Related papers: On the need to enhance physical insight via mathem…
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…
Developing expertise in physics entails learning to use mathematics effectively and efficiently as applied to the context of physical situations. Doing so involves coordinating a variety of concepts and skills including mathematical…
The large number of published articles in physics journals under the title "Comments on ..." and "Reply to ..." is indicative that the conceptual understanding of physical phenomena is very elusive and hard to grasp even to experts, but it…
Learning to use math in science is a non-trivial task. It involves many different skills (not usually taught in a math class) that help blend physical knowledge with mathematical symbology. One of these is the idea of quantification: that…
Highly successful students, as measured by grades and by scores on the Force Concept Inventory, still struggle with fundamental concepts in mathematics and physics. These difficulties, which turn physics into parrot learning and include…
The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even to the way we interpret mathematical…
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…
A primary goal of physics is to create mathematical models that allow both predictions and explanations of physical phenomena. We weave maths extensively into our physics instruction beginning in high school, and the level and complexity of…
In their study of physics beyond the first year of University -- termed upper-division in the US, many of students' primary learning opportunities come from working long, complex back-of-the-book style problems, and from trying to develop…
One finding of cognitive research is that people do not automatically acquire usable knowledge by spending lots of time on task. Because students' knowledge hierarchy is more fragmented, "knowledge chunks" are smaller than those of experts.…
Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. It does not merely…
How students use mathematics in their physics classes has been studied extensively in the physics education literature. In addition to specific mathematical methods in specific physics contexts, possible effects of more general "cultural"…
We investigated the common difficulties that students have with concepts related to rotational and rolling motion covered in the introductory physics courses. We compared the performance of calculus- and algebra-based introductory physics…
The decline in student interest in science and technology is a major concern in the western world. One approach to reversing this decline is to introduce modern physics concepts much earlier in the school curriculum. We have used the…
A priority of physics instruction is to help students make the connection between the formulae they think they are required to memorize and the real world in which they interact every day. If you ask students to describe a situation in real…
In recent years there has been growing evidence that even after teaching designed to address the learning difficulties dictated by literature, many physics learners fail to create the proper reasoning chains that connect the fundamental…
As we typically teach in an introductory mechanics course, choosing a "good" reference frame with convenient axes may present a major simplification to a problem. Additionally, knowing some conserved quantities provides an extremely…
Pre-college mathematics modeling instruction often frames mathematics as being separated from reasoning about the real world -- and commonly treats reasoning mathematically and reasoning about the real-world context as separate stages of a…
Learning to use math in physics involves combining (blending) our everyday experiences and the conceptual ideas of physics with symbolic mathematical representations. Graphs are one of the best ways to learn to build the blend. They are a…
This chapter sets the stage for the rest of the book by exploring the role of intuition as a tool to deepen understanding in Einsteinian physics. Drawing on examples from the history of general relativity, we argue that the development of…