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In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures.…
A new class of Semantic Numeration Systems, namely, positive rational Semantic Numeration Systems is introduced. For cardinal semantic operators, differences in the formation of carry (common carry) and remainders are defined. The…
In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved:…
This work addresses critical challenges to academic integrity, including plagiarism, fabrication, and verification of authorship of educational content, by proposing a Natural Language Processing (NLP)-based framework for authenticating…
"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics.…
Discrete mathematics is the foundation of computer science. It focuses on concepts and reasoning methods that are studied using math notations. It has long been argued that discrete math is better taught with programming, which takes…
We consider the problem of sequential prediction and provide tools to study the minimax value of the associated game. Classical statistical learning theory provides several useful complexity measures to study learning with i.i.d. data. Our…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
Jupyter notebooks are increasingly being adopted by teachers to deliver interactive practical sessions to their students. Notebooks come with many attractive features, such as the ability to combine textual explanations, multimedia content,…
This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the…
To represent anything from mathematical concepts to real-world objects, we have to resort to an encoding. Encodings, such as written language, usually assume a decoder that understands a rich shared code. A semantic embedding is a form of…
This study aims to determine what type of student characteristic most considered by teachers in primary school and their influence on student learning achievement. For this purpose, 37 class teachers of grade 4th to 6th from seven primary…
How best to quantify the information of an object, whether natural or artifact, is a problem of wide interest. A related problem is the computability of an object. We present practical examples of a new way to address this problem. By…
Despite the frequent challenges posed by ambiguity when representing meaning via natural language, it is often ignored or deliberately removed in tasks mapping language to formally-designed representations, which generally assume a…
Common research tasks ask students to identify a correct answer and justify their answer choice. We propose expanding the array of research tasks to access different knowledge that students might have. By asking students to discuss answers…
Equations are about more than computing physical quantities or constructing formal models; they are also about understanding. The conceptual systems physicists use to think about nature are made from many different resources, formal and…
With the rapid rise of generative AI in higher education and the unreliability of current AI detection tools, developing policies that encourage student learning and critical thinking has become increasingly important. This study examines…
Missing data are an unavoidable complication in many machine learning tasks. When data are `missing at random' there exist a range of tools and techniques to deal with the issue. However, as machine learning studies become more ambitious,…
Signed languages are the primary means of communication for many deaf and hard of hearing individuals. Since signed languages exhibit all the fundamental linguistic properties of natural language, we believe that tools and theories of…
We learn mathematics subjectively and must apply it objectively. But sometimes, we apply it subjectively by using wrong intuitions which may be elusive to our eyes. The aim of this note is to disclose the secretes of two kinds of these…