Lessons I wish I had learned before teaching differential equations [pdf] (1997)
Many commenters argue that undergraduate differential equations and calculus are taught as dry “bags of tricks,” with excessive rigor and outdated curricula that obscure intuition, applications, and the historical motivations behind the math. Others push back, saying some level of abstraction and proof is essential for real mathematical understanding, and that the main bottlenecks are student preparation, institutional inertia, and time constraints rather than malice or gatekeeping. Across fields—from physics and engineering to computer science—participants call for better pedagogy, open and iteratively improved textbooks, richer use of numerical and visual tools, and eventually AI-driven, personalized instruction to bridge the gap between formal theory and real-world problem solving.
Infinitesimals and Foundations of Calculus/ODEs
- Discussion of nonstandard analysis as a rigorous way to use infinitesimals; some see it as conceptually attractive and computer-friendly, others note a lack of didactic, step‑by‑step examples.
- One commenter works through the derivative of sin(x) with an infinitesimal “e” and a standard‑part operator, then runs into the issue of justifying approximations like sin(e) ≈ e, cos(e) ≈ 1 − e².
- Tension between classical nonstandard analysis, intuitionistic foundations, and alternative frameworks like synthetic differential geometry and smooth infinitesimal analysis.
- Infinitesimals are also connected to modern tools such as automatic differentiation.
Pedagogy, Gatekeeping, and Textbooks
- Many describe differential equations and physics teaching as ritualized “hazing” or an IQ filter: opaque textbooks and courses that weed out students rather than teach them.
- Others, including practicing academics, deny any deliberate gatekeeping and attribute difficulty to the intrinsic hardness of subjects, poor preparation, and limited time.
- Strong desire for better pedagogy: more conceptual framing (e.g., why DEs matter), better examples, and explicit focus on linear systems with constant coefficients rather than scattered tricks.
- Debate over videos vs text: videos (e.g. popular math channels) give powerful intuition but often don’t suffice for solving textbook‑level problems; many see them as complements to books and problem sets.
- Calls for open‑licensed textbooks that can be iteratively improved, but practical issues are raised: governance, credit, licensing, avoiding low‑quality forks, and sustaining non‑profit use.
Rigor vs Intuition and Applications
- Persistent tension: some argue rigorous proof is central to mathematics and must be taught early; others say most non‑mathematicians need intuition, modeling skill, and numerical methods more than epsilon‑delta proofs.
- Physics and engineering courses are often criticized for hand‑waving math (e.g., differentials, wavefunctions), while math courses are criticized for ignoring physical motivation.
- Several anecdotes: students finally “get” calculus, Fourier analysis, or DEs only when they encounter concrete physical or computational examples (mechanics, circuits, audio, numerical simulation).
Scope and Role of Differential Equations Today
- Many real‑world ODE/PDEs are only solvable numerically; analytic techniques are framed as a limited “bag of tricks” plus theory that underpins numerical methods.
- Some think a tricks‑heavy intro course is acceptable for breadth; others insist courses should focus on a small number of deep ideas students will remember (e.g., exponential behavior, stability, phase planes, Laplace transforms).
- Examples shared: compact code for damped oscillators, and recommendations for DE courses that start with numerical and qualitative viewpoints.