How Nodeledge Works

Most courses move on with the calendar. Nobody checks you got this week before you get next week. At best there's a handful of questions at the end of the chapter, covering a fraction of it. In technical subjects knowledge is cumulative, so you build on fragile foundations and gaps compound.

The answer is mastery learning, set out by Benjamin Bloom in 1968: you don't move on until you've mastered what comes next depends on (Bloom, 1968). Across 108 studies it raised exam results, most for the students who had been struggling (Kulik et al., 1990). Nodeledge does this strictly. You can't start a topic until you've mastered the ones it depends on.

Read more: Technical learning needs a knowledge graph

When you're stuck on a topic, it feels like the topic is the problem. You re-read the chapter and it still won't go in, so you decide you can't do this subject. That's almost always wrong. Most people stuck on a topic are stuck on a prerequisite for it, often one from years ago, and they're grinding on the wrong layer.

David Ausubel made the point in 1968: the most important single factor in learning is what the learner already knows (Ausubel, 1968). New ideas only hold if they attach to ones you already have. So Nodeledge doesn't ask whether you're good at maths. It finds the prerequisite you're missing and fills it.

Read more: You're not bad at maths

Your knowledge frontier is the set of topics you're ready for: you've mastered everything they depend on, but not the topics themselves. That's where learning is fastest. Anything easier teaches you nothing new. Anything harder is trying to build on knowledge that doesn't exist yet.

Vygotsky called this band the zone of proximal development: what you can't do alone yet, but can do with help (Vygotsky, 1978). For a novice, the help that works is explicit instruction: a short explanation, a worked example, then you try it and get feedback on every answer (Kirschner et al., 2006). Nodeledge keeps you on your frontier. You choose the order.

Everyone is time poor. If you've already got some of a learning path, the fastest route is to prove it and start at your knowledge frontier, not to sit through lessons on it. There's a second reason. Instruction built for a beginner, with every step spelled out, gets in the way of someone who can already do it: they learn less from it than from a bare problem (Kalyuga et al., 2003). So anything you show you can do is marked as mastered and taken off your learning path.

Read more: Custom learning paths and diagnostic tests

Knowledge in technical subjects is hierarchical: every topic rests on the ones below it. That structure is what makes a short test possible. A right answer on an advanced topic is evidence you know what it depends on, and a wrong answer on a basic topic is evidence against everything built on it. This is knowledge space theory, the research behind adaptive placement tests (Doignon & Falmagne, 1985).

It's a trade-off. The more a test infers, the shorter it is and the more it can get wrong, and the two mistakes aren't equal. Mark a topic you know as a gap and you lose one lesson. Mark a gap as known and you build on it. We take progression seriously, so we ask for strong evidence before crediting a topic, and we'd rather err towards caution. In Nodeledge a topic is unknown until proven known.

Testing out of a topic is evidence you know it, not proof. To keep the diagnostic test short, we don't ask about every topic. A right answer also credits the topics it depends on, using careful inference built on the structure of the knowledge graph, and even careful inference is sometimes wrong. So every topic the test credits is still reviewed, and the ones it credited without asking are the first to be checked. If a review shows a gap, the lesson goes back on your learning path.

Working memory is the fundamental limit on how fast anyone can learn. It's where thinking happens, it holds about four things at once, and nothing reaches long-term memory without passing through it. When a topic asks for more than four, you don't slow down, you stop. The feeling of being overwhelmed by a page, or switching off halfway through, is usually just working memory blown.

Cognitive load theory is the study of this limit (Sweller et al., 1998), and it gives two ways to work within it. Introduce new things gradually, so each step fits. And make sure what came before is fluent, because anything you have to stop and work out is competing for the same four slots as the new idea. Nodeledge does both: every step teaches one thing, and the prerequisites, the diagnostic test and the reviews exist to keep what's beneath it fluent.

Read more: The ABCDE of cognitive load theory

Most people believe you learn best by struggling with a problem on your own. For a novice, you don't. You flounder. Sometimes you get there, but it takes far longer than being shown, and what you pick up on the way is often wrong and has to be unlearned later. Being shown the method first, solved line by line, then trying one yourself, is faster and more reliable, and the evidence for it is about as strong as anything in learning research (Atkinson et al., 2000; Kirschner et al., 2006). This is called direct instruction, and every step in Nodeledge starts with it: a worked example before you're asked anything.

Nobody expects to get fit by watching their coach work out. Learning is the same. You don't get better at maths, or code, or anything technical, by watching a lecturer fill a whiteboard, and most courses are still built that way: chalk and talk, a sage on the stage.

You learn what you practise, not what you watch. Psychologists call this transfer-appropriate processing (Morris et al., 1977). And answering is better practice than rereading: pulling an idea out of your head builds memory far better than reading it again, however much harder it feels (Adesope et al., 2017).

So most of your time in a lesson goes on answering questions at the right level, thousands of them across the platform. And the moment you answer, right or wrong, you see the full solution. Feedback that explains the answer beats being told whether you were right, and the effect is largest in maths (Van der Kleij et al., 2015).

