An Ontology-Based Approach to Optimizing Geometry Problem Sets for Skill Development

Lielle Rifkin; Matthew Chen; Michael Bouzinier; Sergey Trifonov; Tarun Venkatesh

REVIEW 4 major objections 4 minor 39 references

An Ontology-Based Approach to Optimizing Geometry Problem Sets for Skill Development

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Proof graphs pick problems that force the right geometry skill

desk verdict A useful retrospective of a 1990s geometry problem-annotation framework, but the effectiveness claims are unsupported and a few references look shaky. read the letter →

arxiv 2509.02758 v3 pith:MFDS2VOQ submitted 2025-09-02 math.HO cs.AI

Michael Bouzinier , Sergey Trifonov , Matthew Chen , Tarun Venkatesh , Lielle Rifkin This is my paper

classification math.HOcs.AI MSC 97D4097D50

keywords Euclideangeometrymathematicseducationontologysolutiongraphsskilldevelopmentproblemselectionautomatedproofvalidationsemanticparsing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the reason geometry education is hard to automate is that a problem’s knowledge is procedural: it lives in the many ways a solution can be built, not just in static facts about figures. To capture that, the authors present an ontology from the 1990s that classifies geometric knowledge into facts, objects, and methods, and represents each problem’s possible solutions as a solution graph—a directed acyclic graph whose nodes are skills and whose paths are solution routes. They claim that selecting problems whose solution graphs force passage through a target skill makes problem sets train exactly the intended skill and lets teachers track ability across a curriculum. The same graphs, they argue, point toward automated validation: if a student’s proof can be mapped onto a known solution graph, correct, irrelevant, and incorrect steps become machine-readable. The framework was tested in practice on thousands of problems over three decades, and the authors frame automated annotation and proof validation as the open milestones.

What carries the argument

The solution graph is a directed acyclic graph whose nodes are skills—classified in the ontology as facts, objects, or methods—and whose edges encode which skills a later step depends on. The geometry ontology itself supplies the vocabulary of nodes. The graph’s work is to make the idea that a target skill is essential a structural property: a well-designed problem has no path from givens to conclusion that avoids the target skill, so any successful solution must exercise it. The paper’s second key idea is treating the ontology’s skill catalog as a fixed schema that makes problem annotation consistent across thousands of problems.

What would settle it

A concrete check: take a problem the framework tags as making a skill essential, then collect valid solutions from students or solvers who were not taught that skill. If a complete, non-cumbersome proof that avoids the skill appears, the no-shortcut criterion is violated. A second check: have two annotators construct solution graphs for the same 100-problem sample; stable outputs require an objective rule for what counts as a reasonable alternative solution, which the paper leaves open.

Watch

Extended reading notes

Core claim

The paper’s central claim is that a geometry problem’s solution space can be represented as a solution graph—a directed acyclic graph whose nodes are skills (facts, geometric objects, methods) and whose paths are valid solution routes—and that selecting problems whose graphs have no shortcut path around the target skill makes problem sets align with instructional goals. The ontology classifies roughly 200 geometric facts, 130 objects, and over 80 methods, and the authors report three decades of annotation of thousands of problems. They argue that because a solution graph encodes dependencies among skills, a teacher can not only choose problems that exercise a target skill but can track how t

Load-bearing premise

The framework depends on being able to decide, objectively, whether a solution that skips the target skill is reasonable or significantly simpler; the paper states this criterion but does not define it computationally.

