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The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a…

The question of the definition of what is an algorithm is recurrent. It is found in teaching, at different levels and particularly in secondary education because of the recent evolutions in high school, with immediate consequences in higher…

Categorization axioms have been proposed to axiomatizing clustering results, which offers a hint of bridging the difference between human recognition system and machine learning through an intuitive observation: an object should be assigned…

Curriculum Learning - the idea of teaching by gradually exposing the learner to examples in a meaningful order, from easy to hard, has been investigated in the context of machine learning long ago. Although methods based on this concept…

In an educational landscape dramatically altered by the swift proliferation of Large Language Models, this essay interrogates the urgent this essay interrogates the urgent pedagogical modifications required in secondary schooling. Anchored…

Good problems grab us. They invite us to find patterns, make conjectures, and prove-or perhaps disprove-a conjecture. When I first taught, I saw my work as tantalizing students with structures just beyond their reach, so that I could elicit…

Following the processing of individual topics of elementary school mathematics as content of empirical theories the question is adressed wether the associated conception of mathematics finds itself under established concepts, and how it can…

At the start of the higher education curriculum, the conceptualization of local approximation objects of a function requires the articulation of knowledge and skills from Functional Analysis and Topology. In the study of functions, a number…

In this work, we present two results: The first result is the formalization of Tutte's theorem in Lean, a key theorem concerning matchings in graph theory. As this formalization is ready to be integrated in Lean's mathlib, it provides a…

Our goal in this work is to present some mean value type theorems that are not studied in classic calculus and analysis courses. They are simple theorems yet with large applicability in mathematical analysis (for example, in the study of…

At the University of Colorado Boulder, as part of our broader efforts to transform middle- and upper-division physics courses, we research students' difficulties with particular concepts, methods, and tools in classical mechanics,…

Most educational literature on conceptual change concerns the process by which introductory students acquire scientific knowledge. However, with modern developments in science and technology, the social significance of learning successive…

We prove an intermediate value theorem of an arithmetical flavor, involving the consecutive averages of sequences with terms in a given finite set A. For every such set we completely characterize the numbers x ("intermediate values") with…

This is a brief review of the experimental and theoretical quantum computing. The hopes for eventually building a useful quantum computer rely entirely on the so-called "threshold theorem". In turn, this theorem is based on a number of…

Since there are different ways of axiomatizing and developing a mathematical theory, knowledge about a such a theory may reside in many places and in many forms within a library of formalized mathematics. We introduce the notion of a realm…

As artificial intelligence systems become increasingly prevalent in education, a fundamental challenge emerges: how can we verify if an AI truly understands how students think and reason? Traditional evaluation methods like measuring…

Curriculum learning (CL), motivated by the intuition that learning in increasing order of difficulty should ease generalization, is commonly adopted both in pre-training and post-training of large language models (LLMs). The intuition of CL…

We show that strategies implemented in automatic theorem proving involve an interesting tradeoff between execution speed, proving speedup/computational time and usefulness of information. We advance formal definitions for these concepts by…

When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them,…

This is a pedagogical and (almost) self-contained introduction into the theorem of Groenewold and van Howe, which states that a naive transcription of Dirac's quantisation rules cannot work. Some related issues in quantisation theory are…