Building a Structured Problem Map

Case 1 — Study plan under pressure

A learner has three exams in five days and only eight hours of total study time available. Exam A is the hardest and worth the most points, but the learner already has a strong foundation. Exam B is medium difficulty but the learner is weakest on it. Exam C is easy but worth a lot. The learner must also sleep at least six hours each night and has one meeting they cannot move.

What are the three candidate study plans? Which criteria should you use to compare them?

Case 2 — Moving to a new city

A graduate student is relocating for an internship. They have $800, three weeks, a car of uncertain long-distance reliability, and a week's couch at a friend's place in the destination city. They need to find housing, transport their possessions, wrap up their current lease, and arrive rested before the internship starts.

Generate strategies that vary in cost, speed, and risk.

Case 3 — Product bug triage

A production system is returning incorrect results for 2% of requests. The bug has been present for three weeks. Reverting the last deployment would remove customer-requested fixes. The on-call engineer has never seen this code. The team has four engineers available for the next 48 hours and a customer escalation comes in every two hours.

What's the problem state? What's the goal? What candidate triage strategies exist?

Question 1 — Diagnose

A student makes the following mistake: "Listing options without comparing them." Explain specifically what is wrong with this reasoning and what the student should have done instead.

Can the student identify the flaw and articulate the correction?

Question 2 — Apply

You encounter a new argument that you have never seen before. Walk through exactly how you would build problem map, starting from scratch. Be specific about each step and explain why the order matters.

Can the student transfer the skill of build problem map to a genuinely new case?

Question 3 — Distinguish

Someone confuses problem state with goal state. Write a short explanation that would help them see the difference, and give one example where getting them confused leads to a concrete mistake.

Does the student understand the boundary between the two concepts?

Question 4 — Transfer

The worked example "Choosing a Study Plan" showed one way to handle a specific case. Describe a situation where the same method would need to be adjusted, and explain what you would change and why.

Can the student adapt the demonstrated method to a variation?

Problem 1

A hospital needs to reduce emergency room wait times from 4 hours to under 1 hour. Budget increases are not possible. Staff are already working overtime.

Problem 2

A software team discovers that 80% of their bugs come from 20% of their modules. They have 6 weeks before release and need to cut the bug rate in half.

Problem 3

A city wants to reduce traffic congestion but cannot build new roads. Public transit ridership is declining despite subsidies.

Scenario A

A school must improve test scores within one academic year using only existing staff. The principal believes smaller class sizes are the answer, but the building has no additional rooms.

Scenario B

A restaurant wants to add delivery service but the kitchen is already at capacity during peak hours, and hiring new cooks would take months of training.

Problem

A university library is running out of physical shelf space. Solution A: Digitize all books and remove shelves. Solution B: Build an off-site storage facility and retrieve books on request. Solution C: Remove books not checked out in 10 years.

Problem

A factory's error rate is 5%. Solution A: Add a second quality inspector. Solution B: Retrain all workers. Solution C: Replace the oldest 20% of machines.

Scenario A

A school district must reduce its budget by 15% without laying off teachers or closing schools. Current spending: 60% salaries, 20% facilities, 10% supplies, 10% transportation. Student enrollment is increasing 3% annually.

Scenario B

A software startup has 4 developers and 6 months to build a product that typically takes a team of 10 one year. They cannot hire more staff but can use open-source tools. The product must be secure enough for healthcare data.

Scenario C

A coastal town's only freshwater source (a river) is projected to decrease by 30% due to upstream damming. The population is growing. Desalination costs 5x more than river water. The town budget cannot increase.

Community challenge

A neighborhood has a growing problem with package theft from doorsteps. Solutions tried so far (security cameras, signed delivery) have not reduced theft. The neighborhood association has a small budget. Develop a creative solution.

Workplace challenge

A team of 8 remote workers in 4 time zones needs to collaborate on a project with a 3-month deadline. Previous remote projects have suffered from miscommunication and duplicated work. Design a coordination strategy.

Item 1

A food bank receives 30% more donations in December than it can store. Much of it spoils before distribution.

Item 2

A school bus route takes 90 minutes because it serves three neighborhoods in sequence. Parents are complaining about early pickup times.

Item 3

A non-profit website gets 10,000 visits per month but only 50 donations. The donation button is on a separate page from the stories.

Item 4

A clinic has a 3-week wait for appointments, but 20% of scheduled patients do not show up.

Item 5

A warehouse ships 500 orders daily but the error rate is 8%. Most errors involve wrong quantities rather than wrong items.

Student A's solution

Problem: A library wants to increase youth attendance. Student A wrote: 'Solution: Add a gaming lounge with consoles and high-speed internet. Young people love gaming, so this will bring them in. Cost estimate: $15,000 for equipment. Expected result: 50% increase in youth visits.'

