soft question - How Do You Go About Learning Mathematics?

"What is the most effective way to learn mathematics?"

I have been trying to answer this question for myself, and one measure I've taken towards this goal is to record all of my mathematical reading, work, and random thoughts in a journal. I highly recommend the practice as it has been very illuminating to me since I started a few months ago. Reviewing my previous readings allows me to ascertain how much math I actually end up retaining from my study sessions, and keeping all of my work in one place (as opposed to throwaway scrap paper) allows me to spot any particularly common mistakes.

So far, I've found that my memory is far more tenuous than I had previously assumed. I'd look at last month's entries and realize that I'd only retained 20% of what I had learned; fine details being especially prone to slippage. Yet from analyzing my mistakes, I've also found that those very details are much more crucial than I had thought.

The result of all of this is that I've started to shift my focus from "learning new math rapidly" (which has been my focus since I am still an undergraduate) to "winning the uphill battle against memory loss." From this new perspective, the old adage: "the only way to learn mathematics is through doing" begins to make a lot more sense. While active learning is far from any cure to forgetfulness, given my own mnemonic capabilities I have come to see that it would probably be a better long-term investment to spend a month on fully working and understanding a chapter, than to spend the same time blazing through several chapters but skipping the exercises (having done both.)

I emphasize again that this is my own conclusion based on my own characteristics, and that is precisely why I recommend everyone to find their own answer to this question by keeping their own math notebook.