Raising Calculus to the Surface

Project No.
PI Name
Aaron Wangberg
Winona State University


Abstract 1

Raising Calculus to the Surface

Presentation Type
Elizabeth Gire, Oregon State University Jason Samuels, City University of New York - BMCC Brian Fisher, Lubbock Christian University


The mathematics of calculus utilized by STEM majors can often be discovered as geometric relationships between different quantities, but students often see the mathematics as symbol pushing and memorization of procedures. This project allows students to discover the geometry behind the mathematical concepts and extend their knowledge from first semester calculus to the multivariable setting.


The project developed contextualized student-centered active engagement activities for use with dry-erasable surface manipulatives and contour mats representing abstract functions. Students work together to measure quantities and discover relationships between various mathematical concepts as they answer meaningful questions. Their discoveries uncover geometric relationships which hold in all coordinate systems.


Professional development workshops train instructors to utilize the materials in their class in a mode that encourages student discovery prior to lecture. Through utilizing the materials, students measure meaningful quantities and interpret them as they move between various representations of functions. The project was implemented with over 30 instructors representing high school, two-year colleges, and private and public four year colleges and universities.


The project materials increase student engagement in the classroom, help students visualize calculus concepts and organize and coordinate new mathematical topics. Instructors implementing the materials notice a difference in student engagement in the classroom and place greater emphasis on more varied representations of function.

Broader Impacts

This project will directly impact STEM disciplines which utilize calculus concepts with multivariable data. The investigation of mathematical concepts in context helps students make sense of their computations, a skill which helps students transfer their mathematics to other disciplines. Institutions keep the project manipulatives allowing future instructors to incorporate the materials into their courses. The project website contains an online version of the instructors guide and electronic versions of the activities which are modifiable in order to encourage instructor adoption of the materials for their particular course.

Unexpected Challenges

There was a lot of unexpected paperwork related to participants involved in the project. This required more time by the PI.


Aaron Wangberg and Ben Johnson. Discovering Calculus on the Surface. PRIMUS Vol. 23, Iss. 7, 2013. (https://www.tandfonline.com/doi/abs/10.1080/10511970.2013.775202#.UkkHWob_nL9)

Aaron Wangberg. 'Raising Students' Calculus Understandings to the Surface in Multivariable Calculus.' Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education, 2012, Portland, OR, p589-594. (https://sigmaa.maa.org/rume/crume2012/RUME_Home/RUME_Conference_Papers_files/RUME_XV_Conference_Papers.pdf)

Accepted for Proceedings:
Aaron Wangberg, Brian Fisher, Elizabeth Gire, Jason Samuels. モA case for whole class discussions: Two case studies of the interaction between instructor role and instructor experience with a research-informed curriculum.ヤ Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education, 2016, Pittsburgh, PA.

Brian Fisher, Aaron Wangberg, Jason Samuels. モStudent conceptions of definite integration and accumulation functions.ヤ Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education, 2016, Pittsburgh, PA.

Submitted manuscripts:
Aaron Wangberg, Brian Fisher, Jason Samuels, and Eric Weber. The geometry of directional derivatives. Submitted to College Mathematics Journal. Submitted June 2015.

Aaron Wangberg, Brian Fisher, Elizabeth Gire, Jason Samuels. A case study on the impact of investigating multivariable calculus concepts through geometry and multiple representations. Submitted to the 13th Annual International Congress on Mathematics Education.

Project Page