Students interpret grades as signals of their strengths, and grades inform students' decisions about majors and courses of study. Differences in grades that are not related to learning can impact this judgment and have real-world impact on course-taking and careers. Existing work has examined how an overemphasis on high-stakes exams can create equity gaps where female students and Black, Hispanic, and Native American students earn lower grades than male students and Asian and white students, respectively. Yet, minimal work has examined how the weighting of individual midterm exam scores can also contribute to equity gaps. In this work, we examine how three midterm exam score aggregation methods for final grades affect equity gaps. We collected midterm exam data from approximately 6,000 students in an introductory physics course over 6 years at a large, research-intensive university. Using this data set, we applied common midterm exam score aggregation methods to determine their impact on aggregated midterm exam grades: dropping the lowest midterm exam score, replacing the lowest midterm exam score with the final exam score if higher, and counting the highest midterm exam score more
The spread of generative AI (GenAI) in higher education has raised concerns that students offload cognitive effort to AI, earning high grades without learning. If this "GenAI substitution hypothesis" is true, grades should rise disproportionately in GenAI-susceptible courses--those relying more on assessments like take-home problem sets and essays rather than in-class exams. Substitution could also affect student satisfaction, measured here as self-reported understanding and interest in the subject, which prior research links to assessments. We test the substitution hypothesis using syllabus and administrative data from a large U.S. university (2016-2025; 138,386 students; 72,730 course offerings). We measure courses' GenAI susceptibility using a human-validated LLM pipeline to extract assessment types from syllabi, and use a differences-in-differences design comparing outcomes across courses before and after ChatGPT's release, while modeling COVID-19 pandemic effects as either persistent or transient. We find no significant differential effect of GenAI availability on grades overall or among previously lower-performing students. Effects on self-reported understanding are likewise
Perceptions of disciplinary recognition and identity can be shaped by various forms of feedback and experiences. Here we focus on the potential effects of course grades on the perceievd recognition and physics identity of students. We analyze patterns in changes in physics identity and perceived recognition from pre course to post course across three cohorts of university students enrolled in calculus-based Physics 1 (N=1,681). Students not receiving A grade, on average, showed declines in physics identity and perceived recognition. Even a B grade resulted in declines, and the declines were nonlinear across lower grades. Changes in perceived recognition fully mediated the changes in identity. Importantly, women showed significantly larger declines in identity and perceived recognition, compared to men, if they got less than A grade. The gender moderation was specifically localized to changes in perceived recognition, with no further gender effects on identity beyond the cascading effects on perceived recognition.
In this study, we investigate student performance using grades and grade anomalies across periods before, during, and after COVID-19 remote instruction in courses for bioscience and health-related majors. Additionally, we explore gender equity in these courses using these measures. We define grade anomaly as the difference between a student's grade in a course of interest and their overall grade point average (GPA) across all other courses taken up to that point. If a student's grade in a course is lower than their GPA in all other courses, we refer to this as a grade penalty. Students received grade penalties in all courses studied, consisting of twelve courses taken by the majority of bioscience and health-related majors. Overall, we found that both grades and grade penalties improved during remote instruction but deteriorated after remote instruction. Additionally, we find more pronounced gender differences in grade anomalies than in grades. We hypothesize that women's decisions to pursue STEM careers may be more influenced by the grade penalties they receive in required science courses than men's, as women tend to experience larger penalties across all periods studied. Furtherm
Student repetition in secondary education imposes significant resource burdens, particularly in resource-constrained contexts. Addressing this challenge, this study introduces a unified machine learning framework that simultaneously predicts pass/fail outcomes and continuous grades, a departure from prior research that treats classification and regression as separate tasks. Six models were evaluated: Logistic Regression, Decision Tree, and Random Forest for classification, and Linear Regression, Decision Tree Regressor, and Random Forest Regressor for regression, with hyperparameters optimized via exhaustive grid search. Using academic and demographic data from 4424 secondary school students, classification models achieved accuracies of up to 96%, while regression models attained a coefficient of determination of 0.70, surpassing baseline approaches. These results confirm the feasibility of early, data-driven identification of at-risk students and highlight the value of integrating dual-task prediction for more comprehensive insights. By enabling timely, personalized interventions, the framework offers a practical pathway to reducing grade repetition and optimizing resource allocat
