Breadth first, depth first: how to order a syllabus

There are two ways to walk a syllabus, and most students pick one without noticing they have chosen.

Two ways to walk a syllabus

Borrow the names from graph search. Breadth-first learning means getting a basic understanding of every topic before deepening any of them: skim all eight exam topics, learn the core method for each, and only then start on the harder questions. Depth-first learning means mastering one topic fully before moving to the next: branch and bound from the basics, through the proofs, to the hardest exam questions, before you touch network flows.

Both are real strategies with real advantages. The mistake is running either one on its own.

Why pure depth-first fails

Depth-first feels productive because progress inside a topic is visible: yesterday you could not do the hard questions and today you can. The failure is in the calendar. A course with eight topics and a fixed exam date gives each topic a share of your time whether you plan it or not, and depth-first spends that share on the topics you happen to start with. The common outcome is a student who is excellent at forty percent of the course and has never seen the rest.

There is a second cost. Many exam questions combine topics, and a topic studied in isolation is stored in isolation. You can be fluent in duality and fluent in sensitivity analysis and still not recognise a question that needs both.

Why pure breadth-first fails

Breadth-first fixes the calendar problem and creates a different one. Skimming every topic produces recognition rather than recall: you can follow a worked solution for any topic, and you can start none of them from a blank page. Recognition feels like understanding right up to the moment the exam removes the worked solution.

Recognition feels like understanding until the exam takes the worked solution away.

Pure breadth also has no stopping rule. Every topic can always be covered a little more, so the plan never reaches the hard questions that decide the grade.

Breadth, then depth, then breadth

For exam preparation, the strongest order is usually breadth-first, then depth-first, then breadth-first again.

1. Breadth, to establish minimum competence. Cover the whole syllabus to the level of solving the standard question for each topic. Not the hard one: the one the examiner expects everyone to get. This pass tells you the shape of the course and, more importantly, which topics are weak.
2. Depth, on the weakest and the most valuable. Go deep on the topics where the first pass hurt most, and on the topics that carry the most marks. This is where hard questions, proofs and past-paper problems belong.
3. Breadth again, with mixed questions. Finish with sets that mix topics without telling you which method applies. The skill being trained here is choosing the method, which neither of the earlier passes practises.

The order matters because each pass produces the information the next pass needs. The first pass finds the weaknesses; the second fixes them; the third checks that the fixes survive contact with a question that does not announce its topic.

How Memoza fits

Memoza's recommender does not follow the syllabus order. It scores every concept in a course on the gap between your evidence and mastery, on how uncertain that estimate still is, on the risk that you have forgotten it, on how much the assessment weights it, and on whether a misconception keeps recurring. Two penalties pull the other way: one for concepts whose prerequisites you have not yet shown, and one for anything practised recently. The prerequisite penalty is the breadth-first rule, in code: it keeps you off a topic before its foundations exist. The gap and forgetting terms are the depth-first rule. The recency penalty is what stops it drilling one topic into the ground. It is the three-pass order above, recomputed as new evidence arrives.