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Cognitive Flexibility

Principle · All principles
Evidence · 23 claims (9 for, 12 mixed, 2 against) · 28 studies (10 causal, 6 theoretical, 3 quant-synthesis, 3 review, 3 qualitative, 2 design, 1 associational), q1–q4 · 4 of 28 report an effect size · 17 claims rest on one study

Conditional relationship

This principle is about a learner who already holds some knowledge of a domain in one form (one rule, one textbook case, one representation, one solution method) and must now use it in situations where it plays out differently. The starting response to look for is rigid use: applying the familiar rule or method to a case it does not fit, failing to recognise the concept when it appears in an unfamiliar form, or being able to state it but not adapt it. The expectation is that revisiting the same concepts across several cases, perspectives or representations that differ in how the concept plays out, with explicit prompts to say what stays the same and what changes, helps the learner assemble the knowledge afresh for a new situation. The outcome is near-transfer or far-transfer to a new case (an essay, a solution, a judgement), at short horizons in the evidence here. The target knowledge is a principle, a concept or a complex-skill whose application varies across situations; the theory's home is ill-structured domains such as medicine, law, policy and history.

The wiki tests only part of this. One small randomized experiment tests the bundled approach (revisiting short cases under several themes) against drill, with first- and second-year undergraduates, and finds better transfer essays by the fourth session and worse factual recall. The case-comparison evidence (a meta-analysis and experiments) tests comparing provided cases, not multiple perspectives or ill-structured domains as such, and its benefit was smaller at delay. No claim here tests the expectation that the advantage grows at delayed-retention, how many cases are needed, or whether the approach suits novice learners; the theory itself says novices may be overwhelmed, and one experiment on part-task practice supports simplifying first for them. Whether the learner's knowledge is in fact rigid, and whether the cases really differ in how the concept applies, must be established locally.

The converted sibling Analogical Reasoning owns a narrower relationship: mapping the relations of one provided or familiar case onto a new one, with case comparison as its best-evidenced configuration. This page owns the more general one, holding the same knowledge in several forms so it can be reorganised, and places analogical case comparison inside it as one way to build that flexibility; multiple representations and multiple conceptual perspectives are the others. The Cognitive Flexibility Theory page describes the theory and its criss-crossing design; it is not rewritten here. The case-based learning pattern holds a converted policy for teaching through cases.

"Cognitive flexibility" also names an executive function (switching between rules or sets), and the wiki's claims about training it in kindergarten and first grade (for example cognitive-flexibility-training-increases-cognitive-flexibility-growth and embedded-cognitive-flexibility-no-overall-advantage-winter-kindergarten) are about that construct. They do not test this page's relationship and are not used here.

Default design, while the relationship is untested

The page's earlier guidance, kept as a concrete default a designer can act on and revise. Each step is a design proposal with its evidence status stated beside it. Most of it is untested as written.

  1. Name the throughline first. Write down the one to three concepts that will recur across every case, so variation does not become fragmentation. Untested; from the earlier page ("learners still need a coherent throughline").
  2. For learners new to the domain, start with one clear case or one representation and the parts of the task separately; add variability once they can complete the simple version unaided. Evidence: part-task practice helped novices on a high-interactivity task before the whole task (see below). That study did not vary cases, so it supports simplifying first, not a particular point at which to add variety. A training-design argument makes the same split from the other side: high variability and limited guidance favour transfer once intrinsic load is lowered for novices; it is an argument, not a test. If learners cannot complete the simple version unaided, do not add cases yet: give a worked example or a part task instead.
  3. Choose several short cases in which the same concept plays out differently, not cases that only look different. Revisit each case under more than one theme or question rather than covering it once. Evidence: the criss-crossing experiment used short cases reread under different combinations of themes, and its transfer advantage appeared by the fourth session (see below). No number of cases or sessions is established; the earlier page and the theory page give none either.
  4. Prompt the comparison explicitly: "What stays the same across these cases? What changes, and why does the rule apply differently here?" Put the cases side by side, and state the principle after learners have compared them. Evidence: in the case-comparison meta-analysis, benefits were larger when learners were asked to find similarities and when the principle came after the comparison.
  5. Add a second representation (a diagram, a simulation, a formula) only with instruction in how the representations connect, and check that learners can read each one. Evidence: multiple representations helped in one undergraduate physics field study, and a review finds the benefit depends on learners' skill at connecting representations. Let learners move back and forth between the representations rather than retiring the first one: in one design study of integer arithmetic, students stuck on a desk-scale number line returned to a walk-on number line to check direction and answers (qualitative, no comparison group).
  6. Have learners explain why the familiar rule cannot be applied mechanically to a new case. Evidence: prompting self-explanation improves learning on immediate tests by a moderate average amount (meta-analysis, g = .55), with "why" prompts most useful in mathematics and little evidence on delayed or classroom gains. No claim here tests self-explanation during case comparison specifically, so this step borrows the general effect.
  7. Keep fact learning separate, and judge the design on a new case. Practise the facts the objective needs by recall; assess flexibility with an unfamiliar case scored apart from recall, and repeat it after a delay if the objective needs durable use. Evidence: the criss-crossing group transferred better and recalled facts worse than the drill group.

