Design and validation of a test for representational fluency of 9th grade students in physics and mathematics: The case of linear functions

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.

Figure 1

Graphical representation of the six possible answer options. Full lines are the function types under study, dashed lines are buffer distractors. Abbreviations denote the sign ($N$ for negative, $Z$ for zero, and $P$ for positive) of the $y$ intercept and the slope, respectively.

Figure 2

Example questions from the test. Question P1 is a physics question in which the TG transition from table to graph has to be made. Question M21 is in the context of mathematics and requires a GF transition from graph to formula. Question P9 and M9 are isomorphic questions concerning the formula to table (FT) transition in the context of physics and mathematics, respectively.

Figure 3

Data from the 2-way interaction effect of representational transition (TG, GT, FT, TF, FG and GF) and function type (NP, ZP, PP and PN). The data points connected by lines represent the pairs of transitions which are each other’s opposite. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. There are two significant asymmetries for ZP function type. The first between FT and TF transition with $p<0.01$, and the second between FG and GF transitions with $p<0.05$.

Figure 4

Data from the 3-way interaction effect of representational transition (TG, GT, FT, TF, FG and GF), function type (NP, ZP, PP, and PN) and context for the context of physics. The data points connected by lines represent the pairs of transitions which are each other’s opposite. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. There is a significant ($p<0.05$) asymmetry for the ZP function type between FT and TF transitions in the physics context.

Figure 5

Data from the 3-way interaction effect of representational transition (TG, GT, FT, TF, FG, and GF), function type (NP, ZP, PP, and PN) and context for the context of mathematics. The data points connected by lines represent the pairs of transitions which are each other’s opposite. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. In the mathematics context there are three significant asymmetries: for the ZP function type there is one between FT and TF transitions with $p<0.001$ and one between FG and GF transitions with $p<0.001$; for NP there is a significant asymmetry between FT and TF with $p<0.05$.

Figure 6

Data from the 2-way interaction effect of function type and gender. There are no significant differences between female and male for any function type.

Figure 7

Data from the 3-way interaction effect of function type, context, and gender. There are no significant differences between physics and mathematics for any gender and function type combination.

Figure 8

Data from the 4-way interaction effect of representational transition, function type, context and gender. (a) Mathematics*Female; (b) Mathematics*Male; (c) Physics*Female; (d) Physics*Male. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. The only significant asymmetry is for the ZP function type between FG and GF transitions in the mathematics context with males.

Figure 9

Comparison of the data from De Bock, Van Dooren, and Verschaffel [15] (DB) and our data (Test) for the ZP function type for each of the representational transitions.

Figure 10

Comparison of the data from De Bock, Van Dooren, and Verschaffel [15] (DB) and our data (Test) for the PN function type for each of the representational transitions.

Figure 11

Comparison of the data from De Bock, Van Dooren, and Verschaffel [15] (DB) and our data (Test) for the PP function type for each of the representational transitions.

Figure 12

Hypothetical examples of (a) an all-register synergy; (b) a semiotic chain; (c) a synergy of registers with missing transitions and a central base representation (Formula). A formula can be any algebraic expression, and a diagram could be a free body diagram or a phase diagram, for example.