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IES Grant

Title: Coordinating Multiple Representations: A Comparison of Eye Gaze Patterns of High School Students Who Do and Do Not Enroll in Calculus
Center: NCER Year: 2012
Principal Investigator: Cromley, Jennifer Awardee: University of Illinois, Urbana-Champaign
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years (7/1/2012-6/30/2015) Award Amount: $906,433
Type: Exploration Award Number: R305A140700

Previous Award Number: R305A120471
Previous Awardee: Temple University

Co-Principal Investigators: Julie Booth, Darin Kapanjie, and Thomas Shipley

Purpose: Calculus is a critical "gateway" course for science, technology, engineering, and mathematics (STEM) learning in the undergraduate years. Better preparation at the high school level is associated with better achievement at the undergraduate level and increased persistence in STEM majors. Recent research 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, graph/table skills, background knowledge, cognitive capacities such as working memory and visuo-spatial working memory, prior math ability, and various motivational variables. The overall purpose of this project is to explore the ways in which individual differences such as these are associated with students' ability to coordinate multiple representations (CMR) of problems, a potentially malleable component of many calculus classrooms and texts.

Project Activities: During the first two of the three years of the project, researchers will gather data about the eye tracking behavior of students while they solve calculus problems that involve multiple representations. The researchers will also assess verbal and visuo-spatial working memory, prior math achievement, and student grades. In Year 3, researchers will compare eye gaze patterns between different types of textbook layouts and conventions. In addition, researchers will compare how students in different levels of mathematics instruction respond to these different layouts/conventions over the course of the school year.

Products: Information will be gathered about the ways in which individual differences in CMR contribute to mathematics performance and recommendations will be generated about ways to use this information to improve mathematics instruction. Peer reviewed publications will also be produced.

Structured Abstract

Setting: This study will take place in a high-achieving high school in Pennsylvania, where math scores on the state test are at the 84th percentile. Demographics include 87 percent White students, with 5 percent Black, 4 percent Asian, and 3 percent Hispanic students, with an equal number of males and females, and relatively high socio-economic status. The engineering majors who will participate in the study during Year 1 will be drawn from Temple University.

Sample: The primary sample includes high school students from three groups, with approximately 50 high school students per group per year: (1) high school calculus students; (2) calculus-eligible high school seniors who completed pre-calculus but are not taking calculus; and (3) calculus-ineligible high school seniors—students taking a non-remedial math course other than calculus (such as pre-calculus or trigonometry). In Year 1, a secondary sample will include approximately 50 experts in CMR, who are expected to be college junior and senior engineering majors, ages 20–21.

Intervention: No intervention is being developed in this exploratory study.

Research Design and Methods: In many calculus textbooks, students encounter the same function in different formats: presented graphically, numerically, symbolically, and verbally (such as a line graph, a data table, a formula, and a text passage). To solve problems correctly, students are expected to coordinate across these different representations. In a series of studies, researchers will gather eye tracking data to determine what students in different instructional levels (defined by their level of calculus coursework) look at and how they engage in CMR skills with graphical, symbolic, and tabular representations during mathematical problem solving. The research team will explore:

  1. the extent to which high school seniors at different levels of math proficiency coordinate multiple representations—which is a malleable skill,
  2. the relationship between coordinating multiple representations and scores on practice AP calculus questions,
  3. whether experience in a calculus class is related to students' ability to coordinate multiple representations, and
  4. whether minor text manipulations can increase the extent to which high school seniors coordinate multiple representations.

Comparison Conditions: High school seniors in calculus will be compared to students who are calculus-eligible (Year 1), calculus-ineligible (Years 1–3), and experts (Year 1).

Key Measures: Researchers will use the Coordinating Multiple Representations (CMR) subscale of the Pre-Calculus Concept Assessment. Eye gaze patterns will be measured while students work on problems from the CMR subscale. Researchers will use existing assessments to measure individual differences variables, such as graph/table skills, spatial ability, and visuospatial working memory using the Spatial Span Task. Classroom observations will be conducted and researchers will also create and use a measure of students' conceptual understanding of key components of calculus.

Data Analytic Strategy: Researchers will primarily use analysis of variance (ANOVA) to evaluate the data over the course of the project. Analyses include ANOVAs comparing scores from 3 or 4 groups or independent-sample t tests comparing scores for 2 groups. For example, in Year 3, to answer the question, "Do differences in features included in the representations (e.g., arrows, text descriptions, etc.) alter student eye gaze patterns and/or increase the extent to which students coordinate multiple representations?", the researchers will compare Year 3 pre- and posttest scores for students who are enrolled in calculus and pre-calculus assigned to scaffolded (representations including critical features) vs. non-scaffolded (standard representations) conditions using a repeated-measures (mixed) ANOVA.


Journal article, monograph, or newsletter

Chang, B.L., Cromley, J.G., and Tran, N. (2015). Coordinating Multiple Representations in a Reform Calculus Textbook. International Journal of Science and Mathematics Education: 1–23.

Cromley, J.G., Booth, J.L., Wills, T.W., Chang, B.L., Tran, N., Madeja, M., Shipley, T.F., and Zahner, W. (2017). Relation of Spatial Skills to Calculus Proficiency: A Brief Report. Mathematical Thinking and Learning, 19(1): 55–68.

Miller, B.W., Cromley, J.G., and Newcombe, N.S. (2016). Improving Diagrammatic Reasoning in Middle School Science Using Conventions of Diagrams Instruction. Journal of Computer Assisted Learning, 32(4): 374–390.

Zahner, W., Dai, T., Cromley, J.G., Wills, T.W., Booth, J.L., Shipley, T.F., and Stepnowski, W. (2017). Coordinating Multiple rRepresentations of Polynomials: What do Patterns in Students' Solution Strategies Reveal?. Learning and Instruction, 49: 131–141.


Booth, J., Chang, B., Cromley, J., Shipley, T., and Wills, T. (2014). Calculus Expertise and Strategy Use when Comparing Multiple Representations. In Proceedings of the Cognitive Science Society (pp. 1935–1939).

Wills, T., Shipley, T., Chang, B., Cromley, J., and Booth, J. (2014). What Gaze Data Reveal about Coordinating Multiple Mathematical Representations. In Proceedings of the Cognitive Science Society (pp. 3113–3118).