Reform approaches to teaching calculus (e.g., Hughes-Hallett et al., 2011) deliberately present students with multiple representations of the same function, such as a formula, a data table, a text passage, and a line graph. Despite this emphasis on “the ‘Rule of Four,’ [in which] ideas are presented graphically, numerically, symbolically and verbally” (p. ix), students are not explicitly prepared to map between these different canonical representations. We propose to conduct an exploration project in which we will gather eye tracking and other individual difference data from high school seniors in Calculus (Grant Years 1-3) and compare it to data from three groups: Calculus-eligible—seniors who completed pre-calculus but did not enroll in calculus (GY 1), Calculus-ineligible—seniors taking a non-remedial math course (not calculus) such as pre-calculus or trigonometry (GY 1-3), and one small group of undergraduate engineer-ing students (experts) in GY3. The results from these comparisons can be used to inform the preliminary development of aids to coordinating multiple representations (CMR) in Year 3. The end goal is to better prepare students for calculus and to enhance their learning during calculus.

 

Why calculus? Calculus is a critical “gateway” course for STEM learning in the undergraduate years, and better preparation at the high school level is associated with better achievement at the undergraduate level and less dropout from STEM majors (Adelman, 2006). In 2009, 17% of American students had a calculus course on their transcript (Nord et al., 2011). Another reason for studying calculus might be the wealth of research on how background knowledge influences later learning, which has led to an emphasis on mathematics learning in the early and middle grades. While this work is clearly important for helping students to gain the prerequisite knowledge to qualify them for advanced courses, background knowledge in mathematics may be a necessary but not sufficient prerequisite for success in calculus classes.

 

Recent research in STEM education has identified some potential individual difference variables that can explain why some students do not succeed and persist whereas others do. Prominent among these individual differences are spatial abilities, background knowledge, cognitive capacities such as working memory, and various motivational variables. One potential cognitive prerequisite that seems particularly crucial for calculus is the ability to coordinate multiple representations, (CMR; e.g., coordinating a data table and a graph of a cubic function). This skill is crucial because curriculum, assessments, and real-world problems are all posed using a variety of (combinations of) representations; teachers may lack both skills and time to teach CMR; and the complexity of content and tasks—and therefore representations—increases over the grades. A limited body of research has examined CMR for pre-calculus and calculus students using think-aloud (e.g., Carlson et. al, 2002) and eye tracking (Andra et al., 2009) methodology. If the skill of CMR is associated with success in calculus after accounting for prior math ability, we will be able to identify calculus students who could be more successful in the course with some training on coordinating multiple representations.

 

In GY 1-2, we have collected data on eye tracking while students solve multiple representations problems, assess verbal and visuospatial working memory, and prior math achievement and grades for students in each of the three groups towards the beginning of the school year. Now that we have a sufficient understanding of the interrelationships among these variables, we are prepared to make predictions about the types and features of representations and tasks that are likely to cause difficulty for low-CMR students and low-knowledge students. In GY 3, we will compare eye gaze patterns between different types of textbook layouts and uses of conventions, and will compare effects of these different layouts/conventions among the calculus and low-prepared groups from beginning to end of school year. Results will be informative about whether changing instructional materials might be sufficient to improve calculus learning.