I think there’s a perverse incentive around the learning and usage of math. This is majorly coloured by my experiences in calculus class where it seemed like most students were interested in memorizing how to use the formulas to get the correct answer to get good grades, without necessarily understanding anything about what the calculations they were doing represented or were used for.
Maybe the problem here is fully from Goodharting on grades, but I wonder if there may be a broader phenomenon.
There’s three objects here:
Understanding the symbolic language of some math
Understanding the point of that math: what does it relate to? What can it help you do? Where does it fit in the broader research community?
Social signalling of values and competence
The dynamic I’m hypothesizing is:
In general, (2) is more important than (1), and (1) is held in higher esteem than (2). Ideally people have a strong grasp of both (1) and (2), but people are intrinsically and extrinsically motivated to seek (3), and (2) is easier to fake than (1), so people are motivated to fake (2) and put their effort into signalling competence in (1) which would, ideally imply competence in (2), but doesn’t necessarily.
I feel this relates to what Grant Sanderson of 3b1b talked about in Math’s pedagogical curse. But of course I also worry this is something I’m imagining because I feel motivated to try to understand math that is too difficult for my level of skill and I want to rationalize away my incompetencies. Is it imposter syndrome or honest self knowledge?
What do you think? Is (2) more important than (1), or maybe it’s not important to be skilled at both, and we need people better at (1) and people better at (2)? Do you agree the dynamic I described exists, and does it feel common, or marginal? Or maybe my entire framing is flawed in some way?
Ever read Surely You’re Joking, Mr. Feynmann? Plenty of the stories in there involve someone (or an entire student body) not really understanding extremely basic things about what the hell they were talking about, despite having memorized some formulas and being adept at manipulating them.
Examples include:
Students who can recite the formula for Brewster’s angle but don’t realize what that would have to do with polaroids and light reflecting off of water,
Stedents being able to calculate the displacement of a ray of light shined through glass but thinking that if a book is rotated underneath a glass table that the image would rotate by twice the angle
Students that can recite the definition of diamagnetism but that can’t actually name a single diamagnetic material.
A bunch of engineering students who clearly didn’t understand that the derivative of a curve is the slope of the tangent line.
An assistant to Einstein that could not answer a simple question (whose result should’ve been well known to him) about length minimizing geodesics when phrased in terms of a rocket path that takes the least time to go up and back down again (aka the meaning of (a special case of) ‘length minimizing geodesic’)
a man adept at calculating cube roots with an abacus that had to use the device to confirm that 2^3 = 8.
To paraphrase from memory: Most people’s knowledge is so fragile!
Understanding the symbolic language of math is difficult to master, and easy to evaluate (for someone who has the same skill). That makes it a convenient status marker.
I think there’s a perverse incentive around the learning and usage of math. This is majorly coloured by my experiences in calculus class where it seemed like most students were interested in memorizing how to use the formulas to get the correct answer to get good grades, without necessarily understanding anything about what the calculations they were doing represented or were used for.
Maybe the problem here is fully from Goodharting on grades, but I wonder if there may be a broader phenomenon.
There’s three objects here:
Understanding the symbolic language of some math
Understanding the point of that math: what does it relate to? What can it help you do? Where does it fit in the broader research community?
Social signalling of values and competence
The dynamic I’m hypothesizing is:
I feel this relates to what Grant Sanderson of 3b1b talked about in Math’s pedagogical curse. But of course I also worry this is something I’m imagining because I feel motivated to try to understand math that is too difficult for my level of skill and I want to rationalize away my incompetencies. Is it imposter syndrome or honest self knowledge?
What do you think? Is (2) more important than (1), or maybe it’s not important to be skilled at both, and we need people better at (1) and people better at (2)? Do you agree the dynamic I described exists, and does it feel common, or marginal? Or maybe my entire framing is flawed in some way?
Ever read Surely You’re Joking, Mr. Feynmann? Plenty of the stories in there involve someone (or an entire student body) not really understanding extremely basic things about what the hell they were talking about, despite having memorized some formulas and being adept at manipulating them.
Examples include:
Students who can recite the formula for Brewster’s angle but don’t realize what that would have to do with polaroids and light reflecting off of water,
Stedents being able to calculate the displacement of a ray of light shined through glass but thinking that if a book is rotated underneath a glass table that the image would rotate by twice the angle
Students that can recite the definition of diamagnetism but that can’t actually name a single diamagnetic material.
A bunch of engineering students who clearly didn’t understand that the derivative of a curve is the slope of the tangent line.
An assistant to Einstein that could not answer a simple question (whose result should’ve been well known to him) about length minimizing geodesics when phrased in terms of a rocket path that takes the least time to go up and back down again (aka the meaning of (a special case of) ‘length minimizing geodesic’)
a man adept at calculating cube roots with an abacus that had to use the device to confirm that 2^3 = 8.
To paraphrase from memory: Most people’s knowledge is so fragile!
Understanding the symbolic language of math is difficult to master, and easy to evaluate (for someone who has the same skill). That makes it a convenient status marker.
Yeah. I think because of (3) there might be a perverse incentive to seek (1) at the detriment of (2). What do you think?
Yes.
Also, if you learn the symbolic language of math, you join the ranks of people fluent in math.
If you speculate about a purpose, you join a more diffuse set, which includes some deep thinkers but also many crackpots.
So it’s like the wisest people are in the latter group, but the people in the former group are smarter on average.