Problem-First Learning

Problem-first learning is a way to study a formula-heavy subject. The learner first works out which problem the topic exists to solve, and every way that problem shows up. Only after that does the learner open the equations.

The usual order runs the other way. A learner reads a little theory, jumps to the equations, and starts working practice problems. Practising a problem this way teaches the learner how to answer that one type of question and nothing more. The result of an attempt never goes back into the learner’s picture of how the topic fits together. Each attempt is a guess about which stored procedure to run. A correct answer arrives without the learner knowing why that procedure applied. A question that puts the same maths inside a real situation stays out of reach, because that situation was never part of the practised pattern.

Core takeaways

• Working practice problems before knowing what the topic is for teaches one question type per problem. Questions that wrap the same maths in a situation stay out of reach.
• The first pass is a flat list of every name in the topic. Each name is asked three things in order, what it is, why it matters, and how it relates, before any equation is opened.
• The pass stops when purpose is locked: every equation not yet learned already has a named job. The equations then install fast from worked examples.
• After that, each practice item is a hypothesis: this job should take this tool. A miss updates the map. A hit confirms it.
• The test of the map is writing your own hard applied item that combines several tools. Being unable to write one means the map is thin. The repair is to rebuild the map; working more found problems leaves the gap in place.
• The same split applies to coding with agents. The agent can hold the syntax and the steps. The human holds the map of what each tool is for.

Why the usual order fails

The trap is common in formula-heavy subjects. A little theory, a jump to the equations, and the grinding starts. Applying knowledge early is genuinely good for learning, and that is why the grinding feels productive. What is missing is the return to the knowledge: asking what the result means for the map of the topic, and not only whether the steps ran. Without that return, each new situation that wraps the same maths looks like a new procedure to memorise. Jumping to the equations is the trap. Worked examples are early application with guidance. They belong after the map, and they do not replace it.

The first pass: list the names, then ask why

The first pass produces a flat list of names: the main concepts, the equation types, and the keywords. The names are pulled from a survey of the topic, the curriculum, and a reference, and they go into one list. Two reactions are normal at this point: the urge to open the equations, and feeling overwhelmed by the size of the list.

Every name on the list is then asked three questions, in a fixed order, before any equation is opened: what it is, why it is important, and how it relates to the other names. In formula subjects the second and third questions do most of the work. This is the same order that Layers of Learning describes as big picture before details.

The learner holds one question at a time while searching. A resource that answers slowly, or at the wrong level of detail, is skipped. Once the question is answered, the search stops. Fine detail is left for a later pass. Having a page open is not a reason to keep reading it.

The pass starts slow and then speeds up. The first names on the list take the longest. Answering them covers relationships that later names depend on. By the time the derived terms come up, the learner has already met their relations while finding out what the earlier names were for.

The loop every few minutes

The reading and the questioning alternate in short loops. Every three to five minutes, the learner takes what was just learned and asks it against everything still on the list: is it the same, is it different, does it cause this, does it follow from this. The figure of three to five minutes is a working number for this system. It is not a research finding. In the terms of Bear Hunter System, the list is a standing set of aims and the reading is the shooting. This technique is one specialised use of that cycle.

The relations found are mapped as simply as the page allows. The product is a logical framework of the topic. It is never a transcript of the reading. Survive and Thrive explains why that map of relations is what memory keeps.

What a locked map looks like, and when to stop

Here is a purpose-locked map for a topic about change and accumulation. The topic splits into a change operation and its inverse. The change operation produces a rate, and the rate is used for slopes, extrema, and smoothness. The inverse rebuilds the original quantity, and that is used for accumulated totals, averages, and areas. No calculation has been run. Every equation the learner meets later arrives as a tool for one of those jobs.

The pass stops when purpose is locked. Purpose is locked when, for every equation not yet learned, the job it will do already has a name. After that, the equations install fast, and practice becomes hypothesis testing. Ease at this stage means knowing when a tool applies, because its job is already known. Speed at running the steps is a different thing.

What the word “problem” means here

The problem in “problem first” is the purpose frame: what this tool is for, and every way it applies. It is not an unsolved calculation that the learner opens cold. For the same reason, this is not classroom problem-based learning, which is a different design and often gives minimal guidance. The name can be read as either of those, and neither is meant.

An application learned without a purpose is stored as one more separate note to memorise. An application learned as a use of a tool the learner already wants carries over to new problems. Novices learn procedures better from worked examples than from solving first. So after the map is built, worked examples are how the equations install. Starting from a blank worksheet with no guidance is the version the evidence contradicts.

Two kinds of practice item

Once purpose is locked, practice starts to transfer. Later exercises stand for a whole class of problems instead of adding one more routine. This is a claim about direction; there is no counted ratio behind it.

Formula practice produces two kinds of item. The first kind has the same form as the drill: plug the numbers in and run the steps. The second kind wraps the same maths in a situation, such as a vehicle changing speed along a road, or a quantity accumulating over time. The second kind is where learners fail. Success on the first kind is not evidence that the map exists. Cave Theory explains why purpose before information is what makes material stay: isolated facts slip away, and facts tied to a purpose and connected to each other stay.

Write your own hard item

The wrapped item is also something the learner has to write. Questions get authored, as hard as possible, and always applied. Once answering found items feels comfortable, the next items wrap several concepts together in a realistic combination. They should be hard enough that the learner would not want to sit them. Combining and comparing sit one level above applying any single tool, and that level only exists if the map exists.

