We design, validate, and administer a 24-item test to study student understanding of linear functions in 1D kinematics [$x(t)$] and mathematics [$y(x)$] in the 9th grade. The items assess identification and comparison of initial position and velocity in 1D kinematics and of the $y$ intercept and slope in mathematics using a graph or an algebraic formula. Results show that students’ performance on most mathematics items is significantly better than on their isomorphic kinematic counterparts, but also that most of the easiest as well as the most difficult items are kinematics items. Students achieve the highest accuracies on graphical questions in which they must compare two positive slopes, and they achieve the lowest accuracies on questions in which they must determine or compare a negative slope. We find that students have more difficulties with the $y$ intercept in mathematics and with the slope in kinematics. Furthermore, questions in symbolic representation result in far lower accuracies compared to questions in graphical representation, particularly when the $y$ intercept or the slope has to be determined instead of compared. We also analyze the results qualitatively by categorizing the students’ strategies and errors. We find frequent confusion between the $x$ intercept and the $y$ intercept in mathematics, but far less in kinematics. Negative velocities in kinematics are by far the largest pitfall, whereas negative slope in mathematics is rarely an issue. The results also show a significant frequency of interval or point confusions in kinematics but very little in mathematics. We reaffirm the occurrence of the interval or point confusion in questions with graphs and discuss three different cases of interval or point confusions in questions with algebraic expressions: numerical, algebraic, and unit based. Our results indicate a weak link between kinematics and mathematics and we suggest that closer integration between these two contexts during education could benefit student understanding of linear functions and linear phenomena in kinematics.

This study reports on the development, validation, and administration of a 48-item multiple-choice test to assess students’ representational fluency of linear functions in a physics context (1D kinematics) and a mathematics context. The test includes three external representations: graphs, tables, and formulas, which result in six possible representational transitions between them. Moreover, four linear function types are included: negative $y$ intercept and positive slope, zero $y$ intercept and positive slope, positive $y$ intercept and negative slope, and positive $y$ intercept and positive slope. The test is administered to 385 students aged 14–15 in the 9th grade from 13 schools in Flanders (Belgium) and—after validation—is analyzed by means of generalized estimating equations (GEE). The results show a significant main effect for all design factors and a significant interaction effect between representational transition and function type, as well as additional interaction effects with the gender of the respondents. Furthermore, representational transitions which include a formula prove to be significantly more difficult; in particular for the directly proportional function type, the transition to a formula stands out from the analysis. Mean accuracies in physics are significantly lower compared to mathematics. Function types with negative values for either $y$ intercept or slope also result in significantly lower mean accuracies indicating the difficulty students have with negative numbers in linear functions. A distractor analysis of the incorrect answers chosen by the students reveals three distinct dominant errors: concept switching, sign switching, and switching of the directly proportional function type and the function type with positive $y$ intercept and positive slope.

To improve the teaching and learning materials for a curriculum it is important to incorporate the findings from educational research. In light of this, we present creative exercises and experiments to elicit, confront and resolve misconceptions in geometrical optics. Since ray diagrams can be both the cause and the solution for many misconceptions we focus strongly on improving understanding of this tool to solve and understand optical phenomena. Through a combination of a conceptual understanding programme (CUP) and provocative exercises with ray diagrams we aim to elicit conceptual or cognitive conflict and exploit this to tackle misconceptions and increase students’ conceptual understanding through inquiry. We describe exercises for image formation by a plane mirror, image formation by a convex lens and indirect and direct observation of a real image formed by a convex lens as examples of our approach.

The awareness that many problems in our society are interdisciplinary in nature and require the integration of multiple STEM (Science, Technology, Engineering, Mathematics) concepts to solve them has given rise to a new instructional approach, called “integrated STEM education”. Integrated STEM education aims to remove the barriers from the STEM fields and has the potential to increase students’ interest and motivation for learning, as well as to lead to improved achievement. It is important to assess the effectiveness of educational STEM initiatives in terms of students’ integrating ability, but to date, no such instruments are available. This study provides a definition of “integrating ability” and establishes a framework for understanding its components. Based on this definition and framework, a multiple-choice instrument for testing integrated physics and mathematics in the ninth grade (IPM9) was developed and validated. The definition and framework for integrating ability and the construction guidelines for an integrated test, can be used by researchers to assess students’ ability to integrate STEM subjects.