Knowledge in Pieces

Theory · All theories
Evidence · 10 claims (5 for, 5 mixed) · 11 studies (4 associational, 2 design, 1 causal, 1 quant-synthesis, 1 review, 1 qualitative, 1 theoretical), q1–q4 · 0 of 11 report an effect size · 9 claims rest on one study

Description

Knowledge in pieces is an account of what a misconception is. It holds that a learner's intuitive understanding of a domain is made of many small knowledge elements, each cued by particular contexts and each serving well in some of them, rather than of a single coherent naive theory. Andrea diSessa developed it as an epistemology of intuitive physics (diSessa, 1993). Smith, diSessa and Roschelle (1994) extended it into a critique of misconceptions research: they accept that research's empirical findings but reject the view that students hold flawed ideas that are strongly held, interfere with learning and must be confronted and replaced. On their account, students' prior conceptions are resources continuous with expert knowledge, learning is the refinement and reorganisation of many elements rather than replacement, and a misconception is both flawed and productive Misconceptions as knowledge elements refined, not replaced [+W].

The contrasting position is the coherent-model or framework-theory view. Vosniadou and Brewer (1992) found that children's apparently inconsistent answers about the earth's shape were largely explained by each child's consistent use of one of five alternative mental models, constrained by presuppositions from everyday experience, with synthetic models reconciling a flat initial model with the taught sphere Children's earth answers reflect coherent mental models [~W]. On this view conceptual change happens when the presuppositions are reinterpreted. The two views agree that learners' prior ideas shape what they learn; they disagree on whether those ideas are unified and stable or fragmented and context-dependent, and so on what instruction should do with them.

A third tradition reached a similar question from procedural skills. Brown and Burton (1978) modelled students' arithmetic errors as "bugs" in a procedural network, a deep-structure model of why a student makes a mistake, and used it to diagnose 1,300 students from a database of 20,000 test items. Brown and VanLehn's (1980) repair theory derived those bugs from problem solving on incomplete procedures and made predictions about their frequency and stability. Later studies of error stability found that error rules are used inconsistently below mastery: algebra mal-rules were very unstable, severely skewed in frequency and school-specific Algebra mal-rules are unstable, skewed and school-specific [+W], and eighth graders who had not mastered signed-number arithmetic switched rules between identical items within one test while masters did not Rule use inconsistent below mastery, stable at mastery [+W]. These findings fit the knowledge-in-pieces reading of errors as context-dependent, though none was designed as a test between the two views.

Implications

Context

• Diagnosis needs more than one occasion. If misconceptions are context-cued, one wrong answer, or one pass through a diagnostic test, is weak evidence of a stable belief. Rule diagnoses of non-masters were unstable even across identical items in one sitting Rule use inconsistent below mastery [+W]. Ask the same idea in several contexts before concluding a learner holds a misconception.
• Error catalogues are local. Mal-rules that explained errors well in one school did not in another Algebra mal-rules are school-specific [+W], so a misconception inventory or a set of distractors built in one population may not carry over to another.
• A rise in an alternative conception is not necessarily a failure. On distractor-driven astronomy items, instruction first strengthened students' attraction to alternative conceptions before the preference declined, which the author reads as such ideas marking progress Attraction to alternative conceptions can rise before it falls [+W].
• Confront-and-replace versus refine-and-connect. Instruction built on the coherent-model view elicits a misconception and confronts it; conflict-based instruction and refutation text produce conceptual change more often than standard exposition Confronting misconceptions produces conceptual change more often than exposition [~M]. Knowledge in pieces does not deny this result but reinterprets it: what works is building on the elements of a learner's thinking that are productive in the target context, and connecting them, not erasing them. Which framing a design follows changes whether a learner's idea is treated as an error to remove or a resource to refine.
• Simulated learners and generated distractors. If real misconceptions are unstable and population-specific, a model of a student as a fixed set of buggy rules, or an AI that is meant to "hold" a misconception, is an idealisation. In practice the problem runs the other way: LLM student simulators abandon a simulated misconception under any corrective feedback, whether or not it addresses the misconception LLM simulators flip under any feedback [~M], and LLM-generated distractors are judged mathematically valid more often than plausible to students LLM distractors more valid than plausible [~M], with model likelihoods correlating only moderately with students' choices LLM distractor likelihood moderately tracks student choices [~M].