Being able to answer found questions while being unable to invent a hard applied one is a sign of a conceptual gap. The repair is at the map: rebuild how everything fits and what each piece is for, then try to author again. Working more found practice leaves the gap where it is.

After purpose is locked, a practice item is a hypothesis: this job should take this tool, perhaps combined with that one. A miss updates the map. Either a third element was needed, or this pairing does not apply here. A hit confirms the map. This is a different activity from memorising a fresh procedure for each new situation.

There are two ways the practice can be out of balance, and each has its own repair. If the learner cannot run the steps on a found item, the procedure is thin, and the repair is practice against the map. If the learner can run found items but cannot invent a hard applied item that combines several tools, the map is thin, and the repair is to rebuild purpose and relations and then author again. Using both found questions and authored questions guards against both. Swapping hard items with a partner puts both learners’ maps under test in the same sitting.

What the research says

Routine practice without a map of the principles transfers poorly. Asking why and how-it-relates questions before learning procedures helps the concept layer. The order used here, what, then why, then how it relates, is a default of this system. It is not a research finding.

The same split in coding with agents

The same split between the map and the equations now governs coding and agent work. An agent can hold the syntax, the API details, and the step-by-step execution. Those are the equivalent of the equations. That leaves the human with the map: what each tool is, why it exists, and how the pieces relate, with purpose locked before implementation starts. Someone who can operate the tools but cannot author a maximally hard, realistic problem for the system being built is missing the map. The repair is to rebuild the conceptual layer before doing more repetitions through the tool. Agentic Engineering covers that move of the split into coding work. The split between map and equations is the one Declarative, Procedural, and Conditional Knowledge defines.

The cost, and when not to use this

The first pass is slow and feels overwhelming. It is a flat list of names with no equations yet, and the urge to start calculating is constant. Going back over a topic to lock its purpose is a real second pass, and it takes that time. Against that cost, the benefit is this: later practice stops teaching one question type per item, and situations that wrap the maths stop arriving as new procedures.

This technique is not unguided solving before instruction, and it is not classroom problem-based learning. A reader whose subject is a set of drills with few connections between them should not reach for this first.

Signs the purpose pass was skipped

If, after the inventory-and-why pass, an unlearned equation still has no named job, the map is not locked and the drill book does not start. If the learner can finish the found set and still cannot invent one hard applied item, grinding stops and the map is rebuilt. If two sessions go by in which each new situation needs its own new steps, the purpose pass did not happen.

Purpose locked means every unlearned tool has a named job. The first calculation pass after that checks what the map already says. It should not feel like meeting a new subject. Practice, once purpose is locked, is a test of the map. A miss updates it. Practising an item then teaches more than that one item.

How to practice this

1. List the topic’s names: main concepts, equation types, keywords, from a survey, the curriculum, and a reference. Notice the urge to open the equations; it is normal. Notice that the list feels too big; that is normal too.
2. Ask each name, in order, what it is, why it matters, and how it relates, before opening any equation. Hold one question while you search, and stop when it is answered. Notice that the pass starts slow and speeds up as later names are already partly covered.
3. Every three to five minutes, ask what you learned against everything still on the list: same, different, cause, consequence. Write each relation down as simply as the page allows. Notice that the result is a framework of the topic and holds no transcript.
4. Stop the pass when every equation you have not yet learned has a named job. Notice whether any unlearned equation still has no named job; if so, the map is not locked. Do not start the drill book until it is.
5. Learn the equations from worked examples, then start practice. Treat each item as a hypothesis: this job should take this tool, perhaps with that one. Notice that a miss updates the map and a hit confirms it.
6. Once found items feel comfortable, write your own item that wraps several concepts in a realistic combination. Make it hard enough that you would not want to sit it yourself. Notice if you cannot write one: the map is thin, so rebuild it before more found practice.

Related pages

• Bear Hunter System: the aim–shoot–skin cycle this specialises
• Cave Theory: why purpose-before-information is the keep-signal
• Schema: the relation-map that memory actually keeps
• Layers of Learning: big-picture-before-details as the same sequence
• Declarative, Procedural, and Conditional Knowledge: the split this page spends, map vs equations

1 more related pages

• Agentic Engineering: where the AI-absorbs-procedure angle goes

Open questions

How does the purpose pass run on a subject that is almost entirely procedure, such as a syntax or a technique catalog, where there is no equation to delay?

Sources

• Hatano, G., & Inagaki, K. (1986). Two courses of expertise. In Child Development and Education in Japan. Routine versus adaptive expertise: practice without a principle map stays on the practiced path.
• Singley, M. K., & Anderson, J. R. (1989). The Transfer of Cognitive Skill. Harvard University Press. Single-path productions transfer poorly off the practiced item type.
• Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques. Psychological Science in the Public Interest. Elaborative interrogation, why and how-it-relates, supports the concept layer.
• Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science. Generation and self-explanation as the class this page’s authoring step belongs to.
• Sweller, J., van Merriënboer, J. J. G., & Paas, F. (1998/2019). Cognitive architecture and instructional design. Educational Psychology Review. Worked-example effect: novices learn procedures better from worked examples than from solving first.
• Kapur, M. (2014). Productive failure in learning math. Cognition and Instruction. A bounded unsolved problem before instruction can help conceptual transfer. Neighbour, not identity.
• Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work. Educational Psychologist. Classroom problem-based learning is a different, often minimally guided, design.