A Mathematician's Lament
Imagine a world where music education from K-12 is solely based on learning to write notes on staff paper neatly, without once listening or composing music (15-18). Imagine an art class where all students do is identify the names of colors without actually painting (18-20).
This is the current state of our math education (20-21).
Mathematics and Culture
Mathematics is an art about simple, imaginary objects (22-24). Most of all, it’s an art about playing (24). You define an imaginary pattern, and that pattern can talk back (25).
That’s the art of it, creating these little beautiful poems of thought, these sonnets of pure reason (27).
Mathematics is the art of explanation (29).
The problem is cultural. People believe they know what math is (arithmetic), when it isn’t. They don’t believe math can be criticized when it can. Math happens to be practical sometimes, but that isn’t why we do it (30-33).
Mathematics in School
Math doesn’t need reform, it needs a complete overhaul. Math doesn’t need to be made “more interesting” or more “cutesy” to combat math anxiety, it’s already interesting. The problem is that we aren’t teaching it (36-38).
People enjoy fantasy, and that is just what mathematics can provide (39).
There are exercises in school, but not natural problems. Even “real world” examples are a huge stretch. There are no need for practical problems, just natural mathematical inquiry, a good puzzle (40-41).
Mathematics is an art form done by humans for pleasure (50)!
The Mathematics Curriculum
Math is not being taught in schools. Discovery is replaced with rules and rigidity. Unnecessary notation and definitions (e.g. sec) are being taught to and memorized by students instead solving actual problems (55-61).
High School Geometry: Instrument of the Devil
The teaching of the interesting bits, the argumentation (proof) of mathematical fact is purported to be taught, but is helplessly butchered by rules and notation. Absurd proof format is introduced to obscure natural reasoning (67-75).
Lockhard mentions how you’re ordered to put bars on top of lines and omit then when referring to length and how all of this is completely dumb and unrelated to math (70). I specifically remember this fact from Geometry and how you would lose massive points for mistaking the two.
Proofs should be a struggle, a journey, and refined over time with guidance from a teacher. Things such as rigid theorem names and definitions have no place. These come as a result of answering a question, not introduced beforehand (76-80).
There should be no curriculum, math is a free-form excursion and should be taught as such (81).
Mathematics contains structures which are inspired by reality, but are abstracted. We can observe the behavior of these objects and see what they do (92-96).
This is the Frankenstein aspect of mathematics - we have the authority to define our creations, to instill in them whatever features or properties we choose, but we have no say in what behaviors may then ensue as a consequence of our choices (108).
We are trying both to create a logically consistent argument, but also an elegant one. This is the challenge of math (111-112).
It’s an adventure of exploration. It need no practical purpose other than to be fun. Just play (135-138)!