When you finally understand something, it just clicks. It fits. You can re-derive it, explain it, use it.

That moment captures the hallmark of what I call common-sense learning: an understanding that fuses new ideas into your existing model of the world until they feel self-evident.

At its core, learning means modifying your mental model of reality, the web of ideas that forms your internal common sense. There are two ways to do this:

1. Integrate new information into what you already know.
2. Expand your model to include something genuinely new.

The first path is easier and more stable. When you learn, your first move should be to see how the new idea fits. Fitting it is an art: reframe the problem, look at it from different angles until you find a perspective that feels intuitive to you. Once it fits, it stabilizes. Your brain no longer treats it as foreign information but as a structure within your existing model.

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Example: summing numbers

Take something simple:

$$1+2+3+\dots +n$$

Here is the usual derivation:

$$S=1+2+3+\dots +n$$ $$S=n+(n-1)+(n-2)+\dots +1$$ $$2S=n(n+1)\Rightarrow S=\frac{n(n+1)}{2}$$

But are you able to recall at a glance if it is $n(n+1)$ or $n(n-1)$?

The formula is a result stored in memory. To get to it, you replay a series of symbolic manipulations. It works, but it does not fuse with your intuition. It is not common-sense integrated. It’s simply a series of steps that happens to be easy to remember, but hard to extend.

If I asked you to sum the series $11+15+19+\dots +207$, would you instantly know what to do? You’d probably reconstruct the pattern symbolically again. It never became part of your model of how numbers behave.

Many of us remember a mnemonic for arithmetic progressions: first plus last, times how many, divided by two. You might recall it instantly and use it flawlessly, but that still isn’t common-sense learning. You can’t re-derive it on sight, nor see the boundaries of where it holds, unless you also mechanically recall that it reliably works for arithmetic progressions… but why does it?

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How to find a common-sense perspective

Start watching your own reasoning unfold:

That is the moment understanding fuses with your internal common sense. You have found an invariant, a pattern that does not care about the specific numbers. It now fits naturally into your mental model of how the world works.

The general rule becomes:

$$\text{sum}=\text{average element}\times \text{number of elements}$$

The number of elements is $n$. Also, for any evenly spaced sequence of numbers, the average element is indeed $\frac{\text{first}+\text{last}}{2}$.

You can see this the common-sense way too: pair the smallest and largest numbers, then the next smallest and next largest. Each pair gives the same sum, whether the spacing is 1, 4, or any constant step.

Or, my favorite: start with one or two numbers, then add equally spaced terms to the left and right. Inductively, you’ll see that the average stays the same - the first plus the last, divided by two.

Hence:

$$S=n\cdot \frac{n+1}{2}$$

It’s the same result, but now you own it.

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Try another one

$$1+3+5+\dots +n$$

Apply the same thought process. Average is $\frac{1+n}{2}$. Now count how many elements there are. $1\to 1$, $3\to 2$, $5\to 3$, so in general $n\to \frac{n+1}{2}$.

So:

$$S=\frac{(n+1{)}^2}{4}$$

If $n=2k-1$, then $S=k^2$.

Notice how the same reasoning applies automatically.

This is common-sense learning. You find invariants that are general, simple for you and that unlock whole classes of problems at once. They stick because they fit. You will have a hard time forgetting them and an easy time applying them anywhere the same structure appears.

Want a bonus? This same little pattern (average × count) shows up everywhere.

Work is average force × distance.

Distance is average speed × time.

Area is average height × width.

Probability is average weight × possible outcomes.

This principle works whenever the idea of an average element makes sense: arithmetic or smoothly varying progressions. Discovering when a model holds and when it breaks is a key part of making it your own.

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How to corrupt your common sense

A tempting shortcut for a counterintuitive result is to memorize it as a brute-force exception. You frame it in your mind by its contradiction to what you know, thinking: “That is the one that feels wrong. That is how I will remember it.”

It works in the short term and breaks everything in the long run.

When you rely on contradictions as anchors, you stop building structure and start collecting exceptions. Your internal compass no longer points anywhere consistent.

A deeper problem arises because learning also involves expanding your mental model, a process that happens even unconsciously. Your mind adapts to what you do. So, if you keep acting on inconsistent patterns, your common sense reshapes itself to accommodate them. Every patch you add drifts your model further from coherence, until intuition itself becomes completely unreliable (quite different from George Costanza: perfectly reliable instincts, consistently wrong)

One example is when people move to the UK and think, “Back home I looked left then right when crossing the street, so I’ll remind myself to do what feels weird, I’ll look right then left.” But hold two opposite rule sets at once and soon enough you will be right half the time and dangerously wrong the rest.

This is what it means to corrupt your common sense: holding two contradictory models at once doesn’t make you flexible; it makes you unreliable.

Now, what to do instead? Try to anchor to a new, coherent cue, not a negated one. When I first visited the UK, I’d joke to myself when crossing the street in an exaggerated accent: “Roighty-toighty, lefty-loosey!” Silly, but stable. It gave my mind a clean model to converge toward. The key is not to memorize the contradiction. The key is to build a consistent alternative your brain can trust.

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Why this matters

Every deep idea feels like this when it settles.
It starts as surprise and ends as compression.
The steps melt and what remains is a clear internal model you can rebuild from almost nothing.

Once a pattern fits that tightly, it becomes part of your intuition.
It is no longer something you remember. In a way, it becomes something you are.

This way of learning is personal. I’ve come to rely on it so deeply that I struggle to absorb anything I can’t make intuitive. I no longer try to memorize what I can’t re-derive.

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Where this leads

Once you start seeing through this lens, whole subfields begin to wobble. Probability “paradoxes” vanish the moment you anchor them in real, common-sense reasoning.

Take the classic puzzle: “A family has two children, and at least one is a boy. What is the probability that both are boys?” Now half say it’s 1/2, half say 1/3. People call it a riddle. It is not. It is bookkeeping confusion disguised as mystery. Treat it as an actual situation instead of a formula, and the paradox dissolves instantly. Common sense here means reasoning about how the cases are generated (hint: see probability as a sampling process, not as symbolic algebra).

The same applies across domains that celebrate mental stretching. Math drills, logic puzzles and coding challenges often train you to brute-force familiarity instead of integration. You’re told to solve a thousand problems until intuition appears. I suggest you solve the same problem in a thousand different ways instead.

Real insight comes when you find several solutions to the same problem and see how they converge into a single invariant. That is when your model compresses and real mastery begins.

Do this long enough and you start noticing hidden symmetries everywhere. Methods that looked different turn out to be expressions of the same idea. What once felt advanced becomes obvious, and what once felt obvious becomes profound.

This essay is the foundation. If you have thoughts, counter-examples, or ways this model fails, I’d love to hear them. The next steps I’m exploring involve testing how this kind of integration applies to probabilistic reasoning and creative skill formation.

If those explorations prove useful, I’ll turn them into follow-up notes on practicing and applying common-sense learning deliberately.

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Exercises

1. Compute the sum of the series $11+15+19+\dots +207$ using the “average × count” method. Notice how you can reason through it without a formal $n$-th term formula.
2. Think about examples of common-sense learning in your own life. When has a complex topic suddenly clicked for you? Would you say it was because you found a framing that made it feel simple and intuitive, fitting into what you already understood about the world?
3. Johnny is training to recognize music intervals. He can recognize most diatonic intervals but struggles with the major seventh. To remember it, he anchors it as “the one that sounds weird”. What happens when he starts listening to jazz and meets the minor ninth or augmented eleventh? How does his “it is the weird one” anchor break in this case and what does this tell you about relying on dissonance instead of structure?