Fitting the design to a situation

The three facts that most change the decision:

  • How rigid or how thin the learner's current knowledge is. A learner who has nothing to be rigid about needs a first, simple version (step 2); a learner applying a rule mechanically needs varied cases (step 3). If unstated, ask: can they complete a standard case of this kind unaided, and what do they do with a case where the usual rule does not fit?
  • Whether the domain really varies across situations. In a well-structured procedure with one right method, varied cases add load for little gain (the theory and the earlier page both say so; no claim here tests it). If unstated, ask: do experts in this field disagree about cases, or apply the same rule differently in different settings?
  • What the assessment rewards. The one direct test found a trade-off: better transfer, worse factual recall. If the stakes sit on recall, the design must protect recall separately. If unstated, ask: what will the learner be judged on, a new case, or what they remember?
If the brief says… Then change… Basis
Expertise novice in the domain, high element interactivity Start with one case and part tasks; add the second and third case only after the simple version is completed unaided; prompt the named dimension of comparison. Part-task practice reduces cognitive load for absolute novices during initial skill acquisition. [~M]; population not reported on the claim page
Expertise intermediate or advanced, an ill-structured domain (clinical reasoning, law, policy) Use criss-crossing: revisit each concept in several short cases under different themes across sessions; drop the step-by-step part tasks. Presenting multiple cases from different perspectives supports transfer in ill-structured domains [+M]; tested with first- and second-year undergraduates, untested with professionals
Age child (under 12) Use two cases at a time, side by side, with the one dimension that differs named by the teacher; no free navigation among cases. untested proposal; no claim here tests case comparison with children
Age child, early years (about 2–5), a number concept reachable by several methods (estimating, counting, lining up) Embed the comparison in a purposeful choice ("which pile do you want?"); model one method and ask what its result means for the choice, then show that a second method gives the same answer; expect the child to propose the cross-check unprompted only later, across months. Numerical routines individualize through a three-stage sequence of vertical and horizontal bonding [+W]: a teaching experiment with four children, no comparison group; it describes a sequence, not an effect
Learners fluent in one solution method must take up a second (for example, undergraduate STEM procedures) Show both methods as routes through the same ideas, side by side; give cases where the familiar rule of thumb fails (a non-zero starting value), and ask what each part of the new method stands for; expect the familiar method to persist after one modelled demonstration. Intermediate mechanics students start the semester more familiar with the integration constants (+C) method than with the limits method, and the +C method remains preferred by at least one student even after instruction modeling the limits method [~W]; Students using the limits method initially choose unphysical integration limits, failing to consider the functional relationship between limits on either side of the equal sign [+W]: one qualitative study of two student groups in one course; see also the adolescent algebra row below, which found comparison raised method flexibility
Age adolescent, goal a mathematics procedure Have learners compare alternative solution methods side by side; expect gains in choosing methods, and add a separate item asking why each method works. Comparing cases side by side improves learning and transfer by a moderate average amount, though in one algebra study it raised procedural knowledge and flexibility but not conceptual knowledge [~M]; tested with seventh-graders, procedural flexibility rose, conceptual knowledge did not
Goal verbal-association (facts, terms), or recall-weighted high-stakes assessment Do not make criss-crossing the main activity; give facts their own retrieval practice, and use varied cases only for the transfer part of the objective. Presenting multiple cases from different perspectives supports transfer in ill-structured domains [~M]: the drill group recalled more facts
Goal a STEM concept taught through representations Add a second representation (sketch, simulation, an embodied or walk-on model) beside the formula, with tasks that connect them; keep the first available so learners can switch back to check; monitor reported load and representational skill. Multiple representations improve learning [~M]; one undergraduate physics field study and a review; benefit depends on representational competence. Students who got stuck on the desk-scale number line proactively returned to the body-scale number line to validate their answers, achieving perspectival mutuality [+W]: one middle-school design study, no comparison group
Setting online-self-paced or hypertext Give a guided route through the cases (a theme per pass) rather than free browsing, and build the comparison prompts into the pages. A flexible hypertext course group averaged 76% on achievement tests against 59% for a direct-design group that studied different software, so the contrast confounds design with content [~W]: groups studied different software, so it does not isolate the design; navigation burden is untested, from the theory page
Learners use a generative AI assistant to produce cases, explanations or perspectives Do not let the assistant supply the cases unprompted; ask learners to request deliberately contrasting cases or viewpoints, name whose view each represents, and compare them as for any other case set. Unstructured LLM use contributes to epistemic narrowing by privileging convergent outputs and dominant viewpoints [-W]: a theoretical argument, untested; no claim here tests AI-generated case sets
Duration single-session Use two or three contrasting cases with a similarities prompt and the principle stated after; test on a new case the same day. If the course allows, revisit the cases later under a new theme. Comparing cases side by side improves learning and transfer by a moderate average amount, though in one algebra study it raised procedural knowledge and flexibility but not conceptual knowledge [+M]: comparison helped in single experiments, more at immediate tests
Duration days-weeks or term-plus Return to the same concepts in each session through new cases and new themes; schedule a delayed transfer case. Presenting multiple cases from different perspectives supports transfer in ill-structured domains [+M]: the advantage appeared by the fourth session; delayed benefit untested
Constraint: cases in a second language, or heavy reading load Keep cases short, give them in accessible language, and present the comparison as a table learners fill in, so reading demand does not swamp comparison. untested proposal

