Related papers: Learning Mathematics through incorrect Problems
Drawing appropriate diagrams is a useful problem solving heuristic that can transform a given problem into a representation that is easier to exploit for solving it. A major focus while helping introductory physics students learn problem…
If students are to successfully grapple with authentic, complex biological problems as scientists and citizens, they need practice solving such problems during their undergraduate years. Physics education researchers have investigated…
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…
In this short review, I will summarize my research experience in three fields in applied mathematics: mathematical biology, applied probability, and applied discrete mathematics. Specifically, I will show how each project was initiated, and…
University students taking introductory physics are generally successful executing mathematical procedures in context, but often struggle with the use of mathematical concepts for sense making. Physics instructors note that their students…
In this companion paper, we explore the use of isomorphic problem pairs (IPPs) to assess introductory physics students' ability to solve and successfully transfer problem-solving knowledge from one context to another in mechanics. We call…
We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge in dynamical systems and numerical methods…
This paper investigates the capabilities of large language models (LLMs) in formulating and solving decision-making problems using mathematical programming. We first conduct a systematic review and meta-analysis of recent literature to…
In this study, we examine introductory physics students' ability to perform analogical reasoning between two isomorphic problems which employ the same underlying physics principles but have different surface features. Three hundred and…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
Mathematics is the language of science. Fluent and productive use of mathematics requires one to understand the meaning embodied in mathematical symbols, operators, syntax, etc., which can be a difficult task. For instance, in algebraic…
Evaluating a neural network on an input that differs markedly from the training data might cause erratic and flawed predictions. We study a method that judges the unusualness of an input by evaluating its informative content compared to the…
Showing incorrect answers to Large Language Models (LLMs) is a popular strategy to improve their performance in reasoning-intensive tasks. It is widely assumed that, in order to be helpful, the incorrect answers must be accompanied by…
Studying worked examples has been shown by extensive research to be an effective method for learning to solve well-structured problems in physics and mathematics. The effectiveness of learning with worked examples has been demonstrated and…
Most students struggle when faced with complex and ill-structured tasks because the strategies taught in schools and universities simply require finding and applying the correct formulae or strategy to answer well-structured, algorithmic…
This paper outlines a deceptively complex problem in classical mechanics which the paper names the "Falling Astronaut Problem," and it explores a method for teachers to implement this problem in an undergraduate classroom. The paper…
Several different ways exist for approaching hard optimization problems. Mathematical programming techniques, including (integer) linear programming-based methods and metaheuristic approaches, are two highly successful streams for…
Those of us who teach Mathematics for Liberal Arts (MLA) courses often underestimate the mathematical abilities of the students enrolled in our courses. Despite the fact that many of these students suffer from math anxiety and will admit to…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…