I concur with the points made by SarahC and Unnamed. My experience has been that with very few exceptions, learning substantive subject for the first time is a struggle. I’ve often felt bewildered and disoriented for a significant period of time before things started to gel and coalesce.
Mathematics is amazingly compressible: you may struggle a long time, step by
step, to work through some process or idea from several approaches. But once you
really understand it and have the mental perspective to see it as a whole, there
is often a tremendous mental compression. You can file it away, recall it quickly
and completely when you need it, and use it as just one step in some other mental
process. The insight that goes with this compression is one of the real joys of
mathematics.
After mastering mathematical concepts, even after great effort, it becomes very
hard to put oneself back in the frame of mind of someone to whom they are mysterious.
I remember as a child, in fifth grade, coming to the amazing (to me) realization
that the answer to 134 divided by 29 is 134⁄29 (and so forth). What a tremendous
labor-saving device! To me, ‘134 divided by 29’ meant a certain tedious chore,
while 134⁄29 was an object with no implicit work. I went excitedly to my father to
explain my major discovery. He told me that of course this is so, a/b and a divided
by b are just synonyms. To him it was just a small variation in notation.
One of my students wrote about visiting an elementary school and being asked
to tutor a child in subtracting fractions. He was startled and sobered to see how
much is involved in learning this skill for the first time, a skill which had condensed
to a triviality in his mind.
Mathematics is full of this kind of thing, on all levels. It never stops.
[...]
Similarly, students at more advanced levels know many things which less advanced
students don’t yet know. It is very intimidating to hear others casually toss
around words and phrases as if any educated person should know them, when you
haven’t the foggiest idea what they’re talking about. Less advanced students have
trouble realizing that they will (or would) also learn these theories and their associated
vocabulary readily when the time comes and afterwards use them casually
and naturally. I remember many occasions when I felt intimidated by mathematical
words and concepts before I understood them: negative, decimal, long division,
infinity, algebra, variable, equation, calculus, integration, differentiation, manifold,
vector, tensor, sheaf, spectrum, etc. It took me a long time before I caught on to
the pattern and developed some immunity.
My experience has been that with very few exceptions, learning substantive subject for the first time is a struggle. I’ve often felt bewildered and disoriented for a significant period of time before things started to gel and coalesce.
One of my professors once told me that you only really understand the content of one math course when you’re taking the next one that builds upon it…
I have always wondered exactly how this works. Whenever there’s some maths area I’m struggling with, I only struggle with it while that area is the focus of study. Eigenvalues were a mildly unspeakable horror when I was learning specifically about Eigenvalues, but they suddenly became trivial when the I was asked to use them on a new unspeakable horror.
I don’t remember any progressive improvement. It’s as if I was asked to step up my game, and all the prerequisite knowledge just fell into line without any complaint. This pattern has existed throughout my maths education all the way back to basic algebra and I can’t really find a satisfactory explanation for it. No other subject of study behaves like this.
I wonder if that’s because you’re not trying to “understand” something, you’re just using it as part of a separate algorithm? You stop caring about what it means to find an eigenvalue, and just think about how to get what the number you need to solve the problem in front of you.
Also, oddly enough, when taking the math course that builds on something, you get a lot more practice at it than when you were taking the original course. In other words, you probably do a lot more algebra in calculus class than in algebra class.
I’m starting to wonder if, when first tackling a troublesome topic, I think of it as an enemy, but when given a different enemy I suddenly start treating it like a reluctant ally. The “falling into place” phenomenon also happens when the New Unspeakable Horror is a real-world issue I need to deal with, like a problem at work, or an exam.
I should have thought of this before, but I can already relate to the vocabulary issue described in the second quote. I often use electronics or computer jargon when talking to my wife, not realizing I’m at home and not in the shop. Her arched eyebrow usually cues me to realize I need to swap my terminology around.
I was always a huge fan of the “Cosmos” series, but it never dawned on me to look for books written by Carl Sagan. Thanks!
I’m glad to hear that you’ve been enjoying your studies and am happy that you’re here.
You might find the References & Resources For Less Wrong posting useful.
Aside from the ones mentioned there, one book that I had a favorable impression of upon browsing through it and that looked pretty accessible to me is Carl Sagan’s The Demon-Haunted World: Science as a Candle in the Dark.
I concur with the points made by SarahC and Unnamed. My experience has been that with very few exceptions, learning substantive subject for the first time is a struggle. I’ve often felt bewildered and disoriented for a significant period of time before things started to gel and coalesce.
Quoting from Sections 7 and 8 of William Thurston’s Mathematical Education essay :
[...]
One of my professors once told me that you only really understand the content of one math course when you’re taking the next one that builds upon it…
I have always wondered exactly how this works. Whenever there’s some maths area I’m struggling with, I only struggle with it while that area is the focus of study. Eigenvalues were a mildly unspeakable horror when I was learning specifically about Eigenvalues, but they suddenly became trivial when the I was asked to use them on a new unspeakable horror.
I don’t remember any progressive improvement. It’s as if I was asked to step up my game, and all the prerequisite knowledge just fell into line without any complaint. This pattern has existed throughout my maths education all the way back to basic algebra and I can’t really find a satisfactory explanation for it. No other subject of study behaves like this.
I always said, “I’ve always been bad at math. I’m just bad at different math every year.”
Math divides into “way too hard” and “trivial”—the good news is that the “trivial” pile grows over time.
I wonder if that’s because you’re not trying to “understand” something, you’re just using it as part of a separate algorithm? You stop caring about what it means to find an eigenvalue, and just think about how to get what the number you need to solve the problem in front of you.
Also, oddly enough, when taking the math course that builds on something, you get a lot more practice at it than when you were taking the original course. In other words, you probably do a lot more algebra in calculus class than in algebra class.
I’m starting to wonder if, when first tackling a troublesome topic, I think of it as an enemy, but when given a different enemy I suddenly start treating it like a reluctant ally. The “falling into place” phenomenon also happens when the New Unspeakable Horror is a real-world issue I need to deal with, like a problem at work, or an exam.
I should have thought of this before, but I can already relate to the vocabulary issue described in the second quote. I often use electronics or computer jargon when talking to my wife, not realizing I’m at home and not in the shop. Her arched eyebrow usually cues me to realize I need to swap my terminology around.
I was always a huge fan of the “Cosmos” series, but it never dawned on me to look for books written by Carl Sagan. Thanks!