Numerical Methods Guy

April 05, 2017

In an article titled The Hidden Costs of Active Learning by Thomas Mennalla, an associate professor of biology at Bay Path University, mentions that “Flipped and active learning truly are a better way for students to learn, but they also may be a fast track to instructor burnout.”  He starts his article – “I am an active learning college instructor and I’m tired. I don’t mean end-of-the-semester and need-some-sleep tired. I mean really, weary, bone-deep tired.”

I applaud the effort by Professor Manella as he is using effective evidence-based pedagogy, and I can relate to the burn-out when I first used the flipped classroom. However, I am personally concerned that flipped learning and active learning are considered to be somewhat synonymous.  We do not have to teach a class flipped to incorporate active learning.  Just small exercises at 15-minute intervals in a 50-minute class, like asking two to four clicker questions, a 25-word muddiest point essay, or something akin,  have been found to be effective.

I implore all my higher-ed colleagues and administrators not to push flipped classes just because that is the only way you know or have heard of incorporating active learning. Before embarking on a flipped class, try a blended class first with all the resources you would have developed for a flipped class; couple it with effective learning strategies like distributed practice (algorithmic problems and question banks presented through learning management systems will save you time to grade and curb cheating), giving practice tests, and assigning interleaved practice.

If someone has convinced you that flipped class (learning) is the only answer, ask them, “Have you compared it with a blended class where the active learning items done in class are of low cost – both for class time and professor prep time?”  Comparing the flipped class with the traditional lecture class should be seriously questioned as a research question.  The same meta-study which is quoted by many advocates of active learning and unrelatedly the “death of the lecture” tend to conveniently ignore the last sentence in its abstract – “The results raise questions about the continued use of traditional lecturing as a control in research studies, …..”

I teach my classes flipped as well as blended.  In the blended class, I have several in-class active learning exercises (clicker questions, short exercises, think-pair-share, etc), and therefore a reasonable amount of content has to be pushed out-of-class and online. Are they hence not getting the benefit of self-regulated and deep learning by doing that? Are they not using the group space during active learning exercises for “dynamic, interactive and engaging activities”?

Full Disclosure: I see the benefits of flipped learning but we do not have to flip every topic. Flipping does not have to create a burn-out either.  Even good medicine has to be taken in prescribed doses.  Currently, I am a lead PI on two NSF grants – one on comparing flipped and blended classes and another on using adaptive learning to improve the pre-class experience of flipped classes.  Let the data do the talking.

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Tampa, FL (December 20, 2016) – Ever since Autar Kaw started teaching at the University of South Florida in 1987, he has used evidence-based pedagogies such as interactive instructional software and active learning in the classroom.  Since 2001, he has been conducting federally funded research in transforming undergraduate engineering education through development and assessment of open resources for undergraduate courses such as Numerical Methods.

With his success of using blended learning for more than a decade and having been the lead developer of a comprehensive open courseware for Numerical Methods, the National Science Foundation (NSF) granted a four-institution (University of South Florida, Arizona State University, Alabama A&M, and University of Pittsburgh) grant in 2013 to determine the effectiveness of flipped learning classroom.  As per the Flipped Learning Network, “Flipped Learning is a pedagogical approach in which direct instruction moves from the group learning space to the individual learning space, and the resulting group space is transformed into a dynamic, interactive learning environment where the educator guides students as they apply concepts and engage creatively in the subject matter.”  The research from the grant indicated positive improvement in student learning gains through flipped learning but challenges persisted in some students coming inadequately prepared for the classroom meeting.  Although before class they were required to study through their preference of video lectures or textbook, and take an automatically-graded online quiz, it still was a single recipe prescribed for all.

To address this concern of pre-class preparation, NSF has given Professor Kaw and his colleague Professor Mary Besterfield-Sacre of University of Pittsburgh an Engaged Student Learning Exploration and Design grant that will use the adaptive platform of Knewton, Inc to personalize the pre-class activities for every learner.  Knewton’s mission is to support and challenge every student to meet their learning goals through personalized learning.

