Math's pedagogical curse | Grant Sanderson JPBM Award Lecture, JMM 2023

Episode Overview

• Grant Sanderson of 3Blue1Brown accepts the JPBM Communications Award and discusses his philosophy on making mathematics more accessible and intuitive.
• He introduces the concept of "Math's pedagogical curse," where the pursuit of absolute rigor can unintentionally distract from the goal of clear, intuitive explanation.
• Using the Riemann zeta function as an example, he demonstrates how powerful visualizations can motivate complex concepts like analytic continuation.
• He advocates for an "examples-first" teaching approach, where students grapple with concrete problems before being introduced to a formal abstraction, a method supported by studies on "productive failure."

Key Concepts

• Math's Pedagogical Curse: The idea that the satisfying, clean nature of mathematical rigor can serve as a distraction, leading educators to prioritize formal correctness over the messier but essential work of building pedagogical clarity and intuition.
• Rigor vs. Clarity: A core distinction is made that being rigorous does not inherently mean being unclear. The challenge lies in balancing the pursuit of rigor with the need for accessible explanation.
• A Checklist for Pedagogy: Sanderson proposes creating a set of guiding principles for effective math communication, analogous to the established checklists for rigor, to ensure explanations are intuitive and engaging.
• Visualization for Intuition: Visuals should do more than just illustrate known facts; they should be designed to elicit new insights and intuitions. The animation of the Riemann zeta function is used as a prime example, where the visual itself "screams" for the concept of analytic continuation.
• Examples-First Pedagogy (Class A vs. Class B): A contrast between two teaching styles. "Class B" starts with a formal abstraction and then applies it, which can be disengaging. "Class A" starts with concrete examples, allowing students to explore and build a foundation before the abstraction is introduced as a powerful, unifying concept.
• Productive Failure: A learning principle suggesting that students who are allowed to struggle with and explore problems before being taught the formal method develop a deeper and more robust conceptual understanding.

Quotes

• At 1:25 - "The video that Grant does with that really sort of makes that dynamic and shows you why folding is such a great word for convolution." - Michael Pearson shares a personal anecdote about how Sanderson's visualization of convolution helped him understand the concept more deeply.
• At 5:37 - "But there's another turn of phrase that I've heard... that in some sense is even more worrying than that one, which is the 'I thought I liked math until...'" - Sanderson points out a troubling sentiment where individuals who once enjoyed math were later turned off by it.
• At 11:53 - "What I do think is possibly the case is that this allure of such a notion of exactness... means that rigor serves kind of as a distraction." - Sanderson explains his concept of "Math's pedagogical curse," where the clean nature of rigor can overshadow the work of creating clear explanations.
• At 17:41 - "Is it possible to have something that's analogous for those of us who want to popularize or educate... a checklist for pedagogy?" - Sanderson proposes creating a set of guiding principles for pedagogical clarity, similar to the established standards for mathematical rigor.
• At 28:33 - "The whole diagram is just sort of screaming at you how strange it is that there's an abrupt stop." - He explains that the visual mapping of the zeta function's convergent region naturally motivates the idea of analytic continuation.
• At 31:00 - "Only one extension has a derivative everywhere...that actually locks you into only one choice." - The speaker explains the uniqueness of analytic continuation, which is constrained by the requirement that the extended function must be differentiable.
• At 33:25 - "Do those visuals elicit new intuitions beyond what text can?" - This is a key question from his "Checks for pedagogy" list, emphasizing that visuals should provide deeper insight, not just illustrate.
• At 33:48 - "Are new abstractions preceded with concrete examples?" - The speaker introduces another core tenet of his pedagogical checklist: building up to abstraction from a foundation of specific examples.
• At 39:50 - "It's so tempting to do...which is that you start with the most powerful thing. You state what the powerful thing is that you're going to do. Usually that takes the form of an abstraction." - He critiques the common "abstraction-first" teaching method which he argues is less effective.
• At 41:43 - "By the time you get to the abstraction, it feels so cool and so powerful because you have a sense for all of the different things underneath that it's touching." - He describes the payoff of the "examples-first" approach, where the final abstract concept serves as a powerful unifying principle.

Takeaways

• Prioritize building intuition by leading with concrete examples before introducing formal, abstract formulas.
• Embrace "productive failure" by allowing learners to grapple with problems on their own before being shown the solution, as this fosters deeper understanding.
• Use visualizations to generate new insights and motivate complex topics, not just to illustrate concepts that are already understood.
• Consciously balance the need for mathematical rigor with the equally important goal of pedagogical clarity to avoid alienating learners.
• Frame abstract concepts as the powerful culmination of explored ideas, making them feel like a rewarding discovery rather than an arbitrary rule.
• Recognize that teaching methods are a critical factor in whether a student's initial interest in math is nurtured or extinguished.