The Illustrated Guide To

# The 10 Commandments of Higher Mathematical Learning

Higher math operates a bit differently than the math we see in school. As a result, techniques such as **rote memorization** and **algorithmic learning** rarely work out in the long term...

However, with the **10 commandments** introduced in this guide, there's a good chance that you might be able to survive and thrive as a true math whiz.

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## Table of Commandments

"*What thou shalt and shalt not do...*" Thus the Gods of math hath spoken...

### What?

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.

### Why?

Because individual differences demand **individualized measures** — a fact that usually requires little attention until higher mathematics puts it under spotlight.

### How?

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

COMMANDMENT

II

## Thou Shalt Keep The Big Picture In Mind

### What?

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**.

### Why?

Because the human brain achieves understanding and retention primarily through **associations** and **organisational hierarchy**.

### How?

- 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

## Thou Shalt Operate Within Thy Proximal Zone

### What?

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.

### Why?

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.

### How?

- 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**instead.**representations**

COMMANDMENT

IV

## Thou Shalt Isolate Til Mastery Is Attained

### What?

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.

### Why?

Because higher mathematics is notoriously **cumulative**, and without a solid foundation, any future progress or growth can become very difficult — if not impossible.

### How?

COMMANDMENT

V

## Thou Shalt Think And Act Proactively

### What?

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.

### Why?

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**.

### How?

- 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

## Thou Shalt Try Doing Most Things Inside Thy Head

### What?

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.

### Why?

Because over-relying on tools and technologies can have some nasty side effects — and those include **mathematical atrophy** and **higher math illiteracy**.

### How?

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

COMMANDMENT

VII

## Thou Shalt Apply The Scientific Method Creatively

### What?

The needs to **explore**, **question**, **conjecture** and **test** are fundamental to higher mathematics, and is what separates mathematical experience from schooling.

### Why?

Because when creativity and the scientific method are out of the equation, math learning can become incredibly **domesticated** and **contrived**.

### How?

- 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

## Thou Shalt Not Fret About Real-Life Applicability

### What?

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.

### Why?

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.

### How?

- 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

## Thou Shalt Scale Up By Going Social

### What?

While one could pursue higher math solely through self-studies, they should also consider using **group learning** to further enhance their mathematical experiences.

### Why?

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.

### How?

- 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

## Thou Shalt Embrace Thy Mathematical Experience

### What?

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.

### Why?

Because **mathematical appreciation** determines our mathematical well-being, and **mathematical resilience** determines the future of this great discipline.

### How?

- 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 taking our Higher Math Proficiency Test, or by grabbing the **PDF version** of this guide along with other higher math goodies.

## ~ The End ~

(*Thank you!*)

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