Editorial extensions

If this is right

If solution graphs capture all reasonable solution routes, problem selection can be automated: a curriculum generator can search for problems whose graphs pass through newly taught skills and fail any path avoiding them.
Student solutions can be evaluated against the known graph, so teachers see not just right or wrong but which skill a mis-step skipped or misapplied.
The same graphs make automated validation tractable for statement-reason proofs, since a student’s proof steps can be matched to nodes and edges in the graph.
The triadic ontology scales to large problem banks: because facts, objects, and methods are shared catalog entries, adding a problem means annotating links, not re-describing concepts.
Self-learners could receive hints at the point where their path in the graph diverges from known correct routes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open a computable definition of a reasonable alternative solution; a bounded notion—say, any proof path in a teacher-supplied corpus of accepted solutions—would make the no-shortcut guarantee testable.
The same solution-graph representation could generalize beyond Euclidean geometry to other proof-based subjects, such as algebra or combinatorics, where a skill is a method and a solution is a dependency path.
If large-language-model parsing can map natural-language proofs onto graph nodes, the ontology would provide a natural error taxonomy for feedback: irrelevant, correct but unsupported, or wrong.
Because the ontology separates stable objects from evolving facts and methods, it suggests a way to maintain curriculum alignment as new problem-solving techniques are added: insert a new method node and reconnect existing solution graphs through it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a readable retrospective of a 1990s Soviet/Russian geometry problem-annotation system, not a new research contribution. The useful parts are the documentation of the ontology (facts, objects, methods) and the solution-graph idea; the soft parts are the unsupported effectiveness claims and a couple of apparently dubious references.

What it does well: it gives the most detailed English-language description I've seen of a mature, field-tested ontology for geometry problems — around 200 facts, 130 objects, 80 methods, and 16,000 annotated problems. The solution-graph DAG notion is sensible and connects naturally to recent work like AlphaGeometry and Lean autoformalization. The historical context (Konstantinov, van Hiele in the USSR) is interesting. The authors are also honest in places, framing automated validation as a hypothesis and a research agenda.

Where it falls short: the central claim that the approach 'addresses longstanding challenges' is asserted, not demonstrated. There is no formal model, no experimental evaluation, and no operational definition of the key criterion that there be 'no reasonable alternative solution' that bypasses the target skill. That phrase is doing real work in the optimization, but it is never defined. If this is meant as a human-judgment heuristic, fine — but then it shouldn't be presented as a scalable, computable foundation. The stress-test note is right: as stated, the no-shortcut guarantee is not checkable, and the proposal to automate annotation with LLMs would require a much tighter definition.

I also noticed citation integrity issues: at least one reference has a DOI ending in 999999, and a couple of entries look like they might be product of an LLM. The paper acknowledges using LLMs for style and consistency checking, so the authors need to verify their bibliography.

On self-citation: for a retrospective that's understandable, but it doesn't substitute for independent evidence. The paper would be strengthened by either a small evaluation (e.g., re-annotating a sample of problems with independent raters) or a clear repositioning as historical documentation.

Who is this for? People working on geometry tutoring systems, autoformalization of geometry proofs, or math education ontologies. It's a useful context piece, not a foundational result.

My recommendation: if it lands at a math education venue, send it to referees — but explicitly request major revision: define the criteria, add or omit evaluation claims, and clean up the references. I wouldn't desk-reject it.

Referee Report

4 major / 4 minor

Summary. The paper describes an ontology-based framework for annotating Euclidean geometry problems and their solutions, originally developed in the early 1990s for a Russian educational software tool. The ontology decomposes geometry knowledge into facts, objects, and methods, and augments problem annotations with attributes such as difficulty, provenance, and pedagogical type. The central representational device is the 'solution graph,' a DAG intended to encode multiple solution pathways and their skill dependencies. The authors state three criteria—existing skills, target skill integration, and skill necessity/efficiency—that are supposed to make problem-set selection aligned with instructional objectives. They claim the framework has been tested on thousands of problems over three decades, and they propose that it could eventually support automated annotation and proof validation, possibly through LLM-based semantic parsing. The paper is primarily a descriptive and programmatic presentation; it contains no formal model, no equations, no empirical evaluation, and no algorithmic details.

