Problem Comprehension

Problem Comprehension

Most problem-solving failures happen before any solving begins. A problem that is misunderstood cannot be solved accurately, and time spent on an incorrect interpretation is time lost. Comprehension is the disciplined first phase of Polya's method ("Understand the Problem"): restate, identify, classify, decompose. This skill documents the operations needed to leave comprehension with a clean problem representation that the rest of the pipeline can operate on.

Agent affinity: polya-ps (overall framing), jonassen (problem typology), simon (state-space representation)

Concept IDs: prob-problem-representation, prob-knowns-unknowns, prob-problem-types, prob-goal-decomposition

The Comprehension Toolbox at a Glance

Operation 1 — Restate in Your Own Words

Pattern: Read the problem, then state it without looking at the original. If the restatement is incomplete or wrong, comprehension has failed and no solving should begin.

Worked example. "A train leaves Boston at 8 a.m. traveling 60 mph east. A second train leaves New York at 9 a.m. traveling 50 mph west on the same track. The cities are 220 miles apart. When do they meet?"

Restatement: "Two trains on the same track, one from each end of a 220-mile segment. The eastbound train has a one-hour head start at 60 mph. The westbound starts at 50 mph. Find the time at which they occupy the same point on the track."

The restatement names every entity, every rate, every distance, and the success criterion. If any of these are missing, read the problem again.

Operation 2 — Identify Knowns

Pattern: List every quantity, condition, or relationship the problem provides. Treat this as an inventory, not a narrative.

Worked example (continued). Knowns:

• Distance between cities: 220 miles
• Eastbound train: speed 60 mph, starts 8 a.m.
• Westbound train: speed 50 mph, starts 9 a.m.
• Both on the same track (single-track constraint — also a known)

Operation 3 — Identify Unknowns

Pattern: List every value the problem asks you to find. If the problem asks for multiple things, list each separately.

Worked example (continued). Unknowns:

• Time of meeting (clock time or elapsed time from 8 a.m.)
• Implicitly: position along the track where they meet

The implicit unknown matters because it is required to verify the answer.

Operation 4 — Identify Constraints

Pattern: List every limit on the solution space. Constraints are sometimes stated, sometimes implied by physical reality, and sometimes embedded in the problem type.

Worked example (continued). Constraints:

• Both trains are on the same track (physical collision)
• Both speeds are constant (no acceleration)
• The distance is fixed at 220 miles

Operation 5 — State the Goal

Pattern: Write the success criterion as a sentence. "I will have solved this problem when I know [X]." This prevents drift during solving.

Worked example (continued). Goal: "I will have solved this problem when I know the clock time at which both trains are at the same mile marker."

Operation 6 — Classify the Problem Type

Pattern: Match the problem to a known category. The category determines which solving strategies are appropriate.

Primary categories:

• Well-defined problems have known initial state, goal state, and legal operators. The train problem is well-defined: everything is specified and the answer is a number.
• Ill-defined problems have ambiguous initial or goal states. "How should our company respond to climate change?" is ill-defined because the goal itself is open.
• Optimization problems ask for the best value of a quantity under constraints. "Minimize shipping cost subject to..."
• Decision problems ask for a choice among alternatives. "Should we build plant A or plant B?"
• Puzzles are well-defined problems with no practical consequences — good for strategy practice.
• Wicked problems have no definitive formulation, no stopping rule, and stakeholders who disagree about what counts as a solution.

Recognizing the type narrows the strategy space before any work begins.

Operation 7 — Build a Problem Representation

Pattern: Produce an external representation that makes the structure visible. Diagrams, tables, state-spaces, and equations all qualify.

Worked example (continued). A simple diagram:

Boston                                         New York
  |<---- 220 miles ---->|
  |-- 60 mph ----------->
                        <---- 50 mph --|
  8 a.m.                9 a.m.

This immediately shows that the eastbound train has a head start, which is the central difficulty of the problem.

Operation 8 — Decompose into Sub-Problems

Pattern: Break the problem into pieces that can be solved independently, then combined. If the problem is small, skip this step.

Worked example (continued). Sub-problems:

1. How far has the eastbound train traveled when the westbound starts?
2. What is the remaining distance at 9 a.m.?
3. How fast are they closing on each other after 9 a.m.?
4. At what time after 9 a.m. does the closing distance reach zero?

Each is solvable with simple arithmetic. The combined answer falls out.

Operation 9 — Check for Hidden Assumptions

Pattern: List what you are taking for granted. Hidden assumptions are the most common cause of comprehension errors.

Worked example (continued). Hidden assumptions:

• The trains move at constant speed (not stated but assumed)
• Boston and New York are at fixed known positions (not stated)
• Time zones are the same (true in practice but should be checked for real problems)

If any assumption fails, the answer changes.

Operation 10 — Restate with All of the Above

Pattern: Produce a final restatement that a peer could solve without the original problem text. If the peer cannot, comprehension is incomplete.

Worked example (final). "Between Boston and New York, 220 miles apart on a single track, an eastbound train leaves Boston at 8 a.m. at 60 mph and a westbound train leaves New York at 9 a.m. at 50 mph. Both move at constant speed. Find the clock time at which they occupy the same point."

This restatement is solvable. Comprehension is complete.

Standard Comprehension Procedure

1. Read the problem once for gist.
2. Read again slowly, applying Operations 1-5 in parallel (restate, knowns, unknowns, constraints, goal).
3. Apply Operation 6 (classify) to narrow the strategy space.
4. Apply Operation 7 (representation) to make the structure visible.
5. Apply Operation 8 (decompose) if the problem is large.
6. Apply Operation 9 (check hidden assumptions) and list any that were not explicit.
7. Apply Operation 10 (final restatement) and verify it is peer-solvable.
8. Only now hand off to strategy selection.

When Comprehension Fails

Failures to watch for:

• Jumping to solving before comprehension is complete. Almost always produces the wrong answer or a solution to the wrong problem.
• Missing a constraint. Common when constraints are implicit (physical, legal, social).
• Misclassifying the type. Treating an ill-defined problem as well-defined produces false precision; treating a well-defined problem as ill-defined wastes effort.
• Skipping representation. Verbal-only comprehension often misses structural features that a diagram makes obvious.
• Failing to decompose. Tackling a large problem monolithically often leads to overwhelm and backtracking.

Cross-References

• strategy-selection uses the problem representation and type to choose an approach
• metacognitive-monitoring checks that comprehension was actually sufficient after strategy selection begins
• mathematical-problem-solving relies on operations 1-10 as its own first phase
• design-thinking-ps extends comprehension to ill-defined problems via reframing
• collaborative-problem-solving distributes comprehension work across team members