Target Learners

• Learners with informal experience of a domain (motion, heat, the shape of the earth, arithmetic procedures), whose intuitions are the elements the theory describes
• Learners below mastery, whose error patterns are least stable; masters' rule use is stable Rule use stable at mastery [+W]

Target Learning Objectives

• Conceptual understanding in science and mathematics, where misconceptions research began
• Procedural skill in arithmetic and algebra, where the stability of error rules has been measured
• Diagnostic assessment: what a learner's wrong answer can and cannot tell a teacher or a system

Claims

• Student misconceptions can be read as knowledge elements that serve in some contexts and are refined and reorganised during learning, rather than as strongly held flawed ideas that instruction must confront and replace [+W]
• Children's apparently inconsistent answers about the earth's shape are largely explained by consistent use of alternative mental models [~W]
• Students' algebra mal-rules are used very irregularly, are severely skewed in frequency, and differ in explanatory power from school to school [+W]
• Students' rules of operation for signed-number arithmetic are applied inconsistently by non-masters and stably by masters [+W]
• On distractor-driven science items, instruction can first strengthen students' attraction to an alternative conception before that preference declines [+W]
• Instruction that confronts science misconceptions (conceptual-conflict interventions, refutation text) produces conceptual change more often than standard exposition [~M]
• Walking the body-scale number line disrupted students' prior misconceptions and erroneous rules, which resurfaced when they returned to the desk-scale number line [+W]
• LLM student simulators exhibit near-zero Selective Flip Score, flipping at similarly high rates under targeted, misaligned, and generic feedback [~M]
• LLM-generated distractors for math multiple-choice questions are judged mathematically valid more often than plausible to students, and rarely reproduce the full set of human-written distractors [~M]
• LLMs' generation probabilities for multiple-choice distractors correlate moderately with how often students choose them [~M]

Related Theories

• Multiple local conceptual coherences: student understanding as locally stable patterns requiring stability explanations — a development of the same view, explaining the local stabilities of variable thinking
• Multiple local coherences: stability of student thinking as sustained by real-time activity — related
• Mental models framework: children's conceptual understandings as coherent, theory-like structures constrained by presuppositions — the contrasting coherent-model view, applied to the day and night cycle
• Constructivism — the premise, that learners build new knowledge from prior understanding, on which Smith et al. rest their critique

Examples

• Misconception Probes — the diagnostic element whose reading depends on which view is taken: a probe answer is a sample of context-cued thinking, not necessarily a stable belief
• Conceptual Change — the elicit-confront-resolve strategy built on the coherent-model view

Key Sources

• diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10(2–3), 105–225. doi:10.1080/07370008.1985.9649008
• Smith, J. P., III, diSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163. doi:10.1207/s15327809jls0302_1
• Vosniadou, S., & Brewer, W. F. (1992). Mental models of the earth: A study of conceptual change in childhood. Cognitive Psychology, 24(4), 535–585. doi:10.1016/0010-0285(92)90018-W
• Brown, J. S., & Burton, R. R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2(2), 155–192. doi:10.1207/s15516709cog0202_4
• Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4(4), 379–426. doi:10.1207/s15516709cog0404_3
• Payne, S. J., & Squibb, H. R. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14(3), 445–481. doi:10.1207/s15516709cog1403_4
• Tatsuoka, K. K., Birenbaum, M., & Arnold, J. (1989). On the stability of students' rules of operation for solving arithmetic problems. Journal of Educational Measurement, 26(4), 351–361. doi:10.1111/j.1745-3984.1989.tb00339.x
• Sadler, P. M. (1998). Psychometric models of student conceptions in science: Reconciling qualitative studies and distractor-driven assessment instruments. Journal of Research in Science Teaching, 35(3), 265–296. doi:10.1002/(SICI)1098-2736(199803)35:3<265::AID-TEA3>3.0.CO;2-P
• Feng, W., Lee, J., McNichols, H., Scarlatos, A., Smith, D., Woodhead, S., Ornelas, N. O., & Lan, A. (2024). Exploring automated distractor generation for math multiple-choice questions via large language models. Findings of NAACL 2024, 3067–3082. doi:10.18653/v1/2024.findings-naacl.193
• Do, H., Sonkar, S., & Sachan, M. (2026). Simulating students or sycophantic problem solving? On misconception faithfulness of LLM simulators. arXiv preprint. doi:10.48550/arXiv.2605.12748
• Liu, N., Sonkar, S., & Baraniuk, R. (2025). Do LLMs make mistakes like students? Exploring natural alignments between language models and human error patterns. AIED 2025, LNCS, 364–377. doi:10.1007/978-3-031-98459-4_26

Page metadata

Field	Value
type	theory
title	Knowledge in Pieces
description	Knowledge in pieces holds that learners' intuitive ideas, including misconceptions, are many small context-cued knowledge elements that are refined and reorganised during learning, in contrast to the view that misconceptions are coherent, stable theories to be replaced.
status	review
author	Andrea diSessa; John P. Smith III, Andrea diSessa & Jeremy Roschelle
canonical	True
generated	by claude/unspecified on 2026-10-21
trust	unverified — no one has confirmed this page's content

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