Relation of Spatial Skills to Calculus Proficiency: A Brief Report

Spatial skills have been shown in various longitudinal studies to be related to multiple science, technology, engineering, and math (STEM) achievement and retention. The specific nature of this relation has been probed in only a few domains, and has rarely been investigated for calculus, a critical topic in preparing students for and in STEM majors and careers. We gathered data on paper-and-pencil measures of spatial skills (mental rotation, paper folding, and hidden figures); calculus proficiency (conceptual knowledge and released Advanced Placement [AP] calculus items); coordinating graph, table, and algebraic representations (coordinating multiple representations); and basic graph/table skills. Regression analyses suggest that mental rotation is the best of the spatial predictors for scores on released AP calculus exam questions (β = 0.21), but that spatial skills are not a significant predictor of calculus conceptual knowledge. Proficiency in coordinating multiple representations is also a significant predictor of both released AP calculus questions (β = 0.37) and calculus conceptual knowledge (β = 0.47). The spatial skills tapped by the measure for mental rotation may be similar to those required to engage in mental animation of typical explanations in AP textbooks and in AP class teaching as tested on the AP exam questions. Our measure for calculus conceptual knowledge, by contrast, did not require coordinating representations. Scores on spatial skills measures are predictive of success in multiple science, technology, engineering, and math (STEM) tasks, majors, and careers (Harle & Towns, 2010; Höffler, 2010; Sorby, 2009; Wai, Lubinski, Benbow, & Steiger, 2010). Why are spatial skills related to STEM? For some STEM fields, the connections are obvious—structural geology involves the study of deformations in three dimensions, chemistry involves reasoning about interactions among electrons that take a spatial configuration around atomic nuclei. For many STEM fields, however, the relative importance of different spatial skills, and the relation of spatial skills to different skills in the domain have been underexplored (Hegarty, Crookes, Dara-Abrams, & Shipley, 2010). One such under-researched area of STEM is mathematics, and the key course calculus, in particular. The role of spatial skills in calculus proficiency might be explained by the spatial nature of Cartesian graphs, which are the most frequently used visualization in calculus teaching and textbooks (Chang, Tran, & Cromley, 2016). For example, Bektasli (2006) found a significant relation between spatial skills and graph skills specific to interpreting slope (r = 0.28 for simpler spatial problems on the Purdue Spatial Visualizations Test (PSVT) and 0.40 for two-step problems on the PSVT; the PSVT is a 3D mental rotations measure very similar to the Mental Rotations Test we used (described next). The spatial skills-calculus proficiency relation would also be expected because CONTACT Jennifer G. Cromley [email protected] Department of Educational Psychology, College of Education, University of Illinois Urbana-Champaign, 1310 S. Sixth St., MC-708, Champaign, IL 61820, USA. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/hmtl. © 2017 Taylor & Francis MATHEMATICAL THINKING AND LEARNING 2017, VOL. 19, NO. 1, 55–68 http://dx.doi.org/10.1080/10986065.2017.1258614 central calculus tasks are inherently spatial; for example, visualizing the slope of a tangent to a curve as x values change for the derivative or imagining accumulating the area of “slices under a curve” for a range of x values for integration (see Bremigan, 2005; Sorby, Casey, Veurink, & Dulaney, 2013; Zimmerman, 1991). Coordinating multiple mathematical representations and spatial skills Decades of work by mathematics education researchers have documented that mathematics teachers can foster student understanding through developing students’ proficiency using and coordinating multiple representations (Bell & Janvier, 1981; Brenner et al., 1997; Hiebert & Carpenter, 1992; Kaput, 1991; Leinhardt, Zaslavsky, & Stein, 1990; Roschelle et al., 2010; Yerushalmy, 1991). In fact, mathematical understanding is often defined in terms of fluency connecting representations. For example, the US National Research Council report Adding it Up stated: A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. To find one’s way around the mathematical terrain, it is important to see how the various representations connect with each other, how they are similar, and how they are different. The degree of students’ conceptual understanding is related to the richness and extent of the connections they have made. (National Research Council, 2001, p. 119) In many cases, mathematical problem solving draws heavily on spatial skills (e.g., Ganley & Vasilyeva, 2011). However, there has been little empirical work investigating relations between spatial skills and calculus proficiency. Samuels (2010) administered the “Development” and “Rotations” subscales from the Purdue Spatial Visualization Test and had students solve calculus problems, which were coded for proficiency in two subscales—limits and the derivative. Using these measures of spatial skill, the only significant correlation was between Development and scores on calculus problems involving the derivative, r = 0.53. Constituent spatial skills Although there is not clear agreement about constituent spatial skills, evidence suggests that these skills may be considered as varying on multiple dimensions. One challenge in assessing the effects of spatial skills is therefore the selection of appropriate spatial measure(s). In one typology, two important dimensions are reasoning about spatial relations within an object—intrinsic—versus between two objects—extrinsic—and reasoning about static objects versus moving (or dynamic) objects (Chatterjee, 2008; Newcombe & Shipley, 2015; Uttal et al., 2013). In the present research, we gave measures of static and dynamic reasoning with three different spatial skills measures—the Mental Rotations Test (Peters et al., 1995; an intrinsic and dynamic spatial measure that taps mental rotations), the Hidden Figures Test (Ekstrom, French, Harman, & Derman, 1976; an intrinsic and static spatial measure that taps disembedding), and the Paper Folding Test (Ekstrom et al., 1976; an intrinsic and extrinsic dynamic spatial measure that taps spatial visualization). In addition, spatial skills are malleable (Sorby, 2009; Uttal et al., 2013), which means that finding a relation between spatial skills and calculus proficiency would suggest an avenue for improving calculus proficiency at the high school level. Developing spatial skills could potentially keep talented students in the STEM pipeline and perhaps decrease the need for undergraduate mathematics remediation. In a meta-analysis of spatial training studies, Uttal and colleagues found large effects of training on the three spatial measures we use in the present study: the effect of spatial training on scores on the Hidden Figures measure was Hedges’ g = 0.48, on the Paper Folding measure was g = 0.65, and on the Mental Rotations measure was g = 0.82. Specifically in the domain of calculus, Sorby and colleagues (2013) found that practice reasoning about 3D objects and rotation improved calculus grades in low-spatial engineering students (d = 0.20). 56 J. G. CROMLEY ET AL.