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Courses in mathematical methods for physics students are not known for including too much in the way of mathematical rigour and, in some ways, understandably so. However, the conditions under which some quite commonly used mathematical…
Mathematics has been used in the exploration and enumeration of juggling patterns. In the case when we catch and throw one ball at a time the number of possible juggling patterns is well-known. When we are allowed to catch and throw any…
This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important…
Embedding words in high-dimensional vector spaces has proven valuable in many natural language applications. In this work, we investigate whether similarly-trained embeddings of integers can capture concepts that are useful for mathematical…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
In this paper we present our experience in using visualization in mathematics education. The experience with our university courses: "Computer tools in matematics" and "Symbolic algebra" provides the basis for mathematics teacher education…
In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.
Significant obstacles prevent large, university-level, introductory physics courses from effectively teaching problem-solving skills. We describe our program for integrating three internet-based "teaching-while-quizzing" tools to address…
Decisions about how the population of the United States should be divided into legislative districts have powerful and not fully understood effects on the outcomes of elections. The problem of understanding what we might mean by "fair…
In this article, we present our findings from ten years of research on intelligent educational games. We discuss the architecture of our training environments for learning spelling and mathematics, and specifically focus on the…
We provide a theoretical framework for analysing and comparing different forms of organizing introductions to mathematical analysis at the secondary level, then illustrate it by two characteristic examples from certain periods of change in…
Can AI solve all math? What do we actually mean by doing mathematics? How do we communicate mathematics? What is mathematics beyond problem solving? This essay is my attempt to answer these questions.
The status of our understanding of the mechanisms of color confinement is reviewed, in particular the results of numerical simulations on the lattice.
University students taking introductory physics are generally successful executing mathematical procedures in context, but often struggle with the use of mathematical concepts for sense making. Physics instructors note that their students…
These are the notes from my courses on the arithmetic of quadratic forms.
In this paper, we consider the problem of machine teaching, the inverse problem of machine learning. Different from traditional machine teaching which views the learners as batch algorithms, we study a new paradigm where the learner uses an…
We describe the development of a junior-senior level course for Physics majors designed to teach Mathematica skills in support of their undergraduate coursework, but also to introduce students to modern research level results. Standard…
In these lectures I will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques. After presenting the general results, in order to…
The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even to the way we interpret mathematical…
This paper shares a classroom story from Fall 2022 to Spring 2025 about a learner centered routine in undergraduate mathematics. I use four steps: an opening question, a short mini lecture about meaning, structured small group work, and a…