Global statistic of students’ understandings of Newtonian mechanics represented by a score-state vector of FCI

In this paper, we describe student understanding of the “force concept” (basic concepts of Newtonian mechanics) by representing it as a vector-valued quantity, which we refer to as the “score-state vector,” in a (30-dimensional) Force Concept Inventory (FCI) score space. We use a large ensemble of FCI results collected throughout Japan to explore the global statistical structure of the vector. We used principal component analysis to project the score-state vector into a three-dimensional subspace and found that the student score states follow a non-normal distribution. The distribution exhibits two snakes and is also skewed like a screw. This implies that the correlation of the second order (such as the commonly used correlation coefficients) is insufficient to describe the global structure of student score states. We find the distribution to be roughly divided into two subdistributions corresponding to novice respondents and intermediate (transient) and expert respondents. This indicates that “novice” and/or “expert” are not additional qualities given to the ensemble of samples but are understood as a label distinguishing the characteristic areas of the entire distribution. In addition, we observe, as a result of the non-normal distribution, the deviation of the second-order correlation of the score-state vector from that of an artificial interitem-uncorrelated ensemble of samples. The observed deviations realize many previously obtained results and present us with many other new results. Although ours is only a first step, it is a necessary process toward constructing a more accurate model of the student’s reasoning about Newtonian mechanics.