Peer grading is an educational system in which students assess each other's work. It is commonly applied under Massive Open Online Course (MOOC) and offline classroom settings. With this system, instructors receive a reduced grading workload, and students enhance their understanding of course materials by grading others' work. Peer grading data have a complex dependence structure, for which all the peer grades may be dependent. This complex dependence structure is due to a network structure of peer grading, where each student can be viewed as a vertex of the network, and each peer grade serves as an edge connecting one student as a grader to another student as an examinee. This paper introduces a latent variable model framework for analyzing peer grading data and develops a fully Bayesian procedure for its statistical inference. This framework has several advantages. First, when aggregating multiple peer grades, the average score and other simple summary statistics fail to account for grader effects and, thus, can be biased. The proposed approach produces more accurate model parameter estimates and, therefore, more accurate aggregated grades, by modeling the heterogeneous grading b
The introduction of computing education into curricula worldwide requires multi-year assessments to evaluate the long-term impact on learning. However, no single Computational Thinking (CT) assessment spans primary school, and no group of CT assessments provides a means of transitioning between instruments. This study therefore investigated whether the competent CT test (cCTt) could evaluate learning reliably from grades 3 to 6 (ages 7-11) using data from 2709 students. The psychometric analysis employed Classical Test Theory, Item Response Theory, Measurement Invariance analyses which include Differential Item Functioning, normalised z-scoring, and PISA's methodology to establish proficiency levels. The findings indicate that the cCTt is valid, reliable and gender-fair for grades 3-6, although more complex items would be beneficial for grades 5-6. Grade-specific proficiency levels are provided to help tailor interventions, with a normalised scoring system to compare students across and between grades, and help establish transitions between instruments. To improve the utility of CT assessments among researchers, educators and practitioners, the findings emphasise the importance of
Graded labels are ubiquitous in real-world learning-to-rank applications, especially in human rated relevance data. Traditional learning-to-rank techniques aim to optimize the ranked order of documents. They typically, however, ignore predicting actual grades. This prevents them from being adopted in applications where grades matter, such as filtering out ``poor'' documents. Achieving both good ranking performance and good grade prediction performance is still an under-explored problem. Existing research either focuses only on ranking performance by not calibrating model outputs, or treats grades as numerical values, assuming labels are on a linear scale and failing to leverage the ordinal grade information. In this paper, we conduct a rigorous study of learning to rank with grades, where both ranking performance and grade prediction performance are important. We provide a formal discussion on how to perform ranking with non-scalar predictions for grades, and propose a multiobjective formulation to jointly optimize both ranking and grade predictions. In experiments, we verify on several public datasets that our methods are able to push the Pareto frontier of the tradeoff between ra
Programming courses can be challenging for first year university students, especially for those without prior coding experience. Students initially struggle with code syntax, but as more advanced topics are introduced across a semester, the difficulty in learning to program shifts to learning computational thinking (e.g., debugging strategies). This study examined the relationships between students' rate of programming errors and their grades on two exams. Using an online integrated development environment, data were collected from 280 students in a Java programming course. The course had two parts. The first focused on introductory procedural programming and culminated with exam 1, while the second part covered more complex topics and object-oriented programming and ended with exam 2. To measure students' programming abilities, 51095 code snapshots were collected from students while they completed assignments that were autograded based on unit tests. Compiler and runtime errors were extracted from the snapshots, and three measures -- Error Count, Error Quotient and Repeated Error Density -- were explored to identify the best measure explaining variability in exam grades. Models ut
There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas, and draws connections with definability theory.
Evaluating the performance of students in higher education is essential for gauging the effectiveness of teaching methods and achieving greater equality of opportunities for all. In this study, we investigate the correlation between two teachers' grading practices in a deep learning course at the master's level, offered at CentraleSupélec. The two teachers, who have distinct teaching styles, were responsible for marking the final project oral presentation. Our results indicate a significant positive correlation (0.76) between the two teachers' grading practices, suggesting that their assessments of students' performance are consistent. Although consistent with each other, grades do not seem to be fully reproducible from one examiner to the other suggesting serious drawbacks of only using one examiner for oral projects. Furthermore, we observed that the maximum difference between the grades assigned by the two examiners was 12.5%, with a mean of 6.3\% (and median of 5.0\%), highlighting the potential impact of inter-examiner variability on students' final grades.
