I understood calculus only when Arash asked me to explain it.
Fifteen years after passing the exam, supposedly mastering the material, I finally comprehended what derivatives actually meant. Not the formulas—those I’d memorized. But the concept itself.
Something alchemical happens in transformation from student to teacher.
Arash was twelve, struggling with pre-calculus. “Baba, what’s a derivative?”
I opened my mouth. Nothing came out.
I’d taken calculus. Passed with decent grades. Used it in university courses. But explain it simply? No idea where to start.
“It’s… the rate of change,” I managed.
“What does that mean?”
“How fast something changes.”
“Like speed?”
“Yes, exactly like speed!”
And suddenly I understood. Derivative was just speed but for any function, not just distance. The formula I’d memorized made sense for the first time.
Teaching demands simplification without oversimplification. Distilling complexity into accessible language while preserving meaning.
This compression reveals whether we truly understand or merely think we understand.
I’d thought I understood calculus. Turned out I’d memorized procedures without grasping concepts.
“So if I have this function,” Arash drew a curve, “the derivative tells me how steep it is at each point?”
“Yes.”
“Why didn’t my teacher just say that?”
Good question. Why do we wrap simple ideas in complex language?
I spent that evening relearning calculus properly. Not formulas—concepts. Building understanding from ground up so I could explain it clearly.
The Feynman Technique: if you can’t explain something simply, you don’t understand it well enough.
Teaching forces rebuilding knowledge from first principles, ensuring foundation can support someone else’s learning.
Next day, tried again with Arash. Drew pictures, used analogies, broke down each concept into digestible pieces.
“That makes sense!” he said.
It made sense to me too. Finally.
His questions illuminated blind spots. “Why does the power rule work?” “What if the function isn’t smooth?” “Can you have negative derivatives?”
Questions I’d never asked. Concepts I’d never questioned. My education had been about memorizing, not understanding.
The eight-year-old who asks “why” until we reach limits of knowledge performs valuable service. Their innocent curiosity exposes our sophisticated ignorance.
Started applying this elsewhere. Tried explaining my work to Happy in simple terms.
“What exactly do you do all day?” she’d asked.
Realized I couldn’t explain it clearly. Which meant I didn’t understand my own job as well as I thought.
Teaching activates multiple learning pathways simultaneously. Process information verbally while constructing visual aids. Engage analytical thinking while accessing creative explanation strategies.
Brain fires across domains that passive learning never engages.
My colleague Rashid mentioned teaching programming to his daughter. “I’ve been coding for fifteen years. But explaining loops to a ten-year-old made me realize I’d been doing them mechanically without really understanding.”
Same phenomenon. Teaching reveals gaps.
Started volunteering at Arash’s school, helping with math. Not because I was expert—because teaching forced me to become one.
Each student’s confusion revealed different aspects I hadn’t fully grasped. Their questions made me dig deeper, understand better, explain clearer.
Teaching requires emotional intelligence too. Reading comprehension signals, adjusting pace, maintaining patience when explanation fails.
These interpersonal skills deepened my relationship with knowledge itself. Transformed abstract information into human connection.
“You’re better at explaining this than my teacher,” one student said.
Not because I knew more. Because I still remembered not knowing. Recently struggled with same confusion. Could bridge that gap.
The preparation phase taught as much as presentation. Anticipating potential confusion forced examining concepts from multiple angles.
Creating examples demanded thorough understanding. Building analogies required deep structural comprehension of both source and target domains.
For calculus, I had to understand not just math but physics, geometry, real-world applications. Had to see connections I’d missed in formal education.
“Why are you spending so much time on this?” Happy asked, watching me prepare lesson plans for volunteer teaching.
“Because I’m learning more than they are.”
Perhaps most importantly, teaching revealed knowledge as living system rather than static collection. Information became dynamic when shared, evolving through interaction with questions, growing through collaborative exploration.
Knowledge multiplies when divided.
Arash asked about integrals next. I realized I didn’t understand those either. Knew the formulas, could solve problems. But conceptual understanding? Missing.
So I learned. Properly this time. Then taught him.
“Integrals are just adding up infinite tiny pieces,” I explained.
“Like summing up infinite slices?”
“Exactly.”
We both got it. Together.
Started seeking opportunities to teach what I was learning. Not just mathematics—everything.
Learning photography? Taught Happy and Arash basic principles. Forced me to understand composition, exposure, light in ways watching tutorials never had.
Learning about investment? Explained it to colleagues. Revealed gaps in my understanding I could then fix.
Every explanation sharpened comprehension. Every question from students illuminated blind spots. Every teaching opportunity became learning opportunity.
“You’ve become obsessed with teaching,” Happy observed.
“I’ve become obsessed with actually understanding things.”
“Same thing?”
“Same thing.”
The teacher learns twice—once when studying, once when sharing.
My formal education had given me once. Teaching gave me twice. Made all the difference.
Tonight helped Arash with trigonometry. Concepts I’d “learned” twenty-five years ago, never truly understood until I had to explain them.
“Why is sine opposite over hypotenuse?” he asked.
I drew pictures, explained ratios, connected it to circles, showed real applications. By the end, I understood trigonometry for the first time.
At forty, finally learning what I should have learned at fifteen.
Better late than never.
The act of teaching transformed me from someone who knew formulas into someone who understood concepts. From passive recipient of information into active constructor of knowledge.
Arash benefits. But I benefit more.
He’s learning math. I’m learning how to learn.
And that’s the deeper lesson—teaching isn’t just knowledge transfer. It’s knowledge creation. For both parties.
Every explanation forces clarity. Every question demands depth. Every misunderstanding reveals opportunities for better understanding.
The teacher learns twice. The student once. But both learn together.
And that’s how education should work—not hierarchical transfer but collaborative discovery.
Tonight I seek more opportunities to teach, knowing each will deepen my own understanding. Because the best way to learn something is to teach it.
The student in me thanks the teacher in me. And the teacher in me thanks the student asking questions.
We learn together. We grow together.
That’s the alchemy.