Linear Algebra

One example of course design for active learning can be seen in Colorado School of Mines’ Linear Algebra course.  The two instructors of this course collaborated with a colleague at the University of Denver to develop a partially flipped[1] course in order to create more time for active learning opportunities when students are face to face. The case study below describes some of their strategies for incorporating active learning in this partially flipped class, and the educational research that supports these practices.

Mines faculty Debra Carney and Rebecca Swanson published a paper about their work.  You can learn more about their efforts by reading the paper here.



In the flipped classroom model adopted here in Linear Algebra, students are presented with new material prior to coming to class by viewing a video.  For each course topic, the lesson sequence follows the following tasks:​

  1. a ten minute informational video to be watched before class
  2. a 15-20 minute in-class group activity based upon the video content
  3. a more traditional lecture / discussion built upon the activity (approximately 30 minutes).

By introducing new content to students before class, students are able to access the information at their own pace.  Students who might be familiar with the material can move quickly through the video, while students who need extra time can pause and replay the video, take notes, explore extra resources taking the time they need to get the gist of the information.



It’s important to explain to students the rationale for your course design, and to let students know that you’ll be expecting them to actively engage in their learning, and why.  Richard Felder, a highly regarded expert in Engineering Education, says, “When you use a proven teaching method that makes students uncomfortable, it’s important to let them know why you’re doing it.  If you can convince them that it’s not for your own selfish or lazy purposes but to try to improve their learning and grades, they tend to ramp down their resistance long enough to see the benefits for themselves” (Felder, 2007, p. 183).

Here is a video that illustrates Rebecca Swanson providing the rationale for the flipped class with active in-class learning to her Linear Algebra class at the beginning of the semester –



Once students have reviewed the new content on their own, they come to class and immediately begin working on an activity related to the video.  Click here to see this in action –



When students are engaging in active learning, the instructor’s role changes away from that of a lecturer.  It is important the instructors clarify their new role to the students.  They let students know that their job is to help and support them as they struggle to construct new understandings in the class.  Some metaphors that people use to think of this new role are that of coach, guide-on-the-side, facilitator, or manager of learning experiences (Barkley, 2010, p. 95). The way this looks in class is you walk around the room to listen in and observe student progress. Based on what you see and hear, you ask questions to advance their thinking or to get a better sense of where they are – assessing their understandings. You also provide just-in-time feedback to learners as they struggle.  Watch a clip here of Deb Carney doing this in her Linear Algebra class –

Related to walking around and providing learners with just-in-time support, questions, and redirections, something else good educators do is NAME what students are doing, and model the use of the discipline-specific language and vocabulary, to help reinforce the concepts under study.  You can see professor Carney do this here, as she helps clarify the difference between Eigenvalues and Eigenvectors –



There are two ways that the instructors of Linear Algebra collect information daily about student understanding to formatively assess their learning: through video review questions and group activities in class.  In both cases, simple tasks are assigned to students to determine if they understand the content under study.  After watching ten-minute videos before class, students answer a couple questions to help the faculty assess the students’ general understandings of the new concepts.  The professors review student responses as a whole class to look for patterns that indicate areas the instructors need to help students reconsider or address problematic areas during the class meeting.

Once in class, students are provided an activity to work on as a group.  They turn in to the professor the work they create as a group. Feedback is provided to the entire group to correct any mistakes or misunderstandings displayed on the activity.  The work is scored on a 0-1-2 scale for effort, not correctness.  But more importantly, the students receive feedback from the professor as they learn the material.



Sometimes when we’re teaching content-specific, discrete problem solving, the way Deb Carney and Rebecca Swanson are here in their lesson on Eigenvalues and Eigenvectors, it can be easy to forget to explain to learners how today’s lesson fits in to the overall goals in this class, or how this learning matters in the real world, for the rest of their lives.  Rebecca Swanson does a nice job of explaining how the current day’s problem solving challenges with Eigenvectors and Eigenvalues matter for the rest of the course, and how they are helpful in real life.  See the clip here –

There is so much research about the value and usefulness of helping to explain to students how the day’s lesson connects to the overall unit of study, the goals of the course, and students’ current or future lives outside of school.  Elizabeth Barkley calls this “teaching things worth learning (Barkley, 2010, p.86).”  Professor Swanson explains clearly how the current day’s work connects to other courses at Mines as well as their future careers.  Students are more motivated to engage in the material if they understand its relevance in the world.  In How Learning Works, authors Ambrose, Bridges et. al. explain that showing “relevance to students’ current academic lives and future professional lives (2010, p. 84)” are important strategies professors can undertake to motivate students to learn.



There are many features of Colorado School of Mines’ Linear Algebra course that enable active learning to take place.  First of all, the instructors have created a partially flipped course to present new learning in the form of a video in advance of every class meeting.  In addition, the professors have shifted the way time is spent during class meetings to engage students in activities that have the students wrestling with the content of the lesson, rather than passively listening to the professor, or watching the professor solve problems on the board.  During these activities, the instructors walk around the class and listen in to what students are doing, and provide just-in-time teaching and support to advance learning.  In addition, faculty shares with students the rationale for the course design, to help empower them as active learners. As a result, the professors have lowered the D-F-W rate and increased successful end-of-course grades, as well as the professor’s end-of-course evaluations, which have improved too.​

[1] “A flipped classroom is one in which students’ initial contact with new material occurs outside of class and time in class is spent working on challenging problems, usually through coordinated activities.  In the project described here, faculty members opted for a partially flipped model which incorporated the flipped classroom design for some but not all of the content” (Carney, Ormes, & Swanson, 2015, p. 641)​


Ambrose, S. A. (2010). How learning works: Seven research-based principles for smart teaching. San Francisco, CA: Jossey-Bass.

Barkley, E. F. (2010). Student engagement techniques: A handbook for college faculty. San Francisco, CA: Jossey-Bass.

Carney, D., Ormes, N., & Swanson, R. (2015). Partially Flipped Linear Algebra: A Team–Based Approach. PRIMUS, 25(8), 641-654.

Felder, R. M. (2007). Sermons for grumpy campers. Chemical Engineering Education, 41(3), 183-184.