We study letter grading schemes, which are routinely employed for evaluating student performance. Typically, a numerical score obtained via one or more evaluations is converted into a letter grade (e.g., A+, B-, etc.) by associating a disjoint interval of numerical scores to each letter grade. We propose the first model for studying the (de)motivational effects of such grading on the students and, consequently, on their performance in future evaluations. We use the model to compare uniform letter grading schemes, in which the range of scores is divided into equal-length parts that are mapped to the letter grades, to numerical scoring, in which the score is not converted to any letter grade (equivalently, every score is its own letter grade). Theoretically, we identify realistic conditions under which numerical scoring is better than any uniform letter grading scheme. Our experiments confirm that this holds under even weaker conditions, but also find cases where the converse occurs.
In this paper, we investigate the potential of open-source Large Language Models (LLMs) for grading Unified Modeling Language (UML) class diagrams. In contrast to existing work, which primarily evaluates proprietary LLMs, we focus on non-proprietary models, making our approach suitable for universities where transparency and cost are critical. Additionally, existing studies assess performance over complete diagrams rather than individual criteria, offering limited insight into how automated grading aligns with human evaluation. To address these gaps, we propose a grading pipeline in which student-generated UML class diagrams are independently evaluated by both teaching assistants (TAs) and LLMs. Grades are then compared at the level of individual criteria. We evaluate this pipeline through a quantitative study of 92 UML class diagrams from a software design course, comparing TA grades against assessments produced by six open-source LLMs. Performance is measured across individual criteria, highlighting areas where LLMs diverge from human graders. Our results show per-criterion accuracy of up to 88.56\% and a Pearson correlation coefficient of up to 0.78, representing a substantial i
We present a graded modal type theory, a dependent type theory with grades that can be used to enforce various properties of the code. The theory has $Π$-types, weak and strong $Σ$-types, natural numbers, an empty type, and a universe, and we also extend the theory with weak and strong unit types and graded $Σ$-types. The theory is parameterized by a modality structure, a kind of partially ordered semiring, whose elements (grades) are used to track the usage of variables in terms and types. Different modalities are possible. We focus mainly on quantitative properties, in particular erasure: with the erasure modality one can mark function arguments as erasable. The theory is fully formalized in Agda. The formalization, which uses a syntactic Kripke logical relation at its core and is based on earlier work, establishes major meta-theoretic properties such as subject reduction, consistency, normalization, and decidability of definitional equality. We also prove a substitution theorem for grade assignment, and preservation of grades under reduction. Furthermore we study an extraction function that translates terms to an untyped $λ$-calculus and removes erasable content, in particular f
We introduce Graded Large Language Models (GLLMs), an algebraic framework that equips the representation space of a transformer with a grading and propagates the induced weighted scalar action through embeddings, self-attention, and the training objective. The construction extends the theory of graded neural networks and graded transformers to autoregressive language models while preserving expressive power, asymptotic computational complexity, and inference cost. The governing geometric picture is that of geometric invariant theory. The benefit of a grading is expressed by a Kempf--Ness functional on the grading torus; the grades that improve upon the uniform architecture form an open convex cone whose membership is decided by a Hilbert--Mumford-type criterion pairing a grade direction against two measurable profiles of the target and the data; the optimal grades are the coincidence point of two moment maps, given in closed form; and the ordinary transformer appears as a semistable isotropic point on the boundary of the cone: one member of a larger graded family rather than a distinguished optimum. Separately, for level-stratified targets we prove a minimax separation between the
