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.

10 Commandments Icon


Prefer the PDF Version Instead?

COMMANDMENT

I

Thou Shalt Choose Thy Materials Judiciously

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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, online math materials based on one's interests and level.

COMMANDMENT

II

Thou Shalt Keep The Big Picture In Mind

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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 representations instead.

COMMANDMENT

IV

Thou Shalt Isolate Til Mastery Is Attained

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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?

  • 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

Thou Shalt Think And Act Proactively

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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 brainstorming.
  • 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

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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) 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

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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...

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning
[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

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 ~

(Thank you!)

Brought to you with ❤ by:

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

The Gods of Mathematics

[Illustrated Guide] The 10 Commandments of Higher Mathematical Learning

Additional Readings