Active Learning Resources for
Mathematical Methods in Engineering and Physics
by Gary N. Felder and Kenny M. Felder
"The only way a skill is developed—skiing, cooking, writing, critical thinking, or solving thermodynamics problems—is practice: trying something, seeing how well or poorly it works, reflecting on how to do it differently, then trying it again and seeing if it works better."-Richard Felder
The practice of "active learning" uses student-centered activities in class to encourage them to adopt an engaged and reflective attitude. See www.pnas.org/doi/10.1073/pnas.1319030111 for a general introduction to active learning, and a list of relevant publications.
Research has validated the success of this approach for student understanding and retention, but as an instructor you may have found that supporting students in active learning requires a lot of work. Traditional textbooks and other available resources can be at cross-purposes with active learning and require not just your own creativity but many long hours to rework these materials into an active form. The materials that have been developed to support active learning are mostly at the introductory level.
Thanks to a grant from NSF we have been able to develop a full set of active-learning based "motivating" and "discovery" exercises and computer-based problems for math methods courses for physicists and engineers.
CLICK HERE FOR ACTIVE LEARNING EXERCISES
CLICK HERE FOR COMPUTER PROBLEMS
CLICK HERE FOR USER-CREATED SUPPLEMENTS
These exercises and computer problems are all incorporated in our math methods textbook. Click here for information about the book.
Exercises
An "exercise" is not the same as a problem. The biggest difference is that an exercise is designed to be done before students learn a topic, in order to help prepare them for it; problems are generally assigned after a topic has been discussed in class, to give the students practice and/or deepen their understanding.
We distinguish two different kinds of exercises.
- A "motivating exercise" sets up the physical motivation for a given mathematical technique. When you are done with the "Taylor Series" motivating exercise you don't know how to build a Taylor series, but you know why you might want one.
- A "discovery exercise" steps through a technique. When you are done with the "Taylor Series" discovery exercise you have built your first Taylor series. The assumption is that you have all the prerequisite skills (such as taking a derivative), but have never been taught how to do this before.
All of the individual exercises are listed below, but you can also download the entire set as a zip file.
- Introduction to Ordinary Differential Equations
Motivating Exercise: The Simple Harmonic Oscillator
Discovery Exercise: Overview of Differential Equations
Discovery Exercise: Arbitrary Constants
Discovery Exercise: Slope Fields
Discovery Exercise: Separation of Variables
Discovery Exercise: Guess and Check and Linear Superposition
Discovery Exercise: Coupled Equations
Discovery Exercise: Differential Equations on a Computer - Taylor Series and Series Convergence
Motivating Exercise: Vibrations in a Crystal
Discovery Exercise: Linear Approximations - Complex Numbers
Motivating Exercise: The Underdamped Harmonic Oscillator
Alternate Motivating Exercise: An RLC Circuit - Partial Derivatives
Motivating Exercise: The Wave Equation
Discovery Exercise: Partial Derivatives - Integrals in Two or More Dimensions
Motivating Exercise: Newton's Problem, (or) The Gravitational Field of a Sphere
Discovery Exercise: Sums This exercise doesn't involve integrals, but is a quick warm up for a discussion of interpreting 1D integrals. - Linear Algebra
All I Really Need To Know About Matrices I Learned From the Three-Spring Problem This is different from most motivating exercises. It's an entire section with an explanation and problems designed to set up a problem that you can refer back to throughout a unit on linear algebra. See below for more information about this exercise.
Discovery Exercise: The Easy Matrix Stuff
Discovery Exercise: Multiple Value Problems This exercise doesn't involve matrix multiplication, but is designed as a quick warm up before introducing the rule for multiplying square matrices by column matrices.
