When learning a new mathematical operator, such as Σ, a student typically goes through a series of steps:
Understand what it’s called and what the different parts mean.
See how the operator is used in a bunch of example problems.
Learn some theorems relevant to or using the operator.
Do a bunch of example problems.
Understand what the operator is doing when they encounter it “in the wild” in future math reading.
I’ve only taken a little bit of proof-based math, and I’m sure that the way one relates with operators depends a lot on the type of class one is in and also their natural talent and drive.
Over time, I’ve come to see how much time spent understanding math on a deeper level compounds in value over time. Lately, I’ve started to understand that there’s a gap in my mathematical intuitions.
The gap is that I treat operators like instructions in a technical manual written in a foreign language. When I encounter them in a derivation, here’s the thought process I go through:
I look at them and recall what they mean.
I inspect them to see what I’d have to do in order to take the integral, compute the first several terms of the summation, etc.
I consider how the use of the operator connects with the adjacent steps in the proof—how it was obtained from the previous step, and how it’s used in future steps.
If it’s in an applied context, I confirm that the use of the operator “makes sense” for the physical phenomenon it is modeling.
This is all well and good—I can read the “instruction manual” and (usually) do the right thing. However, this never teaches me how to speak the language. I never have learned how to express mathematical thoughts in terms of summations and series, in terms of integrals, or in terms of dot and cross products, for example.
This is in contrast to how naturally I express thoughts in terms of basic arithmetic, statistical distributions, and geometry. I can pull Fermi estimates out of my butt, casually refer to statistical concepts as long as they’re no more complex than “normal distribution” or “standard error” or “effect size,” and I can model real world situations with basic Euclidean geometry, no problem.
I learned arithmetic as a child, geometry in middle and high school, statistics in college, and Fermi estimates as an adult. I learned calculus, differential equations, and linear algebra in college, but have never found a practice equivalent to Fermi estimation that requires me to use summations, integrals, and so on in order to articulate my own thoughts.
lsusr has a nice example of developing a blog post around developing a differential equation. Von Neumann clearly could think in terms of complex operators as thought they were his own natural language. So I think it is possible and desirable to develop this facility. And if I gained the ability to think in terms of Fermi estimates and do other forms of intuitive mathematical reasoning as an adult, I expect it is possible to learn to use these more advanced operators to express more advanced thoughts.
I think it’s probably also possible to gain more facility with performing substitutions and simplifications, assigning letters to variables, and so on. Ultimately, I’d like to be able to read mathematics with some of the same fluency I can bring to bear on natural language. Right now, I think that contending with these more advanced operators is the bottleneck.
Operator fluency
When learning a new mathematical operator, such as Σ, a student typically goes through a series of steps:
Understand what it’s called and what the different parts mean.
See how the operator is used in a bunch of example problems.
Learn some theorems relevant to or using the operator.
Do a bunch of example problems.
Understand what the operator is doing when they encounter it “in the wild” in future math reading.
I’ve only taken a little bit of proof-based math, and I’m sure that the way one relates with operators depends a lot on the type of class one is in and also their natural talent and drive.
Over time, I’ve come to see how much time spent understanding math on a deeper level compounds in value over time. Lately, I’ve started to understand that there’s a gap in my mathematical intuitions.
The gap is that I treat operators like instructions in a technical manual written in a foreign language. When I encounter them in a derivation, here’s the thought process I go through:
I look at them and recall what they mean.
I inspect them to see what I’d have to do in order to take the integral, compute the first several terms of the summation, etc.
I consider how the use of the operator connects with the adjacent steps in the proof—how it was obtained from the previous step, and how it’s used in future steps.
If it’s in an applied context, I confirm that the use of the operator “makes sense” for the physical phenomenon it is modeling.
This is all well and good—I can read the “instruction manual” and (usually) do the right thing. However, this never teaches me how to speak the language. I never have learned how to express mathematical thoughts in terms of summations and series, in terms of integrals, or in terms of dot and cross products, for example.
This is in contrast to how naturally I express thoughts in terms of basic arithmetic, statistical distributions, and geometry. I can pull Fermi estimates out of my butt, casually refer to statistical concepts as long as they’re no more complex than “normal distribution” or “standard error” or “effect size,” and I can model real world situations with basic Euclidean geometry, no problem.
I learned arithmetic as a child, geometry in middle and high school, statistics in college, and Fermi estimates as an adult. I learned calculus, differential equations, and linear algebra in college, but have never found a practice equivalent to Fermi estimation that requires me to use summations, integrals, and so on in order to articulate my own thoughts.
lsusr has a nice example of developing a blog post around developing a differential equation. Von Neumann clearly could think in terms of complex operators as thought they were his own natural language. So I think it is possible and desirable to develop this facility. And if I gained the ability to think in terms of Fermi estimates and do other forms of intuitive mathematical reasoning as an adult, I expect it is possible to learn to use these more advanced operators to express more advanced thoughts.
I think it’s probably also possible to gain more facility with performing substitutions and simplifications, assigning letters to variables, and so on. Ultimately, I’d like to be able to read mathematics with some of the same fluency I can bring to bear on natural language. Right now, I think that contending with these more advanced operators is the bottleneck.