# 2012-2014: Quod erat demonstrandum: Deepening the understanding of formulas in physics education

The deep interrelation between physics and mathematics – confirmed both by historical and philosophical studies – poses a major challenge for the teaching and learning of physics. Over the past decades, many physics education researchers have dedicated strong effort to enhance students’ “conceptual” understanding of physics with the goal of avoiding mathematical formulations. However, a deep incursion in the nature of physics shows that separating the “conceptual” and “mathematical” parts of physical theories can be quite problematic.

The use of formulas in physics lessons is an important topic in order to investigate how the relation between mathematics and physics is typically approached in the context of education. Instead of being derived and conceptually explained, physical equations tend to be given without further justification and the students’ task when solving problems is frequently to locate the correct one among a list. With this perspective, mathematics is viewed as a “toolbox” that provide “magical” relations between physical magnitudes.

In fact, explaining why a formula has its particular structure involves recognizing physical assumptions (principles, idealizations, etc.) and identifying the reasons for their mathematical representations. This level of understanding is hardly assessed by the standard view of a formula as a calculation tool to solve quantitative problems or as a regularity extracted from experiment. Considering the deductive character of physics, formulae derivations should enhance the understanding about the origin of physics’ equations, allow students to penetrate into the inner structure of physics and avoid the rote memorization of meaningless equations.

Nevertheless, from a didactical perspective, this issue is far from being trivial. In a derivation process conducted by a physics teacher, for example, every single step should be justified by clearly mentioning the connections between physical assumptions and mathematical representations. If the main focus is given to the mathematical rigor, it is likely that it would become an artificial set of logical steps to be memorized by the students. On the other hand, a pure qualitative or too experiment-oriented approach could stimulate an inductive view of physics and give mathematics a mere technical role. In this sense, the proper balance between physical and mathematical arguments in the explanation of physics formulas remains an open question to be investigated. Therefore, a systematic research effort is necessary in order to deepen students’ understanding of formulas in physics education.

With the purpose of achieving this major goal, there are mainly three levels of action: 1) A historical-philosophical reflection is needed to better understand the role of formulas in physics and their peculiarities in each specific context; 2) Physics teachers need to be prepared and sensitized to focus the understanding of formulas in their lessons; 3) The limits and possibilities of approaching this issue with high school students must be known.

The main focus of this research project is on the second, however the first and third levels, as indispensible parts of the process, will also be taken into account. Our proposal is to elaborate, apply and analyze a course grounded on the understanding of physics formulas – from both a historical-philosophical and an educational perspective – with physics teacher students. In order to investigate how to implement this theme in real didactic situations, the application and analysis of “teaching experiments” with high school students are also planed.

### Publications:

Karam, R. & Krey, O. (2015). Quod erat demonstrandum: Understanding and Explaining Equations in Physics Teacher Education. Science & Education (DOI 10.1007/s11191-015-9743-0)