Discovery Exercise: Basis Vectors
Discovery Exercise: Matrix Times Matrix
Discovery Exercise: The Identity and Inverse Matrices
Discovery Exercise: Linear Dependence and the Determinant
Discovery Exercise: Eigenvectors and Eigenvalues
Discovery Exercise: Geometric Transformations
Discovery Exercise: Tensors
Discovery Exercise: Linear Programming and the Simplex Method
- Vector Calculus
Motivating Exercise: Flowing Fluids
Discovery Exercise: Scalar and Vector Fields - Fourier Series and Transforms
Motivating Exercise: Discovering Extrasolar Planets
Discovery Exercise: Introduction to Fourier Series - Methods of Solving Ordinary Differential Equations
Motivating Exercise: A Damped, Driven Oscillator
Discovery Exercise: Guess and Check - Partial Differential Equations
Motivating Exercise: The Heat Equation
Discovery Exercise: Overview of Partial Differential Equations - Special Functions and ODE Series Solutions
Motivating Exercise: The Circular Drum This exercise will work best for students who have already learned to solve PDEs using separation of variables.
Discovery Exercise: Solving Differential Equations with Power Series - Calculus with Complex Numbers
Motivating Exercise: Laplace's Equation
Discovery Exercise: Functions of Complex Numbers - Calculus of Variations
Motivating Exercise: Rescuing the Swimmer
Discovery Exercise: Variational Problems and the Euler-Lagrange Equation - Appendix L: Answers to "Check Yourself" in Exercises
Discovery Exercise: Maclaurin Series
Discovery Exercise: Geometric Series
Discovery Exercise: Infinite Series
Discovery Exercise: Imaginary Numbers
Discovery Exercise: The Complex Plane
Discovery Exercise: Euler's Formula I - The Complex Exponential Function
Discovery Exercise: Euler's Formula II - Modeling Oscillations
Discovery Exercise: The Chain Rule
Discovery Exercise: Implicit Differentiation This exercise doesn't involve implicit differentiation, but is a quick warm up for it.
Discovery Exercise: Directional Derivatives
Discovery Exercise: The Gradient
Discovery Exercise: Tangent Plane Approximations
Discovery Exercise: Optimization and the Gradient
Discovery Exercise: Lagrange Multipliers
Discovery Exercise: Cartesian Double Integrals on a Rectangular Region
Discovery Exercise: Cartesian Double Integrals on a Non-Rectangular Region
Discovery Exercise: Double Integrals in Polar Coordinates
Discovery Exercise: Cylindrical and Spherical Coordinates
Discovery Exercise: Line Integrals
Discovery Exercise: Parametrically Expressed Surfaces
Discovery Exercise: Surface Integrals
Discovery Exercise: Potential in One Dimension
Discovery Exercise: From Potential to Gradient
Discovery Exercise: Divergence
Discovery Exercise: Vectors in Curvilinear Coordinates
Discovery Exercise: The Divergence Theorem
Discovery Exercise: Conservative Vector Fields
Discovery Exercise: Different Periods and Finite Domains
Discovery Exercise: Fourier Series with Complex Exponentials
Discovery Exercise: Fourier Transforms
Discovery Exercise: Multivariate Fourier Series This exercise is a series of short computer problems designed to get students to think about products of sines and cosines in two dimensions, in preparation for introducing multivariate Fourier series.
Discovery Exercise: Linear First Order Differential Equations
Discovery Exercise: Exact Differential Equations This exercise doesn't use exact differential equations, but it gives some practice with differentials that is useful as a warm-up.
Discovery Exercise: Linearly Independent Solutions and the Wronskian
Discovery Exercise: Variable Substitution
Discovery Exercise: Reduction of Order
Discovery Exercise: Heaviside and Dirac
Discovery Exercise: Green's Functions
Discovery Exercise: Normal Modes
Discovery Exercise: Separation of Variables - The Basic Method
Discovery Exercise: Separation of Variables - More than Two Variables
Discovery Exercise: Separation of Variables - Polar Coordinates and Bessel Functions
Discovery Exercise: Inhomogeneous Boundary Conditions
Discovery Exercise: The Method of Eigenfunction Expansion
Discovery Exercise: The Method of Fourier Transforms
Discovery Exercise: Legendre Polynomials
Discovery Exercise: The Method of Frobenius
Discovery Exercise: Bessel Functions
Discovery Exercise: Sturm-Liouville Theory and Series Expansions
Discovery Exercise: The Quantum Harmonic Oscillator and Ladder Operators
Discovery Exercise: Derivatives and Analytic Functions
Discovery Exercise: Mapping Curves and Regions
At home or in class? Alone or in groups?
