Why education is organised into subjects when knowledge isn’t

We separated knowledge because it made education easier to organise.

Schools need departments, teachers need subjects, textbooks need chapters and exams need a syllabus. Drawing boundaries around mathematics, physics, biology, computer science and statistics makes the whole system manageable.

The problem is that knowledge itself doesn't respect those boundaries.

Probability appears in genetics. Calculus runs through physics and engineering. Linear algebra sits underneath computer graphics and machine learning. Statistics becomes difficult to separate from medicine the moment you try to understand how a clinical trial works.

Once you start with a real question instead of a school subject, the lines get blurry very quickly.

That matters because most education is still organised around courses. A course has a fixed starting point, a fixed sequence and a rough idea of who the learner is supposed to be. Real learners are much messier than that. They arrive with years of knowledge, forgotten knowledge, uneven foundations and goals that often cross several subjects at once.

AI has made this mismatch more obvious because explanations are becoming abundant. You can already ask for a simpler explanation of calculus, ten examples of Bayes' theorem or a walkthrough of gradient descent and get an answer in seconds.

The harder problem is knowing what you need to learn in the first place.

A real learning goal rarely belongs to one subject

Say you want to understand clinical trials.

You could reasonably think of that as a medical topic, but it quickly pulls you into probability, distributions, sampling, hypothesis testing, confidence intervals, bias, randomisation and experimental design. Then you need enough biology and medicine to understand what is actually being measured.

There isn't a neat line where the statistics ends and the medicine begins.

Machine learning works the same way. A proper understanding may draw on algebra, functions, vectors, matrices, linear algebra, calculus, optimisation, probability, statistics, algorithms and programming.

The usual response is to turn those areas into separate courses and ask the learner to work through them.

A better question is: which concepts does this particular goal depend on?

That changes the shape of the problem. Instead of choosing a curriculum first, you start with the thing the person wants to understand and work backwards.

Two people can have the same goal and need completely different routes

Imagine two people who both want to learn machine learning.

One is a software engineer. She is comfortable with programming, algorithms and building systems, but her calculus and statistics are rusty.

The other studied physics. He is comfortable with derivatives, matrices, probability and optimisation, but has very little practical programming experience.

Calling both of them "machine learning beginners" is technically correct and almost useless.

The software engineer doesn't need an introduction to variables and loops. The physics graduate probably doesn't need six lessons on vectors. Their destination is the same, but the missing knowledge between them and that destination is different.

A normal course has to choose one starting point. That's a reasonable compromise when you're teaching a room full of people.

Software doesn't have to make quite the same compromise.

If a learning system has a decent picture of what someone already knows, it can build from there rather than pretending everyone enters at the same point.

That's a more interesting version of personalised learning than simply making the next quiz harder or easier.

The thing you're struggling with may not be the thing you're missing

Anyone who has taught something difficult has seen this happen.

A student looks like they have a calculus problem, but after a few questions you realise the real issue is algebra. Someone struggles with probability because fractions never became intuitive. A physics problem feels impossibly complicated because rearranging equations still takes too much concentration.

The difficulty shows up at one level, while the gap sits somewhere underneath it.

This is why prior knowledge matters so much. New ideas have to attach to something. When the foundations are shaky, the later concept gets harder for reasons that aren't always obvious to the learner.

A good teacher naturally probes backwards.

They ask how the student reached an answer. They look at the previous step. They test a simpler version of the idea. Eventually they find the point where understanding becomes uncertain.

The interesting question is whether we can represent enough of those relationships explicitly that a learning system can do something useful with them too.

Prerequisites turn a syllabus into a network

Take a concept like quadratic equations.

Before they become comfortable, a learner probably needs some combination of algebraic notation, arithmetic, variables, equations, powers and factorisation.

Those ideas have dependencies of their own.

Follow the chain forwards and quadratic equations connect into functions, graphs, optimisation, mechanics and other areas of mathematics and physics.

Now zoom out further.

Probability connects into statistics, genetics and medicine. Vectors connect into mechanics, computer graphics and machine learning. Exponential functions show up in finance, population growth, radioactive decay and computing.

The same concept can support many different goals.

At that point, knowledge starts looking less like a list of lessons and more like a network.

What a knowledge graph gives you

A knowledge graph represents concepts and the relationships between them.

For learning, that could include which concepts depend on others, which ideas are closely related and where concepts from different subjects connect.

A tiny section of a maths graph might look roughly like this:

fractions → ratios → proportions → similar triangles → trigonometry

Real knowledge is obviously more tangled. Concepts can have several prerequisites, one idea can support hundreds of later topics, and branches often split and reconnect.

That complexity is useful rather than inconvenient.

Once the structure exists, you can start asking much better questions:

• Which concepts does this goal depend on?
• Which of those does the learner already understand?
• Where is the first important gap?
• Which parts can be skipped?
• Which knowledge from another subject already applies here?

Add a learning goal and some evidence about what the person knows, and you can start turning the graph into a route.

Start with the goal and work backwards

Imagine the learner says:

"I want to understand clinical trials."

Instead of sending them to lesson one of a statistics course, start with the goal.

Work backwards through the concepts that support it. The resulting route might contain probability, distributions, sampling, expected values, variance, hypothesis testing, confidence intervals, statistical power, randomisation, experimental design, bias and some relevant biology.

Then compare that route with what the learner already understands.

If their probability is solid, there is no reason to teach it again. If hypothesis testing is new, it stays. If their answers reveal a weaker foundation than expected, the route can extend backwards until it reaches something stable.

What remains is the knowledge between the person and the goal.

That is far more specific than saying, "Take a statistics course."

The same idea works for machine learning. One person's route may spend heavily on calculus and linear algebra. Another may already know the mathematics and need more computer science. Someone else may know both and mainly be missing probability.

The route changes because the learner changes.

Courses are useful, but they are still one route

A good course contains a huge amount of judgement.

Someone has decided what belongs together, what should come first, which examples are useful and how much depth a learner probably needs.

That's valuable.

A course still has to assume a starting point, though.

The learner may know half the early material already. They may have forgotten one concept the course assumes is secure. They may only need one branch of the subject for the thing they're trying to do.

This gets especially obvious with adults.

By the time you're thirty or forty, your knowledge has come from school, university, work, books, videos, projects and random problems you had to solve because life demanded it.

Then you enter a new learning platform and most of that history disappears.

The platform knows which lessons you've completed there. It usually knows very little about what you actually understand.

So you start again.

Learning should carry forward

Suppose you learn enough statistics to understand clinical trials.

A year later, you decide to study machine learning.

You shouldn't have to relearn probability simply because you've entered through a different subject.

The same goes for calculus learned during an engineering degree, programming learned at work or algebra learned years ago and still remembered perfectly well.

Knowledge should carry forward.

Most digital learning systems record progress through content: lesson completed, module finished, course passed.

Those records are useful, but they aren't the same thing as a model of what someone knows.

A learner can finish a calculus course while remaining weak at integration. They can pass an algebra exam and forget half of it two years later. They can understand a concept deeply without ever having taken a formal course that includes it.

A concept-level map gives you a much better picture.

Instead of a pile of completed courses, you get an evolving model of what you know.