Read more: Why performance is not learning

Beautiful visualisations are cool. They also help you learn: an idea you've seen as a picture and read in words is stored twice, and remembered better than either alone (Clark & Paivio, 1991). Attention is finite, though, so every diagram is built around the one idea the step is teaching, with nothing else on it.

The maths lessons carry hundreds of diagrams, and many of them are interactive. You can drag, slide, rotate and step through them, which makes the diagram part of the practice rather than something to look at.

We believe that everyone, with the right support, can reach the same high standard. It just takes some people longer than others. So the bar on every step is set high and it never moves.

The support is in the size of the steps, not in help over the bar. Most apps keep offering hints until they've lifted you over the bar themselves, and nothing has changed in your head. Nodeledge does the opposite. Every step is small, small enough that anyone can take it, but you have to take it yourself, and you have to take every one. A mountain is climbed one small step at a time.

This is mastery learning inside a lesson (Bloom, 1968): the standard is fixed, and the time is what varies. This can be jarring. Most apps are kind in the moment, and that turns out to be unkind in the long run. We'd rather be tough in the moment and kinder in the long run.

Memory is the residue of thought (Willingham, 2009). What you think about during a lesson is what you keep, and attention is finite, so anything on the screen that isn't the idea is taking attention from it. Fancy videos and animations are cool, and what you remember afterwards is the video.

For most of history, learning maths meant a problem, a pen and a blank page. That's still what it needs. Nodeledge is the modern version: text, diagrams, questions, and nothing else moving. Python runs in the browser with nothing to install, and the reference sheet and calculator sit beside the lesson, so the only thing left to think about is the work.

Read more: Why you should learn to code in the browser

You forget most of what you learn within a day or two of learning it, and after that the loss slows down. This is the forgetting curve, first measured in 1885 and still holding up (Murre & Dros, 2015). Every time you recall something before it's gone, the curve gets shallower: it fades more slowly, so the next review can wait longer. This is called spaced repetition. It beats cramming by a wide margin (Cepeda et al., 2006), and the result holds for maths problems as well as for facts (Rohrer & Taylor, 2006).

Read more: Introducing hierarchical spaced repetition

Ordinary spaced repetition treats every item as separate, so a thousand topics means a thousand review schedules, and the load grows until people quit. Technical subjects aren't separate items. Differentiating a product uses the product rule, the power rule and algebra, so a review of the product rule is practice of all three. Nodeledge's implementation credits that: each review you pass gives weighted credit to the topics it depends on, and each one you fail flags the topics built on it. The result is that a few reviews a day keep hundreds of topics fresh, where reviewing each one on its own would take several times as long.

Read more: Introducing hierarchical spaced repetition

Two things make getting stuck rare here. You never sit a lesson you aren't prepared for, because you've mastered everything it builds on. And every step inside it is small enough to take. So most lessons pass first time, and the ones that don't usually pass on the second. Failing one isn't a wall. You've seen the full solution to every question you got wrong, and often all you need is a break and a fresh mind. So there's a short wait, then the lesson comes back before anything new.

A lesson that keeps failing isn't a verdict on you. It means one of two things. Usually a topic it builds on isn't as solid as the record says. A lesson is the hardest test its prerequisites get, because you have to use them, and the fix is to go back to that topic and rebuild it. We're building the questions that trace a failure back to the prerequisite behind it and teach that first. The other cause is the lesson itself: a step too big, or too much to hold in your head at once. When that happens we change the lesson, split the step and cut the load. Most courses are written once and left. Ours is live, and the lessons people fail are the ones we fix first.

Forgetting is real for everyone, and fluency takes time to build. Passing a lesson once is the start of knowing something, not the end. Most apps give you a tick and never take it away. Nodeledge does. Recall varies from day to day, so one failed review is weak evidence that you've forgotten the topic, and you get a second check a couple of days later. Fail that as well and the topic is no longer counted as mastered. It goes back on your learning path and you learn it again. That can feel unfair in the moment. But everything after that topic is built on it, and building on shaky knowledge is the slow way. It's quicker to relearn one topic now than to fail the ones that depend on it later.

Read more: What happens when you forget?

Life gets in the way. You come back after a month, everything you learned is fading, and the temptation is to press on with new material so it feels like progress. That's the worst order. New topics built on fading prerequisites are harder to learn, and the prerequisites fade further while you're busy with them. So when reviews pile up, Nodeledge puts them before new lessons. And if there are too many, you don't have to grind through them one at a time. Retaking the diagnostic test finds what you still know in one sitting, faster than working through every review.

Consistency is king. Little and often wins the race, because learning doesn't happen in the sitting. The brain needs time between sessions to consolidate what you've met, which is why spacing beats cramming and why the reviews only work if you keep coming back. So what gets you through a learning path isn't a heroic weekend. It's a small amount most days, kept up for months. That takes a habit, and a habit needs a goal: specific, small enough to hit today, and with the result in front of you straight away. That's the daily XP target. You earn XP for the questions you get right, and hitting the target keeps your streak going. The rules are the ones that build any habit: make it easy, make it satisfying, and never miss twice.