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The amount of visual communication we are facing is rapidly increasing, and skills to process, understand, and generate visual representations are in high demand. Especially students focusing on computer graphics and visualization can…
Vector Symbolic Architectures combine a high-dimensional vector space with a set of carefully designed operators in order to perform symbolic computations with large numerical vectors. Major goals are the exploitation of their…
Predicting student performance is a fundamental task in Intelligent Tutoring Systems (ITSs), by which we can learn about students' knowledge level and provide personalized teaching strategies for them. Researchers have made plenty of…
Graph Neural Networks (GNNs) have achieved significant success across various applications. However, their complex structures and inner workings can be challenging for non-AI experts to understand. To address this issue, this study presents…
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…
We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II_1 factor,…
We study the problem of computer-assisted teaching with explanations. Conventional approaches for machine teaching typically only provide feedback at the instance level e.g., the category or label of the instance. However, it is intuitive…
Tensors are ubiquitous in statistics and data analysis. The central object that links data science to tensor theory and algebra is that of a model with latent variables. We provide an overview of tensor theory, with a particular emphasis on…
We introduce a new OpenMath content dictionary, named tensor1, containing symbols for the expression of tensor formulas. These symbols support the expression of non-Cartesian coordinates and invariant, multilinear expressions in the context…
We describe a way to introduce physics high school students with no background in programming to computational problem-solving experiences. Our approach builds on the great strides made by the Modeling Instruction reform curriculum. This…
Vector representations have been pivotal in advancing natural language processing (NLP), with prior research focusing on embedding techniques for mathematical expressions using mathematically equivalent formulations. While effective, these…
Symbolic equations are one of the many representations used in physics. Understanding these representations is important for students because they are how students access knowledge in physics. In this paper I build off of the work by Redish…
The theorems of vector analysis (divergence theorem, etc.) are typically first applied in the undergraduate physics curriculum in the context of the electromagnetic field and the differential forms of Maxwell's equations. However, these…
The computation of distance measures between nodes in graphs is inefficient and does not scale to large graphs. We explore dense vector representations as an effective way to approximate the same information: we introduce a simple yet…
The division between two vectors belonging to the same vector space is obtained by elementary procedures of vector algebra and is defined by a matrix. This representation is obtained for two and three dimensional vector spaces. A new vector…
In a real-world setting, visual recognition systems can be brought to make predictions for images belonging to previously unknown class labels. In order to make semantically meaningful predictions for such inputs, we propose a two-step…
Teaching assistants (TAs) are often responsible for grading in introductory physics courses at large research universities. Their grading practices can shape students' approaches to problem solving and learning. Physics education research…
A simple Neural Network model is presented for end-to-end visual learning of arithmetic operations from pictures of numbers. The input consists of two pictures, each showing a 7-digit number. The output, also a picture, displays the number…
In recent years, computing has become an important part of the way we teach and learn physics. Teachers, both at high school and college levels, now use computational activities in many of their courses. Physics departments are offering…
Human-annotated datasets with explicit difficulty ratings are essential in intelligent educational systems. Although embedding vector spaces are widely used to represent semantic closeness and are promising for analyzing text difficulty,…