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Structure-Mapping-Guided Self-Explanation for Learning Mathematical Procedures

Shinhaeng Lee, Christopher J. MacLellan, Daniel Weitekamp

arXiv:2609.33079v1cs.AI

TL;DR

A worked-example change can have many numerically correct explanations, leaving open how learners select explanations that capture the demonstrated procedure rather than coincidence. The model maps typed relational structures and uses each changed target’s corresponding source region as an anchor for ordered, progressively broadened abductive search. Across 70 transformations, the intended complete procedure ranked first in 56 versus an unguided median of 15.5; preceding alternatives fell from 81,116 to 133.

  • Problem

    A worked-example change can have many numerically correct explanations, leaving open how learners select explanations that capture the demonstrated procedure rather than coincidence.

    Source: §1 · pp. 1–2

  • Method

    The model maps typed relational structures and uses each changed target’s corresponding source region as an anchor for ordered, progressively broadened abductive search.

    Source: §3.2 · p. 6

  • Results

    Across 70 transformations, the intended complete procedure ranked first in 56 versus an unguided median of 15.5; preceding alternatives fell from 81,116 to 133.

    Source: §7 · p. 18

  • Takeaways & Limitations

    The findings support structural correspondence as a way to make abductive procedure learning from worked examples computationally tractable and interpretable, while predicting that people favor corresponding and local evidence.

    Source: §7 · p. 18

    The experiments show improved computational search but do not establish that people search in exactly the same way.

    Source: §6.1 · p. 17

Abstract

from arXiv · show

Worked examples are a powerful form of instruction, but learners must infer how the demonstrated steps were produced. A naive simulation of this self-explanation process can generate thousands of numerical explanations that reproduce one observed change without capturing its underlying procedure. We propose structure-mapping-guided self-explanation as a computational account of the cognitive biases that reduce search effort and make this inference tractable. The model represents mathematical expressions as typed relational structures and uses structure mapping to identify corresponding source and target regions. For each changed target value, the corresponding source region serves as an anchor: it guides abductive search toward structurally relevant values and operations before broader alternatives, yielding ordered, executable candidate procedures with inspectable source evidence. Across 70 mathematical transformations containing 120 changed numeric components, the model recovered every intended procedure. It returned the intended procedure before any other computation producing the same target value in 104 subproblems (86.7%), compared with a median of 32 (26.7%) across 100 unguided runs that tested candidate calculations in random order. Our proposed model also tested 93.8% fewer combinations of values and operations than unguided search before reaching the intended procedures. These results provide an efficient, interpretable account of how relational structure can guide procedural learning and a testable hypothesis about human self-explanation from worked examples.

1. Introduction

Learning from worked examples requires learners to infer procedures, but many numerical explanations can reproduce a change without preserving its underlying operation and operand roles. The paper proposes using structural correspondences to order this search and evaluates whether that guidance recovers intended procedures with less computation.

1. Introduction

  • Worked-example learning requires learners to connect steps to underlying principles, yet a target number can have many correct calculations that do not preserve the demonstrated procedure.The motivating question is how learners can abduce useful explanations rather than coincidental ones.Source: §1 · pp. 1–2
  • For the running case, guided search tested two arithmetic operations, compared with a median of 884 for unguided random ordering.The proposed account aims to explain tractable and interpretable learner search, not to outperform quantitative-reasoning solvers.Source: §1 · p. 2
  • Structure mapping is hypothesized to prioritize values and operations near corresponding source regions before broadening search to unrelated alternatives.The model represents expressions as typed ASTs, expands search by AST distance, and retains executable candidates that reconstruct the target.Source: §1 · p. 2
  • The evaluation tests full-mechanism recovery and search cost, then examines component controls and composition into procedures that apply to new values.The supplied evaluation overview identifies visible-first staging, mapping-relative distance, and anchoring as the component controls.Source: §1 · pp. 2–3
  • Across 70 transformations, the model recovered all 120 intended local procedures, ranked the intended procedure first in 86.7% of subproblems versus 26.7% for unguided search, and tested 93.8% fewer combinations.Unguided search used the median across 100 random search orders.Source: §1 · p. 3

2. Background and Related Work

The background motivates self-explanation as a way learners connect worked steps to principles, then places this paper among computational accounts of procedure induction and analogy. Its specific focus is using structure mapping to guide a symbolic search for the missing computation.

