Linking number of monotonic cycles in random book embeddings of complete graphs

A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is $\frac{i}{6}$, where $i$ is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of $n$-cycles in $K_{2n}$ grows linearly in $n$.

math.GT↗

arXiv2022

The Game of Cycles for Grids and Select Theta Graphs

We are investigating who has the winning strategy in a game in which two players take turns drawing arrows trying to complete cycle cells in a graph. A cycle cell is a cycle with no chords. We examine game boards where the winning strategy was previously unknown. Starting with a $C_{5}$ sharing two consecutive edges with a $C_{7}$ we solve multiple classes of graphs involving "stacked" polygons. We then expand upon and improve previous theorems and conjectures, and offer some new directions of research related to the Game of Cycles. The original game was described by Francis Su in his book Mathematics for Human Flourishing. The first results on the game were published in The Game of Cycles arXiv:arch-ive/04.00776.

math.CO↗

arXiv2022

Absence of a COVID-induced academic drop in high-school physics learning

At the start of the COVID 19 pandemic, the majority of secondary instruction in the United States transitioned to an online environment. In many parts of the country, online schooling continued for upwards of two years. Many experts have hypothesized an "academic slide" (a reduction in student learning) following this period of online instruction. We investigated the change in student preparation for introductory college physics in incoming Stanford university students between the Fall term of 2019 to the Fall term of 2021. We did this by looking at the performance on a validated physics diagnostic exam that all Stanford students intending to take a physics course took before enrolling in an introductory physics course. We found no statistically or educationally significant change in scores. Despite many anecdotal faculty reports, at least for this population, the level of student preparation in physics and related math appears to be unchanged.

physics.ed-ph↗

arXiv2022

Preparation for future active learning

It is well documented that students sometimes resist active learning techniques. A recent study showed how students believed that they learned less in active learning classrooms than they learned in lectures, even though they learned more. In this article, we describe a method for introducing active learning methods to college students that is based on a preparation for future learning approach. The students who received this introduction to active learning appeared to be more receptive to group work in the classroom than students who started the course with an explanation of the reasons and values of active learning.

physics.ed-ph↗

arXiv2021

An equitable and effective approach to introductory mechanics

Introductory mechanics ("physics 1") is a critical gateway course for students desiring to pursue a STEM career. A major challenge with this course is that there is a large spread in the students' incoming physics preparation, and this level of preparation is strongly predictive of a students' performance. The level of incoming preparation is also largely determined by a student's educational privilege, and so this course can amplify inequities in K-12 education and provide a barrier to a STEM career for students from marginalized groups. Here, we present a novel introductory course design to address such equity challenges in physics 1. We designed the course based on the concept of deliberate practice to give students targeted, scaffolded, and repeated opportunities to engage in research-identified practices and decisions required for effective problem-solving. We used real-world problems, as they carry less resemblance to physics high school problems, and so even the students with the best high school physics instruction have little experience or skill in solving them. The students learned the physics content knowledge they needed in future courses, particularly in engineering, and their problem-solving skills improved substantially. Furthermore, the success in the course was not correlated with incoming physics preparation, in stark contrast to the outcomes from traditional physics 1 courses. These findings suggest that we made physics 1 more equitable by employing a deliberate practice approach in the context of real-world problem-solving.

physics.ed-ph↗

arXiv2021

A Detailed Characterization of the Expert Problem-Solving Process in Science and Engineering; Guidance for Teaching and Assessment

A primary goal of science and engineering (S & E) education is to produce good problem solvers, but how to best teach and measure the quality of problem-solving remains unclear. The process is complex, multifaceted, and not fully characterized. Here we present a theoretical framework of the S & E problem-solving process as a set of specific interlinked decisions. This theory is empirically grounded and describes the entire process. To develop this theory, we interviewed 52 successful scientists and engineers (experts) spanning different disciplines, including biology and medicine. They described how they solved a typical but important problem in their work, and we analyzed the interviews in terms of decisions made. Surprisingly, we found that across all experts and fields, the solution process was framed around making a set of just twenty-nine specific decisions. We also found that the process of making those discipline-general decisions (selecting between alternative actions) relied heavily on domain-specific predictive models that embodied the relevant disciplinary knowledge. This set of decisions provides a guide for the detailed measurement and teaching of S & E problem-solving. This decision framework also provides a more specific, complete, and empirically based theory describing the practices of science.

physics.ed-ph↗

arXiv2019

The impact of incoming preparation and demographics on performance in Physics I: a multi-institution comparison

We have studied the impact of incoming preparation and demographic variables on student performance on the final exam in physics 1, the standard introductory, calculus-based mechanics course This was done at three different institutions using multivariable regression analysis to determine the extent to which exam scores can be predicted by a variety of variables that are available to most faculty and departments. We have found that the results are surprisingly consistent across the institutions, with the only two variables that have predictive power being math SAT/ACT scores and concept inventory pre-scores. The importance of both variables is comparable and fairly similar across the institutions. They explain 20 - 30 percent of the variation in students' performance on the final exam. Most notably, the demographic variables (gender, under-represented minority, first generation to attend college) are not significant. In all cases, although there appear to be gaps in exam performance if one considers only the demographic variable, once these two proxies of incoming preparation are included in the model, there is no longer a demographic gap. There is only a preparation gap that applies equally across the entire student population. This work shows that to properly understand differences in student performance across a diverse population, and hence to design more effective instruction, it is important to do statistical analyses that take multiple variables into account. It also illustrates the importance of having measures that are sensitive to both subject specific and more general preparation. The results suggest that better matching of the course design and teaching to the incoming student preparation will likely be the most effective way to eliminate observed performance gaps across demographic groups while also improving the success of all students.

physics.ed-ph↗