One of the reasons I was successful teaching mathematics is that I never learned it the way most mathematicians did.

I did not come through an academic pipeline. I studied mathematics while driving a taxi. Between fares I worked problems, read books, and slowly built an understanding of the subject. Later I passed the qualifying test and found myself teaching mathematics despite having less formal education than anyone else in the department.

I was probably the least educated teacher in the math department.

I was not the least effective.

Students often asked the same question.

“Why am I doing this? I’ll never use any of this.”

I told them mathematics is like weightlifting.

Athletes do not lift weights because football games involve barbells. They lift weights because the strength developed in the gym helps them perform better on the field.

Mathematics works much the same way.

Most students will never use quadratic equations. Many will never calculate a derivative. Some will never need more than basic arithmetic.

That is not the point.

Mathematics strengthens the ability to reason, to recognize patterns, to think through a chain of logic, and to solve problems. Those abilities transfer to almost everything else in life.

A stronger mind is useful regardless of what you do for a living.

What I discovered as a teacher is that many students are not actually bad at mathematics. They are being taught in a way that does not match how they learn.

Most math instruction begins with formulas and procedures.

I discovered that many students need a picture first.

They need to understand what the machine does before they can understand its parts.

Recently I was thinking about the chain rule from calculus. I never taught calculus, and for years the chain rule never completely made sense to me. I understood the formula. I could see the symbols. What I did not see was the process.

Then it finally clicked.

The chain rule is simply change moving through a sequence of steps.

If time affects miles driven, and miles driven affects taxi fare, then the change in fare depends on both rates of change.

The effects multiply as they pass through the chain.

Suddenly the formula was no longer a formula. It became a description of something happening.

Once I could see the machinery, the mathematics became obvious.

That experience reminded me of countless students I taught over the years.

Some students learn quickly. They can manipulate symbols almost immediately. They move through a course at high speed.

Other students move more slowly.

The educational system often assumes the fast students are smarter.

I am not so sure.

The slower student frequently asks a different question.

“Why does this work?”

A fast student may learn a procedure in ten minutes.

A slow student may need three days.

But when the slower student finally understands the underlying process, he often possesses something far more durable than a memorized procedure.

He possesses understanding.

Many mathematics courses move at a pace designed for the students who grasp abstractions immediately. The students who need time to build a mental picture are left behind.

That is unfortunate because some of those students become the best problem solvers later in life.

They cannot move on until they understand the machine.

Once they do, they never forget it.

Teaching mathematics taught me a lesson that applies far beyond mathematics.

People rarely learn well when they are handed formulas.

People learn when they understand what the formulas are describing.

Before a student can learn a formula, he must first see the machine behind it.

Once he sees the machine, the formula almost teaches itself.