Related papers: Some instructive mathematical errors
Concave mirrors are fundamental optical elements, yet some easily observed behaviors are rarely addressed in standard textbooks, such as the formation of multiple reflected images. Here we investigate self-imaging -- where the observer is…
Previous theoretical results pertaining to meta-learning on sequences build on contrived assumptions and are somewhat convoluted. We introduce new information-theoretic tools that lead to an elegant and very general decomposition of error…
Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical…
Papert's (1980) work with Turtle Geometry offered an early and provocative vision of how digital technologies could be used with young learners. Since then, research on digital technology use has focused on the middle and high school…
Error type information has been widely used to improve the performance of grammatical error correction (GEC) models, whether for generating corrections, re-ranking them, or combining GEC models. Combining GEC models that have complementary…
In this chapter, I review the main methods and techniques of complex systems science. As a first step, I distinguish among the broad patterns which recur across complex systems, the topics complex systems science commonly studies, the tools…
We formalize an interpretational error that is common in statistical causal inference, termed identity slippage. This formalism is used to describe historically-recognized fallacies, and analyse a fast-growing literature in statistics and…
In many large multiple testing problems the hypotheses are divided into families. Given the data, families with evidence for true discoveries are selected, and hypotheses within them are tested. Neither controlling the error-rate in each…
In this review, we present some advanced algorithms and programs used in our scientific school with short description of types of astrophysical systems, which we study. However, we discuss mainly mathematical methods, which may be applied…
Prior research in computational argumentation has mainly focused on scoring the quality of arguments, with less attention on explicating logical errors. In this work, we introduce four sets of explainable templates for common informal…
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs…
This paper discusses digital online mathematics examinations -- a discussion ranging from high school to university level examinations. In particular, we consider the nature of mathematical writing, what is distinctive about mathematical…
We consider fits to two or more datasets for which results from the sa me experiment share a common systematic uncertainty in addition to their individ ual statistical errors. This is important in extracting the maximum information from a…
Statistics experiences a storm around the perceived misuse and possible abuse of its methods in the context of the so-called reproducibility crisis. The methods and styles of quantification practiced in mathematical modelling rarely make it…
Spreadsheets are ubiquitous in business with the financial sector particularly heavily reliant on the technology. It is known that the level of spreadsheet error can be high and that it is often necessary to review spreadsheets based on a…
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…
Structure or projectional editors are a well-studied concept among researchers and some practitioners. They have the huge advantage of preventing syntax and in some cases type errors, and aid the discovery of syntax by users unfamiliar with…
The use of computer-based mathematics tools is widespread in learning. Depending on the way that these tools assess the learner's solution paths, one can distinguish between automatic assessment tools and semi-automatic assessment tools.…
Empirical studies on formal methods and tools are rare. In this paper, we provide guidelines for such studies. We mention their main ingredients and then define nine different study strategies (laboratory experiments with software and human…
New technologies have led to vast troves of large and complex datasets across many scientific domains and industries. People routinely use machine learning techniques to not only process, visualize, and make predictions from this big data,…