2.1 Worked Examples and Self-Explanation

  • Worked-example studies link successful learning with learners’ explanations connecting example steps to domain principles, while documenting individual differences in self-explanation.Those findings motivate self-explanation but do not specify an algorithm for choosing among candidate explanations.Source: §2.1 · p. 3

2.2 Simulated Learners and Procedure Induction

  • Procedure-learning systems model how learners acquire computations, and the how/where/when framework distinguishes inferring a computation from locating its arguments and selecting when to apply it.This paper focuses on how-learning and constructs inspectable symbolic candidates rather than storing only input–output associations.Source: §2.2 · pp. 3–4

2.3 Structure Mapping and Analogy

  • Unlike direct analogical transfer, structure mapping here aligns two states within one demonstrated step, identifying changed targets and guiding a separate symbolic search for their computation.The mapping is a relevance heuristic, not a complete solution by itself.Source: §2.3 · p. 4

2.4 From Observations to Executable Knowledge

  • The model infers missing executable knowledge from an observed application, rather than beginning with a complete declarative procedure or supplied production rules.Its approach makes representation, search control, operators, and acceptance tests explicit for generating explanatory knowledge.Source: §2.4 · p. 4

3. Structure-Mapping-Guided Self-Explanation

The method maps relational structure between source and target expressions, then uses each changed value’s corresponding source region to prioritize abductive search. Local explanations are composed into executable candidate procedures, with search progressively broadened when nearby evidence is insufficient.

3. Structure-Mapping-Guided Self-Explanation

  • The method represents expressions as typed relational structures, maps corresponding components, orders numerical explanations using those correspondences, and composes local explanations into executable candidate procedures.This sequence is specified as an executable, testable cognitive hypothesis.Source: §3 · p. 4

3.1 Relational Representation and Structure Mapping

  • Typed ASTs make operators, nesting, and ordered argument roles explicit; mapping identifies which target components changed and the corresponding source regions, while distinguishing equal-valued occurrences in different roles.SMTB evaluates type- and role-compatible correspondences using agreement and preservation of surrounding relational structure.Source: §3.1 · pp. 5–6

3.2 Anchors and Structural Distance

  • Each changed target’s corresponding source region becomes an initial anchor; search tests executable compound anchors first, then adds source-visible evidence in structural-distance order before fallback knowledge.Progressive relaxation allows search to move beyond a poor initial correspondence rather than treating it as an inviolable constraint.Source: §3.2 · pp. 6–7

3.3 Prior Knowledge and Progressive Abductive Search

  • The candidate space uses arithmetic operations and integers from −10 through 10, with absent operations or constants available only during fallback; each candidate applies one operation to two inputs and must use source evidence.Candidate identity preserves both the operation and exact source occurrences, so numerically equivalent derivations remain distinct.Source: §3.3 · pp. 7–8

3.4 From Local Explanations to Procedures

  • Exact components are copied into a target template, while explanatory functions fill changed-value holes to form executable transformations that are checked against the demonstration.When multiple candidates reconstruct the target, they remain provisional alternatives for later evidence to distinguish.Source: §3.4 · p. 8

3.5 Extending the Search Beyond Local Evidence

  • In the like-terms example, 3 + 12 = 15 emerged after five calculations versus a median of 479; the derivative case tested 62 combinations versus 327.5, an 81.1% reduction.For the derivative case, the search recruits subtraction and 1 only after source-visible evidence is insufficient.Source: §3.5 · pp. 8–9

4. Evaluation Method

The evaluation tests whether search can recover procedures behind changed numbers in adjacent worked-example steps. Its design specifies intended procedures, compares candidate-ordering strategies, and measures local explanations, complete procedures, transfer, and search effort.

4.1 Transformation Corpora

  • 70 adjacent-step transformations across primary (47) and extended (23) corpora yield 120 changed-number subproblems, spanning four mathematical domains and varied structural distances and notation.Each changed number has an intended explanation using one arithmetic operation; a separate test applies procedures to new numerical instances when source roles are supplied.Source: §4.1 · pp. 9–10

4.2 Intended Procedures

  • Intended procedures specify the operation tree, exact source-number occurrences, and any absent operation or constant; matching preserves occurrence identity and canonicalizes only commutative addition and multiplication.Producing the target value alone does not count as matching the demonstrated procedure.Source: §4.2 · p. 10

4.3 Controlled Search Conditions

  • Six configurations compare full guidance with unguided, visible-first, anchor-only, value-distance, and operation-distance search, holding the candidate space fixed and changing candidate order.Randomized conditions use 100 seeded orders; full guidance is deterministic, and the intended procedure does not influence candidate ordering.Source: §4.3 · pp. 10–11

4.4 Complete Procedures and Transfer to New Values

  • Complete procedures combine one local candidate per changed value; transfer tests reuse the demonstration’s top-ranked procedure, then use a new instance to resolve ambiguity before another reuse.Transfer comprises 16 families with one demonstration and two new instances each, holding input roles constant.Source: §4.4 · p. 12, §4.5 · p. 13

4.5 Measures and Analysis

  • Search effort counts grounded value–operation combinations examined before the intended procedure, while rank measures its position among successful explanations and totals summarize whole-corpus outcomes.Randomized-condition totals are medians across 100 orders; effort reductions use transformation-level bootstrap resamples to estimate confidence intervals.Source: §4.5 · pp. 12–13

5. Results

Structure-mapping guidance recovered intended local and complete procedures while prioritizing them over alternatives and reducing search. The results also show how value ordering, operation ordering, and supplied examples affect search and procedure reuse.

