We perform a factor analysis on a ''Force Concept Inventory'' (FCI) data set collected from 2109 respondents. We address two questions: the appearance of conceptual coherence in student responses to the FCI and some consequences of this factor analysis on the teaching of Newtonian mechanics. We will highlight the apparent conflation of Newton's third law with Newton's first law in one of the FCI questions and suggest an approach to teaching that may avoid this issue. We also note the absence of a distinct factor interpretable as relating specifically to kinematics. Furthermore, we identify and discuss some of the technical difficulties which may be encountered when performing factor analysis on categorical data sets, such as is the case with FCI data sets.

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.

In this paper we present a series of item response models of data collected using the Force Concept Inventory. The Force Concept Inventory (FCI) was designed to poll the Newtonian conception of force viewed as a multidimensional concept, that is, as a complex of distinguishable conceptual dimensions. Several previous studies have developed single-trait item response models of FCI data; however, we feel that multidimensional models are also appropriate given the explicitly multidimensional design of the inventory. The models employed in the research reported here vary in both the number of fitting parameters and the number of underlying latent traits assumed. We calculate several model information statistics to ensure adequate model fit and to determine which of the models provides the optimal balance of information and parsimony. Our analysis indicates that all item response models tested, from the single-trait Rasch model through to a model with ten latent traits, satisfy the standard requirements of fit. However, analysis of model information criteria indicates that the five-trait model is optimal. We note that an earlier factor analysis of the same FCI data also led to a five-factor model. Furthermore the factors in our previous study and the traits identified in the current work match each other well. The optimal five-trait model assigns proficiency scores to all respondents for each of the five traits. We construct a correlation matrix between the proficiencies in each of these traits. This correlation matrix shows strong correlations between some proficiencies, and strong anticorrelations between others. We present an interpretation of this correlation matrix.

The Force Concept Inventory is one of the most popular and most analyzed multiple-choice concept tests used to investigate students' understanding of Newtonian mechanics. The correct answers poll a set of underlying Newtonian concepts and the coherence of these underlying concepts has been found in the data. However, this inventory was constructed after several years of research into the common preconceptions held by students and using these preconceptions as distractors in the questions. Their sole purpose is to deflect non-Newtonian candidates away from the correct answer. Alternatively, one can argue that the responses could also be treated as polling these preconceptions. In this paper we shift the emphasis of the analysis away from the correlation structure of the correct answers and look at the latent traits underlying the incorrect responses. Our analysis models the data employing exploratory factor analysis, which uses regularities in the data to suggest the existence of underlying structures in the cognitive processing of the students. This analysis allows us to determine whether the data support the claim that there are alternate non-Newtonian worldviews on which students' incorrect responses are based. The existence of such worldviews, and their coherence, could explain the resilience of non-Newtonian preconceptions and would have significant implications to the design of instruction methods. We find that there are indeed coherent alternate conceptions of the world which can be categorized using the results of the research that led to the construction of the Force Concept Inventory.