Traditional vs Flipped Classroom (Photo Courtesy of University of Washington – Center for Teaching and Learning and Office of the Provost)

To improve the pre-class activities of the flipped class, the instructional and assessment modules from prior NSF support are being augmented and revised to conform to the Knewton adaptive platform. Specifically, the project will be comparing the effectiveness of three instructional approaches: a flipped class with adaptive learning, a flipped class without adaptive learning, and a blended class without adaptive learning, based on student conceptual gains, procedural knowledge, higher-order problem solving, and affective learning. These comparisons will be conducted statistically at a granular level using the factors of GPA, gender, race, age, transfer status, socioeconomic status, working hours, course topic, proficiency levels, and time on task, as well as qualitatively via focus groups and interviews.  This work will hence advance the understanding of the impact of the combined flipped and adaptive approaches on cognitive and affective learning gains of students representing diverse populations.

This project will specifically provide materials and best practices for teaching Numerical Methods in a flipped setting on an adaptive platform and provide new viable opportunities to how STEM courses are taught. This effort will also take us closer to meeting the NAE’s 21st Century Grand Challenge of “Advance Personalized Learning”.  Dissemination avenues include the freely-available Knewton adaptive platform, an open education resource portal from prior support, and social media.

For more information about the project, send an email to kaw@usf.edu.

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This post is brought to you by

• Holistic Numerical Methods Open Course Ware:
  • Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;
  • Introduction to Matrix Algebra for the STEM undergraduate at http://ma.MathForCollege.com
• the textbooks on
  • Numerical Methods with Applications
  •  Introduction to Programming Concepts Using MATLAB
• the Massive Open Online Course (MOOCs) available at
  • Numerical Methods MOOC
  • Numerical Methods MOOC
  • Introduction to Matrix Algebra

In the previous blogs (Part 1, Part 2, Part 3, Part 4), we clarified the difference and similarities between diagonally dominant matrices, weakly diagonal dominant matrices, strongly diagonally dominant matrices, and irreducibly diagonally dominant matrices.  In this blog, we enumerate what implications these classifications have.

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If a square matrix is strictly diagonally dominant

• then the matrix is non-singular [1].
• then if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite [1].
• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge [2].
• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge [2].
• then if the diagonal entries of the matrix are positive, the real parts of the matrix eigenvalues are positive [1].
• then if the diagonal entries of the matrix are negative, the real parts of the matrix eigenvalues are negative [1].
• then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [2].
• then if the matrix is column dominant, no pivoting is needed for LU factorization [2].

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If a square matrix is irreducible diagonally dominant

1. then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge.
2. then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge.
3. the matrix is non-singular [2].

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If a square matrix is diagonally dominant (also called weakly diagonally dominant)

1. then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [3].
2. then if the matrix is column dominant, no pivoting is needed for LU factorization [3].

References

1. Briggs, Keith. “Diagonally Dominant Matrix.” FromMathWorld–A Wolfram Web Resource, created by Eric W. Weisstein.  http://mathworld.wolfram.com/DiagonallyDominantMatrix.html

2. Diagonally Dominant Matrix, see https://en.wikipedia.org/wiki/Diagonally_dominant_matrix, Last accessed on November 4, 2016.

3. “Lecture 4: A Gaussian Elimination Example”, see http://www.cs.yale.edu/homes/spielman/BAP/lect4.pdf, last accessed on November 4, 2016.

You can view the above document as a pdf file as well.

Other blogs on diagonally dominant matrices
Clearing up the confusion about diagonally dominant matrices – Part 1

Clearing up the confusion about diagonally dominant matrices – Part 2

Clearing up the confusion about diagonally dominant matrices – Part 3

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This post is brought to you by

• Holistic Numerical Methods Open Course Ware:
  • Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com;
  • Introduction to Matrix Algebra for the STEM undergraduate athttp://ma.MathForCollege.com
• the textbooks on
  • Numerical Methods with Applications
  •  Introduction to Programming Concepts Using MATLAB
• the Massive Open Online Course (MOOCs) available at
  • Numerical Methods MOOC
  • Numerical Methods MOOC
  • Introduction to Matrix Algebra