My learning progress skyrocketed after adopting a new standard: Intuition Isn't Optional.
Imagine a chef who follows a new recipe to the letter. No matter how it looks, no matter the reviews the recipe has, if the dish doesn't taste good we know something is wrong. A sense of taste is the ultimate cooking tool.
When learning, we defer to external indicators (tests, teachers) to inform us we've learned something. External standards are made to be objective and easily-verified (Did you pick the correct answer?), but the important, subjective question is how well a concept sits in your mind. Did you actually experience it?
My checklist of truly learning a topic means it is:
Understandable: Did I have an aha! moment? Can I explain the concept in simple language? Does it connect to other topics I know?
Memorable: Do I have an analogy, diagram, or example that will stick with me for months or years?
Enjoyable: Do I want to revisit or use this knowledge? Don't study literature in a way that makes you hate reading.
That's my current definition of "intuitive understanding", and for subjects I care about, I keep digging until I have all three aspects.
It's ok to take your time (calculus took years to become enjoyable) and it's ok to not care about everything equally (biology isn't particularly compelling for me). I firmly believe any subject can become intuitive if I put in the effort to find analogies, diagrams, examples, plain-english descriptions, and technical details (the ADEPT method).
So, how do you set your own learning standard?
Step 1: Study Famous Learners
Let's not recreate the wheel: famous learners have already described their thinking process, which we can adopt. It's not about memorizing Einstein's Theory of Relativity, it's about internalizing the mindset that could lead to that idea.
Here's a few viewpoints that resonated for me:
"Education is what remains after one has forgotten what one has learned in school." —Albert Einstein
“The only real valuable thing is intuition.” —Albert Einstein
- True learning goes beyond memorized facts. While I can forget the equation of a circle, I can't forget that it's round. And knowing it's perfectly round quickly leads me back to the equation.
"The noblest pleasure is the joy of understanding." —Da Vinci
- True understanding implies joy. And practically, you'll only continue studying what you like.
"To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.” —George Póyla
“Education is the kindling of a flame, not the filling of a vessel.” —Socrates
- We aren't robots, and we should embrace the subjective aspects of learning. A teacher's goal goes beyond knowledge-transfer to enjoyment-transfer.
- There are deeper, richer levels of understanding than what's traditionally used. Explore a higher standard.
"I think most people can learn a lot more than they think they can. They sell themselves short without trying. One bit of advice: it is important to view knowledge as sort of a semantic tree — make sure you understand the fundamental principles, ie the trunk and big branches, before you get into the leaves/details or there is nothing for them to hang on to." —Elon Musk
- Your own standards greatly influence your understanding. External tests won't check if facts are comfortably connected.
I have a larger collection of quotes that help align my thinking.
Step 2: Ask Questions That Check Your Standards
After rummaging through quotes that resonate, build a set of questions that capture your standard. For me, it became:
- Do I have a visceral, ingrained analogy? Can it help solve problems?
- Can I explain the concept to others? Do they want to explain it to their friends afterwards?
- Will I remember the essential idea after a few months or years?
- Can I find something to enjoy in the topic? Will I return after I inevitably forget 95% of it?
Questions seem to prompt more interest than a statement: "Do I have an analogy?" vs. "I must have an analogy".
With this approach, strange corners of math I didn't previously enjoy (like Euler's Formula) became mysteries to solve: what is the insight here? Can I express it in a plain-English sentence? (Here's a shot: Continuous rotation means you're moving in a circle.)
Setting new standards helps take control of your education and overcome longstanding demons.
When people say "I hate math" I doubt they actually hate numbers (arithmetic), patterns & relationships (algebra), or shapes (geometry). They hate lessons that don't contain insight, enjoyment, and basic human empathy. It's fine to be disinterested in Ancient Egyptian Civilization, but hate comes from getting lost on a tour and spending the night near a sarcophagus.
These are the questions that helped me: what are your standards for learning?
Other Posts In This Series
- Developing Your Intuition For Math
- Why Do We Learn Math?
- How to Develop a Mindset for Math
- Learning math? Think like a cartoonist.
- Math As Language: Understanding the Equals Sign
- Avoiding The Adjective Fallacy
- Finding Unity in the Math Wars
- Brevity Is Beautiful
- Learn Difficult Concepts with the ADEPT Method
- Intuition, Details and the Bow/Arrow Metaphor
- Learning To Learn: Intuition Isn't Optional
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Learning to Learn: Math Abstraction
- Learning Tip: Fix the Limiting Factor
- Honest and Realistic Guides for Learning
- Empathy-Driven Mathematics
- Studying a Course (Machine Learning) with the ADEPT Method
- Math and Analogies
- Colorized Math Equations
- Analogy: Math and Cooking
- Learning Math (Mega Man vs. Tetris)