Learning a mathematics course better, LATER

[symbolipoint]

What can you say about struggling hard to learn a Mathematics course while you are "young", not succeeding, but learning the same course effectively, later, when you are "not as young"? Meanwhile, you studied other Mathematics courses with acceptable success, but NOT that particular one with which you struggled at the time that you were "young".
ASSUME that the teaching quality was the same for all courses and that the course content of that particular course was the same both when you were "young" and when "not as young".

What could account for this?
Increased brain growth?
Increased study skills?
Increased subject-matter development?
What --- anything els?

 

[marcflores]

I'm also fascinated with this topic. I feel like I was a late bloomer. Though I was considered relatively intelligent in high school, I felt like I was a fake and that I didn't have a true grasp of things until I was much older. Even now, I'll read things I wrote about psychology and cognitive science two years ago and think to myself, "Wow, was I really that naive? Did I think I was smart? Was I convinced that I knew what I was talking about?"

Sometimes I feel like learning later was the best thing for me because there was no way I was going to absorb a lot of information when I was younger. Everyone is different and brain development and changes occur during different times, but I feel like when I turned 22 or 23, I started having a better understanding, and hence appreciation, for a lot of disciplines of knowledge.

I think that, aside from individual temperaments and talents, there is no substitute for time. I felt like my understanding for certain subjects, whether it was the sciences or humanities, were greatly influenced by a multitude of other aspects of my life and learning. Steven Pinker and Edward O. Wilson believe it's important to have a well-rounded education and to be knowledgeable in fields outside of our concentration. They argue that it is essential to know how the arts, humanities, and sciences influence each other. I strongly believe that knowledge in other disciplines gave me a better understanding of math and science. Even learning about the history of Newton's work or Euler's work helped me to understand what they were trying to accomplish. As a result, this knowledge trickled down to the other math courses I was or am studying and has helped me a great deal.

In the end, I think everyone's experience will be different but I think that it might be universal that we learn and understand more as we get a little older as long as the education process never stops.

 

[Defennder]

Well this should vary from person to person, but for me I'll say it's mostly learning the best way to learn maths which increases with age which largely aids studying it later in life.

Earlier on, before I discovered Physicsforums.com, I looked through a number of maths texts, not knowing what each of them covered, in which fields of science and engineering they would be employed, and the pre-requisites needed to understand them. That, and not knowing which were the classic texts or even good texts like Stewart's Calculus, to best learn maths from effectively hindered my learning. That said, I'm still not sure if this factor was that instrumental in determining whether I could understand the maths then.

Of course I'm referring to self-study of maths, as opposed to studying maths courses in Uni, because by then most would have grasped, at least in outline, of how best to study maths.

 

[mathwonk]

some topics in which i was poorly taught proved very hard and long for me to learn, like physics, english, and analysis. so sometimes a good teacher who teaches you well early gives you a head start, and otherwise it takes a long time to catch on alone and unguided.

i was a weak algebra student and strong in advanced calculus and geometry - topology at first as a researcher, but eventually while working with a colleague who was very strong in algebra and very patient I became more sophisticated. But I was good at the aspects of algebra I learned from Maurice Auslander, one of the best teachers I ever had.

It also helps if you go to class, review lecture notes, do the hw, and read the book, some practices I long disdained.

 

[JasonRox]

I noticed that I struggled with some 1st and 2nd year stuff, but when I look at it now being in 4th year, it's really easy stuff and just wonder how I found that hard.

I think just doing mathematics over the years improves your intuition of mathematics and just plain general knowledge.

 

[SRode]

I always found I never really understood a subject until it was applied to a more advanced course later on. When I was taking courses I could solve the problems, but never picked up how to solve them intuitively until one or several semesters later.

I think that it mostly comes down to seeing countless variations on problems, and solving systems over and over until it becomes second nature. I had a professor once and he was teaching mathematical physics and said "this is the point where math goes from being mechanical to being an art form, the only way to understand this is from experience and practice" and he was right, I never fully understood the theorms until later when I had a lot more experience.

 

[mathwonk]

at first you don't see the forest for the trees. after the technical details become second nature, you can see the structure of the amterial. niote after 40 years i was abkle tow rite a 15 page linear algebra book, and 10 years afterw riting a 400 page algebra book i was able to write a 100 page one.

and i can sum up now all of the founDATIONS OF calculus in one principle: namely deciding when a function is constant. (this is contained in the Mean Value Thm which implies the Fundamental Theorem of Calc).

or the riemann roch theorem which says that the residue principle is a complete obstructiion to thw existence of meromorphic functions with given pole divisor, i.e. in a sense, it says that the converse of the residue theorem is true too.

 

[Darkiekurdo]

Three months ago I just could not grasp the concept of derivatives. Now I ask myself why. It isn't that hard. Now it is something else, but I am confident I will understand that subject when I'm older.

 

[ice109]

hindsight is 20/20

 

[SRode]

hindsight is 20/20

Nailed it.

 

[FrancisZ]

As a teacher I can tell you that I have learned a lot more by teaching than I ever did as a student. Just about all of the educational research I am exposed to as a grad student supports this as well. Apparently: to serve humanity in this respect is also beneficial to yourself.

When you forget about trusting in your professor to be a good teacher and struggle with it yourself--for someone else's benefit--you in turn learn the material better than you would have, merely for selfish purposes.

 

[kaisxuans]

As a teacher I can tell you that I have learned a lot more by teaching than I ever did as a student. Just about all of the educational research I am exposed to as a grad student supports this as well. Apparently: to serve humanity in this respect is also beneficial to yourself.

When you forget about trusting in your professor to be a good teacher and struggle with it yourself--for someone else's benefit--you in turn learn the material better than you would have, merely for selfish purposes.

Whao! first time i saw a teacher reply!
ea very specific and all.
hmm..all logical
i surpport her!
visit my blog!

 

[FrancisZ]

Whao! first time i saw a teacher reply!
ea very specific and all.
hmm..all logical
i surpport her!
visit my blog!

Not for anything (it isn't important I understand) but that is...cough..."Francis" with an "i"

The "i"<-----stands for "dude"

"Frances" with an "e" is the female form of Francis.

:)