5.1 Finding Intended Explanations with Less Search

  • The model placed the intended explanation first for 104 of 120 numeric changes (86.7%), versus 32 under unguided search (26.7%), while testing 93.8% fewer calculations.Table 3 reports results for the 120 numeric changes; randomized results are medians across 100 search orders.Source: §5.1 · pp. 13–14, Table 3 · p. 14

5.2 Combining and Reusing Learned Procedures

  • For complete transformations, the model ranked the intended procedure first in 56 of 70 cases (80.0%), versus an unguided median of 15.5 (22.1%).It encountered 133 alternative complete procedures before the intended ones, compared with 81,116 under unguided search.Source: §5.2 · p. 14
  • With input roles supplied, the model’s first candidate solved 28 of 32 new instances, and it solved the second instance in 14 of 16 families without using the first.Using one additional example to select among existing candidates then succeeded in all 16 families.Source: §5.2 · p. 15
  • Value-distance guidance matched the model’s 104 intended-first results while testing a median of 853.5 calculations, compared with 2,442 for full guidance.Anchoring alone yielded a median of 98 intended-first results using 3,897 calculations; mapping-relative distance changed priority rather than excluding distant evidence.Source: §5.1 · p. 14, §5.3 · p. 15
  • Expanding the operation vocabulary by 61.7% kept the model at 104 intended-first results but raised its calculation count to 3,889, versus 1,317.5 for value-distance guidance.The larger vocabulary produced a modest selection benefit, not a search-cost advantage.Source: §5.3 · p. 16
  • In 23 of 120 subproblems, the intended calculation required an absent operation, which visible-first search cannot prioritize structurally.Preferring visible operations adds work when the required operation is absent from the source.Source: §5.3 · p. 16
  • On 27 changes with competing visible operations, the model returned all 27 intended explanations first and tested 149 calculations, 43.5% fewer than value-distance guidance.Value-distance guidance returned a median of 26 intended explanations first and tested 263.5 calculations.Source: §5.3 · p. 16, Table 4 · p. 16

6. Discussion

The discussion presents structural locality as an efficient way to prioritize explanations and reports that the local priorities persist when candidates are composed and tested on new values. It frames the account as a testable cognitive hypothesis and situates it within a broader learning architecture, while identifying human behavior and reusable procedure hierarchies as unevaluated extensions.

6.1 A Computational Hypothesis About Learning

  • The mechanism examined 93.8% fewer candidate combinations; against randomized visible-first search, it ranked 104 first versus a median of 62 while examining 63.6% fewer combinations.Source: §6.1 · p. 17
  • After composition, the intended complete procedure ranked first in 56 of 70 transformations versus an unguided median of 15.5.The mechanism encountered 99.8% fewer preceding complete alternatives.Source: §6.1 · p. 17
  • Top-ranked candidates produced 28 of 32 new targets, and one additional instance selected a procedure that transferred in all 16 families.The transfer test supplied input roles.Source: §6.1 · p. 17
  • The computational results do not establish that people search the same way; the model predicts that they prefer corresponding, nearby evidence.A proposed behavioral test would vary evidence location while holding the intended arithmetic relationship fixed.Source: §6.1 · p. 17

6.2 Broader Architectural Context

  • The broader DIPL account assigns where-learning to identifying and binding procedure inputs and when-learning to acquiring application conditions.The present mechanism supplies ordered candidates that a broader learner could retain across examples and refine with later evidence; extending it could remove the supplied-role assumption and support contextual selection and multi-step learning.Source: §6.2 · pp. 17–18
  • Learning reusable procedure hierarchies was not evaluated, and testing less structured problems would require identifying the transformation itself.The discussion proposes storing learned procedures as reusable building blocks within longer solutions.Source: §6.2 · p. 18

7. Conclusion

The conclusion presents structure-guided abduction as a way to make procedure learning from worked transformations tractable and interpretable. Its results include local and composed procedure comparisons, while the predicted human search preference remains a hypothesis rather than an established account of behavior.

7. Conclusion

  • The model recovered all 120 intended local procedures, ranked 104 first versus an unguided median of 32, and examined 93.8% fewer value–operation combinations.Against randomized visible-first search, it ranked 104 first versus a median of 62 and examined 63.6% fewer combinations.Source: §7 · p. 18
  • Value-distance guidance matched full guidance’s intended-first count at lower cost, while operation distance reduced search by 43.5% when visible operators competed.The conclusion says operation distance’s benefit depends on evidence available in the expression.Source: §7 · p. 18
  • The intended complete procedure ranked first in 56 of 70 transformations versus an unguided median of 15.5, and preceding alternatives fell from 81,116 to 133.Top-ranked candidates produced 28 of 32 new targets, and one additional example selected a candidate that transferred in all 16 families; the predicted human preference for local evidence remains testable, not established.Source: §7 · p. 18
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