Supporting Transition to Proof Courses with Calculus Context
By Audrey Malagon
Moving from the calculus sequence to abstract mathematics can be challenging for students. The ASPIRE team seeks to build on students’ previous experiences in calculus with intentionally designed, inquiry-oriented materials that support students as they transition to upper level mathematics. In this post, Dr. Tenchita Alzaga Elizondo, Research Associate at Portland State University, tells us about the materials they have developed and how this large project team works together to improve student learning.
1) What inspired you to start this project and apply for NSF funding?
The proposal was a natural next step following several years of research focused on developing an approach to teaching real analysis at Portland State University that intentionally builds on students’ extensive prior experiences in calculus. We wrote the NSF proposal to turn our work into a shareable set of instructional materials complete with instructor support resources.
2) Describe some of the materials you have developed for these transition courses.
We have developed inquiry-oriented course materials for transition to proof courses in the context of real analysis and group theory. Instructors can choose between task sequences that differ in length, context (real analysis and/or group theory), and emphasized math practices (proof comprehension and/or mathematical language). Students get templates to develop a custom Wiki-text for sharable permanent records of the mathematical progress of the classroom community.
By engaging with these materials, students reinvent some foundational concepts of upper level mathematics while also engaging in authentic mathematical practices like defining and conjecturing. In one set of materials, students develop and explore a root approximation method which leads to identifying several sequences and conjecturing about their properties. Motivation to prove these conjectures leads to defining unbounded above and sequence convergence that support eventual proof of theorems like the Monotone Convergence Theorem.
3) What were your goals for supporting students in transitioning to more advanced mathematics with these materials?
The transition from calculus to more advanced proof-based courses can be jarring for students. They are introduced to formal language, logic, and ways of conceptualizing mathematics that are very different from their previous experiences. Our materials aim to ease this transition by presenting students with approachable tasks that use knowledge developed during the calculus sequence. Our approach to supporting students’ learning around formal mathematics and proof is steeped in the calculus context rather than being presented context free through direct instruction in logic.
4) Other NSF projects have highlighted the importance of professional development and support for instructors when implementing new methods. How has your project incorporated instructor support for these courses as well?
A primary way we provide support is through our instructor support website which has additional information about each task regarding the rationale for the task, common student thinking, and implementation suggestions. We include samples of student work and class videos to provide instructors with illustrative examples.
We have also designed and implemented several professional development workshops for instructors in consultation with the Teachers Development Group. These aim to support instructors in using student thinking and developing inclusive teaching practices. We are now working on developing asynchronous professional development resources and a series of tutorial videos to help instructors make the most of these technology rich support materials.
5) How have you adapted or redesigned materials based on what you learned? Was anything surprising to you as you tried out the materials?
Our materials are developed using design research methodology which involves iterative cycles where we implement our task sequence, analyze the implementation, and make changes according to what we have learned. As such, we have been continuously adapting and redesigning our materials throughout the project. Often a redesign is the result of the project team identifying better ways to support students through a complicated topic, finding that tasks didn’t result in the desired mathematical activity, or noting points where instructors could use additional support.
Due to the pandemic we had to learn how to adapt our materials to work in a new, remote environment. We were surprised by some of the great benefits that technology had on students’ collaborative work. Because of this, we have begun to explore the impact of incorporating technology in face-to-face courses for both instructors and students.
6) Your grant involves several institutions. How do you manage a big project across institutions? What have you learned about effective collaboration?
One thing we have found to be particularly effective has been the regular “newsletters” that our formative evaluator creates to provide regular feedback to the team about the project’s progress. The project team is kept informed of all the different aspects of the project and can initiate productive discussions between team members.
7) What impact do you hope the project will have on mathematics education, equitable outcomes in student learning, or other goals you've set for the project?
We hope that our project will support the reform of transition-to-proof courses to include more inquiry-based approaches that support students’ understanding as well as introduce them to authentic mathematical practices. Further, we hope that our project encourages instructors to incorporate inquiry approaches in other courses, subsequently expanding the impact of our project.
You can learn more and access materials at https://aspireinmath.org/Main_Page
Learn more about NSF DUE (#) 1916490
Full Project Name: Advancing Students' Proof Practices in Mathematics through Inquiry, Reinvention, and Engagement
Abstract Link: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1916490&HistoricalAwards=false
Project Contact: Tenchita Alzaga Elizondo <halzaga@pdx.edu>