14. Appendix: Math for research - sample lesson plans

Teaching the “math for research” material can clearly be sliced up in many different ways. I have taught it several times, and I always followed a similar logistical format: class length of 1 hour and 15 minutes, and new material once/week, with catch-up sessions twice/week.

But the order in which I teach the various pieces, and the choice of which pieces to teach, depends largely on the makeup of the group of students I work with.

Similarly, the amount of extra background material and the choice of “visions into advanced material” anecdotes depend on the group of students I have.

In this appendix I collect some of the specific lesson plan sequences I have chosen. At this time the most complete listing I have collected is from Spring 2026.

14.1. Winter/Spring 2026

We had a total of ?? lessons, from 2025-02-02 to 2025-??-??.

Opening lesson 1:

  • Introduce myself, discuss how researchers see mathematics

  • Dive into visualizing \(\sin(x) \approx 9^{th}\) degree Taylor polynomial

  • Approximating functions with series. From that chapter we do “Sequences and sums”, “Do sums converge?”, “Approximating pi with series”, up to and including A digression on the factorial.

Lesson 2:

Lesson 3:

Lesson 4:

  • Review again how we calculate the series for sin() and cos() (just because repetition is good).

  • Taylor series for \(e^x\)

  • Reflections on \(e^x = \cos(x) + i\sin(x)\)

  • Taylor series for \(\log(1 - x)\)

  • Taylor series for \(\log(1 + x)\)

From Chapter More taylor series

Lesson 6:

  • Microscopic introduction to physics - Newton’s first two laws

  • Microscopic introduction to differential equations

  • The simple harmonic oscillator with a mass on a sprint

From Chapter Taylor series – applications and intuition

Lesson 7:

  • Review the microscopic introductions to physics and differential equations

  • Physics application - the linearized pendulum

This concludes the segment on Taylor Series - discuss what we have just learned.

From Chapter Taylor series – applications and intuition

Lesson 8:

  • Start Fourier Series

Make this a very visual introduction with \({\rm square}(x)\) and the terms \(\frac{4}{\pi}(\sin(x) + \frac{1}{3}\sin(3x) + \dots)\)

Discuss some reflections on these plots, prompted by the reflections in the book chapter.

From Chapter Fourier series: the “bones” of a function