Related papers: Making mathematical claims
We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed…
Many undergraduate students of engineering and the exact sciences have difficulty with their mathematics courses due to insufficient proficiency in what we in this paper have termed clear thinking. We believe that this lack of proficiency…
Discrete mathematics and probability theory contain foundational material for computer scientists. Despite their importance, instructors often worry that students will find these courses to be too abstract and seemingly disconnected from…
We illustrate the concept of mathematical proof.
Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice…
The example of the calculus is used to explain how simple, practical math was made enormously complex by imposing on it the Western religiously-colored notion of mathematics as "perfect". We describe a pedagogical experiment to make math…
It is rare to succeed in getting mathematics into ordinary conversation without meeting all kinds of reservations. In order to raise public awareness of mathematics effectively, it is necessary to modify such attitudes. In this paper, we…
Thirty original and collected problems, puzzles, and paradoxes in mathematics and physics are explained in this paper, taught by the author to the elementary and high school teachers at the University of New Mexico - Gallup in 1997-8 and…
A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into…
Doing mathematics implies three levels of manipulation: manipulating the abstract, manipulating symbols and manipulating logic. Teaching mathematics therefore involves the teacher proposing situations in which pupils can explore a small…
The ability to read, write, and speak mathematics is critical to students becoming comfortable with statistical models and skills. Faster development of those skills may act as encouragement to further engage with the discipline. Vocabulary…
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…
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.
How does the mathematical community accept that a given proof is correct? Is objective verification based on explicit axioms feasible, or must the reviewer's experiences and prejudices necessarily come into play? Can automated provers avoid…
Beyond the difficulty to give true practical motives to introduce a theorem, the didactic transposition of scholarly knowledge in a school setting often leads to freezing a technical tool into a theorem given under minimal assumptions.…
Mathematicians occasionally discover interesting truths even when they are playing with mathematical ideas with no thoughts about possible consequences of their actions. This paper describes two specific instances of this phenomenon. The…
Mathematical modelling and ethics have more touching points than most of us would like to admit. Everyday decisions are often reasoned by mathematical arguments. Mathematics teachers belong to those mathematically literate, who must point…
In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This…
We re-examine the old question to what extent mathematics may be compared with a game. Mainly inspired by Hilbert and Wittgenstein, our answer is that mathematics is something like a rhododendron of language games, where the rules are…
A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I…