Significance. The paper has genuine archival value: it documents a large, practitioner-developed ontology (roughly 200 facts, 130 objects, and 80 methods), links it to public GitHub catalogs, and connects it to current proof-graph research such as AlphaGeometry and Lean-based formalization. If the framework were made precise and validated, it could be a useful bridge between pedagogical skill taxonomies and modern automated geometry reasoning. However, as written, the central claims that the approach 'addresses longstanding challenges' and 'enables' alignment of problem selection are asserted rather than demonstrated. The absence of formal definitions and empirical evidence means the present significance is mostly historical and programmatic, not a demonstrated contribution to educational technology or ontology engineering.

major comments (4)

[Problem Solutions Graph Representation] Criterion 3 (Skill Necessity or Efficiency) is the load-bearing condition for the no-shortcut guarantee that the framework is claimed to provide. Yet 'reasonable alternative solution' and 'significantly simplify' are undefined, and no procedure is given for enumerating the relevant alternative solution space. Without a precise definition of the class of solutions over which 'reasonable' ranges, the condition cannot be checked manually or computationally, so the 'optimization' is not a well-defined decision procedure. The authors should formalize the notion relative to a fixed annotated database, or state explicitly that Criterion 3 is a heuristic expert judgment, and they should provide a worked example showing how a problem is admitted or rejected on this basis.
[Problem Solutions Graph Representation] The solution graph is described as a DAG whose nodes correspond to skills (facts, objects, methods), but the paper does not specify the edge relation, the interpretation of paths, what it means for a path to 'pass through' a skill, or how global no-shortcut properties are verified. Figure 1 is illustrative only and is not accompanied by an explained example. This under-specification blocks the claim that solution graphs 'enable' curriculum alignment and granular tracking; a reader cannot construct such a graph from the description. Please define the graph formally—node types, edge types, projection from actual proofs to graph paths—and give at least one non-trivial annotated example.
[Introduction / Discussion] The paper repeatedly grounds its central claim in experience: 'tested with thousands of problems over multiple decades' and 'our experience over the past thirty years has shown...'. No data, annotation statistics, inter-rater agreement, student outcomes, or even a representative sample from the problem database are presented. The GitHub catalogs list terms but do not show annotations of problems or solution graphs. As a result, the claimed demonstration that solution graphs 'systematically support curriculum design and skill development tracking' is not evidenced. A case study or a small quantitative summary—for example, the number of problems annotated per attribute, or a sample problem set generated from the criteria—would be needed to support the claim; without this, these statements remain anecdotal.
[Ontologies in Euclidean Geometry / Discussion] The paper asserts that the framework addresses 'longstanding challenges' in representing procedural and inferential knowledge, scalability, and integration of multiple representations. However, it does not provide a formal knowledge representation (OWL/RDF, types, constraints, inference rules), nor does it explain how adding a new configuration is easier than in existing ontologies. The phrase 'scalable, adaptable foundation' is unsupported. To make this claim defensible, the authors should either specify the representation language and inference requirements, or explicitly narrow the claim to a pedagogical annotation system rather than a solution to ontology expressivity/efficiency problems.

minor comments (4)

[Discussion] Typo: 'SIncalir' should be 'Sinclair'. Also, the citation 'Gordin et al., 2000' in the Historical Context does not match the reference list entry 'Gordin, R. K., & Sharygin, I. F. (2000)', and 'Lu et al., 2024' for AlphaGeometry is not in the references; the corresponding entry is 'Trinh et al., 2024'.
[Problem Solutions Graph Representation] Figure 1 is referenced but not actually present in the text provided; please ensure a legible, labeled diagram is included and that it is described in enough detail for the accompanying discussion to be understood.
[Supplementary Note 1] The 1–40 difficulty scale is explicitly subjective. This is acceptable if treated as an expert rating, but the paper should state whether the scale has been calibrated in any way or whether it is used only within the original authors' internal tool.
[References] A few references, such as 'Elizrova et al. (2022)' and 'Zhang et al. (2024)', appear to be to conference proceedings with minimal bibliographic detail; please add full venue/DOI information where available.