Student B's solution

Problem: A rural clinic needs more doctors. Student B wrote: 'Offer student loan forgiveness for doctors who commit to 3 years. This addresses the financial barrier. However, the clinic budget is limited. We could partner with the state medical school for funding. Risk: doctors may leave after the commitment period.'

Student C's solution

Problem: Plastic waste in a coastal town. Student C wrote: 'Ban all single-use plastics immediately. This will eliminate the source of waste. Similar bans worked in other countries. Implementation: pass a local ordinance next month and begin enforcement.'

Task 1

A mid-sized city wants to reduce food waste by 50% within three years. Currently, 40% of food waste comes from restaurants, 35% from households, and 25% from grocery stores. Design a comprehensive solution addressing all three sources.

Task 2

A university wants to reduce cheating on online exams without making the exam experience stressful for honest students. Current proctoring software has a 15% false-positive rate. Design a better system.

Task 3

An island community of 5,000 people wants to become energy-independent within five years. They currently import all fuel by ship. The island has strong winds, moderate sunlight, and geothermal activity. Design an energy transition plan.

Solution 1

Problem: Reducing traffic congestion. Proposed solution: Make public transit free. Find a scenario where this backfires or fails to reduce congestion.

Solution 2

Problem: Improving employee retention. Proposed solution: Increase salaries by 20%. Find a scenario where this does not solve the retention problem.

Solution 3

Problem: Reducing hospital readmissions. Proposed solution: Schedule follow-up appointments before patients are discharged. Find a scenario where this approach fails.

Solution 4

Problem: Reducing food waste in school cafeterias. Proposed solution: Let students choose their portion sizes. Find a scenario where this makes the problem worse.

Scenario 1

A city council must decide whether to invest in a new water treatment plant. Data: current system serves 50,000 people and is at 95% capacity. Population is growing at 2% annually. The new plant would cost $80 million. An alternative is to reduce per-capita usage through conservation programs. Apply problem-solving frameworks, use inductive projections for population growth, and construct a deductive argument for or against the investment.

Scenario 2

A tech company's AI model shows bias in hiring recommendations. The team must fix the bias without reducing the model's overall accuracy. Apply: (1) problem-solving to define constraints, (2) abductive reasoning to explain why the bias emerged, (3) deductive reasoning to determine what fairness criteria logically require.

Scenario 3

A school district must allocate a fixed budget between math tutoring and reading programs. Research shows both are effective but reading programs show stronger results for younger students while math tutoring benefits older students more. Use inductive evidence to evaluate program effectiveness, deductive reasoning to apply budget constraints, and problem-solving frameworks to design an optimal allocation.

Misconception 1

A student says: 'The first step in problem-solving is to generate solutions. You should brainstorm answers immediately before the problem gets worse.'

Misconception 2

A student claims: 'A solution that worked in one context will always work in a similar context. Best practices are universal.'

Misconception 3

A student writes: 'The optimal solution is always the one that completely solves the problem. Partial solutions are failures.'

Misconception 4

A student argues: 'Constraints are always obstacles. The more constraints you have, the worse your solution will be.'

Misconception 5

A student says: 'Once you have found a working solution, there is no reason to look for alternatives. Efficiency means going with the first thing that works.'

Scaffold 1

Problem: A community health clinic has a 40% no-show rate. Stage 1: Define the problem precisely -- what are the actual costs and impacts? Stage 2: Identify at least 4 constraints (budget, staff, technology, patient demographics). Stage 3: Generate 5 potential solutions. Stage 4: Evaluate each solution against your constraints. Stage 5: Design an implementation plan for your top solution.

Scaffold 2

Problem: A company's customer support response time is 48 hours, but the industry standard is 4 hours. Stage 1: Decompose the problem -- where does time go? Stage 2: Identify root causes (staffing, tools, processes, training). Stage 3: Generate solutions targeting each root cause. Stage 4: Evaluate trade-offs (cost vs. speed vs. quality). Stage 5: Propose a phased implementation plan.

Comprehensive 1

A mid-sized city (population 300,000) wants to become carbon-neutral by 2040. Current state: 50% of energy comes from natural gas, 30% from coal, 20% from renewables. Transportation is 90% fossil-fuel vehicles. The city's budget cannot increase, but state and federal grants are available. There are three coal-dependent neighborhoods where 15,000 jobs are at stake. Apply a complete problem-solving framework: define sub-problems, map constraints, generate solutions per sector, evaluate trade-offs, and create a phased 15-year plan.

Comprehensive 2

A regional hospital network (5 hospitals, 2,000 beds) must prepare for a potential surge of 3x normal patient volume while maintaining standard care. Constraints: cannot build new facilities in time, staff are already near burnout, supply chains are strained, and rural hospitals have fewer resources than urban ones. Design a comprehensive surge plan with prioritization criteria, resource allocation strategies, and contingency options.