Map the subject before you study it: concepts and prerequisites as a graph

A course arrives as a list: twelve lectures, a reading list, a syllabus with bullet points. You study it the same way, top to bottom.

But the material is not a list. A confidence interval depends on the sampling distribution, which depends on the standard deviation. If the standard deviation is shaky, the next three lectures will feel hard for reasons that have nothing to do with those lectures.

Before an exam it pays to spend an hour turning the list into a graph: concepts as nodes, and links that say how they depend on each other. This post explains what such a map is, what the research does and does not support, and how to make one that helps.

Why structure matters

Joseph Novak's research group at Cornell developed concept maps in 1972, building on David Ausubel's psychology of learning. In Novak and Cañas's summary, the central idea is that learning happens when new concepts and propositions are assimilated into the framework of concepts the learner already holds.

In practice that means two things. What you can learn next depends on what you already know. And when something does not make sense, the cause is often an earlier concept, not the one in front of you.

There is a related finding about experts. Chi, Feltovich and Glaser (1981) studied how physics experts and novices sort and approach problems. Novices started from the problem's literal features, the things the problem mentions. Experts started from the physics principle that applies. A map that is organised around principles rather than surface topics pushes you toward the expert view.

What a concept map is

Novak and Cañas (2008) describe the parts:

• Concepts, usually short noun phrases in boxes.
• Links between concepts, labelled with a verb or phrase ("is computed from", "is a special case of").
• Propositions: two concepts plus their link form a statement you could check, for example "standard error is computed from the standard deviation".
• A focus question the map answers, so you know what belongs on it.
• Cross-links between different parts of the map, where much of the insight sits.

They also give practical advice. Start from a list of key concepts, usually 15 to 25 is enough. Keep the ones you have not placed yet in a "parking lot" next to the map. Expect to revise: good maps usually take several versions.

What the research supports

Nesbit and Adesope (2006) analysed 55 studies with 5,818 students, from Grade 4 to university. Using concept maps was associated with better knowledge retention across many settings, with effects ranging from small to large depending on how maps were used and what they were compared with.

Schroeder, Nesbit, Anguiano and Adesope (2018) updated the picture with 142 effect sizes and 11,814 participants. The overall effect was moderate (g = 0.58). Creating maps (g = 0.72) helped more than studying maps someone else made (g = 0.43).

So drawing your own map beats reading a finished one.

What the research does not support

There is an important limit. Karpicke and Blunt (2011) compared concept mapping with retrieval practice, matched for time. One week later, the retrieval group did better, even on a final test where students had to draw a concept map.

The lesson is not "skip maps". It is that a map drawn while looking at the textbook is mostly a reorganisation exercise. It shows the structure of the course, which is useful for planning. It does not replace practising recall.

The fix is simple: once your map exists, redraw it from memory. That turns the map into a retrieval exercise (see the post on retrieval practice).

Prerequisite links

Most concept maps use many link types. For exam planning, one type matters most: "A is a prerequisite of B", meaning you cannot really understand B without A.

This idea has a formal side. Knowledge space theory, introduced by Doignon and Falmagne (1985), describes a field as a set of questions or problems and a learner's knowledge state as the subset of them the learner can solve. Because some problems build on others, not every combination is a realistic state. The ALEKS assessment system uses this theory to estimate what a student knows and what the student is ready to learn next.

You do not need the mathematics to use it. Draw prerequisite links as arrows from the prerequisite to the concept that depends on it. Then:

• Study order follows the arrows: a concept is ready to study when its prerequisites are solid.
• Diagnosis goes against the arrows: when you are stuck on a concept, walk back to its prerequisites and test yourself on them first.

How to build one in an hour

Here is the procedure, with an introductory statistics course as the example.

1. Write the focus question. Make it about the exam, not the course: "What do I need to be able to do in the statistics final?" Past papers and the syllabus learning outcomes are the best sources.

2. Fill the parking lot. List 15 to 25 concepts from lecture titles, past papers and the glossary. Use concepts, not chapter titles. For statistics: random variable, mean, standard deviation, normal distribution, sampling distribution, standard error, confidence interval, null hypothesis, p-value, t-test, chi-square test, correlation, regression.

3. Draw the prerequisite arrows first. For each concept, ask "what must I understand before this makes sense?"

4. Add a few labelled links. "Chi-square test is used for categorical data", "regression slope is related to correlation". Keep it to links that would answer an exam question.

5. Mark what you know, by testing. For each concept, try to explain it on a blank page or solve one problem. Mark it solid, shaky or unknown. Do not mark it by how familiar it looks; familiarity is a poor guide.

6. Pick the frontier. Your next study targets are the shaky or unknown concepts whose prerequisites are all solid. In the table above, if standard deviation is solid but sampling distribution is shaky, fix the sampling distribution before you touch confidence intervals.

7. Redraw from memory every few days. Compare with the previous version. Missing nodes and links are your next review list, and new cross-links are a sign you understand more than before.

Common mistakes

• Too big. A map with 80 nodes is a textbook index. Stay near 15 to 25 per exam, and make sub-maps for dense areas.
• Unlabelled lines. A line without a label does not say anything you could check. Label it, or use only the prerequisite arrow.
• Copying the table of contents. Chapters are an order of presentation, not of dependency. Several prerequisites usually cross chapter boundaries.
• Decorating. Colours and layout are not learning. A pencil sketch you redraw three times is worth more than one beautiful diagram.
• Mapping instead of practising. The map tells you what to practise and in what order. The practice itself is still recall and problems.

The short version

Turn the syllabus into a small graph of 15 to 25 concepts. Draw prerequisite arrows, test yourself on each node, and study the frontier: concepts that are not solid yet but whose prerequisites are. Redraw the map from memory to check what stuck.

Learnode builds this kind of map as you learn with it, with prerequisite links between concepts and a record of what you have practised; a sheet of paper does the same job by hand.