Circularity Check

2 steps flagged · score 6.0 of 10

Criterion 3's no-shortcut guarantee is definitional (encoded in the graph), and the 'tested over three decades' validation is self-citational; the central alignment claim is partially circular.

self definitional [Methodology, 'Problem Solutions Graph Representation']
"3. Skill Necessity or Efficiency: There should be no reasonable alternative solution that excludes the target skill, or else the skill should significantly simplify the solution. ... Effective problem design requires a path through the graph that passes through required skills and does not allow shortcuts that bypass newly introduced skills."

The promised pedagogical guarantee—that a target skill cannot be bypassed—is not derived from an independent enumeration of all possible proofs. Instead, it is encoded in the Solution Graph itself: the graph is constructed by the annotators' choice of which solution pathways count as 'reasonable.' Thus 'no reasonable alternative' is defined to mean 'no path in the graph bypasses the skill,' and the alignment property is true by construction rather than by verification. The undefined terms 'reasonable' and 'significantly simplify' make the criterion unfalsifiable, so the framework's central optimization claim reduces to an assumption built into the graph.
self citation load bearing [Abstract; Discussion, 'Today's Relevance: from building sets to validating proofs']
"The framework has been tested in practice through the annotation of thousands of problems over three decades. ... Our experience over the past thirty years has shown that, in some cases, this methodology empowered self-motivated students to develop significant problem-solving skills independently, with only occasional teacher consultation."

The only evidence offered for the framework's effectiveness is the authors' own experience and their own prior publications (Gordin & Sharygin 2000; Gordin 2003, 2006; Sharygin & Gordin 2001). These works are not machine-checked, code-reproduced, or independently benchmarked, and they originate from the same team that now presents the framework. Because 'tested in practice over three decades' is a load-bearing validation of the paper's central claim, the argument relies on self-citation rather than independent evidence, making the claimed practical success an assertion rather than a derived or externally grounded result.

full rationale

The paper is a methodology/position paper rather than a formal derivation, so much of its content is descriptive. However, two load-bearing moves exhibit circularity. First, the 'Skill Necessity or Efficiency' criterion is operationalized through Solution Graphs: the no-shortcut property is built into the graph by the annotators' decisions about which solutions to include. Consequently, the claim that problem selection 'ensures' a skill cannot be bypassed is equivalent to the assumption that the graph contains all reasonable solutions—an assumption that is neither stated precisely nor independently verified. This is self-definitional: the framework's central guarantee is true by construction of the graph. Second, the paper's validation of the framework rests on the authors' own three-decade experience and their own prior publications, with no external benchmarks or independent evaluation. This self-citation is load-bearing because the paper explicitly presents 'tested in practice' as evidence of the methodology's value. These two issues do not make the entire ontology useless—it may still be a useful classification scheme—but they do mean the core claims of 'no shortcuts' and 'proven in practice' are not independently established. The score of 6 reflects partial circularity: one central guarantee reduces to construction, and the practical validation reduces to self-citation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central claim rests on domain assumptions about the completeness and decomposability of geometry knowledge, the finite DAG representability of solution spaces, and the objectivity of the paper's problem-selection criteria. These are asserted rather than proven, and the paper's own evaluation is absent. The difficulty rating is the only explicit free parameter, but it is subjective and uncalibrated.

free parameters (1)

Difficulty rating (1-40) = integer 1-40, subjective
The paper assigns each problem a difficulty score from 1 to 40 based on the tool authors' judgment (Supplementary Note 1). It is used to filter and select problems but is not calibrated against student outcomes or any external metric.

assumptions (4)

domain assumption Euclidean geometry facts can be exhaustively enumerated and organized (around 200 facts, 130 objects, 80 methods).
The ontology depends on fixed catalogs of facts, objects, and methods, listed in the supplementary note. If the catalogs are incomplete or unstable, the skill tracking and graph construction lose validity. No proof of completeness is given.
domain assumption Student skills can be decomposed into discrete facts, objects, and methods.
The granular tracking of abilities assumes skills are atomic and can be learned and measured independently. The paper does not validate this pedagogical assumption empirically.
domain assumption A geometry problem's solution space is finite and can be represented as a DAG with skill nodes.
Solution graphs require enumerating all solution pathways and their dependencies. The paper gives no argument that this is finite or practically enumerable for the problems in its database.
ad hoc to paper The three criteria (existing skills, target skill integration, and skill necessity/efficiency) are sufficient and objective for problem selection.
These criteria are introduced by the authors in 'Problem Solutions Graph Representation' without operational definitions. Criterion 3 relies on 'no reasonable alternative solution', which is not formally defined.

