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I think everyone should learn calculus.
Or at least, I think everyone should have the chance to see what calculus reveals about the world. There is beauty in that part of mathematics: motion, change, accumulation, and slope becoming parts of the same language.
So I am not suspicious of theory. I like theory.
But my first teaching role, shortly after finishing my MBA, showed me something I had not understood before: learning theory is not the same thing as understanding what theory explains.
I had been hired to teach accounting at Northern Lights College in northern British Columbia. After years of tests in school, this was my first real one.
Thirty students.
I scribbled my notes from the text and faithfully replicated them on the board. T-accounts. Debits. Credits. And Generally Accepted Accounting Principles. Just as I had been taught.
I had done well in academic settings. Well enough to trust, perhaps too easily, that if I could learn something from a textbook, I could teach it from one.
But I was shocked when, after filling the board with T-accounts, one of the bookkeepers asked me where the rest of the record was.
I looked at the T-account with its debit value, then at another T-account with its credit value. I had labeled them correctly. They balanced.
I looked back at the student.
“The rest of the record?” I asked.
That was the moment I realized there was a difference between learning accounting and understanding the system accounting was trying to describe.
I had learned the formal structure. They had lived inside the procedure, a procedure I had never learned despite taking accounting courses to the advanced level.
And then I discovered something.
When I explained the matching principle, they did not hear it as an abstract rule. They recognized it. They had been making those entries for years because someone had told them to do it that way. Now they knew why.
The theory did not create understanding from nothing. It organized experience they already had.
Generally Accepted Accounting Principles were absorbed almost instantly by the bookkeepers in the class, while the other students were still trying to figure out what had happened.
I thought about that accounting class again recently after a conversation with a colleague about Engineering education. Her position was familiar: teach the theory first, then let students apply it. My view, shaped by the classroom, was less orthodox. Sometimes the application should come first, because it creates the question the theory answers.
The disagreement was not about whether theory matters. It was about sequence. In Engineering education, the sequence can feel almost fixed. Learn the principles. Enter the lab. Apply the model. Confirm the calculation.
I understand the appeal of that order. It is tidy. It is defensible. It feels rigorous. It is also how many of us were taught.
But my experience at Northern Lights made me less certain that it is always the best order.
Sometimes it is better to have someone perform the task first and then teach the theory afterwards.
The “aha” moment comes later, when theory gives a name to something the learner has already experienced.
The theory-first instinct is not wrong. There are places where application-first is irresponsible. Some mistakes are too expensive, too dangerous, or too misleading to be allowed as discovery exercises. No one wants students wiring high-voltage systems, designing bridges, or tuning control loops entirely by intuition. Some theory must come before some forms of practice.
But that does not mean all theory must come before all practice.
A student who has watched a control loop overshoot a setpoint, correct itself, overshoot again, and slowly settle may be more ready to understand damping than a student who has only copied the definition.
In many learning situations, theory works less like a foundation and more like a map.
But students cannot stand on a map.
They need some ground beneath their feet before the map can mean anything.
Application-first teaching is not the same as turning students loose and hoping confusion becomes insight. The instructor still chooses the room, the tools, the limits, and the failure. The voltage is safe. The system is bounded. The mistake teaches something specific. The theory arrives before the wrong lesson is learned.
The accounting class at Northern Lights took a different path than I expected. I learned bookkeeping from the students, and they learned accounting from me. Everyone learned more than expected.
I still believe in theory. I may believe in it more than many people do. I still think everyone should have the chance to see what calculus reveals about the world.
I even perceived calculus differently. In Grade 11 physics, I had already worked with motion, change, and slope through algebra, not calculus. The principles arrived first. Later, calculus gave them a stronger framework than algebra alone could.
Teaching changed my sense of when theory should arrive. Sometimes theory is the beginning. Sometimes it is the answer to a question the student has not yet learned to ask.
And when it arrives after the problem, after the work, after the frustration, it can land with surprising force.
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