Problems live in two worlds. Externally we see the task; internally we build a “problem space,” a mental map of the current state, the goal state, plus possible moves in-between. When the space is huge, random searching stalls. “We need not be concerned with how large the haystack is, if we can identify a small part of it in which we are quite sure to find a needle” (Newell & Simon).
Heuristics beat blind luck. Knowing where to start and what action to try next (e.g., “solve one layer of a Rubik’s Cube first”) shrinks the haystack. The classic heuristic here is means–end analysis. Continually comparing present state with goal state and choosing moves that reduce the gap.
Knowledge is jet fuel. Expert chess players don’t scan every legal move; they “progressively deepen” examining a narrow, promising branch, then resetting to the base position before exploring another (de Groot, 1965). Prior knowledge prunes the search tree.
Practice isn’t “drill-and-kill”; it’s drill-to-skill. “Nothing flies more in the face of the last 20 years of research than the assertion that practice is bad” (Anderson, Reder, & Simon). Deliberate, feedback-rich practice automates lower-level moves so students can think strategically.
Definitions
- The Problem space is the mental playground where we keep track of where we are, where we’re going, and all the possible moves we might try. When a fifth-grader sketches “Start,” “Halfway,” and “Goal” boxes before writing a persuasive essay, she’s mapping her problem space.
- An operator is a move that changes the current state like swapping two numbers while simplifying a fraction.
- Means–end analysis is when a student repeatedly asks, “What’s the difference between now and the goal, and what step shrinks that gap?” During Lego robotics, students compare the current robot path with the desired route and adjust coding blocks that turn the wheels an exact number of degrees.
Why This Matters for Teaching
- Model first, release second. Like a Rubik’s Cube solving guide, give novices worked examples before “productive struggle.”
- Chunk the task. Break a multi-step problem into visible steps so students can return to a safe checkpoint if they hit a dead end.
- Build background knowledge early. Automatic times-table recall lets students devote working memory to multi-step problem solving in later grades.
Concrete Moves You Can Make
- Fourth-Grade Science
“Explain events, procedures, ideas, or concepts in a historical, scientific, or technical text, including what happened and why, based on specific information in the text.”
- Implementation: Before a lab on evaporation, project a flowchart missing key steps. Students use the text to fill in each step explicitly building the problem space then rehearse the procedure in pairs.
- Implementation: Before a lab on evaporation, project a flowchart missing key steps. Students use the text to fill in each step explicitly building the problem space then rehearse the procedure in pairs.
- Fifth-Grade Information Literacy
“Integrate information from several texts on the same topic in order to write or speak about the subject knowledgeably.”
- Implementation: Teams investigate coral bleaching via three different articles. They sort facts onto a board labeled Current State (ocean temps), Desired State (healthy reefs), and Operators (policy actions). Their final slide deck must show how each operator or action narrows the gap. This is a live example of means–end analysis.
- Implementation: Teams investigate coral bleaching via three different articles. They sort facts onto a board labeled Current State (ocean temps), Desired State (healthy reefs), and Operators (policy actions). Their final slide deck must show how each operator or action narrows the gap. This is a live example of means–end analysis.
- Second-Grade Math Fluency
- After explicit demonstrations, students practice targeted fact families in daily two-minute sprints. Mastery frees cognitive resources so, come word-problem time, they’re not “counting on fingers” while also choosing the right operation.
- After explicit demonstrations, students practice targeted fact families in daily two-minute sprints. Mastery frees cognitive resources so, come word-problem time, they’re not “counting on fingers” while also choosing the right operation.
- Whole-Class Reflection Routine
- End problem-solving lessons with two prompts:
- Where did you start? (Identifying the first step.)
- Which move shrank the gap the most? (Surfacing effective actions.)
Over time, students internalize this metacognitive script.
- Where did you start? (Identifying the first step.)
- End problem-solving lessons with two prompts:
The Challenge
Choose one challenging task. Maybe a multi-paragraph writing assignment or a tricky lab. Explicitly draw the first two steps and one action/operator on the board. Then watch how much further (and faster) your students travel through the haystack.
APA References
Newell, A., & Simon, H. A. (1972). Human problem solving. Prentice-Hall.
Anderson, J. R., Reder, L. M., & Simon, H. A. (1999). Applications and misapplications of cognitive psychology to mathematics education. Texas Education Review, 1, 29–49.
de Groot, A. D. (1965). Thought and choice in chess (2nd ed.). Mouton Publishers.
For more information on this concept, read How Learning Happens: Seminal Works in Educational Psychology and What They Mean in Practice (https://a.co/d/a0tZSMR) This post is a summary of concepts from How Learning Happens.
School Intellect 