Thinking: search strategies, framing and base rates

Thinking: search strategies, framing and base rates

Compare efficient heuristics with systematic search and check a judgment against the relevant reference group.

Question and prediction

Can a vivid description outweigh the size of the group it came from?

Predict whether a category match is enough to identify a randomly selected member of a mixed group.

Mechanism and evidence

Problem solving starts with a representation: what is the goal, what operations are allowed and what constraints matter? An algorithm can systematically search a bounded space; a heuristic reduces work by using a plausible shortcut. Heuristics can be effective without guaranteeing correctness. Functional fixedness and a mental set can make an established interpretation hard to revise when the situation changes.

Judgment can overemphasize an easily recalled example or a description that resembles a category. Availability concerns ease of retrieval; representativeness concerns resemblance. Base rates describe the prevalence of categories in the relevant population. A vivid description is evidence only to the extent that its likelihood differs across groups; resemblance alone is not a probability calculation.

In the fictional club, 90 members belong to group A and 10 to group B. Six A members and four B members wear blue badges. Among blue-badge members, B’s share is 4/(6+4)=40%, despite blue being much more common within B. The appropriate denominator depends on the question. This uses invented neutral groups and avoids stereotypes about occupations or identities.

Framing can change how equivalent information is presented. Calling a result 80% success or 20% failure should preserve the numerical complement when the same denominator and outcome definitions apply. Whether actual choices change is an empirical question. A careful experiment holds outcomes fixed, randomizes wording and records decisions rather than assuming that every person must show the same bias.

Inspect the relationships

Group	Blue	Not blue	Total
A	6	84	90
B	4	6	10
Total	10	90	100

Worked example

There are 10 blue-badge members, of whom 4 are in B: 40%. The within-B blue rate is 4/10=40%, but it answers a different question and happens to match here. A’s within-group rate is 6/90≈6.7%. Never select a denominator merely because it produces a plausible number.

Your investigation

Draw a two-by-two count table and circle the reference group for each conditional question. Write 180 words correcting ‘Most likely B because B members often wear blue.’ Then specify a framing comparison that holds probabilities and outcome definitions fixed.

Unfamiliar transfer

A shortcut works on familiar puzzles but fails when one rule changes. What is the best next step? Explain your reasoning before checking feedback.

Reading

OpenStax Psychology 2e: problem solving. The records below are original fictional teaching cases, not research results from this source.

Fictional membership Group A: 90 total, 6 blue badges. Group B: 10 total, 4 blue badges. One blue-badge member is selected uniformly at random.

Fictional wording proposal Version 1 says 80 of 100 attempts succeed. Version 2 says 20 of the same 100 attempts fail. Both use the same success definition; no participant choices have been collected.

Investigate mental set rather than label a mistake

A mental set is a tendency to reuse a familiar approach even when another approach would better fit the problem. Prior success can make a strategy worth trying; the difficulty is failing to reconsider it when the task changes. A single error does not reveal its cause: misunderstanding, attention or arithmetic can also explain failure. Functional fixedness concerns a narrower representational constraint: treating an object as usable only for its familiar purpose. It is not identical to every repeated arithmetic strategy. These distinctions follow the linked OpenStax overview; the cases below are original teaching designs.

When should you change a successful strategy?

Original arithmetic puzzles, not participant data or a psychological test. A, B and C are number cards. Choose a signed count from −2 to +2 for each: positive adds copies, negative subtracts copies, zero omits a card. This is algebra, not a physical pouring procedure.

Start with B − A − 2C. Predict whether it will still reach the target in the next case before changing the case. Keep your first prediction in the Experiment notebook.

Puzzle casePractice 1Practice 2Compare efficiencyChanged targetCheck the search boundary

Target: 16. A = 7, B = 29, C = 3. Counts stay unchanged when you switch cases so you can test the same strategy.

Signed A count-2-1012 Signed B count-2-1012 Signed C count-2-1012

A: -1 × 7 = -7

B: 1 × 29 = 29

C: -2 × 3 = -6

Sum: 16. Target: 16. Target reached. Card uses: 4.

Card uses count every added or subtracted copy: B − A − 2C uses four. This is a declared puzzle cost, not measured thinking time, effort or intelligence. Can you meet the target with fewer uses?

Audit all allowed combinations

Explain the difference between a correct but costly plan, a failed plan and exhaustive absence of a solution. Your own preference after these examples does not establish a mental-set effect in a population. After seeing the audit, the case is exposed practice.

Restore original strategy

Controls and audit visibility are temporary and reset on reload. Save your prediction, expressions and revised explanation in the Experiment stage. No completion or mastery is awarded by this workbench.

Design the comparison needed for a psychological claim

Build a two-group study on paper. One group practices several cases solved by the same expression; a comparison group receives equally long, similarly difficult practice with varied expressions. Both then receive the same new cases, instructions and time allowance. Define a first-strategy measure, successful solution and card-use cost separately. Random assignment helps distinguish prior practice from pre-existing group differences. Do not infer persistence solely from whether an answer is right: a correct but costly expression may still show reuse. Include a check of instruction comprehension and arithmetic accuracy.

Use a supplied fictional observation to practice interpretation: 12 of 20 repeated-strategy learners first choose the old expression on the changed target, compared with 5 of 20 varied-practice learners. Calculate the contrast, explain what it supports under the proposed assignment, and name an alternative explanation requiring further evidence. These are invented outcomes, not the results of an actual study or your workbench actions.

Check the strategy-choice comparison

The first-choice rates are 60% and 25%, a 35-percentage-point difference. Under the stated randomized design this is evidence consistent with an effect of practice type on first strategy choice in this fictional sample. It does not mean that 60% failed, that nobody later switched, or that familiar approaches always harm problem solving. Record final success, switching and uncertainty before making broader claims.

Transfer to an unfamiliar representation

A fictional museum task asks for a label stand. An empty box can serve as a support, but an initial picture shows it only holding supplies. Propose two versions that change its presentation while preserving the available materials and task instructions. Explain why that tests an object’s perceived use more directly than the arithmetic workbench. Keep this a supplied-design exercise; no live participant recruitment is required.

Save your first strategy, audit comparison, study design and museum-task explanation in the Experiment evidence response. High-school work can use the provided comparison; AP work should distinguish mental set from functional fixedness using specific observations; college work should specify assignment, scoring, uncertainty and a competing explanation. None of these guided cases substitutes for independently reviewed project evidence.