Concepts and problem solving

Problem solving includes representing the goal and selecting a method. A shortcut can save effort but may fail outside the conditions where it works.

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Concept and Algorithm

Objective: Explain concept and algorithm, distinguish their meanings, and apply them to a described case.

Concept: A concept is a mental category organizing related examples or properties. Algorithm: An algorithm is a specified procedure for solving a class of problems. Thinking uses representations and strategies that can be efficient yet fallible. A heuristic is not always wrong, and a named bias is a tendency rather than a diagnosis of irrationality. Judgment should be compared with a specified standard and relevant information. Language supports and can influence thought without fixing every possible idea. Distinguish a demonstrated task effect from an unrestricted claim about all human reasoning. Application: A learner groups several instances under one idea. This most directly illustrates concept. A student follows a complete calculation method. This most directly illustrates algorithm.

Worked examples

Try itOriginal teaching scenario: A learner groups several instances under one idea. Which idea from this section does it most directly illustrate?Show answer

Step 1 — Read the observation: A learner groups several instances under one idea. Step 2 — Recall the relevant idea: A concept is a mental category organizing related examples or properties. Step 3 — Connect the observation to the idea: The scenario is an illustration of concept. Step 4 — Check the limit: This short scenario illustrates the concept; it does not establish every detail or exclude every other influence in a real situation.

Try itOriginal teaching scenario: A student follows a complete calculation method. Which idea from this section does it most directly illustrate?Show answer

Step 1 — Read the observation: A student follows a complete calculation method. Step 2 — Recall the relevant idea: An algorithm is a specified procedure for solving a class of problems. Step 3 — Connect the observation to the idea: The scenario is an illustration of algorithm. Step 4 — Check the limit: This short scenario illustrates the concept; it does not establish every detail or exclude every other influence in a real situation.

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Concept and Algorithm

Objective: Explain concept and algorithm, distinguish their meanings, and apply them to a described case.

Worked examples

Try itOriginal teaching scenario: A learner groups several instances under one idea. Which idea from this section does it most directly illustrate?Show answer

Try itOriginal teaching scenario: A student follows a complete calculation method. Which idea from this section does it most directly illustrate?Show answer

Heuristic and Mental set

Objective: Explain heuristic and mental set, distinguish their meanings, and apply them to a described case.

Heuristic: A heuristic is a simplifying strategy that does not guarantee a correct outcome. Mental set: A mental set favors a familiar approach even when another may work better. Thinking uses representations and strategies that can be efficient yet fallible. A heuristic is not always wrong, and a named bias is a tendency rather than a diagnosis of irrationality. Judgment should be compared with a specified standard and relevant information. Language supports and can influence thought without fixing every possible idea. Distinguish a demonstrated task effect from an unrestricted claim about all human reasoning. Application: A solver uses a quick rule to choose an option. This most directly illustrates heuristic. A solver repeats an old method despite a changed problem. This most directly illustrates mental set.

Worked examples

Try itOriginal teaching scenario: A solver uses a quick rule to choose an option. Which idea from this section does it most directly illustrate?Show answer

Step 1 — Read the observation: A solver uses a quick rule to choose an option. Step 2 — Recall the relevant idea: A heuristic is a simplifying strategy that does not guarantee a correct outcome. Step 3 — Connect the observation to the idea: The scenario is an illustration of heuristic. Step 4 — Check the limit: This short scenario illustrates the concept; it does not establish every detail or exclude every other influence in a real situation.

Try itOriginal teaching scenario: A solver repeats an old method despite a changed problem. Which idea from this section does it most directly illustrate?Show answer

Step 1 — Read the observation: A solver repeats an old method despite a changed problem. Step 2 — Recall the relevant idea: A mental set favors a familiar approach even when another may work better. Step 3 — Connect the observation to the idea: The scenario is an illustration of mental set. Step 4 — Check the limit: This short scenario illustrates the concept; it does not establish every detail or exclude every other influence in a real situation.

Functional fixedness and Problem representation

Objective: Explain functional fixedness and problem representation, distinguish their meanings, and apply them to a described case.

Functional fixedness: Functional fixedness restricts thinking about alternative uses of an object. Problem representation: Representation organizes a problem's information, constraints, and goal. Thinking uses representations and strategies that can be efficient yet fallible. A heuristic is not always wrong, and a named bias is a tendency rather than a diagnosis of irrationality. Judgment should be compared with a specified standard and relevant information. Language supports and can influence thought without fixing every possible idea. Distinguish a demonstrated task effect from an unrestricted claim about all human reasoning. Application: A person overlooks a tool's unconventional useful role. This most directly illustrates functional fixedness. A student redraws a word problem to reveal its structure. This most directly illustrates problem representation.

Worked examples

Try itOriginal teaching scenario: A person overlooks a tool's unconventional useful role. Which idea from this section does it most directly illustrate?Show answer

Step 1 — Read the observation: A person overlooks a tool's unconventional useful role. Step 2 — Recall the relevant idea: Functional fixedness restricts thinking about alternative uses of an object. Step 3 — Connect the observation to the idea: The scenario is an illustration of functional fixedness. Step 4 — Check the limit: This short scenario illustrates the concept; it does not establish every detail or exclude every other influence in a real situation.

Try itOriginal teaching scenario: A student redraws a word problem to reveal its structure. Which idea from this section does it most directly illustrate?Show answer

Step 1 — Read the observation: A student redraws a word problem to reveal its structure. Step 2 — Recall the relevant idea: Representation organizes a problem's information, constraints, and goal. Step 3 — Connect the observation to the idea: The scenario is an illustration of problem representation. Step 4 — Check the limit: This short scenario illustrates the concept; it does not establish every detail or exclude every other influence in a real situation.

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