Mix it up. See what works for you. We sometimes assign them as homework due on the day we are going to cover the material, and sometimes as an in-class exercise to begin the lecture. You can have students do them individually or in groups, or a mix of the two. One professor we spoke to starts them in class, and then has her students finish them at home—an approach we never even thought of. You will probably keep your students' interest better if you vary your approach.
Do I need to assign all the exercises?
No. If you are uncomfortable with the process, you may want to try only one or two. We hope you will find them easy to use and valuable, and over time you will use them more, but you will probably never use them all.
How long do they take?
Some are five minutes or less; some are twenty minutes or even more. Very few of them should take the students more than half an hour.
That was all pretty noncommittal. Do you have any solid advice at all?
Actually, we do. First, we hope you will use at least some of the exercises, because we believe they contribute a valuable part of the learning process. Second, exercises should almost always be used before you introduce a particular topic—not as a follow-up. You can start your lecture by taking questions and finding out where the students got stuck.
This exercise is very different from the rest. It starts with an Explanation (with nothing for the student to do except read it), and then a set of Problems meant to help the students explore the Explanation. A professor would assign a carefully chosen subset of those Problems, not all of them. In all those senses this is like a typical section of our book, rather than being like an exercise.
But the entire section does not teach the first thing about matrices! It sets up a problem and steps through the solution, leaving holes that will be filled in with matrices. In that sense, it is very much like most of our other motivating exercises.
We find this to be a powerful way to introduce linear algebra and tie together many of the most important topics in that field, but it only makes sense to use this if you keep referring back to it throughout the unit. When you teach them how to use matrices to change bases, have them use that to convert between initial positions and amplitudes of normal modes. When you teach eigenvectors and eigenvalues, have them derive the normal modes of the coupled springs as eigenvectors of the matrix of coefficients. The exercise sets this up by pointing out which linear algebra topics will be used for various parts of the solution.
At the end of the unit you can come back and tie it all together with a section where we revisit the three spring problem, using linear algebra in every step of the solution.
Computer Problems
Computers can be used in a number of ways in math methods courses: to illustrate complicated math problems, to apply techniques to problems too complicated to solve by hand, or to skip tedious algebra steps and focus on the math you're trying to teach. In addition, computer skills may be one of the things you want to teach in your class.
The problems below are all platform independent. The instructions simply tell the students what to do, but the details of how to do it will of course be different if they are using Mathematica, Matlab, Maple, or some other platform.
The computer problems for all of the topics are listed below, but you can also download the entire set as a zip file.
- Computer Problems for Introduction to Ordinary Differential Equations
- Computer Problems for Taylor Series and Series Convergence
- Computer Problems for Complex Numbers
- Computer Problems for Partial Derivatives
- Computer Problems for Integrals in Two or More Dimensions
- Computer Problems for Linear Algebra
- Computer Problems for Vector Calculus
- Computer Problems for Fourier Series and Transforms
- Computer Problems for Methods of Solving Ordinary Differential Equations
- Computer Problems for Partial Differential Equations
- Computer Problems for Special Functions and ODE Series Solutions
- Computer Problems for Calculus with Complex Numbers
User-Created Supplements
David Slavsky of Loyola University Chicago wrote to us that "I think you have a rare opportunity to use your text to generate a community of learners all contributing to this field." He wrote the following supplement to our book and gave us permission to post it to this site. We appreciate his help, and encourage other users to do the same.
- Mathematica assignments showing the convergence of Fourier series
- Mathematica animations showing combinations of normal modes
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Click here for the home page for the book, http://www.felderbooks.com/mathmethods. |
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If you have questions, we would love to hear from you. Click here to email the authors at GFelder@Smith.edu. |