Studying MAthematics: Does it become easier to pick up the broad ideas

[Bourbaki1123]

We all are likely familiar with the notion of mathematical maturity, you sort of get an intuition for how to put mathematical ideas into context and make sense out of them as you do/read more and more math.

I've noticed that it's much easier for me to pick up a graduate level math text and grasp the material now than it was, say, a little over a year ago. Over the four years that I've been serious about studying pure math, I've noticed this trend continue and there has clearly been a great deal of accumulation.

My questions are (to those who wish to answer): (a) What is your experience with this sort of thing? (b) Does ease tend to vary logarithmically with how much time you put in studying mathematics? That is, I assume it can't continue to feel easier and easier to get through dense material without dropping off as the complexity of the material goes up, but the more I do/ read math, the easier it gets to look at another area of mathematics.

 

[chiro]

We all are likely familiar with the notion of mathematical maturity, you sort of get an intuition for how to put mathematical ideas into context and make sense out of them as you do/read more and more math.

I've noticed that it's much easier for me to pick up a graduate level math text and grasp the material now than it was, say, a little over a year ago. Over the four years that I've been serious about studying pure math, I've noticed this trend continue and there has clearly been a great deal of accumulation.

My questions are (to those who wish to answer): (a) What is your experience with this sort of thing? (b) Does ease tend to vary logarithmically with how much time you put in studying mathematics? That is, I assume it can't continue to feel easier and easier to get through dense material without dropping off as the complexity of the material goes up, but the more I do/ read math, the easier it gets to look at another area of mathematics.

I will be graduating next year just so you know where I am coming from.

I have found like you that things do become easier as things go along, but only in the context of studying things that build on in some way of concepts that I have met before.

For example if I just picked up a book on advanced logic, I don't think that I would be able to see the forest from the trees in the same way that I would do in the case of understanding more statistics. I will probably have a better chance of grasping quicker than I would say a year before, but I don't think I would get the same kind of effect if I had not even explored (or at least thought about) basic ideas that the particular subject builds upon.

One thing that I have noticed is that within mathematics itself, there is a pattern to the heuristics of understand it that is invariant to the topic.

For example every area of mathematics involves assumptions and constraints. It doesn't matter if you are talking about a pivotal quantity like the t-distribution or chi-square distribution in statistics, or whether you are talking about a basic results of standard calculus (like taylor series to give an example), every result is based on assumptions. You can't use Taylor series for example if you don't have specific convergence properties, and you need some kind of normal approximations in many statistical tests.

Once you understand that, then you start to look at everything in assumptions: why is this result true under these assumptions? What other things does that imply? How flexible are these assumptions? (that is, can you use something that doesn't exactly meet the assumptions and still make some kind of sense of it?)

Without this insight into using this assumption view as a template, a lot of the learning is seemingly disconnected (or at least it has the tendency to be) and real understanding can potentially be lost.

With regards to the second question, I would say yes to that too.

It's really amazing about how our brain works. It's almost like it's tuning into knowledge like a person adjusting the tuning knob of a radio to get the right station. The fact that given enough time, effort, and conscious/unconscious thought, a lot of us can obtain a type of understanding that is really incredible: it's like magic if you ask me.

I don't think this is just limited to mathematics though. If you ask anyone who has been doing something for a long time who has done a lot of varied work (i.e. not been doing the same sort of problem day in day out), then you'll find that they see things that many amateurs would miss or not even conceive of. It's like a programmer knowing intuitively how to fix a bug and identify it quickly without too much effort in comparison to someone that is still learning to program.

One thing I do want to mention though is that especially with things like math, you are always learning different perspectives of how to view mathematics. There are possibly infinitely (or at least a large number of) ways to interpret mathematics in many contexts.

For example, I did a teaching unit where I taught high school for a few weeks and the supervisor was explaining how the second derivative explained concavity of the function. The teacher was explaining in a visual manner how the slope of the slope was changing for some function. Now it makes perfect sense when you look at it in that way, but I just never really thought about how to explain it in that way using that perspective.

