There is a moment in teaching that I have come to treasure above all others.
It is not the moment when a learner correctly recalls a formula. It is not the moment when someone scores well on a test. It is the moment when a learner looks up from their work with a slightly surprised expression and says: I think I see it.
That pause. That look. That is what I am always trying to create.
The Problem with How We Usually Teach
Most education begins at the wrong end.
A student opens a textbook and finds: The Pythagorean theorem states that a² + b² = c². An explanation follows. Some worked examples. Then practice problems.
This approach is efficient. In forty minutes, you can deliver a concept to thirty people. But something important is missing: the experience of discovery.
When a formula is handed to you, you receive it as a package you must store and retrieve. You may understand it intellectually. But you do not know it the way a geologist knows rock — through touch, through repeated encounter, through being surprised by it in the field.
More importantly: when you receive a pre-formed answer, you are implicitly told something about yourself. You are told that knowledge comes from outside you. You are positioned as a recipient. And after years of this experience, many people come to believe a genuinely harmful thing: that they are simply not the kind of person who understands mathematics, or physics, or algorithms.
I have seen this belief in adult learners over and over again. Intelligent, capable people who are certain, before they begin, that the material will be beyond them.
A Different Starting Point
What happens if we begin differently?
Instead of stating the Pythagorean theorem, give a learner a piece of squared paper. Ask them to draw a right triangle. Ask them to draw a square on each side. Ask them to count the squares inside each shape.
They do this for a triangle with sides of 3, 4 and 5. They count 9, 16 and 25.
Ask them: do you notice anything?
Some will not see it immediately. That is fine. Give them another triangle — 5, 12 and 13. They count 25, 144 and 169. Ask again.
Something happens. A hesitant observation: the big one always seems to equal the other two together?
Yes. Precisely. Now they have made a discovery.
Their relationship to this idea is completely different from someone who read it in a textbook. They found it. It is theirs.
What Discovery Does That Instruction Cannot
Discovery creates several things that instruction alone cannot reliably produce.
Understanding of the why. When you discover a pattern, you immediately want to know why it is true. Your curiosity is already pointed in the right direction. An explanation at this moment lands completely differently from an explanation that precedes the observation.
Memory. Things we discover are remembered differently from things we are told. The moment of surprise, the sequence of problems, the sensation of recognition — these are emotionally and cognitively vivid in a way that passive reception rarely is.
Confidence. This is the most important one. When a learner discovers something, even something small, they experience their own mind working. They feel what it is like to figure something out. This is radically different from being told something correct.
There is a story I return to often. A learner who had spent years believing she was simply not good at mathematics sent me a message after one of my courses. She said she had gained enough confidence in her own thinking that she had started teaching mathematics to her young niece.
She was not teaching because she had memorised more facts. She was teaching because she had learned to trust her own mind.
The Design Challenge
Learning by Inventing is not simply a matter of attitude or intention. It requires careful design.
You cannot just present a problem and hope the learner discovers the right thing. The sequence matters enormously. Each problem must be small enough to be approachable but designed to point attention in exactly the right direction. The gap between steps must be wide enough to require genuine thought but narrow enough to be crossable.
This is hard work. Designing a good discovery sequence for, say, gradient descent in machine learning takes more thought than writing a clear lecture. You have to deeply understand the concept yourself — not just its final form, but how a mind might move toward it.
I think of it as the difference between building a path and simply standing at the destination. Any teacher can stand at the destination and describe what they see. Far fewer can build a path that leads someone there while feeling like they are exploring freely.
Machine Learning Through Discovery
When I teach machine learning, I do not begin with neural networks or backpropagation.
I begin with a simple prediction problem. I ask learners to guess, by hand, some numbers that are unknown but clearly related to numbers they can see. I ask them to try to improve their guesses. I ask them what they would change if they were wrong.
Through this process, learners begin to invent the core ideas of machine learning — not because I have told them what to invent, but because those ideas are the natural response to the problems I have set. They discover that iterating toward a better answer is more powerful than trying to get it right the first time. They discover that measuring error is the key to improvement.
When we eventually name these ideas — loss function, gradient, parameter update — the names feel like labels for something the learner already understands, rather than terms they must memorise.
What This Means for the AI Teacher
I have spent some time thinking about what an AI teacher based on this philosophy would look like.
Not an AI that answers questions — there are plenty of those. An AI that asks the right question at the right moment. One that can assess where a learner is and design the next small step accordingly. One that knows when to be silent and let the learner struggle, and when to offer a carefully chosen nudge.
This is a harder problem than building a question-answering system. But I believe it is one of the most important things we could build with AI.
Because the goal is not a learner who knows more. The goal is a learner who believes in their own capacity to discover.
If you are an educator, a learner or a builder interested in this philosophy, I would be glad to exchange ideas. You can find more of my thinking on education at the Learning by Inventing section of this site.