L Mahadevan, Lola England de Valpine Professor of Applied Mathematics in SEAS, and Professor of Organismic and Evolutionary Biology, and of Physics in FAS used a 2017-2018 SEAS Learning Incubator LInc Faculty Fellowship to emphasize active learning in his Mathematical Modeling course. He implemented a flipped classroom approach to enable students to come to class with problems and questions to collaborate on, time to develop their own problems from scratch, and work on modeling with peers. The foundational arc supporting this process has students move from observations through abstraction, analysis and communication, and iteration.
Students develop quantitative capabilities for analyzing the world and learn that when they ask questions appropriately, they can answer them quantitatively. “In most courses, we aim to teach tools that students need to answer questions, but we are not so good at teaching them how to think about new questions. The goal in redesigning this course was to move the focus towards enabling students to think from scratch about new problems and be able to determine and apply the right quantitative tools to the problem at hand.”
Mahadevan finds that students are not comfortable making mistakes, but says, “I’m happy to be an idiot in front of my students—then work out with them how, where, and why I went wrong and learn to correct the mistakes.” This approach to modeling demonstrates to students that it is perfectly fine to make mistakes while learning and encourages them to feel more comfortable working collaboratively on problems by being constructively critical.
Takeaways and best practices
Use the board.
As part of their class time, students are divided into teams of three to work through problems on a board. The teaching staff circulates around the room, observing the work and asking questions. “When we ask students to work problems in front of peers, their misconceptions are in plain sight for everyone to be able to critique, but in a constructive way.” This allows students to talk through problems collaboratively to determine how to proceed, while providing a lens for instructors to assess the gaps in both their knowledge and their reasoning.
Using past problems and projects, students share critiques about problem formulation, analysis, and conclusions, as well as how to communicate mathematical ideas. This is done in a jigsaw so students move from one project to the next. When students are working on their own projects, they have a model for self-assessment as well as for offering constructive criticism and asking questions of their classmates.
Demonstrate a way of thinking through analogy.
Mahadevan divides his course into four modules that highlight mathematical principles across fields using deterministic, stochastic, continuous and discrete ways of thinking: equilibrium and homeostasis, flow and transport, diffusion and reaction, and control and design. These modules move from description to prediction and eventually to design. Each conveys to students that, at a deep level, mathematics is the science of uncovering patterns and discovery using analogy. Demonstrating how the same kinds of methods and tools can transfer to physical and biological sciences or technology is powerful to students’ learning.
Providing students with more class time to actively work on problems and learn from mistakes in real-time helps them develop the skills and confidence to think originally about problems and solutions.