Eleven of the thirteen rows rest on a claim, each only as far as its tested population and setting; the rows for children under 12 (case comparison) and for reading load are untested proposals. The early-years, second-method and AI rows rest on qualitative or theoretical sources only and are the weakest of the eleven.

Observation, state and explanation

What can be observed is a response to a particular case under stated conditions: which rule or method the learner applied, whether they noticed that the case differed, what they said changed and what stayed the same, whether a hint was needed. "Rigid knowledge", "flexible understanding" and "oversimplified schema" are inferred states. Record which cases and representations the learner had already met, every prompt and hint, and whether the test case shares surface features with a taught one.

Response Plausible explanations Observation that could change the interpretation
Applies the familiar rule to a case it does not fit Knowledge held as one rigid schema; did not notice the case differs; noticed but has no alternative to apply Ask what is different about this case before asking for a solution; give a matched case where the difference is made salient and see whether the choice changes.
Solves taught cases well, fails a new case in the same domain Learned the cases, not the concept; new case needs knowledge not taught; test case harder in ways unrelated to flexibility Give a new case with the taught surface and a different principle, and one with a new surface and the taught principle; score which one fails.
Lists differences between cases but cannot say what stays the same Surface comparison only; too little knowledge of the concept to see it; prompt asked for differences Ask for the shared idea explicitly, then for a prediction in a third case; compare with a learner given the principle after the comparison.
Recalls facts poorly after several case sessions, writes good case analyses Time moved from facts to cases; facts not needed for the analyses; analysis uses general reasoning rather than domain knowledge Score fact items and case items separately; check whether the case analyses use the facts the objective requires.
Keeps using the familiar method after a second has been modelled, or uses the new one with rule-of-thumb errors (pairing the wrong starting values, leaving the variable out) The new method was learned as a ritual, not connected to what its parts mean; the familiar method is simply more fluent; the task never required the new method Give a case where only the new method's meaning gets it right (a non-zero starting value) and ask what each limit stands for; if the learner can say it but still reverts, fluency rather than understanding is the gap. Observed in one qualitative physics study (+C persistence [~W], unphysical limits [+W], numeric-only limits [+W]).
Stuck in the abstract representation, goes back to the concrete one to check Has not yet linked the two; is using the concrete one as a check, which is the link forming Ask the learner to predict in one representation and confirm in the other, in both directions; if they never go back once the abstract one works, the link may be made. Observed in one design study (returning to the walk-on number line [+W]).
Overwhelmed by several cases or representations: stops, picks one at random Working memory exceeded; cannot read one of the representations; unclear what the comparison is for Reduce to two cases or one representation and repeat; ask the learner to read each representation alone; give the comparison question before the cases.