invented entities (2)

Solution Graph
purpose: To represent multiple solution pathways and skill dependencies for a geometry problem, enabling automated problem selection and validation.
The paper defines solution graphs as DAGs with skill nodes and edges, but provides no empirical demonstration that they faithfully capture real student solution spaces. It is the paper's own construction, not validated against external data.
Skill (as triadic decomposition into facts, objects, methods)
purpose: To serve as the atomic unit for tracking student abilities and aligning problem sets.
The skill construct is a pedagogical assumption, not measured independently. No evidence is provided that skills in actual students neatly decompose this way.

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Cite this review

Pith. "Pith review of An Ontology-Based Approach to Optimizing Geometry Problem Sets for Skill Development." pith.science (2026). https://pith.science/paper/MFDS2VOQ

@misc{pith2026250902758,
  author       = {Pith},
  title        = {Pith review of: An Ontology-Based Approach to Optimizing Geometry Problem Sets for Skill Development},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFDS2VOQ}},
  note         = {Machine review of arXiv:2509.02758}
}

read the original abstract

Euclidean geometry has historically played a central role in cultivating logical reasoning and abstract thinking within mathematics education, but has experienced waning emphasis in recent curricula. The resurgence of interest, driven by advances in artificial intelligence and educational technology, has highlighted geometry's potential to develop essential cognitive skills and inspired new approaches to automated problem solving and proof verification. This article presents an ontology-based framework for annotating and optimizing geometry problem sets, originally developed in the 1990s. The ontology systematically classifies geometric problems, solutions, and associated skills into interlinked facts, objects, and methods, supporting granular tracking of student abilities and facilitating curriculum design. The core concept of 'solution graphs': directed acyclic graphs encoding multiple solution pathways and skill dependencies enables alignment of problem selection with instructional objectives. The framework has been tested in practice through the annotation of thousands of problems over three decades. We contend that our approach addresses longstanding challenges in representing dynamic, procedurally complex mathematical knowledge. We conclude by articulating a research agenda: the open problems of automated problem annotation and solution validation, whose resolution would reduce the time teachers spend validating student work and enable interactive feedback for self-learners.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Knowledge Engineering and the Semantic Web (KESW 2014) , editor =

Nevzorova, Olga and Zhiltsov, Nikita and Kirillovich, Alexander and Lipachev, Evgeny , title =. Knowledge Engineering and the Semantic Web (KESW 2014) , editor =. 2014 , pages =

work page 2014

[2] [2]

Education and Information Technologies , year =

Stancin, Kristian and Poscic, Patrizia and Jaksic, Danijela , title =. Education and Information Technologies , year =

work page

[3] [3]

Elizarov, A. M. and Kirillovich, A. V. and Lipachev, E. K. and Nevzorova, O. A. , title =. Doklady Mathematics , year =

work page

[4] [4]

Automated Deduction -- CADE 28 , series =

de Moura, Leonardo and Ullrich, Sebastian , title =. Automated Deduction -- CADE 28 , series =. 2021 , pages =

work page 2021

[5] [5]

Educational Studies in Mathematics , year =

Duval, Raymond , title =. Educational Studies in Mathematics , year =

work page

[6] [6]

1956 , publisher =

Euclid , title =. 1956 , publisher =

work page 1956

[7] [7]

1973 , publisher =

Freudenthal, Hans , title =. 1973 , publisher =

work page 1973

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1983 , publisher =

Freudenthal, Hans , title =. 1983 , publisher =

work page 1983