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Higher Mathematical Learning

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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.

- Choose math books by personally
**skimming**through them in a public or an university library. - Maintain an
**active****list**of well-structured, online math materials based on one's interests and level.

COMMANDMENT

II

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**.

- Get into the habit of mentally
**summarizing**the recently-learned notions and techniques every few minutes or so. - Use
**bullet points**and**tree diagrams**to facilitate mental representations of mathematical concepts and entities.

COMMANDMENT

III

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.

- When stuck in a math problem, consider temporarily solving a
**simplified version**of the same problem instead. - When a math concept seems hard to grasp, consider thinking about some of its
**examples**or**alternative representations**instead.

COMMANDMENT

IV

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.

- When solving problems, aim not to master a problem by itself — but all its
**related variants**as well. - When studying a topic, opt for having several similar materials at hand — so that they can be used in
**alternation**.

COMMANDMENT

V

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**.

- When learning a new subtopic, always keep up with some form of
**exploratory thinking**and**brainstormin**g. - While working on problems, make an effort to come up with some
**novel questions**or**theory**of your own.

COMMANDMENT

VI

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**.

- Make an effort to perform most
**symbolic manipulation tasks**(e.g., algebra, proofs, calculus) off the top of the head. - Beware of the use of
**stationery**,**calculator**and**textbook**for tasks which can be done mentally.

COMMANDMENT

VII

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**.

- When doing math, always keep an eye out for the invisible, abstract
**patterns**. - When in doubt about your hunches, consider formulating a
**claim**and putting it to test somehow.

COMMANDMENT

VIII

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.

- Refrain from asking questions such as "
**why do I need this math?**" or "**how can this math help me in real life?**" - Learn to stop seeing math as a set of algorithms — but as a gateway to
**abstraction**,**deduction**,**creation**and**mental exploration**.

COMMANDMENT

IX

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.

- Join a local
**math circle**to get involved in both competitions and collaborative activities. - Attend
**seminars**and**gatherings**hosted by math institutions around the corner to keep you in the loop.

COMMANDMENT

X

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.

- Establish a
**math routine**to keep the mathematical thread going — whether things are good or bad. - Get used to trying
**new approaches**repeatedly — when viable solutions are not readily available.

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.

**~ The End ~**

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