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As math complexity increases, choosing materials based on one's own interests and background level becomes more important than passively accepting what's being given.
Because individual differences demand individualized measures — a fact that usually requires little attention until higher mathematics puts it under spotlight.
While technical details and rigor matter in higher mathematics, it's even more important to connect each concept and technique to an overarching picture or theme.
Because the human brain achieves understanding and retention primarily through associations and organisational hierarchy.
It is vital to constantly re-adjust the difficulty level to keep things both challenging and manageable — especially when a material or a problem gets surprisingly easy or tough.
Because optimal learning occurs right at the edge of comfort and discomfort — where the level of cognitive stress is neither too overwhelming or too little.
Instead of dipping into a few math tidbits here and there, it's better to focus on mastering a math technique or a subtopic first — before moving on to the next cool thing.
Because higher mathematics is notoriously cumulative, and without a solid foundation, any future progress or growth can become very difficult — if not impossible.
Instead of waiting for an institution or a textbook to tell us what math to do and how to do it, it's better to learn to think for ourselves and proactively seek out the math on our own.
When proactivity and thinking are absent, our mathematical growth is capped by what the environment has to offer, but when they are present, the growth can be unstoppable.
If a mathematical task can be done off the top of the head with a reasonable amount of training, then that's probably the right way to do it.
Because over-relying on tools and technologies can have some nasty side effects — and those include mathematical atrophy and higher math illiteracy.
The needs to explore, question, conjecture and test are fundamental to higher mathematics, and is what separates mathematical experience from schooling.
Because when creativity and the scientific method are out of the equation, math learning can become incredibly domesticated and contrived.
The primary value of mathematics lies not in its applications or other functional purposes. As such, learning math for the sake of them only can deem to be a losing fight.
Because when our math interest is purely tied to an external reason, that extrinsic motivation can also set up a strict barrier on how much we can enjoy and achieve in higher mathematics.
While one could pursue higher math solely through self-studies, they should also consider using group learning to further enhance their mathematical experiences.
Because working in groups can give us tremendous leverages in mathematics — both in terms of the amount of new information received and the amount of mutual support gathered.
To embrace the mathematical experience is to enjoy prolonged mathematical thinking, problem solving and other challenges — while doing so with great fascination and exploratory instinct.
Because mathematical appreciation determines our mathematical well-being, and mathematical resilience determines the future of this great discipline.
Looking For More?
Hey. Looks like the Gods of Math have just finished delivering their awesome speech, but there's still more to be said about higher mathematics...
But if you're interested in taking things to the next level, you can always start by grabbing the PDF version of this guide — along with other higher math goodies.