Grading German legal exam solutions faces growing volumes and a shortage of qualified graders, delaying feedback and creating a bottleneck. At the same time, it is a high-stakes expert task, since state exam grades strongly influence career outcomes in Germany. Despite this practical relevance, literature lacks systematic studies on effective methods for grading legal exams. To address this gap, we investigate whether large language models (LLMs) can support the automated grading of German legal case solutions in criminal and public law, thereby enabling scalable feedback and student self-testing. We present a systematic evaluation of 27 proprietary and open-source LLMs, benchmarking prompting strategies that incrementally add task-related information, such as a sample solution and a grading rubric. Using quadratic weighted kappa (QWK), reasoning-oriented LLMs can approximate expert grading in public law when given a sample solution and a grading rubric (up to 0.91), compared to 0.60 in criminal law, suggesting a harder grading task in criminal law. Beyond single-model grading, ensembling improves agreement by up to 0.15 over its best member and can offer an alternative to stronger
Existing AGIQA models typically estimate image quality by measuring and aggregating the similarities between image embeddings and text embeddings derived from multi-grade quality descriptions. Although effective, we observe that such similarity distributions across grades usually exhibit multimodal patterns. For instance, an image embedding may show high similarity to both "excellent" and "poor" grade descriptions while deviating from the "good" one. We refer to this phenomenon as "semantic drift", where semantic inconsistencies between text embeddings and their intended descriptions undermine the reliability of text-image shared-space learning. To mitigate this issue, we draw inspiration from psychometrics and propose an improved Graded Response Model (GRM) for AGIQA. The GRM is a classical assessment model that categorizes a subject's ability across grades using test items with various difficulty levels. This paradigm aligns remarkably well with human quality rating, where image quality can be interpreted as an image's ability to meet various quality grades. Building on this philosophy, we design a two-branch quality grading module: one branch estimates image ability while the ot
We introduce the Graded Transformer framework, a new class of sequence models that embeds algebraic inductive biases through grading transformations on vector spaces. Extending Graded Neural Networks (GNNs), we propose two architectures: the Linearly Graded Transformer (LGT) and the Exponentially Graded Transformer (EGT). These models apply parameterized scaling operators, governed by fixed or learnable grading tuples and in the case of EGT exponential factors, to encode hierarchical structure in attention and representation layers and to improve efficiency for structured data. We establish rigorous guarantees, including universal approximation theorems for continuous and Sobolev functions, reduced sample complexity via effective VC dimension bounds, Lipschitz continuity of graded operations, and robustness to perturbations. A graded loss ensures gradient stability and alignment with domain priors during optimization. By treating grades as differentiable parameters, the framework enables adaptive feature prioritization, overcoming limitations of fixed grades in earlier models. The Graded Transformer provides a mathematically principled approach to hierarchical learning and neuro-sy
The operations to be performed by a quantum computer are almost invariably given in the form of a quantum circuit. In the final stage of compilation, a quantum circuit must be translated into the input signals accepted by the quantum hardware itself. For a quantum computer based on superconducting qubits, this will be a sequence of microwave control pulses to be sent to the various input channels. A pulse schedule gives a full specification for which pulse should be applied to which channel at what time. There is as yet no language for these pulse schedules that is very amenable to formal semantics. In this paper, we propose such a language called GRAMPUS (GRAded Modal type theory for PUlse Schedules). It is a graded modal type theory, where the grades represent timing information: a variable $x :^{50} Q_1$ will represent a state of qubit $Q_1$ that will exist 50 nanoseconds in the future, and a variable $y :^{-75} Q_2$ will represent a state of qubit $Q_2$ that existed 75 nanoseconds in the past. We give the syntax for two type theories, one with grades (the annotated language) and one without (the plain language). We prove some metatheoretic properties, and describe the semantics
Automated essay scoring (AES) has advanced significantly with neural language models, yet most systems remain opaque, offering little visibility into how grades are produced. In educational settings, instructors must be able to understand, trust, and occasionally override the automated grading decisions. We introduce EssayCBM, a rubric-aligned concept bottleneck framework that decomposes essay evaluation into eight interpretable writing concepts before computing the final score. Unlike direct LLM-based grading approaches, EssayCBM learns an explicit and auditable mapping from writing concepts to grades, allowing instructors to inspect and adjust rubric-level predictions during grading. EssayCBM matches neural AES baselines while making grading decisions transparent and directly editable at the rubric level. We further present an interactive system that demonstrates this capability by allowing instructors to inspect and modify concept predictions in real time.