Related papers: Bad Foundations and Manipulable Objects
Usually the first course in mathematics is calculus. Its a core course in the curriculum of the Business, Engineering and the Sciences. However many students face difficulties to learn calculus. These difficulties are often caused by the…
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 basic requirement for a mathematical model is often that its solution (output) shouldn't change much if the model's parameters (input) are perturbed. This is important because the exact values of parameters may not be known and one would…
Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with $p < 0.05$, we consider matrices $A \in \mathbb{R}^{n \times n}$ whose rows are normalized in $\ell^2$ and for which $2^{-n}\sum_{x \in…
Starting from the well-known and elementary problem of inscribing the rectangle of the greatest area in an ellipse, we look at possible, gradually more and more complicated variants of this problem. Our goal is to demonstrate to an average…
Counterfactual explanations (CEs) offer a human-understandable way to explain decisions by identifying specific changes to the input parameters of a base or present model that would lead to a desired change in the outcome. For optimization…
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…
It is the aim of this work to identify and illustrate the potential and weaknesses of the computer algebra system Maple in the area of the Calculus of Variations: a classical area of mathematics that studies the methods for finding maximum…
Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…
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…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
Despite the impressive performance in a variety of complex tasks, modern large language models (LLMs) still have trouble dealing with some math problems that are simple and intuitive for humans, such as addition. While we can easily learn…
Curriculum learning--ordering training examples in a sequence to aid machine learning--takes inspiration from human learning, but has not gained widespread acceptance. Static strategies for scoring item difficulty rely on indirect proxy…
Teaching logic effectively requires an understanding of the factors which cause logic students to struggle. Formalization exercises, which require the student to produce a formula corresponding to the natural language sentence, are a good…
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…
Much of the controversy about methods for automated decision making has focused on specific calculi for combining beliefs or propagating uncertainty. We broaden the debate by (1) exploring the constellation of secondary tasks surrounding…
The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained…
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…
Effective caching is crucial for the performance of modern-day computing systems. A key optimization problem arising in caching -- which item to evict to make room for a new item -- cannot be optimally solved without knowing the future.…
Context: Students often misunderstand programming problem descriptions. This can lead them to solve the wrong problem, which creates frustration, obstructs learning, and imperils grades. Researchers have found that students can be made to…