It is a trivial example no doubt, but it does give you a hint about how many things in math have a variety of perspectives and explanations in a variety of contexts.

 

[Kindayr]

I definitely think it gets easier as it goes. But the learning curve adjusts this as that, we learn harder things faster and faster as we go, which is interesting.

I mean I just happened to look at my first year calculus exams, and I laughed, straight up, at how inexperienced and how 'stupid' I was back then, and that was only 2 years ago. (ie. i though differentiability was how since a function covers every point over an interval, that each point has a specific 'rate of change' associated with it. that was my legit definition. and the scary part, was that I finished in the top 3 of my class.)

However, today, I just met with a prof to discuss some subjects that I've been thinking about recently. And he was able to introduce me to the p-adics, early measure theory, and some interesting number-theoretic results all in one day.

I love how I am now able to pick up, with ease, new topics and ideas with the ability to understand the rigor behind them. I'm glad that I've picked up this ability, and I couldn't be more thankful. And I hope this skill can only grow with further and further introduction into mathematics.

So I am definitely on the side that things get easier to grasp. The topics at hand aren't necessarily easier, its just that we've slowly trained our minds to be able to adjust and adapt quickly to new areas of interest.

Those are my 2 cents atleast.

 

[nickadams]

I wish I could develop this "mathematical intuition" you guys speak of. I am so slow to pick up any concept in math. :(


Maybe someday... But unfortunately I am currently in the last math class for my degree plan but I have still not developed any math skill!

 

[aspiring_one]

i like what chiro said. I think he/she pretty much sums it up for any science really. Science is based on assumptions and inductions relative to and limited to our observations. This isn't necessarily bad though, i think i may be going off topic.

Anyway, the way they taught me science and math in elementary and high school was very superficial. Either that or I wasn't paying as much attention as I should have. Then again I live in the U.S.

 

[mathwonk]

if you love it pursue it; it does get easier.

 

[nickadams]

if you love it pursue it; it does get easier.

What are some good math courses to take to develop that improved learning ability for math that you guys speak of?


I may take some courses outside of my degree plan ;)

 

[chiro]

I wish I could develop this "mathematical intuition" you guys speak of. I am so slow to pick up any concept in math. :(


Maybe someday... But unfortunately I am currently in the last math class for my degree plan but I have still not developed any math skill!

When I learned probability concepts in my senior year of high school, I had absolutely no idea what was going on. Anything beyond simple probability was just to perplexing.

When I did the course in university, it was actually really simple, when the lecturer broke it down into what it was and it was actually pretty simple. Using a variety of techniques ranging from the Kolmogorov axioms, to intuitive perspectives like tree diagrams, it become simple when things were broken down into atomic events in which simple laws could be applied.

In high school I was using formulas that I didn't really understand, and I somehow was able to coast on them for the majority of problems.

I am thankful that I actually (for the most part) know what I am doing.

It took someone with a lot of understanding and great communication skills for me to get there, and hopefully the same can happen for you too.

 

[Bourbaki1123]

if you love it pursue it; it does get easier.

I've had many a romantic, candlelit evening twirling the pages of my copy of Eisenbud's "Commutative Algebra with a View Towards Algebraic Geometry", but in retrospect it only took a few of those before I realized how easy it was. Then one day I found it letting someone else clumsily thumb through its pages... my heart was broken that day.

 

[Bourbaki1123]

What are some good math courses to take to develop that improved learning ability for math that you guys speak of?I may take some courses outside of my degree plan ;)

Learning how proofs work is important. Other than that, I would say reading lots of mathematics and re-reading it multiple times. That's what I do, and it feels like I periodically 'level up' and suddenly it's easier to pick up new math.

My main question in this is whether that subective feeling of 'leveling up' slows down over time like it does with chess or Go or other skill based tasks.

 

[leontd]

There is a huge difference in learning concepts and applying those concepts to do research. Over time you will start to cover all learned fundamental tasks by heart and it will applied rapidly, but researching new concepts will slow down your subjective feeling of "leveling up" tremendously. Just like chess, there's so much thing you can do once your a grand master.