Goals for teaching numerical integration in Calculus or Differential Equations Courses
Alanna HoyerLeitzel, Mathematics and Statistics, Mount Holyoke CollegeThere's a joke about mathematicians where the punch line involves a mathematician in a room with a burning fire, a sink and a bucket. The mathematician doesn't put out the fire, but simple points to the objects in the room and states that "A solution exists." I love this joke because it points out the practicality of proving that a solution exists before solving for it, but as an applied mathematician, I know that actually finding the solution (or a good approximation of the solution) is the part of the problem that can actually be applied to a situation. Often the only way to find a solution is computationally.
I've seen (and I've done it myself) where rather than teaching computation, students are given a computational tool like a calculator or a builtin solver for a software program, and they use it without understanding how it works. This leads to a dependency on the computer or calculator, and a blind assumption that the answer given by the computer or calculator is always correct. I've started actively working to counteract this, and to show students what goes on behind the scenes in the tools that they use.
I've mostly taught the ideas of computation in the context of numerical integration, either in a calculus course or in differential equations. In either class, the goals are for students to

Understand the algorithm by writing their own code and then to test their results with example that can be solved analytically.

Verify that the algorithm gives a good approximation of the analytical solution. De pending on the sophistication of the students, the class would learn about error bounds for the algorithm or conceptually discuss how to improve results (smaller stepsize, dif ferent algorithms).

Implement the algorithm to solve a problem that could not be solved analytically
 Discuss what a reasonable error tolerance might be for a particular application.
Following this, I will show students the builtin tools for numerical integration or differ ential equation solvers in the software program we are using. This gives them a dependable and faster way to find a numerical solution (in most cases) so that we can further our discus sion to other topics in the course. But the result of starting from the barebones algorithm is that students have respect for the tool, and a curiosity about why it works so well. Given the opportunity to do an expository presentation later in the semester, students are excited to talk more about other techniques for numerical integration.
Despite having taught this a few times, I'm still exploring how to improve the conversation and activities I have with my students. I am curious about ways to assess student learning of computation. Is it only possible with a big project or are there shorter assignments where this assessment is possible? I'm also unsure how to address other computational issues with my students, for example finding problems within differential equations where there may not be a unique solution, and where it is necessary to find an optimal solution.