These are proposals, untested with learners; the two rows drawn from qualitative studies describe responses those studies saw, not tested probes. A probe can shift confidence between explanations; it does not identify a cause, and probing itself is practice (a "what is different here" prompt is also a comparison prompt).

Evidence and qualifications

The Alfieri et al. case-comparison meta-analysis stands under two of these titles, so count studies, not claims: nine distinct studies stand behind the six core claims.

Taken together: revisiting concepts through varied, compared cases has randomized and synthesis-experimental support for transfer at short horizons, mostly with undergraduates; the one direct test of the bundled approach is small and traded recall for transfer. The flexibility gained may be procedural rather than conceptual, multiple representations help only when learners can connect them, and novices may need a simpler start. Delayed transfer, professional learners, children, and the number of cases are not established. Comparators (drill, separate cases, traditional tasks, a different course) and outcomes (essays, negotiations, algebra items, a physics test) differ, so do not rank these results by effect label.

How far each may be carried: the criss-crossing experiment supports steps 3 and 7 of the default design for undergraduates in one ill-structured topic. The comparison evidence supports step 4 for provided cases. The part-task study supports step 2's "simplify first" for novices, not when to add variety. The hypertext study supports nothing on its own. The self-explanation meta-analyses support step 6 as a general effect, transported to case comparison. The qualitative and design studies (integration methods, number lines, early numeracy) describe what rigid and emerging flexible use look like and supply the diagnostic rows; none tests a design against an alternative.

The evidence by question

Further evidence, not yet read against this model

Claims read against the model above (2026-10-19) that are relevant but do not change it; each says why.

Objective and learner-valued goal

The designer's objective names the concept that should travel, the kinds of new case it should be used on (near or far, with or without a hint), what counts as adapting it well, and the horizon. Elicit separately what the learner wants from it. Agreement, divergence and uncertainty are observations.

For example, a second-year law student may want to pass a doctrinal exam that rewards stating rules precisely, while the designer's objective is to apply a negligence standard to fact patterns that differ from the leading cases. Criss-crossing cases serves the designer's aim and, on the one direct test, may cost some recall the student values. Record both aims, give the rule statements their own practice, and show the student a new fact pattern where the stated rule alone gives the wrong answer, so the value of the case work is visible rather than asserted.

What would revise this model?

The model should weaken if well-controlled comparisons with comparable learners find no transfer advantage for revisiting concepts across varied, compared cases over studying the same material once in one form, or find the advantage gone at the delay the objective needs. If the advantage is consistently procedural (choosing methods) and never conceptual, keep that distinction rather than calling both flexibility. If novices given varied cases from the start do as well as those given a simple start, step 2 should be dropped; if they do worse, it becomes a requirement. If the recall cost seen in the one direct test persists and matters to the objective, the design needs separate fact practice by default. Do not protect the model by calling every transfer failure a sign that the cases were not varied enough.

Noticing that a case differs, saying what changes, choosing an appropriate method, applying a concept in a new case, doing so in another domain, and remembering it later are separate claims. The present evidence does not establish how many cases or perspectives to use, how different they should be, or whether multiple perspectives add anything beyond case comparison.

Claims

Examples

  • Intermediate Mechanics Tutorials
  • Math Talks
  • Math Talks / Number Talks
  • A longitudinal teaching experiment in which a child ("Milo") was asked "Where Is There More?" using piles of blocks or snacks across five interview rounds; vertical bonding first appeared when the interviewer counted and aligned two piles and asked "What does it mean?", and horizontal bonding first appeared when the child spontaneously proposed an alignment check to confirm a visual estimate.
  • Multiple-text comparison: Learners read contrasting accounts of the same event and identify differences in framing, evidence, and implication.
  • Stakeholder analysis: Learners compare how different stakeholders would interpret or be affected by a decision.
  • Perspective interview synthesis: Learners interview others about an issue and compare their own assumptions to what they heard.
  • Structured debate preparation: Learners prepare to argue a position they do not initially hold in order to understand it more precisely.
  • Quinmester humanities course 'Ideas of Western Man: Expressionism and Realism' integrating literature, philosophy, art, and music
  • Debates that require learners to articulate the strongest case for an opposing position.
  • Historical inquiry that compares how different groups experienced the same event.

Illustrative

Cognitive Flexibility Theory — Learners revisit concepts across multiple cases, perspectives, and representations instead of mastering one linear explanation.

Cross-case comparison in medicine or law — Students compare superficially similar cases with different underlying structures, then explain why the same rule cannot be applied mechanically.

