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Working with letters that represent unknown constants, i.e., parameters, has been historically challenging for students. This is an important skill for their success in many future quantitative settings, and yet it appears this topic is…
We investigate if the visual representation of vectors can affect which methods students use to add them. We gave students one of four questions with different graphical representations, asking students to add the same two vectors. For…
The aim of this study is to construct and compose an instructional design in combinatorial learning, particularly in the concept of counting. A composed design is expected to optimize students' combinatorial-thinking skill. This research…
In order to describe natural phenomena, science develops sophisticated models that use mathematical and formal languages which seem, and often are, very far from common experience. When a phenomenon is not accessible to our senses, its…
Over the past thirty years or so the authors have been teaching various programming for mathematics courses at our respective Universities, as well as incorporating computer algebra and numerical computation into traditional mathematics…
This paper intends to introduce a game-based 3D Virtual Learning Environment (VLE) to second graders. The impetus arose from the need to make learning in mathematics more effective and interesting through multimedia. Applied in a game, the…
We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations,…
The authors investigate how teaching art and astronomy together has the potential to inspire new art forms, enhance scientific public outreach, and promote art and science education. The authors teach an astro-animation class at the…
We study the computational complexity of computing role colourings of graphs in hereditary classes. We are interested in describing the family of hereditary classes on which a role colouring with k colours can be computed in polynomial…
A vector $t$-coloring of a graph is an assignment of real vectors $p_1, \ldots, p_n$ to its vertices such that $p_i^Tp_i = t-1$ for all $i=1, \ldots, n$ and $p_i^Tp_j \le -1$ whenever $i$ and $j$ are adjacent. The vector chromatic number of…
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…
Understanding the stroke-based evolution of visual artworks is useful for advancing artwork learning, appreciation, and interactive display. While the stroke sequence of renowned artworks remains largely unknown, formulating this sequence…
Beck's conjecture on coloring of graphs associated to various algebraic objects has generated considerable interest in the community of discrete mathematics and combinatorics since its inception in the year 1988. The version of this…
The paper describes methodology of math education for students interested in chemistry. Suppose we have mathematical circle 2 hours per week. What can be done? We can not provide any systematic study, but we can choose one subject and show…
We introduce an axiomatic theory of spherical diagrams as a tool to study certain combinatorial properties of polyhedra in $\mathbb R^3$, which are of central interest in the context of Art Gallery problems for polyhedra and other…
In education there exists a tension between two modes of learning: traditional lecture-based instruction and more tinkering-based creative learning. In this paper, we outline our efforts as two Ph.D. students (who are skilled in…
In 1966, David Mumford created a drawing of $\operatorname{Proj} \mathbb Z [X,Y]$ in his book, "Lectures on Curves on an Algebraic Surface". In following, he created a photo of a so-called 'arithmetic surface' $\operatorname{Spec} \mathbb Z…
Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the…
In this paper, we revisit the classical problem of determining osculating conics and sextactic points for a given algebraic curve. Our focus is on a particular family of plane cubic curves known as the Hesse pencil. By employing classical…
A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer…