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IES Grant

Title: Fostering Reliance on Visuospatial Representations to Enhance High School Students' Success in Pre-Calculus Trigonometry
Center: NCER Year: 2015
Principal Investigator: McClelland, James Awardee: Stanford University
Program: Cognition and Student Learning      [Program Details]
Award Period: 4 years (7/1/2015 – 6/30/2019) Award Amount: $1,543,138
Type: Exploration Award Number: R305A150453

Purpose: This project explored whether encouraging high school and community college students to use visuospatial representations and providing them with support to enhance familiarity with these representations contributes to success in trigonometry, a difficult subject that sits at the gateway to entry into university-level science, technology, engineering, and mathematics (STEM) coursework and ultimately into careers in these disciplines. It is common to view mathematics as a formal system of abstract symbols and rules, but an alternate view is that mathematical and scientific reasoning operates on idealized objects humans can manipulate in their minds, similar to the ways humans manipulate real objects. A visuospatial representation may provide the type of grounded conceptual framework necessary to support such manipulations. In this project, the research team sought to identify ways to enhance students' use of visuospatial representations (specifically the unit circle) in solving pre-calculus trigonometry problems, establish whether teaching students using visuospatially-oriented training improves students' ability to master trigonometry relationships, and identify moderators of the relationship between the use of visuospatial representations and performance in trigonometry.

Project Activities: The research team conducted three studies with community college and undergraduate students who had not had prior exposure to material covered in pre-calculus trigonometry, and with high school students who were enrolled in pre-calculus courses. In Study 1, the team explored the role of visuospatial grounding and rules in understanding pre-calculus trigonometry. In Study 2, the team tested whether unit-circle based lessons improved students' understanding of pre-calculus trigonometry. In Study 3, the team explored what pre-requisite knowledge was needed for learning unit-circle trigonometry.

Pre-registration site: The analysis for Study 2 was preregistered on the Open Science Framework at

Key Outcomes: The main findings of the project are:

  1. Trigonometry knowledge and understanding was correlated with reliance on the unit circle.
  2. Lessons that used the unit-circle increased students' understanding of trigonometric identities relative to a no-lesson baseline. Students' prior knowledge significantly predicted performance on the lessons and posttest.

Structured Abstract

Setting: This research took place at suburban universities in California and Wisconsin as well as at a suburban community college and suburban high school in California

Sample: In Study 1, approximately 200 undergraduates participated. In Study 2, 1908 community college students who had not taken pre-calculus or trigonometry at the community-college level completed trigonometry knowledge assessments. Of those, 53 students completed the study, which included lessons and a post-test. Additionally, 48 high school students who were enrolled in a pre-calculus course but had not yet been exposed to trigonometric identities completed the study. In Study 3, 46 undergraduate students participated.

Intervention: This was an exploration project, so the goal was to identify malleable factors of instruction that improved students' trigonometry learning. This research explored whether grounding instruction in visuospatial thinking would improve student learning. In order to conduct the research, the research team had to develop lessons reflecting visuospatially-oriented instruction and rule-based instruction for teaching trigonometric identities. Additionally, the researchers developed beginner-level triangle trigonometry lessons that encouraged perceptuo-motor engagement with triangles to help build intuition about sine and cosine values.

Research Design and Methods: In Study 1, researchers randomly assigned students to either experience a visuospatially grounded trigonometry lesson or a formal rule-based lesson to a no-lesson control. Researchers collected both pre- and post-test measures. In Study 2, researchers randomly assigned students to experience either pre-calculus trigonometry instruction using a meaningful visuospatial representation (i.e., the unit circle) or a no-lesson baseline condition. Researchers collected both pre- and post-test measures, and students in the visuospatial condition completed a set of six lessons. In Study 3, researchers randomly assigned students to experience a lesson with minimal written instruction and heavy emphasis on perceptuo-motor engagement with enhanced understanding of triangle trigonometry or to a no-lesson baseline condition. Researchers collected pre- and post-test measures.

Control Condition: The comparison conditions varied across studies. Study 1 included a no-lesson control condition as well as a rule-based training condition as a comparison. Studies 2 and 3 included no-lesson control groups.

Key Measures: The primary outcome measure for all three studies was participants' change in scores from pre- to post-test on researcher-developed tests of trigonometric concepts.

Data Analytic Strategy: Researchers analyzed data using logistic regressions with bootstrapped p-values.

Project Website:

Products and Publications

Journal article, monograph, or newsletter

Lampinen, A. K., & McClelland, J. L. (2018). Different presentations of a mathematical concept can support learning in complementary ways. Journal of Educational Psychology, 110(5), 664–682.

Book chapter

Mickey, K. & McClelland, J. L. (2017). The unit circle as a grounded conceptual structure in pre-calculus trigonometry. In D. C. Geary, D. B. Berch, R. Ochsendorf and K. Mann Koepke (Eds.), Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts. Elsevier/Academic Press.

Doctoral dissertation

Mickey, K.W. (2018). Understanding trigonometric relationships by grounding rules in a coherent conceptual structure. Stanford University.

Research brief

A different tangent to teaching trigonometry (2018), Scientia.

Publicly available data

Data and code for Study 1 can be found on the Open Science Framework at Data and code for Study 2 can be found at