Multiple-solution-method analysis — Learners study several valid approaches to the same problem and articulate what each method reveals or obscures.

Key Sources

  • Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1991). Cognitive flexibility, constructivism, and hypertext. Educational Technology, 31(5), 24-33.
  • Jacobson, M. J., & Spiro, R. J. (1995). Hypertext learning environments, cognitive flexibility, and the transfer of complex knowledge: An empirical investigation. Journal of Educational Computing Research, 12(4), 301-333. https://doi.org/10.2190/4t1b-hbp0-3f7e-j4pn
  • Black, K.E. and Wittmann, M.C. (2009). Understanding the use of two integration methods on separable first order differential equations. https://perlnet.umaine.edu/imt/
  • Lavie, I., & Sfard, A. (2019). How Children Individualize Numerical Routines: Elements of a Discursive Theory in Making. Journal of the Learning Sciences, 28(4-5), 419-461. https://doi.org/10.1080/10508406.2019.1646650
  • Brookfield, S. D. (1997). Assessing critical thinking. New Directions for Adult and Continuing Education, 1997(75), 17-29. https://doi.org/10.1002/ace.7502
  • Eyal, T., Steffel, M., & Epley, N. (2018). Perspective mistaking: Accurately understanding the mind of another requires getting perspective, not taking perspective. Journal of Personality and Social Psychology, 114(4), 547-571. https://doi.org/10.1037/pspa0000115
  • LaRusso, M., Kim, H. Y., Selman, R., Uccelli, P., Dawson, T., Jones, S., Donovan, S., & Snow, C. (2016). Contributions of academic language, perspective taking, and complex reasoning to deep reading comprehension. Journal of Research on Educational Effectiveness, 9(2), 201-222. https://doi.org/10.1080/19345747.2015.1116035
  • Hall, Peggy; Knappe, Shirley. (1971). Ideas of Western Man: Expressionism and Realism. Dade County Public Schools. https://eric.ed.gov/?id=ED064728
  • van Merriënboer, J. J. G., Kester, L., & Paas, F. (2006). Teaching complex rather than simple tasks: Balancing intrinsic and germane load to enhance transfer of learning. Applied Cognitive Psychology, 20(3), 343–352. https://doi.org/10.1002/acp.1250
  • Bisra, K., Liu, Q., Nesbit, J. C., Salimi, F., & Winne, P. H. (2018). Inducing Self-Explanation: a Meta-Analysis. Educational Psychology Review, 30(3), 703–725. doi:10.1007/s10648-018-9434-x
  • Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education, 49(4), 599–611. https://doi.org/10.1007/s11858-017-0834-z
  • Jacqueline Anton, Dor Abrahamson. (2025). Walking the number line: towards an enactive understanding of integer arithmetic. Instructional Science. https://doi.org/10.1007/s11251-025-09707-w
  • Mireia Vendrell, Samantha-Kaye Johnston. (2026). Scaffolding critical thinking with generative AI: Design principles for integrating large language models in higher education. Computers and Education: Artificial Intelligence. https://doi.org/10.1016/j.caeai.2026.100572
  • Kuhn, D. (1991). The skills of argument. Cambridge University Press. doi:10.1017/cbo9780511571350
  • Nussbaum, E. M., & Schraw, G. (2007). Promoting argument-counterargument integration in students' writing. The Journal of Experimental Education, 76(1), 59-92. https://doi.org/10.3200/JEXE.76.1.59-92
Page metadata
Field Value
type principle
id cognitive-flexibility
aliases represent-procedures-as-shared-resources-to-clarify-thinking, vertical-and-horizontal-bonding-in-numerical-routines, perspective-seekingmultiple-perspectives, multiple-open-ended-definitions-over-fixed-labels, perspective-taking
title Cognitive Flexibility
description Revisiting the same concepts across several cases, perspectives or representations that differ in how the concept applies, with explicit prompts to compare them, is expected to help learners with some but rigid knowledge of an ill-structured domain transfer it to a new case at a short horizon; it may cost factual recall, novices may need a simpler start, and delayed effects are untested here.
status review
canonical True
generated by claude/unspecified on 2026-10-19
trust unverified — no one has confirmed this page's content
sources 7 — jacobson-1995, black-2009, brookfield-1997, eyal-2018, larusso-2016, hall-1971, nussbaum-2007

Frontmatter is the page's machine-readable record — see the Schema & Guide.