Math That Can Take You Higher

Two versions of a circular diagram, with a small round object in the center and a pattern of waves wrapping to the right. Upper image is a grid of lines with varying density; lower image renders this density pattern in smooth colors.

An adaptive grid (top) used to compute the supersonic flow around a cylinder (Mach number = 2). The methods taught in this class form the foundations for computational fluid dynamic analyses such as this. (Image courtesy of Prof. David Darmofal. Used with permission.)

By Joe Pickett, OCW Publication Director

Airliners crisscrossing the globe, rockets hurtling into space, satellites orbiting distant planets—it’s where the fantastic meets the familiar—it’s aerospace! It’s where dreams of soaring above the clouds come true. And those dreams are made real by—mathematical models!

The methods underpinning those models are the focus of 16.90 Computational Methods in Aerospace Engineering, a course just published on OCW.

The OCW site showcases the materials for this course as it was taught in Spring 2014 by Professors Karen Willcox and Qiqi Wang. To improve student learning, the instructors used the Residential MITx platform to flip their classroom, requiring students to work through the assigned readings and problem sets before coming to class, so that class time could be devoted to problem-solving, small group exercises, and project work.

Learning Units, Measurable Outcomes, and Content Types

The course site is arranged in three main learning units, each with readings and simple, interactive assessments that allow students to test their understanding. The reading materials and assessments in each unit are linked to measurable outcomes, the skills that students are expected to demonstrate to pass the course. Thus, students can easily identify exactly what they are supposed to get out of a given session in the course, and they can immediately find the resources necessary to master that very topic.

The site also features many lecture videos, lecture notes, and homework assignments—these resources are assembled together by content type (rather than linearly) for user convenience. As the course was running, the lecture videos were broadcast live over WebEx so as to allow students to engage in remote activities, such as presenting at conferences, without falling behind.

Three programming projects were required in Spring 2014, and descriptions of these projects, along with sample student projects, are also included on the site.

What Happens When You Flip

Professor Willcox discusses the challenges of teaching a flipped classroom in her Instructor Insights on her This Course at MIT page, and the challenges were considerable. Creating the online materials, such as the class notes and the assessments, took a lot of thought and effort.  When you’re used to standing at the front of a lecture hall and taking the students where you want to go, it can be unsettling to find yourself winging it. It takes some getting used to:

“I was used to going into a classroom and delivering a great blackboard lecture on a particular topic. I learned it can be overwhelming to walk into a classroom and not know how exactly how the session will go because it will depend on how well students grasped the material they read on their own. This took some getting used to. It’s harder to plan for these types of sessions. You have to be willing to be flexible, and you need to be prepared to facilitate the session in several different ways depending on students’ learning needs.”

Professor Willcox also discusses what led her to create the measurable outcomes, why she prefers giving oral exams, how the programming projects get students to apply their skills to real problems, and the advantages of co-teaching the course.

Connections that Give Students More Control

A long-standing champion of OCW, Professor Willcox is one of MIT’s leading educational innovators. She is a main force behind MIT Crosslinks, a project that links topics in the MIT curriculum to online educational resources (including but not limited to OCW). Among her other educational initiatives is the MIT Undergraduate Curriculum Map, which shows the relations between subjects in the MIT undergraduate curriculum as well as which of these subjects have been published on OCW.

And (as if all this weren’t all) Professor Willcox is piloting Fly-by-Wire, a blended learning technology to help at-risk students in secondary schools stick with their studies and apply themselves, so they can graduate and take off.

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